A nonperturbative Real-Space Renormalization Group scheme

TL;DR

This paper introduces a rigorous, nonperturbative real-space renormalization group scheme that extends the density matrix renormalization group (DMRG) to finite temperatures and higher dimensions using auxiliary spaces.

Contribution

It provides a new mathematical formulation for real-space RG based on auxiliary spaces, overcoming DMRG limitations at finite temperature and in higher dimensions.

Findings

Overcomes DMRG limitations to zero temperature and one dimension.

Uses auxiliary Hilbert spaces to define embedding and truncation maps.

Provides a rigorous, nonperturbative RG framework.

Abstract

Based on the original idea of the density matrix renormalization group (DMRG), i.e. to include the missing boundary conditions between adjacent blocks of the blocked quantum system, we present a rigorous and nonperturbative mathematical formulation for the real-space renormalization group (RG) idea invented by L.P. Kadanoff and further developed by K.G. Wilson. This is achieved by using additional Hilbert spaces called auxiliary spaces in the construction of each single isolated block, which is then named a superblock according to the original nomenclature. On this superblock we define two maps called embedding and truncation for successively integrating out the small scale structure. Our method overcomes the known difficulties of the numerical DMRG, i.e. limitation to zero temperature and one space dimension.