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. 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. arXiv:1505.02290v2 [cond-mat.str-el] 15

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 non-trivial kinetics arises only beyond the local-induction approximation and is governed by three-kelvon collisions; corresponding kinetic equation is derived. On the basis of the kinetic equation, 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.PACS numbers: 67.40. Vs, 03.75.Lm, 47.32.Cc The distortion waves on a vortex filament-Kelvin waves (KW)-have been known for more than a century [1]. Superfluids with their topological (quantized) vorticity form a natural domain for KW [2]. Nowadays there is a strong interest to the non-linear aspects of KW associated with studying low-temperature superfluid turbulence of 4 He [3,4,5,6,7,8,9,10], as well as vortex dynamics in ultra-cold atomic gases [11,12].The superfluid turbulence [2, 13] is a chaotic tangle of vortex lines. In the absence of the normal component (T → 0 limit), KW play a crucial part in the vortex tangle relaxational dynamics. In contrast to a normal fluid, the quantization of the velocity circulation in a superfluid makes it impossible for the vortex line to relax by gradually slowing down. The only allowed way of relaxation is reducing the total line length. At T = 0 even this generic scenario becomes non-trivial, as the total line length is, to a very good approximation, a constant of motion. In the scenario proposed by one of us [3], the vortex line length-in the form of KW generated in the process of vortex line reconnections-cascades from the main length scale (typical interline separation, R 0 ) to essentially lower length scales; ultimately decaying into phonons, as it was pointed out by Vinen [4,8].A very specific feature of KW cascade is that the intrinsic vortex line dynamics in the local-induction approximation (LIA) (for an introduction, see, e.g., [2,13]) controlled by the small parameter 1/ ln(R 0 /ξ), with ξ the vortex core radius, is subject to a specific curvatureconservation constraint rendering it unable to support the cascade process [3] (see also below). Within LIA, an "external" ingredient of the vortex line dynamics-the vortex line crossings with subsequent reconnections-is required to push the KW cascade down towards arbitrarily small wavelengths. The most characteristic feature of this LIA scenario, distinguishing it from typical non-linear cascades, is the fragmentation of the vortex lines due to local self-crossings [3]; we will thus refer to this scenario as fragmentational scenario.Experimentally, the main consequence of the existence of a cascade regime, no matter what is its microscopic nature, is independence of the relaxation time of superfluid turbulence on temperature in the T → 0 limit. Davis A general question arises, however, of ho...

Precise understanding of strongly interacting fermions, from electrons in modern materials to nuclear matter, presents a major goal in modern physics. However, the theoretical description of interacting Fermi systems is usually plagued by the intricate quantum statistics at play. Here we present a cross-validation between a new theoretical approach, Bold Diagrammatic Monte Carlo (BDMC), and precision experiments on ultra-cold atoms. Specifically, we compute and measure with unprecedented accuracy the normal-state equation of state of the unitary gas, a prototypical example of a strongly correlated fermionic system. Excellent agreement demonstrates that a series of Feynman diagrams can be controllably resummed in a non-perturbative regime using BDMC. This opens the door to the solution of some of the most challenging problems across many areas of physics.

The Luttinger-Ward functional Φ [G], which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's 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 non-interacting Green's function G0 converges to the correct physical branch of Σ 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.

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. Experimental progress in the field of atomic quantum gases has led to a new approach to quantum many-body physics. In particular, the combination of quantum degenerate and strongly interacting Fermi gases [1,2] with optically induced lattice potentials [3] now allows the study of a centerpiece of quantum condensed matter physics, the Fermi-Hubbard model [4]. The high level of control over the atomic systems has led to the concept of quantum simulation, which for the case of the Fermi-Hubbard model is expected to provide answers to intriguing open questions of frustrated magnetism and d-wave superfluidity [5]. Recent experiments [6,7] have indeed demonstrated that the strongly correlated regime of the repulsive FermiHubbard model is experimentally accessible and the emergence of a Mott insulating state has been observed. In this Letter, we succeed in performing a quantitative simulation of the Fermi-Hubbard model using cold atoms. The level of precision of the experiment enables us to determine the entropy and the temperature of the system, and thereby to quantify the approach to the low temperature phases.The main challenge for the quantum simulation of the Fermi-Hubbard model is a further reduction in temperature. Here the lack of a quantitative thermometry method in the lattice is a key obstacle. For strongly correlated bosonic systems thermometry has recently been demonstrated by direct comparison with quantum Monte Carlo simulations [8] or by using the boundary of two spin polarized clouds [9]. In the fermionic case, previous methods to determine the temperature could be used only in limiting regimes of the Hubbard model, namely, the noninteracting [10,11] and zero-tunneling [6,12] regimes. However, intermediate interactions are most interesting for quantum simulation of the Fermi-Hubbard model and no reliable thermometry method has been available up to now.In both the metallic and Mott insulating regimes an accurate measurement of the double occupancy provides direct access to thermal excitations. We analyze the crossover from thermal creation of double occupancies to thermal depletion which is unique to a trapped system (see Fig. 1). The variability of the d...

