# Anomalous localization in spin-chain with tilted interactions

Arindam Mallick<EMAIL_ADDRESS>Instytut Fizyki Teoretycznej,
Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland  Jakub
Zakrzewski<EMAIL_ADDRESS>Instytut Fizyki Teoretycznej,
Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland  Mark Kac
Complex Systems Research Center, Uniwersytet Jagielloński, Kraków, Poland

###### Abstract

The localization properties of a disorder-free spin chain with inhomogeneous
interactions are studied. In particular, we consider interaction strength
growing linearly along the chain for systems with different interaction
ranges. Using exact diagonalization we find the participation ratio of all
eigenstates which allows us to quantify the localization volume in the Hilbert
space. Surprisingly the localization volume changes nonmonotonically with the
interaction range. The model for the infinite interaction range resembles the
Schwinger model of lattice gauge theory in staggered formalism. The model
studied may be implemented in state-of-the-art cold atomic devices and could
reveal hidden features in disorder-free confinement phenomena.

##  I Introduction

Localization properties or in general non-ergodicity at the quantum many-
particle level have been attracting physicists for more than half of a century
[1], as it breaks the conventional understanding of thermalization in
classical physics. One major branch deals with the random [2, 3, 1, 4] or
quasi-periodic disorder [5, 6, 7, 8] induced localization. Recently another
branch—localization without disorder—is catching a lot of attention. The
latter happens in the presence of energy constraints e.g. linear tilt (DC
field) or harmonic traps [9, 10, 11, 12, 13, 14, 15], gauge field induced
confinement [16, 17, 18, 19, 20, 21, 22]; in the presence of other species of
particles which effectively induce random environment [23, 24]; in presence of
flatbands [25]; etc. The study of long-range e.g. dipolar interaction effects
on such models has also a rich history [26, 27, 28, 29, 30, 31, 32, 33] with
recent progress being related to the experimental progress in cooling and
trapping of dipolar matter [34, 35, 36].

In this article, we introduce a one-dimensional (1D) spin chain model in which
the interaction strength is inhomogeneous and, in particular, increases with
the distance along the lattice. While typically the interaction strength
decays as a power law or exponentially with the distance [31, 37] we consider
the case where it grows linearly, not only with respect to the relative
distance but also with the explicit location of spins. In that way, it also
differs from the 1D Coulomb model in the continuum where interaction strength
grows with the relative distance between charge densities only [38]. By
numerical full diagonalization using the Quspin python library [39, 40] we
study the localization volume, i.e. the participation ratio (PR) in the
Hilbert space spanned by the eigenstates of spin-half $\sigma^{z}$ operator,
as a function of the range of interactions and their strength. Therefore our
study captures not only the near zero temperature case but also all
temperature cases. The nearest neighbor case of our model is similar to the
recently studied Hamiltonian in Ref. [41].

Stronger interactions turn the hopping or spin-flip less probable, and
Hamiltonian eigenstates appear more similar to the spin-aligned states or the
eigenstates of $\sigma^{z}$ operators. The alignment becomes more robust and
locally conserved as we move from left to right of the chain with linearly
growing interaction strength. This phenomenon is similar to the Hilbert space
fragmentation leading to the presence of conserved local operators [42, 43,
44]. Therefore a stronger interaction reduces the Hilbert space occupation,
the localization volume—as observed in our work and as reported recently [41].
This happens irrespective of the range of interactions. On the other hand, the
longer the range of interactions is, induces longer the correlations and one
expects to see always an increase in the localization volume. But,
surprisingly, we observe that the localization volume reduces upon increasing
the range and after an intermediate range, it again starts to grow. Such a
nonmonotonic behavior was not reported before in any clean (disorder-free)
lattice model. We explain this phenomenon by moving from a spin picture to the
particle occupation picture. We show that there is an interplay between the
fragmentation induced by two-body interactions and an effective tilt potential
induced by longer-ranged interaction.

The paper is organized as follows. We introduce our model and measures studied
in Section II. Section III is dedicated to our numerical full
diagonalizations, the extraction of relevant results for clean systems with
various ranges of interactions, and an explanation of the results. The section
IV is dedicated to the effect of quasi-periodic additive disorder on clean
systems. We conclude in section V. Appendix A describes the connection of our
clean Hamiltonian with the lattice gauge theory.

##  II Model

Here we define the spin-half Hamiltonian on a 1D finite chain with various
ranges of tilted ZZ interaction

$\displaystyle\mathcal{H}_{\mathcal{R}}=-t\sum_{n=0}^{L-2}(\sigma^{+}_{n}\sigma^{-}_{n+1}+\sigma^{-}_{n}\sigma^{+}_{n+1})$
$\displaystyle+U\sum_{l=1}^{\mathcal{R}}\sum_{n=0}^{L-(l+1)}(n+l)\sigma^{z}_{n}\sigma^{z}_{n+l}$
(1)

where the range $\mathcal{R}$ can take any value from the integer set
$\\{1,2,3,\cdots,L-2,L-1\\}$. At $\mathcal{R}=L-1$ our model is similar to the
lattice Schwinger model of staggered spinless fermions after integrating out
the electromagnetic gauge degrees of freedom—see Appendix A for details. We
use the notation $\sigma^{+}_{n}$ = $\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)$ = $(\sigma^{-}_{n})^{\dagger}$,
$\sigma^{z}_{n}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ for all lattice site $n$. $U$ is the interaction
strength parameter while $t$ is the hopping amplitude put to unity from now
on.

