# Approach to the lower critical dimension of the $\varphi^{4}$ theory in the
derivative expansion of the Functional Renormalization Group

Lucija Nora Farkaš<EMAIL_ADDRESS>Department of Physics, University of Zagreb,
Bijenička c. 32, 10000 Zagreb, Croatia  LPTMC, CNRS-UMR 7600, Sorbonne
Université, 4 Place Jussieu, 75252 Paris cedex 05, France  Gilles Tarjus
<EMAIL_ADDRESS>LPTMC, CNRS-UMR 7600, Sorbonne Université, 4 Place
Jussieu, 75252 Paris cedex 05, France  Ivan Balog<EMAIL_ADDRESS>Institute of
Physics, P.O.Box 304, Bijenička cesta 46, HR-10001 Zagreb, Croatia

###### Abstract

We revisit the approach to the lower critical dimension $d_{\rm lc}$ in the
Ising-like $\varphi^{4}$ theory within the functional renormalization group by
studying the lowest approximation levels in the derivative expansion of the
effective average action. Our goal is to assess how the latter, which provides
a generic approximation scheme valid across dimensions and found to be
accurate in $d\geq 2$, is able to capture the long-distance physics associated
with the expected proliferation of localized excitations near $d_{\rm lc}$. We
show that the convergence of the fixed-point effective potential is nonuniform
when $d\to d_{\rm lc}$ with the emergence of a boundary layer around the
minimum of the potential. This allows us to make analytical predictions for
the value of the lower critical dimension $d_{\rm lc}$ and for the behavior of
the critical temperature as $d\to d_{\rm lc}$, which are both found in fair
agreement with the known results. This confirms the versatility of the
theoretical approach.

###### pacs:

11.10.Hi, 75.40.Cx

##  I Introduction

Collective behavior characterized by an emergent scale invariance is
encountered in a wide variety of physical situations where many degrees of
freedom are correlated over long distances. Since its introduction, the
Renormalization Group has been the theoretical tool of choice for
understanding and describing this phenomenon.wilson-kogut  It provides a
powerful conceptual framework but, exact results being scarce, the search for
generic and efficient approximation schemes has been very active from the very
beginning.wilson-wavelet ; wilson-fisher72 ; ma72 ; migdal-kadanoff  One
relatively recent line of research starts from an exact formulation of the
Renormalization Group, in the form of a functional Renormalization Group (FRG)
for scale-dependent generating functionals of correlation functions,wegner-
houghton ; polchinski ; wetterich93  and introduces potentially
nonperturbative approximations through ansatzes for the scale-dependent
generating functional under study. The question we want to address is to what
extent such generic approximation schemes are able to describe specific
problems in which the long-distance behavior involves strongly nonuniform
configurations with localized excitations.

An example of such an approximation scheme within the FRG is the so-called
derivative expansion of the effective average action (coarse-grained Gibbs
free energy in the language of magnetic systems), which amounts to truncating
the functional form of the latter in powers of the external momenta or
equivalently in gradients of the fields.morris94  The versatility and the
effectiveness of the approach have been discussed in several reviews: see
Refs. [berges02, ; dupuis21, ]. One key advantage of such an approach is that
space dimension $d$ (as well as number of components of the fields, etc.) can
be continuously varied at will, allowing one to describe critical behavior
from the upper dimension $d_{\rm uc}$ where spatial fluctuations of the local
order parameter are easily tamed and classical (mean-field) exponents are
observed down to the lower critical dimension $d_{\rm lc}$ below which
fluctuations become so strong that no phase transition is possible.

The derivative-expansion approximation focuses on the long-distance properties
and, in terms of coarse-grained configurations of the system, works about
uniform configurations. One may therefore wonder if such a scheme is able to
capture the physics associated with nonuniform configurations containing,
e.g., domain walls, spin waves, or localized defects. The answer appears to be
positive in the case of configurations involving extended defects, i.e.,
defects whose energy scales with the system size but in a subextensive way.
For instance, the effect of spin waves or domain walls which are associated
with the return to convexity of the free energy of an O($N$) model in its low-
temperature ordered phase when spatial fluctuations are taken into
account,ringwald-wetterich ; tetradis92 ; berges02 ; pelaez-wschebor  or the
role of singular avalanche events and of scale-free droplet excitations in the
critical random-field Ising modeltissier06 ; tissier12 ; tarjus20  are all
properly accounted for by the truncated derivative expansion even at the
lowest orders.

Yet, the jury is still out when the relevant coarse-grained configurations
that control the large-scale behavior involve localized excitations such as
the kinks and anti-kinks found in the instanton analysis of the 1-dimensional
Ising model.rulquin15  As the approach to the lower critical dimension for
systems with a discrete symmetry is expected to be controlled by the
proliferation of such localized excitations,bruce81 ; bruce83 ; wallace84
describing the long-distance physics in, say, a model in the Ising
universality class such as the $\varphi^{4}$ theory in $d=1+\epsilon$ when
$\epsilon\to 0$ is thus a more demanding task for the nonperturbative but
approximate FRG than describing the O($N>2$) universality class near
$d=2$.footnote_2d

In this paper we investigate how low orders of the derivative expansion in the
FRG describe the approach to the lower critical dimension of the $\varphi^{4}$
theory. The lowest order is known as the Local Potential Approximation
(LPA)wegner-houghton  and is clearly unphysical in low dimensions as it
predicts $d_{\rm lc}=2$. Indeed, field renormalization is not accounted for in
the LPA, implying that the anomalous dimension of the field is always
$\eta=0$. This then misses a crucial ingredient for investigating Ising
criticality in dimensions less than $2$. We thus consider the simplest
approximation beyond the LPA that incorporates this effect and is often
referred to as the LPA’.berges02 ; dupuis21  Working at this level allows us
to provide a detailed analytical treatment of the problem.

We stress that the issue per se is not to provide another characterization of
the $\varphi^{4}$ theory near $d=1$, as for instance Bruce and Wallacebruce81
; bruce83 ; wallace84  have already developed an efficient approach in terms a
specifically tailored droplet theory. The issue is to assess the ability of a
generic nonperturbative approximation scheme within the FRG to quantitatively
describe the long-distance physics of a model across the whole range of space
dimensions from $d_{\rm lc}$ to $d_{\rm uc}$ without a priori knowledge of the
relevant real-space coarse-grained configurations. This also involves the
question of the continuity of the critical behavior in the dimensionality of
space, which was first investigated by Ballhausen et al.ballhausen04  At odds
with the latter work we show that convergence of the critical behavior of the
$\varphi^{4}$ theory when $d\to d_{\rm lc}$ within the FRG is nonuniform.

The outline of the paper is as follows. In Sec. II we summarize the FRG
framework and the derivative expansion scheme for the scalar $\varphi^{4}$
theory. We also introduce the LPA’ approximation and the approach to the lower
critical dimension $d_{\rm lc}$. We then show in Sec. III that the convergence
of the fixed-point effective potential to the lower critical dimension is
nonuniform in the field and involves a boundary (or interior) layer around the
minimum of the potential. We detail the singular perturbation treatment that
allows us to find the solution at leading order over the whole range of field.
We next present in Sec. IV the results that we obtain for the value the lower
critical dimension, which we find close to the exact value $d_{\rm lc}=1$, as
well as for the critical temperature and for the critical exponents as $d\to
d_{\rm lc}$. We finally give some concluding remarks and provide additional
details on the method and the solution in several appendices.

##  II Functional RG, derivative expansion, and the LPA’

We are interested in the critical behavior of the Ising universality class,
which can be represented at a field-theoretical level by a a scalar
$\varphi^{4}$ theory,

$S[\varphi]=\int_{x}\big{(}\frac{1}{2}(\partial_{x}\varphi(x))^{2}+\frac{r}{2}\varphi(x)^{2}+\frac{u}{4!}\varphi(x)^{4}\big{)},$
(1)

where $\int_{x}\equiv\int d^{d}x$. To do so, we use the FRG approach which is
a modern version of Wilson’s RG in which fluctuations are progressively
incorporated in the calculation of the partition function of the model through
the addition to the action of an infrared (IR) regulator,berges02

$\Delta S_{k}[\varphi]=\frac{1}{2}\int_{xy}R_{k}(x-y)\varphi(x)\varphi(y),$
(2)

where $R_{k}$ is an IR cutoff function that suppresses integration of modes
with momenta less than $k$ without altering that of modes with momenta larger
than $k$. Typical choices of $R_{k}$ will be discussed below. The modified
partition function

$Z_{k}[J]=\int\mathcal{D}\varphi\exp(-S[\varphi]-\Delta S_{k}[\varphi])$ (3)

is the scale-dependent generating functional of correlation functions and via
a modified Legendre transform,

$\Gamma_{k}[\phi]=-\ln Z_{k}[J]+\int_{x}J(x)\phi(x)-\Delta S_{k}[\phi]),$ (4)

one can introduce the effective average action $\Gamma_{k}[\phi]$, with
$\phi(x)=\langle\varphi(x)\rangle=\delta\ln Z_{k}[J]/\delta J(x)$, which is
the scale-dependent generating functional of the 1-particle irreducible (1-PI)
correlation functions. It obeys an exact functional RG equation that describes
its evolution with the IR scale $k$,wetterich93

$\partial_{t}\Gamma_{k}[\phi]=\frac{1}{2}\int_{xy}\partial_{t}R_{k}(x-y)\big{[}(\Gamma_{k}^{(2)}[\phi]+R_{k})^{-1}\big{]}_{xy},$
(5)

where $\Gamma_{k}^{(2)}$ is the second functional derivative of $\Gamma_{k}$
and $t=\ln(k/\Lambda)$ with $\Lambda$ a UV cutoff.

