arXiv subjects

Bruno Bertini

Publications and source records attributed to Bruno Bertini.

At least 55 records · Page 3Linked to original sources

arXiv2022

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

Rényi entropies are conceptually valuable and experimentally relevant generalisations 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 characterisation 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.

cond-mat.stat-mech↗

arXiv2022

Integrable quenches in the Hubbard model

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.

cond-mat.stat-mech↗

arXiv2022

BBGKY Hierarchy and Generalised Hydrodynamics

We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally non-local and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solutions to the first of these conditions are the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model) and the Calogero-Sutherland potentials. We use our construction to show how Generalized Hydrodynamics (GHD) emerges from the Bogoliubov--Born--Green--Kirkwood--Yvon hierarchy, and argue that GHD in the weak interaction regime is robust under non-integrable perturbations.

cond-mat.stat-mech↗

arXiv2022

Exact Spectral Statistics in Strongly Localised Circuits

Since the seminal work of Anderson, localisation has been recognised as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea acquired even more prominence in the last decade as it has been argued that localisation -- dubbed many-body localisation (MBL) in this context -- can sometimes survive local interactions in the presence of sufficiently strong disorder. A conventional signature of localisation is in the statistical properties of the spectrum -- spectral statistics -- which differ qualitatively from those in the ergodic phase. Although features of the spectral statistics are routinely used as numerical diagnostics for localisation, they have never been derived from first principles in the presence of non-trivial interactions. Here we fill this gap and provide the example of a simple class of quantum many-body systems -- which we dub strongly localised quantum circuits -- that are interacting, localised, and where the spectral statistics can be characterised exactly. Furthermore, we show that these systems exhibit a cascade of three different regimes for spectral correlations depending on the energy scale: at small, intermediate, and large scales they behave as disconnected patches of three decreasing sizes. We argue that these features appear in generic MBL systems, with the difference that only at the smallest scale they do become Poissonian.

nlin.CD↗

arXiv2022

Duality between weak and strong interactions in quantum gases

In one dimensional quantum gases there is a well known "duality" between hard core bosons and non-interacting fermions. However, at the field theory level, no exact duality connecting strongly interacting bosons to weakly interacting fermions is known. Here we propose a solution to this long standing problem. Our derivation relies on regularizing the only point-like interaction between fermions in 1D that induces a discontinuity in the wave function proportional to its derivative. In contrast to all known regularisations our potential is weak for small interaction strengths. Crucially, this allows one to apply standard methods of diagrammatic perturbation theory to strongly interacting bosons. As a first application we compute the finite temperature spectral function of the Cheon-Shigehara model, the fermionic model dual to the celebrated Lieb-Liniger model.

quant-ph↗

arXiv2021

Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54

We study the out-of-equilibrium dynamics of the quantum cellular automaton Rule 54 using a time-channel approach. We exhibit a family of (non-equilibrium) product states for which we are able to describe exactly the full relaxation dynamics. We use this to prove that finite subsystems relax to a one-parameter family of Gibbs states. We also consider inhomogeneous quenches. Specifically, we show that when the two halves of the system are prepared in two different solvable states, finite subsystems at finite distance from the centre eventually relax to the non-equilibrium steady state (NESS) predicted by generalised hydrodynamics. To the best of our knowledge, this is the first exact description of the relaxation to a NESS in an interacting system and, therefore, the first independent confirmation of generalised hydrodynamics for an inhomogeneous quench.

cond-mat.stat-mech↗

arXiv2021

Generalized-Hydrodynamic approach to Inhomogeneous Quenches: Correlations, Entanglement and Quantum Effects

We give a pedagogical introduction to the Generalized Hydrodynamic approach to inhomogeneous quenches in integrable many-body quantum systems. We review recent applications of the theory, focusing in particular on two classes of problems: bipartitioning protocols and trap quenches, which represent two prototypical examples of broken translational symmetry in either the system initial state or post-quench Hamiltonian. We report on exact results that have been obtained for generic time-dependent correlation functions and entanglement evolution, and discuss in detail the range of applicability of the theory. Finally, we present some open questions and suggest perspectives on possible future directions.

