# A note on the quasiperiodic many-body localization transition in dimension
$d>1$

Utkarsh Agrawal  Department of Physics, University of Massachusetts, Amherst,
MA 01003, USA  Romain Vasseur  Department of Physics, University of
Massachusetts, Amherst, MA 01003, USA  Sarang Gopalakrishnan  Department of
Physics, The Pennsylvania State University, University Park, PA 16802, USA

###### Abstract

The nature of the many-body localization (MBL) transition and even the
existence of the MBL phase in random many-body quantum systems have been
actively debated in recent years. In spatial dimension $d>1$, there is some
consensus that the MBL phase is unstable to rare thermal inclusions that can
lead to an avalanche that thermalizes the whole system. In this note, we
explore the possibility of MBL in quasiperiodic systems in dimension $d>1$. We
argue that (i) the MBL phase is stable at strong enough quasiperiodic
modulations for $d=2$, and (ii) the possibility of an avalanche strongly
constrains the finite-size scaling behavior of the MBL transition. We present
a suggestive construction that MBL is unstable for $d\geq 3$.

## Introduction.

Generic isolated many-body systems are expected to thermalize under their
intrinsic unitary dynamics: at long times, _local_ observables and correlation
functions exhibit equilibrium behavior Deutsch (1991); Srednicki (1994); Rigol
_et al._ (2007); Nandkishore and Huse (2015). As already suggested in
Anderson’s seminal paper Anderson (1958), thermalization is not fully generic:
systems subject to strong quenched randomness or quasiperiodic modulations can
instead exhibit a many-body localized (MBL) phase, in which thermalization
fails, and the system instead retains a local memory of its initial conditions
to arbitrarily late times Basko  _et al._ (2006); Gornyi  _et al._ (2005);
Basko  _et al._ ; Oganesyan and Huse (2007); Nandkishore and Huse (2015);
Abanin  _et al._ (2019a); Imbrie (2016a). This memory is due to the existence
in the MBL phase of (quasi-)local operators that are _exact_ integrals of
motion Serbyn  _et al._ (2013); Huse  _et al._ (2014); Ros  _et al._ (2015);
Imbrie (2016a); Imbrie  _et al._ (2017). Despite the existence of a
mathematical proof, under minimal assumptions Imbrie (2016a, b), of the
existence of an MBL phase in 1d random quantum systems with short-range
interactions, the existence and nature of the MBL phase as observed in
numerics and in experiments remain the subject of active debate Šuntajs  _et
al._ (2019); Sels and Polkovnikov (2020); Abanin  _et al._ (2019b); Panda  _et
al._ (2020); Crowley and Chandran (2020); Morningstar  _et al._ (2021);
Garratt  _et al._ (2021); LeBlond  _et al._ (2021); Sels and Polkovnikov
(2021). Asymptotically, the MBL transition in 1d (assuming it exists) is
believed to be driven by “avalanches” caused by rare thermal inclusions De
Roeck and Huveneers (2017). This leads to a Kosterlitz-Thouless-like picture
of the asymptotic transition Goremykina  _et al._ (2019); Dumitrescu  _et al._
(2019); Morningstar and Huse (2019); Morningstar  _et al._ (2020); Šuntajs
_et al._ (2020); Laflorencie  _et al._ (2020), while the finite-size MBL
crossover observed in numerics is driven by many-body resonances Crowley and
Chandran (2020); Morningstar  _et al._ (2021); Garratt  _et al._ (2021).
Assuming the existence of the avalanche mechanism, the MBL phase is always
unstable to thermalization in spatial dimensions $d>1$ De Roeck and Huveneers
(2017); Potirniche  _et al._ (2019).

Many experiments on the MBL phase treat systems subject to _quasiperiodic_
(QP) rather than random potentials Schreiber  _et al._ (2015); Bordia  _et
al._ (2017); Iyer  _et al._ (2013); Khemani  _et al._ (2017); Setiawan  _et
al._ (2017); Zhang and Yao (2018); Weidinger  _et al._ (2018); Žnidarič and
Ljubotina (2018); Doggen and Mirlin (2019); Doggen  _et al._ (2021); Pöpperl
_et al._ (2021); Singh  _et al._ (2021); Aramthottil  _et al._ (2021).
Noninteracting QP systems exhibit Anderson localization, and MBL appears at
least perturbatively stable for weak enough interactions Iyer  _et al._
(2013). For QP systems, unlike random systems, the avalanche instability does
not rule out the possibility of an MBL phase in $d>1$. In the random case, the
existence of rare locally thermal regions is inevitable in a large enough
sample; in QP systems, however, there is the possibility that the avalanche
_does not get started_ because there are strictly no thermal regions in the
sample. On the other hand, very little is known with certainty about MBL in QP
systems: even proving that it exists in 1d has yet not been possible, because
of the lack of relevant analytical tools.

