# Reentrant localization transition in the Su-Schrieffer-Heeger model with
random-dimer disorder

Zheng-Wei Zuo  Dawei Kang  School of Physics and Engineering, and Henan Key
Laboratory of Photoelectric Energy Storage Materials and Applications, Henan
University of Science and Technology, Luoyang 471023, China

###### Abstract

The question of whether the topological insulators can host a stable
nontrivial phase in the presence of spatially correlated disorder is a
fundamental question of interest. Based on theoretical investigations, we
analyze the effect of random-dimer disorder on the quantum phase transitions
of the Su-Schrieffer-Heeger model. We explicitly demonstrate that, due to the
absence of symmetry, there are no in-gap edge states in certain disordered
topological nontrivial gapped phase and the bulk-boundary correspondence
breaks down. However, the fractionalized end charges still appear at the ends
of the chain. The energy distribution possibility of the in-gap edge states
and the possible values of fractionalized end charges are dependent on the
concentration of random-dimer disorder. On the other hand, random-dimer
disorder and dimer hopping intertwine in an interesting manner. Reentrant
localization transition behavior appears, which is evidenced by the
fingerprint of the inverse participation ratio, normalized participation
ratio, and tunneling conductivity.

###### pacs:

71.10.Fd,71.10.Hf,73.90.+f,64.60.Ej

##  I Introduction

Research on topological quantum states has been one of the fascinating and
influential fields in condensed-matter physics[1, 2, 3, 4, 5, 6, 7]. As is
well known, the discoveries of the integer and fractional quantum Hall
effects[8, 9] have triggered numerous theoretical explanations and
experimental explorations of topological quantum matters. In particular,
findings on topological insulators, topological superconductors, and
topological semi-metals comprise the major themes at present. More recently,
extending the notions of topological quantum states to the high-order
topological phases[10, 11, 12, 13, 14], topological crystalline phases[15, 16,
17, 18, 19], and non-Hermitian topological phases[7] has become the broad
frontier of current research. Usually, topological quantum matter exhibits
nontrivial edge or surface states, which have potential applications from
spintronics to topological quantum computation[20].

Disorders are almost unavoidable in real quantum materials, which dramatically
affects physical properties such as conductivity due to impurity scattering.
Thus, the Anderson transitions between localized and metallic phases in the
disordered systems have been intensively studied[21, 22, 23]. For the
dimensions $D\leq 2$, the scaling theory of localization[22] predicts that all
single-particle states are localized for non-interacting fermions with
uncorrelated disorder, regardless of the amount of disorder. For spatially
correlated disorders, exception situations can occur. For example, the random-
dimer model[24, 25, 26], in which the onsite potentials for pairs of lattice
sites are assigned one of two values at random, exhibits surprising
localization-delocalization transitions, which has also been demonstrated
experimentally[27, 28]. On the other hand, the effect of correlated disorder
on topological insulators[29, 30, 31, 32, 33] has been vividly investigated
recently. Strong disorder can transform such systems into topological Anderson
insulators[34, 35, 36, 37, 38, 39]. As a one-dimensional topological
insulator, the Su-Schrieffer-Heeger (SSH) model[40] provides a paradigmatic
example of the most spectacular topological phases. There are two nontrivial
in-gap edge states and a fractionalized end charge $e/2$ at half filling in
the topological phase of the finite SSH model. The effect of diagonal and off-
diagonal disorder on nontrivial quantum phase transitions of the SSH model has
also been analyzed[37, 41, 42, 43, 44, 45].

It is interesting to investigate the effect of spatially correlated disorder
such as random-dimer disorder on the SSH model, which has not been explored
and can be easily realized in an optical waveguide array. Our numerical
calculations demonstrate that the interplay of dimer hopping and correlated
disorder can induce various unexpected features. Our results reveal that (i)
correlated disorder can cause the topological phase without the in-gap edge
states; (ii) the energy distribution possibility of the in-gap edge states and
possible values of the fractionalized end charge depend on the concentration
of random-dimer disorder; (iii) the quantum transition is not tied to a
delocalization-localization transition of the eigenstates; (iv) the reentrant
localization transition behaviors appear in the non-topological parameter
region; and (v) the topological phase transition regions depend on the
concentration of random-dimer disorder, but the localization-delocalization
transition does not. The rest of this paper is organized as follows. First, in
Sec.II, we present the system model. Then, in Sec. II.1, the quantum phase
transitions at half filling are discussed. In Sec. II.2, the localization-
delocalization transition is investigated. The conclusion is given in Sec.
III.

