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Anderson Topological Superconductor and Entanglement Entropy

Jan Borchmann, Aaron Farrell, T. Pereg-Barnea

Published Nov 12, 2015• 0 citations
OpenAlex

Abstract

In this paper we study the phase diagram of a disordered, spin-orbit coupled superconductor with $s$-wave or $d+id$-wave pairing symmetry in symmetry class $D$. We analyze the topological phase transitions by applying three different methods which include a disorder averaged entanglement entropy, a disorder averaged real-space Chern number, and an evaluation of the momentum space Chern number in a disorder averaged effective model. We find evidence for a disorder-induced topological state. While in the clean limit there is a single phase transition from a trivial phase with a Chern number $C=4$ to a topological phase with $C=1$, in the disordered system there is an intermediate phase with $C=3$. The phase transition from the trivial $C=4$ phase into the intermediate phase with $C=3$ is seen in the real-space calculation of the Chern number. In spite of this, this phase transition is not detectable in the entanglement entropy. A second phase transition from the disorder induced $C=3$ into the $C=1$ phase is seen in all three quantities.

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Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional ppp-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the kkk-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the kkk-space fidelity operator.
We analyze the entanglement entropy properties of a 2D p-wave superconductor with Rashba spin-orbit coupling, which displays a rich phase-space that supports non-trivial topological phases, as the chemical potential and the Zeeman term are varied. We show that the entanglement entropy and its derivatives clearly signal the topological transitions and we find numerical evidence that for this model the derivative with respect to the magnetization provides a sensible signature of each topological phase. Following the area law for the entanglement entropy, we systematically analyze the contributions that are proportional to or independent of the perimeter of the system, as a function of the Hamiltonian coupling constants and the geometry of the finite subsystem. For this model, we show that even though the topological entanglement entropy vanishes, it signals the topological phase transitions in a finite system. We also observe a relationship between a topological contribution to the entanglement entropy in a half-cylinder geometry and the number of edge states, and that the entanglement spectrum has robust modes associated with each edge state, as in other topological systems.
We analyze the entanglement entropy properties of a 2D p-wave superconductor with Rashba spin-orbit coupling, which displays a rich phase-space that supports non-trivial topological phases, as the chemical potential and the Zeeman term are varied. We show that the entanglement entropy and its derivatives clearly signal the topological transitions and we find numerical evidence that for this model the derivative with respect to the magnetization provides a sensible signature of each topological phase. Following the area law for the entanglement entropy, we systematically analyze the contributions that are proportional to or independent of the perimeter of the system, as a function of the Hamiltonian coupling constants and the geometry of the finite subsystem. For this model, we show that even though the topological entanglement entropy vanishes, it signals the topological phase transitions in a finite system. We also observe a relationship between a topological contribution to the entanglement entropy in a half-cylinder geometry and the number of edge states, and that the entanglement spectrum has robust modes associated with each edge state, as in other topological systems.
We investigate the behavior of a topological Chern insulator in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information about the topological universality class revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases. The phase diagram of a Chern insulator is explicitly computed as function of Fermi level (EF) and disorder strength using the level statistics of the EtS and energy spectrum, together with a computation of the Chern number (C) via a new, efficient real-space formula.
We study the two-dimensional disordered topological superconductor with Hubbard interactions. When the magnitude of the pairing potential is tuned to special values, this interacting model is exactly solvable even when disorders are imposed on the potential term or coupling constants. The topology of this model is investigated in detail by the real space Chern number formula, which computes the topological index of disordered systems to high precisions. It is found that the disorders can drive the system from topological trivial phase to a non-trivial phase, which generalizes the topological Anderson phenomena to interacting models. The self-consistent Born approximation is also employed to understand the influence of the disorders on the parameters of the interacting topological superconductor. It provide an alternative way to understand the topological transitions at weak disordered region.
In this work we study a two-dimensional topological superconductor on a square lattice. It contains hopping, spin-orbit coupling and a time reversal symmetry breaking Zeeman term. This realizes, in combination with a singlet pairing term, an effective triplet pairing that leads to a topological superconductor. As previously found numerically, the transitions are seen in the entanglement entropy as cusps as a function of model parameters. We complement this through studying the entanglement entropy analytically by keeping only its most important components. Our study is based on the intuition that the number of Fermi surfaces in the system controls the topological invariant. With our approximate expression for the entanglement entropy we are able to extract the divergent entanglement entropy derivative close to the phase transition.