# Effects of a rotating periodic lattice on coherent quantum states in a ring
topology: The case of positive nonlinearity

Hongyi Huang 1 and Kunal K. Das1,2 1Department of Physics and Astronomy, Stony
Brook University, New York 11794-3800, USA  2Department of Physical Sciences,
Kutztown University of Pennsylvania, Kutztown, Pennsylvania 19530, USA

###### Abstract

We study the landscape of solutions of the coherent quantum states in a ring
shaped lattice potential in the context of ultracold atoms with an effective
positive nonlinearity induced by interatomic interactions. The exact
analytical solutions in the absence of lattice are used as a starting point
and the transformation of those solutions is mapped as the lattice is
introduced and strengthened. This approach allows a simple classification of
all the solutions into states with periods commensurate/incommensruate with
the lattice period and those with/without nodes. Their origins are traced to
the primary dispersion curve and the swallowtail branches of the lattice-free
spectrum. The commensurate states tend to remain delocalized with increasing
lattice depth, whereas the incommensurate ones may be localized. The symmetry
and stability properties of the solutions are examined and correlated with
branch energies. The crucial importance of rotation is highlighted by its
utility in continuously transforming solutions and accessing in a finite ring
with a few sites the full spectrum of nonlinear Bloch waves on an infinite
lattice.

##  I Introduction

A defining feature of Bose Einstein condensates (BEC) has been the effective
nonlinear behavior induced by interatomic interactions, and a substantial
literature exists on the subject [1]. The interplay of such interactions with
a periodic lattice potential [2] has led to realizing significant phenomena
such as the Mott insulator to superfluid transition [3]. The ubiquity of
harmonic trapping in experiments [4] dictated that the bulk of the studies of
ultracold atoms in periodic lattices have been in the context of open systems
with trivial topology. In recent years, ultracold atoms have been trapped in
ring configurations proving convenient for the study of superfluid features,
like persistent flow and quantization of angular momentum [5, 6]. However,
such efforts have not extended to include an azimuthal periodic lattice,
although the capability exists [7, 8].

The non-trivial topology of a ring combined with the periodic lattice
structure has already been shown to present a wealth of physical phenomena,
whether examined with continuum [9, 10, 11, 12, 13, 14, 15, 16, 17] or
discrete [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] lattice models. Adding
nonlinearity via atomic interactions substantially increases the complexity of
the system and hence the range of possible behavior. The system offers the
possibility of studying nonlinear dynamics in a lattice system which is
closed, finite and is naturally without boundaries. A necessary prelude to
such a study is mapping out the space of allowed solutions and the effect of
possible rotation. That is the specific goal of this study.

The nonlinear states of a BEC in a lattice have been examined in several
studies, experimentally [29, 30] and theoretically [31, 32, 33, 34, 35, 36,
37, 38, 39]. However, the primary focus of these studies have been systems
with trivial topology and in the limit of large lattices. It is only recently
that there has been a thorough study of nonlinear states focussed entirely on
a continuum ring lattice configuration with a few periods [40]. The approach
taken there was to assume the presence of a lattice of fixed strength and
examine the effect of introducing the nonlinearity. We take a different but
complementary approach, where we examine the effects of introducing the
lattice into the nonlinear system. There are multiple motivations for this.
Prime among them is that exact analytical solutions exist for such nonlinear
systems, in the absence of a lattice [41, 42]. They provide concrete insights
when the lattice is introduced with increasing strength. Secondly, in
experiments adiabatic introduction of the lattice leads to some interesting
dynamics in the linear regime as one of us recently showed [11], and this
current work provides the essential basis for a study of similar dynamics in
the nonlinear regime. Finally, and significantly, this approach leads to a
simpler and more transparent description of the spectrum and states, and their
classifications and interconnections. This will be particularly relevant when
examining the complex dynamics of such a system.

Our approach leads to several useful insights about coherent modes in ring
lattices. The relation of the lattice period with that of the eigenmodes in
the absence of the lattice determines the degree of localization of the modes
when the lattice is present, and the origins are tied to the nature of the
nonlinear spectrum in the absence of lattices. Gap solitons are found to be
not a very distinctive feature for finite ring lattices. We identify certain
symmetries of the lattice-free solutions, which persist even when the lattice
is introduced and strengthened. Effects of the ring topology is particularly
prominent for small lattice sizes, in the quantized modes. Increasing the
number of lattice sites leads to emergence of proportionately more soliton
branches. Rotation is an essential tool with ring lattices, allowing access to
a continuous range of solutions, for a set lattice period and size.

In Sec. II we set up our physical model, then identify the key effects of the
ring boundary conditions and possible rotation in Sec. III. We set the
template for examining the states with an azimuthal lattice in Sec. IV with a
brief derivation and classification of nonlinear analytical solutions in a
ring without a lattice. Section V provides a comprehensive description of how
the spectrum of nonlinear solutions transforms as the lattice is introduced
and strengthened. The persistent symmetries of the eigenstates are identified
in Sec. VI and used to classify and characterize them, and effects of the
lattice on the eigenstates are explained in Sec. VII, specifically identifying
why certain solutions remain delocalized and others tend to localize. Section
VIII compares our approach and results to those obtained by introducing
nonlinearity with an existing lattice of fixed strength. We analyze the
stability of the solutions by considering small fluctuations about the mean
field solutions in Sec. IX, examine the behavior in the limit of rings with
large number of lattice sites in Sec. X, and conclude in Sec. XI with a brief
summary and outlook for continuing work.

##  II Physical Model

We consider a BEC in a toroidal trap similarly our prior work [11, 10]. We
take the minor radius to be much smaller than the major radius $R$ so that the
system can be treated as a cylinder ${\bf r}=(z,r,\varphi)$ with periodic
boundary condition on $z$. We assume the confinement along $(r,\varphi)$,
transverse to the ring circumference to be sufficiently strong to keep the
atoms in the ground state $\psi(r,\varphi)$ for those degrees of freedom, so
that the three-dimensional bosonic field operator can be written in the
effective form $\hat{\Psi}(z)\psi(r,\varphi)$. Integrating out the transverse
degrees of freedom, the dynamics can be described by an effective one
dimensional (1D) Hamiltonian

$\displaystyle\hat{H}(t)$ $\displaystyle=$ $\displaystyle\int_{0}^{2\pi R}{\rm
d}z\hat{\Psi}^{\dagger}(z,t)\times\left[-\frac{\hbar}{2m}\partial_{z}^{2}+V(z,t)\right.$
$\displaystyle\left.+\frac{g_{3D}N}{4\pi
l^{2}}\hat{\Psi}^{\dagger}(z,t)\hat{\Psi}(z,t)\right]\hat{\Psi}(z,t).$

where $g_{3D}=4\pi\hbar^{2}a/m$ is the interaction strength defined by $a$ the
$s$-wave scattering length , $m$ is the mass of individual atoms, $N$ is the
total number of atoms, and $l=\sqrt{\hbar/m\omega_{T}}$ is the harmonic
oscillator length for the transverse confinement along the cross-section of
the torus. We assume a sinusoidal lattice potential
$V(z,t)=V_{0}\sin^{2}\left(\frac{q}{2}(z/R-\Omega t)\right)$ rotating with
angular velocity $\Omega$ with respect to the laboratory frame.

