Plaquette Phase Research Articles

Plaquette order and deconfined quantum critical point in the spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice

We have precisely determined the ground state phase diagram of the quantum spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice using the tensor renormalization group method. We find that the ferromagnetic, ferroquadrupolar, and a large part of the antiferromagnetic phases are stable against quantum fluctuations. However, around the phase where the ground state is antiferroquadrupolar ordered in the classical limit, quantum fluctuations suppress completely all magnetic orders, leading to a plaquette order phase which breaks the lattice symmetry but preserves the spin SU(2) symmetry. On the evidence of our numerical results, the quantum phase transition between the antiferromagnetic phase and the plaquette phase is found to be either a direct second order or a very weak first order transition.

Phase diagram of the fully frustrated transverse-field Ising model on the honeycomb lattice

Motivated by the current interest in the quantum dimer model on the triangular lattice, we investigate the phase diagram of the closely related fully frustrated transverse-field Ising model on the honeycomb lattice using classical and semiclassical approximations. We show that, in addition to the fully polarized phase at a large field, the classical model possesses a multitude of phases that break the translational symmetry which, in the dimer language, correspond to a plaquette phase and a columnar phase separated by an infinite cascade of mixed phases. The modification of the phase diagram by quantum fluctuations has been investigated in the context of linear spin-wave theory. The extrapolation of the semiclassical energies suggests that the plaquette phase extends down to zero field for spin 1/2, in agreement with the $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ phase of the quantum dimer model on the triangular lattice with only kinetic energy.

Generalized quantum dimer model applied to the frustrated Heisenberg model on the square lattice: Emergence of a mixed columnar-plaquette phase

Aiming to describe frustrated quantum magnets with nonmagnetic singlet ground states, we have extended the Rokhsar-Kivelson (RK) loop expansion to derive a generalized quantum dimer model containing only connected terms up to arbitrary order. For the square-lattice frustrated Heisenberg antiferromagnet (${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ model), an expansion up to eighth order shows that the leading correction to the original RK model comes from dimer moves on length-6 loops. This model free of the original sign problem is treated by advanced numerical techniques. The results suggest that a rotationally anisotropic plaquette phase [A. Ralko, D. Poilblanc, and R. Moessner, Phys. Rev. Lett. 100, 037201 (2008)] is the ground state of the Heisenberg model in the parameter region of maximum frustration.

Plaquette order in theJ1−J2−J3model: Series expansion analysis

Series expansion based on the flow equation method is employed to study the zero temperature properties of the spin-1/2 J1-J2-J3 antiferromagnet in two dimensions. Starting from the exact limit of decoupled plaquettes in a particular generalized J1-J2-J3 model we analyze the evolution of the ground state energy and the elementary triplet excitations in powers of all three inter-plaquette couplings up to fifth order. We find the plaquette phase to remain stable over a wide range of exchange couplings and to connect adiabatically up to the case of the plain J1-J2-J3 model, however not to the J1-J2 model at J3=0. Besides confirming the existence of such a phase, recently predicted by Mambrini, et al. [Phys. Rev. B 74, 144422 [2006)], we estimate its extent by Dlog-Pade analysis of the critical lines that result from closure of the triplet gap.

Resonating singlet valence plaquettes

We consider the simplest generalizations of the valence bond physics of SU(2) singlets to $\mathrm{SU}(N)$ singlets that comprise objects with $N$ sites---these are $\mathrm{SU}(N)$ singlet plaquettes with $N=3$ and $N=4$ in three spatial dimensions. Specifically, we search for a quantum mechanical liquid of such objects---a resonating singlet valence plaquette phase that generalizes the celebrated resonating valence bond phase for SU(2) spins. We extend the Rokhsar-Kivelson construction [Phys. Rev. Lett. 61, 2376 (1988)] of the quantum dimer model to the simplest SU(4) model for valence plaquette dynamics on a cubic lattice. The phase diagram of the resulting quantum plaquette model is analyzed both analytically and numerically. We find that the ground state is solid everywhere, including at the Rokhsar-Kivelson point where the ground state is an equal amplitude sum. By contrast, the equal amplitude sum of SU(3) singlet triangular plaquettes on the face-centered cubic lattice is liquid and is thus a candidate for describing a resonating single valence plaquette phase, given a suitably defined local Hamiltonian.

