# Many-body localization properties of fully frustrated Heisenberg spin-1/2
ladder model with next-nearest-neighbor interaction

Jiameng Hong  Center for Quantum Sciences and School of Physics, Northeast
Normal University, Changchun 130024, China  Taotao Hu<EMAIL_ADDRESS>Center for Quantum Sciences and School of Physics, Northeast Normal
University, Changchun 130024, China

###### Abstract

Many-body localization (MBL) is an intriguing physical phenomenon that arises
from the interplay of interaction and disorder, allowing quantum systems to
prevent thermalization. In this study, we investigate the MBL properties of
the fully frustrated Heisenberg spin-1/2 ladder model with next-nearest-
neighbor hopping interaction along the leg direction and compare it with the
Heisenberg spin-1/2 single-chain model with next-nearest-neighbor hopping
interaction. We explore the MBL transition using random matrix theory and
study the characteristics of entanglement entropy and its variance. Our
results show that for the single-chain model, the critical point $w_{1}\sim$
7.5 ± 0.5, whereas for the frustrated ladder model, $w_{2}\sim$ 10.5 ± 0.5.
Moreover, we observe the existence of a many-body mobility edge in the
frustrated ladder model. We also investigate the dynamical properties of the
frustrated ladder model and identify the logarithmic growth of entanglement
entropy, high fidelity of initial information, and magnetic localization
phenomenon in the localized phase. Finally, we explore the finite-size scaling
of the two models. Our findings suggest that interpreting MBL transition as a
continuous second-order phase transition yields a better scaling solution than
the Kosterlitz-Thouless type transition for our two models, and this
difference is more pronounced in the frustrated ladder model compared with the
single-chain model.

††preprint: APS/123-QED

##  I Introduction

The eigenstate thermalization hypothesis (ETH) [1, 2, 3] suggests that over
long periods of time, all microscopic states of the system are visited with
equal probability, i.e. a single many-body eigenstate possesses thermodynamic
observables. However, the phenomenon of many-body localization (MBL) [4, 5, 6,
7, 8, 8] challenges the ETH by hindering the approach to equilibrium in
strongly disordered interacting quantum systems. This is attributed to the
emergence of local integrals of motion (LIOM) [9, 10, 11, 12, 13], which
enables the system to retain information about its initial state, impeding the
dissemination of information [14, 15] and decelerating the diffusion of
entanglement [16, 17]. Therefore, this is also why MBL is considered a
potential candidate for constructing quantum memories. Moreover, this
transition depends on the position of the energy spectrum known as the
mobility edge, causing mid-spectrum eigenstates to localize at higher disorder
values than the edge eigenstates.

Experiments on MBL have also been realized on artificial platforms such as
ultracold atoms [18, 19, 20], ion traps [21, 22], and superconducting qubits
[23, 24]. Additionally, researchers are also exploring the existence of many-
body localization in real materials. Furthermore, the MBL properties in one-
dimensional spin systems [25, 26, 27], bosonic systems [28, 29], and fermionic
systems [30, 31], based on Heisenberg and Ising interactions, have achieved
consistency between theory and experiments. In addition, significant progress
has been made in the study of many-body localization based on the
phenomenological renormalization group [32, 33, 34, 35]. This provides an
important reference value for distinguishing the universal classes of many-
body localization.

The Heisenberg spin-1/2 ladder model [36, 37, 38] has recently attracted great
interest. Excitingly, in the experimental realm, H. Yamaguchi et al. have
realized ladder models with exchange interactions exhibiting both
ferromagnetic and antiferromagnetic behavior [39, 40, 41]. Then, in
theoretical aspects, Zheng-Hang Sun et al. have investigated the MBL
transition and dynamic properties of the ladder model [42].

More recently, the frustrated (with crossed interactions) Heisenberg spin
ladder model [43] has attracted significant attention among researchers.
Dominik Hahn et al. have explored the dynamics of the model through out-of-
time-ordered correlators (OTOC) and entanglement entropy. Sk Saniur Rahaman
and Rojas studied the quantum phases and thermodynamic properties of
frustrated spin-1/2 ladders with alternate Ising-Heisenberg exchange
interactions, providing a phase diagram for this model [44]. In contrast to
their work, our study focuses more on the many-body localization properties of
this model in the presence of disorder.

