All publications from Pasquale Calabrese
More on symmetry resolved operator entanglement
Murciano S., Dubail J., Calabrese P.
The ‘operator entanglement’ of a quantum operator O is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global U(1) conservation law, and on operators O with a well-defined U(1) charge, for which it is possible to resolve the operator entanglement of O according to the U(1) symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in Rath et al (2023 PRX Quantum 4 010318) and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix ρ β = e − β H and of charged local operators evolving in Heisenberg picture O = e i t H O e − i t H . Our main results are: i) the SROE of ρ β obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.
Time evolution of entanglement entropy after quenches in two-dimensional free fermion systems: A dimensional reduction treatment
Yamashika S., Ares F., Calabrese P.
We study the time evolution of the Rényi entanglement entropies following a quantum quench in a two-dimensional (2D) free fermion system. By employing dimensional reduction, we effectively transform the 2D problem into decoupled chains, a technique applicable when the system exhibits translational invariance in one direction. Various initial configurations are examined, revealing that the behavior of entanglement entropies can often be explained by adapting the one-dimensional quasiparticle picture. However, intriguingly, for specific initial states the entanglement entropy saturates to a finite value without the reduced density matrix converging to a stationary state. We discuss the conditions necessary for a stationary state to exist and delve into the necessary modifications to the quasiparticle picture when such a state is absent.
Non-equilibrium entanglement asymmetry for discrete groups: the example of the XY spin chain
Ferro F., Ares F., Calabrese P.
Entanglement asymmetry is a novel quantity that, using entanglement methods, measures how much a symmetry is broken in a part of an extended quantum system. So far, it has only been used to characterise the breaking of continuous Abelian symmetries. In this paper, we extend the concept to cyclic Z N groups. As an application, we consider the XY spin chain, in which the ground state spontaneously breaks the Z 2 spin parity symmetry in the ferromagnetic phase. We thoroughly investigate the non-equilibrium dynamics of this symmetry after a global quantum quench, generalising known results for the standard order parameter.
Entanglement entropy along a massless renormalisation flow: the tricritical to critical Ising crossover
Rottoli F., Ares F., Calabrese P., Horváth D.X.
We study the Rényi entanglement entropies along the massless renormalisation group flow that connects the tricritical and critical Ising field theories. Similarly to the massive integrable field theories, we derive a set of bootstrap equations, from which we can analytically calculate the twist field form factors in a recursive way. Additionally, we also obtain them as a non-trivial ‘roaming limit’ of the sinh-Gordon theory. Then the Rényi entanglement entropies are obtained as expansions in terms of the form factors of these branch point twist fields. We find that the form factor expansion of the entanglement entropy along the flow organises in two different kind of terms. Those that couple particles with the same chirality, and reproduce the entropy of the infrared Ising theory, and those that couple particles with different chirality, which provide the ultraviolet contributions. The massless flow under study possesses a global ℤ2 spin-flip symmetry. We further consider the composite twist fields associated to this group, which enter in the study of the symmetry resolution of the entanglement. We derive analytical expressions for their form factors both from the bootstrap equations and from the roaming limit of the sinh-Gordon theory.
Symmetry resolution of the computable cross-norm negativity of two disjoint intervals in the massless Dirac field theory
Bruno A., Ares F., Murciano S., Calabrese P.
We investigate how entanglement in the mixed state of a quantum field theory can be described using the cross-computable norm or realignment (CCNR) criterion, employing a recently introduced negativity. We study its symmetry resolution for two disjoint intervals in the ground state of the massless Dirac fermion field theory, extending previous results for the case of adjacent intervals. By applying the replica trick, this problem boils down to computing the charged moments of the realignment matrix. We show that, for two disjoint intervals, they correspond to the partition function of the theory on a torus with a non-contractible charged loop. This confers a great advantage compared to the negativity based on the partial transposition, for which the Riemann surfaces generated by the replica trick have higher genus. This result empowers us to carry out the replica limit, yielding analytic expressions for the symmetry-resolved CCNR negativity. Furthermore, these expressions provide also the symmetry decomposition of other related quantities such as the operator entanglement of the reduced density matrix or the reflected entropy.
Entanglement asymmetry and quantum Mpemba effect in the XY spin chain
Murciano S., Ares F., Klich I., Calabrese P.