##  II Model and Result

We consider a one-dimensional model of spinless fermions with nearest-neighbor
dimer hoppings and random-dimer disorder on an $L$ sites chain. The tight-
binding Hamiltonian of the system is

$H=\sum_{j=1}^{L-1}t\left[1-(-1)^{j}\Delta\right](c_{j}^{\dagger}c_{j+1}+\mathrm{H.c.})+\sum_{j=1}^{L}\epsilon_{i}n_{j}$
(1)

where $t$ is the hopping amplitude (set to unit of energy hereafter), and
$\Delta$ denotes the dimerization parameter. $c_{j}^{\dagger}(c_{j})$ creates
(annihilates) a fermionic particle at site $j$. $n_{j}=c_{j}^{\dagger}c_{j}$
is the electronic occupation number operator. For the onsite potential,
$\epsilon_{b}$ ($\epsilon_{a}$) is assigned at random to pairs of lattice
sites (that is, two sites in succession) with probabilities $q$ and $1-q$,
respectively. The model proposed herein can be easily realized in an optical
waveguide array[28]. The one-dimensional waveguide array contains two types of
waveguides with propagation constants $\epsilon_{a}$ and $\epsilon_{b}$, which
are assigned at random to pairs of dimer lattice sites (two adjacent
waveguides with the same onsite potential). The probability of onsite
potential $\epsilon_{b}$ ($\epsilon_{a}$) is $q$ ($1-q$). The two staggered
values of the inter-waveguide separation are set, so that the coupling
strengths $t(1-\Delta)$ and $t(1+\Delta)$ can be easily achieved. For
$\epsilon_{i}=0$, the system reduces to the SSH model[40]. The system becomes
the conventional random-dimer model[24, 25] when the dimerization $\Delta=0$.
For simplicity, in the following we set $q=0.5$ and $\epsilon_{a}=0$.

###  II.1 Phase transitions at half filling

As is well known, the SSH model at half filling is in the topological (non-
topological) phase when the dimerization strength $\Delta<0$ ($>0$)[46, 47].
The appearance of random-dimer disorder breaks the translation invariance.
Thus, we investigate the topological properties of the system in real space.
First, the energy gap under periodic boundary condition as a function of
dimerization parameter and random-dimer disorder is shown in Fig.1a when the
system size is $L=2000$ at half filling under one realization. In the center
red region, the system is gapless because of bulk-gap closing. In other
regions, the system is in the insulating phase. Here, we use Resta’s formula
of polarization[48] in real space as the topological invariant with which to
characterize the topological phase transitions in this disordered system,
which is defined as

$P=\frac{1}{2\pi}\operatorname{Im}\left[\ln\left\langle\Psi\left|e^{2\pi
i\sum_{j}\frac{n_{j}}{L}}\right|\Psi\right\rangle\right]\quad\text{(mod 1)}$
(2)

where the operator $n_{j}$ is electron number operator at site $j$, and $\Psi$
is the many-body ground state.

Figure 1: (a) Energy gap of the system as a function of dimerization parameter
($\Delta$) and onsite potential ($\epsilon_{b}$) under periodic boundary
condition. (b) Phase diagram according to polarization calculated in real
space. (c) Energy spectrum as a function of onsite potential $\epsilon_{b}$
with dimerization amplitude $\Delta=-0.7$. Blue regions are the energy
spectrum with periodic boundary condition. Red dots are edge states under open
boundary condition. (d) Electron density at $\epsilon_{b}=0.5$ and $1.5$ with
$\Delta=-0.7$. Other parameters are $L=2000$ at half filling under each
disorder configuration.

According to the numerical calculations of the polarization, we can obtain the
phase diagram at half filling, as shown in Fig.1b. For the lower left (blue)
region, the polarization $P=0.5$, which indicates that the system is in the
nontrivial topological phase. In the center (red) region, the system is in the
metal phase. In the lower right region, the system is in the trivial
insulating phase with polarization $P=0$. The polarization $P$ does not take a
fixed value for the upper region ($\epsilon_{b}>4$) under more than 200
realizations. Compared with Fig1a, it can be seen that the topological phase
transitions coincide with the energy-gap closing. The topological states are
robust against weak disorder, and the more strongly the dimer parameter
$\Delta$ evolves, the larger the threshold of disorder strength $\epsilon_{b}$
at the phase transition becomes.