The frustrated spin ladder model, often described as a “ quasi-one-dimensional
” or “ weakly two-dimensional ” system, serves as an intermediate model
between 1D and 2D systems [45]. In this study, we investigate the many-body
localization properties of the fully frustrated Heisenberg spin-1/2 ladder
model (hereafter referred to as “ the frustrated ladder model ”) using exact
diagonalization. To facilitate a comparison with the Heisenberg spin-1/2
single-chain model with next-nearest-neighbor hopping interaction (hereafter
referred to as “ the single-chain model ”), we also consider next-nearest-
neighbor hopping interaction along the leg direction in our frustrated ladder
model.

The rest of this article is structured as follows. In Sec. II, we introduce
our model, including its parameters and the study methods. In Sec. III, we
compare the spectrum and eigenstate properties of the two models and
approximately determine the critical points. Moreover, we observe the
existence of a many-body mobility edge in the frustrated ladder model. Sec. IV
shows the dynamics of the frustrated ladder model, encompassing the time
evolution of entanglement entropy, fidelity, and magnetization for initial
states. In Sec. V, we perform a finite-size scaling analysis based on two
types of MBL transitions. Finally, Sec. VI presents the conclusion.

##  II model

### The single-chain model

The Hamiltonian of the Heisenberg spin-1/2 single-chain model with next-
nearest-neighbor hopping interactions can be expressed as follows:

$\displaystyle H_{1}$
$\displaystyle=\sum_{i=1}^{N-1}J_{1}\left(S_{i}^{x}S_{i+1}^{x}+S_{i}^{y}S_{i+1}^{y}+S_{i}^{z}S_{i+1}^{z}\right)$
$\displaystyle+\sum_{i=1}^{N-2}J_{2}\left(S_{i}^{x}S_{i+2}^{x}+S_{i}^{y}S_{i+2}^{y}\right)+W\sum_{i=1}^{N}h_{i}S_{i}^{z}.$
(1)

### The frustrated ladder model

The Hamiltonian of the fully frustrated Heisenberg spin-1/2 ladder model with
next-nearest neighbor hopping interaction can be written as follows:

$\displaystyle
H_{2}=J_{1}H_{\mid\mid}+J_{2}H_{\mid\mid\mid}+J_{3}H_{\perp}+J_{4}H_{\times}+H_{z},$
(2)

where

$\displaystyle H_{\mid\mid}$
$\displaystyle=\sum_{j=1,2}\sum_{i=1}^{M-1}\left(S_{j,i}^{x}S_{j,i+1}^{x}+S_{j,i}^{y}S_{j,i+1}^{y}+S_{j,i}^{z}S_{j,i+1}^{z}\right),$
$\displaystyle H_{\mid\mid\mid}$
$\displaystyle=\sum_{j=1,2}\sum_{i=1}^{M-2}\left(S_{j,i}^{x}S_{j,i+2}^{x}+S_{j,i}^{y}S_{j,i+2}^{y}\right),$
$\displaystyle H_{\bot}$
$\displaystyle=\sum_{i=1}^{M}\left(S_{1,i}^{x}S_{2,i}^{x}+S_{1,i}^{y}S_{2,i}^{y}+S_{1,i}^{z}S_{2,i}^{z}\right),$
(3) $\displaystyle H_{\times}$
$\displaystyle=\sum_{i=1}^{M-1}(S_{1,i}^{x}S_{2,i+1}^{x}+S_{1,i}^{y}S_{2,i+1}^{y})$
$\displaystyle+\sum_{i=1}^{M-1}(S_{1,i+1}^{x}S_{2,i}^{x}+S_{1,i+1}^{y}S_{2,i}^{y}),$
$\displaystyle H_{z}$
$\displaystyle=W\sum_{i=1}^{M}h_{i}\left(S_{1,i}^{z}+S_{2,i}^{z}\right),$

where $S_{i}^{\left\\{x/y/z\right\\}}$denotes the spin components in the $x$,
$y$, and $z$ directions of a spin-1/2 system located at site $i$. We set
$J_{1}$ = $J_{2}$ = $J_{3}$ = $J_{4}$ = 1, which corresponds to the fully
frustrated ladder model as described in Ref. [43]. Simultaneously, we consider
$J_{1}$ and $J_{2}$ as the nearest-neighbor and next-nearest-neighbor coupling
constants along the leg direction, while $J_{3}$ and $J_{4}$ along the ladder
direction, as shown in Fig. 1. $h_{i}$ is a random magnetic field with a
uniform and independent distribution $[-1,1]$, and $W$ is the strength of the
random field. In this work, we set $N$ = $2M$, $N$ represents the number of
lattice sites, and $M$ denotes the length of the ladder. We investigate the
properties of our models in Eq. (1) and Eq. (2) with a size of $N$ =
10,12,…,18 by exact diagonalization within the $S_{\rm tot}^{z}$ = 0 sector.

