The 17/5 spectrum of the Kelvin-wave cascade

Direct numeric simulation of the Biot-Savart equation readily resolves the 17/5 spectrum of the Kelvin-wave cascade from the 11/3 spectrum of the non-local (in the wavenumber space) cascade scenario by L'vov and Nazarenko. This result is a clear-cut visualisation of the unphysical nature of the 11/3 solution, which was established earlier on the grounds of symmetry.Comment: 2 pages, 1 figur

Kolmogorov and Kelvin-Wave Cascades of Superfluid Turbulence at T=0: What is in Between?

As long as vorticity quantization remains irrelevant for the long-wave physics, superfluid turbulence supports a regime macroscopically identical to the Kolmogorov cascade of a normal liquid. At high enough wavenumbers, the energy flux in the wavelength space is carried by individual Kelvin-wave cascades on separate vortex lines. We analyze the transformation of the Kolmogorov cascade into the Kelvin-wave cascade, revealing a chain of three distinct intermediate cascades, supported by local-induction motion of the vortex lines, and distinguished by specific reconnection mechanisms. The most prominent qualitative feature predicted is unavoidable production of vortex rings of the size of the order of inter-vortex distance.Comment: 4 RevTex pages, 1 figure. Quantitative analysis of the regime 2 has been revise

Comment on "Dispersive bottleneck delaying thermalization of turbulent Bose-Einstein Condensates" by Krstulovic and Brachet [arXiv:1007.4441]

We reveal the connection of the recent numerical observations of Krstulovic and Brachet [arXiv:1007.4441] with the general theory of relaxation kinetics of the strongly non-equilibrium Bose-Einstein condensates.Comment: comment on arXiv:1007.4441, published version, minor stylistic change

Combinatorial summation of Feynman diagrams: Equation of state of the 2D SU(N) Hubbard model

We introduce a universal framework for efficient summation of connected or skeleton Feynman diagrams for generic quantum many-body systems. It is based on explicit combinatorial construction of the sum of the integrands by dynamic programming, at a computational cost that can be made only exponential in the diagram order. We illustrate the technique by an unbiased diagrammatic Monte Carlo calculation of the equation of state of the $2D$ $SU(N)$ Hubbard model in an experimentally relevant regime, which has remained challenging for state-of-the-art numerical methods.Comment: 8 pages, 4 figure

Non-existence of the Luttinger-Ward functional and misleading convergence of skeleton diagrammatic series for Hubbard-like models

The Luttinger-Ward functional $\Phi[\mathbf{G}]$, which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function $\mathbf{G}$, is found to be ill-defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy $\mathbf{\Sigma}[\mathbf{G}] \propto \delta\Phi[\mathbf{G}]/\delta \mathbf{G}$ is not a single-valued functional of $\mathbf{G}$: in addition to the physical solution for $\mathbf{\Sigma}[\mathbf{G}]$, 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 $\mathbf{\Sigma}$ in terms of $\mathbf{G}$ 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 non-interacting Green's function $\mathbf{G}_0$ converges to the correct physical branch of $\mathbf{\Sigma}$ in all cases currently accessible by diagrammatic Monte Carlo. Besides their conceptual importance, these observations have important implications for techniques based on the explicit summation of diagrammatic series.Comment: 5 pages, 5 figure

Strange Metal to Insulator Transitions in the Lowest Landau Level

We study the microscopic model of electrons in the partially-filled lowest Landau level interacting via the Coulomb potential by the diagrammatic theory within the GW approximation. In a wide range of filling fractions and temperatures, we find a homogeneous non-Fermi liquid (nFL) state similar to that found in the Sachdev-Ye-Kitaev (SYK) model, with logarithmic corrections to the anomalous dimension. In addition, the phase diagram is qualitatively similiar to that of SYK: a first-order transition terminating at a critical end-point separates the nFL phase from a band insulator that corresponds to the fully-filled Landau level. This critical point, as well as that of the SYK model -- whose critical exponents we determine more precisely -- are shown to both belong to the Van der Waals universality class. The possibility of a charge density wave (CDW) instability is also investigated, and we find the homogeneous nFL state to extend down to the ground state for fillings $0.2 \lesssim \nu \lesssim 0.8$, while a CDW appears outside this range of fillings at sufficiently low temperatures. Our results suggest that the SYK-like nFL state should be a generic feature of the partially-filled lowest Landau level at intermediate temperatures.Comment: 11+6 pages, 6 figure

Ground state phase diagram of the repulsive fermionic t−t′t-t^{\prime} Hubbard model on the square lattice from weak-coupling

We obtain a complete and exact in the weak-coupling limit ($U \rightarrow 0$) ground state phase diagram of the repulsive fermionic Hubbard model on the square lattice for filling factors $0 < n < 2$ and next-nearest-neighbour hopping amplitudes $0 \le t^{\prime} \le 0.5$. Phases are distinguished by the symmetry and the number of nodes of the superfluid order parameter. The phase diagram is richer than may be expected and typically features states with a high --- higher than that of the fundamental mode of the corresponding irreducible representation --- number of nodes. The effective coupling strength in the Cooper channel $\lambda$, which determines the critical temperature $T_c$ of the superfluid transition, is calculated in the whole parameter space and regions with high values of $\lambda$ are identified. It is shown that besides the expected increase of $\lambda$ near the Van Hove singularity line, joining the ferromagnetic and antiferromagnetic points, another region with high values of $\lambda$ can be found at quarter filling and $t^{\prime}=0.5$ due to the presence of a line of nesting at $t^{\prime} \ge 0.5$. The results can serve as benchmarks for controlled non-perturbative methods and guide the ongoing search for high-$T_c$ superconductivity in the Hubbard model.Comment: 11 Pages, 9 Figure

Fulde-Ferrell-Larkin-Ovchinnikov pairing as leading instability on the square lattice

We study attractively interacting spin-1/2 fermions on the square lattice subject to a spin population imbalance. Using unbiased diagrammatic Monte Carlo simulations we find an extended region in the parameter space where the Fermi liquid is unstable towards formation of Cooper pairs with non-zero center-of-mass momentum, known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. In contrast to earlier mean-field and quasi-classical studies we provide quantitative and well-controlled predictions on the existence and location of the relevant Fermi-liquid instabilities. The highest temperature where the FFLO instability can be observed is about half of the superfluid transition temperature in the unpolarized system.Comment: 7 pages, 4 figures; v2: improved references and discussion, added calculations with larger cutoff order that corroborate our earlier result

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. The 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 [1] and recently developed analytic theory [2]. [1] W.F. Vinen, M. Tsubota and A. Mitani, Phys. Rev. Lett. 91, 135301 (2003). [2] E.V. Kozik and B.V. Svistunov, Phys. Rev. Lett. 92, 035301 (2004).Comment: 4 pages, RevTe