The antiferromagnetic Ising model on a checkerboard lattice has an ice-like ground state manifold with extensive degeneracy. and, to leading order in ${J}_{xy}$, deconfined spinon excitations. We explore the role of cyclic exchange arising at order ${J}_{xy}^{2}∕{J}_{z}$ on the ice states and their associated spinon excitations. By mapping the original problem onto an equivalent quantum six-vertex model, we identify three different phases as a function of the chemical potential for flippable plaquettes---a phase with long range N\'eel order and confined spinon excitations, a nonmagnetic state of resonating square plaquettes, and a quasicollinear phase with gapped but deconfined spinon excitations. The relevance of the results to the square-lattice quantum dimer model is also discussed.
The antiferromagnetic Ising model on a checkerboard lattice has an ice-like ground state manifold with extensive degeneracy. and, to leading order in Jxy, deconfined spinon excitations. We explore the role of cyclic exchange arising at order Jxy2∕Jz on the ice states and their associated spinon excitations. By mapping the original problem onto an equivalent quantum six-vertex model, we identify three different phases as a function of the chemical potential for flippable plaquettes---a phase with long range N'eel order and confined spinon excitations, a nonmagnetic state of resonating square plaquettes, and a quasicollinear phase with gapped but deconfined spinon excitations. The relevance of the results to the square-lattice quantum dimer model is also discussed.
We study the dynamics of spinons in quantum square ice, as captured by the strongly anisotropic spin-1/2 XXZ model on the checkerboard lattice. We analyze in detail the kinetics of quantum-mechanical spin-flip excitations called spinons, which is governed by the transverse term. We formulate effective free theories of spinon pair propagation, either deconfined or artificially confined to nearest-neighbor distance. We identify the fingerprint of coherent spinon propagation in the dynamic structure factor of the full interacting model, evaluated numerically for finite clusters. We thus provide unbiased evidence for the formation of free coherent quasiparticles in the correlated vacuum of quantum square ice.
We study the effect of quantum fluctuations by means of a transverse magnetic field (Γ) on the antiferromagnetic J1−J2 Ising model on the checkerboard lattice, the two dimensional version of pyrochlore lattice. A plaquette operator formalism has been implemented to bosonise the model, in which a single boson is associated to each eigenstate of a plaquette and the inter-plaquette interactions define an effective Hamiltonian. The excitations of a plaquette would represent an-harmonic fluctuations of the model, which lead not only to lower the excitation energy compared with a single-spin flip but also to lift the extensive degeneracy in favor of a plaquette ordered state, which breaks lattice rotational symmetry for J2>J1 (the collinear phase). The bosonic excitation gap vanishes at the critical boundary to N\'{e}el (J2J1) phases, respectively. At the homogeneous coupling (J1=J2) a (canted) resonating plaquette state (RPS) is established from an-harmonic fluctuations, which lasts for low fields, Γ/J1≲0.4, followed by a crossover to the fully polarized phase. We will show that our results predict an order-by-disorder as a results of transverse field at the critical point (J2=0.5J1) of the antiferromagnetic J1−J2 Ising square lattice.
Spin excitation spectra of two dimensional S=1/2 Heisenberg model with checkerboard structures.
We study the excitation spectra of two-dimensional spin-1/2 Heisenberg model with checkerboard structures using stochastic analytic continuation of imaginary-time correlation function obtained from quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions J1 and J2, and the tuning parameter g is defined as J2/J1. The dynamic structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, wave theory is employed to explain some features of numerical results, especially for the low-energy part. When g is close to 1, the features of the excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them are discussed. In contrast to the other checkerboard structures investigated in this paper, the 3×3 checkerboard model has distinctive excitation features, like a gap between a low-energy gapless branch and a gapped high-energy part exists when g is small. The gapless branch in this case can be regarded as wave in Neˊel order formed by block spin in each 3×3 plaquette with effective exchange interaction originated from renormalization. One finding is that the continuum also appears in this low-energy branch, which may originated from nearly deconfined process in block spin.
Quantum phase transition from the N\'eel to the dimer states in an antiferromagnetic (AF) Heisenberg model on square lattice is studied. We introduce a control parameter $\ensuremath{\alpha}$ for the exchange coupling that connects the N\'eel $(\ensuremath{\alpha}=0)$ and the dimer $(\ensuremath{\alpha}=1)$ states. We employ the CP1 representation of the s=21 spin operator and integrate out the half of the CP1 variables at odd sites to obtain a CP1 nonlinear $\ensuremath{\sigma}$ model. The effective coupling constant is a function of $\ensuremath{\alpha}$, and at $\ensuremath{\alpha}=0$, the CP1 model is in the ordered phase that corresponds to the N\'eel state of the AF Heisenberg model. A phase transition to the dimer state occurs at a certain critical value of ${\ensuremath{\alpha}}_{C}$ as $\ensuremath{\alpha}$ increases. In the N\'eel state, the dynamical composite U(1) gauge field in the CP1 model is in a Higgs phase, and low-energy excitations are gapless spin waves. In the dimer phase, a confinement phase of the gauge theory with s=1 excitations is realized. For the critical point, we argue that a deconfinement phase, which is similar to the Coulomb phase in three spatial dimensions, is realized and s=21 spinons appear as low-energy excitations.
We study the spin-1/2 Heisenberg model on the square lattice with first- and second-neighbor antiferromagnetic interactions J(1) and J(2), which possesses a nonmagnetic region that has been debated for many years and might realize the interesting Z(2) spin liquid. We use the density matrix renormalization group approach with explicit implementation of SU(2) spin rotation symmetry and study the model accurately on open cylinders with different boundary conditions. With increasing J(2), we find a Néel phase and a plaquette valence-bond (PVB) phase with a finite spin gap. From the finite-size scaling of the magnetic order parameter, we estimate that the Néel order vanishes at J(2)/J(1)≃0.44. For 0.5<J(2)/J(1)<0.61, we find dimer correlations and PVB textures whose decay lengths grow strongly with increasing system width, consistent with a long-range PVB order in the two-dimensional limit. The dimer-dimer correlations reveal the s-wave character of the PVB order. For 0.44<J(2)/J(1)<0.5, spin order, dimer order, and spin gap are small on finite-size systems, which is consistent with a near-critical behavior. The critical exponents obtained from the finite-size spin and dimer correlations could be compatible with the deconfined criticality in this small region. We compare and contrast our results with earlier numerical studies.
We study the spin excitation spectra of the two-dimensional spin-21 Heisenberg model with a checkerboard structure using stochastic analytic continuation of the imaginary-time correlation function obtained from a quantum Monte Carlo simulation. The checkerboard models have two different antiferromagnetic nearest-neighbor interactions J1 and J2, and the tuning parameter g is defined as J2/J1. The dynamic spin structure factors are systematically calculated in all phases of the models as well as at the critical points. To give a full understanding of the dynamic spectra, spin wave theory is employed to explain some features of numerical results, especially for the low-energy part. When g is close to 1, the features of the spin excitation spectra of each checkerboard model are roughly the same as those of the original square lattice antiferromagnetic Heisenberg model, and the high-energy continuum among them is discussed. In contrast to the other checkerboard structures investigated in this paper, the $3\ifmmode\times\else\texttimes\fi{}3$ checkerboard model has distinctive excitation features, such as a gap between a low-energy gapless branch and a gapped high-energy part that exists when g is small. The gapless branch in this case can be regarded as a spin wave in N\'eel order formed by a ``block spin'' in each $3\ifmmode\times\else\texttimes\fi{}3$ plaquette with an effective exchange interaction originating from renormalization. One unexpected finding is that the continuum also appears in this low-energy branch.