††thanks: equally contributed††thanks: equally contributed

# Scaling of Fock space propagator in quasiperiodic many-body localizing
systems

Soumi Ghosh1,4<EMAIL_ADDRESS>Jagannath Sutradhar1,2,3
<EMAIL_ADDRESS>Subroto Mukerjee1<EMAIL_ADDRESS>Sumilan Banerjee1
<EMAIL_ADDRESS>1Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India.
2Department of Physics, Bar-Ilan University, 52900, Ramat Gan, Israel.
3Department of Physics, Faculty of Natural Sciences, Ariel University, Ariel
40700, Israel
4International Centre for Theoretical Sciences, Tata Institute of Fundamental
Research, Bengaluru 560089, India.

###### Abstract

Recently many body localized systems have been treated as a hopping problem on
a Fock space lattice with correlated disorder, where the many-body eigenstates
exhibit multi-fractal character. The many-body propagator in Fock space has
been shown to be useful for capturing this multifractality and extracting a
Fock-space localization length for systems with random disorder in real space.
Here we study a one-dimensional interacting system of spinless Fermions in the
presence of a deterministic quasiperiodic potential using the Fock-space
propagator. From the system-size scaling of the self-energy associated with
the diagonal elements and the scaling of the off-diagonal elements of the
propagator, we extract fractal characteristics and FS localization lengths,
respectively, which behave similarly to that in the random system. We compute
the sample-to-sample fluctuations of the typical self-energy and the off-
diagonal propagator over different realizations of the potential and show that
the fluctuations in the self-energy distinguish quasiperiodic and random
systems, whereas the fluctuations of the off-diagonal elements cannot
demarcate the two types of potential.

##  I Introduction

Localization in isolated quantum systems in the presence of disorder and
interaction has received a lot of interest in recent times owing to its
fascinating non-equilibrium properties Basko  _et al._ (2006); Gornyi  _et
al._ (2005); Vosk  _et al._ (2015); Abanin and Papić (2017); Alet and
Laflorencie (2018); Abanin  _et al._ (2019). While most many-body systems in
the presence of interactions thermalize Deutsch (1991); Srednicki (1994,
1999), many body localized (MBL) systems in the presence of strong enough
disorder violate the eigenstate thermalization hypothesis (ETH), show area law
entanglement entropy even for high energy states, and retain their memory of
initial conditions for arbitrarily long times Basko  _et al._ (2006);
Oganesyan and Huse (2007); Bauer and Nayak (2013); Žnidarič  _et al._ (2008);
Nandkishore and Huse (2015); Abanin and Papić (2017); Alet and Laflorencie
(2018); Abanin  _et al._ (2019). MBL systems have been argued to have emergent
quasi-local constants of motion called the local integrals of motion (LIOMs)
or l-bits that prevent thermalization Vosk and Altman (2013); Serbyn  _et al._
(2013a); Huse  _et al._ (2014); Chandran  _et al._ (2015). The existence of
the MBL phase in the presence of strong disorder has been corroborated in one
dimension (1D) through a mathematical proof with a weak assumption Imbrie
(2016), a large number of numerical studies Oganesyan and Huse (2007); Pal and
Huse (2010); Kjäll  _et al._ (2014); Luitz  _et al._ (2015); Serbyn  _et al._
(2015), e.g., using exact diagonalization (ED), as well as experiments on cold
atoms and trapped ions Schreiber  _et al._ (2015); Rispoli  _et al._ (2019);
Lukin  _et al._ (2019). Phenomenological real-space renormalization group
(RSRG) approaches Vosk  _et al._ (2015); Potter  _et al._ (2015); Morningstar
and Huse (2019); Goremykina  _et al._ (2019); Dumitrescu  _et al._ (2019),
that can access much larger systems compared to ED, have also been developed
to study the critical properties of the MBL-to-thermal transition. However, in
recent years, arguments have been put forth about the instability of the MBL
phase in systems with random disorder in dimension $d>1$ due to so-called
_avalanches_ De Roeck and Huveneers (2017) mediated via ‘rare regions’ of weak
disorder. The stability of MBL even in 1D random systems has been called into
question Šuntajs  _et al._ (2020); Sels and Polkovnikov (2021); Sels (2022),
and evidence of long-range many-body resonances Morningstar  _et al._ (2022),
that can potentially destroy MBL in the thermodynamic limit, at least, over a
much larger range of disorder than previously deduced Luitz  _et al._ (2015),
has been shown.

Starting with the pioneering experiment of Schreiber et. al. Schreiber  _et
al._ (2015), strong evidence of MBL has also been found in systems with
quasiperiodic potentials in 1D Iyer  _et al._ (2013); Modak and Mukerjee
(2015); Li  _et al._ (2015); Setiawan  _et al._ (2017); Modak  _et al._
(2018); Deng  _et al._ (2017); Aramthottil  _et al._ (2021). The quasiperiodic
potential being deterministic does not allow large rare regions of weak
disorder on its own. Thus, naively, the avalanche instability should be absent
in quasiperiodic systems, unlike the random case. However, in the presence of
interactions, Hartree-Fock shifts, either in the many-body states or in the
initial condition for the non-equilibrium time evolution, can still lead to
effective rare regions Agrawal  _et al._ (2022). Based on ED studies Khemani
_et al._ (2017), the MBL transitions in the random and quasiperiodic systems
have been argued to be in different universality classes due to differences in
‘inter-sample’ (different disorder realizations), and ‘intra-sample’
(different eigenstates for the same realization) fluctuations of entanglement
entropy. Phenomenological RSRG studies Zhang and Yao (2018, 2019), though
consistent with two different universality classes, reach very different
conclusions regarding the critical properties, e.g., the critical exponent, of
the quasiperiodic MBL transition and the relevance of weak random
perturbations on the quasiperiodic potential. Since the ED studies are limited
to small finite-size systems, and as the phenomenological RSRG rules cannot be
derived _ab initio_ from the microscopic models, the questions on the
fundamental differences between MBL phenomena in interacting random and
quasiperiodic systems remain largely unresolved.

In this work, we construct a Fock-space (FS) picture, complementary to the
above real-space perspective, for the quasiperiodic MBL and the transition. In
particular, we ask how the differences between the random and quasiperiodic
potentials in real space manifest in the FS localization properties, namely in
the many-body FS propagators Sutradhar  _et al._ (2022); Roy  _et al._ (2023).
The interacting problem in real space can be viewed as a non-interacting
hopping problem on the complex FS graph or lattice with correlated disorder
Altland and Micklitz (2017); Welsh and Logan (2018); Logan and Welsh (2019);
Roy and Logan (2020a, b); Ghosh  _et al._ (2019); Sutradhar  _et al._ (2022).
This has led to approximate analytical approaches to describe MBL phenomena
Pietracaprina  _et al._ (2016); Logan and Welsh (2019), insights into the
importance of strong FS correlations to realize MBL Altland and Micklitz
(2017); Welsh and Logan (2018); Logan and Welsh (2019); Roy and Logan (2020a,
b); Ghosh  _et al._ (2019), the demonstration of multifractality of the MBL
eigenstates De Luca and Scardicchio (2013); Luitz  _et al._ (2015); Macé  _et
al._ (2019); Roy and Logan (2021); De Tomasi  _et al._ (2021), and numerical
scaling theories of the MBL transition in terms of the FS inverse
participation ratio (IPR) and FS propagator Macé  _et al._ (2019); Roy and
Logan (2021); Sutradhar  _et al._ (2022). The FS perspective has been useful
Roy  _et al._ (2023) to distinguish thermal and MBL states from other types of
non-ergodic states and understand the MBL proximity effect Nandkishore (2015)
in a quasiperiodic system with a single-particle mobility edge.

