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

Jonathan Tekverk 1, Christopher Siebor 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 spectrum and stationary states in a ring-shaped lattice potential
in the context of ultracold atoms with attractive interatomic interactions. We
determine analytical solutions in the absence of a lattice by mapping them to
those for repulsive interactions, and then we numerically follow the
transformation of those solutions as the lattice is introduced and
strengthened. Several features emerge that are specific to negative
nonlinearity, that include: Soliton branches detaching to create new ground
states; gaps opening up at the bottom of the primary spectral branch; multiple
splitting and rejoining of some branches. We correlate the spectral features
with the behavior of the density and phase of the corresponding eigenstates,
and track them along branches and as various system parameters change. We find
that the phase is sensitive to how a specific point in the spectrum is
approached, particularly relevant at certain persistent gaps in the spectrum.
The symmetry and stability properties are generally found to be opposite of
that found for repulsive interactions.

##  I Introduction

This paper is the second of a sequence of two papers that examines the
stationary states of a self-interacting coherent medium in a ring-shaped
lattice potential. The first paper considered repulsive interaction [1], and
in this current one, we consider attractive interaction. Our analyses are
framed in the context of a Bose-Einstein condensate (BEC) in a toroidal trap
[2], described with a mean field picture [3] based on a nonlinear Schrödinger
equation (NLSE).

We are motivated by a rich array of physical phenomena available when a BEC is
confined within the non-trivial topology of a ring trap with an added
azimuthal periodic lattice structure; which have been studied with both
continuum [4, 5, 6, 7, 8, 9, 10, 11, 12] and discrete [13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23] models. Toroidal traps have been utilized in several
experiments to explore physics associated with the superfluidity of BEC [24,
25, 26] but usage of an azimuthal periodic lattice is pending despite existing
capabilities [27, 28].

Interactions are naturally present in ultracold atomic systems [29], and makes
the physics richer and more complex. Specifically, in the mean field limit,
the setup we consider can be used to probe and simulate nonlinear dynamics in
a lattice system which is closed, finite and unbounded. A necessary
preliminary to exploring the dynamics is a thorough understanding of the full
landscape of stationary solutions. We do that here for attractive interaction
to complement our prior analysis with repulsive interaction [1], to present a
comprehensive picture.

In lattices with trivial topology, experiments [30, 31] have explored
interacting BEC, complemented by theoretical studies [32, 33, 34, 35, 36, 37,
38, 39, 40]. However, in the context of ring lattices, apart from our recent
work [1], there has only been one other detailed study [41], but none for
attractive interactions. Our approach here will be built on the analytical
solutions that exist for such nonlinear systems in the absence of a lattice
[42, 43] and determine the effects of introducing a lattice of increasing
strength. This will draw on a convenient mapping between solutions for
positive and negative nonlinearities that one of us recently demonstrated for
the relevant NLSE [44].

Our results will have relevance in a broader context since the nonlinear
Schrödinger equation we use is also relevant in the field of nonlinear and
fiber optics [45, 46, 47]. Thus, it is well established that attractive
interaction leads to significantly different behavior, for example, in three
dimensions sufficiently strong interactions and large number of particles can
lead to wave-function collapse [48], but though not necessarily in one
dimension (1D) [49]. There have been no comprehensive study of how attractive
interactions in an NLSE impact the states in the presence of a lattice in a
ring configuration. As we will show in the paper, there are indeed
dramatically different behaviors manifest when the interaction becomes
attractive that we did not observe in our prior study of the same
configuration with repulsive nonlinearity.

In Sec. II, we summarize our physical model; and then we determine the
analytical solutions in the absence of a lattice in Sec. III, focussing on the
essential features that distinguish having attractive interactions instead of
repulsive. We discuss the effects of introducing the lattice, on the spectrum
in Sec. IV; and proceed to examine the impact on the corresponding stationary
states, first on the density in Sec. V and then on the phase in Sec. VI. The
interplay between varying the strength of the lattice and that of the
interaction-induced nonlinearity is explored in Sec. VII as regards the
effects on the spectrum. In Sec. VIII, we focus on several anomalous spectral
features we observe. Section IX presents a stability analysis for the various
stationary states. We conclude with a summary of our results and outlook in
Sec. X.

Figure 1: (a) 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, $R\gg r$. Features of phase
space curves are compared for (b,c) positive nonlinearity $g>0$ with those for
(d,e) negative nonlinearity $g<0$. (b,d) Schematics of the cubic function $f$
that sets the density variation, shown for all three real roots $\\{r_{i}\\}$;
the dotted parts lie in the shaded forbidden region. (c,e) Corresponding phase
space plot of $\rho^{\prime}=\pm\sqrt{f}$ versus $\rho$ has a wing-loop
structure; the orientation of the wing depends on the sign of the nonlinearity
$g$.

##  II Physical Model

The details of our physical model are presented in our prior paper focussed on
repulsive interactions [1]; here we will only recapitulate the salient points.
Consider a BEC in a toroidal trap with its minor radius $r$ much smaller than
its major radius $R$ so that the system is treated as a cylinder ${\bf
r}=(z,r,\varphi)$ with periodic boundary condition on $z$. Strong confinement
along $(r,\varphi)$ justifies integrating them out for an effective 1D
Hamiltonian

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

where $g_{3D}=4\pi\hbar^{2}a/m$ is the interaction strength, $a$ is 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 minor radius.
Taking the major radius $R$ as the length unit, the azimuthal distance is
rescaled $z/R=\theta\in[0,2\pi)$; the lowest energy scale in the ring
$E_{R}=\frac{\hbar^{2}}{mR^{2}}$ is set as the energy unit; and the
corresponding frequency $\omega_{R}=\frac{\hbar}{mR^{2}}$ is set as the unit
for frequency as well as for angular velocity, and sets the time scale
$\tau=\omega_{R}^{-1}$. The effective nonlinear constant in 1D then has the
form $g=2a\omega_{T}N$. 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)

with normalization $\int_{0}^{2\pi}d\theta\left|\psi(\theta,t)\right|^{2}=1$.
The lattice $V(z,t)=V_{0}\sin^{2}\left(\frac{q}{2}(z/R-\Omega t)\right)$ can
rotate with angular velocity $\Omega$ with respect to the laboratory frame,
and in Eq. (2) we transformed to the frame rotating with the lattice, so the
Hamiltonian itself is time-independent. 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
$\mu$ defines the chemical potential.