The exact FRG equation in Eq. (5) is a convenient starting point for devising
nonperturbative approximation schemes in the form of ansatzes for the
functional dependence of the effective average action. One such scheme used to
capture the long-distance physics is the derivative expansion in which the
Lagrangian associated with $\Gamma_{k}$ is expanded in gradients of the
fields,

$\Gamma_{k}[\phi]=\int_{x}\big{[}U_{k}(\phi(x))+\frac{1}{2}Z_{k}(\phi(x))(\partial_{x}\phi(x))^{2}+{\rm
O}(\partial\partial\partial\partial)\big{]}.$ (6)

When inserted in Eq. (5) the above ansatz provides a hierarchy of coupled FRG
equations for the functions $U_{k}(\phi)$, $Z_{k}(\phi)$, etc., where the
field configurations involved are now uniform, i.e., $\phi(x)=\phi$.

Scale invariance associated with criticality is described by a fixed point in
the FRG equations once the latter have been cast in a dimensionless form via
the use of scaling dimensions. One defines dimensionless quantities $\varphi$,
$u_{k}$, $z_{k}$, etc., through

$\phi=k^{D_{\phi}}\varphi,\;\;U_{k}(\phi)=k^{d}u_{k}(\varphi),\;\;Z_{k}(\phi)=Z_{k}z_{k}(\varphi),$
(7)

etc., where the dimension of the field is related to the anomalous dimension
$\eta$ by $D_{\phi}=(d-2+\eta)/2$ and where the field renormalization constant
$Z_{k}$ goes as $k^{-\eta}$ in the vicinity of the fixed point. (Note that we
have used the same notation $\varphi$ for the bare variable in Eq. (1) and the
dimensionless average field, as the former will no longer appear in what
follows.)

The hierarchy of FRG equations when expressed in terms of dimensionless
quantities takes the form

$\displaystyle\partial_{t}u_{k}(\varphi)=-du_{k}(\varphi)+\frac{(d-2+\eta_{k})}{2}\varphi
u_{k}^{\prime}(\varphi)+\beta_{u}(\varphi;\eta_{k})$ (8)
$\displaystyle\partial_{t}z_{k}(\varphi)=\eta_{k}z_{k}(\varphi)+\frac{(d-2+\eta_{k})}{2}\varphi
z_{k}^{\prime}(\varphi)+\beta_{z}(\varphi;\eta_{k}),$

etc., where a prime indicates a derivative with respect to the argument of the
function; $\beta_{u}$, $\beta_{z}$, etc. are functionals of
$u_{k}^{\prime\prime}$, $z_{k}$, etc., and are given in Appendix A. The fixed
points of the flow equations are reached when $t\to-\infty$ (i.e., $k\to 0$)
and the left-hand sides go to zero.

As already stressed, a proper description of the approach to the lower
critical dimension should incorporate field renormalization and a nonzero
anomalous dimension $\eta$. The lowest order of the derivative expansion that
achieves this is the so-called LPA’ in which one retains on top of the
renormalized potential $U_{k}(\phi)$ a field independent but scale dependent
$Z_{k}$. In explicit form, the dimensionless equation for the fixed-point
potential is now

$\displaystyle 0=-d\,u(\varphi)+\frac{d-2+\eta}{2}\varphi
u^{\prime}(\varphi)+2v_{d}\ell_{0}^{(d)}(u^{\prime\prime}(\varphi);\eta),$ (9)

where $v_{d}^{-1}=2^{d+1}\pi^{d/2}\Gamma(d/2)$ and $\ell_{0}^{(d)}$ is a
(strictly positive) dimensionless threshold function which enforces the
decoupling of the low-momentum and high-momentum modes; it is defined in terms
of the dimensionless IR cutoff function
$r(y=q^{2}/k^{2})=R_{k}(q^{2})/(Z_{k}q^{2})$ by

$\displaystyle\ell_{0}^{(d)}(w;\eta)=-\frac{1}{2}\int_{0}^{\infty}dyy^{\frac{d}{2}}\frac{\eta
r(y)+2yr^{\prime}(y)}{(y[1+r(y)]+w)}$ (10)

and is described in more detail in Appendix B. We have dropped the subscript
$k\to 0$ for quantities at the fixed point in the above equation to simplify
the notation.

Deriving Eq. (LABEL:eq_FPpotential) gives an equation for
$u^{\prime}(\varphi)$ from which one extracts the equation for its minima
$\pm\varphi_{\rm m}$ [through $u^{\prime}(\pm\varphi_{\rm m})=0$],

$0=\frac{(d-2+\eta)}{2}\varphi_{{\rm
m}}+2v_{d}\frac{u^{\prime\prime\prime}(\varphi_{\rm
m})}{u^{\prime\prime}(\varphi_{{\rm
m}})}\partial_{w}\ell_{0}^{(d)}(w;\eta)|_{w=u^{\prime\prime}(\varphi_{\rm
m})},$ (11)

and deriving one more time gives an equation for the “squared mass”
$u^{\prime\prime}(\varphi)$. Both equations will be useful below.

In the LPA’ the field renormalization constant $Z_{k}$ is chosen such that at
the minimum of the potential $z_{k}(\pm\varphi{\rm m})=1$.berges02 ;
ballhausen04  From Eq. (LABEL:eq_ERGEdimensionless) and the explicit form of
$\beta_{z}$ given in Appendix A one then obtains that

$\eta=\frac{4v_{d}}{d}u^{\prime\prime\prime}(\varphi_{\rm
m})^{2}m_{4,0}^{(d)}(u^{\prime\prime}(\varphi_{\rm m});\eta),$ (12)

where $m_{4,0}^{(d)}$ is another (strictly positive) dimensionless threshold
function defined by

$\displaystyle
m_{4,0}^{(d)}(w;\eta)=\frac{1}{2}\int_{0}^{\infty}dyy^{\frac{d}{2}}\frac{1+(yr(y))^{\prime}}{(y[1+r(y)]+w)^{4}}\bigg{[}2\eta(yr(y))^{\prime}$
(13)
$\displaystyle+4(y^{2}r^{\prime}(y))^{\prime}-4\,\frac{y[1+(yr(y))^{\prime}][\eta
r(y)+2yr^{\prime}(y)]}{y[1+r(y)]+w}\bigg{]}$

and discussed in Appendix B. Once a specific form for the dimensionless IR
cutoff function $r(y)$ has been chosen, the solution of Eqs.
(LABEL:eq_FPpotential-LABEL:eq_threshold_m40) fully characterizes the LPA’
fixed point. In what follows we will use two much studied forms of $r(y)$:

$\displaystyle r(y)=\alpha\Theta(1-y)(1-y)/y$ (14) $\displaystyle r(y)=\alpha
e^{-y}/y$

where $\Theta$ is the Heaviside step function and $\alpha$ is a variational
parameter of O(1) that can be determined, e.g., by the principle of minimum
sensitivity.litim01 ; dupuis21 ; balog19  We will refer to these two choices
as Theta and Exponential cutoff functions.

We illustrate the results for the fixed point at LPA’ and the choice of the
Theta cutoff function with $\alpha=1$ (similar results are obtained with other
choices, see the discussion further below) in Figs. 1 and 2. Fig 1 displays
the evolution of the dimensionless potential $u(\varphi)$ and the “square
mass” function $u^{\prime\prime}(\varphi)$ as the space dimension $d$
decreases, and Fig. 2 that of the field scaling dimension
$D_{\phi}=(d-2+\eta)/2$. To show that a similar behavior in low dimension is
also expected for higher orders of the derivative expansion, so that the LPA’
level is not atypical, we plot the evolution with $d$ of the dimensionless
potential and of the field dimension $D_{\phi}$ at the second order of the
derivative expansion for which the field renormalization is now a full
function of the field [see Eq. (LABEL:eq_ERGEdimensionless)] in Fig. 3

a)b) Figure 1: Dimensionless effective potential $u(\varphi)$ (a) and its
second derivative $u^{\prime\prime}(\varphi)$ (b) at the LPA’ fixed point for
several dimensions $d$ between $1.6$ and $1.1$. We have used the Theta IR
cutoff function with $\alpha=1$ and a numerical resolution of the FRG
equations.  Figure 2: Variation with the space dimension $d$ of the field
scaling dimension $D_{\phi}=(d-2+\eta)/2$ at the LPA’ fixed point as obtained
from a numerical resolution and the Exponential IR cutoff function. a)b)
Figure 3: Dimensionless effective potential $u(\varphi)$ (a) and field scaling
dimension $D_{\phi}=(d-2+\eta)/2$ (b) at the fixed point obtained of the
second order of the derivative expansion for several dimensions $d$ between
$1.8$ and $1.11$. We have used the Exponential IR cutoff function with
$\alpha=1$ and a numerical resolution of the FRG equations.

A defining property of the lower critical dimension is that $(d-2+\eta)|_{d\to
d_{l}}\to 0$. This is equivalent to stating that the scaling dimension
$D_{\phi}$ of the field vanishes (see above and Fig. 2). If the field does not
rescale, its fluctuations along the RG flow remain of order $1$ in terms of
the dimensionful field and ordering associated with a nonzero dimensionful
average field in zero applied source is impossible. We thus find it convenient
to define

$\tilde{\epsilon}(d)=\frac{d-2+\eta}{2(2-\eta)},$ (15)

which goes to $0$ as $d\to d_{\rm lc}$. We use the notation $\tilde{\epsilon}$
to avoid confusion with $\epsilon=d-d_{\rm lc}$ (with $d_{\rm lc}=1$ in an
exact treatment.)