In this note, we study the nature of avalanche instabilities in QP systems,
and argue that the MBL phase appears to be stable at strong enough QP
modulations (though we cannot rule out other instabilities that have yet to be
uncovered). We also argue that the avalanche picture strongly constrains the
nature of the MBL transition in spatial dimensions $d=2$, as it reduces it to
a nucleation problem. This mechanism puts strong constraints on the finite
size scaling near such transitions, which we derive on general grounds.

## Setup.

We consider a two-dimensional interacting spin-$\frac{1}{2}$ system (or
alternatively, a system of interacting spinless fermions) defined on a square
lattice. The exact details of the microscopic model will not be important for
our purposes, and we will denote the strength of the interactions given by
$J$. A quasiperiodic (QP) field, $h_{xy}$, is applied to the system

$\displaystyle
h_{xy}=W_{1}\cos\left(2\pi\varphi\mathbf{k_{1}.r}+2\pi\phi_{1}\right)+W_{2}\cos\left(2\pi\varphi\mathbf{k_{2}.r}+\phi_{2}\right),$
(1)

where $\mathbf{r}=\left(x,y\right)$, $\mathbf{k_{1,2}}$ are orthonormal
vectors, $(\phi_{1},\phi_{2})$ are phase shifts, and $\varphi$ is the QP
frequency which we take to be the golden ratio for concreteness. $W_{1,2}$
determines the strength of the QP potential and for simplicity we assume
$W_{1}=W_{2}=W$. We assume $\mathbf{k_{1,2}}$ are generic vectors and not fine
tuned; for certain special values of $\mathbf{k_{1,2}}$ there are extensive
lines of resonances in the system which destroy MBL for any strength $W$ Szabó
and Schneider (2020). We will denote by $L$ the linear size of the system. For
a given QP system, we are interested in localization properties averaged over
$\phi_{1,2}$. Different $\phi_{1,2}$ are equivalent to (approximate)
translations over the lattice. The MBL to thermal transition, if it exists, is
tuned by the ratio of $W$ and interaction strength $J$. For large $g=W/J$ we
expect a MBL phase, whereas for small $g$ the system should thermalize under
its own dynamics.

## Avalanche instability in $d>1$.

We first briefly review how an avalanche spreads in $d>1$ starting from an
initial spherical thermal region (“bubble”) of radius $\ell$ embedded in a
putatively MBL bulk De Roeck and Huveneers (2017). (As we will discuss below
this is not the dominant thermal region, but it is the simplest to analyze.)
This spreading process is essentially the same in random and QP systems. We
first imagine cutting all the bonds connecting the thermal region to the bulk.
Then by assumption the bulk has a complete set of exponentially localized
integrals of motion (LIOMs), with localization length $\xi$, while the bubble
has a random-matrix like level structure with level spacing $2^{-\ell^{d}}$.
We now recouple the bubble to the bulk. LIOMs a distance $L$ from the center
of the bubble couple to the bubble with matrix element $\exp(-|L-\ell|/\xi)$.
Provided that the associated Golden-Rule decay rate is larger than the level
spacing of the bubble, these LIOMs get absorbed into the bubble; crucially,
each time the bubble absorbs a LIOM, its level spacing goes down by a factor
of two. If the bubble is initially large enough, it is sure to absorb nearby
shells of LIOMs, and thus decrease its level spacing. After the bubble has
grown to a size $x$, its level spacing is $2^{-x^{d}}$; meanwhile, the
coupling to the next shell of LIOMs is $\exp(-|x-\ell|/\xi)$. Asymptotically,
the level spacing decreases faster than the matrix element, so once the bubble
has started growing it continues to grow. (In one dimension, by contrast, the
two quantities scale the same way, allowing the MBL phase to remain stable for
small enough $\xi$.)

