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arXiv:2404.10576 · Physical Review A

Quantum phase transition and critical behavior between the gapless topological phases

Hao-Long Zhang, Han-Ze Li, Sheng Yang, Xue-Jia Yu

Published Jun 27, 2024• 22 citations
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Abstract

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 $XXZ$ 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 $XXZ$-type anisotropy parameter $\mathrm{\ensuremath{\Delta}}$ varies, both the critical points ${h}_{c}$ and correlation length exponent $\ensuremath{\nu}$ remain the same as in the $\mathrm{\ensuremath{\Delta}}=0$ case, characterized by $c=3/2$ (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.

DOI
10.1103/physreva.109.062226

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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 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.
Superfluids and superconductors can be either fully gapped or gapless with nodal structures in momentum space. Both the gapped and nodal phases can be topologically protected and possess nontrivial topological invariants. Topological quantum phase transitions exist at zero temperature between different gapped phases or between gapped and nodal phases. These phase transitions can be driven by chemical potential and/or spin exchange fields. The two phases separated by the phase transition can have the same local order but differ in global topology. Thus, these topological quantum phase transitions cannot be described by the Landau paradigm of symmetry breaking. Although some aspects of these transitions, such as the change of topological invariants and gapless boundary states, have been discussed before, a complete theory of these transitions has yet to be developed. In this dissertation, we construct effective field theories to study the universality and thermodynamic signatures of these transitions. We find four different universality classes in superfluids and superconductors. Certain thermodynamic quantities, such as compressibility or spin susceptibility, change non-analytically across the transitions. For certain time-reversal symmetry breaking fields that lead to bulk phase transitions, there also exist topological phase transitions on the surface. All the topological phase transitions studied in this dissertation only exist at zero temperature. At finite temperature, different states are connected by smooth crossovers. There exists a quantum critical region at finite temperature near the quantum critical point (QCP). In this quantum critical region, thermodynamic quantities have universal scaling dependence on temperature dictated by the universality class of the QCP. We argue that these scaling properties can be used to probe and differentiate these QCPs. These bulk and surface topological quantum phase transitions are discussed in various concrete models, such as chiral and time-reversal invariant p-wave superfluids, topological superconductors of emergent Dirac fermions, and topological superconducting model of CuₓBi₂Se₃.
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.
Recently topological states of matter have witnessed a new physical phenomenon where both edge modes and gapless bulk coexist at topological quantum criticality. The presence and absence of edge modes on a critical line can lead to an unusual class of topological phase transition between the topological and nontopological critical phases. We explore the existence of this new class of topological phase transitions in a generic model representing the topological insulators and superconductors and we show that such transition occurs at a multicritical point, i.e., at the intersection of two critical lines. To characterize these transitions we reconstruct the theoretical frameworks, which include bound state solution of the Dirac equation, winding number, correlation factors, and scaling theory of the curvature function to work for the criticality. Critical exponents and scaling laws are discussed to distinguish between the multicritical points, which separate the critical phases. Entanglement entropy and its scaling in the real space provide further insights into the unique transition at criticality revealing the interplay between fixed point and critical point at the multicriticalities.
Long-range interactions can fundamentally alter properties in gapped topological phases such as emergent massive edge modes. However, recent research has shifted attention to topological nontrivial critical points or phases, and it is natural to explore how long-range interactions influence them. In this work, we investigate the topological behavior at the quantum critical point of extended Kitaev chains with long-range interactions, which can be derived from the critical Ising model via the Jordan-Wigner transformation in the short-range limit. Specifically, we analytically find that the critical edge modes at the critical point remain stable against long-range interactions. More importantly, we observe that these critical edge modes remain massless even when long- range interactions become very strong. As a by-product, we numerically find that the critical behavior of the long-range model belongs to the free-Majorana-fermion universality class, which is entirely different from the long-range universality class in the usual long-range spin models. Our work could shed new light on the interplay between long-range interactions (frustrated) and the gapless topological phases of matter.