This thesis presents theoretical groundwork for the investigation of particular phenomena of instability and decay in out-of-equilibrium many body systems. By employing mathematical methods of integrability, we succeed in gaining new insights into problems not traditionally associated with an integrable realm. The family of homogeneous sine-Gordon models provides a rare example of an integrable quantum field theory where both stable and unstable excitations are present in the spectrum. We studied the effect of unstable particles in different out-of-equilibrium settings and have shown that their presence is associated with specific signatures in many quantities of physical interest. Adopting the generalized hydrodynamic approach, we investigated the non-equilibrium dynamics following a quench and observed hallmarks of the creation and decay of unstable particles. Furthermore, we explored in considerable detail their quantitative and qualitative dependence on the parameters of the problem. Decay processes due to environmental interactions are an experimentally relevant problem, yet the theory of open systems is substantially less developed than for their closed counterparts. We concentrate on Markovian open quantum systems described by a Lindblad time evolution equation and design a framework for analytic investigation of Lindblad dynamics via establishing a connection with integrability. We identify a number of Lindblad equations that can be mapped onto non-Hermitian interacting Yang-Baxter integrable models. The key ingredient of the proposed map is the structural similarity between the operators considered. We present examples of mappings between integrable Hamiltonians and Lindblad generators and underline several models of physical interest which can be investigated within this framework. By employing Bethe ansatz techniques, we study the dynamics of some of these models.
This thesis presents theoretical groundwork for the investigation of particular phenomena of instability and decay in out-of-equilibrium many body systems. By employing mathematical methods of integrability, we succeed in gaining new insights into problems not traditionally associated with an integrable realm. The family of homogeneous sine-Gordon models provides a rare example of an integrable quantum field theory where both stable and unstable excitations are present in the spectrum. We studied the effect of unstable particles in different out-of-equilibrium settings and have shown that their presence is associated with specific signatures in many quantities of physical interest. Adopting the generalized hydrodynamic approach, we investigated the non-equilibrium dynamics following a quench and observed hallmarks of the creation and decay of unstable particles. Furthermore, we explored in considerable detail their quantitative and qualitative dependence on the parameters of the problem. Decay processes due to environmental interactions are an experimentally relevant problem, yet the theory of open systems is substantially less developed than for their closed counterparts. We concentrate on Markovian open quantum systems described by a Lindblad time evolution equation and design a framework for analytic investigation of Lindblad dynamics via establishing a connection with integrability. We identify a number of Lindblad equations that can be mapped onto non-Hermitian interacting Yang-Baxter integrable models. The key ingredient of the proposed map is the structural similarity between the operators considered. We present examples of mappings between integrable Hamiltonians and Lindblad generators and underline several models of physical interest which can be investigated within this framework. By employing Bethe ansatz techniques, we study the dynamics of some of these models.
In the context of quantum field theory (QFT), unstable particles are associated with complex-valued poles of two-body scattering matrices in the unphysical sheet of rapidity space. The Breit-Wigner formula relates this pole to the mass and life-time of the particle, observed in scattering events. In this paper, we uncover new, dynamical signatures of unstable excitations and show that they have a strong effect on the non-equilibrium properties of QFT. Focusing on a 1+1D integrable model, and using the theory of Generalized Hydrodynamics, we study the formation and decay of unstable particles by analysing the release of hot matter into a low-temperature environment. We observe the formation of tails and the decay of the emitted nonlinear waves, in sharp contrast to the situation without unstable excitations. We also uncover a new phenomenon by which a wave of a stable population of unstable particles may persist without decay for long times. We expect these signatures of the presence of unstable particles to have a large degree of universality. Our study shows that the out-of-equilibrium dynamics of many-body systems can be strongly affected not only by the spectrum, but also by excitations with finite life-times.
