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arXiv:1210.6249 · Physical Review B

Superconductivity in the two-dimensional Hubbard model: Gutzwiller wave function solution

Jan Kaczmarczyk, Józef Spałek, Tobias Schickling, Jörg Bünemann

Published Sep 16, 2013• 64 citations
arXivHTMLPDF

Abstract

A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF) is formulated and used for the description of superconducting (SC) ground state in the two-dimensional Hubbard model with electron-transfer amplitudes $t$ (and ${t}^{\ensuremath{'}}$) between nearest (and next-nearest) neighbors. The method is numerically very efficient and allows for a detailed analysis of the phase diagram as a function of all relevant parameters ($U$, $\ensuremath{\delta}$, ${t}^{\ensuremath{'}}$) and a determination of the kinetic-energy driven pairing region. SC states appear only for substantial interactions, $U/t\ensuremath{\gtrsim}3$, and for not too large hole doping, $\ensuremath{\delta}\ensuremath{\lesssim}0.32$ for ${t}^{\ensuremath{'}}=0.25t$; this upper critical doping value agrees well with experiment for the cuprate high-temperature superconductors. We also obtain other important features of the SC state: (i) the SC gap at the Fermi surface resembles ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave only around the optimal doping and the corrections to this state are shown to arise from the longer range of the pairing; (ii) the nodal Fermi velocity is almost constant as a function of doping and agrees quantitatively with the experimental results; (iii) the SC transition is driven by the kinetic-energy lowering for low doping and strong interactions.