Figure 1: Envelope plots of the histograms of the number of states
$\mathcal{N}(\varepsilon)$ as a function of $\varepsilon$ [Eq. (4)] for
different $\mathcal{R}$ as indicated in the panels, while $U=1.2$ and $L=18$
for all subplots. 100 bins are used during histogram analysis for all cases.
The insets are the same plots as in the main panels but on a log-linear scale.

Note that the interaction strength in the model (1) depends not only on the
relative distance $l=(n+l)-n$ between spins but also depends on the spins’
location. The Hamiltonian possesses two obvious symmetries: It commutes with
and hence dynamically conserves associated eigenvalues of (i) total
magnetization operator $\sum_{n=0}^{L-1}\sigma^{z}_{n}$ and (ii) product of
all spin-flip operators $\prod_{n=0}^{L-1}\sigma^{x}_{n}$. We consider an even
number of lattice sites and only zero magnetization sector
$\sum_{n=0}^{L-1}\sigma^{z}_{n}=0$ such that the total number of spin-up = the
total number of spin-down. This corresponds to the maximum Hilbert space
dimension compared to other magnetization sectors, and it is similar to the
half-filled spinless fermions. The zero magnetization sector can be divided
into two mutually orthogonal vector spaces concerning the $\mathbb{Z}_{2}$
symmetry induced by the product of spin-flip operators: they consist of
“linear superposition” of $\prod_{n=0}^{L-1}\sigma^{z}_{n}$ eigenstates which
are either symmetric having eigenvalue $+1$ or antisymmetric having eigenvalue
$-1$ under the action of $\prod_{n=0}^{L-1}\sigma^{x}_{n}$. The total number
of basis states $\ket{s_{j}}$ in the symmetric sector is equal to that in the
antisymmetric sector. Therefore the Hilbert space dimension of one of the
sectors is

$\displaystyle\mathcal{D}=\frac{L!}{2[(L/2)!]^{2}}\;.$ (2)

We consider the symmetric sector only while analyzing the clean system. For
the system size $L=18$ and $16$ the $\mathcal{D}$ = 24310 and 6435,
respectively. We numerically diagonalize the full Hamiltonian in this sector
for different $\mathcal{R}$, $U$, and for different chain lengths $L$ with
open boundary conditions. We concentrate mainly on the second participation
entropy [45] for the eigenvector $\ket{\psi_{E}}$ at each eigen-energy $E$

$\displaystyle\mathcal{S}(E)=-\ln\left(\sum_{j=1}^{\mathcal{D}}\left|\langle
s_{j}|\psi_{E}\rangle\right|^{4}\right),$ (3)

calculated in the $\\{\ket{s_{j}}\\}$ basis defined above. For mutually
unbiased basis $\\{\ket{s_{j}}\\}$ and the set of eigenvectors [46] one may
expect that for a fully ergodic system with equal probability amplitude at
$\ket{s_{j}}$ for all $j$, $\mathcal{S}(E)=\ln\mathcal{D}$. While for a
nonergodic system with singly occupied $\ket{s_{j}}$ state $\mathcal{S}(E)=0$.
This implies $\mathcal{S}(E)/\ln\mathcal{D}\in[0,1]$. Therefore the
$\mathcal{S}(E)$ captures the localization volume in symmetric spin-space.

##  III Numerical results

To take care of the energy dependence of our problem we follow the standard by
now procedure [47] and scale the energy eigenvalues, $E$, as

$\displaystyle\varepsilon=(E-E_{\min})/(E_{\max}-E_{\min})\in[0,1]\;.$ (4)

Figure 1 shows the number of states at different energies with the variation
of range for $L=18$ and $U=1.2$. Two observations follow. Firstly, with the
increase of the range of interactions, the maximum of the density shifts first
to smaller $\epsilon$ values then it comes back near to the center again and,
for still larger $\mathcal{R}$ moves towards right $\varepsilon=1$. Secondly,
for intermediate $\mathcal{R}$ distinct peaks appear in the $\mathcal{N}$
profile which is a clear sign of a Hilbert space fragmentation-like effect.
For longer $\mathcal{R}$ the peak heights as well as the number of peaks in a
distribution reduce with an increase of $\mathcal{R}$.

Figure 2: Participation entropy $\mathcal{S}/\ln\mathcal{D}$ averaged over the
energies within each bin in energy histogram [Fig. 1], as a function of scaled
energy and range. $U=1.2$, $L=18$.

Figure 3: (a) Participation entropy
$\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$, i.e. $S/\ln\mathcal{D}$ averaged
over the set $\mathcal{Z}$ [Eq. (5)], as a function of $U$ and $\mathcal{R}$.
$L=18$. (b) $\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$ for different system
sizes
$L=14~{}(\mathcal{D}=1716),16~{}(\mathcal{D}=6435),18~{}(\mathcal{D}=24310)$,
and different $U$.

To analyze the energy-resolved participation entropy we consider 10 bins in
$\epsilon$. Note, parenthetically, that we consider a disorder-free system so
no disorder average is possible. That explains some of the fluctuations
observed in different figures. We average the entropy over all eigenvalues
within each bin. Figure 2 depicts the result which shows maximum $\mathcal{S}$
near the maximum peak in Fig. 1 for any $\mathcal{R}$. Interestingly for
almost all energies at some intermediate values of $\mathcal{R}$ the
localization volume in spin Hilbert space i.e. the the participation entropy
is lower compared to the shorter and longer ranges.

