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arXiv:cond-mat/0004171 · Physical review. B, Condensed matter

t-U-Wmodel of a dx2−y2 superconductor in the proximity of an antiferromagnetic Mott insulator: Diagrammatic studies versus quantum Monte Carlo simulations

Thomas Eckl, Enrico Arrigoni, Werner R. Hanke, Fakher F. Assaad

Published Nov 1, 2000• 1 citation
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Abstract

We examine the competition and relationship between an antiferromagnetic (AF) Mott insulating state and a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconducting (SC) state in two dimensions using semianalytical, i.e., diagrammatic calculations of the t-U-W model. The AF Mott insulator is described by the ground state of the half filled Hubbard model on a square lattice with on-site Coulomb repulsion U and nearest-neighbor single-particle hopping t. To this model, an extra term W is added, which depends upon the square of the single-particle nearest-neighbor hopping. Staying at half band filling and a constant value of U, it was previously shown with quantum Monte Carlo (QMC) simulations that one can generate a quantum transition as a function of the coupling strength W between an AF Mott insulating state and a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}\mathrm{SC}$ state. Here we complement these earlier QMC simulations with physically more transparent diagrammatic calculations. We start with a standard Hartree-Fock (HF) calculation to capture the ``high-energy'' physics of the t-U-W model. Then, we derive and solve the Bethe-Salpeter equation, i.e., we account for fluctuation effects within the time-dependent HF or generalized random-phase approximation scheme. Spin and charge susceptibility as well as the effective interaction vertex are calculated and systematically compared with QMC data. Finally, the corresponding BCS gap equation obtained for this effective interaction is solved.

