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Geometrical phases and entanglement in real space for 1D SSH topological insulator: effects of first and second neighbor-hoppings and intra-cell modulation

Leonardo A. Navarro-Labastida, Francisco A. Domínguez-Serna, F. Rojas

Published Jun 5, 2021• 0 citations
OpenAlex

Abstract

We study the hybrid atoms-cell site entanglement in one-dimensional Su-Schrieffer-Heeger (SSH) topological insulator with first, second neighbor-hopping interaction and intra-cell modulation, in space representation of finite chains. We determine the geometric phases by the Resta electric polarization and the entanglement in the atomic basis by the Schmidt number as metric of the degree of non-classicity, the maximally entangled quantum states are characterized by the inverse participation ratio IPR. A relation between entanglement and the topological phase transitions (TPT) are found, since the Schmidt number presented critical points of maximally entangled (ME) states just in the singularities of the change of the winding number for the nearest states to the zero mode energy and shifted for higher energy state.We present the general conditions that has to be satisfy to produce a ME hybrid Bell states. We obtain that the states with second neighbors, with two phases, shows a higher degree of entanglement than the first order hopping. We obtain parameter conditions with broad regions for non null topological phases that can have maximum entanglement. We find a robust relation between entanglement and the inverse participation ratio (IPR) for the the eigenstates with ME which are associated to maximum delocalization in both models. The amplitude distribution for each atom are spatially localized in cell site showing a bunching in cell sets. The intra-cell modulation provides a better connection between entanglement and electric polarization with a periodical driven period. The study of topological properties through entanglement is an complementary tool for determining macroscopic properties and analyzing more complex systems with many-body particles or higher dimensions.

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Topological phases and entanglement in real space for 1D SSH topological insulator: effects of first and second neighbor-hoppings

Leonardo A. Navarro-Labastida, Francisco A. Domínguez-Serna, F. Rojas

The hybrid atoms-cell site entanglement in a one-dimensional Su-Schrieffer-Heeger (SSH) topological insulator with first and second neighbor hopping interaction in space representation of finite chains is analyzed. The geometric phase is calculated by the Resta electric polarization and the entanglement in the atomic basis by the Schmidt number. A relation between entanglement and the topological phase transitions (TPT) is given since the Schmidt number has critical points of maximal entangled (ME) states in the singularities of the geometrical phase. States with second-neighbors have higher entanglement than first-order hopping. The general conditions to produce ME hybrid Bell states and the localization-entanglement relation are given.

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We study the hybrid atoms-cell site entanglement in one-dimensional Su-Schrieffer-Heeger (SSH) topological insulator with first, second neighbor-hopping interaction and intra-cell modulation, in space representation of finite chains. We determine the geometric phases by the Resta electric polarization and the entanglement in the atomic basis by the Schmidt number as metric of the degree of non-classicity, the maximally entangled quantum states are characterized by the inverse participation ratio IPR. A relation between entanglement and the topological phase transitions (TPT) are found, since the Schmidt number presented critical points of maximally entangled (ME) states just in the singularities of the change of the winding number for the nearest states to the zero mode energy and shifted for higher energy state.We present the general conditions that has to be satisfy to produce a ME hybrid Bell states. We obtain that the states with second neighbors, with two phases, shows a higher degree of entanglement than the first order hopping. We obtain parameter conditions with broad regions for non null topological phases that can have maximum entanglement. We find a robust relation between entanglement and the inverse participation ratio (IPR) for the the eigenstates with ME which are associated to maximum delocalization in both models. The amplitude distribution for each atom are spatially localized in cell site showing a bunching in cell sets. The intra-cell modulation provides a better connection between entanglement and electric polarization with a periodical driven period. The study of topological properties through entanglement is an complementary tool for determining macroscopic properties and analyzing more complex systems with many-body particles or higher dimensions.
The hybrid atoms-cell site entanglement in a one-dimensional Su-Schrieffer-Heeger (SSH) topological insulator with first and second neighbor hopping interaction in space representation of finite chains is analyzed. The geometric phase is calculated by the Resta electric polarization and the entanglement in the atomic basis by the Schmidt number. A relation between entanglement and the topological phase transitions (TPT) is given since the Schmidt number has critical points of maximal entangled (ME) states in the singularities of the geometrical phase. States with second-neighbors have higher entanglement than first-order hopping. The general conditions to produce ME hybrid Bell states and the localization-entanglement relation are given.
The hybrid atoms-cell site entanglement in a one-dimensional Su-Schrieffer-Heeger (SSH) topological insulator with first and second neighbor hopping in space representation of finite chains is analyzed. The geometrical phase is calculated by the Resta electric polarization and the entanglement in the atomic basis by the Schmidt number. A relation between entanglement and the topological phase transitions (TPT) is given since the Schmidt number has local critical points of maximal entangled (ME) states in the singularities of the geometrical phase. States with second neighbors have higher entanglement than first-neighbors hopping. The general conditions to produce ME hybrid Bell states and the localization-entanglement relation are given.
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.
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.
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.
We investigate quantum correlations and entanglement in a non-Hermitian quantum system modeled by a 2D stacked Su–Schrieffer–Heeger (SSH) lattice incorporating nonreciprocal hopping amplitudes and balanced on-site gain and loss. Focusing on the interplay between non-Hermiticity, topology, and open-boundary conditions, we analyze how these factors shape bipartite entanglement properties, characterized through the entanglement spectrum and entanglement negativity ( E N ). By examining the complex energy spectrum under open boundary conditions, we identify critical points associated with the closing of complex energy gaps, which significantly influence entanglement structure and signal topological phase transitions. Our findings reveal that boundary-localized non-Bloch modes and nonreciprocal couplings can induce highly tunable and unconventional entanglement features, absent under periodic boundary conditions. These results highlight the breakdown of conventional bulk-boundary correspondence and underscore the utility of entanglement measures in diagnosing non-Hermitian topological phases. Beyond advancing the fundamental understanding of quantum correlations in open systems, our study suggests potential pathways to engineer quantum devices and topological materials where gain, loss, and nonreciprocity are intrinsic elements.