Spherical model

Spherical model

The spherical model in statistical mechanics is a model of ferromagnetism similar to the Ising model, which was solved in 1952 by T.H. Berlin and M. Kac. It has the remarkable property that when applied to systems of dimension "d" greater than four, the critical exponents which govern the behaviour of the system near the critical point are independent of "d" and the geometry of the system. It is one of the few models of ferromagnetism that can be solved exactly in the presence of an external field.

Formulation

The model describes a set of particles on a lattice mathbb{L} which contains "N" sites. For each site "j" of mathbb{L} , a spin sigma_j which interacts only with its nearest neighbours and an external field "H". It differs from the Ising model in that the sigma_j are no longer restricted to sigma_j in {1,-1} , but can take all real values, subject to the constraint that

: sum_{j=1}^N sigma_j^2 = N

which in a homogenous system ensures that the average of the square of any spin is one, as in the usual Ising model.

The partition function generalizes from that of the Ising model to

: Z_N = int_{-infty}^{infty} ldots int_{-infty}^{infty} dsigma_1 ldots dsigma_N exp left [ K sum_{langle jl angle} sigma_j sigma_l + h sum_j sigma_j ight] delta left [N - sum_j sigma_j^2 ight]

where delta is the Dirac delta function, langle jl angle are the edges of the lattice, and K=J/kT and h= H/kT , where "T" is the temperature of the system, "k" is Boltzmann's constant and "J" the coupling constant of the nearest-neighbour interactions.

Berlin and Kac saw this as an approximation to the usual Ising model, arguing that the sigma-summation in the Ising model can be viewed as a sum over all corners of an "N"-dimensional hypercube in sigma-space. The becomes an integration over the surface of a hypersphere passing through all such corners.

It was rigorously proved by Kac and C.J. Thompson that the spherical model is a limiting case of the N-vector model.

Equation of state

Solving the partition function and using a calculation of the free energy yields an equation describing the magnetization "M" of the system

: 2J(1-M^2) = kTg'(H/2JM)

for the function "g" defined as

: g(z) = (2 pi)^{-d} int_0^{2pi} ldots int_0^{2pi} domega_1 ldots domega_d ln [z+d -cos omega_1 - ldots - cos omega_d]

The internal energy per site is given by

: u =frac{1}{2} kT - Jd - frac{1}{2}H(M+M^{-1})

an exact relation relating internal energy and magnetization.

Critical behaviour

For d leq 2 the critical temperature occurs at absolute zero, resulting in no phase transition for the spherical model. For "d" greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite Curie temperature where ferromagnetism ceases. The critical behaviour of the spherical model was derived in the completely general circumstances that the dimension "d" may be a real non-integer dimension.

The critical exponents alpha, eta, gamma and gamma' in the zero-field case which dictate the behaviour of the system close to were derived to be

: alpha = egin{cases} - frac{4-d}{d-2} & mathrm{if} 2 4 end{cases}

: eta = frac{1}{2}

: gamma = egin{cases} frac{2}{d-2} & if 2 4 end{cases}

: delta = egin{cases} frac{d+2}{d-2} & if 2 < d < 4 \ 3 & if d > 4 end{cases}

which are independent of the dimension of "d" when it is greater than four, the dimension being able to take any real value.

References


* R.J. Baxter, "Exactly solved models in statistical mechanics", London, Academic Press, 1982

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