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arXiv:2502.06612 · Journal of Physics A Mathematical and Theoretical

Time evolution of the symmetry resolved entanglement entropy after a mass quench

Federico Rottoli, Michele Mazzoni, Fabio Sailis, Olalla A. Castro-Alvaredo

Published Jul 2, 2025• 0 citations
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Abstract

Abstract In this paper we investigate the properties of the symmetry resolved entanglement entropy after a mass quench in the Ising field theory. Since the theory is free and the post-quench state known explicitly, the one-point function of the relevant (composite) branch point twist field can be computed using form factor techniques, similar to previous work on the branch point twist field and the magnetisation, respectively. We find that the symmetry resolved entropy grows linearly in time at the same rate as the total entropy, and that there are sub-leading oscillatory corrections. This result provides the first explicit computation of the out-of-equilibrium dynamics of the symmetry resolved entropy employing twist fields in quantum field theory and is consistent with existing results based on the quasiparticle picture.

DOI
10.1088/1751-8121/adeb17

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Abstract In this paper we investigate the properties of the symmetry resolved entanglement entropy after a mass quench in the Ising field theory. Since the theory is free and the post-quench state known explicitly, the one-point function of the relevant (composite) branch point twist field can be computed using form factor techniques, similar to previous work on the branch point twist field and the magnetisation, respectively. We find that the symmetry resolved entropy grows linearly in time at the same rate as the total entropy, and that there are sub-leading oscillatory corrections. This result provides the first explicit computation of the out-of-equilibrium dynamics of the symmetry resolved entropy employing twist fields in quantum field theory and is consistent with existing results based on the quasiparticle picture.
A bstract We extend the branch point twist field approach for the calculation of entanglement entropies to time-dependent problems in 1+1-dimensional massive quantum field theories. We focus on the simplest example: a mass quench in the Ising field theory from initial mass m 0 to final mass m . The main analytical results are obtained from a perturbative expansion of the twist field one-point function in the post-quench quasi-particle basis. The expected linear growth of the Rényi entropies at large times mt ≫ 1 emerges from a perturbative calculation at second order. We also show that the Rényi and von Neumann entropies, in infinite volume, contain subleading oscillatory contributions of frequency 2 m and amplitude proportional to ( mt ) − 3 / 2 . The oscillatory terms are correctly predicted by an alternative perturbation series, in the pre-quench quasi-particle basis, which we also discuss. A comparison to lattice numerical calculations carried out on an Ising chain in the scaling limit shows very good agreement with the quantum field theory predictions. We also find evidence of clustering of twist field correlators which implies that the entanglement entropies are proportional to the number of subsystem boundary points.
In this paper I propose a branch point twist field approach to computing the temporal entropy, that is, an entanglement measure across different time regions, as opposed to the usual spacial measures. Considering only the ground state of a gapped theory, I discuss how the shift to time-dependence manifests in form factor calculations. I show that the generalization of the spacial measures to temporal ones reproduces expected features of the temporal entanglement: the entropy is complex, oscillatory and reminiscent of the evolution of entanglement following a global quench. Considering the temporal von Neumann entropy, I argue that spacial and temporal entropies are two sides of the same coin. They both encapsulate universal information about the theory, in particular its mass spectrum. Also in both cases, a quasiparticle picture can be employed to interpret results. This picture connects features of the temporal entropy with those of the spacial entropy following a quantum quench.
The time evolution of the entanglement entropy is a key concept to understand the structure of a nonequilibrium quantum state. In a large class of models, such evolution can be understood in terms of a semiclassical picture of moving quasiparticles spreading the entanglement throughout the system. However, it is not yet known how the entanglement splits between the sectors of an internal local symmetry of a quantum many-body system. Here, guided by the examples of conformal field theories and free-fermion chains, we show that the quasiparticle picture can be adapted to this goal, leading to a general conjecture for the charged entropies whose Fourier transform gives the desired symmetry-resolved entanglement Sn(q){S}_{n}(q)Sn​(q). We point out two physically relevant effects that should be easily observed in atomic experiments: a delay time for the onset of Sn(q){S}_{n}(q)Sn​(q) which grows linearly with $|\mathrm{\ensuremath{\Delta}}q|$ (the difference between the charge qqq and its mean value) and an effective equipartition when $|\mathrm{\ensuremath{\Delta}}q|$ is much smaller than the subsystem size.
A bstract We investigate the non-equilibrium time evolution of symmetry-resolved entanglement entropy (SREE) following an inhomogeneous quench in a critical one-dimensional free fermion system. Using conformal field theory, we derive an exact expression for the SREE and analyze its behavior. We find that, at leading order in the long-time limit, the SREE grows logarithmically as log t . While the equipartition of entanglement holds at leading order, we identify subleading corrections that break it. Our numerical simulations corroborate the analytical predictions with excellent agreement.
In this short review, we present the key definitions, ideas and techniques involved in the study of symmetry resolved entanglement measures, with a focus on the symmetry resolved entanglement entropy. In order to be able to define such entanglement measures, it is essential that the theory under study possess an internal symmetry. Then, symmetry-resolved entanglement measures quantify the contribution to a particular entanglement measure that can be associated to a chosen symmetry sector. Our review focuses on conformal (gapless/massless/critical) and integrable (gapped/massive) quantum field theories, where the leading computational technique employs symmetry fields known as (composite) branch point twist fields.
The charged moments of a reduced density matrix provide a natural starting point for deriving symmetry-resolved Rényi and entanglement entropies, which quantify how entanglement is distributed among symmetry sectors in the presence of a global internal symmetry in a quantum many-body system. In this work, we study charged moments within the framework of Ballistic Fluctuation Theory (BFT). This theory describes large-scale ballistic fluctuations of conserved charges and associated currents and, by exploiting the height-field formulation of twist fields, gives access to the asymptotic behaviour of their two-point correlation functions. In Del Vecchio Del Vecchio et al. [1][1][1], this approach was applied to the special case of branch-point twist fields used to compute entanglement entropies within the replica approach. Here, we extend those results by applying BFT to composite branch-point twist fields, obtained by inserting an additional gauge field. Focusing on free fermions, we derive analytic expressions for charged Rényi entropies both at equilibrium, in generalized Gibbs ensembles, and out of equilibrium following a quantum quench from U(1)U(1)U(1) preserving pair producing integrable initial states. In the latter case, our results agree with the conjecture arising from the quasiparticle picture.