We propose an exactly soluble Ising model with two kinds of exchange integrals on a square lattice. We assume that the number of bonds on which the exchange integrals take a value is even around each elementary plaquette; we name such a plaquette an even-bonds plaquette. We find two phase transition temperatures in the Ising model. We have a magnetic phase transition at one of them and a phase transition relating to the ordering of a dual spins introduced by a dual transformation of bond variables at the other. The latter phase transition is investigated in detail and turned out to have a relation with the confinement of odd-bonds plaquettes in a generalized system. Both phase transitions are of Onsager type.We propose an exactly soluble Ising model with two kinds of exchange integrals on a square lattice. We assume that the number of bonds on which the exchange integrals take a value is even around each elementary plaquette; we name such a plaquette an even-bonds plaquette. We find two phase transition temperatures in the Ising model. We have a magnetic phase transition at one of them and a phase transition relating to the ordering of a dual spins introduced by a dual transformation of bond variables at the other. The latter phase transition is investigated in detail and turned out to have a relation with the confinement of odd-bonds plaquettes in a generalized system. Both phase transitions are of Onsager type.

Journal of the Physical Society of JapanSep 15, 19891 citation

Phase transitions in anisotropic lattice spin systems

Jürg Martin Fröhlich, Élliott H. Lieb

Communications in Mathematical PhysicsOct 1, 1978128 citations

Tripartite entangled plaquette state in a cluster magnet

Juan Carrasquilla, Gang Chen, Roger G. Melko

Using large-scale quantum Monte Carlo simulations we show that a spin- $\frac{1}{2}\phantom{\rule{1em}{0ex}}XXZ$ model on a two-dimensional anisotropic kagome lattice exhibits a tripartite entangled plaquette state that preserves all of the Hamiltonian symmetries. It is connected via phase boundaries to a ferromagnet and a valence-bond solid that break $U(1)$ and lattice translation symmetries, respectively. We study the phase diagram of the model in detail, in particular the transitions to the tripartite entangled plaquette state, which are consistent with conventional order-disorder transitions. Our results can be interpreted as a description of the charge sector dynamics of a Hubbard model applied to the spin liquid candidate ${\mathrm{LiZn}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{8}$ , as well as a model of strongly correlated bosonic atoms loaded onto highly tunable trimerized optical kagome lattices.Using large-scale quantum Monte Carlo simulations we show that a spin-$\frac{1}{2}\phantom{\rule{1em}{0ex}}XXZ$ model on a two-dimensional anisotropic kagome lattice exhibits a tripartite entangled plaquette state that preserves all of the Hamiltonian symmetries. It is connected via phase boundaries to a ferromagnet and a valence-bond solid that break $U(1)$ and lattice translation symmetries, respectively. We study the phase diagram of the model in detail, in particular the transitions to the tripartite entangled plaquette state, which are consistent with conventional order-disorder transitions. Our results can be interpreted as a description of the charge sector dynamics of a Hubbard model applied to the spin liquid candidate ${\mathrm{LiZn}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{8}$, as well as a model of strongly correlated bosonic atoms loaded onto highly tunable trimerized optical kagome lattices.

arXiv:1704.03478Physical review. B./Physical review. BAug 4, 201717 citations

Global phase diagram for the spin-1 antiferromagnet with uniaxial anisotropy on the kagome lattice

Cenke Xu, Joel E. Moore

The phase diagram of the $XXZ$ spin-1 quantum magnet on the kagome lattice is studied for all cases where the $z$ -component nearest-neighbor spin interaction ${J}_{z}$ is antiferromagnetic. Besides ${J}_{z}$ and the nearest-neighbor in-plane spin interaction ${J}_{xy}$ , the system is also parametrized by an on-site anisotropic term $D{({S}^{z})}^{2}$ . In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the ``superfluid'' (ferromagnetic) phase with an in-plane global U(1) symmetry broken when ${J}_{xy}<0$ ; the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order when ${J}_{xy}>0$ ; the nematic phase when $D<0$ and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound $m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}$, or optical lattices of atoms with three degenerate states on every site.The phase diagram of the $XXZ$ spin-1 quantum magnet on the kagome lattice is studied for all cases where the $z$-component nearest-neighbor spin interaction ${J}_{z}$ is antiferromagnetic. Besides ${J}_{z}$ and the nearest-neighbor in-plane spin interaction ${J}_{xy}$, the system is also parametrized by an on-site anisotropic term $D{({S}^{z})}^{2}$. In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the ``superfluid'' (ferromagnetic) phase with an in-plane global U(1) symmetry broken when ${J}_{xy}<0$; the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order when ${J}_{xy}>0$; the nematic phase when $D<0$ and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound $m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}$, or optical lattices of atoms with three degenerate states on every site.

arXiv:cond-mat/0701635Physical Review BSep 24, 200715 citations

Using large-scale quantum Monte Carlo simulations we show that a spin-

\frac{1}{2}\phantom{\rule{1em}{0ex}}XXZ

model on a two-dimensional anisotropic kagome lattice exhibits a tripartite entangled plaquette state that preserves all of the Hamiltonian symmetries. It is connected via phase boundaries to a ferromagnet and a valence-bond solid that break

U(1)

and lattice translation symmetries, respectively. We study the phase diagram of the model in detail, in particular the transitions to the tripartite entangled plaquette state, which are consistent with conventional order-disorder transitions. Our results can be interpreted as a description of the charge sector dynamics of a Hubbard model applied to the spin liquid candidate

{\mathrm{LiZn}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{8}

, as well as a model of strongly correlated bosonic atoms loaded onto highly tunable trimerized optical kagome lattices.

The phase diagram of the

XXZ

spin-1 quantum magnet on the kagome lattice is studied for all cases where the

z

-component nearest-neighbor spin interaction

{J}_{z}

is antiferromagnetic. Besides

{J}_{z}

and the nearest-neighbor in-plane spin interaction

{J}_{xy}

, the system is also parametrized by an on-site anisotropic term

D{({S}^{z})}^{2}

. In the zero magnetic field case, the six previously introduced phases, found using various methods, are the nondegenerate gapped photon phase, which breaks no space symmetry or spin symmetry; the sixfold degenerate phase with plaquette order, which breaks both time-reversal symmetry and translational symmetry; the ``superfluid'' (ferromagnetic) phase with an in-plane global U(1) symmetry broken when

{J}_{xy}<0

; the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order when

{J}_{xy}>0

; the nematic phase when

D<0

and large; and a phase with resonating dimers on each hexagon. We obtain all of these phases and partial information about their quantum phase transitions in a single framework by studying condensation of defects in the sixfold plaquette phases. The transition between nematic phase and the sixfold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, another ordered phase with translation symmetry broken is opened up in the nematic phase. Due to the breaking of time-reversal symmetry by the field, a supersolid phase emerges between the sixfold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, organic compound $m\text{\ensuremath{-}}\mathrm{MPYNN}∙\mathrm{B}{\mathrm{F}}_{4}$, or optical lattices of atoms with three degenerate states on every site.