This paper describes a group-theoretical classification of the mean-field solutions of the t-J model and the extended t-J model on a planar square lattice. It is shown that the mean-field solutions can be classified into four categories, spin density wave (SDW), charge density wave (CDW), bond order wave (BOW) and charve current wave (CCW) with distinct behavior under spatial inversion, spin rotation and time reversal. We derived systematically all types of broken-symmetry solutions, each of which is characterized by a single order parameter, for the ordering vector q= Q≡(π, π) and those for q= q1≡(π, 0). It is shown that for the t-J model there are four states (flux, Peierls, kite and SDW) with q= Q and six states (two CDWs, two BOWs and two CCWs) with q= Q and six states (two CDWs, two BOWs and two CCWs) with q= q1. For the t-J-J′ model including next-nearest-neighbour Heisenberg coupling, other four states (one CDW, one BOW and two CCWs) with q= Q and seven states (three SDWs, two BOWs and two CCWs) with q= q1 are possible. These phases are derived naturally from the irreducible spectral decomposition of the Σ JSiSj term and the self-consistent condition of the renormalized mean-field theory. The role of local SU(2) symmetry present at half filling is also discussed.This paper describes a group-theoretical classification of the mean-field solutions of the t-J model and the extended t-J model on a planar square lattice. It is shown that the mean-field solutions can be classified into four categories, spin density wave (SDW), charge density wave (CDW), bond order wave (BOW) and charve current wave (CCW) with distinct behavior under spatial inversion, spin rotation and time reversal. We derived systematically all types of broken-symmetry solutions, each of which is characterized by a single order parameter, for the ordering vector q= Q≡(π, π) and those for q= q1≡(π, 0). It is shown that for the t-J model there are four states (flux, Peierls, kite and SDW) with q= Q and six states (two CDWs, two BOWs and two CCWs) with q= Q and six states (two CDWs, two BOWs and two CCWs) with q= q1. For the t-J-J′ model including next-nearest-neighbour Heisenberg coupling, other four states (one CDW, one BOW and two CCWs) with q= Q and seven states (three SDWs, two BOWs and two CCWs) with q= q1 are possible. These phases are derived naturally from the irreducible spectral decomposition of the Σ JSiSj term and the self-consistent condition of the renormalized mean-field theory. The role of local SU(2) symmetry present at half filling is also discussed.

Progress of Theoretical PhysicsDec 1, 19904 citations

Broken symmetry solutions of the extended Hubbard model

Masa-aki Ozaki

Abstract This paper describes a group theoretical classification of the unrestricted Hartree–Fock (UHF) solution of the extended Hubbard model including nearest‐neighbor repulsion and exchange interactions. We derived systematically all types of the broken symmetry solution with a single‐order parameter for the ordering vector Q = (π,π). It is shown that there are four nonmagnetic states: a charge density wave (CDW), a charge current wave (CCW), and two bond‐order waves (BOW), and five magnetic states: an ordinary antiferromagnetic state (AF), a spin current wave (SCW), two axial spin bond‐order waves (ASBOW), and an helical spin bond‐order wave (HSBOW). We showed that these phases can be derived naturally from the irreducible decomposition of the Fourier transformed interaction and the self‐consistent field (SCF) condition. Some numerical results are given for the half‐filled case.Abstract This paper describes a group theoretical classification of the unrestricted Hartree–Fock (UHF) solution of the extended Hubbard model including nearest‐neighbor repulsion and exchange interactions. We derived systematically all types of the broken symmetry solution with a single‐order parameter for the ordering vector Q = (π,π). It is shown that there are four nonmagnetic states: a charge density wave (CDW), a charge current wave (CCW), and two bond‐order waves (BOW), and five magnetic states: an ordinary antiferromagnetic state (AF), a spin current wave (SCW), two axial spin bond‐order waves (ASBOW), and an helical spin bond‐order wave (HSBOW). We showed that these phases can be derived naturally from the irreducible decomposition of the Fourier transformed interaction and the self‐consistent field (SCF) condition. Some numerical results are given for the half‐filled case.