We take the major radius $R$ as the length unit so that the linear distance
along the ring coincides with the angular distance $z/R=\theta\in[0,2\pi)$; we
take the lowest energy scale in the ring $E_{R}=\frac{\hbar^{2}}{mR^{2}}$ as
the energy unit; we use the associated frequency
$\omega_{R}=\frac{\hbar}{mR^{2}}$ as unit for frequency as well as the angular
velocity, use it to set our time scale $\tau=\omega_{R}^{-1}$. The effective
nonlinear constant in 1D then has the form $g=2a\omega_{T}N$, note we include
the particle number in the constant. Using these units leads to the equation
of motion which, in the mean field limit $\langle\hat{\Psi}\rangle=\psi$, is a
nonlinear Schrödinger equation:

$\left[{\textstyle\frac{1}{2}}(i\partial_{\theta}+\Omega)^{2}+V_{0}\sin^{2}({\textstyle\frac{1}{2}}q\theta)+g|\psi|^{2}\right]\psi=i\partial_{t}\psi$
(2)

The explicit presence of $N$ implies the normalization
$\int_{0}^{2\pi}d\theta\left|\psi(\theta,t)\right|^{2}=1$. The presence of the
angular velocity $\Omega$ in above equation represents transformation to the
frame rotating with the lattice, which remove explicit dependence on time in
the lattice potential. A centrifugal term $\propto\Omega^{2}$ is left out as
being a constant offset for fixed radius $R$. The stationary solutions
$\varphi(\theta)=\psi(\theta,t)e^{i\mu t}$ satisfy the time-independent
version of Eq. (2) with $i\partial_{t}\rightarrow\mu$ where the eigenvalues
$\mu$ of the equation defines the chemical potential. That equation can be
considered the stationary solution of the energy functional

$\displaystyle E\left[\varphi\right]=\int_{0}^{2\pi}d\theta\times$ (3)
$\displaystyle\left[{\textstyle\frac{1}{2}}\varphi^{*}(i\partial_{\theta}+\Omega)^{2}\varphi+V_{0}\sin^{2}({\textstyle\frac{1}{2}}q\theta)|\varphi|^{2}+{\textstyle\frac{1}{2}}g\left|\varphi\right|^{4}\right]$

with the chemical potential as the Lagrange multiplier for the normalization
$\int_{0}^{2\pi}d\theta|\psi|^{2}=1$ condition, and related to the mean field
energy as
$\mu=E\left[\varphi\right]+{\textstyle\frac{1}{2}}g\int_{0}^{2\pi}d\theta|\varphi|^{4}$.

Figure 1: (Color online) The atoms are trapped in a toroidal trap with an
azimuthal lattice potential of period $2\pi/q$, its variation of depth shown
schematically as a thick sinusoidal line. The torus is taken as a wrapped
cylinder with our choice of co-ordinates ${\bf r}=(z,r,\varphi)$ shown,
assuming the major radius to be much larger than the minor radius, $R\gg r$.

##  III Boundary Conditions and Rotation

The behavior of the physical observables can be understood best by writing the
mean field stationary state in the polar amplitude-angle form, referred to as
the hydrodynamic picture,
$\varphi(\theta)=\sqrt{\rho(\theta)}e^{i\phi(\theta)}$, leading to an equation
for the density $\rho$

${\textstyle\frac{1}{8}}(\partial_{\theta}\rho)^{2}-{\textstyle\frac{1}{4}}\rho\partial_{\theta}^{2}\rho+{\textstyle\frac{1}{2}}\alpha^{2}+V_{0}\sin^{2}({\textstyle\frac{1}{2}}q\theta)\rho^{2}+g\rho^{3}-\mu\rho^{2}=0$
(4)

and a phase equation that provides an integral of motion,

$\displaystyle\textstyle\rho\partial_{\theta}\phi-\Omega\rho$ $\displaystyle=$
$\displaystyle\alpha$ (5)
$\displaystyle\Delta\phi(\theta)=\phi(\theta)-\phi(0)$ $\displaystyle=$
$\displaystyle\Omega\theta+\int_{0}^{\theta}\frac{\alpha}{\rho(\theta^{\prime})}d\theta^{\prime}.$

This sets the current density $J=N\alpha$, the superfluid velocity
$v=\alpha/\rho(\theta)$ and angular momentum per particle
$L=\hbar\Omega+2\pi\hbar\alpha$.

The single-valuedness of the quantum wave function and the closed topology of
the ring impose the following boundary conditions

$\displaystyle\rho(0)=\rho(2\pi)\hskip
14.22636pt\rho^{\prime}(0)=\rho^{\prime}(2\pi)\hskip
14.22636pt\Delta\phi(2\pi)=2\pi n,$ (6)

with the integer $n$ being the winding number. The total phase change contains
the effect of rotation. We will use the _bare_ phase change around the ring
neglecting rotation

$\displaystyle\delta\phi\equiv\delta\phi(2\pi)=\int_{0}^{2\pi}\frac{\alpha}{\rho(\theta^{\prime})}d\theta^{\prime}=\Delta\phi(2\pi)-2\pi\Omega.$
(7)

Figure 2: (Color online)(a) Shape of the cubic function that sets the density
variation, displaying its roots; the dotted parts are forbidden, (b) Loop in
phase space that satisfies the boundary conditions on a ring, and corresponds
to the solid sections in panel (a). (c) The mean field chemical potential
$\mu$ in the absence of a lattice, $V_{0}=0$, for two interaction strengths
$g=3$ and $g=10$; the soliton branches indexed by $j$ lengthen as nonlinearity
increases. Quantization of the total phase around the ring only allows
solutions marked by intersections with vertical dashed lines, which can be
continuously shifted by rotation, by any value of the angular momentum
$\Omega\neq 0$ indicated schematically by the solid lines. (d) The case of
$g=10$ in periodic zone scheme as a function of rotation. Lowest panels
display the eigenstates for $\mu$ marked by circles in (c): Their densities
are shown in (e,g) with plane wave (flat green line) at the right edge of the
branch, solutions with nodes (red line) at the left edge, and one example of
solutions without nodes (dotted blue) that lie in between. Corresponding bare
phase variations $\delta\phi$ are shown in (f,h).

We can understand the system behavior by tracking the chemical potential $\mu$
as function of the bare phase change $\delta\phi$. Insisting on the density
boundary conditions, as we vary $\alpha$, yields a continuum of solutions such
as illustrated in Fig. 2(c) for the case when the lattice is absent. However,
the phase boundary condition for a finite size ring picks out only discrete
points on that spectrum as physically relevant, marked on the plot by dashed
vertical lines. Rotation shifts those vertical lines by adding a phase ramp,
allowing access to the complete spectrum. Owing to the complementary relation
$2\pi\Omega=-\delta\phi+2\pi n$, the angular velocity $\Omega$ can replace the
bare phase, in the representation of the spectrum as shown in a periodic zone
scheme in Fig. 2(d). This can compared to the limit of no potentials presented
in a recent work [43].

##  IV Exact solutions without lattice

In the absence of a lattice $V_{0}=0$, solutions of Eq. (2) can be found
analytically [41, 42] which will provide the framework for our description
once the lattice is introduced. A first integration of Eq. (4) yields

$\displaystyle\partial_{\theta}\rho=\pm\sqrt{f\left(\rho\right)};\hskip
8.53581ptf\left(\rho\right)=4g\rho^{3}-8\mu\rho^{2}+8\beta\rho-4\alpha^{2}$
(8)

with integral of motion $\beta$. The nature of the solutions can be related to
the root structure of the function $f(\rho)$ which has a general shape shown
in Fig. 2(a) for repulsive nonlinearity, $g>0$. It is clearly required that
$f(\rho)\geq 0$, corresponding to the colored solid segments in the plot, and
the roots of $f(\rho)=0$, labelled $r_{1}\leq r_{2}\leq r_{3}$, need to be
real to be physically relevant. Viewing $\rho^{\prime}$ and $\rho$ as
parametric functions of $\theta$ in a phase space representation, their joint
periodic boundary conditions in Eq. (6) require that $\pm\sqrt{f}$ plotted as
a function of $\rho$ forms one or more complete circuit around the loop drawn
in Fig. 2(b) as $\theta$ is varied from $[0,2\pi)$. With the cubic form for
$f$ this is only possible if we combine $+\sqrt{f}$ and $-\sqrt{f}$ to form
the two halves of the loop meeting at the roots $r_{1}$ and $r_{2}$.

