Ground State Energy of Unitary Fermion Gas with the Thomson Problem Approach

Abstract

The dimensionless universal coefficient ξ defines the ratio of the unitary fermions energy density to that for the ideal non-interacting ones in the non-relativistic limit with T = 0. The classical Thomson Problem is taken as a nonperturbative quantum many-body arm to address the ground state energy including the low energy nonlinear quantum fluctuation/correlation effects. With the relativistic Dirac continuum field theory formalism, the concise expression for the energy density functional of the strongly interacting limit fermions at both finite temperature and density is obtained. Analytically, the universal factor is calculated to be ξ = With the further developments of the Bardeen-CooperSchrieffer theory, the possibility about the existence of the fermions superfluidity in the dilute gas system motivates widely theoretical studies and experimental efforts. Since DeMarco and Jin achieved the Fermi degeneracy[1], the ultra-cold fermion atoms gas has stirred intense interest about the fundamental Fermi-Dirac statistical physics in the strongly interacting limit.Across the Feshbach resonance regime, the interaction changes from weakly to strongly attractive according to the magnitude of the magnetic field. At the midpoint of this crossover unitary regime from Bardeen-CooperSchrieffer(BCS) to Bose-Einstein condensation(BEC), the scattering length will diverge due to the existence of a zero-energy bound state for the two-body system. In this limit, the only dimensionful parameter is the Fermi momentum k f at T = 0. The corresponding energy scale is the Fermi kinetic energy ε f = k 2 f /(2m), while m is the fermion mass. According to dimensional analysis, the system details do not contribute to the thermodynamics properties, i.e., the thermodynamics properties are universal [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The energy density should be proportional to that of a free Fermi gas E/V = ξ (E/V ) f ree = ξ How to approach the exact value of ξ analytically is a bewitching topic in the Fermi-Dirac statistical physics. To attack this intriguing topic is a seriously difficult problem in many-body theory. The essential task is how to in- * chenjs@iopp.ccnu.edu.cn corporate the nonlinear quantum fluctuation/correlation effects into the thermodynamics by going beyond any naive loop diagram expansions or the lowest order mean field theory. To our knowledge, the hitherto considerations looking for ξ have been solely limited in the nonrelativistic frameworks and with quite different results. How about a relativistic Dirac phenomenology attempt?Motivation: Essentially, the unitary physics with infinite scattering lengths is quite similar to the universal strongly instantaneous Coulomb correlation thermodynamics in a compact nuclear confinement environment resulting from the competition of long and short range forces [27]. At the crossover point, the cross-section between the two-body particle is limited by σ ∼ 4π/k 2 (k is the relative wavevector of the colliding particles), while the gauge vector b...

Lattice Monte Carlo calculations for unitary fermions in a finite box

“…The fit result over the interval L = [10,16] is shown in Fig. 7, and at infinite volume we obtain E/E F ree = 0.2130 (26). Both our N O = 1 and N O = 5 results for the ground state energy of four unitary fermions are consistent with the benchmark calculation reported [69], within the given uncertainties.…”

“…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.…”

Quantum Many-Body Calculations using Body-Centered Cubic Lattices

“…The constant ξ is sometimes called the Bertsch parameter and has been measured in many experiments using ultracold trapped atoms [30][31][32][33][34][35][36][37][38][39][40][41]. It has also been calculated by analytical methods [42][43][44][45][46][47][48][49][50][51][52][53][54]. In addition, a substantial number of numerical studies have been made using lattice and continuum quantum Monte Carlo simulations as well as other techniques .…”

Generalized Simulated Annealing for Global Optimization: The GenSA Package

“…The objective of the Thomson problem is to find the global energy minimum of N equal point charges on a sphere (Thomson, 1904). The Thomson model has been widely studied in physics (Erber and Hockney, 1991;Altschuler et al, 1994;Morris et al, 1996;Xiang et al, 1997;Xiang and Gong, 2000a;Levin and Arenzon, 2003;Wales and Ulker, 2006;Altschuler and Pérez-Garrido, 2006;Ji-Sheng, 2007;Wales et al, 2009). In the Thomson model, the energy of N point charges constrained on a sphere is…”