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arXiv:1309.6490 · Physical Review B

Plaquette valence-bond solid in the square-lattice J1 - J2 antiferromagnet Heisenberg model: A bond operator approach

R. L. Doretto

Published Mar 19, 2014• 55 citations
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Abstract

We study the plaquette valence-bond solid phase of the spin-$1/2$${J}_{1}$-${J}_{2}$ antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four $S=1/2$ spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the ${J}_{1}$-${J}_{2}$ model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region $0.34<{J}_{2}/{J}_{1}<0.59$, which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings ${J}_{2}=0.34{J}_{1}$ and ${J}_{2}=0.59{J}_{1}$. Interestingly, for ${J}_{2}<0.48{J}_{1}$, the excitation gap corresponds to a singlet-triplet excitation at the $\ensuremath{\Gamma}$ point while, for ${J}_{2}>0.48{J}_{1}$, it is related to a singlet-singlet excitation at the $\mathbf{X}=(\ensuremath{\pi}/2,0)$ point of the tetramerized Brillouin zone.

DOI
10.1103/physrevb.89.104415

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We study the plaquette valence-bond solid phase of the spin-1/2$${J}_{1}-J2{J}_{2}J2​ antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four S=1/2S=1/2S=1/2 spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the J1{J}_{1}J1​-J2{J}_{2}J2​ model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region 0.34<J2/J1<0.590.34<{J}_{2}/{J}_{1}<0.590.34<J2​/J1​<0.59, which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings J2=0.34J1{J}_{2}=0.34{J}_{1}J2​=0.34J1​ and J2=0.59J1{J}_{2}=0.59{J}_{1}J2​=0.59J1​. Interestingly, for J2<0.48J1{J}_{2}<0.48{J}_{1}J2​<0.48J1​, the excitation gap corresponds to a singlet-triplet excitation at the \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ point while, for J2>0.48J1{J}_{2}>0.48{J}_{1}J2​>0.48J1​, it is related to a singlet-singlet excitation at the X=(\ensuremathπ/2,0)\mathbf{X}=(\ensuremath{\pi}/2,0)X=(\ensuremathπ/2,0) point of the tetramerized Brillouin zone.
The spin-1/21/21/2 J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.
The plaquette valence bond solid (VBS) phase of the J1−J2J_1-J_2J1​−J2​ antiferromagnetic (AFM) Heisenberg honeycomb model has been studied using plaquette operator theory. Considering six S=12S =\frac{1}{2}S=21​ spins on a single plaquette the plaquette operator representation for the spin operators in terms of plaquette-singlet and plaquette-triplet boson operators has been developed. The formalism has been applied to the J1−J2J_1-J_2J1​−J2​ honeycomb antiferromagnet to derive an effective interacting boson model in terms of those singlets and triplets. The effective model has been analyzed within the harmonic approximation and results have been compared with the exact diagonalization data. The evolution of Dirac and Weyl nodes with respect to J2/J1J_2/J_1J2​/J1​ for this model has been studied.
We study the phase diagram and the dynamical spin structure factor of the spin-1/2 J1−−J3{J}_{1}--{J}_{3}J1​−−J3​ Heisenberg model on the square lattice using density matrix renormalization group, exact diagonalization (ED), and cluster perturbation theory (CPT). By extrapolating the order parameters and studying the level crossings of the low-lying energy and entanglement spectra, we obtain the phase diagram of this model and identify a narrow region of quantum spin liquid (QSL) phase followed by a plaquette valence-bond solid (PVBS) state in the intermediate region, whose nature has been controversial for many years. More importantly, we use CPT and ED to study the dynamical spin structure factor in the QSL and the PVBS phase. In the QSL phase, the high-energy magnon mode completely turns into some dispersive weak excitations around the XXX and MMM points. For the PVBS phase, the low-energy spectrum is characterized by a gapped triplet excitation, and at the high energy, we find another branch of dispersive excitation with broad continua, which is unlike the plaquette phase in the $2\ifmmode\times\else\texttimes\fi{}2$ checkerboard model. In the latter case, the second branch of excitation is nearly flat due to the weak effective interactions between the local excitations of the plaquettes. And in the J1−−J3{J}_{1}--{J}_{3}J1​−−J3​ Heisenberg model, the uniform interactions and the spontaneously translational symmetry breaking of the PVBS phase make the difference in the excitation spectra.
