††thanks: Corresponding author

# Disorder-free localization transition in a two-dimensional lattice gauge
theory

Nilotpal Chakraborty<EMAIL_ADDRESS>Max-Planck-Institut für Physik
komplexer Systeme, Nöthnitzer Straße 38, Dresden 01187, Germany  Markus Heyl
Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38,
Dresden 01187, Germany  Theoretical Physics III, Center for Electronic
Correlations and Magnetism, Institute of Physics, University of Augsburg,
D-86135 Augsburg, Germany  Petr Karpov  Max-Planck-Institut für Physik
komplexer Systeme, Nöthnitzer Straße 38, Dresden 01187, Germany  Roderich
Moessner  Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße
38, Dresden 01187, Germany

###### Abstract

Disorder-free localization represents a novel mechanism for ergodicity
breaking in lattice gauge theories (LGTs) which can even occur in two spatial
dimensions (2D). It has been shown that the U(1) quantum link model (QLM) can
localize due to an emergent classical percolation transition fragmenting the
system into disconnected real-space clusters. While the nature of the quantum
localization transition (QLT) is still debated for conventional many-body
localization, here we provide the first comprehensive characterization of the
QLT for the QLM in 2D for a disorder-free case. In this work we find
compelling evidence that the QLT in the 2D QLM is continuous and we determine
its universality class. We base our considerations on a spectral analysis of
finite-size clusters in the percolation problem which exhibits two regimes -
one in which large clusters effectively behave non-ergodically, a result
naturally accounted for as an interference phenomenon in configuration space,
and the other in which all large clusters behave ergodically.Our analysis can
also be applied to other 2D U(1) LGTs potentially including also matter
degrees of freedom.

The quantum localization transition (QLT) has been of great interest since
Anderson and co-workers' discovery that an electron hopping on a one- or two-
dimensional lattice can localize in presence of disorder Anderson (1958);
Abrahams  _et al._ (1979). Pioneering work has found such a transition even in
presence of weak interactions implying the presence of an extended nonergodic
so called many body localized (MBL) phase Basko  _et al._ (2006). While the
MBL phase is believed to be stable in one dimension (1D) Imbrie (2016), the
opposite has been argued in two dimensions (2D) De Roeck and Huveneers (2017);
Potirniche  _et al._ (2019). However, the nature of the QLT in 1D is still
debated and it is unclear if it is continuous such as the noninteracting
Anderson transition Oganesyan and Huse (2007); Pal and Huse (2010); Potirniche
_et al._ (2019); De Roeck and Huveneers (2017); Šuntajs  _et al._ (2020);
Morningstar  _et al._ (2021); Abanin  _et al._ (2021).

Recent works show that localization also occurs in models of lattice gauge
theories (LGTs) without quenched disorder - a phenomenon dubbed disorder free
localization (DFL) Smith  _et al._ (2017); Brenes  _et al._ (2018).
Localization in these LGTs is due to the emergence of so-called gauge
superselection sectors, which induce an effective internal quenched disorder
for the dynamics. Most models for DFL are in 1D, where quantum interference
induces localization similar to conventional Anderson localization Smith  _et
al._ (2017); Brenes  _et al._ (2018); Papaefstathiou  _et al._ (2020); Halimeh
_et al._ (2021a, b); Li  _et al._ (2021); Hodson  _et al._ (2021). Recently,
certain 2D LGTs have also shown to exhibit localization due to the presence of
a non-percolating phase in an emerging classical percolation problem
describing the gauge charges Karpov  _et al._ (2021); Verdel  _et al._ (2021).
However, the characterization of the resulting disorder-free QLT has remained
elusive. Settling the nature of the transition is particularly demanding in
interacting many-body systems, such as these LGTs, where a treatment beyond
perturbative arguments is necessary.