Abstract The generalized hydrodynamic (GHD) approach has been extremely successful in describing the out-of-equilibrium properties of a great variety of integrable many-body quantum systems. It naturally extracts the large-scale dynamical degrees of freedom of the system, and is thus a particularly good probe for emergent phenomena. One such phenomenon is the presence of unstable particles, traditionally seen via special analytic structures of the scattering matrix. Because of their finite lifetime and energy threshold, these are especially hard to study. In this paper we apply the GHD approach to a model possessing both unstable excitations and quantum integrability. The largest family of relativistic integrable quantum field theories known to have these features are the homogeneous sine-Gordon models. We consider the simplest non-trivial example of such theories and investigate the effect of an unstable excitation on various physical quantities, both at equilibrium and in the non-equilibrium state arising from the partitioning protocol. The hydrodynamic approach sheds new light onto the physics of the unstable particle, going much beyond its definition via the analytic structure of the scattering matrix, and clarifies its effects both on the equilibrium and out-of-equilibrium properties of the theory. Crucially, within this dynamical perspective, we identify unstable particles as finitely-lived bound states of co-propagating stable particles of different types, and observe how stable populations of unstable particles emerge in large-temperature thermal baths.
In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator eiβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.
Abstract Unstable particles rarely feature in conjunction with integrability in 1+1D quantum field theory. However, the family of homogenous sine-Gordon models provides a rare example where both stable and unstable bound states are present in the spectrum whilst the scattering matrix is diagonal and solves the usual bootstrap equations. In the standard scattering picture, unstable particles result from complex poles of the S-matrix located in the unphysical sheet of rapidity space. Since they are not part of the asymptotic spectrum, their presence is only felt through the effect they have on physical quantities associated either to the theory as a whole (i.e. scaling functions, correlation functions) or to the stable particles themselves (i.e. energy/particle density). In two recent publications, the effect of unstable particles in different out-of-equilibrium settings has been studied. It has been shown that their presence is associated with specific signatures in many quantities of physical interest. A good way to select those quantities is to adopt the generalised hydrodynamic approach and to consider the effective velocities and particle densities of the stable particles in the theory. For an initial state given by a spacial gaussian profile of temperatures peaked at the origin, time evolution gives rise to particle and spectral particle densities that exhibit hallmarks of the creation and decay of unstable particles. While these signatures have been observed numerically elsewhere, this paper explores their quantitative and qualitative dependence on the parameters of the problem. We also consider other initial states characterised by “inverted gaussian” and “double gaussian” temperature profiles.
In this Thesis I present the results of four research projects I carried out during my Ph.D. training at SISSA. The broad topic of my research was the mechanism of relaxation in isolated, integrable quantum systems. Roughly speaking, integrable quantum systems are one-dimensional lattice models or field theories that feature an extensively large number of conservation laws beyond energy and momentum conservation, which greatly influence their dynamical behaviour. Integrable systems have enjoyed a considerable interest in the last decade, mainly due to the development of ultracold atomic techniques. These techniques made it possible to implement experimentally a wealth of one-dimensional quantum models whose significance had been purely theoretical before. The theoretical interest in integrable systems stems in part from the fact that in many aspects they are exactly solvable. Exact analytical formulas are available for their spectrum and their equilibrium thermodynamic properties, creating an incentive to study analytically their non-equilibrium behaviour as well. Exact results for the dynamics of integrable systems may serve as a benchmark for confirming theories in statistical mechanics or testing approximative methods. To this day, computing analytically the entire time evolution of observables in a genuinely interacting integrable system is prohibitively difficult in general. In this Thesis, I concentrate on the description of the system at long times after it is brought out of equilibrium. Homogeneous, translationally invariant integrable systems are expected to relax to a stationary state, in which local observables are described by a generalization of the Gibbs ensemble that takes into consideration all the conservation laws beyond the energy. Two of the projects presented here concern this relaxed state. In Chapter 2, a method is presented for computing the Rényi entropies of the Generalized Gibbs Ensemble describing the relaxed state in the integrable Heisen- berg XXZ spin chain. Rényi entropies provide information