Next, we focus on the peak density of states by considering the following set
of eigenvalues

$\displaystyle\mathcal{Z}=\\{\varepsilon:\ln[\mathcal{N}(\varepsilon)]\geq
0.9\max\\{\ln[\mathcal{N}(\varepsilon)]\\}\\}\;.$ (5)

The natural logarithm is taken to smoothen the distributions—compare the
insets and main plots in Fig. 1 for details. The participation entropy
averaged over all eigenstates that belong to $\mathcal{Z}$, labeled as
$\mathcal{S}_{\mathcal{Z}}$ is plotted as a function of $U$ and $\mathcal{R}$
in Fig. 3. The nonmonotonic or anomalous behavior of the localization volume
as a function of range for almost all $U$ and all system sizes is visible. The
intermediate range $\mathcal{R}=\mathcal{R}_{c}$ for which the minimum occurs
shifts to larger values with an increase of system size $L$ at a fixed $U$. On
the other hand for a fixed $L$ increase of $U$ shifts the $\mathcal{R}_{c}$ to
smaller values. Larger system size $L$, at a fixed range $\mathcal{R}$ and
fixed $U$ leads to smaller $\mathcal{S}_{\mathcal{Z}}$ indicating that the
behavior observed is not due to the small system size.

###  III.1 Localization transition analysis

We now compare the participation entropies for different system sizes in units
of $\ln\mathcal{D}$ to extract possible system size independent crossing
points similar to the analysis done in [45].

Figure 4: Plots (a)-(d): $\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$ as a
function of interaction parameter $U\geq 0.05$ for various $L$ and
$\mathcal{R}$. The crossing point at each subplot is associated with the
ergodic to nonergodic transitions. (e) The transition interaction parameter
$U_{c}$ as function of range extracted from different $L$ combinations. (f)
Averaged $U_{c}$ over all combinations in subfigure (e).

Figure 5: (a)-(c) Spectral gap ratio $r$ averaged over the set $\mathcal{Z}$
as a function of $U$ for various $L$ and $\mathcal{R}$. 100 bins are used for
all cases.

We plot $\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$ as a function of
interaction parameter $U$ for different ranges and system sizes in Fig.
4(a)-(d): only four $\mathcal{R}$ cases are shown for convenience. The
crossing point for different $L$ appears approximately at the same point
$U=U_{c}$ for any fixed $\mathcal{R}$. A closer inspection reveals that it
also shows a nonmonotonic behavior with $\mathcal{R}$. For small $\mathcal{R}$
the crossing point happens for smaller values of $U$, it shifts to larger $U$
at intermediate ranges, and for longer ranges it again moves to smaller $U$
values. The crossing points are associated with the ergodic to nonergodic
transitions [45]. We conjecture that the crossing point approaches $U\to 0$
when $\mathcal{R}\to\infty$ which is possible at $L\to\infty$ which signifies
the appearance of nonergodic behavior for arbitrarily small $U$ in massless
clean Schwinger like models. To calculate the value of $U_{c}$ for each
$\mathcal{R}$ we consider three crossing points of three pairs of curves: (i)
$\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$ for $L=14$ and $L=16$ when
$\mathcal{R}\leq 13$, (ii) $\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$ for
$L=14$ and $L=18$ when $\mathcal{R}\leq 13$, and (iii)
$\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$ for $L=16$ and $L=18$ when
$\mathcal{R}\leq 15$. Numerically the crossing point is defined as the data
coordinate $(U,\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D})$ where the absolute
value of the difference of $\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$ for two
different $L$ is minimum. The result is depicted at Figs. 4(e)-(f) which
clearly captures a nonmonotonic behavior with $\mathcal{R}$. For stronger $U$
we have shown the participation entropy plots in the appendix B.

The anomalous effect is also visible in the spectral gap ratio

$\displaystyle
r_{\alpha}=\frac{\min\\{\varepsilon_{\alpha+1}-\varepsilon_{\alpha},\varepsilon_{\alpha}-\varepsilon_{\alpha-1}\\}}{\max\\{\varepsilon_{\alpha+1}-\varepsilon_{\alpha},\varepsilon_{\alpha}-\varepsilon_{\alpha-1}\\}}$
(6)

where $\alpha$ is the index of the sorted eigenvalues in ascending order.
Figure 5 shows the spectral gap ratio $r$ averaged over all $\alpha$ for which
$\varepsilon_{\alpha},\varepsilon_{\alpha\pm 1}$ belong to the set
$\mathcal{Z}$ as a function of $U$ for different ranges. We labeled it as
$r_{\mathcal{Z}}$. The crossing points $U_{c}$ in Figs. 4(e)-(f) correspond to
the $r_{\mathcal{Z}}$ values corresponding to the entrance of the system from
the ergodic regime to the nonergodic regime.

