Sign-free stochastic mean-field approach to strongly correlated phases of ultracold fermions

Abstract

We propose a new projector quantum Monte-Carlo method to investigate the ground state of ultracold fermionic atoms modeled by a lattice Hamiltonian with on-site interaction. The many-body state is reconstructed from Slater determinants that randomly evolve in imaginary-time according to a stochastic mean-field motion. The dynamics prohibits the crossing of the exact nodal surface and no sign problem occurs in the Monte-Carlo estimate of observables. The method is applied to calculate ground-state energies and correlation functions of the repulsive two-dimensional Hubbard model. Numerical results for the unitary Fermi gas validate simulations with nodal constraints.PACS. 03.75.Ss, 05.30.Fk, 71.10.Fd 1. Introduction. Since the experimental achievement of Fermi degeneracy [1] with an atomic vapor, a considerable attention has been attracted by the physics of dilute ultracold fermions. The ability to tune many parameters, such as temperature, density or inter-particle interactions, makes atomic Fermi gases ideal candidates to understand a wealth of phenomena relevant for physical systems ranging from nuclear matter to high-temperature superconductors. Of particular interest is the strongly interacting regime in the transition from Bardeen-Cooper-Schrieffer (BCS) superfluidity of Cooper pairs to Bose-Einstein condensation (BEC) of atomic dimers. In the crossover regime, Fermi condensates have been observed on both the BCS and the BEC sides of a magnetically controlled Feshbach resonance [2][3][4]. The unitary regime, where the scattering length diverges, is particularly studied [5-7] to investigate the universal features of the fermionic quantum many-body problem. Using several standing laser beams, ultracold atoms can also be loaded in optical lattices where they experience all the strong many-body correlations described by the Hubbard model of solid-state physics [8,9]. Optical lattice setups may allow for engineering quantum spin models [10], fractional quantum Hall effect [11], non-Abelian gauge potentials [12] or quantum information processing [13]. In this paper, we investigate a new Monte-Carlo scheme to study strongly correlated ground states of ultracold fermions interacting on a lattice. The projection onto the ground state is performed through a reformulation of the imaginary-time Schrödinger equation in terms of Slater determinants undergoing a Brownian motion driven by the Hartree-Fock Hamiltonian. Such exact stochastic extensions of the mean-field approaches have been recently proposed for boson systems [14,15]. Up to now, the fermionic counterpart uses Slater determinants whose orbitals evolve under their own mean-field, supplemented with a stochastic one-particle-one-hole excitation [16,17]. Unfortunately, the sampling generally suffers from negative weight trajectories that cause an exponential decay of the signal-tonoise ratio, which is known as the sign problem. The convergence issue of such a Monte-Carlo calculation plagued by negative "probabilities" belongs to the class of NP hard problems and a ...

Thermodynamics of Ultracold Fermi Gases

“…Interestingly, using the experimental value ∆ 0 = 0.44 E F from [51] (which agrees with a recent calculation using an unbiased Monte-Carlo computation [158]), we obtain 5/8(∆ 0 /E F ) 2 0.12, showing that equation (4.16) approximately remains valid even for resonant interactions. It would be interesting to investigate more deeply this behavior and how it depends on the interaction strength.…”

“…In Fig.5.9 we compare the values of η c extracted from our data to lower bounds 1 − 2∆ 0 /µ 1 , using values of the gap measured by the MIT group in [166], or calculated in [179,60,182,158]. Quite surprisingly, the § In the BEC limit, it is possible to add extra majority atoms to a super uid without breaking it.…”

“…Comparison between the experimental η c values determined from our work (black dots) to the lower bounds 1 − 2∆ 0 /µ 1 provided by the single-particle excitation gap. The stars are given by the experimental ∆ 0 values from the MIT group [166], the open squares from diagrammatic calculations [179], the open circles from the most recent Fixed Node Monte Carlo calculations [60], the crosses from a Quantum Monte Carlo calculation extrapolated at T = 0 [182], and the open triangle from a zerotemperature Quantum Monte Carlo calculation [158].…”

“…In the unitary limit, the single-particle gap ∆ 0 = 0.44 E F is known more precisely, from experiment [166] and unbiased Quantum Monte Carlo calculations [158]. Using η c = 0.065 (2) and the relation µ = ξ s E F , ξ s = 0.41(1), we obtain:…”

Lattice Monte Carlo calculations for unitary fermions in a finite box

“…Several experimental groups have measured the Bertsch parameter using a variety techniques involving ultra-cold trapped atoms [2][3][4][5][6][7][8][9][10][11][12][13]. On the theoretical side, in addition to analytical calculations [14][15][16][17][18][19][20][21][22][23][24][25][26][27], there has also been a substantial number of numerical studies of unitary fermions from the microscopic theory using quantum Monte Carlo and other techniques [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. Many of these studies use the Bertsch parameter as a benchmark calculation.…”