In this thesis, we first present a review on the background and research related to our published works. We then present our first work, which investigates entanglement entropy free-fermionic systems of D-spatial dimensions using an effective quasiparticle description. In particular, we consider a quantum quench to the D-dimensional tight-binding model from an initial state invariant under translation by nu sites. We develop an efficient Monte Carlo scheme to implement the quasiparticle solution to arbitrary connected subsystems, and show that, in two dimensions, the shape and orientation of the subsystem with respect to the underlying lattice is able to effect qualitative behaviour of the entanglement growth. In our second work, we investigate the breaking and restoration of translation symmetry in the same quench setup restricted to the one-dimensional setting. We establish the presence of the so-called quantum Mpemba effect in this setting, which had hitherto been almost exclusively considered with respect to continuous U(1) symmetry breaking. We also apply a quasiparticle picture to our chosen measure of symmetry breaking and find that this picture does not agree with the solution obtained by exact diagonalisation. From this we conclude that the restoration of discrete symmetries in integrable systems goes beyond the leading hydrodynamic order. Finally, in this thesis we also present a brief review related to our most recent work, which considers the recently proposed hierarchical dual-unitary circuit in two dimensions. We then present the current iteration of this work, which defines the set of DU2 conditions on the two-dimensional gates that may be used to fully simplify two-point spatio-temporal correlation functions in terms of the repeat action of quantum channels. Furthermore, we identify directions of these correlations along the edge of the causal lightcone with no 1D counterpart that are generically non-vanishing with respect to these DU2 conditions. This is demonstrated for a particular family of gate with site dimension d=6.
In this thesis, we first present a review on the background and research related to our published works. We then present our first work, which investigates entanglement entropy free-fermionic systems of D-spatial dimensions using an effective quasiparticle description. In particular, we consider a quantum quench to the D-dimensional tight-binding model from an initial state invariant under translation by nu sites. We develop an efficient Monte Carlo scheme to implement the quasiparticle solution to arbitrary connected subsystems, and show that, in two dimensions, the shape and orientation of the subsystem with respect to the underlying lattice is able to effect qualitative behaviour of the entanglement growth. In our second work, we investigate the breaking and restoration of translation symmetry in the same quench setup restricted to the one-dimensional setting. We establish the presence of the so-called quantum Mpemba effect in this setting, which had hitherto been almost exclusively considered with respect to continuous U(1) symmetry breaking. We also apply a quasiparticle picture to our chosen measure of symmetry breaking and find that this picture does not agree with the solution obtained by exact diagonalisation. From this we conclude that the restoration of discrete symmetries in integrable systems goes beyond the leading hydrodynamic order. Finally, in this thesis we also present a brief review related to our most recent work, which considers the recently proposed hierarchical dual-unitary circuit in two dimensions. We then present the current iteration of this work, which defines the set of DU2 conditions on the two-dimensional gates that may be used to fully simplify two-point spatio-temporal correlation functions in terms of the repeat action of quantum channels. Furthermore, we identify directions of these correlations along the edge of the causal lightcone with no 1D counterpart that are generically non-vanishing with respect to these DU2 conditions. This is demonstrated for a particular family of gate with site dimension d=6.
The study of symmetry restoration has recently emerged as a fruitful means to extract high-level information on the relaxation of quantum many-body systems. However, while the restoration of internal symmetries has been investigated intensively, that of spatial symmetries has hitherto only been considered in the context of random unitary circuits. Here, we present a complementary study of translation symmetry restoration in integrable systems. In particular, we consider a one-dimensional chain of spinless, noninteracting fermions quenched from a ν > 1 shift invariant state, and follow the local restoration of one-site shift invariance using the Frobenius distance Δ F A between the state on a subsystem and its symmetrized counterpart. Distinct from the case of random unitary circuits, where symmetry restoration occurs abruptly for times proportional to the subsystem size, we find that symmetry here is restored smoothly and over timescales of the order of the subsystem size squared. We also find that the so-called ‘‘quantum Mpemba effect’’ is readily observed. Most importantly, we show that—in contrast to the case of continuous internal symmetries—this discrete symmetry restoration is not qualitatively described by a quasiparticle picture for Δ F A , and therefore goes beyond the hydrodynamic description. Our results can be directly extended to higher dimensions.
