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.Comment: 30 pages, 6 figure

Nonequilibrium Quantum States of Matter

Among the few lessons that Physics teaches us on a daily basis, one in particular is hard to miss: thermalization processes are ubiquitous. With the same degree of certainty, one can predict an apple to fall onto the ground if its stalk is cut, or a cup of tea to be found cold if left on the table for too long. In this respect, generic classical and quantum mechanical systems seem to behave in the same way. This is also one of the main difficulties in the experimental observation of the bizarre effects predicted from the quantum theory, as several of these are known to disappear at finite temperatures. It has been known for a long time that thermalization is associated with \u201cchaotic\u201d behavior at the microscopic level, but recently systematic efforts, both theoretical and experimental, have allowed us to understand its mechanisms at an unprecedented level of accuracy. Indeed, it has been realized that thermalization generally occurs also in isolated systems, where the absence of interactions with the environment allows for first principle theoretical investigations1. In this case thermalization takes place locally, as the whole system acts as a thermal bath for its own subsystems. Theoretical research has been motivated and paralleled by exciting experimental progress in cold-atom physics, which has provided robust, nearly ideal realizations of several theoretical models2. As an established piece of knowledge, recent research has now confirmed that two outstanding exceptions actually exist in the quantum realm, where thermalization is prevented to occur. These are many-body localized systems3, where disorder plays a crucial role, and integrable ones, which are protected by the existence of higher conservation laws. By their own nature, these systems display exceptional non-equilibrium features, which are not washed out by the onset of thermalization as relaxation occurs. At the core of the present thesis lie the remarkable properties of integrable systems out of equilibrium, with a special attention to physical effects which could be observed in cold-atom realizations. The work collected here is part of a large theoretical effort4 which in the past decade has focused on the study of relatively simple protocols to bring a quantum system out of equilibrium, such as the so-called quantum quench. In these \u201ctheoretical laboratories\u201d it has been possible to provide quantitative predictions which helped us to develop an intuition on general questions regarding non-equilibrium and thermalization processes. Furthermore, in some cases these studies have highlighted interesting physical effects which could be directly probed experimentally within cold-atom settings. The contribution of the present thesis is two-fold. On the one hand, we have developed new technical tools for the study of integrable systems out of equilibrium, and for the computation of measurable physical quantities such as correlation functions. The techniques employed are mainly analytical and rooted within the mathematical structures of integrability. On the other hand, we have singled out physically relevant nonequilibrium situations where exotic, non-thermal states of matter emerge after relaxation occurs, and provided quantitative analytical predictions in these cases. A comprehensive discussion of the motivations and results of our work will be presented in Chapter 1, where we provide a complete overview of this thesis and discuss the organization of its content

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 R\'enyi-$2$ 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.Comment: 29 pages, 7 figure

Stabilizer entropies and nonstabilizerness monotones

We study different aspects of the stabilizer entropies (SEs) and compare them against known nonstabilizerness monotones such as the min-relative entropy and the robustness of magic. First, by means of explicit examples, we show that, for R\'enyi index $0\leq n<2$, the SEs are not monotones with respect to stabilizer protocols which include computational-basis measurements, not even when restricting to pure states (while the question remains open for $n\geq 2$). Next, we show that, for any R\'enyi index, the SEs do not satisfy a strong monotonicity condition with respect to computational-basis measurements. We further study SEs in different classes of many-body states. We compare the SEs with other measures, either proving or providing numerical evidence for inequalities between them. Finally, we discuss exact or efficient tensor-network numerical methods to compute SEs of matrix-product states (MPSs) for large numbers of qubits. In addition to previously developed exact methods to compute the R\'enyi SEs, we also put forward a scheme based on perfect MPS sampling, allowing us to compute efficiently the von Neumann SE for large bond dimensions.Comment: 14 pages, 5 figure

Quantum Cellular Automata, Tensor Networks, and Area Laws

Quantum Cellular Automata are unitary maps that preserve locality and respect causality. We identify them, in any dimension, with simple tensor networks (PEPU) whose bond dimension does not grow with the system size. As a result, they satisfy an area law for the entanglement entropy they can create. We define other classes of non-unitary maps, the so-called quantum channels, that either respect causality or preserve locality. We show that, whereas the latter obey an area law for the amount of quantum correlations they can create, as measured by the quantum mutual information, the former may violate it. We also show that neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size.Comment: 5+2 pages, 2 figures; v3: minor revisio