cond-mat.stat-mech↗

arXiv2021

Entanglement Barriers in Dual-Unitary Circuits

After quantum quenches in many-body systems, finite subsystems evolve non-trivially in time, eventually approaching a stationary state. In typical situations, the reduced density matrix of a given subsystem begins and ends this endeavour as a low-entangled vector in the space of operators. This means that if its operator space entanglement initially grows (which is generically the case), it must eventually decrease, describing a barrier-shaped curve. Understanding the shape of this "entanglement barrier" is interesting for three main reasons: (i) it quantifies the dynamics of entanglement in the (open) subsystem; (ii) it gives information on the approximability of the reduced density matrix by means of matrix product operators; (iii) it shows qualitative differences depending on the type of dynamics undergone by the system, signalling quantum chaos. Here we compute exactly the shape of the entanglement barriers described by different Rényi entropies after quantum quenches in dual-unitary circuits initialised in a class of solvable matrix product states (MPS)s. We show that, for free (SWAP-like) circuits, the entanglement entropy behaves as in rational CFTs. On the other hand, for completely chaotic dual-unitary circuits it behaves as in holographic CFTs, exhibiting a longer entanglement barrier that drops rapidly when the subsystem thermalises. Interestingly, the entanglement spectrum is non-trivial in the completely chaotic case. Higher Rényi entropies behave in an increasingly similar way to rational CFTs, such that the free and completely chaotic barriers are identical in the limit of infinite replicas (i.e. for the so called min-entropy). We also show that, upon increasing the bond dimension of the MPSs, the barrier maintains the same shape. It simply shifts to the left to accommodate for the larger initial entanglement.

cond-mat.stat-mech↗

arXiv2021

Thermalisation Dynamics and Spectral Statistics of Extended Systems with Thermalising Boundaries

We study thermalisation and spectral properties of extended systems connected, through their boundaries, to a thermalising Markovian bath. Specifically, we consider periodically driven systems modelled by brickwork quantum circuits where a finite section (block) of the circuit is constituted by arbitrary local unitary gates while its complement, which plays the role of the bath, is dual-unitary. We show that the evolution of local observables and the spectral form factor are determined by the same quantum channel, which we use to characterise the system's dynamics and spectral properties. In particular, we identify a family of fine-tuned quantum circuits -- which we call strongly non-ergodic -- that fails to thermalise even in this controlled setting, and, accordingly, their spectral form factor does not follow the random matrix theory prediction. We provide a set of necessary conditions on the local quantum gates that lead to strong non-ergodicity, and in the case of qubits, we provide a complete classification of strongly non-ergodic circuits. We also study the opposite extreme case of circuits that are almost dual-unitary, i.e., where thermalisation occurs with the fastest possible rate. We show that, in these systems, local observables and spectral form factor approach respectively thermal values and random matrix theory prediction exponentially fast. We provide a perturbative characterisation of the dynamics and, in particular, of the time-scale for thermalisation.

cond-mat.stat-mech↗

arXiv2021

Real-Time Evolution in the Hubbard Model with Infinite Repulsion

We consider the real-time evolution of the Hubbard model in the limit of infinite coupling. In this limit the Hamiltonian of the system is mapped into a number-conserving quadratic form of spinless fermions, i.e. the tight binding model. The relevant local observables, however, do not transform well under this mapping and take very complicated expressions in terms of the spinless fermions. Here we show that for two classes of interesting observables the quench dynamics from product states in the occupation basis can be determined exactly in terms of correlations in the tight-binding model. In particular, we show that the time evolution of any function of the total density of particles is mapped directly into that of the same function of the density of spinless fermions in the tight-binding model. Moreover, we express the two-point functions of the spin-full fermions at any time after the quench in terms of correlations of the tight binding model. This sum is generically very complicated but we show that it leads to simple explicit expressions for the time evolution of the densities of the two separate species and the correlations between a point at the boundary and one in the bulk when evolving from the so called generalised nested Néel states.

