This thesis consists of two projects which are seemingly disconnected, yet closely related. The first part explores the effects of Bose-Einstein condensation at temperatures close to, but slightly above, criticality. Following a general introduction into bosonic condensation we justify why a phenomenological theory, similar to the Ginzburg-Landau theory for fermions, holds for weakly interacting Bose gases. From this theory we predict the divergence of certain observables, in particular the quasi-magnetic susceptibility, and discuss the effects of a trapping potential. The divergence of the magnetic susceptibility motivates the introduction of an original scheme in order to measure it, published in Phys. Rev. A 93, 041602(R). The scheme uses modulated laser fields to create well- controlled gradients of artificial magnetic fields. In addition we discuss how rotational schemes might be helpful in detecting different quantum phases by exploiting different signatures in their moments of inertia. \n \nThe second part investigates binary mixtures in one dimension. We show that in certain limits such systems behave like two simply coupled Luttinger liquids, which effectively describe polaronic modes. We study and calculate explicitly how an impurity immersed in the one dimensional system creates two depletion clouds and a phase drop in each of the liquids. After arguing that these clouds and phase drops necessitate a coupling of the impurity to the low-lying excitation modes of the Luttinger liquids, we derive the edge-state singularities of the bosonic and fermionic dynamical structure factors which depend on the coupling between the liquids.
This thesis consists of two projects which are seemingly disconnected, yet closely related. The first part explores the effects of Bose-Einstein condensation at temperatures close to, but slightly above, criticality. Following a general introduction into bosonic condensation we justify why a phenomenological theory, similar to the Ginzburg-Landau theory for fermions, holds for weakly interacting Bose gases. From this theory we predict the divergence of certain observables, in particular the quasi-magnetic susceptibility, and discuss the effects of a trapping potential. The divergence of the magnetic susceptibility motivates the introduction of an original scheme in order to measure it, published in Phys. Rev. A 93, 041602(R). The scheme uses modulated laser fields to create well- controlled gradients of artificial magnetic fields. In addition we discuss how rotational schemes might be helpful in detecting different quantum phases by exploiting different signatures in their moments of inertia. \n \nThe second part investigates binary mixtures in one dimension. We show that in certain limits such systems behave like two simply coupled Luttinger liquids, which effectively describe polaronic modes. We study and calculate explicitly how an impurity immersed in the one dimensional system creates two depletion clouds and a phase drop in each of the liquids. After arguing that these clouds and phase drops necessitate a coupling of the impurity to the low-lying excitation modes of the Luttinger liquids, we derive the edge-state singularities of the bosonic and fermionic dynamical structure factors which depend on the coupling between the liquids.
This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional.
This Thesis discusses approaches to calculating static and dynamic correlation functions in onedimensional gases of interacting bosons. The first part of the Thesis deals with the momentummomentum correlation function in the weakly interacting 1D Bose gas. In the regime of phase fluctuating quasicondensate this correlator is found to differ qualitatively from the form predicted by Bogoliubov theory of the true condensate in that correlations between any two values of momentum become finite. Linear hydrodynamics used to calculate the quasicondensate correlation function prove to be adequate. A classical field approximation is used to smoothly interpolate between the quasicondensate and strongly degenerate gas regimes. In the second part the focus is shifted to the dynamical structure factor and it is shown that hydrodynamics is generally inapplicable to calculating the dynamical correlators. The hydrodynamic treatment is enhanced using impurity theory and an exact model-dependent boundary is obtained for the region of applicability of hydrodynamics on the momentum-energy plane. Numerical estimates show that non-hydrodynamic behavior should be observable for the currently available values of experimental parameters.
The recent experimental advances in the field of quantum spin materials have led to well-controlled systems which can be used in order to simulate complex quantum models in a clean fashion. Here we discuss how fascinating effects normally known from bosonic systems such as the Bose-Einstein condensation can be investigated with the help of such materials. We discuss as an example how a crossover between an effectively one-dimensional geometry described by a Luttinger liquid to a three-dimensional Bose-Einstein condensation can be induced. Additionally, we point out how more complex phases as the Bose-glass phase could be realized in these quantum magnets. Introduction Although it can seem paradoxical, the quantum magnetic states found in insulators offer an interesting route to study Bose-Einstein condensation (BEC) and more generally phenomena related to itinerant, interacting bosonic systems. In many materials, the repulsion between the electrons causes the ground state to be a Mott insulator with one electron localized per site. In such a case, the charge degrees of freedom are completely locked by large charge gaps (usually of the order of tens of electron volts). However, one can study the fascinating physics of the spin degrees of freedom. As was shown a long time ago, the localized spins are coupled by a so-called superexchange which may be ferro- or antiferromagnetic. The properties of the exchange coupling between the spins depend mainly on the underlying atomic structure of the material. The resulting physics is extremely rich and such materials can have a host of phases ranging from ordered magnetic phases to so-called spin-liquid phases in which spin–spin correlations decay extremely fast with distance. The field of research concerning these materials, dubbed quantum magnetism, is thus an extremely active field in its own right [1]. Additionally, one can exploit the formal mapping that exists between spin operators and bosonic ones, as will be described in the next section, to connect the magnetic materials to itinerant bosonic systems with interactions between the bosons. This opens the way to use such quantum spin systems as quantum simulators. A quantum simulator means that the experimental system can be employed in order to “solve” in a controlled way a problem/Hamiltonian known from a different less controlled physical realization, e.g., inspired by fundamental many-body physics, quantum information or computational applications.