The roots define the state of the system. To start with the symmetry of the
loop assures that the state will always have at least two points of reflection
symmetry along the ring, which will be relevant when the lattice is turned on
as we will discuss in Sec. VI. The density varies between the roots $r_{1}$
and $r_{2}$ suggesting parametrization as $\rho=r_{1}+(r_{2}-r_{1})t^{2}$.
Choosing $\rho(\theta=0)=r_{1}$, Eq. (8) can be integrated as

$\displaystyle{\textstyle\theta=\frac{1}{\sqrt{g(r_{3}-r_{1})}}}\int_{0}{{}^{\sqrt{\frac{\rho(\theta)-r_{1}}{r_{2}-r_{1}}}}\frac{dt}{\sqrt{(1-t^{2})(1-mt^{2})}}}$
(9)

with the relevant parameters also in terms of the roots

$\displaystyle\textstyle m$ $\displaystyle=$
$\displaystyle\frac{r_{2}-r_{1}}{r_{3}-r_{1}},\hskip
14.22636pt\mu=\frac{g}{2}(r_{1}+r_{2}+r_{3})$ $\displaystyle\alpha^{2}$
$\displaystyle=$ $\displaystyle gr_{1}r_{2}r_{3},\hskip
14.22636pt\beta=\frac{g}{2}(r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}).$ (10)

Inversion of Eq. (9) yields a Jacobi elliptic function [44], and using it in
Eq. (7) determines the phase in terms of an incomplete elliptic integral of
the third kind [45],

$\displaystyle\rho(\theta)$ $\displaystyle=$ $\displaystyle
r_{1}+(r_{2}-r_{1}){\rm sn}^{2}(\textstyle{\sqrt{g(r_{3}-r_{1})}}\ \theta,m)$
$\displaystyle\delta\phi(\theta)$ $\displaystyle=$
$\displaystyle\frac{\alpha}{r_{1}\sqrt{g(r_{3}-r_{1})}}\Pi(1-\frac{r_{2}}{r_{1}},\varphi,m)$
(11)

where $\varphi=\sin^{-1}[{\rm sn}(\sqrt{g(r_{3}-r_{1})}\ \theta,m)]$. Taking
the smallest value of $\theta\geq 0$ such $\rho(\theta)=r_{2}$, Eq. (9) yields
$\theta=K(m)/\sqrt{g(r_{3}-r_{1})}$ which therefore sets the period of any
density modulation. This however corresponds to only half of the loop in phase
space. To satisfy the density boundary condition, the _complete_ phase space
loop in Fig. 2(b) has to be traversed by an integer number of turns we denote
by $j$, leading to the condition for a complete circuit of the ring,

$jK(m)=\pi\sqrt{g(r_{3}-r_{1})}.$ (12)

The bare phase change around the ring is given by complete elliptic integrals
of the first and the third kinds, $K(m)$ and $\Pi(m)$

$\displaystyle\delta\phi=\delta\phi(2\pi)$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha}{K(m)r_{1}}\Pi(1-\frac{r_{2}}{r_{1}},m).$ (13)

Figure 3: (Color online) The chemical potential $\mu$ plotted as a function of
the lattice depth $V_{0}$ for the number of lattice periods $q=1,2,$ and $3$
in panels (a,b,c). The nonlinear strength is fixed at $g=10$. The specific
bare phase values accumulated around the ring $\delta\phi$ are indicated in
each panel and correspond to the vertical dotted lines in Fig. 4. The vertical
dashed lines that appear here in turn indicate the values of $V_{0}$ that
correspond to the plots of $\mu$ versus $\delta\phi$ that appear in Fig. 4.
The labels for the different branches follow the convention defined in Eq.
(20) also in the context of the next figure. The dashed lines correspond to
the intra-band soliton branches in the next figure, with blue and red marking
upper and lower ones respectively of the split branches.  Figure 4: (Color
online) The chemical potential $\mu$ plotted as a function of the phase
$\delta\phi$ for three different lattice periods $q=1,2,$ and $3$, one in each
row. The nonlinear strength is fixed at $g=10$. The three panels in each row
are for different values of the lattice depth $V_{0}$ corresponding to the
vertical dotted lines in Fig. 3. The vertical dotted lines here indicate the
phase values $\delta\phi$ that correspond to the plots in Fig. 3. The inset in
panel (b) shows that the rounded hook structures appear as swallowtails when
total energy rather than chemical potential is plotted, here illustrated for
the gray highlighted region. The labels for the different branches follow the
convention defined in Eq. (20). The intra-band soliton branches are shown in
dashed lines, and when they split, the upper branches are shown in blue and
the lower ones in red.

The normalization of the density provides an additional constraint involving
the complete elliptic integral of the first and second kinds $K(m)$ and $E(m)$

$\displaystyle\textstyle\int_{0}^{2\pi}\rho d\theta=2\pi
r_{1}+2\pi(r_{3}-r_{1})[1-E(m)/K(m)]=1.$ (14)

The definition of $m$ in Eq. (IV) along with Eqs. (12) and (14) can be used to
express the roots in terms of the complete elliptic integrals of the first and
second kinds

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2\pi}+(E-K)\frac{j^{2}K}{\pi^{2}g}}$
$\displaystyle r_{2}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2\pi}+(E-K+mK)\frac{j^{2}K}{\pi^{2}g}}$
$\displaystyle r_{3}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2\pi}+\frac{j^{2}EK}{\pi^{2}g}}.$ (15)

When $j$ and $g$ are specified, these are completely determined by the value
of $m$, which can be determined by imposing the phase boundary condition,
yielding the complete solution.

The resulting spectrum is illustrated in Fig. 2(c) comprising of a parabolic
dispersion curve and a sequence of branches given the fanciful name of
swallowtails due to their appearance in the periodic zone scheme 2(d). The
spectrum represents three different behaviors as discussed below. Examples for
$j=1$ and $j=2$ are shown in Fig. 2(e-h).

Plane waves: Spectral values on the parabolic dispersion curve correspond to
plane wave solutions as would be the case in the linear regime. The solutions
arise when two or more of the roots of $f(\rho)$ are real and degenerate, and
both $f(\rho)=0$ and $f^{\prime}(\rho)=0$ in the phase space plot. The density
being uniform, these are without nodes

$\displaystyle{\textstyle\psi(\theta)=\sqrt{\frac{1}{2\pi}}e^{in\theta},\ \
\alpha=\frac{n}{2\pi},\ \ \mu=\frac{n^{2}}{2}+\frac{g}{2\pi}}$ (16)

with $n=0,\pm 1,\pm 2,\cdots$. These appear with uniform density and a phase
ramp in Fig. 2(e-h).