The characterization of the dynamical structure of quantum many-body systems is an important and very active endeavor both experimentally and theoretically. Several tools, such as quantum-information theoretic observables and spectral measures, provide a powerful lens for characterizing correlations and their dynamics in quantum many-body systems. These quantities can reveal important features of experimentally realizable systems and provide insight into their control and dynamical behaviour. This thesis investigates nonequilibrium settings in which the spreading and redistribution of quantum correlations exhibit unconventional behaviour, with particular emphasis on the role of extended-range correlations. We consider two main classes of systems: lattice models undergoing quantum quenches and central-spin models motivated by experimental applications. For lattice systems, we study the dynamics of entanglement and local symmetry breaking in free fermionic and bosonic models, interacting integrable systems, and quantum circuits. We first investigate symmetry restoration following quenches from ground states of long-range fermionic Hamiltonians. We show that two states undergoing symmetry restoration can exhibit multiple inversions, manifested through multiple crossings of the distance measure, implying either the evasion of the quantum Mpemba effect or a more intricate route to its occurrence. This demonstrates that anomalous relaxation is not restricted to short times and can be understood through the momentum-dependent distribution and velocities of quasiparticle excitations. We then study quenches from initial states possessing extensive, spatially organized entanglement, moving beyond the conventional setting of weakly entangled initial states. For crosscap states, characterized by antipodal correlations and volume-law entanglement entropy, we uncover unconventional entanglement dynamics and formulate an antipodal quasiparticle picture in which initially entangled degrees of freedom originate from macroscopically separated points. We subsequently generalize this construction to entangled multipodal states, whose polygon-like correlation structure produces richer equilibrium and dynamical entanglement patterns. We further investigate charge-deformed crosscap states and show that their local symmetry-breaking dynamics can display the unusual phenomenon of dynamical symmetry breaking under symmetry-preserving evolution. Finally, we extend the crosscap construction to bosonic systems and describe the dynamics of the entanglement entropy, mutual information, and logarithmic negativity within a corresponding quasiparticle framework. The second class of systems considered concerns central-spin models and their application to quantum memories in semiconductor quantum dots. We investigate how disorder in couplings and frequencies modifies collective many-body dynamics and affects the storage and retrieval of quantum information. Using spectral, dynamical, and information-theoretic diagnostics, we characterize the resulting dynamical regimes and relate the loss of collectivity and the spreading of information through the spin bath to the performance of quantum-memory protocols. Overall, the results demonstrate how extended-range correlations, whether encoded in an initial state or generated by the interaction structure, can qualitatively reshape entanglement spreading, symmetry dynamics, and information retention in nonequilibrium quantum many-body systems.
We begin a systematic investigation of quench dynamics in higher-dimensional lattice systems considering the case of noninteracting fermions with conserved particle number. We prepare the system in a translational-invariant nonequilibrium initial state, the simplest example being a classical configuration with fermions at fixed positions on the lattice, and let it evolve in time. We characterize the system's dynamics by measuring the entanglement between a finite connected region and its complement. We observe the transmutation of entanglement entropy into thermodynamic entropy and investigate how this process depends on the shape and orientation of the region with respect to the underlying lattice. Interestingly, we find that irregular regions display a distinctive multislope entanglement growth, while the dependence on the orientation angle is generically fairly weak. This is particularly true for regions with a large (discrete) rotational symmetry group. The main tool of our analysis is the celebrated quasiparticle picture of Calabrese and Cardy, which we generalize to describe the case at hand. Specifically, we show that for generic initial configurations (even when restricting to classical ones) one has to allow for the production of multiplets involving n > 2 quasiparticles and carrying nondiagonal correlations. We obtain quantitatively accurate predictions, tested against exact numerics, and propose an efficient Monte Carlo based scheme to evaluate them for arbitrary connected regions of generic higher-dimensional lattices. Published by the American Physical Society 2024
The quantum Mpemba effect is the counterintuitive nonequilibrium phenomenon wherein the dynamic restoration of a broken symmetry occurs more rapidly when the initial state exhibits a higher degree of symmetry breaking. The effect has been recently discovered theoretically and observed experimentally in the framework of global quantum quenches, but so far it has only been investigated in one-dimensional systems. Here we focus on a two-dimensional free-fermion lattice employing the entanglement asymmetry as a measure of symmetry breaking. Our investigation begins with the ground-state analysis of a system featuring nearest-neighbor hoppings and superconducting pairings, the latter breaking explicitly the U(1) particle-number symmetry. We compute analytically the entanglement asymmetry of a periodic strip using dimensional reduction, an approach that allows us to adjust the extent of the transverse size, achieving a smooth crossover between one and two dimensions. Further applying the same method, we study the time evolution of the entanglement asymmetry after a quench to a Hamiltonian with only nearest-neighbor hoppings, preserving the particle-number symmetry which is restored in the stationary state. We find that the quantum Mpemba effect is strongly affected by the size of the system in the transverse dimension, with the potential to either enhance or spoil the phenomenon depending on the initial states. We establish the conditions for its occurrence based on the properties of the initial configurations, extending the criteria found in the one-dimensional case.