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

The phase diagram of the XXZ spin-1 quantum magnet on the kagome lattice is studied for all cases where the $J_z$ coupling is antiferromagnetic. 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 six-fold 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 $\sqrt{3}\times\sqrt{3}$ 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 six-fold plaquette phases. The transition between nematic phase and the six-fold degenerate plaquette phase is potentially an unconventional second-order critical point. In the case of a nonzero magnetic field along $\hat{z}$, 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 six-fold plaquette order and the superfluid phase. This phase diagram might be accessible in nickel compounds, BF$_4$ salts, or optical lattices of atoms with three degenerate states on every site.

Exotic Mott phases of the extendedt−Jmodel on the checkerboard lattice at commensurate densities

Coulomb repulsion between electrons moving on a frustrated lattice can give rise, at simple commensurate electronic densities, to exotic insulating phases of matter. Such a phenomenon is illustrated using an extended $t\text{\ensuremath{-}}J$ model on a planar pyrochlore lattice for which the work on the quarter-filled case [D. Poilblanc et al., Phys. Rev. B 75, 220503 (2007)] is complemented and extended to $1∕8$ and $3∕8$ fillings. The location of the metal-insulator transition as a function of the Coulomb repulsion is shown to depend strongly on the sign of the hopping. Quite generally, the metal-insulator transition is characterized by lattice symmetry breaking but the nature of the insulating Mott state is more complex than a simple charge density wave. Indeed, in the limit of large Coulomb repulsion, the physics can be described in the framework of (extended) quantum fully packed loop or dimer models carrying extra spin degrees of freedom. Various diagonal and off-diagonal plaquette correlation functions are computed and the low-energy spectra are analyzed in detail in order to characterize the nature of the insulating phases. We provide evidence that, as for an electronic density of $n=1∕2$ (quarter filling), the system at $n=1∕4$ or $n=3∕4$ exhibits also plaquette order by forming a (lattice rotationally invariant) resonant singlet pair crystal, although with a quadrupling of the lattice unit cell (instead of a doubling for $n=1∕2$) and a fourfold degenerate ground state. Interestingly, qualitative differences with the bosonic analog (e.g., known to exhibit columnar order at $n=1∕4$) emphasize the important role of the spin degrees of freedom in, e.g., stabilizing plaquette phases with respect to rotational symmetry-breaking phases.

Exotic phenomena in doped quantum magnets

We investigate the properties of a two-dimensional frustrated quantum antiferromagnet on asquare lattice, especially at infinitesimal doping. We find that next nearest neighbour (NN)J2 andnext–next NN J3 interactions together destroy the antiferromagnetic long-range order and stabilize aquantum disordered valence bond crystalline plaquette phase. A static vacancy or adynamic hole doped into this phase liberates a spinon. From the profile of the spinonwavefunction around the (static) vacancy we identify an intermediate behaviour betweencomplete deconfinement (behaviour seen in the kagome lattice) and strong confinement(behaviour seen in the checkerboard lattice) with the emergence of two length scales, aspinon confinement length larger than the magnetic correlation length. When finite holehopping is introduced, this behaviour translates into an extended (mobile) spinon–holonbound state with a very small quasiparticle weight. These features provide clear evidencefor a nearby ‘deconfined critical point’ in a doped microscopic model. Finally, we givearguments to support the idea that the doped plaquette phase has superconductingproperties.

Plaquette bond order wave in the quarter-filled extended Hubbard model on the checkerboard lattice

An extended Hubbard model (including nearest-neighbor repulsion and antiferromagnetic spin exchange) is investigated on the frustrated checkerboard lattice, a two-dimensional analog of the pyrochlore lattice. Combining Gutzwiller renormalized mean-field (MF) calculations, exact diagonalization (ED) techniques, and a weak-coupling renormalization group (RG) analysis we provide strong evidence for a crystalline valence bond plaquette phase at quarter-filling. The ground state is twofold degenerate and breaks translation symmetry. The bond energies show a staggering while the charge distribution remains uniform.

