Topological Phase Transitions: Criticality, Universality, and Renormalization Group Approach

Chen, Wei; Sigrist, Manfred

doi:10.1002/9781119407317.ch7

Cited by 9 publications

(31 citation statements)

References 39 publications

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“…In this work, we aim to study TPTs in both the static and periodically driven 2D Kitaev models hosting MMs. We classify these transitions from the perspective of universality using two measures: (i) a Majorana version of a recently proposed stroboscopic Wannier state correlation function [38,[55][56][57], which encodes the notion of a correlation length that diverges at the TPT and (ii) a momentum-dependent fidelity susceptibility that measures the distance between Bloch states in the momentum space [58]. Our classification reveals the existence of cross-dimensional universality classes.…”

Section: Introductionmentioning

confidence: 99%

Crossdimensional universality classes in static and periodically driven Kitaev models

et al. 2021

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The Kitaev model on the honeycomb lattice is a paradigmatic system known to host a wealth of nontrivial topological phases and Majorana edge modes. In the static case, the Majorana edge modes are nondispersive. When the system is periodically driven in time, such edge modes can disperse and become chiral. We obtain the full phase diagram of the driven model as a function of the coupling and the driving period. We characterize the quantum criticality of the different topological phase transitions in both the static and driven model via the notions of Majorana-Wannier state correlation functions and momentum-dependent fidelity susceptibilities. We show that the system hosts crossdimensional universality classes: although the static Kitaev model is defined on a two-dimensional (2D) honeycomb lattice, its criticality falls into the universality class of one-dimensional (1D) linear Dirac models. For the periodically driven Kitaev model, in addition to the universality class of prototype 2D linear Dirac models, an additional 1D nodal loop type of criticality exists owing to emergent time-reversal and mirror symmetries, indicating the possibility of engineering multiple universality classes by periodic driving. The manipulation of time-reversal symmetry allows the periodic driving to control the chirality of the Majorana edge states.

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Section: Introductionmentioning

confidence: 99%

Crossdimensional universality classes in static and periodically driven Kitaev models

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“…Recently, a framework based on scaling theory was proposed to discuss the criticality of TPTs [20][21][22][23]. The theory relies on the topological invariant C being generally an integration of a certain "curvature function" F over the momentum or flux space.…”

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“…This feature allows the classification of TPTs according to standard concepts of critical exponents and universality classes. Furthermore, due to the conservation of the topological invariant, scaling laws linking the exponents naturally emerge [22][23][24]. Based on this notion of divergence, a simple renormalization group (RG) approach is proposed to analyze TPTs.…”

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“…In this review, we outline this unified description of TPTs and its applications to a variety of static, periodip-1 arXiv:1912.08819v1 [cond-mat.stat-mech] 18 Dec 2019 cally driven, weakly and strongly interacting systems. We mainly focus on one (1D) and two-dimensional (2D) Dirac models, although higher dimensional cases and more exotic topology have also been discussed [20][21][22][23][24][25][26][27][28]. Correlation functions, critical exponents, and universality classes.…”

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“…With the Wannier wave function W n (r − R) = r|Rn centered at the home cell R, Eq. (7) indicates that the correlation functioñ F (R) = R|R|0 is a measure of the overlap of Wannier functions centered at two home cells that are distance R apart [22][23][24]. Combining the Lorentzian form in Eq.…”

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Unifying topological phase transitions in non-interacting, interacting, and periodically driven systems

Molignini

Читра

Chen

2020

EPL

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Topological phase transitions occur in real materials as well as quantum engineered systems, all of which differ greatly in terms of dimensionality, symmetries, interactions, and driving, and hence require a variety of techniques and concepts to describe their topological properties. For instance, topology may be accessed from single-particle Bloch wave functions, Green's functions, or many-body wave functions. We demonstrate that despite this diversity, all topological phase transitions display a universal feature: namely, a divergence of the curvature function that composes the topological invariant at the critical point. This feature can be exploited via a renormalization-group-like methodology to describe topological phase transitions. This approach serves to extend notions of correlation function, critical exponents, scaling laws and universality classes used in Landau theory to characterize topological phase transitions in a unified manner.

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Multi-critical topological transition at quantum criticality

Kumar

Kartik

Sinha

et al. 2021

Sci Rep

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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.

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Topological Phase Transitions: Criticality, Universality, and Renormalization Group Approach

Cited by 9 publications

References 39 publications

Crossdimensional universality classes in static and periodically driven Kitaev models

Crossdimensional universality classes in static and periodically driven Kitaev models

Unifying topological phase transitions in non-interacting, interacting, and periodically driven systems

Multi-critical topological transition at quantum criticality

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