Local reversibility and entanglement structure of many-body ground states

The low-temperature physics of quantum many-body systems is largely governed by the structure of their ground states. Minimizing the energy of local interactions, ground states often reflect strong properties of locality such as the area law for entanglement entropy and the exponential decay of correlations between spatially separated observables. In this letter we present a novel characterization of locality in quantum states, which we call `local reversibility'. It characterizes the type of operations that are needed to reverse the action of a general disturbance on the state. We prove that unique ground states of gapped local Hamiltonian are locally reversible. This way, we identify new fundamental features of many-body ground states, which cannot be derived from the aforementioned properties. We use local reversibility to distinguish between states enjoying microscopic and macroscopic quantum phenomena. To demonstrate the potential of our approach, we prove specific properties of ground states, which are relevant both to critical and non-critical theories.Comment: 12 revtex pages, 2 pdf figs; minor changes, typos corrected. To be published in Quantum Science and Technolog

Quantum Quenches in Extended Systems

We study in general the time-evolution of correlation functions in a extended quantum system after the quench of a parameter in the hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d+1 dimensions. For d=1 this allows to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the gaussian (mean-field) approximation. These predictions are checked against the real-time evolution of some solvable models that allows also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long-time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments.Comment: 24 Pages, 4 figure

Quantum Information Theory in Condensed Matter Physics

2015 - 2016Inthe“standard”Gizburg-Landauapproach,aphasetransitionisintimately connected to a local order parameter, that spontaneously breaks some symmetries. In addition to the “traditional” symmetry-breaking ordered phases, a complex quantum system exhibits exotic phases, without classical counterpart, that can be described, for example, by introducing non-local order parameters that preserve symmetries. In this scenario, this thesis aims to shed light on open problems, such as the localdistinguishabilitybetweengroundstatesofasymmetry-breakingordered phase and the classification of one dimensional quantum orders, in terms of entanglement measures, in systems for which the Gizburg-Landau approach fails. In particular, I briefly introduce the basic tools that allow to understand the nature of entangled states and to quantify non-classical correlations. Therefore, I analyze the conjecture for which the maximally symmetry-breaking ground states (MSBGSs) are the most classical ones, and thus the only ones selected in real-world situations, among all the ground states of a symmetry-breaking ordered phase. I make the conjecture quantitatively precise, by proving that the MSBGSs are the only ones that: i) minimize pairwise quantum correlations, as measured by the quantum discord; ii) are always local convertible, by only applying LOCC transformations; iii) minimize the residual tangle, satisfying at its minimum the monogamy of entanglement. Moreover,Ianalyzehowevolvesthedistinguishability,afterasuddenchange of the Hamiltonian parameters. I introduce a quantitative measure of distinguishability, in terms of the trace distance between two reduced density matrices. Therefore, in the framework of two integrable models that falls in two different classes of symmetries, i.e. XY models in a transverse magnetic field and the N-cluster Ising models, I prove that the maximum of the distinguishability shows a time-exponential decay. Hence, in the limit of diverging time, all the informations about the particular initial ground state disappear, even if a system is integrable. Far away from the Gizburg-Landau scenario, I analyze a family of fullyanalyticalsolvableonedimensionalspin-1/2models,namedtheN-clustermodels in a transverse magnetic field. Regardless of the cluster size N + 2, these modelsexhibitaquantumphasetransition,thatseparatesaparamagneticphase from a cluster one. The cluster phase coresponds to a nematic ordered phase or a symmetry-protected topological ordered one, for even or odd N respectively. Using the Jordan-Wigner transformations, it is possible to diagonalize these models and derive all their spin correlation functions, with which reconstruct their entanglement properties. In particular, I prove that these models have only a non-vanishing bipartite entanglement, as measured by the concurrence, between spins at the endpoints of the cluster, for a magnetic field strong enough. Moreover, I introduce the minimal set of nonlinear ground-states functionals to detect all 1-D quantum orders for systems of spin-1/2 and fermions. I show that the von Neumann entanglement entropy distinguishes a critical systemfromanoncriticalone,becauseofthelogarithmicdivergenceataquantum critical point. The Schmidt gap detect the disorder of a system , because it saturates to a constant value in a paramagnetic phase and