Symmetry-resolved entanglement in critical non-Hermitian systems

Fossati M., Ares F., The study of entanglement in the symmetry sectors of a theory has recently attracted a lot of attention since it provides better understanding of some aspects of quantum many-body systems. In this paper, we extend this analysis to the case of non-Hermitian models, in which the reduced density matrix ρA may be nonpositive definite and the entanglement entropy negative or even complex. Here we examine in detail the symmetry-resolved entanglement in the ground state of the non-Hermitian Su-Schrieffer-Heeger chain at the critical point, a model that preserves particle number and whose scaling limit is a bc-ghost nonunitary conformal field theory (CFT). By combining bosonization techniques in the field theory and exact lattice numerical calculations, we analytically derive the charged moments of ρA and |ρA|. From them, we can understand the origin of the nonpositiveness of ρA and naturally define a positive-definite reduced density matrix in each charge sector, which gives a well-defined symmetry-resolved entanglement entropy. As a by-product, we also obtain the analytical distribution of the critical entanglement spectrum.

Entanglement of stationary states in the presence of unstable quasiparticles

Horváth D.X., The effect of unstable quasiparticles in the out-of-equilibrium dynamics of certain integrable systems has been the subject of several recent studies. In this paper we focus on the stationary value of the entanglement entropy density, its growth rate, and related functions, after a quantum quench. We consider several quenches, each of which is characterised by a corresponding squeezed coherent state. In the quench action approach, the coherent state amplitudes K(θ) become input data that fully characterise the large-time stationary state, thus also the corresponding Yang-Yang entropy. We find that, as function of the mass of the unstable particle, the entropy growth rate has a global minimum signalling the depletion of entropy that accompanies a slowdown of stable quasiparticles at the threshold for the formation of an unstable excitation. We also observe a separation of scales governed by the interplay between the mass of the unstable particle and the quench parameter, separating a non-interacting regime described by free fermions from an interacting regime where the unstable particle is present. This separation of scales leads to a double-plateau structure of many functions, where the relative height of the plateaux is related to the ratio of central charges of the UV fixed points associated with the two regimes, in full agreement with conformal field theory predictions. The properties of several other functions of the entropy and its growth rate are also studied in detail, both for fixed quench parameter and varying unstable particle mass and viceversa.

Entanglement resolution of free Dirac fermions on a torus

Foligno A., Murciano S., Whenever a system possesses a conserved charge, the density matrix splits into eigenspaces associated to the each symmetry sector and we can access the entanglement entropy in a given subspace, known as symmetry resolved entanglement (SRE). Here, we first evaluate the SRE for massless Dirac fermions in a system at finite temperature and size, i.e. on a torus. Then we add a massive term to the Dirac action and we treat it as a perturbation of the massless theory. The charge-dependent entropies turn out to be equally distributed among all the symmetry sectors at leading order. However, we find subleading corrections which depend both on the mass and on the boundary conditions along the torus. We also study the resolution of the fermionic negativity in terms of the charge imbalance between two subsystems. We show that also for this quantity, the presence of the mass alters the equipartition among the different imbalance sectors at subleading order.

Domain wall melting across a defect

Capizzi L., Scopa S., Rottoli F., We study the melting of a domain wall in a free-fermionic chain with a localised impurity. We find that the defect enhances quantum correlations in such a way that even the smallest scatterer leads to a linear growth of the entanglement entropy contrasting the logarithmic behaviour in the clean system. Exploiting the hydrodynamic approach and the quasiparticle picture, we provide exact predictions for the evolution of the entanglement entropy for arbitrary bipartitions. In particular, the steady production of pairs at the defect gives rise to non-local correlations among distant points. We also characterise the subleading logarithmic corrections, highlighting some universal features.

Solution of the BEC to BCS Quench in One Dimension

Rylands C., A gas of interacting fermions confined in a quasi one-dimensional geometry shows a BEC to BCS crossover upon slowly driving its coupling constant through a confinement-induced resonance. On one side of the crossover the fermions form tightly bound bosonic molecules behaving as a repulsive Bose gas, while on the other they form Cooper pairs, whose size is much larger than the average interparticle distance. Here we consider the situation arising when the coupling constant is varied suddenly from the BEC to the BCS value. Namely, we study a BEC-to-BCS quench. By exploiting a suitable continuum limit of recently discovered solvable quenches in the Hubbard model, we show that the local stationary state reached at large times after the quench can be determined exactly by means of the quench action approach. We provide an experimentally accessible characterization of the stationary state by computing local pair correlation function as well as the quasiparticle distribution functions. We find that the steady state is increasingly dominated by two-particle spin singlet bound states for stronger interaction strength, but that bound state formation is inhibited at larger BEC density. The bound state rapidity distribution displays quartic power-law decay suggesting a violation of Tan's contact relations.