DOI
10.1103/physrevb.88.115127

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A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF) is formulated and used for the description of superconducting (SC) ground state in the two-dimensional Hubbard model with electron-transfer amplitudes ttt (and t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′) between nearest (and next-nearest) neighbors. The method is numerically very efficient and allows for a detailed analysis of the phase diagram as a function of all relevant parameters (UUU, \ensuremathδ\ensuremath{\delta}\ensuremathδ, t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′) and a determination of the kinetic-energy driven pairing region. SC states appear only for substantial interactions, U/t\ensuremath≳3U/t\ensuremath{\gtrsim}3U/t\ensuremath≳3, and for not too large hole doping, \ensuremathδ\ensuremath≲0.32\ensuremath{\delta}\ensuremath{\lesssim}0.32\ensuremathδ\ensuremath≲0.32 for t\ensuremath′=0.25t{t}^{\ensuremath{'}}=0.25tt\ensuremath′=0.25t; this upper critical doping value agrees well with experiment for the cuprate high-temperature superconductors. We also obtain other important features of the SC state: (i) the SC gap at the Fermi surface resembles dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave only around the optimal doping and the corrections to this state are shown to arise from the longer range of the pairing; (ii) the nodal Fermi velocity is almost constant as a function of doping and agrees quantitatively with the experimental results; (iii) the SC transition is driven by the kinetic-energy lowering for low doping and strong interactions.
A systematic diagrammatic expansion for Gutzwiller wavefunctions (DE-GWFs) proposed very recently is used for the description of the superconducting (SC) ground state in the two-dimensional square-lattice t – J model with the hopping electron amplitudes t (and ) between nearest (and next-nearest) neighbors. For the example of the SC state analysis we provide a detailed comparison of the methodʼs results with those of other approaches. Namely, (i) the truncated DE-GWF method reproduces the variational Monte Carlo (VMC) results and (ii) in the lowest (zeroth) order of the expansion the method can reproduce the analytical results of the standard Gutzwiller approximation (GA), as well as of the recently proposed 'grand-canonical Gutzwiller approximation' (called either GCGA or SGA). We obtain important features of the SC state. First, the SC gap at the Fermi surface resembles a wave only for optimally and overdoped systems, being diminished in the antinodal regions for the underdoped case in a qualitative agreement with experiment. Corrections to the gap structure are shown to arise from the longer range of the real-space pairing. Second, the nodal Fermi velocity is almost constant as a function of doping and agrees semi-quantitatively with experimental results. Third, we compare the doping dependence of the gap magnitude with experimental data. Fourth, we analyze the k -space properties of the model: Fermi surface topology and effective dispersion. The DE-GWF method opens up new perspectives for studying strongly correlated systems, as it (i) works in the thermodynamic limit, (ii) is comparable in accuracy to VMC, and (iii) has numerical complexity comparable to that of the GA (i.e., it provides the results much faster than the VMC approach).
In nonperturbative regimes, the superfluid instability in the two-dimensional (2D) Hubbard model can be described by an emergent BCS theory with small effective pairing constants. We compute the effective couplings using a controlled bold-line diagrammatic Monte Carlo approach, which stochastically sums all skeleton Feynman diagrams dressed in the one- and two-particle channels to high expansion orders, and map out the resulting superfluid ground-state phase diagram in a range of next-nearest-neighbor hopping $0\ensuremath{\le}{t}^{\ensuremath{'}}\ensuremath{\le}0.3t$, interaction strength $0\ensuremath{\le}U\ensuremath{\le}3t$, and lattice filling $0\ensuremath{\le}n\ensuremath{\le}2$. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and ${t}^{\ensuremath{'}}$. At ${t}^{\ensuremath{'}}=0.3$, the weak-coupling picture with the dominant triplet pairing sharply peaked at $n\ensuremath{\approx}0.73$ due to the Van Hove singularity is replaced by a plateau of the singlet ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ paring, while for $U\ensuremath{\gtrsim}3t$ the effective couplings are consistent with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.
The principal purpose of this topical review is to single out some of the universal features of high-temperature (high-Tc) and related strongly-correlated systems, which can be compared with experiment in a quantitative manner. The description starts with the concept of exchange-interaction-mediated (real-space) pairing, combined with strong correlations among narrow band electrons, for which the reference state is that of the Mott–Hubbard insulator. The physical discussion of concrete properties relies on variational approach, starting from generalized renormalized mean-field theory (RMFT) in the form of statistically-consistent Gutzwiller approximation (SGA), and its subsequent generalization, i.e., a systematic Diagrammatic Expansion of the Variational (Gutzwiller-type) Wave Function (DE-GWF). The solution leads to the two energy scales, one involving unconventional quasiparticles close to the Fermi energy, and the other reflecting the fully correlated state, involving electrons deeper below the Fermi surface. Those two regimes are separated by a kink in the dispersion relation, which is observed in photoemission. As a result, one obtains both the doping dependent properties in the correlated state, and effective renormalized quasiparticles. The reviewed ground-state characteristics for high-Tc systems encompass high-Tc superconductivity, nematicity, charge- (and pair-) density-wave effects, as well as non-BCS kinetic energy gain in the paired state. The calculated dynamic properties are: the universal Fermi velocity, the Fermi wave-vector, the effective mass enhancement, the pseudogap; and the d-wave gap magnitude. We discuss that, within the variational approach, the minimal realistic model is represented by the so-called t-J-U Hamiltonian. Inadequacy of the t-J and Hubbard models, particularly of their RMFT versions, is discussed explicitly. For heavy fermion systems, modeled by the Anderson lattice model and with the DE-GWF approach, we discuss the phase diagram, encompassing superconducting, Kondo insulating, ferro- and anti-ferromagnetic states. The superconducting state is then a two-d-wave gap system. If the orbital degeneracy of f-electrons is included, the coexistent ferromagnetic- (spin–triplet) superconducting phases appear and match those observed for UGe2 in a semiquantitative manner. Finally, in the second part, we generalize our approach to the collective spin and charge fluctuations in high-Tc systems, starting from variational approach, defining the saddle-point state, and combining it with 1/N expansion. The present scheme differs essentially from that starting from the saddle-point Hartree–Fock approximation, and incorporating the fluctuations in the random phase approximation (RPA). The spectrum of collective spin and charge excitations is determined for the Hubbard and t-J-U models, and subsequently compared quantitatively with recent experiments. The Appendices provide formal details to make this review self-contained.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
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.
The ground state of the two-dimensional (2D) Hubbard model is investigated by adopting improved wave functions that take into account intersite electron correlation beyond the Gutzwiller ansatz. The ground-state energy is lowered considerably, giving the best estimate of the ground-state energy for the 2D Hubbard model. There is a crossover from weakly to strongly correlated regions as the on-site Coulomb interaction UUU increases. The antiferromagnetic correlation induced by UUU is reduced for hole doping when UUU is large, being greater than the bandwidth, thus increasing the kinetic energy gain. The spin and charge fluctuations are induced in the strongly correlated region. These antiferromagnetic and kinetic charge fluctuations induce electron pairings, which would result in high-temperature superconductivity.