Universal finite-size amplitude and anomalous entangment entropy of $z=2$ quantum Lifshitz criticalities in topological chains
Ke Wang, Tigran Sedrakyan
Abstract
We consider Lifshitz criticalities with dynamical exponent z=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>L</mml:mi> </mml:math> , as \sim L^{-2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, \epsilon \propto \pm k^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>∝</mml:mo> <mml:mo>±</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . We also show that the entanglement entropy exhibits at the criticality a non-logarithmic dependence on l/L <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mi>/</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> , where l <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>l</mml:mi> </mml:math> is the length of the sub-system. In the limit of l\ll L <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>≪</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> , the maximally-entangled ground state has the entropy, S(l/L)=S_0+(l/L)\log(l/L) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>l</mml:mi> <mml:mi>/</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>l</mml:mi> <mml:mi>/</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>log</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>l</mml:mi> <mml:mi>/</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . Here S_0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> 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.