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Physical review. B, Condensed matter

Frustrated spin-1/2 model in two dimensions with a known ground state

Indrani Bose, Partha P. Mitra

Published Jul 1, 1991• 21 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.443

Related papers

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

Uma Bhaumik, Indrani Bose

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.

Physical review. B, Condensed matter•Nov 1, 1995•6 citations

Spin-1/2 frustrated antiferromagnet on a spatially anisotopic square lattice: contribution of exact diagonalizations

Philippe Sindzingre

The phase diagram of a spin-1/2 $J-J'-J_2$ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the $J_1-J_2$ model on the square lattice with two different nearest-neighbor couplings $J,J'$ 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 $J_2=0.5J' \ll J$ 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 side by a columnar dimer long range order. This columnar order could extends for $J'\to J$ which correspond to the $J_1-J_2$ model.

•Jan 1, 2004•0 citations

Spin-1/2 frustrated antiferromagnet on a spatially anisotropic square lattice: Contribution of exact diagonalizations

Philippe Sindzingre

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 ${J}_{1}--{J}_{2}$ 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 ${J}_{1}--{J}_{2}$ model.

arXiv:cond-mat/0307118•Physical Review B•Mar 19, 2004•46 citations

Quantum phases of the 1D anisotropic spin-1/2 frustrated ferromagnetic model: view point of quantum correlations

F. Khastehdel Fumani, Mostafa Motamedifar, S. Mahdavifar

Abstract We have studied the ground state phases of the one-dimensional (1D) spin-1/2 anisotropic frustrated ferromagnetic model, using the numerical Lanczos method. We have focused on the quantum correlations as the concurrence and the quantum discord (QD) between the nearest neighbor (NN) and the next-nearest neighbor (NNN) spins. Numerical results show that the Tomonaga-Luttinger liquid (TLL), the even-parity dimer, and the vector chiral phases can be distinguished from each other using the long-distance quantum correlations. Specially, the critical points are explicitly detected in the first derivative of the concurrence between the nearest neighbor spins with respect to the frustration parameter.

Physica Scripta•Feb 7, 2020•5 citations

Exact columnar dimer ground state and quantum phase transitions in a frustrated coupled spin ladder model

M. Mahapatra, Rakesh Kumar

We study a spin-half frustrated coupled ladder system, in which ladders with leg, rung, and diagonal interactions are linked via nearest-neighbor coupling. By introducing a leg-symmetric inter-ladder interaction that connects the left-to-left and right-to-right legs of adjacent ladders, the model is found to possess an exact dimer ground state, characterized by a product of two-spin singlets forming a columnar dimer phase. We analyze this model using bond-operator mean-field theory (BOMFT) and the density matrix renormalization group (DMRG) to probe the phase transitions that occur as one traverses the coupling space. The BOMFT reveals three distinct phases: a double-stripe ordered phase, a Néel ordered phase, and a quantum disordered dimerized phase. The critical points for the transitions are at $ J_1 = -0.81 $ (double-stripe to dimerized) and at $ J_1 = 2.81 $ (dimerized to Néel phase). Further, the DMRG results corroborate the exact ground state and refine the critical points to $ J_1 = -0.79 $ and $ J_1 = 2.29 $ for the respective transitions. Additionally, another transition is identified as the Néel order vanishes for $ J_1 \ge 4.5 $. The model can alternatively be represented as a network of orthogonal zigzag and fully frustrated spin ladders, offering a structural framework conducive to quantum materials engineering.

arXiv:2602.13406•Feb 13, 2026•0 citations

Fourfold degenerate columnar-dimer ground state in square lattice antiferromagnets

Rakesh Kumar, Brijesh Kumar

We construct and study the two frustrated quantum spin-$1∕2$ 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.

arXiv:0803.3306•Physical Review B•Apr 10, 2008•11 citations
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.
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.
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.
Abstract We have studied the ground state phases of the one-dimensional (1D) spin-1/2 anisotropic frustrated ferromagnetic model, using the numerical Lanczos method. We have focused on the quantum correlations as the concurrence and the quantum discord (QD) between the nearest neighbor (NN) and the next-nearest neighbor (NNN) spins. Numerical results show that the Tomonaga-Luttinger liquid (TLL), the even-parity dimer, and the vector chiral phases can be distinguished from each other using the long-distance quantum correlations. Specially, the critical points are explicitly detected in the first derivative of the concurrence between the nearest neighbor spins with respect to the frustration parameter.
We study a spin-half frustrated coupled ladder system, in which ladders with leg, rung, and diagonal interactions are linked via nearest-neighbor coupling. By introducing a leg-symmetric inter-ladder interaction that connects the left-to-left and right-to-right legs of adjacent ladders, the model is found to possess an exact dimer ground state, characterized by a product of two-spin singlets forming a columnar dimer phase. We analyze this model using bond-operator mean-field theory (BOMFT) and the density matrix renormalization group (DMRG) to probe the phase transitions that occur as one traverses the coupling space. The BOMFT reveals three distinct phases: a double-stripe ordered phase, a Néel ordered phase, and a quantum disordered dimerized phase. The critical points for the transitions are at $ J_1 = -0.81 $ (double-stripe to dimerized) and at $ J_1 = 2.81 $ (dimerized to Néel phase). Further, the DMRG results corroborate the exact ground state and refine the critical points to $ J_1 = -0.79 $ and $ J_1 = 2.29 $ for the respective transitions. Additionally, another transition is identified as the Néel order vanishes for $ J_1 \ge 4.5 $. The model can alternatively be represented as a network of orthogonal zigzag and fully frustrated spin ladders, offering a structural framework conducive to quantum materials engineering.
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.