The critical behaviour of the d -dimensional Ising model in a transverse field H ⊥ at temperatures near T = 0 and for H ⊥ → H c ⊥ , are predicted to be the same as for a ( d + 1)-dimensional Ising model in zero field as a function of temperature for T → T c . We present evidence for such a dimensionality cross-over by studying the field-dependent critical behaviour of the staggered magnetization of the d = 3 Ising-type antiferromagnet MnCl 2 ·4H 2 O at low temperatures.
The critical behaviour of the d -dimensional Ising model in a transverse field H ⊥ at temperatures near T = 0 and for H ⊥ → H c ⊥ , are predicted to be the same as for a ( d + 1)-dimensional Ising model in zero field as a function of temperature for T → T c . We present evidence for such a dimensionality cross-over by studying the field-dependent critical behaviour of the staggered magnetization of the d = 3 Ising-type antiferromagnet MnCl 2 ·4H 2 O at low temperatures.
The field theoretical renormalisation group method, which correctly describes the critical as well as the multicritical (at zero temperature) behaviours of the quantum mechanical d-dimensional Ising model in a transverse magnetic field, is proposed. Due to the introduction of the additional renormalisation the singularities associated with zero temperature have been removed. The crossover scaling form of the inverse correlation length and longitudinal susceptibility are considered for the following spatial dimensions d:2<d<3, d=3 and 3<d<4. For 2<d<3, d=3 and 3<d<4 the calculations are performed to the first order in 3-d and 4-d, respectively.
The quantum-classical crossover critical behaviour of the Ising model in a transverse field Gamma is first studied in one dimension where exact results are obtained. The correspondence between the quantum-classical crossover behaviour of this d-dimensional quantum model and the finite-size crossover behaviour of an equivalent (d+1)-dimensional classical model exact for d=1 is extended to higher dimensions. The crossover depends strongly on the relative magnitude of the crossover exponent phi and of the critical line-shift exponent Psi . In particular, if Pi > Psi approaches the critical line along a path of constant transverse field less than the critical field Gamma c and a low temperature, only classical behaviour is seen, quantum effects do not show up. There is some controversy about the value of Psi . Experiments are suggested to give an answer.
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.
Ten terms have been obtained in the high-temperature series for the fluctuation in long-range order of the Ising model in a transverse magnetic field on the three cubic lattices. This model is relevant to real materials such as singlet-singlet magnetic systems, hydrogen-bonded ferroelectrics and cooperative Jahn-Teller systems. The critical temperature of the system was obtained over essentially the entire range of the transverse field for which a phase transition exists. No crossover to the expected d=4 critical behaviour was observed at T=0.
The d-dimensional, quantum-mechanical Ising model in a transverse field Gamma displays a critical line, Gamma = Gamma c (T), which terminates in a multicritical point, ( Gamma ,T)=( Gamma c (0),0), with the exponents of the (d+1)-dimensional, nonquantal Ising model. Analogous behaviour is found for a nonquantal, (d+1)-dimensional n-component model of finite thickness, L in the last dimension, with the replacement Gamma to T,T to 1/L. A field theoretic method is used to study crossover behaviour in the vicinity of the multicritical point, for 3<d<4, where the multicritical exponents are mean-field-like. Multicritical behaviour is found to be governed by a Gaussian-like fixed point of the renormalisation group, yielding the crossover exponent, phi T = 1 / 2 (4-d). Crossover scaling functions are constructed to first order in epsilon =4-d for the longitudinal susceptibility and specific heat.
We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the nonthermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"uneisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.