The FS propagator can be computed using an efficient recursive technique
Sutradhar  _et al._ (2022) that can reach systems sizes comparable to those
accessible by state-of-the art ED methods Sierant  _et al._ (2020);
Pietracaprina  _et al._ (2018). It has been shown that the Feenberg self-
energy Logan and Welsh (2019) derived from the diagonal elements of the FS
propagator acts as a probabilistic order parameter for the thermal-MBL
transition and captures information about the multi-fractal nature of the MBL
states Sutradhar  _et al._ (2022). The off-diagonal elements of the FS
propagator have also been studied in Ref. Roy  _et al._ (2023); Sutradhar  _et
al._ (2024) for random systems and a quasiperiodic systems with single-
particle mobility edges. This has led to the identification of an FS
localization length that remains finite in the MBL phase and at the MBL-
thermal transition. Thus, the FS localization length mimics the behaviour of
the length scale associated with the size of the LIOMs Serbyn  _et al._
(2013a); Huse  _et al._ (2014); Chandran  _et al._ (2015) namely, the local
support of a LIOM in real space.

Here we calculate the diagonal and off-diagonal elements of the FS propagator
using the recursive technique for the interacting quasiperiodic one-
dimensional Aubry-Andre-Harper (AAH) model Aubry and André (1980). We show
that the typical values of the Feenberg self-energy and the off-diagonal
elements of the FS propagator, extracted from different parts of the FS
lattice and over many disorder samples, show features very similar to the
random case Sutradhar  _et al._ (2022, 2024). For example, a fractal dimension
$D_{s}$ extracted from the Feenberg self-energy asymptotically approaches $1$
in the thermal phase, and varies between $0$ and $1$ in the MBL phase,
signifying the multifractal nature of the many body eigenstates. Based on an
asymmetric finite-size scaling ansatz Macé  _et al._ (2019); Sutradhar  _et
al._ (2022), we show that a _non-ergodic_ volume scale in the thermal phase
and a correlation length in the MBL phase diverge with a Kosterlitz-Thouless-
like essential and power-law singularities, respectively, approaching the MBL
transition. On the other hand, the FS localization length extracted from the
off-diagonal elements of the FS propagator in the MBL phase shows a variation
with potential strength very similar to that of the localization length
associated with LIOMs in the MBL phase. While these features are alike in the
presence of quasiperiodic and random disordered potentials, the sample-to-
sample fluctuations of the Feenberg self-energy seem to differentiate between
random and quasiperiodic disorders. We discuss the possible connection between
the fluctuation of self-energy, related to an inverse time scale, and rare-
region effects, or their absence, for random and quasiperiodic disorder. We
show that the inter-sample fluctuations in off-diagonal elements associated
with the FS localization length cannot distinguish between random and
quasiperiodic systems, and thus, presumably remains largely unaffected by
these rare region effects.

##  II Model and method

In this paper, we consider the standard microscopic model Oganesyan and Huse
(2007) for MBL systems, namely interacting spinless fermions on a one-
dimensional lattice with either quasiperiodic or random onsite potential

$\hat{H}=t\sum\limits_{i=1}^{L}\left(\hat{c}_{i}^{\dagger}\hat{c}_{i+1}+h.c.\right)+\sum\limits_{i=1}^{L}W_{i}\hat{n}_{i}+V\sum\limits_{i=1}^{L-1}\hat{n}_{i}\hat{n}_{i+1}$
(1)

where the operator $\hat{c}_{i}^{\dagger}$ creates a spinless fermion at site
$i$. The model can be equivalently described in terms of hard-core bosons or
as the XXZ model for spin-1/2 via Jordan-Wigner transformation. The
quasiperiodic potential is taken to be the Aubry-Andre-Harper potential Aubry
and André (1980), $W_{i}=W\mathrm{cos}(2\pi\alpha i+\phi)$, where $\phi$ is a
random phase chosen from the uniform distribution $\mathcal{U}(-\pi,\pi)$ and
the irrational number $\alpha=(\sqrt{5}-1)/2$, the inverse of golden ratio.
The hopping parameter $t$ and the nearest-neighbor interaction strength $V$
are set to $0.5$ and $1$, respectively, to be consistent with numerical
studies of the XXZ spin model Luitz  _et al._ (2015). For the random
potential, $W_{i}\in[-W,W]$ follows a uniform distribution with strength $W$.
Both random and quasiperiodic models show signatures of thermal (ergodic) to
MBL (non-ergodic) transition in ED studies for finite systems as the potential
strength $W$ is tuned Luitz  _et al._ (2015); Iyer  _et al._ (2013);
Aramthottil  _et al._ (2021). The critical potential strengths for these
transitions have been estimated as $W_{c}\sim 3-4$ and $W_{c}\sim 2-3$, for
random and quasiperiodic systems, respectively.

Figure 1: Schematic representation of the Fock space (FS) lattice for system
size $L=8$. The blue dots represent different basis elements, and the black
lines denote the hopping connections generated by the nearest-neighbor hopping
in the real space one-dimensional lattice. The schematic illustrates the
layered structure described in the text. The two apexes denote the minimally
connected FS sites $|11110000\rangle$ and $|00001111\rangle$. The FS sites
within the middle slice (largest slice) are highlighted in red.

Most of the numerical studies leading to the current understanding of these
systems across the thermal-MBL transition have been obtained by analyzing
different properties of the many-body energy spectrum, eigenstates, and their
entanglement characteristics Oganesyan and Huse (2007); Pal and Huse (2010);
Bardarson  _et al._ (2012); Serbyn  _et al._ (2013b); Kjäll  _et al._ (2014);
Serbyn  _et al._ (2015); Nandkishore and Huse (2015); Luitz  _et al._ (2015),
as well as dynamical correlations Žnidarič  _et al._ (2008); Mierzejewski  _et
al._ (2016); Bera  _et al._ (2017), obtained via ED. Here, we study the many-
body resolvent or the Green’s function defined as

$\mathbf{G}(E)=\left[E^{+}\mathbf{\mathbb{I}}-\mathbf{H}\right]^{-1}.$ (2)

Here, $E^{+}=E+\mathrm{i}\eta$, where $E$ represents the energy of interest
and $\eta>0$ acts as a broadening parameter for the delta function in the
calculation. We use a recursive technique MacKinnon and Kramer (1983) for
computing the many-body Green’s function following Refs. Sutradhar  _et al._
(2022); Roy  _et al._ (2023); Sutradhar  _et al._ (2024). For implementing the
recursive technique, we first write the Hamiltonian in Fock space (FS), which
is a connected graph representing the Hilbert space of the many-body problem
Welsh and Logan (2018). Each FS site corresponds to a set of occupation
numbers for all the real space sites:
$|I\rangle=|n_{1}^{(I)},n_{2}^{(I)},\dots,n_{L}^{(I)}\rangle$. For the case of
half-filling considered, there are $\mathcal{N}_{F}=\binom{L}{L/2}$ sites in
the many-body Hilbert space of a one-dimensional lattice with $L$ sites. When
expressed in the FS basis, the Hamiltonian [Eq. (1)] takes the form of a
tight-binding model Welsh and Logan (2018); Ghosh  _et al._ (2019); Logan and
Welsh (2019) with correlated onsite disorder,

