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

Rakesh Kumar, Brijesh Kumar

Published May 19, 2011• 0 citations
OpenAlex

Abstract

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.

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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.
We look into the quantum phase diagram of a spin- antiferromagnet on the square lattice with degenerate Shastry–Sutherland ground states, for which only a schematic phase diagram is known so far. Many exotic phases were proposed in the schematic phase diagram by the use of exact diagonalization on very small system sizes. In our present work, an important extension of this antiferromagnet is introduced and investigated in the thermodynamic limit using triplon mean-field theory. Remarkably, this antiferromagnet shows a stable plaquette spin-gapped phase like the original Shastry–Sutherland antiferromagnet, although both of these antiferromagnets differ in the Hamiltonian construction and ground state degeneracy. We propose a sublattice columnar dimer phase which is stabilized by the second and third neighbor antiferromagnetic Heisenberg exchange interactions. There are also some commensurate and incommensurate magnetically ordered phases, and other spin-gapped phases which find their places in the quantum phase diagram. Mean-field results suggest that there is always a level-crossing phase transition between two spin-gapped phases, whereas in other situations, either a level-crossing or a continuous phase transition happens.
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 consider the zero-temperature properties of the spin-half two-dimensional Shastry-Sutherland antiferromagnet by using a high-order coupled cluster method treatment. We find that this model demonstrates various ground-state phases (N\'eel, magnetically disordered, orthogonal dimer), and we make predictions for the positions of the phase transition points. In particular, we find that the orthogonal-dimer state becomes the ground state at ${J}_{2}^{d}∕{J}_{1}\ensuremath{\sim}1.477$. For the critical point J2c∕J1{J}_{2}^{c}∕{J}_{1}J2c​∕J1​ where the semiclassical N\'eel order disappears we obtain a significantly lower value than J2d∕J1{J}_{2}^{d}∕{J}_{1}J2d​∕J1​, namely, J2c∕J1{J}_{2}^{c}∕{J}_{1}J2c​∕J1​ in the range 1.14--1.39. We therefore conclude that an intermediate phase exists between the N\'eel and the dimer phases. An analysis of the energy of a competing spiral phase yields clear evidence that the spiral phase does not become the ground state for any value of J2{J}_{2}J2​. The intermediate phase is therefore magnetically disordered but may exhibit plaquette or columnar dimer ordering.
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.
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.
Frustrated spin-1/2 model with all-antiferromagnetic exchange on Shastry-Sutherland lattice (SSL) has been extensively explored, and in this work we study the quantum phase diagram of isotropic Heisenberg spin- 1 / 2 model on the SSL with ferromagnetic exchange interactions along leg, and antiferromagnetic couplings along the diagonal bonds ( J ). Using density matrix renormalization group (DMRG) and exact diagonalization calculations, we identify four distinct ground-state (gs) phases for this frustrated spin-1/2 model: stripe ( π , 0 ) , stripe ( 0 , π ) , ferromagnetic, and spiral-dimer phases. In the spiral-dimer phase the gs can exhibit dimer formation along the diagonal bonds with or without spins spiral along the x axis or y axis in different parameter regime and these states are connected through crossovers. There is a continuous transition between stripe with a long-range spin-spin correlations and spiral-dimer phase with a short-range correlations. An effective spin parameter is introduced to accurately determine the phase boundaries. Our DMRG results differ significantly from previous Schwinger boson mean-field theory predictions. The spiral phase occurs in a narrow range of the parameter space and shows short-range correlations in the system.