Quantum decimation renormalization group method for one dimensional spin-1/2 systems

展也 前島

J. Szṅajd

Chen Xi-Yao, G. F. Tuthill

We have extended the range of application of the decimation renormalization-group (RG) method, using a combination of analytic and numerical techniques in calculating thermal and magnetic properties of a wide variety of one-dimensional quantum spin-(1/2) systems. Efforts to improve the accuracy of the approach include increasing the spatial rescaling factor and investigating the effect of free versus periodic boundary conditions for each renormalization cluster. We have used the RG method to calculate the specific heat and susceptibility for chains with uniaxial exchange anisotropy (ranging from the Ising to the XY limits) in both longitudinal and transverse magnetic fields, as well as for the Heisenberg antiferromagnet with alternating couplings, and have compared our results, when possible, with either exact solutions or with numerical extrapolations from finite-chain calculations. Decimation is a high-temperature approximation, but its range of reliability is found in some cases (e.g., transverse susceptibility for the Ising chain, or susceptibility for the Heisenberg ferromagnet in a field) to extend well into the strong-coupling regime. We note that the decimation results for magnetic properties are, in general, more accurate than those for the thermal response.

Naobumi Honda

Jenő Sólyom

The zero-temperature renormalization-group transformations for quantum spin systems are analyzed. The block transformation and the decimation-type transformation used in the study of the one-dimensional Ising model in transverse field are extended to the quantum version of the Potts model and Ashkin-Teller model. It is shown that for these self-dual models the two kinds of renormalization-group transformations are dual to each other and therefore give the same result for the critical behavior. The duality persists even if the higher-lying states are taken into account in a perturbational way.

Gregory S. Um

We have applied a decimation (dedecoration) method to a quantum system in order to obtain a critical exponent of the two-dimensional Ising model. Because the renormalization-group transformation is not exact, we have used a perturbation similar to the one used by Niemeijer and Van Leeuwen. The calculations were done to third order. Contrary to the 3-spin-cell method, the second order generates only next-nearest interaction, while the third order generates the next-next-nearest neighbor, etc. The numerical results are compared and analyzed. There are oscillations in our calculation, and the perturbation seems to be at least asymptotic.