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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Comment on the SLAC renormalization-group approach to the Ising chain in a transverse magnetic field

Amalio F. Pacheco

Published May 15, 1979• 38 citations
Publisher

Abstract

It is shown that the critical parameters obtained for the Ising chain in a transverse field by using the SLAC naive renormalization-group approach can be easily and considerably improved. This is obtained after doing a convenient trick so that site and link terms of the Hamiltonian of the theory are treated in a symmetric way. In particular, the critical coupling constant and the critical thermal exponent obtained with our method are the exact known results of this model.

DOI
10.1103/physrevd.19.3173

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It is shown that the critical parameters obtained for the Ising chain in a transverse field by using the SLAC naive renormalization-group approach can be easily and considerably improved. This is obtained after doing a convenient trick so that site and link terms of the Hamiltonian of the theory are treated in a symmetric way. In particular, the critical coupling constant and the critical thermal exponent obtained with our method are the exact known results of this model.
A zero-temperature real-space renormalization-group method is presented and applied to the quantum Ising model with a transverse field in one dimension. The transition between the low-field and high-field regimes is studied. Magnetization components, spin correlation functions, and critical exponents are derived and checked against the exact results. It is shown that increasing the size of the blocks in the iterative procedure yields more accurate results, especially for the critical "magnetic" exponents near the transition.
The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
Fisher's [Phys. Rev. B 51,51,51, 6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.
We apply a recently proposed real-space renormalization-group method to the two-dimensional square Ising model in a transverse magnetic field at zero temperature. We do the calculation to first and second order in the intercell coupling; in both cases we find a nontrivial fixed point. We compare the obtained critical exponents with accepted values for the three-dimensional Ising model from high-temperature series expansions. In first order the agreement is poor, in second order we obtain good agreement with the expected values.