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arXiv:2112.05137 · Physical review. B./Physical review. B

Another exact ground state of a two-dimensional quantum antiferromagnet

Pratyay Ghosh, Tobias Müller, Ronny Thomale

Published May 24, 2022• 30 citations
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Abstract

We present the exact dimer ground state of a quantum antiferromagnet on the maple-leaf lattice. A coupling anisotropy for one of the three inequivalent nearest-neighbor bonds is sufficient to stabilize the dimer state. Together with the Shastry-Sutherland Hamiltonian, we show that this is the only other model with an exact dimer ground state for all two-dimensional lattices with uniform tilings.

DOI
10.1103/physrevb.105.l180412

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We present the exact dimer ground state of a quantum antiferromagnet on the maple-leaf lattice. A coupling anisotropy for one of the three inequivalent nearest-neighbor bonds is sufficient to stabilize the dimer state. Together with the Shastry-Sutherland Hamiltonian, we show that this is the only other model with an exact dimer ground state for all two-dimensional lattices with uniform tilings.
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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.
Abstract The spin- S Heisenberg antiferromagnet on the two-dimensional lattice is investigated for S = 1/2 and S = 1. We consider interaction at isolated dimers ( J d ) and interaction bonds that form the bounce lattice ( J b ). For J d = J b , the system is reduced to the maple-leaf-lattice antiferromagnet. We primarily conduct highly parallelized numerical diagonalization to examine the spin excitation gap above the ground state for various J b / J d cases. For S = 1/2, we report calculations for a 42-site cluster that has not been previously treated. The S = 1 case is examined for the first time for clusters up to 24 sites. Regardless of whether S = 1/2 or 1, we find that the system has a gapped nature for small J d / J b and becomes gapless at J d / J b ∼ 1.4. For S = 1, we also find that another gapped region appears between the gapless case at J d / J b ∼ 1.4 and the boundary of the exact-dimer phase.
Inspired by the exact solution of the Majumdar-Ghosh model, a family of one-dimensional, translationally invariant spin Hamiltonians is constructed. The exchange coupling in these models is antiferromagnetic, and decreases linearly with the separation between the spins. The coupling becomes identically zero beyond a certain distance. It is rigorously proved that the dimer configuration is an exact, superstable ground-state configuration of all the members of the family on a periodic chain. The ground state is twofold degenerate, and there exists an energy gap above the ground state. The Majumdar-Ghosh Hamiltonian with a twofold degenerate dimer ground state is just the first member of the family. The scheme of construction is generalized to two and three dimensions, and illustrated with the help of some concrete examples. The first member in two dimensions is the Shastry-Sutherland model. Many of these models have exponentially degenerate, exact dimer ground states.
The Heisenberg antiferromagnet on the maple-leaf lattice has recently gathered a great deal of attention. Competition between three nonequivalent bond interactions results in various ground-state quantum phases, with the exact dimer-product singlet ground state being among them. The thermodynamic properties of this model are much less understood. We used high-temperature expansion up to the 18th order to study the thermodynamics of the S = 1 / 2 Heisenberg model on the uniform maple-leaf lattice with the ground state exhibiting a six-sublattice 120 ∘ long-range magnetic order. Padé approximants allow us to get reliable results up to the temperatures of about T ≈ 0.4 . To study thermodynamics for arbitrary temperatures, we made the interpolation using the entropy method. Based on the analysis of close Padé approximants, we find ground-state energy e 0 = − 0.53064 ... − 0.53023 in good agreement with numerical results. The specific heat c ( T ) has a typical maximum at rather low temperatures T ≈ 0.379 and the uniform susceptibility χ ( T ) at T ≈ 0.49 . We also estimate the value of χ ( T ) at zero temperature χ 0 ≈ 0.05 ... 0.06 . The ground-state order manifests itself in the divergence of the so-called generalized Wilson ratio.