One of the most powerful and prized concepts in physics is that of universal behaviour. Universality allows us to make predictions for entire classes of systems without requiring knowledge of the microscopics, and can be found in classical and quantum systems in both equilibrium and in their dynamics. Often in many-body systems, this universal behaviour is found in regimes where effects at macroscopic length scales dominate over microscopic fluctuations, which is particularly true at a phase transition. In this thesis, we will address universality in quantum many-body physics, and its connection to the branch of mathematics known as catastrophe theory (CT). In CT, singularities in a theory take on several universal forms, known as catastrophes, which can be shown to manifest themselves in classical mechanical trajectories. We extend the concept of catastrophes to their wave variants, known as diffraction integrals, and identify how these universal features appear in many-body wavefunctions and observables. Specifically, in Chapter 2, we examine how the wavefunction of a $\delta$-kicked Hamiltonian can be mapped exactly onto the Pearcey function, along with the effects of a phase transition on the diffraction. In Chapter 3, we examine the free-Fermion representation of the one-dimensional transverse-field Ising model in a similar vein, and identify the presence of catastrophes away from and through the critical point.
One of the most powerful and prized concepts in physics is that of universal behaviour. Universality allows us to make predictions for entire classes of systems without requiring knowledge of the microscopics, and can be found in classical and quantum systems in both equilibrium and in their dynamics. Often in many-body systems, this universal behaviour is found in regimes where effects at macroscopic length scales dominate over microscopic fluctuations, which is particularly true at a phase transition. In this thesis, we will address universality in quantum many-body physics, and its connection to the branch of mathematics known as catastrophe theory (CT). In CT, singularities in a theory take on several universal forms, known as catastrophes, which can be shown to manifest themselves in classical mechanical trajectories. We extend the concept of catastrophes to their wave variants, known as diffraction integrals, and identify how these universal features appear in many-body wavefunctions and observables. Specifically, in Chapter 2, we examine how the wavefunction of a δ-kicked Hamiltonian can be mapped exactly onto the Pearcey function, along with the effects of a phase transition on the diffraction. In Chapter 3, we examine the free-Fermion representation of the one-dimensional transverse-field Ising model in a similar vein, and identify the presence of catastrophes away from and through the critical point.
As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-body wave functions after a quench, focusing on two-mode systems that include the two-site Bose Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space plus time certain characteristic structures generically appear that we identify as cusp caustics. In the vicinity of such a catastrophe the wave function takes on a universal form described by the Pearcey function and obeys scaling relations which depend on the total number of particles N. In the thermodynamic limit (N→∞) the cusp becomes singular, but at finite N it is decorated by an interference pattern. This pattern contains an intricate network of vortex-antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench takes the form of a δ-kick we show how to analytically map the wave function onto the Pearcey function and hence obtain the scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern.
Abstract As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-particle wave functions following a quench, focusing on two-mode systems that include the two-site Bose–Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space certain characteristic shapes, that we identify as cusp catastrophes, appear under generic conditions. In the vicinity of a cusp the wave function takes on a universal structure described by the Pearcey function and obeys scaling relations which depend on the total number of particles N . In the thermodynamic limit ( N → ∞ ) the cusp becomes singular, but at finite N it is decorated by an interference pattern. This pattern contains an intricate network of vortex–antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench is a δ -kick the problem can be solved analytically and we obtain scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern. Finally, we use these scalings to describe the wave function in the critical regime of a Z 2 symmetry-breaking dynamical phase transition.