$H=\sum\limits_{I,J}T_{IJ}|I\rangle\langle
J|+\sum\limits\mathcal{E}_{I}|I\rangle\langle I|,$ (3)

where $T_{IJ}$ indicates hopping connections between FS sites $I$ and $J$, and
$\mathcal{E}_{I}$ denotes onsite potential. Specifically, $T_{IJ}$ is equal to
$t$ if $I$ and $J$ are connected by a nearest-neighbour hop in real space
[Eq.(1)], and zero otherwise. The onsite disorder potentials,
$\mathcal{E}_{I}=\sum_{i=1}^{L}W_{i}n_{i}^{(I)}$, are highly correlated, i.e.,
$\left<\mathcal{E}_{I}\mathcal{E}_{J}\right>\neq 0$ for typical $I,J$ pairs
Altland and Micklitz (2017); Ghosh  _et al._ (2019); Roy and Logan (2020a).
These correlations are essential for localization on the FS lattice,
especially given its typical coordination number diverges in the thermodynamic
limit. For the Hamiltonian in Eq.(1), Fig. 1 illustrates the layered structure
of the FS, with $(1+L^{2}/4)$ number of slices, each containing different FS
sites Welsh and Logan (2018); Sutradhar  _et al._ (2022). In such a layered
arrangement, an FS site in one slice connects only to sites in adjacent
slices. Leveraging this layered structure and the tight-binding nature of the
Hamiltonian, the recursive technique, detailed in Ref. Sutradhar  _et al._ ,
2022, computes the different Green’s function elements $G_{IJ}(E)=\langle
I|\mathbf{G}(E)|J\rangle$ iteratively. Each iteration adds a single FS layer,
facilitating computations for system sizes $L\lesssim 22-24$, comparable to
the _state-of-the-art_ ED calculations Sierant  _et al._ (2020); Pietracaprina
_et al._ (2018), especially when prioritizing specific elements.

We shift the energy eigenvalues for each realization of the Hamiltonian using
the transformation $H\rightarrow
H-\left(\mathrm{Tr}H/\mathcal{N}_{F}\right)\mathbb{I}$ to keep the middle of
the spectrum aligned for different disorder realizations. The many-body
density of states for such a system is a Gaussian distribution with the
variance $\mu_{E}$ scaling as $\propto L$ with increasing system sizes Logan
and Welsh (2019). Consequently, we rescale the Hamiltonian as $H\rightarrow
H/\sqrt{L}$ to make the density of states invariant to system sizes Logan and
Welsh (2019); Sutradhar  _et al._ (2022).

For our analysis, we calculate the diagonal elements $G_{II}$ and also several
off-diagonal elements $G_{IJ}$ and $G_{1I}$ of the Green’s function (Eq. 2).
Here both $I$ and $J$ belong to the middle slice of the FS lattice (Fig. 1).
Thus, the particular choice of $G_{IJ}$ characterizes the non-local FS
propagator between generic sites at the middle slice with connectivity
$\propto L$. On the contrary, $G_{1I}$ captures the propagation between a site
in the middle slice $I$ and a _non-generic_ FS site $1\equiv|1111\dots
0000\dots\rangle$ located at the apex of the FS graph, characterized by the
minimal connectivity (exactly $1$). Given our focus on the spectrum’s
midpoint, we set $E=0$. The broadening $\eta$ is chosen as the mean many-body
level spacing $\delta=\sqrt{2\pi}\mu_{E}/\sqrt{L}\mathcal{N}_{F}$. For
statistical purposes, we average all the quantities over different
realizations of the quasiperiodic potential by choosing different phases
$\phi$. Specifically, we choose $10000$, $5000$, $2000$, $1000$, and $100$
samples for system sizes $12$, $14$, $16$, $18$, and $20$, respectively. For
the random disorder discussed in Sec. III, we consider identical system sizes
and disorder realizations as in the quasiperiodic case.

##  III Results

Here we first discuss different aspects of the diagonal and the off-diagonal
elements of Green’s function for the quasiperiodic potential in Eq.(1). The
distinct system size scalings of these diagonal and off-diagonal elements in
the thermal and MBL phases are discussed in subsections III.1 and III.3. We
also discuss some characteristic features of the distribution of the diagonal
elements of Green’s function in subsection III.2 Finally, in subsection III.4,
we discuss the sample-to-sample fluctuations of these Green’s function
elements, drawing comparisons with their counterparts in the random disordered
case.

###  III.1 On-site Feenberg self-energy

We calculate the on-site self-energy, i.e., the Feenberg self-energy, Logan
and Welsh (2019) denoted as $\Sigma_{I}(E)$ corresponding to site $I$ from the
diagonal element of the Green’s function as $G_{II}(E)=\langle
I|\mathbf{G}(E)|I\rangle=\left[E+\mathrm{i}\eta-\mathcal{E}_{I}-\Sigma_{I}(E)\right]^{-1}$.
We are specifically interested in the imaginary part of the self-energy
$\Delta_{I}(E)=\mathrm{Im}\Sigma_{I}(E)$, henceforth referred to as the
Feenberg self-energy for brevity. We calculate the typical value of the
Feenberg self-energy defined as
$\ln\Delta_{t}=\langle\ln\Delta_{I}\rangle_{I,\phi}$, where
$\langle\cdots\rangle_{I,\phi}$ represents averaging over all FS sites
belonging to the middle slice and different values of $\phi$. The self-energy
$\Delta_{I}(E)$ represents the inverse lifetime of an _excitation_ of energy
$E$ created at the FS site $I$ Sutradhar  _et al._ (2022). Therefore, in the
thermal phase, we expect the excitation to decay in some finite time and
consequently $\Delta_{t}\sim O(1)$ in the thermodynamic limit. On the other
hand, in the MBL phase, where we expect the state to survive for an infinitely
long time, the typical value $\Delta_{t}\rightarrow 0$ in the thermodynamic
limit. It has been shown Sutradhar  _et al._ (2022); Roy  _et al._ (2023) that
due to the multifractal nature Macé  _et al._ (2019) of the MBL states, the
typical value $\Delta_{t}$ decreases with increasing system sizes as
$\Delta_{t}\sim\mathcal{N}_{F}^{-(1-D_{s})}$ for
$\eta\sim\mathcal{N}_{F}^{-1}\ll\eta_{c}$, where
$\eta_{c}\sim\mathcal{N}_{F}^{-z}$ ($0<z<1$) is an emergent energy scale and
$0<D_{s}<1$ is a spectral fractal dimension Altshuler  _et al._ (2016).

Figure 2: (a) Variation of the typical value $\Delta_{t}$ with increasing
system size. In the thermal phase ($W<W_{c}$), $\Delta_{t}\sim O(1)$, while in
the MBL phase ($W>W_{c}$), $\Delta_{t}$ decreases following a power-law
$\Delta_{t}\propto\mathcal{N}_{F}^{-(1-D_{s})}$, with $D_{s}$ being the
fractal dimension. The black dashed line corresponds to the critical potential
strength $W_{c}$. (b) Variation of fractal dimension $D_{s}$ as a function of
the potential strength where $0<D_{s}<1$ in the MBL phase including the
critical point. $D_{s}$ extracted from the power-law fit of the raw data of
panel (a) [circle] and from finite-size scaling [square] are both shown. The
critical potential strength $W_{c}$ is indicated by the vertical dashed line.

The multifractal nature and the underlying scaling of the self-energy in both
the thermal and MBL phases have been investigated in Ref. Sutradhar  _et al._
(2022) for an interacting one-dimensional system of spinless fermions in the
presence of random disorder. Here, we perform a similar analysis in the
presence of a quasiperiodic potential and study whether the nature of the
disorder in real space, namely random vs. quasiperiodic, influences the
multifractal characteristics of the states in the Fock space.