Next, we investigate the fate of the in-gap edge states in the topological
phases. In Fig.1c, the energy spectra as functions of disorder strength
$\epsilon_{b}$ with the dimerization amplitude $\Delta=-0.7$ are plotted. As
shown in Ref.[49], the edge states in one-dimensional band insulators have
inherent fragility because they are separated from bulk bands, except those
protected by symmetry. In the system studied herein, the chiral and inversion
symmetries are lost because of the spatially random dimer-disorder and the
zero-energy in-gap states of the clean system become energetic. From Fig.1c,
it can be seen that for small disorder the two in-gap edge states survive in
the energy gap at half filling, and the degeneracy is lifted due to random-
dimer disorder. The two in-gap edge states begin to assimilate into the bulk
states when the disorder strength $\epsilon_{b}$ passes a critical value
(approximately $1.4$). In other words, there are no in-gap edge states in the
nontrivial topological gapped phase and the bulk–boundary correspondence
breaks down. The correlated disorder provides another route to the breakdown
of bulk–boundary correspondence. According to the presence or absence of in-
gap edge states, the green dashed curve in Fig.1b divides the topological
phase region into two parts. For the upper region, there are no in-gap edge
states in this topological phase. As the disorder strength $\epsilon_{b}$
increases ($1.4<\epsilon_{b}<4$), the energy gap closes and the system remains
in the metal phase, as shown in Fig.1c. When the on-site potential
$\epsilon_{b}$ goes beyond $4$, the system enters the trivial insulating phase
without in-gap edge states at half filling.

Figure 2: Energy spectra, polarization and left fractionalized end charges at
half filling under 200 realizations at $\epsilon_{b}=0.5$ [(a) and (b)] and
$1.5$ [(c) and (d)]. Other parameters are $L=2000$ and $\Delta=-0.7$.

The fractionalized charges appear at the ends of the chain as a hallmark of
the topological states in the one-dimensional chain. It is interesting to
analyze the end-charge properties of the SSH model with correlated disorder.
The electron density with $\epsilon_{b}=0.5$ and $1.5$ at half filling under
the open boundary condition is shown in Fig.1d. According to the formula
$Q_{L}=\sum_{j=1}^{L/2}(\langle c_{j}^{\dagger}c_{j}\rangle-\bar{\rho})$
($\bar{\rho}$ is the bulk charge density) of the left end charge, we
numerically found that the end charges in the topological phase are fractional
at a nonzero disorder strength of $\epsilon_{b}$. In Fig.2, we plot the energy
spectrum, polarization, and left fractionalized end charge at half filling
under 200 realizations at $\epsilon_{b}=0.5$ and $1.5$. For the in-gap edge
states of topological states (red dots in Fig.2a), numerical simulations
indicate that the possibility of the high-energy (low-energy) in-gap edge
state approaches the possibility $q$ ($1-q$) of $\epsilon_{b}$
($\epsilon_{a}$) under extreme realizations. In contrast, the possibility of
the fractionalized end charge $e/2$ tends towards the possibility $(1-q)$ of
$\epsilon_{a}$, while the total possibility of the small and large
fractionalized end charge (not equal to $e/2$) approaches the possibility $q$
of $\epsilon_{b}$ (see Fig.2c for detail). This behavior of the fractionalized
end charge also applies to the case of the topological phase without in-gap
edge states (Fig.2d). The possibility properties of in-gap edge states and
fractionalized end charge are unique to the topological phase of the system
studied here. The average value of the left fractionalized end charge under
200 realizations approaches the stable value $e/2$. In addition, it should be
stressed that the other edge states, due to the on-site disorder, appear when
the filling fraction deviates from the half filling case. However, the
numerical calculations of polarization do not take the fixed value under
distinct realizations and the average value exhibits uncertainty.

The probability-dependent energy distribution of the in-gap edge states can be
understood by the following argument: When the system is in topological
nontrivial states at half filling, the energy location of an in-gap (assumed
to be locating at the left edge) state depends on the dimer disorder
$\epsilon_{a}$ or $\epsilon_{b}$ sited at the two left end sites under a
disorder configuration. When the dimer disorder $\epsilon_{b}$ is located at
the two left end sites, the in-gap left edge state occupies the high-energy
region of the energy gap. Thus, the energy distribution possibility of the
high-energy (low-energy) in-gap edge state depends on the concentration of
random-dimer disorder $\epsilon_{b}$ ($\epsilon_{a}$). As the onsite potential
$\epsilon_{b}$ increases further (larger then the critical value), the in-gap
edge states assimilate into the bulk states. As long as the energy gap does
not close, the system still remains in the topological phase, while there are
no in-gap edge states.

In short, the numerical simulations demonstrate the existence of the
topological insulating phase without the in-gap edge states at half filling
induced by random-dimer disorder. The energy distribution possibility of the
in-gap edge states and possible values of the fractionalized end charge depend
on the concentration of random-dimer disorder.