The stationary solutions can be determined in the hydrodynamic picture,
expressing $\varphi(\theta)=\sqrt{\rho(\theta)}e^{i\phi(\theta)}$, leading to
coupled equations for the density $\rho$ and the phase $\phi$; the latter can
be formally integrated:

$\displaystyle{\textstyle\frac{1}{8}}(\partial_{x}\rho)^{2}-{\textstyle\frac{1}{4}}\rho\partial_{x}^{2}\rho+{\textstyle\frac{1}{2}}\alpha^{2}+V_{0}\sin^{2}({\textstyle\frac{1}{2}}q\theta)\rho^{2}+g\rho^{3}-\mu\rho^{2}=0$
$\displaystyle\Delta\phi(\theta)=\phi(\theta)-\phi(0)=\Omega\theta+\int_{0}^{\theta}\frac{\alpha}{\rho(\theta^{\prime})}d\theta^{\prime}.\hskip
28.45274pt$ (3)

The integral of motion is associated with the current density $J=N\alpha$ and
the superfluid velocity $v=\alpha/\rho(\theta)$. The ring topology imposes
periodic boundary conditions

$\displaystyle\rho(0)=\rho(2\pi)$
$\displaystyle\rho^{\prime}(0)=\rho^{\prime}(2\pi)$
$\displaystyle\Delta\phi(2\pi)=2\pi n,$
$\displaystyle\delta\phi\equiv\delta\phi(2\pi)=\Delta\phi(2\pi)-2\pi\Omega,$
(4)

with the integer $n$ being the winding number. We will plot the chemical
potential $\mu$ as a function of the bare phase change $\delta\phi$ acquired
around the ring neglecting rotation. The phase boundary condition for a finite
size ring picks out only discrete points on the continuum spectrum as
physically relevant but, as can be seen above, the entire spectrum can be
accessed via rotation [1].

##  III Exact solutions without lattice

In the absence of a lattice, $V_{0}=0$, analytical solutions can be found for
Eq. (II) [1, 44, 42, 43], and they will provide the framework for our
description once the lattice is introduced. We will show that the solutions
for $g<0$ can be related intuitively to those for $g>0$ we presented earlier
[1]. Integration of Eq. (II) yields

$\displaystyle\partial_{x}\rho=\pm\sqrt{f\left(\rho\right)};\hskip
8.53581ptf\left(\rho\right)=4g\rho^{3}-8\mu\rho^{2}+8\beta\rho-4\alpha^{2}$
(5)

with integral of motion $\beta$. All the stationary states of the system can
be expressed in terms of the three roots of this cubic function $f(\rho)$
[44]. The function is compared for $g>0$ and $g<0$ in Fig. 1(b,d). With the
forbidden negative density regimes of the function (shown in dotted lines)
left out, the phase space plots of $\rho^{\prime}=\pm\sqrt{f(\rho)}$ versus
$\rho$ in Fig. 1 (c,d) generally form a “loop-wing” structure, with a closed
loop that corresponds to oscillating solutions and an open wing shape tied to
solutions that diverge. Special cases occur when some of the roots are
degenerate or complex [44]. In terms of the roots, the coefficients are

$\displaystyle\textstyle\mu=\frac{g}{2}(r_{1}+r_{2}+r_{3}),\hskip
14.22636pt\alpha^{2}$ $\displaystyle=$ $\displaystyle gr_{1}r_{2}r_{3}$
$\displaystyle\beta=\frac{g}{2}(r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}).$ (6)

For real roots, without loss of generality, we assume ascending order of the
roots $r_{1}\leq r_{2}\leq r_{3}$.

Since $\alpha^{2}\geq 0$, one root has to be negative or zero when $g<0$; but
two and three negative roots are not allowed; and neither are complex roots
since a conjugate pair would have a positive product and the whole phase space
curve would be in the unphysical negative density regime. All the roots
therefore have to be real with $r_{1}\leq 0$, and $r_{2},r_{3}\geq 0$; and the
wing intersects with the $\rho=0$ axis at $r_{1}$. The wing, being open on the
left, lies entirely in the $\rho\leq 0$ regime. Therefore solutions on the
wing are forbidden, and only those on the loop are allowed. Furthermore, in a
ring with uniform potential, solutions that grow or decay will not satisfy the
periodic boundary conditions. So, only oscillatory solutions are possible,
with planes waves being a limiting case.

Figure 2:  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 $g=-1$ and the bare phase
accumulated around the ring $\delta\phi=0.5$ correspond to the vertical dashed
lines in Fig. 3. 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. 3.

The general shape of the phase space curves for $g>0$ and $g<0$ are mirror
reflections across some symmetry axis, switching the roles of $r_{1}$ and
$r_{3}$. As a result, we find that solutions for $g<0$ can obtained from
corresponding solutions for $g>0$ by simply switching $r_{1}\leftrightarrow
r_{3}$. We proved this rigorously using the properties of Jacobi elliptic
functions in Ref. [44]. This insight allows us to summarize the solutions for
the attractive interactions as direct counterparts of the solutions we found
for the repulsive interactions in our prior work [1], with the density and
phase now given by

$\displaystyle\rho(\theta)$ $\displaystyle=$ $\displaystyle
r_{3}+(r_{2}-r_{3}){\rm sn}^{2}(\textstyle{\sqrt{g(r_{1}-r_{3})}}\
\theta,m^{\prime})$ $\displaystyle\delta\phi(\theta)$ $\displaystyle=$
$\displaystyle\frac{\alpha}{r_{3}\sqrt{g(r_{1}-r_{3})}}\Pi(1-\frac{r_{2}}{r_{3}},\varphi,m^{\prime})$
(7)

where $\Pi$ is an incomplete elliptic integral of the third kind [50] and
$\varphi=\sin^{-1}[{\rm sn}(\sqrt{g(r_{1}-r_{3})}\ \theta,m^{\prime})]$. The
parameter $m^{\prime}$ can be obtained from its counterpart, $m$ for $g>0$ by
switching $r_{1}\leftrightarrow r_{3}$ as well

$\displaystyle\textstyle m^{\prime}=1-m$ $\displaystyle=$
$\displaystyle\frac{r_{2}-r_{3}}{r_{1}-r_{3}}.$ (8)

The density oscillates with period
$\theta=K(m^{\prime})/\sqrt{g(r_{1}-r_{3})}$. To satisfy the density boundary
condition, the _complete_ phase space loop has to be traversed by an integer
number of turns $j$, leading to the condition for a complete circuit of the
ring,

$jK(m^{\prime})=\pi\sqrt{g(r_{1}-r_{3})}.$ (9)

The bare phase in a circuit of the ring and the normalization of the density
constrain the solutions

$\displaystyle\delta\phi=\delta\phi(2\pi)=\frac{2\pi\alpha}{K(m^{\prime})r_{3}}\Pi(1-\frac{r_{2}}{r_{3}},m^{\prime})\hskip
28.45274pt$ (10) $\displaystyle\textstyle\int_{0}^{2\pi}\rho d\theta=2\pi
r_{3}+2\pi(r_{1}-r_{3})[1-E(m^{\prime})/K(m^{\prime})]=1.$

Here, $K(m^{\prime})$ and $E(m^{\prime})$ are complete elliptic integrals of
the first and the second kinds, respectively.