We find that, similar to the random disordered potential, $\Delta_{t}$ remains
$O(1)$ in the thermal phase, and in the MBL phase, it decays with increasing
system sizes following a power-law
$\Delta_{t}\sim\mathcal{N}_{F}^{-(1-D_{s})}$, as shown in Fig.2(a). We extract
the fractal dimension $D_{s}$ from the power-law fit to
$\Delta_{t}(\mathcal{N}_{F})$ [Fig.2(b)]. The value of $D_{s}$ estimated from
power-law fit is consistent with its value extracted from finite-size scaling
as discussed below. $D_{s}$ obtained from finite-size scaling changes
discontinuously across the critical potential strength $W_{c}$ from $1$ in the
thermal phase to a value between $0$ and $1$ in the MBL phase, where $W_{c}$
is also estimated from finite-size scaling.

Figure 3: Scaling collapse for the typical value of the Feenberg self-energy
in (a) the thermal phase and (b) the MBL phase. The FS volume scale $\Lambda$
and the length scale $\xi$ extracted from the scaling collapse are shown as a
function of $|\delta W|=|W-W_{c}|$ in the respective insets. The critical
potential strength $W_{c}=2.25$ is estimated from the quality of the finite-
size scaling collapse (see main text).

We further analyze the finite-size results for the self-energy using an
asymmetric scaling ansatz given by Sutradhar  _et al._ (2022); Macé  _et al._
(2019); Roy and Logan (2021); García-Mata  _et al._ (2017)

$\ln\left(\dfrac{\Delta_{t}}{\Delta_{c}}\right)=\begin{cases}\mathcal{G}_{\mathrm{vol}}\left(\ln\frac{\mathcal{N}_{F}}{\Lambda}\right)&\mathrm{for}~{}~{}W<W_{c}\\\
\mathcal{G}_{\mathrm{lin}}\left(\frac{\ln\mathcal{N}_{F}}{\xi}\right)&\mathrm{for}~{}~{}W>W_{c}\\\
\end{cases}$ (4)

where $\Delta_{c}$ is the typical value of the self-energy at the critical
point $W=W_{c}$. In the thermal phase, $\mathcal{G}_{\mathrm{vol}}$ is a
‘volumic’ scaling function associated with a Fock-space volume scale $\Lambda$
while in the MBL phase, $\mathcal{G}_{\mathrm{lin}}$ is a ‘linear’ scaling
function with a Fock-space length scale $\xi$. Fig. 3 shows the scaling
collapse of the finite size data up to $L=20$ with $W_{c}$ chosen as $2.25$,
which gives the best quality of collapse, i.e., the error of fit to the finite
size scaling forms [Eq.(4)] to the data on both thermal and MBL sides is
minimal. This estimate is also consistent with $W_{c}$ obtained from the many-
body level-spacing ratio(not shown). Here, the scaling functions
$\mathcal{G}_{\mathrm{vol}}$ and $\mathcal{G}_{\mathrm{lin}}$ provide
satisfactory collapse in the thermal and MBL phases, respectively.

In the thermal phase, the FS volume scale $\Lambda$ obtained from finite-size
scaling diverges near the critical point following a KT-like essential
singularity $\Lambda\sim\mathrm{exp}\left[b/(W_{c}-W)^{a}\right]$ with $b\sim
O(1)$ and $a\simeq 0.5$ [as shown in the inset of Fig. 3(a)], similar to the
observation for the case of random disordered potential Sutradhar  _et al._
(2022). Since $\Delta_{t}\sim O(1)$ in the thermal phase for large enough
system sizes $\mathcal{N}_{F}\gg\Lambda$, it can be shown Sutradhar  _et al._
(2022) that in the asymptotic limit of $x=\mathcal{N}_{F}/\Lambda\gg 1$,
$\mathcal{G}_{vol}\left(x\right)\sim-(1-D_{c})\ln x$, $D_{c}$ being the
fractal dimension at the critical point $W_{c}$. As seen in Fig. 3(a),
$D_{c}=0.46$ is found from the slope of the linear fit in the asymptotic limit
($x\gg 1$). This is in agreement with $D_{c}$ extracted directly from the
scaling of $\Delta_{t}$ with system size,
$\Delta_{t}\sim\mathcal{N}_{F}^{-(1-D_{s})}$, shown in Fig. 2(a).

In the MBL phase, the Fock-space length scale $\xi$ obtained from finite-size
scaling diverges near the critical point [see the inset of Fig. 3(b)]
following a power law $\xi\sim|W-W_{c}|^{-\beta}$ with $\beta\simeq 0.5$ while
approaching the critical point from the MBL side. This again is very similar
to what we see in the random disorder case Sutradhar  _et al._ (2022). In the
limit of $x=(\ln\mathcal{N}_{F})/\xi\gg 1$ the scaling function
$\mathcal{G}_{lin}(x)$ asymptotically approaches Sutradhar  _et al._ (2022)
$\sim-(1-D_{c})x$, as seen in Fig.3(b). Again, $D_{c}=0.46$ is consistent with
that obtained from asymptotic scaling in the thermal side, discussed above, as
well as with $D_{c}$ estimated in Fig.2. The scaling form $\mathcal{G}_{lin}$
along with the power-law $\Delta_{t}\sim\mathcal{N}_{F}^{-(1-D_{s})}$ for
$W>W_{c}$ implies $\xi=(1-D_{c})/(D_{c}-D_{s})$, and thus a diverging $\xi$ as
$D_{s}\to D_{c}$ for $W\to W_{c}^{+}$.

In summary, we find that the differences between random and quasiperiodic
disorder in real space do not give rise to any qualitative differences in
fractal properties of the states across the MBL transition and the associated
finite-size scaling of the typical FS Feenberg self-energy for the system
sizes accessed in our study.

###  III.2 Distribution of Feenberg self-energy

Since the typical Feenberg self-energy cannot distinguish between random and
quasiperiodic systems in the Fock space, in this section, we look into the
statistics of $\Delta_{I}$ in terms of its distribution over different FS
sites $I$ in the middle slice and over different disorder realizations. An
approximate self-consistent theory Logan and Welsh (2019) for the self-energy
predicts a log-normal distribution in the thermal phase, while deep in the MBL
phase, the ratio of the self-energy to the width $\eta$ is predicted to follow
a Lévy distribution with a characteristic tail $\sim(\Delta/\eta)^{-3/2}$.

Figure 4: Distribution of the self-energy ($\mathrm{ln}\Delta_{I}$) for
potential strengths $W=1.0$, $1.5$, $2.25$. The Gaussian fit shows the
deviation from the expected log-normal functional form for $\Delta_{I}$.
Figure 5: Binder cumulant for the distribution $P(\mathrm{ln}\Delta_{I})$ as a
function of the potential strength for (a) random disorder and (b) the
quasiperiodic potential. Near the transition for both models, the distribution
approaches a Gaussian as indicated by $\mathcal{B}\simeq 0$. The insets show
the variation of the skewness $\mathcal{S}$ of the distributions for the
respective cases. The skewness changes sign close to the transition. The
critical potential strength $W_{c}$ is indicated in the plots via vertical
dashed lines for reference.

We find that for both potentials the distribution of self-energies
$\Delta_{I}$ follows slightly different functional forms than those predicted
by the self-consistent calculations in the thermal phase. While deep in the
thermal phase the distribution of $\ln\Delta_{I}$ is expected to be Gaussian,
we see non-Gaussian characteristics as well as a lack of symmetry in the
distribution, as shown in Fig.4 for quasiperiodic potential. Similar results
are found for random system (not shown).

