Abstract Symmetry resolved entanglement and entanglement asymmetry are two measures of quantum correlations sensitive to symmetries of the system. Here we discuss their non-equilibrium dynamics in the Rule 54 cellular automaton, a simple, yet interacting, integrable model. Both quantities can be expressed in terms of the more analytically tractable ‘charged moments’, i.e. traces of powers of a suitably deformed density matrix, via a replica trick. We express them in terms of a tensor network, which we contract in space using a system of local algebraic relations. This gives the asymptotic form for the charged moments, valid in the regime of large but finite time that is shorter than all the relevant subsystem sizes. In this regime the charge moments decay exponentially with the rate given by the leading solution to a cubic equation.
Abstract Symmetry resolved entanglement and entanglement asymmetry are two measures of quantum correlations sensitive to symmetries of the system. Here we discuss their non-equilibrium dynamics in the Rule 54 cellular automaton, a simple, yet interacting, integrable model. Both quantities can be expressed in terms of the more analytically tractable ‘charged moments’, i.e. traces of powers of a suitably deformed density matrix, via a replica trick. We express them in terms of a tensor network, which we contract in space using a system of local algebraic relations. This gives the asymptotic form for the charged moments, valid in the regime of large but finite time that is shorter than all the relevant subsystem sizes. In this regime the charge moments decay exponentially with the rate given by the leading solution to a cubic equation.
Owing to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution-or its Fourier transform known as full counting statistics (FCS)-contains much more information than say the mean value of the measured observable, and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables-the charged moments-that characterize how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a nontrivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out of equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain.
We investigate the entanglement dynamics of the Rule 54 quantum cellular automaton subject to stochastic projective measurements. Rule 54 is an interacting integrable model whose closed dynamics are exactly described by a semiclassical quasiparticle picture: entangled soliton pairs propagate ballistically and scatter elastically, generating entanglement at a rate determined analytically by the thermodynamic entropy density and the dressed quasiparticle velocity. We study how this picture is modified when measurements are interleaved with the unitary evolution, comparing two protocols that differ in their relationship to the conserved soliton charge. For measurements in the computational basis (Z-basis), which commute with the conserved charge, we derive an exact analytical prediction for the entanglement decay: the quasiparticle ansatz is modified by a measurement-induced exponential envelope exp(−4γt) , where γ is the measurement rate. This prediction is validated through finite-size scaling collapse across system sizes and measurement rates, with excellent agreement between analytics and Matrix Product State simulations. For measurements in the X-basis, which do not commute with the conserved charge, measurements act simultaneously as a source and a sink of entanglement through a quasiparticle reseeding mechanism. Our numerical results are consistent with a measurement-induced phase transition at a finite critical rate, in sharp contrast to the Z-basis case where any nonzero measurement rate drives the system to an area-law phase. A preliminary finite-size scaling analysis yields γc = 0.370 ± 0.004 (pc ≈ 0.31) and ν = 0.97 ± 0.03, though a complete characterization of the universality class requires larger system sizes.
We investigate symmetry-resolved entanglement in non-relativistic quantum field theories, including complex Lifshitz scalar chains and Lifshitz fermionic models. Using charged moments and the correlator method, we compute symmetry-resolved Renyi and von Neumann entropies and analyze their dependence on subsystem size, charge, mass, and the dynamical exponent z. Our results reveal distinct features of non-relativistic entanglement. In Lifshitz scalar theories, approximate equipartition among charge sectors emerges in the large-z regime, with configurational entropy dominating, whereas Lifshitz fermionic models exhibit genuine equipartition only in the relativistic limit, with fluctuation entropy prevailing. These findings highlight a rich interplay between conserved charges, subsystem size, mass, and dynamical scaling, and provide a framework to explore operationally accessible entanglement in non-relativistic systems. Our study offers insights relevant to experimental platforms such as cold atom setups and mesoscopic systems, where particle-number-resolved measurements can probe symmetry-resolved entanglement.
A bstract We investigate symmetry-resolved entanglement in non-relativistic quantum field theories, including complex Lifshitz scalar chains and Lifshitz fermionic models. Using charged moments and the correlator method, we compute symmetry-resolved Rényi and von Neumann entropies and analyze their dependence on subsystem size, charge, mass, and the dynamical exponent z . Our results reveal distinct features of non-relativistic entanglement. In Lifshitz scalar theories, approximate equipartition among charge sectors emerges in the large- z regime, with configurational entropy dominating, whereas Lifshitz fermionic models exhibit genuine equipartition only in the relativistic limit, with fluctuation entropy prevailing. These findings highlight a rich interplay between conserved charges, subsystem size, mass, and dynamical scaling, and provide a framework to explore operationally accessible entanglement in non-relativistic systems. Our study offers insights relevant to experimental platforms such as cold atom setups and mesoscopic systems, where particle-number-resolved measurements can probe symmetry-resolved entanglement.
Characterising the universal aspects of non-equilibrium quantum many-body dynamics is one of the key goals of this century's physics research. Progress, however, is hindered by the lack of general theoretical frameworks for studying interacting quantum matter far from equilibrium. A recent breakthrough has been the realization that several key non-equilibrium quantities, such as the rate of growth of entanglement or the fluctuations of conserved charges within finite subsystems, can be related to equilibrium properties through a space-time duality that effectively exchanges the roles of space and time. This observation effectively enables the study of non-equilibrium phenomena using tools and concepts borrowed from equilibrium statistical mechanics and thermodynamics. A first proof of principle of this framework, dubbed space-time duality approach (SDA), was provided by interacting integrable systems, where thermodynamic properties can often be characterized exactly, while dynamical quantities typically remain beyond analytical reach. Subsequent developments, however, revealed that the SDA suffered from an intrinsic ambiguity, restricting its applicability to homogeneous quenches and to charge fluctuations arising from symmetric initial states. Here we resolve this ambiguity from first principles and derive closed-form predictions for entanglement growth and charge fluctuations after general quantum quenches. We benchmark our results against the exact analytical solution of the Rule 54 quantum cellular automaton and extensive TEBD simulations of the XXZ chain. Moreover we show that, when specialised to the entanglement entropy, our framework naturally reproduces the predictions of the quasiparticle picture.
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], 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) preserving pair producing integrable initial states. In the latter case, our results agree with the conjecture arising from the quasiparticle picture.