Another anticipated feature of the approach to the lower critical dimension is
the fact that the propagator of the theory approaches a pole. Indeed, the
lower critical dimension corresponds to the merging of the critical fixed
point and the zero-temperature fixed point associated with the symmetry-broken
ordered phase, and the return to convexity of the effective potential along
the FRG flow is controlled in the latter by the presence of a pole in the
propagator.tetradis92 ; berges02 ; pelaez-wschebor  In the LPA’ the
dimensionless propagator is given by

$p(y;\varphi)=\frac{1}{y[1+r(y)]+u^{\prime\prime}(\varphi)}$ (16)

and must of course be positive. With the choice of the Theta cutoff function
in Eq. (LABEL:eq_cutoff_choices) the pole in either in $y=q^{2}/k^{2}=0$ and
$u^{\prime\prime}(\varphi)=-\alpha$ when $\alpha<1$ or in $y=1$ and
$u^{\prime\prime}(\varphi)=-1$ when $\alpha>1$. With the Exponential cutoff
function the pole is either in $y=0$ and $u^{\prime\prime}(\varphi)=-\alpha$
when $\alpha<1$ or in $y=\ln\alpha$ and
$u^{\prime\prime}(\varphi)=-(1+\ln\alpha)$ when $\alpha>1$: see Appendix B.

In the following we will investigate in more detail the structure of the
fixed-point solution at LPA’ when $d\to d_{\rm lc}$. As the numerical solution
becomes harder if not impossible in this limit, progress should be made
through an analytical treatment. We stress again that contrary to what happens
for the O($N>2$) models where there are Goldstone modes associated with the
breaking of a continuous symmetry which quite straightforwardly imply that
$d_{\rm lc}=2$,zinn-justin89 ; berges02 ; delamotte04 ; dupuis21  the LPA’ or
any level of truncation of the derivative expansion within the FRG need not
predict the exact value of the lower critical dimension, $d_{\rm lc}=1$, for
the present model with a discrete $Z_{2}$ symmetry. The approximate $d_{\rm
lc}$ must then be computed.

##  III Nonuniform convergence to the lower critical dimension

###  III.1 nonuniform convergence and boundary layer

Consider the LPA’ fixed-point equation for the second derivative of the
potential,

$\displaystyle 0=-u^{\prime\prime}(\varphi)+\tilde{\epsilon}(d)\varphi
u^{\prime\prime\prime}(\varphi)+\frac{2v_{d}}{2-\eta(d)}\partial^{2}_{\varphi}\ell_{0}^{(d)}(u^{\prime\prime}(\varphi);\eta(d)),$
(17)

where $\eta=2-d+d\tilde{\epsilon}+{\rm O}(\tilde{\epsilon}^{2})$. As the
dependence of $\tilde{\epsilon}$ on $d$ is expected to be monotonic, one can
study the above equation at fixed $\tilde{\epsilon}$ instead of fixed $d$, and
when $\tilde{\epsilon}\to 0$,

$\displaystyle 0=-u^{\prime\prime}(\varphi)+\tilde{\epsilon}\varphi
u^{\prime\prime\prime}(\varphi)+\frac{2v_{d}}{d}\partial^{2}_{\varphi}\ell_{0}^{(d)}(u^{\prime\prime}(\varphi);2-d),$
(18)

where $d\equiv d(\tilde{\epsilon})\to d_{\rm lc}$ and $d_{\rm lc}$ a priori
unknown.

If $u^{\prime\prime}(\varphi)$, its first derivatives, and $\varphi$ are of
O(1) the second term in Eq. (18) can be set to zero as a leading approximation
in which $\tilde{\epsilon}=0$ altogether. This describes a uniform convergence
toward the lower critical dimension, as was assumed in Ref. [ballhausen04, ].
Our claim, which we substantiate below, is that the second term leads to a
singular-perturbation problem and that the limit $\tilde{\epsilon}\to 0$ is
nonuniform in the field.

From the shape of the fixed-point potential in Fig. 1 one can see that three
domains of field values can be distinguished: the large-field region,
$|\varphi|\to+\infty$, where $u(\varphi)$ and its first derivatives blow up,
the region of the minima where the first derivative is zero but higher-order
derivatives grow large as $d$ decreases, and the inner region of fields of
order O(1) in which the potential and its derivatives appear of O(1). The
latter region should be describable by the $\tilde{\epsilon}=0$ equation (plus
regular perturbation in $\tilde{\epsilon}$) and the large-field one
corresponds to the situation where the square mass $u^{\prime\prime}(\varphi)$
diverges and the nontrivial beta functions go to zero to only leave the
scaling part of the equation: here,
$0=-u^{\prime\prime}(\varphi)+\tilde{\epsilon}\varphi
u^{\prime\prime\prime}(\varphi)$, which leads to

$u^{\prime\prime}(\varphi)\sim|\varphi|^{\frac{1}{\tilde{\epsilon}}}\;\;{\rm
when}\;\;\varphi\to\pm\infty.$ (19)

The region of the close vicinity of the minima needs more care and entails a
boundary-layer treatment. (Note that interior layer would be more adequate in
this case than boundary layer because the region in which variation is very
fast is away from the boundaries, but with this caveat we will nonetheless
keep using the term boundary layer.) A unique global solution valid for all
fields is finally obtained by “matching” the partial solutions obtained in
each domain in the intermediate regions of field over which they overlap. This
matching procedure is a key element of the singular perturbation
treatment.singular-perturbation

The potential $u(\varphi)$ being $Z_{2}$ symmetric, we choose to restrict our
analysis to positive fields, $\varphi\geq 0$.

###  III.2 The solution of the $\tilde{\epsilon}=0$ equation

Consider first the LPA’ equation for $\tilde{\epsilon}=0$. Introducing for
simplicity the notation $w(\varphi):=u^{\prime\prime}(\varphi)$, one has

$\displaystyle
w(\varphi)=\frac{2v_{d}}{d}\partial^{2}_{\varphi}\ell_{0}^{(d)}(w(\varphi);2-d),$
(20)

where $d=d_{\rm lc}$, which we assume in the following to be strictly less
than $2$, and the initial conditions are $w(0)=w_{0}$ and all the odd
derivatives of $w$ are zero in $\varphi=0$. Let also define
$\Phi(\varphi):=\ell_{0}^{(d)}(w(\varphi);2-d)$. The function
$\ell_{0}^{(d)}(w;2-d)$ being monotonically increasing with $w$, one can
invert it and define $w=F(\Phi)$ with $F$ such that
$F(\ell_{0}^{(d)}(w;2-d)=w$. Eq. (20) can then be rewritten as

$\displaystyle\partial^{2}_{\varphi}\Phi(\varphi)=\frac{d}{2v_{d}}F(\Phi(\varphi)),$
(21)

which is the equation of motion of an anharmonic oscillator $\Phi(\varphi)$
with $\varphi$ playing the role of time and $(d/(2v_{d}))F(\Phi)$ being the
force. The solution for $\Phi(\varphi)$ is a periodic function starting in
$\Phi_{0}=\ell_{0}^{(d)}(w_{0};2-d)$ with a velocity
$\partial_{\varphi}\Phi|_{0}=0$. The half-period $\varphi_{*}$ corresponds to
the first time at which the velocity is again equal to $0$. By using the
energy balance equation associated with Eq. (21),

$\displaystyle\int_{\Phi_{0}}^{\Phi(\varphi)}d\Phi^{\prime}\frac{d}{2v_{d}}F(\Phi^{\prime})=\frac{1}{2}[\partial_{\varphi}\Phi(\varphi)]^{2},$
(22)

one derives that $\varphi_{*}$ is obtained from

$\displaystyle\int_{0}^{\varphi_{*}}d\varphi^{\prime}\partial_{\varphi^{\prime}}\Phi(\varphi^{\prime})w(\varphi^{\prime})=0.$
(23)

Note that $\varphi_{*}$ and the solution $\Phi(\varphi)$ are parametrized by
the initial value $w_{0}$.

Because of the monotonic relation between $\Phi$ and $w$, the solution for
$w(\varphi)$ is also a periodic function of half-period $\varphi_{*}$ that
oscillates between a minimum value $w_{0}$ and a maximum one
$w_{*}=w(\varphi_{*})$, the two values being uniquely related. Clearly, this
solution cannot be that of the full problem (which is not periodic: see Fig.
1(b)) when $\varphi$ is close to and larger than $\varphi_{*}$. As alluded to
above, a boundary-layer type of solution must replace the solution of the
$\tilde{\epsilon}=0$ equation. Since $w(\varphi)$ is very large in the close
vicinity of the (exact) minimum of the potential, $\varphi_{\rm m}$, a
potential matching between the two types of solution must take place for
$\varphi<\varphi_{\rm m}\lesssim\varphi_{*}$, which requires that
$w(\varphi_{*})$ very large. In this limit, it can be shown from the
properties of the $\tilde{\epsilon}=0$ solution (see Appendix C) that

$\varphi_{*}\sim\sqrt{\ln w_{*}}.$ (24)

The matching requirement and the constraint it puts on the value of
$w(\varphi_{0})=w_{0}$ will be considered in more detail below.

###  III.3 The inner solution within the layer

The $\tilde{\epsilon}=0$ equation ceases to be the proper description when the
second term of Eq. (18) becomes of the same order as the other terms and a new
solution must be found. Guided by the numerical solution and by physical
intuition, we argue that a new solution takes place within a “boundary layer”
(actually, an “interior layer”: see the above comment) around the minimum
$\varphi_{\rm m}$ of the potential. In this region, $w\gg 1$, and Eq. (18)
becomes

$\displaystyle 0=-w(\varphi)+\tilde{\epsilon}\varphi
w^{\prime}(\varphi)+\frac{2v_{d}}{d}\alpha
A_{d}\partial^{2}_{\varphi}\left[\frac{1}{w(\varphi)}\right],$ (25)

where we have used that for large square mass $w$, Eq. (10) gives

$\displaystyle\ell_{0}^{(d)}(w;2-d)\sim\frac{\alpha A_{d}}{w}+{\rm
O}(\frac{1}{w^{2}})$ (26)

with, after some manipulations, $A_{d}$ obtained as

$\displaystyle
A_{d}=d\int_{0}^{\infty}dyy^{\frac{d}{2}}\left[\frac{r(y)}{\alpha}\right].$
(27)

More details are given in Appendix B. It is then convenient to rescale the
field $\varphi$ by a multiplicative factor $\sqrt{d/(2\alpha v_{d}A_{d})}$, so
that Eq. (25) reads

$\displaystyle 0=-w(\varphi)+\tilde{\epsilon}\varphi
w^{\prime}(\varphi)+\partial^{2}_{\varphi}\left[\frac{1}{w(\varphi)}\right],$
(28)

with no explicit dependence on $d$ (which we recall should be taken as $d_{\rm
lc}$ in the limit $\tilde{\epsilon}\to 0$ that we consider).

