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Physics Letters A

On symmetry-protected topological phase transition in one-dimensional fermion systems at finite temperature

X. Y. Cui

Published Aug 12, 2025• 0 citations
Publisher
DOI
10.1016/j.physleta.2025.130886

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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.
Abstract We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in Z2×H1(G,Z2)×H2(G,U(1)p)\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})Z2​×H1(G,Z2​)×H2(G,U(1)p​) with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.