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arXiv:1807.01339 · Journal of Physics Condensed Matter

Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii–Moriya interaction

Alireza Habibi, Rasoul Ghadimi, S A Jafari

Published Aug 30, 2019• 5 citations
arXivHTMLPDF

Abstract

Abstract The topological index of a system determines its edge physics. However, in situations such as strong disorder where due to level repulsion the spectral gap closes, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined by the transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the ‘relative’ Hamiltonians. As a case study, we study a generalized XY model (2XY model) further augmented by a generalized Dziyaloshinskii–Moriya-like (DM) interaction parameterized by that after fermionization breaks the time-reversal invariance. The parent Hamiltonian at which belongs to the BDI class is indexed by an integer winding number while the daughter Hamiltonian which belongs to class D is specified by a Z 2 index . We show that the localization length, in addition to determining Z 2 , can count the number of Majorana zero modes leftover at the boundary of the daughter Hamiltonian—which are not protected by the winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.

DOI
10.1088/1361-648x/ab401c

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Abstract The topological index of a system determines its edge physics. However, in situations such as strong disorder where due to level repulsion the spectral gap closes, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined by the transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the ‘relative’ Hamiltonians. As a case study, we study a generalized XY model (2XY model) further augmented by a generalized Dziyaloshinskii–Moriya-like (DM) interaction parameterized by that after fermionization breaks the time-reversal invariance. The parent Hamiltonian at which belongs to the BDI class is indexed by an integer winding number while the daughter Hamiltonian which belongs to class D is specified by a Z 2 index . We show that the localization length, in addition to determining Z 2 , can count the number of Majorana zero modes leftover at the boundary of the daughter Hamiltonian—which are not protected by the winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.
We elucidate the reduction of the winding number caused by the on-site disorder in a next-nearest-neighbor XY model. When disorder becomes strong enough, Majorana edge modes become critically extended, beyond which they collapse into Anderson localized (AL) states in the bulk, resulting in a topological Anderson insulating state. We identify a resilience threshold Wt{W}_{t}Wt​ for every pair of Majorana fermions (MFs). In response to increasing disorder, every pair of MFs collapse into AL states in the bulk beyond their resilience threshold. For very strong disorder, all Majorana fermions collapse, and a topologically trivial state is obtained. We show that the threshold values are related to the localization length of Majorana fermions, which can be efficiently calculated by an appropriate modification of the transfer-matrix method. At the topological transition point, the localization length of the zero modes diverges, and the system becomes scale invariant. The number of peaks in the localization length as a function of disorder strength determines the number of zero modes in the clean state before disorder is introduced. This finding elevates the transfer-matrix method to the level of a tool for the determination of the topological index of both clean and disordered systems.
We present an exactly solvable extension of the quantum XY chain with longer range multi-spin interactions. Topological phase transitions of the model are classified in terms of the number of Majorana zero modes which are in turn related to an integer winding number. We further find a general relation between the winding number and the number of Majorana fermions at the ends of an open chain. The present class of exactly solvable models belong to the BDI class in the Altland-Zirnbauer classification (A. Altland, M. R. Zirnbauer, Phys. Rev. B (1997) 55 1142) of topological superconductors. We show that time reversal (TR) symmetry of the spin variables translates into a peculiar particle-hole transformation in the language of Jordan-Wigner (JW) fermions that is accompanied by a {\pi} shift in the wave vector (π\piπPH). Presence of π\piπPH symmetry restricts the winding number of TR symmetric extensions of XY to odd integers. The {\pi}PH operator may serve in further detailed classification of topological superconductors.