At each stage of this procedure, the bubble grows by an amount proportional to
its surface area. Since bubbles with $N$ spins are exponentially rare in $N$,
the most dangerous bubbles in any dimension (i.e., the ones likeliest to start
an avalanche) are those that maximize the surface area to volume ratio, i.e.,
one-dimensional bubbles. The level spacing of a one-dimensional bubble goes
down to $2^{-(z-1)N}$ after it absorbs one layer of neighbors, where $z$ is
the typical coordination number; however, its coupling to the next layer of
spins is only down by a constant factor $\exp(-1/\xi)$. Thus, naively, we
might estimate that $N_{c}\sim 1/\xi$. In fact this is an underestimate of the
critical bubble size since it neglects the fact that at large $W/J$ the bubble
becomes an increasingly ineffective bath on account of its narrow bandwidth
Gopalakrishnan and Nandkishore (2014); a more careful estimate yields
$N_{c}\sim 1/\xi^{2}$ in all $d>1$ Gopalakrishnan and Huse (2019). Bubbles
with fewer than $N_{c}$ spins are ineffective as baths, and absorb at most
O(1) spins before they stop growing. In a random system the probability of
finding two spins to be in resonance scales as $J/W\sim 1/|\log(\xi)|$; thus
the density of super-critical bubbles scales as
$P(N_{c})\sim\exp(-1/\xi^{3})$. This is nonzero for any $\xi>0$ in the
thermodynamic limit, so a random system always thermalizes.

For a quasiperiodic system, the same estimate of $N_{c}$ applies; however,
since the potential is deterministic, the density of super-critical bubbles
need not scale in the same way with $\xi$: in fact, it is entirely possible
for this density to remain strictly zero in the thermodynamic limit, leaving
the MBL phase stable. We now argue that this is indeed the case.

Figure 1: (a) Lattice points and $\delta$-resonances on $\phi_{1}-\phi_{2}$
torus. The plot is a schematic and is not drawn to scales. The lattice points
are defined as $\phi_{1}(\mathbf{r})=\varphi\mathbf{k_{1}.r}+\phi_{0,1}$ and
$\phi_{2}(\mathbf{r})=\varphi\mathbf{k_{2}.r}+\phi_{0,2}$. Bold red points and
lines correspond to locations of perfect 3 and 2 site resonances respectively;
these bold regions do not form a dense set. Light red regions around the bold
red points and lines correspond to regions where interactions will hybridize 3
and 2 sites to form a resonance; the radius of the light red region
$\delta_{\epsilon}$ increases with the interaction strength $J=\epsilon$ (see
main text). To form bigger resonances, nearby lattice points should lie within
the red regions. In the plot above there are pairs of neighboring 2-site
resonances at (0,0) and (0,1) which will be combined to a 4-site resonance by
the interactions. Such 4-site resonance can be shown to always exist. Forming
5-site or higher resonances needs fine-tuning at small $\delta_{\epsilon}$.
(b) Sketch of the finite-size scaling of the thermalization probability
$P_{T}(L,g)$ in the nucleation scenario, see eq. (2).

## Stability of the MBL phase.

We consider an infinite system with weak interactions and try to identify the
largest bubble that can be created by the interactions. We map the lattice
points to the $\phi_{1}-\phi_{2}$ torus by identifying $\mathbf{r}$ with
$\phi_{1}(\mathbf{r})=\varphi\mathbf{k_{1}.r}+\phi^{0}_{1}$ and
$\phi_{2}(\mathbf{r})=\varphi\mathbf{k_{2}.r}+\phi^{0}_{2}$. The lattice
points form a dense subset on the torus. We next identify what we call
$\delta$-resonances: a connected path of $n$ sites is called $n$-site
$\delta$-resonance if the difference between neighboring $h_{xy}$ along the
path is within $\delta$. That is, two neighboring points $P,Q$ are part of a
$\delta$-resonance if $|h_{P}-h_{Q}|<\delta$. We argue that for a fixed set of
generic $\mathbf{k_{1,2}}$, and low enough $\delta$, we can only have 2- and
3-site $\delta$-resonances. To see this let us take $\delta=0$ and without
loss of generality consider conditions on $\phi^{0}_{1},\phi^{0}_{2}$ for
having a $\delta$-resonance at the origin. Demanding a two-site perfect
resonance gives a constraint on $\phi^{0}_{1,2}$, and since we have two
degrees of freedom, $\phi^{0}_{1,2}$, for a given value of $\phi^{0}_{1}$ (or
$\phi^{0}_{2}$) there exists a $\phi^{0}_{2}$ (or $\phi^{0}_{1}$) for which
there is a two-site perfect resonance at the origin. Thus these type of
resonances form lines on the torus (Fig. 1). Similarly for three-site
$\delta$-resonance we have two constraints which lead to isolated pairs of
$(\phi_{1},\phi_{2})$. Note that the values of phases
$(\phi^{0}_{1},\phi^{0}_{2})$ for a $\delta=0$-resonance at the origin form a
measure zero subset on the torus. There is a perfect resonance at $\mathbf{r}$
if $(\phi_{1}(\mathbf{r}),\phi_{2}(\mathbf{r}))$ lies in this set. Since these
points form a non-dense subset, in general there can only be at most one
${\delta=0}$-resonance in a system. On slightly increasing $\delta>0$, the
points on the torus consistent with $\delta=0$-resonances become balls of
small radius proportional to $\delta$; see Fig. 1. There is $\delta$-resonance
at $\mathbf{r}$ if $(\phi_{1}(\mathbf{r}),\phi_{2}(\mathbf{r}))$ lies in one
of these balls. Since lattice points are dense on the torus, there are non-
zero densities of 2- and 3-site $\delta$-resonances which go up with $\delta$.
But for a higher site $\delta$-resonance to form we require overlapping
2-sites and 3-sites resonances which is generically not possible without fine-
tuning $\mathbf{k}$.

