# Strong pinning transition with arbitrary defect potentials

Filippo Gaggioli  Institut für Theoretische Physik, ETH Zürich, CH-8093
Zürich, Switzerland  Gianni Blatter  Institut für Theoretische Physik, ETH
Zürich, CH-8093 Zürich, Switzerland  Martin Buchacek  Institut für
Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland  Vadim B.
Geshkenbein  Institut für Theoretische Physik, ETH Zürich, CH-8093 Zürich,
Switzerland

###### Abstract

Dissipation-free current transport in type II superconductors requires
vortices, the topological defects of the superfluid, to be pinned by defects
in the underlying material. The pinning capacity of a defect is quantified by
the Labusch parameter $\kappa\sim f_{p}/\xi\bar{C}$, measuring the pinning
force $f_{p}$ relative to the elasticity $\bar{C}$ of the vortex lattice, with
$\xi$ denoting the coherence length (or vortex core size) of the
superconductor. The critical value $\kappa=1$ separates weak from strong
pinning, with a strong defect at $\kappa>1$ able to pin a vortex on its own.
So far, this weak-to-strong pinning transition has been studied for isotropic
defect potentials, resulting in a critical exponent $\mu=2$ for the onset of
the strong pinning force density $F_{\mathrm{pin}}\sim
n_{p}f_{p}(\xi/a_{0})^{2}(\kappa-1)^{\mu}$, with $n_{p}$ denoting the density
of defects and $a_{0}$ the intervortex distance. This result is owed to the
special rotational symmetry of the defect producing a finite trapping area
$S_{\mathrm{trap}}\sim\xi^{2}$ at the strong-pinning onset. The behavior
changes dramatically when studying anisotropic defects with no special
symmetries: the strong pinning then originates out of isolated points with
length scales growing as $\xi(\kappa-1)^{1/2}$, resulting in a different force
exponent $\mu=5/2$. Our analysis of the strong pinning onset for arbitrary
defect potentials $e_{p}(\mathbf{R})$, with $\mathbf{R}$ a planar coordinate,
makes heavy use of the Hessian matrix describing its curvature and leads us to
interesting geometrical structures: the strong pinning onset is characterized
by the appearance of unstable areas of elliptical shape whose boundaries mark
the locations where vortices jump. The associated locations of asymptotic
vortex positions define areas of bistable vortex states; these bistable
regions assume the shape of a crescent with boundaries that correspond to the
spinodal lines in a thermodynamic first-order transition and cusps
corresponding to critical end-points. Both, unstable and bistable areas grow
with $\kappa>1$ and join up into larger domains; for a uniaxially anisotropic
defect, two face to face crescents merge into the ring-shaped area previously
encountered for the isotropic defect. Both, onset and merger points are
defined by local differential properties of the Hessian’s determinant
$D(\mathbf{R})$, specifically, its minima and saddle points. Extending our
analysis to the case of a random two-dimensional pinning landscape, we discuss
the topological properties of unstable and bistable regions as expressed
through the Euler characteristic, with the latter related to the local
differential properties of $D(\mathbf{R})$ through Morse theory.

##  I Introduction

Vortex pinning by material defects Campbell and Evetts (1972) determines the
phenomenological properties of all technically relevant (type II)
superconducting materials, e.g., their dissipation-free transport or magnetic
response. Similar applies to the pinning of dislocations in metals Kassner
(2015) or domain walls in magnets Gorchon  _et al._ (2014), with the
commonalities found in the topological defects of the ordered phase being
pinned by defects in the host material: these topological defects are the
vortices Abrikosov (1957), dislocations Burgers (1940), or domain walls Bloch
(1932); Landau and Lifshitz (1935) appearing within the respective ordered
phases—superconducting, crystalline, or magnetic. The theory describing the
pinning of topological defects has been furthest developed in superconductors,
with the strong pinning paradigm Labusch (1969); Larkin and Ovchinnikov (1979)
having been strongly pushed during the last decade Blatter  _et al._ (2004);
Thomann  _et al._ (2012); Willa  _et al._ (2015); Buchacek  _et al._ (2019a).
In its simplest form, it boils down to the setup involving a single vortex
subject to one defect and the cage potential Ertas and Nelson (1996); Vinokur
_et al._ (1998) of other vortices. While still exhibiting a remarkable
complexity, it produces quantitative results which benefit the comparison
between theoretical predictions and experimental findings Buchacek  _et al._
(2019b). So far, strong pinning has focused on isotropic defects, with the
implicit expectation that more general potential shapes would produce small
changes. This is not the case, as first demonstrated by Buchacek et al.
Buchacek  _et al._ (2020) in their study of correlation effects between
defects that can be mapped to the problem of a string pinned to an anisotropic
pinning potential. In the present work, we generalize strong pinning theory to
defect potentials of arbitrary shape. We find that this simple generalization
has pronounced (geometric) effects near the onset of strong pinning that even
change the growth of the pinning force density
$F_{\mathrm{pin}}\propto(\kappa-1)^{\mu}$ with increasing pinning strength
$\kappa>1$ in a qualitative manner, changing the exponent $\mu$ from $\mu=2$
for isotropic defects Labusch (1969); Blatter  _et al._ (2004) to $\mu=5/2$
for general anisotropic pinning potentials.

The pinning of topological defects poses a rather complex problem that has
been attacked within two paradigms, weak-collective- and strong pinning. These
have been developed in several stages: originating in the sixties of the last
century, weak pinning and creep Larkin and Ovchinnikov (1979) has been further
developed with the discovery of high temperature superconductors as a subfield
of vortex matter physics Blatter  _et al._ (1994). Strong pinning was
originally introduced by Labusch Labusch (1969) and by Larkin and Ovchinnikov
Larkin and Ovchinnikov (1979) and has been further developed recently with
several works studying critical currents Blatter  _et al._ (2004),
current–voltage characteristics Thomann  _et al._ (2012, 2017), magnetic field
penetration Willa  _et al._ (2015, 2016); Gaggioli  _et al._ (2022), and creep
Buchacek  _et al._ (2018, 2019a); Gaggioli  _et al._ (2022); results on
numerical simulations involving strong pins have been reported in Refs. Kwok
_et al._ , 2016; Willa  _et al._ , 2018a, b. The two theories come together at
the onset of strong pinning: an individual defect is qualified as weak if it
is unable to pin a vortex, i.e., a vortex traverses the pin smoothly. Crossing
a strong pin, however, the vortex undergoes jumps that mathematically
originate in bistable distinct vortex configurations, ‘free’ and ‘pinned’.
Quantitatively, the onset of strong pinning is given by the Labusch criterion
$\kappa=1$, with the Labusch parameter
$\kappa\equiv\max[-e_{p}^{\prime\prime}]/\bar{C}\sim f_{p}/\xi\bar{C}$, the
dimensionless ratio of the negative curvature $e_{p}^{\prime\prime}$ of the
isotropic pinning potential and the effective elasticity $\bar{C}$ of the
vortex lattice. Strong pinning appears for $\kappa>1$, i.e., when the lattice
is soft compared to the curvatures in the pinning landscape.

So far, the strong pinning transition at $\kappa=1$ has been described for
defects with isotropic pinning potentials; it can be mapped Blatter  _et al._
(2004) to the magnetic transition in the $h$-$T$ (field–temperature) space,
with the strong-pinning phenomenology at $\kappa>1$ corresponding to the
first-order Ising magnetic transition at $T<T_{c}$ and the critical point at
$T=T_{c}$ corresponding to the strong pinning transition at $\kappa=1$. The
role of the reduced temperature $T/T_{c}$ is then assumed by the Labusch
parameter $\kappa$ and the bistabilities associated with the ferromagnetic
phases at $T/T_{c}<1$ translate to the bistable pinned and free vortex states
at $\kappa>1$, with the bistability disappearing on approaching the critical
point, $T/T_{c}=1$ and $\kappa=1$, respectively.

A first attempt to account for correlations between defects has been done in
Ref. Buchacek  _et al._ , 2020. The latter analysis takes into account the
enhanced pinning force excerted by pairs of isotropic defects that can be cast
in the form of anisotropic effective pinning centers. Besides shifting the
onset of strong pinning to $\kappa=1/2$ (with $\kappa$ defined for one
individual defect), the analysis unravelled quite astonishing (geometric)
features that appeared as a consequence of the symmetry reduction in the
pinning potential. In the present paper, we take a step back and study the
transition to strong pinning for anisotropic defect potentials $e_{p}({\bf
R})$, with $\mathbf{R}$ a planar coordinate, see Fig. 1. Note that collective
effects of many weak defects can add up to effectively strong pins that
smoothen the transition at $\kappa=1$, thereby turning the strong pinning
transition into a weak-to-strong pinning crossover.

We find that the onset of strong pinning proceeds quite differently when going
from the isotropic defect to the anisotropic potential of a generic defect
without special symmetries and further on to a general random pinning
landscape. The simplest comparison is between an isotropic and a uniaxially
anisotropic defect, acting on a vortex lattice that is directed along the
magnetic field ${\bf B}\parallel{\bf e}_{z}$ chosen parallel to the $z$-axis;
for convenience, we place the defect at the origin of our coordinate system
${\bf r}=({\bf R},z)$ and have it act only in the $z=0$-plane. In this setup,
see Fig. 1, the pinning potential $e_{p}({\bf R})$ acts on the nearest vortex
with a force ${\bf f}_{p}({\bf R})=-\nabla_{\bf R}e_{p}|_{z=0}$ attracting the
vortex to the defect; the presence of the other vortices constituting the
lattice renormalizes the vortex elasticity $\bar{C}$. With the pinning
potential acting in the $z=0$ plane, the vortex is deformed with a pronounced
cusp at $z=0$, see Fig. 1; we denote the tip position of the vortex where the
cusp appears by $\tilde{\bf R}$, while the asymptotic position of the vortex
at $z\to\pm\infty$ is fixed at $\bar{\bf R}$. With this setup the problem can
be reduced to a planar one, with the tip coordinate $\tilde{\bf R}$ and the
asymptotic coordinate $\bar{\bf R}$ determining the location and full shape
(and hence the pinning force) of the vortex line.

Figure 1: Sketch of a vortex interacting with a defect located at the origin.
The vortex approaches the asymptotic position $\bar{\bf R}$ at $z\to\pm\infty$
and is attracted to the defect residing at the origin; the cusp at $z=0$
defines the tip position $\tilde{\bf R}$ and its angle quantifies the pinning
strength.

In the case of an isotropic pin, e.g., produced by a point-like defect Thomann
_et al._ (2012), strong pinning first appears on a circle of finite radius
$R_{m}\sim\xi$, typically of order of the vortex core radius $\xi$, see left
panel of Fig. 2(a). This is owed to the fact that, given the radial symmetry,
the Labusch criterion $\kappa=\max_{R}[-e_{p}^{\prime\prime}(R)]/\bar{C}=1$ is
satisfied on a circle $R=R_{m}$ where the (negative) curvature
$-e_{p}^{\prime\prime}>0$ is maximal. Associated with the radius $R_{m}$ where
the tip is located at $\kappa=1$,
$\tilde{R}(\kappa=1)\equiv\tilde{R}_{m}=R_{m}$, there is an asymptotic vortex
position $\bar{R}(\kappa=1)=\bar{R}_{m}>\tilde{R}_{m}$. Increasing the Labusch
parameter beyond $\kappa=1$, the circle of radius $\bar{R}_{m}$ transforms
into a ring $\bar{R}_{-}<\bar{R}<\bar{R}_{+}$ of finite width. Vortices placed
inside the ring at small distances $\bar{R}<\bar{R}_{-}$ near the defect are
qualified as ‘pinned’, while vortices at large distances $\bar{R}>\bar{R}_{+}$
away from the pin are described as ‘free’, see right panel in Fig. 2(a);
physically, we denote a vortex configuration as ‘free’ when it is smoothly
connected to the asymptotic undeformed state, while a ‘pinned’ vortex is
localized to a finite region around the defect. Vortices placed inside the
bistable ring at $\bar{R}_{-}<\bar{R}<\bar{R}_{+}$ acquire two possible
states, pinned and free (colored magenta in Fig. 2, the superposition of red
(pinned state) and blue (free state) colors).

The onset of strong pinning for the uniaxially anisotropic defect proceeds in
several stages. Let us consider an illustrative example and assume a defect
with an anisotropy aligned with the axes and a steeper potential along $x$. In
this situation, strong pinning as defined by the criterion $\kappa_{m}=1$,
with a properly generalized Labusch parameter $\kappa_{m}$, appears out of two
points $(\pm\bar{x}_{m},0)$ where the Labusch criterion $\kappa_{m}=1$ is met
first, see Fig. 2(b) left. Increasing $\kappa_{m}>1$ beyond unity, two
bistable domains spread around these points and develop two crescent-shaped
areas (with their large extent along $\bar{y}$) in asymptotic $\bar{\bf
R}$-space, see Fig. 2(b) right. Vortices with asymptotic positions within
these crescent-shaped regions experience bistability, while outside these
regions the vortex state is unique. Classifying the bistable solutions as
‘free’ and ‘pinned’ is not possible, with the situation resembling the one
around the gas–liquid critical point with a smooth crossover (from blue to
white to red) between phases. With $\kappa_{m}$ increasing further, the cusps
of the crescents approach one another. As the arms of the two crescents touch
and merge at a sufficiently large value of $\kappa_{m}$, the topology of the
bistable area changes: the two merged crescents now define a ring-like
geometry and separate $\bar{\bf R}$-space into an inside region where vortices
are pinned, an outside region where vortices are free and the bistable region
with pinned and free states inside the ring-like region. As a result, the
pinning geometry of the isotropic defect is recovered, though with the perfect
ring replaced by a deformed ring with varying width. Using the language
describing a thermodynamic first-order transition, the cusps of the crescents
correspond to critical points while its boundaries map to spinodal lines; the
merging of critical points changing the topology of the bistable regions of
the pinning landscape goes beyond the standard thermodynamic analogue of phase
diagrams.

Figure 2: Illustration of bistable regions in asymptotic $\bar{\bf R}$-space
for a vortex pinned to a defect located at the origin. (a) For an isotropic
defect (Lorentzian shape with $\kappa=1,~{}1.5$), pinning appears at
$\kappa=1$ along a ring with radius $\bar{R}_{m}$, with the red area
corresponding to pinned states and free states colored in blue. With
increasing pinning strength $\kappa$, see right panel at $\kappa=1.5$, a
bistable region (in magenta) appears in a ring geometry, with vortices
residing inside, $\bar{R}<\bar{R}_{-}$, being pinned and vortices outside,
$\bar{R}>\bar{R}_{+}$, remaining free. Vortices with asymptotic positions
inside the ring ($\bar{R}_{-}<\bar{R}<\bar{R}_{+}$) exhibit bistable states,
pinned and free. The dashed circle $\bar{R}_{0}$ marks the crossing of pinned
and free branches, see Fig. 4. (b) For a uniaxially anisotropic defect, see
Eq. (94) with $\epsilon=0.3$ and largest (negative) curvature along $x$,
pinning appears in two points $(\pm\bar{x}_{m},0)$ along the $x$-axis. As the
pinning strength increases beyond unity, see right panel, bistable regions
(magenta) develop in a crescent-shape geometry. Pinned- and free-like states
are smoothly connected as indicated by the crossover of colors (see Sec. III.3
for the precise description of coloring in terms of an ‘order parameter’). As
$\kappa_{m}$ further increases, the cusps of the two crescents merge on the
$y$-axis, changing the topology of the $\bar{\bf R}$-plane through separation
into inner and outer regions (not shown). A ring-like bistable region appears
as in $(\mathrm{a})$, with the inner (outer) region corresponding to unique
vortex states that are pinned (free), while vortices residing inside the ring-
shaped domain exhibit bistable states, pinned and free.

The bistable area is defining the trapping area where vortices get pinned to
the defect; this trapping area is one of the relevant quantities determining
the pinning force density $F_{\mathrm{pin}}$, the other being the jumps in
energy associated with the difference between the bistable states Labusch
(1969); Blatter  _et al._ (2004), see the discussion in Secs. II.3, II.5, and
III.7 below. It is the change in the bistable- and hence trapping geometry
that modifies the exponent $\mu$ in $F_{\mathrm{pin}}\propto(\kappa-1)^{\mu}$,
replacing the exponent $\mu=2$ for isotropic defects by the new exponent
$\mu=5/2$ for general anisotropic pinning potentials.

While the existence of bistable regions $\mathcal{B}_{\bar{\bf R}}$ in the
space of asymptotic vortex positions $\bar{\bf R}$ is an established element
of strong pinning theory by now, in the present paper, we introduce the new
concept of unstable domains $\mathcal{U}_{\tilde{\bf R}}$ in tip-space. The
two coordinates $\tilde{\bf R}$ and $\bar{\bf R}$ represent dual variables in
the sense of the thermodynamic analog, with the asymptotic coordinate
$\bar{\bf R}$ corresponding to the driving field $h$ in the Ising model and
the tip position $\tilde{\bf R}$ replacing the magnetic response $m$; from a
thermodynamic perspective it is then quite natural to change view by going
back and forth between intensive ($h$) and extensive ($m$) variables. In tip
space $\tilde{\bf R}$, the onset of pinning appears at isolated points
$\tilde{\bf R}_{m}$ that grow into ellipses as $\kappa$ is increased beyond
unity. These ellipses describe unstable areas $\mathcal{U}_{\tilde{\bf R}}$ in
the $\tilde{\bf R}$-plane across which vortex tips jump when flipping between
bistable states; they relate to the bistable crescent-shaped areas
$\mathcal{B}_{\bar{\bf R}}$ in asymptotic space through the force balance
equation; the latter determines the vortex shape with elastic and pinning
forces compensating one another. The unstable regions $\mathcal{U}_{\tilde{\bf
R}}$ in tip space are actually more directly accessible than the bistable
regions $\mathcal{B}_{\bar{\bf R}}$ in asymptotic space and play an equally
central role in the discussion of the strong pinning landscape.

The simplification introduced by the concept of unstable domains
$\mathcal{U}_{\tilde{\bf R}}$ in tip space $\tilde{\bf R}$ is particularly
evident when going from individual defects as described above to a generic
pinning landscape. Here, we focus on a model pinning potential landscape (or
short pinscape) confined to the two-dimensional (2D) $\mathbf{R}$ plane at
$z=0$; such a pinscape can be produced, e.g., by defects that reside in the
$z=0$ plane. The pinned vortex tip $\tilde{\bf R}$ then still resides in the
$z=0$ plane as well and the strong pinning problem remains two-dimensional.
For a 2D random pinscape, unstable ellipses appear sequentially out of
different (isolated) points and at different pinning strength $\kappa_{m}$;
their assembly defines the unstable area $\mathcal{U}_{\tilde{\bf R}}$, with
each newly appearing ellipse changing the topology of $\mathcal{U}_{\tilde{\bf
R}}$, specifically, its number of components. Increasing $\kappa_{m}$, the
ellipses first grow in size, then deform away from their original elliptical
shapes, and finally touch and merge in a hyperbolic geometry. Such mergers
change, or more precisely reduce, the number of components in
$\mathcal{U}_{\tilde{\bf R}}$ and hence correspond again to topological
transitions as described by a change in the Euler characteristic $\chi$
associated with the shape of $\mathcal{U}_{\tilde{\bf R}}$. Furthermore, these
mergers tend to produce $\mathcal{U}_{\tilde{\bf R}}$ shapes that are non-
simply connected, again implying a topological transition in
$\mathcal{U}_{\tilde{\bf R}}$ with a change in $\chi$. Such non-simply
connected parts of $\mathcal{U}_{\tilde{\bf R}}$ separate the tip space into
‘inner’ and ‘outer’ regions that allows to define proper ‘pinned’ states
(localized near a potential minimum) in the ‘inner’ of
$\mathcal{U}_{\tilde{\bf R}}$, while ‘free’ states (smoothly connected to
asymptotically undeformed vortices) occupy the regions outside of
$\mathcal{U}_{\tilde{\bf R}}$.

