Condensation temperature of interacting Bose gases with and without disorder

Abstract

The momentum-shell renormalization group ͑RG͒ is used to study the condensation of interacting Bose gases without and with disorder. First of all, for the homogeneous disorder-free Bose gas the interactioninduced shifts in the critical temperature and chemical potential are determined up to second order in the scattering length. The approach does not make use of dimensional reduction and is thus independent of previous derivations. Secondly, the RG is used together with the replica method to study the interacting Bose gas with delta-correlated disorder. The flow equations are derived and found to reduce, in the high-temperature limit, to the RG equations of the classical Landau-Ginzburg model with random-exchange defects. The random fixed point is used to calculate the condensation temperature under the combined influence of particle interactions and disorder.The realization of Bose-Einstein condensation in ultracold atomic gases has renewed the interest in the quantumstatistical properties of interacting Bose gases around the phase transition. One particular focus of theoretical work in this area concerns the calculation of the critical temperature T c of the homogeneous Bose gas as a function of the scattering length a. Due to its nonperturbative character, this apparently simple problem turned out to present a considerable theoretical challenge. For a long time, the leading-order functional dependence of the shift in T c on the scattering length remained highly controversial. It was even unclear whether the interactions lead to an upward or downward shift in the critical temperature ͓1͔. Only recently, a generally accepted result has emerged ͓2-4͔. It was shown that the critical temperature of an interacting Bose gas with spatial density n obeys the relation T c T c 0 = 1 + c 1 an 1/3 + ͓c 2 Ј ln͑an 1/3 ͒ + c 2 Љ͔a 2 n 2/3 + O͑a 3 n͒, ͑1͒where T c 0 denotes the critical temperature of an ideal Bose gas with the same density. To calculate the coefficients c i , one can make use of dimensional reduction ͓1,5͔ and perturbatively match the full quantum-mechanical problem to a classical field theory with a lower-͑i.e., three-͒ dimensional action ͓4͔. The coefficient c 2 Ј can then be determined by perturbation theory alone, whereas the evaluation of c 1 and c 2 Љ requires nonperturbative input from the classical theory. This input was calculated in Refs. ͓6-8͔ with so far unsurpassed accuracy by means of Monte Carlo lattice simulations.Even after the qualitative and quantitative behavior of the T c shift had been solidly established ͓2-4,6,8͔, a large number of papers continued to appear that presented further calculations of critical properties of interacting Bose gases. The main purpose of these works was not to correct or improve the previous results, but to introduce, test, and refine alternative approaches. The applied methods comprise, e.g., linear ␦ expansion ͓9-14͔, variational perturbation theory ͓15-17͔, and the exact renormalization group ͓18-20͔.The present paper follows this research line of investigat...

Effect of random on-site energies on the critical temperature of a lattice Bose gas

“…(1) K is the strength of the disorder: the disorder potential u( r) has averages u( r) = 0 and u( r, r ′ ) = Kδ( r − r ′ ) (the bar denotes the average all disorder configurations). The result (1) has been confirmed in [33] by one-loop Wilson renormalization-group calculations. The shift of the critical temperature for the continuous Bose gas has been also recently computed in [35] using the Popov method, obtaining a value for the relative shift δT c /T (0) c which differs by a factor of 1/2 from the result (1).…”

“…In presence of random on-site energies, T c increases for large filling, but much less than for for bond-disordered lattices, resulting in very small shift of T c for small disorder. These results should be compared with the findings for a continuous (i.e., without optical lattice) Bose gas in presence of a disordering potential [32,33,34,35,36]: without any confining potential, it has been shown that the critical temperature decreases with disorder [32]. In Refs.…”

Bose-Einstein-condensed systems in random potentials

“…At asymptotically small ν → 0, numerical estimates give the shift of the critical temperature δT c ∼ −2ν/9π, which is close to the shift found in Refs. [24,25].…”

“…Their results were recovered by Giorgini et al [23] using the hydrodynamic approximation, which is mathematically equivalent to the Bogolubov approximation. Lopatin and Vinokur [24] estimated the shift of the critical temperature due to weak disorder in a weakly interacting gas, which also was studied by Zobay [25], using renormalization group techniques. If the results obtained for asymptotically weak disorder are formally extended to strong disorder, then one comes [22,26] to the state, where n 0 = 0 but n s = 0, which corresponds to the Bose glass phase.…”

Bose systems in spatially random or time-varying potentials

“…Experimentally, such random potentials are realized by means of optical speckles [14][15][16][17]. Physical properties of these Bose systems in an external random potential have been theoretically studied for the case of weak disorder [18][19][20][21] and for arbitrarily strong disorder [22,23].…”