Entanglement under the renormalization-group transformations on quantum states and in quantum phase transitions

TL;DR

This paper investigates how entanglement in quantum states evolves under renormalization-group transformations, revealing critical behavior and universality near quantum phase transitions through geometric entanglement measures.

Contribution

It introduces a method to quantify entanglement under RG transformations and analyzes its behavior in ground states of MPS Hamiltonians at quantum critical points.

Findings

Entanglement exhibits singular behavior near critical points.

Finite RG steps follow a scaling hypothesis revealing correlation length.

Infinite RG steps show universal behavior linked to conformal field theory.

Abstract

We consider quantum states under the renormalization-group (RG) transformations introduced by Verstraete et al. [Phys. Rev. Lett. 94, 140601 (2005)] and propose a quantification of entanglement under such RG (via the geometric measure of entanglement). We examine the resulting entanglement under RG for the ground states of "matrix-product-state" (MPS) Hamiltonians constructed by Wolf et al. [Phys. Rev. Lett. 97, 110403 (2006)] that possess quantum phase transitions. We find that near critical points, the ground-state entanglement exhibits singular behavior. The singular behavior within finite steps of RG obeys a scaling hypothesis and reveals the correlation length exponent. However, under the infinite steps of RG transformation, the singular behavior is rendered different and is universal only when there is an underlying conformal-field-theory description of the critical point.