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arXiv:2403.03716 · Physical review. B./Physical review. B

Fidelity susceptibility at the Lifshitz transition between the noninteracting topologically distinct quantum critical points

Xue-Jia Yu, Weilin Li

Published Jul 11, 2024• 22 citations
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Abstract

By constructing an exactly solvable spin model, we investigate the critical behaviors of transverse-field Ising chains interpolated with cluster interactions, which exhibit various types of topologically distinct Ising critical points. Using fidelity susceptibility as an indicator, we establish the global phase diagram, including ferromagnetic, trivial paramagnetic, and symmetry-protected topological phases. Different types of critical points exist between these phases, encompassing both topologically trivial and nontrivial Ising critical points, as well as Gaussian critical points. Importantly, we demonstrate the existence of a Lifshitz transition between these topologically distinct Ising critical points, with central charge and critical exponents determined through finite-size scaling. This work serves as a valuable reference for further research on phase transitions within the gapless quantum phase of matter.

DOI
10.1103/physrevb.110.045119

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By constructing an exactly solvable spin model, we investigate the critical behaviors of transverse-field Ising chains interpolated with cluster interactions, which exhibit various types of topologically distinct Ising critical points. Using fidelity susceptibility as an indicator, we establish the global phase diagram, including ferromagnetic, trivial paramagnetic, and symmetry-protected topological phases. Different types of critical points exist between these phases, encompassing both topologically trivial and nontrivial Ising critical points, as well as Gaussian critical points. Importantly, we demonstrate the existence of a Lifshitz transition between these topologically distinct Ising critical points, with central charge and critical exponents determined through finite-size scaling. This work serves as a valuable reference for further research on phase transitions within the gapless quantum phase of matter.
The phase transition between gapped topological phases represents a class of unconventional criticality beyond the Landau paradigm. However, recent research has shifted attention to topological phases without a bulk gap, where the phase transitions between them are still elusive. In this work, based on large-scale density-matrix renormalization-group techniques, we investigate the critical behaviors of the extended quantum XXZXXZXXZ model obtained by the Kennedy-Tasaki transformation. Using fidelity susceptibility as a diagnostic, we obtain a complete phase diagram, which includes both topological nontrivial and trivial gapless phases. Furthermore, as the XXZXXZXXZ-type anisotropy parameter $\mathrm{\ensuremath{\Delta}}$ varies, both the critical points hc{h}_{c}hc​ and correlation length exponent $\ensuremath{\nu}$ remain the same as in the $\mathrm{\ensuremath{\Delta}}=0$ case, characterized by c=3/2c=3/2c=3/2 (Ising plus free boson) conformal field theory. Our results indicate that fidelity susceptibility can effectively detect and reveal a stable unconventional critical line between the topologically distinct gapless phases for general $\mathrm{\ensuremath{\Delta}}$. This work serves as a valuable reference for further research on phase transitions within the gapless topological phase of matter.
The investigation and characterization of topological quantum phase transition between gapless phases is one of the recent interest of research in topological states of matter. We consider transverse field Ising model with three spin interaction in one dimension and observe a topological transition between gapless phases on one of the critical lines of this model. We study the distinct nature of these gapless phases and show that they belong to different universality classes. The topological invariant number (winding number) characterize different topological phases for the different regime of parameter space. We observe the evidence of two multi-critical points, one is topologically trivial and the other one is topologically active. Topological quantum phase transition between the gapless phases on the critical line occurs through the non-trivial multi-critical point in the Lifshitz universality class. We calculate and analyze the behavior of Wannier state correlation function close to the multi-critical point and confirm the topological transition between gapless phases. We show the breakdown of Lorentz invariance at this multi-critical point through the energy dispersion analysis. We also show that the scaling theories and curvature function renormalization group can also be effectively used to understand the topological quantum phase transitions between gapless phases. The model Hamiltonian which we study is more applicable for the system with gapless excitations, where the conventional concept of topological quantum phase transition fails.