To quantify these further, we calculate the Binder cumulant or kurtosis for
$x=\ln\Delta_{I}$ defined as $\mathcal{B}=\langle\delta
x^{4}\rangle/\langle\delta x^{2}\rangle^{2}-3$ with $\delta x$ defined as
$\delta x=x-\langle x\rangle$. A Binder cumulant $\mathcal{B}=0$ indicates a
perfect Gaussian distribution of the underlying variable, and the deviation
from $\mathcal{B}=0$ characterizes non Gaussianity of the distribution. We
also calculate the skewness defined by $\mathcal{S}=\langle\delta
x^{3}\rangle/\langle\delta x^{2}\rangle^{3/2}$. Since a Gaussian distribution
is symmetric around the mean, the skewness is zero. Any non-zero skewness
again indicates deviation from Gaussianity.

As shown in Fig. 5 (a), the Binder cumulant of the distribution
$P(\ln\Delta_{I})$ for the random disordered potential approaches zero near
the critical point for the thermal-MBL transition. However, the potential
strength at which the cumulant becomes $0$ is somewhat below the typical
estimate of the critical disorder $W_{c}\simeq 3.5-4.0$ Luitz  _et al._
(2015); Sutradhar  _et al._ (2022); Sierant and Zakrzewski (2022) for the
random case considered here. The skewness of the distribution [inset of Fig.5]
also changes sign close to the transition indicating that the distribution
$P(\ln\Delta_{I})$ changes from being left skewed to right skewed as the
disorder strength is changed. However, curves corresponding to different
system sizes cross at a value of $W$ where the skewness $\mathcal{S}$ is non-
zero but small. This together with the Binder cumulant implies the
distribution becomes Gaussian close to the transition. But the values of $W$
at which this Gaussianity emerges seem to lie within the thermal side,
somewhat below the estimated range of critical disorder for the putative MBL
transition Luitz  _et al._ (2015); Sutradhar  _et al._ (2022); Sierant and
Zakrzewski (2022).

Similar features are seen for the quasiperiodic potential in Fig. 5(b). Here,
although a decrease in the Binder cumulant is observed close to the thermal-
MBl transition, the Binder cumulant never actually reaches zero, especially
for the largest system size ($L=20$), unlike in the case of the random
potential. Accordingly the system size variation of skewness shows a crossing
near the thermal-MBL transition. But, the value of skewness at the crossing
point is substantially larger than that for the random case. Nevertheless, the
overall variation of both quantities appears qualitatively similar to the case
of the random disordered potential, namely the distribution $P(\ln\Delta_{I})$
approaches a Gaussian [or, $P(\Delta_{I})$ a log-normal] distribution in the
thermal phase before the MBL transition, whereas non-Gaussianity increases
with system size deep in the thermal and MBL phase. Thus, the distribution
$P(\Delta_{I})$ never truly becomes log-normal even deep inside the thermal
phase. This is in contrast to tight-binding model on regular lattices with
uncorrelated disorder, where the log-normal distribution is ubiquitous, both
in the localized and delocalized phases Schubert  _et al._ (2010), and
naturally arises in the non-linear $\sigma$ model description of such systems
Mirlin (1996, 2000). On the FS lattice, however, the disorder is highly
correlated Altland and Micklitz (2017); Logan and Welsh (2019); Roy and Logan
(2020b); Ghosh  _et al._ (2019), and there is no reason for such descriptions
for an uncorrelated disorder to be applicable, as clearly suggested by our
results.

###  III.3 Fock space localization length from off-diagonal elements of the
Green’s function

Here we discuss the scaling of the non-local elements of the Green’s function.
We are again interested in the middle slice of the FS lattice. As already
mentioned, we consider two types of off-diagonal elements of the Green’s
function: (i) a typical $G_{IJ}=\langle I|\mathbf{G}|J\rangle$, where both $I$
and $J$ are FS sites belonging to the middle slice, and correspond to generic
states, and (ii) a non-generic $G_{1I}=\langle 1|\mathbf{G}|I\rangle$, where
$I$ belongs to the middle slice and $|1\rangle$ is an atypical FS site having
the minimal connectivity in the FS; $|1\rangle$ corresponds to a single domain
wall state where all the sites on the left are occupied and the sites on the
right are empty. Thus, this is a non-generic state for energy $E=0$ in the
middle of the many-body spectrum. The rationale behind the choice of only two
types of off-diagonal elements of FS propagator are (a) computational
convenience since the recursive Green’s function method is most efficient when
only a few types of elements are targeted, and (b) the fact the above two
types of elements represent two extreme cases, namely propagator involving two
typical sites and that between an atypical and a typical one.

To quantify $G_{IJ}$, we use the hopping distance $r_{IJ}$ as a notion of
distance in FS. The hopping distance $r_{IJ}$ is given by the minimum number
of hops that connect FS sites $I$ and $J$. Due to the construction of FS (Fig.
1), the hopping distance from $1$ to site $I$ is the same for all sites $I$
belonging to the same slice. Thus, $r_{1I}\propto L^{2}$ for all sites
belonging to the middle slice of FS. On the other hand, the hopping distance
$r_{IJ}$ between typical FS sites $I$ and $J$ can vary from $2$ to $r_{max}$
where $I$ and $J$ both belong to the middle slice. The maximum hopping
distance $r_{max}$ depends on the system size $L$ and the number of particles,
as discussed in Appendix A.

Figure 6: (a) Variation of the typical value of the off-diagonal elements
$G_{IJ}$ of the Green’s function as a function of the hopping distance $r$ for
different values of the potential strength $W$ for system size $L=20$. (b)
Scaling of the Fock-space localization length $\xi_{F}^{(1)}$ as a function of
the potential strength for $W>W_{c}$. The inset shows the correlation length
$\zeta^{(1)}$ that varies as $\zeta^{(1)}\sim|W-W_{c}|^{-\nu}$ with $\nu\sim
0.9$.

Typical FS localization length.— We calculate the typical value of off-
diagonal elements $G_{IJ}$ for $I$ and $J$ separated by hopping distance
$r_{IJ}=r$ as $\ln
G_{MM}(r)=\langle\ln|G_{IJ}(r_{IJ})|\rangle_{r_{IJ}=r;\phi}$ where
$\langle\cdots\rangle_{r_{IJ}=r;\phi}$ denotes averaging over different pairs
of $I$ and $J$ with the same hopping distance $r$ and over different
realizations of the offset angle $\phi$. Fig. 6(a) shows the variation of the
typical value $G_{MM}(r)$ as a function of $r$. In the thermal phase
($W<W_{c}$), $\ln G_{MM}$ is almost independent of $r$, while, in contrast,
$G_{MM}$ decreases exponentially with $r$ in the MBL phase.

Figure 7: Variation of the typical value of off-diagonal elements $G_{1I}$ of
the Green’s function as a function of Fock space dimension $\mathcal{N}_{F}$
for different values of the potential $W$. (b) Scaling collapse of the typical
value $\ln G_{1M}$ assuming a functional form
$F(\ln\mathcal{N}_{F}/\xi_{F}^{(2)})$ for $W>W_{c}$. The inset shows the
variation of the localization length $\xi_{F}^{(2)}$ as a function of the
potential strength $W$.

In the MBL phase, the eigenstates have non-zero amplitude on
$\mathcal{N}_{F}^{D}$ number of sites that diverge in the thermodynamic limit.
Here $D$ ($0<D<1)$ is a fractal dimension that can be extracted, e.g., from
inverse participation ratio (IPR) Macé  _et al._ (2019); Roy and Logan (2021),
and is expected to be close to the spectral fractal dimension $D_{s}$
Altshuler  _et al._ (2016); Roy  _et al._ (2023) discussed in Sec.III.1.
However, the fraction ($\sim\mathcal{N}_{F}^{-(1-D)}$) of such FS sites goes
to zero in the thermodynamic limit indicating the multifractality of these
states. Nevertheless, we find that the typical off-diagonal propagator
$G_{MM}$, which captures the correlations of amplitudes at different sites,
decreases exponentially with the hopping distance $r$ in the MBL phase. Thus,
we can define a Fock space localization length $\xi_{F}^{(1)}(W,L)$ associated
with the exponential decay of $G_{MM}(r)$ through

$G_{MM}(r)\sim\exp{\left(-\frac{r}{\xi_{F}^{(1)}}\right)}$

We call $\xi_{F}^{(1)}$ a typical FS localization length since it involves
correlations of amplitudes between two generic sites.