Entanglement and symmetries are two pillars of our understanding of modern Physics, but surprisingly the study of the interplay between these two fundamental concepts has become the subject of an intense research area only in the last few years. In particular, the resolution of entanglement in the various symmetry sectors of a quantum many-body system with a global conserved charge gained a lot of interest within the entanglement community. As shown in some experimental papers, the study of this symmetry resolved entanglement is necessary for a better understanding of the non-equilibrium dynamics of quantum many-body systems. This Thesis is devoted to the study of the symmetry resolution of the entanglement measures in free fermionic systems and Conformal Field Theories. It is organised as follows. In Chap. 1 we give an overview of the main definitions and concepts related to the entanglement and its measures in quantum mechanics and quantum field theory. Then, we specialize to the case of conformal field theories, to derive expressions of the entanglement entropy and negativity in all the cases that will be examined in the second part of the Thesis. We finally illustrate the main free-fermionic techniques that will be used to perform the analytical as well as numerical lattice calculations. In Chap. 2, we consider a 1-dimensional system of free fermions hopping on a lattice, in which the conserved U(1) charge is the total particle number. We evaluate the symmetry resolved R'enyi and entanglement entropies determining exactly all the non-universal constants and logarithmic corrections to the scaling, thanks to the so-called Fisher-Hartwig conjecture. In Chap. 3, we study the symmetry resolved entanglement entropies in 1-dimensional systems in presence of boundaries, considering the cases of an interval starting from the boundary and disconnected from it. We evaluate the charged and symmetry resolved entropies for conformal invariant theories and a semi-infinite chain of free fermions, finding entanglement equipartition. Using a generalization of the lattice calculations performed in Chap. 2, we also evaluate the first term breaking equipartition for the free fermionic chain. In Chap. 4, we study the symmetry resolution of the entanglement negativity in free fermionic systems, which admits a decomposition in terms of the charge imbalance between the two subsystems. We use conformal field theory to derive the universal behavior of the charge imbalance resolved negativity for a free Dirac field at finite temperature and size. For the lattice version of the Dirac filed, we also determine exactly the model-dependent terms using the Fisher-Hartwig conjecture. We find that at leading order the charge imbalance resolved negativity verifies equipartition. For the lattice model, we also evaluate the first corrections to this equipartition. In Chap. 5, we study the time evolution of the symmetry resolved entanglement entropy and a suitably defined symmetry resolved mutual information in a system of free fermions after a quench. In the context of free fermions and conformal field theories, we are able to provide analytical results for the time-dependent charged and resolved entropies, and shed light on two new physical phenomena. First, the symmetry resolved R'enyi entropies start evolving with a calculable time delay that depends on the charge sector. Second, there is an effective equipartition of the entanglement in the limit of large subsystem size. For free fermions, we also find that effective equipartition is verified by the mutual information. We expect that these phenomena occur in more general quantum systems, as they can be understood from the quasiparticle picture for the spreading of entanglement. In the Appendix, we give an overview of conformal field theory, reporting the main concepts and formulas that are needed for a more complete understanding of the previous Chapters.
Quantum systems display counterintuitive features, escaping our everyday-life intuition, whose origin is related to entanglement. This phenomenon arises when different parts of the same systems display correlations, which cannot be understood via classical mechanics, and they are dubbed as quantum. To characterize the correlations above, it is necessary to find good entanglement measures which are able to capture quantitatively the quantum features of a given system. This motivates the investigation and the computation of quantities such as the von Neumann entropy of regions, which provides information about how a system is entangled. Our focus is quantum one-dimensional systems close to their critical point, where quantum correlations are enhanced, and a characterization via Quantum Field Theory (QFT) can be provided. In particular, we aim to understand the role of symmetries and localized defects on the entanglement properties. Those aspects display an intimate connection. First, via replica trick, the entropy is expressed via a replica permutation symmetry. Then, single defects break symmetries that are present in the corresponding homogeneous systems. Due to these reasons, the characterization of symmetry actions on subsystems can be viewed as a unifying framework for quantifying entanglement. We split this thesis into two parts. The first part is devoted to the symmetry-resolved entanglement. We first compute it in massive integrable field theories, giving analytical predictions for free fermions/bosons and the 3-state Potts model. For instance, we characterize the universal corrections to area law via form factor bootstrap. We also study excited states of field theories, and we characterize both low-energy states of Conformal Field Theory (CFT) and quasi-particle states. The second part deals with quantum correlations across defects. We provide a CFT framework to compute Rényi entropy and negativity in the ground state of conformal junctions, giving analytical predictions for free fermions and bosons. We find logarithmic growth of the entanglement measures above, and we relate the universal prefactor to the boundary conditions, encoded in a boundary scattering matrix. Then, we focus on the entanglement dynamics, and we study the time evolution of correlations for two paradigmatic classes of quench protocols. For a global quench, we find a linear growth of entropy across the defect whose slope is computed both in CFT, for thermal states, and for free fermions, via a modification of the quasi-particle picture. For a domain wall melting protocol, also known as inhomogeneous quench, we find that, while local observables are easily understood via hydrodynamics and local generalized Gibbs ensembles, the same is not true for the entropy. In contrast, by employing the quasi-particle picture, we provide precise predictions for the dynamics of entanglement, revealing the influence of long-range correlations between points that are specular with respect to the defect.