Scaling behavior in a multicritical one-dimensional topological insulator

Abstract

A class of Aubry-André-Harper models of spin-orbit coupled electrons exhibits a topological phase diagram where two regions belonging to the same phase are split up by a multicritical point. The critical lines which meet at this point each defines a topological quantum phase transition with a second-order nonanalyticity of the ground-state energy, accompanied by a linear closing of the spectral gap with respect to the control parameter; except at the multicritical point which supports fourth-order transitions with parabolic gap-closing. Here both types of criticality are characterized through a scaling analysis of the curvature function defined from the topological invariant of the model. We extract the critical exponents of the diverging curvature function at the nonhigh symmetry points in the Brillouin zone where the gap closes, and also apply a renormalization group approach to the flattening curvature function at high symmetry points. We also derive a basis-independent correlation function between Wannier states to characterize the transition. Intriguingly, we find that the critical exponents and scaling law defined with respect to the spectral gap remain the same regardless of the order of the transition.

Crossdimensional universality classes in static and periodically driven Kitaev models

“…which is consistent with the scaling laws γ = ν x + ν y for 2D linear Dirac gap closures [38,47,[55][56][57][95][96][97][98][99][100]. Note that driving restores the full two-dimensional nature of the TPT as opposed to TPTs in the static model which belonged to the 1D universality class.…”

A supervised learning algorithm for interacting topological insulators based on local curvature

“…To summarize, the phase diagram correctly captures all phases and phase boundaries, and moreover indicates the appearance of a multicritical point as a function of coupling u around M = −2.0, indicating that electron-phonon interaction can also serve as a mechanism to induce multicriticality. Thus, many-body interactions are added to the list of several recently uncovered mechanisms that can trigger topological multicriticality, including periodic driving or quantum walk protocols [17][18][19]58], long range hopping or pairing [12,13,59], spin-orbit coupling [11,60], topological insulator/topological superconductor hybridization [61], as well as more complicated mechanisms in the spin liquid [62] and toric code models [63]. We close this section by making a comparison between the CRG [8-15, 17, 18] and the ML scheme proposed here.…”

“…These aspects include the notion of critical exponents, scaling laws, universality classes, and correlation functions. These notions form the basis of the curvature renormalization group (CRG) method which can capture the TPTs solely based on the renormalization of the curvature function near the HSP [8], regardless of whether the system is noninteracting [9][10][11][12][13] or interacting [14,15] or periodically driven [16][17][18][19].…”

A supervised learning algorithm for interacting topological insulators based on local curvature

“…To summarize, the phase diagram correctly captures all phases and phase boundaries, and moreover indicates the appearance of a multicritical point as a function of coupling u around M = −2.0, indicating that electron-phonon interaction can also serve as a mechanism to induce multicriticality. Thus, many-body interactions are added to the list of several recently uncovered mechanisms that can trigger topological multicriticality, including periodic driving or quantum walk protocols, [17][18][19]58 long range hopping or pairing, 12,13,59 spin-orbit coupling, 11,60 topological insulator/topological superconductor hybridization, 61 as well as more complicated mechanisms in the spin liquid 62 and toric code models. 63 We close this section by making a comparison between the CRG [8][9][10][11][12][13][14][15]17,18 and the ML scheme proposed here.…”

“…This includes the notion of critical exponents, scaling laws, universality classes, and correlation functions. This forms the basis of the curvature renormalization group (CRG) method which can capture the TPTs solely based on the renormalization of the curvature function near the HSP, 8 regardless of whether the system is noninteracting [9][10][11][12][13] or interacting 14,15 or periodically driven. [16][17][18][19] The CRG method demonstrates that, although topology is a global property of the entire manifold of the D-dimensional Brillouin zone (BZ), the knowledge about topology can be entirely encoded in the curvature function near a HSP.…”