Fig. 6(b) shows the variation of $\xi_{F}^{(1)}$ as a function of $W$ for the
largest system size, $L=20$. We find $\xi_{F}^{(1)}$ to have little system
size dependence (see Appendix C). The localization length increases as one
approaches the critical point $W_{c}$ from the MBL side, but seems to remain
finite at the critical point with
$\xi_{F,c}^{(1)}=\xi_{F}^{(1)}|_{W=W_{c}}\simeq 2.5$. This behaviour is
similar to the FS localization length extracted from a somewhat different
correlation function involving the square of the modulus of the FS amplitudes
in Ref.Roy and Logan (2021). The length scale $\xi_{F}^{(1)}$, which remains
finite at the MBL transition, also mimics the length scale associated with
either the local support or the associated decay of matrix elements, of the
LIOMs or the $l$-bits Serbyn  _et al._ (2013a); Huse  _et al._ (2014);
Chandran  _et al._ (2015). Furthermore, the value of $\xi_{F,c}^{(1)}$ is
quite close to the critical value $\sim 2/\ln(2)$ expected for the LIOM length
scale at the MBL transition De Roeck and Huveneers (2017); Potirniche  _et
al._ (2019); Crowley and Chandran (2022), although the direct connection
between the $l$-bit length scale and the FS length $\xi_{F}^{(1)}$ is not
clear at this stage. Nevertheless, the similarity motivates us to define
another length scale $\zeta^{(1)}=[1/\xi_{F}^{(1)}-1/\xi_{F,c}^{(1)}]^{-1}$,
which diverges at the transition. We find
$\zeta^{(1)}\sim|W-W_{c}^{+}|^{-\nu}$ with $\nu=0.9$, as shown in the inset of
Fig.6(b). Following Ref.Crowley and Chandran (2022), $\nu\approx 1$ suggests a
similarity of $\zeta^{(1)}$ with one of the length scales associated with the
probability of finding resonance between two eigenstates which differ by
rearrangements of $r$ local $l$ bits.

Atypical FS localization length.— We now discuss the localization length that
can be extracted from the non-generic off-diagonal propagator $G_{1I}$ where
$I$ belongs to the the middle of FS graph (Fig.1). Fig. 7(a) shows the
variation of the typical value of $G_{1I}$ with system sizes. The typical
value $G_{1M}$ for $G_{1I}$ is calculated as $\ln G_{1M}=\left\langle\ln
G_{1I}\right\rangle_{I;\phi}$ where $\langle~{}.~{}\rangle_{I;\phi}$ denotes
averaging over all FS sites belonging to the middle slice and different
realizations of offset angle $\phi$. Since all sites $I$ belong to the middle
slice, the hopping distances of these sites $I$ from the apex site $1$ are the
same. Thus, we cannot extract a length scale from the variation of the
$G_{1M}$ with hopping distance, as in the case of $G_{MM}$ discussed above.

However, a length scale can be extracted from the system size dependence of
$G_{1M}$. As shown in Fig.7(a), in the thermal phase, the typical value
$G_{1M}$ is almost independent of the system size. On the other hand, in the
MBL phase, the typical value decreases with the Hilbert space dimension
$\mathcal{N}_{F}$. We assume a scaling form $\ln G_{1M}\sim
f(\ln\mathcal{N}_{F}/\xi_{F}^{(2)})$ to collapse $\ln G_{1M}$ for different
$W$ and different system sizes. Here $\xi_{F}^{(2)}$ denotes an atypical FS
localization length. The good quality of the scaling collapse shown in Fig.
7(b) indicates that $\ln G_{1M}$ indeed is function of
$\ln\mathcal{N}_{F}/\xi_{F}^{(2)}$. The extracted localization length is shown
in the inset of Fig. 7(b). Very similar to $\xi_{F}^{(1)}$, $\xi_{F}^{(2)}$
also decreases with increasing potential strength in the MBL phase
($W>W_{c}$), approaching a finite value $\xi_{F,c}^{(2)}$ at $W_{c}$. However,
$\xi_{F,c}^{(2)}$ substantially smaller than $\xi_{F,c}^{(1)}$. Again, a
length scale $\zeta^{(2)}$ defined by
$\zeta^{(2)}=[1/\xi_{F}^{(2)}-1/\xi_{F,c}^{(2)}]^{-1}$ diverges near the
critical point $W_{c}$ following a power law $|W-W_{c}|^{-\nu}$ with $\nu=0.9$
(not shown). Therefore, the atypical length scales $\xi_{F}^{(2)}$ and
$\zeta^{(2)}$ exhibit similar critical properties as $\xi_{F}^{(1)}$ and
$\zeta^{(1)}$, respectively, and might also be related to length scales
associated with LIOMs. However, the exponent $\nu\approx 1$ associated with
divergence of the length scales $\zeta^{(1)},~{}\zeta^{(2)}$ at $W_{c}^{+}$
appear to be discernibly different from the exponent $\beta\approx 0.5$ for
the length scale $\xi$ extracted from the finite-size scaling of the typical
value of Feenberg self-energy in Sec.III.1. We also note that both the
exponents, particularly $\beta$, associated with diverging FS length scales
violate the Harris-Luck bound Luck (1993), which requires $\beta,\nu\geq 1/d$
for $d$-dimensional quasiperiodic systems. This is believed to be a common
artifact of small system sizes accessed in all the _ab initio_ numerical
studies of MBL systems, e.g., via ED. The behaviours of the FS localization
lengths for quasiperiodic disorder discussed here are found to be very similar
to that for the random disorder case Sutradhar  _et al._ (2024).

Figure 8: Sample to sample fluctuations in the typical value of the Feenberg
self-energy $\ln\Delta_{t}$ for (a) random disorder and (b) the quasiperiodic
potential. The peak in the fluctuations increases with system size and becomes
sharper near the thermal-MBL transition in the random disorder case. In
contrast, in the case of the quasiperiodic potential, while the fluctuation
peaks near the transition, the peak height decreases with system size instead
of increasing. The critical potential strength is indicated by the vertical
dashed line in the plots for reference.

###  III.4 Sample-sample fluctuations of FS propagators for quasiperiodic and
random disorders

In the previous subsection, we have shown that the typical values associated
with distributions of quantities related to the local and the non-local parts
of FS Green’s function over various FS sites and disorder realizations exhibit
quite sharp signatures of MBL transition in terms of different diverging
length scales, at least for the finite systems accessed in this work. These
analyses, though give important insights into the structure of the Green’s
function in the FS, do not provide any distinctions between random and
quasiperiodic disorders from the perspective of Fock space. However, the
quasiperiodic potential is very different from the true random disorder owing
to the deterministic nature of the former. Since the typical values fail to
capture any difference between the two, we analyze the quantities of interest
further by calculating the sample-to-sample fluctuations. In Ref.Khemani  _et
al._ , 2017, the MBL transitions in the random and quasiperiodic systems have
been argued to be in different universality classes due to differences in
fluctuations of entanglement entropy from one realization of potential (_a
sample_) to other, and fluctuations within a given sample over different
eigenstates.

