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Revista Mexicana de Física

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

Published Apr 22, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.31349/revmexfis.68.031404

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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.
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.
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.
We investigate the quantum phase transition of the Su-Schrieffer-Heeger (SSH) model by inspecting the two-site entanglements in the ground state. It is shown that the topological phase transition of the SSH model is signified by a nonanalyticity of local entanglement, which becomes discontinuous for finite even system sizes, and that this nonanalyticity has a topological origin. Such a peculiar singularity has a universal nature in one-dimensional topological phase transitions of noninteracting fermions. We make this clearer by pointing out that an analogous quantity in the Kitaev chain exhibiting the identical nonanalyticity is the local electron density. As a byproduct, we show that there exists a different type of phase transition, whereby the pattern of the two-site entanglements undergoes a sudden change. This transition is characterised solely by quantum information theory and does not accompany the closure of the spectral gap. We analyse the scaling behaviours of the entanglement in the vicinities of the transition points.
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.
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 study two-particle states in a Su-Schrieffer-Heeger (SSH) chain with periodic boundary conditions and nearest-neighbor (NN) interactions. The system is mapped into a problem of a single particle in a two-dimensional (2D) SSH lattice with potential walls along specific edges. The 2D SSH model has a trivial Chern number but a non-trivial Zak's phase, the one-dimensional (1D) topological invariant, along specific directions of the lattice, which allow for the presence of topological edge states. Using center-of-mass and relative coordinates, we calculate the energy spectrum of these two-body states for strong interactions and find that, aside from the expected appearance of doublon bands, two extra in-gap bands are present. These are identified as bands of topological states localized at the edges of the internal coordinate, the relative distance between the two particles. As such, the topological states reported here are intrinsically many-body in what concerns their real space manifestation, having no counterpart in single-particle states derived from effective models. Finally, we compare the effect of Hubbard interactions with that of NN interactions to show how the presence of the topological bound states is specific to the latter case.