International Journal of Quantum ChemistryApr 5, 199237 citations

Group theoretical classification of the Hartree-Fock solutions of the Hubbard model on a triangular lattice

Atsushi Ikawa, Makiko Ozaki

Group theoretically possible spin density waves (SDWs) of the Hubbard model on a triangular lattice are found and classified for ordering vectors q of K and M in the Brillouin zone by finding the maximal subgroups of the symmetry group of the Hamiltonian. It is found that classifications of SDWs with these ordering vectors are reduced to classifications of SDWs on molecules with D 3 and T d symmetries, respectively. From this reduction the authors found two axial-type ASDWs (ASDW) and one plane-type SDW (PSDW) for point K and three ASDWs, two PSDWs and one cubic- (three-dimensional-) type SDW (CSDW) for point M. The standard mean-field Hamiltonians and their invariance groups are listed for each class.Group theoretically possible spin density waves (SDWs) of the Hubbard model on a triangular lattice are found and classified for ordering vectors q of K and M in the Brillouin zone by finding the maximal subgroups of the symmetry group of the Hamiltonian. It is found that classifications of SDWs with these ordering vectors are reduced to classifications of SDWs on molecules with D 3 and T d symmetries, respectively. From this reduction the authors found two axial-type ASDWs (ASDW) and one plane-type SDW (PSDW) for point K and three ASDWs, two PSDWs and one cubic- (three-dimensional-) type SDW (CSDW) for point M. The standard mean-field Hamiltonians and their invariance groups are listed for each class.

Journal of Physics Condensed MatterApr 13, 19925 citations

Pair density wave solution for a self-consistent model

Sergey I. Matveenko, Sergei I. Mukhin

In the self-consistent approximation for the two-dimensional mean-field model we found an analytic solution for the ground state with a coexisting $d$ -wave symmetric bond ordered pair density wave (PDW) and spin (SDW) or charge (CDW) density waves, as observed in some high-temperature superconductors. In particular, the solution gives the same periodicity for CDW and PDW, and a pseudogap in the Fermi-excitation spectrum.In the self-consistent approximation for the two-dimensional mean-field model we found an analytic solution for the ground state with a coexisting $d$-wave symmetric bond ordered pair density wave (PDW) and spin (SDW) or charge (CDW) density waves, as observed in some high-temperature superconductors. In particular, the solution gives the same periodicity for CDW and PDW, and a pseudogap in the Fermi-excitation spectrum.

arXiv:2102.00144Physical review. B./Physical review. BMar 25, 20250 citations

PHASE DIAGRAM OF THE ONE-DIMENSIONAL t–U–J MODEL WITH ON-BOND REPULSION AT HALF FILLING

Hanqin Ding, Yanshen Wang

By using the bosonization approach and the renormalization group (RG) technique, we study the half-filled band one-dimensional t–U–J model with additional on-bond repulsion (W>0) in the weak-coupling regime. The presence of on-bond repulsion is responsible for realization of a metallic phase in the system, and the phase diagram is strongly controlled by the symmetry of the model. By analyzing the RG flow diagram and comparing order parameters, the phase boundaries are determined and the structure of the phase diagram is clarified. In the case of SU (2) ⊗ SU (2) symmetry, the phase diagram consists of a metallic phase characterized by a Luttinger liquid (LL) and two insulting phases characterized by the degenerate spin-density-wave (SDW) and the bond-charge-density-wave (BCDW). In the SU (2) ⊗ U(1)-symmetric case, the phase diagram contains two metallic phases: a LL and a Luther–Emery phase, and three insulating phases: the transverse SDW ( SDW ±), the longitudinal SDW ( SDW z) and the dimerized BCDW. The insulating charge-density-wave and bond-spin-density-wave (BSDW) phases are always suppressed in the ground state. In addition, the system show a long-ranged order in the BCDW and SDW z phases.By using the bosonization approach and the renormalization group (RG) technique, we study the half-filled band one-dimensional t–U–J model with additional on-bond repulsion (W>0) in the weak-coupling regime. The presence of on-bond repulsion is responsible for realization of a metallic phase in the system, and the phase diagram is strongly controlled by the symmetry of the model. By analyzing the RG flow diagram and comparing order parameters, the phase boundaries are determined and the structure of the phase diagram is clarified. In the case of SU (2) ⊗ SU (2) symmetry, the phase diagram consists of a metallic phase characterized by a Luttinger liquid (LL) and two insulting phases characterized by the degenerate spin-density-wave (SDW) and the bond-charge-density-wave (BCDW). In the SU (2) ⊗ U(1)-symmetric case, the phase diagram contains two metallic phases: a LL and a Luther–Emery phase, and three insulating phases: the transverse SDW ( SDW ±), the longitudinal SDW ( SDW z) and the dimerized BCDW. The insulating charge-density-wave and bond-spin-density-wave (BSDW) phases are always suppressed in the ground state. In addition, the system show a long-ranged order in the BCDW and SDW z phases.

