[Paper Review] A universal formula for the entanglement asymmetry of matrix product states

Luca Capizzi, Vittorio Vitale|arXiv (Cornell University)|Oct 3, 2023

Quantum Information and CryptographyComputer Science3 citations

TL;DR

This paper establishes a universal formula for the entanglement asymmetry of matrix product states (MPS) with finite bond dimension in the thermodynamic limit, showing that the Rényi entanglement asymmetry depends solely on the symmetry-breaking pattern $G \to H$, not on microscopic details. For discrete groups, $\Delta S_n \simeq \log(|G|/|H|)$, and for compact Lie groups, $\Delta S_n \simeq \frac{1}{2}(\dim\mathfrak{g} - \dim\mathfrak{h})\log|A|$, confirmed numerically via iDMRG simulations on the XXZ model.

ABSTRACT

Symmetry breaking is a fundamental concept in understanding quantum phases of matter, studied so far mostly through the lens of local order parameters. Recently, a new entanglement-based probe of symmetry breaking has been introduced under the name of extit{entanglement asymmetry}, which has been employed to investigate the mechanism of dynamical symmetry restoration. Here, we provide a universal formula for the entanglement asymmetry of matrix product states with finite bond dimension, valid in the large volume limit. We show that the entanglement asymmetry of any compact -- discrete or continuous -- group depends only on the symmetry breaking pattern, and is not related to any other microscopic features.

Motivation & Objective

• To establish a universal formula for entanglement asymmetry in matrix product states (MPS) with finite bond dimension in the thermodynamic limit.
• To prove that the Rényi entanglement asymmetry depends exclusively on the symmetry-breaking pattern $G \to H$, not on microscopic details of the state.
• To extend the previously conjectured formula for finite groups to compact Lie groups, including continuous symmetries such as $U(1)$.
• To validate the theoretical predictions through numerical simulations using infinite DMRG (iDMRG) on the XXZ spin chain.

Proposed method

• Define the entanglement asymmetry as the Rényi entropy difference between the original reduced density matrix $\rho_A$ and its symmetrized version $\tilde{\rho}_A$ using the group action.
• Use the Haar measure for compact Lie groups and group averaging to construct $\tilde{\rho}_A$, ensuring it is $G$-invariant.
• Derive the charged moments $\text{Tr}(\tilde{\rho}_A^n)$ via group character theory and spectral properties of the local MPS tensors.
• Prove that in the large volume limit, $\Delta S_n \simeq \log(|G|/|H|)$ for finite groups and $\Delta S_n \simeq \frac{1}{2}(\dim\mathfrak{g} - \dim\mathfrak{h})\log|A|$ for compact Lie groups.
• Perform numerical simulations using iDMRG to compute the Rényi-2 entanglement asymmetry for the antiferromagnetic ground state of the XXZ model with $\Delta > 1$, confirming the predicted asymptotic values.

Experimental results

Research questions

• RQ1 Does the entanglement asymmetry of a large subsystem in a matrix product state depend only on the symmetry-breaking pattern $G \to H$, independent of microscopic details?
• RQ2 Can the conjectured universal formula $\Delta S_n \simeq \log(|G|/|H|)$ for finite groups be rigorously proven in the thermodynamic limit?
• RQ3 How does the entanglement asymmetry scale for continuous compact Lie groups, and does it follow a universal form involving the dimension of the Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$?
• RQ4 Is the predicted universal scaling of entanglement asymmetry numerically observable in realistic quantum spin chains such as the XXZ model?
• RQ5 What happens to the entanglement asymmetry in critical systems or in the presence of boundary-induced symmetry breaking?

Key findings

• The Rényi entanglement asymmetry for large subsystems in matrix product states is universally determined by the symmetry-breaking pattern $G \to H$, with $\Delta S_n \simeq \log(|G|/|H|)$ for finite groups.
• For compact Lie groups, the entanglement asymmetry scales as $\Delta S_n \simeq \frac{1}{2}(\dim\mathfrak{g} - \dim\mathfrak{h})\log|A|$, confirming universal scaling behavior.
• Numerical simulations of the XXZ model with $\Delta > 1$ show $\Delta S_2$ converging to $\log 2$, consistent with the prediction $\log(|G|/|H|) = \log 2$ for $|G|=4$, $|H|=2$, validating the formula in a physical model.
• The convergence is slower near the critical point $\Delta = 1$, attributed to diverging correlation length and dynamical symmetry restoration.
• The results are robust for translational-invariant MPS with finite bond dimension and are compatible with prior observations in global quench dynamics and ground states of integrable field theories.