###  III.2 Discussion

To explain the anomalous localization behavior, it is convenient to switch
from the spin language to an equivalent spinless fermions description. We
define the particle number operator at site $n$ as
$\hat{N}_{n}=\sigma^{z}_{n}+\frac{1}{2},$ relating the spin-up (spin-down)
state to the presence (absence) of a particle. Only single occupancy at a
lattice site is possible. The $\sigma_{n}^{\pm}$ operators are transferred
into creation/annihilation operators for particles and the first line in (1)
gives the standard kinetic energy term (describing shopping). The interactions
part becomes

$\displaystyle\mathcal{H}_{\mathrm{in}}=$
$\displaystyle~{}U\sum_{l=1}^{\mathcal{R}}\sum_{n=0}^{L-(l+1)}(n+l)\sigma^{z}_{n}\sigma^{z}_{n+l}$
$\displaystyle=$
$\displaystyle~{}U\sum_{l=1}^{\mathcal{R}}\sum_{n=0}^{L-(l+1)}(n+l)\hat{N}_{n}\hat{N}_{n+l}$
$\displaystyle-\frac{U}{2}\sum_{l=1}^{\mathcal{R}}\sum_{n=0}^{L-(l+1)}(n+l)\big{(}\hat{N}_{n}+\hat{N}_{n+l}\big{)}$
(7)

where we neglected the additional constant interaction energy term. The first
term above is the two-body interaction operator. The second line strongly
resembles the tilted potential present in recent studies of disorder-free
many-body localization [9, 10, 11, 15, 48]. We use the term “tilted potential”
referring to the single particle potential term in the last line of Eq. (7)
and “two-body interaction part” to discuss the first term in (7). The tilted
interaction stands for the full ZZ-interactions term.

Two-body interaction part: The important fact about the two-body interacting
term is that any particle at right position $(n+l)$ has the same energy of
interaction with any particle at left site $(n+r)<(n+l)$ for various $r$ ($r$
can be negative or positive) while the relative distance between them is
$(l-r)$. Therefore the absolute position of the left particle does not
influence the two-body interaction term for a fixed position of the right
particle at any relative distance. Thus all the “left” particles belonging to
different pair configurations have the same interaction energy that may lead
to resonance features. The interaction energy only changes if the location of
the “right” particle changes. For an interaction range $\mathcal{R}$ any site
(except the sites $n=0,1,\cdots,\mathcal{R}-1$) possesses $\mathcal{R}$ number
of degenerate/resonating pairs. A particle at $n=1$ possesses only one pair
with $n=0$, a particle at $n=2$ possesses two such pairs with $n=0$ and $n=1$,
etc. Such different sets of pairs corresponding to particles at different
right locations $(n+l)$ fragment the whole accessible Hilbert space into
disconnected domains. The number of possible pairs for a particle at fixed
right position $(n+l)$ grows as $\mathcal{R}$ grows. Therefore even if the
two-body interaction causes Hilbert space fragmentation the growth of
individual fragmented domains could cause an increase in localization volume.
Moreover, there may exist some configurations of the particle distributions
that correspond to the presence of extra particles in between the lattice
sites that formed a pair. In that case, the extra particles will be connected
with the particle forming the pair and among themselves by shorter range
$<\mathcal{R}$ of interactions, and they change the energy of the
configuration. Such configurations induce more fragmentation in the Hilbert
space.

Tilted potential part: The tilted effective potential part in the particle
picture for a general $\mathcal{R}$ reads

$\displaystyle V_{\mathcal{R}}=$
$\displaystyle-\frac{U}{2}\sum_{l=1}^{\mathcal{R}}\sum_{n=0}^{L-(l+1)}(n+l)\big{(}\hat{N}_{n}+\hat{N}_{n+l}\big{)}$
$\displaystyle=$
$\displaystyle-\frac{U}{2}\sum_{l=1}^{\mathcal{R}}\Bigg{[}\sum_{n=0}^{l-1}(n+l)\hat{N}_{n}+\sum_{n=L-l}^{L-1}n\hat{N}_{n}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{n=l}^{L-(l+1)}(2n+l)\hat{N}_{n}\Bigg{]}$
$\displaystyle=$
$\displaystyle-U\sum_{n=0}^{L-1}(\mathcal{R}n)\hat{N}_{n}+\frac{U}{2}\sum_{n=0}^{\mathcal{R}-1}n(\mathcal{R}-n)\hat{N}_{n}$
$\displaystyle+\frac{U}{4}\sum_{n=0}^{\mathcal{R}}\left[2nL+n(n-1)\right]\hat{N}_{L-\mathcal{R}+n-1}\;.$
(8)

For convenience we denote $V_{\mathcal{R}}/U$ as
$\sum_{n=0}^{L-1}\mathcal{V}(n)\hat{N}_{n}$.

Figure 6: The tilted potential [Eq. (8)] as a function of lattice site $n$.
The plots are for various ranges in units of $U$ for fixed system size $L=18$.

In the last two lines of the above expression (8) while changing the double
index sum to a single index sum we have omitted the extra additive term =
$-(UL/4)$$\sum_{l=1}^{\mathcal{R}}l$ which comes from the total magnetization
or particle number conservation: $\sum_{n=0}^{L-1}\hat{N}_{n}=L/2$ for even
$L$. For detailed derivation see Appendix C.

