Entanglement topological invariants for one-dimensional topological superconductors

Abstract

Entanglement is known to serve as an order parameter for true topological order in two-dimensional systems. We show how entanglement of disconnected partitions defines topological invariants for one-dimensional topological superconductors. These order parameters quantitatively capture the entanglement that is possible to distill from the ground state manifold, and are thus quantized to 0 or log 2. Their quantization property is inferred from the underlying lattice gauge theory description of topological superconductors, and is corroborated via exact solutions and numerical simulations. Transitions between topologically trivial and non-trivial phases are accompanied by scaling behavior, a hallmark of genuine order parameters, captured by entanglement critical exponents. These order parameters are experimentally measurable utilizing state-of-the-art techniques. Panel b): quadripartite von Neumann entropy S D as a function of µ/t, U/t, at fixed ∆ = 1 and LA = LB = 12. Black lines as from Ref. [22]. The colour plot is obtained via interpolation on a 6 x 8 grid.-that can be distilled from the ground state manifold; (ii) display scaling behavior when approaching quantum phase transitions, and thus allow for the definition of entanglement critical exponents that describe the build-up of non-local quantum correlations across such transitions; (iii) are experimentally measurable in-and out-of-equilibrium utilizing recently introduced [14,15] and demonstrated [25] techniques based on arXiv:1909.04035v1 [cond-mat.supr-con]

Topological entanglement properties of disconnected partitions in the Su-Schrieffer-Heeger model

“…A similar generic proof is missing for S D . Instead, simulations, exact solutions, or a gauge theory analogy validate its success for some examples [29,31,50].…”

“…Similarly to the bipartite entanglement entropy, it is possible to define and use the disconnected entropy using the Rényi-α entanglement entropies [31]. These extensions are useful for two reasons.…”

“…Within the topological phase, S D remains quantized on average in the presence of disorder. Such a scaling behavior and critical exponents are different with respect to the Kitaev wire [31] (the only other occurrence of such an analysis for fermionic systems to our knowledge) and to bosonic cluster models realized as instances of random unitary circuits (e.g., Ref. [39]).…”

“…Ref. [31] used the Kitaev wire [32] to prove that S D is also valid for 1D topological superconducting phases, where it is captured within a lattice gauge theory framework. In the Kitaev wire, S D is traced back to the entanglement necessarily present by construction between the only two fractional (Majorana) modes of the model, even when the modes are localized on each edge of the chain.…”

Deep learning of topological phase transitions from entanglement aspects

“…In other words, the spectrum preserves topological information. Moreover, for any system with a quadratic Hamiltonian, the whole situation can be simplified to compute the eigenvalues of the correlation function matrix, also known as the one-particle entanglement spectrum [17][18][19]. Majorana zero modes are then shown in terms of doubly degenerate eigenvalues, 1/2, in the spectrum [18].…”