arXiv subjects

Bruno Bertini

Publications and source records attributed to Bruno Bertini.

At least 73 records · Page 4Linked to original sources

arXiv2020

Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits

The entanglement in operator space is a well established measure for the complexity of the quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case. Here we study local-operator entanglement in dual-unitary quantum circuits, a class of "statistically solvable" quantum circuits that we recently introduced. We identify a class of "completely chaotic" dual-unitary circuits where the local-operator entanglement grows linearly and we provide a conjecture for its asymptotic behaviour which is in excellent agreement with the numerical results. Interestingly, our conjecture also predicts a "phase transition" in the slope of the local-operator entanglement when varying the parameters of the circuits.

cond-mat.stat-mech↗

arXiv2020

Operator Entanglement in Local Quantum Circuits II: Solitons in Chains of Qubits

We provide exact results for the dynamics of local-operator entanglement in quantum circuits with two-dimensional wires featuring ultralocal solitons, i.e. single-site operators which, up to a phase, are simply shifted by the time evolution. We classify all circuits allowing for ultralocal solitons and show that only dual-unitary circuits can feature moving ultralocal solitons. Then, we rigorously prove that if a circuit has an ultralocal soliton moving to the left (right), the entanglement of local operators initially supported on even (odd) sites saturates to a constant value and its dynamics can be computed exactly. Importantly, this does not bound the growth of complexity in chiral circuits, where solitons move only in one direction, say to the left. Indeed, in this case we observe numerically that operators on the odd sublattice have unbounded entanglement. Finally, we present a closed-form expression for the local-operator entanglement entropies in circuits with ultralocal solitons moving in both directions. Our results hold irrespectively of integrability.

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arXiv2020

Exact dynamics in dual-unitary quantum circuits

We consider the class of dual-unitary quantum circuits in $1+1$ dimensions and introduce a notion of ``solvable'' matrix product states (MPSs), defined by a specific condition which allows us to tackle their time evolution analytically. We provide a classification of the latter, showing that they include certain MPSs of arbitrary bond dimension, and study analytically different aspects of their dynamics. For these initial states, we show that while any subsystem of size $\ell$ reaches infinite temperature after a time $t\propto \ell$, irrespective of the presence of conserved quantities, the light-cone of two-point correlation functions displays qualitatively different features depending on the ergodicity of the quantum circuit, defined by the behavior of infinite-temperature dynamical correlation functions. Furthermore, we study the entanglement spreading from such solvable initial states, providing a closed formula for the time evolution of the entanglement entropy of a connected block. This generalizes recent results obtained in the context of the self-dual kicked Ising model. By comparison, we also consider a family of non-solvable initial mixed states depending on one real parameter $β$, which, as $β$ is varied from zero to infinity, interpolate between the infinite temperature density matrix and arbitrary initial pure product states. We study analytically their dynamics for small values of $β$, and highlight the differences from the case of solvable MPSs.

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arXiv2019

Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions

We consider a class of quantum lattice models in $1+1$ dimensions represented as local quantum circuits that enjoy a particular "dual-unitarity" property. In essence, this property ensures that both the evolution "in time" and that "in space" are given in terms of unitary transfer matrices. We show that for this class of circuits, generically non-integrable, one can compute explicitly all dynamical correlations of local observables. Our result is exact, non-pertubative, and holds for any dimension $d$ of the local Hilbert space. In the minimal case of qubits ($d = 2$) we also present a classification of all dual-unitary circuits which allows us to single out a number of distinct classes for the behaviour of the dynamical correlations. We find "non-interacting" classes, where all correlations are preserved, the ergodic and mixing one, where all correlations decay, and, interestingly, also classes that are are both interacting and non-ergodic.

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arXiv2019

Entanglement evolution and generalised hydrodynamics: interacting integrable systems

We investigate the dynamics of bipartite entanglement after the sudden junction of two leads in interacting integrable models. By combining the quasiparticle picture for the entanglement spreading with Generalised Hydrodynamics we derive an analytical prediction for the dynamics of the entanglement entropy between a finite subsystem and the rest. We find that the entanglement rate between the two leads depends only on the physics at the interface and differs from the rate of exchange of thermodynamic entropy. This contrasts with the behaviour in free or homogeneous interacting integrable systems, where the two rates coincide.

