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Annalen der Physik

Dynamic Field‐Theoretic Approach to a Disordered Model Superconductor

E. Kolley, Winfried Kolley

Published Jan 1, 1989• 0 citations
Publisher

Abstract

Abstract On the basis of the attractive Hubbard model with a Gaussian random potential we describe superconductivity in the presence of quenched disorder. A dynamic path‐integral formalism with an 8‐component Keldysh‐Nambu spinor is shown to be convenient for solving the averaging problem without the recourse to the replica trick. The effective action is bosonized in terms of collective variables, namely local Q‐matrices and real space pair fields. Saddle point values are calculated in two simplified limits.

DOI
10.1002/andp.19895010106

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Abstract On the basis of the attractive Hubbard model with a Gaussian random potential we describe superconductivity in the presence of quenched disorder. A dynamic path‐integral formalism with an 8‐component Keldysh‐Nambu spinor is shown to be convenient for solving the averaging problem without the recourse to the replica trick. The effective action is bosonized in terms of collective variables, namely local Q‐matrices and real space pair fields. Saddle point values are calculated in two simplified limits.
Abstract A real‐time finite‐temperature field theory on the Keldysh contour is used to describe the dynamics of an electron system within a random Hubbard model. The effective, i.e. quenched‐averaged, fermionic action under the functional integral is bosonized by introducing both a matrix field and an exchange field via Hubbard‐Stratonovich transformations. The resulting collective action permits the loop expansion, giving rise to a systematic approximation scheme. The saddle point and Gaussian fluctuations around it are calculated.
Abstract We consider bosonic atoms with a repulsive contact interaction in a trap potential for a Bose‐Einstein condensation (BEC) and additionally include a random potential. The ensemble averages for two models of static (I) and dynamic (II) disorder are performed and investigated in parallel. The bosonic many body systems of the two disorder models are represented by coherent state path integrals on the Keldysh time contour which allow exact ensemble averages for zero and finite temperatures. These ensemble averages of coherent state path integrals therefore present alternatives to replica field theories or super‐symmetric averaging techniques. Hubbard‐Stratonovich transformations (HST) lead to two corresponding self‐energies for the hermitian repulsive interaction and for the non‐hermitian disorder‐interaction. The self‐energy of the repulsive interaction is absorbed by a shift into the disorder‐self‐energy which comprises as an element of a larger symplectic Lie algebrasp(4M) the self‐energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product ofM×sp(2); ‘M’ is the number of discrete time intervals of the disorder‐self‐energy in the generating function ). After removal of the remaining Gaussian integral for the self‐energy of the repulsive interaction, the first order variations of the coherent state path integrals magnified image result in the exact mean field or saddle point equations, solely depending on the disorder‐self‐energy matrix. These equations can be solved by continued fractions and are reminiscent to the `Nambu‐Gorkov' Green function formalism in superconductivity because anomalous terms or pair condensates of the bosonic atoms are also included into the selfenergies. The derived mean field equations of the models with static (I) and dynamic (II) disorder are particularly applicable for BEC ind= 3 spatial dimensions because of the singularity of the density of states at vanishing wavevector. However, one usually starts out from restricted applicability of the mean field approach ford= 2; therefore, it is also pointed out that one should consider different HST's ind= 2 spatial dimensions with the block diagonal densities as ‘hinge’ functions and that one has to introduce a coset decompositionSp(4M)\U(2M) into densities and anomalous terms of the total disorder‐self‐energysp(4M) for deriving a nonlinear sigma model.
— The real-space single-particle Green’s function formalism is extended to the attractive disordered Hubbard Hamiltonian by matrix Nambu notation, thus providing a general approach to the superconductive instability of a disordered system in quantum mean-field approximation. Configurational averages may be performed by a recent method, already employed for the repulsive model, thus allowing for spatial fluctuations of the local order parameter. The method is shown to be a natural way of investigating the validity limit of the Anderson theorem. An exact analytical solution is provided for a Bethe lattice with no diagonal disorder and infinitely large connectivity.
Abstract On the basis of an extended attractive Hubbard model with nearest‐neighbour interaction of opposite spins two functional methods are adopted for deriving a self‐consistent equation for the superconducting order parameter. The functional‐integral formalism is exploited in saddle‐point approximation. The functional‐derivative technique using Schwinger source fields leads via the self‐energy in Hartree‐Fock‐Bogolubov approximation to a nonlinear gap equation. The superconducting transition temperature is investigated for the case of s‐wave pairing.
Abstract We investigate a system of interacting bosons with random intersite tunnelling amplitudes. We describe these by introducing Gaussian-distributed hopping integrals into the standard Bose–Hubbard model. This system has been recently shown to exhibit a quantum phase transition to a glassy state. The latter is characterized by a quenched disorder of boson wave-function phases. In this aspect, the system resembles quantum spin-glass systems that attracted much attention. By exploiting this analogy, we employ the well-established methodology originated by Sherrington and Kirkpatrick, which bases on the replica trick and the Trotter–Suzuki expansion. This treatment transforms the original quantum problem into an effective classical one with an additional time-like dimension. Here, we focus on autocorrelation functions of canonical variables of the effective system in the time-like domain. Deep in the disordered phase, we find a highly dynamical nature of correlations in agreement with the expected short memory of the system. This behaviour weakens while approaching and passing the phase boundary, where in the glassy phase asymptotically non-vanishing correlations are encountered. Thus, the state features infinite memory, which is consistent with the quenched nature of glassy disorder with random but frozen boson phases.