In the vicinity of $\varphi_{\rm m}$ we introduce a rescaled variable
$x=(\varphi-\varphi_{\rm m})/\delta(\tilde{\epsilon})$ with $x={\rm O}(1)$ as
$\tilde{\epsilon}\to 0$. By balancing the first two terms of Eq. (28) one
obtains that $\delta(\tilde{\epsilon})=\tilde{\epsilon}\varphi_{\rm m}$, i.e.,

$x=\frac{\varphi-\varphi_{\rm m}}{\tilde{\epsilon}\varphi_{\rm m}},$ (29)

where we assume for now, and check later on, that
$\tilde{\epsilon}\varphi_{\rm m}\to 0$ when $\tilde{\epsilon}\to 0$. By also
requiring that the third term is of the same order of magnitude as the first
two, one is further led to introduce a function $g(x)$ which is defined by

$w(\varphi)=\frac{g(x)}{\tilde{\epsilon}\varphi_{\rm m}}$ (30)

and which is of O(1) when $x$ is of O(1). The LPA’ equation in the boundary
layer can then be expressed as

$-g(x)+g^{\prime}(x)+\partial^{2}_{x}\left[\frac{1}{g(x)}\right]=0.$ (31)

This is complemented by the equation for the minimum, Eq. (11), which leads to

$g^{\prime}(0)=g(0)^{3}.$ (32)

The boundary layer equation can be solved in an implicit form by first
introducing the auxiliary function

$X(x)=g(x)+\partial_{x}\left[\frac{1}{g(x)}\right]$ (33)

which satisfies $X(0)=0$ because of Eq. (32). Then, one has that

$g(x)=X^{\prime}(x),$ (34)

and the solution for $g(x)$ as a function of $X(x)$ is given by

$\displaystyle
g(x)=g(0)\frac{e^{-\frac{X(x)^{2}}{2}}}{\left[1-\sqrt{\frac{\pi}{2}}g(0)+g(0)\int_{X(x)}^{+\infty}dt\,e^{-\frac{t^{2}}{2}}\right]}\,.$
(35)

The interest of the above expression is that it allows us to study the limit
$x\to+\infty$ and require matching with the outer solution at large field,
$w(\varphi\to\infty)\sim\varphi^{1/\tilde{\epsilon}}\to+\infty$, already
derived. (This is the standard method of matched asymptotic expansions used in
singular perturbation problems.) Choosing the matching region such that
$1/\tilde{\epsilon}\gg x\gg 1$, the latter then imposes that $g(x)$ diverges
as $\exp(x)$ at large positive $x$. From Eqs. (35) and (33) it is
straightforward to see that one must have

$1-\sqrt{\frac{\pi}{2}}g(0)=0,$ (36)

which fixes $g(0)$ and via Eq. (32) $g^{\prime}(0)$.

We plot in Fig. 4 the inverse of $g(x)$, as obtained from the solution of the
above equations. It is a monotonically decreasing function. When $x$ is
negative and $|x|$ very large, it behaves as one finds that

$\displaystyle\frac{1}{g(x)}\sim\sqrt{2}|x|\sqrt{\ln|x|}\Big{[}1+{\rm
O}\left(\frac{\ln(\ln|x|)}{\ln|x|}\right)\Big{]},$ (37)

an expression which will be useful when considering matching with the outer
solution obtained from the equation with $\tilde{\epsilon}=0$ (see Sec.
III.2).

Figure 4: Inverse of the inner solution $g(x)$ giving the square-mass function
within the layer around the minimum of the potential. The latter corresponds
to $x=0$.

###  III.4 Matching the inner solution with that of the $\tilde{\epsilon}=0$
equation

At least at the present LPA’ level of approximation the solution $g(x)$ within
the layer around $\varphi_{\rm m}$ is fully determined by matching with the
outer solution obtained at large field. However, the relation between
$\varphi_{\rm m}$ and $\tilde{\epsilon}$ is yet to be determined. It is also
crucial to check that matching with the outer solution corresponding to fields
less than $\varphi_{\rm m}$ and to the $\tilde{\epsilon}=0$ equation can be
enforced, so that a solution can be constructed for all field values.

In Sec. III.2 we have argued that matching should take place for fields
$\varphi<\varphi_{\rm m}\lesssim\varphi_{*}$, where furthermore the outer
solution associated with the $\tilde{\epsilon}=0$ equation should be such that
$1\ll w(\varphi)\ll w(\varphi_{\rm m}),w_{*}$. We can choose the matching
region where the two solutions overlap such that

$\varphi_{\rm m}-\varphi={\rm O}((\tilde{\epsilon}\varphi_{\rm
m})^{a})\;\;{\rm with}\;0<a<1,$ (38)

and, as a result, $x\sim-(\tilde{\epsilon}\varphi_{\rm m})^{1-a}$ is negative
and very large. The asymptotic limit of the inner solution is then given by
Eq. (37), which implies that

$w_{*}\gg w(\varphi)={\rm O}\big{(}\tilde{\epsilon}\varphi_{\rm
m})^{-a}|\ln(\tilde{\epsilon}\varphi_{\rm
m})|^{-\frac{1}{2}}\big{)}\to+\infty.$ (39)

From Eq. (23) one can obtain the relation between $w_{*}$ and $w_{0}$ as

$\displaystyle 0$
$\displaystyle=\int_{w_{0}}^{w_{*}}dw^{\prime}w^{\prime}\partial_{w^{\prime}}\Phi(w^{\prime})$
(40)
$\displaystyle=w_{*}\Phi(w_{*})-w_{0}\Phi(w_{0})-\int_{w_{0}}^{w_{*}}dw^{\prime}\Phi(w^{\prime}),$

where, we recall, $\Phi(w)=\ell_{0}^{(d)}(w;2-d)$ with the latter given in Eq.
(10). To evaluate the quantities in the above equation we split the integral
by introducing an intermediate value $w_{c}$, which we choose positive and of
O(1). Taking into account that $w_{*}$ diverges and using the property that
the function $\Phi(w)$ is monotonically decreasing and asymptotically goes to
zero as $\alpha A_{d}/w+{\rm O}(1/w^{2})$, we transform Eq. (40) into

$\displaystyle\alpha A_{d}\ln
w_{*}=-w_{0}\Phi(w_{0})+\int_{w_{0}}^{w_{c}}dw^{\prime}\Phi(w^{\prime}),$ (41)

up to O(1) terms, and, since $w_{0}$ and $w_{c}$ are of O(1), this implies
that $\Phi(w_{0})$ diverges. This can only occur if $w_{0}$ approaches the
pole of the propagator, which we call $w_{P}$ and can be either $-1$,
$-\alpha$, or $-(1+\ln\alpha)$ depending on the IR cutoff function and on
$\alpha$ [see Eq. 16) and below]. Notwithstanding the precise asymptotic
behavior of $\Phi(w)\equiv\ell_{0}^{(d)}(w;2-d)$ when the pole is approached
(this depends on the IR cutoff function, see Appendix B), the second term of
the right-hand side is subdominant compared to the first one and one has

$\displaystyle\Phi(w_{0})\sim\frac{\alpha A_{d}}{|w_{P}|}\ln w_{*}\to+\infty.$
(42)

Matching thus entails that the square mass in zero field $w_{0}\to w_{P}^{+}$,
which, as argued above, is one of the expected hallmarks of the approach to
the lower critical dimension. Note that in the limit process $w_{0}$ must
remain strictly larger than the pole $w_{P}$ by a quantity that goes to zero
with $\tilde{\epsilon}$: this again illustrates the highly singular and
nonuniform approach to the lower critical dimension.