Here, we define the sample-to-sample fluctuation of the Feenberg self-energy
as $\mathcal{F}[\ln\Delta_{t}]=\sigma~{}[\ln\Delta_{t}^{(s)}]/\ln\Delta_{t}$,
where $\ln\Delta_{t}^{(s)}=\left\langle\ln\Delta_{I}\right\rangle_{I}^{(s)}$
is the typical value for a given sample $s$ for the distribution of
$\Delta_{I}$ over different sites in the middle slice (Fig. 1), and
$\sigma~{}[\ln\Delta_{t}^{(s)}]$ is the standard deviation of these typical
values over different samples.

For the non-local part of the Green’s function, we analyze the fluctuation in
the quantity $G_{IJ}$ for $r_{IJ}=L-2$. The fluctuation is defined as
$\mathcal{F}\left[\ln G_{MM}\right]=\sigma~{}[\ln G_{MM}^{(s)}]/\ln G_{MM}$,
where $\ln G_{MM}^{(s)}=\left\langle\ln G_{IJ}\right\rangle_{I,J}^{(s)}$ is
the typical value calculated for each sample $s$ by averaging over all the
pairs $I,J$ with hopping distance $r_{IJ}=L-2$. Similar to self-energy,
$\sigma[\ln G_{MM}^{(s)}]$ is the standard deviation of the distribution of
$\ln G_{MM}^{(s)}$ over different samples. Here, we only consider the FS site
pairs with hopping distance $L-2$. This choice is not special; any $r_{IJ}<L$
gives similar results. The statistics become poor for FS site pairs with
hopping distances larger than $L-2$, as the number of such pairs becomes very
small for all the system sizes accessed here [see Appendix A]. Fluctuations of
the atypical element $G_{1M}$ give very similar results, as shown in Appendix
B.

Figure 9: Sample to sample fluctuation in the typical value of off-diagonal
elements $G_{IJ}$ ($\ln G_{MM}$) for the random disorder (a) and the
quasiperiodic potential (b). The fluctuation changes almost monotonically as a
function of potential strength $W$. The critical potential strength is
indicated by the vertical dashed line in the plots for reference.

Fig. 8 shows the fluctuation in self-energy as a function of the potential
strength $W$. Deep in the thermal phase and MBL phase, the fluctuations are
very small and they decrease with $L$. The fluctuation peaks near, but
somewhat below, the critical point $W_{c}$ on the thermal side. Incidentally,
for both quasiperiodic and random systems, the peak position is very close to
the potential strength where the distribution of Feenberg self-energy becomes
nearly Gaussian, as discussed in Sec.III.2. In the presence of true random
disorder [Fig. 8(a)], the fluctuation’s peak height increases rapidly with
increasing system size and the peak becomes sharper. On the contrary, in the
case of the quasiperiodic potential [Fig. 8(b)], though the sample-to-sample
fluctuation of the Feenberg self-energy exhibits a peak near the transition,
the peak height diminishes with increasing system size.

This increasing fluctuation in the random disorder case might be attributed to
Griffiths-like rare region effects Gopalakrishnan  _et al._ (2016); Agarwal
_et al._ (2015, 2017), known to occur in the presence of true random disorder.
These are related to the presence of rare localized (thermal) regions in an
otherwise thermal (localized) system due to finite but large regions with
atypical strong (weak) disorder near the transition for $W<W_{c}$ ($W>W_{c}$).
In the thermal phase, these rare regions hinder transport and strongly affect
characteristic time scales related to relaxation, whereas, in the MBL phase,
they act like a finite-size bath Agarwal  _et al._ (2017); De Roeck and
Huveneers (2017). In fact, these rare region effects are known to dominate
transport over an extended range of disorder near the MBL transition, but
somewhat below it, on the thermal side Agarwal  _et al._ (2015, 2017). Thus,
in the thermal phase near $W_{c}$, disorder realizations with such rare
regions are expected to have a very small typical value of the self-energy
compared to a typical disorder realization. Therefore, the sample-to-sample
fluctuations of the typical Feenberg self-energy may become very large
approaching $W_{c}$ from the thermal phase with increasing system size in the
case of true random disorder. In contrast, for the deterministic quasiperiodic
potential, naively, one does not expect any rare regions to exist and no such
enhancement of the fluctuations of Feenberg self-energy.

Fig. 9 shows that the sample-to-sample fluctuations in $G_{MM}$ vary almost
monotonically with increasing potential strength $W$, unlike the fluctuations
in Feenberg self-energy. However, in close inspection, a weak _hump_ like
feature seems to develop for larger systems in the thermal region near the
transition. Nevertheless, there is not much qualitative difference between the
random disordered and quasiperiodic potentials when viewed through the lens of
fluctuations in the off-diagonal elements of Green’s functions. Similar
variations are seen in the off-diagonal elements $G_{1I}$ as shown in Appendix
B. Thus, the FS localization lengths $\xi_{F}^{(1)},~{}\xi_{F}^{(2)}$, which
remain finite at the transition, seem to be more or less unaffected by the
putative rare region effects, in contrast to the characteristic rate or time
scale obtained from the Feenberg self-energy.

##  IV Conclusion

In this paper, we provide a characterization of the thermal and MBL phases,
and the phase transitions between them, in the Fock space for a one-
dimensional quasiperiodic system in terms of the many-body FS propagator or
Green’s function. In particular, we focus on how the difference of real-space
correlations, or lack thereof, associated with quasiperiodic and random
systems manifests in Fock space. To this end, we compute the diagonal and off-
diagonal elements of the FS propagator at an energy in the middle of the many-
body spectrum using an efficient recursive Green’s function method that can
access system sizes comparable to state-of-the-art exact diagonalization
studies Sierant  _et al._ (2020); Pietracaprina  _et al._ (2018). We show that
the Feenberg self-energy extracted from the diagonal part of the Fock space
propagator follows different finite-size scaling forms in the thermal and the
MBL phases. In the MBL phase for the quasiperiodic system, the self-energy
captures characteristic features of the multifractality of the MBL
eigenstates, previously seen for the random case Sutradhar  _et al._ (2022).
The typical off-diagonal elements of the FS propagator also vary quite
differently in the two phases. We extract typical and atypical FS localization
lengths from the off-diagonal propagator connecting different parts of the FS
graph and show that these length scales obey similar critical properties. In
particular, the FS localization lengths increase approaching the transition
from the MBL phase, but remain finite at the transition, closely mimicking the
real-space length scales associated with $l$-bits that appear in the effective
description of the MBL phase Serbyn  _et al._ (2015); Huse  _et al._ (2014);
Chandran  _et al._ (2015). However, these features are very similar to the
observations in the case of random disorder Sutradhar  _et al._ (2022); Roy
_et al._ (2023), and, thus, the typical diagonal and off-diagonal FS
propagators cannot distinguish quasiperiodic and random systems in Fock space.
However, we show that the sample-to-sample fluctuations of the typical
Feenberg self-energy vary differently as a function of potential strength and
system size for the quasiperiodic and random systems.

The fluctuation in self-energy is found to be small deep in the thermal phase
and MBL phases, whereas the fluctuation exhibits a peak on the thermal side
near the transition. For the random system, the peak becomes higher and
sharper with increasing system size, although the fluctuation decreases with
system size deep in the phases. In contrast, the peak and the overall
fluctuation for the entire range of potential strength monotonically decreases
with system size for the quasiperiodic system. We discuss the plausible
connection of the different system size dependence of the inter-sample
fluctuation of self-energy for random and quasiperiodic systems with the
presence and absence of rare regions, respectively. We also show that the
sample-to-sample fluctuation in the off-diagonal elements of the FS propagator
associated with the FS localization lengths does not capture any difference
for the two different potentials.