We studied the ground-state phase diagram of spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model on the square lattice with first-neighbor J1{J}_{1}J1​ and second-neighbor J2{J}_{2}J2​ antiferromagnetic interactions using both infinite density matrix renormalization group (iDMRG) and DMRG approaches. We show that a plaquette valence bond phase is realized in an intermediate region $0.50\ensuremath{\le}{J}_{2}/{J}_{1}<0.54$ between a N\'eel magnetic ordered phase at J2/J1<0.50{J}_{2}/{J}_{1}<0.50J2​/J1​<0.50 and a stripy magnetic ordered phase at ${J}_{2}/{J}_{1}\ensuremath{\ge}0.54$. The plaquette valence bond phase is characterized by finite dimer orders in both the horizontal and vertical directions. Contrary to the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, we do not find numerical evidence for a quantum spin liquid phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model.
The nature of quantum spin liquids is studied for the spin-1/21/21/2 antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, J1J_1J1​, and those between next-nearest-neighbor sites, J2J_2J2​. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region 0.4<J2/J1≤0.60.4< J_2/J_1\le 0.60.4<J2​/J1​≤0.6 sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of 0.4<J2/J1≤0.50.4 < J_2/J_1 \le 0.50.4<J2​/J1​≤0.5, while the gapped valence-bond-crystal (VBC) phase is stabilized for 0.5<J2/J1≤0.60.5 < J_2/J_1 \le 0.60.5<J2​/J1​≤0.6. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the C4vC_{4v}C4v​ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation ⟨S(0)S(r)⟩∝1/rz+η\langle S(0)S(r)\rangle \propto 1/r^{z+η}⟨S(0)S(r)⟩∝1/rz+η at a long distance rrr appears to increase from z+η∼1z+η\sim 1z+η∼1 at J2/J1∼0.4J_2/J_1\sim0.4J2​/J1​∼0.4 toward the continuous transition to the VBC phase at J1/J1∼0.5J_1/J_1\sim0.5J1​/J1​∼0.5. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.
We study a system of N‾\overline{N}N interacting triplons (the elementary excitations of a valence-bond solid) described by an effective interacting boson model derived within the bond operator formalism in order to determine the stability and the properties of many-triplon states. In particular, we consider the square lattice spin-$1/2\phantom{\rule{4pt}{0ex}}{J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Heisenberg model, focus on the intermediate parameter region where a quantum paramagnetic phase sets in, and consider the columnar valence-bond solid as a reference state. Within the bond operator theory, the Heisenberg model is mapped into an effective boson model in terms of triplet operators ttt. The effective boson model is studied at the harmonic approximation and the energy of the triplons and the expansion of the triplon operators bbb in terms of the triplet operators ttt are determined. Such an expansion allows us to perform a second mapping and, therefore, determine an effective interacting boson model in terms of the triplon operators bbb. We then consider systems with a fixed number N‾\overline{N}N of triplons and study the stability of many-triplon states within a mean-field approximation. We show that many-triplon states are stable, the lowest-energy ones are constituted by a small number of triplons, and the excitation gaps are finite. For J2=0.48J1{J}_{2}=0.48{J}_{1}J2​=0.48J1​ and 0.52J10.52{J}_{1}0.52J1​, we also calculate spin-spin and dimer-dimer correlation functions, dimer order parameters, and the bipartite von Neumann entanglement entropy within our mean-field formalism in order to determine the properties of the many-triplon state as a function of the triplon number N‾\overline{N}N. We find that the spin and the dimer correlations decay exponentially and that the entanglement entropy obeys an area law, regardless of the triplon number N‾\overline{N}N. Moreover, only for J2=0.48J1{J}_{2}=0.48{J}_{1}J2​=0.48J1​, the spin correlations indicate that the many-triplon states with large triplon number N‾\overline{N}N might display a more homogeneous singlet pattern than the columnar valence-bond solid. We also comment on possible relations between the many-triplon states with large triplon number N‾\overline{N}N and gapped spin-liquid states.