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arXiv:0801.0744 · Physical Review B

Resonating plaquette phases in SU(4) Heisenberg antiferromagnet

Cenke Xu, Congjun Wu

Published Apr 30, 2008• 78 citations
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Abstract

Large spin cold atom systems can exhibit magnetic properties that do not appear in usual spin-$1∕2$ systems. We investigate the SU(4) resonating plaquette state in the three-dimensional cubic optical lattice with spin-$3∕2$ cold fermions. A gauge field formalism is constructed to describe the Rokhsar--Kivelson type of Hamiltonian and a duality transformation is used to study the phase diagram. Due to the proliferation of topological defects, the system is generally gapped for the whole phase diagram of the quantum model, which agrees with the recent numerical studies. The classical plaquette model on the cubic lattice is also studied, and a critical phase is predicted by tuning one parameter in the low energy field theory.

DOI
10.1103/physrevb.77.134449

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Large spin cold atom systems can exhibit magnetic properties that do not appear in usual spin-1∕21∕21∕2 systems. We investigate the SU(4) resonating plaquette state in the three-dimensional cubic optical lattice with spin-3∕23∕23∕2 cold fermions. A gauge field formalism is constructed to describe the Rokhsar--Kivelson type of Hamiltonian and a duality transformation is used to study the phase diagram. Due to the proliferation of topological defects, the system is generally gapped for the whole phase diagram of the quantum model, which agrees with the recent numerical studies. The classical plaquette model on the cubic lattice is also studied, and a critical phase is predicted by tuning one parameter in the low energy field theory.
We consider the simplest generalizations of the valence bond physics of SU(2) singlets to SU(N)\mathrm{SU}(N)SU(N) singlets that comprise objects with NNN sites---these are SU(N)\mathrm{SU}(N)SU(N) singlet plaquettes with N=3N=3N=3 and N=4N=4N=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.
The SU(4) Heisenberg model can serve as a low-energy model of the Mott insulating state in materials where the spins and orbitals are highly symmetric or in systems of alkaline-earth atoms on an optical lattice. Recently, it has been argued that on the honeycomb lattice, the model exhibits a unique spin-orbital liquid phase with an algebraic decay of correlations [P. Corboz et al., Phys. Rev. X 2, 041013 (2012)]. Here we study the instability of the algebraic spin-orbital liquid toward the spontaneous formation of SU(4) singlet plaquettes (tetramerization). Using a variational Monte Carlo approach to evaluate the projected wave function of fermions with $\ensuremath{\pi}$-flux state, we find that the algebraic liquid is robust and that a small finite value of the next-nearest exchange is needed to induce tetramerization. We also studied the phase diagram of a model which interpolates between the nearest-neighbor Heisenberg model and a Hamiltonian for which the singlet-plaquette-product state is an exact ground state.
We examine the topological order in the resonating singlet valence plaquette (RSVP) phase of the hard-core quantum plaquette model (QPM) on the face centered cubic (FCC) lattice. To do this, we construct a Rohksar-Kivelson type Hamiltonian of local plaquette resonances. This model is shown to exhibit a Z3{\mathbb{Z}}_{3}Z3​ topological order, which we show by identifying a Z3{\mathbb{Z}}_{3}Z3​ topological constant (which leads to a 33{3}^{3}33-fold topological ground state degeneracy on the 3-torus) and topological pointlike charge and looplike magnetic excitations which obey Z3{\mathbb{Z}}_{3}Z3​ statistics. We also consider an exactly solvable generalization of this model, which makes the geometrical origin of the Z3{\mathbb{Z}}_{3}Z3​ order explicitly clear. For other models and lattices, such generalizations produce a wide variety of topological phases, some of which are novel fracton phases.
We revisit the phase diagram of Rokhsar-Kivelson models, which are used in fields such as superconductivity, frustrated magnetism, cold bosons, and the physics of Josephson junction arrays. From an extended height effective theory, we show that one of two simple generic phase diagrams contains a mixed phase that interpolates continuously between columnar and plaquette states. For the square lattice quantum dimer model we present evidence from exact diagonalization and Green's function Monte Carlo techniques that this scenario is realized, by combining an analysis of the excitation gaps of different symmetry sectors with information on plaquette structure factors. This presents a natural framework for resolving the disagreement between previous studies.
We extend the definition of the Jordan-Wigner transformation to three dimensions using the generalization of ideas that were used in the two-dimensional case by one of the present authors. Under this transformation, the 3D XY Hamiltonian is transformed into a system of spinless fermions coupled to a gauge field with only two components. We calculate the flux per plaquette for the 3 elementary perpendicular plaquettes of a cubic lattice, and find that it is nonzero for only two of the plaquettes. We provide a simple interpretation for the average phase-per-plaquette being πππ on the plaquettes where it is nonzero. Then we apply these findings to the investigation of the Heisenberg bilayer antiferromagnet.
antiferromagnetic Heisenberg honeycomb model when the ground state of the system lies in a spin-disordered plaquette-valence-bond-solid phase. A six-spin plaquette operator theory of this honeycomb model has been developed for this purpose by using the eigenstates of an isolated Heisenberg hexagonal plaquette. Spin-1/2 operators are thus expressed in the Fock space spanned by the plaquette operators those are obtained in terms of exact analytic form of eigenstates for a single frustrated Heisenberg hexagon. Ultimately, an effective interacting boson model of this system is obtained on the basis of low energy singlets and triplets plaquette operators by employing a mean-field approximation. The values of ground state energy and spin gap of this system have been estimated and the validity of this formalism has been tested upon comparison with the known results. Emergence of topological point and line nodes on the basis of spin-disordered ground state noted in this investigation is very rare on any frustrated system as well as the presence of triplet flat band. Evolution of those topological nodes is studied throughout the full frustrated regime. Finally, emergence of topological phases has been reported upon adding a time-reversal-symmetry breaking term to the Hamiltonian. Coexistence of spin gap with either topological nodes or phases turns this honeycomb model an interesting one.