Universal finite-size amplitude and anomalous entanglement entropy of $z=2$ quantum Lifshitz criticalities in topological chains

TL;DR

This paper investigates $z=2$ Lifshitz criticalities in topological chains, revealing universal finite-size energy corrections and a novel non-logarithmic entanglement entropy behavior linked to long-range correlations.

Contribution

It uncovers the universal finite-size energy scaling and the unique entanglement entropy form at Lifshitz critical points in topological chains, highlighting long-range correlation effects.

Findings

Finite-size energy correction scales as $L^{-2}$ with a universal coefficient.

Entanglement entropy shows a non-logarithmic dependence on $l/L$, with a linear term in $l/L$.

Long-range correlations mediated by a zero mode cause the novel entanglement behavior.

Abstract

We consider Lifshitz criticalities with dynamical exponent $z=2$ that emerge in a class of topological chains. There, such a criticality plays a fundamental role in describing transitions between symmetry-enriched conformal field theories (CFTs). We report that, at such critical points in one spatial dimension, the finite-size correction to the energy scales with system size, $L$, as $\sim L^{-2}$, with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, $\epsilon \propto \pm k^2$. We also show that the entanglement entropy exhibits at the criticality a non-logarithmic dependence on $l/L$, where $l$ is the length of the sub-system. In the limit of $l\ll L$, the maximally-entangled ground state has the entropy, $S(l/L)=S_0+2n(l/L)\log(l/L)$. Here $S_0$ is some non-universal entropy originating from short-range correlations and $n$ is half-integer or integer depending on the degrees of freedom in the model. We show that the novel entanglement originates from the long-range correlation mediated by a zero mode in the low energy sector. The work paves the way to study finite-size effects and entanglement entropy around Lifshitz criticalities and offers an insight into transitions between symmetry-enriched criticalities.