goes to zero otherwise. The mutual information, between two subsystems macroscopically separated, identifiesthesymmetry-breakingorderedphases,becauseofitsdependenceon the order parameters. The topological order phases, instead, via their deeply non-locality, can be characterized by analyzing all three functionals. [edited by author]In aggiunta alle tradizionali fasi ordinate con rottura spontanea di simmetria, ben descritte con un approccio alla Gizburg-Landau, dove una transizione di fase `e intimamente connessa alla rottura spontanea di qualche simmetria e ad un parametro d’ordine locale, un sistema quantistico presenta anche fasi esotiche,senzaanalogoclassico,chesonoperesempiocaratterizzatedaparametri d’ordine non locali, senza una necessaria rottura di simmetria. Partendo da questi presupposti, questa tesi si pone come obiettivo quello di fare luce su alcuni problemi ancora aperti, come la distinguibilit`a tra stati fondamentaliinsistemiquantisticiconrotturaspontaneadisimmetriaelaclassificazionedituttelefasipresentiinsistemiunidimensionalidispin-1/2efermioni, per i quali l’approccio alla Gizburg-Landau non fornisce una descrizione adeguata. Inparticolare,sid`aunaspiegazioneall’ipotesisecondolaqualeglistatifondamentali che rompono massimamente la simmetria sono quelli pi`u classici, e quindi selezionati dalla decoerenza dell’ambiente, tra tutti gli stati fondamentali,edenergeticamenteequivalenti,diunafaseordinataconrotturaspontanea di simmetria. Si dimostra, infatti, che gli stati che rompono massimamente la simmetria sono gli unici stati che soddisfano tre criteri di classicalit`a: i) minimizzano l’entanglement bipartito, come quantificato dalla discord; ii) sono gli uniciversocuituttiglialtristatifondamentalisonolocalmenteconvertibili,mediante LOCC; iii) minimizzano il tangle residuo, soddisfacendo al minimo la monogamia dell’entanglement. Viene analizzato, inoltre, come evolve la distinguibilit`a tra stati fondamentali, dopo un quench dei parametri Hamiltoniani. Dopo aver introdotto una misura quantitativa della distinguibilit`a, in termini della distanza tra due matrici densit`a ridotte, si dimostra, per due sistemi integrabili con diverse classi di simmetria, nel dettaglio il modello XY in campo magnetico e i modelli NclusterIsing,cheladistinguibilit`adecadeesponenzialmenteneltempoequindi, nel limite di tempi lunghi, tutte le informazioni sullo stato fondamentale di partenza si perdono, anche per sistemi integrabili, nei quali la termalizzazione non si verifica. LontanodalloscenarioGizburg-Landau,sianalizzaunafamigliadimodelli di spin-1/2 esattamente risolvibili, nel dettaglio i modelli N-cluster in campo magnetico, che mostrano una transizione tra una fase disordinata e una di tipo cluster, che pu`o essere nematica o topologica, rispettivamente per N pari o dispari. Usando le trasformazioni di Jordan-Wigner `e possibile diagonalizzare questi modelli, ricavare lo stato fondamentale, le funzioni di correlazione fermioniche e tutte le loro propriet`a di entanglement di. Si dimostra che questi modellinonhannoentanglementmultipartito,masoloentanglementbipartito, come misurato dalla concurrence, tra due spin alle estremit`a del cluster, per un campo magnetico sufficientemente intenso. Inoltre, sidimostrachel’entropiadivonNeumann,loSchmidtgapelamutualinformationrappresentanoilsetminimodifunzionalinonlinearidellamatrice densit`a ridotta, mediante le quali caratterizzare tutte le fasi presenti in sistemi unidimensionali di spin -1/2 e fermioni. In particolare, l’entropia di von Neumann caratterizza la criticalit`a del sistema, per la sua divergenza logaritmica al punto critico; lo Schmidt gap caratterizza il disordine di un sistema, perch´e satura ad un valore costante nelle fasi disordinate e va rapidamente a zero altrove; la mutual information cattura le fasi ordinate con rottura spontanea di simmetria, per le quali cio` e `e possibile definire un parametro d’ordine diverso da zero su un supporto finito. Le fasi topologiche, per via della loro natura fortemente non locale, necessitano di tutte e tre i funzionali per essere individuate. [a cura dell'autore]XV n.s

Dynamics of a Quantum Phase Transition and Relaxation to a Steady State

We review recent theoretical work on two closely related issues: excitation of an isolated quantum condensed matter system driven adiabatically across a continuous quantum phase transition or a gapless phase, and apparent relaxation of an excited system after a sudden quench of a parameter in its Hamiltonian. Accordingly the review is divided into two parts. The first part revolves around a quantum version of the Kibble-Zurek mechanism including also phenomena that go beyond this simple paradigm. What they have in common is that excitation of a gapless many-body system scales with a power of the driving rate. The second part attempts a systematic presentation of recent results and conjectures on apparent relaxation of a pure state of an isolated quantum many-body system after its excitation by a sudden quench. This research is motivated in part by recent experimental developments in the physics of ultracold atoms with potential applications in the adiabatic quantum state preparation and quantum computation.Comment: 117 pages; review accepted in Advances in Physic

