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arXiv:1207.1618 · Physical Review Letters

Evidence for Columnar Order in the Fully Frustrated Transverse Field Ising Model on the Square Lattice

S. C. Wenzel, Tommaso Coletta, S. E. Korshunov, Frédéric Mila

Published Nov 2, 2012• 23 citations
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Abstract

Using extensive classical and quantum Monte Carlo simulations, we investigate the ground-state phase diagram of the fully frustrated transverse field Ising model on the square lattice. We show that pure columnar order develops in the low-field phase above a surprisingly large length scale, below which an effective U(1) symmetry is present. The same conclusion applies to the quantum dimer model with purely kinetic energy, to which the model reduces in the zero-field limit, as well as to the stacked classical version of the model. By contrast, the 2D classical version of the model is shown to develop plaquette order. Semiclassical arguments show that the transition from plaquette to columnar order is a consequence of quantum fluctuations.

DOI
10.1103/physrevlett.109.187202

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We study the effect of quantum fluctuations by means of a transverse magnetic field ($\Gamma$) on the antiferromagnetic $J_1-J_2$ 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 $J_2>J_1$ (the collinear phase). The bosonic excitation gap vanishes at the critical boundary to N\'{e}el ($J_2 J_1$) phases, respectively. At the homogeneous coupling ($J_1=J_2$) a (canted) resonating plaquette state (RPS) is established from an-harmonic fluctuations, which lasts for low fields, $\Gamma/J_1\lesssim 0.4$, 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 ($J_2=0.5 J_1$) of the antiferromagnetic $J_1-J_2$ Ising square lattice.

•May 6, 2014•0 citations
Using extensive classical and quantum Monte Carlo simulations, we investigate the ground-state phase diagram of the fully frustrated transverse field Ising model on the square lattice. We show that pure columnar order develops in the low-field phase above a surprisingly large length scale, below which an effective U(1) symmetry is present. The same conclusion applies to the quantum dimer model with purely kinetic energy, to which the model reduces in the zero-field limit, as well as to the stacked classical version of the model. By contrast, the 2D classical version of the model is shown to develop plaquette order. Semiclassical arguments show that the transition from plaquette to columnar order is a consequence of quantum fluctuations.
We investigate the zero-temperature phase diagram of the fully frustrated transverse-field Ising model on the square lattice both in the classical limit and in the presence of quantum fluctuations. At the classical level (the limit of infinite spin SSS), we find that upon decreasing the transverse field $\ensuremath{\Gamma}$ this model exhibits a phase transition from the fully polarized state into an eightfold degenerate translational symmetry breaking state. This phase can be identified to correspond to plaquette order in the dimer language and remains the lowest-energy state in the entire range of fields below the critical one, ${\ensuremath{\Gamma}}_{c}$. The eightfold degenerate solution that corresponds to columnar order in the dimer language is a saddle point of the classical energy. It is degenerate with the plaquette solution at $\ensuremath{\Gamma}=0$ and is only slightly higher in energy in the whole interval $0<\ensuremath{\Gamma}<{\ensuremath{\Gamma}}_{c}$. The effect of quantum fluctuations is investigated in the context of a large S expansion both for the plaquette and columnar structures. For this purpose we employ an approximate method allowing to estimate from above the fluctuation-induced correction to the energy of a configuration which at the classical level is a saddle point of the energy, not a local minimum, and find that the harmonic quantum fluctuations show a clear tendency to overcome the energy difference between the two states. For relatively high fields, the transition from the plaquette to the columnar state takes place at values of SSS large enough to be in the domain of validity of the harmonic approximation.
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.
Using quantum Monte Carlo simulations and field-theory arguments, we study the fully frustrated transverse-field Ising model on the square lattice for the purpose of quantitatively relating two different order parameters to each other. We consider a ``primary'' spin order parameter and a ``secondary'' dimer order parameter, which both lead to the same phase diagram but detect Z8{\mathbb{Z}}_{8}Z8​ and Z4{\mathbb{Z}}_{4}Z4​ symmetry breaking, respectively. While at T>0T>0T>0 their scaling exponents are simply related to each other, as explained by a mapping to a height model, we show that at T=0T=0T=0 they correspond to different charge sectors of the O(2) model in 2+12+12+1 dimensions with nontrivial exponents that are not simply related to each other. Our insights are transferrable to a broad class of Ising models whose low-energy physics involves dimer degrees of freedom, and also serve as a guide to treating primary and secondary order parameters more generally.
We investigate the quantum phase diagram of the J 1 − J 2 − J 3 antiferromagnetic transverse-field Ising model on the ruby lattice. In the low-field limit we derive an effective quantum dimer model, analyzing how the extensive ground-state degeneracy at zero field is lifted by an order-by-disorder scenario. We support our analysis by studying the gap closing of the high-field phase using series expansions. For J 2 > J 3 , we find a columnar phase at low fields, followed by a clock-ordered phase stabilized by resonating plaquettes at intermediate field values, and an emergent three-dimensional (3D)-XY quantum phase transition to the polarized high-field phase. For J 3 > J 2 , an order-by-disorder mechanism stabilizes a distinct k = ( 0 , 0 ) order and a quantum phase transition in the 3D-Ising universality class is observed. Further, we discuss the possible implementation of the columnar- and clock-ordered phase in existing Rydberg atom quantum simulators. When taking into account the full algebraically decaying long-range interactions on the ruby lattice, we find that long-range interactions favor the same ground state as the quantum fluctuations induced by a transverse field, which could make the ruby lattice a promising candidate for the realization of a clock-ordered phase. Published by the American Physical Society 2024
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.
We study the effect of quantum fluctuations by means of a transverse magnetic field (Γ\GammaΓ) on the antiferromagnetic J1−J2J_1-J_2J1​−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>J1J_2>J_1J2​>J1​ (the collinear phase). The bosonic excitation gap vanishes at the critical boundary to N\'{e}el (J2J1J_2 J_1J2​J1​) phases, respectively. At the homogeneous coupling (J1=J2J_1=J_2J1​=J2​) a (canted) resonating plaquette state (RPS) is established from an-harmonic fluctuations, which lasts for low fields, Γ/J1≲0.4\Gamma/J_1\lesssim 0.4Γ/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=0.5J1J_2=0.5 J_1J2​=0.5J1​) of the antiferromagnetic J1−J2J_1-J_2J1​−J2​ Ising square lattice.