Interactions will hybridize the $\delta$-resonant sites. For small interaction
strength $J=\epsilon$ all $\delta<\delta_{\epsilon}$ resonances will be
hybridized with $\delta_{\epsilon}$ depending on $\epsilon$ and
$\delta_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$; we will refer to
the $\delta$-resonances hybridized by the interactions simply as resonances.
The exact relation between $\delta_{\epsilon}$ and $\epsilon$ is not
important. Moreover, interactions will also cause nearby resonances to
hybridize Gopalakrishnan and Huse (2019). For example, if two 3-site
resonances are, let’s say, 2 lattice sites111The distance on which the
interactions are able to merge two resonances scale as $-\frac{1}{\ln J/W}$
which for small $J$ is a small number. This can also be thought of as the
localization length. away then there is a chance that interactions will
hybridize them to form a 6-site resonance; see Fig. 2. We can show that there
always exists a pair of neighboring 2-site $\delta$-resonances which will be
hybridized to a 4-site resonance by the interactions. To get even bigger
resonances we need the 2,3,4-site resonances to get close to each other. Again
for small enough $\epsilon$, we expect this to require fine-tuning of
$\varphi,\mathbf{k_{1,2}}$, so we conjecture that there are no 5-site or
higher resonances for weak enough interactions.

Figure 2: Here we demonstrate a hypothetical scenario where a small $\epsilon$
increase in the interaction strength $J$ leads to formation of two 3-site
resonances separated by 2 lattice sites.

Note that 2,3,4-site resonances are not strong enough thermal bubbles to
thermalize nearby single spins. Thus the only way we can get bigger
resonances/baths is by having multi-site resonances close together which, as
mentioned above, requires fine-tuning of $\mathbf{k_{1,2}}$ due to the
correlated nature of the QP potential. The absence of bubbles of size greater
than $5$ for low enough $J$ implies no avalanche instability to the MBL as the
critical bath size, which scales as $\sim 1/\xi^{2}$, can be made arbitrarily
large by choosing lower $J$ or higher $W$. Thus we argue that the MBL phase is
stable for large enough $g=W/J$.

We remark, finally, that it is possible to choose quasiperiodic potentials for
which even two-site resonances are absent Prange  _et al._ (1984). Thus the
size of the largest bubble is nonuniversal and depends on the details of the
quasiperiodic potential; what is generic, however, is that the size is $O(1)$,
and therefore is inadequate to create a super-critical bubble when $g$ is
large enough.

## Constraints on the MBL transition.

Having argued for the stability of the MBL phase for strong enough QP
potentials $g\gg 1$, we now discuss possible scenarios for its transition to
the thermal phase. As $g$ is decreased, the criterion for two sites (or pairs
of sites, etc.) to be resonant becomes less stringent so the density of
resonances increases, and some of them overlap forming larger resonances [Fig.
2]. Eventually these resonances will form a bath and thermalize the whole
system. We discuss one natural scenarios by which this might happen: the
_nucleation_ scenario. If this scenario holds we can strongly constrain the
properties of the transition. Of course, the true transition might take place
through an entirely different instability, and if it does so our constraints
will not apply.

