Physics Letters A 167 (1992) 84-88 North-Holland PHYSICS LETTERS A Correlated finite temperature BCS approximation in finite systems R. Rossignoli Departamento de Fisica, UniversidadNacional de La Plata, 1900 La Plata, Argentina R.M. Quick, H.G. Miller and F. Solms Department of Physics, Universityof Pretoria, Pretoria 0002, South Africa Received 8 April 1992; accepted for publication 9 May 1992 Communicated by L.J. Sham We examine the solutions of a correlated finite temperature BCS approximation within the context of a general pairing Hamil- tonian. It is shown that a new class of superconducting solutions, characterized by equal absolute values of the BCS transformation parameters u and v in a finite region around the Fermi surface exists at all temperatures. Furthermore, solutions of this type may provide a lower free energy than the ordinary normal solution above the BCS transition temperature for sufficiently strongly coupled finite systems. Comparison with the exact results in a particular finite case is made. Recently it has been shown that a microscopic cor- related finite temperature mean field approximation [ 1-3 ], based on the addition of diagonal two-body terms in the density operator, can lead to an im- proved picture of the thermal behavior of finite sys- tems, compared to that given by ordinary finite tem- perature mean field methods. Sharp transitions occurring in the ordinary mean field approximation (see for instance ref. [ 4 ] ), which usually reflect the classical or thermodynamic limit of the system, may become smooth in the correlated treatment for finite systems, in agreement with the exact results. A sig- nificant improvement in the prediction of physical quantities both in transitional regions and for tem- peratures beyond the corresponding mean field crit- ical temperature is obtained, in particular for the specific heat and the expectation values of different interaction terms [ 1-3]. In the present work we examine the behavior of the correlated treatment within the context of a fi- nite superconducting system. We shall show that a new type of superconducting solution exists in the correlated treatment for quite general Hamiltonians. This solution leads to an approximate restoration of symmetries associated with particle number conser- vation and is characterized by a vanishing pairing tensor. We consider the general BCS-like pairing Hamiltonian H= Z ek(C~Ck+C~.C~)-- ~, Gkk, C~C~Ct;,Ck,, (1) k k,k' where/~ denotes the time reversed state of k. Both the conventional BCS Hamiltonian [5,6] and the more recent Hubbard based superconducting Ham- iltonians [7,8] are of the form given by ( 1 ). We shall consider a finite system, with the total number of states given by 2K. Given the standard real BCS transformation ak=UkCk--VkC~, a~=UkC~+VkC~, (2) with u~,+v$=l, it is convenient to define the operators Qk =a~ak+a~ae-- 1 . (3) The diagonal part of H' - H- p.N in the quasiparti- cle representation can then be written as t ~v /'1 /U2 V2"~ Ha=Ho+Y~ k~k~ k-- kJ k - E Gkk'[QkQk'UkVkUk'Vk'+~kk'Q2(½--U2V2)], k,k' (4) where Ho= 52k (E~--g) and ~;,=~k-#- ½Okk. (5) 84 0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.