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Physical Review Letters

Flux quantization in the two-dimensional repulsive and attractive Hubbard models

Fakher F. Assaad, Werner R. Hanke, Douglas J. Scalapino

Published Sep 20, 1993• 35 citations
Publisher

Abstract

In order to provide a numerical criterion for superconductivity, the ground state energy ${\mathit{E}}_{0}$(\ensuremath{\Phi}) is calculated for the attractive and repulsive Hubbard models on a cylinder geometry threaded by a flux \ensuremath{\Phi}. From the functional form of ${\mathit{E}}_{0}$(\ensuremath{\Phi}), superconducting ground states may be identified without prior knowledge of the pairing symmetry. ${\mathit{E}}_{0}$(\ensuremath{\Phi}) is calculated with Monte Carlo methods. For the attractive Hubbard model, our results confirm the existence of a superconducting ground state. In contrast, our results show that the quarter-filled repulsive Hubbard model is not superconducting and that a Hartree-Fock paramagnetic approximation fits the data very well.

DOI
10.1103/physrevlett.71.1915

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In order to provide a numerical criterion for superconductivity, the ground state energy E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}) is calculated for the attractive and repulsive Hubbard models on a cylinder geometry threaded by a flux \ensuremath{\Phi}. From the functional form of E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}), superconducting ground states may be identified without prior knowledge of the pairing symmetry. E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}) is calculated with Monte Carlo methods. For the attractive Hubbard model, our results confirm the existence of a superconducting ground state. In contrast, our results show that the quarter-filled repulsive Hubbard model is not superconducting and that a Hartree-Fock paramagnetic approximation fits the data very well.
Based on extensions of quantum Monte Carlo algorithms to incorporate magnetic fields, two criteria to detect superconductivity in two-dimensional Hubbard models are investigated. In order to provide such criteria, we calculate both the internal energy E(\ensuremath{\Phi},T) as well as the ground-state energy, E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}), for Hubbard models on a cylinder geometry threaded by a flux \ensuremath{\Phi}. The temperature derivative of the superfluid density, \ensuremath{\partial}\ensuremath{\beta}Ds{\mathit{D}}_{\mathit{s}}Ds​(\ensuremath{\beta})/\ensuremath{\partial}\ensuremath{\beta}, is obtained from the difference in internal energy of systems which differ by a phase twist \ensuremath{\pi}/2 in the boundary condition along one lattice direction. In the framework of a Kosterlitz-Thouless transition, \ensuremath{\partial}\ensuremath{\beta}Dx{\mathit{D}}_{\mathit{x}}Dx​(\ensuremath{\beta})/\ensuremath{\partial}\ensuremath{\beta} scales to a Dirac \ensuremath{\delta} function at the transition temperature. On finite-sized lattices, \ensuremath{\partial}\ensuremath{\beta}Ds{\mathit{D}}_{\mathit{s}}Ds​(\ensuremath{\beta})/\ensuremath{\partial}\ensuremath{\beta} shows a response which increases with lattice size. Flux quantization is a T=0 method. From the functional form of E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}), superconducting or nonsuperconducting ground states may be identified. In both approaches, superconductivity may be detected without prior knowledge of the symmetry and nature of the pairing correlations. For single-band Hubbard models, our main quantum Monte Carlo results include numerical data which (a) confirm the existence and pin down the transition temperature of a Kosterlitz-Thouless-type transition in the attractive Hubbard model away from half-band filling and (b) show that the quarter-filled repulsive Hubbard model is not superconducting. For the three-band Hubbard model we consider two parameter sets which take into account the differences in static magnetic structure and Fermi surfaces between La-Sr-Cu-O and Y-Ba-Cu-O materials. For both parameter sets, the finite-temperature approach showed no sign of a Kosterlitz-Thouless-type transition up to inverse temperatures \ensuremath{\beta}=17.5, in units of the Cu-O hopping, and hole doping \ensuremath{\delta}=0.25. Flux quantization results for the Y-Ba-Cu-O parameters on clusters up to 8\ifmmode\times\else\texttimes\fi{}8 unit cells equally showed no sign of superconductivity at a hole doping \ensuremath{\delta}=0.25.
In this paper we compare numerical results for the ground state of the Hubbard model obtained by Quantum-Monte-Carlo simulations with results from exact and stochastic diagonalizations. We find good agreement for the ground state energy and superconducting correlations for both, the repulsive and attractive Hubbard model. Special emphasis lies on the superconducting correlations in the repulsive Hubbard model, where the small magnitude of the values obtained by Monte-Carlo simulations gives rise to the question, whether these results might be caused by fluctuations or systematic errors of the method. Although we notice that the Quantum-Monte-Carlo method has convergence problems for large interactions, coinciding with a minus sign problem, we confirm the results of the diagonalization techniques for small and moderate interaction strengths. Additionally we investigate the numerical stability and the convergence of the Quantum-Monte-Carlo method in the attractive case, to study the influence of the minus sign problem on convergence. Also here in the absence of a minus sign problem we encounter convergence problems for strong interactions.
The ground-state energy of rings with 10 or 16 sites and 2 or 4 fermions (holes or electrons) is computed as a function of the flux \ensuremath{\Phi} created by a vector potential constant along the ring (Aharonov-Bohm setup) for three different model Hamiltonians: the attractive (U0) and repulsive (U>0) Hubbard and a t-J-like Hamiltonian. In all cases the flux is found to be quantized in units hc/2e, showing that the charge carriers are pairs. The possible role of phonon fluctuations in discriminating among the different models is also investigated.
We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.
We obtain the ground-state energy E0{\mathit{E}}_{0}E0​(\ensuremath{\Phi}) exactly as a function of flux \ensuremath{\Phi} in a one-dimensional ``copper-oxide model'' on rings of finite circumference L, including on-site interactions and a nearest-neighbor interaction V. For V of the order of the charge-transfer gap or larger, the model extrapolated to large L exhibits flux quantization with charge 2e, and a slow algebraic decay of the singlet superconducting correlation function on oxygen sites. The extrapolated superfluid stiffness appears, however, finite only for not too large V. These results suggest a superconductive state at V of order the charge-transfer gap of the model, but a paired and phase-separated state at larger V.
A nonzero superfluid density is equivalent to the occurrence of a Meissner effect and therefore signals superconductivity. A recent theorem shows that in the case of a spectrum with a gap the superfluid density is equivalent to the Drude weight. This theorem is employed to compare approximate calculations of the superfluid density in the two-dimensional attractive Hubbard model using the Hartree-Fock approximation with exact diagonalization calculations of the Drude weight. Direct comparison of the approximate results with recent finite-temperature quantum Monte Carlo calculations is also made. The approximate results are found to be quantitatively accurate for all fillings, except close to half-filling.