The Density Matrix Renormalization Group Method applied to Interaction Round a Face Hamiltonians

TL;DR

This paper introduces IRF-DMRG, a variant of the density matrix renormalization group method tailored for IRF Hamiltonians, reducing computational complexity by exploiting symmetry and connecting to integrable systems.

Contribution

It develops a new IRF-DMRG formulation that leverages symmetry factorization, enabling more efficient analysis of IRF Hamiltonians in statistical mechanics and quantum models.

Findings

IRF-DMRG reduces Hilbert space dimensions compared to vertex-DMRG.

Demonstrated effectiveness on SOS and RSOS models.

Connects IRF-DMRG with integrable systems and conformal field theory.

Abstract

Given a Hamiltonian with a continuous symmetry one can generally factorize that symmetry and consider the dynamics on invariant Hilbert Spaces. In Statistical Mechanics this procedure is known as the vertex-IRF map, and in certain cases, like rotational invariant Hamiltonians, can be implemented via group theoretical techniques. Using this map we translate the DMRG method, which applies to 1d vertex Hamiltonians, into a formulation adequate to study IRF Hamiltonians. The advantage of the IRF formulation of the DMRG method ( we name it IRF-DMRG), is that the dimensions of the Hilbert Spaces involved in numerical computations are smaller than in the vertex-DMRG, since the degeneracy due to the symmetry has been eliminated. The IRF-DMRG admits a natural and geometric formulation in terms of the paths or string algebras used in Exactly Integrable Systems and Conformal Field Theory. We illustrate the IRF-DMRG method with the study of the SOS model which corresponds to the spin 1/2 Heisenberg chain and the RSOS models with Coxeter diagram of type A, which correspond to the quantum group invariant XXZ chain.