Enhancing entanglement asymmetry in fragmented quantum systems

Abstract

Entanglement asymmetry provides a quantitative measure of symmetry breaking in many-body quantum states. Focusing on inhomogeneous $U(1)$ charges, such as dipole and multipole moments, we show that the typical asymmetry is bounded by a specific fraction of its maximal value, and verify this behavior in several settings, including random matrix product states. Within the latter ensemble, by identifying the bond dimension with an effective time, we qualitatively reproduce recent findings on the entanglement asymmetry dynamics in random quantum circuits, thereby suggesting a universal dynamical structure of the asymmetry of $U(1)$ charges in local ergodic systems. Multipole charges naturally arise in systems with Hilbert-space fragmentation, where the dynamics splits into exponentially many disconnected sectors. Using the commutant algebra formalism, we generalize entanglement asymmetry to account for fragmentation. We derive general upper bounds for both conventional and fragmented symmetries and identify states that saturate them. While the asymmetry grows logarithmically for conventional symmetries, it can scale extensively in fragmented systems, providing a probe that distinguishes classical from genuinely quantum fragmentation.

Figures (3)

• Figure 1: Schematic representation of the random states that form the two classes of ensembles considered in Sec. \ref{['sec:typ_states']} to study the typical multipole entanglement asymmetry: (a) Haar random states; to a set of $L$ qudits (with local Hilbert space $\mathbb{C}^d$, represented by blue lines) in the state $\ket{0}$, we apply a $d^L\times d^L$ random unitary matrix drawn from the Haar measure. (b) Inhomogeneous random MPS with bond dimension $D$; we consider again a set of $L$ qudits in the state $\ket{0}$ and we sequentially apply an independent $dD\times dD$ Haar random unitary matrix to each of them as shown in the figure. Red lines represent the bond space $\mathbb{C}^D$, which is initialized in the state $\ket{0}_D$.
• Figure 2: Average Rényi-2 entanglement asymmetry in the inhomogeneous random MPS calculated using the charged moments \ref{['eq:av_charged_mom_mps_transfer_main']}. In the upper panels, we plot it as a function of the bond dimension $D$, parametrized as $D = e^t$, for two subsystem sizes $\ell_A$. In the lower panels, it is shown as a function of $\ell_A$ for different values of $t$. In the left panels, we consider a charge $p = 0$, while in the right panels we take $p = 1$. The black curves in the upper panels correspond to the saddle point approximation in Eq. \ref{['eq:av_asymm_mps']}. In the limit $t\to\infty$ ($D\to\infty$), we recover the results for Haar random states.
• Figure 3: Rényi-2 entanglement asymmetry as a function of the variance of the charge $\hat{Q}_p$ for different values of $p$ in the ground state of the Kitaev chain (see main text). Since this state is Gaussian, the symbols have been calculated numerically using Eqs. \ref{['eq:charged_mom_gauss']} and \ref{['eq:var_gauss']} for different subsystem sizes and values of the parameters of the Kitaev chain $h$ and $\gamma$. Dashed and solid curves correspond to Eqs. \ref{['eq:asymm_var_gauss']} and \ref{['eq:asymm_var_gauss_p']}, respectively.