Analysis of Quantum Spin Systems by a Nonlinear Sigma Model Method

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In a quantum spin system with antiferromagnetic interactions, strong quantum fluctuation may destroy the spin order so as to form a disordered ground state. Then a spin excitation from the ground state has a finite excitation energy or spin-gap. We formulated a novel nonlinear σ model method and, by using it, we examined disordered states for quantum spin systems.Especially, we established a nonlinear σ model method for a spin system with first- and second-neighbor exchange interactions on a square lattice, which is called the J1-J2 Heisenberg model. Based on the formulation and assisted by some known results, we found that the ground state of this model is the plaquette state in the disordered phase.The formulation can be extended, in priciple, to another two-dimensional spin system, if its classical version has an antiferromagnetic spin order. We examined the cases of a triangular lattice with square-lattice-like anisotropy and a honycomb lattice. The characters of the disorderd ground states are issues. The ground state for an anisotropic triangular lattice near a square lattice continues to that for a sqare lattice, meaning that their disordered state is basically the same.On the other hand, the nonlinear σ model method for a system with strong frustration is not established yet. In the case of strong frustration, the classical version of a quantum spin system has, for example, an order of 120-degree structure and not an antiferromagnetic order. The extention of the nonlinear σ model method to such a system is not easy. We have done some trials for a general formulation.For one-dimensional quantum spin systems, we extend the nonlinear σ model method to the case of spins with side chains. The condition for having gapless spin excitations is changed from the case of no side chains.