To complete the proof, we note that in the chosen matching region, the leading
behavior of $w(\varphi)$ in the boundary layer and that corresponding to the
$\tilde{\epsilon}=0$ solution obey the same equation,
$-w(\varphi)+\partial^{2}_{\varphi}[1/w(\varphi)]=0$. The difference is in the
boundary condition at large field: The $\tilde{\epsilon}=0$ is limited by
$\varphi_{*}$ while it is convenient to consider the boundary-layer one up to
$\varphi_{\rm m}$. Taking this into account, the solution can then be obtained
either as

$\varphi_{\rm m}-\varphi(w)=\tilde{\epsilon}\varphi_{\rm
m}|x|\approx\frac{\sqrt{2}}{2}\frac{1}{w\sqrt{\ln(\frac{1}{\tilde{\epsilon}\varphi_{\rm
m}w})}}$ (43)

or as

$\varphi_{*}-\varphi(w)\approx\frac{\sqrt{2}}{2}\frac{1}{w\sqrt{\ln(\frac{w_{*}}{w})}}.$
(44)

Matching between the two solutions is then enforced at leading order if

$\displaystyle\varphi_{*}\sim\varphi_{\rm m},$ (45) $\displaystyle
w_{*}\sim\frac{1}{\tilde{\epsilon}\varphi_{\rm m}}\sim w_{\rm m},$

which, since $\varphi_{*}$ diverges as $\sqrt{\ln w_{*}}$ (see Eq. (24) and
Appendix C), immediately leads to

$\displaystyle w_{\rm m}\sim
w_{*}\sim\frac{1}{\tilde{\epsilon}\sqrt{\ln(\frac{1}{\tilde{\epsilon}})}},$
(46) $\displaystyle\varphi_{\rm
m}\sim\varphi_{*}\sim\sqrt{\ln(\frac{1}{\tilde{\epsilon}})}+{\rm
O}(\ln\ln(\frac{1}{\tilde{\epsilon}})).$

So, as anticipated the location of the minimum of the potential $\varphi_{\rm
m}$ diverges when $\tilde{\epsilon}\to 0$ but the width of the layer
$\tilde{\epsilon}\varphi_{\rm m}$ goes to $0$. This is supported by the
numerical resolution of the LPA’ flow equation for values of $d$ approaching
as close as possible the lower critical dimension: see Fig. 5. This result is
different than the prediction of the previous FRG analysis of the approach to
the lower critical dimension within the truncated derivative expansion in Ref.
[ballhausen04, ]. The latter missed the emergence of the boundary layer near
the minimum of the potential, which led to the scaling $\varphi_{\rm m}\sim
1/\sqrt{\tilde{\epsilon}}$ that does not fit the data as shown in Fig. 5a.

a)b) Figure 5: (a) Location of the minimum of the potential $\varphi_{\rm m}$
divided by $\sqrt{\ln(1/\tilde{\epsilon})}$ as a function of
$\tilde{\epsilon}$. (b) Same but for the square mass $w_{\rm m}\equiv
u^{\prime\prime}(\varphi_{\rm m})$ multiplied by
$\delta(\tilde{\epsilon})=\tilde{\epsilon}\sqrt{\ln(1/\tilde{\epsilon})}$. As
predicted, both curves appear to essentially go to finite nonzero values as
$\tilde{\epsilon}\to 0$, in agreement with our singular perturbation analysis
(loglog corrections are not detectable with our data). On the other hand the
prediction from [ballhausen04, ] (dashed black line in (a)) clearly does not
fit the data. The data points in (a) and (b) are obtained from the numerical
resolution of the LPA’ fixed-point equations for the Exponential cutoff
functions with several different values of the prefactor $\alpha$.

Collecting all of the above results allows one to build a fixed-point solution
$w(\varphi)\equiv u^{\prime\prime}(\varphi)$ that is valid over the whole
range of field values when $\tilde{\epsilon}\to 0$. One can note the peculiar
form of the present singular perturbation problem in which neither the initial
condition in $\varphi=0$ nor the location of the layer in $\varphi_{\rm m}$
are determined a priori and must be determined through the matching procedure.

We now discuss the consequences for the LPA’ prediction of the lower critical
dimension $d_{\rm lc}$, the behavior of the critical temperature $T_{c}$, and
the critical exponents as $\tilde{\epsilon}\to 0$.

##  IV Results

###  IV.1 Determination of the lower critical dimension

To determine the value of the lower critical dimension $d_{\rm lc}$ we
consider the last of the LPA’ equations that we have not yet used, i.e., Eq.
(12) for the anomalous dimension of the field. This equation involves the
square mass $w(\varphi)$ in the boundary layer only. When $\tilde{\epsilon}\to
0$, $w(\varphi_{\rm m})\to\infty$, $\eta\to 2-d$, and the threshold function
$m_{4,0}^{(d)}(w;\eta)$ can be replaced in Eq. (12) by its asymptotic form,

$\displaystyle m_{4,0}^{(d)}(w;2-d)\sim\frac{\alpha
dA_{d}-\alpha^{2}B_{d}}{w^{4}}$ (47)

where $A_{d}$ is given in Eq. (27) and, as derived in Appendix B,

$\displaystyle
B_{d}=\frac{3d-2}{2}\int_{0}^{\infty}dyy^{\frac{d}{2}}[(y\frac{r(y)}{\alpha})^{\prime}]^{2}\,.$
(48)

As before a prime indicates a derivative with respect to the argument of the
function and the IR cutoff functions that we use are defined in Eq.
(LABEL:eq_cutoff_choices).

With the rescaling of the field and the notations introduced in Sec. III.3,
one can then rewrite Eq. (12) as

$2-d=2(d-\alpha\,\frac{B_{d}}{A_{d}})\frac{g^{\prime}(0)^{2}}{g(0)^{4}}=\frac{4}{\pi}(d-\alpha\,\frac{B_{d}}{A_{d}})$
(49)

where we have used that the solution within the layer around the minimum
satisfies Eqs. (32) and (36). The solution of the above equation gives
$d=d_{\rm lc}$.

For the Theta cutoff function, $B_{d}/A_{d}=(3d-2)/4$ (see Appendix B), so
that we obtain an explicit analytical expression for the lower critical
dimension:

$d_{\rm lc}(\alpha)=2\frac{\pi-\alpha}{\pi+4-3\alpha},$ (50)

which for instance predicts $d_{\rm lc}=1.03419\cdots$ for $\alpha=1$. Note
that the solution derived in the previous section required $d<2$. This entails
that $\alpha<2$, so that the pole in $\alpha=(\pi+4)/3$ is not attained. The
variation of $d_{\rm lc}$ with $\alpha$ is shown in Fig. 6.

Figure 6: Lower critical dimension $d_{\rm lc}(\alpha)$ predicted by the
solution of the LPA’ with the Theta (upper blue curve) and Exponential (lower
red curve) IR cutoff functions.

For the Exponential cutoff function, one finds that
$B_{d}/A_{d}=2^{-(1+d/2)}(3d-2)/4$ (see Appendix B), so that $d_{\rm lc}$ is
solution of the implicit equation

$d_{\rm lc}(\alpha)=2\frac{\pi-\alpha_{*}(d)}{\pi+4-3\alpha_{*}(d)}$ (51)

with $\alpha_{*}(d)=2^{-(1+d/2)}\alpha$. The outcome is plotted in Fig. 6.

We therefore obtain that for a reasonable range of the variational parameter
$\alpha$ (which cannot be fixed by any principle of minimum sensitivity on the
variation of $d_{\rm lc}(\alpha)$ because the latter does not display a
minimum at the present level of approximation) the predicted lower critical
dimension is indeed close to the exact result, $d_{\rm lc}=1$ (within $10\%$
for the Exponential cutoff function; the result found with the Theta regulator
appears less well-behaved than that of the Exponential cutoff).

###  IV.2 Critical temperature $T_{c}$ as $\tilde{\epsilon}\to 0$

One of the many defining properties of the lower critical dimension is that
the critical temperature $T_{c}$ goes to zero. This is a bare quantity which
is not easily retrieved from the RG flow. However when it goes to zero, a
simple reasoning based on the Boltzmann form of the distribution suggests that
the field scales as the square root of the inverse temperature. As the
dimension of the field at criticality goes to zero at the lower critical
dimension, one therefore expects that $T_{c}\sim 1/\varphi_{\rm m}^{2}$. This
is indeed what is found in the correspondance between the Wilsonian
dimensionless action of the O($N>2$) model and the nonlinear sigma model near
the lower critical dimension $d=2$.??? Together with Eq.
(LABEL:eq_matchingfinal), this scaling leads to

$T_{c}\sim\frac{1}{\ln(\frac{1}{\tilde{\epsilon}})}\to 0$ (52)

when $\tilde{\epsilon}\to 0$.

Recast in terms of the field dimension $D_{\phi}=(d-2+\eta)/2$ the above
expression is equivalent to

$T_{c}\propto\frac{1}{\ln(\frac{1}{D_{\phi}})}.$ (53)

This relation is similar to that obtained by Bruce and Wallace from a detailed
droplet theory.bruce81 ; bruce83  In the latter, the expansion is performed in
$\epsilon=d-d_{\rm lc}$ with $d_{\rm lc}=1$. The outcome is that $T_{c}$ has a
simple expansion in powers of $\epsilon$, $T_{c}\propto\epsilon+{\rm
O}(\epsilon^{2})$, but $D_{\phi}$ has instead a singular behavior, with
$D_{\phi}\sim e^{-2/\epsilon}/\epsilon$. Combining the two gives Eq. (53).
Note that this relation is not verified by the prediction of Ref.
[ballhausen04, ].

###  IV.3 Stability of the fixed point, essential scaling, and correlation-
length critical exponent as $\tilde{\epsilon}\to 0$

The stability of the fixed point can be studied by looking at perturbations
around it and the resulting eigenvalue equation equation obtained in linear
order of the perturbation. For the present LPA’ approximation, after
introducing small perturbations around the fixed point as
$w_{k}(\varphi)=w(\varphi)+k^{\lambda}\delta w(\varphi)$,
$\eta_{k}=\eta+k^{\lambda}\delta\eta$, $\varphi_{\rm mk}=\varphi_{\rm
m}+k^{\lambda}\delta\varphi_{\rm m}$, etc., with $\lambda$ an eigenvalue to be
determined, the linearized equation for $\delta w(\varphi)$ reads

$\displaystyle\lambda\delta w(\varphi)=-(2-\eta)\delta
w(\varphi)+(2-\eta)\tilde{\epsilon}\varphi\delta w^{\prime}(\varphi)+$ (54)
$\displaystyle
2v_{d}\partial^{2}_{\varphi}[\partial_{w}\ell_{0}^{(d)}(w(\varphi);\eta)\delta
w(\varphi)]+\big{(}w(\varphi)+\frac{1}{2}\varphi w^{\prime}(\varphi)$
$\displaystyle+2v_{d}\partial^{2}_{\varphi}[\partial_{\eta}\ell_{0}^{(d)}(w(\varphi);\eta)]\big{)}\delta\eta,$

and the expressions for $\delta\eta$ and $\delta\varphi_{\rm m}$ are given in
Appendix D. We are especially interested in finding the relevant eigenvalue
that gives the correlation length exponent $\nu$ which is known to diverge at
the lower critical dimension in an exact treatment.

