Recent ground-breaking experiments have realized strongly interacting quantum degenerate Fermi gas in a cold atomic system with tunable interactions. This has provided a table-top system which is extremely hydrodynamic in nature. This experimental realization helps us to investigate several aspects such as the interplay between nonlinearity, dissipation and dispersion. We find, for instance, that the dynamics in such a system shows near perfect agreement with a hydrodynamic theory. In collaboration with the group of John Thomas at Duke we interpreted studies of collision of two strongly interacting Fermi gases that led to shock waves which are a hallmark of nonlinear physics. Due to reasons such as the nature of interactions, higher dimensionality, these cold atomic systems are non-integrable and moreover the underlying field theory construction is mostly phenomenological in nature. On the other hand there are certain one-dimensional systems which are not only integrable but also facilitate more formal and rigorous ways of deriving the corresponding integrable field theories. One such family of models is the family of Calogero models (and their generalizations). They provide an extraordinary insight into the field of strongly correlated systems and hydrodynamics. We study the collective field theory of such models and address aspects of nonlinear physics such as Spin-Charge Interaction, Emptiness Formation Probability, Solitons etc; We derive a two-component nonlinear, nonlocal, integrable field theory. We also show that the Calogero family which is integrable even in an external harmonic trap (usually unavoidable in cold atom setups) is relatively short ranged thereby qualifying as a toy model for cold atom experiments. Transport in certain strongly correlated systems (impurity models) was studied using few low-dimensional techniques such as a 1/N diagrammatic expansion, Slave Boson Mean Field Theory and the Bethe Ansatz. A mesoscopic setup such as parallel quantum dots forms an ideal platform for such an investigation and comparison between different low-dimensional techniques. We studied transport, correlations and nature of the ground state of double quantum dots. We probed several non-perturbative aspects of this double-impurity model. For example, we showed that the RKKY interaction in closely spaced dots can be non-ferromagnetic due to its non-pertubative nature. This study helped us to point some discrepancies between different methods (such as the Numerical Renormalization Group). We give possible reasons for these discrepancies.
Recent ground-breaking experiments have realized strongly interacting quantum degenerate Fermi gas in a cold atomic system with tunable interactions. This has provided a table-top system which is extremely hydrodynamic in nature. This experimental realization helps us to investigate several aspects such as the interplay between nonlinearity, dissipation and dispersion. We find, for instance, that the dynamics in such a system shows near perfect agreement with a hydrodynamic theory. In collaboration with the group of John Thomas at Duke we interpreted studies of collision of two strongly interacting Fermi gases that led to shock waves which are a hallmark of nonlinear physics. Due to reasons such as the nature of interactions, higher dimensionality, these cold atomic systems are non-integrable and moreover the underlying field theory construction is mostly phenomenological in nature. On the other hand there are certain one-dimensional systems which are not only integrable but also facilitate more formal and rigorous ways of deriving the corresponding integrable field theories. One such family of models is the family of Calogero models (and their generalizations). They provide an extraordinary insight into the field of strongly correlated systems and hydrodynamics. We study the collective field theory of such models and address aspects of nonlinear physics such as Spin-Charge Interaction, Emptiness Formation Probability, Solitons etc; We derive a two-component nonlinear, nonlocal, integrable field theory. We also show that the Calogero family which is integrable even in an external harmonic trap (usually unavoidable in cold atom setups) is relatively short ranged thereby qualifying as a toy model for cold atom experiments. Transport in certain strongly correlated systems (impurity models) was studied using few low-dimensional techniques such as a 1/N diagrammatic expansion, Slave Boson Mean Field Theory and the Bethe Ansatz. A mesoscopic setup such as parallel quantum dots forms an ideal platform for such an investigation and comparison between different low-dimensional techniques. We studied transport, correlations and nature of the ground state of double quantum dots. We probed several non-perturbative aspects of this double-impurity model. For example, we showed that the RKKY interaction in closely spaced dots can be non-ferromagnetic due to its non-pertubative nature. This study helped us to point some discrepancies between different methods (such as the Numerical Renormalization Group). We give possible reasons for these discrepancies.
This thesis considers out-of-equilibrium dynamics of strongly interacting non-relativistic Fermi gases in several two and three dimensional geometries. The tools of second-order hydrodynamics and gauge-gravity duality will be utilized to address this system. Many of the themes of this work are motivated by the observed similarities in transport properties between strongly interacting Fermi gases and other very different strongly interacting quantum fluids such as the quark-gluon plasma, high temperature superconductors, and quantum field theories described by gauge-gravity duality. In particular, these systems all nearly saturate the conjectured lower bound on the ratio of shear viscosity to entropy density η/s≥ℏ/(4πkB) coming from the AdS/CFT correspondence. Among other things, this observation, in conjunction with current experiment and data analysis in atomic, condensed matter, and nuclear physics lends itself to the following questions: How perfect of a fluid is the strongly interacting Fermi gas, and can one find a more stringent constraint on η/s in Fermi gases? Do the similarities in transport properties among strongly interacting quantum systems extend beyond dynamics controlled by the hydrodynamical shear viscosity? In regards to the first question, by utilizing second-order hydrodynamics, it will be demonstrated that higher-order collective modes of a harmonically trapped Fermi gas may serve as a more sensitive probe of the shear viscosity. For the second question, both second-order hydrodynamics and a gravity dual theory are used to make predictions about dynamics occurring on short timescales where hydrodynamics is expected to break down. In particular the appearance of a class of "non-hydrodynamic" collective modes not contained within a Navier-Stokes description of the strongly interacting Fermi gas will be discussed.
