Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0803.3306 · Physical Review B

Fourfold degenerate columnar-dimer ground state in square lattice antiferromagnets

Rakesh Kumar, Brijesh Kumar

Published Apr 10, 2008• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.77.144413

Related papers

Two-dimensional quantum antiferromagnet with a fourfold degenerate dimer ground state

Alain Gellé, Andreas Martin Läuchli, Brijesh Kumar +1 more

We study the competition between antiferromagnetic order and valence-bond-crystal formation in a two-dimensional frustrated spin-$1∕2$ model. The ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model on the square lattice is further frustrated by introducing products of three-spin projectors which stabilize four dimer-product states as degenerate ground state. These four states are reminiscent of the dimerized singlet ground state of the Shastry-Sutherland model. Using exact diagonalization, we study the evolution of the ground state by varying the ratio of interactions. For a large range of parameters $({J}_{2}\ensuremath{\gtrsim}0.25{J}_{1})$, our model shows a direct transition between the valence-bond-crystal phase and the collinear antiferromagnetic phase. For small values of ${J}_{2}$, several intermediate phases appear which are also analyzed.

arXiv:0704.2352•Physical Review B•Jan 15, 2008•14 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

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

Triplon mean-field theory of an antiferromagnetic model with an exact fourfold degenerate Shastry-Sutherland ground state

Rakesh Kumar, Brijesh Kumar

In a recent work [Phys. Rev. B {\bf 77}, 014419 (2008)], a quantum spin-1/2 model having an exact fourfold degenerate Shastry-Sutherland ground state was constructed, and studied using finite-size numerical exact diagonalization. There, a schematic quantum phase diagram suggesting many competing phases was also proposed. In the present work, an extended version of this model is investigated using dimer and plaquette triplon mean-field theories. The quantum phase diagram of the extended model (in the thermodynamic limit) is found to have many interesting phases: commensurate and incommensurate magnetically ordered phases, and columnar dimer, sublattice columnar dimer, Shastry-Sutherland and plaquette spin-gapped phases. Among themselves, these phases undergo a continuous or a level-crossing transition.

•May 19, 2011•0 citations

Bond operators and triplon analysis for spin- S dimer antiferromagnets

Brijesh Kumar

The mean-field triplon analysis is developed for spin-$S$ quantum antiferromagnets with dimerized ground states. For the spin-1/2 case, it reduces to the well-known bond-operator mean-field theory. It is applied to a columnar dimer model on square lattice and to a model on honeycomb lattice with spontaneous dimerization in the ground state. Different phases in the ground state are investigated as a function of spin. It is found that under suitable conditions (such as strong frustration), a quantum ground state (dimerized singlet phase in the present study) can survive even in the limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$. Two quick extensions of this representation are also presented. In one case, it is extended to include the quintet states. In another, a similar representation is worked out on a square plaquette. A convenient procedure for evaluating the total-spin eigenstates for a pair of quantum spins is presented in the appendix.

arXiv:1003.5261•Physical Review B•Aug 3, 2010•29 citations

Ground-state phase diagram of the spin- 12 square-lattice J1 - J2 model with plaquette structure

Oliver Götze, Sven E. Krüger, F. Fleck +2 more

Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-$\frac{1}{2}$ ${J}_{1}$-${J}_{2}$ model on the square lattice with plaquette structure. We consider antiferromagnetic (${J}_{1}>0$) as well as ferromagnetic (${J}_{1}<0$) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction ${J}_{2}>0$. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform ${J}_{1}$-${J}_{2}$ model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic ${J}_{1}$) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case $s=1/2$. For the antiferromagnetic case (${J}_{1}>0$) N\'eel antiferromagnetic long-range order at small ${J}_{2}/{J}_{1}$ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large ${J}_{2}/{J}_{1}$ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (${J}_{1}<0$) we have the trivial ferromagnetic ground state at small ${J}_{2}/|{J}_{1}|$. By increasing ${J}_{2}$ this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length $s=2$ are antiferromagnetically long-range ordered. Further increasing of ${J}_{2}$ then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.

arXiv:1205.0681•Physical Review B•Jun 21, 2012•16 citations
We construct and study the two frustrated quantum spin-1∕21∕21∕2 models on square lattice, which are like the antiferromagnetic J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ 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 competition between antiferromagnetic order and valence-bond-crystal formation in a two-dimensional frustrated spin-1∕21∕21∕2 model. The ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model on the square lattice is further frustrated by introducing products of three-spin projectors which stabilize four dimer-product states as degenerate ground state. These four states are reminiscent of the dimerized singlet ground state of the Shastry-Sutherland model. Using exact diagonalization, we study the evolution of the ground state by varying the ratio of interactions. For a large range of parameters $({J}_{2}\ensuremath{\gtrsim}0.25{J}_{1})$, our model shows a direct transition between the valence-bond-crystal phase and the collinear antiferromagnetic phase. For small values of J2{J}_{2}J2​, several intermediate phases appear which are also analyzed.
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.
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.
In a recent work [Phys. Rev. B {\bf 77}, 014419 (2008)], a quantum spin-1/2 model having an exact fourfold degenerate Shastry-Sutherland ground state was constructed, and studied using finite-size numerical exact diagonalization. There, a schematic quantum phase diagram suggesting many competing phases was also proposed. In the present work, an extended version of this model is investigated using dimer and plaquette triplon mean-field theories. The quantum phase diagram of the extended model (in the thermodynamic limit) is found to have many interesting phases: commensurate and incommensurate magnetically ordered phases, and columnar dimer, sublattice columnar dimer, Shastry-Sutherland and plaquette spin-gapped phases. Among themselves, these phases undergo a continuous or a level-crossing transition.
The mean-field triplon analysis is developed for spin-SSS quantum antiferromagnets with dimerized ground states. For the spin-1/2 case, it reduces to the well-known bond-operator mean-field theory. It is applied to a columnar dimer model on square lattice and to a model on honeycomb lattice with spontaneous dimerization in the ground state. Different phases in the ground state are investigated as a function of spin. It is found that under suitable conditions (such as strong frustration), a quantum ground state (dimerized singlet phase in the present study) can survive even in the limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$. Two quick extensions of this representation are also presented. In one case, it is extended to include the quintet states. In another, a similar representation is worked out on a square plaquette. A convenient procedure for evaluating the total-spin eigenstates for a pair of quantum spins is presented in the appendix.
Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-12\frac{1}{2}21​ J1{J}_{1}J1​-J2{J}_{2}J2​ model on the square lattice with plaquette structure. We consider antiferromagnetic (J1>0{J}_{1}>0J1​>0) as well as ferromagnetic (J1<0{J}_{1}<0J1​<0) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction J2>0{J}_{2}>0J2​>0. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform J1{J}_{1}J1​-J2{J}_{2}J2​ model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic J1{J}_{1}J1​) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case s=1/2s=1/2s=1/2. For the antiferromagnetic case (J1>0{J}_{1}>0J1​>0) N\'eel antiferromagnetic long-range order at small J2/J1{J}_{2}/{J}_{1}J2​/J1​ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large J2/J1{J}_{2}/{J}_{1}J2​/J1​ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (J1<0{J}_{1}<0J1​<0) we have the trivial ferromagnetic ground state at small J2/∣J1∣{J}_{2}/|{J}_{1}|J2​/∣J1​∣. By increasing J2{J}_{2}J2​ this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length s=2s=2s=2 are antiferromagnetically long-range ordered. Further increasing of J2{J}_{2}J2​ then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.