Figure 1: Schematics of our models. (a) The single-chain model. (b) The
frustrated ladder model. $J_{1}$ and $J_{2}$ are the nearest-neighbor and
next-nearest-neighbor coupling constants along the leg direction, while
$J_{3}$ and $J_{4}$ are the nearest-neighbor and next-nearest-neighbor
coupling constants along the ladder direction.

##  III  spectrum and eigenstate

In this section, we analyze several important observables that indicate the
occurrence of the MBL transition and compare the frustrated ladder model with
the single-chain model.

### the adjacent gap ratio

$\bar{r}$ is a common tool based on the random matrix theory [46, 47] for
spectrum analysis, used to distinguish the localized phase from the
thermalized phase and determine the critical point. The adjacent gap ratio
[48, 49, 46] is defined as

$r_{n}\equiv\rm{min}(s_{n},s_{n+1})/\rm{max}(s_{n},s_{n+1}),$ (4)

where $s_{n}=E_{n}-E_{n-1}$. It is important to note that the spectrum is
ordered $E_{n}>E_{n-1}$, to determine the critical point. As in previous
studies [48, 50], the target energy density is defined as

$\displaystyle\varepsilon=\frac{E-E_{\rm min}}{E_{\rm max}-E_{\rm min}},$ (5)

where $E_{\rm max}(E_{\rm min})$ is the maximum (minimum) value in the energy
spectrum $\left\\{E\right\\}$. In our study, we choose the eigenenergies in
the window of $\varepsilon$ = 0.5 $\pm$ 0.01, which is at the middle of the
energy spectrum [48, 23]. In the thermalized phase, the system demonstrates a
Wigner-Dyson (WD) distribution with $\left\langle r\right\rangle_{\rm W}$ =
0.5307. This is caused by the overlap of energy eigenstates, which results in
repulsion between energy levels. On the contrary, it conforms to a Poisson
distribution with $\left\langle r\right\rangle_{\rm P}$ = 0.3863 in the
localized phase, where there is no level repulsion [51].

Figure 2:  Spectrum and eigenstate properties of our two models. [$(a)$ and
$(b)$] show the adjacent gap ratio $\bar{r}$ for different $N$ as a function
of $W$ for the single-chain model and the frustrated ladder model. They both
illustrate the variation of $\bar{r}$ from $\left\langle r\right\rangle_{\rm
W}$ to $\left\langle r\right\rangle_{\rm P}$ as $W$ increases in the two
models. The critical points are $w_{1}\sim 7.5\pm 0.5$ and $w_{2}\sim 10.5\pm
0.5$ (indicated by the red arrows). The insets of [(a) and (b)] depict the
level distribution $P(s)$ for different $W$ as a function of $s$. They display
the transition of $P(s)$ from the WD distribution to the Poisson distribution
as $W$ increases for both models when considering system size $N$ = 16. [(c)
and (d)] display $\overline{S/S_{T}}$ for different $N$ as a function of $W$.
They exhibit the transition of $\overline{S/S_{T}}$ from volume-law
entanglement in the thermalized phase to area-law entanglement in the
localized phase. Notably, the estimated critical points (indicated by the red
arrows) align with the values obtained from $\bar{r}$. The insets of [(c) and
(d)] show the variance $\bigtriangleup S$ for different $N$ as a function of
$W$.

### level-spacing distribution

$P(s)$ is a useful tool for understanding the short-range spectral
correlations in a system during its long-term evolution. Specifically,
$P(s)ds$ stands for the probability that $s_{n}$ is in $(s,s+ds)$. $P(s)$
obeys the WD distribution $P(s)$ = $\frac{\pi s}{2}e^{-\frac{\pi s^{2}}{4}}$
in the thermalized phase and the Poisson distribution $P(s)$ = $e^{-s}$ in the
localized phase [50, 52]. However, near the critical point, the localization
and thermalization features of the system begin to mix, and there are
significant quantum fluctuations presented in the system. As a consequence,
$P(s)$ exhibits intermediate behavior between the two distributions.

