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Physical Review B

Current-carrying state of a nodal correlated superconductor

F. Horváth, Manfred Sigrist, R. Hlubina

Published May 29, 2012• 2 citations
Publisher

Abstract

In the limit of vanishing temperatures, we study the $d$-wave superconducting state of the two-dimensional Hubbard model subject to a finite supercurrent. Making use of the recently developed variational approach to Kohn-Luttinger superconductors, we take into account both the change of the pairing interaction as well as the change of the gap function in presence of a finite Cooper-pair momentum $2\mathbf{q}$. We study the evolution of the superconducting state with increasing $\mathbf{q}$ and we find that the order of the transition to the normal state depends on the direction of $\mathbf{q}$.

DOI
10.1103/physrevb.85.184527

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In the limit of vanishing temperatures, we study the ddd-wave superconducting state of the two-dimensional Hubbard model subject to a finite supercurrent. Making use of the recently developed variational approach to Kohn-Luttinger superconductors, we take into account both the change of the pairing interaction as well as the change of the gap function in presence of a finite Cooper-pair momentum 2q2\mathbf{q}2q. We study the evolution of the superconducting state with increasing q\mathbf{q}q and we find that the order of the transition to the normal state depends on the direction of q\mathbf{q}q.
An effective Hamiltonian for the Kohn-Luttinger superconductor is constructed and solved in the BCS approximation. The method is applied to the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model in two dimensions with the following results: (i) The superconducting phase diagram at half filling is shown to provide a weak-coupling analog of the recently proposed spin liquid state in the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. (ii) In the parameter region relevant for the cuprates, we have found a nontrivial energy dependence of the gap function in the dominant d-wave pairing sector. The hot spot effect in the angular dependence of the superconducting gap is shown to be quite weak.
Applying the recently developed variational approach to Kohn-Luttinger superconductivity to the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model in two dimensions, we have found, for sizeable next-nearest-neighbor hopping, an electron-density-controlled quantum phase transition between a d-wave superconducting state close to half filling and an s-wave superconductor at lower electron density. The transition occurs via an intermediate time-reversal breaking d+isd+isd+is superconducting phase, which is characterized by nonvanishing chirality and density-current correlation. Our results suggest the possibility of a bulk time-reversal symmetry breaking state in overdoped cuprates.
Recently developed numerical methods have enabled the explicit construction of the superconducting state of the Hubbard model of strongly correlated electrons in parameter regimes where the model also exhibits a pseudogap and a Mott insulating phase. d(x(2)-y(2)) symmetry superconductivity is found to occur in proximity to the Mott insulator, but separated from it by a pseudogapped nonsuperconducting phase. The superconducting transition temperature and order parameter amplitude are found to be maximal at the onset of the normal-state pseudogap. The emergence of superconductivity from the normal state pseudogap leads to a decrease in the excitation gap. All of these features are consistent with the observed behavior of the copper-oxide superconductors.
We study the superconducting order in a two-dimensional square lattice Hubbard model with weak repulsive interactions, subject to a Zeeman field and weak Rashba spin-orbit interactions. Diagonalizing the noninteracting Hamiltonian leads to two separate bands, and by deriving an effective low-energy interaction we find the mean field gap equations for the superconducting order parameter on the bands. Solving the gap equations just below the critical temperature, we find that superconductivity is caused by Kohn-Luttinger-type interaction, while the pairing symmetry of the bands is indirectly affected by the spin-orbit coupling. The dominating attractive momentum channel of the Kohn-Luttinger term depends on the filling fraction nnn of the system, and it is therefore possible to change the momentum dependence of the order parameter by tuning nnn. Moreover, nnn also determines which band has the highest critical temperature. Rotating the magnetic field changes the momentum dependence from states that for small momenta reduce to a chiral ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ type state for out-of-plane fields, to a nodal ppp-wave-type state for purely in-plane fields.
We consider possible superconducting instabilities in a two-dimensional Fermi system with short-ranged repulsive interactions between electrons. The possibility of an unusual superconducting paring due to the Kohn-Luttinger mechanism is examined. The quasiparticle scattering amplitude is shown to possess an attractive harmonic in second-order perturbation theory for finite values of the energy transfer. The corresponding singularity in the pairing vertex leads to a superconducting pairing of the electron excitations with finite energies. We identify the energy transfer in the Cooper channel as the binding energy of the excited pair. At low enough temperatures, the Fermi system is a mixture of normal electron excitations and fluctuating d-wave Cooper pairs possessing a finite gap.
We analyze solutions to a superconducting gap equation based on the two-dimensional Hubbard model with nearest and next-to-nearest neighbor hopping. The Cooper pair potential can be calculated exactly and expressed in terms of elliptic functions. The Fermi surfaces at finite temperature and chemical potential are calculated based on the exact two-body S-matrix of the Hubbard model using the formalism we recently developed, which allows variation of hole doping. The resulting solutions to the gap equation are strongly anisotropic, namely largest in the anti-nodal direction, and zero in the nodal directions of the Brillouin zone. For U/t = 13 and t' /t =-0.3, appropriate to BSCO, and a physically natural choice for the cut-off, our self-contained analytic calculations yield the gap in the anti-nodal direction Delta/t = 0.06 and a maximum Tc/t = 0.04 at hole doping h=0.15. For phenomenological fits to the Fermi surfaces for cuprates, we obtain the comparable value Tc/t = 0.03 at optimal doping, both in good agreement with experiments. The superconducting gap is non-zero for all hole-doping h < 0.35, and increases all the way down to zero doping, suggesting that it evolves into the pseudogap.