One central result of this work is the precise characterization of the QLT in
the 2D U(1) quantum link model (QLM) on a square lattice, involving an
accurate estimate of the quantum critical point and its associated
universality class including critical exponents. Besides the above central
result, we also make the following advances: i) Despite the lattice being
fragmented into disconnected clusters in the underlying classical percolation
problem, we find evidence that quantum interference can, in addition, play a
crucial role in the dynamics. We show via level-spacing statistics that the
larger clusters emerging in the vicinity of the classical percolation
transition, and on the non-percolating side, experience a crossover from
ergodic to non-ergodic on tuning the ratio of the potential and kinetic
couplings. This result implies that the long-time dynamics on the ergodic side
of the QLT will eventually thermalize only for very large systems and very
long times. Similar claims of non-ergodicity in experimentally relevant sample
sizes with a slow drift towards ergodicity have been made in a recent work
about the MBL regime Morningstar  _et al._ (2021). ii)Another important result
of our work is a concrete prediction on the dynamical properties of
observables close to the QLT. We show that for a space- and time-dependent
spin correlator the asymptotic long-time value reflects critical exponents of
classical percolation.

2D U(1) QLM.— The QLM is a discrete version of a Wilsonian LGT where instead
of continuous U(1) parallel transporters, the degrees of freedom on links are
discrete spins Wiese (2013); Chandrasekharan and Wiese (1997). Due to its
discrete nature the QLM is amenable to numerics and experiment Wiese (2013).
We consider the following spin-$1/2$ Hamiltonian with a plaquette flip term
(kinetic energy) with coupling constant $J$ and a Rokhsar-Kivelson potential
Rokhsar and Kivelson (1988) with coupling constant $\lambda$, which counts the
number of flippable plaquettes:

$\hat{H}=\lambda\sum_{\square}(\hat{U}_{\square}+\hat{U}_{\square}^{\dagger})^{2}-J\sum_{\square}(\hat{U}_{\square}+\hat{U}_{\square}^{\dagger})\,,$
(1)

where
$\hat{U}_{\square}=\hat{S}^{+}_{\textbf{r},i}\hat{S}^{+}_{\textbf{r}+\textbf{i},j}\hat{S}^{-}_{\textbf{r}+\textbf{j},i}\hat{S}^{-}_{\textbf{r},j}$
flips all the spins on a plaquette if they are oriented clockwise or
anticlockwise. If this condition is met, we say that the plaquette is
flippable; for an illustration see Fig. 1a. Here,
$\hat{S}^{+/-}_{\textbf{r},\mu}$ denotes the spin raising/lowering operators
for the spin joining site r and $\textbf{r}+\hat{\mu}$, where
$\hat{\mu}=\hat{i}/\hat{j}$ is the lattice vector. $\hat{H}$ in Eq. (1)
satisfies local constraints due to the U(1) gauge symmetry, generators of
which have the form
$\hat{G}_{\mathbf{r}}=\sum_{\hat{\mu}}(\hat{S}^{z}_{\mathbf{r},\hat{\mu}}-\hat{S}^{z}_{\mathbf{r}-\hat{\mu},\hat{\mu}})$
and can be interpreted as the net electric field flowing into the site. These
generators commute with $\hat{H}$ and with each other on different sites,
hence eigenvalues of $G_{\mathbf{r}}$, $q_{\mathbf{r}}\in\\{-2,-1,0,1,2\\}$,
can label eigenstates of $\hat{H}$. The set
$\mathbf{q}=\\{q_{\mathbf{r}_{1}},q_{\mathbf{r}_{2},...}\\}$ for all lattice
sites $\mathbf{r}_{i}$ defines a superselection sector in Hilbert space with
states satisfying
$G_{\mathbf{r}}\ket{\psi(\mathbf{q})}=q_{\mathbf{r}}\ket{\psi(\mathbf{q})}$.
These states are kinematically disconnected from states with different
$\mathbf{q}$. Due to the net electric field interpretation of
$G_{\mathbf{r}}$, $q_{\mathbf{r}}$ can be interpreted as a static background
charge satisfying Gauss' law. Such discrete link models have been widely
studied in condensed matter theory, with examples ranging from the toric code
Kitaev (2003) and quantum spin ice Hermele  _et al._ (2004); Shannon  _et al._
(2004) to quantum dimer models Rokhsar and Kivelson (1988); Moessner and
Sondhi (2001). Experimental proposals to realize such models in cold atom
settings have also been made Glaetzle  _et al._ (2014); Celi  _et al._ (2020)
with a recent proposal implemented for finite sizes in a 2D Rydberg atom setup
Semeghini  _et al._ (2021), however, their non-equilibrium dynamics Debashish
(2021) are still widely unknown due to the numerical challenges associated
with treating 2D interacting quantum matter .