International Journal of Modern Physics BDec 30, 20102 citations

Itinerant fermions on a triangular lattice: Unconventional magnetism and other ordered states

Mengxing Ye, Andrey V. Chubukov

We consider a system of 2D fermions on a triangular lattice with well separated electron and hole pockets of similar sizes, centered at certain high-symmetry points in the Brillouin zone. We first analyze Stoner-type spin-density-wave (SDW) magnetism. We show that SDW order is degenerate at the mean-field level. Beyond mean-field, the degeneracy is lifted and is either ${120}^{\ensuremath{\circ}}$ ``triangular'' order (same as for localized spins), or a collinear order with antiferromagnetic spin arrangement on two-thirds of sites, and nonmagnetic on the rest of sites. We also study a time-reversal symmetric directional spin bond order, which emerges when some interactions are repulsive and some are attractive. We show that this order is also degenerate at a mean-field level, but beyond mean-field the degeneracy is again lifted. We next consider the evolution of a magnetic order in a magnetic field starting from an SDW state in zero field. We show that a field gives rise to a canting of an SDW spin configuration. In addition, it necessarily triggers the directional bond order, which, we argue, is linearly coupled to the SDW order in a finite field. We derive the corresponding term in the free energy. Finally, we consider the interplay between an SDW order and superconductivity and charge order. For this, we analyze the flow of the couplings within parquet renormalization group (pRG) scheme. We show that magnetism wins if all interactions are repulsive and there is little energy space for pRG to develop. However, if system parameters are such that pRG runs over a wide range of energies, the system may develop either superconductivity or an unconventional charge order, which breaks time-reversal symmetry.We consider a system of 2D fermions on a triangular lattice with well separated electron and hole pockets of similar sizes, centered at certain high-symmetry points in the Brillouin zone. We first analyze Stoner-type spin-density-wave (SDW) magnetism. We show that SDW order is degenerate at the mean-field level. Beyond mean-field, the degeneracy is lifted and is either ${120}^{\ensuremath{\circ}}$ ``triangular'' order (same as for localized spins), or a collinear order with antiferromagnetic spin arrangement on two-thirds of sites, and nonmagnetic on the rest of sites. We also study a time-reversal symmetric directional spin bond order, which emerges when some interactions are repulsive and some are attractive. We show that this order is also degenerate at a mean-field level, but beyond mean-field the degeneracy is again lifted. We next consider the evolution of a magnetic order in a magnetic field starting from an SDW state in zero field. We show that a field gives rise to a canting of an SDW spin configuration. In addition, it necessarily triggers the directional bond order, which, we argue, is linearly coupled to the SDW order in a finite field. We derive the corresponding term in the free energy. Finally, we consider the interplay between an SDW order and superconductivity and charge order. For this, we analyze the flow of the couplings within parquet renormalization group (pRG) scheme. We show that magnetism wins if all interactions are repulsive and there is little energy space for pRG to develop. However, if system parameters are such that pRG runs over a wide range of energies, the system may develop either superconductivity or an unconventional charge order, which breaks time-reversal symmetry.

arXiv:1803.04977Physical review. B./Physical review. BJun 7, 20183 citations

In the self-consistent approximation for the two-dimensional mean-field model we found an analytic solution for the ground state with a coexisting

d

-wave symmetric bond ordered pair density wave (PDW) and spin (SDW) or charge (CDW) density waves, as observed in some high-temperature superconductors. In particular, the solution gives the same periodicity for CDW and PDW, and a pseudogap in the Fermi-excitation spectrum.