Note, importantly, that, after representing the single-particle potential in
terms of single index sums it contains both the linear and square terms in
site number. That complicates its analysis. The presence of the linearly
tilted potential may induce another type of fragmentation in Hilbert space due
to an approximate conservation of the dipole moment as extensively discussed
already [43, 42]. Such a clean situation almost occurs for $\mathcal{R}=1$,
compare Fig. 6(a). However, the potential becomes quite different for a
general $\mathcal{R}$ forming first a triangular-like well with the minimum
shifting close to the center of the lattice and changing its shape completely
for longer ranges c.f. Fig. 6. The $V_{\mathcal{R}}$ effectively consists of
three parts: the central region, and two sets associated with the sites near
the two edges. Three different tilted parts become prominent for
$\mathcal{R}\geq 9$. Instead of linear tilt, they resemble two concave curves
(left and central) and one convex curve (right). For longer $\mathcal{R}\geq
13$ the central region grows and the potential resembles a single concave
quadratic curve.

Simultaneous effect of two-body interaction and the tilted potential:

Consider small $\mathcal{R}$ with a triangular well structure. The tilt of the
potential is quite strong (for $U$ of the order of unity) resulting locally in
a very strong localization. It has been shown elsewhere [49, 15, 48] that the
localization properties for sufficient tilt depend on the local effective
“electric field” inducing the tilt. Thus increasing $\mathcal{R}$ in this
region should enhance localization leading to smaller $U_{c}$ being needed for
a nonergodic behavior. As depicted in Fig. 4(e) the situation is the opposite.
Apparently, the effect of increased tilt is overcome by the increased range of
interactions that mix the dynamics. The trend is reversed for $\mathcal{R}\geq
5$, the two-body interaction term becomes dominant and it induces Hilbert-
space fragmentation lowering $U_{c}$. This qualitatively explains the
nonmonotonic behavior of $U_{c}$ as a function of $\mathcal{R}$ in Fig.
4(e)-(f).

Hilbert space fragmentation divides the whole system into several disconnected
parts, while correlations exist among states inside each fragmented part. An
analysis of possible degenerate pairs suggests that the presence of a positive
gradient part ($\partial\mathcal{V}/\partial n>0$) in tilted potential reduces
the number of states inside each fragmented part of the Hilbert space, i.e.,
number of disconnected parts increases. Beyond $\mathcal{R}\geq 7$ an increase
of $\mathcal{R}$ shrinks the positive gradient part in the tilted potential,
which increases the number of states inside each fragmented part and the total
number of fragmented parts reduces. This is visible from the growing subpeak
volumes in the number of state distributions in Fig. 1(b)-(c). This
qualitatively explains the growing participation entropy
$\mathcal{S}_{\mathcal{Z}}$ beyond $\mathcal{R}\geq 7$ with an increase of
$\mathcal{R}$, e.g. Fig. 3 for a fixed $U$, even if the system is in the
nonergodic regime.

##  IV The presence of quasi-periodic disorder

To understand the anomalous localization behavior better and to check its
robustness we add quasiperiodic disorder on the original clean Hamiltonian
(1). The new Hamiltonian reads

$\displaystyle\mathcal{H}_{\mathrm{dis}}=\mathcal{H}_{\mathcal{R}}+\frac{W}{2}\sum_{n=0}^{L-1}\cos(2\pi\beta
n+\pi\varphi)\sigma^{z}_{n}$ (9)

where $\beta$ is the golden ratio = $(\sqrt{5}+1)/2$ and the site independent
phase $\varphi$ is a random number picked from a uniform distribution
$\in[0,1]$. This model is related to the truncated Schwinger model in the
presence of disordered mass or chemical potential [Eq. (10)]. Note that such
onsite disorder breaks the spin-flip symmetry that was present in the clean
Hamiltonian. Therefore in the disorder case instead of considering only the
symmetric sector, we consider symmetric and antisymmetric blocks together
i.e., $\mathcal{D}=(L!)/[(L/2)!]^{2}$. The additive disorder term changes the
structure of the potential function c.f. $\mathcal{V}_{\mathcal{R}}$ [Eq.
(8)].

Figure 7: Participation entropy $\mathcal{S}_{\mathcal{Z}}$ which is averaged
over the set $\mathcal{Z}$ and over 150 random realizations of angle
$\varphi$, as a function of $W$ and $\mathcal{R}$. $L=14,U=1.2$.
$\mathcal{D}=3432$. In the last subfigure (d) we exclude the $W=0$ cases.

For computational convenience, we choose $L=14$ and $U=1.2$ and show the
participation entropies $\mathcal{S}$ as a function of disorder strength $W$
for variation of interaction ranges at Fig. 7. The shown plots of
$\mathcal{S}$ are averaged over the set $\mathcal{Z}$ [Eq. (5)] and scaled by
the logarithm of the Hilbert space dimension $\ln\mathcal{D}$.

Figure 7 reveals that the same anomalous behavior of localization volume
observed for the clean system is preserved at weaker disorder for all ranges.
Introducing disorder with $W=0.1$ on the clean system reduces the localization
volume or $\mathcal{S}$ for all ranges of interaction. For stronger disorder
system comes up with more exotic anomalous behaviors. For $\mathcal{R}\leq 8$
[Fig. 7(a)-(b)] we see an initial increase of localization volume with an
increase of $W$, and a decrease of it upon further increase of $W.$ For such a
fragmented system, the disorder induces a transition between different parts
of the fragmented Hilbert space, which results in growing correlations between
them and the corresponding enhancement of the localization volume. The peaks
occur when the disorder induces resonance. The increase of $\mathcal{S}$ is
more prominent for the ranges $\mathcal{R}\in\\{1,3,4,\cdots,8\\}$ when the
fragmentation induced localization is observed in Fig. 3. The required $W$
increases for longer $\mathcal{R}$ as the energy required to induce resonance
between different fragmented sectors increases. For a very strong disorder
system starts to show monotonically decreasing localization volume which is
similar to the many-body localization.

