††thanks: Those authors contributed equally to this work.††thanks: Those
authors contributed equally to this work.

# Disorder in interacting quasi-one-dimensional systems: flat and dispersive
bands

Mi-Ji Liang  Key Laboratory of Quantum Theory and Applications of MoE $\&$
Lanzhou Center for Theoretical Physics $\&$ Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
Yong-Feng Yang  Key Laboratory of Quantum Theory and Applications of MoE $\&$
Lanzhou Center for Theoretical Physics $\&$ Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
Chen Cheng<EMAIL_ADDRESS>Key Laboratory of Quantum Theory and
Applications of MoE $\&$ Lanzhou Center for Theoretical Physics $\&$ Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou, Gansu 730000, China  Rubem Mondaini<EMAIL_ADDRESS>Beijing
Computational Science Research Center, Beijing 100193, China

###### Abstract

We use the density-matrix renormalization group method to investigate the
superconductor-insulator transition (SIT) in disordered quasi-one-dimensional
systems. Focusing on the case of an interacting spinful Hamiltonian at
quarter-filling, we contrast the differences arising in the SIT when the
parent non-interacting model features flat or dispersive bands. Furthermore,
we unveil the critical disorder amplitude that triggers insulating behavior by
comparing disorder distributions that preserve or not SU(2)-symmetry. While
scaling analysis suggests the transition be of a Berezinskii-Kosterlitz-
Thouless type for all models (two lattices and two disorder types), only in
the flat-band model with Zeeman-like disorder the critical disorder is
nonvanishing. In this sense, the flat-band structure does strengthen
superconductivity in the presence of attractive interactions. For both flat
and dispersive band models, i) in the presence of SU(2)-symmetric random
chemical potentials, the disorder-induced transition is from superconductor to
insulator of singlet pairs; ii) for the Zeeman-type disorder, the transition
is from superconductor to insulator of unpaired fermions. In all cases, our
numerical results suggest no intermediate disorder-driven metallic phase.

##  I Introduction

Band dispersion naturally affects the physics of quantum systems. Compared to
regular dispersive bands, systems exhibiting flat bands support abundant
phenomena such as topological insulating/superconducting physics, various edge
states, and exotic superfluid phases. In the noninteracting case, a purely
flat band has constant energy as a function of quasimomentum. For a particle
loaded in a flat band, the high degeneracy causes it to localize in a compact
form within a few sites whose geometry depends on the details of the
Hamiltonian [1, 2, 3, 4]. Any finite interaction will be much larger than the
bandwidth, leading to rich strongly-correlated physics at any value of the
interaction strength.

Among the many interesting aspects of such systems, one of the interests lies
in the interplay of the flat band structure, and superconductivity [5, 6, 7,
8, 9, 10, 11, 12, 13]. Studies on topological models suggest that the isolated
flat bands have much higher superconducting transition temperature [5, 6, 7].
In lattice models with flat bands, the preformed pairs dominate transport even
above the critical temperature of the transition to a superfluid state [8].
Compared to a standard two-leg fermionic ladder, recent work argues that the
Creutz lattice [14] exhibiting a flat dispersion in the non-interacting regime
has longer-ranged pairing correlation function, suggesting a more robust
pairing and superconductivity [9].

One way to further probe whether pairing and superconductivity are enhanced in
systems with flat dispersion is to estimate their robustness against disorder.
In the absence of interactions, disorder generically localizes all single-
particle eigenstates and induces Anderson localization [15]; in flat-band
systems, such localization phenomenon still occurs but with characteristic
critical exponents that depend on specific details [16, 17], including the
existence of coupling to dispersive bands [18].

At high energies, the interplay between disorder and interactions can lead to
disorder-free flat-band localization at weak disorder, and conventional
disorder-induced many-body localization in the strong-disorder regime [19, 20,
21, 22, 23]. Turning to the low-energy but yet interacting scenario,
sufficient disorder destroys the phase coherence associated with
superfluid/superconducting order, leading to a superconductor-insulator
transition (SIT) 111SIT can also occur in clean systems through the tuning of
Hamiltonian parameters that govern the single-particle dispersion, e.g., see
Refs. [68, 79, 76, 69].. However, whether the route to insulating behavior
proceeds through the direct localization of Cooper pairs [25] or via the
destruction of Cooper pairing then followed by the standard localization of
single electrons is still unsettled [26, 27]. Although disorder-induced
ground-state transitions have been extensively studied for either spin or
bosonic lattices [28, 29, 30, 31], the transition type and the universality
class of disorder-driven SIT in the presence of both charge and spin degrees
of freedom remains elusive.

In this work, we aim to systematically investigate the disorder-induced SIT in
a fully interacting setting from the perspective of how robust is the pairing
and superconductivity against disorder in systems with either flat or
dispersive bands. Additionally, we are also interested in details of the SIT,
including its universality class, via proper finite-size scaling from
numerically exact calculations of systems with different sizes. Specifically,
we focus on the attractive Fermi-Hubbard model on the Creutz lattice with flat
dispersion in the noninteracting regime, and carefully examine the pairing and
superconductivity via energies, superfluid densities, and correlation
functions. We benchmark our results with a regular two-leg ladder with
dispersive bands to study how band dispersion affects pairing and
superconductivity under the influence of disorder.

The rest of the paper is organized as follows. In Sec. II, we introduce the
Hamiltonian on two lattice types with different dispersions, two types of
disorder, and also our numerical method. Section III is devoted to the finite-
size scaling of the superfluid weight, where we discuss the universality class
of disorder-induced SIT. In Sec. IV, we further analyze the ground-state phase
transition and corresponding phases via correlation functions. The summary of
the results is presented in Sec. V.

