Capacity of entanglement in Lifshitz theories

We study the capacity of entanglement in certain integrable scale invariant theories that exhibit Lifshitz scaling symmetry with a generic dynamical exponent <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>z</a:mi></a:math> at the critical point. This measure characterizes the width of the eigenvalue spectrum of the reduced density matrix and is a quantum informational counterpart of heat capacity. We explore various aspects of capacity of entanglement, such as the corresponding universal terms for the ground state, its <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>z</c:mi></c:math> dependence, and its temporal evolution, after a global quantum quench in two dimensions. We carefully examine the existence of a convenient entropic <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>c</e:mi></e:math>-function based on this quantity both in bosonic and fermionic theories. While in the relativistic case the corresponding <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>c</g:mi></g:math>-function displays a monotonic behavior under the renormalization group flow, this is not the case for the nonrelativistic theories. We also investigate the dynamics of capacity of entanglement after a mass quench and show that it follows the quasiparticle interpretation for the spreading of entanglement. Finally, we discuss how these results are consistent with the behavior of other entanglement measures, including the entanglement entropy.