The discussion below is dominated by three mathematical tools: for one, it is
the Hessian matrix $\mathrm{H}({\bf R})$ of the pinning potential Buchacek
_et al._ (2020); Willa  _et al._ (2022) $e_{p}({\bf R})$, its eigenvalues
$\lambda_{\pm}({\bf R})$ and eigenvectors ${\bf v}_{\pm}({\bf R})$, its
determinant $\det[\mathrm{H}]({\bf R})$ and trace $\tr[\mathrm{H}]({\bf R})$.
The Hessian matrix involves the curvatures
$\mathrm{H}_{ij}=\partial_{i}\partial_{j}e_{p}({\bf R})$, $i,j\in\\{x,y\\}$,
of the pinning potential, that in turn are the quantities determining strong
pinning, as can be easily conjectured from the form of the Labusch parameter
$\kappa\propto-e_{p}^{\prime\prime}$ for the isotropic defect. The second tool
is the Landau-type expansion of the total pinning energy near the strong-
pinning onset around $\tilde{\bf R}_{m}$ at $\kappa_{m}=1$ (appearance of a
critical point) as well as near merging around $\tilde{\bf R}_{s}$ at
$\kappa(\tilde{\bf R}_{s})\equiv\kappa_{s}=1$ (disappearance of a pair of
critical points); the standard manipulations as they are known from the
description of a thermodynamic first-order phase transition produce most of
the new results. Third, the topological structure of the unstable domain
$\mathcal{U}_{\tilde{\bf R}}$ associated with a generic 2D pinning landscape,
i.e., its components and their connectedness, is conveniently described
through its Euler characteristic $\chi$ with the help of Morse theory.

The structure of the paper is as follows: In Section II, we briefly introduce
the concepts of strong pinning theory with a focus on the isotropic defect.
The onset of strong pinning by a defect of arbitrary shape is presented in
Sec. III; we start with a translation and extension of the strong pinning
ideas from the isotropic situation to a general anisotropic one, that leads us
to the Hessian analysis of the pinning potential as our basic mathematical
tool. Close to onset, we find (using a Landau-type expansion, see Sec. III.1)
that the unstable (Sec. III.2) and bistable (Sec. III.3) domains are
associated with minima of the determinant of the Hessian curvature matrix and
assume the shape of an ellipse and a crescent, respectively. Due to the
anisotropy, the geometry of the trapping region depends non-trivially on the
Labusch parameter and the critical exponent for the pinning force is changed
from $\mu=2$ to $\mu=5/2$, see Sec. III.7. The analytic solution of the strong
pinning onset for a weakly uniaxial defect presented in Sec. IV leads us to
define new hyperbolic points associated with saddle points of the determinant
of the Hessian curvature matrix. These hyperbolic points describe the merging
of unstable and bistable domains, see Sec. V.1, and allow us to relate the new
results for the anisotropic defect to our established understanding of
isotropic defects. In a final step, we extend the local perspective on the
pinscape, as acquired through the analysis of minima and saddles of the
determinant of the Hessian curvature matrix, to a global description in terms
of the topological characteristics of the unstable domain
$\mathcal{U}_{\tilde{\bf R}}$: in Sec. VI, we discuss strong pinning in a two-
dimensional pinning potential of arbitrary shape, e.g., as it appears when
multiple pinning defects overlap (though all located in one plane). We follow
the evolution of the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ with
increasing pinning strength $\kappa_{m}$ and express its topological
properties through the Euler characteristic $\chi$; the latter is related to
the local differential properties of the pinscape’s curvature, its minima,
saddles, and maxima, through Morse theory. Finally, in Appendix A, we map the
two-dimensional Landau-type theories (involving two order parameters)
describing onset and merging, to effective one-dimensional Landau theories and
rederive previous results following standard statistical mechanics
calculations as they are performed in the analysis of the critical point in
the van der Waals gas.

##  II Strong pinning theory

We start with a brief introduction to strong pinning theory, keeping a focus
on the transition region at moderate values of $\kappa>1$. We consider an
isotropic defect (Sec. II.1) and determine the unstable and bistable ring
domains for this situation in Sec. II.2. We derive the general expression for
the pinning force density $F_{\mathrm{pin}}$ in Sec. II.3, determine the
relevant scales of the strong pinning characteristic near the crossover in
Sec. II.4, and apply the results to derive the scaling
$F_{\mathrm{pin}}\propto(\kappa-1)^{2}$ for the isotropic defect (Sec. II.5).
In Sec. II.6, we relate the strong pinning theory for the isotropic defect to
the Landau mean-field description for the Ising model in a magnetic field.

###  II.1 Isotropic defect

The standard strong-pinning setup involves a vortex lattice directed along $z$
with a lattice constant $a_{0}$ determined by the induction
$B=\phi_{0}/a_{0}^{2}$ that is interacting with a dilute set of randomly
arranged defects of density $n_{p}$. This many-body problem can be reduced
Blatter  _et al._ (2004); Willa  _et al._ (2016); Buchacek  _et al._ (2019a)
to a much simpler effective problem involving an elastic string with effective
elasticity $\bar{C}$ that is pinned by a defect potential $e_{p}({\bf R})$
acting in the origin, as described by the energy function

$e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})=\frac{\bar{C}}{2}(\tilde{\bf
R}-\bar{\bf R})^{2}+e_{p}(\tilde{\bf R})$ (1)

depending on the tip- and asymptotic coordinates $\tilde{\bf R}$ and $\bar{\bf
R}$ of the vortex, see Fig. 1. The energy (or Hamiltonian)
$e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ of this setup involves an
elastic term and the pinning energy $e_{p}({\bf R})$ evaluated at the location
$\tilde{\bf R}$ of the vortex tip. We denote the depth of the pinning
potential by $e_{p}$. A specific example is the point-like defect that
produces an isotropic pinning potential which is determined by the form of the
vortex Thomann  _et al._ (2012) and assumes a Lorentzian shape
$e_{p}(R)=-e_{p}/(1+R^{2}/2\xi^{2})$ with $R=\absolutevalue{{\bf R}}$; in Sec.
III below, we will consider pinning potentials of arbitrary shape $e_{p}({\bf
R})$ but assume a small (compared to the coherence length $\xi$) extension
along $z$. ‘Integrating out’ the vortex lattice, the remaining string or
vortex is described by the effective elasticity
$\bar{C}\approx\nu\varepsilon(a_{0}^{2}/\lambda_{\rm\scriptscriptstyle
L})\sqrt{c_{66}c_{44}(0)}\sim\varepsilon\varepsilon_{0}/a_{0}$. Here,
$\varepsilon_{0}=(\phi_{0}/4\pi\lambda_{\rm\scriptscriptstyle L})^{2}$ is the
vortex line energy, $\lambda_{\rm\scriptscriptstyle L}$ denotes the London
penetration depth, $\varepsilon<1$ is the anisotropy parameter for a uniaxial
material Blatter  _et al._ (1994), and $\nu$ is a numerical, see Refs. Kwok
_et al._ , 2016; Willa  _et al._ , 2018b.

Figure 3:  Graphical illustrationBuchacek  _et al._ (2019a) of the self-
consistent solution of the microscopic force-balance equation Eq. (2) for a
Lorentzian potential with $\kappa=2.5$. The vortex coordinates $\tilde{x}$ and
$\bar{x}$ are expressed in units of $\xi$. When moving the asymptotic vortex
position $\bar{x}$ across the bistable interval $[\bar{x}_{-},\bar{x}_{+}]$,
we obtain three solutions describing pinned
$\tilde{x}_{\mathrm{p}}\lesssim\xi$, free $\tilde{x}_{\mathrm{f}}$ close to
$\bar{x}$, and unstable $\tilde{x}_{\mathrm{us}}$ states; they define the
corresponding pinned (red), free (blue), and unstable (black dotted) branches.
The tip-positions at the edges of the bistable interval denoted by
$\tilde{x}_{\mathrm{p+}}$ and $\tilde{x}_{\mathrm{f-}}$ denote jump points
where the vortex tip turns unstable, see Eq. (3); they are defined by the
condition
$f^{\prime}_{p}(\tilde{x}_{\mathrm{p+}})=f^{\prime}_{p}(\tilde{x}_{\mathrm{f-}})=\bar{C}$
(black solid dots). The associated positions $\tilde{x}_{\mathrm{f+}}$ and
$\tilde{x}_{\mathrm{p-}}$ denote the tip landing points after the jump (open
circles); they are given by the second solution of Eq. (2) at the same
asymptotic position $\bar{x}$. The open red/blue circles and the cross mark
the positions of metastable minima and the unstable maximum in Fig. 4. The
lower right inset shows the weak-pinning situation at $\kappa<1$, here
implemented with a larger $\bar{C}$, where the tip solution $\tilde{x}$ is
unique for all $\bar{x}$.

The most simple pinning geometry is for a vortex that traverses the defect
through its center. Given the rotational symmetry of the isotropic defect, we
choose a vortex that impacts the defect in a head-on collision from the left
with asymptotic coordinate $\bar{\bf R}=(\bar{x},0)$ and increase $\bar{x}$
along the $x$-axis; finite impact parameters $\bar{y}\neq 0$ will be discussed
later. The geometry then simplifies considerably and involves the asymptotic
vortex position $\bar{x}$ and the tip position $\tilde{x}$ of the vortex,
reducing the problem to a one-dimensional one; the full geometry of the
deformed string can be determined straightforwardly Willa  _et al._ (2016)
once the tip position $\tilde{x}$ has been found. The latter follows from
minimizing (1) with respect to $\tilde{x}$ at fixed asymptotic position
$\bar{x}$ and leads to the non-linear equation

$\bar{C}(\tilde{x}-\bar{x})=-\partial_{x}e_{p}|_{x=\tilde{x}}=f_{p}(\tilde{x}).$
(2)

This can be solved graphically, see Fig. 3, and produces either a single
solution or multiple solutions—the appearance of multiple tip solutions is the
signature of strong pinning. The relevant parameter that distinguishes the two
cases is found by taking the derivative of (2) with respect to $\bar{x}$ that
leads to

$\partial_{\bar{x}}\tilde{x}=\frac{1}{1-f_{p}^{\prime}(\tilde{x})/\bar{C}},$
(3)

where prime denotes the derivative,
$f^{\prime}_{p}(x)=\partial_{x}f_{p}(x)=-\partial_{x}^{2}e_{p}(x)$. Strong
pinning involves vortex instabilities, i.e., jumps in the tip coordinate
$\tilde{x}$, that appear when the denominator in (3) vanishes; this leads us
to the strong pinning parameter $\kappa$ first introduced by Labusch Labusch
(1969),

$\kappa=\max_{\tilde{x}}\frac{f^{\prime}_{p}(\tilde{x})}{\bar{C}}=\frac{f^{\prime}_{p}(\tilde{x}_{m})}{\bar{C}},$
(4)

with $\tilde{x}_{m}$ defined as the position of maximal force derivative
$f_{p}^{\prime}$, i.e., $f_{p}^{\prime\prime}(\tilde{x}_{m})=0$, or maximal
negative curvature $-e_{p}^{\prime\prime}$ of the defect potential. Defining
the force scale $f_{p}\equiv e_{p}/\xi$ and estimating the force derivative or
curvature $f_{p}^{\prime}=-e_{p}^{\prime\prime}\sim f_{p}/\xi$ produces a
Labusch parameter $\kappa\sim e_{p}/\bar{C}\xi^{2}$; for the Lorentzian
potential, we find that $f_{p}^{\prime}(\tilde{x}_{m})=e_{p}/4\xi^{2}$ at
$\tilde{x}_{m}=\sqrt{2}\,\xi$ and hence $\kappa=e_{p}/4\bar{C}\xi^{2}$. We see
that strong pinning is realized for either large pinning energy $e_{p}$ or
small effective elasticity $\bar{C}$.

As follows from Fig. 3 (inset), for $\kappa<1$ (large $\bar{C}$) the solution
to Eq. (2) is unique for all values of $\bar{x}$ and pinning is weak, while
for $\kappa>1$ (small $\bar{C}$), multiple solutions appear in the vicinity of
$\tilde{x}_{m}$ and pinning is strong. These multiple solutions appear in a
finite interval $\bar{x}\in[\bar{x}_{-},\bar{x}_{+}]$ and we denote them by
$\tilde{x}=\tilde{x}_{\mathrm{f}},\tilde{x}_{\mathrm{p}},\tilde{x}_{\mathrm{us}}$,
see Fig. 3; they are associated with free (weakly deformed vortex with
$\tilde{x}_{\mathrm{f}}$ close to $\bar{x}$), pinned (strongly deformed vortex
with $\tilde{x}_{\mathrm{p}}<\xi$), and unstable vortex states.

Inserting the solutions
$\tilde{x}(\bar{x})=\tilde{x}_{\mathrm{f}}(\bar{x}),\tilde{x}_{\mathrm{p}}(\bar{x}),\tilde{x}_{\mathrm{us}}(\bar{x})$
of Eq. (2) at a given vortex position $\bar{x}$ back into the pinning energy
$e_{\mathrm{pin}}(\tilde{x};\bar{x})$, we find the energies of the
corresponding branches,

$e^{\mathrm{i}}_{\mathrm{pin}}(\bar{x})\equiv
e_{\mathrm{pin}}[\tilde{x}_{\mathrm{i}}(\bar{x});\bar{x}],\quad\mathrm{i}=\mathrm{f,p,us}.$
(5)

The pair $e_{p}(\tilde{x})$ and $e^{\mathrm{i}}_{\mathrm{pin}}(\bar{x})$ of
energies in tip- and asymptotic spaces then has its correspondence in the
force: associated with $f_{p}(\tilde{x})$ in tip space are the force branches
$f^{\mathrm{i}}_{\mathrm{pin}}(\bar{x})$ in asymptotic $\bar{x}$-space defined
as

$f^{\mathrm{i}}_{\mathrm{pin}}(\bar{x})=f_{p}[\tilde{x}_{\mathrm{i}}(\bar{x})],\quad\textrm{i}=\mathrm{f,p,us}.$
(6)

Using Eq. (2), it turns out that the force $f_{\mathrm{pin}}$ can be written
as the total derivative of $e_{\mathrm{pin}}$,

$f_{\mathrm{pin}}(\bar{x})=-\frac{de_{\mathrm{pin}}[\tilde{x}(\bar{x});\bar{x}]}{d\bar{x}}.$
(7)

The multiple branches $e^{\mathrm{i}}_{\mathrm{pin}}$ and
$f^{\mathrm{i}}_{\mathrm{pin}}$ associated with a strong pinning situation at
$\kappa>1$ are shown in Figs. 4 and 5$(\mathrm{b})$.

Figure 4:  Multi-valued pinning energy landscape
$e_{\mathrm{pin}}^{\mathrm{i}}(\bar{x})$ for a defect producing a Lorentzian-
shaped potential with $\kappa=2.5$; the branches $\mathrm{i}=\mathrm{p,f,us}$
correspond to the pinned (red), free (blue), and unstable (black dotted)
vortex states. The bistability extends over the intervals
$|\bar{x}|\in\left[\bar{x}_{-},\bar{x}_{+}\right]$ where the different
branches coexist; pinned and free vortex branches cut at the branch crossing
point $\bar{x}=\bar{x}_{0}$. A vortex traversing the defect from left to right
assumes the free and pinned states marked with thick colored lines and
undergoes jumps $\Delta e_{\mathrm{pin}}^{\mathrm{fp}}$ and $\Delta
e_{\mathrm{pin}}^{\mathrm{pf}}$ in energy (vertical black solid lines) at the
boundaries $-\bar{x}_{-}$ and $\bar{x}_{+}$. The asymmetric occupation of
states produces a finite pinning force density $F_{\mathrm{pin}}$. Inset:
Total energy $e_{\mathrm{pin}}(\tilde{x};\bar{x})$ versus vortex tip position
$\tilde{x}$ for a fixed vortex position $\bar{x}$ (vertical dashed line in the
main figure). The points $\tilde{x}_{\mathrm{f}}$, $\tilde{x}_{\mathrm{p}}$,
and $\tilde{x}_{\mathrm{us}}$ mark the free, pinned, and unstable solutions of
the force-balance equation (2); they correspond to local minima and the
maximum in $e_{\mathrm{pin}}(\tilde{x};\bar{x})$ and are marked with
corresponding symbols in Fig. 3.

###  II.2 Unstable and bistable domains $\mathcal{U}_{\tilde{\bf R}}$ and
$\mathcal{B}_{\bar{\bf R}}$

Next, we identify the unstable (in $\tilde{x}$) and bistable (in $\bar{x}$)
domains of the pinning landscape that appear as signatures of strong pinning
when $\kappa$ increases beyond unity. Figure 5(a) shows the force profile
$f_{p}(\tilde{x})$ as experienced by the tip coordinate $\tilde{x}$. A vortex
passing the defect on a head-on trajectory from left to right undergoes a
forward jump in the tip from $-\tilde{x}_{\mathrm{f-}}$ to
$-\tilde{x}_{\mathrm{p-}}$; subsequently, the tip follows the pinned branch
until $\tilde{x}_{\mathrm{p+}}$ and then returns back to the free state with a
forward jump from $\tilde{x}_{\mathrm{p+}}$ to $\tilde{x}_{\mathrm{f+}}$. The
jump positions (later indexed by a subscript ‘$\mathrm{jp}$’) are determined
by the two solutions of the equation

$f_{p}^{\prime}(x)\Big{|}_{-\tilde{x}_{\mathrm{f-}},\tilde{x}_{\mathrm{p+}}}=\bar{C}$
(8)

that involves the curvature of the pinning potential $e_{p}(x)$; the landing
positions $-\tilde{x}_{\mathrm{p-}}$ and $\tilde{x}_{\mathrm{f+}}$ (later
indexed by a subscript ‘$\mathrm{lp}$’), on the other hand, are given by the
second solution of the force-balance equation (2) that involves the driving
term $\bar{C}(\tilde{x}-\bar{x})$ and hence depends on the asymptotic position
$\bar{x}$. Finally, the positions in asymptotic space $\bar{x}$ where the
vortex tip jumps are obtained again from the force balance equation (2),

$\displaystyle\bar{x}_{-}$ $\displaystyle=$
$\displaystyle\tilde{x}_{\mathrm{f-}}-f_{p}(\tilde{x}_{\mathrm{f-}})/\bar{C},$
(9) $\displaystyle\bar{x}_{+}$ $\displaystyle=$
$\displaystyle\tilde{x}_{\mathrm{p+}}-f_{p}(\tilde{x}_{\mathrm{p+}})/\bar{C}.$

Note that the two pairs of tip jump and landing positions,
$\tilde{x}_{\mathrm{p+}},~{}\tilde{x}_{\mathrm{f+}}$ and
$\tilde{x}_{\mathrm{f-}},~{}\tilde{x}_{\mathrm{p-}}$ are associated with only
two asymptotic positions $\bar{x}_{+}$ and $\bar{x}_{-}$.