Integrability of 1D1D Lindbladians from operator-space fragmentation

We introduce families of one-dimensional Lindblad equations describing open many-particle quantum systems that are exactly solvable in the following sense: $(i)$ the space of operators splits into exponentially many (in system size) subspaces that are left invariant under the dissipative evolution; $(ii)$ the time evolution of the density matrix on each invariant subspace is described by an integrable Hamiltonian. The prototypical example is the quantum version of the asymmetric simple exclusion process (ASEP) which we analyze in some detail. We show that in each invariant subspace the dynamics is described in terms of an integrable spin-1/2 XXZ Heisenberg chain with either open or twisted boundary conditions. We further demonstrate that Lindbladians featuring integrable operator-space fragmentation can be found in spin chains with arbitrary local physical dimension.Comment: 8 pages, no figures; v2: minor revisio

Quench quantistici in sistemi integrabili e Bethe ansatz

Negli ultimi anni alcuni esperimenti fondamentali nel campo degli atomi freddi hanno dimostrato la possibilità di realizzare in laboratorio sistemi quantistici a molti corpi quasi perfettamente isolati e di osservarne l'evoluzione temporale coerente. Questi lavori sperimentali hanno rinnovato l'interesse teorico nello studio dei problemi aperti inerenti alla dinamica del non equilibrio in sistemi isolati, riportando all'attenzione, in particolare, le domande riguardanti la caratterizzazione del raggiungimento dello stato stazionario e della sua descrizione. In questo quadro, il protocollo di quench, cioè la procedura di portare un sistema lontano dall'equilibrio attraverso una variazione improvvisa di un parametro nella sua hamiltoniana, ha ricevuto molta attenzione: da una parte per la sua relativa semplicità, dall'altra perché permette diverse realizzazioni sperimentali. Un filone di ricerca che negli ultimi anni ha raccolto sforzi significativi si è concentrato sullo studio di quench nei sistemi integrabili, cioè sistemi esattamente risolubili; sia per la necessità di chiarire il loro carattere speciale, sia per la possibilità di studiarli utilizzando tecniche analitiche esatte fornendo un controllo teorico impossibile in casi generici. La mia tesi si inserisce in questo discorso, focalizzando in particolare l'attenzione sullo studio di quench in sistemi integrabili risolubili tramite il cosiddetto Bethe ansatz, una tecnica introdotta da Bethe nel 1931 per risolvere il modello isotropo di Heisenberg (o catena di spin XXX); questi sistemi rappresentano modelli a molti corpi completamente risolubili genuinamente interagenti (non mappabili cioè in sistemi di particelle libere). Nello specifico, l'oggetto di studio della mia tesi sono stati il modello di Lieb-Liniger e quello anisotropo di Heisenberg (conosciuto anche come catena XXZ); il primo è un modello continuo unidimensionale di bosoni con interazione di contatto repulsiva introdotto negli anni Sessanta, mentre il secondo consiste in un reticolo discreto di spin unidimensionale con accoppiamento a primi vicini. Questi sistemi sono introdotti nei primi capitoli, e le hamiltoniane corrispondenti vengono diagonalizzate attraverso Bethe ansatz. Nello studio dell'evoluzione temporale del valore di aspettazione sullo stato del sistema di un'osservabile d'interesse a seguito di un quench, un approccio utilizzato è quello dell'espansione in fattori di forma; i building blocks necessari per usare questo metodo sono gli autostati normalizzati del sistema, gli elementi di matrice dell'osservabile che si sta analizzando e i prodotti scalari (overlap) tra autostati di hamiltoniane corrispondenti a due diversi valori del parametro di quench. Nei sistemi integrabili è possibile calcolare la norma degli autostati e gli elementi di matrice attraverso il cosiddetto algebraic Bethe ansatz, un formalismo potente introdotto negli anni Settanta e che è presentato nella tesi. Grazie a questa tecnica, si possono ottenere formule compatte per queste quantità in termini del determinante di matrici le cui dimensioni scalano linearmente con quelle del sistema, e questo consente di effettuare calcoli esatti anche nel limite termodinamico. Per contro, in generale il calcolo degli overlap è tipicamente molto difficile; il problema rimane aperto per sistemi integrabili generici ed è stato preso in considerazione nella mia tesi per i modelli di Lieb-Liniger e XXZ. Il lavoro originale della tesi si divide principalmente in due parti. Nella prima, proponiamo un metodo per calcolare gli overlap nel modello XXZ tra autostati dell'hamiltoniana e stati fattorizzati, attraverso la derivazione di formule ricorsive. Le formule ricorsive per gli overlap sono ottenute utilizzando in modo sistematico tecniche di algebraic Bethe ansatz. Questo approccio è generalizzato per un caso particolare al modello di Lieb-Liniger, ed è naturalmente generalizzabile a modelli integrabili su reticolo. Partendo da una formula ricorsiva dimostrata, otteniamo una formula compatta per gli overlap tra autostati del modello XXZ e il cosiddetto stato di Néel, ritrovando in modo più semplice un risultato recentemente dimostrato in letteratura. Nella seconda parte originale della tesi, studiamo un quench nel modello di Lieb-Liniger. Il problema generico corrispondente ad una variazione arbitraria della costante d'interazione c che compare nell'hamiltoniana rimane aperto e difficile da attaccare, per l'assenza di una formula di overlap tra autostati di hamiltoniane con parametri d'interazione diversi. Nella tesi si è considerato il problema più semplice (ma ancora non studiato) di variazione al prim'ordine del parametro c, con c arbitrario, ottenendo una formula per l'overlap in questo caso, sempre attraverso l'uso dell'algebraic Bethe ansatz. Questa formula ha permesso uno studio numerico dell'evoluzione temporale di alcune quantità del sistema; abbiamo in particolare considerato il Loschmidt echo, il valore di aspettazione dell'osservabile $\psi^{\dagger}(x)\psi^{\dagger}(x)\psi(x)\psi(x)$ e la funzione di correlazione a due punti a tempi uguali, confrontando i risultati con uno studio numerico precedente sul modello di Lieb-Liniger in cui l'hamiltoniana dopo il quench è quella libera (in cui cioè la costante d'interazione viene improvvisamente posta uguale a zero). I risultati esatti presentati nella tesi riguardanti le formule analitiche degli overlap nel modello XXZ e quella corrispondente al quench al prim'ordine nel modello di Lieb-Liniger, si uniscono a quelli comparsi nella letteratura dell'ultimo anno, che costituiscono un punto di partenza verso la soluzione del problema generale di quench in sistemi integrabili. Trovare un formalismo che permetta di ricavare formule compatte per gli overlap tra autostati di hamiltoniane corrispondenti a parametri di quench diversi per sistemi integrabili generici renderebbe possibile il controllo analitico per molti problemi teorici, fornendo calcoli esatti per una grande classe di potenziali esperimenti

