# A scaling hypothesis for matrix product states

## TL;DR

This paper develops a scaling hypothesis for matrix product states to analyze critical spin systems and quantum field theories, enabling data collapse and precise determination of critical exponents and central charge.

## Contribution

It introduces a novel scaling framework connecting transfer matrix eigenvalues, lattice spacing, and critical phenomena in matrix product states, with applications to quantum field theories.

## Key findings

- Successful benchmarking on Ising and Potts models.
- Derived scaling ansatz for correlation length and entanglement entropy.
- Demonstrated double data collapse for $\

## Abstract

We revisit the question of describing critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. Critical exponents and central charge are determined by optimizing the exponents such as to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $\lambda\phi^4$ with mass $\mu^2$ and lattice spacing $a$, we demonstrate a double data collapse for the correlation length $ \delta \xi(\mu,\lambda,D)=\tilde{\xi} \left((\alpha-\alpha_c)(\delta/a)^{-1/\nu}\right)$ with $D$ the bond dimension, $\delta$ the gap between eigenvalues of the transfer matrix, and $\alpha_c=\mu_R^2/\lambda$ the parameter which fixes the critical quantum field theory.