Let us generalize the geometry and consider a vortex moving parallel to
$\bar{x}$, impacting the defect at a finite distance $\bar{y}$. We then have
to extend the above discussion to the entire $z=0$ plane, see Fig. 5. For an
isotropic defect, the jump- and landing points now define jump circles with
radii $\tilde{R}_{\mathrm{jp}}$ given by
$\tilde{R}_{\mathrm{f-}}=\tilde{x}_{\mathrm{f-}}$ and
$\tilde{R}_{\mathrm{p+}}=\tilde{x}_{\mathrm{p+}}$ (solid circles in Fig.
5$(\mathrm{c})$) and landing circles with radii $\tilde{R}_{\mathrm{lp}}$
given by $\tilde{R}_{\mathrm{f+}}=\tilde{x}_{\mathrm{f+}}$,
$\tilde{R}_{\mathrm{p-}}=\tilde{x}_{\mathrm{p-}}$ (dashed circles in Fig.
5$(\mathrm{c})$). Their combination defines an unstable ring
$\tilde{R}_{\mathrm{p+}}<\tilde{R}<\tilde{R}_{\mathrm{f-}}$ in tip space where
tips cannot reside. The existence of unstable domains $\mathcal{U}_{\tilde{\bf
R}}$ in tip space is a signature of strong pinning.

Figure 5: (a) and (b): Force profiles $f_{p}(\tilde{x})$ and
$f_{\mathrm{pin}}(\bar{x})$ in tip- and asymptotic coordinates for a
Lorentzian-shaped potential with $\kappa=2.5$. The tip of a vortex moving from
left to right along the $x$-axis approaches the defect on the free branch
(thick blue line) undergoes a jump (arrow) from $-\tilde{x}_{\mathrm{f-}}$ to
$-\tilde{x}_{\mathrm{p-}}$, follows the pinned branch (red) until
$\tilde{x}_{\mathrm{p+}}$ and then jumps back (arrow) to the free (blue) state
at $\tilde{x}_{\mathrm{f+}}$. Extending these jump positions to the
$(\tilde{x},\tilde{y})$-plane, see (c), defines jump (solid) and landing
(dashed) circles, with the jump circles enclosing an unstable domain
$\mathcal{U}_{\tilde{\bf R}}$ characteristic of strong pinning. The force
profile $f_{\mathrm{pin}}(\bar{x})$ in $(\mathrm{b})$ includes free (blue),
pinned (red), and unstable branches (black dotted). (d) Extending the bistable
intervals $[-\bar{x}_{+},-\bar{x}_{-}]$ and $[\bar{x}_{-},\bar{x}_{+}]$ to the
$[\bar{x},\bar{y}]$-plane defines a bistable ring $\mathcal{B}_{\bar{\bf R}}$
(magenta), again a strong pinning characteristic. The dashed circle of radius
$\bar{R}_{0}$ in (d) marks the branch crossing point. Vortices passing the
defect with a finite impact parameter $\bar{y}\neq 0$ move on a straight line
in asymptotic space, see (d); the associated trajectory in tip space is
nontrivial, see $(\mathrm{c})$ and undergoes jumps at pinning (circle
$\tilde{R}_{\mathrm{f-}}$) and depinning (circle $\tilde{R}_{\mathrm{p+}}$).

Figures 5$(\mathrm{b})$ and $(\mathrm{d})$ show the corresponding results in
asymptotic coordinates $\bar{x}$ and $\bar{\bf R}$, respectively. The pinning
force $f_{\mathrm{pin}}(\bar{x})=f_{p}[\tilde{x}(\bar{x})]$ shown in
$(\mathrm{b})$ is simply an ‘outward tilted’ version of $f_{p}(\tilde{x})$,
with $S$-shaped overhangs that generate bistable intervals
$[-\bar{x}_{+},-\bar{x}_{-}]$ and $[\bar{x}_{-},\bar{x}_{+}]$. Extending them
to the asymptotic $\bar{\bf R}$-plane with radii
$\bar{R}_{-}\equiv\bar{x}_{-}$ and $\bar{R}_{+}\equiv\bar{x}_{+}$, see Fig.
5$(\mathrm{d})$, we obtain a ring $\bar{R}_{-}<\bar{R}<\bar{R}_{+}$ that marks
the location of bistability. Again, the appearance of bistable domains
$\mathcal{B}_{\bar{\bf R}}$ in asymptotic space is a signature of strong
pinning. Both, the size of the unstable- and bistable rings depend on the
Labusch parameter $\kappa$; they appear out of circles with radii
$\tilde{R}=\tilde{x}_{m}$ and
$\bar{R}=\bar{x}_{m}=\tilde{x}_{m}-f_{p}(\tilde{x}_{m})/\bar{C}$ at $\kappa=1$
and grow in radius and width when $\kappa$ increases. The unstable and
bistable domains $\mathcal{U}_{\tilde{\bf R}}$ and $\mathcal{B}_{\bar{\bf R}}$
(see Ref. Buchacek, 2020) will exhibit interesting non-trivial behavior as a
function of $\kappa$ when generalizing the analysis to defect potentials of
arbitrary shape.

####  II.2.1 Alternative strong pinning formulation

An alternative formulation of strong pinning physics is centered on the local
differential properties of the pinning energy
$e_{\mathrm{pin}}(\tilde{x};\bar{x})$, i.e., its extremal points in
$\tilde{x}$ at different values of the asymptotic coordinate $\bar{x}$. We
start from equation (1) restricted to one dimension and rearrange terms to
arrive at the expression

$e_{\mathrm{pin}}(\tilde{x};\bar{x})=e_{\mathrm{eff}}(\tilde{x})-\bar{C}\bar{x}\>\tilde{x}+\bar{C}\bar{x}^{2}/2$
(10)

with the effective pinning energy

$e_{\mathrm{eff}}(\tilde{x})=e_{p}(\tilde{x})+\bar{C}\tilde{x}^{2}/2$ (11)

involving both pinning and elastic terms. Equation (10) describes a particle
at position $\tilde{x}$ subject to the potential $e_{\mathrm{eff}}(\tilde{x})$
and the force term $f\>\tilde{x}=-\bar{C}\bar{x}\>\tilde{x}$, see also Ref.
Willa  _et al._ , 2022. The potential $e_{\mathrm{eff}}(\tilde{x})$ can trap
two particle states if there is a protecting maximum with negative curvature
$\partial_{\tilde{x}}^{2}e_{\mathrm{eff}}=\partial_{\tilde{x}}^{2}e_{\mathrm{pin}}<0$,
preventing its escape from the metastable state at forces
$f=\pm\bar{C}\bar{x}$ with $\bar{x}\in[\bar{x}_{+},\bar{x}_{-}]$; the maximum
in $e_{\mathrm{pin}}$ at $\tilde{x}_{\mathrm{us}}$ then separates two minima
in $e_{\mathrm{pin}}$ defining distinct branches with different tip
coordinates $\tilde{x}_{\mathrm{p}}$ and $\tilde{x}_{\mathrm{f}}$, see the
inset of Fig. 4.

As the asymptotic position $\bar{x}$ approaches the boundaries
$\bar{x}_{\pm}$, one of the minima joins up with the maximum to define an
inflection point with

$[\partial_{\tilde{x}}^{2}e_{\mathrm{eff}}]_{\tilde{x}_{\mathrm{jp}}}=[\partial_{\tilde{x}}^{2}e_{\mathrm{pin}}]_{\tilde{x}_{\mathrm{jp}}}=0,$
(12)

that corresponds to the instability condition (8) where the vortex tip jumps;
the persistent second minimum in $e_{\mathrm{pin}}(\tilde{x};\bar{x})$ defines
the landing position $\tilde{x}_{\mathrm{lp}}$ and the condition for a flat
inflection point
$[\partial_{\tilde{x}}e_{\mathrm{pin}}]_{\tilde{x}_{\mathrm{jp}}}=0$ defines
the associated asymptotic coordinate $\pm\bar{x}_{\pm}$.

Finally, strong pinning vanishes at the Labusch point $\kappa=1$, with the
inflection point in $e_{\mathrm{eff}}(\tilde{x})$ coalescing with the second
minimum at $\tilde{x}_{m}$, hence

$\displaystyle[\partial_{\tilde{x}}^{2}e_{\mathrm{eff}}]_{\tilde{x}_{m}}$
$\displaystyle=$ $\displaystyle 0\quad\textrm{and}$ (13)
$\displaystyle[\partial_{\tilde{x}}^{3}e_{\mathrm{eff}}]_{\tilde{x}_{m}}$
$\displaystyle=$
$\displaystyle[\partial_{\tilde{x}}^{3}e_{p}]_{\tilde{x}_{m}}=0.$

Note the subtle use of $e_{\mathrm{pin}}$ versus $e_{\mathrm{eff}}$ versus
$e_{p}$ in the above discussion; as we go to higher derivatives, first the
asymptotic coordinate $\bar{x}$ turns irrelevant in the second derivative
$\partial_{\tilde{x}}^{2}e_{\mathrm{pin}}=\partial_{\tilde{x}}^{2}e_{\mathrm{eff}}$
and then all of the elastic response, i.e., $\bar{C}$, drops out in the third
derivative
$[\partial_{\tilde{x}}^{3}e_{\mathrm{pin}}]=[\partial_{\tilde{x}}^{3}e_{p}]$.

The above alternative formulation of strong pinning turns out helpful in
several discussions below, e.g., the derivation of strong pinning
characteristics near the transition in Secs. II.4 and III.1 and in the
generalization of the instability condition to an anisotropic defect in Sec.
III and furthermore provides an inspiring link to the Landau theory of phase
transitions discussed below in Sec. II.6.

###  II.3 Pinning force density $F_{\mathrm{pin}}$

Next, we determine the pinning force density $F_{\mathrm{pin}}$ at strong
pinning, assuming a random homogeneous distribution of pins with a small
density $n_{p}$, $n_{p}a_{0}\xi^{2}\ll 1$, see Refs. Willa  _et al._ , 2016;
Buchacek  _et al._ , 2019a. The derivation of $F_{\mathrm{pin}}$ is
conveniently done in asymptotic $\bar{\bf R}$ coordinates where vortex
trajectories follow simple straight lines. Vortices approach the pin by
following the free branch until its termination, jump to the pinned branch to
again follow this to its termination, and finally jump back to the free
branch. This produces an asymmetric pinned-branch occupation $p_{c}(\bar{\bf
R})$ that leads to the pinning force density (we assume vortices approaching
the defect along $\bar{x}$ from the left; following convention, we include a
minus sign)

$\displaystyle\mathbf{F}_{c}$ $\displaystyle=-n_{p}\int\frac{d^{2}\bar{\bf
R}}{a_{0}^{2}}\bigl{[}p_{c}(\bar{\bf
R})\mathbf{f}^{\mathrm{p}}_{\mathrm{pin}}(\bar{\bf R})+(1-p_{c}(\bar{\bf
R}))\mathbf{f}^{\mathrm{f}}_{\mathrm{pin}}(\bar{\bf R})\bigr{]}$
$\displaystyle=-n_{p}\int\frac{d^{2}\bar{\bf R}}{a_{0}^{2}}p_{c}(\bar{\bf
R})[\partial_{x}\Delta e^{\mathrm{fp}}_{\mathrm{pin}}(\bar{\bf
R})]\,\mathbf{e}_{\bar{x}},$ (14)

with the energy difference $\Delta e^{\mathrm{fp}}_{\mathrm{pin}}(\bar{\bf
R})=e^{\mathrm{f}}_{\mathrm{pin}}(\bar{\bf
R})-e^{\mathrm{p}}_{\mathrm{pin}}(\bar{\bf R})$ and $\mathbf{e}_{\bar{x}}$ the
unit vector along $\bar{x}$; the $\bar{y}$-component of the pinning force
density vanishes due to the antisymmetry in $f_{\mathrm{pin},\bar{y}}$. For
the isotropic defect, the jumps $\Delta
e^{\mathrm{fp}}_{\mathrm{pin}}(\bar{\bf R})$ in energy appearing upon changing
branches are independent of angle and the average in (II.3) separates in
$\bar{x}$ and $\bar{y}$ coordinates; note that the energy jumps are no longer
constant for an anisotropic defect and hence such a separation does not occur.
Furthermore, i) all vortices approaching the defect within the transverse
length $|\bar{y}|<\bar{R}_{-}$ get pinned, see Fig. 5(d), while those passing
further away follow a smooth (weak pinning) trajectory that does not undergo
jumps and hence do not contribute to the pinning force, and ii) all vortices
that get pinned contribute the same force that is most easily evaluated for a
head-on vortex–defect collision on the $\bar{x}$-axis with
$p_{c}(\bar{x})=\Theta(\bar{x}+\bar{x}_{-})-\Theta(\bar{x}-\bar{x}_{+})$ and

$\displaystyle\langle f_{\mathrm{pin}}\rangle$
$\displaystyle=-\\!\\!\int_{-a_{0}/2}^{a_{0}/2}\frac{d\bar{x}}{a_{0}}\>\bigl{[}p_{c}(\bar{x})f^{\mathrm{p}}_{\mathrm{pin}}(\bar{x})+(1-p_{c}(\bar{x}))f^{\mathrm{f}}_{\mathrm{pin}}(\bar{x})\bigr{]}$
$\displaystyle=\frac{\Delta
e^{\mathrm{fp}}_{\mathrm{pin}}(-\bar{x}_{-})+\Delta
e^{\mathrm{pf}}_{\mathrm{pin}}(\bar{x}_{+})}{a_{0}},$ (15)

where we have replaced $-\Delta e^{\mathrm{fp}}_{\mathrm{pin}}(\bar{x}_{+})$
by $\Delta e^{\mathrm{pf}}_{\mathrm{pin}}(\bar{x}_{+})>0$. Hence, the average
pinning force $\langle f_{\mathrm{pin}}\rangle$ is given by the jumps in the
pinning energy $e_{\mathrm{pin}}^{\mathrm{i}}(\bar{x})$ associated with
different branches $\mathrm{i}=\mathrm{p,f}$, see Fig. 4.

Finally, accounting for trajectories with finite impact parameter
$|\bar{y}|<\bar{R}_{-}$, we arrive at the result for the pinning force density
$F_{\mathrm{pin}}$ acting on the vortex system,

$F_{\mathrm{pin}}=n_{p}\frac{2\bar{R}_{-}}{a_{0}}\langle
f_{\mathrm{pin}}\rangle=n_{p}\frac{2\bar{R}_{-}}{a_{0}}\frac{\Delta
e^{\mathrm{fp}}_{\mathrm{pin}}+\Delta e^{\mathrm{pf}}_{\mathrm{pin}}}{a_{0}},$
(16)

where the factor $2\bar{\bf R}_{\mathrm{-}}/{a_{0}}$ accounts for the
averaging of the pinning force along the $y$-axis. As strong pins act
independently, a consequence of the small defect density $n_{p}$, the pinning
force density is linear in the defect density, $F_{\mathrm{pin}}\propto
n_{p}$. If pinning is weak, i.e., $\kappa<1$, we have no jumps, $\langle
f_{\mathrm{pin}}\rangle=0$, and $F_{\mathrm{pin}}|_{\mathrm{strong}}=0$. A
finite pinning force then only arises from correlations between pinning
defects and scales in density as Larkin and Ovchinnikov (1979); Blatter  _et
al._ (2004) $F_{\mathrm{pin}}|_{\mathrm{weak}}\propto n_{p}^{2}$. This
contribution to the pinning force density $F_{\mathrm{pin}}$ continues beyond
$\kappa=1$, hence, while the strong pinning onset at $\kappa=1$ can be
formulated in terms of a transition, weak pinning goes to strong pinning in a
smooth crossover.

Knowing the pinning force density $F_{\mathrm{pin}}$, the motion of the vortex
lattice follows from the bulk dynamical equation

$\eta\mathbf{v}=\mathbf{F}_{\rm\scriptscriptstyle
L}(\mathbf{j})-\mathbf{F}_{\mathrm{pin}}.$ (17)

Here, $\eta=BH_{c2}/\rho_{n}c^{2}$ is the Bardeen-Stephen viscosity Bardeen
and Stephen (1965) (per unit volume; $\rho_{n}$ is the normal state
resistivity) and $\mathbf{F}_{\rm\scriptscriptstyle
L}=\mathbf{j}\times\mathbf{B}/c$ is the Lorentz force density driving the
vortex system. The pinning force density $\mathbf{F}_{\mathrm{pin}}$ is
directed along $\mathbf{v}$, in our case along $x$.

Next, we determine the strong pinning characteristics $\bar{x}_{-}$,
$\bar{x}_{+}$, $\tilde{x}_{\mathrm{f}\pm}$, $\tilde{x}_{\mathrm{p}\pm}$,
$\Delta e^{\mathrm{fp}}_{\mathrm{pin}}$ and $\Delta
e^{\mathrm{pf}}_{\mathrm{pin}}$ as a function of the Labusch parameter
$\kappa$ close to the strong pinning transition, i.e., $\kappa\gtrsim 1$.

###  II.4 Strong pinning characteristics near the transition

Near the strong pinning transition at $\kappa\gtrsim 1$, we can derive
quantitative results for the strong pinning characteristics by expanding the
pinning energy $e_{\mathrm{pin}}(\tilde{x};\bar{x})$ in $\tilde{x}$ at fixed
$\bar{x}$; this reminds about the Landau expansion of the free energy
$f(\phi,h)$ in the order parameter $\phi$ at a fixed field $h$ in a
thermodynamic transition, see Sec. II.6 below for a detailed discussion.

We expand $e_{\mathrm{pin}}(\tilde{x};\bar{x})$ in $\tilde{x}$ around the
point of first instability $\tilde{x}_{m}$ by introducing the relative tip and
asymptotic positions $\tilde{u}=\tilde{x}-\tilde{x}_{m}$ and
$\bar{u}=\bar{x}-\bar{x}_{m}$ and make use of our alternative strong pinning
formulation summarized in Sec. II.2.1. At $\tilde{x}_{m}$ and close to
$\kappa=1$, we have
$[\partial_{\tilde{x}}^{2}e_{\mathrm{pin}}]_{\tilde{x}_{m}}=[\partial_{\tilde{x}}^{2}e_{p}]_{\tilde{x}_{m}}+\bar{C}=\bar{C}(1-\kappa)$
and $[\partial_{\tilde{x}}^{3}e_{\mathrm{pin}}]_{\tilde{x}_{m}}=0$, hence,

$\displaystyle
e_{\mathrm{pin}}(\tilde{x};\bar{x})\approx\frac{\bar{C}}{2}(1-\kappa)\>\tilde{u}^{2}+\frac{\gamma}{24}\>\tilde{u}^{4}-\bar{C}\bar{u}\tilde{u},$
(18)

where we have introduced the shape parameter
$\gamma=[\partial^{4}_{x}e_{p}]_{\tilde{x}_{m}}$ describing the quartic term
in the expansion and we have made use of the force balance equation (2) to
rewrite $f_{p}(\tilde{x}_{m})=\bar{C}(\tilde{x}_{m}-\bar{x}_{m})$;
furthermore, we have dropped all irrelevant terms that do not depend on
$\tilde{u}$.