We first comment on the nucleation scenario. The QP MBL phase in $d>1$ is
reminiscent of a supercooled liquid: it is unstable to the introduction of any
thermal bubbles, but in the absence of thermal bubbles the instability cannot
get started. In the nucleation scenario, the transition happens because a
local configuration somewhere in the system is able to get large and well-
coupled enough to act as a heat bath and seed the avalanche. The key
assumption is that when this happens the seed is of finite extent, i.e., there
is some $L$ such that regions of linear size $\geq L$ have some probability
$P_{T}(L)$ of thermalizing. In this scenario the seed is a discrete but “good”
(i.e., finite-bandwidth) bath at the point when it first comes into existence.
If $P_{T}(L)>0$ for some $L$, then a large enough system will necessarily
thermalize, so in the MBL phase we require that $P_{T}(L)=0$ for all $L$.

Moreover, in the thermal phase, near the critical point, we expect
$P_{T}(L,g)$ to scale as $\sim\rho_{c}(g)L^{2}$ where $\rho_{c}(g)$ is the
density of the critical thermal bubbles. Due to the deterministic and almost
periodic nature of the QP potential, the critical bubbles appear in a QP
pattern with a characteristic length scale $\zeta$ Agrawal  _et al._ (2020a,
b) which is also equal to the correlation length,
$\left(g_{c}-g\right)^{-\nu}$, inside the thermal phase. Thus,
$\rho_{c}\sim\zeta^{-2}\sim\left(g_{c}-g\right)^{2\nu}$. Based on the
discussion above, we propose following scaling ansatz for $P_{T}(L,g)$,

$\displaystyle P_{T}(L,g)$
$\displaystyle=f\left((g-g_{c})L^{\frac{1}{\nu}}\right)$
$\displaystyle=\begin{cases}\sim 1&g<g_{c},\
L\gg\left(g_{c}-g\right)^{-\nu}\\\ \sim
L^{2}\left(g_{c}-g\right)^{2\nu}&g<g_{c},\ L\ll\left(g_{c}-g\right)^{-\nu}\\\
0&g>g_{c}\end{cases}.$ (2)

Note that this particular finite size scaling form forbids the existence of a
finite size crossing: $P_{T}(L,g)$ must be an increasing function of $L$. The
exponent $\nu\geq 1/2$ is constrained by the Harris-Luck criterion Luck
(1993).

The nucleation scenario assumes that thermalization is due to a finite-size
resonant configuration that becomes _possible_ when $g$ decreases to some
critical value. This is not the only possible scenario: a possible alternative
is that as one comes from the thermal phase, increasing $g$, the nucleus of
the avalanche becomes increasingly sparse in real space. The spacing of sites
on this sparse resonant cluster might go as $\ell\sim|g-g_{c}|^{-\gamma}$. As
it does so, its bandwidth shrinks as $\delta\sim\exp(-\ell/\ell_{0})$, and it
becomes an increasingly ineffective bath. In this scenario, a thermal seed
will repeat at a distance scale where the quasiperiodic potential repeats to
exponential accuracy. This would give Kosterlitz-Thouless-like scaling of the
form $\zeta\sim\exp(|g-g_{c}|^{-\gamma})$.

##  Weak randomness and Hartree effects.

We now briefly discuss the fate of the QP MBL phase when a weak random
potential of bandwidth $w_{r}$ is added to the quasiperiodic potential. When
the random potential is sufficiently weak, it does not affect the counting of
few-spin resonances that we gave above. However, at distances $\sim
1/\sqrt{w_{r}}$, one can typically find pairs of sites that are detuned by
less than $w_{r}$. Rare fluctuations of the random potential can then put
sequences of sites at this distance scale exactly on shell, creating a sparse
thermalizing network of very narrow bandwidth Gopalakrishnan and Nandkishore
(2014) that can initiate an avalanche.

This observation raises a natural concern about our analysis of resonances:
while checking for resonance, we ignored the random “Hartree” shifts of the
splitting of a LIOM due to its interactions with other LIOMs that (at high
temperature) are in a random configuration. Could these Hartree shifts act
like emergent randomness and create a resonant network? The answer to this
question, as we will now discuss, differs between $d=2$ and $d>2$.

We first consider $d=3$ (though our considerations extend naturally to $d>3$).
Let us consider two sites $A=(0,0,0)$ and $B=(L,0,0)$ that are detuned by an
amount $\leq J$ before including Hartree effects. Here we expect $L\sim
J^{-1/3}$. We now attempt to bring these sites closer to resonance by
adjusting the occupation pattern of LIOMs on the planes $(0,x,y)$ and
$(L,x,y)$. The result of Ref. Nandkishore  _et al._ (2014) shows that one can
get the two energies arbitrarily close to one another. This is a consequence
of the fact that the thermally averaged spectral function of a LIOM in $d\geq
2$ forms a continuum. Moreover, for large $L$ (i.e., small $J$), the two
planes are far apart so “crosstalk” between the two planes is negligible. This
procedure clearly allows us to construct an arbitrarily long chain of
resonances, which can then act as a thermalizing bath. These considerations
strongly suggest that in $d\geq 3$ it is impossible to construct stable LIOMS,
even for QP potentials.

