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arXiv:2007.10669 · Physical review. B./Physical review. B

Fidelity susceptibility near topological phase transitions in quantum walks

S. Panahiyan, Wei Chen, S. Fritzsche

Published Oct 16, 2020• 32 citations
arXivHTMLPDF

Abstract

The notion of fidelity susceptibility, introduced within the context of quantum metric tensor, has been an important quantity to characterize the criticality near quantum phase transitions. We demonstrate that for topological phase transitions in Dirac models, provided the momentum space is treated as the manifold of the quantum metric, the fidelity susceptibility coincides with the curvature function whose integration gives the topological invariant. Thus the quantum criticality of the curvature function near a topological phase transition also describes the criticality of the fidelity susceptibility, and the correlation length extracted from the curvature function also gives a momentum scale over which the fidelity susceptibility decays. To map out the profile and criticality of the fidelity susceptibility, we turn to quantum walks that simulate one-dimensional class BDI and two- dimensional class D Dirac models and demonstrate their accuracy in capturing the critical exponents and scaling laws near topological phase transitions.

DOI
10.1103/physrevb.102.134111

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The notion of fidelity susceptibility, introduced within the context of quantum metric tensor, has been an important quantity to characterize the criticality near quantum phase transitions. We demonstrate that for topological phase transitions in Dirac models, provided the momentum space is treated as the manifold of the quantum metric, the fidelity susceptibility coincides with the curvature function whose integration gives the topological invariant. Thus the quantum criticality of the curvature function near a topological phase transition also describes the criticality of the fidelity susceptibility, and the correlation length extracted from the curvature function also gives a momentum scale over which the fidelity susceptibility decays. To map out the profile and criticality of the fidelity susceptibility, we turn to quantum walks that simulate one-dimensional class BDI and two- dimensional class D Dirac models and demonstrate their accuracy in capturing the critical exponents and scaling laws near topological phase transitions.
The fidelity per site between two ground states of a quantum lattice system corresponding to different values of the control parameter defines a surface embedded in a Euclidean space. The Gaussian curvature naturally quantifies quantum fluctuations that destroy orders at transition points. It turns out that quantum fluctuations wildly distort the fidelity surface near the transition points, at which the Gaussian curvature is singular in the thermodynamic limit. As a concrete example, the one-dimensional quantum Ising model in a transverse field is analyzed. We also perform a finite size scaling analysis for the transverse Ising model of finite sizes. The scaling behavior for the Gaussian curvature is numerically checked and the correlation length critical exponent is extracted, which is consistent with the conformal invariance at the critical point.
We apply the fidelity metric approach to analyze two recently introduced models that exhibit a quantum phase transition to a topologically ordered phase. These quantum models have a known connection to classical statistical mechanical models; we exploit this mapping to obtain the scaling of the fidelity metric tensor near criticality. The topological phase transitions manifest themselves in divergences of the fidelity metric across the phase boundaries. These results provide evidence that the fidelity approach is a valuable tool to investigate novel phases lacking a clear characterization in terms of local order parameters.
For DDD-dimensional weakly interacting topological insulators in certain symmetry classes, the topological invariant can be calculated from a DDD- or (D+1)(D+1)(D+1)-dimensional integration over a certain curvature function that is expressed in terms of single-particle Green's functions. Based on the divergence of curvature function at the topological phase transition, we demonstrate how a renormalization group approach circumvents these integrations and reduces the necessary calculation to that for the Green's function alone, rendering a numerically efficient tool to identify topological phase transitions in a large parameter space. The method further unveils a number of statistical aspects related to the quantum criticality in weakly interacting topological insulators, including correlation function, critical exponents, and scaling laws, that can be used to characterize the topological phase transitions driven by either interacting or noninteracting parameters. We use 1D class BDI and 2D class A Dirac models with electron-electron and electron-phonon interactions to demonstrate these principles and find that interactions may change the critical exponents of the topological insulators.
We investigate quantum phase transitions in one-dimensional quantum disordered lattice models, the Anderson model and the Aubry-Andr\'e model, from the fidelity susceptibility approach. First, we find that the fidelity susceptibility and the generalized adiabatic susceptibility are maximum at the quantum critical points of the disordered models, through which one can locate the quantum critical point in disordered lattice models. Second, finite-size scaling analysis of the fidelity susceptibility and of the generalized adiabatic susceptibility show that the correlation length critical exponent and the dynamical critical exponent at the quantum critical point of the one-dimensional Anderson model are respectively 2/3 and 2 and of the Aubry-Andr\'e model are respectively 1 and 2.375. Thus the quantum phase transitions in the Anderson model and in the Aubry-Andr\'e model are of different universality classes. Because the fidelity susceptibility and the generalized adiabatic susceptibility are directly connected to the dynamical structure factor which are experimentally accessible in the linear response regime, the fidelity susceptibility in quantum disordered systems may be observed experimentally in the near future.
Quantum phase transitions that take place between two distinct topological phases remain an unexplored area for the applicability of the fidelity approach. Here, we apply this method to spin systems in two and three dimensions and show that the fidelity susceptibility can be used to determine the boundary between different topological phases particular to these models, while at the same time offer information about the critical exponent of the correlation length. The success of this approach relies on its independence on local-order parameters or breaking symmetry mechanisms, with which nontopological phases are usually characterized. We also consider a topological insulator-superconducting phase transition in three dimensions and point out the relevant features of fidelity susceptibility at the boundary between these phases.
Abstract In topological insulators and topological superconductors, the discrete jump of the topological invariant upon tuning a certain system parameter defines a topological phase transition. A unified framework is employed to address the quantum criticality of the topological phase transitions in one to three spatial dimensions, which simultaneously incorporates the symmetry classification, order of band crossing, m -fold rotational symmetry, correlation functions, critical exponents, scaling laws, and renormalization group approach. We first classify higher-order Dirac models according to the time-reversal, particle-hole, and chiral symmetries, and determine the even–oddness of the order of band crossing in each symmetry class. The even–oddness further constrains the rotational symmetry m permitted in a symmetry class. Expressing the topological invariant in terms of a momentum space integration over a curvature function, the order of band crossing determines the critical exponent of the curvature function, as well as that of the Wannier state correlation function introduced through the Fourier transform of the curvature function. The conservation of topological invariant further yields a scaling law between critical exponents. In addition, a renormalization group approach based on deforming the curvature function is demonstrated for all dimensions and symmetry classes. Through clarification of how the critical quantities, including the jump of the topological invariant and critical exponents, depend on the nonspatial and the rotational symmetry, our work introduces the notion of universality class into the description of topological phase transitions.