cond-mat.stat-mech↗

arXiv2021

Entanglement dynamics in Rule 54: Exact results and quasiparticle picture

We study the entanglement dynamics generated by quantum quenches in the quantum cellular automaton Rule $54$. We consider the evolution from a recently introduced class of solvable initial states. States in this class relax (locally) to a one-parameter family of Gibbs states and the thermalisation dynamics of local observables can be characterised exactly by means of an evolution in space. Here we show that the latter approach also gives access to the entanglement dynamics and derive exact formulas describing the asymptotic linear growth of all Rényi entropies in the thermodynamic limit and their eventual saturation for finite subsystems. While in the case of von Neumann entropy we recover exactly the predictions of the quasiparticle picture, we find no physically meaningful quasiparticle description for other Rényi entropies. Our results apply to both homogeneous and inhomogeneous quenches.

cond-mat.stat-mech↗

arXiv2021

Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits

We investigate a class of brickwork-like quantum circuits on chains of $d-$level systems (qudits) that share the so-called `dual unitarity' property. Namely, these systems generate unitary dynamics not only when propagating in the time direction, but also when propagating in the space direction. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over ${\rm SU}(d)$, e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time $t$ in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of $t$ sites. For general dual unitary circuits of qubits $(d=2)$ and a family of their extensions to higher $d>2$, we provide explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all $t$, if only the local 2-qubit gates are different from a SWAP (non-interacting gate). We discuss and partly prove possible extensions of our results to a weaker (more singular) forms of disorder averaging, as well as to quantum circuits with time-reversal symmetry, and to computing higher moments of the spectral form factor.

math-ph↗

arXiv2021

Exact thermalization dynamics in the "Rule 54" Quantum Cellular Automaton

We study the out-of-equilibrium dynamics of the quantum cellular automaton known as "Rule 54". For a class of low-entangled initial states, we provide an analytic description of the effect of the global evolution on finite subsystems in terms of simple quantum channels, which gives access to the full thermalization dynamics at the microscopic level. As an example, we provide analytic formulae for the evolution of local observables and Rényi entropies. We show that, in contrast to other known examples of exactly solvable quantum circuits, Rule 54 does not behave as a simple Markovian bath on its own parts, and displays typical non-equilibrium features of interacting integrable many-body quantum systems such as finite relaxation rate and interaction-induced dressing effects. Our study provides a rare example where the full thermalization dynamics can be solved exactly at the microscopic level.

cond-mat.stat-mech↗

arXiv2021

Chaos and Ergodicity in Extended Quantum Systems with Noisy Driving

We study the time evolution operator in a family of local quantum circuits with random fields in a fixed direction. We argue that the presence of quantum chaos implies that at large times the time evolution operator becomes effectively a random matrix in the many-body Hilbert space. To quantify this phenomenon we compute analytically the squared magnitude of the trace of the evolution operator -- the generalised spectral form factor -- and compare it with the prediction of Random Matrix Theory (RMT). We show that for the systems under consideration the generalised spectral form factor can be expressed in terms of dynamical correlation functions of local observables in the infinite temperature state, linking chaotic and ergodic properties of the systems. This also provides a connection between the many-body Thouless time $τ_{\rm th}$ -- the time at which the generalised spectral form factor starts following the random matrix theory prediction -- and the conservation laws of the system. Moreover, we explain different scalings of $τ_{\rm th}$ with the system size, observed for systems with and without the conservation laws.

cond-mat.stat-mech↗

arXiv2020

Statistics of the Spectral Form Factor in the Self-Dual Kicked Ising Model

We compute the full probability distribution of the spectral form factor in the self-dual kicked Ising model by providing an exact lower bound for each moment and verifying numerically that the latter is saturated. We show that at large enough times the probability distribution agrees exactly with the prediction of Random Matrix Theory if one identifies the appropriate ensemble of random matrices. We find that this ensemble is not the circular orthogonal one - composed of symmetric random unitary matrices and associated with time-reversal-invariant evolution operators - but is an ensemble of random matrices on a more restricted symmetric space (depending on the parity of the number of sites this space is either ${Sp(N)/U(N)}$ or ${O(2N)/{O(N)\!\times\!O(N)}}$). Even if the latter ensembles yield the same averaged spectral form factor as the circular orthogonal ensemble they show substantially enhanced fluctuations. This behaviour is due to a recently identified additional anti-unitary symmetry of the self-dual kicked Ising model.