Enhanced entanglement negativity in boundary-driven monitored fermionic chains

We investigate entanglement dynamics in continuously monitored open quantum systems featuring current-carrying nonequilibrium states. We focus on a prototypical one-dimensional model of boundary-driven noninteracting fermions with monitoring of the local density, whose average Lindblad dynamics features a well-studied ballistic to diffusive crossover in transport. Here we analyze the dynamics of the fermionic negativity, mutual information, and purity along different quantum trajectories. We show that monitoring this boundary-driven system enhances its entanglement negativity at long times, which otherwise decays to zero in the absence of measurements. This result is in contrast with the case of unitary evolution where monitoring suppresses entanglement production. For small values of gamma, the stationary-state negativity shows a logarithmic scaling with system size, transitioning to an area-law scaling as gamma is increased beyond a critical value. Similar critical behavior is found in the mutual information, while the late-time purity shows no apparent signature of a transition, being O(1) for all values of gamma. Our work unveils the double role of weak monitoring in current-driven open quantum systems, simultaneously damping transport and enhancing entanglement

Exact local correlations and full counting statistics for arbitrary states of the one-dimensional interacting Bose gas

We derive exact analytic expressions for the n-body local correlations in the one-dimensional Bose gas with contact repulsive interactions (Lieb-Liniger model) in the thermodynamic limit. Our results are valid for arbitrary states of the model, including ground and thermal states, stationary states after a quantum quench, and nonequilibrium steady states arising in transport settings. Calculations for these states are explicitly presented and physical consequences are critically discussed. We also show that the n-body local correlations are directly related to the full counting statistics for the particle-number fluctuations in a short interval, for which we provide an explicit analytic result

Universal broadening of the light cone in low-temperature transport

We consider the low-temperature transport properties of critical one-dimensional systems that 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 \u3b6=x/t. Here, using a nonlinear Luttinger liquid description, we show that for generic observables this picture is spoiled as soon as a nonlinearity in the spectrum is present. In correspondence of the transition points x/t=\ub1v, 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 \u3b6. 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