##  II Model and Method

We first consider the attractive Hubbard model on the Creutz lattice described
by the Hamiltonian:

$\displaystyle\hat{\cal H}_{C}=$ $\displaystyle-{\rm
i}t\sum_{j,\sigma}\left(\hat{c}^{A\dagger}_{j,\sigma}\hat{c}^{A}_{j+1,\sigma}-\hat{c}^{B\dagger}_{j,\sigma}\hat{c}^{B}_{j+1,\sigma}-{\rm
H.c.}\right)$
$\displaystyle-t\sum_{j,\sigma}\left(\hat{c}^{A\dagger}_{j,\sigma}\hat{c}^{B}_{j+1,\sigma}+\hat{c}^{B\dagger}_{j,\sigma}\hat{c}^{A}_{j+1,\sigma}+{\rm
H.c.}\right)$
$\displaystyle+U\sum_{j,\alpha}\hat{n}^{\alpha}_{j,\uparrow}\hat{n}^{\alpha}_{j,\downarrow},$
(1)

where $\hat{c}^{\alpha\dagger}_{j,\sigma}$ ($\hat{c}^{\alpha}_{j,\sigma}$)
creates (annihilates) a fermion with spin $\sigma=\uparrow,\downarrow$ on the
$j$-th unit cell with chain index $\alpha=A,B$ [see cartoon in Fig. 1(a)];
$\hat{n}^{\alpha}_{j,\sigma}=\hat{c}^{\alpha\dagger}_{j,\sigma}\hat{c}^{\alpha}_{j,\sigma}$
is the corresponding the number-density operator. For comparison, we also
examine the attractive Hubbard model on the regular two-leg ladder:

$\displaystyle\hat{\cal H}_{L}=$
$\displaystyle-t\sum_{j,\sigma}\left(\hat{c}^{A\dagger}_{j,\sigma}\hat{c}^{A}_{j+1,\sigma}+\hat{c}^{B\dagger}_{j,\sigma}\hat{c}^{B}_{j+1,\sigma}+{\rm
H.c.}\right)$
$\displaystyle-t\sum_{j,\sigma}\left(\hat{c}^{A\dagger}_{j,\sigma}\hat{c}^{B}_{j,\sigma}+\hat{c}^{B\dagger}_{j,\sigma}\hat{c}^{A}_{j,\sigma}+{\rm
H.c.}\right)$
$\displaystyle+U\sum_{j,\alpha}\hat{n}^{\alpha}_{j,\uparrow}\hat{n}^{\alpha}_{j,\downarrow}\
,$ (2)

where a schematic representation is shown in Fig. 1(b). For both models, the
linear lattice size is $L$, and the hopping amplitudes are proportional to $t$
(in the Creutz lattice, intrachain hoppings gain a phase, being purely
imaginary). The interactions are attractive, $U<0$, wherein fermions with
opposite spins form pairs to lower the total energy, further condensing to
form a superfluid state, provided the parent state is metallic.

Figure 1: Pair excitation spectrum,
$E_{0}(N_{\uparrow}+1,N_{\downarrow}+1)-E_{0}(N_{\uparrow},N_{\downarrow})$,
in the presence of the negative $U$ for (a) the Creutz lattice and (b) a
regular two-leg ladder. Here $E_{0}(N_{\uparrow},N_{\downarrow})$ stands for
the groundstate energy in a system with $N_{\uparrow}$ spin-up and
$N_{\downarrow}$ spin-down fermions. Insets show the lattice geometry: the red
ellipse denotes a unit cell labeled by $j$, $A$ and $B$ label two legs. For
the Creutz lattice, the arrows depict the sign of the hopping in the
intrachain bonds. For the regular two-leg ladder, all hoppings are real
without extra phases. Here the results are obtained from numerical
calculations of systems with $L=64$ with open boundary conditions.

The Creutz lattice features a flat dispersion in the noninteracting case
($U=0$), and the two-leg ladder has, on the other hand, dispersive bands. An
immediate question that arises is how the interaction affects the band
structure, which can be inferred by the pair-excitation spectrum (i.e., the
chemical potential to introduce a singlet pair) obtained from many-body
numerical calculations, as shown in Fig. 1 (direct comparison of single-
particle excitations is given in Appendix A). For the Creutz lattice, there
are two highly degenerated bands at $\pm 4t$ ($\pm 2t$ in the single-particle
picture), each with zero bandwidth at $U=0$. Their bandwidth grows in the
presence of finite interactions, but the two bands are still relatively
narrow. On the contrary, the bandwidth of the two-leg ladder is much larger.
Therefore, we assume that the difference in the band structure in the
noninteracting case would also affect the pairing and superconductivity in the
presence of interactions. Notice that the model on both lattices at half-
filling displays a vanishing superfluid weight ${\cal D}_{s}$; we focus on
filling
$\langle\hat{n}\rangle=\sum_{j,\alpha,\sigma}\langle\hat{n}_{j,\sigma}^{\alpha}\rangle/2L=1/4$,
to ensure we start from a robust superfluid state. Besides that, in what
follows, we fix the interaction strength at $U=-8$ [$t=1$ sets the energy
scale], at which ${\cal D}_{s}$ for the Creutz lattice is close to its maximum
in clean cases [9].