This thesis explores the effects singularities have on stationary and dynamical properties of many-body quantum systems. In papers I and II we find that the ground state suffers a Z2 symmetry breaking phase transition (PT) when a single impurity is added to a Bose-Einstein condensate (BEC) in a double well (bosonic Josephson junction). The PT occurs for a certain value of the BEC-impurity interaction energy, Λc . A result of the PT is the mean-field dynamics undergo chaotic motion in phase space once the symmetry is broken. We determine the critical scaling exponents that characterize the divergence of the correlation length and fidelity susceptibility at the PT, finding that the BEC-impurity system belongs to the same universality class as the Dicke and Lipkin-Meshkov-Glick models (which also describe symmetry breaking PTs in systems of bosons). In paper III we study the dynamics of a generic two-mode quantum field following a quench where one of the terms in the Hamiltonian is flashed on and off. This model is relevant to BECs in double wells as well as other simple many-particle systems found in quantum optics and optomechanics. We find that when plotted in Fock-space plus time, the semiclassical wave function develops prominent cusp-shaped structures after the quench. These structures are singular in the classical limit and we identify them as catastrophes (as described by the Thom-Arnold catastrophe theory) and show that they arise from the coalescence of classical (mean-field) trajectories in a path integral description. Furthermore, close to the cusp the wave function obeys a remarkable set of scaling relations signifying these structures as examples of universality in quantum dynamics. Within the cusp we find a network of vortex-antivortex pairs which are phase singularities caused by interference. When the mean-field Hamiltonian displays a Z2 symmetry breaking PT modelled by the Landau theory of PTs we calculate scaling exponents describing how the separation distance between the members of each pair diverges as the PT is approached. We also find that the cusp becomes infinitely stretched out at the PT due to critical slowing down. In paper IV we investigate in greater detail the morphology of the vortex network found within cusp catastrophes in many-body wave functions following a quench. In contrast to the cusp catastrophes studied so far in the literature, these structures live in Fock space which is fundamentally granular. As such, these cusps represent a new iii type of catastrophe, which we term a ‘quantum catastrophe’. The granularity of Fock space introduces a new length scale, the quantum length lq = N −1 which effectively removes the vortex cores. Nevertheless, a subset of the vortices persist as phase singularities as can be shown by integrating the phase of the wave function around circuits in Fock-space plus time. Whether or not the vortices survive in a quantum catastrophe is governed by the separation of the vortex-antivortex pairs lv ∝ N −3/4 in comparison to lq , i.e. they survive if lv lq . When particle numbers are reached such that lq ≈ lv the vortices annihilate in pairs.
We describe a new class of nonequilibrium quantum many-body phenomena in the form of networks of caustics that dominate the many-body wave function in the semiclassical regime following a sudden quench. It includes the light cone-like propagation of correlations as a particular case. Caustics are singularities formed by the birth and death of waves and form a hierarchy of universal patterns whose natural mathematical description is via catastrophe theory. Examples in classical waves range from rainbows and gravitational lensing in optics to tidal bores and rogue waves in hydrodynamics. Quantum many-body caustics are discretized by second quantization (``quantum catastrophes'') and live in Fock space, which can potentially have many dimensions. We illustrate these ideas using the Bose Hubbard dimer and trimer models, which are simple enough that the caustic structure can be elucidated from first principles and yet run the full range from integrable to nonintegrable dynamics. The dimer gives rise to discretized versions of fold and cusp catastrophes whereas the trimer allows for higher catastrophes including the codimension-3 hyperbolic and elliptic umbilics, which are organized by, and projections of, an 8-dimensional corank-2 catastrophe known as X9. These results describe a hitherto unrecognized form of universality in quantum dynamics organized by singularities that manifest as strong fluctuations in mode population probabilities.
This thesis reports the results obtained during my PhD research in the field of out of equilibrium quantum many-body systems. Chapter 1 consists in a brief introduction of the field and the introduction of concept that are useful for the following chapters. In Chapter 2 the statistics of the work as a tool for characterizing the dynamics of many-body quantum systems is introduced its general features discussed. Then, such a statistics is computed for generic time-dependent protocols (both global and local) in the quantum Ising chain and in the Gaussian field theory, showing, in particular, that in its low-energy part there are features that are independent of the details of the specific chosen protocol. Chapter 3 is devoted to the study of the dynamical phase transition in the O(N) quantum vector model in the N→∞ limit, whose critical properties in generic dimensions are characterized. Moreover, a strong connection between such a transition and the statistics of excitations produced in a double quench as a function of the waiting time is showed. The chapter ends by studying the fate of the dynamical transition and the its critical properties when a ramp of finite duration τ is applied to the system instead of a sudden quench. In particular, we will show that when τ→∞ the critical point tends to the equilibrium critical point (at zero temperature) in a power-law fashion and that for every finite τ the critical properties are always the same (and different from the equilibrium critical properties). Finally in Chapter 4 we will discuss the emergence of a non adiabatic behavior in the dynamics of the order parameter for a low dimensional quantum system driven within a gapped phase by considering in detail the case of a quantum Ising chain subject to a linear variation in time of the transverse field, showing that, no matter how slowly the ramp is performed, such a change leads eventually to the disruption of the order.