###  II.2 Localization-delocalization transition

We next investigate the localization-delocalization transition of interest due
to the spatially correlated disorder. A way of discerning between localized
and extended states is provided by investigating the inverse participation
ratio (IPR) and normalized participation ratio (NPR), which are given by

$\mathrm{IPR_{n}}=\sum_{j=1}^{L}\left|\phi_{j}^{n}\right|^{4},\mathrm{NPR_{n}}=(L\sum_{j=1}^{L}\left|\phi_{j}^{n}\right|^{4})^{-1}$
(3)

respectively, where $\phi_{j}^{n}$ is the $n$th eigenstate. The IPR of an
extended state scales as $1/L$, thereby vanishing in the large-$N$ limit,
while remaining finite for a localized state. For the intermediate states, the
mean $\langle\mathrm{IPR}\rangle$ (the average of IPR for all the eigenstates)
and $\langle\mathrm{NPR}\rangle$ are finite. The localization-delocalization
phase diagram can be obtained by computing the $\eta$ quantity[50]:

$\eta=\log_{10}[\langle\mathrm{IPR}\rangle\times\langle\mathrm{NPR}\rangle]$
(4)

Figure 3: (a) $\eta$ quantity of the system as a function of dimerization
parameter ($\Delta$) and onsite potential ($\epsilon_{b}$) under periodic
boundary condition. Mean $\langle\mathrm{IPR}\rangle$ and
$\langle\mathrm{NPR}\rangle$ as a function of $\epsilon_{b}$ with
$\Delta=-0.5$ (b) and 0.5 (c). System size in simulation is $L=2000$.

From Fig.3a, it can be seen that away from the clean limit the localization
transition occurs through a small disorder and the system hosts the mobility
edge in both the topological trivial and nontrivial regions. We first
investigate the topological regime($\Delta<0$). Although the occupied states
of the system are topological at half filling, the random-dimer order can
change the location properties of the occupied states as shown in Fig. 3b. The
finite $\langle\mathrm{IPR}\rangle$ and $\langle\mathrm{NPR}\rangle$ signify
the presence of both the localized and extended states(see the green shaded
region in Fig.3b), and the system enters into the coexisting phase. When the
on-site potential $\epsilon_{b}$ goes beyond the critical values, all the
states of the system will be localized. For example,
$\langle\mathrm{IPR}\rangle\neq 0$ and $\langle\mathrm{NPR}\rangle=0$ for all
the onsite potentials $\epsilon_{b}>1$ when the dimer strength
$\Delta=-0.5$(see Fig3b). Moreover, as the dimer strength $|\Delta|$
increases, the critical value of the on-site potential $\epsilon_{b}$
decreases. In addition, compared with Figs.3a and 1b, it can be concluded that
topological quantum transition is not tied to a localization-delocalization
transition of the eigenstates.

In the non-topological regime($\Delta>0$), the system first undergoes the
localization-delocalization transition due to the very small disorder, similar
to the behavior in the topological regions. All the eigenstates of the system
are localized when the onsite potential $\epsilon_{b}$ goes beyond small
critical values, as shown in Figs.3a and 3c. Surprisingly, as the onsite
potential $\epsilon_{b}$ increases, the system again undergoes the
localization-delocalization transition, which is a counter-intuitive result.
The second green region (at approximately $\epsilon_{b}=3$) of Fig.3c
indicates the reentrant localization phenomenon, in which the
$\langle\mathrm{IPR}\rangle$ and $\langle\mathrm{NPR}\rangle$ are nonzero.
Next, we use the IPR, NPR, and transport properties to analyze the coexisting
phase of the localized and extended states in the reentrant regime. In Fig4c,
the $\mathrm{IPR_{n}}$ and $\mathrm{NPR_{n}}$ for all eigenstates are plotted.
Clearly, for some eigenstates, $\mathrm{IPR_{n}}=0$ and $\mathrm{NPR_{n}}\neq
0$, with which the extended properties of the eigenstates are encoded. As
discussed in the random-dimer model[24], the numerical simulations demonstrate
that $\sqrt{L}$ of all eigenstates in the coexisting phase are extended and
unscattered, which is attributed to the single-dimer impurity resonance. To
further investigate the quantum-transport properties of these extended states,
we calculated the tunneling conductivity by using the scattering theory[51].
To this end we attach the first and last sites of the chain to two different
perfect leads. The hoppings between the chain and leads are set to $t=1$. The
tunneling conductance is shown in Fig.4d, when the parameters $\Delta=0.5$ and
$\epsilon_{b}=2.9$. Obviously, at the Fermi energy of the extended states, the
tunneling conductance approaches the quantized conductance $G_{0}=2e^{2}/h$.
For comparison, the IPR, NPR, and tunneling conductance for the parameters
$\Delta=-0.5$ and $\epsilon_{b}=0.5$ in the coexisting region of topological
regime are also plotted in Fig.4a and 4b. In brief, the IPR, NPR, and
tunneling conductance provide compelling evidence for the localization-
delocalization transition in the non-topological regions. This reentrant
localization transition behavior is due to the interplay of dimer hopping and
random-dimer disorder.

