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arXiv:0710.1356 · Physical Review B

Sign change of the extended s -wave pairing vertex in the dynamic Hubbard model: A quantum Monte Carlo study

Karim Bouadim, Matthew Enjalran, F. Hébert, G. G. Batrouni, Richard T. Scalettar

Published Jan 22, 2008• 9 citations
arXivHTMLPDF

Abstract

The ``dynamic'' Hubbard Hamiltonian describes interacting fermions on a lattice whose on-site repulsion is modulated by a coupling to a fluctuating bosonic field. We investigate one such model, introduced by Hirsch, using the determinant quantum Monte Carlo method. Our key result is that the extended $s$-wave pairing vertex, repulsive in the usual static Hubbard model, becomes attractive as the coupling to the fluctuating Bose field increases. The sign problem prevents us from exploring a low enough temperature to see if a superconducting transition occurs. We also observe a stabilization of antiferromagnetic correlations and the Mott gap near half-filling, and a near linear behavior of the energy as a function of particle density which indicates a tendency toward phase separation.

DOI
10.1103/physrevb.77.014516

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The ``dynamic'' Hubbard Hamiltonian describes interacting fermions on a lattice whose on-site repulsion is modulated by a coupling to a fluctuating bosonic field. We investigate one such model, introduced by Hirsch, using the determinant quantum Monte Carlo method. Our key result is that the extended sss-wave pairing vertex, repulsive in the usual static Hubbard model, becomes attractive as the coupling to the fluctuating Bose field increases. The sign problem prevents us from exploring a low enough temperature to see if a superconducting transition occurs. We also observe a stabilization of antiferromagnetic correlations and the Mott gap near half-filling, and a near linear behavior of the energy as a function of particle density which indicates a tendency toward phase separation.
The extended Hubbard model describes fermions on a lattice coupled through on-site, UUU, and first-neighbor, VVV, interactions. In the context of high-Tc{T}_{c}Tc​ cuprates, antiferromagnetic fluctuations may lead to an attractive channel, hence to superconductivity. Despite interest in the two-dimensional version of the model, the current knowledge about the phase diagram is still far from complete. Here, we report on the results of extensive determinant quantum Monte Carlo simulations for this model at half filling, in which we have used the average sign of the product of fermionic determinants as an additional observable to locate critical points. We arrive at a ground state phase diagram in the $U\text{\ensuremath{-}}V$ plane in which the boundaries involving antiferromagnetic, charge-ordered, sss- and ddd-wave superconductivity, and phase-separated phases are quantitatively set with good accuracy. We have also proposed a partial phase diagram, Tc(U,V){T}_{c}(U,V)Tc​(U,V), featuring critical temperatures for the charge-density wave and sss-wave superconducting phases.
We investigate the phase diagram of the extended Hubbard model on a half-filled honeycomb lattice through determinant quantum Monte Carlo simulations. At zero temperature, the repulsive Hubbard model undergoes a transition to an antiferromagnetic state only for sufficiently strong on-site coupling UUU. However, the inclusion of a repulsive extended interaction VVV between fermions in nearest-neighbor orbitals induces a charge-ordered phase. For attractive interactions, we also identify superconducting and phase-separated phases. We have determined a complete phase diagram with the aid of structure factors associated with various orderings, combined with measurements of double occupancy and the average sign of fermionic determinants. We have established that the semimetallic phase forms a zone from which ordered phases are excluded, preventing the stabilization of a superconducting phase with ddd-wave symmetry, i.e., only sss-wave pairing is allowed.
