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arXiv:2505.08297 · Physical review. D/Physical review. D.

Capacity of entanglement in Lifshitz theories

Sare Khoshdooni, Komeil Babaei Velni, M. Reza Mohammadi Mozaffar

Published Jul 14, 2025• 2 citations
arXivHTMLPDF

Abstract

We study the capacity of entanglement in certain integrable scale invariant theories that exhibit Lifshitz scaling symmetry with a generic dynamical exponent z 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 z dependence, and its temporal evolution, after a global quantum quench in two dimensions. We carefully examine the existence of a convenient entropic c -function based on this quantity both in bosonic and fermionic theories. While in the relativistic case the corresponding c -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.

DOI
10.1103/7cg6-m7dn

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We study the capacity of entanglement in certain integrable scale invariant theories that exhibit Lifshitz scaling symmetry with a generic dynamical exponent z 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 z dependence, and its temporal evolution, after a global quantum quench in two dimensions. We carefully examine the existence of a convenient entropic c -function based on this quantity both in bosonic and fermionic theories. While in the relativistic case the corresponding c -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.
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In recent years, the study of entanglement properties of quantum field theories has led to deep insights in fields as diverse as quantum gravity and condensed matter physics. Originating as effective field theories for certain quantum dimer models, the Quantum Lifshitz Model (QLM) and its generalizations are bosonic quantum field theories with anisotropic scaling symmetry between space and time. Being closely related to conformal field theories, they provide a fruitful playground, where diverse entanglement calculations can be performed analytically. \n \nIn this thesis, we concentrate on two entanglement measures, the entanglement entropy and logarithmic negativity. Motivated to extract sub-leading universal behavior, we perform analytic calculations in two and higher even dimensions. In order to make the calculations tractable, we put the QLM on compact manifolds, such as spheres and tori, where the spectrum of a certain operator appearing in the ground state of the theory is explicitly known. Mostly by means of the replica method, we then find analytic expressions for the finite sub-leading terms of the entanglement entropy and logarithmic negativity of the ground state, as well as the entanglement entropy of the excited states of the QLM. In the case of the ground state entanglement entropy, we provide analytic expressions for the sub-leading terms as functions of the dimension and the dynamical critical exponent. For the excited states we provide analytic formulae of the sub-leading coefficients as functions of the excitation numbers.
Abstract We study the time evolution of the entanglement entropy after quantum quenches in Lifshitz free scalar theories, with the dynamical exponent z > 1, by using the correlator method. For quantum quenches we consider two types of time-dependent mass functions: end-critical-protocol (ECP) and cis-critical-protocol (CCP). In both cases, at early times the entanglement entropy is independent of the subsystem size. After a critical time ( t c ), the entanglement entropy starts depending on the subsystem size significantly. This critical time t c for z = 1 in the fast ECP and CCP has been explained well by the fast quasi-particle of the quasi-particle picture. However, we find that for z > 1 this explanation does not work and t c is delayed. We explain why t c is delayed for z > 1 based on the quasiparticle picture: in essence, it is due to the competition between the fast and slow quasiparticles. At late times, in the ECP, the entanglement entropy slowly increases while, in the CCP, it is oscillating with a well defined period by the final mass scale, independently of z . We give an interpretation of this phenomena by the correlator method. As z increases, the entanglement entropy increases, which can be understood by long-range interactions due to z .
We study several aspects of symmetry resolved capacity of entanglement in certain nonrelativistic quantum field theories which are scale invariant. In particular, we focus on scalar theories with nontrivial dynamical exponent. This study aims to develop a deeper understanding of how each symmetry sector contributes to the capacity of entanglement of the corresponding ground state. Using the correlator method, we compute symmetry-resolved capacity of entanglement and analyze its dependence on the charge and the dynamical exponent. Our results reveal that in these theories, approximate equipartition of entanglement among charge sectors emerges in the large-z regime.