The robustness of the superconductivity is estimated by examining the critical
disorder to break the pairing coherence and corresponding superconducting
state. We first consider the spin-independent random chemical potentials which
do not break local singlet pairs:

$\displaystyle\hat{\cal H}_{\mu}=\sum_{i}\mu_{i}\hat{n}_{i},$ (3)

where $i$ labels a single site and $\mu_{i}\in[-W,W]$ is taken from an
uncorrelated, uniform distribution with disorder strength $W$. Alternatively,
random Zeeman-like fields introduce another kind of disorder

$\displaystyle\hat{\cal H}_{h}=\sum_{i}h_{i}\hat{S}^{z}_{i}$ (4)

with $h_{i}\in[-W,W]$ and
$S^{z}_{i}=\hat{n}_{i,\uparrow}-\hat{n}_{i,\downarrow}$. The latter breaks
SU(2)-symmetry and tends to disassemble pairs (local or not).

To solve Eqs. (II) and (II) in the presence of disorder, we numerically employ
the density matrix renormalization group (DMRG) [32, 33] method, which is
extremely powerful in (quasi-) one-dimensional systems, to obtain the ground
state of different lattices, including under disordered settings. The two
Hamiltonians have $U(1)$ symmetry with conserved total particle number
$N_{\sigma}=\sum_{i}\langle\hat{n}_{i,\sigma}\rangle$ for spin species
$\sigma$ even in the presence of disorder, thus we perform DMRG calculations
in the sector with fixed good quantum numbers $N_{\sigma}$. Observables such
as superfluid weight, pair binding energy, and correlation functions are
computed to characterize the ground state properties. In calculations aiming
to obtain the superfluid weight, twisted boundary conditions are used; in the
rest of the simulations, we implement open boundary conditions to reduce the
computational cost. Up to 2000 DMRG kept states are used in all calculations,
and the largest truncation error is about $10^{-6}$. For disordered cases, all
observables are obtained from the average over calculations of many disorder
samples as indicated in what follows [see Appendix B for a benchmark against
exact results in small systems.].

##  III BKT Scaling of Drude weight

The pairing of electrons is one of the necessary preconditions of
superconductivity, which is a macroscopically coherent state of pairs. We
estimate the pair formation via the singlet-pair binding energy

$\displaystyle E_{b}\equiv~{}$ $\displaystyle
E_{0}(N_{\uparrow}+1,N_{\downarrow}+1)+E_{0}(N_{\uparrow},N_{\downarrow})$
$\displaystyle-2E_{0}(N_{\uparrow}+1,N_{\downarrow}).$ (5)

A negative $E_{b}$ in the thermodynamic limit denotes that the energy cost for
adding two interacting particles (or holes, depending on the filling) with
opposite spins is lower than that of two noninteracting ones. As a result, the
system exhibits a tendency toward the singlet-pair formation to lower the
total energy. We first display the pairing binding energy in the presence of
random chemical potentials in Fig. 2. In this case, the binding energy $E_{b}$
is negative at zero disorder, and remains so with $W>0$, for both the Creutz
lattice and regular two-leg ladder. Therefore, from the energetic
consideration, fermions in the presence of random chemicals still tend to form
pairs, despite the inclusion of disorder.

Figure 2: Pairing binding energy $E_{b}$ as a function of disorder $W$ for (a)
the Creutz lattice and (b) regular two-leg ladder with different $L$. Here the
disorder is introduced by the random chemical potentials.

Whereas useful to characterize singlet pair formation, the binding energy
cannot quantify the coherence of such pairs, being thus unable to discern the
superconducting state. For that, we examine the superfluid weight, which in
one-dimension (1D) is equivalent to the Drude weight [34, 35, 36, 37, 38, 39,
40]:

$\displaystyle{\cal D}_{s}=\pi
L\frac{\partial^{2}E_{0}(\Phi)}{\partial\Phi^{2}}\bigg{|}_{\Phi=0},$ (6)

where $E_{0}(\Phi)$ is the ground state in the presence of a threaded magnetic
flux $\frac{\hbar c}{e}\Phi$ [41, 42]. Such flux is equivalent to the
introduction of twisted boundary conditions [43] via the replacement
$\hat{c}_{j,\sigma}\rightarrow e^{\mathrm{i}\phi j}\hat{c}_{j,\sigma}$, where
$\phi=\Phi/L$ is the phase gradient per unit cell. In actual calculations, we
use an approximant ${\cal D}_{s}\approx 2\pi
L[E_{0}(\delta\Phi)-E_{0}(0)]/(\delta\Phi)^{2}$ [29], choosing
$\delta\Phi=\pi/2$ to minimize the numerical error. Thus the superfluid weight
is obtained by

$\displaystyle{\cal
D}_{s}\approx\frac{8L}{\pi}\left[E_{0}(\pi/2)-E_{0}(0)\right].$ (7)

The approximation in Eq. 7, while seemingly crude for such a large
$\delta\Phi$, has been numerically confirmed in the clean case, resulting in
an absolute error of the order of $10^{-3}$ [see Appendix C].

Figure 3: Data collapse of the superfluid weight ${\cal D}_{s}$ as a function
of $L/\xi$ [$-L/\xi$ if $W<W_{c}$] for (a) the Creutz lattice and (b) regular
two-leg ladder with random chemical potentials. Each inset shows ${\cal
D}_{s}$ as a function of the disorder $W$ for different lattice lengths $L$.
The optimal parameters $b$ and $W_{c}$ in Eq. 8 are determined by minimizing a
cost function of the data collapse [see Appendix D for detailed information].
Numerical results for $L=12$/$16$/$20$/$24$ are obtained by the average over
320/256/160/96 disorder realizations.