nlin.CD↗

arXiv2020

Correlations in Perturbed Dual-Unitary Circuits: Efficient Path-Integral Formula

Interacting many-body systems with explicitly accessible spatio-temporal correlation functions are extremely rare, especially in the absence of integrability. Recently, we identified a remarkable class of such systems and termed them dual-unitary quantum circuits. These are brick-wall type local quantum circuits whose dynamics are unitary in both time and space. For these systems the spatio-temporal correlation functions are non-trivial only at the edge of the causal light cone and can be computed in terms of one-dimensional transfer matrices. Dual-unitarity, however, requires fine-tuning and the degree of generality of the observed dynamical features remained unclear. Here we address this question by introducing arbitrary perturbations of the local gates. Considering fixed perturbations, we prove that for a particular class of unperturbed elementary dual-unitary gates the correlation functions are still expressed in terms of one-dimensional transfer matrices. These matrices, however, are now contracted over generic paths connecting the origin to a fixed endpoint inside the causal light cone. The correlation function is given as a sum over all such paths. Our statement is rigorous in the "dilute limit", where only a small fraction of the gates is perturbed, and in the presence of random longitudinal fields, but we provide theoretical arguments and stringent numerical checks supporting its validity even in the clean case and when all gates are perturbed. As a byproduct, in the case of random longitudinal fields -- which turns out to be equivalent to certain classical Markov chains -- we find four types of non-dual-unitary(and non-integrable) interacting many-body systems where the correlation functions are exactly given by the path-sum formula.

cond-mat.stat-mech↗

arXiv2020

Prethermalisation and Thermalisation in the Entanglement Dynamics

We investigate the crossover of the entanglement entropy towards its thermal value in nearly integrable systems. We employ equation of motion techniques to study the entanglement dynamics in a lattice model of weakly interacting spinless fermions after a quantum quench. For weak enough interactions we observe a two-step relaxation of the entanglement entropies of finite subsystems. Initially the entropies follow a nearly integrable evolution, approaching the value predicted by the Generalized Gibbs Ensemble (GGE) of the unperturbed model. Then, they start a slow drift towards the thermal stationary value described by a standard Gibbs Ensemble (GE). While the initial relaxation to the GGE is independent of the interaction, the slow drift from GGE to GE values happens on time scales proportional to the inverse interaction squared. For asymptotically large times and subsystem sizes the dynamics of the entropies can be predicted using a modified quasiparticle picture that keeps track of the evolution of the fermionic occupations caused by the integrability breaking. This picture gives a quantitative description of the results as long as the integrability-breaking timescale is much larger than the one associated with the (quasi) saturation to the GGE. In the opposite limit the quasiparticle picture still provides the correct late-time behaviour, but it underestimates the initial slope of the entanglement entropy.

cond-mat.stat-mech↗

arXiv2020

Scrambling in Random Unitary Circuits: Exact Results

We study the scrambling of quantum information in local random unitary circuits by focusing on the tripartite information proposed by Hosur et al. We provide exact results for the averaged Rényi-$2$ tripartite information in two cases: (i) the local gates are Haar random and (ii) the local gates are dual-unitary and randomly sampled from a single-site Haar-invariant measure. We show that the latter case defines a one-parameter family of circuits, and prove that for a "maximally chaotic" subset of this family quantum information is scrambled faster than in the Haar-random case. Our approach is based on a standard mapping onto an averaged folded tensor network, that can be studied by means of appropriate recurrence relations. By means of the same method, we also revisit the computation of out-of-time-ordered correlation functions, re-deriving known formulae for Haar-random unitary circuits, and presenting an exact result for maximally chaotic random dual-unitary gates.

cond-mat.stat-mech↗