about the relaxed state and its entanglement properties, going beyond the ordinary von Neumann thermo- dynamic entropy. The method presented there is very general; in principle it works in any integrable model and any translationally invariant thermodynamic macrostate. An alternative to the GGE in describing relaxed states after a quantum quench is the Quench Action approach. This approach is based on the explicit analytical knowledge of the overlaps between the initial state and the Hamiltonian eigenstates. In Chapter 3, this approach is generalized to a spin-1 integrable spin chain, the SU(3) invariant Lai–Sutherland model. The evolution of entanglement is then studied using the quasiparticle picture, which features ballistically propagating classical particle pairs carrying entanglement. If the requirement of translational invariance of the initial state is relaxed, various transport phenomena arise in integrable systems. The evolution of slowly varying, smoothly distributed states is described by a generalization of hydrodynamics, which takes into consideration the infinitely many local conservation laws. At the scale of non-diffusive, ballistic transport, the main feature of this generalized hydrodynamics (GHD) is the existence of continuity equations for the occupation number of each mode. Further two projects are presented in this Thesis which are concerned with GHD.I n Chapter 4, a numerical method is presented for computing the evolution of entanglement entropies within GHD. This method combines the quasiparticle picture of entanglement with a molecular dynamics simulation of GHD, the so-called fleagas. Numerical evidence is shown that this method reproduces previously known analytical results in GHD. Its versatility is demonstrated by computing another entanglement-related quantity, the mutual information, for which no analytical results are available. Finally, in Chapter 5, the GHD framework is generalized to a system of spin-1/2 fermions with repulsive contact interaction, the Yang–Gaudin model. It is shown that at low temperatures, the dynamics of the system is described by the superposition of two uncoupled, inhomogeneous conformal field theories. This fact can be interpreted as a separation of the spin and the charge degrees of freedom of the system.
In this thesis, we investigated the evolution of low-dimensional quantum many-body systems through the lenses of their dynamics. The main tools we used are unitary dynamics and kinetic theory. The interest in this class of systems derives from the fact that they are dominated by the effects of interactions between fundamental constituents. In one dimension, the situation is even more dramatic: even the smallest interaction leads to a description that cannot be cast into a small perturbation of the initial model. Even though, a particular set of systems still allows for exact solutions; hence, they are deemed integrable and they allow for an infinite number of conserved currents. A simple way, that can also be performed experimentally, to drive many-body systems outside of the equilibrium is called quantum quench. It consists in switching instantaneously the parameters of the theory after the system is at the equilibrium and measuring an observable after the driving has occurred. Our contributions to the field amount to providing a full characterization of the evolution of the observables of an integrable system after a global quantum quench. We were also able to extend these results, valid in principle for relativistic theories, to the Lieb-Liniger model, prominent for the description of experimental setups, by the means of non-relativistic limit. For two-dimensional models, interactions are not changing the system from its foundations, although there are unexpected effects. A paradigmatic example is graphene: in a two-dimensional system where carriers are charged and massless, namely their dispersion relation does not have any gap and particles and holes can be freely excited from the thermal state, it is predicted that they are interacting via Coulomb force hundreds of times more than normal electrons in void. Hence, the system behaves in a completely different way from electrons in quantum electrodynamics: for instance, there are additional modes which are called plasmons that are also massless but neutral. They represent collective excitations of the model, as oscillations in the local density of charge. This system is not integrable but still strongly interacting; solving it as we do for one-dimensional systems is not possible. The role of conservation laws is a common theme of this thesis. In the example above exposed, total charge, momentum and energy are conserved in the pristine model, but in presence of impurities and defects they dissipate. The evolution of such quantities can though be derived exactly by the means of hydrodynamics. We were able to derive the equations that rule their evolution in time for any electronic system with two bands touching at the Dirac point. They are generalizations of the celebrated Navier-Stokes equations, and they differ from them only outside the compressible regime. We are able to show that the transport properties in the hydrodynamic regime strongly differ from the usual ballistic picture popularized by Drude and his model; this is captured for instance by violations of the Wiedemann-Franz ratio. Moreover, we prove that the role of interaction is prominent in the viscosity of these fluids.