Solutions with nodes: In the extended zone scheme, the swallowtail branches
terminate at multiples of the Brillouin zone (multiples of $\pm\pi$ in our
units). These termination points correspond to solutions with nodes, and the
stationary states can be taken to be real valued everywhere, which implies as
we approach these nodes, $\alpha\rightarrow 0$ and the bare phase
$\delta\phi\rightarrow\pm j\pi$ so that at each node there is a phase slip of
$\pm\pi$ [46]. The states and the corresponding chemical potentials are
limiting cases of the general solution above when $r_{1}\rightarrow 0$

$\begin{array}[]{c}\psi(\theta)=\sqrt{\frac{m}{2\pi\left(1-E/K\right)}}{\rm
sn}\left(\frac{jK\theta}{\pi}|m\right),\
\mu=\frac{(1+m)j^{2}K^{2}}{2\pi^{2}}\end{array}$ (17)

The index $j$ enumerates the nodes of the wave function. Since the phase can
jump at the nodes, in the absence of rotation, the periodic boundary condition
requires even integers $j=2,4,6,...$, but odd values can be accessed by
rotation. Eqs. (12) and (14) lead to the condition

$\frac{\pi g}{2j^{2}}=K(m)^{2}-K(m)E(m).$ (18)

Examples of such solutions are shown in Fig. 2(e-h), in solid curves with
density vanishing at $j$ points, with phase jumps of $\pi$ occuring at those
points. The nodes occur at the tips of the swallowtails which correspond to
extrema of the chemical potential with respect to both $\Omega$ and
$\delta\phi$, due to the relation between them, and evident as symmetry points
in the periodic zone view Fig. 2(d). Assuming the conditions for nodes
$\rho=\rho^{\prime}=0$, it can be easily seen that $\partial
E/\partial\Omega=0$ at those points, so the energy has extrema, as well as,
$\mu=\partial E/\partial N$. These continue to be true with the lattice on.

Nodeless Density Modulations: The remaining stretch of each swallowtail branch
is described by the general solution defined above, and correspond to nodeless
complex-valued states with density modulation with $j$ dips. These are
intermediate between the nodeless plane waves at one end of the branch and the
solutions with nodes at the other end. The bare phase change is a
monotonically _decreasing_ function of the elliptic parameter $m$, and for
each specific branch labelled by $j$, the phase lies in the range
$[j\pi,\sqrt{j^{2}\pi^{2}+2\pi g}]$ corresponding to $m\in[0,m_{c}]$. The
upper limit of the elliptic parameter $m_{c}$ set by Eq. (18) marks the
solutions with nodes and the lower limit $m=0$ corresponds to plane waves. In
Fig. 2(e-h), we see examples plotted in dotted lines; the density is
periodically modulated, but it never vanishes, and the bare phase has a more
complicated position dependence.

##  V Spectrum with Lattice

We now examine how the nonlinear spectrum in the ring, as discussed in the
last section, is impacted by the introduction of the lattice potential. In
general, this needs numerical determination of the stationary solutions of Eq.
(2). We do a Fourier expansion of the wave function in a plane wave basis
$\psi=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}c_{n}e^{in\theta}$ which
leads to a set of coupled nonlinear equations in the momentum space amplitudes
[11]

$\displaystyle{\textstyle\frac{1}{2}}\\{(n-\Omega)^{2}+V_{0}-2\mu\\}c_{n}-{\textstyle\frac{1}{4}}V_{0}c_{n-q}-{\textstyle\frac{1}{4}}V_{0}c_{n+q}$
$\displaystyle\hskip 56.9055pt+g\sum\limits_{j,k}c_{k}^{*}c_{l}c_{n+k-l}=0.$
(19)

They are solved iteratively by Newton’s method, until convergence criteria are
met. In practice, only a finite number of modes are needed, typically between
$10$ and $20$, with larger values of $g$ and $V_{0}$ requiring more.

As the lattice potential is turned on, Fig. 3 shows that the chemical
potential splits and at higher values of the lattice potential,$V_{0}$ new
degeneracies can also appear as some of the split branches come together
again. This occurs for any value of latticed period, as seen for the three
values of $q=1,2,3$. Notably due to the ring configuration, even one lattice
site $q=1$ constitutes a periodic lattice. However, more insight can be gained
by plotting the chemical potential as a function of the bare phase at
different strengths of the lattice potential, shown in Fig. 4 for specific
values of $V_{0}$ that are marked by vertical dashed lines in Fig. 3.

As for a linear system, turning on the lattice has the usual effect of opening
up gaps in the spectrum. But, in addition, when the lattice is still weak, the
swallowtail branches split as well, as seen in Fig. 4(a,d,g). Two distinct
behavior emerge: At the band edge, such splitting and the band gaps create
hook like structures that terminate at the band edge. The intra-band
swallowtails, shown as colored dashed lines in that figure, each simply splits
into two with the ends still terminating at the main dispersion curve, shown
as solid black lines, which marked the plane wave solutions in the absence of
the lattice, and transform to nonlinear Bloch wave solutions with the lattice
on. The special case of $q=1$ obviously has no intraband swallowtails.

As the lattice depth increases, several things happen. The band gaps for the
main branches widen as can be expected even in the linear case, but
additionally the hooks shorten and can eventually vanish, as the lattice depth
overwhelms the effect of the nonlinearity [47]. The splitting of the intra-
band swallowtails widen as well as can be seen in Fig. 4(d-f), (g-i). Some of
them eventually separate from the main branch and inherit existing hook
structures in the main band (panels (e) and (h)), and they then follow the
same shrinking trend with increasing lattice depth. As the band gaps widen, in
panels (e,f,h,i) we see that some of the detached intra-band swallowtails end
up in the band gap. They can be thought to correspond to what are called gap
solitons [31, 29, 48] in the case of large open lattices.

The structure of the analytical solutions without lattice allows us to
classify the solutions with the lattice on. We use a three part label

$\displaystyle[j\mbox{-}index]\ [subbranch]\ [location],\hskip
14.22636ptj=0,1,2,\cdots$ $\displaystyle
subbranch=\\{m,u/u^{\prime},d/d^{\prime}\\},\hskip
14.22636ptlocation=\\{l,r\\}$ (20)

with $j$ being the swallowtail branch index based on the analytical solutions,
with $0$ representing the ground band section of the main dispersion curve
once gaps open; the _sub-branch_ labels are $m$ for main branch, $u,d$ for
up/down marking upper/lower branches of the split swallowtails and
$u^{\prime},d^{\prime}$ for the hook section of detached intra-band
swallowtails; and the _location_ labels $l,r$ for left or right end of each
branch terminating at some multiple of $\pi$. Note,when a hook disappears, the
location label can switch, as for example $2dl\rightarrow 2dr$ in Fig.
4(e)$\rightarrow$(f) and $3dl\rightarrow 3dr$ in Fig. 4(h)$\rightarrow$ (i).

Our representation provides a different and more unified perspective on the
various solutions and how they are related. Exclusive focus on the spectrum
and states in the absence of rotation is limited to a zero measure subset of
the full range of solutions and hides the relations between various solutions.
For example in Fig. 4(h), in the absence of rotation, the lowest energy
eigenstates would include the Bloch wave state $0ml$ at $\delta\phi/(2\pi)=0$
along with states that are on the vertical slice at $\delta\phi/(2\pi)=1$,
which in our notation, includes the soliton solutions with nodes,
$1u^{\prime}l,2dl,2ul$ and solutions without nodes lying on the $1d,1u$
branches, along with a nonlinear Bloch wave solution on the main branch. Of
course, by symmetry the counterparts at $\delta\phi/(2\pi)=-1$ would be
included. But, in our representation, we can also see that all the soliton
solutions $1dl,1ul,1u^{\prime}l$ originate from the splitting of the $j=1$
swallowtail branch and $2dl,2ul$ from the splitting of $j=2$ branch in the
absence of the lattice; and that $1ul,1u^{\prime}l$ have a continuous range of
nodeless states connecting them, all accessible by rotation.