The strongly-correlated regime of the BCS-BEC crossover 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/εF for the transition temperature between the normal and the superfluid states as a function of the scattering length with the maximum on the BEC side. At unitarity, we confirm that Tc/εF = 0.152(7). PACS numbers:In the area of ultracold gases, the problem of the crossover between the Bardeen-Cooper-Schrieffer pairing and the Bose-Einstein condensation of composite molecules (the so-called BCS-BEC crossover) has recently received a lot of theoretical and experimental attention [1]. A dilute two-component Fermi gas, where the inter-particle distance is much larger than the interaction range, features a remarkable universality at low temperatures. Since the interaction is completely described by the s-wave scattering length a, the only physically relevant coupling parameter is κ = 1/k F a, where k F is the Fermi momentum. One thus obtains a unified and universal description of systems as diverse as ultracold fermionic gases in magnetic or optical traps [1], fermions in optical lattices, inner crusts of neutron stars [2,3], and, plausibly, excitonic condensates [4].In the limit κ → −∞, the Fermi gas is described by the BCS theory, while for κ → +∞ the fermions pair into compact bosonic molecules which then form a BEC state below the critical temperature. Separating these extreme states is a strongly correlated regime which features the so-called unitary point κ = 0. At unitarity, the scattering length is infinite and the interaction thus drops out of the relations between different thermodynamic potentials making these relations formally identical to those of a non-interacting Fermi gas [5]. On the experimental side, using the technique of a (wide) Feshbach resonance in a system of cold atoms, one can traverse the whole range of parameter κ from the BEC to the BCS limit [1].Despite considerable recent investigation, the quantitative description of the BEC-BCS crossover is far from being complete, even for the simplest case of the equal mixture of two components. Due to the strongly correlated nature of the problem, analytical mean-field-type calculations (e.g. [6,7,8]) unavoidably involve approximations, the accuracy of which is difficult to access unless the exact result is known. Renormalization group treatments can be carried out as expansions in either ǫ = 4−d [9], or 1/N F (where N F is the number of fermion species) [10,11], but the applicability of these calculations to the physically relevant case of d = 3 and N F = 2 is not known a priori.Numerical studies of fermionic systems are computationally demanding and further complicated by the need to study the limit of small densities to access the universal regime. Some numerical techniques avoid the fermionic sign problem with a help of uncontrollable approximat...

Diagrammatic Monte Carlo (DiagMC) is a numeric technique that allows one to calculate quantities specified in terms of diagrammatic expansions, the latter being a standard tool of many-body quantum statistics. The sign problem that is typically fatal to Monte Carlo approaches, appears to be manageable with DiagMC. Starting with a general introduction to the principles of DiagMc, we present a detailed description of the DiagMC scheme for interacting fermions (Hubbard model), as well as the first illustrative results for the equations of state

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 Feynman's 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 expansion and with single-site and cluster dynamical mean-field theory.PACS numbers: 02.70. Ss, 05.10.Ln, 71.10.Fd, 05.30.Fk Advancing first-principle simulations of fermionic manyparticle systems is notoriously difficult due to an exponential computational complexity [1]. One of the most famous examples is the Hubbard model [2, 3] whose phase diagram remains highly controversial,whereĉ † iσ creates a fermion with spin projection σ =↑, ↓ on site i,n iσ =ĉ † iσĉiσ , . . . restricts summation to neighboring lattice sites, and t, U and µ are the hopping amplitude, the on-site repulsion, and the chemical potential respectively.The Hubbard model is of great technological and scientific importance since it is believed by many to be the central model for high-temperature superconductors [3]. Moreover, it can nowadays be realized with ultracold atoms in optical lattices [4] where Mott physics [5,6] has been observed recently. The prospect of using cold atomic gases as building blocks of quantum simulators is thus becoming a reality. However, claiming the equation of state, either numerically or experimentally, can only be substantiated on the basis of unbiased tools. Not only numerical methods need to be tested against known unbiased results, but also quantum emulators need to be validated by benchmarking them at ever lower temperatures. In the bosonic case this can be done with high precision [7], but for fermions it is still an open question what method can be used to accurately describe the lowtemperature regime of the model. Precise numerical solutions of the Hubbard model can only be obtained in the particlehole symmetric case n σ = 1/2, where the sign problem is absent in quantum Monte Carlo (QMC) simulations. In all other cases circumventing exponential scaling necessarily involves approximations with systematic errors that are hard to assess and control.In general terms, Diagrammatic Monte Carlo (DiagMC) [8] is a numerical approach based on stochastic sampling of Feynman diagrams [9, 10] to increasingly higher orders in the coupling constant-rather than evaluating integrals over internal variables for each diagram, one samples their momenta, imaginary times and expansion orders stochastically. So far, DiagMC algorithms have been successfully used for obtaining unbiased solutions for polaron problems (for the latest work, see Ref.[11]).Nevertheless, the crucial question remained unclear of whether the DiagMC approach could be applied to true manybody systems in the thermodynamic limit at physically interesting temper...