Excitation spectrum of the Lieb-Liniger model

We study the integrable model of one-dimensional bosons with contact repulsion. In the limit of weak interaction, we use the microscopic hydrodynamic theory to obtain the excitation spectrum. The statistics of quasiparticles changes with the increase of momentum. At lowest momenta good quasiparticles are fermions, while at higher momenta they are Bogoliubov bosons, in accordance with recent studies. In the limit of strong interaction, we analyze the exact solution and find exact results for the spectrum in terms of the asymptotic series. Those results undoubtedly suggest that fermionic quasiparticle excitations actually exist at all momenta for moderate and strong interaction, and also at lowest momenta for arbitrary interaction. Moreover, at strong interaction we find highly accurate analytical results for several relevant quantities of the Lieb-Liniger model.Comment: seven pages and two figure

Order and transport in interacting disordered low-dimensional systems

The subject of this thesis is order and transport in interacting disordered low-dimensional systems, namely Luttinger liquids and thin superconducting films. The disorder is produced by point impurities in Luttinger liquids and by magnetic dots in thin films. In the second chapter we consider the influence of dissipation on the transport properties of a Luttinger liquid. In the presence of a single impurity we find an exponential suppression of the conductance as a function of applied voltage and temperature at sufficiently low voltages and temperatures. In the third chapter we study a Luttinger liquid in the presence of two impurities and of an external magnetic field. The magnetic field splits the electrons with respect to their spin. We show that this system exhibits spin-filtering effect due to the resonance tunneling effect, which can also be achieved in this particular case, similarly to the case without the field. By tuning a single parameter, one may reach the resonance points when the transmission of spins with one spin-direction is dominating at low temperatures and voltages. The fourth chapter is related to the question of the competition of a periodic and random potential in a Luttinger liquid. The periodic potential alone produces a Mott-insulating state at sufficiently small amount of quantum fluctuations, measured by the interaction parameter K. The random potential alone leads to the Anderson-insulating state for small values of K. In the presence of both potentials and when the interaction is short-ranged, we do not find an intermediate Mott-glass phase, contrary to some studies. The fifth chapter deals with a thin superconducting film with a magnetic dot placed on top of it. We calculate the configurations of vortex-antivortex ground states for such system, finding a diversity of vortex-antivortex states as a function of parameters of the dot. The sixth chapter studies a thin superconducting film covered by magnetic dots of random magnetization. We show that this system is a realization of the two-dimensional XY model with random phase shifts. The latter model helps us to determine the phase diagram of our system

Spectrum of elementary excitations in Galilean-invariant integrable models

The spectrum of elementary excitations in one-dimensional quantum liquids is generically linear at low momenta. It is characterized by the sound velocity that can be related to the ground state energy. Here we study the spectrum at higher momenta in Galilean invariant integrable models. Somewhat surprisingly, we show that the spectrum at arbitrary momentum is fully determined by the properties of the ground state. We find general exact relations for the coefficients of several terms in the expansion of the excitation energy at low momenta and arbitrary interaction and express them in terms of the Luttinger liquid parameter. We apply the obtained formulas to the Lieb-Liniger model and obtain several new results.Comment: 5 page

Method of difference-differential equations for some Bethe ansatz solvable models

In studies of one-dimensional Bethe ansatz solvable models, a Fredholm integral equation of the second kind with a difference kernel on a finite interval often appears. This equation does not generally admit a closed-form solution and hence its analysis is quite complicated. Here we study a family of such equations concentrating on their moments. We find exact relations between the moments in the form of difference-differential equations. The latter results significantly advance the analysis, enabling one to practically determine all the moments from the explicit knowledge of the lowest one. As applications, first we study the moments of the quasimomentum distribution in the Lieb-Liniger model and find explicit analytical results. The latter moments determine several basic quantities, e.g., the $N$-body local correlation functions. We prove the equivalence between different expressions found in the literature for the three-body local correlation functions and find an exact result for the four-body local correlation function in terms of the moments of the quasimomentum distributions. We eventually find the analytical results for the three- and four-body correlation functions in the form of asymptotic series in the regimes of weak and strong interactions. Next, we study the exact form of the low-energy spectrum of a magnon (a polaron) excitation in the two-component Bose gas described by the Yang-Gaudin model. We find its explicit form, which depends on the moments of the quasimomentum distributions of the Lieb-Liniger model. Then, we address a seemingly unrelated problem of capacitance of a circular capacitor and express the exact result for the capacitance in the parametric form. In the most interesting case of short plate separations, the parametric form has a single logarithmic term. This should be contrasted with the explicit result that has a complicated structure of logarithms.Comment: 12 page

Field-theoretical approach to the Casimir-like interaction in a one-dimensional Bose gas

We study the fluctuation-induced interaction between two impurities in a weakly-interacting one-dimensional Bose gas using the field theoretical approach. At separations between impurities shorter and of the order of the healing length of the system, the induced interaction has a classical origin and behaves exponentially. At separations longer than the healing length, the interaction is of a quantum origin and scales as the third power of the inverse distance. Finite temperature destroys the quasi-long-range order of the Bose gas and, accordingly, the induced interaction becomes exponentially suppressed beyond the thermal length. We obtain analytical expressions for the induced interaction at zero and finite temperature that are valid at arbitrary distances. We discuss experimental realizations as well as possible formation of bound states of two impurities, known as bipolarons.Comment: 14 pages, 1 figure; accepted to Physical Review

Exact Results for the Moments of the Rapidity Distribution in Galilean-Invariant Integrable Models

We study a class of Galilean-invariant one-dimensional Bethe ansatz solvable models in the thermodynamic limit. Their rapidity distribution obeys an integral equation with a difference kernel over a finite interval, which does not admit a closed-form solution. We develop a general formalism enabling one to study the moments of the rapidity distribution, showing that they satisfy a difference-differential equation. The derived equation is explicitly analyzed in the case of the Lieb-Liniger model and the moments are analytically calculated. In addition, we obtained the exact information about the ground-state energy at weak repulsion. The obtained results directly enter a number of physically relevant quantities.Comment: 6 page

Thermal conductivity of the degenerate one-dimensional Fermi gas

We study heat transport in a gas of one-dimensional fermions in the presence of a small temperature gradient. At temperatures well below the Fermi energy there are two types of relaxation processes in this system, with dramatically different relaxation rates. As a result, in addition to the usual thermal conductivity, one can introduce the thermal conductivity of the gas of elementary excitations, which quantifies the dissipation in the system in the broad range of frequencies between the two relaxation rates. We develop a microscopic theory of these transport coefficients in the limit of weak interactions between the fermions.Comment: 17 page