Thermodynamics of the quantum Ising model in the two-dimensional kagome lattice

Abstract

In the classical Ising model on the kagome lattice, there are macroscopically degenerate classical states with exponentially-decayed spin correlation. As the singular quantum perturbation is introduced by applying the transverse magnetic field, the quantum ground state remains disordered, dubbed by "disorder-by-disorder". Here we compute the temperature dependence of the spin susceptibility and the specific heat for difference strength of the field to understand the effect of the singular quantum perturbation in this system. The relevance to the ZnCu3(OH)6Cl2 will be also commented.PACS numbers: 75.30. Gw, 75.40.Cx, 75.40.Mg Macroscopically degenerate states often occur in the classical level in the anti-ferromagnetic systems that the exchange energy can not be perfectly shared by all the nearest neighbor spins. It happens naturally if spins are placed in the triangular geometry in the two-dimensional lattices. In two dimensions, triangles can share the edges to form the triangular network or share the corners to form the kagome lattice. If we consider the classical Ising dynamics in these two systems, both of them show macroscopically degenerate classical ground states. However, the spin correlation in these two systems are very different, which determines the fate of the consequent quantum ground states when the quantum dynamics is introduced. When the transverse magnetic field is introduced by the following Hamiltonian,where S k = σ k /2 and σ k are the Pauli spin matrices and J is taken to be positive, the quantum ground state in the triangular lattice favors an spin-ordered ground state, the maximally-flippable state, because the spin correlation is critical in the classical model. On the other hand, the quantum ground state in the kagome case remains disordered because the spin correlation is exponentiallydecayed [1,2]. In this paper, we focus on the kagome case and compute its thermodynamic properties with various Γ. At Γ = 0, the spin susceptibility shows dramatic upturn in the low temperature and diverges at the zero temperature. For non-zero Γ, we also see the significant upturn but it saturates at the zero temperature. Our calculation is the first results on the thermodynamic quantities to distinguish the classical disordered state and the quantum disorder states. Furthermore, as the macroscopically degenerate classical states lead to the residual entropy at T = 0, the quantum dynamics which lifts the degeneracy results in additional peak in the specific heat in the low temperature at the order of Γ. Due to these properties, our model might be relevant to the recent discovered ZnCu 3 (OH) 6 Cl 2 , which shows the abnormal upturn in the spin susceptibility and saturates at T = 0. There have been several theoretical papers trying to describe this unusual property. A straightforward interpretation is the presence of the impurity[3] that contributes additionally to the susceptibility. However, a naive inclusion of the contribution from the impurity can explain the data only above 20K [4]. Different ang...

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