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arXiv:2004.11050 · Physical review. B./Physical review. B

Topological and nontopological features of generalized Su-Schrieffer-Heeger models

N. Ahmadi, Jahanfar Abouie, D. Baeriswyl

Published May 11, 2020• 34 citations
arXivHTMLPDF

Abstract

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 ${t}_{a}$ and ${t}_{b}$ for the $A$ and $B$ sublattices, respectively. For ${t}_{a}={t}_{b}$, 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 ${t}_{a}$ 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.

DOI
10.1103/physrevb.101.195117

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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{t}_{a}ta​ and tb{t}_{b}tb​ for the AAA and BBB sublattices, respectively. For ta=tb{t}_{a}={t}_{b}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, ta=\ensuremath−tb{t}_{a}=\ensuremath{-}{t}_{b}ta​=\ensuremath−tb​, is also remarkable, as it breaks both charge-conjugation CCC and parity PPP but conserves the product CPCPCP. 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{t}_{a}ta​ measures the disparity between bond strengths on AAA and BBB 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 investigate the quantum phases of a half-filled generalized interacting Su-Schrieffer-Heeger model with intracell, nearest-neighbor, and next-nearest-neighbor intercell hoppings, together with an on-site inter-sublattice interaction. In the noninteracting limit, the model hosts one topologically trivial phase and two symmetry-protected topological (SPT) phases, distinguished under periodic boundary conditions by different winding numbers and under open boundary conditions by two-fold and four-fold entanglement-spectrum degeneracies, respectively. When interactions are introduced, these free-fermion SPT phases evolve into distinct interacting topological phases that retain characteristic signatures such as entanglement-spectrum degeneracy structures, boundary modes, and nonzero string order parameters. For strong repulsive interactions, a symmetry-breaking phase with unequal but spatially uniform sublattice densities appears between the trivial and topological regimes. For strong attractive interactions, period-2 and period-4 charge-density-wave phases emerge from particle clustering. At intermediate attractive interactions, the competition between interaction-induced localization and hopping-induced delocalization gives rise to a Luttinger liquid phase, a paired Luttinger liquid phase, and a gapless symmetry-protected topological (gSPT) phase. The gSPT phase is characterized by a gapless charge mode together with symmetry-protected current-carrying edge states. We further characterize the gapless phases and the associated quantum phase transitions through central charges and critical exponents.
We address the effect of nearest-neighbor (NN) interactions on the topological properties of the Su-Schrieffer-Heeger (SSH) chain, with alternating hopping amplitudes t1{t}_{1}t1​ and t2{t}_{2}t2​. Both numerically and analytically, we show that the presence of interactions induces phase transitions between topologically different regimes. In the particular case of one-hole excitations in a half-filled SSH chain, the V/t2V/{t}_{2}V/t2​ versus t1/t2{t}_{1}/{t}_{2}t1​/t2​ phase diagram has topological phases at diagonal regions of the phase plane. The interaction acts in this case as a passivation potential. For general filling of the SSH chain, different eigensubspaces of the SSH Hamiltonian may be classified as topologically trivial and nontrivial. The two-hole case is studied in detail in the large interaction limit, and we show that a mapping can be constructed of the two-hole SSH eigensubspaces into one-particle states of a noninteracting one-dimensional (1D) tight-binding model, with interfaces between regions with different hopping constants and local potentials. The presence of edge states of topological origin in the equivalent chain can be readily identified, as well as their correspondence to the original two-hole states. Of these states only some, identified by us, are protected and, therefore, truly topological. Furthermore, we found that the presence of the NN interaction generates a state where two holes occupy two consecutive edge states. Such many-body states should also occur for arbitrary filling leading to the possibility of a macroscopic hole gathering at the surface (at consecutive edge states).
The topological properties of the Su-Schrieffer-Heeger (SSH) model in the presence of nearest-neighbor interaction are investigated by means of a topological marker, generalized from a noninteracting one by utilizing the single-particle Green's function of the many-body ground state. We find that despite the marker not being perfectly quantized in the presence of interactions, it always remains finite in the topologically nontrivial phase while converging to zero in the trivial phase when approaching the thermodynamic limit, and hence correctly judges the topological phases in the presence of interactions. The marker also correctly captures the interaction-driven, second-order phase transitions between a topological phase and a Landau-ordered phase, which is a charge-density wave order in our model with a local-order parameter, as confirmed by the calculation of entanglement entropy and the many-body Zak phase. Our paper thus points to the possibility of generalizing topological markers to interacting systems through Green's function, which may be feasible for topological insulators in any dimension and symmetry class.
Despite extensive studies on the one-dimensional Su-Schrieffer-Heeger-Hubbard (SSHH) model, the variant incorporating second-nearest neighbor hopping remains largely unexplored. Here, the topological classification of the extended SSH model is analyzed within the framework of the BDI symmetry class, using the winding number as the corresponding topological invariant. We investigate the ground-state properties of this extended SSHH model using the constrained-path auxiliary-field quantum Monte Carlo (CP-AFQMC) method. We show that this model exhibits rich topological phases, characterized by robust edge states against interaction. We quantify the properties of these edge states by analyzing spin correlation and second-order Rényi entanglement entropy. The system exhibits long-range spin correlation and near-zero Rényi entropy at half-filling. Besides, there is an anti-ferromagnetic order at quarter-filling. Interestingly, an external magnetic field disrupts this anti-ferromagnetic order, restoring long-range spin correlation and near-zero Rényi entropy. Furthermore, our work provides a paradigm for studying topological properties in large interacting systems via the CP-AFQMC algorithm.
Symmetry plays a crucial role in understanding topological phases in materials. In one-dimensional systems, such as the Su-Schrieffer-Heeger (SSH) model, chiral symmetry is thought to ensure the quantization of the Zak phase and the nontrivial topological phase. However, our work demonstrates that the one-dimensional lattice system with broken chiral symmetry can still possess quantized Zak phase and nontrivial topological phase. Specifically, we use a Bose-Einstein condensate of 87Rb atoms in a momentum space lattice of ultracold atoms to effectively simulate a one-dimensional Zigzag model of 26 sites, which intrinsically breaks the chiral symmetry by additional next-nearest-neighbor coupling. To ensure the existence of the nontrivial topological phase, where the Zak phase can be measured from the time-averaged displacement during the system’s evolution, we need to preserve the inversion symmetry by modulating laser power so that all next-nearest-neighbor coupling strengths are equal. Furthermore, by changing the ratio of nearest-neighbor coupling strengths, we observe a topological phase transition from a nontrivial topological phase to a trivial topological phase at the point where the ratio equals 1. Our work demonstrates that the ultracold atom system provides a controllable platform for studying the symmetrical phase and topological phase, with the potential to explore nonlinear topological phenomena by increasing the interactions among atoms. In addition, our system can be used to investigate other interesting topological phenomena with more complex models, such as critical phenomena at the phase transitions and flat band structures in the extended SSH model with long-range coupling. By controlling the coupling strengths, we can also explore the influence of different symmetries on the topological properties of extended SSH models in the future. Moreover, our platform makes it possible to studythe models with more energy bands, such as the Aharonov-Bohm caging model with a three-level structure, which shows peculiar flat-band properties. This work provides opportunities for various studies in the fields of symmetry, topology, and the interaction of controllable quantum systems.
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.