### entanglement entropy

$S$ is a valuable tool for providing crucial information about the
entanglement structure and quantum coherence of the system [53, 54]. We divide
the system into two parts, $A$ and $B$, with equal lattice size $M$. The
entanglement entropy of subsystem $A$ can be expressed as follows:

$\displaystyle S=-{\rm Tr}(\rho_{A}{\rm ln}\rho_{A}),$ (6)

where $\rho_{A}$ = ${\rm Tr}_{B}\rho$ = ${\rm
Tr}_{B}|\psi\rangle\langle\psi|$. In the thermalized phase, the system obeys
the volume law, while in the localized phase, it adheres to the area law [49].

### the variance of the entanglement entropy

$\bigtriangleup S$ is defined as

$\displaystyle\left(\bigtriangleup S\right)^{2}=\left\langle
S^{2}\right\rangle-\left\langle S\right\rangle^{2}.$ (7)

This well-defined quantity serves as an excellent indicator of phase
transitions [49, 54]. The variance reaches its maximum value near the critical
point.

We now move to the results analysis. We average the gap ratio $\bar{r}$ =
$\left\langle r_{n}\right\rangle$ over eigenvalues included in
$\varepsilon\in$ [0.49, 0.51] from each disorder realization and 1000 to 20000
disorder realizations depending on $N$ for each $W$. In Figs. 2[(a) and (b)],
we observe that the curves for different $N$ approximately cross at the
critical point $w_{1}\sim 7.5\pm 0.5$ (The value is slightly larger than that
in Ref. [53] because we choose a smaller energy window) for the single-chain
model, and $w_{2}\sim 10.5\pm 0.5$ for the frustrated ladder model. This
implies that the frustrated ladder model requires larger $W$ to undergo the
MBL transition. Moreover, we notice that as $N$ increases, the critical points
shift towards larger $W$, and this trend is more pronounced in the frustrated
ladder model due to the broader distribution of the curves in Fig. 2(b). We
can attribute these phenomena to the increased complexity and stronger
interactions present in the frustrated ladder model.

The insets of Figs. 2[(a) and (b)] demonstrate that for small and large values
of $W$, the $P(s)$ of our two models follow the WD distribution and the
Poisson distribution, respectively, which are characteristic of the
thermalized phase and localized phase [50]. However, the insets depict an
intermediate behavior between these two distributions near the critical
points. Although the $P(s)$ distribution is closer to the Poisson distribution
(which can be attributed to finite size effects, similar to Ref. [52]), it
also exhibits a level repulsion for small energy separations s $\ll$ 0.1.

Figs. 2[(c) and (d)] illustrate $\overline{S/S_{T}}$ as a function of $W$ for
different system sizes $N$, where $S_{T}$ = 0.5 ($N$ln2 - 1) is the Page value
for random pure states [54, 53]. The number of disorder realizations is 500 to
10000 for $N$ = 18 to 10. With $W$ increase, we observe a transition in
$\overline{S/S_{T}}$ going from 1 in the ETH phase characterized by volume-law
[55] entanglement to 0 in the MBL phase characterized by area-law
entanglement. Moreover, the crossings of the curves for the two models align
closely with the critical points derived from $\bar{r}$. Additionally, it is
worth mentioning that the size effect of entanglement entropy is not as
pronounced as that of the adjacent gap ratio.

In addition, as shown in the insets of Figs. 2[(c) and (d)], when $W\to 0$,
$\bigtriangleup S\to 0$, this is consistent with the ETH. Near the critical
point, $\bigtriangleup S$ exhibits its maximum fluctuations. The peak values
of the curves are slightly smaller than the previously critical points
$w_{1}(w_{2})$, thus serving as a lower bound for the occurrence of the phase
transition, which is consistent with previous studies in one-dimensional
system [56] and Heisenberg ladder model [48].

