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arXiv:1002.0548 · Physical Review B

Superconductivity generated by coupling to a cooperon in a two-dimensional array of four-leg Hubbard ladders

Robert Konik, Thomas Maurice Rice, Alexei M. Tsvelik

Published Aug 2, 2010• 14 citations
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Abstract

Starting from an array of four-leg Hubbard ladders weakly doped away from half-filling and weakly coupled by interladder tunneling, we derive an effective low-energy model which contains a partially truncated Fermi surface and a well-defined cooperon excitation formed by a bound pair of holes. An attractive interaction in the Cooper channel is generated on the Fermi surface through virtual scattering into the cooperon state. Although the model is derived in the weak coupling limit of a four-leg ladder array, an examination of exact results on finite clusters for the strong coupling $t\text{\ensuremath{-}}J$ model suggests the essential features are also present for a strong coupling Hubbard model on a square lattice near half-filling.

DOI
10.1103/physrevb.82.054501

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Starting from an array of four-leg Hubbard ladders weakly doped away from half-filling and weakly coupled by interladder tunneling, we derive an effective low-energy model which contains a partially truncated Fermi surface and a well-defined cooperon excitation formed by a bound pair of holes. An attractive interaction in the Cooper channel is generated on the Fermi surface through virtual scattering into the cooperon state. Although the model is derived in the weak coupling limit of a four-leg ladder array, an examination of exact results on finite clusters for the strong coupling t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model suggests the essential features are also present for a strong coupling Hubbard model on a square lattice near half-filling.
It is shown, using asymptotically exact methods, that the two-dimensional repulsive Hubbard model with strongly modulated interactions exhibits ``high-temperature superconductivity.'' Specifically, the explicit modulation, which has the same symmetry as period 4 bond-centered stripes, breaks the system into an alternating array of more and less heavily hole-doped, nearly decoupled two-leg ladders. It is shown that this system exhibits a pairing scale determined by the spin gap of the undoped two-leg ladder, and a phase ordering temperature proportional to a low positive power of the interladder coupling.
We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of N/2N/2N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band (N+1)/2(N+1)/2(N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair (N/2,N/2+1)(N/2,N/2+1)(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases N=3N=3N=3 and 4.
The Hubbard model provides a simple framework in which one can study how certain aspects of the electronic structure of strongly interacting systems can be tuned to optimize the superconducting pairing correlations and how these changes affect the mechanisms giving rise to them. Here we use a weak-coupling random phase approximation to study a two-dimensional Hubbard model with a unidirectional modulation of the hopping amplitudes as the system evolves from the uniform square lattice to an array of weakly coupled two-leg ladders. We find that the pairing correlations retain their dominant ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wavelike structure and that they are significantly enhanced for a slightly modulated lattice. This enhancement is traced backed to an increase in the strength of the spin-fluctuation pairing interaction due to favorable Fermi surface nesting in the modulated system. We then use a random-phase approximation BCS framework to examine the evolution of the neutron resonance in the superconducting state. We find that it changes only weakly for moderate modulations, but breaks up into two distinct resonances at incommensurate wave vectors in the limit of weakly coupled ladders.
We use exact diagonalization to determine the spectrum of reduced Hamiltonians based on renormalization group flows to strong coupling. For the half-filled two-leg Hubbard ladder we reproduce the known insulating d-Mott ground state with spin and charge gaps. For the saddle point regions of the two-dimensional Hubbard model near half filling we find a crossover to a similar strong coupling state, which truncates the Fermi surface near the saddle points. At lower scales d-wave superconductivity appears on the remaining Fermi surface.
The weak-coupling form of the effective interaction between two holes in the antiferromagnetic state of the two-dimensional Hubbard model near half-filling is derived. It is shown to predict a Cooper instability in the singlet ddd-wave channel, with binding temperature Tc>100{T}_{c}>100Tc​>100 K. Our results differ from those of the "spin-bag" theory of Schrieffer, Wen, and Zhang.
We study a two-dimensional Fermi liquid with a Fermi surface containing the saddle points $(\ensuremath{\pi},0)$ and $(0,\ensuremath{\pi})$. Including Cooper and Peierls channel contributions leads to a one-loop renormalization group flow to strong coupling for short range repulsive interactions. In a certain parameter range the characteristics of the fixed point, opening of a spin and charge gap, and dominant pairing correlations are similar to those of a two-leg ladder at half-filling. We argue that an increase of the electron density leads to a truncation of the Fermi surface to only four disconnected arcs.