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arXiv:1806.05814 · Scientific Reports

Exact dimer phase with anisotropic interaction for one dimensional magnets

Hong-Ze Xu, Shun-Yao Zhang, Guang-Can Guo, Ming Gong

Published Mar 19, 2021• 11 citations
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Abstract

We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic region. We demonstrate that this phase can be realized even in models when antiferromagnetic interaction along one of the three directions. This model also supports three different ferromagnetic (FM) phases, denoted as x-FM, y-FM and z-FM, polarized along the three directions. The boundaries between the exact dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry breaking or [Formula: see text] symmetry breaking leads to exact dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge [Formula: see text] for free fermions. We characterize the properties of these boundaries using entanglement entropy, excitation gap, and long-range spin-spin correlation functions. These results are relevant to a large number of one dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in realistic materials with magnetic field along different directions and show that the anisotropy may resolve the disagreement between theory and experiments based on isotropic spin-spin interactions.

DOI
10.1038/s41598-021-85483-0

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We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic region. We demonstrate that this phase can be realized even in models when antiferromagnetic interaction along one of the three directions. This model also supports three different ferromagnetic (FM) phases, denoted as x-FM, y-FM and z-FM, polarized along the three directions. The boundaries between the exact dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry breaking or [Formula: see text] symmetry breaking leads to exact dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge [Formula: see text] for free fermions. We characterize the properties of these boundaries using entanglement entropy, excitation gap, and long-range spin-spin correlation functions. These results are relevant to a large number of one dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in realistic materials with magnetic field along different directions and show that the anisotropy may resolve the disagreement between theory and experiments based on isotropic spin-spin interactions.
We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by direct generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic regime. This model also supports three different ferromagnetic (FM) phases, denoted as xxx-FM, yyy-FM and zzz-FM. The boundaries between the dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry or Z2\mathbb{Z}_2Z2​ symmetry lead to dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge c=1c=1c=1. We characterize the properties of these boundaries using entanglement entropy, excitation gap and long-range spin-spin correlations. These results are relevant to a large number of quasi-one-dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in true materials with magnetic field along different directions, and show that the anisotropy may resolve the long-standing disagreements between theory and experiments.
Interacting spin chains and ladders are known to support a plethora of quantum phases with complex ground-state phase diagrams. In this work, we study a large family of such models and determine precise, explicit conditions under which an exact dimer state is guaranteed to be the ground state. These general conditions are validated for various generalizations of the Majumdar-Ghosh model using exact diagonalization. Our results provide exact reference points in the phase diagrams of a wide class of spin chains and ladders, including those with anisotropic and arbitrary-range interactions.
We show that spin SSS Heisenberg spin chains with an additional three-body interaction of the form $({\mathbf{S}}_{\mathbf{i}\mathbf{\ensuremath{-}}\mathbf{1}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{\mathbf{i}})({\mathbf{S}}_{\mathbf{i}}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{\mathbf{i}+\mathbf{1}})+\mathrm{H}.\mathrm{c}.$ possess fully dimerized ground states if the ratio of the three-body interaction to the bilinear one is equal to $1/[4S(S+1)\ensuremath{-}2]$. This result generalizes the Majumdar-Ghosh point of the ${J}_{1}\ensuremath{-}{J}_{2}$ chain, to which the present model reduces for S=1/2S=1/2S=1/2. For S=1S=1S=1, we use the density matrix renormalization group method to show that the transition between the Haldane and the dimerized phases is continuous with a central charge c=3/2c=3/2c=3/2. Finally, we show that such a three-body interaction appears naturally in a strong-coupling expansion of the Hubbard model, and we discuss the consequences for the dimerization of actual antiferromagnetic chains.
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.
Using the quantum Monte Carlo method, we study, under external magnetic fields, the ground state phase diagram of the two-dimensional spin S=1/2 dimer model with an anisotropic intraplane antiferromagnetic coupling. With the anisotropy 4 greater/approximately Delta greater/approximately 3, a supersolid phase characterized by a nonuniform Bose condensate density that breaks translational symmetry is found. The rich phase diagram also contains a checkerboard solid, an antiferromagnet in the z axis, and a superfluid phase formed by S(z)= +1 spin triplets which has a finite staggered magnetization in the in-plane direction. As we show, the model can be realized as a consequence of including the next nearest neighbor coupling among dimers and our results suggest that spin dimer systems may be an ideal model system to study the supersolid phase.
Interacting spin chains and ladders are known to support a plethora of quantum phases withcomplex ground-state phase diagrams. In this work, we study a large family of such models anddetermine precise, explicit conditions under which an exact dimer state is guaranteed to be theground state. These general conditions are validated for various generalizations of the Majumdar-Ghosh model using exact diagonalization. Our results provide exact reference points in the phasediagrams of a wide class of spin chains and ladders, including those with anisotropic and arbitrary-range interactions.