The definition of $m^{\prime}$ in Eq. (8) along with Eqs. (9) and (10) can be
used to express the roots in terms of the complete elliptic integrals of the
first and second kinds, exact counterparts of those we derived for $g>0$ after
applying the mapping we mention above. When $j$ and $g$ are specified, these
are completely determined by the value of $m^{\prime}$, which can be
determined by imposing the phase boundary condition, yielding the complete
solution.

The spectrum plotted as a function of the bare phase comprises of a parabolic
dispersion curve and a sequence of swallowtail branches [51] that mark
solitonic solutions. As with positive nonlinearity, we find three distinct
types of the solutions: (i) Plane waves corresponding to the spectral values
on the parabolic dispersion curve; (ii) Density modulations with nodes that
correspond to the tips of the swallowtails and (iii) Nodeless density
modulations that mark the rest of the swallowtail branches.

We discussed the features of these solutions for $g>0$ in some detail [1],
here we will discuss the differences that emerge for $g<0$. For plane wave
solutions, $g<0$ shifts the parabola down
$\mu=\frac{n^{2}}{2}+\frac{g}{2\pi}$, which translates to a downward shift for
the entire spectrum; so the ground state has negative energy. The bare phase
in Eq. (10) as function of $m^{\prime}$ is monontically increasing for $g<0$
and so, for the swallowtail branches, the phase lies in the range
$[\sqrt{j^{2}\pi^{2}+2\pi g},j\pi]$ corresponding to
$m^{\prime}\in[0,m^{\prime}_{c}]$. The $m=0$ value corresponds to the lower
limit of the bare phase for negative $g$ and corresponds to the end attached
to the parabolic curve, where it tends to a plane wave. The value
$m_{c}^{\prime}$ correspond to bare phase multiples of $\pi$, the tips of the
swallowtails which mark the solutions with nodes. Since the upper value of the
bare phase marks the tip, the swallowtails are now on the outside of the
parabola as we will see in Fig. 3 discussed in the next section. This is in
contrast to those for positive nonlinearity, where the swallowtails extend on
the inside concave side of the parabola [1]. This is the most striking
difference in the spectrum between $g>0$ and $g<0$, in the absence of lattice.

There is another distinguishing feature for these swallowtails, that remains
significant even when the lattice is present: Clearly for $g<-j^{2}\pi/2$, the
range for the bare phase becomes $[0,j\pi]$ which, as we will see, marks a
transition point where the swallowtail branch of index $j$ detaches from the
parabola and slides below the lowest energy possible for the plane wave
solutions associated with the parabola; and the ground state then corresponds
to a swallowtail. This behavior is not seen for repulsive interactions when
$g>0$.

The solutions with nodes cannot carry current, and therefore can be made real-
valued everywhere, and they can be readily obtained from the general solution
in Eq. (III) setting $r_{2}=0$

$\begin{array}[]{c}\psi(\theta)=\sqrt{\frac{m^{\prime}}{2\pi\left(m^{\prime}+E/K-1\right)}}{\rm
cn}\left(\frac{jK\theta}{\pi}|m^{\prime}\right),\
\mu=\frac{(1+m^{\prime})j^{2}K^{2}}{2\pi^{2}}.\end{array}$ (11)

This differs in having an elliptic ${\rm cn}$ rather than ${\rm sn}$ that is
present for $g>0$, and in that the elliptic parameter is $m^{\prime}$ instead
of $m$.

Figure 3:  The chemical potential $\mu$ plotted as a function of the bare
phase $\delta\phi$ for three different lattice periods $q=1,2,$ and $3$, one
in each row. The nonlinear strength is fixed at $g=-1$. The three panels in
each row are for different values of the lattice depth $V_{0}$ corresponding
to the vertical dashed lines in Fig. 2. The vertical dashed lines here
indicate the bare phase values $\delta\phi$ that correspond to the plots in
Fig. 2.

##  IV Spectrum with Lattice

In the presence of attractive interatomic interactions, $g<0$, even with no
lattice, there can be a multitude of solutions. Keeping the interaction
strength fixed at $g=-1$ and the bare phase gathered around the ring fixed at
$\delta\phi=0.5$, we track the progression of the chemical potential in Fig. 2
as the lattice potential is turned on, for three different lattice periods,
$q=1,2,3$. The qualitative behavior here resembles that for $g>0$: Some of the
branches split and then some of them rejoin again.

Significant differences in behavior emerge when we plot the chemical potential
as a function of the bare phase in Fig. 3. In these and all other similar
spectrum plots in the paper, we reiterate that for a fixed value of the
angular velocity (including $\Omega=0$) only specific points on each trace are
allowed that meet the phase boundary condition in Eq. (II); those points lie
on equally spaced vertical slices in the spectral plots such as in Fig. 3 and
any angular velocity $\Omega$ shifts those lines continuously [1] allowing
access to the entire spectrum. Even when there is no lattice, the obvious
difference in changing the sign of the nonlinear is that for $g<0$ the soliton
branches are on the outside of the main parabola which marks plane wave
solutions; for $g>0$ they are on the inside. As the lattice is turned on, this
difference leads to several interesting features that differentiate negative
nonlinearity from positive. What remains similar is that the lattice splits
the soliton branches, but with gaps progressively smaller for higher $j$,
requiring stronger lattices to create noticeable gaps. Gaps open up in the
parabola as well at the soliton branches with index $j$ that are commensurate,
as in exact multiples of the lattice periodicity, $q$.

When $q=1$, there is only one lattice period around the ring, and every
soliton index $j$ is commensurate. So in Fig. 3(a-c), as the lattice is
introduced, band gaps open up on the parabola at every soliton branch. For
$q=2$ in Fig. 3(d-f) and $q=3$ in Fig. 3(g-i), distinctions emerge between
inter-band and intra-band soliton branches, corresponding to the soliton index
being commensurate and incommensurate respectively. As lattice strength
increases, the splits appear in all the soliton branches,though progressively
less conspicuous for higher ones. For commensurate branches, the split soliton
branches blend with the main branch segments above and below the gap to form
hook-like structures which, for negative nonlinearity, lie on the _outer_
convex side of the parabola, as shown by Fig. 3(b-c). With increasing lattice
depth, the spectral lines flatten out, causing the commensurate soliton
branches to become visibly indistinguishable from the main branches at band
gaps.

The intraband branches highlight a different behavior: For positive $g$, with
increasing lattice strength those branches slide upward towards higher
chemical potential. With negative $g$, the opposite happens and they slide
downward along the parabola. In the case of the lowest band, at sufficiently
high lattice strength, those spectral lines eventually ‘fall off’ the main
branch and connect directly to the vertical axis, creating a new ground state
in the process. This behavior can be seen for both $q=2$ and $q=3$ in Fig. 3,
with the $j=1$ branch detaching and falling below the main branch in both
cases.

We noted in the previous section that this behavior occurs even without a
lattice, for any branch when $g\leq-j^{2}\pi/2$. Here we see that even at
values of $g$ that does not meet that criterion, increasing the lattice
strength induces the same qualitative behavior. The strength of the
nonlinearity $g$ determines the number of intra-band soliton branches that
detach even with no lattice.