Entanglement asymmetry is a quantity recently introduced to measure how much a symmetry is broken in a part of an extended quantum system. It has been employed to analyze the non-equilibrium dynamics of a broken symmetry after a global quantum quench with a Hamiltonian that preserves it. In this work, we carry out a comprehensive analysis of the entanglement asymmetry at equilibrium taking the ground state of the XY spin chain, which breaks the U(1) particle number symmetry, and provide a physical interpretation of it in terms of superconducting Cooper pairs. We also consider quenches from this ground state to the XX spin chain, which preserves the U(1) symmetry. In this case, the entanglement asymmetry reveals that the more the symmetry is initially broken, the faster it may be restored in a subsystem, a surprising and counter-intuitive phenomenon that is a type of a quantum Mpemba effect. We obtain a quasi-particle picture for the entanglement asymmetry in terms of Cooper pairs, from which we derive the microscopic conditions to observe the quantum Mpemba effect in this system, giving further support to the criteria recently proposed for arbitrary integrable quantum systems. In addition, we find that the power law governing symmetry restoration depends discontinuously on whether the initial state is critical or not, leading to new forms of strong and weak Mpemba effects.
Entanglement asymmetry as a probe of symmetry breaking
Ares F., Murciano S., Calabrese P.
Symmetry and symmetry breaking are two pillars of modern quantum physics. Still, quantifying how much a symmetry is broken is an issue that has received little attention. In extended quantum systems, this problem is intrinsically bound to the subsystem of interest. Hence, in this work, we borrow methods from the theory of entanglement in many-body quantum systems to introduce a subsystem measure of symmetry breaking that we dub entanglement asymmetry. As a prototypical illustration, we study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global U(1) symmetry is restored dynamically. We adapt the quasiparticle picture for entanglement evolution to the analytic determination of the entanglement asymmetry. We find, expectedly, that larger is the subsystem, slower is the restoration, but also the counterintuitive result that more the symmetry is initially broken, faster it is restored, a sort of quantum Mpemba effect, a phenomenon that we show to occur in a large variety of systems.
Symmetry-resolved entanglement in fermionic systems with dissipation
Murciano S., Calabrese P., Alba V.
We investigate symmetry-resolved entanglement in out-of-equilibrium fermionic systems subject to gain and loss dissipation, which preserves the block-diagonal structure of the reduced density matrix. We derive a hydrodynamic description of the dynamics of several entanglement-related quantities, such as the symmetry-resolved von Neumann entropy and the charge-imbalance-resolved fermionic negativity. We show that all these quantities admit a hydrodynamic description in terms of entangled quasiparticles. While the entropy is dominated by dissipative processes, the resolved negativity is sensitive to the presence of entangled quasiparticles, and it shows the typical ‘rise and fall’ dynamics. Our results hold in the weak-dissipative hydrodynamic limit of large intervals, long times and weak dissipation rates.
Transport and Entanglement across Integrable Impurities from Generalized Hydrodynamics
Rylands C., Calabrese P.
Quantum impurity models (QIMs) are ubiquitous throughout physics. As simplified toy models they provide crucial insights for understanding more complicated strongly correlated systems, while in their own right are accurate descriptions of many experimental platforms. In equilibrium, their physics is well understood and have proven a testing ground for many powerful theoretical tools, both numerical and analytical, in use today. Their nonequilibrium physics is much less studied and understood. However, the recent advancements in nonequilibrium integrable quantum systems through the development of generalized hydrodynamics (GHD) coupled with the fact that many archetypal QIMs are in fact integrable presents an enticing opportunity to enhance our understanding of these systems. We take a step towards this by expanding the framework of GHD to incorporate integrable interacting QIMs. We present a set of Bethe-Boltzmann type equations which incorporate the effects of impurity scattering and discuss the new aspects which include entropy production. These impurity GHD equations are then used to study a bipartioning quench wherein a relevant backscattering impurity is included at the location of the bipartition. The density and current profiles are studied as a function of the impurity strength and expressions for the entanglement entropy and full counting statistics are derived.
Nonequilibrium Full Counting Statistics and Symmetry-Resolved Entanglement from Space-Time Duality
Bertini B., Calabrese P., Collura M., Klobas K., Rylands C.
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.
