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arXiv:1503.05154 · Physical Review B

Entanglement modes and topological phase transitions in superconductors

T. P. Oliveira, Pedro D Sacramento

Published Mar 13, 2014• 22 citations
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Abstract

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 $p$-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 $k$-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 $k$-space fidelity operator.

DOI
10.1103/physrevb.89.094512

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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.
Using bulk gapless topological superconductors in both 1D and 2D as free fermion model examples, we demonstrate the power of subsystem correlation spectrum (the spectrum of correlation matrix), or equivalently the entanglement spectrum for the case of free fermions, in characterizing the topology of the nontrivial ground state. For the systems considered, we show that signatures of the low-energy spectrum, including both the edge modes and the bulk modes, appear in the correlation spectrum, albeit with different behaviors. This work generalizes the 2D Li-Haldane entanglement spectrum characterization of topological edge states to 2D topological systems with gapless bulk.
In this paper we study the phase diagram of a disordered, spin-orbit coupled superconductor with sss-wave or d+idd+idd+id-wave pairing symmetry in symmetry class DDD. 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=4C=4C=4 to a topological phase with C=1C=1C=1, in the disordered system there is an intermediate phase with C=3C=3C=3. The phase transition from the trivial C=4C=4C=4 phase into the intermediate phase with C=3C=3C=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=3C=3C=3 into the C=1C=1C=1 phase is seen in all three quantities.
Recently topological states of matter have witnessed a new physical phenomenon where both edge modes and gapless bulk coexist at topological quantum criticality. The presence and absence of edge modes on a critical line can lead to an unusual class of topological phase transition between the topological and nontopological critical phases. We explore the existence of this new class of topological phase transitions in a generic model representing the topological insulators and superconductors and we show that such transition occurs at a multicritical point, i.e., at the intersection of two critical lines. To characterize these transitions we reconstruct the theoretical frameworks, which include bound state solution of the Dirac equation, winding number, correlation factors, and scaling theory of the curvature function to work for the criticality. Critical exponents and scaling laws are discussed to distinguish between the multicritical points, which separate the critical phases. Entanglement entropy and its scaling in the real space provide further insights into the unique transition at criticality revealing the interplay between fixed point and critical point at the multicriticalities.
Entanglement signatures of gapless topological phases in a ppp-wave superconductor
We explore the gapless topological phases of a ppp-wave superconductor, probing its rich topologically ordered phases and underlying quantum phenomena. The topological order of the system is characterized by studying its entanglement properties. This study confirms the bulk-boundary correspondence in the entanglement spectrum, even without a full bulk gap. For contractible bipartitions, the entanglement entropy varies non-monotonically with the chemical potential, displaying pronounced peaks at points where the bulk gap closes and reopens, signaling topological quantum phase transitions. This behavior remains robust in the thermodynamic limit. The entanglement entropy grows with system size for non-contractible bipartitions, indicating long-range entanglement in the gapless phase. These findings reveal the subtle interplay between symmetry, entanglement, and topology in gapless systems, and emphasize the role of entanglement-based diagnostics in identifying unconventional topological phases beyond the gapped paradigm.