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arXiv:2111.15500 · Physical Review A

Critical phase boundary and finite-size fluctuations in the Su-Schrieffer-Heeger model with random intercell couplings

D. S. Shapiro, S. V. Remizov, A. V. Lebedev, Бабухин Данила Валерьевич, Ramil S. Akzyanov, Andrei A. Zhukov, L. V. Bork

Published Feb 22, 2022• 2 citations
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Abstract

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 ${\mathbb{Z}}_{2}$-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.

DOI
10.1103/physreva.105.023321

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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{\mathbb{Z}}_{2}Z2​-topological invariant \ensuremathν\ensuremath∈0;1\ensuremath{\nu}\ensuremath{\in}{0;1}\ensuremathν\ensuremath∈0;1 in terms of a non-Hermitian part of the total Hamiltonian, we calculate \ensuremath⟨\ensuremathν\ensuremath⟩\ensuremath{\langle}\ensuremath{\nu}\ensuremath{\rangle}\ensuremath⟨\ensuremathν\ensuremath⟩ 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 \ensuremathν\ensuremath{\nu}\ensuremathν 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.
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=0,1w=0,\phantom{\rule{4pt}{0ex}}1w=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{U}_{c}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{U}_{c}^{\mathrm{SCBA}}UcSCBA​ and that for first Born approximation (FBA) UcFBA{U}_{c}^{\mathrm{FBA}}UcFBA​ are numerically calculated. UcSCBA{U}_{c}^{\mathrm{SCBA}}UcSCBA​ is found to match with Uc{U}_{c}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{U}_{c}^{\mathrm{FBA}}UcFBA​ is quantitatively closer to Uc{U}_{c}Uc​ than UcSCBA{U}_{c}^{\mathrm{SCBA}}UcSCBA​ as long as the renormalized band gap is zero within FBA.
We investigate the topological phase in a one-dimensional Su-Schrieffer-Heeger (SSH) chain in the presence of a finite concentration of non-Hermitian substitutional impurities. Our primary focus is the discovery of a deterministic mechanism to actively reconfigure the topological phase boundaries by introducing site-dependent, non-reciprocal intracell and intercell hopping amplitudes. Specifically, breaking the internal structural symmetry of the non-reciprocal impurity hoppings induces a targeted macroscopic displacement of the critical point. This mechanism allows for the controlled expansion or compression of the topological phase strictly via local defect asymmetry. We validate this boundary displacement through finite-size scaling analysis. The extracted correlation length critical exponents (ν≈1.455ν\approx 1.455ν≈1.455 and ν≈1.397ν\approx 1.397ν≈1.397 for symmetric and asymmetric configurations, respectively) confirm a disorder-induced departure from the pristine SSH universality class, with the exponent diverging at high defect densities. We also report an impurity-induced real-to-complex spectral transition geometrically characterized by the Non-Hermitian Skin Effect. Ultimately, our results establish how isolated non-Hermitian asymmetries cooperatively interact to deterministically shift macroscopic topological protection.
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.
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=0w=0w=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.
Topological phases at quantum criticality have attracted much attention recently. Here we numerically study the interaction-induced phase transitions at around the topological quantum critical points of an extended Su-Schrieffer-Heeger (SSH) chain with next-nearest-neighbor hopping. This extended SSH model shows topological phase transitions between the topologically trivial and nontrivial critical phases when interaction is absent. So long as the interaction terms are turned on, the topologically nontrivial (trivial) critical phases are driven into topologically nontrivial (trivial) insulator phases with finite energy gaps. Particularly, we find the trivial insulator phase is further driven to the nontrivial insulator phase, through interaction-induced topological phase transition, although interaction generally is harmful to nontrivial topology. The stability of the trivial insulator phase against interaction tends to vanish at the multicritical point that separates the trivial and nontrivial critical phases. Our work provides a concrete example for manifesting the impact of interaction on topological quantum criticality.
We investigate interacting Su-Schrieffer-Heeger (SSH) chains with two- and three-site unit cells using density matrix renormalization group (DMRG) simulations. By selecting appropriate filling fractions and sweeping across interaction strength Jz{J}_{z}Jz​ and dimerization $\ensuremath{\delta}$, we map out their phase diagrams and identify transition lines via entanglement entropy and magnetization measurements. In the two-site model, we observe the emergence of an interaction-induced antiferromagnetic intermediate phase between the topologically trivial and nontrivial regimes, as well as a critical region at negative Jz{J}_{z}Jz​ with suppressed magnetization and finite-size scaling of entanglement entropy. In contrast, the three-site model lacks an intermediate phase and exhibits asymmetric edge localization and antiferromagnetic ordering in both positive and negative Jz{J}_{z}Jz​ regimes. We further examine the response of edge states to Ising perturbations. In the two-site model, zero-energy edge modes are topologically protected and remain robust up to a finite interaction strength. However, in the three-site model, where the edge states reside at finite energy, this protection breaks down. Despite this, the edge-localized nature of these states survives in the form of polarized modes whose spatial profiles reflect the noninteracting limit.