Functional renormalization for the Bardeen–Cooper–Schrieffer to Bose–Einstein condensation crossover

Abstract

We review the functional renormalization group (RG) approach to the Bardeen-CooperSchrieffer to Bose-Einstein condensation (BCS-BEC) crossover for an ultracold gas of fermionic atoms. Formulated in terms of a scale-dependent effective action, the functional RG interpolates continuously between the atomic or molecular microphysics and the macroscopic physics on large length scales. We concentrate on the discussion of the phase diagram as a function of the scattering length and the temperature, which is a paradigm example for the non-perturbative power of the functional RG. A systematic derivative expansion provides for both a description of the many-body physics and its expected universal features as well as an accurate account of the few-body physics and the associated BEC and BCS limits.

Solving functional flow equations with pseudospectral methods

“…The index n counts the energy level. Evaluating (21) for each model, we obtain for the first potential, (14),…”

“…In particular, this work makes use of the formulation of the exact renormalisation group by Wetterich [2], which has been applied to a wide range of systems, e.g. scalar field theories [3][4][5][6][7][8][9][10][11][12], fermionic systems [13][14][15][16][17][18][19], critical phenomena [20][21][22][23][24][25][26], gauge theories [27][28][29][30][31][32][33][34] and quantum gravity [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]…”

Physics and the choice of regulators in functional renormalisation group flows

“…To understand this, let us re-evaluate the example of the flows with r A and r B leading to the circular flow (12). In the spirit of the discussion above it seems to be natural to compare the two flows from k = Λ to k = 0 with the regulators r s=0 = r A and r s=1 = r B , respectively, while the s-flow in this example simply switches the regulator at a fixed scale k = Λ.…”

“…Pictorial representation of the integrability condition (12) in the theory space of action functionals. By means of (11), we can map the two actions at the initial scale Λ onto each other.…”

“…Applications range from quantum dots and wires, statistical models, condensed matter systems in solid state physics and cold atoms over quantum chromodynamics to the standard model of particle physics and even quantum gravity. For reviews on the various aspects of the functional RG see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”