Figure 3:  Disorder strength ($W$)—Energy density ($\varepsilon$) phase
diagram of the frustrated ladder model for $N$ = 16. The color bar represents
the $\bar{r}$ value ranging from 0.3863 to 0.5307. The pink rectangle mark
corresponds to a visual estimate of the boundary between the WD distribution
and the Poisson distribution of $\bar{r}$. The “D-shape” structure in the
phase diagram exhibits the well-known mobility edge.

Furthermore, in Fig. 3, we plot the adjacent gap ratio $\bar{r}$ as a function
of $W$ and $\varepsilon$ for $N$ = 16. To achieve this, we sample energies
only in $\varepsilon\in$ [0.1,0.9], because the density of states is
particularly low at the edge of the energy spectrum. We partition
$\varepsilon$ into 16 equal segments, resulting in a value of
$\bigtriangleup\varepsilon$ = 0.05. We then perform 3000 disorder realizations
for each $\bigtriangleup\varepsilon$ at different $W$. With $W$ increase, the
phase diagram shows a continuous transition from the thermalized phase (red
region) to the localized phase (dark blue region). Moreover, the pink
rectangle mark corresponds to a visual estimate of the boundary between the WD
distribution and the Poisson distribution of $\bar{r}$.

Specifically, at small $W$, the eigenstates within intermediate energy
densities are found to be thermalized, while the eigenstates for edge energy
densities exhibit localization. The “D-shaped” structure in the phase diagram
of the frustrated ladder model is consistent with observations in one-
dimensional models [49], ladder models, and triangular models [50]. However,
its boundary corresponds to larger $W$.

##  IV Dynamics

In the previous sections, we compared the properties of the two models. Here,
our focus shifts to the dynamic evolution of the frustrated ladder model,
which represents an additional significant approach for studying MBL
transition. We choose the $\rm{N\acute{e}el}$ state as the initial state
$\left|\psi(0)\right\rangle$ for numerical simulations in our model. We
average 300 disorder realizations on the system with $N$ = 14 at different
$W$. Additionally, we analyze the size effects for different $N$ with the
fixed disorder strength $W$ = 3. The number of disorder realizations is 100 to
5000 for $N$ = 16 to 10.

### entanglement entropy

The growth behavior of entanglement entropy $S(t)$ serves as a significant
criterion for distinguishing between the thermalized phase and the localized
phase [23, 55, 57]. We divide the system into two parts, $A$ and $B$, with
equal lattice size $M$. The evolution of entanglement entropy is defined by
setting

$\displaystyle S(t)=-{\rm Tr}\left[\rho_{A}(t){\rm ln}\rho_{A}(t)\right],$ (8)

where $\rho_{A}(t)$ = ${\rm Tr}_{B}\rho(t)$ = ${\rm
Tr}_{B}|\psi(t)\rangle\langle\psi(t)|$ is the reduced density matrix of the
system after tracing out subsystem $B$, while $\left|\psi(t)\right\rangle$ =
$e^{-iHt}\left|\psi(0)\right\rangle$ is the state obtained after the evolution
of the initial state $\left|\psi(0)\right\rangle$ for time $t$.

Fig. 4(a) illustrates the evolution of entanglement entropy with time for $N$
= 14. In the thermalized phase ($W$ = 1), a rapid power-law growth of the
entanglement entropy is evident, which quickly saturates to a large value
within a short period of time. This value is close to the random pure state
Page value $S_{T}$ = 0.5 ($N$ ln2 - 1) of the volume law. In contrast, in the
deep localized phase ($W$ = 20), the entropy exhibits a distinct slow
logarithmic growth (related to the integration of local integrals of motion)
and saturates to a small value over a long period of time. It is worth noting
that for intermediate disorder strengths ($W$ = 5), the entanglement entropy
initially undergoes a short period of power-law growth, where information
spreads rapidly. However, unlike in the thermalized phase, the spread of
information almost comes to a halt within a short period of time. Instead, it
exhibits a long-lasting slow logarithmic growth trend. This is consistent with
the behavior observed in one-dimensional systems [58, 49].