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.
Low-dimensional systems are an important field of current theoretical and experimental research. Recent technological developments provide many possible realizations of effectively one-dimensional systems. These devices promise to give us access to a new range of phenomena. It is therefore very interesting to develop theoretical methods specific for such systems to model their behavior and calculate the correlators of the resulting theory. Incidentally, one such method exists and is known as Bosonization. It can be applied to one-dimensional systems and effectively describes low energy excitations in a universal way. We use the example of a correlator known as the Emptiness Formation Probability to show that Bosonization fails to describe some long range correlators corresponding to large disturbances (the EFP measures the probability for the ground state of the system to develop a region without particles). We trace this failure to the fact that Bosonization is constructed as a linear approximation of the full theory and we set up to develop a collective description with the required non-linearity. The resulting scheme is essentially a Hydrodynamic paradigm for quantum systems. We show how to construct such a hydrodynamic description for a variety of exactly integrable models and illustrate how it can be used to make new predictions. For the special case of the spin-1/2 XY model we take advantage of the structure of the model to express the EFP as a determinant of a very special type of matrix, known as Toeplitz Matrix. We use the theory of Toeplitz determinants to calculate the asymptotic behavior of the EFP in the XY model and discuss its relation with the criticality of the theory.
The quantum physics of one dimensional systems exhibits many remarkable and exotic physical phenomena. These are the result of the strong correlations which exist in such systems due to the reduced dimensionality and in many cases can be described by Luttinger Liquid theory. Likewise quantum impurity systems, where a bath of particles couples to the same impurity, exhibit well known non perturbative phenomena like the Kondo effect. In this thesis we study the physics of quantum impurities in interacting environments - the intersection of these two areas. Such Luttinger-impurity systems can be realized experimentally in a range of different settings from quantum wires and carbon nano tubes to edges of fractional quantum hall systems and cold atom gases. We show that many of the models used to describe these experiments are integrable and can be studied by means of the Bethe Ansatz. This powerful and exact method provides us with the exact eigenstates and energy levels of the model Hamiltonian. The models considered include two which describe a Lutttinger liquid coupled to a back scattering impurity, the Kane-Fisher and weak tunnelling model as well as Luttinger liquids coupled a quantum dot in various geometries: at the boundary, sidecoupled and embedded. To incorporate both the backscattering nature of the impurities and the interacting bulk particles a new formulation of the Bethe Ansatz is developed and implemented. The eigenstates and spectra of the models are found and used to investigate their ground state and thermodynamic properties.
We study non-equilibrium and strongly-correlated dynamics in two contexts. We begin by analysing quantum many-body systems out of equilibrium through the lens of cold atomic impurities in Bose gases. Such highly-imbalanced mixtures provide a controlled arena for the study of interactions, dissipation, decoherence and transport in a many-body quantum environment. Specifically we investigate the oscillatory dynamics of a trapped and initially highly-localised impurity interacting with a weakly-interacting trapped quasi low-dimensional Bose gas. This relates to and goes beyond a recent experiment by the Inguscio group in Florence. We witness a delicate interplay between the self-trapping of the impurity and the inhomogeneity of the Bose gas, and describe the dissipation of the energy of the impurity through phononic excitations of the Bose gas. We then study the transport of a driven, periodically-trapped impurity through a quasi one-dimensional Bose gas. We show that placing the weakly-interacting Bose gas in a separate periodic potential leads to a phononic excitation spectrum that closely mimics those in solid state systems. As a result we show that the impurity-Bose gas system exhibits phonon-induced resonances in the impurity current that were predicted to occur in solids decades ago but never clearly observed. Following this, allowing the bosons to interact strongly, we predict the effect of different strongly-correlated phases of the Bose gas on the motion of the impurity. We show that, by observing the impurity, properties of the excitation spectrum of the Bose gas, e.g., gap and bandwidth, may be inferred along with the filling of the bosonic lattice. In other words the impurity acts as a probe of its environment. To describe the dynamics of such a strongly-correlated system we use the powerful and near-exact time-evolving block decimation (TEBD) method, which we describe in detail. The second part of this thesis then analyses, for the first time, the performance of this method when applied to simulate non-equilibrium classical stochastic processes. We study its efficacy for a well-understood model of transport, the totally-asymmetric exclusion process, and find it to be accurate. Next, motivated by the inefficiency of sampling-based numerical methods for high variance observables we adapt and apply TEBD to simulate a path-dependent observable whose variance increases exponentially with system size. Specifically we calculate the expected value of the exponential of the work done by a varying magnetic field on a one-dimensional Ising model undergoing Glauber dynamics. We confirm using Jarzynski's equality that the TEBD method remains accurate and efficient. Therefore TEBD and related methods complement and challenge the usual Monte Carlo-based simulators of non-equilibrium stochastic processes.
Dynamics of interacting cold atomic gases have recently become a focus of both experimental and theoretical studies. Often cold-atom systems show hydrodynamic behavior and support the propagation of nonlinear dispersive waves. Although this propagation depends on many details of the system, great insight can be obtained in the rather universal limit of weak nonlinearity, dispersion, and dissipation. In this limit, using a reductive perturbation method we map some of the hydrodynamic models relevant to cold atoms to well-known chiral one-dimensional equations such as the Korteweg--de Vries (KdV), Burgers, KdV-Burgers, and Benjamin-Ono equations. These equations have been thoroughly studied in the literature. The mapping gives us a simple way to make estimates for the original hydrodynamic equations and to study the interplay between nonlinearity, dissipation, and dispersion which are the hallmarks of nonlinear hydrodynamics.