Universal Asymptotic Eigenvalue Distribution of Density Matrices and the Corner Transfer Matrices in the Thermodynamic Limit

TL;DR

This paper derives a universal asymptotic eigenvalue distribution for density and corner transfer matrices in large quantum and classical systems, supported by analytical results for integrable models and numerical verification for non-integrable ones.

Contribution

It establishes a universal asymptotic form for eigenvalue distributions of DM and CTM in large systems, bridging integrable and non-integrable models.

Findings

Derived exact asymptotic eigenvalue distribution for integrable models.

Numerically verified the universal form in non-integrable models.

Suggested universality of eigenvalue distribution across diverse systems.

Abstract

We study the asymptotic behavior of the eigenvalue distribution of the Baxter's corner transfer matrix (CTM) and the density matrix (DM) in the White's density-matrix renormalization group (DMRG), for one-dimensional quantum and two-dimensional classical statistical systems. We utilize the relationship ${\rm DM}={\rm CTM}^4$ which holds for non-critical systems in the thermodynamic limit. Using the known diagonal form of CTM, we derive exact asymptotic form of the DM eigenvalue distribution for the integrable $S=1/2$ XXZ chain (and its related integrable models) in the massive regime. The result is then recast into a ``universal'' form without model-specific quantities, which leads to $\omega_{m}\sim \exp[-{\rm const.}(\log m)^2]$ for $m$-th DM eigenvalue at larg $m$. We perform numerical renormalization group calculations (using the corner-transfer-matrix RG and the product-wavefunction RG) for non-integrable models, verifying the ``universal asymptotic form'' for them. Our results strongly suggest the universality of the asymptotic eigenvalue distribution of DM and CTM for a wide class of systems.