In either 1D or quasi-1D lattices, the quantum phase transition between the
Mott insulating phase and the superfluid phase at fixed commensurate lattice
filling is known to be of the Berezinskii-Kosterlitz-Thouless (BKT) type [28,
44, 45, 46]. In the case of disordered systems, such scaling form persists for
bosonic systems when transitioning from a superfluid to a Bose glass [47, 46].
A low-energy effective theory (bosonization) has also been developed for the
superfluid-disordered insulating transition in the case of fermionic two-leg
ladders [48, 49, 50], suggesting a BKT-type phase transition for the
dispersive model. In particular, we notice that the attractive Hubbard model
displays the formation of increasingly local Cooper pairs when $|U|\gg 1$,
which are mimicked by hardcore bosons in this limit [51, 52, 53]. Moreover, we
recall that in the clean case, the results of the superfluid weight at the
strong, attractive interactions we use, $U=-8$, steadily approach the ones for
the corresponding bosonic model [9], lending further support for the same type
of transition similarly occurring here 222Similar arguments are valid for
clean SIT transitions [68, 69]. There, on the other hand, the transition is of
second order [$(d+1)-XY$ universality class], and the fermionic transition
shares the same scaling properties of the corresponding mapped bosonic model
[80, 81]. Thus assuming a BKT scaling form for the superconductor to the
disorder-induced insulator transition (whether the bands are dispersive or
not), the disorder-dependent correlation length scales as [55, 56, 57, 58, 59]

$\displaystyle\xi=\exp\left\\{\frac{b_{\pm}}{\sqrt{|W-W_{c}|}}\right\\}.$ (8)

Here $W_{c}$ is the critical disorder in the thermodynamic limit, and $b_{+}$
($b_{-}$) is a nonuniversal parameter for $W>W_{c}(W<W_{c})$. For numerical
convenience, we make the approximation that $b_{+}=b_{-}\equiv b$. Then the
critical disorder and the parameter $b$ can be determined by the best data
collapse of ${\cal D}_{s}(L,W)$ as a function of $L/\xi$.

Figure 4: Pairing binding energy $E_{b}$ as a function of disorder $W$ for (a)
the Creutz lattice and (b) regular two-leg ladder with different $L$. Here the
disorder is introduced by the random Zeeman field.  Figure 5: Data collapse of
the superfluid weight ${\cal D}_{s}$ versus $L/\xi$ [$-L/\xi$ if $W<W_{c}$]
for (a) the Creutz lattice and (b) regular two-leg ladder with random Zeeman
fields. Other parameters are the same as in Fig. 3.

We first consider the disorder introduced by random chemical potentials; the
corresponding results of the superfluid weight and its scaling are displayed
in Fig. 3. The data collapse of the ${\cal D}_{s}(L)$ versus $L/\xi$ indicates
that the critical disorder $W_{c}=0^{+}$ in the thermodynamic limit for both
the Creutz lattice and the regular two-leg ladder, within system sizes
amenable to our calculations. In this sense, the lattice geometry and band
dispersion do not qualitatively affect the (lack of) robustness of the
superconductivity against this SU(2)-symmetric disorder. Moreover, the
superconducting state is so fragile that an infinitesimal disorder strength
destroys the superconductivity. On the other hand, since random chemical
potentials do not necessarily break pairs but rather their phase coherence,
the system is an (interacting) Anderson insulator of singlet pairs in the
disordered phase.

In contrast to the SU(2)-symmetric random chemical potential, the random
magnetic field can break the singlet pairs induced by the local attraction
$U$. In this case, the introduction of random Zeeman fields with growing
disorder strength results in the amplitude of $E_{b}$ gradually decreasing to
zero with growing $W$, regardless of the lattice geometry used, as shown in
Fig. 4. While this result immediately points out the differences arising from
the symmetry-type of disorder used on the pair robustness [see Fig. 2 for a
comparison], it does not make it clear if pairs are more resilient if
contrasting dispersive or dispersionless systems under $\hat{\cal H}_{h}$.
Owing to the lack of a proper and systematic scaling procedure for $E_{b}$,
extracting the critical disorder that breaks pairs only from the binding
energy is challenging. Consequently, the results in Fig. 4 do not conclude
whether the singlet pair is more robust against disorder in the Creutz lattice
with flat bands than a regular lattice with dispersive bands.

To solve this question, we again resort to the superfluid weight, which
further probes the phase coherence of the formed pairs. The finite-size
scaling of the superfluid weight ${\cal D}_{s}$ is reported in Fig. 5. Here,
the Creutz lattice and regular two-leg ladder results are qualitatively
different under this type of random Zeeman-like disorder. In contrast to the
case with random chemical potentials, the superconducting state survives in
the Creutz lattice up until disorder strengths of $W_{c}\simeq 4.8$, as shown
in Fig. 5(a). However, the superconductivity is still fragile and destroyed by
an arbitrarily small disorder in the regular two-leg ladder, as shown in Fig.
5(b). In this sense, the flat dispersion has dramatically enhanced the
robustness of the superfluidity, even in the presence of substantial disorder.

##  IV correlation functions

Figure 6: Pairing correlations functions $P(r)$ versus distance $r$ for (a)
the Creutz lattice and (b) the regular two-leg ladder. The solid lines give an
estimation of a power-law fitting $\propto r^{\alpha}$ extracted for the clean
case that guides the interpretation (note the log-log scale). Here the lattice
length $L=128$ and the disorder is introduced via random chemical potentials
$\hat{\cal H}_{\mu}$.

