The classical Ising model: A quantum renormalization group approach

Subir Sachdev A

Michael Levin, Cody P. Nave

We describe a simple real space renormalization group technique for two-dimensional classical lattice models. The approach is similar in spirit to block spin methods, but at the same time it is fundamentally based on the theory of quantum entanglement. In this sense, the technique can be thought of as a classical analogue of the density matrix renormalization group method. We demonstrate the method - which we call the tensor renormalization group method - by computing the magnetization of the triangular lattice Ising model.

Geoffrey Grimmett

M. Kolb

A real-space renormalization-group transformation for quantum systems is proposed, based on an equivalent classical representation. Classical real-space renormalization groups and in particular Monte Carlo renormalization groups can then be used to study quantum systems. This formulation is valid for zero temperature as well as finite temperature. The Ising model in a transverse field in one and two dimensions is treated by Monte Carlo renormalization group.

Miguel Angel Martin-Delgado, Germán Sierra

We apply real space renormalization group (RG) methods to study two quantum group invariant Hamiltonians, that of the XXZ model and the Ising model in a transverse field (ITF) defined in an open chain with appropriate boundary terms. The quantum group symmetry is preserved under the RG transformation except for the appearance of a quantum group anomalous term which vanishes in the classical case. We obtain correctly the line of critical XXZ models. In the ITF model the RG flow coincides with the tensor product decomposition of cyclic irreducible representations of ${\mathrm{SU}}_{q}\left(2\right)$ with ${q}^{4}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.

Geoffrey Grimmett