We find the jump and landing positions $\tilde{x}_{\mathrm{jp}}$ and
$\tilde{x}_{\mathrm{lp}}$ exploiting the differential properties of
$e_{\mathrm{pin}}(\tilde{x})$ at a fixed $\bar{x}$: As discussed above, the
vortex tip jumps at the boundaries $\bar{x}_{\pm}$ of the bistable regime,
where $e_{\mathrm{pin}}$ develops a flat inflection point at
$\tilde{x}_{\mathrm{jp}}$ with one minimum joining up with the unstable
maximum and the second minimum at the landing position
$\tilde{x}_{\mathrm{lp}}$ staying isolated. Within our fourth-order expansion
the jump positions at (de)pinning are placed symmetrically with respect to the
onset at $\tilde{x}_{m}$,

$\tilde{x}_{\mathrm{p+}}=\tilde{x}_{m}+\tilde{u}_{\mathrm{jp}},~{}~{}~{}\tilde{x}_{\mathrm{f-}}=\tilde{x}_{m}-\tilde{u}_{\mathrm{jp}}$
(19)

and imposing the condition
$[\partial_{\tilde{u}}^{2}e_{\mathrm{pin}}]_{\tilde{x}_{\mathrm{jp}}}=0$ (that
is equivalent to the jump condition
$f_{p}^{\prime}[\tilde{x}_{\mathrm{f-}}]=f_{p}^{\prime}[\tilde{x}_{\mathrm{p+}}]=\bar{C}$
of Eq. (8), see also Fig. 3), we find that

$\tilde{u}_{\mathrm{jp}}\approx-\sqrt{\frac{2\bar{C}}{\gamma}}(\kappa-1)^{1/2}.$
(20)

In order to find the (symmetric) landing positions, it is convenient to shift
the origin of the expansion to the jump position,
$\tilde{u}\to\tilde{u}-\tilde{u}_{\mathrm{jp}}\equiv\tilde{u}^{\prime}$, and
define the jump distance $\Delta\tilde{u}$,

$\displaystyle\tilde{x}_{\mathrm{f+}}=\tilde{x}_{\mathrm{p+}}+\Delta\tilde{u},~{}~{}~{}\tilde{x}_{\mathrm{p-}}=\tilde{x}_{\mathrm{f-}}-\Delta\tilde{u}.$
(21)

At the jump position, the linear and quadratic terms in $\tilde{u}^{\prime}$
vanish, resulting in the expansion (up to an irrelevant constant)

$e_{\mathrm{pin}}(\tilde{x}_{\mathrm{p+}}+\tilde{u}^{\prime};\bar{x}_{+})\approx\frac{\gamma}{6}\tilde{u}_{\mathrm{jp}}\tilde{u}^{\prime\,3}+\frac{\gamma}{24}\tilde{u}^{\prime\,4}$
(22)

and similar at $\tilde{x}_{\mathrm{f-}}$ and $\bar{x}_{-}$ for a left moving
vortex. This expression is minimal at the landing position
$\tilde{x}_{\mathrm{f+}}$, i.e., at $\tilde{u}^{\prime}=\Delta\tilde{u}$,
$[\partial_{\tilde{u}^{\prime}}e_{\mathrm{pin}}]_{\Delta\tilde{u}}=0$, and we
find the jump distance

$\Delta\tilde{u}=-3\tilde{u}_{\mathrm{jp}}.$ (23)

Inserting this result back into (22), we obtain the jump in energy $\Delta
e_{\mathrm{pin}}^{\mathrm{pf}}=e_{\mathrm{pin}}(\tilde{x}_{\mathrm{p+}};\bar{x}_{+})-e_{\mathrm{pin}}(\tilde{x}_{\mathrm{f+}};\bar{x}_{+})$,

$\Delta
e_{\mathrm{pin}}^{\mathrm{pf}}(\bar{x}_{+})\approx\frac{\gamma}{72}(\Delta\tilde{u})^{4}\approx\frac{9\bar{C}^{2}}{2\gamma}(\kappa-1)^{2},$
(24)

and similar at $\bar{x}_{-}$. Note that all these results have been obtained
without explicit knowledge of the asymptotic coordinates $\bar{x}_{\pm}$ where
these tip jumps are triggered. The latter follow from the force equation (2)
that corresponds to the condition
$[\partial_{\tilde{x}}e_{\mathrm{pin}}]_{\tilde{x}_{\mathrm{jp}}}=0$ for a
flat inflection point. Using the expansion (18) of the pinning energy, we find
that

$\bar{x}_{\pm}-\bar{x}_{m}=\mp\frac{2}{3}\tilde{u}_{\mathrm{jp}}(\kappa-1)=\pm\frac{2}{3}\sqrt{\frac{2\bar{C}}{\gamma}}(\kappa-1)^{3/2}.$
(25)

The pair $\bar{x}_{m}$ and $\tilde{x}_{m}$ of asymptotic and tip positions
depends on the details of the potential; while $\tilde{x}_{m}$ derives solely
from the shape $e_{p}(\tilde{x})$, $\bar{x}_{m}$ as given by (2) involves
$\bar{C}$ and shifts $\propto(\kappa-1)$. For a Lorentzian potential, we find
that

$\tilde{x}_{m}=\sqrt{2}\xi,\quad\bar{x}_{m}=2\sqrt{2}\xi+\sqrt{2}\xi(\kappa-1).$
(26)

The shape coefficient is $\gamma=3e_{p}/4\xi^{4}$ and the Labusch parameter is
given by $\kappa=e_{p}/4\bar{C}\xi^{2}$ (hence
$\bar{C}^{2}/\gamma=e_{p}/12\kappa^{2}$), providing us with the results

$\tilde{u}_{\mathrm{jp}}\approx-\xi\,[2(\kappa-1)/3]^{1/2}\mathrm{~{}~{}and~{}~{}}\Delta
e_{\mathrm{pin}}^{\mathrm{pf}}\approx\frac{3}{8}e_{p}(\kappa-1)^{2}.$ (27)

###  II.5 Pinning force density for the isotropic defect

Using the results of Sec. II.4 in the expression (16) for the pinning force
density, we find, to leading order in $\kappa-1$,

$F_{\mathrm{pin}}=9n_{p}\frac{\bar{x}_{m}}{a_{0}}\frac{\bar{C}^{2}}{\gamma
a_{0}}(\kappa-1)^{2}.$ (28)

The scaling $F_{\mathrm{pin}}\sim n_{p}(\xi/a_{0})^{2}f_{p}(\kappa-1)^{2}$
(with $\bar{C}\xi^{2}/e_{p}\sim 1/\kappa$, up to a numerical) uniquely derives
from the scaling $\propto(\kappa-1)^{2}$ of the energy jumps in (24), as the
asymptotic trapping length $\bar{x}_{-}\sim\xi$ remains finite as $\kappa\to
1$ for the isotropic defect; this will change for the anisotropic defect.

###  II.6 Relation to Landau’s theory of phase transitions

The expansion (18) of the pinning energy $e_{\mathrm{pin}}(\tilde{x};\bar{x})$
around the inflection point $\tilde{x}_{m}$ of the force takes the same form
as the Landau free energy of a phase transitionBlatter  _et al._ (2004),

$\displaystyle f(\phi;h)$ $\displaystyle=$
$\displaystyle\frac{r_{0}}{2}(T/T_{c}-1)\phi^{2}+u\phi^{4}-h\phi,$ (29)

with the straightforward transcription $\tilde{u}\leftrightarrow\phi$,
$\bar{C}(1-\kappa)\leftrightarrow r_{0}(T/T_{c}-1)$, $\gamma/24\leftrightarrow
u$ and the conjugate field $\bar{C}\bar{u}\leftrightarrow h$. The functional
(29) describes a one-component oder parameter $\phi$ driven by $h$, e.g., an
Ising model with magnetization density $\phi$ in an external magnetic field
$h$. This model develops a mean-field transition with a first-order line in
the $h$–$T$ phase diagram that terminates in a critical point at $T=T_{c}$ and
$h=0$. The translation to strong pinning describes a strong pinning region at
large $\kappa$ that terminates (upon decreasing $\kappa$) at $\kappa=1$. The
ferromagnetic phases with $\phi=\pm\sqrt{r_{0}(1-T/T_{c})/4u}$ correspond to
pinned and unpinned states, the paramagnetic phase at $T>T_{c}$ with $\phi=0$
translates to the unpinned domain at $\kappa<1$. The spinodals associated with
the hysteresis in the first-order magnetic transition correspond to the
termination of the free and pinned branches at $\bar{x}_{\pm}$; indeed, the
flat inflection points appearing in $e_{\mathrm{pin}}(\tilde{x};\bar{x})$ at
the boundaries of the bistable region $\mathcal{B}_{\bar{\bf R}}$ as discussed
in Sec. II.2 correspond to the disappearance of metastable magnetic phases in
(29) at the spinodals of the first-order transition where
$\partial_{\phi}f(\phi;h)=\partial_{\phi}^{2}f(\phi;h)=0$. When including
correlations between defects, the unpinned phase at $\kappa<1$ transforms into
a weakly pinned phase that continues beyond $\kappa=1$ into the strongly
pinned phase. Including such correlations, the strong-pinning transition at
the onset of strong pinning at $\kappa=1$ transforms into a weak-to-strong
pinning crossover.

##  III Anisotropic defects

Let us generalize the above analysis to make it fit for the ensuing discussion
of an arbitrary pinning landscape or short, pinscape. Central to the
discussion are the unstable and bistable domains $\mathcal{U}_{\tilde{\bf R}}$
and $\mathcal{B}_{\bar{\bf R}}$ in tip- and asymptotic space. The boundary of
the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ in tip space is determined
by the jump positions of the vortex tip. The latter follows from the local
differential properties of $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ at
fixed asymptotic coordinate $\bar{\bf R}$, for the isotropic defect, the
appearence of an inflection point
$[\partial_{\tilde{x}}^{2}e_{\mathrm{pin}}(\tilde{x},\bar{x})]=0$, see Eq.
(12). In generalizing this condition to the anisotropic situation, we have to
study the Hessian matrix of $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$
defined in Eq. (1),

$\bigl{[}\mathrm{Hess}\bigl{[}e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf
R})|_{\bar{\bf
R}}\bigr{]}\bigr{]}_{ij}=\bar{C}\delta_{ij}+\mathrm{H}_{ij}(\tilde{\bf R})$
(30)

with

$\mathrm{H}_{ij}(\tilde{\bf
R})=\partial_{\tilde{x}_{i}}\partial_{\tilde{x}_{j}}e_{p}(\tilde{\bf
R};\bar{\bf R})$ (31)

the Hessian matrix associated with the defect potential $e_{p}(\tilde{\bf
R})$. The vortex tip jumps when the pinning landscape
$e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ at fixed $\bar{\bf R}$ opens up
in an unstable direction, i.e., develops an inflection point; this happens
when the lower eigenvalue $\lambda_{-}(\tilde{\bf R})<0$ of the Hessian matrix
$\mathrm{H}_{ij}(\tilde{\bf R})$ matches up with $\bar{C}$,

$\displaystyle\lambda_{-}(\tilde{\bf R})+\bar{C}=0,$ (32)

and strong pinning appears in the location where this happens first, say in
the point $\tilde{\bf R}_{m}$, implying that the eigenvalue
$\lambda_{-}(\tilde{\bf R})$ has a minimum at $\tilde{\bf R}_{m}$.
Furthermore, the eigenvector $\mathbf{v}_{-}(\tilde{\bf R}_{m})$ associated
with the eigenvalue $\lambda_{-}(\tilde{\bf R}_{m})$ provides the unstable
direction in the pinscape $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ along
which the vortex tip escapes.

Defining the reduced curvature function

$\displaystyle\kappa(\tilde{\bf R})\equiv\frac{-\lambda_{-}(\tilde{\bf
R})}{\bar{C}},$ (33)

we find the generalized Labusch parameter

$\displaystyle\kappa_{m}\equiv\kappa(\tilde{\bf R}_{m}),$ (34)

and the Labusch criterion takes the form

$\displaystyle\kappa_{m}=1.$ (35)

The latter has to be read as a double condition: i) find the location
$\tilde{\bf R}_{m}$ where the smaller eigenvalue $\lambda_{-}(\tilde{\bf R})$
is negative and largest, from which ii), one obtains the critical elasticity
$\bar{C}$ where strong pinning sets in.

A useful variant of the strong pinning condition (32) is provided by the
representation of the determinant of the Hessian matrix,

$\displaystyle D(\tilde{\bf R})$
$\displaystyle\equiv\det\\{\mathrm{Hess}\bigl{[}e_{\mathrm{pin}}(\tilde{\bf
R};\bar{\bf R})|_{\bar{\bf R}}\bigr{]}\bigr{\\}},$ (36)

in terms of its eigenvalues $\lambda_{\pm}(\tilde{\bf R})$, $D(\tilde{\bf
R})=[\bar{C}+\lambda_{-}(\tilde{\bf R})][\bar{C}+\lambda_{+}(\tilde{\bf R})]$;
near onset, the second factor $\bar{C}+\lambda_{+}(\tilde{\bf R})$ stays
positive and the strong pinning onset appears in the point $\tilde{\bf R}_{m}$
where $D(\tilde{\bf R})$ has a minimum which touches zero for the first time,
i.e., the two conditions $\nabla D(\tilde{\bf R})|_{\tilde{\bf R}_{m}}=0$ and
$D(\tilde{\bf R}_{m})=0$ are satisfied simultaneously. The latter conditions
make sure that the minima of $\lambda_{-}(\tilde{\bf R})$ and $D(\tilde{\bf
R})$ line up at $\tilde{\bf R}_{m}$. Note that the Hessian determinant
$D(\tilde{\bf R})$ does not depend on the asymptotic coordinate $\bar{\bf R}$
as it involves only second derivatives of $e_{\mathrm{pin}}(\tilde{\bf
R};\bar{\bf R})$.

The Labusch criterion defines the situation where jumps of vortex tips appear
for the first time in the isolated point $\tilde{\bf R}_{m}$. Increasing the
pinning strength, e.g., by decreasing the elasticity $\bar{C}$ for a fixed
pinning potential $e_{p}({\bf R})$ (alternatively, the pinning scale $e_{p}$
could be increased at fixed $\bar{C}$) the condition (32) is satisfied on the
boundary of a finite domain and we can define the unstable domain
$\mathcal{U}_{\tilde{\bf R}}$ through (see also Ref. Buchacek, 2020)

$\displaystyle\mathcal{U}_{\tilde{\bf R}}=\left\\{\tilde{\bf
R}~{}~{}|~{}~{}\lambda_{-}(\tilde{\bf R})+\bar{C}\leq 0\right\\}.$ (37)

Once the latter has been determined, the bistable domain
$\mathcal{B}_{\bar{\bf R}}$ follows straightforwardly from the force balance
equation

$\displaystyle\bar{C}(\tilde{\bf R}-\bar{\bf R})={\bf f}_{p}(\tilde{\bf
R})={\bf f}_{\mathrm{pin}}(\bar{\bf R}),$ (38)

i.e.,Buchacek (2020)

$\displaystyle\mathcal{B}_{\bar{\bf R}}=\left\\{\bar{\bf R}=\tilde{\bf R}-{\bf
f}_{p}(\tilde{\bf R})/\bar{C}~{}~{}|~{}~{}\tilde{\bf
R}\in\mathcal{U}_{\tilde{\bf R}}\right\\}.$ (39)

In a last step, one then evaluates the energy jumps appearing at the boundary
of $\mathcal{B}_{\bar{\bf R}}$ and proper averaging produces the pinning force
density $\mathbf{F}_{\mathrm{pin}}$.

Let us apply the above generalized formulation to the isotropic situation.
Choosing cylindrical coordinates $(r,\varphi)$, the Hessian matrix
$\mathrm{H}_{ij}$ is already diagonal; close to the inflection point
$\tilde{R}_{m}$, where $e_{p}^{\prime\prime\prime}(\tilde{R}_{m})=0$, the
eigenvalues are $\lambda_{-}(\tilde{R})=e_{p}^{\prime\prime}(\tilde{R})<0$ and
$\lambda_{+}(\tilde{R})=e_{p}^{\prime}(\tilde{R})/\tilde{R}>0$, producing
results in line with our discussion above.

###  III.1 Expansion near strong pinning onset

With our focus on the strong pinning transition near $\kappa(\tilde{\bf
R}_{m})=1$, we can obtain quantitative results using the expansion of the
pinning energy $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$, Eq. (1), close
to $\tilde{\bf R}_{m}$, cf. Sec. II.4. Hence, we construct the Landau-type
pinning energy corresponding to (29) for the case of an anisotropic pinning
potential, i.e., we generalize (18) to the two-dimensional situation.

When generalizing the strong pinning problem to the anisotropic situation, we
are free to define local coordinate systems $(\tilde{u},\tilde{v})$ and
$(\bar{u},\bar{v})$ in tip- and asymptotic space centered at $\tilde{\bf
R}_{m}$ and $\bar{\bf R}_{m}$, where the latter is associated with $\tilde{\bf
R}_{m}$ through the force balance equation (38) in the original laboratory
system. Furthermore, we fix our axes such that the unstable direction
coincides with the $u$-axis, i.e., the eigenvector ${\bf v}_{-}(\tilde{\bf
R}_{m})$ associated with $\lambda_{-}(\tilde{\bf R}_{m})$ points along $u$; as
a result, the mixed term $\propto\tilde{u}\tilde{v}$ is absent from the
expansion. Keeping all potentially relevant terms up to fourth order in
$\tilde{u}$ and $\tilde{v}$ in the expansion, we then have to deal with an
expression of the form

$\displaystyle e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf
R})=\frac{\bar{C}+\lambda_{-}}{2}\,\tilde{u}^{2}+\frac{\bar{C}+\lambda_{+}}{2}\,\tilde{v}^{2}-\bar{C}\,\bar{u}\tilde{u}-\bar{C}\,\bar{v}\tilde{v}$
$\displaystyle\quad+\frac{a}{2}\,\tilde{u}\tilde{v}^{2}+\frac{a^{\prime}}{2}\,\tilde{u}^{2}\tilde{v}+\frac{b^{\prime}}{6}\,\tilde{u}^{3}+\frac{b^{\prime\prime}}{6}\,\tilde{v}^{3}$
(40)
$\displaystyle\qquad+\frac{\alpha}{4}\,\tilde{u}^{2}\tilde{v}^{2}+\frac{\beta}{6}\,\tilde{u}^{3}\tilde{v}+\frac{\beta^{\prime\prime}}{6}\,\tilde{u}\tilde{v}^{3}+\frac{\gamma}{24}\,\tilde{u}^{4}+\frac{\gamma^{\prime\prime}}{24}\,\tilde{v}^{4},$

with $\lambda_{\pm}=\lambda_{\pm}(\tilde{\bf R}_{m})$,

$\displaystyle\tilde{\bf R}$ $\displaystyle=\tilde{\bf R}_{m}+\delta\tilde{\bf
R},\quad\delta\tilde{\bf R}=(\tilde{u},\tilde{v}),$ (41)
$\displaystyle\bar{\bf R}$ $\displaystyle=\bar{\bf R}_{m}+\delta\bar{\bf
R},\quad\delta\bar{\bf R}=(\bar{u},\bar{v}),$

and coefficients given by the corresponding derivatives of $e_{p}(\bf R)$,
e.g., $a\equiv\partial_{u}\partial_{v}^{2}e_{p}({\bf R})|_{\tilde{\bf
R}_{m}}$, $\dots$, $\gamma^{\prime\prime}\equiv\partial_{v}^{4}e_{p}({\bf
R})|_{\tilde{\bf R}_{m}}$. As we are going to see, the primed terms in this
expansion vanish due to the condition of a minimal Hessian determinant at the
onset of strong pinning, while double-primed terms will turn out irrelevant to
leading order in the small distortions $\tilde{u}$ and $\tilde{v}$.