DOI
10.1103/physrevb.62.12395

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We examine the competition and relationship between an antiferromagnetic (AF) Mott insulating state and a dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ superconducting (SC) state in two dimensions using semianalytical, i.e., diagrammatic calculations of the t-U-W model. The AF Mott insulator is described by the ground state of the half filled Hubbard model on a square lattice with on-site Coulomb repulsion U and nearest-neighbor single-particle hopping t. To this model, an extra term W is added, which depends upon the square of the single-particle nearest-neighbor hopping. Staying at half band filling and a constant value of U, it was previously shown with quantum Monte Carlo (QMC) simulations that one can generate a quantum transition as a function of the coupling strength W between an AF Mott insulating state and a dx2\ensuremath−y2SC{d}_{{x}^{2}\ensuremath{-}{y}^{2}}\mathrm{SC}dx2\ensuremath−y2​SC state. Here we complement these earlier QMC simulations with physically more transparent diagrammatic calculations. We start with a standard Hartree-Fock (HF) calculation to capture the ``high-energy'' physics of the t-U-W model. Then, we derive and solve the Bethe-Salpeter equation, i.e., we account for fluctuation effects within the time-dependent HF or generalized random-phase approximation scheme. Spin and charge susceptibility as well as the effective interaction vertex are calculated and systematically compared with QMC data. Finally, the corresponding BCS gap equation obtained for this effective interaction is solved.
We consider a Hubbard model on a square lattice with an additional interaction, WWW, which depends upon the square of the near-neighbor hopping. At half filling and a constant value of the Hubbard repulsion, increasing the strength of the interaction WWW drives the system from an antiferromagnetic Mott insulator to a ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ superconductor. This conclusion is reached on the basis of zero temperature quantum Monte Carlo simulations on lattice sizes up to $16\ifmmode\times\else\texttimes\fi{}16$.
To the Hubbard model on a square lattice we add an interaction WWW that depends upon the square of a near-neighbor hopping. We use zero-temperature quantum Monte Carlo simulations on lattice sizes up to $16\ifmmode\times\else\texttimes\fi{}16,$ to show that at half-filling and constant value of the Hubbard repulsion, the interaction WWW triggers a quantum transition between an antiferromagnetic Mott insulator and a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor. With a combination of finite-temperature quantum Monte Carlo simulations and the maximum entropy method, we study spin and charge degrees of freedom in the superconducting state. We give numerical evidence for the occurrence of a finite-temperature Kosterlitz-Thouless transition to the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconducting state. Above and below the Kosterlitz-Thouless transition temperature, TKT,{T}_{\mathrm{KT}},TKT​, we compute the one-electron density of states $N(\ensuremath{\omega}),$ the spin relaxation rate 1/T1,{1/T}_{1},1/T1​, as well as the imaginary and real part of the spin susceptibility $\ensuremath{\chi}(q\ensuremath{\rightarrow},\ensuremath{\omega}).$ The spin dynamics are characterized by the vanishing of 1/T1{1/T}_{1}1/T1​ and divergence of $\mathrm{Re}\ensuremath{\chi}(q\ensuremath{\rightarrow}=(\ensuremath{\pi},\ensuremath{\pi}),\ensuremath{\omega}=0)$ in the low-temperature limit. As TKT{T}_{\mathrm{KT}}TKT​ is approached $N(\ensuremath{\omega})$ develops a pseudogap feature and below ${T}_{\mathrm{KT}}\mathrm{Im}\ensuremath{\chi}(q\ensuremath{\rightarrow}=(\ensuremath{\pi},\ensuremath{\pi}),\ensuremath{\omega})$ shows a peak at finite frequency.
With high-Tc cuprates in mind, properties of correlated dx2-y2-wave superconducting (SC) and antiferromagnetic (AF) states are studied for the Hubbard (t-t'-U) model on square lattices, using a variational Monte Carlo method. We employ simple trial wave functions including only crucial parameters, such as a doublon-holon binding factor indispensable to describe correlated SC and normal states as doped Mott insulators. U/t, t'/t and δ(doping rate) dependence of relevant quantities are systematically calculated. As U/t increases, a sharp crossover of SC properties occurs at U_co/t \sim 10 from a conventional BCS type to a kinetic-energy-driven type for any t'/t. As δdecreases, U_co/t is smoothly connected to the Mott transition point at half filling. For U/t\lsim 5, steady superconductivity corresponding to the cuprates is not found, whereas the d-wave SC correlation function Pd^\infty rapidly increases for U/t\gsim 6 and becomes maximum at U=U_co. Comparing the δdependence of Pd^\infty with experimentally observed dome-shaped Tc and condensation energy, we find that the effective value of UUU for the cuprates should be larger than the band width, for which the t-J model is valid. Analyzing the kinetic energy, we reveal that for U>U_co only doped holes (electrons) become charge carriers, which will make a small Fermi surface (hole pocket), but for UU_co is determined by two factors, i.e., the AF spin correlation, which creates singlet pairs (pseudogap), and the charge mobility dominated by Mott physics.
Cooperation and competition between the antiferromagnetic, ddd-wave superconducting, and Mott-insulating states are explored for the two-dimensional Hubbard model including nearest and next-nearest-neighbor hoppings at zero temperature. Using the variational cluster approach with clusters of different shapes and sizes up to 10 sites, it is found that the doping-driven transition from a phase with microscopic coexistence of antiferromagnetism and superconductivity to a purely superconducting phase is discontinuous for strong interaction and accompanied by phase separation. At half-filling the system is in an antiferromagnetic Mott-insulating state with vanishing charge compressibility. Upon decreasing the interaction strength UUU below a certain critical value of roughly ${U}_{c}\ensuremath{\sim}4$ (in units of the nearest-neighbor hopping), however, the filling-dependent magnetic transition changes its character and becomes continuous. Phase separation or, more carefully, the tendency towards the formation of inhomogeneous states disappears. This critical value is in contrast to previous studies, where a much larger value was obtained. Moreover, we find that the system at half-filling undergoes the Mott transition from an insulator to a state with a finite charge compressibility at essentially the same value. The weakly correlated state at half-filling exhibits superconductivity microscopically admixed to the antiferromagnetic order. This scenario suggests a close relation between phase separation and the Mott-insulator physics.
Based on the known phenomenology of high-Tc cuprates and the available numerical calculations of the t − J model, a two-dimensional effective fermionic model with the nearest neighbor attraction is proposed. Numerical calculations suggest that the model has the dx2−y2 superconductivity (SC) in the ground state at a low fermionic density. We argue that this model captures the important physics of the dx2−y2 superconducting correlations found earlier in the t − J model by the exact diagonalization approach. Within a self-consistent RPA diagrammatic study, the density and the coupling strength dependence of the critical temperature is calculated. We also investigate the influence of the impurities on our results and show that the suppression of the superconductivity is insignificant, when the retardation effects are accounted for as opposed to the Hartree–Fock approximation.
In connection with recent experiments on excitation in which Mott insulators change to conductors, we study the properties of excited states beyond the Mott gap as quasi-stationary states for a two-dimensional Hubbard (t-t'-U) model at half filling. A variational Monte Carlo method is used with trial wave functions for paramagnetic or normal (PM), superconducting with dx2-y2-wave (d-SC), isotropic s-wave, and extended s-wave symmetries, and antiferromagnetic (AF) states. The excited states are generated by imposing a minimum number of doubly occupied sites (doublons) D_L on the lowest-energy states. For U>W (W: band width), d_L=D_L/Ns (Ns: number of sites) corresponds to the excitation intensity. It is found that the AF state is the most stable among the states we treated for d_L<0.14 and insulating. The PM and d-SC states become conductive over a threshold d_Lc, and the conduction is caused by unbound doublons and holons (empty sites). The PM state arises for d_L>0.14, but the d-SC state is always hidden by the AF state. The s-wave-type superconducting states are not stabilized for any parameter set.