Entanglement Barrier and its Symmetry Resolution: Theory and Experimental Observation

Rath A., Vitale V., Murciano S., Votto M., Dubail J., Kueng R., Branciard C., The operator entanglement (OE) is a key quantifier of the complexity of a reduced density matrix. In out-of-equilibrium situations, e.g., after a quantum quench of a product state, it is expected to exhibit an entanglement barrier. The OE of a reduced density matrix initially grows linearly as entanglement builds up between the local degrees of freedom; it then reaches a maximum and ultimately decays to a small finite value as the reduced density matrix converges to a simple stationary state through standard thermalization mechanisms. Here, by performing a new data analysis of the published experimental results of Brydges et al. [Science 364, 260 (2019)], we obtain the first experimental estimation of the OE of a subsystem reduced density matrix in a quantum many-body system. We employ the randomized-measurements toolbox and we introduce and develop a new efficient method to postprocess experimental data in order to extract higher-order density-matrix functionals and access the OE. The OE thus obtained displays the expected barrier as long as the experimental system is large enough. For smaller systems, we observe a barrier with a double-peak structure, the origin of which can be interpreted in terms of pairs of quasiparticles being reflected at the boundary of the qubit chain. As U(1) symmetry plays a key role in our analysis, we introduce the notion of symmetry-resolved operator entanglement (SROE), in addition to the total OE. To gain further insights into the SROE, we provide a thorough theoretical analysis of this new quantity in chains of noninteracting fermions, which, in spite of their simplicity, capture most of the main features of OE and SROE. In particular, we uncover three main physical effects: the presence of a barrier in any charge sector, a time delay for the onset of the growth of SROE, and an effective equipartition between charge sectors.

Entanglement and negativity Hamiltonians for the massless Dirac field on the half line

Rottoli F., Murciano S., Tonni E., We study the ground-state entanglement Hamiltonian of several disjoint intervals for the massless Dirac fermion on the half-line. Its structure consists of a local part and a bi-local term that couples each point to another one in each other interval. The bi-local operator can be either diagonal or mixed in the fermionic chiralities and it is sensitive to the boundary conditions. The knowledge of such entanglement Hamiltonian is the starting point to evaluate the negativity Hamiltonian, i.e. the logarithm of the partially transposed reduced density matrix, which is an operatorial characterisation of entanglement of subsystems in mixed states. We find that the negativity Hamiltonian inherits the structure of the corresponding entanglement Hamiltonian. We finally show how the continuum expressions for both these operators can be recovered from exact numerical computations in free-fermion chains.

Erratum: Exact quench dynamics of symmetry resolved entanglement in a free fermion chain (J. Stat. Mech. (2021) 093102 DOI: 10.1088/1742-5468/ac21d7)

Parez G., Bonsignori R., The study of the entanglement dynamics plays a fundamental role in understanding the behaviour of many-body quantum systems out of equilibrium. In the presence of a globally conserved charge, further insights are provided by the knowledge of the resolution of entanglement in the various symmetry sectors. Here, we carry on the program we initiated in Parez et al (2021 Phys. Rev. B 103 L041104), for the study of the time evolution of the symmetry resolved entanglement in free fermion systems. We complete and extend our derivations also by defining and quantifying a symmetry resolved mutual information. The entanglement entropies display a time delay that depends on the charge sector that we characterise exactly. Both entanglement entropies and mutual information show effective equipartition in the scaling limit of large time and subsystem size. Furthermore, we argue that the behaviour of the charged entropies can be quantitatively understood in the framework of the quasiparticle picture for the spreading of entanglement, and hence we expect that a proper adaptation of our results should apply to a large class of integrable systems. We also find that the number entropy grows logarithmically with time before saturating to a value proportional to the logarithm of the subsystem size.