The scaling of superfluid weight suggests a BKT transition from the
superconducting state to the insulating state for both lattice geometries and
disorder types. However, the physical characteristics of the disordered phase
can be, in principle, different. For example, a large disorder can lead to an
Anderson insulator of singlet pairs or even of unpaired fermions. In this
section, we calculate the correlation functions to characterize these states
further. Specifically, we compute the pairing correlation function,

$\displaystyle
G^{P}_{ij}=\langle\hat{\Delta}^{A\dagger}_{i}\hat{\Delta}^{A}_{j}\rangle,$ (9)

where
$\hat{\Delta}^{A}_{i}=\hat{c}^{A}_{i,\uparrow}\hat{c}^{A}_{i,\downarrow}$
annihilates a local singlet pair on the $i$-th unit cell with chain index $A$,
and the single-particle Green’s function,

$\displaystyle
G^{\sigma}_{ij}=\langle\hat{c}^{A}_{i,\sigma}\hat{c}^{A\dagger}_{j,\sigma}\rangle.$
(10)

Computing correlations along one of the chains is sufficient since both
lattices have mirror symmetry across the rungs. As mentioned before, we use
open boundary conditions for calculations of correlation functions to reduce
the computational cost. In this case, for a generic two-point correlation
function $X_{ij}$ between sites $i$ and $j$, one can extract the averaged
correlation decay as a function of distance as [60, 61]

$\displaystyle X(r)=\frac{1}{\cal N}\sum_{|i-j|=r}X_{ij}\ ,$ (11)

where $\cal N$ is the total number of pairs $\\{i,j\\}$ satisfying $|i-j|=r$.
Based on this, we define the average pairing correlation function [single-
particle Green’s function] as $P(r)$ [$G(r)$].

For the clean case, the system described by the attractive Hubbard model on
both lattices features superconductivity, with power-law decaying of pair
correlations denoting quasi-long range order. When the disorder is
sufficiently strong, the system is in an Anderson-insulating ground state, and
the corresponding pairing correlation function decays exponentially. Compared
to the Creutz lattice with the flat band, $P(r)$ decays slightly faster in the
regular two-leg ladder [9]. However, the differences in pairing correlations
responses to disorder between the two lattices are marginal when random
chemical potentials are introduced, as shown in Fig. 6. In this case, $P(r)$
turns to an exponential decay as soon as a minor disorder appears for both
lattices, in agreement with the scaling of superfluid weight in Sec. III.

Those results are not fortuitous, as they are precisely aligned with analytic
ones from Refs. [48, 49, 50], using bosonization. The Tomonaga-Luttinger
exponent of the symmetric charge mode $K_{\rho+}$ describes the stability of
the superconducting phase upon inclusion of (non-magnetic) disorder. In
particular, when $K_{\rho+}>3/2\ (K_{\rho+}<3/2)$ the superconducting phase is
stable (unstable) to $W$. The singlet pair-pair correlations have a general
power-law decay of the form, $P(r)\propto r^{-\alpha}=r^{-1/2K_{\rho+}}$ [62].
In the clean case, the fittings in Fig. 6 are compatible with $K_{\rho+}\simeq
1.1$ and 0.65, for the Creutz and the regular ladder, respectively. As a
result, they both fall into the regime of unstable superconducting behavior
towards including an arbitrarily small disorder, as seen in the results of the
superfluid weight [Fig. 3] and predicted in Refs. [48, 49, 50] for dispersive
bands.

Differences between the results of the two lattices are much more prominent
when random Zeeman fields introduce the disorder, and in this case, no-
analytical results stemming from a bosonization analysis exist for such type
of magnetic disorder. As shown in Fig. 7, while $P(r)$ decays exponentially in
the presence of a weak disorder strength in the regular two-leg ladder, the
pairing correlation function in the Creutz lattice preserves a power-law form
even at very large values $W\simeq 6$, for the same system size. Notice that
although one cannot directly compare the critical disorder from correlation
functions at a finite system size with the scaling result of superfluid
weights in the thermodynamic limit, these results show qualitatively the same
conclusion: the flat-band dispersion dramatically enhances the robustness of
the superconductivity against spin-dependent disorder.

Figure 7: Similar to Fig. 6 but for disorder introduced via random Zeeman
fields $\hat{\cal H}_{h}$. The pairing correlation functions $P(r)$ roughly
keep their power-law up decay at $W\lesssim 6$ for the Creutz lattice (a); the
regular ladder (b) shows a much less resilient power-law dependence of $P(r)$.
As before, the lattice length is $L=128$ and the solid line gives an
estimation of a power-law fitting $\propto r^{\alpha}$ that guides the
interpretation. Figure 8: Single-particle Green’s functions $G(r)$ versus
distance $r$ for (a) the Creutz lattice and (b) the regular two-leg ladder,
respectively. Here lattice length $L=128$ and the disorder is introduced by
random chemical potentials $\hat{\cal H}_{\mu}$; note the vertical log-scale.
Figure 9: The same as Fig. 8 but for the case of disorder introduced by random
Zeeman fields $\hat{\cal H}_{h}$. Note the robust exponential decay for a wide
range of disorder amplitudes even if significantly far from the critical value
$W_{c}$ extracted from the scaling of ${\cal D}_{s}$.

