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.

Dissipative Dynamics of a Heavy Impurity in a Bose Gas in the Strong Coupling Regime.

We study the motion of a heavy impurity in a one-dimensional Bose gas. The impurity experiences the friction force due to scattering off thermally excited quasiparticles. We present detailed analysis of an arbitrarily strong impurity-boson coupling in a wide range of temperatures within a microscopic theory. Focusing mostly on weakly interacting bosons, we derive an analytical result for the friction force and uncover new regimes of the impurity dynamics. Particularly interesting is the low-temperature T^{2} dependence of the friction force obtained for a strongly coupled impurity, which should be contrasted with the expected T^{4} scaling. This new regime applies to systems of bosons with an arbitrary repulsion strength. We finally study the evolution of the impurity with a given initial momentum. We evaluate analytically its nonstationary momentum distribution function. The impurity relaxation towards the equilibrium is a realization of the Ornstein-Uhlenbeck process in momentum space.

Quasiparticle Energy Relaxation in a Gas of One-Dimensional Fermions with Coulomb Interaction.

We consider a system of charged one-dimensional spin-1/2 fermions at low temperature. We study how the energy of a highly excited quasiparticle (or hole) relaxes toward the chemical potential in the regime of weak interactions. The dominant relaxation processes involve collisions with two other fermions. We find a dramatic enhancement of the relaxation rate at low energies, with the rate scaling as the inverse sixth power of the excitation energy. This behavior is caused by the long-range nature of the Coulomb interaction.

Exact Results for the Boundary Energy of One-Dimensional Bosons.

We study bosons in a one-dimensional hard-wall box potential. In the case of contact interaction, the system is exactly solvable by the Bethe ansatz, as first shown by Gaudin in 1971. Although contained in the exact solution, the boundary energy in the thermodynamic limit for this problem is only approximately calculated by Gaudin, who found the leading order result at weak repulsion. Here we derive an exact integral equation that enables one to calculate the boundary energy in the thermodynamic limit at an arbitrary interaction. We then solve such an equation and find the asymptotic results for the boundary energy at weak and strong interactions. The analytical results obtained from the Bethe ansatz are in agreement with the ones found by other complementary methods, including quantum Monte Carlo simulations. We study the universality of the boundary energy in the regime of a small gas parameter by making a comparison with the exact solution for the hard rod gas.

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.

Dynamics of a Mobile Impurity in a One-Dimensional Bose Liquid.

We develop a microscopic theory of a quantum impurity propagating in a one-dimensional Bose liquid. As a result of scattering off thermally excited quasiparticles, the impurity experiences the friction. We find that, at low temperatures, the resulting force scales either as the fourth or the eighth power of temperature, depending on the system parameters. For temperatures higher than the chemical potential of the Bose liquid, the friction force is a linear function of temperature. Our approach enables us to find the friction force in the crossover region between the two limiting cases. In the integrable case, corresponding to the Yang-Gaudin model, the impurity becomes transparent for quasiparticles and thus the friction force is absent. Our results could be further generalized to study other kinetic phenomena.

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.

Multigrid algorithms for tensor network states.

The widely used density matrix renormalization group (DRMG) method often fails to converge in systems with multiple length scales, such as lattice discretizations of continuum models and dilute or weakly doped lattice models. The local optimization employed by DMRG to optimize the wave function is ineffective in updating large-scale features. Here we present a multigrid algorithm that solves these convergence problems by optimizing the wave function at different spatial resolutions. We demonstrate its effectiveness by simulating bosons in continuous space and study nonadiabaticity when ramping up the amplitude of an optical lattice. The algorithm can be generalized to tensor network methods and combined with the contractor renormalization group method to study dilute and weakly doped lattice models.

Phase transition of interacting disordered bosons in one dimension.

Interacting bosons generically form a superfluid state. In the presence of disorder it can get converted into a compressible Bose glass state. Here we study such a transition in one dimension at moderate interaction using bosonization and renormalization group techniques. We derive the two-loop scaling equations and discuss the phase diagram. We find that the correlation functions at the transition are characterized by universal exponents in a finite region around the fixed point.

Super-rough glassy phase of the random field XY model in two dimensions.

We study both analytically, using the renormalization group (RG) to two loop order, and numerically, using an exact polynomial algorithm, the disorder-induced glass phase of the two-dimensional XY model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e., below the critical temperature T(c), the disorder and thermally averaged correlation function B(r) of the phase field θ(x), B(r)=([θ(x)-θ(x+r)](2)) behaves, for r >> a, as B(r) is approximately equal to A(τ)ln(2)(r/a) where r=|r| and a is a microscopic length scale. We derive the RG equations up to cubic order in τ=(T(c)-T)/T(c) and predict the universal amplitude A(τ)=2τ(2)-2τ(3)+O(τ(4)). The universality of A(τ) results from nontrivial cancellations between nonuniversal constants of RG equations. Using an exact polynomial algorithm on an equivalent dimer version of the model we compute A(τ) numerically and obtain a remarkable agreement with our analytical prediction, up to τ≈0.5.

Spin filtering by field-dependent resonant tunneling.

We consider theoretically transport in a spinful one-channel interacting quantum wire placed in an external magnetic field. For the case of two pointlike impurities embedded in the wire, under a small voltage bias the spin-polarized current occurs at special points in the parameter space, tunable by a single parameter. At sufficiently low temperatures complete spin polarization may be achieved, provided repulsive interaction between electrons is not too strong.

Transport in a dissipative Luttinger liquid.

We study theoretically the transport through a single impurity in a one-channel Luttinger liquid coupled to a dissipative (Ohmic) bath. For nonzero dissipation, the single impurity is always a relevant perturbation which suppresses transport strongly. At zero temperature, the current voltage relation of the link is I approximately exp(-E0/eV), where E0 approximately eta/kappa and kappa denotes the compressibility. At nonzero temperature T, the linear conductance is proportional to exp(-sqrt(CE0/kBT)). The decay of Friedel oscillation saturates for a distance larger than L(eta) approximately 1/eta from the impurity.

Absence of the Mott glass phase in 1D disordered fermionic systems.

We study the competition between the Mott and the Anderson insulating state in a one-dimensional disordered fermionic system. The notorious difficulties associated with strong coupling phases are avoided by using a new description in terms of the kink energies (or instanton surface tension) of the electronic displacement pattern. The approach has some similarities to the description of the flat phase of surfaces undergoing a roughening transition. Tracing back both a finite compressibility and a nonzero ac conductivity to vanishing kink energy we exclude the existence of an intermediate Mott-glass phase in systems with short range interaction only. In systems with long range interaction the Anderson insulating phase exhibits the features of a Mott glass however. The phase diagram is constructed from combining the information from the renormalization group flow, the kink energy, and simple rescaling analysis.