Plaquette phase of the square-lattice quantum dimer model: Quantum Monte Carlo calculations

The plaquette phase of the square lattice quantum dimer model is studied using a continuous-time reptation quantum Monte Carlo method for lattices of sizes up to 48x48 sites. We determine the location of the phase transition between the columnar and plaquette phases to occur at V_c/J=0.60 +- 0.05 which is significantly larger than inferred from previous exact diagonalization studies on smaller lattices. Offdiagonal correlation functions are obtained. They exhibit long-range order in the plaquette phase but not at the Rokhsar-Kivelson point. We also observe significant finite-size corrections to scaling for the transition between the plaquette phase and the critical resonating valence bond liquid. This study demonstrates the importance of understanding finite-size effects when considering critical properties of the square lattice quantum dimer model.

Resonating Plaquette Phase of a Quantum Six-Vertex Model

The simplest quantum generalization of the six-vertex model describes fluctuations of the order parameter of the d-density wave (DDW), believed to compete with superconductivity in the high-T(c) superconductors. The ground state of this model undergoes a first order transition from the DDW phase to a resonating plaquette phase as the quantum fluctuations are increased, which is explored with the help of quantum Monte Carlo simulations and analytic considerations involving the n-vector (n = 2) model with cubic anisotropy. In addition to finding a new quantum state, we show that the DDW is robust against a class of quantum fluctuations of its order parameter. The inferred finite temperature phase diagram contains unsuspected multicritical points.

Geometric criticality for transitions between plaquette phases in integer-spin kagome XXZ antiferromagnets

The phase diagram of the uniaxially anisotropic $s=1$ antiferromagnet on the kagom\'e lattice includes a critical line exactly described by the classical three-color model. This line is distinct from the standard geometric classical criticality that appears in the classical limit ($s \to \infty$) of the 2D XY model; the $s=1$ geometric T=0 critical line separates two unconventional plaquette-ordered phases that survive to nonzero temperature. The experimentally important correlations at finite temperature and the nature of the transitions into these ordered phases are obtained using the mapping to the three-color model and a combination of perturbation theory and a variational ansatz for the ordered phases. The ordered phases show sixfold symmetry breaking and are similar to phases proposed for the honeycomb lattice dimer model and $s=1/2$ $XXZ$ model. The same mapping and phase transition can be realized also for integer spins $s \geq 2$ but then require strong on-site anisotropy in the Hamiltonian.

First-Order Phase Transitions in Frustrated Spin Systems

We give a short review of our recent works on the first-order quantum phase transitions in frustrated spin chains with orthogonal-dimer structure. When the ratio of the competing antiferromagnetic exchange couplings is varied, a first-order transition occurs between the dimer phase and the plaquette phase, which is accompanied by the discontinuity in the spin excitation gap. We further show that strong frustration triggers the phase transitions in a magnetic field, which exhibit plateaus and jumps in the magnetization curve. The hole-doping effect is also addressed for the orthogonal-dimer chain with linked-tetrahedra structure. It is found that the competing antiferromagnetic interactions result in a first-order metal-insulator transition upon hole doping.

Phase diagram of the hexagonal lattice quantum dimer model

We discuss the phase diagram of the quantum dimer model on the hexagonal (honeycomb) lattice. In addition to the columnar and staggered valence-bond solids which have been discussed in previous work, we establish the existence of a plaquette valence-bond solid. The transition between the plaquette and columnar phases at $v/t=\ensuremath{-}0.2\ifmmode\pm\else\textpm\fi{}0.05$ is argued to be first order. We note that this model should describe valence-bond-dominated phases of frustrated Heisenberg models on the hexagonal lattice and discuss its relation to recent exact diagonalization work by Fouet et al. on the ${J}_{1}\ensuremath{-}{J}_{2}$ model on the same lattice. Our results also shed light on the properties of the transverse field Ising antiferromagnet on the triangular lattice and the classical Ising antiferromagnet on the stacked triangular lattice, which are related to dimer models by duality.