Figure 4: The time evolution of quantities in the frustrated ladder model.
[(a), (b), and (c)] depict the distinct dynamical evolution behaviors of
entanglement entropy, fidelity, and magnetization, for different $W$ at $N$ =
14. At small $W$, (a) displays the rapid power-law growth of entanglement
entropy, (b) illustrates the loss of initial information, and (c) depicts the
behavior of free magnetization, which are characteristics of the ETH phase.
However, at large $W$, (a) depicts the slow logarithmic growth of entanglement
entropy, (b) shows the high fidelity of initial information, and (c) displays
the presence of initial magnetization, which are characteristics of the MBL
phase. Additionally, [(d), (e), and (f)], show the size effects for $N$ = 10
to 16 at $W$ = 3. For the same disorder strength $W$, as $N$ decreases, the
properties of the system become more similar to those of the MBL phase.

In Fig. 4(d) we also analyze the finite-size effects for systems with sizes
ranging from $N$ = 10 to 16 and $W$ = 3. We find that as $N$ increases, the
entanglement entropy saturates to a larger value. More importantly, as $N$
increases, the time for power-law growth of the entanglement entropy
increases, while the time for logarithmic growth decreases. This suggests that
systems with larger $N$ will localize at larger $W$, which is consistent with
the discussion in Fig. 2(d) that as $N$ increases, the crossing point shifts
to larger $W$.

### fidelity

$F(t)$ is an important observable in quantum information, used to measure the
similarity between the initial state and the evolved state, allowing us to
determine the properties of the system. The evolution of fidelity [59, 60]is
defined as

$\displaystyle
F(t)=\rm{Tr}\left[\rho(t)^{1/2}\rho_{0}\rho(t)^{1/2}\right]^{1/2},$ (9)

where $\rho(t)$ = $|\psi(t)\rangle\langle\psi(t)|$ and $\rho(0)$ =
$|\psi(0)\rangle\langle\psi(0)|$. The evolved state
$\left|\psi(t)\right\rangle$ = $e^{-iHt}\left|\psi(0)\right\rangle$.

In Fig. 4(b), we observe that at small disorder strength ($W$ = 1), the
fidelity stabilizes at a relatively small value after long-time evolution,
indicating that the initial state information is almost completely lost,
consistent with the ETH phase. On the other hand, at large disorder strength
($W$ = 20), the fidelity remains constant with time, indicating that the
initial state information is fully preserved, in line with the MBL phase.
Additionally, we find that as $W$ increases, the preservation of initial state
information becomes higher.

Furthermore, in Fig. 4(e), we also analyze the influence of $N$ on the system
properties at $W$ = 3. We find that for the same $W$, as $N$ increases from 10
to 16, the fidelity takes a longer time to stabilize, and the stability value
is smaller. We believe that in smaller systems, there are fewer interactions
between spins, resulting in fewer paths required for information transmission.
This makes the system more sensitive to random perturbations and thus enables
faster information propagation and retention.

Figure 5: Upper panels: the finite-size scaling of quantities as a function of
$sN/\xi_{SO}$, where $\xi_{SO}$ is the correlation strength in Eq. (11) and
$s$ = ${\rm{sgn}}[W-w]$. [(a) and (b)] $\bar{r}$ and $\overline{S/S_{T}}$ for
the single-chain model, [(c) and (d)] $\bar{r}$ and $\overline{S/S_{T}}$ for
the frustrated ladder model. Lower panels: the finite-size scaling of
quantities as a function of $sN/\xi_{KT}$, where $\xi_{KT}$ is the correlation
strength in Eq. (12). [(e) and (f)] $\bar{r}$ and $\overline{S/S_{T}}$ for the
single-chain model, [(g) and (h)] $\bar{r}$ and $\overline{S/S_{T}}$ for the
frustrated ladder model. In each panel, we provide the parameters obtained by
minimizing $Q$ for optimizing the best data collapse.

### magnetization

$M_{I}(t)$ is an important property used to characterize MBL transition. [61,
62, 63]. By calculating the average magnetization of a given lattice site
overall eigenstates, we can study its time evolution and analyze system
properties. The definition of the expectation value of magnetization at the
site $I$ is provided by

$M_{I}(t)=\langle\psi(t)|S_{I}^{z}|\psi(t)\rangle,$ (10)

where $\left|\psi(t)\right\rangle$ = $e^{-iHt}\left|\psi(0)\right\rangle$, and
here we let $I$ = 1.

