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arXiv:2401.02270 · Physical review. B./Physical review. B

Robustness of topological order against disorder

Lucas A. Oliveira, Wei Chen

Published Mar 8, 2024• 22 citations
arXivHTMLPDF

Abstract

A universal topological marker has been proposed recently to map the topological invariants of Dirac models in any dimension and symmetry class to lattice sites. Using this topological marker, we examine the conditions under which the global topological order, represented by the spatially averaged topological marker, remains unchanged in the presence of disorder for 1D and 2D systems. We find that if an impurity corresponds to varying a nonzero-matrix element of the lattice Hamiltonian, regardless the element represents hopping, chemical potential, pairing, etc., then the average topological marker is conserved. However, if there are many strong impurities and the average distance between them is shorter than a correlation length, then the average marker is no longer conserved. In addition, strong and dense impurities can be used to continuously interpolate between one topological phase and another. A number of prototype lattice models including Su-Schrieffer-Heeger model, Kitaev chain, Chern insulators, Bernevig-Hughes-Zhang model, and chiral $p$-wave superconductors are used to elaborate the ubiquity of these statements.

DOI
10.1103/physrevb.109.094202

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A universal topological marker has been proposed recently to map the topological invariants of Dirac models in any dimension and symmetry class to lattice sites. Using this topological marker, we examine the conditions under which the global topological order, represented by the spatially averaged topological marker, remains unchanged in the presence of disorder for 1D and 2D systems. We find that if an impurity corresponds to varying a nonzero-matrix element of the lattice Hamiltonian, regardless the element represents hopping, chemical potential, pairing, etc., then the average topological marker is conserved. However, if there are many strong impurities and the average distance between them is shorter than a correlation length, then the average marker is no longer conserved. In addition, strong and dense impurities can be used to continuously interpolate between one topological phase and another. A number of prototype lattice models including Su-Schrieffer-Heeger model, Kitaev chain, Chern insulators, Bernevig-Hughes-Zhang model, and chiral ppp-wave superconductors are used to elaborate the ubiquity of these statements.
The topological order of Dirac models in any dimension and symmetry class can be mapped to lattice sites by a universal topological marker. From the topological operator that extracts this marker, one can further define a nonlocal marker that detects topological phase transitions. Several prototype models such as the Su-Schrieffer-Heeger model, Majorana chain, Chern insulator, Bernevig-Hughes-Zhang model, chiral and helical ppp-wave superconductors, three-dimensional time-reversal symmetric topological insulators, among many others, are employed to demonstrate the ubiquity of these markers.
Topological markers provide an efficient and powerful characterization of topological features of many systems, especially when the translation symmetry is broken. Recently, a universal topological marker applicable in different symmetry classes of topological systems is proposed. However, it suffers from irregular behaviors at the boundary and its connection to other topological indexes remains elusive. In this work, we construct regularized universal topological markers that apply to Dirac systems by utilizing position operators that are compatible with periodic boundary conditions. The regularized markers eliminate the obstructive boundary irregularities successfully and give rise to the desired global topological invariants, such as the Chern number, consistently when integrated over all the lattice sites. Furthermore, the regularized form allows us to establish an explicit connection between the markers and some other known topological indices in two dimensions. For instance, it turns out to be equivalent to the Bott index in classes A, D, and C and equivalent to the spin Chern number in classes DIII and AII. We further examine the utility and stability of this marker in disordered scenarios. We find that its variance shows peaks at the phase boundaries, which promotes it as a useful indicator for detecting disorder-induced topological phase transitions.
We investigate disorder-driven transitions between trivial and topological insulator (TI) phases in two-dimensional (2D) systems. Our study primarily focuses on the Bernevig-Hughes-Zhang (BHZ) model with Anderson disorder, while other standard 2D TI models exhibit equivalent features. The analysis is based on the local Chern marker (LCM), a local quantity that allows for the characterization of topological transitions in finite and disordered systems. Our simulations indicate that disorder-driven trivial to topological insulator transitions are nicely characterized by C0{\mathcal{C}}_{0}C0​, the disorder-averaged LCM near the central cell of the system. We show that C0{\mathcal{C}}_{0}C0​ is characterized by a single-parameter scaling, namely, ${\mathcal{C}}_{0}(M,W,L)\ensuremath{\equiv}{\mathcal{C}}_{0}(z)$, with $z=[{W}^{\ensuremath{\mu}}\ensuremath{-}{W}_{c}^{\ensuremath{\mu}}(M)]L$, where MMM is the Dirac mass, WWW is the disorder strength, and LLL is the system size, while ${W}_{c}(M)\ensuremath{\propto}\sqrt{M}$ and $\ensuremath{\mu}\ensuremath{\approx}2$ stand for the critical disorder strength and the critical exponent, respectively. Our numerical results are in agreement with a theoretical prediction based on a first-order Born approximation analysis. These observations lead us to speculate that the universal scaling function we have found is rather general for amorphous and disorder-driven topological phase transitions.
Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although they are well understood, a thermodynamic description of these materials remained elusive, firstly because the edges yield a non-extensive contribution to the thermodynamic potential, and secondly because topological field theories involve non-local order parameters, and cannot be captured by the Ginzburg-Landau formalism. Recently, this challenge has been overcome: by using Hill thermodynamics to describe the Bernevig-Hughes-Zhang model in two dimensions, it was shown that at the topological phase transition the thermodynamic potential does not scale extensively due to boundary effects. Here, we extend this approach to different topological models in various dimensions (the Kitaev chain and Su-Schrieffer-Heeger model in one dimension, the Kane-Mele model in two dimensions and the Bernevig-Hughes-Zhang model in three dimensions) at zero temperature. Surprisingly, all models exhibit the same universal behavior in the order of the topological-phase transition, depending on the dimension. Moreover, we derive the topological phase diagram at finite temperature using this thermodynamic description, and show that it displays a good agreement with the one calculated from the Uhlmann phase. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter.
Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker-the Chern number expressed in terms of the Fourier transformed Chern character-is an easily applicable local marker in even dimensions, but there are no analogous expressions for odd dimensions. We provide general analytic expressions for local markers for free-fermion topological states in odd dimensions protected by local symmetries: a Chiral marker, a local Z marker which in case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local Z_{2} marker characterizing all nonchiral phases in odd dimensions. We achieve this by introducing a one-parameter family P_{ϑ} of single-particle density matrices interpolating between a trivial state and the state of interest. By interpreting the parameter ϑ as an additional dimension, we calculate the Chern marker for the family P_{ϑ}. We demonstrate the practical use of these markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions: a topological superconductor (Z classification) and a topological insulator (Z_{2} classification).
The topological properties of the Su-Schrieffer-Heeger (SSH) model in the presence of nearest-neighbor interaction are investigated by means of a topological marker, generalized from a noninteracting one by utilizing the single-particle Green's function of the many-body ground state. We find that despite the marker not being perfectly quantized in the presence of interactions, it always remains finite in the topologically nontrivial phase while converging to zero in the trivial phase when approaching the thermodynamic limit, and hence correctly judges the topological phases in the presence of interactions. The marker also correctly captures the interaction-driven, second-order phase transitions between a topological phase and a Landau-ordered phase, which is a charge-density wave order in our model with a local-order parameter, as confirmed by the calculation of entanglement entropy and the many-body Zak phase. Our paper thus points to the possibility of generalizing topological markers to interacting systems through Green's function, which may be feasible for topological insulators in any dimension and symmetry class.