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arXiv:cond-mat/9310025 · Physical review. B, Condensed matter

Temperature derivative of the superfluid density in the attractive Hubbard model

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

Published Feb 1, 1994• 24 citations
arXivHTMLPDF

Abstract

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}${\mathit{D}}_{\mathit{s}}$(\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}${\mathit{D}}_{\mathit{s}}$(\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.

DOI
10.1103/physrevb.49.4327

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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.
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.
We study the superconducting Kosterlitz-Thouless (KT) transition of the attractive Hubbard model on a two-dimensional triangular lattice using auxiliary field quantum Monte Carlo method for system sizes up to $14\ifmmode\times\else\texttimes\fi{}14$ sites. Combining three methods to analyze the numerical data, we find, for the attractive interaction of $U=\ensuremath{-}4t$, that the transition temperature is broadly maximized around the band filling of 1.2<n<1.61.2<n<1.61.2<n<1.6, while it is found to be rather low in the range of n<1n<1n<1 and n>1.8n>1.8n>1.8. We discuss that this band filling dependence may be explained in terms of the band filling dependence of the superfluid density and the effective mass. Our analysis also teaches us that the estimation of the KT transition temperature requires some caution when the density of states near the Fermi level is small and the calculation is restricted to small system sizes.
We study the attractive fermionic Hubbard model on a honeycomb lattice using determinantal quantum Monte Carlo simulations. By increasing the interaction strength UUU (relative to the hopping parameter ttt) at half filling and zero temperature, the system undergoes a quantum phase transition at 5.0<Uc/t<5.15.0<{U}_{c}/t<5.15.0<Uc​/t<5.1 from a semimetal to a phase displaying simultaneously superfluid behavior and density order. Doping away from half filling, and increasing the interaction strength at finite but low temperature TTT, the system always appears to be a superfluid exhibiting a crossover between a BCS and a molecular regime. These different regimes are analyzed by studying the spectral function. The formation of pairs and the emergence of phase coherence throughout the sample are studied as UUU is increased and TTT is lowered.
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.
The helicity modulus, which is the stiffness associated with a twisted order parameter, for the two-dimensional Hubbard model is calculated for the equivalent cases of (i) attractive on-site interaction (negative U) with arbitrary strength, arbitrary electron density, and zero magnetic field and (ii) repulsive on-site interaction (positive U) with arbitrary strength, at half-filling and in an arbitrary magnetic field. An explicit formula for the helicity modulus is derived using the Bogoliubov-Hartree-Fock approximation. An improved value for the helicity modulus is obtained by performing variational Monte Carlo calculations using a Gutzwiller projected trial wave function. To within a small correction term the helicity modulus is found to be given by -1/8 the average kinetic energy. The variational Monte Carlo calculation is found to increase the value of the helicity modulus by a small amount (about 5% for intermediate values of the interaction strength) compared to the results from the Bogoliubov-Hartree-Fock approximation. In the case of attractive interaction, from a comparison with the Kosterlitz-Thouless relation between critical temperature and helicity modulus, the critical temperature for a Kosterlitz-Thouless transition is calculated and a phase diagram is obtained. An optimal critical temperature is found for an intermediate value of U. We discuss connections of our results with results in the literature on the Hubbard model using the random-phase approximation and quantum Monte Carlo calculations.
We study the two-dimensional attractive Hubbard model using the mapping onto the half-filled repulsive Hubbard model in a uniform magnetic field coupled to the fermion spins. The low-energy effective action for charge and pairing fluctuations is obtained in the hydrodynamic regime. We recover the action of a Bose superfluid where half the fermion density is identified as the conjugate variable of the phase of the superconducting order parameter. By integrating out charge fluctuations, we obtain a phase-only action. In the zero-temperature superconducting state, this action describes a collective phase mode smoothly evolving from the Anderson-Bogoliubov mode at weak coupling to the Bogoliubov mode of a Bose superfluid at strong coupling. At finite temperature, the phase-only action can be used to extract an effective XYXYXY model and thus obtain the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also identify a renormalized classical regime of superconducting fluctuations above the BKT phase transition, and a regime of incoherent pairs at higher temperature. Special care is devoted to the nearly half-filled case where the symmetry of the order parameter is enlarged to SO(3) due to strong $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge fluctuations. The low-energy effective action is then an SO(3) nonlinear $\ensuremath{\sigma}$ model with a (symmetry breaking) magnetic field proportional to the doping. In the strong-coupling limit, the attractive Hubbard model can be mapped onto the Heisenberg model in a magnetic field, which reduces to the quantum XYXYXY model (except for a weak magnetic field, i.e., in the low-density limit of the attractive model). In the low-density limit, the Heisenberg model allows one to recover the action of a Bose superfluid, including the ${(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}$ term (with $\ensuremath{\rho}$ the density), and in turn the Gross-Pitaevskii equation.