The first term in (40) drives the strong pinning transition as it changes sign
when $\lambda_{-}=-\bar{C}$. Making use of the Labusch parameter $\kappa_{m}$
defined in (34), we can replace (see also (18))

$\bar{C}+\lambda_{-}\to\bar{C}(1-\kappa_{m}).$ (42)

In our further considerations below, the quantity $\kappa_{m}-1\ll 1$ acts as
the small parameter; it assumes the role of the distance $1-T/T_{c}$ to the
critical point in the Landau expansion of a thermodynamic phase transition.

The second term in (40) stabilizes the theory along the $v$ direction as
$\bar{C}+\lambda_{+}>0$ close to the Labusch point, while the sign of the
cubic term $a\,\tilde{u}\tilde{v}^{2}/2$ determines the direction of the
instability along $x$, i.e., to the right ($a>0$) or left ($a<0$). The quartic
terms $\propto\alpha,\gamma>0$ bound the pinning energy at large distances,
while the term $\propto\beta$ determines the skew angle in the shape of the
unstable domain $\mathcal{U}_{\tilde{\bf R}}$, see below. Finally, we have
used the force balance equation (38) in the derivation of the driving terms
$\bar{C}\,\bar{u}\tilde{u}$ and $\bar{C}\,\bar{v}\tilde{v}$.

The parameters in (40) are constrained by the requirement of a minimal
determinant $D(\tilde{\bf R})$ at the strong pinning onset $\tilde{\bf
R}=\tilde{\bf R}_{m}$ and $\kappa_{m}=1$, i.e., its gradient has to vanish,

$\mathbf{\nabla}_{\tilde{\bf R}}\,D(\tilde{\bf R})\big{|}_{\tilde{\bf
R}_{m}}=0,$ (43)

and its Hessian $\mathrm{Hess}[D(\tilde{\bf R})]$ has to satisfy the relations

$\displaystyle\mathrm{det}\bigl{[}\mathrm{Hess}\bigl{[}D(\tilde{\bf
R})\bigr{]}\bigr{]}\big{|}_{\tilde{\bf R}_{m}}$ $\displaystyle>0,$ (44)
$\displaystyle\mathrm{tr}\bigl{[}\mathrm{Hess}\bigl{[}D(\tilde{\bf
R})\bigr{]}\bigr{]}\big{|}_{\tilde{\bf R}_{m}}$ $\displaystyle>0.$ (45)

Making use of the expansion (40), the determinant $D(\tilde{\bf R})$ reads

$D(\tilde{\bf
R})=\big{\\{}[\partial_{\tilde{u}}^{2}e_{\mathrm{pin}}][\partial_{\tilde{v}}^{2}e_{\mathrm{pin}}]-[\partial_{\tilde{u}}\partial_{\tilde{v}}e_{\mathrm{pin}}]^{2}\big{\\}}_{\tilde{\bf
R}}$ (46)

with

$\displaystyle\partial_{\tilde{u}}^{2}e_{\mathrm{pin}}=\bar{C}\left(1\\!-\\!\kappa_{m}\right)+a^{\prime}\tilde{v}+b^{\prime}\tilde{u}+\alpha\tilde{v}^{2}\\!/2+\beta\tilde{u}\tilde{v}+\gamma\tilde{u}^{2}\\!/2,$
$\displaystyle\partial_{\tilde{v}}^{2}e_{\mathrm{pin}}=\bar{C}+\lambda_{+}+a\tilde{u}+b^{\prime\prime}\tilde{v}+\alpha\tilde{u}^{2}/2+\beta^{\prime\prime}\tilde{u}\tilde{v}+\gamma^{\prime\prime}\tilde{v}^{2}/2,$
$\displaystyle\partial_{\tilde{u}}\partial_{\tilde{v}}e_{\mathrm{pin}}=a\tilde{v}+a^{\prime}\tilde{u}+\alpha\tilde{u}\tilde{v}+\beta\tilde{u}^{2}/2+\beta^{\prime\prime}\tilde{v}^{2}/2,$

and produces the gradient

$\displaystyle\mathbf{\nabla}_{\tilde{\bf R}}\,D(\tilde{\bf
R})\Big{|}_{\tilde{\bf R}_{m}}=(\bar{C}+\lambda_{+})(b^{\prime},a^{\prime}),$
(47)

hence the primed parameters indeed vanish, $a^{\prime}=0$ and $b^{\prime}=0$.
The Hessian then takes the form

$\displaystyle\mathrm{Hess}\bigl{[}D(\tilde{\bf R})\bigr{]}\Big{|}_{\tilde{\bf
R}_{m}}$
$\displaystyle=(\bar{C}+\lambda_{+})\begin{bmatrix}\gamma&~{}~{}~{}\beta\\\
\beta&~{}~{}~{}\delta\end{bmatrix}$ (48)

at the Labusch point $\kappa_{m}=1$, where we have introduced the parameter

$\delta\equiv\alpha-\frac{2a^{2}}{\bar{C}}\frac{1}{1+\lambda_{+}/\bar{C}}.$
(49)

The stability conditions (44) and (45) translate, respectively, to

$\gamma\delta-\beta^{2}>0$ (50)

(implying $\delta>0$) and

$\gamma+\delta>0.$ (51)

The Landau-type theory (40) involves the two ‘order parameters’ $\tilde{u}$
and $\tilde{v}$ and is driven by the dual coordinates $\bar{u}$ and $\bar{v}$.
This $n=2$ theory involves a soft order parameter $\tilde{u}$ and the stiff
$\tilde{v}$, allowing us to integrate out $\tilde{v}$ and reformulate the
problem as an effective one-dimensional Landau theory (154) of the van der
Waals kind—the way of solving the strong pinning problem near onset in this 1D
formulation is presented in Appendix A.1.

###  III.2 Unstable domain $\mathcal{U}_{\tilde{\bf R}}$

Next, we determine the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ in tip
space as defined in (37). We will find that, up to quadratic order, the
boundary of $\mathcal{U}_{\tilde{\bf R}}$ has the shape of an ellipse with the
semiaxes lengths scaling as $\sqrt{\kappa_{m}-1}$.

####  III.2.1 Jump line $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$

We find the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ by determining its
boundary $\partial\mathcal{U}_{\tilde{\bf R}}$ that is given by the set of
jump positions $\tilde{\bf R}_{\mathrm{jp}}$ making up the jump line
$\mathcal{J}_{\mathrm{\tilde{\bf R}}}$. The boundary
$\partial\mathcal{U}_{\tilde{\bf R}}$ is determined by the condition
$\bar{C}+\lambda_{-}=0$ or, equivalently, the vanishing of the determinant

$D(\tilde{\bf R}_{\mathrm{jp}})\equiv 0.$ (52)

The latter condition guarantees the existence of an unstable direction
parallel to the eigenvector $\mathbf{v}_{-}(\tilde{\bf R}_{\mathrm{jp}})$
associated with the eigenvalue $\lambda_{-}(\tilde{\bf R}_{\mathrm{jp}})$
where the energy (40) turns flat, cf. our discussion in Sec. II.2. The edges
of the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ therefore correspond to a
line of inflection points in $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$
along which one of the bistable tip configurations of the force balance
equation (38) coalesces with the unstable solution. Near the onset of strong
pinning, the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ is closely confined
around the point $\tilde{\bf R}_{m}$ where $\mathbf{v}_{-}(\tilde{\bf
R}_{m})\parallel\hat{\mathbf{u}}$. The unstable direction
$\mathbf{v}_{-}(\tilde{\bf R}_{\mathrm{jp}})$ is therefore approximately
homogeneous within the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ and is
parallel to the $u$ axis. This fact will be of importance later, when
determining the topological properties of the unstable domain
$\mathcal{U}_{\tilde{\bf R}}$.

Inspection of the condition (52) with $D(\tilde{\bf R})$ given by Eq. (46)
shows that the components of $\delta\tilde{\bf R}_{\mathrm{jp}}$ scale as
$\sqrt{\kappa_{m}-1}$: in the product
$[\partial_{\tilde{u}}^{2}e_{\mathrm{pin}}][\partial_{\tilde{v}}^{2}e_{\mathrm{pin}}]$,
the first factor involves the small constant $\bar{C}(1-\kappa_{m})$ plus
quadratic terms (as $a^{\prime}=0$ and $b^{\prime}=0$), while the second
factor comes with the large constant $\bar{C}+\lambda_{+}$ plus corrections.
The leading term in
$[\partial_{\tilde{u}}\partial_{\tilde{v}}e_{\mathrm{pin}}]$ is linear in
$\tilde{v}$ with the remaining terms providing corrections. To leading order,
the condition of vanishing determinant then produces the quadratic form

$[\gamma\,\tilde{u}^{2}+2\beta\,\tilde{u}\tilde{v}+\delta\,\tilde{v}^{2}]_{\tilde{\bf
R}_{\mathrm{jp}}}=2\bar{C}\left(\kappa_{m}-1\right).$ (53)

With $\gamma$ and $\delta$ positive, this form is associated with an elliptic
geometry of extent $\propto\sqrt{\kappa_{m}-1}$. For later convenience, we
rewrite Eq. (53) in matrix form

$\delta\tilde{\bf
R}_{\mathrm{jp}}^{\mathrm{T}}M_{\mathrm{jp}}\,\delta\tilde{\bf
R}_{\mathrm{jp}}=\bar{C}(\kappa_{m}-1)$ (54)

with

$\displaystyle M_{\mathrm{jp}}$
$\displaystyle=\begin{bmatrix}\gamma/2&~{}~{}~{}\beta/2\\\
\beta/2&~{}~{}~{}\delta/2\end{bmatrix}$ (55)

and $\det M_{\mathrm{jp}}=(\gamma\delta-\beta^{2})/4>0$, see Eq. (50). The
jump line $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ can be expressed in the
parametric form

$\begin{split}\tilde{u}_{\mathrm{jp}}(|\tilde{v}|<\tilde{v}_{c})&=-\frac{1}{\gamma}\Bigl{[}\beta\tilde{v}\\\
&\pm\sqrt{2\gamma\bar{C}(\kappa_{m}-1)-(\gamma\delta-\beta^{2})\tilde{v}^{2}}\Bigr{]},\end{split}$
(56)

with

$\tilde{v}_{c}=\sqrt{2\gamma\,\bar{C}(\kappa_{m}-1)/(\gamma\delta-\beta^{2})}$
(57)

and is shown in Fig. 6 for the example of an anisotropic potential inspired by
the uniaxial defect in Sec. IV with 10 % anisotropy. The associated unstable
domain $\mathcal{U}_{\tilde{\bf R}}$ assumes a compact elliptic shape, with
the parameter $\beta$ describing the ellipse’s skew. Comparing with the
isotropic defect, this ellipse assumes the role of the ring bounded by solid
lines in Fig. 5(c), see Sec. III.5 for a discussion of its different topology.

An additional result of the above discussion concerns the terms that we need
to keep in the expansion of the pinning energy (40): indeed, dropping
corrections amounts to dropping terms with double-primed coefficients and we
find that the simplified expansion

$\displaystyle e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf
R})=\frac{\bar{C}}{2}(1-\kappa_{m})\,\tilde{u}^{2}+\frac{\bar{C}+\lambda_{+}}{2}\,\tilde{v}^{2}+\frac{a}{2}\,\tilde{u}\tilde{v}^{2}$
$\displaystyle\quad+\frac{\alpha}{4}\,\tilde{u}^{2}\tilde{v}^{2}+\frac{\beta}{6}\,\tilde{u}^{3}\tilde{v}+\frac{\gamma}{24}\,\tilde{u}^{4}-\bar{C}\,\bar{u}\tilde{u}-\bar{C}\,\bar{v}\tilde{v}$
(58)

produces all of our desired results to leading order.

Figure 6: Jump line $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ (solid red/blue,
see Eq. (54)) and landing line (dashed red/blue, see Eq. (63))
$\mathcal{L}_{\mathrm{\tilde{\bf R}}}$ in tip space $\tilde{\bf R}$ (in units
of $\xi$), with the ellipse $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$
representing the edge $\partial\mathcal{U}_{\tilde{\bf R}}$ of the unstable
domain $\mathcal{U}_{\tilde{\bf R}}$. We choose parameters
$\kappa_{m}-1=10^{-2}$, with
$\lambda_{-}=-0.25\,e_{p}/\xi^{2},\lambda_{+}=0.05\,e_{p}/\xi^{2}$, and
$a=0.07\,e_{p}/\xi^{3}$,
$\alpha=0.1\,e_{p}/\xi^{4},\beta=0,\gamma=0.75\,e_{p}/\xi^{4}$ inspired by the
choice of the uniaxial defect with 10 % anisotropy in Sec. IV; the dotted
ellipse shows the effect of a finite skew parameter
$\beta=0.05\,e_{p}/\xi^{4}$ on the jump ellipse
$\mathcal{J}_{\mathrm{\tilde{\bf R}}}$. Along the edges of
$\mathcal{U}_{\tilde{\bf R}}$, one of the stable tip configurations coalesces
with the unstable solution of (38) and the total pinning energy
$e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ develops an inflection line in
the tip coordinate $\tilde{\bf R}$. Crosses correspond to the contact points
(67) between the two ellipses $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ and
$\mathcal{L}_{\mathrm{\tilde{\bf R}}}$. Blue and red colors identify different
types of vortex deformations upon jump and landing. Pairs of solid and open
circles connected via long arrows are, respectively, examples of pairs of
jumping- and landing tip positions for vortices approaching the defect from
the left (top) and right (bottom), see Fig. 5(c) for the isotropic problem’s
counterpart. The unstable direction $\mathbf{v}_{-}(\tilde{\bf
R}_{\mathrm{jp}})$, shown as short black arrows for different points on the
ellipse, always points in the $u-$direction and are parallel to the tangent
vector of the unstable ellipse at the contact points (67).

####  III.2.2 Landing line $\mathcal{L}_{\mathrm{\tilde{\bf R}}}$

We find the landing positions $\tilde{\bf R}_{\mathrm{lp}}$ by extending the
discussion of the isotropic situation in Sec. II.4 to two dimensions: we shift
the origin of the expansion (III.2.1) to the jump point $\tilde{\bf
R}_{\mathrm{jp}}$ and find the landing point $\tilde{\bf
R}_{\mathrm{lp}}=\tilde{\bf R}_{\mathrm{jp}}+\Delta\tilde{\bf R}$ by
minimizing the total energy $e_{\mathrm{pin}}(\Delta\tilde{\bf R})$ at the
landing position. Below, we use $\Delta\tilde{\bf R}$ both as a variable and
as the jump distance to avoid introducing more coordinates.

We exploit the differential properties of $e_{\mathrm{pin}}$ at the jump and
landing positions. At landing, $e_{\mathrm{pin}}(\tilde{\bf
R}_{\mathrm{jp}}+\Delta\tilde{\bf R})$ has a minimum, hence, the configuration
is force free, in particular along $\tilde{v}$,

$\displaystyle\partial_{\tilde{v}}e_{\mathrm{pin}}(\tilde{\bf
R}_{\mathrm{jp}}+\Delta\tilde{\bf R})$
$\displaystyle\approx[\partial_{\tilde{v}}\partial_{\tilde{u}}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}\Delta\tilde{u}$
$\displaystyle\qquad+[\partial_{\tilde{v}}^{2}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}\Delta\tilde{v}=0,$

from which we find that $\Delta\tilde{u}$ and $\Delta\tilde{v}$ are related
via

$\Delta\tilde{v}\approx-\frac{[\partial_{\tilde{v}}\partial_{\tilde{u}}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}}{[\partial_{\tilde{v}}^{2}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}}\Delta\tilde{u}.$ (59)

Here, we have dropped higher order terms in the expansion, assuming that the
jump is mainly directed along the unstable $u$-direction—indeed, using the
expansion (III.2.1), we find that

$\Delta\tilde{v}\approx-\frac{a\tilde{v}_{\mathrm{jp}}}{\bar{C}+\lambda_{+}}\,\Delta\tilde{u}\propto\sqrt{\kappa_{m}-1}\>\Delta\tilde{u}.$
(60)

Note that we cannot interchange the roles of $\tilde{u}$ and $\tilde{v}$ in
this force analysis, as higher order terms in the expression for the force
along $\tilde{u}$ cannot be dropped.

