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arXiv:cond-mat/9904213 · Journal of the Physical Society of Japan

Possible Ordered States in the 2D Extended Hubbard Model

Masakazu Murakami

Published Apr 15, 2000• 30 citations
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Abstract

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 $V<0$, a state where d_{x^{2}-y^{2}}-wave superconductivity (dSC), commensurate spin-density-wave (SDW) and $π$-triplet pair coexist is seen to be stabilized. Here, the importance of $π$-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, ($π$,0) and (0,$π$), is applied. The crucial difference from the mean field result is that superconductivity can arise even for U>0 and $V\geq0$, 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.

DOI
10.1143/jpsj.69.1113

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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.
A one-dimensional model of interacting electrons with on-site U, nearest- neighbor V, and pair-hopping interaction W is studied at half-filling using the continuum limit field theory approach. The ground state phase diagram is obtained for a wide range of coupling constants. In addition to the insulating spin-density wave (SDW) and charge-density wave (CDW) phases for large U and V, respectively, we identify a bond-charge-density-wave (BCDW) phase W < 0, \U - 2V\ < \2W\ and a bond-spin-density-wave (BSDW) for W > 0, \U - 2V\ < W. The possibility of bond-located ordering results from the site-off-diagonal nature of the pair-hopping term and is a special feature of the half-filled band case. The BCDW phase corresponding to an enhanced Peierls instability in the system. The BdSDW is an unconventional insulating magnetic phase, characterized by a gapless spin excitation spectrum and a staggered magnetization located on bonds between sites. The general ground state phase diagram including insulating, metallic, and superconducting phases is discussed. A transition to the eta(pi) -superconducting phase at \U - 2V\ much less than 2t less than or equal to W is briefly discussed.
This paper describes group theoretical classification of superconducting states (SC) in the extended Hubbard model with on-site repulsion (U), nearest neighbor attraction (V) and nearest neighbour exchange interaction (J) on the two-dimensional square lattice using the mean field approach. By decomposing the pairing interaction into irreducible parts; A1g, B1g and Eu of D4h point symmetry, we have derived two singlet SCs (s-wave and d-wave) from A1g and B1g, eight triplet SCs from Eu. The first three types of triplet SC have pairing by electrons with antiparallel spin, the second two types have pairing by electrons with equal spin and the last three types are non-unitary and have pairing by only up-spin electrons. We showed that three non-unitary states have to be accompanied with a ferromagnetic order from the structure of the maximal little groups. We performed numerical studies for these SCs. For parameters and electron density favorable for the ferromagnetic order, a non-unitary SC coexistent with ferromagnetism is most stable.
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 study the phase diagram of the extended Hubbard model on a two-dimensional square lattice, including on-site (UUU) and nearest-neighbor (VVV) interactions, at weak couplings. We show that the charge-density wave phase that is known to occur at half filling when 4V>U4V>U4V>U gives way to a dxy{d}_{xy}dxy​-wave superconducting instability away from half filling, when the Fermi surface is not perfectly nested, and for sufficiently large repulsive VVV and a range of on-site repulsive interaction UUU. In addition, when nesting is further suppressed and in the presence of a nearest-neighbor attraction, a triplet time-reversal breaking (px+ipy)({p}_{x}+i{p}_{y})(px​+ipy​)-wave pairing instability emerges, competing with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing state that is known to dominate at fillings just slightly away from half. At even smaller fillings, where the Fermi surface no longer presents any nesting, the (px+ipy)({p}_{x}+i{p}_{y})(px​+ipy​)-wave superconducting phase dominates in the whole regime of on-site repulsions and nearest-neighbor attractions, while dxy{d}_{xy}dxy​ pairing occurs in the presence of on-site attraction. Our results suggest that zero-energy Majorana fermions can be realized on a square lattice in the presence of a magnetic field. For a system of cold fermionic atoms on a two-dimensional square optical lattice, both an on-site repulsion and a nearest-neighbor attraction would be required, in addition to rotation of the system to create vortices. We discuss possible ways of experimentally engineering the required interaction terms in a cold atom system.
We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH.
We revisit the ground-state phase diagram of the one-dimensional half-filled extended Hubbard model with on-site (U) and nearest-neighbor (V) repulsive interactions. In the first half of the paper, using the weak-coupling renormalization-group approach (g(g(g-ology) including second-order corrections to the coupling constants, we show that bond-charge-density-wave (BCDW) phase exists for $U\ensuremath{\approx}2V$ in between charge-density-wave (CDW) and spin-density-wave (SDW) phases. We find that the umklapp scattering of parallel-spin electrons disfavors the BCDW state and leads to a bicritical point where the CDW-BCDW and SDW-BCDW continuous-transition lines merge into the CDW-SDW first-order transition line. In the second half of the paper, we investigate the phase diagram of the extended Hubbard model with either additional staggered site potential $\ensuremath{\Delta}$ or bond alternation $\ensuremath{\delta}.$ Although the alternating site potential $\ensuremath{\Delta}$ strongly favors the CDW state (that is, a band insulator), the BCDW state is not destroyed completely and occupies a finite region in the phase diagram. Our result is a natural generalization of the work by Fabrizio, Gogolin, and Nersesyan [Phys. Rev. Lett. 83,83,83, 2014 (1999)], who predicted the existence of a spontaneously dimerized insulating state between a band insulator and a Mott insulator in the phase diagram of the ionic Hubbard model. The bond alternation $\ensuremath{\delta}$ destroys the SDW state and changes it into the BCDW state (or Peierls insulating state). As a result the phase diagram of the model with $\ensuremath{\delta}$ contains only a single critical line separating the Peierls insulator phase and the CDW phase. The addition of $\ensuremath{\Delta}$ or $\ensuremath{\delta}$ changes the universality class of the CDW-BCDW transition from the Gaussian transition into the Ising transition.