Figure 5: (a) The lattice period (dashed red line) does not typically match
the period of the density modulation of the eigenstates (solid blue line)
before the lattice is introduced. (b) When the lattice is barely introduced,
the allowed eigenstates adapt to match the lattice period so that a subset of
their separate maxima and minima line up at certain symmetry points in the
ways shown. (c) There are three distinct ways how maximum/minimum of the
lattice and of the eigenstates _without_ the lattice line up at _adjacent_
symmetry points, as the lattice is barely introduced. Each case has two
subcases with _Type-A_ always containing a minimum of the density lining up
with a maximum of the lattice.

The full spectrum as displayed here also provides an alternate perspective on
gap solitons as arising from intra-band swallowtail branches $j\neq nq$ (for
integer $n$) as the lattice opens gaps among states on the main band. The
intra-band swallowtails can detach from the main branch and end up in the gap,
for example $1u^{\prime}l$ in plot Fig. 4(e) and $1ur,2dl,2ul$ in plot (i). We
can also see clearly the sensitivity of their appearance on the lattice depth.
As the lattice depth increases, new gap solitons can appear with different
properties, for example, progressing from plot (e) to (f), a new nodeless
solution on the branch $1ul-1ur$ satisfies the boundary condition in the first
gap. The widening gap also can accommodate more solitons, for example in panel
(i) has $1ul,1ur$ that were not in the gap in (h).

The above statements need to be taken with the caveat that gap solitions are
less well defined in a small ring lattice of few sites, inasmuch as, for any
fixed angular velocity (including $\Omega=0$) the bands themselves are not
continuum and comprise of $q$ discrete states allowing for positive and
negative quasi-momenta, so the distinction between states in the ‘bands’ and
states in the ‘band gaps’ is not as well demarcated . Here, we designate as
gap solitons any state with its energy lying in the gaps of the main
dispersion curve (solid black lines in Fig. 4) for specific nonlinearity and
lattice depth.

Rotation can clearly affect the set of allowed solutions dramatically in a
ring lattice: Imagine sliding a set of vertical lines at
$\delta\phi/(2\pi)=1,2,3,\cdots$ on any of the plots in Fig. 4 and sliding
them together continuously, and wherever they intersect the spectrum are
allowed solutions. Thus gap solitons can emerge and disappear, and states with
nodes (at the tips of the branches) morph into nodeless solutions and
eventually into a solution with different number of nodes, for instance in
plot (i), transitioning a solution with two nodes at $1ur$ for $\Omega=0$ to a
solution with one node at $1ul$ for $\Omega=\pi$ with a continuum of nodeless
solutions in between.

Our representation also highlights certain interesting features that were not
sufficiently stressed in earlier studies. An intraband hook structure can
actually split into two separate branches, creating two distinct solutions, as
seen with $1ur$ in plot (e) splitting into $1u^{\prime}r$ and $1ur$ in plot
(f). Also, it is evident from their swallowtail origins that soliton branches
can have the tips at any integer and half-integer multiples of
$\delta\phi/(2\pi)$ and not just at the edge of Brillouin zone as sometimes
implied.

##  VI Symmetry of Eigenstates

It is obvious that the lattice, as well as the modulated eigenstates in the
_absence_ of the lattice, such as plotted in Fig. 2, have reflection symmetry
about their respective maxima and minima. The index $j$ enumerates the density
nodes in the allowed eigenstates in the absence of the lattice, and in the
rest of the branch these nodes transform into uniformly spaced density dips
that get shallower and eventually vanish on merging into the main branch that
corresponds to plane waves. The modulatated solutions without a lattice are a
manifestation of spontaneously broken symmetry [49]. When the lattice is
turned on, the translational invariance is broken up to a lattice period, and
the eigenstates adapt to have their points of reflection symmetry line up with
those of the lattice, which we will call symmetry points. Notably, even a weak
lattice is sufficient to cause this physical realignment, we have observed
this for lattice strength as low as $V_{0}=0.01$ in our units, relative to
kinetic energy $\sim 1$ and nonlinearity $g=10$.

This indicates four possible pairings of lattice max/min with the density
max/min. We find there is a symmetry of the density modulation that remains
invariant when the lattice is turned on and even in the presence of rotation,
as we now describe. Consider the smallest pair of mutually prime integers,
$n_{s}$ and $n_{l}$ that enumerate the half periods of an eigenstate and of
the lattice respectively that separate adjacent symmetry points, so that

$\frac{\pi}{j}n_{s}=\frac{\pi}{q}n_{l}$ (21)

Even values for either integer would indicate repetition of maximum or minimum
at every symmetry point and odd values indicate alternating of maximum and
minimum. The case of both being even is clearly left out. That leaves us with
three possibilities for when the adjacent symmetry points are different: (I)
$n_{l}$ is even and $n_{l}$ is odd (II) $n_{s}$ is even and $n_{l}$ is odd
(III) $n_{s}$ is odd and $n_{l}$ is odd. Each of these has two sub-cases: A:
when a lattice maximum and a state minimum coincide at one of the symmetry
points and B: when a lattice maximum and state maximum never coincide at any
of the symmetry points. All of these cases are illustrated in Fig. 5.

As the lattice is turned on, and even if there is rotation to access to all
points on a branch, the number of dips and crests in the density can change,
and even local minima and maxima can switch. However, the key point is that
the location of the symmetry points remain unchanged, meaning that those
points continue to have reflection symmetry, even in the presence of the
lattice and rotation.

We illustrate this in Figs. 6 and 7 with eigenstates for lattice with $q=2$
and $q=3$ sites corresponding to spectra shown in Fig. 4. It is clear that
where the branch index is a multiple of the lattice period $j=n\times q$, band
gaps open up and the mutual commensurateness leads to nonlinear Bloch waves,
as seen for panels (a) and (b) in both figures. The remaining panels show
instances of the different symmetry cases mentioned above and are labelled as
such. The states without lattice are shown as filled gray shapes and the
lattice itself in long dashed lines, and the symmetry points lie at the
_center_ and the _edges_ of each panel. When each soliton branch splits with
the introduction of the lattice, we observe that the lower branch (labelled by
second index ‘d’) belong to Type-A, where at least one of the density minimum
coincides with a lattice maximum; this is seen in the left panels (c) and (g)
in Fig. 6 and the left panels (c,e,g,i) in Fig. 7. Likewise, we observe the
upper branch (labelled by second index ‘u’) follows Type-B as seen in the
right panels of both figures.

It is consistent energetically that the lower energy branches correspond to
Type A, where a density minimum is at the lattice maximum. Notably this is the
case for the Bloch wave states as well: At the band gaps which also appear as
splitting of swallowtail branches with $j=n\times q$, each lower branch,
corresponding to the top of the lower band, is of Type A, with density minima
matching lattice maximum; and each upper branch corresponding to the bottom of
the upper band is of Type B, which in the case of Bloch waves also ensure that
density maxima match lattice maxima. This can be seen with states $2dr$ and
$2ul$ in Fig. 6(a,b) and states $3dr$ and $3ul$ in Fig. 7(a,b).

In Fig. 6 we also illustrate the evolution of the states with increasing
lattice depth, first at $V_{0}=0.75$ in panel (d) and then at $V_{0}=5$ in
panels (e) and (f). The symmetry points remain fixed, even though the shape of
the states change substantially and even as new states with nodes $1ur$ and
$1u^{\prime}r$ emerge as the hook structure of upper branch breaks at the
right edge, as noted at the end of the previous section.