Competing Spin-Gap Phases in a Frustrated Quantum Spin System in Two Dimensions

We investigate quantum phase transitions among the spin-gap phases and the magnetically ordered phases in a two-dimensional frustrated antiferromagnetic spin system, which interpolates several important models such as the orthogonal-dimer model as well as the model on the 1/5-depleted square lattice. By computing the ground state energy, the staggered susceptibility and the spin gap by means of the series expansion method, we determine the ground-state phase diagram and discuss the role of geometrical frustration. In particular, it is found that a RVB-type spin-gap phase proposed recently for the orthogonal-dimer system is adiabatically connected to the plaquette phase known for the 1/5-depleted square-lattice model.

First-order quantum phase transition in the orthogonal-dimer spin chain

We investigate the low-energy properties of the orthogonal-dimer spin chain characterized by a frustrated dimer-plaquette structure. When the competing antiferromagnetic couplings are varied, the first-order quantum phase transition occurs between the dimer and the plaquette phases, which is accompanied by nontrivial features due to frustration: besides the discontinuity in the lowest excitation gap at the transition point, a sharp level crossing occurs for the spectrum in the plaquette phase. We further reveal that the plateau in the magnetization curve at 1/4 of the full moment dramatically changes its character in the vicinity of the critical point. It is argued that the first-order phase transition in this system captures some essential properties found in the two-dimensional orthogonal-dimer model proposed for ${\mathrm{SrCu}}_{2}({\mathrm{BO}}_{3}{)}_{2}.$

Dimer order with striped correlations in theJ1−J2Heisenberg model

Ground-state energies for plaquette and dimer order in the ${J}_{1}{\ensuremath{-}J}_{2}$ square-lattice spin-half Heisenberg model are compared using series expansion methods. We find that these energies are remarkably close to each other at intermediate values of ${J}_{2}{/J}_{1},$ where the model is believed to have a quantum disordered ground state. They join smoothly with those obtained from the Ising expansions for the two-sublattice N\'eel state at ${J}_{2}{/J}_{1}\ensuremath{\approx}0.4,$ suggesting a second-order transition from a N\'eel state to a quantum disordered state, whereas they cross the energy for the four-sublattice ordered state at ${J}_{2}{/J}_{1}\ensuremath{\approx}0.6$ at a large angle, implying a first-order transition to the four-sublattice magnetic state. The strongest evidence that the plaquette phase is not realized in this model comes from the analysis of the series for the singlet and triplet excitation spectra, which suggest an instability in the plaquette phase. Thus our study supports the recent work of Kotov et al., which presents a strong picture for columnar dimer order in this model. We also discuss the striped nature of spin correlations in this phase, with substantial resonance all along columns of dimers.

Competition between plaquette and dimer phases in Heisenberg chains

We consider a class of one-dimensional Heisenberg spin models (plaquette chains) related to the recently found 1 5 - depleted square-lattice Heisenberg system CaV 4O 9. A number of exact and exact-numerical results concerning the properties of the competing dimer and resonating plaquette phases are presented.

Application of guided random walks to U(1)2 Hamiltonian lattice gauge theory.

In this paper we study U(1${)}_{2}$ Hamiltonian lattice gauge theory as a model application of a guided-random-walk algorithm. We numerically evaluate the mean plaquette, vacuum energy per plaquette, and plaquette-plaquette correlation functions using the algorithm. The correlation-function measurements indicate that adjacent plaquette phases are weakly anticorrelated in the vacuum state. To our knowledge this is the first measurement of interplaquette correlations using a stochastic Hamiltonian method. We find that our numerical measurements are generally in good agreement with the results of U(1${)}_{2}$ weak- and strong-coupling expansions in the appropriate limits.