As shown in Fig. 4(c), when the disorder strength is small ($W$ = 1), the
long-term evolution of the magnetization of the site $I$ tends to a very small
value. This corresponds to the prediction of the ETH for an effectively free
spin, indicating the loss of initial information. However, when the disorder
strength is large ($W$ = 20), the average magnetization stabilizes at 1/2
after long-time evolution, indicating good preservation of the initial state
information. The behavior for intermediate disorder strengths $W$ falls
between the localized phase and the thermalized phase. The reason behind these
phenomena is the spins in the thermalized phase are correlated over long
distances in space, which causes the magnetization to decay exponentially as
time progresses. Conversely, in the localized phase, the system shows magnetic
localization, where spins are randomly distributed in space and do not exhibit
long-range order. As a result, the system only maintains strong spin
correlations within a limited range, leading to a more stable evolution of
magnetization over time. Furthermore, we also observe that as $W$ decreases,
the decay of magnetization is faster. The reason behind this is that when the
disorder strength is small, the spins on individual lattice sites are less
affected, leading to a faster convergence of their evolution towards a stable
state.

In Fig. 4(f), we also analyze the size effect for $W$ = 3, which shows
consistency with the dynamic properties of fidelity. Specifically, as $N$
increases, the average magnetization decays faster and stabilizes at a smaller
value after a longer period. Based on our analysis, we can conclude that in
smaller systems, the limited range of interactions between spins leads to a
stronger localization effect. Additionally, in smaller systems, the energy
level spacing is smaller and the degeneracy of energy levels is higher. This
affects the thermal fluctuation behavior of the spins, which can cause the
spins in smaller systems to remain consistent for longer periods in more
states, resulting in a slower decay rate for the average magnetization.

##  V  finite-size scaling analysis

Now, let’s compare the finite-size scaling properties of our two models.
Currently, the description of the MBL transition is mainly based on two
hypotheses, both of which suggest that the length scale $\xi$ diverges near
the critical point. Most literature suggests that MBL follows a continuous
second-order phase transition [53, 52, 64, 65], with its power-law divergent
correlation strength

$\xi_{SO}\sim{|W-w|^{-\nu}},$ (11)

where $w$ is the critical point and $\nu$ is the universal critical exponent.
In this hypothesis, the famous Harris criterion states that $\nu\geq 2/d$ and
$d$ is the dimension of the system. Nevertheless, most numerical simulations
based on one-dimensional systems show $\nu\sim 1$ [49, 64], which contradicts
the Harris bound $\nu\geq 2$. Two explanations have been put forward to
account for this discrepancy. One is that the simulated system sizes are
insufficient to emulate the thermodynamic limit. Alternatively, the hypothesis
proposing a continuous second-order phase transition for MBL is valid.

Figure 6: Disorder strength $W$—Universal critical exponent $\nu$
(Nonuniversal parameter $b$) phase diagram. The color bar represents the value
of quality function $Q$. Upper panels: The $Q$ values of quantities as a
function of $W$ and $\nu$ with the correlation strength $\xi_{SO}$. [(a) and
(b)] $\bar{r}$ and $\overline{S/S_{T}}$ for the single-chain model. [(c) and
(d)] $\bar{r}$ and $\overline{S/S_{T}}$ for the frustrated ladder model. Lower
panels: The $Q$ values of quantities as a function of $W$ and $b$ with the
correlation strength $\xi_{KT}$. [(e) and (f)] $\bar{r}$ and
$\overline{S/S_{T}}$ for the single-chain model, [(g) and (h)] $\bar{r}$ and
$\overline{S/S_{T}}$ for the frustrated ladder model.

Recently, an improved real-space renormalization group (RG) scheme has
proposed a Kosterlitz-Thouless (KT) type MBL transition [66, 67, 68, 69], in
which the correlation length

$\xi_{KT}\sim{\rm{exp}}({{b|W-w|^{-0.5}}}),$ (12)

where $w$ is the critical point and $b$ is a nonuniversal parameter. Although
several articles have compared the MBL transition with the two length scales
$\xi_{SO}$ and $\xi_{KT}$, no conclusive answer has been provided regarding
which one is more suitable for finite-size scaling. Some have concluded that
$\xi_{SO}$ provides better data collapse than $\xi_{KT}$ [70], while others
have suggested the opposite [71, 72]. Therefore, motivated by these results we
perform finite-size scaling on our two models with the two correlation lengths
$\xi_{SO}$ and $\xi_{KT}$.

