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| Fig. 1: Phase diagram of the undriven Kitaev chain. (Source: S. Mishra, after Thakurathi et al [8]) |
The use of topological invariants to identify and classify phases of matter has been an active and exciting area of research in condensed matter physics. [1-3] Typically, this technique has been applied to systems that have gaps in the bulk spectrum to understand and explain the emergence of robust zero energy modes at the boundaries. The topological invariant, generally an integer, counts the number of species of the boundary zero energy modes. The range of the topological invariant is decided by the dimension of the physical system and the symmetries it possesses. The strength and generality of the topological invariant as a classification technique arises from the fact that the invariant does not change under perturbations to the system as long as the bulk spectrum remains gapped and the symmetries are preserved.
Seminal work in the 1980s explaining the topological origin of the quantization of conductance in the quantum Hall effect germinated this entire field and since then this technique has been applied to many other systems like two- and three-dimensional topological insulators and wires with p-wave superconductivity. [4,5] In this paper, we will concern ourselves with the latter, i.e. one-dimensional p-wave superconducting wires.
Among topological systems, there has been significant excitement about many-body systems with a particular kind of zero energy boundary mode called Majorana modes. Such modes are manifest in a toy model called Kitaev chain modelling a one-dimensional p-wave superconducting wire. [6] The interest in studying Majorana end modes (MMs) was spurred on by a proposal by Kitaev and Preskill outlining a way to realise quantum computing with topological qubits that are based on non-Abelian anyons and are protected against decoherence. [7] Non-Abelian anyons can be realised in topological states of matter, specifically those that support Majorana end modes. Fig. 1 shows the topologically classified phases of the Kitaev chain. [8] Phases I and II are topologically non-trivial.
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| Fig. 2: Plot showing the process of determining topological invariant. (Source: S. Mishra, after Thakurathi et al [8]) |
Over time, topological classification has evolved from a technique used to provide elegant explanations for profound physics to being used as a guide to construct phases that display non-trivial and exciting physics. In the recent years, driven and out-of-equilibrium many-body quantum systems have garnered a lot of interest and, quite naturally, the topological properties of these systems have also been studied. The general problem of understanding many-body quantum systems with arbitrary time-dependence is quite difficult and remains unsolved but inroads have been made into understanding a particular class of time-dependent systems - periodic systems. Such systems are handily analysable by Floquet theory and realisable with modern experimental systems.
Topology in Floquet systems can arise in many flavours, just like in time-independent systems. [9] For example, it has been shown that it is possible in theory to induce anomalous edge states in Chern insulators as well as Majorana modes in otherwise topologically trivial phase of the Kitaev chain. [10,11] Thakurathi et al. demonstrate the latter, i.e., generation of Majorana end modes by time-periodic driving of the Kitaev chain. [8] The authors construct a novel topological invariant that counts the number of induced end-modes separately at the Floquet eigenvalues +1, -1 (shown in Fig. 2). These induced modes can persist even in the presence of electron-phonon interactions at non-zero temperature and random noise in the chemical potentials (given that these effects are sufficiently weak).
© Sattwik Deb Mishra. The author warrants that the work is the authors own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute, and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All of the rights, including commercial rights, are reserved to the author.[1] X.-L. Qi and S.-C. Zhang, "Topological Insulators and Superconductors," Rev. Mod. Phys. 83, 1057 (2011).
[2] L. Fidkowski and A. Kitaev, "Topological Phases of Fermions in One Dimension," Phys. Rev. B 83, 075103 (2011).
[3] M. Z. Hasan and C. L. Kane, "Colloquium: Topological Insulators," Rev. Mod. Phys. 82, 3045 (2010).
[4] B. I. Halperin, "Quantized Hall Conductance, Current-Carrying Edge States, and the Existence of Extended States in a Two-dimensional Disordered Potential," Phys. Rev. B 25, 2185 (1982).
[5] R. B. Laughlin, "Quantized Hall Conductivity in Two Dimensions," Phys. Rev. B 23, 5632 (1981).
[6] A. Y. Kitaev, "Unpaired Majorana Fermions in Quantum Wires," Phys.-Usp. 44, 131 (2001).
[7] A. Kitaev and J. Preskill, "Topological Entanglement Entropy," Phys. Rev. Lett. 96, 110404 (2006).
[8] M. Thakurathi et al., "Floquet Generation of Majorana End Modes and Topological Invariants," Phys. Rev. B 88, 155133 (2013).
[9] F. Harper et al., "Topology and Broken Symmetry in Floquet Systems," Annu. Rev. Condens. Matter Phys. 11, 345 (2020).
[10] M. S. Rudner et al., "Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems," Phys. Rev. X 3, 031005 (2013).
[11] B. Foutty, "Edge States and Topology in Floquet Systems," Physics 470, Stanford University, Spring 2020.