This underscores the intricate interplay of the lattice and the nonlinearity
that defines the behavior of this system. That interplay is responsible for
other interesting variations of this behavior, some seen in Fig. 3: In the
case of $q=2$, the $j=1$ branch splits. Labelling the upper and lower branches
$u$ and $d$ for ‘up’ and ‘down’, only the lower branch $1d$ detaches from the
main branch, while the upper one $1u$ shortens and blends with the main branch
as the branches flatten with increasing lattice depth. On the other hand, for
$q=3$, both $1u$ and $1d$ detach and remain so as the lattice depth increases.
However, we do see the $j=2$ branch vanish into the main branch. Although the
lattice splits the soliton branches, the effect is not quite uniform. The
commensurate branches split significantly as they coincide with the band gaps
opening up. The intraband solitons splittings are smaller. Some branches are
surprisingly resistant to any noticeable splitting; for example the $j=2$
branch for $q=3$ in Fig. 3(g-i) simply disappears at sufficiently deep
lattice.

Figure 4:  The top row plots the chemical potential $\mu$ as a function of the
phase $\delta\phi$, for lattice period $q=2$, and shows the progression as the
lattice strength is increased from $V_{0}=0$ to $V_{0}=1.7$, keeping the
nonlinear strength fixed at $g=-6.5$. The insets show details of the upper
splitting of $j=2$, labelled $2u$, and the unsplit $j=3$ branch. The solid
circles mark values for which the states are plotted in the rows below, with
labels as described in Eq. (12). The plots of the density and phases of the
state are lined up with the corresponding spectral plots in the top row, with
the middle rows for soliton branch $3d$ and one point from the main branch,
and the bottom rows for the soliton branch $2u$. Labels on the density traces
mark which point they correspond to in the spectrum, the phase plots have
matching traces. The lattice is indicated in filled gray.

##  V Effect of Lattice on Density

In order to get a better insight into the correlation of the various spectral
features and ultimately the behavior of the system, we have to examine the
associated eigenstates. We do so in the context of $q=2$ and $q=3$, fixing the
nonlinear strength and then switching on and progressively increasing the
lattice depth.

As the spectrum evolves with the introduction of the lattice, it becomes
harder to classify branches as being on the main branch or a specific soliton
branch indexed by $j$ as the branches blend, split and merge. Certain features
of the eigenstates help identify and classify them: (i) In the presence of a
lattice, the main branch solutions transform from plane waves to Bloch waves
with the same period as the lattice. (ii) Soliton branches can be identified
by the number of density peaks that should equal the $j$ index. (iii) When
soliton branches split, the eigenstates corresponding to the upper and lower
branches display different symmetry properties relative to the lattice.

The symmetry properties were discussed in detail in the case of positive
nonlinearity [1], here we note the differences: For $g<0$, for each splitting,
the energetically lower branches have at least one density maximum line up
with a lattice minimum, while the upper branches _never_ have any state
maximum line up with a lattice minimum. This is in contrast for $g>0$ wherein
for a lower branch, at least one of the density minimum lines up with a
lattice maximum; and for the upper branch, lattice maxima and state maxima
never coincide.

The eigenstates as plotted in Fig. 4 and 5 provide more insights. Densities
and phases of the states are plotted in vertical alignment with the
corresponding spectra in the top row. On the spectra, solid dots mark the
points that correspond to states plotted below, and labelled by:

$\displaystyle[j\mbox{-}index]\ [sub\mbox{-}branch]\ [location],\hskip
14.22636ptj=1,2,\cdots$ $\displaystyle
sub\mbox{-}branch=\\{m,u/u^{\prime},d/d^{\prime}\\},\hskip
14.22636ptlocation=\\{l,c,r\\}$ (12)

with $j$ being the swallowtail branch index based on the analytical solutions
with no lattice; the _sub-branch_ labels are $m$ for main branch, $u,d$ for
up/down marking upper/lower in a split and if there are further splittings of
those branches, we label them $u^{\prime},d^{\prime}$; and the _location_
labels $l,c,r$ indicate left, somewhere near the center, and right end of each
branch.

Figure 5:  The top row plots the chemical potential $\mu$ as a function of the
phase $\delta\phi$, for lattice period $q=3$, and shows the progression as the
lattice strength is increased from $V_{0}=0$ to $V_{0}=2.5$, keeping the
nonlinear strength fixed at $g=-15$. The inset shows details of the upper
split of $j=3$, labelled $3u$ and the unsplit $j=4$ branches. The solid
circles mark values for which the states are plotted in the rows below, with
labels as described in Eq. (12). The density and phase are not plotted for
$V_{0}=0$, so the lower four rows line up and correspond to only the non-zero
potentials in the top row, as indicted by arrows. The middle rows are for
soliton branch $4$ and a point from the main branch, and the bottom rows are
for the soliton branch $3u$ with the last two columns including states from
the newly emergent $3u^{\prime}$ branch as well. Labels on the density traces
mark which point they correspond to in the spectrum, the phase plots have
matching traces. The lattice is indicated in filled gray.

###  V.1 q=2

We examine the case of $q=2$ lattice periodicity in Fig. 4 at fixed $g=-6.5$.
This nonlinearity is sufficiently strong to have the $j=1$ and $j=2$ soliton
branches to be detached and below the main branch even with no lattice. The
lattice splits them both, with the gaps getting wider with increasing lattice
depth. But, the $j=2$ branch behaves differently: The upper branch, $2u$ and
the main branch both detach from the vertical axis; and their left tips
approach each other, tending to form a hook-like structure. However, the left
tips never actually meet in our simulations, always leaving a gap, as shown in
the insets in the top row of Fig. 4. As the lattice is strengthened, the $2u$
branch shrinks, eventually disappearing (not shown). The general behavior
resembles the opening of a band gap at commensurate soliton branches, but with
some differences.

We observe another interesting feature with the $j=3$ branch. We do not see
any visible splitting when the lattice is introduced but, due to the observed
symmetry features, we label it $3d$. Increasing lattice strength causes it to
slide downwards along the main branch, until it does something surprising: It
extends out beyond the now much contracted main branch, as shown for
$V_{0}=1.5$ in Fig. 4. At even stronger lattice, the branch extends out
farther and slides farther down to intersect with the shrunken $2u$ branch. An
interesting characteristic of these branches is that they have termination
points both on the left and the right; the left termination points are
atypically not at multiples of $\pi$ as we can see in Fig. 4 while the right
termination points are.

On each soliton branch, the number of density peaks equals its $j$ index. The
main branch densities start off as plane waves but, with the introduction of
the lattice, they transform to uniformly modulated Bloch waves similar to
states on the commensurate $j=2$ branch. Specifically, as the left tips,
labelled $2ul$ and $ml$, of the detached $2u$ and the main $m$ branches
approach each other, the corresponding densities shown in thick red lines,
converge to the same form. They appear to be mutual continuations, although we
could never get the gap to close. At stronger lattice, the main branch
continues to shrink, approaching termination at $\theta=2\pi$ on the left, for
a solution with two nodes; and the densities all along the shrinking $2u$
branch also tend towards that two node solution.

