Hartree–Fock mean-field theory for trapped dirty bosons

Abstract

Here we work out in detail a non-perturbative approach to the dirty boson problem, which relies on the Hartree-Fock theory and the replica method. For a weakly interacting Bose gas within a trapped confinement and a delta-correlated disorder potential at finite temperature, we determine the underlying free energy. From it we determine via extremization self-consistency equations for the three components of the particle density, namely the condensate density, the thermal density, and the density of fragmented local Bose-Einstein condensates within the respective minima of the random potential landscape. Solving these self-consistency equations in one and three dimensions in two other publications has revealed how these three densities change for increasing disorder strength.

Analytical and numerical study of dirty bosons in a quasi-one-dimensional harmonic trap

“…In the one-dimensional case and at T = 0, the mean-field Hartree-Fock theory with the help of the replica method leads to the free energy [50]: Thus, from equation (B6) we conclude that our result agrees qualitatively with the Huang-Meng theory. But quantitatively the comparison of equations (B5) and (B6) reveals that a factor of 2 3 2 is missing in our result (B6).…”

“…The disorder correlation function is given by equation (15), where we assume ( ) ( ) d -¢ = -¢ D x x D x x , and D denotes the disorder strength. Within the Hartree-Fock mean-field approximation with the replica method, [50] obtains self-consistency equations, which determine the particle density n(x) as well as the order parameter of the superfluid ( ) n x 0 , which represents the condensate density, and the order parameter of the Bose-glass phase q(x), which stands for the density of the particles being condensed in the respective minima of the disorder potential.…”

“…More precisely, the two order parameters ( ) n x 0 and q(x) of our mean-field theory at T = 0 are defined by following the notion of spin-glass theory [70][71][72]. On the one hand, the off-diagonal long-range limit of the twopoint correlation function defines the condensate density [73], Note that the validity of equations (A3) and (A4) within the Hartree-Fock approximation was analysed in detail in [50]. There it is also shown that two-and four-point correlations decay exponentially on a length scale which can be physically interpreted as the localisation length named after Larkin [37,75].…”

“…Furthermore, from the Gaussian fit in figure 1 (dotted, blue line), we remark that the original data of the total density approach a Gaussian form in the intermediate disorder regime much better than in the weak disorder regime. This can be explained with the argument that increasing the disorder reduces effectively the repulsive interaction between the particles [50] and, thus, approaches the case of noninteracting bosons, where the total density is given by a Gaussian.…”

“…As a result, the corresponding phase diagram for the occurrence of the superfluid, the Bose-glass, and the normal phase was determined in the control parameter plane spanned by disorder strength and temperature. This Hartree-Fock theory is extended in [50] to a harmonic confinement, and is applied in the following to onedimensional systems.…”