Determinant quantum Monte Carlo (DQMC) is used to determine the pairing and magnetic response for a Hubbard model built up from four-site clusters---a two-dimensional square lattice consisting of elemental $2\ifmmode\times\else\texttimes\fi{}2$ plaquettes with hopping ttt and on-site repulsion UUU coupled by an interplaquette hopping ${t}^{\ensuremath{'}}\ensuremath{\le}t$. Superconductivity in this geometry has previously been studied by a variety of analytic and numeric methods, with differing conclusions concerning whether the pairing correlations and transition temperature are raised near half filling by the inhomogeneous hopping or not. For U/t=4U/t=4U/t=4, DQMC indicates an optimal ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.4$ at which the pairing vertex is most attractive. The optimal ${t}^{\ensuremath{'}}/t$ increases with U/tU/tU/t. We then contrast our results for this plaquette model with a hamiltonian which instead involves a regular pattern of site energies whose large site energy limit is the three-band CuO2{\mathrm{CuO}}_{2}CuO2​ model; we show that there the inhomogeneity rapidly, and monotonically, suppresses pairing.
We utilize numerical linked-cluster expansions (NLCEs) and the determinantal quantum Monte Carlo algorithm to study pairing correlations in the square-lattice Hubbard model. To benchmark the NLCE, we first locate the finite-temperature phase transition of the attractive model to a superconducting state away from half filling. We then explore the superconducting properties of the repulsive model for the ddd-wave and extended sss-wave pairing symmetries. The pairing structure factor shows a strong tendency to ddd-wave pairing and peaks at an interaction strength comparable to the bandwidth. The extended sss-wave structure factor and correlation length are larger at higher temperatures but clearly saturate as temperature is lowered, whereas the ddd-wave counterparts, which start off lower at high temperatures, continue to rise near half filling. This rise is even more dramatic in the ddd-wave susceptibility. The convergence of NLCEs breaks down as the susceptibilities and correlation lengths become large, so we are unable to determine the onset of long-range order. However, our results extend the conclusion, previously restricted to only magnetic and charge correlations, that NLCEs offer a unique window into pairing in the Hubbard model at strong coupling.
The Hubbard and extended Hubbard models on the honeycomb lattice can be seen as prototype models of a single-layer graphene placed in a high dielectric constant environment that screens the Coulomb interaction. Taking advantage of the absence of a sign problem at half-filling, we study this problem with clusters up to 96 sites with the determinant quantum Monte Carlo method as an impurity solver for the dynamical cluster approximation at finite temperatures. After determining the stability of the semimetallic phase to interaction-induced spin-density wave (SDW), charge-density wave (CDW), and Mott insulating phases, we study the single-particle dynamics of the Dirac fermions. We show that when spontaneous symmetry breaking is avoided, the semimetallic phase is a stable Fermi liquid in the presence of repulsive interactions and that Kondo screening dominates the low temperature regime, even though there is a $\ensuremath{\rho}(\ensuremath{\omega})=|\ensuremath{\omega}|$ type local density of states. We also investigate the impact of the correlation effects on the renormalization of the Fermi velocity vF{v}_{F}vF​. We find that vF{v}_{F}vF​ is not renormalized when only on-site repulsion UUU is present, but that near-neighbor repulsion VVV does renormalize vF{v}_{F}vF​. This may explain the variations between different measurements of vF{v}_{F}vF​ in graphene.
We design a novel formalism of determinant quantum Monte Carlo method for the half-filled Hubbard model with on-site Hubbard interaction UUU and nearest-neighbor density-density interaction VVV on the square lattice. The formalism is free of sign problem for $|U|\ensuremath{\ge}4|V|$, and is achieved by introducing discrete auxiliary fields on the nearest-neighbor bonds alone. Based on this formalism, we study the ground-state phase diagram of the model systematically using the projector algorithm. Within the sign-problem free parameter space $|U|\ensuremath{\ge}4|V|$, we obtain antiferromagnetism for $U\ensuremath{\ge}4|V|>0$, charge density wave for $U<\ensuremath{-}4V$ and V>0,sV>0, sV>0,s-wave superconductivity for U<0U<0U<0 and small negative VVV, and phase separation for U<0U<0U<0 and a larger negative VVV. We also obtain the single particle gap and spin excitation spectra, and by comparison to mean field results, as well as available literatures, we discuss the possible phase boundary beyond the sign-problem free region. The unbiased numerically exact results can be taken as benchmarks in future studies. Finally, we discuss possible extension for longer-range interactions in the model.