Figure 1: a) Sample initial configuration for Monte Carlo flip process,
$\uparrow$ and $\rightarrow$ represent spin $\ket{\uparrow}$ whereas
$\downarrow$ and $\leftarrow$ represent spin $\ket{\downarrow}$. The ticked
plaquettes are flippable due to their clockwise/anticlockwise orientation,
background charges are indicated by the circles on the sites. The crossed
plaquettes are completely frozen due to charge 2 sites. The box indicates the
flip process where the $\hat{U}_{\square}$ operator flips all the spins
(coloured blue) on plaquette P. b) Disconnected clusters of flipped plaquettes
(in black) in the non-percolating phase for the state $\ket{\psi(\alpha=0)}$
on a $40\times 40$ lattice. Frozen plaquettes are shown in white.

Figure 2: Results for the percolation problem on square lattices of linear
dimension $L=20,30,40$ and $80$. We average over 1000 sectors for each
$\alpha$ a) Probability that a cluster wraps around either the x or the y axes
vs $\alpha$. The dotted line represents the percolation threshold $\alpha_{c}$
(Inset) Fraction of plaquettes occupied by the largest cluster vs $\alpha$ b)
Scaling collapse for the order parameter with the critical exponents
$\alpha_{c}=0.261$, $\nu=4/3$ and $\beta=5/36$. Here
$\alpha_{r}=(\alpha-\alpha_{c})/\alpha_{c}$

Bounding the QLT through a classical percolation problem.— DFL in LGTs arises
due to an effective disorder average due to background charges in typical
superselection sectors Brenes  _et al._ (2018). To characterize the QLT we
choose the following parameterizaiton for the initial state which allows us to
access different superselection sectors Karpov  _et al._ (2021).

$\ket{\psi_{0}(\alpha)}=\otimes_{i}\big{[}\text{sin}(\alpha+\pi/4)\ket{\mathrm{FF}_{i}}+\text{cos}(\alpha+\pi/4)\ket{\overline{\mathrm{FF}}_{i}}\big{]}$
(2)

where $\alpha\in[0,\pi/4]$ controls the density of background charges.
$\ket{\mathrm{FF}_{i}}$ and $\ket{\overline{\mathrm{FF}}_{i}}$ denote the
orientation of the $i^{\text{th}}$ spin in two possible fully flippable states
(all plaquettes flippable – checkerboard configurations of clockwise and
anticlockwise plaquettes). For $\alpha=0$ we get the product state where each
spin can be $\ket{\uparrow}$ or $\ket{\downarrow}$ with equal probability,
such a state is distributed over all super-selection sectors. For
$\alpha=\pi/4$ we get the product state from the fully flippable, i.e. zero
background charge sector. LGTs like in Eq. (1) with finite local Hilbert
spaces can be analyzed via a classical correlated percolation problem Karpov
_et al._ (2021), which we summarize for completeness. For every $\alpha$ in
Eq. (2) we get an initial state distributed over certain superselection
sectors. One can choose a random basis state (sector) from this distribution,
perform an infinite temperature Monte Carlo (MC) search for a plaquette and
then flip it if flippable. We then define a site percolation problem on a
square lattice, where every plaquette is considered occupied if flippable.
Clusters in the problem are defined as a set of connected plaquettes, each of
which has been flipped more than 100 times (this defines the exit criteria for
the MC search; if this criteria is not met we exit after $10^{8}$ searches, we
have checked for convergence on varying number of searches). Static background
charges $q_{\mathbf{r}}$ are unchanged throughout the plaquette flip process.
We study the percolation transition in system sizes of up to linear dimension
$L=80$. To determine the percolation threshold we go beyond Ref. Karpov  _et
al._ (2021) and calculate the percolation probability using wrapping
probabilities Newman and Ziff (2001), in which percolation occurs if there
exists a cluster which wraps onto itself. This can occur in topologically
distinct ways and in Fig. 2a we calculate the probability that a cluster wraps
around either in the $x$ or in the $y$-direction. This approach is better
compared to percolation probabilities calculated for open boundary conditions
as in Karpov  _et al._ (2021) since the former has O($1/L^{2}$) finite size
corrections as compared to O($1/L$) of the latter. Ref.Newman and Ziff (2001)
also showed that $\alpha_{c}$ estimated from finite size wrapping
probabilities converges to the critical value for the infinite system as
$|\alpha_{c}(L)-\alpha_{c}|\sim L^{-11/4}$. From Fig. 2a we infer that for a
finite parameter regime in $\alpha$ we get a non-percolating phase comprising
disconnected clusters (Fig. 1b), in agreement with Ref Karpov  _et al._
(2021).Using our improved percolation model we go beyond the analysis in ref
Karpov  _et al._ (2021) and determine the nature of the transition via a
scaling collapse of the order parameter $S_{L}$ (inset of Fig. 2a) defined as
the fraction of plaquettes occupied by the largest cluster, using the value
for $\alpha_{c}=0.261$ . We choose critical exponents $\nu=4/3$ and
$\beta=5/36$, which gives a good collapse as in Fig. 2(b). Hence, our
percolation problem is in the universality class of the standard 2D site
percolation problem with short-range or no spatial correlations. In the non-
percolating phase it is clear that real-space fragmentation due to background
charges induces localization. Such a classical threshold sets a bound for the
QLT, the percolation algorithm, however, is insufficient to determine the
properties of the quantum transition and the exact quantum transition point
which depend on quantum dynamics of finite-size clusters. There could be
further localization within larger clusters,which would make the quantum and
classical transitions differ.

