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

<jats:p>We consider Lifshitz criticalities with dynamical exponent <jats:inline-formula><jats:alternatives><jats:tex-math>z=2</jats:tex-math><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></jats:alternatives></jats:inline-formula> 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, <jats:inline-formula><jats:alternatives><jats:tex-math>L</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>L</mml:mi></mml:math></jats:alternatives></jats:inline-formula>, as <jats:inline-formula><jats:alternatives><jats:tex-math>\sim L^{-2}</jats:tex-math><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></jats:alternatives></jats:inline-formula>, with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, <jats:inline-formula><jats:alternatives><jats:tex-math>\epsilon \propto \pm k^2</jats:tex-math><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></jats:alternatives></jats:inline-formula>. We also show that the entanglement entropy exhibits at the criticality a non-logarithmic dependence on <jats:inline-formula><jats:alternatives><jats:tex-math>l/L</jats:tex-math><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></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>l</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>l</mml:mi></mml:math></jats:alternatives></jats:inline-formula> is the length of the sub-system. In the limit of <jats:inline-formula><jats:alternatives><jats:tex-math>l\ll L</jats:tex-math><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></jats:alternatives></jats:inline-formula>, the maximally-entangled ground state has the entropy, <jats:inline-formula><jats:alternatives><jats:tex-math>S(l/L)=S_0+(l/L)\log(l/L)</jats:tex-math><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></jats:alternatives></jats:inline-formula>. Here <jats:inline-formula><jats:alternatives><jats:tex-math>S_0</jats:tex-math><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></jats:alternatives></jats:inline-formula> 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. </jats:p>