Lack of symmetry restoration after a quantum quench: An entanglement asymmetry study
Ares F., Murciano S., Vernier E., Calabrese P.
We consider the quantum quench in the XX spin chain starting from a tilted Néel state which explicitly breaks the U(1) symmetry of the post-quench Hamiltonian. Very surprisingly, the U(1) symmetry is not restored at large time because of the activation of a non- Abelian set of charges which all break it. The breaking of the symmetry can be effectively and quantitatively characterised by the recently introduced entanglement asymmetry. By a combination of exact calculations and quasi-particle picture arguments, we are able to exactly describe the behaviour of the asymmetry at any time after the quench. Furthermore we show that the stationary behaviour is completely captured by a non-Abelian generalised Gibbs ensemble. While our computations have been performed for a noninteracting spin chain, we expect similar results to hold for the integrable interacting case as well because of the presence of non-Abelian charges also in that case.
One-particle density matrix and momentum distribution of the out-of-equilibrium one-dimensional Tonks-Girardeau gas: Analytical results at large N
Scopa S., Ruggiero P., Calabrese P., Dubail J.
In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms N, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large N, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix (ψ †(x)ψ(x′)) of the out-of-equilibrium TG gas in the N→∞ limit, valid on distances |x-x′| much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well, which mimics a "quantum Newton cradle"setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.
Full counting statistics and symmetry resolved entanglement for free conformal theories with interface defects
Capizzi L., Murciano S., Calabrese P.
We consider the ground state of two species of one-dimensional critical free theories coupled together via a conformal interface. They have an internal U(1) global symmetry and we investigate the quantum fluctuations of the total charge on one side of the interface, giving analytical predictions for the full counting statistics, the charged moments of the reduced density matrix and the symmetry resolved Rényi entropies. Our approach is based on the relation between the geometry with the defect and the homogeneous one, and it provides a way to characterize the spectral properties of the correlation functions restricted to one of the two species. Our analytical predictions are tested numerically, finding a perfect agreement.
A hydrodynamic approach to Stark localization
Capizzi L., Vanoni C., Calabrese P., Gambassi A.
When a free Fermi gas on a lattice is subject to the action of a linear potential it does not drift away, as one would naively expect, but it remains spatially localized. Here we revisit this phenomenon, known as Stark localization, within the recently proposed framework of generalized hydrodynamics. In particular, we consider the dynamics of an initial state in the form of a domain wall and we recover known results for the particle density and the particle current, while we derive analytical predictions for relevant observables such as the entanglement entropy and the full counting statistics. Then, we extend the analysis to generic potentials, highlighting the relationship between the occurrence of localization and the presence of peculiar closed orbits in phase space, arising from the lattice dispersion relation. We also compare our analytical predictions with numerical calculations and with the available results, finding perfect agreement. This approach paves the way for an exact treatment of the interacting case known as Stark many-body localization.
Finite temperature negativity Hamiltonians of the massless Dirac fermion
Rottoli F., Murciano S., Calabrese P.
The negativity Hamiltonian, defined as the logarithm of a partially transposed density matrix, provides an operatorial characterisation of mixed-state entanglement. However, so far, it has only been studied for the mixed-state density matrices corresponding to subsystems of globally pure states. Here, we consider as a genuine example of a mixed state the one-dimensional massless Dirac fermions in a system at finite temperature and size. As subsystems, we consider an arbitrary set of disjoint intervals. The structure of the corresponding negativity Hamiltonian resembles the one for the entanglement Hamiltonian in the same geometry: in addition to a local term proportional to the stress-energy tensor, each point is non-locally coupled to an infinite but discrete set of other points. However, when the lengths of the transposed and non-transposed intervals coincide, the structure remarkably simplifies and we retrieve the mild non-locality of the ground state negativity Hamiltonian. We also conjecture an exact expression for the negativity Hamiltonian associated to the twisted partial transpose, which is a Hermitian fermionic matrix. We finally obtain the continuum limit of both the local and bi-local operators from exact numerical computations in free-fermionic chains.
Symmetry-resolved entanglement in critical non-Hermitian systems
Fossati M., Ares F., Calabrese P.
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., Calabrese P., Castro-Alvaredo O.A.
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., Calabrese P.
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., Calabrese P.
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., Calabrese P., Bertini B.
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.

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