As we did for the fixed point, one can attempt a singular perturbation
analysis when $\tilde{\epsilon}\to 0$ (and $\eta\to 2-d$) by looking
separately at the $\tilde{\epsilon}=0$ equation for $\varphi$ of O(1) and at
an equation in terms of the scaled variable $x=(\varphi-\varphi_{\rm
m})/(\tilde{\epsilon}\varphi_{\rm m})$ near the minimum $\varphi_{\rm m}$.
However, one immediately sees that if $\lambda={\rm O}(\tilde{\epsilon})$ or
more generally goes to zero when $\tilde{\epsilon}\to 0$, which is the
expected behavior of the relevant eigenvalue(s), working at the leading order
in $\tilde{\epsilon}$ does not allow the determination of $\lambda$ beyond the
fact that it starts as $0$.

This can be illustrated by considering one eigenvalue that can be exactly
obtained together with its eigenfunction. One easily finds that
$\lambda=-(2-\eta)\tilde{\epsilon}$ is a solution of Eq. (LABEL:eq_eigenvalue)
with

$\displaystyle\delta w(\varphi)=\delta K\,w^{\prime}(\varphi)$ (55)
$\displaystyle\delta\eta=0=\delta\tilde{\epsilon}$
$\displaystyle\delta\varphi_{{\rm m}}=-\delta K$

with $\delta K$ a constant that can be taken as infinitesimal to linearize the
RG flow equations. Note that despite the fact that it corresponds to a
relevant direction around the fixed point this eigenvalue is not the one we
are interested in because it is associated with an odd ($Z_{2}$ antisymmetric)
perturbation. We would instead like to determine the relevant eigenvalue
associated with an even ($Z_{2}$ symmetric) perturbation which gives the
correlation-length exponent $\nu$ through $1/\nu=-\lambda$. It is nonetheless
instructive to study how the exact result for
$\lambda=-(2-\eta)\tilde{\epsilon}$ translates into the leading order of the
singular perturbation analysis and we trivially find that only $\lambda=0$ can
be obtained by working at the leading order of the $\tilde{\epsilon}=0$ and of
the boundary-layer equations.

This example confirms that eigenvalues going to zero as $\tilde{\epsilon}\to
0$ cannot be determined from the singular perturbation analysis at the leading
order. One needs to go to the next order. In the present case this seems a
formidable task that we will not undertake. We instead perform a numerical
investigation by solving the LPA’ eigenvalue equation, Eq.
(LABEL:eq_eigenvalue), together with the fixed-point equation at fixed $d$,
trying to reach as low as possible values near the lower critical dimension.
As it should, we find that the critical fixed point has two relevant
eigendirections: one corresponds to an even eigenfunction and gives the
critical exponent $\nu$ and the other is equal to
$-(2-\eta)\tilde{\epsilon}=-D_{\phi}$ and is associated with an odd
eigenfunction related to the magnetic field (the scaling dimension of the
magnetic field is then $d-D_{\varphi}$). All the other eigenvalues are
positive, i.e., irrelevant, and of O(1) when $\tilde{\epsilon}\to 0$, as it
should for a critical fixed point. We show $1/\nu$ as obtained for the
Exponential IR cutoff function and several values of the parameter $\alpha$ in
Fig. 7. It is plotted both versus $\tilde{\epsilon}$ and versus the boundary-
layer width
$\delta(\tilde{\epsilon})\sim\tilde{\epsilon}\sqrt{\ln(1/\tilde{\epsilon})}$.
We observe that $1/\nu$ seems to be heading toward $0$ when
$\tilde{\epsilon}\to 0$, which is the expected behavior for an essential
scaling of the correlation length as one approaches the lower critical
dimension. Over the accessible range of dimensions, it appears to do so slower
than linearly in $\tilde{\epsilon}$, possibly like
$\tilde{\epsilon}\sqrt{\ln(1/\tilde{\epsilon})}$. On the other hand the
behavior is not compatible with the prediction of the droplet theory which
would be $1/\ln(1/\tilde{\epsilon})$.bruce81 ; bruce83 ; wallace84  However,
our numerical results may not yet be in the asymptotic regime near $d_{\rm
lc}$ and the conclusion should therefore be taken with a grain of salt.

a)b) Figure 7: The relevant eigenvalue $1/\nu$ obtained from the numerical
resolution of the LPA’ equations with the Exponential IR cutoff function and
several values of the parameter $\alpha$. It is plotted as a function of
$\tilde{\epsilon}$ (a) and
$\tilde{\epsilon}\sqrt{\ln(1/\tilde{\epsilon})}\sim\delta(\tilde{\epsilon})$
(b).

##  V Conclusion

We have presented a functional renormalization group (FRG) description of the
approach to the lower critical dimension $d_{\rm lc}$ in a scalar
$\varphi^{4}$ theory by using one of the simplest nonperturbative
approximation level obtained as a truncation of the derivative expansion, the
so-called LPA’. Our purpose is to test how a generic approximation scheme that
works across dimensions in a continuous way and has been shown to be accurate
in dimensions $d\geq 2$ for instancedupuis21  is able to describe dimensions
close to the lower critical dimension in a system with a discrete symmetry
where it is known that the long-distance physics is controlled by the
proliferation of localized excitations (in the present case, droplets that
become point-like kinks and anti-kinks at the lower critical dimension $d_{\rm
lc}=1$bruce81 ; bruce83 ; wallace84 ). We show that the limit of $d$ going to
$d_{\rm lc}$ for the fixed-point effective action is nonuniform in the
(average) field, with the emergence of a boundary (or more precisely,
interior) layer around the minimum of the dimensionless potential. The minimum
goes to infinity and the width of the layer goes to zero as $d\to d_{\rm lc}$,
at odds with the outcome of an earlier FRG study.ballhausen04  The behavior of
the critical temperature $T_{c}$ is compatible with the expected exact results
and, although the prediction of $d_{\rm lc}$ is dependent on the infrared
regulator used in the FRG, we find it rather close to the exact value $d_{\rm
lc}=1$ (within $10\%$ for a range of regulators).

The next step is to check that the scenario found at the LPA’ is valid at all
orders of the truncated derivative expansion. Work is now in progress to
investigate the next order, which includes a field-renormalization function in
addition to the potential. Preliminary results indicate that the same
mechanism of a nonuniform convergence to the lower critical dimension with the
emergence of an interior layer around the minimum of the dimensionless
potential is also at play. The next order of the derivative expansion also
seems to more properly describe the form of divergence of the correlation
length exponent $\nu$.

As already stressed, our goal is not to provide yet another theoretical
description of the approach to the lower critical dimension for systems in the
universality class of the Ising model, a question which has been quite well
understood for several decades. It is to benchmark a generic nonperturbative
but approximate FRG approach to later tackle problems that are still open such
as the value of the lower critical dimension of the athermally driven random-
field Ising model (RFIM).sethna  The lower critical dimension of the RFIM in
equilibrium has been rigorously shown to be $d_{\rm lc}=2$aizenman-wehr , but
that for the far-from-equilibrium driven RFIM is debated.shukla ; spasojevic ;
hayden-sethna  Finally one might also hope that the present solution near the
lower critical dimension can suggest new approximation schemes of the FRG that
are not necessarily based on the truncation of the derivative expansion.

###### Acknowledgements.

We thank Adam Rançon for numerous discussions related to this topic during the
years. IB wishes to thank Nicolas Wschebor and Maroje Marohnić for useful
discussions and feedbacks. LNF and IB acknowledge the support of the the
QuantiXLie Centre of Excellence, a project cofinanced by the Croatian
Government and European Union through the European Regional Development Fund -
the Competitiveness and Cohesion Operational Programme (Grant
KK.01.1.1.01.0004).

##  Appendix A FRG flow equations

The beta functional describing the FRG flow of the dimensionless effective
potential $u_{k}(\varphi)$ is given byberges02

$\displaystyle\beta_{u}(\varphi;\eta)=2v_{d}\ell_{0}^{(d)}(u^{\prime\prime}(\varphi);\eta,z(\varphi)),$
(56)

where $v_{d}^{-1}=2^{d+1}\pi^{d/2}\Gamma(d/2)$; $\ell_{n}^{(d)}$ is a
(strictly positive) dimensionless threshold function defined by

$\displaystyle\ell_{n}^{(d)}(w;\eta,z)=-\left(\frac{n+\delta_{n,0}}{2}\right)\int_{0}^{\infty}dyy^{\frac{d}{2}}\frac{\eta
r(y)+2yr^{\prime}(y)}{(y[z+r(y)]+w)^{n+1}}$ (57)

where the dimensionless infrared cutoff function (or IR regulator) $r(y)$ is
obtained from the dimensionful one, $R_{k}(q^{2})$, introduced in Eq. (2)
through

$R_{k}(q^{2})=Z_{k}k^{2}y\,r(y)\;\;\;{\rm with}\;\;y=\frac{q^{2}}{k^{2}}$ (58)

with $k$ is the running IR cutoff and $Z_{k}$ the dimensionful field
renormalization (such that the running anomalous dimension is defined by
$\eta_{k}=-k\partial_{k}Z_{k}$).

