Self-congruent point in critical matrix product states: An effective field theory for finite-entanglement scaling

TL;DR

This paper develops an effective field theory for matrix product states near critical points, revealing a self-congruent point where finite-entanglement effects balance, supported by numerical evidence from the Ising model and DMRG.

Contribution

It introduces a novel effective field theory framework for understanding finite-entanglement scaling in critical MPS, identifying a self-congruent point where relevant perturbations balance.

Findings

Finite bond dimension acts as a relevant perturbation to the critical fixed point.

The universal transfer matrix's gap ratios differ from pure CFT predictions due to finite-entanglement effects.

Numerical evidence from the Ising model and DMRG supports the theoretical framework.

Abstract

We set up an effective field theory formulation for the renormalization flow of matrix product states (MPS) with finite bond dimension, focusing on systems exhibiting finite-entanglement scaling close to a conformally invariant critical fixed point. We show that the finite MPS bond dimension $\chi$ is equivalent to introducing a perturbation by a relevant operator to the fixed-point Hamiltonian. The fingerprint of this mechanism is encoded in the $\chi$-independent universal transfer matrix's gap ratios, which are distinct from those predicted by the unperturbed Conformal Field Theory. This phenomenon defines a renormalization group self-congruent point, where the relevant coupling constant ceases to flow due to a balance of two effects; When increasing $\chi$, the infrared scale, set by the correlation length $\xi(\chi)$, increases, while the strength of the perturbation at the lattice scale decreases. The presence of a self-congruent point does not alter the validity of the finite-entanglement scaling hypothesis, since the self-congruent point is located at a finite distance from the critical fixed point, well inside the scaling regime of the CFT. We corroborate this framework with numerical evidences from the exact solution of the Ising model and density matrix renormalization group (DMRG) simulations of an effective lattice model.