Quantum quenches in Zn symmetric spin chains: an iTEBD study

Lo studio della dinamica dei sistemi quantistici fuori dall’equilibrio ha introdotto delle problematiche ancora irrisolte in fisica. Negli ultimi anni si è assistito a un enorme progesso teorico in questo campo, mosso da incredibili progressi tecnologici sia nell’ambito di gas atomici e molecolari a basse temperature, che hanno reso possibile la manipolazione di sistemi quantistici con molti gradi di libertà, che in quello di algoritmi in grado di simulare l’evoluzione temporale. In questa lavoro rivolgiamo la nostra attenzione su di un semplice paradigma: lo studio della dinamica fuori dall’equilibrio di sistemi quantistici isolati unidimensionali a seguito della variazione di uno o più parametri del sistema (quench quantistico). In particolare viene studiata la dinamica di catene di spin con simmetria Zn e come questa venga modificata dalla rottura esplicita di tale simmetria. La parte originale del lavoro è nello studio della propagazione dell’entanglement nel modello di Potts con campo longitudinale nella sua fase paramagnetica, dove si è osservato, come recentemente nel modello di Ising con campo longitudinale, un repentino aumento nel tasso di produzione di entanglement. Questo si associa alla comparsa di una nuova particella nello spettro dell’Hamiltoniana dopo il quench. Il fenomeno viene spiegato come la versione fuori dall’equilibrio del noto paradosso di Gibbs. Tutti i risultati numerici della tesi sono stati ottenuti con l’algoritmo iTEBD sviluppato dall’autore

Topologically non-trivial states in one- and quasi-one-dimensional frustrated spin systems

Magnetic frustration is a phenomenon arising in spin systems when spin interactions cannot all be satisfied at the same time. A typical example of geometric frustration is a triangle with Ising-spins at its vertices and antiferromagnetic interaction. While we can easily anti-align two neighbouring spins, it is not possible for the third one to simultaneously anti-align with both of them. Another flavour of magnetic frustration is the so called exchange frustration, where different spin components interact in an Ising fashion on different bonds. Moreover, frustrated spin systems give rise to exotic states of matter, such as spin liquids, spin ices and nematic phases. As frustrated systems are rarely analytically solvable, numerical techniques are of the utmost importance in this framework. This dissertation is concerned with a specific class of models, namely one- and quasi-one-dimensional spin systems and studies their properties by making use of the density matrix renormalisation group technique. This method has been shown to be extremely powerful and reliable to study chain and ladder models. We consider examples of both geometric and exchange frustration. For the former, we take into consideration one of the prototypical examples of geometric frustration in one dimension: the J1-J2 model with ferromagnetic nearest-neighbour interaction J10. Our results show the existence of a Haldane gap supported by a special AKLT-like valence bond solid state in a specific region of the coupling ratio. Furthermore, we consider the effect of dimerisation of the first-neighbour coupling. This dimerisation affects the critical point and the ground state underlying the spin gap. These models are of interest in the context of cuprate chain materials such as LiVCuO4, LiSbCuO4 and PbCuSO4(OH)2. Concerning exchange frustration, we consider the celebrated Kitaev-Heisenberg model: it is an extension of the exactly solvable Kitaev model with an additional Heisenberg interaction. The Kitaev-Heisenberg model is currently the minimal model for candidate Kitaev materials. The extended model is not analytically solvable and numerics are needed to study the properties of the system. While both the original Kitaev and the Kitaev-Heisenberg models live on a honeycomb lattice, we here perform systematic studies of the Kitaev-Heisenberg chain and of the two-legged ladder. While the chain cannot support a Kitaev spin liquid state, it shows nevertheless a rich phase diagram despite being a one-dimensional system. The long-range ordered states of the honeycomb can be understood in terms of coupled chains within the Kitaev-Heisenberg model. Following this reasoning, we turn our attention to the Kitaev-Heisenberg model on a two-legged ladder. Remarkably, the phase diagram of the ladder is extremely similar to that of the honeycomb model and the differences can be explained in terms of the different dimensionalities. In particular, the ladder exhibits a topologically non-trivial phase with no long-range order, i.e., a spin liquid. Finally, we investigate the low-lying excitations of the Kitaev-Heisenberg model for both the chain and the ladder geometry