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arXiv:cond-mat/0202492 · Physical review. B, Condensed matter

Symmetry breaking in the Hubbard model at weak coupling

T. Schauerte, P. G. J. van Dongen

Published Feb 7, 2002• 15 citations
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Abstract

The phase diagram of the Hubbard model is studied at weak coupling in two and three spatial dimensions. It is shown that the N\'eel temperature and the order parameter in $d=3$ are smaller than the Hartree-Fock predictions by a factor of $q=0.2599.$ For $d=2$ we show that the self-consistent (sc) perturbation series bears no relevance to the behavior of the exact solution of the Hubbard model in the symmetry-broken phase. We also investigate an anisotropic model and show that the coupling between planes is essential for the validity of mean-field-type order parameters.

DOI
10.1103/physrevb.65.081105

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The phase diagram of the Hubbard model is studied at weak coupling in two and three spatial dimensions. It is shown that the N'eel temperature and the order parameter in d=3d=3d=3 are smaller than the Hartree-Fock predictions by a factor of q=0.2599.q=0.2599.q=0.2599. For d=2d=2d=2 we show that the self-consistent (sc) perturbation series bears no relevance to the behavior of the exact solution of the Hubbard model in the symmetry-broken phase. We also investigate an anisotropic model and show that the coupling between planes is essential for the validity of mean-field-type order parameters.
We study a three-dimensional single-band repulsive Hubbard model at weak coupling. We establish the superconducting phase diagram in the parameter space of the chemical potential and the out-of-plane hopping strength. The model continuously connects the Hubbard model in two and three dimensions. We confirm previously established results in these limits, and identify a rich structure of competing order parameters in between. Specifically, we find five types of ppp- and ddd-wave orders. In several regions of the phase diagram, even when the Fermi surface is a corrugated cylinder, the ground state is a time-reversal-symmetry-breaking superconductor with nodes, i.e., a Weyl superconductor.
The Hubbard model including nearest neighbor interaction is studied at T=0 on a d-dimensional hypercubic lattice (d\ensuremath{\gg}1) close to half filling. For the model in d=\ensuremath{\infty} we derive the exact result that the ground state at weak coupling is phase separated. Results for lower dimensions are then derived in a 1/d expansion. To obtain these results we first consider possible second order transitions. One then finds that the broken-symmetry phase near half filling is incommensurate. However, the corresponding ground state has negative compressibility and is hence thermodynamically unstable. A Maxwell construction is used to construct the actual phase separated ground state, which consists of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. It is shown that both the doping level below which phase separation occurs and the order parameter differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion and turns out to be identical to the renormalization factor previously calculated for the low-temperature thermodynamics at half filling. \textcopyright{} 1996 The American Physical Society.
We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction U , for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small U is mapped on to an effective Hamiltonian with the attractive interaction using the canonical transformation: H eff = e S H e − S . The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling n e and the next nearest-neighbor transfer t '. The d-wave pairing is dominant for the square lattice in a wide range of n e and t '. The d-wave pairing is also stable for the square lattice with anisotropic t '. The three-band d–p model is also investigated, for which the d-wave pairing is stable in a wide range of n e and t pp (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter - t ' could not be so large so that the optimum doping rate is in the range of 0.8 < n e < 0.85.
Abstract We study Hartree–Fock theory at half-filling for the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter t in the x - and y -directions and a possibly different hopping parameter tzt_ztz​ t z in the z -direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases tz=0t_z=0tz​=0 t z = 0 and tz=tt_z=ttz​=t t z = t , respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that t=1t=1t=1 t = 1 , we analyze how the Néel temperature and the antiferromagnetic mean field depend on the coupling parameter, U , and on the hopping parameter tzt_ztz​ t z . We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as tz→0t_z \rightarrow 0tz​→0 t z → 0 . It is found that the asymptotic formulas are qualitatively different for tz=0t_z = 0tz​=0 t z = 0 (the two-dimensional case) and tz>0t_z > 0tz​>0 t z > 0 (the case of nonzero hopping in the z -direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit tz→0t_z \rightarrow 0tz​→0 t z → 0 in which the three-dimensional model reduces to the two-dimensional model.
The half-filled extended Hubbard model (including nearest-neighbor interaction) is considered in perturbation theory at weak coupling on a d-dimensional hypercubic lattice (d\ensuremath{\gtrsim}3). It is shown that the exact critical temperature and the exact order parameter in the weak-coupling limit differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion. The results depend sensitively on dimension and on the relative strength of the nearest-neighbor interaction. The renormalization factor in d=3, estimated from the 1/d expansion, is q\ensuremath{\simeq}0.282 for the pure Hubbard model, q\ensuremath{\simeq}0.179 near the spin-density-wave to charge-density-wave transition and q\ensuremath{\simeq}0.830 in the limit of pure nearest-neighbor interaction.
The phases with spontaneously broken symmetries corresponding to antiferromagnetic and ddd-wave superconducting order in the two-dimensional $t\ensuremath{-}{t}^{'}$-Hubbard model are investigated by means of the functional renormalization group. The introduction of composite boson fields in the magnetic, charge density, and superconducting channels allows an efficient parametrization of the four-fermion vertex and the study of regimes where either the antiferromagnetic or superconducting order parameter, or both, are nonzero. We compute the phase diagram and the temperature dependence of the order parameter below the critical temperature, where antiferromagnetic and superconducting order show a tendency of mutual exclusion.