Figure 8: Spectral gap ratio averaged over the set $\mathcal{Z}$ and over 150
random realizations of angle $\varphi$, as a function of $W$ and
$\mathcal{R}$. $L=14,U=1.2$. We do not include $W=0$ case here.

The spectral gap analysis is shown in Fig. 8. For the longest
$\mathcal{R}=(L-1)$, the system is in the ergodic regime for smaller $W$ while
for larger $W$ it starts to enter into the nonergodic regime. For all other
ranges at some intermediate value of $W$, we see either an increase of
$r_{\mathcal{Z}}$ or a slowdown in the transition from maximum value to
minimum. We relate this phenomenon with the disorder-induced local ergodicity
or correlation between different fragmented sectors. For very large $W$ the
system is near to a many-body localized like regime.

##  V Conclusion

We have discussed the localization properties of a 1D spin model with
interaction strength dependent on the location of interacting spins for
different ranges of interactions. Our model captures a lattice Schwinger model
with a truncated range of interaction [see Appendix A for details]. The work
shows that the localization properties do not change monotonically with the
change of interaction range as one could expect naively. We explain this
nonmonotonic behavior as a competition between Hilbert space fragmentation and
quantum correlations. The effect remains unchanged in the presence of weaker
disorder potentials. Our work contributes to the important question of many-
body system properties under long-range interactions which has an increasing
physical relevance. It helps us to understand the nature of Coulomb-type
interaction as well as the lattice gauge theory models in one dimension. The
truncated or finite range of interactions may be associated with the presence
of a charge screening or shielding effect. We emphasize that the purpose of
this work is not to explore the details of many-body localization transition
from ergodic to nonergodic phases, but to report the nonmonotonic behavior of
the localization volumes.

In the future, we would like to extend to a higher dimensional lattice where
the nature of the interaction could change from linear to logarithmic or in
different ways [38]. Introduction of sublattice structure and gauge potential
flux can introduce different phenomena that need to be investigated.

It is known that the usual XXZ model is simulable by ultracold Rydberg atoms
interacting by van-der-Waals forces [50], where the atom states are mapped to
spin up and down states. Tuning the van-der-Waals coefficients for different
sets of two Rydberg atoms we could map the inhomogeneous linearly growing ZZ
long-range interactions while keeping the hopping term short-ranged [Eq. (1)].
One could use the recently implemented XXZ setups in cold 7Li atoms in optical
lattices [51] where anisotropic ZZ interaction parameters can be set by
locally controlling the applied magnetic field. It may also be possible to
implement our model using a trapped ion-based digital quantum simulator [52].

###### Acknowledgements.

The authors acknowledge the financial support from the National Science
Centre, Poland (grant 2021/03/Y/ST2/00186) within the QuantERA II Programme,
which has received funding from the European Union’s Horizon 2020 research and
innovation programme under Grant Agreement No 101017733. J. Z. acknowledges
also the partial support from the National Science Centre, Poland under grant
2019/35/B/ST2/00034. A. M. thanks Bitan De, Pedro Nicácio Falcão, and Adith
Sai Aramthottil for useful discussions.

##  Appendix A Similarity of our Hamiltonian with modified Lattice Schwinger
model

The lattice Schwinger Hamiltonian for the staggered spinless fermions with the
electromagnetic gauge potential reads [53, 54]

$\displaystyle\mathcal{H}_{\mathrm{sch}}=$
$\displaystyle-i\sum_{n=0}^{L-2}\left(\psi^{\dagger}_{n}e^{i\phi_{n,n+1}}\psi_{n+1}-\psi^{\dagger}_{n+1}e^{-i\phi_{n,n+1}}\psi_{n}\right)$
$\displaystyle+U\sum_{n=-1}^{L-1}\mathcal{E}^{2}_{n,n+1}+\sum_{n=0}^{L-1}m_{n}(-1)^{n}\psi^{\dagger}_{n}\psi_{n}\;.$
(10)

It controls the dynamics of spinless fermion and anti-fermion fields
$\psi_{n}$ coupled to $U(1)$ gauge boson fields $\phi_{n,n+1}$ which acts as
vector potential with the temporal gauge, and $\mathcal{E}_{n,n+1}$ is
corresponding electric field and conjugate momentum of $\phi_{n,n+1}$. In the
lattice representation fermion fields are located at lattice sites $n$ while
the gauge fields are at the links between sites. Let us consider the massless
case: $m_{n}=0$. The presence and absence of particles (or anti-particles)
form two possibilities at each site $n$, i.e. Hilbert space of dimension two.
The dynamical fermionic charge operator is defined as

$\displaystyle\mathcal{F}_{n}=\psi^{\dagger}_{n}\psi_{n}-\frac{1}{2}[1-(-1)^{n}]\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),\psi^{\dagger}_{n}\psi_{n}=\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)$ (15)
$\displaystyle\implies\mathcal{F}_{n=\text{even}}=\psi^{\dagger}_{n}\psi_{n},~{}\mathcal{F}_{n=\text{odd}}=\psi_{n}\psi^{\dagger}_{n}\;.$
(16)

