For pt.I see ibid., vol.20, p.1327 (1987). A block-scaling method, applied previously to the Ising model with a transverse field in one dimension, is extended to two (2D) and three (3D) dimensions at T=0. The model in question has been considered on hexagonal, triangular and square lattices in 2D for small cells (the size n s =4 or 5) up to second order in the inter-cell interactions as well as for a number of larger cells in 2D and 3D up to first order in the perturbation. It is demonstrated that the extension yields a consistent global treatment of quantum and disorder effects and a substantial improvement in the numerical values of the correlation length exponent nu .
For pt.III see ibid., vol.20, p.1841 (1987). The global real-space renormalisation group approach is applied in two and three dimensions to study the pure and diluted transverse Ising model at finite temperatures. A phase diagram in the temperature-field plane is presented for the pure quantum model on the honeycomb, triangular and square lattices in the first-order cumulant expansion. An improvement of the classical Ising fixed-point data in two and three dimensions is considered by having recourse to larger cells and the second-order cumulant expansion. The zero-temperature percolative behaviour of the bond-diluted system is recovered from the fixed-point data analysis. Some shortcomings and relations to other approaches are also discussed.
A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.
For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco (1979 Phys. Rev. D 19 3173 ) is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent ν = 1. Recently, Miyazaki and Nishimori (2013 Phys. Rev. E 87 032154 ) have proposed to study the disordered quantum Ising model in dimensions d > 1 by applying the Fernandez-Pacheco procedure successively in each direction. To avoid the inequivalence of directions of their approach, we propose here an alternative procedure where the d directions are treated on the same footing. For the pure model, this leads to the correlation length exponents ν ≃ 0.625 in d = 2 (to be compared with the 3D classical Ising model exponent ν ≃ 0.63) and ν ≃ 0.5018 (to be compared with the 4D classical Ising model mean-field exponent ν = 1/2). For the disordered model in dimension d = 2, either ferromagnetic or spin-glass, the numerical application of the renormalization rules to samples of linear size L = 4096 yields that the transition is governed by an Infinite Disorder Fixed Point, with the activated exponent ψ ≃ 0.65, the typical correlation exponent ν typ ≃ 0.44 and the finite-size correlation exponent ν FS ≃ 1.25. We discuss the similarities and differences with the Strong Disorder Renormalization results.
A zero-temperature real-space renormalization-group method for quantum systems is applied to the
S=21XY model with a
Z field with the Hamiltonian $H=\ensuremath{-}\frac{1}{2}J\ensuremath{\Sigma}{〈i, j〉}^{}[(1+\ensuremath{\gamma}){S}_{i}^{x}{S}_{j}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{i}^{y}{S}_{j}^{y}]\ensuremath{-}h\ensuremath{\Sigma}{i}^{}{S}_{i}^{z}$ on quadratic, triangular, and hexagonal lattices in two dimensions (2D) and on a cubic lattice in 3D. In all the isotropic ($\ensuremath{\gamma}=0$) 2D cases the method gives no order in the ground state
∣0〉 but changes in the correlation functions at ${(\frac{h}{J})}_{c}=\frac{1}{2}{z}^{\ensuremath{'}}$ [${z}^{\ensuremath{'}}$ is the number of nearest neighbors (NN)] are observed. An argument is presented that at
(Jh)c the method becomes exact in the limit of infinite-size spin blocks. It is suggested that the correlation functions $〈0|{{S}_{0}}^{\ensuremath{\alpha}}{{S}_{R}}^{\ensuremath{\alpha}}|0〉$ ($\ensuremath{\alpha}=x, y$) for large
R have a power-law behavior $\ensuremath{\sim}{R}^{\ensuremath{-}\ensuremath{\eta}}$ for
0< ∣Jh∣<∣(Jh)c∣ with $\ensuremath{\eta}\ensuremath{\approx}1.5$ for
∣Jh∣<(Jh)c∣ and $\ensuremath{\eta}=2$ at
(Jh)c. The dynamical exponent
z is obtained to be \ensuremath{\approx} 0.3 for $|\frac{h}{J}|<\frac{1}{2}{z}^{\ensuremath{'}}$ and at
(Jh)c close to 2. An indirect method of calculating the ground-state energy is developed, and the results are compared with the known estimates. For $\ensuremath{\gamma}\ensuremath{\ne}0$ and
h=0 the $\ensuremath{\gamma}$ dependences of the order in the ground state ${\ensuremath{\mu}}^{x}(\ensuremath{\gamma})$ is calculated to be ${\ensuremath{\mu}}^{x}\ensuremath{\sim}{\ensuremath{\gamma}}^{0.21}$ for $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ and
h=0. In a 3D cubic lattice ($\ensuremath{\gamma}=0$) the analysis of recursion relations indicates the existence of order, in accord with recent exact theorems.
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.
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.