From the exact FRG equation for the 2-point 1-PI correlation function
evaluated for a uniform field configuration one can extract the beta
functional for the dimensionless field renormalization function
$z_{k}(\varphi)$,berges02

$\displaystyle\beta_{z}(\varphi;\eta)=-\frac{4v_{d}}{d}u^{\prime\prime\prime}(\varphi)^{2}m_{4,0}^{(d)}(u^{\prime\prime}(\varphi);\eta,z(\varphi))-$
(59)
$\displaystyle\frac{8v_{d}}{d}u^{\prime\prime\prime}(\varphi)z^{\prime}(\varphi)m_{4,0}^{(d+2)}(u^{\prime\prime}(\varphi);\eta,z(\varphi))$
$\displaystyle-\frac{4v_{d}}{d}z^{\prime}(\varphi)^{2}m_{4,0}^{(d+4)}(u^{\prime\prime}(\varphi);\eta,z(\varphi))-2v_{d}z^{\prime\prime}(\varphi)\times$
$\displaystyle\ell_{1}^{(d)}(u^{\prime\prime}(\varphi);\eta,z(\varphi))+4v_{d}u^{\prime\prime\prime}(\varphi)z^{\prime}(\varphi)\ell_{2}^{(d)}(u^{\prime\prime}(\varphi);\eta,z(\varphi))$
$\displaystyle+2v_{d}\frac{1+2d}{d}z^{\prime}(\varphi)^{2}\ell_{2}^{(d+2)}(u^{\prime\prime}(\varphi);\eta,z(\varphi)),$

where $m_{n,0}^{(d)}$ is another (strictly positive) threshold function
defined as

$\displaystyle
m_{n,0}^{(d)}(w;\eta,z)=\frac{1}{2}\int_{0}^{\infty}dyy^{\frac{d}{2}}\frac{z+(yr(y))^{\prime}}{(y[z+r(y)]+w)^{n}}\bigg{[}2\eta(yr(y))^{\prime}$
(60)
$\displaystyle+4(y^{2}r^{\prime}(y))^{\prime}-n\,\frac{y[z+(yr(y))^{\prime}][\eta
r(y)+2yr^{\prime}(y)]}{y[z+r(y)]+w}\bigg{]}.$

To derive Eq. (LABEL:eq_beta_fieldrenormalization) we have neglected the
higher-order terms in Eq. (6) which involve four spatial derivatives: It
therefore represents the second order of the derivative expansion which is
fully characterized by the two functions $U_{k}(\phi)$ and $Z_{k}(\phi)$.
Three more functions are required at the order O($\partial^{4}$), etc.

##  Appendix B Properties of the threshold functions

The threshold functions introduced in the main text and in the above Appendix
are a strictly positive dimensionless functions that enforce the decoupling of
the low-momentum and high-momentum modes.berges02  We only consider the LPA’
approximation so that the dimensionless field renormalization function
$z(\varphi)\equiv 1$, but this is easily generalizable.

Before discussing some of their generic properties it is illustrative to give
their explicit expression for a specific choice of IR cutoff function,
$r(y)=\Theta(1-y)(1-y)/y$, which is the Theta cutoff function with $\alpha=1$
(also called Litim or optimized regulatorlitim01 ):

$\displaystyle\ell_{n}^{(d)}(w;\eta)=\frac{2(d+2-\eta)}{d(d+2)}\frac{n}{(1+w)^{n+1}}$
(61) $\displaystyle m_{n,0}^{(d)}(w;\eta)=\frac{1}{(1+w)^{n}}.$

One can see that the threshold functions monotonically decrease as $w$
increases, blow up near the pole of the propagator $w_{P}$ (here $w_{P}=-1$)
and go to zero as power laws when $w\to+\infty$. They are defined for
$w>w_{P}$.

We analyze the behavior of the threshold functions in two limiting cases, when
the mass $w$ is large and when it approaches the pole of the propagator, i.e.,
when $w+\min_{y}{\\{y(1+r(y))\\}}\to 0$.

When the mass $w\to\infty$ and $z=1$ one easily finds from Eq. (57) that

$\displaystyle\ell_{n}^{(d)}(w;\eta)\sim\frac{A_{n}^{(d)}(\eta)}{w^{n+1}}+{\rm
O}(\frac{1}{w^{n+2}})$ (62)

where

$\displaystyle A_{n}^{(d)}(\eta)$
$\displaystyle=-\left(\frac{n+\delta_{n,0}}{2}\right)\int_{0}^{\infty}dyy^{\frac{d}{2}}[\eta
r(y)+2yr^{\prime}(y)]$ (63) $\displaystyle=(n+\delta_{n,0})A_{0}^{(d)}(\eta)$

and $A_{0}^{(d)}(\eta)$ can be rewritten as

$\displaystyle
A_{0}^{(d)}(\eta)=\frac{d+2-\eta}{2}\int_{0}^{\infty}dyy^{\frac{d}{2}}r(y).$
(64)

The choices of $r(y)$ that we use in this work are given in Eq.
(LABEL:eq_cutoff_choices) so that $A_{0}^{(d)}(\eta)$ is proportional to
$\alpha$. When $\eta=2-d$ this leads to the expression of $A_{d}$ in Eq. (27).

Similarly, from Eq. (LABEL:eq_threshold_m) one finds

$\displaystyle
m_{n,0}^{(d)}(w;\eta)\sim\frac{-B^{(d)}(\eta)+dA_{0}^{(d)}(\eta)}{w^{n}}+{\rm
O}(\frac{1}{w^{n+1}})\,,$ (65)

with

$\displaystyle
B^{(d)}(\eta)=-\int_{0}^{\infty}dyy^{\frac{d}{2}}(yr(y))^{\prime}[\eta(yr(y))^{\prime}+2(y^{2}r^{\prime}(y))^{\prime}]$
(66)
$\displaystyle=\frac{d+2-2\eta}{2}\int_{0}^{\infty}dyy^{\frac{d}{2}}[(yr(y))^{\prime}]^{2}.$

When $\eta=2-d$ one immediately obtains Eq. (48).

All of the above results of course match with the expansion of the expressions
in Eq. (LABEL:eq_litim).

We now turn to the expression of the threshold functions near the pole of the
propagator, when $w\to w_{P}=-\min_{y}{\\{y(1+r(y))\\}}$. Note that the FRG
equations are well behaved for $w+\min_{y}{\\{y(1+r(y))\\}}>0$. The approach
to the pole is what controls the return to convexity of the effective
potential in the ordered phasetetradis92 ; pelaez-wschebor ; berges02  and is
therefore important in the vicinity of the lower critical dimension where the
critical fixed and the fixed point describing the ordered phase merge.

For the Theta cutoff function and for $z=1$,

$\displaystyle\ell_{n}^{(d)}(w;\eta)=\alpha\frac{n+\delta_{n,0}}{2}\int_{0}^{1}dyy^{\frac{d}{2}-1}\frac{[(2-\eta)+\eta
y]}{[w+\alpha+(1-\alpha)y]^{n+1}}$ (67) $\displaystyle m_{n,0}^{(d)}(w;\eta)=$
(68)
$\displaystyle\alpha\frac{2-\alpha}{[w+1]^{n}}-\alpha\eta(1-\alpha)\int_{0}^{1}dyy^{\frac{d}{2}}\frac{1}{[w+\alpha+(1-\alpha)y]^{n}}+$
$\displaystyle\alpha\frac{n}{2}(1-\alpha)^{2}\int_{0}^{1}dyy^{\frac{d}{2}}\frac{[(2-\eta)+\eta
y]}{[w+\alpha+(1-\alpha)y]^{n+1}}\,.$

The pole of the propagator $1/[w+\alpha+(1-\alpha)y]$ is $w_{P}=-\alpha$ if
$\alpha<1$, and is reached in $y=0$, and is $w_{P}=-1$ if $\alpha>1$, and is
reached in $y=1$. The case $\alpha=1$ corresponds to the expressions in Eq.
(LABEL:eq_litim) and the approach to the pole in $w_{P}=-1$ can be read off
directly. The threshold functions generically diverge as inverse power laws of
$(w+w_{P})$ when $w$ approaches the pole $w_{P}$ which is either $-\alpha$ or
$-1$. An exception is $\ell_{0}^{(d)}(w;\eta)$ which behaves as $-\ln(w+1)$
when $\alpha>1$.

For the Exponential cutoff function (and for $z=1$), a similar behavior is
encountered except that the pole is attained either in $y=0$ and is equal to
$w_{P}=-\alpha$ when $\alpha<1$ or in $y=\ln\alpha$ and is equal to
$w=-(1+\ln\alpha)$ when $\alpha>1$. The divergence of the threshold function
as $w+w_{P}\to 0^{+}$ is generically power-law-like.

##  Appendix C Further analysis of the $\tilde{\epsilon}=0$ solution

To prove Eq. (24) we start from Eq. (22) where we recall that $F(\Phi)\equiv
w(\Phi)$ and $\Phi(w)=\ell_{0}^{(d)}(w;2-d)$. From the analysis of the
threshold functions in the preceding section, one can infer that $w(\Phi)$ is
a monotonically decreasing function that starts from $+\infty$ when $\Phi=0$
and asymptotically goes to the pole $w_{P}<0$ when $\Phi\to+\infty$. In the
regime of interest where $w_{0}=w(0)\to w_{P}^{+}$ and
$w_{*}=w(\varphi_{*})\to+\infty$, the relevant range of $\Phi$ is from
$\Phi_{*}\sim\alpha A_{d}/w_{*}\to 0$ (see Eq. (26) to
$\Phi_{0}\sim\alpha(A_{d}/|w_{P}|)\ln w_{*}\to+\infty$ (see Eq. (42).

