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Physical review. B, Condensed matter

Dependence of the one-body density matrix on the superconducting gap parameter Δ(T)

William L. Clinton, Louis J. Massa, Sipra Pal

Published Oct 1, 1992• 1 citation
Publisher

Abstract

We show how the one-electron thermally averaged density matrix \ensuremath{\rho}(1',1,T)==〈${\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}$(1')\ensuremath{\Psi}(1)〉==N〈Fd(2,...,N)${\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}$ (1',...,n)\ensuremath{\Psi}(1,...,N)〉 varies with the superconducting gap parameter \ensuremath{\Delta}(T). It is suggested how these results can be applied to experiments that probe the density matrix. In particular it is suggested that the x-ray structure factor F(G) should be an ideal probe of the change in \ensuremath{\rho}(r) as a material goes superconducting. We also discuss how our results can be applied to recent analysis of atomic pair distribution functions (PDF) coming from measurements of diffuse scattering of pulsed neutrons on the thallium high-${\mathit{T}}_{\mathit{c}}$ materials.

DOI
10.1103/physrevb.46.9237

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We show how the one-electron thermally averaged density matrix \ensuremath{\rho}(1',1,T)==〈\ensuremathΨ∗{\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}\ensuremathΨ∗(1')\ensuremath{\Psi}(1)〉==N〈Fd(2,...,N)\ensuremathΨ∗{\mathrm{\ensuremath{\Psi}}}^{\mathrm{*}}\ensuremathΨ∗ (1',...,n)\ensuremath{\Psi}(1,...,N)〉 varies with the superconducting gap parameter \ensuremath{\Delta}(T). It is suggested how these results can be applied to experiments that probe the density matrix. In particular it is suggested that the x-ray structure factor F(G) should be an ideal probe of the change in \ensuremath{\rho}(r) as a material goes superconducting. We also discuss how our results can be applied to recent analysis of atomic pair distribution functions (PDF) coming from measurements of diffuse scattering of pulsed neutrons on the thallium high-Tc{\mathit{T}}_{\mathit{c}}Tc​ materials.
Symmetry of the Gap Function in High-TcT_cTc​ Superconductors
One of the most debated issues related to high-TcT_cTc​ superconductivity is the symmetry of the Cooper pair or the gap function. In this report, we present numerical results regarding the gap function in strongly correlated electron systems using t−Jt-Jt−J and Hubbard models in one and two dimensions. To this end, we use exact diagonalization to study the ground states of 8- and 16-site clusters consisting of single or coupled layers. We calculate a reduced two-particle density matrix in momentum space which is a measure of the gap function. We then analyze the eigenvectors of this density matrix, which display the possible Cooper pair symmetries. The eigenvector corresponding to the largest eigenvalue indicates a vanishing gap on the Fermi surface (which is in favour of odd-gap pairing) although dx2−y2d_{x^2-y^2}dx2−y2​ symmetry is seen to be a very close contestant in many of the cases.
In this work we study the superconductivity within an attractive two-dimensional one-band Hubbard model. We consider a d-wave superconducting gap and a Hubbard-I approximation to describe the strongly correlated superconducting regime. We use the Green’s function method to obtain the order parameter Δ and the superconducting critical temperature Tc. The results show that for fixed values of the superconducting attractive potential U(U<0), the gap increases for low temperatures but diminishes abruptly as the temperature increases. The effect of pressure is discussed, varying the next-nearest-neighbor hopping t2, yielding a change in Tc, and also in Δ0, and a tendency to suppress the superconductivity.
A new variational method for studying the equilibrium states of an interacting particles system has been proposed. The statistical description of the system is realized by means of a density matrix. This method is used for description of conduction electrons in metals. An integral equation for the electron distribution function over wave vectors has been obtained. The solutions of this equation have been found for those cases where the single-particle Hamiltonian and the electron interaction Hamiltonian can be approximated by a quite simple expression. It is shown that the distribution function at temperatures below the critical value possesses previously unknown features which allow to explain the superconductivity of metals and presence of a gap in the energy spectrum of superconducting electrons.
An open quantum system (i.e., one that interacts with its environment) is almost always entangled with its environment; it is therefore usually not attributed a wave function but only a reduced density matrix ρ. Nevertheless, there is a precise way of attributing to it a wave function ψ1, called its conditional wave function, which is a random wave function of the system whose probability distribution µ1 depends on the entangled wave function ψ ∈ H1 ⊗ H2 in the Hilbert space of system and environment together. We prove several universality (or typicality) results about µ1; they show that if the environment is sufficiently large then µ1 does not depend much on the details of ψ and is approximately given by one of the so-called GAP measures. Specifically, for most entangled states ψ with given reduced density matrix ρ1, µ1 is close to GAP (ρ1). We also show that, if the coupling between the system and the environment is weak, then for most entangled states ψ from a microcanonical subspace corresponding to energies in a narrow interval [E, E + δE] (and most bases of H2), µ1 is close to GAP (ρβ) with ρβ the canonical density matrix on H1 at inverse temperature β = β(E). This provides the mathematical justification of the claim that GAP (ρβ) is the thermal equilibrium distribution of ψ1.
A universal LDA-type density functional describing the electronic correlations in superconductors is developed from first principles. The functional is constructed from the exchange-correlation free-energy density, fxchom{f}_{\mathrm{xc}}^{\mathrm{hom}}fxchom​, of a homogeneous electron gas exposed to an external translationally invariant pairing field. The quantity fxchom{f}_{\mathrm{xc}}^{\mathrm{hom}}fxchom​, which is a function of the density and a functional of the induced order parameter, is calculated by many-body perturbation theory.
Magnetic interactions are widely believed to play a crucial role in the microscopic mechanism leading to high critical temperature superconductivity. It is therefore important to study the signatures of pairing in the magnetic excitation spectrum of simple models known to show unconventional superconducting tendencies. Using the density matrix renormalization group technique, we calculate the dynamical spin structure factor $S(\mathbf{k},\ensuremath{\omega})$ of a generalized $t\ensuremath{-}U\ensuremath{-}J$ Hubbard model away from half filling in a two-leg ladder geometry. The addition of JJJ enhances pairing tendencies. We analyze quantitatively the signatures of pairing in the magnetic excitation spectra. We found that the superconducting pair-correlation strength, that can be estimated independently from ground state properties, is closely correlated with the integrated low-energy magnetic spectral weight in the vicinity of $(\ensuremath{\pi},\ensuremath{\pi})$. In this wave-vector region, robust spin incommensurate features develop with increasing doping. The branch of the spectrum with rung direction wave vector krung=0{k}_{\mathrm{rung}}=0krung​=0 does not change substantially with doping where pairing dominates and thus plays a minor role. We discuss the implications of our results for neutron scattering experiments, where the spin excitation dynamics of hole-doped quasi-one-dimensional magnetic materials can be measured and also address implications for recent resonant inelastic x-ray scattering experiments.