Besides of the universality class of the disorder-induced SIT in fermion-
Hubbard models, there has been another long-standing question in this topic.
That is, whether the route to insulating behavior proceeds through the direct
localization of Cooper pairs, or by a two-step process in which the Cooper
pairing is first destroyed and then followed by the standard localization of
single electrons [26]. Alternatively, an intermediate (poor) metallic state
exists where the disorder destroys the pairing coherence, but localization
does not yet occur. We try to answer this question by examining the single-
particle Green’s function, distinguishing the metallic phase from other phases
with gapped single-particle excitations, such as the superconducting state and
insulating states.

As previously mentioned, the random chemical potential does not break (local)
singlet pairs. Thus the SIT transition is ought to be direct from the
superconducting to the pair localization state. In that case, the single-
particle Green’s function would decay exponentially as the disorder strength
$W$ grows, indicating that the single-particle gap remains open throughout the
transition. This picture is confirmed in Fig. 8, which displays the $G(r)$
decaying profile for both lattices with random chemical potentials. In
contrast, the disorder introduced by the random Zeeman fields can destroy
singlet pairs. However, such a disorder also destroys the coherence of pairs
according to our numerical results for the superfluid weight ${\cal D}_{s}$.
In Fig. 9, we find no clue of an intermediate metallic state with algebraic
decay single-particle Green’s function for both lattices with different band
dispersions [see Appendix E for energetic analysis of excitations supporting
these results]. Finally, as an addendum, we note that a true uncorrelated
Anderson insulator also exhibits gapless single-particle excitations. The fact
that we observe gapped single-particle excitations across a wide range of
disorder values, even substantially far from the SIT, indicates that
correlation effects are still significantly relevant. If there is a transition
(possibly a crossover) to such a regime, this occurs at values of $W$ where
the interaction strength $|U|$ is an irrelevant perturbation.

##  V Summary and Discussion

We systematically investigate the disorder-induced SIT of the attractive
Hubbard model in two lattices, the Creutz lattice with noninteracting flat
bands and the regular two-leg ladder with noninteracting dispersive bands. Two
disorder types have been considered, random chemical potentials, which do not
break local singlet pairs, and random Zeeman fields that do break pairs in
general. The finite-size scaling of numerically obtained superfluid weights
suggests a BKT-type phase transition for both lattices and disorder types. For
the situation of non-magnetic disorder (i.e., introduced by random chemical
potentials), an infinitesimal disorder drives the superconducting state to a
correlated Anderson insulator of singlet pairs for both lattice geometries.
For the case of the regular ladder, these results are in line with the ones
obtained by a low-energy theory of the model in the regime that the attractive
interactions are sufficiently large [48, 49, 50].

For the disorder introduced by the random Zeeman fields, the superconductivity
is more robust when the noninteracting lattice has flat bands: it requires a
significant disorder strength to break the superconducting state in the Creutz
lattice; in contrast, the critical disorder is zero in the regular two-leg
ladder. The conclusion is that the flat dispersion can enhance the
superconducting state’s resilience (in the presence of attractive
interactions), confirmed by the pairing correlation function calculations.

One aspect that can make this comparison of results between the two types of
ladder elusive is that the coordination number is not the same ($z=3$ for the
regular ladder, and $z=4$ for the Creutz ladder), which can significantly
impact the resilience of the superfluidity to disorder in both cases. A
minimal analysis that can take this difference into account is to normalize
the critical disorder $W_{c}$ by the total (including gaps) non-interacting
bandwidth ($4t$ in the Creutz lattice and $6t$ for the regular ladder). As
stated, the only case in which there is a difference between $W_{c}$ obtained
in the Creutz and the regular ladder is when Zeeman disorder is introduced.
But here, however, $W_{c}^{\rm C}\simeq 4.8t$ with $W_{c}^{\rm L}\simeq 0$,
such that no (small) deviation in the band-structure widths can account for
the different results observed. Nonetheless, we remark that this is certainly
not the case for certain repulsive models, in which the inclusion of the extra
hopping terms can fundamentally change the fate of pairing [63, 64], but
mainly in a scenario of competing orders [65], absent in our investigation.

Turning back to the original model, we also try to answer the long-standing
question in disorder-induced SIT about whether this transition is direct or a
two-step process by carefully examining the single-particle Green’s function.
Our results suggest no intermediate metallic state during the SIT process for
all parameters involved in this work. Lastly, it is worth noting that the
Hamiltonian of a Creutz ladder has already been emulated with ultracold
fermionic atoms via optical potentials [66, 67], which makes our protocol
possible for experimental verification in future investigations.

An outstanding question refers to the generality of the universality class of
disorder-driven SITs in such models. While we find a clear indication of BKT-
type phase transition, further supported by results in related bosonic systems
[46, 47] and in the case of dispersive bands for fermions [48, 49, 50], this
contrasts to SITs in clean systems using similar attractive Hubbard
Hamiltonians [68, 69], which exhibit second-order phase transitions
[$(d+1)$-XY universality class]. Whether this difference carries over to
different dimensionalities is a question that warrants future investigation.

## Acknowledgements

We thank Edmond Orignac for pointing out pertinent references for the two-leg
ladder case. C.C. was supported by the National Natural Science Foundation of
China (grant nos. 11904145, 12174167, 12247101) and the Fundamental Research
Funds for the Central Universities. R.M. thanks George Batrouni and Marcos
Rigol for discussions and collaborations in related contributions. R.M.
acknowledges support from NSFC Grants No. NSAF-U2230402, 12050410263,
12111530010, 11974039, and No. 12222401.

