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Physical review. B, Condensed matter

Frustrated spin-1/2 model in two dimensions with columnar dimer states as possible ground states

Uma Bhaumik, Indrani Bose

Published Nov 1, 1995• 6 citations
Publisher

Abstract

An antiferromagnetic spin S=1/2 model, the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$-${\mathit{J}}_{3}$-${\mathit{J}}_{4}$-${\mathit{J}}_{5}$ model, is considered in two dimensions. ${\mathit{J}}_{1}$,${\mathit{J}}_{2}$,${\mathit{J}}_{3}$,${\mathit{J}}_{4}$,${\mathit{J}}_{5}$ are the strengths of the nearest-neighbor, diagonal, next-nearest-neighbor, knight's-move-distance-away, and further-neighbor-diagonal interactions, respectively. It is known that when ${\mathit{J}}_{1}$:${\mathit{J}}_{2}$:${\mathit{J}}_{3}$:${\mathit{J}}_{4}$:${\mathit{J}}_{5}$=1:1:1/2:1/2:1/4 the four columnar dimer states are the exact eigenstates. In a bosonic mean-field theory, we find that the columnar dimer state is the ground state. We compare the results obtained with those obtained in the same mean-field theory for the one-dimensional Majumdar-Ghosh (MG) model. For special values of the couplings, the dimer state is known to be the exact ground state of the MG model. The comparison of the mean-field results for both the models leads to the conjecture that the columnar dimer states are the exact ground states of the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$-${\mathit{J}}_{3}$-${\mathit{J}}_{4}$-${\mathit{J}}_{5}$ model.

DOI
10.1103/physrevb.52.12489

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An antiferromagnetic spin S=1/2 model, the J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​-J3{\mathit{J}}_{3}J3​-J4{\mathit{J}}_{4}J4​-J5{\mathit{J}}_{5}J5​ model, is considered in two dimensions. J1{\mathit{J}}_{1}J1​,J2{\mathit{J}}_{2}J2​,J3{\mathit{J}}_{3}J3​,J4{\mathit{J}}_{4}J4​,J5{\mathit{J}}_{5}J5​ are the strengths of the nearest-neighbor, diagonal, next-nearest-neighbor, knight's-move-distance-away, and further-neighbor-diagonal interactions, respectively. It is known that when J1{\mathit{J}}_{1}J1​:J2{\mathit{J}}_{2}J2​:J3{\mathit{J}}_{3}J3​:J4{\mathit{J}}_{4}J4​:J5{\mathit{J}}_{5}J5​=1:1:1/2:1/2:1/4 the four columnar dimer states are the exact eigenstates. In a bosonic mean-field theory, we find that the columnar dimer state is the ground state. We compare the results obtained with those obtained in the same mean-field theory for the one-dimensional Majumdar-Ghosh (MG) model. For special values of the couplings, the dimer state is known to be the exact ground state of the MG model. The comparison of the mean-field results for both the models leads to the conjecture that the columnar dimer states are the exact ground states of the J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​-J3{\mathit{J}}_{3}J3​-J4{\mathit{J}}_{4}J4​-J5{\mathit{J}}_{5}J5​ model.
A frustrated spin-1/2 model is constructed on the square lattice, the Hamiltonian of which contains five types of interactions: nearest-neighbor (NN), diagonal (D), next-nearest-neighbor (NNN), knight's-move-distance away (KM), and further-neighbor diagonal (FND). The NN bonds in a particular direction, horizontal or vertical, are of alternating coupling strengths 7J and J. The other interaction strengths are: J (for NN and D), (J), J/2 (for NNN and KM), and J/4 (for FND). We prove that the exact ground state of the Hamiltonian is the columnar-dimer state.
Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction J2{\mathit{J}}_{2}J2​=1/2J1{\mathit{J}}_{1}J1​, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations J1{\mathit{J}}_{1}J1​-2J2{\mathit{J}}_{2}J2​+J3{\mathit{J}}_{3}J3​=0 and J3{\mathit{J}}_{3}J3​=2J4{\mathit{J}}_{4}J4​ hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-j2{\mathit{j}}_{2}j2​ around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.
The phase diagram of a spin-1/2 J−J′−J2J-J'-J_2J−J′−J2​ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the J1−J2J_1-J_2J1​−J2​ model on the square lattice with two different nearest-neighbor couplings J,J′J,J'J,J′ and may be also viewed as an array of coupled Heisenberg chains. The results suggest that the resonnating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B {\bf 67}, 024422 (2003)] for J2=0.5J′≪JJ_2=0.5J' \ll JJ2​=0.5J′≪J is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both side by a columnar dimer long range order. This columnar order could extends for J′→JJ'\to JJ′→J which correspond to the J1−J2J_1-J_2J1​−J2​ model.
The phase diagram of a spin-1/2 $J\ensuremath{-}{J}^{\ensuremath{'}}\ensuremath{-}{J}_{2}$ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ model on the square lattice with two different nearest-neighbor couplings ${J,J}^{\ensuremath{'}}$ and may also be viewed as an array of coupled Heisenberg chains. The results suggest that the resonating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B 67, 024422 (2003)] for ${J}_{2}{=0.5J}^{\ensuremath{'}}\ensuremath{\ll}J$ is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both sides by a columnar dimer long range order. This columnar order could extend for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}J$ which correspond to the J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ model.
We construct and study the two frustrated quantum spin-1∕21∕21∕2 models on square lattice, which are like the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model with some additional four-spin exchange interactions. These models admit an exactly solvable case in which the ground state consists of four degenerate columnar-dimer singlet (CDS) configurations. Away from the exact case, we employ bond-operator mean-field theory to investigate the evolution of the ground state by varying the interaction parameters. The mean-field calculation reveals a quantum phase diagram in which the CDS phase undergoes a continuous phase transition to either N\'eel or collinear ordered antiferromagnetic phases.
We study the stability region for the columnar dimer state proposed as a candidate ground state for the square-lattice quantum antiferromagnet with first- and second-neighbor antiferromagnetic couplings (J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​ model). We use a boson representation of the spin operators suited to the perturbative expansion around a dimer ground state. At lowest order, the columnar dimer is found to be stable only at the classical critical value J2{\mathit{J}}_{2}J2​/J2{\mathit{J}}_{2}J2​=1/2. However, we show that the leading anharmonic corrections stabilize the dimerized phase in a region of a finite width around J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​=1/2. A comparison of the ground-state energies shows that among the possible dimerized states the columnar dimer is the most favorable candidate to separate the two ordered states in the S=1/2 antiferromagnetic with first- and second-neighbor exchange.