In this notation, the particle can only sit at an even site and it is
annihilated by operator $\psi$, while the anti-particle can sit at the odd
site only and it is annihilated by operator $\psi^{\dagger}$. In the presence
of background static charges $q_{n}$ the discrete Gauss law reads

$\displaystyle\left(\mathcal{E}_{n,n+1}-\mathcal{E}_{n-1,n}-\mathcal{F}_{n}\right)\ket{\Psi}=q_{n}\ket{\Psi}\;.$
(17)

where $\ket{\Psi}$ are the only allowed physical states. If we set

$\displaystyle q_{n}=-\frac{(-1)^{n}}{2}$ (18)

the Gauss law turns

$\displaystyle\mathcal{E}_{n,n+1}-\mathcal{E}_{n-1,n}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)=\sigma^{z}_{n}\;.$ (21)

Jordan-Wigner transformation [55]

$\displaystyle\psi_{n}=\prod_{j=0}^{n-1}\left[2ie^{-i\phi_{j,j+1}}\sigma^{z}_{j}\right]\sigma_{n}^{-}$
(22)

with Pauli spin matrices: $\sigma^{+}_{n}=\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)$, $\sigma^{-}_{n}=\left(\begin{array}[]{cc}0&0\\\
1&0\end{array}\right)$ and $\sigma^{z}_{n}$ turns the hopping part of the
Hamiltonian to

$\displaystyle-\sum_{n=0}^{L-2}\left(\sigma^{+}_{n}\sigma^{-}_{n+1}+\sigma^{-}_{n}\sigma^{+}_{n+1}\right)$
(23)

which is the XX part of the usual XXZ spin chain model.

For our work, we modify the gauge field energy term of the lattice Schwinger
model with link $g_{n}$ dependent terms

$\displaystyle U\sum_{n=-1}^{L-1}g_{n}\mathcal{E}^{2}_{n,n+1}\;.$ (24)

If we choose $\mathcal{E}_{-1,0}=0$ from Gauss law [Eq. (21)] we obtain

$\displaystyle\mathcal{E}_{0,1}=\sigma^{z}_{0}\implies\mathcal{E}_{1,2}=\sigma^{z}_{1}+\sigma^{z}_{0}$
$\displaystyle\implies\cdots\implies\mathcal{E}_{n,n+1}=\sum_{j=0}^{n}\sigma^{z}_{j}\;.$
(25)

Therefore the energy term turns

$\displaystyle
U\sum_{n=0}^{L-1}g_{n}\mathcal{E}^{2}_{n,n+1}=\sum_{n=0}^{L-1}g_{n}\left(\sum_{j=0}^{n}\sigma^{z}_{n}\right)^{2}$
$\displaystyle=g_{0}\left(\sigma^{z}_{0}\right)^{2}+g_{1}\left(\sigma^{z}_{0}+\sigma^{z}_{1}\right)^{2}$
$\displaystyle+g_{2}\left(\sigma^{z}_{0}+\sigma^{z}_{1}+\sigma^{z}_{2}\right)^{2}+\cdots+g_{L-1}\left(\sum_{j=0}^{L-1}\sigma^{z}_{j}\right)^{2}\;.$
(26)

We use the fact the $(\sigma^{z}_{j})^{2}$ is proportional to the identity
and, hence can be treated as an additional constant and unimportant, therefore
upto these additional constants

$\displaystyle U\sum_{n=0}^{L-1}g_{n}\mathcal{E}^{2}_{n,n+1}=$
$\displaystyle~{}2\sigma^{z}_{0}\sigma^{z}_{1}\sum_{j=1}^{L-1}g_{j}+2\left[\sigma^{z}_{0}\sigma^{z}_{2}+\sigma^{z}_{1}\sigma^{z}_{2}\right]\sum_{j=2}^{L-1}g_{j}$
$\displaystyle+2\left[\sigma^{z}_{0}\sigma^{z}_{3}+\sigma^{z}_{1}\sigma^{z}_{3}+\sigma^{z}_{2}\sigma^{z}_{3}\right]\sum_{j=3}^{L-1}g_{j}$
$\displaystyle+\cdots+2\left[\sum_{n=0}^{L-2}\sigma^{z}_{n}\sigma^{z}_{L-1}\right]g_{L-1}\;.$
(27)

To produce our model Hamiltonian [Eq. (1)] with longest range of ZZ
interaction $\mathcal{R}=L-1$ we need to set

$\displaystyle\sum_{j=p}^{L-1}g_{j}=\frac{p}{2};~{}~{}p=1,2,\cdots,L-1$
$\displaystyle\implies
g_{1}=g_{2}=\cdots=g_{L-2}=-\frac{1}{2},~{}g_{L-1}=\frac{L-1}{2}\;.$ (28)

This is the gauge energy term in a lattice Schwinger Hamiltonian in the
presence of a defect at a boundary. Therefore for the interaction range
$\mathcal{R}<(L-1)$ the Hamiltonian [Eq. (1)] represents a truncated version
of the modified Schwinger Hamiltonian.

##  Appendix B Plots of $\mathcal{S}_{\mathcal{Z}}$ for the extended $U$

Figure 9: Plots (a)-(d): $\mathcal{S}_{\mathcal{Z}}/\ln\mathcal{D}$ as a
function of interaction parameter $U$ for various $L$ and $\mathcal{R}$ for
the clean system.

