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arXiv:cond-mat/9609162 · Physical review. B, Condensed matter

Strong-coupling phases of the t−t′ Hubbard model

J. González, Jose Vicente Alvarez

Published Jul 1, 1997• 5 citations
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Abstract

We study the strong-coupling regime of the $t$-${t}^{\ensuremath{'}}$ Hubbard model, filled up to the level of the van Hove singularities, by means of an exact diagonalization approach. We characterize the different phases of the model by the different sectors of the Hilbert space with given quantum numbers. By looking for the ground state of the system, we find essentially the competition between a state with incipient ferromagnetism and other mimicking a $d$-wave condensate, which has the lowest energy in a large region of the phase space.

DOI
10.1103/physrevb.56.367

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We study the strong-coupling regime of the ttt-t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ Hubbard model, filled up to the level of the van Hove singularities, by means of an exact diagonalization approach. We characterize the different phases of the model by the different sectors of the Hilbert space with given quantum numbers. By looking for the ground state of the system, we find essentially the competition between a state with incipient ferromagnetism and other mimicking a ddd-wave condensate, which has the lowest energy in a large region of the phase space.
We devise a computational approach to the Hubbard model that captures the strong coupling dynamics arising when the Fermi level is at a Van Hove singularity in the density of states. We rely on an approximate degeneracy among the many-body states accounting for the main instabilities of the system (antiferromagnetism, d-wave superconductivity). The Fermi line turns out to be deformed in a manner consistent with the pinning of the Fermi level to the Van Hove singularity. For a doping rate $\ensuremath{\delta}\ensuremath{\sim}0.2,$ the ground state is characterized by d-wave symmetry, quasiparticles gapped only at the saddle points of the band, and a large peak at zero momentum in the d-wave pairing correlations.
We exactly diagonalize the effective Hamiltonian obtained from the Hubbard model in the strong-coupling limit for a two-dimensional $\sqrt{10}\ifmmode\times\else\texttimes\fi{}\sqrt{10}$ size square lattice. The effective Hamiltonian operates on a restricted Hilbert space containing only states with singly occupied sites, which makes the diagonalization possible within reasonable computational time for the above lattice. The ground-state energy and wave function are obtained for several values of the coupling ratio tU\frac{t}{U}Ut​ and the doping fraction xxx. We find three different phases in the (x,tUx,\frac{t}{U}x,Ut​) phase diagram. The first is characterized by antiferromagnetic order which extends to the longest possible distance in the 10-site system. The second and third phases are characterized by antiferromagnetic and ferromagnetic short-range correlations, respectively. We comment on the possible relevance of the results to the recently discovered high-temperature copper-oxide superconductors.
We apply a Wilsonian renormalization group approach to the continuum limit of the attractive t − t ′ Hubbard model, taken when the Fermi level is at the Van Hove singularity of the density of states. We determine the leading instabilities by computing the different response functions up to t ′ = 0.5 t . The phase diagram shows a large boundary between superconducting and charge-density-wave phases, that merge in a triple point with a phase separation instability. The latter is realized down to very low coupling constant, as the Fermi sea degenerates towards a pair of straight lines near t ′ = 0.5 t .
The phase diagram of the two-dimensional extended one-band $U\ensuremath{-}V\ensuremath{-}J$ Hubbard model is considered within a mean-field approximation and two- and many-patch renormalization-group (RG) approaches near the Van Hove band fillings. At small ${t}^{\ensuremath{'}}$ and J>0,J>0,J>0, mean-field and many-patch RG approaches give similar results for the leading spin-density-wave (SDW) instability, while the two-patch RG approach that predicts a wide region of charge-flux (CF) phase becomes unreliable due to nesting effect. There is a complex competition between SDW, CF phases, and d-wave superconductivity at small ${t}^{\ensuremath{'}}$ in two- and many-patch RG approaches. The spin-flux phase, which is not stable at the mean-field level, is identified as a possible ground state at J<0J<0J<0 in both RG approaches. With increasing ${t}^{\ensuremath{'}}$ the results of all three approaches merge: d-wave superconductivity at J>0J>0J>0 and ferromagnetism at J<0J<0J<0 become the leading instabilities. For large enough V, the charge-density-wave state occurs.
We obtain a complete and numerically exact in the weak-coupling limit $(U\ensuremath{\rightarrow}0)$ ground-state phase diagram of the repulsive fermionic Hubbard model on the square lattice for filling factors 0<n<20<n<20<n<2 and next-nearest-neighbor hopping amplitudes $0\ensuremath{\le}{t}^{\ensuremath{'}}\ensuremath{\le}0.5$. Phases are distinguished by the symmetry and the number of nodes of the superfluid order parameter. The phase diagram is richer than may be expected and typically features states with a high---higher than that of the fundamental mode of the corresponding irreducible representation---number of nodes. The effective coupling strength in the Cooper channel $\ensuremath{\lambda}$, which determines the critical temperature Tc{T}_{c}Tc​ of the superfluid transition, is calculated in the whole parameter space and regions with high values of $\ensuremath{\lambda}$ are identified. It is shown that besides the expected increase of $\ensuremath{\lambda}$ near the Van Hove singularity line, joining the ferromagnetic and antiferromagnetic points, another region with high values of $\ensuremath{\lambda}$ can be found at quarter filling and ${t}^{\ensuremath{'}}=0.5$ due to the presence of a line of nesting at ${t}^{\ensuremath{'}}\ensuremath{\ge}0.5$. The results can serve as benchmarks for controlled nonperturbative methods and guide the ongoing search for high-Tc{T}_{c}Tc​ superconductivity in the Hubbard model.