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arXiv:1503.06213 · Physical Review B

Finite-temperature superconducting correlations of the Hubbard model

Ehsan Khatami, Richard T. Scalettar, Rajiv R. P. Singh

Published Jun 18, 2015• 24 citations
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Abstract

We utilize numerical linked-cluster expansions (NLCEs) and the determinantal quantum Monte Carlo algorithm to study pairing correlations in the square-lattice Hubbard model. To benchmark the NLCE, we first locate the finite-temperature phase transition of the attractive model to a superconducting state away from half filling. We then explore the superconducting properties of the repulsive model for the $d$-wave and extended $s$-wave pairing symmetries. The pairing structure factor shows a strong tendency to $d$-wave pairing and peaks at an interaction strength comparable to the bandwidth. The extended $s$-wave structure factor and correlation length are larger at higher temperatures but clearly saturate as temperature is lowered, whereas the $d$-wave counterparts, which start off lower at high temperatures, continue to rise near half filling. This rise is even more dramatic in the $d$-wave susceptibility. The convergence of NLCEs breaks down as the susceptibilities and correlation lengths become large, so we are unable to determine the onset of long-range order. However, our results extend the conclusion, previously restricted to only magnetic and charge correlations, that NLCEs offer a unique window into pairing in the Hubbard model at strong coupling.

DOI
10.1103/physrevb.91.241107

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We investigate the phase diagram of the extended Hubbard model on a half-filled honeycomb lattice through determinant quantum Monte Carlo simulations. At zero temperature, the repulsive Hubbard model undergoes a transition to an antiferromagnetic state only for sufficiently strong on-site coupling UUU. However, the inclusion of a repulsive extended interaction VVV between fermions in nearest-neighbor orbitals induces a charge-ordered phase. For attractive interactions, we also identify superconducting and phase-separated phases. We have determined a complete phase diagram with the aid of structure factors associated with various orderings, combined with measurements of double occupancy and the average sign of fermionic determinants. We have established that the semimetallic phase forms a zone from which ordered phases are excluded, preventing the stabilization of a superconducting phase with ddd-wave symmetry, i.e., only sss-wave pairing is allowed.
A linked cluster expansion of the Gutzwiller variational wave function is used to study the one- and two-dimensional Hubbard model; in parallel the variational results are compared with quantum Monte Carlo simulations on finite-size lattices. Detailed results are presented for the magnetic and pairing correlations as a function of wave vector, electron density, and interaction strength U. There is no sign for superconducting long-range order although both the d-wave and the extended s-wave pairing fluctuations are substantially enhanced as U increases.
Superconducting correlation of the Hubbard model on a square lattice is reinvestigated by quantum Monte Carlo method. Results from more than one order of magnitude larger Monte Carlo steps than before are presented to discuss the convergence of the data to the ground state. By examining the dependence on the choice of initial trial states as well as on the Trotter number in detail, the superconducting correlation in the ground state is reliably obtained up to 12×12 lattice away from the half-filling and up to 16×16 lattice at the half-filling, which substantially improve former results near the half-filling. Various channels of pairing in the ground state show no system size dependence and remain quite short ranged at any filling including the region close to the half-filling. Numerical data show further evidence that neither extended s - nor d -wave superconducting state is possible in the two-dimensional Hubbard model.
We study the square-lattice extended Hubbard model with on-site UUU and nearest-neighbor VVV interactions by exact diagonalization. We show that nonequilibrium quench dynamics can help determine the equilibrium phase transition boundaries, which agree with the calculations of the fidelity metric, dynamical structure factor, and correlation function. At half filling, the phase diagrams in the strong-coupling regime include spin density wave and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity at large positive UUU, charge density wave (extended s∗{s}^{*}s∗-wave superconductivity) at large positive (negative) VVV, and sss-wave superconductivity at large negative UUU with vanishing VVV. The energies of different particle sectors also help determine the phase separation region. With carrier doping, charge fluctuation results in a strong competition between different orders, making it more difficult to identify the leading instability on finite-size clusters. Nevertheless, the more exotic ppp-wave superconducting pairing is found to be enhanced when the system is heavily overdoped by 37.537.5%--50%37.5 holes, especially in the interaction parameter range relevant to the cuprate superconductors.
The low-temperature properties of the two-dimensional attractive Hubbard model are strongly influenced by the fermion density. Away from half-filling, there is a finite-temperature transition to a phase with sss-wave pairing order. However, Tc{T}_{\mathrm{c}}Tc​ is suppressed to zero at half-filling, where long-range charge-density-wave order also appears, degenerate with superconductivity. This paper presents determinant quantum Monte Carlo simulations of the attractive Hubbard model in the presence of a confining potential Vtrap{V}_{\mathrm{trap}}Vtrap​ which makes the fermion density $\ensuremath{\rho}$ inhomogeneous across the lattice. Pair correlations are shown to be large at low temperatures in regions of the trapped system with incommensurate filling, and to exhibit a minimum as the local density $\ensuremath{\rho}(\mathbit{i})$ passes through one fermion per site. In this ring of $\ensuremath{\rho}(\mathbit{i})=1$, charge order is enhanced. A comparison is made between treating Vtrap{V}_{\mathrm{trap}}Vtrap​ within the local-density approximation (LDA) and in an ab initio manner. It is argued that certain sharp features of the LDA result at integer filling do not survive the proximity of doped sites. The critical temperature of confined systems of fixed characteristic density is estimated.
The ``dynamic'' Hubbard Hamiltonian describes interacting fermions on a lattice whose on-site repulsion is modulated by a coupling to a fluctuating bosonic field. We investigate one such model, introduced by Hirsch, using the determinant quantum Monte Carlo method. Our key result is that the extended sss-wave pairing vertex, repulsive in the usual static Hubbard model, becomes attractive as the coupling to the fluctuating Bose field increases. The sign problem prevents us from exploring a low enough temperature to see if a superconducting transition occurs. We also observe a stabilization of antiferromagnetic correlations and the Mott gap near half-filling, and a near linear behavior of the energy as a function of particle density which indicates a tendency toward phase separation.