Eq. (22) can be rewritten as

$\frac{\partial\Phi}{\partial\varphi}=-\sqrt{\frac{d}{v_{d}}}\sqrt{-\int_{\Phi}^{\Phi_{0}}d\Phi^{\prime}w(\Phi^{\prime})},$
(69)

with $\Phi(\varphi)$ is a monotonically decreasing function between $0$ and
$\varphi_{*}$. (Note that by definition
$\int_{\Phi_{*}}^{\Phi_{0}}d\Phi\,w(\Phi)=0$.) This leads to

$\varphi(\Phi)=\sqrt{\frac{v_{d}}{d}}\int_{\Phi}^{\Phi_{0}}\frac{d\Phi^{\prime}}{\sqrt{-\int_{\Phi^{\prime}}^{\Phi_{0}}d\Phi^{\prime\prime}w(\Phi^{\prime\prime})}}\,,$
(70)

We define $\Phi_{\rm i}$ and the associated field $\varphi_{\rm i}$ such that
$w(\Phi_{\rm i})=w(\varphi_{\rm i})=0$. Then,

$\varphi_{\rm i}=\sqrt{\frac{v_{d}}{d}}\int_{\Phi_{\rm
i}}^{\Phi_{0}}\frac{d\Phi}{\sqrt{-\int_{\Phi}^{\Phi_{0}}d\Phi^{\prime}w(\Phi^{\prime})}}\,,$
(71)

where $\Phi_{\rm i}=\ell_{0}^{(d)}(w=0)$ is of O($1$) and $w(\Phi)$ is
monotonically decreasing and negative in the interval between $\Phi_{\rm i}$
and $\Phi_{0}$. From the properties of the threshold functions it is easily
check that that $w(\Phi)$ is concave: indeed, its second derivative is
$w^{\prime\prime}(\Phi)=-\partial_{w}^{2}\ell_{0}^{(d)}(w)/[\partial_{w}\ell_{0}^{(d)}(w)]^{3}$
with $\partial_{w}\ell_{0}^{(d)}(w)=-\ell_{1}^{(d)}(w)<0$ and
$\partial_{w}^{2}\ell_{0}^{(d)}(w)=\ell_{2}^{(d)}(w)>0$. In consequence, when
$\Phi_{\rm i}\leq\Phi\leq\Phi_{0}$,

$|w_{0}|\geq-w(\Phi)\geq|w_{0}|\frac{\Phi-\Phi_{\rm i}}{\Phi_{0}-\Phi_{\rm
i}}.$ (72)

When inserted in Eq. (71), after some elementary algebra and using the
asymptotic behavior of $\Phi_{0}$ when $w_{*}\to\infty$, this implies that

$\frac{\sqrt{2}}{2}\pi\sqrt{\frac{\alpha v_{d}A_{d}}{d\,w_{P}^{2}}}\sqrt{\ln
w_{*}}\geq\varphi_{\rm i}\geq 2\sqrt{\frac{\alpha
v_{d}A_{d}}{d\,w_{P}^{2}}}\sqrt{\ln w_{*}}\,,$ (73)

which proves that $\varphi_{\rm i}\sim\sqrt{\ln w_{*}}$.

To complete the demonstration for all $\varphi$’s between $\varphi_{\rm i}$
and $\varphi_{*}$ we can rewrite Eqs. (70) and (71) as

$\displaystyle\varphi_{*}-\varphi_{\rm
i}=\sqrt{\frac{v_{d}}{d}}\int_{\Phi_{*}}^{\Phi_{\rm
i}}\frac{d\Phi}{\sqrt{\int_{\Phi_{*}}^{\Phi}d\Phi^{\prime}w(\Phi^{\prime})}}\,.$
(74)

By using the properties of the function $w(\Phi)$ we find that
$\int_{\Phi_{*}}^{\Phi}d\Phi^{\prime}w(\Phi^{\prime})$ is a monotonically
increasing and concave function of $\Phi$ for $\Phi\leq\Phi_{\rm i}$, which
implies that

$\displaystyle\int_{\Phi_{*}}^{\Phi}d\Phi^{\prime}w(\Phi^{\prime})\geq\int_{\Phi_{*}}^{\Phi_{i}}d\Phi
w(\Phi)\left(\frac{\Phi-\Phi_{*}}{\Phi_{i}-\Phi_{*}}\right).$ (75)

Then,

$\displaystyle\varphi_{*}-\varphi_{\rm i}$
$\displaystyle\leq\sqrt{\frac{v_{d}}{d}}\sqrt{\frac{\Phi_{i}-\Phi_{*}}{\int_{\Phi_{*}}^{\Phi_{i}}d\Phi
w(\Phi)}}\int_{\Phi_{*}}^{\Phi_{\rm i}}\frac{d\Phi}{\sqrt{\Phi-\Phi_{*}}}$
(76) $\displaystyle\leq
2\sqrt{\frac{v_{d}}{d}}\frac{\Phi_{i}-\Phi_{*}}{\sqrt{\int_{\Phi_{*}}^{\Phi_{i}}d\Phi
w(\Phi)}},$

where we recall that $\Phi_{\rm i}=\ell_{0}^{(d)}(w=0)={\rm O}(1)$ and
$\Phi_{*}\sim\alpha A_{d}/w_{*}\to 0$. After rewriting
$\int_{\Phi_{*}}^{\Phi_{i}}d\Phi
w(\Phi)=-\int_{0}^{w_{*}}dww\partial_{w}\ell_{0}^{(d)}(w)$, integrating by
part and using the properties of the threshold function $\ell_{0}^{(d)}$, one
obtains that the integral behaves as $\alpha A_{d}\ln w_{*}$ when $w_{*}\gg
1$. This finally leads to

$\displaystyle\varphi_{*}-\varphi_{\rm i}\lesssim 2\sqrt{\frac{v_{d}}{\alpha
dA_{d}}}\frac{\ell_{0}^{(d)}(w=0)}{\sqrt{\ln w_{*}}}\,,$ (77)

so that $\varphi_{*}\sim\varphi_{\rm i}\sim\sqrt{\ln w_{*}}$, as announced.

##  Appendix D Eigenvalue equations

the linearized equation for the perturbation of the square-mass function
$k^{\lambda}\delta w(\varphi)$ in Eq. (LABEL:eq_eigenvalue) should be
complemented by linearized equations for the perturbation of the anomalous
dimension $\delta\eta$ and of the minimum of the potential $\delta\varphi_{\rm
m}$. That for $\delta\eta$ follows directly from Eq. (12) which is valid for
all scales $k$ in the LPA’. It reads

$\displaystyle\delta\eta=$
$\displaystyle\frac{4v_{d}}{d}\Big{(}2w^{\prime}(\varphi_{\rm
m})m_{4,0}^{(d)}(w(\varphi_{\rm m});\eta)[\delta w^{\prime}(\varphi_{\rm
m})+w^{\prime\prime}(\varphi_{\rm m})\delta\varphi_{\rm m}]$ (78)
$\displaystyle+w^{\prime}(\varphi_{\rm
m})^{2}\partial_{w}m_{4,0}^{(d)}(w(\varphi_{\rm m});\eta)[\delta
w(\varphi_{\rm m})+w^{\prime}(\varphi_{\rm m})\delta\varphi_{\rm m}]$
$\displaystyle+w^{\prime}(\varphi_{\rm
m})^{2}\partial_{\eta}m_{4,0}^{(d)}(w(\varphi_{\rm m});\eta)\delta\eta\Big{)}$

and does not involve $\lambda$ explicitly.

We next need the flow equation for the $k$-dependent minimum which is obtained
from that of $u^{\prime}_{k}(\varphi)$ as

$\displaystyle\partial_{t}\varphi_{{\rm m}k}=$
$\displaystyle-\frac{(d-2+\eta_{k})}{2}\varphi_{{\rm
m},k}-2v_{d}\,\frac{w_{k}^{\prime}(\varphi_{{\rm m}k})}{w_{k}(\varphi_{{\rm
m}k})}\times$ (79)
$\displaystyle\partial_{w}\ell_{0}^{(d)}(w;\eta_{k})|_{w=w_{k}(\varphi_{{\rm
m}k})}.$

Linearizing then leads to

$\displaystyle\lambda\delta\varphi_{{\rm
m}}=-\Big{[}\frac{(d-2+\eta)}{2}+2v_{d}\ell_{1}^{(d)}(w(\varphi_{{\rm
m}});\eta)\Big{(}\frac{w^{\prime}(\varphi_{{\rm m}})^{2}}{w(\varphi_{{\rm
m}})^{2}}-$ (80) $\displaystyle\frac{w^{\prime\prime}(\varphi_{{\rm
m}})}{w(\varphi_{{\rm m}})}\Big{)}-4v_{d}\ell_{2}^{(d)}(w(\varphi_{{\rm
m}});\eta)\frac{w^{\prime}(\varphi_{{\rm m}})^{2}}{w(\varphi_{{\rm
m}})}\Big{]}\delta\varphi_{{\rm m}}+$ $\displaystyle
2v_{d}\Big{[}\ell_{1}^{(d)}(w(\varphi_{{\rm m}});\eta)\Big{(}\frac{\delta
w^{\prime}(\varphi_{{\rm m}})}{w(\varphi_{{\rm
m}})}-\frac{w^{\prime}(\varphi_{{\rm m}})\delta w(\varphi_{{\rm
m}})}{w(\varphi_{{\rm m}})^{2}}\Big{)}+$ $\displaystyle
2\ell_{2}^{(d)}(w(\varphi_{{\rm m}});\eta)\frac{w^{\prime}(\varphi_{{\rm
m}})\delta w(\varphi_{{\rm m}})}{w(\varphi_{{\rm
m}})}\Big{]}-\delta\eta\Big{[}\frac{\varphi_{{\rm m}}}{2}+$ $\displaystyle
2v_{d}\partial_{\eta}\ell_{1}^{(d)}(w(\varphi_{{\rm
m}});\eta)\frac{w^{\prime}(\varphi_{{\rm m}})}{w(\varphi_{{\rm m}})}\Big{]}$

where we have used the property of the threshold functions that
$\partial_{w}\ell_{n}^{(d)}(w;\eta)=-(n+1)\ell_{n+1}^{(d)}(w;\eta)$.

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