To obtain more precise estimates of $w$ and other important parameters, we
present a finite-size scaling analysis on the curves shown in Fig. 2. In the
upper panels of Fig. 5, we plot $\bar{r}$ and $\overline{S/S_{T}}$ as a
function of $sN/\xi_{SO}$, where $\xi_{SO}$ is the correlation strength in Eq.
(11), while in the lower panels, we plot them as a function of $sN/\xi_{KT}$,
where $\xi_{KT}$ is the correlation strength in Eq. (12) and $s$ =
${\rm{sgn}}[W-w]$ for our two models. By varying the values of $w$, $\nu$
($b$), the curves for different $N$ collapse on each other. We can achieve the
best possible data collapse by minimizing a quality function

$Q=\frac{\sum_{j=1}^{N_{p}-1}|Y_{j+1}-Y_{j}|}{{\rm{max}}\left\\{Y_{j}\right\\}-{\rm{min}}\left\\{Y_{j}\right\\}}-1,$
(13)

given in Refs. [73, 71, 70, 72, 74]. In Eq. (13), $Y$ =
$\left\\{Y_{j}\right\\}$ represents a quantity with $Np$ values at various $N$
and $W$, and it is sorted in nondecreasing order based on the values of
$sN/\xi_{SO}$ or $sN/\xi_{KT}$. Under the best data collapse, where all curves
collapse onto a single curve, $Q$ = 0, otherwise $Q$ $>$ 0\. In our analysis,
we select the parameters that minimize $Q$.

As shown in Fig. 5, the values of $w$ obtained from the best scaling solution
approximately consist with the crossing points of curves for different $N$ in
Fig. 2 for our two models. Furthermore, the $\nu$ values obtained from our two
models both violate the Harris bound, consistent with previous studies [49,
64] which prevents us from using the universal critical exponent $\nu$ to
classify the MBL transition of our two models as done in Ref. [42]. More
importantly, we obtain smaller values of $Q$ from the scaling solution
quantities as a function of $sN/\xi_{SO}$ than from $sN/\xi_{KT}$ for both
models. We also observe that $\overline{S/S_{T}}$ produces smaller $Q$
compared with $\bar{r}$ for both models and both scaling solutions. To further
support our claims, we plot $Q$ as a function of $w$ and $\nu(b)$ in Fig. 6.
We also observed a smaller $Q$ (more pronounced dark blue) for
$\overline{S/S_{T}}$ as a function of $sN/\xi$ compared with $\bar{r}$. This
is consistent with the smaller size effect of entanglement entropy shown in
Fig. 2. Based on the comparisons, we conclude that $\xi_{SO}$ yields a better
scaling solution than $\xi_{KT}$, as evidenced by the smaller $Q$, and this is
more pronounced in the frustrated ladder model ($\bigtriangleup Q$ =
$Q_{2_{KT}}-Q_{2_{SO}}\approx 0.7$) compared with the single-chain model
($\bigtriangleup Q$ = $Q_{{1_{KT}}}-Q_{2_{SO}}\approx 0.2$).

##  VI conclusion

In this article, the MBL properties of the fully frustrated ladder model (with
the next-nearest neighbor hopping interaction along the leg direction) are
investigated and compared with the Heisenberg spin-1/2 single-chain model with
the next-nearest neighbor hopping interaction. Similar MBL properties were
found in these two models. However, there are differences, as the fully-
frustrated ladder model requires stronger disorder ($w_{2}\sim$ 10.5 ± 0.5) to
undergo the MBL transition and exhibits more pronounced size effects due to
its stronger and more complex interactions. Additionally, intriguing phenomena
were observed in the fully-frustrated ladder model. In the thermalized phase,
the power-law growth of entanglement entropy, loss of initial information, and
the emergence of effective free-magnetization were observed. Conversely, in
the localized phase, logarithmic growth of entanglement entropy, highly
preserved initial information, and magnetic localization phenomenon were
observed. Furthermore, the finite-size scaling properties of our two models
are studied. Our analysis shows that the continuous second-order phase
transition interpretation of the MBL transition provides a better scaling
solution than the Kosterlitz-Thouless (KT) type transition for our two models,
and this effect is even more pronounced in the frustrated ladder model due to
its larger $\bigtriangleup Q$.

## Acknowledgement

This work was supported by the Plan for Scientific and Technological
Development of Jilin Province (No. 20230101018JC).

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