Figure 6: The figure illustrates the dependence of the phase on how a
particular spectral value is approached. (a,b) The chemical potential is
plotted versus the phase for the main branch and $3d$ soliton branch, for
$q=2$ and $V_{0}=1.3$, identically to the center top row panel in Fig. 4: In
panel (a) the left eigenstates are computed starting at the tip of the $3d$
branch as indicated by arrows, while in panel (b) they are computed starting
at the tip of $4u$ branch. The inset in (a) shows that the bottom part of the
main branch actually comprises of closely spaced double lines. (c,d) The
densities are identical for corresponding points in the two different cases.
(e,f) The phase however clearly behaves differently in the two cases. Two
additional traces appear on the left plots (c,e) corresponding to the points
on the $3d$ soliton branch.

The densities on the $j=3$ soliton branch start off as Bloch waves with three
peaks in the absence of a lattice, but due to it being incommensurate, its
starts to get more localized with the introduction of the lattice [1]. As the
lattice deepens, the right tip labelled $3dr$ continues to terminate at
$\delta\phi=3\pi$ and have three nodes as it did without a lattice. As the
branch shoots out and detaches from the main branch, the emergent left tip,
labelled $3dl$ and the corresponding density shown in dotted green line,
displays qualitatively different behavior with only two density minima.
Density at an intermediate point, $3dc$ shown in dashed magenta, indicates a
morphing from three density minima at the right tip branch to only two at the
left. But, the left tip does not feature nodes as it does not terminate at a
phase multiple of $\pi$.

###  V.2 q=3

We next examine the case of $q=3$ lattice periodicity in Fig. 5 at fixed
$g=-15$. We find counterparts of most of the features we noted above for
$q=2$, adjusting for the change of the lattice period. However, since the
lowest commensurate branch has a higher index $j=3$, we need a larger
magnitude nonlinearity to access them. Those shared features includes the
detachment of both the main branch and the $j=3u$ branch from the vertical
axis, the subsequent formation of a persistent gap, as well as, the extension
of the $j=4d$ branch beyond the tip of the shrunken main branch. However,
there is a new feature here not seen with $q=2$: At higher potential strength,
the upper split of the $j=3$ branch splits further to create an additional
branch we label $3u^{\prime}$, with distinctive features that we describe
below.

The $j=4$ branch does not visibly split, or at least our simulations did not
converge to two separate branches. However, based on the symmetry feature that
one density maximum coincides with a lattice minimum, if there is a split we
have converged on the lower branch and so we label it $4d$. With increasing
lattice depth, this branch behaves exactly like the $3d$ branch for $q=2$,
with the density starting off as a uniform Bloch wave and getting
progressively localized. Then, as it extends beyond the main branch, there is
gradual morphing of the density from having $4$ nodes at the right tip to
having only $2$ at the left tip, for respective bare phases of $4\pi$ and
$2\pi$.

As the left tips of the detached main branch and the $3u$ branches approach
each other, the gap never closes, but the densities at the their tips,
labelled $ml$ and $3ul$ and plotted in thick red lines, tend to almost
identical Bloch waves with three maxima, analogous to their $q=2$ counterpart
in Fig. 4. At $V_{0}=2$ and above, the $3u$ branch splits further, creating
the new branch $3u^{\prime}$ with a density profile distinctly different from
that of a Bloch wave. The densities at its left and right tips, plotted in
dotted green and dashed cyan, display localization but still with three
density peaks; the symmetry properties are that of an upper branch with none
of the peaks lining up with a lattice minimum. The tips do not reach multiples
of $\pi$, and hence they do not have nodes in their densities. They just get
very close to having nodes, particularly at the right tip when the lattice
depth is $V_{0}=2.5$ in Fig. 5.

##  VI Sensitivity of Phase

In the previous section we observed that for both $q=2$ and $q=3$, the upper
of the respectively commensurate $j=2$ and $j=3$ branches detach from the
vertical axis, with the left tip tending towards that of the also detached
main branch. The tips never meet even as the lattice depth increases; yet the
densities seem mutual continuations, suggestive of a continuous morphing. This
raises the question whether the gap is real, or just an artifact of numerical
convergence. There are two reasons to suggest that it is indeed physical: The
gap persists as we increase the lattice depth, even as the upper solitonic
branch contracts to eventually disappear as seen for the $2u$ branch in Fig.
4. More significantly, we notice that the phase has different behavior at the
tips of the two relevant branches. In Fig. 4, tip $ml$ of the main branch
corresponds to a solution with phase having half the period as that for the
solution that corresponds to the tip $2ul$ of the upper $j=2$ branch; both
branches are shown in thick red lines for easy comparison; the same pattern is
seen in Fig. 5 with their counterparts $ml$ and $3ul$.

However, the phase issue is a bit more subtle. In our simulations we determine
the solutions along the branches progressively, starting from a specific point
in the spectrum, using that solution to find the next adjacent solution and so
on. We find that the density of the eigenstates are insensitive to which
initial solution we converge from. On the other hand, the phase can be quite
susceptible to the path we take. We illustrate this in Fig. 6 where we assume
lattice periodicty $q=2$ and strength $V_{0}=1.3$, as in the center column in
Fig. 4: On the left column, Fig. 6(a,c,e), we plot the spectrum, density and
phase respectively, at specific points marked with filled circles, starting at
the tip of the $3d$ soliton branch. On the right column, Fig. 6(b,d,f), we
plot their counterparts starting at the tip of the $4u$ branch instead.

When we start at the tip of the $3d$ branch, we progress from localized
solitonic solutions having density profiles with three nodes. They lose the
nodes as we progress towards the main branch; shown in dotted lines of non-
uniform modulation in panel (c). On reaching the main branch, we follow it
down until we reach the tip labelled $ml$; the corresponding density is shown
in thick red line and is a Bloch wave, consistent with Fig. 4. We then follow
the main branch back up all the way to the tip of the $4d$ branch. This path
differs from the path of approach; the inset in panel (a) shows a very small
gap that exists between the left terminal segment of the $3d$ branch and the
main branch in leading down to the tip $ml$. As we follow the main branch up,
the densities remain Bloch waves with two density maxima, shown in thin solid
lines in panel (c). But as we approach the $4d$ branch, the two peaks flatten
out and each develops a new minimum; so the full density profile develops four
minima and eventually four nodes at the tip of the $4d$ branch.

In starting from the $j=4$ branch, for numerical convergence, we have to
initiate in the $4u$ branch and progress along it until the solution drops
down to the $4d$ branch, and then we continue down to the tip, $ml$. We then
converge back up along that branch all the way to the tip of the $4d$ branch
and plot the densities and the phases along the path towards $4d$ at exactly
the same points as in the previous case, only leaving out the two points on
the $3d$ branch, not being present in this path as shown in panel (b). The
densities in panels (c) and (d) for the corresponding points on the spectrum
along the two paths of convergence are identical.