We conclude this section with comments on a couple of special cases. Clearly,
cases with $n_{s}=j,n_{l}=q$ are always a solution for Eq. (21). If $j,q$ are
also co-prime, it means there will be only two symmetric axes, at two opposing
ends of the ring. If $j$ and $q$ are not co-prime and share a non-trivial
common factor $k$, then $n_{s}=j/k,n_{l}=q/k$ is a solution, and the symmetry
points are spaced by $\pi/k$. These states are in the quasi-Bloch form whose
period is $2\pi/k$ instead of $2\pi/q$, a $q/k$ multiple of the lattice period
[37, 39]. For example, $q=6,j=3$ would result in a state with its density
modulated with period $2\pi/3$, although we do not include a plot here. The
above symmetry categories still apply to these cases.

##  VII Effect of Lattice on Eigenstates

We display several states in Figs. 6 and 7, corresponding the spectra shown in
Fig. 4. However, we should reiterate that many of these states are accessible
only by introducing rotation. For example, due to the phase boundary condition
$\delta\phi/(2\pi)=n$, states $2dl$ and $2ul$ are allowed stationary states
without rotation for Fig. 6 whereas, the corresponding states in Fig. 7 are
not.

The states on the main band have the Bloch structure where the periodicity of
the density is commensurate with that of the lattice. On each band the states
vary in shape continuously as we progress along the spectral curve, with
different number of nodes at the two ends, spanned by nodeless solutions in
between. This can be seen in Fig. 6(a,b) as we progress from solution $0ml$
with no nodes to $2dr$ with 2 nodes, and from $2ul$ with 2 nodes to $4dl$ with
4 nodes, and likewise Fig. 7(a,b) from $0ml$ with no nodes to $3dr$ with three
nodes and from $3ul$ with 3 nodes to $6dl$ with 6 nodes. The progressive
morphing of the solutions in between can be seen with solutions $2mr$ and
$3mr$ in the respective figures.

As the lattice splits the swallowtails into two branches, at those branches
with index commensurate with the lattice period, the split corresponds to a
band gap and the lower branch initiates from lattice-free solution with a
density minimum aligned with a lattice maximum, and the upper branch with
density maximum aligned with lattice maximum. This can be see with $2dl$ and
$2ul$ in Fig. 6(a,b) and $3dl$ and $3ul$ in Fig. 7(a,b). The grey filled
shapes outline states in the absence of a lattice, however in panels (a) in
both figures, we show the nodeless solution that morphs to $0ml$ in the
presence of the lattice, but the $2dl$ and $3dl$ solutions can be understood
to emerge from the lattice-free solution shown in panels (b) in both figures
but shifted by half a period to have the density minima line up with the
lattice maxima.

It is interesting to note that this pattern is maintained as we go from the
state with nodes at the lower edge of the band to that at the upper edge. The
number of nodes increase by $q$ by creating new minima. We observe these in
transitions from $2ul$ to $4dl$ in Fig. 6(b) and $3ul$ and $6dl$ in Fig. 7(b).

In the case of intra-band solitons, we still find that the lower branch
originates in a state with at least one density minimum aligned with a lattice
maximum corresponding to Type A in our symmetry classification, whereas the
upper branch originates with the same lattice free state shifted and falls in
with Type B. This can be seen both Figs. 6 and 7. The primary difference with
the non-linear Bloch bands is that for them, the pattern repeats commensurate
with the lattice period. With deepening lattice, the incommensurate solitons
originating with intraband swallowtails can get markedly more localized as
seen in Figs. 6(d,ef,g) and 7(d,e,f,j).

Figure 6: (Color online) The nonlinear eigenstates for lattice periodicity
$q=2$ in the presence of the lattice (long dashed lines). The lattice depth is
$V_{0}=5$ and state labels correspond to those in Fig. 4(f) except for panel
(d) tied to Fig. 4(e). Eigenstates in the absence of the lattice are shown in
filled gray. (a,b) Nonlinear Bloch waves, where state and lattice
periodicities match. The remaining panels are soliton solutions of various
types, with the symmetry type labelled. Panels (e) and (f) represent states at
a higher lattice depth $V_{0}=5$ on spectral branches that arise from a single
branch at a lower lattice depth $V_{0}=0.75$, the states for which are shown
in (d).  Figure 7: The nonlinear eigenstates similar to Fig. 6 but for lattice
periodicity $q=3$. The state labels correspond to Fig. 4(i) with lattice depth
$V_{0}=6$. (a,b) Nonlinear Bloch waves, where state and lattice periodicities
match. For the remaining panels, the left ones belong to symmetry Type A and
right ones belong to symmetry Type B.

The localization of soliton state with incommensurate lattice can be
understood by considering its Fourier modes. The Fourier expansion of soliton
with nodes arising from the branch $j$ in the absence of a lattice, in Eq.
(17) has the form [45],

$\psi_{j}(\theta)=\sqrt{\frac{2\pi}{K^{2}-EK}}\sum\limits_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}}{1-q^{2n+1}}\sin[j(n+\textstyle{\frac{1}{2}})\theta]$
(22)

where the modes included are commensurate with the branch index $j$.
Introduction of the lattice couples each Fourier mode $n$ in a ladder stepped
by multiples of the lattice period $n\pm mq$. If $j$ is a multiple of $q$, the
coupling will fill the modes with step $q$, resulting in a Bloch form
solution. If $j$ is not a multiple of $q$ but shares common factor $k$ with
$q$, the modes will be coupled in steps of $k$, resulting in a quasi-Bloch
state with period $2\pi/k$. If $j$ and $q$ are co-prime, the lattice coupling
can spread the modes over all integers. As it spreads out in the Fourier
space, there will be progressive localization in position space.

We illustrate these considerations in Fig. 8, where we present an
incommensurate case with $j=2$ mode in a ring lattice with $q=3$ minima. The
Fourier expansion coefficients $c_{n}$ are as defined in Eq. (19). As the
lattice depth is increased, more Fourier modes are found to be involved in
panel (a), even though with lesser weights. This results in more localization
in position space as shown in panel (b). The degree of localization can be
quantified by the inverse participation ratio (IPR), $\int
d\theta|\psi_{j}|^{4}$ for normalized states. It is seen to climb up with
deepening lattice in panel (c) indicating more localization, while its
momentum space counterpart $\sum_{n}|c_{n}|^{4}$ in panel (d) gets
progressively smaller as can be expected from the Heisenberg Uncertainty
Principle.

Specifically, we can see that in Fig. 8(a), for the displayed mode $j=2$, in
the absence of a lattice the $V_{0}=0$, only the modes $j(n+\frac{1}{2})=\pm
1,\pm 3,\cdots$ are present. As the lattice is turned on with $q=3$, it
couples each of these modes in ladder steps of $\pm q$ as indicated in Eq.
(19), for example, $\pm 1\rightarrow\pm 1\pm 3=-2,3,\pm 4$ all of which
acquire visible occupation in that plot as the lattice depth increases to
$V_{0}=7.25$ and then to $V_{0}=14.75$.

This also underscores that for small lattice sizes on a ring and weak lattice
strengths, differentiation between localized and delocalized states may not be
very obvious, as can be gathered from the states shown in Figs. 6 and 7. That
may only become prominent as the lattice depth is increased. However, our
criterion above provides a concrete way to distinguish the states that can
become localized versus those that will not.

##  VIII Varying Nonlinearity at Fixed lattice

The nonlinearity and the lattice introduce complementary effects and it is
interesting to compare the results of our approach where we start from the
nonlinear solutions with no lattice, with those of Ref. [40] where the
starting point was linear Bloch waves. We have confirmed that the approach
taken here is completely consistent with all of the results in that study, and
here we summarize a brief comparison.