##  Appendix C DC field potential part [Eq. (8)] of interacting Hamiltonian
for clean system

Here we derive the form of the tilted potential term for general $\mathcal{R}$
after checking the expressions for a few special ranges.

####  C.0.1 $\mathcal{R}=1$

$\displaystyle V_{1}$
$\displaystyle=-\frac{U}{2}\sum_{n=0}^{L-2}(n+1)\big{(}\hat{N}_{n}+\hat{N}_{n+1}\big{)}$
$\displaystyle=-\frac{U}{2}\hat{N}_{0}-\frac{U}{2}(L-1)\hat{N}_{L-1}-\frac{U}{2}\sum_{n=1}^{L-2}(2n+1)\hat{N}_{n}$
$\displaystyle=\frac{UL}{2}\hat{N}_{L-1}-\frac{U}{2}\sum_{n=0}^{L-1}2n\hat{N}_{n}-\frac{UL}{4},$
(29)

where in the last term we used the total spin zero condition which implies a
half-filled lattice $\sum_{n}\hat{N}_{n}=L/2$.

####  C.0.2 $\mathcal{R}=2$

$\displaystyle V_{2}$
$\displaystyle=V_{1}-\frac{U}{2}\sum_{n=0}^{L-3}(n+2)\big{(}\hat{N}_{n}+\hat{N}_{n+2}\big{)}$
$\displaystyle=-\frac{U}{2}\left[2\hat{N}_{0}+3\hat{N}_{1}+(L-1)\hat{N}_{L-1}+(L-2)\hat{N}_{L-2}\right]$
$\displaystyle~{}~{}~{}-\frac{U}{2}\sum_{n=2}^{L-3}(2n+2)\hat{N}_{n}+V_{1}$
$\displaystyle=-\frac{UL}{2}-\frac{U}{2}\left[\hat{N}_{1}+(L-3)\hat{N}_{L-1}+(L-4)\hat{N}_{L-2}\right]$
$\displaystyle~{}~{}~{}-\frac{U}{2}\sum_{n=2}^{L-3}2n\hat{N}_{n}+V_{1}$
$\displaystyle=-\frac{U}{2}\left[3\hat{N}_{1}+(2L-5)\hat{N}_{L-1}+(3L-8)\hat{N}_{L-2}\right]-\frac{U}{2}\sum_{n=2}^{L-3}4n\hat{N}_{n}$
$\displaystyle=\frac{U}{2}\left[\hat{N}_{1}+(2L+1)\hat{N}_{L-1}+L\hat{N}_{L-2}\right]-\frac{U}{2}\sum_{n=0}^{L-1}4n\hat{N}_{n}\;.$
(30)

In the second from the last line, we omitted the constant terms.

####  C.0.3 $\mathcal{R}=3$

$\displaystyle V_{3}$
$\displaystyle=V_{2}-\frac{U}{2}\sum_{n=0}^{L-4}(n+3)\big{(}\hat{N}_{n}+\hat{N}_{n+3}\big{)}$
$\displaystyle=-\frac{U}{2}\left[3\hat{N}_{0}+4\hat{N}_{1}+5\hat{N}_{2}+\sum_{p=1}^{3}(L-p)\hat{N}_{L-p}\right]$
$\displaystyle~{}~{}~{}-\frac{U}{2}\sum_{n=3}^{L-4}(2n+3)\hat{N}_{n}+V_{2}$
$\displaystyle=-\frac{U}{2}\left[\hat{N}_{1}+2\hat{N}_{2}+\sum_{p=1}^{3}(L-p-3)\hat{N}_{L-p}\right]$
$\displaystyle~{}~{}~{}-\frac{U}{2}\sum_{n=3}^{L-4}2n\hat{N}_{n}-\frac{3UL}{4}+V_{2}$
$\displaystyle=-\frac{U}{2}\left[4\hat{N}_{1}+10\hat{N}_{2}\right]-\frac{U}{2}\sum_{n=3}^{L-4}6n\hat{N}_{n}$
$\displaystyle~{}~{}~{}-\frac{U}{2}\big{[}(3L-9)\hat{N}_{L-1}+(4L-13)\hat{N}_{L-2}+(5L-18)\hat{N}_{L-3}\big{]}$
$\displaystyle=-\frac{U}{2}\sum_{n=0}^{L-1}6n\hat{N}_{n}+\frac{U}{2}\big{[}2\hat{N}_{1}+2\hat{N}_{2}+(3L+3)\hat{N}_{L-1}$
$\displaystyle~{}~{}~{}+(2L+1)\hat{N}_{L-2}+L\hat{N}_{L-3}\big{]},$ (31)

where again we omitted the constant term coming from the total magnetization
conservation.

Therefore for a general $\mathcal{R}$ the resulting term reads

$\displaystyle V_{\mathcal{R}}=$
$\displaystyle-U\mathcal{R}\sum_{n=0}^{L-1}n\hat{N}_{n}+\frac{U}{2}\sum_{n=0}^{\mathcal{R}-1}n(\mathcal{R}-n)\hat{N}_{n}$
$\displaystyle+\frac{U}{4}\sum_{n=0}^{\mathcal{R}}\left[2nL+n(n-1)\right]\hat{N}_{L-\mathcal{R}+n-1}\;.$
(32)

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