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Journal of Physics C Solid State Physics

The Ising model with a transverse field. II. Ground state properties

Pierre M. Pfeuty, Roger James Elliott

Published Oct 28, 1971• 214 citations
Publisher

Abstract

For Pt. I see ibid., vol. 4, 2359 (1971). The properties of the Ising model with a transverse field, with the Hamiltonian written H=- Gamma Sigma i S i x - 1 / 2 Sigma i,j J ij S i z S j z , are studied by perturbation expansions at T=0 in the ground state, both below and above the critical field Gamma C , for an hypercubic lattice and short range interactions. The order parameter (S z ) and the correlation function G=) Sigma i S i z Sigma j S j z ) are calculated respectively below and above Gamma C . The gap in the excitation spectrum is also studied above the critical field. The critical behaviour obtained from these expansions at T=0 is different from that obtained at T finite and is shown to be analogous to the critical behaviour of the Ising model for a lattice with one more dimension.

DOI
10.1088/0022-3719/4/15/024

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Journal of Physics A Mathematical and General•Dec 1, 1980•23 citations
For Pt. I see ibid., vol. 4, 2359 (1971). The properties of the Ising model with a transverse field, with the Hamiltonian written H=- Gamma Sigma i S i x - 1 / 2 Sigma i,j J ij S i z S j z , are studied by perturbation expansions at T=0 in the ground state, both below and above the critical field Gamma C , for an hypercubic lattice and short range interactions. The order parameter (S z ) and the correlation function G=) Sigma i S i z Sigma j S j z ) are calculated respectively below and above Gamma C . The gap in the excitation spectrum is also studied above the critical field. The critical behaviour obtained from these expansions at T=0 is different from that obtained at T finite and is shown to be analogous to the critical behaviour of the Ising model for a lattice with one more dimension.
For Pt. I see ibid., vol. 6, no. 15, 2459 (1973). By taking the (1/z) expansion one order beyond that considered in Pt. I, a detailed discussion is provided of the generalized susceptibilities and spectral functions rho alpha beta of the Ising model in a transverse field Gamma . The leading contributions to the damping of the excitations as well as the first-order shift in their energy and in the critical temperature, are obtained. The damping arises from break-up into (i) a spin wave and a spin fluctuation or (ii) a pair of spin waves.
The Ising model (exchange constant J) with a transverse field Gamma can be regarded as a simple model of hydrogen bonded ferroelectrics and of certain magnetic systems. The transition temperature T C decreases as Gamma increases and falls to zero at a critical field Gamma C . This transition is studied in the quadratic and simple cubic lattices with nearest neighbour interactions by a power series expansion of the susceptibility at high T up to fifth order in (J/kT) to all orders in ( Gamma /kT). The coefficients are calculated by diagram counting and time ordering perturbation methods. The critical Gamma against T curve is derived and the critical exponent gamma of the susceptibility. This appears to vary smoothly with Gamma but it is argued that this is an artifact due to the use of a short series and that the critical exponents are probably independent of Gamma as long as T C remains finite but change discontinuity at T C =0.
The critical behaviour of the susceptibility of the Ising model with a transverse field is investigated by means of the perturbation expansions. Exact expressions for the coefficients up to seventh order in J / k T are given as functions of Δ/ k T for the quadratic, triangular, simple cubic, b.c.c. and f.c.c. lattices. The universality of the critical index is concluded from the analysis of the coefficients and the critical temperature is obtained as function of the transverse field for the five kinds of lattices.
The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.