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arXiv:2608.22875

Fidelity susceptibility of Su-Schrieffer-Heeger model with further neighbour hopping term

Surajit Mandal, Asim Kumar Ghosh

Published Aug 24, 2026• 0 citations
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Abstract

In this study, topological phase transition in Su-Schrieffer-Heeger (SSH) model with a further neighbour hopping term has been studied in terms of fidelity susceptibility. Topological phase transition point in the standard SSH model has been identified before by noting the divergence of fidelity susceptibility. The same approach has been employed here where fidelity susceptibility is found to diverge at the phase transition points. Additionally, effect of staggered potential on the fidelity susceptibility has been explored. Analytic expression for fidelity susceptibility has been obtained along with its numerical estimation on finite chains by exact diagonalization. Fidelity susceptibility exhibits sharp peaks at the phase transition points in both approaches. Scaling exponent of this divergence has been obtained numerically which is found to agree to that of the standard SSH model without further neighbour terms.

DOI
10.48550/arxiv.2608.22875

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In this study, topological phase transition in Su-Schrieffer-Heeger (SSH) model with a further neighbour hopping term has been studied in terms of fidelity susceptibility. Topological phase transition point in the standard SSH model has been identified before by noting the divergence of fidelity susceptibility. The same approach has been employed here where fidelity susceptibility is found to diverge at the phase transition points. Additionally, effect of staggered potential on the fidelity susceptibility has been explored. Analytic expression for fidelity susceptibility has been obtained along with its numerical estimation on finite chains by exact diagonalization. Fidelity susceptibility exhibits sharp peaks at the phase transition points in both approaches. Scaling exponent of this divergence has been obtained numerically which is found to agree to that of the standard SSH model without further neighbour terms.
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 present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermions---a simplified Su-Schrieffer-Heeger (SSH) model---with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a scaling analysis of these data for essentially semi-infinite chains we obtain the fidelity susceptibility and show that it contains a boundary contribution which is different in the topologically ordered than in the topologically trivial phase. For the entanglement spectrum and entropy we confirm predictions from massive field theory for a block in the middle of an infinite chain but also consider blocks containing the edge of the chain. For the latter case we show that in the SPT phase additional entanglement---as compared to the trivial phase---is present which is localized at the boundary. Finally, we extend our study to the dimerized chain with a nearest-neighbour interaction using exact diagonalization, Arnoldi, and density-matrix renormalization group methods and show that a phase transition into a topologically trivial charge-density wave phase occurs.
We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermions---a simplified Su-Schrieffer-Heeger (SSH) model---with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a scaling analysis of these data for essentially semi-infinite chains we obtain the fidelity susceptibility and show that it contains a boundary contribution which is different in the topologically ordered than in the topologically trivial phase. For the entanglement spectrum and entropy we confirm predictions from massive field theory for a block in the middle of an infinite chain but also consider blocks containing the edge of the chain. For the latter case we show that in the SPT phase additional entanglement---as compared to the trivial phase---is present which is localized at the boundary. Finally, we extend our study to the dimerized chain with a nearest-neighbour interaction using exact diagonalization, Arnoldi, and density-matrix renormalization group methods and show that a phase transition into a topologically trivial charge-density wave phase occurs.
We use a curvature renormalization group (CRG) approach to study the topological phase transitions in a Su-Schrieffer-Heeger chain and its extensions coming from periodic hopping modulations. A curvature function is defined in terms of system parameters near high-symmetry points where the divergence of this function at critical points, in analogy to usual phase transitions, signals a topological phase transition. According to this theory, the phase transition line for the two-site Su-Schrieffer-Heeger (SSH) model is visible at the critical line Δ= 0 where the curvature function diverges. Our study involves this model and also the modulated one with periodicity of four lattice spacing where the curvature function not only diverges at the topological phase transition point (Dirac-like) |Δ/t| = \sqrt(2) but also shows faster divergence at the non-topological gapless point Δ= 0. We further notice faster divergence of correlation length for Δ-> 0 as compared to that for the |Δ/t| -> \sqrt(2) resulting in two different sets of critical exponents making them lie in different universality classes. The edge state exhibits very slow decay into the bulk near the Δ= 0 point while a much quicker decay from edge into bulk is discernible around the |Δ/t| = \sqrt(2) point. We also continue similar analysis for a SSH model with hopping periodicity of eight lattice spacing.
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 $\ensuremath{\nu}\ensuremath{\in}{0;1}$ 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.
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.