Detlef Lehmann
An Explicit Solution of the BCS Model and
Demonstration of Symmetry Breaking
(117K, TeX)
ABSTRACT. The BCS model, whose partition function is given by Tr e^{-\beta H(\lambda)}
where H(\lambda)= 1/L^d \sum_{k,\sigma} e(k) a_{k\sigma}^+a_{k\sigma}
+ \lambda/L^{3d} \sum_{k,p} a_{k\up}^+a_{-k\down}^+a_{-p\down}a_{p\up}
is solved explicitely as a quartic model without using mean field theory.
The Greens functions and their generating functional can be computed
explicitely in arbitrary dimension d and for arbitrary energy momentum
relation e(k). Renormalization effects as well as symmetry breaking can be
seen explicitely. The more general interaction
1/L^{3d}\sum_{\sigma\tau} \sum_{k,p,q} \sum_{l=0}^N
\lambda_l\bar y_l(k')y_l(p')[\delta_{k,p}+\delta_{k,-p}+\delta_{q,0}]
a_{k\sigma}^+a_{q-k\tau}^+a_{q-p\tau}a_{p\sigma}
which contains a forward, an exchange and a BCS term can also be treated.
For pure BCS, one obtains a 6(N+1) dimensional integral representation for
the two point functions and for the generating functional. The existence and
symmetry of a gap is determined by the global minimum of an effective potential
of 6(N+1) variables. We show, in 3 space dimensions, that the usual mean field
approach may be misleading if the electron electron interaction contains higher
angular momentum terms. In particular, if the interaction is given by a single
attractive even l term \lambda_l/L^{3d} \sum_{m=-l}^l Y_{l m}(k')\barY_{l m}(p')
a_{k\up}^+a_{-k\down}^+a_{-p\down}a_{p\up}, then, if e(k) has SO(3)
symmetry, the expectations also have to have
SO(3) symmetry. The property which makes the model explicitely solvable is the
fact that the interaction is given by a finite sum of products of quadratic factors
\sum_{\sigma\tau}\sum_{l=0}^N (\sum_k \bar y_l(k')a_{k\sigma}^+a_{-k\tau}^+)
(\sum_p y_l(p')a_{-p\tau}a_{p\sigma}).