Electrostatic solution of massless quenches in Luttinger liquids

Ruggiero P., The study of non-equilibrium dynamics of many-body systems after a quantum quench received a considerable boost and a deep theoretical understanding from the path integral formulation in imaginary time. However, the celebrated problem of a quench in the Luttinger parameter of a one dimensional quantum critical system (massless quench) has so far only been solved in the real-time Heisenberg picture. In order to bridge this theoretical gap and to understand on the same ground massive and massless quenches, we study the problem of a gaussian field characterized by a coupling parameter K within a strip and a different one K0 in the remaining two semi-infinite planes. We give a fully analytical solution using the electrostatic analogy with the problem of a dielectric material within a strip surrounded by an infinite medium of different dielectric constant, and exploiting the method of charge images. After analytic continuation, this solution allows us to obtain all the correlation functions after the quench within a path integral approach in imaginary time, thus recovering and generalizing the results in real time. Furthermore, this imaginary-time approach establishes a remarkable connection between the quench and the famous problem of the conductivity of a Tomonaga-Luttinger liquid coupled to two semi-infinite leads: the two are in fact related by a rotation of the spacetime coordinates.

Rényi entropy and negativity for massless complex boson at conformal interfaces and junctions

Capizzi L., Murciano S., We consider the ground state of a theory composed by M species of massless complex bosons in one dimension coupled together via a conformal interface. We compute both the Rényi entropy and the negativity of a generic partition of wires, generalizing the approach developed in a recent work for free fermions. These entanglement measures show a logarithmic growth with the system size, and the universal prefactor depends both on the details of the interface and the bipartition. We test our analytical predictions against exact numerical results for the harmonic chain.

Integrable quenches in the Hubbard model

Rylands C., Bertini B., We study the quench dynamics of the one-dimensional Hubbard model through the quench action formalism. We introduce a class of integrable initial states—expressed as product states over two sites—for which we can provide an exact characterisation of the late-time regime. This is achieved by finding a closed-form expression for the overlaps between our states and the Bethe ansatz eigenstates, which we check explicitly in the limits of low densities and infinite repulsion. Our solution gives access to the stationary values attained by local observables (we show the explicit example of the density of doubly occupied sites) and the asymptotic entanglement dynamics directly in the thermodynamic limit. Interestingly, we find that for intermediate interaction strength Rényi entropies display a double-slope structure.

Generalized hydrodynamics of the repulsive spin- 12 Fermi gas

Scopa S., We study nonhomogeneous quantum quenches in a one-dimensional gas of repulsive spin-1/2 fermions, as described by the integrable Yang-Gaudin model. By means of generalized hydrodynamics (GHD), we analyze in detail the real-time evolution following a sudden change of the confining potential. We consider in particular release protocols and trap quenches, including a version of the quantum Newton's cradle. At zero temperature, we employ a simplified phase-space hydrodynamic picture to characterize the dynamics of the particle- and spin-density profiles. Away from zero temperatures, we perform a thorough numerical study of the GHD equations, and provide quantitative predictions for different values of the temperature, external magnetic field, and chemical potential. We highlight the qualitative features arising due to the multicomponent nature of the elementary excitations, discussing in particular effects of spin-charge separation and dynamical polarization.

Postquantum Quench Growth of Renyi Entropies in Low-Dimensional Continuum Bosonic Systems

Murciano S., The growth of Renyi entropies after the injection of energy into a correlated system provides a window upon the dynamics of its entanglement properties. We develop here a simulation scheme by which this growth can be determined in Luttinger liquids systems with arbitrary interactions, even those introducing gaps into the liquid. We apply this scheme to an experimentally relevant quench in the sine-Gordon field theory. While for short times we provide analytic expressions for the growth of the second and third Renyi entropy, to access longer times, we combine our scheme with truncated spectrum methods.

Multi-charged moments of two intervals in conformal field theory

Ares F., We study the multi-charged moments for two disjoint intervals in the ground state of two 1 + 1 dimensional CFTs with central charge c = 1 and global U(1) symmetry: the massless Dirac field theory and the compact boson (Luttinger liquid). For this purpose, we compute the partition function on the higher genus Riemann surface arising from the replica method in the presence of background magnetic fluxes between the sheets of the surface. We consider the general situation in which the fluxes generate different twisted boundary conditions at each branch point. The obtained multi-charged moments allow us to derive the symmetry resolution of the Rényi entanglement entropies and the mutual information for non complementary bipartitions. We check our findings against exact numerical results for the tight-binding model, which is a lattice realisation of the massless Dirac theory.

Symmetry-resolved Page curves

Murciano S., Given a statistical ensemble of quantum states, the corresponding Page curve quantifies the average entanglement entropy associated with each possible spatial bipartition of the system. In this work, we study a natural extension in the presence of a conservation law and introduce the symmetry-resolved Page curves, characterizing average bipartite symmetry-resolved entanglement entropies. We derive explicit analytic formulas for two important statistical ensembles with a U(1)-symmetry: Haar-random pure states and random fermionic Gaussian states. In the former case, the symmetry-resolved Page curves can be obtained in an elementary way from the knowledge of the standard one. This is not true for random fermionic Gaussian states. In this case, we derive an analytic result in the thermodynamic limit based on a combination of techniques from random-matrix and large-deviation theories. We test our predictions against numerical calculations and discuss the subleading finite-size corrections.

