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Journal of Physics C Solid State Physics

Ising model in a transverse field. II. Spectral functions and damping

Robin B. Stinchcombe

Published Aug 6, 1973• 49 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/6/15/010

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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.
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.
Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .
The Ising Model in a transverse field is investigated employing a field theoretical diagrammatic technique. The diagrams are classified with respect to a high density parameter 1/z, where z is the number of spins interacting with a given spin. The energy spectrum of collective excitations has been obtained both in the disordered phase and in the ordered phase. Summation of the chain diagrams (of order 1) reproduces the previous results. We have calculated the contributions of all diagrams of order 1/z and showed that the RPA commonly used has failed to include consistently a class of diagrams which reflect the kinematics of spin operators. The damping of the excitation waves is also obtained.
An Ising model in a transverse field with general spin S is considered within RPA Green-function theory in the paramagnetic region. Expressions for the temperature renormalised collective excitation spectrum with a gap decreasing as the exchange interaction increases, the transverse correlation function, and the critical exchange-field ratio for magnetic ordering, are derived. Zero-temperature expansions are given in the limit of weak interaction. Critical values at T=0K are calculated for several values of S and an asymptotical expression for S>>1 is given.