Many body methods and effective field theory

Abstract

In the framework of pionless nucleon-nucleon effective field theory we study different approximation schemes for the nuclear many body problem. We consider, in particular, ladder diagrams constructed from particle-particle, hole-hole, and particle-hole pairs. We focus on the problem of finding a suitable starting point for perturbative calculations near the unitary limit (k F a) → ∞ and (k F r) → 0, where k F is the Fermi momentum, a is the scattering length and r is the effective range. We try to clarify the relationship between different classes of diagrams and the large g and large D approximations, where g is the fermion degeneracy and D is the number of space-time dimensions. In the large D limit we find that the energy per particle in the strongly interacting system is 1/2 the result for free fermions.

Resummation of fermionic in-medium ladder diagrams to all orders

“…(6) is equal to the sum of the particle-particle bubble and the hole-hole bubble, R(s, κ) = F pp (s, κ) + F pp (−s, κ), with the unusual feature that the latter is also taken at momenta below the Fermi surface. For comparison the particle-particle bubble reads [7,8]:…”

“…Thus there remains the imaginary part due to the second θ(...) θ(...) term in eq. (8). After visualizing the occurring product of θ-and δ-functions one sees that the imaginary part has a nice geometrical interpretation: namely as | q | times that part of the solid angle of a (centered) sphere of radius | q | which lies inside the intersection region of two spheres of radius k f with their centers displaced by 2| P |.…”

“…The Bertsch parameter following from Steele's approximation is ξ (St) = 4/9, surprisingly close to recent quantum Monte-Carlo results. The validity of Steele's arguments concerning the expansion in 1/D has been critically reassessed in the more elaborate work by Schäfer et al [8]. There it has been shown that if the strong coupling limit a → ∞ is taken after the limit D → ∞ the universal Bertsch parameter is ξ (∞) = 1/2 (see eq.…”

Effective Field Theory and Finite-Density Systems

“…and to determine real time properties that are hard to access numerically. A number of analytical methods have been considered, such as an expansion in the number of fermion species [23,24] or the number of spatial dimensions (which is related to the hole-line expansion of Brueckner, Bethe, and Goldstone) [25,26]. In the following we shall discuss a proposal by Nussinov & Nussinov to perform an expansion around d = 4 − ǫ spatial dimensions.…”

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.…”