Finite Temperature Entanglement Negativity of Fermionic Symmetry Protected Topological Phases and Quantum Critical Points in One Dimension

TL;DR

This paper investigates the finite-temperature entanglement negativity in one-dimensional fermionic SPT phases and quantum critical points, revealing how entanglement measures signal topological and phase transition features.

Contribution

It introduces a detailed analysis of entanglement negativity in fermionic SPT and critical phases at finite temperature, highlighting its role in identifying topological boundary modes and phase transitions.

Findings

Entanglement negativity signals the quantum critical fan boundary.

Sudden negativity decrease indicates fermionic Lifshitz transition.

Boundary mode entanglement vanishes at finite temperature.

Abstract

We study the logarithmic entanglement negativity of symmetry-protected topological (SPT) phases and quantum critical points (QCPs) of one-dimensional noninteracting fermions at finite temperatures. In particular, we consider a free fermion model that realizes not only quantum phase transitions between gapped phases but also an exotic topological phase transition between quantum critical states in the form of the fermionic Lifshitz transition. The bipartite entanglement negativity between adjacent fermion blocks reveals the crossover boundary of the quantum critical fan near the QCP between two gapped phases. Along the critical phase boundary between the gapped phases, the sudden decrease in the entanglement negativity signals the fermionic Lifshitz transition responsible for the change in the topological nature of the QCPs. In addition, the tripartite entanglement negativity between spatially separated fermion blocks counts the number of topologically protected boundary modes for both SPT phases and topologically nontrivial QCPs at zero temperature. However, the long-distance entanglement between the boundary modes vanishes at finite temperatures due to the instability of SPTs, the phases themselves.