##  Appendix A Single-particle excitations: clean case

In the main text, we contrast the pair excitation spectrum of both the Creutz
lattice and the two-leg ladder to infer that the flatness of the Bloch bands
at $U=0$ still influences the regime of strong interactions we investigate.
Here we make a direct assessment, by computing the single-particle spectral
function,

$\displaystyle A_{k_{x},\sigma}(\omega)$
$\displaystyle=\sum_{\alpha}|\langle\phi_{\alpha}|\hat{c}_{k_{x},\sigma}^{\dagger}|\psi_{0}\rangle|\delta\left(\omega-(E_{\alpha}-E_{0})\right)$
$\displaystyle+\sum_{\alpha}|\langle\phi_{\alpha}|\hat{c}_{k_{x},\sigma}^{\phantom{\dagger}}|\psi_{0}\rangle|\delta\left(\omega+(E_{\alpha}-E_{0})\right)\
.$ (12)

In this expression, $|\psi_{0}\rangle$ is the ground-state of either (II) or
(II) at filling $\langle\hat{n}\rangle=1/4$, and the excited eigenstates
states $|\phi_{\alpha}\rangle$ are the ones from the sectors with an added
(removed) particle, $N_{\sigma}+1$ ($N_{\sigma}-1$). The operator in momentum
space is defined as
$\hat{c}_{k_{x},\sigma}^{\dagger}=\frac{1}{\sqrt{L}}\sum_{j,y}e^{{\rm
i}k_{x}j}\hat{c}_{j,\sigma}^{y\dagger}$, which has considered the summation of
the chain index $y=A,B$.

Those are directly computed by means of a Krylov-Schur-based diagonalization
method [70, 71], which we use to compute 400 eigenpairs
($E_{\alpha},|\psi_{\alpha}\rangle$) in the low-lying spectrum of the
corresponding sector, effectively truncating the summation (12). Moreover,
since the number of inequivalent $k$-points is $L$, we further improve the
statistics by averaging each momentum value among a set of 20 equidistant
twisted boundary conditions. As mentioned in the main text, these can be
equivalently interpreted as threading a flux $\Phi$ on the ring ladder and can
be used to mitigate finite-size effects when averaging over many values
$\Phi\in[0,2\pi)$ [43, 72].

We report in Fig. 10 the result for this quantity in the clean case ($W=0$),
contrasting both Hamiltonians at $U=-8$, for a lattice with $L=12$, and taking
into account $\sigma=\uparrow$ [$\sigma=\downarrow$ results are the same owing
to the SU(2) symmetry]. A superconducting gap is clearly seen for both cases,
while significant broadening of the bands occurs due to the presence of the
interactions. We notice, however, that significant weight is accumulated
around the $k_{x}=0$ excitation momentum with a flatter momentum dispersion
for the Creutz ladder in comparison to the two-leg ladders, indicating the
influence of the flat-band physics in this Hamiltonian type even with
substantial interactions. Larger lattices (finer resolution in momentum) can
potentially improve this contrast.

Figure 10: The heatmap of the single-particle spectral function
$A_{k_{x},\uparrow}(\omega)$ for the Creutz (a) and the two-leg ladder (b), in
an $L=12$ lattice with Hubbard interaction $U=-8$ without disorder ($W=0$). A
smaller number of dispersive features in (a) is indicative of the influence of
flat-band physics in this case, even for substantially large interactions.

##  Appendix B Benchmark of DMRG results

The arguments in this work are mainly based on numerical calculations using
DMRG, which is one of the most powerful methods in solving quantum many-body
systems, especially in 1D and quasi-1D quantum lattices. However, in the case
of periodic boundary conditions, which is precisely the case when computing
the superfluid weights, DMRG meets much larger truncation errors. In other
words, achieving the same precision of calculations with open boundary
conditions takes a much more expensive computational effort. Moreover, when
the strong disorder breaks the lattice homogeneity, the DMRG procedure is
likely to be trapped in local minima, even if using an optimized strategy
specially designed for disordered lattices [73]. These difficulties,
accompanied by the fact that extracting information from disordered systems
requires repeating calculations for various disorder samples, restrict our
investigations to relatively small system sizes. To be more rigorous, we also
perform exact diagonalization (ED) calculations as a benchmark. As shown in
Fig. 11, the two methods provide precisely the same results, therefore
confirming the reliability of the numerical results illustrated in this work.

Figure 11: Comparison between ED and DMRG results for (a) Creutz lattice and
(b) regular two-leg ladder in the presence of random chemical potentials. Here
we use 30 disorder realizations for the benchmark.

##  Appendix C Approximation of ${\cal D}_{s}$

In the absence of disorder, the ground-state energy $E_{0}(\Phi)$ is a
quadratic function of $\Phi$ in the range $\Phi\in\left[0,\pi/2\right]$, as
shown in Fig. 12(a). Therefore, the superfluid weight ${\cal D}_{s}$ of the
form in Eq. 6 can be obtained by the following procedure: first, do a second-
order polynomial fitting of several $E_{0}(\Phi)$ with different twisted
$\Phi$, and then compute the ${\cal D}_{s}$ by the second-order derivative of
the previously obtained polynomial. However, this procedure is rather time-
consuming, especially in the disorder case, which requires many disorder
realizations.

Figure 12: (a) The ground state energy $E_{0}(\Phi)$ versus the twisted angle
$\Phi$ for the Creutz lattice in the clean case. The solid blue line denotes a
second-order polynomial fitting. (b) The absolute error between the superfluid
weight $D_{s}$ obtained from Eq. 6 and the approximation ${\cal D}_{s}\approx
2\pi L[E_{0}(\delta\Phi)-E_{0}(0)]/(\delta\Phi)^{2}$ using different
$\delta\Phi$ in Eq. 7. Here results are from DMRG calculation of $L=32$.

