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arXiv:2005.10380 · Physical review. B./Physical review. B

Phase diagram of an interacting staggered Su-Schrieffer-Heeger two-chain ladder close to a quantum critical point

A. A. Nersesyan

Published Jul 6, 2020• 21 citations
arXivHTMLPDF

Abstract

We study the ground-state phase diagram of an interacting staggered Su-Schrieffer-Heeger (SSH) ladder in the vicinity of the Gaussian quantum critical point. The corresponding effective field theory, which nonperturbatively treats correlation effects in the ladder, is a double-frequency sine-Gordon (DSG) model. It involves two perturbations at the Gaussian fixed point: the deviation from criticality, and umklapp scattering processes. While massive phases with broken symmetries are identified by means of local order parameters, a topological distinction between thermodynamically equivalent phases becomes feasible only when nonlocal fermionic fields, parity, and the string order parameter are included in consideration. We prove that a noninteracting fermionic staggered SSH ladder is exactly equivalent to an O(2)-symmetric model of two decoupled Kitaev-Majorana chains, or two one-dimensional $p$-wave superconductors. Close to the Gaussian fixed point, the SSH ladder maps to an Ashkin-Teller-like system when interactions are included. Thus, the topological order in the SSH ladder is related to broken-symmetry phases of the associated quantum spin-chain degrees of freedom. The obtained phase diagram includes a Tomonaga-Luttinger liquid state that, due to umklapp processes, can become unstable against either spontaneous dimerization or the onset of a charge-density wave (CDW). In these gapped phases, elementary bulk excitations are quantum kinks carrying the charge ${Q}_{F}=1/2$. For sufficiently strong, long-range interactions, the phase diagram of the model exhibits a bifurcation of the Gaussian critical point into two outgoing Ising criticalities. The latter sandwich a mixed phase in which dimerization coexists with a site-diagonal CDW. In this phase, elementary bulk excitations are represented by two types of topological solitons carrying different fermionic charges, which continuously interpolate between 0 and 1. This phase has also mixed topological properties with coexisting parity and string order parameters.