At the jump position $\tilde{\bf R}_{\mathrm{jp}}$, the state is force-free,
i.e., the derivatives $[\partial_{\tilde{u}}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}$ and $[\partial_{\tilde{v}}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}$ vanish, and the Hessian determinant vanishes as well.
Therefore, the expansion of $e_{\mathrm{pin}}(\tilde{\bf
R}_{\mathrm{jp}}+\Delta\tilde{\bf R})$ has no linear terms and the second
order terms $[\partial_{\tilde{u}}^{2}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}\Delta\tilde{u}^{2}/2+[\partial_{\tilde{u}}\partial_{\tilde{v}}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}\Delta\tilde{u}\Delta\tilde{v}+[\partial_{\tilde{v}}^{2}e_{\mathrm{pin}}]_{\tilde{\bf
R}_{\mathrm{jp}}}\Delta\tilde{v}^{2}/2$ combined with Eq. (59) can be
expressed through the Hessian determinant,
$\\{[\partial_{\tilde{u}}^{2}e_{\mathrm{pin}}][\partial_{\tilde{v}}^{2}e_{\mathrm{pin}}]-[\partial_{\tilde{u}}\partial_{\tilde{v}}e_{\mathrm{pin}}]^{2}\\}_{\tilde{\bf
R}_{\mathrm{jp}}}\Delta\tilde{u}^{2}/2=0$, that vanishes as well. Therefore,
the expansion of $e_{\mathrm{pin}}$ around $\tilde{\bf R}_{\mathrm{jp}}$
starts at third order in $\Delta\tilde{\bf R}\approx(\Delta\tilde{u},0)$ and
takes the form (we make use of (60), dropping terms $\propto\Delta\tilde{v}$
and a constant)

$e_{\mathrm{pin}}(\tilde{\bf R}_{\mathrm{jp}}+\Delta\tilde{\bf
R})\approx\frac{1}{6}\bigl{(}\gamma\tilde{u}_{\mathrm{jp}}+\beta\tilde{v}_{\mathrm{jp}}\bigr{)}\Delta\tilde{u}^{3}+\frac{\gamma}{24}\Delta\tilde{u}^{4}.$
(61)

Minimizing this expression with respect to $\Delta\tilde{u}$ (as
$e_{\mathrm{pin}}$ is minimal at $\tilde{\bf R}_{\mathrm{lp}}$), we obtain the
result

$\Delta\tilde{u}\approx-3(\gamma\tilde{u}_{\mathrm{jp}}+\beta\tilde{v}_{\mathrm{jp}})/\gamma.$
(62)

Making use of the quadratic form (54), we can show that the equation for the
landing position $\tilde{\bf R}_{\mathrm{lp}}=\tilde{\bf
R}_{\mathrm{jp}}+\Delta\tilde{\bf R}$ can be cast into a similar quadratic
form (with $\delta\tilde{\bf R}_{\mathrm{lp}}$ measured relative to
$\tilde{\bf R}_{m}$)

$\delta\tilde{\bf
R}_{\mathrm{lp}}^{\mathrm{T}}M_{\mathrm{lp}}\,\delta\tilde{\bf
R}_{\mathrm{lp}}=\bar{C}(\kappa_{m}-1),$ (63)

but with the landing matrix now given by

$M_{\mathrm{lp}}=\frac{1}{4}M_{\mathrm{jp}}+\begin{bmatrix}0&0\\\
0&~{}~{}~{}\displaystyle{\frac{3}{4}\Bigl{(}\frac{\delta}{2}-\frac{\beta^{2}}{2\gamma}\Bigr{)}}\end{bmatrix}.$
(64)

In the following, we will refer to the solutions of Eq. (63) as the ‘landing’
or ‘stable’ ellipse $\tilde{\bf R}_{\mathrm{lp}}$ and write the jump distance
in a parametric form involving the shape $\tilde{u}_{\mathrm{jp}}(\tilde{v})$
in Eq. (56) of the jumping ellipse,

$\displaystyle\Delta\tilde{u}(\tilde{v})=-3\left[\gamma\,\tilde{u}_{\mathrm{jp}}(\tilde{v})+\beta\,\tilde{v}\right]/\gamma,$
(65)
$\displaystyle\Delta\tilde{v}(\tilde{v})=-\left[a/(\bar{C}+\lambda_{+})\right]\,\tilde{v}\,\Delta\tilde{u}(\tilde{v}).$
(66)

The landing line derived from (63) is displayed as a dashed line in Fig. 6.
Two tip jumps connected by an arrow are shown for illustration, with solid
dots marking the jump position $\tilde{\bf R}_{\mathrm{jp}}$ of the tip and
open dots its landing position $\tilde{\bf R}_{\mathrm{lp}}$; they describe
tip jumps for a vortex approaching the unstable ellipse once from the left
(upper pair) and another time from the right (lower pair). The different
topologies associated with jumps and landing showing up for the isotropic
defect in Fig. 5(c) (two concentric circles) and for the generic onset in Fig.
6 (two touching ellipses) will be discussed later.

Inspecting the matrix equation (63), we can gain several insights on the
landing ellipse $\mathcal{L}_{\mathrm{\tilde{\bf R}}}$: (i) the matrix
$M_{\mathrm{jp}}/4$ on the right-hand side of (64) corresponds to an ellipse
with the same geometry as for $\mathcal{J}_{\mathrm{\tilde{\bf R}}}$ but
double in size, (ii) the remaining matrix with vanishing entries in the off-
diagonal and the $M_{xx}$ elements leaves the size doubling of the stable
ellipse $\mathcal{L}_{\mathrm{\tilde{\bf R}}}$ at $\tilde{v}=0$ unchanged, and
(iii) the finite $M_{yy}$ component exactly counterbalances the doubling along
the $v-$direction encountered in (i), cf. the definiton (55) of
$M_{\mathrm{jp}}$, up to a term proportional to the skew parameter $\beta$
accounting for deviations of the semiaxis from the $v-$axis. Altogether, the
stable ellipse $\mathcal{L}_{\mathrm{\tilde{\bf R}}}$ extends with a double
width along the $u-$axis and smoothly overlaps with the unstable ellipse at
the two contact points $\tilde{v}_{c,\pm}$. The latter are found by imposing
the condition $\Delta\tilde{u}=\Delta\tilde{v}=0$ in Eqs. (65) and (66); we
find them located (relative to $\tilde{\bf R}_{m}$) at

$\displaystyle\delta\tilde{\bf R}_{c,\pm}$
$\displaystyle=\pm\left(-\beta/\gamma,1\right)\,\tilde{v}_{c},$ (67)

with the endpoint coordinate $\tilde{v}_{c}$ given in Eq. (57), and mark them
with crosses in Fig. 6. As anticipated, the contact points are off-set with
respect to the $v-$axis for a finite skew parameter $\beta$. At these points,
the unstable and the stable tip configurations coincide and the vortex tip
undergoes no jump. Furthermore, the vector tangent to the jump (or landing)
ellipse is parallel to the $u-$direction at the contact points. To see that,
we consider (56) and find that

$\displaystyle\frac{\partial\tilde{u}}{\partial\tilde{v}}\Big{|}_{\tilde{v}\to\pm\tilde{v}_{c}}\\!\\!\\!\\!\\!$
$\displaystyle\approx\pm\left(\sqrt{\tilde{v}_{c}^{2}-\frac{2\gamma\,\bar{C}(\kappa_{m}-1)}{\gamma\beta-\delta^{2}}}\right)^{-1}\\!\\!\\!\to\pm\infty,$
(68)

hence, the corresponding tangents $\partial_{\tilde{u}}\tilde{v}$ vanish.

The asymptotic positions $\bar{\bf R}$ where the vortex tips jump and land
belong to the boundary of the bistable region $\mathcal{B}_{\bar{\bf R}}$; for
the isotropic case in Fig. 5(d) these correspond to the circles with radii
$\bar{R}_{-}$ (pinning) and $\bar{R}_{+}$ (depinning) with jump and landing
radii $\tilde{R}_{\mathrm{f-}}(\bar{R}_{-})$ and
$\tilde{R}_{\mathrm{p-}}(\bar{R}_{-})$ and
$\tilde{R}_{\mathrm{p+}}(\bar{R}_{+})$ and
$\tilde{R}_{\mathrm{f+}}(\bar{R}_{+})$, respectively, see Fig. 5(c). For the
anisotropic defect, we have only a single jump/landing event at one asymptotic
position $\bar{\bf R}$ that we are going to determine in the next section.

###  III.3 Bistable domain $\mathcal{B}_{\bar{\bf R}}$

The set of asymptotic positions $\bar{\bf R}$ corresponding to the tip
positions $\tilde{\bf R}_{\mathrm{jp}}$ along the edges of
$\mathcal{U}_{\tilde{\bf R}}$ forms the boundary
$\partial\mathcal{B}_{\bar{\bf R}}$ of the bistable domain
$\mathcal{B}_{\bar{\bf R}}$; they are related through the force-balance
equation (38), with every vortex tip position $\tilde{\bf
R}_{\mathrm{jp}}\in\partial\mathcal{U}_{\tilde{\bf R}}$ defining an associated
asymptotic position $\bar{\bf R}(\tilde{\bf
R}_{\mathrm{jp}})\in\partial\mathcal{B}_{\bar{\bf R}}$.

At the onset of strong pinning, the bistable domain corresponds to the
isolated point $\bar{\bf R}_{m}$, related to $\tilde{\bf R}_{m}$ through (38).
Beyond the Labusch point, $\mathcal{B}_{\bar{\bf R}}$ expands out of $\bar{\bf
R}_{m}$ and its geometry is found by evaluating the force balance equation
(38) at a given tip position $\tilde{\bf
R}_{\mathrm{jp}}\in\partial\mathcal{U}_{\tilde{\bf R}}$, $\bar{\bf
R}(\tilde{\bf R}_{\mathrm{jp}})=\tilde{\bf
R}_{\mathrm{jp}}-\mathbf{f}_{p}(\tilde{\bf
R}_{\mathrm{jp}})/\bar{C}\in\partial\mathcal{B}_{\bar{\bf R}}$. Using the
expansion (III.2.1) for $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$, this
force equation can be expressed as
$\nabla_{\mathbf{R}}e_{\mathrm{pin}}(\mathbf{R};\bar{\bf
R})\big{|}_{\tilde{\bf R}}=0$, or explicitly (we remind that we measure
$\bar{\bf R}=\bar{\bf R}_{m}+(\bar{u},\bar{v})$ relative to $\bar{\bf
R}_{m}$),

$\displaystyle\bar{C}\bar{u}$
$\displaystyle=\bar{C}(1-\kappa_{m})\tilde{u}+\frac{a}{2}\tilde{v}^{2}+\frac{\gamma}{6}\tilde{u}^{3}+\frac{\beta}{2}\tilde{u}^{2}\tilde{v}+\frac{\alpha}{2}\tilde{u}\tilde{v}^{2},$
$\displaystyle\bar{C}\bar{v}$
$\displaystyle=(\bar{C}+\lambda_{+})\tilde{v}+a\,\tilde{u}\tilde{v}+\frac{\beta}{6}\tilde{u}^{3}+\frac{\alpha}{2}\tilde{u}^{2}\tilde{v}.$
(69)

Inserting the results for the jump ellipse $\mathcal{J}_{\mathrm{\tilde{\bf
R}}}$, Eq. (56), into Eqs. (69), we find the crescent-shape bistable domain
$\mathcal{B}_{\bar{\bf R}}$ shown in Fig. 7; let us briefly derive the origin
of this shape.

Figure 7: (a) Bistable domain $\mathcal{B}_{\bar{\bf R}}$ in asymptotic
$\bar{\bf R}$-space measured in units of $\xi$; the same parameters as in Fig.
6 have been used. Note the different scaling of the axes in $\kappa_{m}-1$;
the right panel (b) shows $\mathcal{B}_{\bar{\bf R}}$ in isotropic scales. The
bistable domain $\mathcal{B}_{\bar{\bf R}}$ is elongated along the transverse
direction $\bar{v}$ and narrow/bent along the unstable direction $\bar{u}$,
giving $\mathcal{B}_{\bar{\bf R}}$ its peculiar crescent-like shape. The
branch crossing line $\bar{\bf R}_{0}$, see (77), is shown as a dashed black
line. Black crosses mark the cusps of $\mathcal{B}_{\bar{\bf R}}$ and are
associated with the contact points of $\mathcal{U}_{\tilde{\bf R}}$ through
the force balance equation (38); they correspond to critical end-points in the
thermodynamic Ising analogue, while the boundaries
$\partial\mathcal{B}_{\bar{\bf R}}$ map to spinodals. Blue and red colors
identify different characters of vortex tip configurations as quantified
through the ‘order parameter’ $\tilde{u}$ of the Landau expansion (at
$\beta=0$), see text, while magenta is associated to the bistable area
$\mathcal{B}_{\bar{\bf R}}$; the blue and red branches extend to the far side
of the crescent and terminate in the blue and red colored boundaries
$\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{b}}$ and
$\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{r}}$, respectively. Thin
horizontal lines show vortex trajectories that proceed smoothly in asymptotic
space, see also Fig. 5(d). Blue and red dots mark the asymptotic positions
associated with vortex tip jumps that happen at the exit of
$\mathcal{B}_{\bar{\bf R}}$; they correspond to the pairs of tip positions in
Fig. 6. (b) Bistable domain $\mathcal{B}_{\bar{\bf R}}$ in isotropic scaled
coordinates $\bar{u}$ and $\bar{v}$ showing the ‘true’ shape of
$\mathcal{B}_{\bar{\bf R}}$. Vortices impacting on the bistable domain with an
angle $|\theta|\leq\theta^{\ast}$ undergo a single jump on the far side of
$\mathcal{B}_{\bar{\bf R}}$, with the pinning force density directed along $u$
and scaling as $F_{\mathrm{pin}}^{\parallel}\propto(\kappa-1)^{5/2}$. Vortices
crossing $\mathcal{B}_{\bar{\bf R}}$ at large angles close to $\pi/2$ jump
either never, once, or twice; at $\theta=\pi/2$ the pinning force density is
small, $F_{\mathrm{pin}}^{\perp}\propto(\kappa-1)^{3}$, and directed along
$v$.

Solving (69) to leading order,
$\bar{C}\bar{u}^{\scriptscriptstyle(0)}\approx(a/2)\tilde{v}^{2}$ and
$\bar{C}\bar{v}^{\scriptscriptstyle(0)}\approx(\bar{C}+\lambda_{+})\tilde{v}$,
we find the parabolic approximation

$\displaystyle\bar{u}^{\scriptscriptstyle(0)}$
$\displaystyle\approx\frac{a}{2\bar{C}}\frac{1}{(1+\lambda_{+}/\bar{C})^{2}}\,\bar{v}^{{\scriptscriptstyle(0)}\,2},$
(70)

telling that the extent of $\mathcal{B}_{\bar{\bf R}}$ scales as
$(\kappa_{m}-1)$ along $\bar{u}$ and $\propto(\kappa_{m}-1)^{1/2}$ along
$\bar{v}$, i.e., we find a flat parabola opening towards positive $\bar{u}$
for $a>0$, see Fig. 7.

In order to find the width of $\mathcal{B}_{\bar{\bf R}}$, we have to solve
(69) to the next higher order,
$\bar{u}=\bar{u}^{\scriptscriptstyle(0)}+\bar{u}^{\scriptscriptstyle(1)}$; for
$\beta=0$, we find the correction

$\bar{u}^{\scriptscriptstyle(1)}=(1-\kappa_{m})\tilde{u}+\frac{\gamma}{6\bar{C}}\tilde{u}^{3}+\frac{\alpha}{2\bar{C}}\tilde{u}\tilde{v}^{2}$
(71)

that produces a $\bar{v}\leftrightarrow-\bar{v}$ symmetric crescent. Inserting
the two branches (56) of the jump ellipse, we arrive at the width of the
crescent that scales as $(\kappa_{m}-1)^{3/2}$. The correction to $\bar{v}$ is
$\propto(\kappa_{m}-1)$ and we find the closed form

$\displaystyle\bar{v}$
$\displaystyle\approx[1+(\lambda_{+}+a\tilde{u})/\bar{C}]\>\tilde{v}$ (72)

with a small antisymmetric (in $\tilde{u}$) correction. For a finite
$\beta\neq 0$, the correction $\bar{u}^{\scriptscriptstyle(1)}$ picks up an
additional term $(\beta/2\bar{C})\,\tilde{u}^{2}\tilde{v}$ that breaks the
$\bar{v}\leftrightarrow-\bar{v}$ symmetry and the crescent is distorted.

Viewing the boundary $\partial\mathcal{B}_{\bar{\bf R}}$ as a parametric curve
in the variable $\tilde{v}$ with
$\tilde{u}=\tilde{u}_{\mathrm{jp}}(\tilde{v})$ given by Eq. (56), we obtain
the boundary $\partial\mathcal{B}_{\bar{\bf R}}$ in the form of two separate
arcs that define the crescent-shaped domain $\mathcal{B}_{\bar{\bf R}}$ in
Fig. 7(a). The two arcs merge in two cusps at $\bar{\bf R}_{c,\pm}$ that are
associated to the touching points (67) in dual space and derive from Eqs.
(69); measured with respect to $\bar{\bf R}_{m}$, these cusps are located at

$\displaystyle\delta\bar{\bf R}_{c,\pm}$ $\displaystyle=$
$\displaystyle(\bar{u}_{c},\pm\bar{v}_{c})$ $\displaystyle\approx$
$\displaystyle\left[\left(a/2\,\bar{C}\right)\,\tilde{v}^{2}_{c},\,\pm(1+\lambda_{+}/\bar{C})\tilde{v}_{c}\right].$

The coloring in Fig. 7 indicates the characters ‘red’ and ‘blue’ of the vortex
states; these are defined in terms of the ‘order parameter’
$\tilde{u}-\tilde{u}_{m}(\bar{v})$ of the Landau functional (III.2.1) that
changes sign at the branch crossing line Eq. (77), with the shift

$\tilde{u}_{m}(\bar{v})=-\frac{\beta}{\gamma}\tilde{v}(\bar{v})\approx-\frac{\beta}{\gamma}\frac{\bar{v}}{1+\lambda_{+}/\bar{C}},$
(74)

$\tilde{u}_{m}(\bar{v})=0$ for our symmetric case with $\beta=0$ in Fig. 7.
Going beyond the cusps (or critical points) at $\bar{\bf R}_{c,\pm}$, the two
states smoothly crossover between ‘red’ and ‘blue’ (indicated by the smooth
blue–white–red transition), as known for the van der Waals gas (or Ising
magnet) above the critical point. Within the bistable region
$\mathcal{B}_{\bar{\bf R}}$, both ‘red’ and ‘blue’ states coexist and we color
this region in magenta.

The geometry of the bistable domain $\mathcal{B}_{\bar{\bf R}}$ is very
different from the ring-shaped geometry of the isotropic problem discussed in
Sec. II.1, see Fig. 5(d); in the discussion of the uniaxial anisotropic defect
below, we will learn how these two geometries are interrelated. Comparing the
overall dimensions of the crescent with the ring in Fig. 5(d), we find the
following scaling behavior in $\kappa_{m}-1$: while the crescent
$\mathcal{B}_{\bar{\bf R}}$ grows along $\bar{v}$ as $(\kappa_{m}-1)^{1/2}$,
the isotropic ring involves the characteristic size $\xi$ of the defect,
$\bar{R}_{-}\sim\xi$ and hence its extension along $\bar{v}$ is a constant. On
the other hand, the scaling of the crescent’s and the ring’s width is the
same, $\propto(\kappa_{m}-1)^{3/2}$. The different scaling of the transverse
width then will be responsible for the new scaling of the pinning force
density, $F_{\mathrm{pin}}\propto(\kappa_{m}-1)^{5/2}$.

###  III.4 Comparison to isotropic situation

Let us compare the unstable domains $\mathcal{U}_{\tilde{\bf R}}$ for the
isotropic and anisotropic defects in Figs. 5(c) and 6, respectively. In the
isotropic example of Sec. II.1, the jump- and landing-circles
$\tilde{R}_{\mathrm{jp}}(\bar{R})$ and $\tilde{R}_{\mathrm{lp}}(\bar{R})$ are
connected to different phases, e.g., free (colored in blue at
$\tilde{R}_{\mathrm{jp}}=\tilde{R}_{\mathrm{f-}}$) and pinned (colored in red
at $\tilde{R}_{\mathrm{lp}}=\tilde{R}_{\mathrm{p-}}$) associated with
$\bar{R}_{-}$. Furthermore, the topology is different, with the unstable ring
domain separating the two distinct phases, free and pinned ones. As a result,
a second pair of jump- and landing-positions associated with the asymptotic
circle $\bar{R}_{+}$ appears along the vortex trajectory of Fig. 5(c); these
are the located at the radii $\tilde{R}_{\mathrm{jp}}=\tilde{R}_{\mathrm{p+}}$
and $\tilde{R}_{\mathrm{lp}}=\tilde{R}_{\mathrm{f+}}$ and describe the
depinning process from the pinned branch back to the free branch (while the
previous pair at radii $\tilde{R}_{\mathrm{f-}}$ and $\tilde{R}_{\mathrm{p-}}$
describes the pinning process from the free to the pinned branch). The pinning
(at $\bar{R}_{-}$) and depinning (at $\bar{R}_{+}$) processes in the
asymptotic coordinates are shown in figure 5(d). The bistable area
$\mathcal{B}_{\bar{\bf R}}$ with coexisting free and pinned states has a ring-
shape as well (colored in magenta, the superposition of blue and red); the two
pairs of jump and landing points in tip space have collapsed to two pinning
and depinning points in asymptotic space.

