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Physical review. B, Condensed matter

Renormalization of the Ising model in a transverse field

A. Drzewiński, J. M. J. van Leeuwen

Published Jan 1, 1994• 23 citations
Publisher

Abstract

The renormalization scheme recently proposed by White is applied to the Ising model in a transverse field. It is found that this scheme is a significant improvement over standarad techniques as far as the computation of the ground state is concerned. It is shown that the errors in the ground-state energy are the largest in the neighborhood of the phase transition. Thus, this method has no special virtue in predicting the location of the phase transition and its associated critical exponents.

DOI
10.1103/physrevb.49.403

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The renormalization scheme recently proposed by White is applied to the Ising model in a transverse field. It is found that this scheme is a significant improvement over standarad techniques as far as the computation of the ground state is concerned. It is shown that the errors in the ground-state energy are the largest in the neighborhood of the phase transition. Thus, this method has no special virtue in predicting the location of the phase transition and its associated critical exponents.
The renormalization scheme recently proposed by White is applied to the d=1 anisotropic XY model in a transverse field (AXY). A flow diagram, critical exponents and energies have been calculated. It is found that this scheme is a distinct improvement over the standard technique as far as the computation of the ground state is concerned. The accuracy increases rapidly, when we keep more states in each renormalization step, but the errors in the ground state energy are always the largest in the neighborhood of the phase transitions. Comparing with the Ising model in a transverse field, on account of more complicated symmetries, the AXY demands more precautions during constructing a renormalization group transformation.
The renormalization scheme recently proposed by White is applied to the d=1 anisotropic XY model in a transverse field (AXY). It is found that this scheme offers a distinct improvement over standard techniques as far as the computation of the ground state is concerned. However, compared to the Ising model in a transverse field, on account of more complicated symmetries the AXY demands more precautions during the construction of a renormalization-group transformation. The new method predicts definitely better the location of the phase transition in the XY-like region than in the Ising-like region, but only in the Ising-like region is there any progress for the critical exponent \ensuremath{\alpha}.
We have adjusted the density-matrix renormalization method to handle two-dimensional systems of limited width. The key ingredient for this extension is the incorporation of symmetries in the method. The advantage of our approach is that we can force certain symmetry properties to the resulting ground-state wave function. Combining the results obtained for system sizes up to $30\ifmmode\times\else\texttimes\fi{}6$ and finite-size scaling, we derive the phase-transition point and the critical exponent for the gap in the Ising model in a transverse field on a two-dimensional square lattice.
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 renormalization group approach to critical phenomena is developed for quantum mechanical problems with non-commuting operators. Applying the theory to the spin 1 / 2 Ising model with a transverse field Gamma it is found that the critical exponents are those of the Ising model with Gamma =0 if the transition occurs at T>0. However, for transitions at T=0 the critical behaviour of the d-dimensional transverse system corresponds to that of the (d+1)-dimensional Ising model with Gamma =0, in agreement with series expansion predictions. At T=0 the dynamic, as well as static, critical behaviour is given by mean field theory for d>3.