Quantum entanglement can be an effective diagnostic tool for probing topological phases protected by global symmetries. Recently, the notion of nontrivial topology in critical systems has been proposed and is attracting growing attention. In this work, as a concrete example, we explore the quantum entanglement properties of fermionic symmetry-enriched quantum critical points by constructing exactly solvable models based on stacked multiple Kitaev chains. We first analytically establish the global phase diagram using entanglement entropy and reveal three topologically distinct gapped phases with different winding numbers, along with three topologically distinct transition lines separating them. Importantly, we unambiguously demonstrate that two transition lines exhibit fundamentally different topological properties despite sharing the same central charge. Specifically, they display nontrivial topological degeneracy in the entanglement spectrum under periodic boundary conditions, thereby generalizing the Li-Haldane bulk-boundary correspondence to a broader class of fermionic symmetry-enriched criticality. Additionally, we identify a novel Lifshitz multicritical point at the intersection of the three transition lines, which also exhibits nontrivial topological degeneracy. This work provides a valuable reference for investigating gapless topological phases of matter from the perspective of quantum entanglement.
Abstract We perform a comprehensive analytical study of the exotic quantum phases and phase transitions emerging from the cluster-Ising model with off-diagonal Gamma interactions. Specifically, we map out the ground-state phase diagram by analyzing both local and nonlocal order parameters, together with the energy spectra. The results reveal one pair of gapped phases—namely antiferromagnetic (AFM) long-range ordered phase and symmetry-protected topological (SPT) phase—as well as two distinct gapless spiral phases induced by the off-diagonal interactions. Remarkably, the phase diagram contains four distinct transition lines. Two of them, separating different gapped or gapless phases, are governed by the Ising and three-copy Ising conformal field theories, respectively. In contrast, the remaining two transition lines—between the gapless spiral and gapped phases—belong to a non-conformal Lifshitz criticality with dynamical critical exponent z=2z = 2z=2 z = 2 . More importantly, the intersection of these four transition lines gives rise to a new Lifshitz multicritical point exhibiting emergent conformal symmetry, marking a fundamental departure from all previously known non-conformal Lifshitz points. This work provides a valuable reference for future investigations of exotic gapless phases and their transitions in exactly solvable many-body systems.
An attempt is made to show the topological transition on conformal manifold for quantum Ising model with longer range interaction. This model Hamiltonian system has different gapped phases with different topological indices and also different quantum critical lines depending on the presence and absence of the transverse field. We also present the central charge for the different regimes of the parameter space. We show explicitly on the interplay of criticality, topology and central charge (C) in the presence and absence of transverse field and also non-universal character of C. We show how the Lifshitz transition occurs in the presence of transverse field and changes the universality class of the central charge in presence of transverse field. We show explicitly the existence of conformal field theory (CFT) criticality and non-CFT criticality. We have presented an explicit calculations to find the relation between the polynomial function and the Anderson-pseudo spin model Hamiltonian. Our results are far more enrich than the existence results of noninteracting fermionic many body system. This work not only provides a new perspective in topological state of conformal field theory but also for low dimensional quantum many body system.
Topology is a fascinating phenomena in condensed-matter physics typically associated with a bulk gap. However, recent research shifts focus to quantum critical points or phases that exhibit nontrivial topological properties. Here we explore a cluster-Ising chain with long-range antiferromagnetic interactions that decay as a power law with distance. Using large-scale density matrix renormalization group simulations, we demonstrate that the nontrivial topology at the critical point remains stable against long-range interactions, resulting in a topologically nontrivial critical line. Moreover, even within the gapped region, the interplay between topology and long-range interaction can give rise to a topological phase featuring algebraically decaying correlations and edge modes, similar to gapless topological phases. We refer to this phase as the algebraic topological phase, which exhibits nontrivial gapless topological behaviors and arises solely from long-range interactions without short-range counterparts. The findings pave the way for more studies on topological states in long-range systems. Long-range interactions play a crucial role in condensed-matter systems and can lead to range of emerging and exotic physical phenomenon. Here using simulations, the authors investigate the non-trivial topological properties that can arise in a cluster Ising chain with long-range antiferromagnetic interactions, finding evidence for what they classify as an algebraic symmetry-protected topological phase.