Density-matrix spectra for integrable models

TL;DR

This paper explains why density-matrix spectra derived from DMRG decay exponentially in integrable spin chains by connecting the reduced density matrix to corner transfer matrices (CTMs) of the associated 2D classical models. Through analytic CTM results for the transverse Ising chain and the XXZ model, it shows that the CTM spectra are exponential with model-dependent spacings and degeneracies, a prediction confirmed by extensive DMRG calculations up to 100 sites. The work provides a clear, analytic origin for the observed spectra and offers practical guidance on how many states to retain in DMRG and how this depends on the correlation length and integrability. It also discusses limitations and open questions for non-integrable and critical systems, where CTM-based predictions are less straightforward and may involve conformal-invariance considerations.

Abstract

The spectra which occur in numerical density-matrix renormalization group (DMRG) calculations for quantum chains can be obtained analytically for integrable models via corner transfer matrices. This is shown in detail for the transverse Ising chain and the uniaxial XXZ Heisenberg model and explains in particular their exponential character in these cases.

Figures (7)

• Figure 1: The strip geometry as discussed in the text with some portions of a square lattice. Also indicated are the corner transfer matrices A, B, C, D and the row transfer matrix (shaded).
• Figure 2: System composed of four corner transfer matrices with standard shape. The arrows indicate the direction of the transfer.
• Figure 3: Density-matrix spectrum of a transverse Ising chain with $\lambda=0.8$ and 120 sites, calculated with $30$ states.
• Figure 4: Density-matrix spectrum of a transverse Ising chain with $\lambda=1.25$ and 120 sites, calculated with $30$ states.
• Figure 5: Density-matrix spectra of transverse Ising chains for three values of $\lambda$ without the degeneracies, calculated with $30$ states for $\lambda=2$ and $1.25$ and with 60 states for $\lambda=1.11$. The $\varepsilon$-values were obtained from the slopes.