A universal formula for the entanglement asymmetry of matrix product states

TL;DR

The work addresses how symmetry breaking is echoed in entanglement by introducing entanglement asymmetry as a universal probe for 1D quantum states. It combines analytic proofs for finite and compact Lie groups with a transfer-operator framework in translationally invariant MPS to derive universal scalings: for finite groups $ΔS_n ≃ log(|G|/|H|)$ and for compact Lie groups $ΔS_n ≃ 0.5 (dim g - dim h) log |A|$, up to $O(1)$ terms, validated by numerical tests. The results reveal that the entanglement-based measure depends only on the symmetry-breaking pattern, not microscopic details, and generalize previous abelian results to non-abelian groups. This provides a practical, experimentally accessible criterion for symmetry breaking and restoration in 1D quantum systems, with potential extensions to MPO, PEPS, and randomized-measurement experiments.

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 \textit{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.

Figures (3)

• Figure 1: Diagrammatic representation of $(i)$$R_{(a,a')(b,b')}$ as in Eq. \ref{['eq:Rdef']}, $(ii)$$(R_g)_{(a,a')(b,b')}$ as in Eq. \ref{['eq:R_g']}, $(iii)$$R^{\ell}_{(a,a')(b,b')}$, and $(iv)$ of the charged moments as in Eq. \ref{['eq:tilderhoA']} where $P$ in $(v)$ is the projector in Eq. \ref{['eq:Pdef']}. We replace the standard trace loop by circles at the end points that virtually connect to each other only horizontally at the same level.
• Figure 2: Rényi-2 entanglement asymmetry $\Delta S_2$ of the ground state of the XXZ model for several values of $\Delta$, as a function of the length of the system considered $\ell$, computed using $(a)$ iDMRG, bond dimension considered $D=128$ and $(b)$ DMRG, $D=160$. The dashed grey line is the asymptotic value $\log 4$, while the dashed black line marks the value $\log 2$. We highlight the exponential fit of $\Delta S_2$ in the critical region, by dashed yellow lines.
• Figure 3: Symmetrization of the reduced density matrix. The blocks that connect equivalent representations, the blue and green ones, become proportional to the identity. The other (yellow) ones are washed out by the symmetrization procedure.