The direct generalization of this argument to $d=2$ does not yield an
instability, since the mechanism would involve trying to tune the LIOM energy
by adjusting occupation numbers on a _line_. In the limit of strong disorder,
the spectral function averaged over the occupation numbers on a line is _not_
a continuum, but instead occupies a measure-zero set of frequencies
Nandkishore  _et al._ (2014). However, one can generalize it Potter  by
considering quasi-1d strips wide enough so that the spectral function of each
strip is a continuum Nandkishore  _et al._ (2014): we hope to come back to
this potential instability in future works et al .

##  Summary.

In this note, we studied stability of the MBL phase in the presence of QP
disorder in $d>1$ many-body quantum systems. In 2D we find that unlike the
random case, for strong enough QP strength or/and weak interactions the QP
potential landscape is devoid of arbitrarily large resonant spots. Bigger
resonances can only be formed by the combination of small resonances. However,
since the set of small resonances is not dense, having multiple such
resonances close to each other requires fine-tuning. We also study the nature
of transition from the MBL phase to the thermal phase using the avalanche
instability. The critical size for the thermal bubble needed to thermalize the
whole system does not scale with the system size. This puts severe constraints
on the finite size scaling ansatz for the transition (eq (2)). In $d\geq 3$,
at the level of counting small resonances, the situation is very similar to
$d=2$. However, Hartree interactions among the putative LIOMs can potentially
destabilize these LIOMs by setting them on resonance with one another.

Though we argue that the nature of the avalanche instability constrains the
behavior of the transition, it would be interesting to verify this
numerically. Finite size numerics based on exact diagonalization may not be
sufficient as we need to access system sizes bigger than the critical bubble
size to observe the scaling ansatz in eq (2). Another possibility would be to
have phenomenological renormalization group (RG) studies similar to those in
1d Vosk  _et al._ (2015); Potter  _et al._ (2015); Dumitrescu  _et al._
(2017); Zhang  _et al._ (2016); Thiery  _et al._ (2017, 2018); Goremykina  _et
al._ (2019); Morningstar and Huse (2019); Morningstar  _et al._ (2020). But
unlike 1d, 2d systems can have non-trivial bipartitioning geometries for
dividing the system into thermal and insulating regions. Nevertheless, as long
as the phenomenological description of the RG contains the essential features
of the avalanche instability highlighted in this paper, we should expect to
see the proposed scaling ansatz near the transition.

Another important direction for the future would be to put the arguments for
stability of the MBL phase on a solid mathematical grounds. Our argument
relied on the non-denseness of the resonance regions (light red regions in
Fig. 1) in the $\phi_{1}-\phi_{2}$ plane which suggests that nearby lattice
points do not simultaneously lie in resonant regions and hence cannot be
hybridized by the interactions. However we cannot rule out these instances due
to some non-obvious/hidden patterns in the correlated nature of the QP
potential, though they seem highly unlikely without fine-tuning of the QP
potential.

Unlike 1d, the peculiar nature of the avalanche instability enabled us to
identify the universal features of the MBL transition in 2d. On the other
hand, a complete understanding of 1d QP MBL transition remain missing Khemani
_et al._ (2017) and is an obvious future direction to explore. The RG schemes
have been very successful for random systems but have failed to give a
consistent understanding for the QP systems. We think that combining ideas
about microscopic QP resonant structures presented in this paper together with
existing phenomenological RGs might help shed some light on the nature of this
transition in 1d.

## Note added:

While preparing this manuscript, we became aware of Ref. Crowley and Chandran
(2022) which also argues, from a complementary perspective, for the stability
of 2d QP MBL phase against avalanches. After posting, we also became aware of
Ref. Štrkalj  _et al._ (2022) which also argues for the stability of 2d QP
MBL.

## Acknowledgments.

We thank D. Huse, V. Khemani, A. Morningstar, A.C. Potter, Z. Shi, H. Singh,
and B. Ware for useful discussions and collaboration on related works. We
acknowledge support from NSF Grants No. DMR-2103938 (S.G.), DMR-2104141
(R.V.), the Alfred P. Sloan Foundation through a Sloan Research Fellowship
(R.V.).

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