Figure 3: a) The gap ratio averaged over the spectrum and over all clusters
with 16, 18 and 20 plaquettes in $N=1500$ superselection sectors of the
$80\times 80$ lattice vs $\lambda/J$ for state $\ket{\psi(\alpha=0)}$. Here,
``rnd'' implies results for the randomized energy model with same
configuration space connectivity. Error bars indicate variation due to
different cluster shapes, (inset) same data for
$\ket{\psi(\alpha=\alpha_{c})}$ with $N=2500$. b) Corresponding probability
distribution for a 20 plaquette cluster. Figure 4: Configuration space graph
for a 16 plaquette cluster. (a,b) Graph with correlated and randomized on-site
energies respectively. The edge colour goes from red to blue as separation
between connecting shells increases. (c,d) Histogram for normalized difference
of the on-site energies $d=|E_{i}-E_{j}|/\lambda$ between nearest neighbours
$i,j$ in (a) and (b) respectively.

Quantum-classical coincidence and non-ergodicity in finite clusters due to
quantum interference.— In the percolation problem localization arises
classically from removing bonds due to static background charges which render
certain plaquettes unflippable. However, plaquettes in each cluster are
kinetically connected through subsequent plaquette flips. For every $\alpha$
we get a certain cluster distribution in our classical percolation problem
with a non-percolating phase for $\alpha<\alpha_{c}$, and at a fixed $\alpha$,
$\lambda/J$ controls the quantum dynamics of finite clusters (Fig. 3a) We find
that finite size clusters for every $\alpha<\alpha_{c}$, experience a
crossover from ergodic to non-ergodic behaviour upon tuning $\lambda/J$ in Eq.
(1). We demonstrate this by calculating adjacent gap ratios, a common
diagnostic for distinguishing between ergodic and non-ergodic behaviour
Oganesyan and Huse (2007). The adjacent gap ratio is defined as
$r_{n}=\text{min}\\{\delta_{n},\delta_{n+1}\\}/\text{max}\\{\delta_{n},\delta_{n+1}\\}$
where $\delta_{n}=E_{n}-E_{n-1}$, $E_{n}$ is the energy of $n^{\text{th}}$
eigenstate and the eigenstates are arranged in ascending order of their
energies. We average $r_{n}$ over the entire spectrum of eigenstates of the
cluster and over all clusters of same size in $1500$ superselection sectors
(disorder realizations) of the $80\times 80$ lattice to get $[\langle
r\rangle]$. For small $\lambda/J$, $\langle r\rangle$ approaches the mean of
the Gaussian Orthogonal Ensemble (GOE) distribution ($\langle
r\rangle_{\mathrm{GOE}}\approx 0.53$) and the probability distribution is GOE-
like, as expected in a thermal phase. However, in the opposite limit, $\langle
r\rangle$ approaches the mean of Poisson distribution ($\langle
r\rangle_{\mathrm{POI}}\approx 0.39$) as we go to larger clusters and the
distribution is Poissonian, as seen in Fig. 3a and b. There is a drift in the
gap ratio on increasing cluster size implying that the infinite percolating
cluster might not show Poissionian statistics and this non-ergodicity is a
finite-size effect. There is also a slight variation depending on cluster
shape as shown by the error bars in Fig. 3a. Since there is no real-space
fragmentation within a cluster, i.e all plaquettes are kinematically
connected, we interpret the nonergodicity as a quantum interference effect. We
confirm that ergodicity for small $\lambda/J$ in finite-size clusters also
persists at the critical point $\alpha_{c}$ of the percolation problem (inset
of Fig. 3a). Ergodicity in all large clusters for small $\lambda/J$ at
$\alpha_{c}$ implies that in the small potential limit, information is spread
out in all large clusters and due to the absence of any localization within
large finite clusters the classical and quantum phase transitions coincide.
Our finding of ergodic behavior for individual clusters allows us to precisely
characterize the disorder free QLT in 2D, which is our central result. We
conclude that, for small $\lambda/J$, the disorder-free QLT is a _continuous_
transition with critical exponents $\nu=4/3$ and $\beta=5/36$ and hence it
belongs to same universality class as conventional percolation.

