A Numerical Renormalization Group for Continuum One-Dimensional Systems

TL;DR

This procedure integrates Wilson's numerical renormalization group with Zamolodchikov's truncated conformal spectrum approach and works naturally on a wide class of interacting one-dimension models based on perturbed (possibly strongly) continuum conformal and integrable models.

Abstract

We present a renormalization group (RG) procedure which works naturally on a wide class of interacting one-dimension models based on perturbed (possibly strongly) continuum conformal and integrable models. This procedure integrates Kenneth Wilson's numerical renormalization group with Al. B. Zamolodchikov's truncated conformal spectrum approach. Key to the method is that such theories provide a set of completely understood eigenstates for which matrix elements can be exactly computed. In this procedure the RG flow of physical observables can be studied both numerically and analytically. To demonstrate the approach, we study the spectrum of a pair of coupled quantum Ising chains and correlation functions in a single quantum Ising chain in the presence of a magnetic field.

Figures (4)

• Figure 1: A schematic of the finite sized system, both in real space and in terms of energy levels, analyzed in the TSA procedure.
• Figure 2: An outline of the NRG algorithm
• Figure 3: Plots showing the behavior of $\Delta_2$ as a function of the truncation energy, $E_{\rm trunc}$ (for $R=5\tilde{\lambda}^{-1}$).
• Figure 4: A plot of the matrix element, $\langle 0| \sigma(0) |A_1(p)A_1(-p)\rangle$ as a function of energy, $\omega = 2(p^2+m_1^2)^{1/2}$.