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

Abstract

We consider Lifshitz criticalities with dynamical exponent z=2z=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, LL, as \sim L^{-2}∼L−2, with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, \epsilon \propto \pm k^2ϵ∝±k2. We also show that the entanglement entropy exhibits at the criticality a non-logarithmic dependence on l/Ll/L, where ll is the length of the sub-system. In the limit of l\ll Ll≪L, the maximally-entangled ground state has the entropy, S(l/L)=S_0+(l/L)\log(l/L)S(l/L)=S0+(l/L)log(l/L). Here S_0S0 is some non-universal entropy originating from short-range correlations. 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.