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arXiv2019

Transport in the sine-Gordon field theory: from generalized hydrodynamics to semiclassics

The semiclassical approach introduced by Sachdev and collaborators proved to be extremely successful in the study of quantum quenches in massive field theories, both in homogeneous and inhomogeneous settings. While conceptually very simple, this method allows one to obtain analytic predictions for several observables when the density of excitations produced by the quench is small. At the same time, a novel generalized hydrodynamic (GHD) approach, which captures exactly many asymptotic features of the integrable dynamics, has recently been introduced. Interestingly, also this theory has a natural interpretation in terms of semiclassical particles and it is then natural to compare the two approaches. This is the objective of this work: we carry out a systematic comparison between the two methods in the prototypical example of the sine-Gordon field theory. In particular, we study the "bipartitioning protocol" where the two halves of a system initially prepared at different temperatures are joined together and then left to evolve unitarily with the same Hamiltonian. We identify two different limits in which the semiclassical predictions are analytically recovered from GHD: a particular non-relativistic limit and the low temperature regime. Interestingly, the transport of topological charge becomes sub-ballistic in these cases. Away from these limits we find that the semiclassical predictions are only approximate and, in contrast to the latter, the transport is always ballistic. This statement seems to hold true even for the so-called "hybrid" semiclassical approach, where finite time DMRG simulations are used to describe the evolution in the internal space.

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arXiv2019

Entanglement spreading in a minimal model of maximal many-body quantum chaos

The spreading of entanglement in out-of-equilibrium quantum systems is currently at the centre of intense interdisciplinary research efforts involving communities with interests ranging from holography to quantum information. Here we provide a constructive and mathematically rigorous method to compute the entanglement dynamics in a class of "maximally chaotic", periodically driven, quantum spin chains. Specifically, we consider the so called "self-dual" kicked Ising chains initialised in a class of separable states and devise a method to compute exactly the time evolution of the entanglement entropies of finite blocks of spins in the thermodynamic limit. Remarkably, these exact results are obtained despite the models considered are maximally chaotic: their spectral correlations are described by the circular orthogonal ensemble of random matrices on all scales. Our results saturate the so called "minimal cut" bound and are in agreement with those found in the contexts of random unitary circuits with infinite-dimensional local Hilbert space and conformal field theory. In particular, they agree with the expectations from both the quasiparticle picture, which accounts for the entanglement spreading in integrable models, and the minimal membrane picture, recently proposed to describe the entanglement growth in generic systems. Based on a novel "duality-based" numerical method, we argue that our results describe the entanglement spreading from any product state at the leading order in time when the model is non-integrable.

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arXiv2019

Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos

The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological success, a general mechanism explaining the emergence of RMT without reference to semiclassical concepts is still missing. Here we provide the example of a quantum many-body system with no semiclassical limit (no large parameter) where the emergence of RMT spectral correlations is proven exactly. Specifically, we consider a periodically driven Ising model and write the Fourier transform of spectral density's two-point function, the spectral form factor, in terms of a partition function of a two-dimensional classical Ising model featuring a space-time duality. We show that the self-dual cases provide a minimal model of many-body quantum chaos, where the spectral form factor is demonstrated to match RMT for all values of the integer time variable $t$ in the thermodynamic limit. In particular, we rigorously prove RMT form factor for odd $t$, while we formulate a precise conjecture for even $t$. The results imply ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization. Our method provides a novel route for obtaining exact nonperturbative results in non-integrable systems.

nlin.CD↗

arXiv2018

Spin-charge separation effects in the low-temperature transport of 1D Fermi gases

We study the transport properties of a one-dimensional spinful Fermi gas, after junction of two semi-infinite sub-systems held at different temperatures. The ensuing dynamics is studied by analysing the space-time profiles of local observables emerging at large distances $x$ and times $t$, as a function of $ζ= x/t$. At equilibrium, the system displays two distinct species of quasi-particles, naturally associated with different physical degrees of freedom. By employing the generalised hy- drodynamic approach, we show that when the temperatures are finite no notion of separation can be attributed to the quasi-particles. In this case the profiles can not be qualitatively distinguished by those associated to quasi-particles of a single species that can form bound states. On the contrary, signatures of separation emerge in the low-temperature regime, where two distinct characteristic ve- locities appear. In this regime, we analytically show that the profiles display a piece-wise constant form and can be understood in terms of two decoupled Luttinger liquids.

