# High-temperature series expansion for the spin-1/2 Heisenberg model of the simple cubic lattice

## Abstract

The Heisenberg model, described by the Hamiltonian

H = − 2J ∑_{〈ij〉}**s**_{i}⋅**s**_{j} − 2μh ∑_{i }s_{i}^{z}

is used to calculate the partition function Z = Tr e^{−βH} from which the thermodynamic functions are obtained as expansions in the variable K = βJ = J/kT. Here, **s**_{i} is a spin operator at the site i of the simple cubic crystal lattice, J the exchange energy between neighboring spins (and positive for ferromagnetic coupling), μ the magnetic moment, and s_{i}^{z} the component of **s**_{i}, in the direction of the external magnetic field h. Formally, from the expression

N^{-1} ln Z = F_{0}(K) + (βμh)^{2}F_{2}(K)/2 + ⋯,

the zero-field specific heat

C = Nk K^{2 }∂^{2}F_{0}(K)/∂ K^{2}

and the susceptibility

χ = Nμ^{2}F_{2}(K)/kT

are found, N being the number of lattice sites. The problem is reduced to calculating the functions F_{0}(K) and F_{2}(K) as power series in K using the finite-lattice method.

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*Proceedings of the Samahang Pisika ng Pilipinas*

**12**, SPP-1994-TP-02 (1994). URL: https://proceedings.spp-online.org/article/view/SPP-1994-TP-02.