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arXiv:cond-mat/0312314 · Physical Review B

Phase diagram of the two-dimensional extended Hubbard model: Phase transitions between different pairing symmetries when charge and spin fluctuations coexist

Seiichiro Onari, Ryotaro Arita, Kazuhiko Kuroki, Hideo Aoki

Published Sep 30, 2004• 93 citations
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Abstract

In order to explore how superconductivity arises when charge fluctuations and spin fluctuations coexist, we have obtained a phase diagram against the off-site repulsion $V$ and band filling $n$ for the extended, repulsive Hubbard model on the square lattice with the fluctuation exchange approximation. We have found the existence of (i) a transition between ${d}_{xy}$ and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing symmetries, (ii) $f$-pairing in between the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and CDW phases for intermediate $0.5<n<1.0$ and large $V$, and (iii) for anisotropic cases the pairing symmetry changing, in agreement with a previously proposed ``generic phase diagram,'' as $d\ensuremath{\rightarrow}f\ensuremath{\rightarrow}s$ when $V$ (hence the charge fluctuations) are increased. All these are consequences of the structure in the charge and spin susceptibilities, which have peaks habitating at common or segregated positions in $k$ space.

DOI
10.1103/physrevb.70.094523

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In order to explore how superconductivity arises when charge fluctuations and spin fluctuations coexist, we have obtained a phase diagram against the off-site repulsion VVV and band filling nnn for the extended, repulsive Hubbard model on the square lattice with the fluctuation exchange approximation. We have found the existence of (i) a transition between dxy{d}_{xy}dxy​ and dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ pairing symmetries, (ii) fff-pairing in between the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ and CDW phases for intermediate 0.5<n<1.00.5<n<1.00.5<n<1.0 and large VVV, and (iii) for anisotropic cases the pairing symmetry changing, in agreement with a previously proposed ``generic phase diagram,'' as d\ensuremath→f\ensuremath→sd\ensuremath{\rightarrow}f\ensuremath{\rightarrow}sd\ensuremath→f\ensuremath→s when VVV (hence the charge fluctuations) are increased. All these are consequences of the structure in the charge and spin susceptibilities, which have peaks habitating at common or segregated positions in kkk space.
The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g{B}_{2g}B2g​ (dxy{d}_{xy}dxy​) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN{V}_{\mathrm{NN}}VNN​, the latter induces pairing in other symmetry channels, including, e.g., A2g{A}_{2g}A2g​ (ggg-wave), nodal A1g{A}_{1g}A1g​ (extended sss-wave), or nodal Eu{E}_{u}Eu​ (ppp-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0{V}_{\mathrm{NN}}<0VNN​<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu{E}_{u}Eu​ (ppp-wave) spin-triplet or B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.
We explore how the superconductivity arising from the on-site electron-electron repulsion changes when the repulsion is made long-ranged, 1∕r1∕r1∕r-like interaction by introducing an extended Hubbard model with the repulsion extended to distant (12th) neighbors. With a simplified fluctuation-exchange approximation, we have found for the square lattice that: (i) As the band filling becomes dilute enough, the charge susceptibility becomes comparable with the spin susceptibility, where ppp- and then sss-wave pairings become relatively dominant, in agreement with the result for the electron gas by Takada, while (ii) the ddd wave, which reflects the lattice structure, dominates well away from the half-filling. All of these can be understood in terms of the spin and charge structures along with the shape and size of the Fermi surface.
Abstract We consider the extended Hubbard model on a two-dimensional square lattice at half-filling. The model is investigated using the strong coupling diagram technique. We sum infinite series of ladder diagrams allowing for full-scale charge and spin fluctuations and the actual short-range antiferromagnetic order for nonzero temperatures. In agreement with earlier results, we find the first-order phase transition in the charge subsystem occurring at v = v c ≳ U/4 with v and U the intersite and on-site Coulomb repulsion constants. The transition reveals itself in an abrupt sign change of a sharp maximum in the zero-frequency charge susceptibility at the corner of the Brillouin. States arising at the transition have alternating deviations of electron occupations from the mean value on neighboring sites. Due to fluctuations, these alternating occupation deviations have short-range order. For the considered parameters, such behavior is found for U ≲ 5t with t the hopping constant. For the insulating case U ≳ 6t, in which the transition is not observed, we find a continuous growth of the Mott gap with v. The evolution of the electron density of states with increasing v is also considered.
Possible ordered states in the 2D extended Hubbard model with on-site (U>0) and nearest-neighbor (V) interaction are examined near half filling, with emphasis on the effect of finite V. First, the phase diagram at absolute zero is determined in the mean field approximation. For V<0V<0V<0, a state where d_{x^{2}-y^{2}}-wave superconductivity (dSC), commensurate spin-density-wave (SDW) and πππ-triplet pair coexist is seen to be stabilized. Here, the importance of πππ-triplet pair on the coexistence of dSC and SDW is indicated. This coexistent state is hampered by the phase separation (PS), which is generally expected to occur in the presence of finite-range attractive interaction, but survives. For V>0, a state where commensurate charge-density-wave (CDW), SDW and ferromagnetism (FM) coexist is seen to be stabilized. Here, the importance of FM on the coexistence of CDW and SDW is indicated. Next, in order to examine the effects of fluctuation on each mean field ordered state, the renormalization group method for the special case that the Fermi level lies just on the saddle points, (πππ,0) and (0,πππ), is applied. The crucial difference from the mean field result is that superconductivity can arise even for U>0 and V≥0V\geq0V≥0, where the superconducting gap symmetry is d_{x^{2}-y^{2}}-wave for U>4V and s-wave for U<4V. Finally, the possibilities that the mean field coexistent states survive in the presence of fluctuation are discussed.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.
The interplay between antiferromagnetism and superconductivity is studied by using the three-dimensional nearly half-filled Hubbard model with anisotropic transfer matrices tz{t}_{z}tz​ and ${t}_{\ensuremath{\perp}}.$ The phase diagrams are calculated for varying values of the ratio ${r}_{z}{=t}_{z}{/t}_{\ensuremath{\perp}}$ using the spin fluctuation theory within the fluctuation-exchange approximation. The antiferromagnetic phase around the half-filled electron density expands while the neighboring phase of the anisotropic ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity shrinks with increasing rz.{r}_{z}.rz​. For small rz{r}_{z}rz​ Tc{T}_{c}Tc​ decreases slowly with increasing rz.{r}_{z}.rz​. For moderate values of rz{r}_{z}rz​ we find a second order transition, with lowering temperature, from the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconducting phase to a phase where an incommensurate spin density wave (SDW) coexists with ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity. Resonance peaks discussed previously for two-dimensional (2D) superconductors are shown to survive in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconducting phase of 3D systems. Soft components of the incommensurate SDW spin fluctuation mode grow as the coexistent phase is approached.