In practice, we adopt the approximation ${\cal D}_{s}\approx 2\pi
L[E_{0}(\delta\Phi)-E_{0}(0)]/(\delta\Phi)^{2}$ [29], from which one can
extract $D_{s}$ from a single value of $E_{0}(\Phi)$. We display the absolute
error from these two procedures in Fig. 12(b), where the error is overall
small ($\sim 10^{-3}$) and decreases as the phase twist $\Phi$ increases to
$\pi/2$. In this work, we choose $\Phi=\pi/2$ and use the approximation in Eq.
7 to compute the superfluid weight ${\cal D}_{s}$. Note that the above test
has been done in the clean case, and the situation can be more complicated in
the presence of a finite disorder strength. As long as $E_{0}(\Phi)$ is
monotonic in the range $[0,\pi/2]$, the extracted ${\cal D}_{s}$ still likely
constitutes a good approximation. Nevertheless, the results from the
approximation appear promising and self-consistent in our investigation.

Figure 13: The cost function $C_{X}$ in the two-dimensional parameter space
$\\{b,W_{c}\\}$ for the Creutz lattice with random chemical potentials as an
example. The red star marks the position ($W_{c}=0$, $b=2.3$) of the minimum
$C_{X}$.

##  Appendix D Cost Function Minimization

The key for obtaining a performant data collapse and scaling of the superfluid
weight is to extract the best critical $W_{c}$ and $b$ in Eq. (8), which can
be determined by minimizing the cost function [58, 59, 74]

$\displaystyle
C_{X}=\frac{\sum_{j}|X_{j+1}-X_{j}|}{\max\\{X_{j}\\}-\min\\{X_{j}\\}}-1,$ (13)

where $X_{j}$ is the $j$-th element of the collection for all $D_{s}(L,W)$
values in the parameter space $\\{L,W\\}$. Here the data collection $X$ has
been sorted in a nondecreasing way with $X_{j}\leq X_{j+1}$. The cost function
$C_{X}$ is close to zero for a perfectly smooth and continuous data
collection. In practice, for each pair of fitted parameters value, one obtains
a parameter-dependent cost function $C_{X}(b,W_{c})$. Repeating this procedure
within proper ranges in the two-dimensional parameter space $\\{b,W_{c}\\}$
one can extract the minimum of $C_{X}$ and find the best fitting. As shown in
Fig. 13, the cost function of the Creutz lattice with random chemical
potentials is a unimodal function in $\\{b,W_{c}\\}$. Therefore, it is not
hard to obtain the unambiguous minimum of $C_{X}$, and the corresponding data
collapse in Fig. 3 in the main text. Similar analysis carries over for the
other lattice geometry and disorder type used.

##  Appendix E One- and two-particle excitation gaps

Figure 14: The $m$-particle excitation gaps $\delta_{m}$ (see text for
definition) for (a) [(b)] Creutz lattice with random chemical potentials
[Zeeman fields] and (c) [(d)] regular two-leg ladder with random chemical
potentials [Zeeman fields]. Here results are from DMRG calculations with
$L=12$.

On top of the observables discussed in the main text, further characterization
of the different phases across the SIT can be made by examining the charge
excitation energy [75, 76, 69]. In particular, the $m$-particle excitation gap
can be defined as [69]

$\displaystyle\delta_{m}\equiv E_{0}(N+m)+E_{0}(N-m)-2E_{0}(N)\ .$ (14)

Here $E_{0}(N)$ is the ground state of $N=N_{\uparrow}+N_{\downarrow}$
particles, as defined in the main text. Our interest in the present work lies
in the spin-balanced sector {$N_{\uparrow}=N_{\downarrow}$} for the case of
pair excitations. In actual calculations, ($N\pm 1$) [($N\pm 2$)] is
explicitly regarded as ($N_{\uparrow}\pm 1,N_{\downarrow}$) [($N_{\uparrow}\pm
1,N_{\downarrow}\pm 1$)]. The one- and two-particle excitations of a small
system size with $L=12$ are displayed in Fig. 14.

With disorder induced by random chemical potentials [Figs. 14(a) and 14(c)],
the one-particle excitation gap $\delta_{1}$ is finite in the whole range of
disorder strengths investigated, irrespective of the lattice geometry (or,
equivalently, the band structure). On the other hand, the two-particle
excitation gap $\delta_{2}$ slowly grows with $W$, denoting the onset of
insulating behavior. Remarkably, since $\delta_{2}<\delta_{1}$, pair
excitations are favored within this regime. In the main text, we refer to it
as an Anderson insulating phase of singlet pairs; in other contexts, this is
also dubbed as a Bose-insulator [77, 78, 75, 76, 68, 69]. In passing, we note
that this analysis also makes clear the inexistence of an intermediate
disorder-induced metallic phase.

Such scenario changes in the presence of the disorder induced by the random
Zeeman fields [Figs. 14(b) and 14(d)]. Now, the single-particle (two-particle)
excitation gap substantially decreases (slightly increases) as $W$ grows.
While having both quantities finite is a precondition for driving insulating
behavior, at disorder values $W\gtrsim 10$ the imminent crossing of
$\delta_{2}$ and $\delta_{1}$ marks the crossover from a Bose to Fermi
insulator [68, 76, 69], where single-particle excitations are favored instead.
This change of character of the insulator phase has been seen in other
contexts for clean SITs [68, 69]. While finite-size effects likely
quantitatively impact the results, they support the main findings in the main
text.

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