cond-mat.quant-gas↗

arXiv2018

Entanglement evolution and generalised hydrodynamics: noninteracting systems

The large-scale properties of homogeneous states after quantum quenches in integrable systems have been successfully described by a semiclassical picture of moving quasiparticles. Here we consider the generalisation for the entanglement evolution after an inhomogeneous quench in noninteracting systems in the framework of generalised hydrodynamics. We focus on the protocol where two semi-infinite halves are initially prepared in different states and then joined together, showing that a proper generalisation of the quasiparticle picture leads to exact quantitative predictions. If the system is initially prepared in a quasistationary state, we find that the entanglement entropy is additive and it can be computed by means of generalised hydrodynamics. Conversely, additivity is lost when the initial state is not quasistationary; yet the entanglement entropy in the large-scale limit can be exactly predicted in the quasiparticle picture, provided that the initial state is low entangled.

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arXiv2018

Entanglement and diagonal entropies after a quench with no pair structure

A typical working condition in the study of quantum quenches is that the initial state produces a distribution of quasiparticle excitations with an opposite-momentum-pair structure. In this work we investigate the dynamical and stationary properties of the entanglement entropy after a quench from initial states which do not have such structure: instead of pairs of excitations they generate $ν$-plets of correlated excitations with $ν>2$. Our study is carried out focusing on a system of non-interacting fermions on the lattice. We study the time evolution of the entanglement entropy showing that the standard semiclassical formula is not applicable. We propose a suitable generalisation which correctly describes the entanglement entropy evolution and perfectly matches numerical data. We finally consider the relation between the thermodynamic entropy of the stationary state and the diagonal entropy, showing that when there is no pair structure their ratio depends on the details of the initial state and lies generically between $1/2$ and $1$.

cond-mat.stat-mech↗

arXiv2018

Universal broadening of the light cone in low-temperature transport

We consider the low-temperature transport properties of critical one-dimensional systems which can be described, at equilibrium, by a Luttinger liquid. We focus on the prototypical setting where two semi-infinite chains are prepared in two thermal states at small but different temperatures and suddenly joined together. At large distances $x$ and times $t$, conformal field theory characterizes the energy transport in terms of a single light cone spreading at the sound velocity $v$. Energy density and current take different constant values inside the light cone, on its left, and on its right, resulting in a three-step form of the corresponding profiles as a function of $ζ=x/t$. Here, using a non-linear Luttinger liquid description, we show that for generic observables this picture is spoiled as soon as a non-linearity in the spectrum is present. In correspondence of the transition points $x/t=\pm v$ a novel universal region emerges at infinite times, whose width is proportional to the temperatures on the two sides. In this region, expectation values have a different temperature dependence and show smooth peaks as a function of $ζ$. We explicitly compute the universal function describing such peaks. In the specific case of interacting integrable models, our predictions are analytically recovered by the generalized hydrodynamic approach.

cond-mat.stat-mech↗

arXiv2018

Low-Temperature Transport in Out-of-Equilibrium XXZ Chains

We study the low-temperature transport properties of out-of-equilibrium XXZ spin-$1/2$ chains. We consider the protocol where two semi-infinite chains are prepared in two thermal states at small but different temperatures and suddenly joined together. We focus on the qualitative and quantitative features of the profiles of local observables, which at large times $t$ and distances $x$ from the junction become functions of the ratio $ζ=x/t$. By means of the generalized hydrodynamic equations, we analyse the rich phenomenology arising by considering different regimes of the phase diagram. In the gapped phases, variations of the profiles are found to be exponentially small in the temperatures but described by non-trivial functions of $ζ$. We provide analytical formulae for the latter, which give accurate results also for small but finite temperatures. In the gapless regime, we show how the three-step conformal predictions for the profiles of energy density and energy current are naturally recovered from the hydrodynamic equations. Moreover, we also recover the recent non-linear Luttinger liquid predictions for low-temperature transport: universal peaks of width $Δζ\propto T$ emerge at the edges of the light cone in the profiles of generic observables. Such peaks are described by the same function of $ζ$ for all local observables.