In the present situation describing the strong pinning onset for a generic
anisotropic potential, the unstable domain $\mathcal{U}_{\tilde{\bf R}}$ grows
out of an isolated point (in fact, $\tilde{\bf R}_{m}$) and assumes the shape
of an ellipse that is simply connected; as a result, a vortex incident on the
defect undergoes only a single jump, see Fig. 6. The bistable domain
$\mathcal{B}_{\bar{\bf R}}$ is simply connected as well, but now features two
cusps at the end-points of the crescent, see Fig. 7. The bistability again
involves two states, but we cannot associate them with separated pinned and
free phases—we thus denote them by ‘blue’-type and ‘red’-type. The two states
approach one another further away from the defect and are distiguishable only
in the region close to bistability; in Fig. 7, this is indicated with
appropriate color coding. Note that the Landau-type expansion underlying the
coloring in Fig. 7 fails at large distances; going beyond a local expansion
near $\tilde{\bf R}_{m}$, the distortion of the vortex vanishes at large
distances and red/blue colors faint away to approach ‘white’.

###  III.5 Topology

The different topologies of unstable and bistable regions appearing in the
isotropic and anisotropic situations are owed to the circular symmetry of the
isotropic defect; we will recover the ring-like topology for the anisotropic
situation later when describing a uniaxially anisotropic defect at larger
values of the Labusch parameter $\kappa_{m}$. Indeed, such an increase in
pinning strength will induce a change in topology with two crescents facing
one another joining into a ring-like shape.

Let us discuss the consequences of the different topologies that we
encountered for the isotropic and anisotropic defects in the discussion above.
Specifically, the precise number and position of the contact points have an
elegant topological explanation. When a vortex tip touches the edges
$\tilde{\bf R}_{\mathrm{jp}}$ of the unstable domain there are two
characteristic directions: one is given by the unstable eigenvector
$\mathbf{v}_{-}(\tilde{\bf R}_{\mathrm{jp}})$ discussed in Sec. III.2 along
which the tip will jump initially. The second is the tangent vector to the
boundary $\partial\mathcal{U}_{\tilde{\bf R}}$ of the unstable domain, i.e.,
to the unstable ellipse. While the former is approximately constant and
parallel to the unstable $u$-direction along $\tilde{\bf R}_{\mathrm{jp}}$,
the latter winds around the ellipse exactly once after a full turn around
$\mathcal{U}_{\tilde{\bf R}}$. The contact points $\tilde{\bf R}_{c,\pm}$ of
the unstable and stable ellipses then coincide with those points on the
ellipse where the tangent vector are parallel and anti-parallel to
$\mathbf{v}_{-}$; at these points, the tip touches the unstable ellipse but
does not undergo a jump any more. Given the different winding numbers of
$\mathbf{v}_{-}$ and of the tangent vector, there are exactly two points along
the circumference of $\mathcal{U}_{\tilde{\bf R}}$ where the tangent vector is
parallel/anti-parallel to the $u$-direction; these are the points found in
(67). This argument remains valid as long as the contour
$\partial\mathcal{U}_{\tilde{\bf R}}$ is not deformed to cross/encircle the
singular point of the $\mathbf{v}_{-}(\tilde{\bf R}_{\mathrm{jp}})$ field
residing at the defect center.

The same arguments allow us to understand the absence of contact points in the
isotropic scenario: For an isotropic potential, the winding number
$n_{\scriptscriptstyle\mathcal{U}}$ of the tangent vector around
$\mathcal{U}_{\tilde{\bf R}}$ remains unchanged, i.e.,
$n_{\scriptscriptstyle\mathcal{U}}=\pm 1$, while the unstable direction
$\mathbf{v}_{-}$ is pointing along the radius and thus acquires a unit winding
number as well. Indeed, the two directions, tangent and jump, then rotate
simultaneously and do not wind around each other after a full rotation,
explaining the absence of contact points in the isotropic situation.

###  III.6 Energy jumps

Within strong pinning theory, the energy jump $\Delta e_{\mathrm{pin}}$
associated with the vortex tip jump between bistable vortex configurations at
the boundaries of $\mathcal{B}_{\bar{\bf R}}$ determines the pinning force
density $F_{\mathrm{pin}}$ and the critical current $j_{c}$, see Eqs. (16) and
(17). Formally, the energy jump $\Delta e_{\mathrm{pin}}$ is defined as the
difference in energy $e_{\mathrm{pin}}(\tilde{\bf R};\bar{\bf R})$ at fixed
asymptotic position $\bar{\bf R}\in\partial\mathcal{B}_{\bar{\bf R}}$ between
vortex configurations with tips in the jump ($\tilde{\bf
R}_{\mathrm{jp}}(\bar{\bf R})$) and landing ($\tilde{\bf
R}_{\mathrm{lp}}(\bar{\bf R})=\tilde{\bf R}_{\mathrm{jp}}(\bar{\bf
R})+\Delta\tilde{\bf R}$) positions,

$\Delta e_{\mathrm{pin}}(\bar{\bf R}\in\partial\mathcal{B}_{\bar{\bf
R}})\equiv e_{\mathrm{pin}}[\tilde{\bf R}_{\mathrm{jp}}(\bar{\bf R});\bar{\bf
R}]\\\ -e_{\mathrm{pin}}[\tilde{\bf R}_{\mathrm{lp}}(\bar{\bf R});\bar{\bf
R}].$ (75)

In Sec. III.2.2 above, we have found that the jump $\Delta\tilde{\bf R}$ is
mainly forward directed along $u$. Making use of the expansion (61) of
$e_{\mathrm{pin}}$ at $\tilde{\bf R}_{\mathrm{jp}}$ and the result (62) for
the jump distance $\Delta\tilde{u}$, we find the energy jumps $\Delta
e_{\mathrm{pin}}$ in tip- and asymptotic space in the form (cf. with the
isotropic result Eq. (24)),

$\displaystyle\Delta e_{\mathrm{pin}}(\bar{\bf R})$
$\displaystyle\approx\frac{\gamma}{72}\Delta\tilde{u}^{4}\approx\left(\frac{9}{8\gamma^{3}}\right)\left[\gamma\,\tilde{u}_{\mathrm{jp}}(\tilde{v})+\beta\,\tilde{v}\right]^{4}$
(76)
$\displaystyle\approx\left(\frac{9}{8\gamma^{3}}\right)\left[(\gamma\delta-\beta^{2})\left(\tilde{v}_{c}^{2}-\tilde{v}^{2}\right)\right]^{2}$
$\displaystyle\approx\left(\frac{9}{8\gamma^{3}}\right)\left[\frac{(\gamma\delta-\beta^{2})}{(1+\lambda_{+}/\bar{C})^{2}}\left(\bar{v}_{c}^{2}-\bar{v}^{2}\right)\right]^{2}.$

Here, we have used the parametric shape $\tilde{u}_{\mathrm{jp}}(\tilde{v})$
in Eq. (56) for the jumping ellipse as well as (69) to lowest order,
$\tilde{v}\approx\bar{v}/(1+\lambda_{+}/\bar{C})$, to relate the tip and
asymptotic positions in the last equation. The energy jump (76) scales as
$(\kappa_{m}-1)^{2}$ and is shown in Fig. 8. It depends on the $v$ coordinate
of the asymptotic (or tip) position only and vanishes at the cusps $\bar{\bf
R}_{c,\pm}$, see Eq. (III.3) (or at the touching points $\tilde{\bf
R}_{c,\pm}$, see Eq. (67)). To order $(\kappa_{m}-1)^{2}$, the energy jumps
are identical at the left and right edges of the bistable domain
$\mathcal{B}_{\bar{\bf R}}$.

Figure 8: Energy jump $\Delta e_{\mathrm{pin}}$ along the edges of the
bistable domain $\mathcal{B}_{\bar{\bf R}}$ as a function of the transverse
coordinate $\bar{v}$; we have used the same parameters as in Fig. 6. The
energy jump vanishes at the cusps $\pm\bar{v}_{c}$, as the bistable tip
configurations become identical and their energies turn equal.

Following the two bistable branches and the associated energy jumps between
them to the inside of $\mathcal{B}_{\bar{\bf R}}$, the latter vanish along the
branch crossing line $\bar{\bf R}_{0}$. In the thermodynamic analogue, this
line corresponds to the first-order equilibrium transition line that is framed
by the spinodal lines; for the isotropic defect, this is the circle with
radius $\bar{R}_{0}=x_{0}$ framed by the spinodal circles with radii
$\bar{R}_{\pm}$, see Figs. 4 and 5(d). For the anisotropic defect with
$\beta=0$, this line is trivially given by the centered parabola of
$\mathcal{B}_{\bar{\bf R}}$, see Eq. (70), and hence

$\bar{u}_{0}\approx\frac{a}{2\bar{C}}\frac{1}{(1+\lambda_{+}/\bar{C})^{2}}\bar{v}_{0}^{2}.$
(77)

The result for a finite skew parameter $\beta\neq 0$ is given by Eq. (175) in
Appendix A.1.

###  III.7 Pinning force density

The pinning force density $F_{\mathrm{pin}}$ is defined as the average force
density exerted on a vortex line as it moves across the superconducting
sample. For the isotropic case described in Sec. II.5, the individual pinning
force $\mathbf{f}_{\mathrm{pin}}(\bar{\bf R})=-\nabla_{\bar{\bf
R}}e_{\mathrm{pin}}(\bar{\bf R})$, see Eq. (7), is directed radially and the
force density $F_{\mathrm{pin}}$ is given by the (constant) energy jump
$\Delta e_{\mathrm{pin}}\propto(\kappa-1)^{2}$ on the edge
$\partial\mathcal{B}_{\bar{\bf R}}$ of the bistable domain and the transverse
length $t_{\perp}\sim\xi$, hence, $F_{\mathrm{pin}}\propto t_{\perp}\Delta
e_{\mathrm{pin}}$ scales as $(\kappa-1)^{2}$.

For an anisotropic defect, the pinning force depends on the vortex direction
of motion $\hat{\mathbf{v}}=(\cos\theta,\sin\theta)$ relative to the axis of
the bistable region: we choose angles $-\pi/2\leq\theta\leq\pi/2$ measured
from the unstable direction $\bar{u}$, i.e., vortices incident from the left;
the case of larger impact angles $|\theta|>\pi/2$ corresponds to vortices
incident from the right and can be reduced to the previous case by inverting
the sign of the parameter $a$ in the expansion (III.2.1), i.e., the curvature
of the parabola (70); to our leading order analysis, the results remain the
same. The pinning force is no longer directed radially but depends on
$\theta$; furthermore, the energy jump (76) is non-uniform along the boundary
$\mathcal{B}_{\bar{\bf R}}$.

In spite of these complications, we can perform some simple scaling estimates
as a first step: let us assume a uniform distribution of identical anisotropic
defects, all with their unstable direction pointing along $x$. The jumps in
energy still scale as $\Delta e_{\mathrm{pin}}\propto(\kappa_{m}-1)^{2}$,
however, the trapping distance is no longer finite but grows from zero as
$\kappa_{m}-1$ increases. Due to their elongated shapes, the bistable domains
$\mathcal{B}_{\bar{\bf R}}$ exhibit different extensions along the $y$ and $x$
directions, i.e., $\propto\bar{v}_{c}\propto\sqrt{\kappa_{m}-1}$ along $y$ and
$\propto\bar{u}_{c}\propto(\kappa_{m}-1)$ along $x$, respectively. These
simple considerations then suggest that the pinning force density exhibits a
scaling $F_{\mathrm{pin}}\propto(\kappa_{m}-1)^{\mu}$ with $\mu>2$, different
from the setup with isotropic defects. Even more, vortices moving along the
$x$ or $y$ directions, respectively, will experience different forces
$F_{\mathrm{pin}}^{\parallel}$ and $F_{\mathrm{pin}}^{\perp}$ scaling as

$F_{\mathrm{pin}}^{\parallel}\propto(\kappa_{m}-1)^{5/2},\quad
F_{\mathrm{pin}}^{\perp}\propto(\kappa_{m}-1)^{3}$ (78)

near the onset of strong pinning. While such uniform anisotropic defects could
be created artificially, a more realistic scenario will involve defects that
are randomly oriented and an additional averaging over angles $\theta$ has to
be performed; this will be done at the end of this section.

We first determine the magnitude and orientation of the pinning force density
$\mathbf{F}_{\mathrm{pin}}(\theta)$ as a function of the vortex impact angle
$\theta$ for randomly positioned but uniformly oriented (along $x$) defects of
density $n_{p}$. The pinning force density is given by the average over
relative positions between vortices and defects (with a minus sign following
convention; $\mathcal{V}_{\bar{\bf R}}$ denotes the vortex lattice unit cell),

$\displaystyle\mathbf{F}_{\mathrm{pin}}(\theta)=-n_{p}\int_{\mathcal{V}_{\bar{\bf
R}}\setminus\mathcal{B}_{\bar{\bf R}}}\\!\\!\frac{\mathrm{d}^{2}\bar{\bf
R}}{a_{0}^{2}}\,\mathbf{f}_{\mathrm{pin}}(\bar{\bf R})$ (79)
$\displaystyle\quad-n_{p}\int_{\mathcal{B}_{\bar{\bf
R}}}\\!\\!\\!\frac{\mathrm{d}^{2}\bar{\bf
R}}{a_{0}^{2}}\left[p_{\mathrm{b}}(\bar{\bf
R};\theta)\,\mathbf{f}^{\mathrm{b}}_{\mathrm{pin}}(\bar{\bf
R})+p_{\mathrm{r}}(\bar{\bf
R};\theta)\,\mathbf{f}^{\mathrm{r}}_{\mathrm{pin}}(\bar{\bf R})\right].$

Outside of the bistable domain, i.e., in $\mathcal{V}_{\bar{\bf
R}}\setminus\mathcal{B}_{\bar{\bf R}}$, a single stable vortex tip
configuration exists and the pinning force $\mathbf{f}_{\mathrm{pin}}(\bar{\bf
R})$ is uniquely defined. Inside $\mathcal{B}_{\bar{\bf R}}$, the branch
occupation functions $p_{\mathrm{b,r}}(\bar{\bf R};\theta)$ are associated
with the tip positions appertaining to the ‘blue’ and the ‘red’ vortex
configurations with different tip positions $\tilde{\bf
R}^{\mathrm{b,r}}(\bar{\bf R})$, cf. Figs. 6 and 7. The pinning forces
$\mathbf{f}^{\mathrm{b,r}}_{\mathrm{pin}}(\bar{\bf R})$ are evaluated for the
corresponding vortex tip positions and are defined as

$\mathbf{f}^{\mathrm{b,r}}_{\mathrm{pin}}(\bar{\bf
R})=-\mathbf{\nabla}_{\bar{\bf R}}e_{\mathrm{pin}}[\tilde{\bf
R}^{\mathrm{b,r}}(\bar{\bf R});\bar{\bf R}].$ (80)

Let us now study how vortex lines populate the bistable domain as a function
of the impact angle $\theta$. Examining Fig. 7, we can distinguish between two
different angular regimes: a _frontal_ -impact regime at angles away from
$\pi/2$, $|\theta|\leq\theta^{\ast}$, where all the vortices that cross the
bistable domain undergo exactly one jump on the far edge of
$\mathcal{B}_{\bar{\bf R}}$, see the blue dot and blue boundary
$\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{b}}$ in Fig. 7; and a _transverse_
regime for angles $\theta^{\ast}\leq|\theta|\leq\pi/2$, where vortices
crossing the bistable domain undergo either no jump, one or two. The angle
$\theta^{\ast}$ is given by the (outer) tangent of the bistable domain at the
cusps $\bar{\bf R}_{c,\pm}$; making use of the lowest order approximation (70)
of the crescent’s geometry, we find that

$\displaystyle\tan(\theta^{\ast})$
$\displaystyle=\frac{\partial\bar{v}^{\scriptscriptstyle(0)}}{\partial\bar{u}^{\scriptscriptstyle(0)}}\Big{|}_{\bar{v}_{c}}=\frac{(\bar{C}+\lambda_{+})}{a}\sqrt{\frac{\gamma\delta-\beta^{2}}{2\gamma\bar{C}(\kappa_{m}-1)}},$
(81)

implying that $\pi/2-\theta^{\ast}\propto\sqrt{\kappa_{m}-1}$ is small,

$\displaystyle\theta^{\ast}\approx\pi/2-\frac{a}{(\bar{C}+\lambda_{+})}\sqrt{\frac{2\gamma\bar{C}(\kappa_{m}-1)}{\gamma\delta-\beta^{2}}}.$
(82)

####  III.7.1 Impact angles $|\theta|<\theta^{\ast}$

For a frontal impact with $|\theta|<\theta^{\ast}$, vortices occupy the ‘blue’
branch and remain there throughout the bistable domain $\mathcal{B}_{\bar{\bf
R}}$ until its termination on the far edge $\partial\mathcal{B}_{\bar{\bf
R}}^{\mathrm{b}}$, see Fig. 7, implying that $p_{\mathrm{b}}(\bar{\bf
R}\in\mathcal{B}_{\bar{\bf R}})=1$ and $p_{\mathrm{r}}(\bar{\bf
R}\in\mathcal{B}_{\bar{\bf R}})=0$, independent of $\theta$. As a consequence,
the pinning force $\mathbf{F}_{\mathrm{pin}}$ does not depend an the impact
angle and is given by the expression

$\mathbf{F}^{<}_{\mathrm{pin}}=-n_{p}\\!\int_{\mathcal{V}_{\bar{\bf
R}}\setminus\mathcal{B}_{\bar{\bf R}}}\\!\\!\\!\\!\frac{\mathrm{d}^{2}\bar{\bf
R}}{a_{0}^{2}}\,\mathbf{f}_{\mathrm{pin}}(\bar{\bf
R})-n_{p}\\!\int_{\mathcal{B}_{\bar{\bf
R}}}\\!\\!\\!\\!\frac{\mathrm{d}^{2}\bar{\bf
R}}{a_{0}^{2}}\,\mathbf{f}^{\mathrm{b}}_{\mathrm{pin}}(\bar{\bf R}).$

Next, Gauss’ formula tells us that for a function $e(\mathbf{x})$, we can
transform

$\int_{\mathcal{V}}\mathrm{d}^{n}x\,\mathbf{\nabla}e(\mathbf{x})=\int_{\partial\mathcal{V}}\mathrm{d}^{n-1}\,\mathbf{S}_{\perp}\,e(\mathbf{x}),$
(83)

with the surface element $\mathrm{d}^{n-1}\,\mathbf{S}_{\perp}$ oriented
perpendicular to the surface and pointing outside of the domain $\mathcal{V}$.
In applying (83) to the first integral of $\mathbf{F}^{<}_{\mathrm{pin}}$, we
can drop the contribution from the outer boundary
$\partial\mathcal{V}_{\bar{\bf R}}$ since we assume a compact defect
potential. The remaining contribution from the crescent’s boundary
$\partial\mathcal{B}_{\bar{\bf R}}$ joins up with the second integral but with
an opposite sign, as the two terms involve the same surface but with opposite
orientations. Altogether, we then arrive at the expression

