Evgeny Kozik

Kelvin-Wave Cascade and Decay of Superfluid Turbulence

Kelvin waves (kelvons), the distortion waves on vortex lines, play a key part in the relaxation of superfluid turbulence at low temperatures. We present a weak-turbulence theory of kelvons. We show that nontrivial kinetics arises only beyond the local-induction approximation and is governed by three-kelvon collisions; a corresponding kinetic equation is derived. We prove the existence of Kolmogorov cascade and find its spectrum. The qualitative analysis is corroborated by numeric study of the kinetic equation. The application of the results to the theory of superfluid turbulence is discussed.

Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms

Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.

Quantitative determination of temperature in the approach to magnetic order of ultracold fermions in an optical lattice.

We perform a quantitative simulation of the repulsive Fermi-Hubbard model using an ultracold gas trapped in an optical lattice. The entropy of the system is determined by comparing accurate measurements of the equilibrium double occupancy with theoretical calculations over a wide range of parameters. We demonstrate the applicability of both high-temperature series and dynamical mean-field theory to obtain quantitative agreement with the experimental data. The reliability of the entropy determination is confirmed by a comprehensive analysis of all systematic errors. In the center of the Mott insulating cloud we obtain an entropy per atom as low as 0.77k(B) which is about twice as large as the entropy at the Néel transition. The corresponding temperature depends on the atom number and for small fillings reaches values on the order of the tunneling energy.

Theory of Decay of Superfluid Turbulence in the Low-Temperature Limit

We review the theory of relaxational kinetics of superfluid turbulence—a tangle of quantized vortex lines—in the limit of very low temperatures when the motion of vortices is conservative. While certain important aspects of the decay kinetics depend on whether the tangle is non-structured, like the one corresponding to the Kibble-Zurek picture, or essentially polarized, like the one that emulates the Richardson-Kolmogorov regime of classical turbulence, there are common fundamental features. In both cases, there exists an asymptotic range in the wavenumber space where the energy flux is supported by the cascade of Kelvin waves (kelvons)—precessing distortions propagating along the vortex filaments.At large enough wavenumbers, the Kelvin-wave cascade is supported by three-kelvon elastic scattering. At zero temperature, the dissipative cutoff of the Kelvin-wave cascade is due to the emission of phonons, in which an elementary process converts two kelvons with almost opposite momenta into one bulk phonon.Along with the standard set of conservation laws, a crucial role in the theory of low-temperature vortex dynamics is played by the fact of integrability of the local induction approximation (LIA) controlled by the parameter Λ=ln (λ/a0), with λ the characteristic kelvon wavelength and a0 the vortex core radius. While excluding a straightforward onset of the pure three-kelvon cascade, the integrability of LIA does not plug the cascade because of the natural availability of the kinetic channels associated with vortex line reconnections.We argue that the crossover from Richardson-Kolmogorov to the Kelvin-wave cascade is due to eventual dominance of local induction of a single line over the collective induction of polarized eddies, which causes the breakdown of classical-fluid regime and gives rise to a reconnection-driven inertial range.

Critical Temperature Curve in BEC-BCS Crossover

The strongly correlated regime of the crossover from Bardeen-Cooper-Schrieffer pairing to Bose-Einstein condensation can be realized by diluting a system of two-component fermions with a short-range attractive interaction. We investigate this system via a novel continuous-space-time diagrammatic determinant Monte Carlo method and determine the universal curve Tc/epsilonF for the transition temperature between the normal and the superfluid states as a function of the scattering length with the maximum on the Bose-Einstein condensation side. At unitarity, we confirm that Tc/epsilonF=0.152(7).

Thermodynamics of the 3D Hubbard Model on Approaching the Neel Transition

We study the thermodynamic properties of the 3D Hubbard model for temperatures down to the Néel temperature by using cluster dynamical mean-field theory. In particular, we calculate the energy, entropy, density, double occupancy, and nearest-neighbor spin correlations as a function of chemical potential, temperature, and repulsion strength. To make contact with cold-gas experiments, we also compute properties of the system subject to an external trap in the local density approximation. We find that an entropy per particle S/N ≈ 0.65(6) at U/t = 8 is sufficient to achieve a Néel state in the center of the trap, substantially higher than the entropy required in a homogeneous system. Precursors to antiferromagnetism can clearly be observed in nearest-neighbor spin correlators.

Scale-separation scheme for simulating superfluid turbulence: Kelvin-wave cascade

A Kolmogorov-type cascade of Kelvin waves-the distortion waves on vortex lines-plays a key part in the relaxation of superfluid turbulence at low temperatures. We propose an efficient numeric scheme for simulating the Kelvin-wave cascade on a single vortex line. This idea is likely to be generalizable for a full-scale simulation of different regimes of superfluid turbulence. With the new scheme, we are able to unambiguously resolve the cascade spectrum exponent, and thus to settle the controversy between recent simulations of Vinen, Tsubota, and Mitani [Phys. Rev. Lett. 91, 135301 (2003)]] and recently developed analytic theory [Phys. Rev. Lett. 92, 035301 (2004)]].

Diagrammatic Monte Carlo for correlated fermions

We show that Monte Carlo sampling of the Feynman diagrammatic series (DiagMC) can be used for tackling hard fermionic quantum many-body problems in the thermodynamic limit by presenting accurate results for the repulsive Hubbard model in the correlated Fermi liquid regime. Sampling Feynmans diagrammatic series for the single-particle self-energy, we can study moderate values of the on-site repulsion (U/t~4) and temperatures down to T/t=1/40. We compare our results with high-temperature series expansions and with single-site and cluster dynamical mean-field theory.

Vortex-phonon interaction

Kelvin waves--helical waves on quantized vortex lines--are the normal modes of vortices in a superfluid. At zero temperature, the only dissipative channel of vortex dynamics is phonon emission. Starting with the hydrodynamic action, we derive the Hamiltonian of vortex-phonon interaction, thereby reducing the problem of the interaction of Kelvin waves with sound to inelastic elementary excitation scattering. On the basis of this formalism, we calculate the rate of sound radiation by superfluid turbulence at zero temperature and estimate the value of short-wavelength cutoff of the turbulence spectrum.

Nonexistence of the Luttinger-Ward functional and misleading convergence of skeleton diagrammatic series for hubbard-like models.

The Luttinger-Ward functional Φ[G], which expresses the thermodynamic grand potential in terms of the interacting single-particle Greens function G, is found to be ill defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy Σ[G]∝δΦ[G]/δG is not a single-valued functional of G: in addition to the physical solution for Σ[G], there exists at least one qualitatively distinct unphysical branch. This result is demonstrated for several models: the Hubbard atom, the Anderson impurity model, and the full two-dimensional Hubbard model. Despite this pathology, the skeleton Feynman diagrammatic series for Σ in terms of G is found to converge at least for moderately low temperatures. However, at strong interactions, its convergence is to the unphysical branch. This reveals a new scenario of breaking down of diagrammatic expansions. In contrast, the bare series in terms of the noninteracting Greens function G0 converges to the correct physical branch of Σ in all cases currently accessible by diagrammatic Monte Carlo calculations. In addition to their conceptual importance, these observations have important implications for techniques based on the explicit summation of the diagrammatic series.