cond-mat.stat-mech↗

arXiv2017

Quantum Quench in the Infinitely Repulsive Hubbard Model: the Stationary State

We use the Quench Action approach to study the non-equilibrium dynamics after a quantum quench in the Hubbard model in the limit of infinite interaction. We identify a variety of low-entangled initial states for which we can directly compute the overlaps with the Hamiltonian's eigenstates. For these initial states, we analytically find the rapidity distributions of the stationary state characterising the expectation values of all local observables. Some of the initial states considered are not reflection symmetric and lead to non-symmetric rapidity distributions. To study such cases, we have to introduce a generalised form for the reduced entropy which measures the entropy restricted to states with non-zero overlap. The initial states considered are of direct experimental realisability and also represent ideal candidates for studying non-equilibrium dynamics in the Hubbard model for finite interactions.

cond-mat.stat-mech↗

arXiv2017

Transport in out-of-equilibrium XXZ chains: non-ballistic behavior and correlation functions

We consider the nonequilibrium protocol where two semi-infinite gapped XXZ chains, initially prepared in different equilibrium states, are suddenly joint together. At large times, a generalized hydrodynamic description applies, according to which the system can locally be represented by space- and time- dependent stationary states. The magnetization displays an unusual behavior: depending on the initial state, its profile may exhibit abrupt jumps that can not be predicted directly from the standard hydrodynamic equations and which signal non-ballistic spin transport. We ascribe this phenomenon to the structure of the local conservation laws and make a prediction for the exact location of the jumps. We find that the jumps propagate at the velocities of the heaviest quasiparticles. By means of tDMRG simulations we show that our theory yields a complete description of the long-time steady profiles of conserved charges, currents, and local correlations.

cond-mat.stat-mech↗

arXiv2017

Exact solution for the quench dynamics of a nested integrable system

Integrable models provide an exact description for a wide variety of physical phenomena. For example nested integrable systems contain different species of interacting particles with a rich phenomenology in their collective behavior, which is the origin of the unconventional phenomenon of spin-charge separation. So far, however, most of the theoretical work in the study of non-equilibrium dynamics of integrable systems has been focusing on models with an elementary (i.e., not nested) Bethe ansatz. In this work we explicitly investigate quantum quenches in nested integrable systems, by generalizing the application of the Quench Action approach. Specifically, we consider the spin-$1$ Lai-Sutherland model, described, in the thermodynamic limit, by the theory of two different species of Bethe-ansatz particles, each one forming an infinite number of bound states. We focus on the situation where the quench dynamics starts from a simple matrix product state for which the overlaps with the eigenstates of the Hamiltonian are known. We fully characterize the post-quench steady state and perform several consistency checks for the validity of our results. Finally, we provide predictions for the propagation of entanglement and mutual information after the quench, which can be used as signature of the quasi-particle content of the model.

cond-mat.stat-mech↗

arXiv2016

Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents

We consider the non-equilibrium time evolution of piecewise homogeneous states in the XXZ spin-1/2 chain, a paradigmatic example of an interacting integrable model. The initial state can be thought as the result of joining chains with different global properties. Through dephasing, at late times the state becomes locally equivalent to a stationary state which explicitly depends on position and time. We propose a kinetic theory of elementary excitations and derive a continuity equation which fully characterizes the thermodynamics of the model. We restrict ourselves to the gapless phase and consider cases where the chains are prepared: 1) at different temperatures; 2) in the ground state of two different models; 3) in the "domain wall" state. We find excellent agreement (any discrepancy is within the numerical error) between theoretical predictions and numerical simulations of time evolution based on TEBD algorithms. As a corollary, we unveil an exact expression for the expectation values of the charge currents in a generic stationary state.

cond-mat.stat-mech↗

arXiv2016

Approximate light cone effects in a non-relativistic quantum field theory after a local quench

We study the spreading of correlations after a local quench in a non-relativistic quantum field theory. We focus on noninteracting non-relativistic fermions and study the time evolution after two identical systems in their ground states are suddenly joined together with a localized impurity at the junction. We find that, even if the quasi-particles of the system have unbounded dispersion, the correlations show light cone effects. We carry out a detailed study of these effects by developing an accurate asymptotic expansion of the two-point function and determining exactly the density of particles at any time after the quench. In particular, we find that the width of the light cone region is $\propto t^{1/2}$. The structure of correlations, however, does not show a pure light cone form - "superluminal corrections" are much larger than in the bounded-dispersion case. These findings can be explained by inspecting the structure of excitations generated by the initial state. We show that a similar picture also emerges in the presence of a harmonic trapping potential and when more than two systems are suddenly joined at a single point.

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