Just a couple decades ago bosonic low-temperature physics was synonymous with one particular liquid at a temperature of several degrees Kelvin. This liquid, helium-4, and its famous Helium-II phase, has revealed remarkable quantum properties such as quantized vortices and second sound. Yet, despite many things learned by investigating helium, its study has certain limitations. On the one hand, helium is a strongly interacting liquid, which makes first-principles theoretical research challenging. To name one example, establishing a connection between the non-interacting condensate state described by Bose and Einstein, and the helium properties observed in the lab took decades. Furthermore, helium is a specific chemical element, and hence there is no surprise that its physical properties are fixed by nature. However, every particular chemical element has particular physical properties, so for a long time it seemed that it was as good as it gets. The picture has changed drastically when the first Bose-Einstein condensate (BEC) was observed in an ultracold atomic gas. “Low temperature” has become synonymous with “ultracold”, and the latter implies temperatures on the order of nanokelvins. Most strikingly, the relationship between what is fixed by nature and what is experimentally changeable has evolved. In particular, systems of different particle statistics, spin, and interactions have been engineered using ultracold vapours of various alkali atoms. This versatile manipulation not only makes emulating various condensed-matter physics models in a controlled system without impurities possible, but also opens a path towards novel phenomena that are at present not achievable in any other manner. In this thesis, we have explored some of the novel phenomena that arise in systems of bosonic particles with internal degrees of freedom, such as (pseudo)spin and electric dipole moment. In the introduction, we describe a ferromagnetically coupled spin-1/2 Bose gas with contact interactions in the mean- field approximation. In particular, we compute and discuss the phase diagram of this gas, since it is a simple system which nevertheless has two order parameters. In the chapters that follow, the ideas touched upon in the introduction are developed in more detail. In Chapter 2 we develop a hydrodynamic description of the ferromagnetic spin-1/2 Bose gas at arbitrary temperatures. We study magnetization relaxation and geometric forces. In particular, we consider the topological Hall effect due to the presence of a skyrmion. In Chapter 3 we investigate the miscible (non-ferromagnetic) spin-1/2 Bose gas at arbitrary temperatures, construct its hydrodynamic description, calculate the thermodynamic properties, and study the collective modes of this system. In Chapter 4 we discuss the influence of the off-diagonal Berry curvature on the Bose-Einstein condensation temperature. Finally, in Chapter 5 we study a Bose-Einstein condensate of dipolar molecules in a weak electric field and find it to be described by a quantum rotor model. Moreover, we show that the molecular Bose-Einstein condensate is a ferroelectric material that is fully disordered by quantum fluctuations.
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.
This thesis concerns ultracold two-dimensional atomic gases and the associated novel superfluid phases in an optical lattice subjected to staggered rotations. In the first part, we study the Berezinskii-Kosterlitz-Thouless (BKT) transition in a two-dimensional Bose gas. Motivated by recent experiments on the cold gas in the BKT regime, we employ a modified Popov theory and a renormalization group theory to describe the system. The two-dimensional renormalization group flow equations display unique features, which we interpret as precursors of the BKT physics. We further calculate higher correlation functions close to the BKT transition, which can serve as observables for detecting effects beyond mean-field. Finally, we obtain the density profile under the conditions of the experiment performed in the group of Dalibard in Paris, France. In the second part of the thesis, we study a recent experimental proposal of a time-dependent optical lattice with staggered rotations. We show that in the tight-binding regime, the system can be effectively described by a (Bose-)Hubbard model subjected to an artificial staggered magnetic field. The staggered magnetic field (flux) in the lattice leads to conical points in the band structure which possesses distinct minima for different strength, and can give rise to Dirac-like dispersions. For bosons, we show that besides the homogenous superfluid phase, a staggered-vortex superfluid phase is stabilized for large flux values. We show that the staggered flux renormalizes the phase boundary and find a multi-critical point separating the two superfluid phases and the Mott insulating phase. Finally, the full phase diagram of the generalized Bose-Hubbard model at zero temperature is determined. We then generalize the model for a mixture of ultracold bosons and spin-1/2 fermions. By using the Bogoliubov theory for the bosonic sector, we show that the bosonic superfluid mediates attractive interactions between fermions up to the nearest-neighbor site. The resulting model is an extended Hubbard model in the presence of a staggered magnetic field, where the on-site and the nearest-neighbor attractive interactions can be tuned independently. We then employ the Bardeen-Cooper-Schrieffer theory for superconductivity, by considering an on-site pairing order parameter competing with a nearest-neighbor resonating-valence-bond order parameter. Due to conical points in the energy band, the system possesses quantum critical property and the resulting phase diagram displays features which are strikingly similar to that of high-critical-temperature superconductors or heavy-fermion superconductors.