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 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.
The energy gap of the one-dimensional (1D) spin-[Formula: see text] AF (antiferromagnetic) XXZ model in a transverse magnetic field (h) is studied by using the exact diagonalization technique. We have computed the energy gap in the region of anisotropy parameter -1<Δ<1 and in the vicinity of the critical line Δ = -1. By introducing a practical finite-size scaling approach, we have computed the correct critical exponent of the energy gap in good agreement with the field theoretical and spin-wave approaches. To check this approach, we have also studied the critical exponent of the energy gap of the 1D AF-Heisenberg chain in a staggered magnetic field (h(s)).
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.
We have calculated numerically the gap exponent of the anisotropic Heisenberg model in the presence of a transverse magnetic field. We have implemented the modified Lanczos method to obtain the excited states of our model with the same accuracy as the ground state. The coefficient of the leading term in the perturbation expansion diverges in the thermodynamic limit $(N\ensuremath{\rightarrow}\ensuremath{\infty})$. We have obtained the relation between this divergence and the scaling behavior of the energy gap. We have found that the opening of the gap in the presence of a transverse field scales with a critical exponent which depends on the anisotropy parameter $(\ensuremath{\Delta})$. Our numerical results are in good agreement with the field theoretical approach in the whole range of the anisotropy parameter $\ensuremath{-}1<\ensuremath{\Delta}<1$.
The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-
Hc), where
Hc is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=
kzf(\ensuremath{\Delta}/
kcphi), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors:
kz for \ensuremath{\Delta}\ensuremath{\ll}
kcphi and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}
kcphi, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.