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 phases 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.
A dimerized fermion chain, described by Su-Schrieffer-Heeger (SSH) model, is a well-known example of a one-dimensional system with a nontrivial band topology. An interplay of disorder and topological ordering in the SSH model is of great interest owing to experimental advancements in synthesized quantum simulators. In this paper, we investigate a special sort of a disorder when intercell hopping amplitudes are random. Using a definition for
Z2-topological invariant $\ensuremath{\nu}\ensuremath{\in}{0;1}$ in terms of a non-Hermitian part of the total Hamiltonian, we calculate $\ensuremath{\langle}\ensuremath{\nu}\ensuremath{\rangle}$ averaged by random realizations. This allows to find: (i) an analytical form of the critical surface that separates phases of distinct topological orders and (ii) finite-size fluctuations of $\ensuremath{\nu}$ for arbitrary disorder strength. Numerical simulations of the edge modes formation and gap suppression at the transition are provided for the finite-size system. In the end, we discuss a band-touching condition derived within the averaged Green's function method for a thermodynamic limit.
Recent studies of moiré physics have unveiled a wealth of opportunities for significantly advancing the field of quantum phase transitions. However, properties of reentrant phase transitions driven by moiré strength are poorly understood. Here, we investigate the reentrant sequence of phase transitions and the invariant of universality class in moiré-modulated extended Su-Schrieffer-Heeger (SSH) model. For the simplified case with intercell hopping
w=0, we analytically derive renormalization relations of Hamiltonian parameters to explain the reentrant phenomenon. For the general case, numerical phase boundaries are calculated in the thermodynamic limit. The bulk boundary correspondence between zero-energy edge modes and entanglement spectrum is revealed from the degeneracy of both quantities. We also address the correspondence between the central charge obtained from entanglement entropy and the change in winding number during the phase transition. Our results shed light on the understanding of universal characteristics and bulk-boundary correspondence for moiré induced reentrant phase transitions in 1D condensed-matter systems.
The (one-dimensional) Su-Schrieffer-Heeger Hamiltonian, augmented by spin-orbit coupling and longer-range hopping, is studied at half filling for an even number of sites. The ground-state phase diagram depends sensitively on the symmetry of the model. Charge-conjugation (particle-hole) symmetry is conserved if hopping is only allowed between the two sublattices of even and odd sites. In this case, we find a variety of topologically nontrivial phases, characterized by different numbers of edge states (or, equivalently, different quantized Zak phases). The transitions between these phases are clearly signalled by the entanglement entropy. Charge-conjugation symmetry is broken if hopping within the sublattices is admitted. We study specifically next-nearest-neighbor hopping with amplitudes
ta and
tb for the
A and
B sublattices, respectively. For
ta=tb, parity is conserved, and also the quantized Zak phases remain unchanged in the gapped regions of the phase diagram. However, metallic patches appear due to the overlap between conduction and valence bands in some regions of parameter space. The case of alternating next-nearest-neighbor hopping, ${t}_{a}=\ensuremath{-}{t}_{b}$, is also remarkable, as it breaks both charge-conjugation
C and parity
P but conserves the product
CP. Both the Zak phase and the entanglement spectrum still provide relevant information, in particular about the broken parity. Thus the Zak phase for small values of
ta measures the disparity between bond strengths on
A and
B sublattices, in close analogy to the proportionality between the Zak phase and the polarization in the case of the related Aubry-Andr\'e model.
We use the Lindblad equation approach to investigate topological phases hosting more than one localized state at each side of a disordered Su-Schrieffer-Heeger chain with properly tuned long-range hoppings. Inducing a nonequilibrium steady state across the chain, we probe the robustness of each phase and the fate of the edge modes looking at the distribution of electrons along the chain and the corresponding standard deviation in the presence of different kinds of disorder either preserving the symmetries of the Hamiltonian or not.
We investigate reentrant topological transitions in a one-dimensional Su-Schrieffer-Heeger chain with generalized Bernoulli disorder in the intradimer hopping amplitudes. Owing to its independently tunable values and probabilities, the multivalued disorder distribution provides a direct way to control the topological phase diagram. We show that increasing the disorder strength can split the nontrivial regime into multiple disconnected topological windows, whose number, widths, and locations are determined by the distribution parameters. The phase boundaries are derived analytically from the zero-mode inverse localization length and are governed by a weighted geometric mean of the disordered hopping amplitudes, in agreement with numerical results from the reflection-matrix topological quantum number and the real-space winding number. We also show that the mean chiral displacement dynamically identifies these reentrant windows. These results demonstrate how multivalued random disorder can organize and tune reentrant topological behavior in one-dimensional chiral lattices.
The long-range Su-Schrieffer-Heeger (SSH) model, in which the second nearest-neighbor hopping is taken into account, exhibits a topological phase diagram that contains winding numbers
w=0,1, and 2. In the clean system, the change in winding number stems from the band-touching phenomenon. In the presence of disorder, the renormalization of energy band and Fermi level results in the nonzero density of states in the energy gap. These midgap states cause the crossover phenomenon and the divergence of localization length at a critical disorder strength
Uc in the finite SSH system. In this study, we numerically computed the mean winding number and localization length for the disordered SSH system. We find that the disorder is able to drive phase transitions between different mean winding numbers: $w=0\ensuremath{\rightarrow}1$, $0\ensuremath{\rightarrow}2$, $1\ensuremath{\rightarrow}2$, and $2\ensuremath{\rightarrow}1$ in the weak disorder regime. By investigating the wave function distribution and the self-energy, the nonzero mean winding numbers correspond to the so-called topological Anderson insulating (TAI) phases. The finite size scaling for the mean winding number in the TAI phase is shown. For describing the phase transitions in the thermodynamic limit, we apply the criterion of band gap closure resulting from the broadening of energy band and Fermi level to determine the critical disorder strength. The critical disorder strength for self-consistent Born approximation (SCBA)
UcSCBA and that for first Born approximation (FBA)
UcFBA are numerically calculated.
UcSCBA is found to match with
Uc qualitatively. Nonetheless, SCBA indicates the different roles of band shifts and Fermi level broadening near the topological phase transitions. Band shift/Fermi level broadening is more dominant for the transitions from low-to-high/high-to-low winding number. Interestingly, for the transition from bulk insulator to TAI,
UcFBA is quantitatively closer to
Uc than
UcSCBA as long as the renormalized band gap is zero within FBA.