Scaling of correlation functions near the QLT.— Our findings on the QLT also
allow for direct predictions on dynamical universal behavior of the U(1) QLM.
An essential property of localized systems is absence of transport of energy
or spin. To quantify transport we consider the following connected correlation
function $A_{\mathbf{r}}(t)$ and its long-time average
$A_{\mathbf{r}}^{\infty}$:

$\displaystyle A_{\mathbf{r}}(t)$
$\displaystyle=\langle\hat{S}_{0}^{z}(0)\hat{S}_{\mathbf{r}}^{z}(t)\rangle-\langle\hat{S}_{0}^{z}(0)\rangle\langle\hat{S}_{\mathbf{r}}^{z}(t)\rangle$
(3) $\displaystyle A_{\mathbf{r}}^{\infty}$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}dtA_{\mathbf{r}}(t)\,.$
(4)

Since $\alpha_{c}$ coincides with the QLT point for small $\lambda/J$, in this
limit we can estimate scaling of the dominant contribution to $A_{r}^{\infty}$
near the transition using critical exponents obtained in Fig. 2. Close to the
QLT, on the ergodic side ($\alpha>\alpha_{c}$), real space lattice fragments
into an infinite cluster, some dynamic finite-size clusters (with
characteristic sizes up to correlation length
$\xi\sim|\alpha-\alpha_{c}|^{-\nu}$ of the standard 2D percolation
universality class Stauffer and Aharony (2018)), and some inert plaquettes
(which are not flipped even once). $A^{\infty}_{\mathbf{r}}$ can be affected
only by clusters of diameter $\geq r$. For $r\gg\xi$, finite cluster
contributions are exponentially suppressed, hence if we first take the limit
$r\rightarrow\infty$ and then $\alpha\rightarrow\alpha_{c}$ only the infinite
cluster contributes to $A^{\infty}_{\mathbf{r}}$. From conventional
percolation theory we know that the probability that a given spin belongs to
the infinite cluster $P\sim(\alpha-\alpha_{c})^{\beta}$, hence

$A_{r\rightarrow\infty}^{\infty}\sim P^{2}\sim(\alpha-\alpha_{c})^{2\beta},$
(5)

for $\alpha\geq\alpha_{c}$ and is zero for $\alpha<\alpha_{c}$. Note that
order of limits is important here: first we took $T\rightarrow\infty$, then we
take $r\rightarrow\infty$, only after which we can take
$\alpha\rightarrow\alpha_{c}$ limit.