However, when we plot the phase in panels (e) and (f), we find that the phases
are completely different for corresponding spectral points along the two
different paths of convergence: The right column has half the period of the
left for most of the points (only near the tip of the $4d$ branch, the phase
on the left path morphs significantly leading up to phase slips at the nodes
there). The difference is particularly notable for the point at the bottom tip
of the main branch, $ml$, highlighted in thick red lines in Fig. 6(e) and (f).
This is the upper edge of the gap between the detached $2u$ and main branch in
Fig. 4.

In Sec. V we noted a difference in the phase of the states that correspond to
the top and bottom of that gap, evident in the columns corresponding to
slightly higher lattice depths $V_{0}=1.5$ and $1.7$ in Fig. 4. At those
lattice depths, the $3d$ branch has already detached from and extended beyond
the main branch, so we could only approach the bottom tip $ml$ of the main
branch along the path starting at $4u$ corresponding to the right column
panels in Fig. 6. Now, from the evidence of Fig. 6 we can conclude that the
difference in the phase we observe for above and below the persistent gap
between the points $ml$ and $2ul$ is actually an artifact of the different
ways we approach those points. This is confirmed by the observation that in
the middle column Fig. 4, for $V_{0}=1.3$, the phase has the same period for
the points above and below that gap. Here the $3d$ branch is still attached to
the main branch, and we converged to the point $ml$ along a path starting at
the tip of the $3d$ branch corresponding to the left panels in Fig. 6; the
phase period matches that at the bottom edge $2ul$ of the gap which we
converged to from the tip of the $2u$ branch.

##  VII Interplay of nonlinearity and lattice

We have so far considered the effects of varying the lattice while keeping the
nonlinearity fixed. As we noted in our earlier study of positive nonlinearity
[1], the two factors have complementary effects and their interplay defines
the spectrum and the states. We re-examine some of the features noted earlier,
as we now vary the nonlinearity at fixed lattice depth. In Fig. 7, we use
lattice periodicity $q=3$, assume a weak lattice $V_{0}=0.1$ in panels (a,b)
and a stronger lattice $V_{0}=2.2$ in panels (c,d) and plot the spectrum as a
function of (i) the bare phase at fixed nonlinearity (left panels) and (ii)
the nonlinearity with bare phase fixed at $\phi=1.4$ (right panels).

As we would expect, the chemical potential monotonically decreases as the
nonlinearity becomes more negative. Comparing each pair of plots, (a,b) and
(c,d), at fixed lattice depth provides different perspectives. In both cases,
the gap between soliton branches of same index, for example $3u$ and $3d$,
appears insensitive to the change of nonlinear strength, with the chemical
potential decreasing at about the same rate. This is in contrast to the mutual
gaps between the main branch $m$ and soliton branches associated with
different $j$ indices which diverge with stronger interactions. Curiously, in
panel (d) we observe the newly emergent $3u^{\prime}$ branch gradually
approach the $3d$ branch as the interaction strengthens.

Figure 7: (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
lattice periodicity is fixed at $q=3$, with the lattice depth $V_{0}=0.1$ for
the top row and $V_{0}=2.2$ for the bottom row. The branches are labeled
according to our convention in Eq. (12). The vertical dotted lines mark the
values in each column that are fixed in the counterpart in the other column.

With both the lattice and the phase fixed in panels (b,d), as a function of
the nonlinearity, several of the branches seem to appear out of the ‘blue sky’
once the nonlinear strength reaches a sufficient strength. But, seen as a
function of phase at fixed nonlinearity in panels (a,c) we can understand most
of them as arising from the splitting of soliton branches as the lattice is
increased. Together these plots show that the strengths of both the lattice
and the nonlinearity work in tandem to create branches at any specific value
of the bare phase.

Figure 8: Plots of chemical potential as a function of bare phase with the
rows having fixed lattice depth $V_{0}$ and the columns having fixed
nonlinearity $g$, with their values shown in the plots to display their
interplay. The left block is for lattice periodicity $q=2$ and the right is
for $q=3$, the branches are labeled according to our convention in Eq. (12).
The two sections highlighted in gray are examined further in the next figure
(Fig. 9): The lower left shows the rejoining of the $1u$ branch with the main
branch, while the other one shows a three-way split of the $j=3$ soliton
branch.

Although in Fig. 7(a,c) we show snapshots only at two values of the lattice,
we tracked the evolution of the spectrum as a function of the bare phase at
other lattice depths not shown here. We found that (i) the $4d$ branch
detaches from the main branch, it flattens out and eventually spans the phase
between $2\pi$ and $4\pi$; (ii) the $3u^{\prime}$ splits from the $3u$ forming
a fork on the right, before separating completely; and (iii) with further
increasing of the lattice strength, both of those branches shrink, and
eventually disappear, never closing the gap with the main branch.

In Fig. 7(b,d) the spectrum as a function of the nonlinearity at fixed phase
and lattice strength gives a different perspective. At weak nonlinearity, only
the main branch remains, underscoring that the soliton branches emerge and
extend out from the parabolic dispersion curve as the nonlinearity is
strengthened. The left extremes of these branches in panels (b,d) of Fig. 7
mark the threshold nonlinearity at which the corresponding spectral lines,
when plotted as a function of $\delta\phi$, reach the value of
$\delta\phi/(2\pi)=1.4$ fixed in these panels.

As a different perspective on our discussions in the previous sections, even
when the spectrum is plotted as a function of the nonlinearity in Fig. 7(d), a
gap seems to emerge between the main branch and the $3u$ branch. Although the
gap is rather small, it is interesting that the $4d$ branch extends into
regimes of lower nolinearity beyond the termination of the main branch and
$3u$ branch, precisely where those two branches fail to meet. This is
consistent with what we observe in the complementary plot in Fig. 7(c) where
that gap is more prominent.

##  VIII Anomalous spectral features

The spectrum for negative nonlinearity displays several unusual features that
we did not observe with positive nonlinearity. We have noted some of these
already, and now we will identify a few others, and also explain the origin of
such features. In order to track the interplay of the lattice and the
nonlinearity, in Fig. 8 we plot the spectrum as a function of the bare phase
in a grid, where we vary the lattice strength and the nonlinearity in turn
while keeping the other fixed, for two different lattice periodicities $q=2$
and $q=3$.

