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Physical review. B, Condensed matter

Temperature derivative of the superfluid density and flux quantization as criteria for superconductivity in two-dimensional Hubbard models

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

Published Nov 1, 1994• 33 citations
Publisher

Abstract

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, ${\mathit{E}}_{0}$(\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}${\mathit{D}}_{\mathit{s}}$(\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}${\mathit{D}}_{\mathit{x}}$(\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}${\mathit{D}}_{\mathit{s}}$(\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 ${\mathit{E}}_{0}$(\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.

DOI
10.1103/physrevb.50.12835

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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 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 the grand-canonical quantum Monte Carlo algorithm to incorporate magnetic fields, we provide numerical data confirming the existence of a Kosterlitz-Thouless transition in the attractive Hubbard model. Here, we calculate the temperature derivative of the superfluid density, [\ensuremath{\partial}\ensuremath{\beta}Ds{\mathit{D}}_{\mathit{s}}Ds​(\ensuremath{\beta})]/\ensuremath{\partial}\ensuremath{\beta}, to pin down the transition. The latter quantity 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. Away from half-band filling, [\ensuremath{\partial}\ensuremath{\beta}Ds{\mathit{D}}_{\mathit{s}}Ds​(\ensuremath{\beta})]/\ensuremath{\partial}\ensuremath{\beta} shows a response which increases with lattice size at the transition temperature. In contrast, such a signal is not observed at half-band filling.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
Susceptibilities in the static limit ω→0 are evaluated to show that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional Hubbard model for optimal band parameters. By using quantum Monte Carlo methods, we have examined the size dependence of the spin susceptibility χ( Q ) at half filling and the pair susceptibilities with d - and s -wave symmetries for the repulsive and attractive interactions, respectively. The numerical results for systems of up to 14×14 sites show that the s -wave pair susceptibility scales as χ pair ∼ L 2 for the attractive case, in agreement with previous quantum Monte Carlo studies. The scaling χ pair ∼ L 2 also holds for the d -wave pair susceptibility for the repulsive Hubbard model if we adjust the band parameter t '. Therefore, it is likely that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional repulsive Hubbard model.
We present recent theoretical results on superconductivity in correlated-electron systems, especially in the two-dimensional Hubbard model and the three-band d-p model. The mechanism of superconductivity in high-temperature superconductors has been extensively studied on the basis of various electronic models and also electron-phonon models. In this study, we investigate the properties of superconductivity in correlated-electron systems by using numerical methods such as the variational Monte Carlo method and the quantum Monte Carlomethod. The Hubbard model is one of basic models for strongly correlated electron systems, and is regarded as the model of cuprate high temperature superconductors. The d-p model is more realistic model for cuprates. The superconducting condensation energy obtained by adopting the Gutzwiller ansatz is in reasonable agreement with the condensation energy estimated for YBa2Cu3O7. We show the phase diagram of the ground state using this method. We have further investigated the stability of striped and checkerboard states in the under-doped region. Holes doped in a half-filled square lattice lead to an incommensurate spin and charge density wave. The relationship of the hole density x and incommensurability δ, δ~x, is satisfied in the lower doping region, as indicated by the variationalMonte Carlocalculations for the two-dimensional Hubbard model. A checkerboard-like charge-density modulation with a roughly period has also been observed by scanning tunneling microscopy experiments in Bi2212 and Na-CCOC compounds. We have performed a variational Monte Carlo simulation on a two-dimensional t-t′-t″- U Hubbard model with a Bi-2212 type band structure and found that the period checkerboard spin modulation, that is characterized by multi Q vectors, is indeed stabilized. We have further performed an investigation by using a quantumMonte Carlomethod, which is a numerical method that can be used to simulate the behavior of correlated electron systems. We present a new algorithm of the quantum Monte Carlo diagonalization that is a method for the evaluation of expectation value without the negative sign problem. We compute pair correlation functions and show that pair correlation is indeed enhanced with hole doping.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.