The theory of statistical mechanics provides a powerful conceptual framework within which the relevant (macroscopic) features of systems at equilibrium can be described. As there is currently no equivalent capable of encompassing the much richer class of non-equilibrium phenomena, research in this direction proceeds mainly on an instance-by-instance basis. The aim of this Thesis is to describe in some detail three such attempts, which involve different dynamical aspects of classical and quantum systems. As summarised below, each of the last three Chapters of this document delves into one of these different topics, while Chapter 2 provides a brief introduction on the study of non-equilibrium dynamics. In Chapter 3 we investigate the purely relaxational dynamics of classical critical ferromagnetic systems in the proximity of surfaces, paying particular attention to the effects that the latter induce on the early stages of the evolution following an abrupt change in the temperature of the sample. When the latter ends close enough to the critical value which separates the paramagnetic from the ferromagnetic phase, it effectively introduces a temporal boundary which can be treated as if it were a surface. Within this picture, we highlight the emergence of novel effects near the effective edge formed by the intersection of the two spatial and temporal boundaries. Our findings are apparently in disagreement with previous predictions which were based on the assumption that the presence of such an edge would not affect the scaling behaviour of observables; in order to explain this discrepancy, we propose an alternative for the original power-counting argument which, at least, correctly predicts the emergence of novel field-theoretical divergences in our one-loop calculations. We show that said singularities are associated with the scaling at the edge. Moreover, by encoding our findings in a boundary renormalisation group framework, we argue that the new predicted behaviour represents a universal feature associated to the short-distance expansion of the order parameter of the transition near the edge; we also calculate explicitly its anomalous dimension at the first-order in a dimensional expansion. As a qualitative feature, this anomalous dimension depends on the type of phase transition occurring at the surface. We exploit this fact in order to provide numerical support to our predictions via Monte Carlo simulations of the dynamical behaviour of a three-dimensional Ising model. The main results reported in Chap. 3 have appeared in Ref. [1]. In Chapter 4 we revisit the Euclidean mapping to imaginary times which has been recently proposed [2, 3] as an alternative for approaching the problem of quantum dynamics following a quench. This is expected to allow one to reformulate the original problem as a static one confined in a film geometry. We show that this interpretation actually holds only if the initial state of the dynamics is pure. Statistical mixtures, instead, intertwine the effects due to the two boundaries, which therefore cannot be regarded as being independent. We emphasize that, although the aforementioned reinterpretation as a confined static problem fails, one is still able, in principle, to write down and solve the corresponding equations. We also discuss in some detail the relation between this approach and the real-time field-theoretical one which makes use of the two-time Keldysh contour. For this purpose, we study the analytical structure of relevant observables — such as correlation functions — in the complex plane of times, identifying a subdivision of this domain into several sectors which depend on the ordering of the imaginary parts of the involved time coordinates. Within each of these subdomains, the analytic continuation to the real axis provides in principle a different result. This feature allows one to reconstruct from the Euclidean formalism all possible non-time-ordered functions, which in particular include all those which can be calculated via the Keldysh two-time formalism. Moreover, we give a prescription on how to retrieve response functions, discussing some simple examples and rationalising some recent numerical data obtained for one of these observables in a one-dimensional quantum Ising chain [4]. We also highlight the emergence of a light-cone effect fairly similar to the one previously found for correlation functions [2], which therefore provides further confirmation to the fact that information travels across the system in the form of the entanglement of quasi-particles produced by the quenching procedure. We have reported part of this analysis in Ref. [5]. Chapter 5 presents part of our recent work on effective relaxation in quantum systems following a quench and on the observed prethermalisation. We analyse the effects caused by the introduction of a long-range integrability-breaking interaction in the early stages of the dynamics of an otherwise integrable quantum spin chain following a quench in the magnetic field. By employing a suitable transformation, we redefine the theory in terms of a fully-connected model of hard-core bosons, which allows us to exploit the (generically) low density of excitations for rendering our model exactly solvable (in a numerical sense, i.e., by numerically diagonalising an exact matrix). We verify that, indeed, as long as the parameters of the quench are not too close to the critical point, the low-density approximation captures the dynamical features of the elementary operators, highlighting the appearance of marked plateaux in their dynamics, which we reinterpret as the emergence of a prethermal regime in the original model. As expected, the latter behaviour is reflected also on extensive observables which can be constructed as appropriate combinations of the mode populations. For these quantities, the typical approach to the quasi-stationary value is algebraic with exponent a ≈ 3, independently of the size of the system, the strength of the interaction and the amplitude of the magnetic field (as long as it is kept far from the critical point). The plateaux mentioned above last until a recurrence time — which can be approximately identified with tR ≈ N/2 for single modes and t′R ≈ N/4 for extensive quantities — after which quantum oscillations due to the finite size of the chain reappear. Our procedure allows us to shed some light over prethermal features without having to considerably limit the size of the system, which we can choose to be quite large, as we discuss in Ref. [6].