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arXiv:cond-mat/9706173 · Physical review. B, Condensed matter

Charge and spin structures of a dx2−y2 superconductor in the proximity of an antiferromagnetic Mott insulator

Fakher F. Assaad, Masatoshi Imada, Douglas J. Scalapino

Published Dec 15, 1997• 54 citations
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Abstract

To the Hubbard model on a square lattice we add an interaction $W$ 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 $W$ 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, ${T}_{\mathrm{KT}},$ we compute the one-electron density of states $N(\ensuremath{\omega}),$ the spin relaxation rate ${1/T}_{1},$ 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 ${1/T}_{1}$ and divergence of $\mathrm{Re}\ensuremath{\chi}(q\ensuremath{\rightarrow}=(\ensuremath{\pi},\ensuremath{\pi}),\ensuremath{\omega}=0)$ in the low-temperature limit. As ${T}_{\mathrm{KT}}$ 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.

DOI
10.1103/physrevb.56.15001

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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×\else\texttimes\fi16,16\ifmmode\times\else\texttimes\fi{}16,16\ifmmode×\else\texttimes\fi16, 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 dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ 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 dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ superconducting state. Above and below the Kosterlitz-Thouless transition temperature, TKT,{T}_{\mathrm{KT}},TKT​, we compute the one-electron density of states N(\ensuremathω),N(\ensuremath{\omega}),N(\ensuremathω), the spin relaxation rate 1/T1,{1/T}_{1},1/T1​, as well as the imaginary and real part of the spin susceptibility \ensuremathχ(q\ensuremath→,\ensuremathω).\ensuremath{\chi}(q\ensuremath{\rightarrow},\ensuremath{\omega}).\ensuremathχ(q\ensuremath→,\ensuremathω). The spin dynamics are characterized by the vanishing of 1/T1{1/T}_{1}1/T1​ and divergence of Re\ensuremathχ(q\ensuremath→=(\ensuremathπ,\ensuremathπ),\ensuremathω=0)\mathrm{Re}\ensuremath{\chi}(q\ensuremath{\rightarrow}=(\ensuremath{\pi},\ensuremath{\pi}),\ensuremath{\omega}=0)Re\ensuremathχ(q\ensuremath→=(\ensuremathπ,\ensuremathπ),\ensuremathω=0) in the low-temperature limit. As TKT{T}_{\mathrm{KT}}TKT​ is approached N(\ensuremathω)N(\ensuremath{\omega})N(\ensuremathω) develops a pseudogap feature and below TKTIm\ensuremathχ(q\ensuremath→=(\ensuremathπ,\ensuremathπ),\ensuremathω){T}_{\mathrm{KT}}\mathrm{Im}\ensuremath{\chi}(q\ensuremath{\rightarrow}=(\ensuremath{\pi},\ensuremath{\pi}),\ensuremath{\omega})TKT​Im\ensuremathχ(q\ensuremath→=(\ensuremathπ,\ensuremathπ),\ensuremathω) shows a peak at finite frequency.
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 $16\ifmmode\times\else\texttimes\fi{}16$.
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.
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.
The Hubbard model on a square lattice in the ground state is investigated. Various aspects of Mott transition at half-filling are clarified from the quantum Monte Carlo study. Critical exponents of the transition for charge and spin correlations are estimated. For the doping concentration δ, the charge susceptibility is proportional to δ -1 indicating the divergence of charge mass for δ→0, while the spin susceptibility is always finite and not singular near δ=0. Several theoretical consequences of the above results are discussed. Incommensurate peak amplitude of spin correlation scales as (δ-δ c ) -1 with δ c <0.01 in disagreement with RPA results ln (δ-δ c ). The antiferromagnetic order at the half-filling also shows strong correlation character and is not sensitive to the shape of the fermi surface, while the incommensurate peak observed away from the half-filling is sensitively suppressed by the loss of partial nesting consistently with the weak coupling picture. Comparisons with experimental indications of high- T c oxides are made.
We present a numerically exact study of the Hubbard model with spin-dependent anisotropic hopping on the square lattice using auxiliary-field quantum Monte Carlo method. At half filling, the system undergoes Ising phase transitions on cooling, leading to the formation of Ising-type antiferromagnetic order for repulsive interactions and charge-density wave order for attractive interactions at finite temperatures. By elegantly implementing the sign-problem-free condition and Hubbard-Stratonovich transformations, we achieve significant improvements in precision control of the numerical calculations and obtain highly accurate results of the transition temperatures from weak to strong interactions across representative anisotropies. We further characterize the system by examining the temperature dependence of various thermodynamic properties, including the energy, double occupancy, specific heat, and charge susceptibility. Specifically, we provide unbiased numerical results of the entropy map on temperature-interaction plane, the critical entropy, and the spin, singlon and doublon correlations, all of which are directly measurable in optical lattice experiments. Away from half filling, we explore the behavior of the sign problem and investigate the possible emergence of stripe spin-density wave order in the system with repulsive interaction.