The spectrum of the noninteracting system with lattice comprises of smooth
monotonic curves with gaps at the band edges as shown in Fig. 9(a,c) for
$g=0$. Increasing the lattice depth would widen the gap and flatten the bands.
The eigenstates are Bloch waves,
$\psi_{n,k}=e^{ik\theta}\sum\limits_{l}c_{n,l}e^{ilq\theta}$, with band index
$n$ and quasimomentum $k$. For fixed lattice depth, once interaction is
introduced, as shown in Fig. 9(a,c), doublets of soliton branches emerge from
the main branch terminating $\frac{\Delta\phi}{2\pi}=j/2$ for $j=1,2,3,\cdots$
except initially at $j=nq$ that are multiple of the lattice sites, where band
gaps open up.

Figure 8: Localization with increasing lattice depth $V_{0}$ is shown for
lattice minima $q=3$ incommensurate with state index $j=2$. With increasing
$V_{0}$,(a) more Fourier components have contributing amplitudes $c_{n}$, (b)
the mode gets more localized in position space, (c) the position space inverse
participation ratio (IPR) gets larger and (d) the momentum space IPR gets
smaller. In panel (b) the lattice is shown in dashed line. Figure 9: The
chemical potential as a function of the phase is plotted for different
interaction strengths for lattice periodicities $q=3$ and $q=4$, for fixed
lattice strength $V_{0}=2.25$. In the left panels (a,c) the soliton branches
appear and lengthen with increasing $g$. At larger nonlinearity, in the right
panels (b,d), the hook like structures appear that extend between the band
edges. Panel (e) is a periodic zone representation of panel (d).

As the nonlinear strength is increased the soliton branches lengthen, and
since they still terminate at the same points, their points of contact with
the main branch slides up. More significantly at the band gaps, once the
interaction strength $g$ exceeds some critical value for each gap, [50, 47],
the primary dispersion curve extends beyond the band edge and forms the
familiar hook like structure as we can see in Fig. 9(b,d) for $g=10$. This
perspective has been analyzed in prior studies [39, 36, 50, 51]. But, as we
have shown in this paper, the introduction of the lattice into the nonlinear
solutions offers a different and more unified perspective: The lattice splits
all the swallowtail branches into two, but when $j=nq$, the split coincides
with that band gap. The hook structures seen at the band edge have the same
origins as with the intra-band swallowtails in the lattice-free limit.

In Fig. 9(e), we also show a periodic zone version of the spectrum that
appears in the previous panel (d) of the same figure. This clearly shows the
band separation. The intraband swallow tail structures appear within the span
of each band with the characteristic shape that gives them their names. The
hook like structures in panel (d) associated with the band edge and the
soliton branches in the band gaps, form loops in the extended zone schemes. If
total energy was plotted instead of chemical potential, these rounded loops
would have pointed edges instead, as illustrated in the inset in Fig. 4(b),
and would have the characteristic swallowtail shape.

Figure 10: (Color online) Comparison is made for the plots of the chemical
potential as a function of the phase (a,c) at fixed interaction strength, with
plots as a function of the interaction strength (b,d) at fixed phase. The
vertical dotted lines mark the values in each set that are used in the other
set, for example lines A and B in (a) mark the phase $\delta\phi$ values used
in (b) and (d) respectively; lines A and B in (b) and (d) mark the value of
$g$ used in (a). The lattice depth for (a-d) is set at $V_{0}=4$. The dashed
lines mark intra-band soliton branches as in Fig. 3. Plots (e) and (f) are
similar comparisons but specifically show the emergence of a saddlepoint
bifurcation which does not originate in any of the soliton branches of the
lattice-free system.

In the weak nonlinearity limit, the solitonic solutions can be viewed as
superpositions of degenerate linear solutions. Under small interaction
strength, such solitonic solutions will merge into nonlinear Bloch waves if
$\Delta\phi$ is increased. Once the interaction strength becomes larger, these
solitonic branches will extend further, until it reaches the edge of the
‘Brilluoin zone’, where they are stopped by lattice scattering and their
velocity becomes zero, and these solutions become real solutions with nodes.
For even number of lattice sites, the solitonic solutions reaching the
Brilluoin zone edge will emerge as new physical solutions. These solutions are
the analogs of gap solitons in literature since they emerge inside the band
gaps. For odd number of lattice sites, the solutions from the solitonic branch
$\delta\phi/(2\pi)=q/2-0.5$ are physical, and they exist with arbitrary small
interaction strength. If the interaction strength is further increased, the
nonlinear Bloch waves and the solitonic branches will extend beyond the edge
of the Brilluoin zone, reaches a phase maximum, then turns back and stops at
the Brilluoin zone, forming the hook structure. The solution at the lower
parts of these hooks at the Brilluoin zone edge then emerge as new physical
solutions with nonzero velocity.

Figure 10 compares our representation of the chemical potential as a function
of the bare phase change around the ring in the left panels, with the
variation of the chemical potential with increasing nonlinearity in the right
panels. Note that the spectrum splits as the nonlinearity increases in a
qualitatively similar manner to when the lattice depth is increased in Fig. 3.
The vertical lines labelled in bold upper case letters mark counterparts to be
matched in the two representations. In the $\mu$ versus $\delta\phi$
representation, we observe the hook structures, at line $C$ and $D$ in panel
(c) and (d) respectively; this manifests itself as splitting of the branch in
$\mu$ versus $g$ representation in panels (b) and (d) respectively.

The comparison shows that the splittings observed as nonlinaerity increases in
Fig. 3(b,d,f), are of three different types. The split branches marked by
solid lines mark the emergence of the equivalent of intra-band swallowtail
branches. The splits marked by dashed and dotted lines correspond to the upper
and lower sections of the hook structures. Curiously, the natural continuation
of each such branch that splits as $g$ increases actually corresponds to the
upper end of the hook in our representation, even though in the left panels it
is the lower section that is the immediate continuation of the main branch.
The most unusual feature appears in Fig. 3(e,f) where a pair of branches
appear disconnected from any of the main branches, in both representations.
This is a so called saddle node bifuration that appears when a pair of fixed
points emerge seemingly out of ‘blue sky’ as such bifurcations are also
sometimes referred to [40].

##  IX Stability of States

As noted in Sec. VII, the lattice breaks the translational symmetry upto its
period the splitting of the spectral branches are associated with the
different relative position with respect to the lattice. Here, we show that
they are associated with different dynamical stability properties. We consider
small perturbation around the mean field stationary states:

$\displaystyle\psi(\theta,t)=\psi_{0}(\theta)+\delta ue^{-i\mu t}e^{-i\omega
t}+\delta v^{*}e^{-i\mu t}e^{i\omega^{*}t}$ (23)

and the solve the Bogoliubov equations [4] for the normal modes of the
fluctuations.

$\displaystyle(H_{0}+2g|\psi_{0}|^{2}-\mu)\delta u+g\psi_{0}^{2}\delta
v=w\delta u$ $\displaystyle-(H_{0}+2g|\psi_{0}|^{2}-\mu)\delta
v+g(\psi^{*}_{0})^{2}\delta u=w\delta u$ (24)

If the angular frequencies $\omega$ of the normal modes have imaginary
components and if ${\rm Im}(\omega)>0$ then the fluctuations would grow
exponentially indicating dynamical instability.