Configuration space mapping.—  As a next step, we show that the nonergodicity
in finite clusters observed in Fig. 3 can be interpreted as a quantum
interference effect in configuration space as done in MBL Basko  _et al._
(2006); Roy and Logan (2020); Macé  _et al._ (2019). We map the Hamiltonian of
a cluster as defined in Eq. (1) (now we take the sum in Eq. (1) to be over all
plaquettes in the cluster) to a tight-binding model in configuration space
with

$\hat{H}_{CS}=\sum_{i}E_{i}\ket{\hat{\mu}_{i}}\bra{\hat{\mu}_{i}}-J\sum_{\braket{ij}}\ket{\hat{\mu}_{i}}\bra{\hat{\mu}_{j}}$
(6)

where $\ket{\hat{\mu}_{i}}$ is a string of $-1$'s, 1's and 0's indicating the
configuration of constituent plaquettes(+/- 1 for clockwise/anticlockwise
orientation and 0 otherwise), $E_{i}=m\lambda$, where $m$ is the sum of
absolute values of all string elements and $J$ is the nearest-neighbour
hopping amplitude connecting configurations differing by a plaquette flip.
Nearest neighbours $i$ and $j$ satisfy $|E_{i}-E_{j}|=d\lambda$ where
$d\in\\{0,1,2,3,4\\}$, depending on cluster geometry. Hence, we get a single-
particle problem in a heterogeneous graph with discrete correlated disorder.
If we treat a pair of nearest-neighbours as a two-level system, eigenstates
will be localized when $|E_{i}-E_{j}|\gg J$ and in resonance when
$|E_{i}-E_{j}|\ll J$. Since $|E_{i}-E_{j}|\leq 4\lambda$, when $\lambda/J\ll
1$ every pair is resonant and the system is delocalized and ergodic. However,
when $\lambda/J\gg 1$, resonances drastically reduce since only pairs with
same on-site energy are resonant and most pairs satisfy $|E_{i}-E_{j}|\gg J$
favoring non-ergodicity.

In Fig. 4(a) we show the configuration space graph of a typical cluster with
16 plaquettes from the state $\ket{\psi(\alpha=0)}$. The nodes are arranged in
equi-energy shells and $E_{i}$ increases outwards. To highlight the importance
of on-site energy correlations we randomly shuffle the energies while keeping
same connectivity. On doing so, we lose signatures of non-ergodicity as shown
in Fig. 3, the gap ratio and distribution no longer resemble Poissonian
behaviour. We speculate that this can be understood by comparing configuration
space graphs in Fig. 4(a) and (b). For correlated disorder most edges connect
nodes with same classical energy or with energies differing by $\lambda$,
which is not the case for the randomized energy model as illustrated by the
histograms in Figs. 4(c) and (d). Hence, correlations in configuration space,
which are wiped out by random reshuffling, are central for nonergodicity at
$\lambda/J\gg 1$. Therefore, localization in this graph is caused by quantum
interference due to these correlations.

Summary and discussion.—Exact critical exponents of a quantum transition in
interacting 2D systems are notoriously hard to find. In MBL, the nature of the
transition is still unclear and exact critical exponents are unknown Šuntajs
_et al._ (2020); Abanin  _et al._ (2021). We have provided a detailed
characterization of the QLT including its critical exponents in the 2D U(1)
QLM. We have also shown the presence of an experimentally relevant finite size
non-ergodic regime in disorder free 2D systems. A further key question is the
influence of matter fields, which introduce a dynamic charge in these 2D LGTs,
thereby inducing a coupling between the previously disconnected superselection
sectors of the pure gauge theory. 3D generalizations of this work leads to
questions of localization of emergent photons in quantum spin-ice, of interest
as a strongly-coupled LGT Pace  _et al._ (2021).

Data availability.—The data for the figures is openly available on Zenodo at
Chakraborty  _et al._ (2021).

Acknowledgements.— This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant Agreement No. 853443), and M.H. further
acknowledges support by the Deutsche Forschungsgemeinschaft via the Gottfried
Wilhelm Leibniz Prize program. This work was in part supported by the Deutsche
Forschungsgemeinschaft under grants SFB 1143 (project-id 247310070) and the
cluster of excellence ct.qmat (EXC 2147, project-id 390858490).

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