Non-BCS pairing in anisotropic strongly correlated electron systems in solids

The problem of pairing in anisotropic electron systems possessing patches of fermion condensate in the vicinity of the van Hove points is analyzed. Attention is directed to opportunities for the occurrence of non-BCS pairing correlations between the states belonging to the fermion condensate. It is shown that the physical emergence of such pairing correlations would drastically alter the behavior of the single-particle Green function, the canonical pole of Fermi-liquid theory being replaced by a branch point. PACS: 71.10.Hf, 71.27.+a, 74.20.Mn The ground state of conventional superconductors at T = 0 is known to be a condensate of Cooper pairs with total momentum P = 0. In Fermi-liquid theory, the familiar BCS structure of the ground state is associated with the logarithmic divergence of the particle-particle propagator at P = 0 and is independent of the details of the pairing interaction. However, a markedly different situation can ensue in strongly correlated systems in which the necessary stability condition for the Landau state is violated and the Landau quasiparticle momentum distribution suffers a rearrangement. Under certain conditions this rearrangement leads to a fermion condensate (FC) – a continuum of dispersionless single-particle (sp) states whose energy ǫ(p) coincides with the chemical potential μ over a finite (and in general disconnected) domain p ∈ Ω in momentum space [1-11]. As a result, the preference for pairing with P = 0 comes into question because of the degeneracy of the FC sp spectrum. In this case, the nature of pairing depends on the configuration assumed by the FC. Here we study a two-dimensional square-lattice system in which the FC is situated in domains adjacent to the van Hove points, while the sp states with ordinary dispersion are concentrated around diagonals of the Brillouin zone [3, 11]. To begin with, we focus on the nature of particle-particle correlations in the FC subsystem and ignore contributions from the sp states with nonzero dispersion. Traditional BCS singlet pairing correlates only the sp states belonging to diagonally opposite patches of the FC; the description therefore involves the single collective operator Cp = ap, − a−p,+ and its adjoint C † p, which connect the ground state with states of N and N ∓ 2 particles. However, in anisotropic electron systems inhabiting crystalline materials manifesting fermion condensation, all four FC patches should be treated on an equal footing. Hence an additional relevant collective operator Qp = ap,+ a−p+Q,−, enters the picture, together with its adjoint Q † p. With Q = (π/l, π/l) where l 1 is the lattice constant, this operator characterizes the pairing correlations affecting sp states located in the neighboring FC patches. If such additional correlations are involved together with the ordinary BCS correlations, the ground-state wave function evidently loses the simple BCS structure. A salient feature of this nonabelian exemplar of the pairing problem is the presence of two degenerate collective modes in the particle-particle channel. The creation of a C-pair (Cooper pair) by the operator C p, followed by subsequent annihilation of a Q-pair by the operator Qp, gives rise to an excited two-particle-two-hole state of the N -fermion system. As we shall demonstrate, this process enmeshes a whole band of many-particle-many-hole states and changes the structure of the single-particle Green function dramatically. We restrict considerations to the simplest, δ-like form of the interaction in the particleparticle channel, with strength parameter λ. Also, we assume that all the particle-hole contributions have already been taken into account in terms of an effective single-particle Hamiltonian having sp spectrum ǫ(p) Accordingly, only pairing contributions should be incorporated in the equation for the Green function Gαβ(x, x ) = −i〈Tψα(x)ψ† β(x)〉. This equation, derived with the aid of equation of motion [ε− ǫ(p)]ψα(x)−λψ γ(x)ψγ(x)ψα(x) = 0, takes the form (ε− ǫ(p))Gαβ(x, x) + iλ〈O|Tψ γ(x)ψγ(xψα(x)ψ† β(x)|O〉 = δ(x− x) . (1) In the ordinary pairing problem, the average 〈O|Tψ† γ(x)ψγ(x)ψα(x)ψ† β(x)|O〉 is decoupled as 〈O|Tap,γ(t)a−p,α(t)|C〉〈C|Tap1,γ(t)a † −p1,β (t+τ)|O〉. In the generalized case being developed, where the ground state is connected with two different states given the labels C and Q, this same average has the extended decomposition 〈O|Tap1,γ(t)a−p1,α(t)|C〉〈C|Tap,γ(t)a†−p,β(t+ τ)|O〉+ +〈O|Tap1,γ(t)a−p1+Q,α(t)|Q〉〈Q|Tap,γ(t)a†−p+Q,β(t+ τ)|O〉 . For simplicity we henceforth omit spin indices α, β, γ, etc. The equation for the Green function then reads G(p, ε) = Go(p, ε) [ (1−∆F 1,0(p, ε)−DF 0,1(p, ε) ] , (2) where Go(p, ε) = [ε− ǫ(p)]. In the time domain, the quantity F 1,0 has the expression F 1,0(p, t) = 〈C|Tap(t)a†−p(t + τ)|O〉 and is interpreted as the transition amplitude between the ground state and a state differing from it by the presence of a single Cpair. Similarly, F 0,1(p, τ) = 〈Q|Tap(t)a†−p+Q(t + τ)|O〉 is the transition amplitude between the ground state and a state differing from it by a single Q-pair. The diagramblock ∆ ∝ 〈O|Tap1,γ(t)a−p1,α(t)|C〉 ∝ F1,0(τ = 0) has the same meaning as the gap order parameter of BCS theory, while the new ingredient D has the corresponding structure D ∝ 〈O|Tap1,γ(t)a−p1+Q,α(t)|Q〉 ∝ F0,1(τ = 0). Employing the complementary equation of motion [ε+ ǫ(p)]ψ α(x)+λψ † α(x)ψ † γ(x)ψγ(x) = 0, one can derive equations for the transition amplitudes F 1,0 and F + 0,1: [ε+ ǫ(p)] 〈C|Tap(t)a†−p(t+ τ)|O〉 = −λ ∑