It was noted that in the absence of a lattice with repulsive interaction, both
plane waves and soliton train solutions are dynamically stable [49]. Here we
studied the dynamical stabilities of the lowest soliton branches in the
presence of lattice for different number lattice sites on the ring, $q=1,2$
and $3$. For each swallowtail branch $j$, the two split branches that emerges
with the introduction of the lattice not only differ in symmetry properties,
but also in the stability properties. In Fig. 11, we plot the imaginary
components for each branch as a function of increasing lattice depth. Rather
surprisingly, for the parameter range we have tested, for each split pair, the
branch with the lower chemical potential labelled as $1dl,2dl$ are seen to be
dynamically unstable as soon as the lattice is turned on indicated by the
presence of imaginary components of $\omega$ even as $V_{0}\rightarrow 0$. In
contrast such instability is manifest only at larger lattice depth for the
branch with the higher chemical potential. We found that the pattern appears
to hold for higher lattice sites and branches as well but the difference is
not as stark and not conclusive. Simple energy considerations makes this
behavior rather non-intuitive, but we do not yet have a satisfactory
explanation for this behavior.

Figure 11: Examples of Bogoliubov analysis of the splitting soliton solutions.
Left panels show the imaginary part of the Bogoliubov excitations as the
lattice is turned on. Middle panels show the density profile of the split
solutions for fixed $V_{0}=1$; dashed lines indicate the lattice potential.
Right panels show the corresponding evolution of the chemical potential as the
lattice is turned on.

##  X Large Lattice limit

The ring topology is particularly significant when the size is small and the
number of lattice sites is few in number. As the number of sites is increased
without changing the lattice period and the number of particles per site, the
ring gets larger and we can consider the infinite lattice limit. This however
requires a different scaling than we have used in the rest of the paper, where
we have scaled length by $R$ and energy by $\hbar^{2}/(mR^{2})$, so that
keeping $V$ and $g$ constant has the effect of diminishing the potential and
the nonlinear term relative to the kinetic energy by a factor $\propto R^{2}$.
This did not impact our observations where comparisons were made for fixed
lattice sites. Now, to compare the variation with changing number of lattice
sites, while keeping the lattice period, depth and the particle density
constant, we scale the potential and the nonlinear strength by the square of
the number of the lattice sites $q^{2}$ in Eq. (2) then divide the resulting
chemical potential by $q^{2}$.

Some of our simulations are shown in Fig. 12 where we plot the chemical
potential versus the bare phase around the ring $\delta\phi$ as before, but
now scaled by a multiple of the recoil energy
$8E_{r}=4\pi^{2}\hbar^{2}/(md^{2})$ and $q\pi$ respectively, where $d=2\pi
R/q$ is the lattice constant. The phase scaling is similar to scaling the
linear wave vectors by the recoil momentum $k_{r}=\pi/d$. We plot two cases,
one set in Fig. 12(a-c) for $q=3,6,9$ for $V_{0}=q^{2}/9$ and $g=q^{2}/9$ in
the units of lattice constant and recoil energy mentioned above, and another
set in Fig. 12(d-f) for $q=2,4,6$ for $V_{0}=3q^{2}/4$ and $g=q^{2}/16$.
Scaled this way, we find that the shape of primary dispersion curve is
identical for any number of lattice sites as long as the lattice depth $V_{0}$
and nonlinearity including particle density remain the same. This can be seen
clearly when we overlay the plots from (a-c) in panel (g) and the plots from
(d-f) in panel (h), the main bands are identical and lie on top of each other.
This behavior is identical to the case without nonlinearity, and specifically
we note from panels (d-f) and (g) that even the distinctive nonlienar feature,
the hook structure, is identical for different number of lattice sites.

Figure 12: The effect of increasing the number $q$ of lattice sites is shown
for two different values of the lattice strength $V_{0}$ and nonlinear
constant $g$ in (a-c) and (d-f). Allowed solutions in the absence of rotation
are marked by vertical dotted lines. For consistent comparison across
different lattice sites, the $V_{0}$ and $g$ are scaled by $q^{2}$, and the
chemical potential $\mu$ and the bare phase $\delta\phi$ are also in recoil
units fixed by the lattice constant. Overlaying the ones from (a-c) in panel
(g) and those from (d-f) in (h) shows that the primary dispersion curve
corresponding to nonlinear Bloch waves are identical for any number of lattice
sites, while soliton branches at specific multiples overlap as new ones emerge
with increasing sites.

On the other hand, new soliton branches emerge with increasing lattice sites.
For weak lattices, the gaps open at multiples of the lattice site number
$j=nq$ and with $q-1$ pairs of intraband swallowtails for each band. When $q$
values have a common factor, they share common soliton branches and the shape
of those branches is also identical for the $q$ values. This can be seen in
panels Fig. 12(a-c), for example with the two pairs of intra-band soliton
branches for $q=3$, which also appear identically for $q=6$ and $q=9$, and
likewise in Fig. 12(d-f), the single pair of intra-band soliton branches for
$q=2$ also appear identically for $q=4$ and $q=6$. This similarity is also
reflected in the quantum states, with the basic shape of the wavefunction for
the fundamental period $q=3$, simply repeating twice for $q=6$ over the
lattice and thrice for $q=9$.

We can expect that as the number of sites in the ring gets larger, keeping the
lattice depth and nonlinear strength and particle density fixed, the main
dispersion curve corresponding to the nonlinear Bloch states will remain
unchanged, but more and more soliton branches will appear. However, with
increasing number of such branches, the soliton branches that slide into the
gap as seen in Fig. 12(c) and (g) become more exceptional leading up to the
well-studied properties of gap solitons.

##  XI Conclusions and Outlook

In a finite ring-shaped lattice, we find that the nonlinear solutions without
the lattice define the landscape of solutions with the lattice. Many of the
standard labels and classifications, such as referring to the solutions as
solitons, are a relic of studies in the context of open lattices in the
infinite and tight binding limit, and are not a particularly useful way of
understanding finite closed lattices that can be possibly weak. In a ring
without a lattice, there are three types of solutions: Plane waves, solutions
with nodes, modulated solutions without nodes. The presence of a lattice
modifies these to the following set of independent classifications of
solutions: (I) Solutions with period commensurate/incommensurate with the
lattice period, and (II) solutions with/without nodes. The first
classification can be sub-grouped into nonlinear Bloch waves that exactly
match the periodicity of the lattice, and their super-periodic counterparts
that have periods that are multiples of the lattice period.

The commensurate solutions emerge from the plane wave solutions and the
swallowtails at multiples of the lattice period. The incommensurate ones
originate from intra-band swallowtails. With increasing lattice strength, the
commensurate ones will remain delocalized, whereas with increasing lattice
depth, the incommensurate ones will tend to localize.

As already noted in Ref. [40], the concept of gap solitons is not very
distinctive in a finite ring lattice. There are solutions that exist in the
band gap that originate with the migration of the intra-band swallowtails with
increasing lattice depth, but their behavior falls into the general
classification above.

The most significant aspect of the solutions of a ring lattice is the ability
to access a full range of solutions of the infinite lattice by using rotation.
This makes the ring lattice an invaluable and unique tool for the study of
nonlinear dynamics. The continuous transition between solutions possible with
rotation can be utilized to probe and modify the dynamics of the system, to
access vastly different kinds of behaviour within the same system. We have
already observed that even in the linear case [11], and in our continuing work
we anticipate even richer dynamics in the nonlinear regime. An exploration of
the negative nonlinearity constitutes a natural continuation of this work.
Finally, there is much potential for exploring the quantum dynamics in ring
lattices beyond the mean field regime, which also forms part of our ongoing
research.

###### Acknowledgements.

We gratefully acknowledge valuable discussions with D. Schneble, and the
support of the NSF under Grant No. PHY-2011767.

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