This figure helps explain the curious detachment of certain branches from the
vertical axes, accompanied by a similar detachment for the main branch, which
we observed earlier in Figs. 4 and 5. We confirm in Fig. 8 that this occurs
only for soliton branches that are commensurate with the lattice period, for
example, $j=2$ for $q=2$ and $j=3$ for $q=3$. More significantly, the relevant
commensurate branch needs to be the lowest one still attached to the main
branch or be right below the main branch after having detached from it. If the
lattice were absent the criterion from Sec. III suggests the range
$-(q+1)^{2}\pi/2<g<-(q-1)^{2}\pi/2$, but the lattice would shift that. As the
magnitude of the nonlinearity is strengthened, branches with higher values of
$j$ start ‘sliding off’ and detaching from the main branch. For $q=2$ with no
lattice and $V_{0}=0$, soliton branches $j=1,2$ and $3$ slide off the main
branch as the nonlinear strength $g$ gets more attractive from $g=-3$ to $-8$
to $-16$. Likewise, for $q=3$ with no lattice at $V_{0}=0$, soliton branches
$j=2,3$ and $4$ and slide off the main branch as the nonlinear strength $g$
gets more attractive from $g=-12$ to $-16$ to $-30$. When the lattice is
introduced, we observe the main branch detach from the vertical axis along
with the $j=q$ upper branch, but only when nonlinearity is in the range to
have the two branches be adjacent, as with $g=-8$ column for $q=2$ and $g=-16$
column for $q=-3$. At stronger or weaker nonlinearity this does not occur, for
example, at $g=-3$ or $-16$ for $q=2$ and at $g=-12$ or $-30$ for $q=3$.

Comparison among the different columns also clarifies why this happens. We
notice that for commensurate soliton branches which are still attached to the
main branch, as the lattice splits the branch, the upper branch forms a hook
like structure with the main branch, for example $g=-3$, $V_{0}=0.1$ for $q=2$
. This is exactly what happens even when the commensurate soliton branch is
detached; when the lattice splits it, the upper branch tries to form a similar
hook like structure with the main branch. We can see this as the lattice
strength is increased from $V_{0}=0.3$ to $0.5$ for $q=2$ and $g=-8$, and
likewise when increasing from $V_{0}=0$ to $0.5$ for $q=3$ and $g=-16$. In a
variation of this same behavior for $q=3$ and $g=-12$, this occurs when the
commensurate branch is the lowest branch still attached to the main branch
when the lattice is absent: As the lattice is introduced, at $V_{0}=0.5$ we
see that the main branch has detached from the vertical axis and inches
towards the $3u$ branch to form a hook.

One feature persists in these anomalous hook structures, which is that the
main branch never seems to join up with the upper of the split commensurate
branches, as we have noted earlier. That remains the case for example with
$q=2,g=-8,V_{0}=0.5$ and for $q=3,g=-12,V_{0}=0.5$ in Fig. 8 This is a point
of contrast with such hook structures that appear when the main branch does
not detach from the vertical axis as for $q=2,g=-3,V_{0}=0.1$ in the same
figure.

In Fig. 8 we use gray shading to highlight two other anomalous spectral
features: First, for $q=2$ and $g=-3$, increasing the lattice strength from
$V_{0}=0$ to $V_{0}=0.6$ causes the $j=1$ branch to split; the gap between the
branches widen, until the upper branch, $1u$ actually rejoins the evolved main
branch. Secondly, for $q=2,g=-16$ and $V_{0}=1.5$, we find the $j=3$ branch
develops multiple splittings that we label $3d,3d^{\prime}$ and $3u$. The $3d$
branch disappears as the lattice is strengthened to $V_{0}=3$, and the two
branches $3u$ and $3d^{\prime}$ remain detached from both the main branch and
the vertical axis. Our labelling of these branches is based their symmetry
properties as defined earlier in Sec. V. This is illustrated in Fig. 9(a1,a2),
where the densities for the $3d$ and $3d^{\prime}$ branches each have one
minimum that coincides with a lattice minimum, characteristic of the
energetically lower of a split branch. In contrast, that feature is absent for
the density of the $3u$ branch shown in Fig. 9(a3), which additionally does
not have any of its density maxima aligned with a lattice minimum,
characteristic of energetically higher of a split branch.

We can surmise that the two branches $3d$ and $3d^{\prime}$ split at what
emerged as an inflection point at a lower lattice strength. We can see
something similar occurring for the $j=4$ branch in Fig. 8 for $q=3$ at
lattice depth of $V_{0}=0.9$. At higher lattice depth, $V_{0}=0.97$, it is
quite likely that there are two separate branches that have split at that
inflection point, but we could only converge to the right branch as shown.

In the lower panels (b-d) of Fig. 9, we examine in further detail the
rejoining of the $1u$ branch highlighted in the lowest left panel of Fig. 8.
We plot the densities of a few points on the main branch as well as along the
$1u$ branch as it rejoins the main branch. We label the main branch as $2d$ in
this figure, since Fig. 8 shows that at these lattice depths, the right
termination of the main branch is the $2d$ branch. Before joining, at
$V_{0}=0.53$, the densities (shown in solid lines) along the $1u$ branch have
one minimum, as we would expect with $j=1$, while the densities along the main
branch (shown in dotted lines) are characteristic of Bloch waves with two
maxima, as it should be with $q=2$. However we note at the left end of the
$1u$ branch still on the vertical axis, the density, shown in thick red line
in panel (b2), is noticeably flattened. This remains so as the branch detaches
from the vertical axis at $V_{0}=0.55$. But, when it joins the main branch at
$V_{0}=0.60$, the flattened peak morphs to develop a new minimum, heralding
that it is now on the main branch. But this point of joining is not a Bloch
wave and is markedly nonuniform in its modulation, having characteristics of
both branches that meet here.

Figure 9: The top row (a1,a2,a3) shows the densities for the three split
branches for $j=3$ for $q=2,g=-16,V_{0}=1.5$, highlighted in Fig. 8. The three
traces in each panel represent the densities at the tips of each branch and
for a point in the middle. The lower panels provide details of the rejoining
of the $1u$ branch for $q=2,g=-3,V_{0}=0.6$, also highlighted in Fig. 8. The
left panels (b1,c1,d1) show the progression of the $1u$ branch as it detaches
from vertical axis and joins the main branch. The right panels (b2,c2,d2) show
the evolution of the densities at the points marked on the left panels: The
dotted lines mark Bloch waves corresponding to the $2d$ branch. The solid
lines correspond to the $1u$ branch, with the thick red one marking the left
tip of that branch. Figure 10: The measure of instability of the eigenstates
as a function of the strength ($\pm V_{0}$) is gauged by the imaginary part of
the normal modes ${\rm Im}(\omega)$ of Bogoliubov fluctuations. The three rows
have lattice periodicity $q=1,2,$ and $3$ respectively, for a low nonlinear
strength $g=-1$. The leftmost panels show a _representative_ spectrum at a
specific potential indicated, with vertical lines indicating the bare phase
values, $\delta\phi/(2\pi)=0.5,1,$ and $1.5$, for which the modes are plotted
in the other three columns. The upper and lower branches are plotted
separately, except for the $j=2$ branch for $q=3$ in bottom row, since that
branch was not observed to split. Non-vanishing ${\rm Im}(\omega)$ marks
instability.