In recent years, the stability of the MBL phase, even in one dimension, has
been questioned and intensely debated Šuntajs  _et al._ (2020); Sels and
Polkovnikov (2021); Sels (2022). Evidence of long-range many-body resonances
Morningstar  _et al._ (2022), that almost certainly push the MBL transition to
a much higher value of critical disorder or destroy the MBL phase altogether
Sels and Polkovnikov (2021); Sels (2022), has been found. In the latter case,
the sharp signatures of the MBL transition found here, or in many other
previous works using, e.g., ED Luitz  _et al._ (2015), for finite-systems
should be understood as a finite-size crossover that would go away when the
system size becomes large enough to accommodate the long-range resonances
Crowley and Chandran (2022). The MBL phase and transition in this case should
be dubbed as _finite-size MBL_ phenomena. Nevertheless, the finite-size MBL
phenomenology will still remain very much relevant for experimentally
accessible system sizes Schreiber  _et al._ (2015), as the destabilization of
the MBL by the long-range resonances will presumably require humongous system
sizes, especially for strong enough disorder.

In the scenario where the resonances only shift the critical disorder to a
higher value, we can broadly classify the mechanisms for the thermal-MBL
transition into two further scenarios, though not mutually exclusive. In one
of them, the rare regions of weak disorder in an otherwise localized system
occur due to the randomness of the disorder potential. These rare regions
proliferate Morningstar and Huse (2019); Goremykina  _et al._ (2019);
Dumitrescu  _et al._ (2019) with decreasing disorder strength finally
destabilizing the MBL phase through the ‘quantum avalanche’ procedure De Roeck
and Huveneers (2017) in random systems. These rare regions are absent in the
presence of deterministic potentials like the quasiperiodic potentials. Thus,
the avalanche instability is not expected to occur in the quasiperiodic
systems, unlike the random case. However, in interacting quasiperiodic
systems, Hartree-Fock shifts in the many-body states or in the initial
condition for non-equilibrium time evolution, can still lead to effective rare
regions Agrawal  _et al._ (2022). Numerical ED studies Khemani  _et al._
(2017), as well as phenomenological real-space RG studies Zhang and Yao (2018,
2019), have shown evidence of different universality classes for MBL
transition in quasiperiodic systems compared to the random ones. Nevertheless,
the phenomenological RSRG studies point to different critical properties for
quasiperiodic MBL transition than the random case Zhang and Yao (2018, 2019).

In the alternative picture Crowley and Chandran (2022) for the MBL transition,
a phenomenological random resonance model has been developed without
explicitly assuming the presence of rare regions. Here many-body resonances
have been argued to destabilize the MBL phase for large systems till much
stronger disorder, e.g., $W\gg 3.7$. For system sizes smaller than the
characteristic length scale Crowley and Chandran (2022) of the resonances,
like the system sizes accessed via ED and other related methods, only finite-
size crossover between the thermal and MBL phases are observed. Since the ED
studies are limited to relatively small finite-size systems and the real-space
RG methods Potter  _et al._ (2015); Vosk  _et al._ (2015); Morningstar and
Huse (2019); Goremykina  _et al._ (2019); Dumitrescu  _et al._ (2019) relies
on phenomenological rules that cannot be derived directly from the microscopic
models, the questions on the mechanism of MBL transitions in both random and
quasiperiodic systems and essential differences between MBL phenomena in the
two types of systems remain elusive. Our work, which provides a Fock space
approach to the quasiperiodic MBL and its similarities and differences from
MBL phenomena in random systems, may be useful to build better insights into
the outstanding unresolved questions through a perspective complimentary to
the real-space approaches.

##  V Acknowldegements

We acknowledge useful discussions with Nilanjan Roy, Sthitadhi Roy and David
Logan during collaboration on related topics. SM and SB thank the QuST project
of the DST, Govt. of India for support. S. B. acknowledges support from SERB
(Grant No. CRG/2022/001062), DST, India.

##  Appendix A Dependence of the off-diagonal element $G_{IJ}$ on hopping
distances

Here we discuss the distribution of the number of pairs of Fock space sites
$I,J$ as a function of the hopping distance $r_{IJ}$. For any system size $L$
and Fock space dimension $\mathcal{N}_{F}=\binom{L}{L/2}$, the number of pairs
peaks at some hopping distance after which it decreases rapidly. The largest
possible hopping distance $r_{max}$ for sites belonging to the middle slice
depends both on system size and the number of particles. Fig. 10 shows the
normalized distribution of these hopping distances. The number of pairs with
$r_{IJ}\geq L$ is negligibly small for all system sizes.

Figure 10: Normalized density of the number of FS lattice pairs as a function
of the hopping distance between them. The number of FS lattice pairs in the
middle slice having a hopping distance larger than $L$ is negligibly small for
all system sizes. The x-axis is rescaled by the system size for a better
comparison between different system sizes.

Fig. 11 shows the distribution of $\ln G_{IJ}$ over different pairs $I,J$ for
different choices of $r_{IJ}$. Both in the thermal phase (a) and the MBL phase
(b), the distribution of $\ln G_{IJ}$ looks very similar for different choices
of $r_{IJ}$. Deep in the thermal phase, the distribution is essentially
independent of the hopping distance $r_{IJ}<L$. For $r_{IJ}\geq L$, the
distribution deviates from that for smaller hopping distances and picks up a
dependence on $r_{IJ}$. Further, the distribution for hopping distances close
to $r_{max}$ becomes ill-defined for some system sizes (not shown here). For
these reasons, we analyze the behavior of typical value $\ln G_{MM}$ only up
to hopping distances $L$.

Figure 11: Distribution of $\ln G_{IJ}$ over different pairs of $I,J$
connected by hopping distances $r_{IJ}$ at system size $L=18$. The solid lines
are for hopping distances $r_{IJ}<L$, while the dashed lines are for hopping
distances $r_{IJ}\geq L$. Panel (a) shows the distributions deep in the
thermal phase ($W=0.75$), while panel (b) shows the same deep in the MBL phase
($W=5.5$).

##  Appendix B Fluctuation in $\ln G_{1M}$

We have discussed the sample-to-sample fluctuations of the typical value of
the generic off-diagonal element $G_{IJ}$ between two FS sites $I,J$ in
Sec.III.4. Here, in Fig.12, we show similar results for the element $G_{1I}$
between the atypical apex site $1$ of the FS graph (Fig. 1) and a site $I$ in
the middle slice.

Figure 12: Sample-sample fluctuation in the typical value of off-diagonal
element $G_{1I}$ ($\ln G_{1M}$) for (a) the random disorder and (b) the
quasiperiodic potential. The fluctuation changes monotonically as a function
of potential strength $W$.

##  Appendix C System size dependence of Fock space localization length

Here, we show the variation of the typical Fock space localization length
$\xi_{F}^{(1)}$ with potential strength $W$ ($>W_{c}$) for two different
system sizes $L=18$, $20$. We extract the $\xi_{F}^{(1)}$ from $G_{MM}(r)$
using $G_{MM}(r)\sim\mathrm{exp}\left(-r/\xi_{F}^{(1)}\right)$. We also show
the atypical FS length $\xi_{F}^{(2)}$ extracted from $G_{1M}$ as discussed in
Sec. III.3. The atypical FS localization length decreases with increasing
potential strength in the same qualitative manner as $\xi_{F}^{(1)}$ but has
significantly smaller values than $\xi_{F}^{(1)}$.

Figure 13: Variation of the FS localization length $\xi_{F}^{(1)}$ extracted
from $G_{MM}$ for different system sizes $L=18$, $20$. It shows similar
qualitative behavior as the other FS localization length scale $\xi_{F}^{(2)}$
extracted from $G_{1M}$.

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