A one-dimensional boundary of a two-dimensional topological superconductor can host a number of topologically protected chiral modes. Combining two topological superconductors with different topological indices, it is possible to achieve a situation when only a given number of channels (mmm) are topologically protected, while others are not and therefore are subject to Anderson localization in the presence of disorder. We study transport properties of such quasi-one-dimensional quantum wires with broken time-reversal and spin-rotational symmetries (class D) and calculate the average conductance, its variance and the third cumulant, as well as the average shot noise power. The results are obtained for arbitrary wire length, tracing a crossover from the diffusive Drude regime to the regime of strong localization where only mmm protected channels conduct. Our approach is based on the nonperturbative treatment of the nonlinear supersymmetric sigma model of symmetry class D with two replicas developed in our recent publication [D. S. Antonenko et al., Phys. Rev. B 102, 195152 (2020)]. The presence of topologically protected modes results in the appearance of a topological Wess-Zumino-Witten term in the sigma-model action, which leads to an additional subsidiary series of eigenstates of the transfer-matrix Hamiltonian. The developed formalism can be applied to study the interplay of Anderson localization and topological protection in quantum wires of other symmetry classes.
We show that topology can protect exponentially localized, zero energy edge modes at critical points between one-dimensional symmetry-protected topological phases. This is possible even without gapped degrees of freedom in the bulk-in contrast to recent work on edge modes in gapless chains. We present an intuitive picture for the existence of these edge modes in the case of noninteracting spinless fermions with time-reversal symmetry (BDI class of the tenfold way). The stability of this phenomenon relies on a topological invariant defined in terms of a complex function, counting its zeros and poles inside the unit circle. This invariant can prevent two models described by the same conformal field theory (CFT) from being smoothly connected. A full classification of critical phases in the noninteracting BDI class is obtained: Each phase is labeled by the central charge of the CFT, c∈1/2N, and the topological invariant, ω∈Z. Moreover, c is determined by the difference in the number of edge modes between the phases neighboring the transition. Numerical simulations show that the topological edge modes of critical chains can be stable in the presence of interactions and disorder.
We study the phases of an exactly solvable one dimensional model with 4−4-4−dimensional Γ−Γ-Γ−matrix degrees of freedom on each site. The Γ−Γ-Γ−matrix model has a large set of competing interactions and displays a rich phase diagram with critical lines and multi-critical points. We work with the model with certain Z2Z_2Z2​ symmetries and identify the allowed symmetry protected topological phases using the winding number as the topological invariant. The model belongs to the CII-class of the 10−10-10−fold classification and allows for integer values of the winding number. We confirm that the system also hosts localized zero energy Majorana edge modes, consistent with the integer value of the winding number of the corresponding phase. We further study scaling and universality behaviour of the various topological phase transitions.
The bulk-boundary correspondence asserts that topological data of a (2+1)-dimensional phase determine the physics of its boundary, but in the setting of anyon condensation the correspondence is usually stated structurally rather than quantitatively. We propose a set of computable indices that quantify the correspondence between anyon condensation in a parent topological order and the emergence of Majorana statistics at a gapped boundary. The first index, κ = dim(A)², measures the "size" of the condensate, where A is the connected (étale) algebra of condensed anyons inside the parent modular tensor category C. For normal condensates — those in which every local A-module sector contributes once to the child, as in Cases I and II — κ coincides with D_C²/D_D², the ratio of squared total quantum dimensions of parent and child; for non-normal condensates (Case III) the two expressions differ, and we report both. The second index, μ = Δc/(1/2) = 2Δc, counts chiral Majorana edge modes forced by the mismatch Δc between the chiral central charges of parent and child phases. A third diagnostic, the non-Abelian excess fraction f_NA, measures the share of boundary quantum dimension carried by sectors with d > 1. We evaluate all indices explicitly for three canonical cases: condensing A = 1 ⊕ ψ in the Ising category (κ = 4, μ = 1, boundary σ defects with two fusion channels — the fingerprint of Majorana zero modes); condensing A = 1 ⊕ e in the toric code (κ = 4, μ = 0, single-channel fusion, no Majorana structure); and condensing A = 1 ⊕ (ψ,ψ) in the doubled Ising category (κ = dim(A)² = 4, child/parent dimension ratio D_D/D_C = √(3/7), f_NA = 1/3, non-Abelian boundary carrier (σ,σ)). The contrast between the first two cases shows that κ alone does not detect Majorana statistics; μ and f_NA do. All numerical results are derived by explicit arithmetic from standard category data, divergences between independent drafts are documented in Appendix A, and limitations of the framework Full text and updates: papers.qnfo.org/papers/a-two-index-framework-for-the-bulk-boundary-correspondence-of-anyon-condensation/