##  IX Stability of States

We will now examine the dynamical stability properties of the solutions we
found, by considering small perturbation around the mean field stationary
states:

$\displaystyle\psi(\theta,t)=\psi_{0}(\theta)+\delta ue^{-i\mu t}e^{-i\omega
t}+\delta v^{*}e^{-i\mu t}e^{i\omega^{*}t}$ (13)

and then solve the Bogoliubov equations [3] for the normal modes of the
fluctuations.

$\displaystyle(H_{0}+2g|\psi_{0}|^{2}-\mu)\delta u+g\psi_{0}^{2}\delta
v=\omega\delta u$ $\displaystyle-(H_{0}+2g|\psi_{0}|^{2}-\mu)\delta
v+g(\psi^{*}_{0})^{2}\delta u=\omega\delta u$ (14)

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.

Figure 10 shows the stability properties for a relatively weak nonlinear
strength of $g=-1$, for three different lattice periodicities $q=1,2,$ and
$3$, and the lowest soliton branches $j=1,2,$ and $3$ along the respective
rows. The left column shows the spectrum for each periodicity at a fixed
lattice depth. The remaining three columns plot the imaginary part ${\rm
Im}(\omega)$ of the eigenmodes of the Bogoliubov fluctuations as the lattice
depth is varied. Dotted lines on the spectrum mark the bare phase values
$\delta\phi/(2\pi)=0.5,1,1.5$ for which the modes are plotted; they correspond
to the tips of branches, the solutions with nodes. The vast majority of the
modes, for every case shown are purely real with ${\rm Im}(\omega)=0$ which is
represented by the horizontal line present in all the plots. In most of the
plots, there are a few modes with non-zero ${\rm Im}(\omega)$, always the
lowest one or two modes.

With no lattice present, all of the solutions displayed are stable with all
the modes being real, consistent with conclusions in previous studies for
repulsive interaction [52, 1]. As the lattice is strengthened, the lower
branches, $j=1d,2d$ and $3d$ remain stable at least for weak lattices. The
only exceptions are the $3d$ branches for $q=1,2$ and the $j=2$ branch for
$q=3$, which remains unsplit as far as we could tell. Curiously, in the former
case the instability appears only for a certain range of lattice depth. This
is reminiscent of periodic patterns we observed for nonlinear stationary
solutions with a barrier, caused by the variation of the density at points of
transition of the potential [44]. We tested the stability up to lattice depths
of $V_{0}=20$ and the lower branches remained stable.

In contrast, the upper branches are unstable as soon as the lattice is turned
on, characterized by at least some nonzero ${\rm Im}(\omega)$. Their pattern
of variation with the lattice depth is diverse. A few key features are
apparent: As the lattice gets very deep the instability seems to decline;
there also appear to be some islands of stability at certain lattice depths,
analogous to the regimes of instability we observe for the lower branches. We
do not yet have an explanation for why those occur at those specific values,
and this deserves further investigation.

Our main observation regarding the dynamical stability behavior is that the
behavior for negative nonlinearity is the opposite of that for positive
nonlinearity [1], where we observed that the upper branches were generally
stable in the presence of the lattice. In fact, the behavior was qualitatively
identical as described above, except that the stability of the upper and lower
branches are switched. This underscores that the sign of the nonlinearity, as
in whether interaction is attractive or repulsive, directly impacts the
stability properties of the soliton branches.

##  X Conclusions and Outlook

Since the calculations with lattice are obtained numerically, we comment
briefly on some of the challenges. We find the solutions using Newton’s method
in Fourier space, where we use a finite number of basis states, which we have
described in our earlier paper [1]. Stronger nonlinearity requires a larger
basis, and curiously we found that negative nonlinearity requires more basis
states than positive nonlinearity of the same magnitude. We first find the
solutions with no angular velocity $\Omega=0$ and then increment $\Omega$
gradually to build the full spectrum. This has generally worked very well, but
when there are strong inflections or a jump in the branch, the method can fail
to find the continuation of a branch. Also, it is possible to miss branches
that may not terminate at a node. We tried to account for these limitations,
but despite our best efforts some spectral features or lines may be missing,
particularly at stronger nonlinearity which are computationally more
demanding.

Our examination of the negative nonlinearity induced by attractive interaction
complements our prior work with positive nonlinearity [1] and completes our
survey of the landscape of stationary states of ultracold atoms trapped in
quasi 1D ring shaped lattices. In the absence of an azimuthal lattice, the
analytical solutions display a convenient mapping as the sign of the
nonlinearity is changed. When the lattice is introduced, the solutions are
found to present symmetry and stability properties that are in many respects
the opposite between positive and negative nonlinearities, but not completely.
While certain features like the splitting of soliton branches by the lattice
and general impact of an angular velocity remain unaffected by the change of
sign of the nonlinearity, there are various other features that are specific
to attractive nonlinearity. The most prominent of those is the progressive
detachment and ‘falling off’ of the soliton branches from the main parabolic
one, leading to new ground states. When a solitonic branch with index
commensurate with the lattice periodicity is the lowest one still attached or
the highest one detached and energetically right below the main parabolic
branch, these branches are found to separate from the vertical axis and tend
to form a hook like spectral pattern. Among other spectral features observed,
soliton branches can undergo multiple splits and even rejoin after detachment.

In the case of attempted rejoining of detached commensurate branches with the
main parabolic branch, we found certain persistent gaps. The phase of the
states at the edge of these gaps is found to be sensitive to the path of
convergence taken to approach those particular spectral points. This can be
directly translated to adiabatic evolution along different paths by
dynamically evolving the system parameters, like angular velocity and lattice
depth and interaction strength. Examining the path dependence of the phase
under such variations is part of our ongoing work.

The symmetry properties are not quite mutually opposite between positive and
negative nonlinearities, but they are distinct. Likewise the stability
features are generally complementary with respect to the behavior of the upper
and lower branches in a lattice-induced splitting. The localization properties
of the states are the same regardless of the sign of the nonlinearity, with
soliton branches that are incommensurate with the lattice periodicity having
solutions that tend to localize on introducing the lattice.

Together with our study of the case of repulsive nonlinearity [1] this paper
will provide a firm theoretical basis for exploring the dynamics of ultracold
atoms in ring shaped traps particularly in the substantially more interesting
case of having an azimuthal lattice. This system offers the possibility of
examining numerous interesting phenomena in a system that brings together
elements of quantum mechanics, topology, nonlinear dynamics and mesoscopic
condensed matter physics. The intrinsic description in terms of a NLSE also
broadens the possibility of applications to fiber optical systems as well. We
are currently working on applying our results to understand the dynamics of
coherent median in ring-shaped lattices, and we hope to motivate experiment
that will explore this very rich system.

###### Acknowledgements.

We gratefully acknowledge the support of the NSF under Grant No. PHY-2011767.
We thank Prof. Dominik Schneble, Hongyi Huang and Allison Brattley for
valuable discussions.

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