Rényi entropy and negativity for massless Dirac fermions at conformal interfaces and junctions

Capizzi L., Murciano S., We investigate the ground state of a (1+1)-dimensional conformal field theory (CFT) built with M species of massless free Dirac fermions coupled at one boundary point via a conformal junction/interface. Each CFT represents a wire of finite length L. We develop a systematic strategy to compute the Rényi entropies for a generic bipartition between the wires and the entanglement negativity between two non-complementary sets of wires. Both these entanglement measures turn out to grow logarithmically with L with an exactly calculated universal prefactor depending on the details of the junction and of the bipartition. These analytic predictions are tested numerically for junctions of free Fermi gases, finding perfect agreement.

Growth of Rényi Entropies in Interacting Integrable Models and the Breakdown of the Quasiparticle Picture

Bertini B., Klobas K., Alba V., Lagnese G., Rényi entropies are conceptually valuable and experimentally relevant generalizations of the celebrated von Neumann entanglement entropy. After a quantum quench in a clean quantum many-body system they generically display a universal linear growth in time followed by saturation. While a finite subsystem is essentially at local equilibrium when the entanglement saturates, it is genuinely out of equilibrium in the growth phase. In particular, the slope of the growth carries vital information on the nature of the system's dynamics, and its characterization is a key objective of current research. Here we show that the slope of Rényi entropies can be determined by means of a spacetime duality transformation. In essence, we argue that the slope coincides with the stationary density of entropy of the model obtained by exchanging the roles of space and time. Therefore, very surprisingly, the slope of the entanglement is expressed as an equilibrium quantity. We use this observation to find an explicit exact formula for the slope of Rényi entropies in all integrable models treatable by thermodynamic Bethe ansatz and evolving from integrable initial states. Interestingly, this formula can be understood in terms of a quasiparticle picture only in the von Neumann limit.

Thermodynamic symmetry resolved entanglement entropies in integrable systems

Piroli L., Vernier E., Collura M., We develop a general approach to compute the symmetry-resolved Rényi and von Neumann entanglement entropies (SREE) of thermodynamic macrostates in interacting integrable systems. Our method is based on a combination of the thermodynamic Bethe ansatz and the Gärtner-Ellis theorem from large deviation theory. We derive an explicit simple formula for the von Neumann SREE, which we show to coincide with the thermodynamic Yang-Yang entropy of an effective macrostate determined by the charge sector. Focusing on the XXZ Heisenberg spin chain, we test our result against iTEBD calculations for thermal states, finding good agreement. As an application, we provide analytic predictions for the asymptotic value of the SREE following a quantum quench.

Symmetry-resolved entanglement in a long-range free-fermion chain

Ares F., Murciano S., We investigate the symmetry resolution of entanglement in the presence of long-range couplings. To this end, we study the symmetry-resolved entanglement entropy in the ground state of a fermionic chain that has dimerised long-range hoppings with power-like decaying amplitude - a long-range generalisation of the Su-Schrieffer-Heeger model. This is a system that preserves the number of particles. The entropy of each symmetry sector is calculated via the charged moments of the reduced density matrix. We exploit some recent results on block Toeplitz determinants generated by a discontinuous symbol to obtain analytically the asymptotic behaviour of the charged moments and of the symmetry-resolved entropies for a large subsystem. At leading order we find entanglement equipartition, but comparing with the short-range counterpart its breaking occurs at a different order and it does depend on the hopping amplitudes.

Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 XXZ model

Scopa S., We investigate the non-equilibrium dynamics of a one-dimensional spin-1/2 XXZ model at zero-temperature in the regime |∆| < 1, initially prepared in a product state with two domain walls i.e, |↓ . . . ↓↑ . . . ↑↓ . . . ↓〉. At early times, the two domain walls evolve independently and only after a calculable time a non-trivial interplay between the two emerges and results in the occurrence of a split Fermi sea. For ∆ = 0, we derive exact asymptotic results for the magnetization and the spin current by using a semi-classical Wigner function approach, and we exactly determine the spreading of entanglement entropy exploiting the recently developed tools of quantum fluctuating hydrodynamics. In the interacting case, we analytically solve the Generalized Hydrodynamics equation providing exact expressions for the conserved quantities. We display some numerical results for the entanglement entropy also in the interacting case and we propose a conjecture for its asymptotic value.