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physica status solidi (b)

Critical Behaviour of the Ising Model with a Transverse Field

Julia Mihailowa Wesselinowa

Published Aug 1, 1985• 5 citations
Publisher

Abstract

Abstract A Green's function technique is developed to study the Ising model with a transverse field. The order parameter and the longitudinal susceptibility are determined. The critical exponents β and γ are numerical calculated for different dimensional lattices. Using the scaling laws, α, δ, η, and v are obtained too. The critical exponents are in very good agreement with other theoretical results and with the experimental data.

DOI
10.1002/pssb.2221300232

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Abstract A Green's function technique is developed to study the Ising model with a transverse field. The order parameter and the longitudinal susceptibility are determined. The critical exponents β and γ are numerical calculated for different dimensional lattices. Using the scaling laws, α, δ, η, and v are obtained too. The critical exponents are in very good agreement with other theoretical results and with the experimental data.
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 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.
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.
The critical behaviour of d dimensional Ising models in a transverse magnetic field, Gamma , is investigated using a quantal generalisation of the Landau-Ginzburg-Wilson free-energy functional introduced by Hertz and Young (1976). A variational approximation of the Hartree type allows a detailed analysis: behaviour near the critical line, Gamma = Gammac(T) is governed by the exponents of the d dimensional spherical model (which are mean-field-like for d>or=4). However, the point ( Gamma ,T)=( Gammac(0),0) is multicritical in nature, and characterised by the exponents of the (d+1) dimensional spherical model. Crossover scaling functions, valid near the multicritical point, are found for the longitudinal susceptibility and free energy. They involve the crossover exponent phiT; for 2<d<or=3 this is equal to the (d+1) dimensional correlation length exponent, nud+1=1/(d-1); for 3<d<or=4, we find phiT=1/2(4-d), while for d>4, no crossover occurs. These results are shown to be consistent, for all d>2, with the finite-size scaling analogy suggested by Suzuki (1976) and others.
The square-lattice J 1 - J 2 transverse-field (TF) Ising model was investigated with the exact diagonalization (ED) method. In order to analyze the TF-driven phase transition, we applied the longitudinal-field fidelity susceptibility χ F ( h ) , which is readily evaluated via the ED scheme. Here, the longitudinal field couples with the absolute value of the magnetic moment | M | rather than the raw M so that the remedied fidelity susceptibility exhibits a peak around the critical point; note that the spontaneous magnetization does not appear for the finite-size systems. As a preliminary survey, the modified fidelity susceptibility χ F ( h ) is applied to the analysis of criticality for J 2 = 0 , where a number of preceding results are available. Thereby, properly scaling the distance from the multi-criticality, η = 0 . 5 − J 2 , the χ F ( h ) data were cast into the crossover-scaling formula, and the multi-critical exponent for χ F ( h ) is estimated. The result is cross-checked by the numerically evaluated β -function behavior.
Susceptibility of the transverse field Ising model on the square lattice is calculated numerically in the paramagnetic phase in a wide range of temperatures and transverse fields. An expression with one constant πππ, that determines both the critical exponent γγγ and the critical transverse field, compellingly represents the data asymptotically near the quantum critical point, except for a narrow classical region close to the phase transition line, and shows two crossovers as temperature varies.