DOI
10.1103/physrevb.102.045108

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We study the ground-state phase diagram of an interacting staggered Su-Schrieffer-Heeger (SSH) ladder in the vicinity of the Gaussian quantum critical point. The corresponding effective field theory, which nonperturbatively treats correlation effects in the ladder, is a double-frequency sine-Gordon (DSG) model. It involves two perturbations at the Gaussian fixed point: the deviation from criticality, and umklapp scattering processes. While massive phases with broken symmetries are identified by means of local order parameters, a topological distinction between thermodynamically equivalent phases becomes feasible only when nonlocal fermionic fields, parity, and the string order parameter are included in consideration. We prove that a noninteracting fermionic staggered SSH ladder is exactly equivalent to an O(2)-symmetric model of two decoupled Kitaev-Majorana chains, or two one-dimensional ppp-wave superconductors. Close to the Gaussian fixed point, the SSH ladder maps to an Ashkin-Teller-like system when interactions are included. Thus, the topological order in the SSH ladder is related to broken-symmetry phases of the associated quantum spin-chain degrees of freedom. The obtained phase diagram includes a Tomonaga-Luttinger liquid state that, due to umklapp processes, can become unstable against either spontaneous dimerization or the onset of a charge-density wave (CDW). In these gapped phases, elementary bulk excitations are quantum kinks carrying the charge QF=1/2{Q}_{F}=1/2QF​=1/2. For sufficiently strong, long-range interactions, the phase diagram of the model exhibits a bifurcation of the Gaussian critical point into two outgoing Ising criticalities. The latter sandwich a mixed phase in which dimerization coexists with a site-diagonal CDW. In this phase, elementary bulk excitations are represented by two types of topological solitons carrying different fermionic charges, which continuously interpolate between 0 and 1. This phase has also mixed topological properties with coexisting parity and string order parameters.
Quasi-one-dimensional or ladder models serve as a bridge between one-dimensional and two-dimensional models, providing a simplified framework for incorporating intricate physics. In this study, we extend the concept of the momentum-state lattice by introducing couplings between next-nearest-neighbor sites throughout the entire momentum chain to simulate the staggered Su-Schrieffer-Heeger (SSH) ladder model. This ladder model is a direct extension of the well-known SSH model, recognized as a fundamental progression in modeling systems. Through this model, we explore compelling physical phenomena and successfully replicate the topological phase diagram of the Kitaev-Majorana chain. Utilizing time-averaged mean displacements of wave packets as a metric, we ascertain the topological invariants of the model. By varying staggered dimerization and interchain hopping parameters, we map out the topological phase diagram of the staggered SSH ladder, elucidating the presence of three distinct phases classified by their winding numbers: W = + 1 , 0, and − 1 , respectively.
We study the phase diagram of the interacting fermionic two-leg ladder, which is featured by pair hopping and interactions of singlet and triplet superconducting channels. By using Abelian bosonization method, we obtain the full phase diagram of our model. The superconducting triplet pairing phase is characterized by a fractional edge spin and interpreted as two Kitaev chains under the mean filed approximation. The pair hopping will give rise to spin-density-wave (SDW) orders and can also support Majorana edge modes in spin channel. At half filling, the resulting Majorana-SDW phase shows additional fractionalization in charge channel, and can be interpreted as two Su–Schrieffer–Heeger (SSH) chains in the mean field regime.
Exploring topological phases in interacting systems is a challenging task. We investigate many-body topological physics of interacting fermions in an extended Su-Schrieffer-Heeger (SSH) model, which extends the two sublattices of the SSH model into four sublattices and thus is dubbed the SSH4 model, based on the density-matrix renormalization-group numerical method. The interaction-driven phase transition from the topological insulator phase to the charge density wave (CDW) phase can be identified by analyzing the variations of the entanglement spectrum, entanglement entropies, energy gaps, CDW order parameter, and fidelity. We map the global phase diagram of the many-body ground state, which contains nontrivial topological insulator, trivial insulator, and CDW phases, respectively. In contrast to the interacting SSH model, in which the phase transitions to the CDW phase are argued to be first-order phase transitions, the phase transitions between the CDW phase and topologically trivial or nontrivial phases are shown to be continuous phase transitions. Finally, we also show the phase diagram of the interacting spinful SSH4 model, where the attractive (repulsive) on-site spin interaction amplifies (suppresses) the CDW phase. The models analyzed here can be implemented with ultracold atoms on optical superlattices.
Topological insulators have attracted numerous attentions recent years, where the Su-Schrieffer-Heeger (SSH) model is one of the most studied models. While the interacting version of it has been explored recently, the interplay between interactions and long-range hoppings merit further investigations. In this work, we uncover a rich phase diagram of the interacting SSH model with extended-range hoppings, in which it consists of several topological phases, two novel superconducting-like (SC-like) phases and five distinct charge-density-wave (CDW) phases. We substantiate that the SC-like and two CDW phases are direct consequences of imbalanced interactions and extended-range hoppings. We derive the order parameters (OPs) for each of the phases and verify them in large-system simulations, finding consistency with the entanglement entropy and the fidelity in capturing the phase transitions. In contrast to the non-interacting case where the favored hoppings are unidirectional in the topological phases, the derived OPs suggest non-unidirectional hoppings are possible under the influence of interactions.
We study the topological properties of hardcore bosons on a two-leg ladder consisting of two Su-Schrieffer-Heeger chains that are coupled via hopping and interaction. We chart out the phase diagram for the system and show that based on the relative hopping dimerization pattern along the legs, distinctly different topological phases and phase transitions can occur. When the dimerization along the legs is uniform, we find that the topological nature vanishes for even the slightest rung hopping. For staggered dimerization, the system exhibits a well-defined topological character and a topological phase transition as a function of rung hopping. While the topological phase shows bond order character, the trivial phase shows the behavior of a rung-Mott insulator. For this case, the topological nature is found to survive even in the presence of finite interleg interactions. Moreover, we find that the critical point of the topological phase transition shifts to a higher or a lower rung hopping strength depending on the attractive or repulsive nature of the interaction. To highlight the marked effects of interactions, we propose a scheme involving a Thouless charge pump that provides insights for the topological phases characterized by a quantized particle transport through a periodic modulation of appropriate system parameters. In our studies, we show an interaction-induced charge pumping following specific pumping protocols in the case of staggered dimerization.
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.