$\mathbf{F}^{<}_{\mathrm{pin}}=n_{p}\int_{\partial\mathcal{B}_{\bar{\bf
R}}^{\mathrm{b}}}\frac{\mathrm{d}\,\mathbf{S}_{\perp}}{a_{0}^{2}}\left(e^{\mathrm{b}}_{\mathrm{pin}}(\bar{\bf
R})-e_{\mathrm{pin}}(\bar{\bf R})\right)\\\
+n_{p}\int_{\partial\mathcal{B}_{\bar{\bf
R}}^{\mathrm{r}}}\frac{\mathrm{d}\,\mathbf{S}_{\perp}}{a_{0}^{2}}\left(e^{\mathrm{b}}_{\mathrm{pin}}(\bar{\bf
R})-e_{\mathrm{pin}}(\bar{\bf R})\right),$ (84)

where we have separated the left and right borders
$\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{r,b}}$ of the bistable domain. Due
to continuity, the stable vortex energy $e_{\mathrm{pin}}(\bar{\bf R})$ will
be equal to $e_{\mathrm{pin}}^{\mathrm{b}}(\bar{\bf R})$ on the left border
$\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{r}}$ and equal to
$e_{\mathrm{pin}}^{\mathrm{r}}(\bar{\bf R})$ on the right border
$\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{b}}$. The expression (84) for
$\mathbf{F}^{<}_{\mathrm{pin}}$ then reduces to

$\displaystyle\mathbf{F}^{<}_{\mathrm{pin}}$
$\displaystyle=n_{p}\int_{\partial\mathcal{B}_{\bar{\bf
R}}^{\mathrm{b}}}\frac{\mathrm{d}\,\mathbf{S}_{\perp}}{a_{0}^{2}}\left(e^{\mathrm{b}}_{\mathrm{pin}}(\bar{\bf
R})-e^{\mathrm{r}}_{\mathrm{pin}}(\bar{\bf R})\right)$
$\displaystyle=n_{p}\int_{-\bar{v}_{c}}^{\bar{v}_{c}}\frac{\mathrm{d}\bar{v}}{a_{0}}\,\frac{\Delta
e_{\mathrm{pin}}(\bar{v})}{a_{0}}\left[1,-\partial{\bar{u}}/\partial{\bar{v}}\right]$
$\displaystyle=n_{p}\left[\frac{2\bar{v}_{c}}{a_{0}}\frac{\langle\Delta
e_{\mathrm{pin}}\rangle}{a_{0}},\,0\right]\equiv[F^{\parallel}_{\mathrm{pin}},0]$
(85)

with $\langle\Delta e_{\mathrm{pin}}\rangle$ the average energy jump evaluated
along the $v$-direction. The force $\mathbf{F}^{<}_{\mathrm{pin}}$ is aligned
with the unstable directed along $u$, with the $v$-component vanishing due to
the antisymmetry in $\bar{v}\leftrightarrow-\bar{v}$ of the derivative
$\partial{\bar{u}}/\partial{\bar{v}}$, and is independent on $\theta$ for
$|\theta|<\theta^{*}$.

####  III.7.2 Impact angle $|\theta|=\pi/2$

Second, let us find the pinning force density
$\mathbf{F}^{\pi/2}_{\mathrm{pin}}$ for vortices moving along the (positive)
$v$-direction, $\theta=\pi/2$. As follows from Fig. 7, vortices occupy the
blue branch and jump to the red one upon hitting the lower half of the
boundary $\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{b}}$; vortices that enter
$\mathcal{B}_{\bar{\bf R}}$ but do not cross $\partial\mathcal{B}_{\bar{\bf
R}}^{\mathrm{b}}$ undergo no jump and hence do not contribute to
$\mathbf{F}^{\pi/2}_{\mathrm{pin}}$. As vortices in the red branch proceed
upwards, they jump back to the blue branch upon crossing the red boundary
$\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{r}}$. While jumps appear on all of
the lower half of $\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{b}}$, a piece of
the upper boundary $\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{r}}$ that
contributes with a second jump is cut away (as vortices to the left of
$\bar{u}^{\scriptscriptstyle(0)}+\bar{u}^{\scriptscriptstyle(1)}$ do not
change branch from blue to red). The length $\Delta\bar{v}$ of this interval
scales as $\Delta\bar{v}/\bar{v}_{c}\propto(\kappa_{m}-1)^{1/4}$; ignoring
this small jump-free region, we determine $\mathbf{F}^{\pi/2}_{\mathrm{pin}}$
assuming that vortices contributing to $\mathbf{F}^{\pi/2}_{\mathrm{pin}}$
undergo a sequence of two jumps, from blue to red on the lower half
$\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{b<}}$ and back from red to blue on
the upper half $\partial\mathcal{B}_{\bar{\bf R}}^{\mathrm{r>}}$ of the
boundary $\partial\mathcal{B}_{\bar{\bf R}}$. Repeating the above analysis, we
find that the $u$-components in $\mathbf{F}^{\pi/2}_{\mathrm{pin}}$ arising
from the blue and red boundaries now cancel, while the $v$-components add up,

$\displaystyle\mathbf{F}^{\pi/2}_{\mathrm{pin}}$
$\displaystyle=n_{p}\int_{\partial\mathcal{B}_{\bar{\bf
R}}^{\mathrm{b<}}}\frac{\mathrm{d}\,\mathbf{S}_{\perp}}{a_{0}^{2}}\left(e^{\mathrm{b}}_{\mathrm{pin}}(\bar{\bf
R})-e^{\mathrm{r}}_{\mathrm{pin}}(\bar{\bf R})\right)$
$\displaystyle+n_{p}\int_{\partial\mathcal{B}_{\bar{\bf
R}}^{\mathrm{r>}}}\frac{\mathrm{d}\,\mathbf{S}_{\perp}}{a_{0}^{2}}\left(e^{\mathrm{r}}_{\mathrm{pin}}(\bar{\bf
R})-e^{\mathrm{b}}_{\mathrm{pin}}(\bar{\bf R})\right)$
$\displaystyle=2n_{p}\int_{0}^{\bar{v}_{c}}\frac{\mathrm{d}\bar{v}}{a_{0}}\,\frac{\Delta
e_{\mathrm{pin}}(\bar{v})}{a_{0}}\left[0,\partial{\bar{u}}/\partial{\bar{v}}\right]$
(86) $\displaystyle=n_{p}\left[0,\frac{2\bar{v}_{c}}{a_{0}}\frac{\langle\Delta
e_{\mathrm{pin}}\partial_{\bar{v}}\bar{u}\rangle}{a_{0}}\right]\equiv[0,F^{\perp}_{\mathrm{pin}}].$

Making use of the result (76) for $\Delta e_{\mathrm{pin}}(\bar{v})$ in
(III.7.1), we find explicit expressions for the pinning force densities for
impacts parallel and perpendicular to the unstable direction $u$,

$\displaystyle F_{\mathrm{pin}}^{\parallel}$
$\displaystyle\approx\left(\frac{9n_{p}}{8\,a_{0}^{2}\gamma^{3}}\right)\\!\int_{-\bar{v}_{c}}^{\bar{v}_{c}}\\!\\!\\!\\!\\!\\!\mathrm{d}\bar{v}\left[\frac{\gamma\delta-\beta^{2}}{(1+\lambda_{+}/\bar{C})^{2}}\left(\bar{v}_{c}^{2}-\bar{v}^{2}\right)\right]^{2}$
(87)
$\displaystyle=\frac{24}{5}n_{p}\frac{\sqrt{2\bar{C}/\gamma}}{a_{0}}\frac{\bar{C}^{2}}{\gamma
a_{0}}\frac{\gamma(1+\lambda_{+}/\bar{C})}{\sqrt{\gamma\delta-\beta^{2}}}(\kappa_{m}-1)^{5/2}$

and

$\displaystyle F_{\mathrm{pin}}^{\perp}$ $\displaystyle\approx
3\frac{\bar{C}^{2}}{\gamma a_{0}}\frac{\gamma
a/a_{0}}{\gamma\delta-\beta^{2}}(\kappa_{m}-1)^{3},$ (88)

that confirm the scaling estimates of Eq. (78). Here, we have made use of the
definition (III.3) of $\bar{v}_{c}$ and have brought the final result into a
form similar to the isotropic result (28) (with the length
$\sqrt{\bar{C}/\gamma}$ and the force $\bar{C}^{2}/\gamma a_{0}$, equal to
$\xi/\sqrt{3}\kappa$ and $e_{p}/12\kappa^{2}$ for a Lorentzian potential). The
result (87) provides the pinning force density
$\mathbf{F}_{\mathrm{pin}}=[F_{\mathrm{pin}}^{\parallel},0]$ for all impact
angles $|\theta|\leq\theta^{\ast}$ (note that (87) depends on the curvature
$a$ of the crescent via $\delta$, Eq. (49), that involves $a^{2}$ only, but
higher-order corrections will introduce an asymmetry between left- and right
moving vortices). Within the interval $\theta^{\ast}<\theta<\pi/2$, the
longitudinal force $F_{\mathrm{pin},u}$ along $u$ decays to zero and the
transverse force $F_{\mathrm{pin},v}$ along $v$ becomes finite, assuming the
value (88) at $\theta=\pi/2$. The two force components have been evaluated
numerically over the entire angular regime and the results are shown in Fig.
9: when moving away from the angle $\theta=\pi/2$, the transition from the
blue to the red boundary is moving upwards, with the relevant boundary turning
fully blue at $\theta=\theta^{\ast}$, thus smoothly transforming (III.7.2)
into (III.7.1) (we have adopted the approximation of dropping the jump-free
interval $\Delta\bar{v}$ that moves up and becomes smaller as $\theta$
decreases from $\pi/2$ to $\theta^{\ast}$).

Figure 9: Top: scaled pinning force densities $F_{\mathrm{pin},u}$ and
$F_{\mathrm{pin},v}$ versus impact angle $\theta$; we have used the same
parameters as in Fig. 6. The longitudinal (along $u$) force
$F_{\mathrm{pin},u}$ remains constant and equal to
$F_{\mathrm{pin}}^{\parallel}$ for all angles $|\theta|<\theta^{\ast}$, while
the transverse (along $v$) component $F_{\mathrm{pin},v}$ vanishes in this
regime. The longitudinal force drops and vanishes over the narrow interval
$\theta^{\ast}<|\theta|<\pi/2$, while the transverse force
$F_{\mathrm{pin},v}$ increases up to $F_{\mathrm{pin}}^{\perp}$. Bottom:
critical force density $F_{c}$ (directed along the Lorentz force
$\mathbf{F}_{\rm\scriptscriptstyle L}=\mathbf{j}\wedge\mathbf{B}/c$) versus
angle $\varphi$ of the Lorentz force; the dashed line shows the upper bound
$F_{c}<F_{\mathrm{pin}}^{\perp}/\sin(\varphi)$.

####  III.7.3 Anisotropic critical force density $\mathbf{F}_{c}$

When the vortex system is subjected to a current density $\mathbf{j}$, the
associated Lorentz force $\mathbf{F}_{\rm\scriptscriptstyle
L}(\varphi)=\mathbf{j}\wedge\mathbf{B}/c$ directed along $\varphi$ pushes the
vortices across the defects. When $\mathbf{F}_{\rm\scriptscriptstyle L}$ is
directed along $u$, we have
$\mathbf{F}_{\mathrm{pin}}=[F_{\mathrm{pin}}^{\parallel},0]$ and the vortex
system gets immobilized at force densities $F_{\rm\scriptscriptstyle
L}<F_{c}=F_{\mathrm{pin}}^{\parallel}$ (or associated current densities
$\mathbf{j}_{c}$). When $\mathbf{F}_{\rm\scriptscriptstyle L}$ is directed
away from $u$, the driving component along $v$ has to be compensated by a
finite pinning force $F_{\mathrm{pin},v}$ that appears only for angles
$\theta^{\ast}<\theta<\pi/2$. Hence, the angles of force and motion, $\varphi$
associated with the Lorentz force $\mathbf{F}_{\rm\scriptscriptstyle
L}(\varphi)$ and $\theta$ providing the direction of the pinning force
$\mathbf{F}_{\mathrm{pin}}(\theta)$, are different. We find them, along with
the critical force density $\mathbf{F}_{c}(\varphi)$, by solving the dynamical
force equation (17) at vanishing velocity $\mathbf{v}=0$,

$\mathbf{F}_{c}(\varphi)=\mathbf{F}_{\mathrm{pin}}(\theta)$ (89)

resulting in a critical force density

$F_{c}(\varphi)=\sqrt{F_{\mathrm{pin},u}^{2}(\theta)+F_{\mathrm{pin},v}^{2}(\theta)}$
(90)

with angles $\varphi$ and $\theta$ related via

$\tan\varphi=\frac{F_{\mathrm{pin},u}(\theta)}{F_{\mathrm{pin},v}(\theta)}.$
(91)

Since $F_{\mathrm{pin},u}(\theta<\theta^{\ast})=0$, the entire interval
$\theta<\theta^{\ast}$ is compressed to $\varphi=0$ and it is the narrow
regime $\theta^{\ast}<\theta<\pi/2$ that determines the angular characteristic
of the critical force density $F_{c}(\varphi)$. The critical force density
$F_{c}(\varphi)$ is peaked at $\varphi=0$ as shown in Fig. 9 (with a
correspondingly sharp peak in $j_{c}$ at right angles). Combing Eqs. (90) and
(91), we can derive a simple expression bounding the function
$F_{c}(\varphi)$,

$F_{c}(\varphi)=F_{\mathrm{pin},v}(\theta)\sqrt{1+\cot^{2}(\varphi)}\leq\frac{F_{\mathrm{pin}}^{\perp}}{\sin(\varphi)},$
(92)

that traces $F_{c}(\varphi)$ over a wide angular region, see the dashed line
in Fig. 9. At small values of $\varphi$ we cannot ignore the angular
dependence in $F_{\mathrm{pin},v}(\theta)$ any more that finally cuts off the
divergence $\propto 1/\sin(\varphi)$ at the value $F_{c}(\varphi\to 0)\to
F_{\mathrm{pin}}^{\parallel}$.

####  III.7.4 Isotropized pinning force density $F_{\mathrm{pin}}$

In a last step, we assume an ensemble of equal anisotropic defects that are
uniformly distributed in space and randomly oriented. In this situation, we
have to perform an additional average over the instability directions
$\hat{\mathbf{u}}_{i}$ associated with the different defects $i=1,\dots N$.
Neglecting the modification of $\mathbf{F}_{\mathrm{pin}}(\theta)$ away from
$[F_{\mathrm{pin}}^{\parallel},0]$ in the small angular regions
$\theta^{\ast}<|\theta|<\pi/2$, we find that the force along any direction
$\hat{\mathbf{R}}$ has the magnitude

$\displaystyle F_{\mathrm{pin}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}|(F_{\mathrm{pin}}^{\parallel}\hat{\mathbf{u}}_{i})\cdot\hat{\mathbf{R}}|$
$\displaystyle\approx$ $\displaystyle
F^{\parallel}_{\mathrm{pin}}\int_{-\pi/2}^{\pi/2}\frac{\mathrm{d}\theta}{\pi}\,\cos\theta=\frac{2}{\pi}F_{\mathrm{pin}}^{\parallel}.$

As a result of the averaging over the angular directions, the pinning force
density is now effectively isotropic and directed against the velocity
$\mathbf{v}$ of the vortex motion.

##  IV Uniaxial defect

In Sec. III, we have analyzed the onset of strong pinning for an arbitrary
potential and have determined the shape of the unstable and bistable domains
$\mathcal{U}_{\tilde{\bf R}}$ and $\mathcal{B}_{\bar{\bf R}}$—with their
elliptic and crescent forms, they look quite different from their ring-shaped
counterparts for the isotropic defect in Figs. 5(c) and (d). In this section,
we discuss the situation for a weakly anisotropic defect with a small uniaxial
deformation quantified by the small parameter $\epsilon$ in order to
understand how our previous findings, the results for the isotropic defect and
those describing the strong-pinning onset, relate to one another.

Our weakly deformed defect is described by equipotential lines that are nearly
circular but slightly elongated along $y$, implying that pinning is strongest
in the $x$-direction. We will find that the unstable (bistable) domain
$\mathcal{U}_{\tilde{\bf R}}$ ($\mathcal{B}_{\bar{\bf R}}$) for the uniaxially
anisotropic defect starts out with two ellipses (crescents) on the $x$-axis as
$\kappa_{m}$ crosses unity. With increasing pinning strength, i.e.,
$\kappa_{m}$, these ellipses (crescents) grow and deform to follow the
equipotential lines, with the end-points approaching one another until they
merge on the $\pm y$-axis. These merger points, we denote them as $\tilde{\bf
R}_{s}$ and $\bar{\bf R}_{s}$, define a second class of important points
(besides the onset points $\tilde{\bf R}_{m}$ and $\bar{\bf R}_{m}$) in the
buildup of the strong pinning landscape: while the onset points $\tilde{\bf
R}_{m}$ are defined as minima of the Hessian determinant $D(\tilde{\bf R})$,
the merger points $\tilde{\bf R}_{s}$ turn out to be associated with saddle
points of $D(\tilde{\bf R})$. Pushing across the merger of the deformed
ellipses (crescents) by further increasing the Labusch parameter $\kappa_{m}$,
the unstable (bistable) domains $\mathcal{U}_{\tilde{\bf R}}$
($\mathcal{B}_{\bar{\bf R}}$) undergo a change in topology, from two separated
areas to a ring-like geometry as it appears for the isotropic defect, see
Figs. 5(c) and (d), thus explaining the interrelation of our results for
isotropic and anisotropic defects.

With this analysis, we thus show how the strong pinning landscape for the
weakly uniaxial defect will finally assume the shape and topology of the
isotropic defect as the pinning strength $\kappa_{m}$ overcomes the anisotropy
$\epsilon$. Second, this discussion will introduce the merger points
$\tilde{\bf R}_{s}$ as a second type of characteristic points of strong
pinning landscapes that we will further study in section V.1 using a Landau-
type expansion as done in section III.1 above; we will find that the geometry
of the merger points $\tilde{\bf R}_{s}$ is associated with hyperbolas, as
that of the onset points was associated with ellipses.

Our uniaxially anisotropic defect is described by the stretched (along the
$y$-axis) Lorentzian

$e_{p}(\tilde{x},\tilde{y})=-e_{p}\left(1+\frac{\tilde{x}^{2}}{2\xi^{2}}+\frac{\tilde{y}^{2}}{2\xi^{2}\left(1+\epsilon\right)^{2}}\right)^{-1},$
(94)

with equipotential lines described by ellipses

$\frac{\tilde{x}^{2}}{\xi^{2}}+\frac{\tilde{y}^{2}}{\xi^{2}\left(1+\epsilon\right)^{2}}=\text{const},$
(95)

and the small parameter $0<\epsilon\ll 1$ quantifying the degree of
anisotropy. At fixed radius $\tilde{R}^{2}=\tilde{x}^{2}+\tilde{y}^{2}$, the
potential (94) assumes maxima in energy and in negative curvature on the
$x-$axis, and corresponding minima on the $y-$axis. Along both axes, the
pinning force is directed radially towards the origin and the Labusch
criterion (34) for strong pinning is determined solely by the curvature along
the radial direction. At the onset of strong pinning, the unstable and