Figure 4: (a) and (c) The IPR (red dots), and NPR (blue open circles) of all
eigenstates with system size $L=2000$ under open boundary condition. (b) and
(d) Tunneling conductance (blue curve) and energy spectrum (red asterisks) as
a function of Fermi energy. Other parameters are $\Delta=-0.5$ and
$\epsilon_{b}=0.5$ [(a) and (b)] and $\Delta=0.5$, and $\epsilon_{b}=2.9$ [(c)
and (d)] .

Herein, we use a random-dimer defect $\epsilon_{b}$ to analyze the two
different behaviors of localization-delocalization transitions in topological
and non-topological regimes. Because both of the on-site potentials
$\epsilon_{a}$ and $\epsilon_{b}$ are random in pairs of lattice sites, each
disordered dimer defect can only stay on a whole unit cell of the clean SSH
model (stretching across the two unit cells is not allowed). The dimer defect
$\epsilon_{b}$ can be placed on lattice sites $2k-1$ and $2k$ ($k$ is a non-
negative integer and smaller than $L/2$ ), and all other pairs of lattice
sites along the chain are assigned to $\epsilon_{a}$. Fig. 5 shows two
geometrical configurations for the SSH model with system size $L=10$ as an
example, where a dimer defect $\epsilon_{b}$ is fixed on lattice sites 5 and
6. We can further derive that in thermodynamic limit there are only two types
of geometrical configurations for the fixed dimer defect and dimerization
hoppings, which depends on the sign of the dimerization hopping parameter
$\Delta$. The two configurations induce the different localization properties
of the quantum states. Thus, the two different behaviors of localization-
delocalization transitions induced by the alternatively dimerized hopping in
topological and non-topological regimes can be understood more
comprehensively.

Figure 5: Two geometrical configurations for a dimer defect (fixed on lattice
sites 5 and 6) in the SSH model with system size $L=10$. The box containing
two lattice sites denotes the unit cell. Solid and dotted blue horizontal
lines are alternatively dimerized hoppings. Red vertical lines indicate the
onsite potentials $\epsilon_{a}$ (short lines) and $\epsilon_{b}$ (long
lines).

A comment is in order at this point. Because the energy gap at half filling
can be changed by the concentration of random-dimer disorder, it is easily
verified that the regions of phase diagram at half filling discussed in Sec.
II.1 depend on the concentration $p$ of the onsite potential $\epsilon_{b}$.
However, the conclusions regarding the localization-delocalization transition
in Sec. II.2 are independent of the concentration $p$ and only rely on the
relative difference of $\left|\epsilon_{a}-\epsilon_{b}\right|$.

##  III Conclusion

in this paper we investigated the effect of random-dimer disorder on quantum
phase transitions in the disordered SSH model, which leads to phenomena
drastically distinct from the clean system. We used the topological invariant,
edge states, and fractional end charge to analyze the existence of the
topological phase without in-gap edge states at half filling. The energy
distribution possibility of the in-gap edge states and possible values of the
fractionalized end charge depend on the concentration of random-dimer
disorder. For the localization feature of the system, the reentrant
localization phenomenon in the topological trivial regime appears and is
revealed in detail by the IPR, NPR, and tunneling conductivity. Our findings
could open up new directions to study and facilitate understanding of the
competition mechanism of the modulated hopping and correlated disorder.
Moreover, the non-Hermiticity could also lead to delocalization in one
dimension[52]. It would be interesting to investigate the quantum phase
transitions of the non-Hermitian SSH model with spatially correlated disorder.

Note added: Recently, we became aware of a related work about the SSH model
with random binary disordered hoppings in Ref. [53].

_Acknowledgements._ –Z.W.Z. is grateful to X. Deng for helpful discussion.
This work was supported by the National Natural Science Foundation of China
(Grant Nos. 12074101, and 11604081). Z.W.Z. is also sponsored by the Natural
Science Foundation of Henan (Grant No. 212300410040).

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