Entanglement Asymmetry and Asymmetry Contour under Boundary Symmetry Breaking

Abstract

We investigate how symmetry-breaking boundary magnetic fields affect magnetization and Rényi-2 entanglement asymmetry (EA) in the transverse-field Ising model. For the ground state, magnetization exhibits qualitatively distinct behaviors in the ordered and disordered phases. EA also exhibits phase-dependent behavior, which is well captured by the lower and upper bounds derived in this work. In the critical regime, both magnetization and EA exhibit behavior qualitatively consistent with the field-theoretical predictions for a related setup. We also study boundary-quench dynamics and find that magnetization propagates from the boundary into the bulk with a velocity consistent with the maximum quasiparticle velocity. In contrast, because EA is a subsystem-integrated quantity, its spatial structure is not directly resolved. We therefore introduce the asymmetry contour, a contour-like finite-difference quantity that provides a spatially resolved characterization of the symmetry-breaking information encoded in the reduced density matrix. The resulting contour is complementary to the local magnetization, retaining sensitivity to nonlocal many-body correlations while its dynamics also reflects the underlying quasiparticle propagation. These results show how EA and its contour can provide complementary information-theoretic characterizations of symmetry breaking and its spatial structure.

The paper studies how symmetry breaking induced by boundary longitudinal fields penetrates the transverse-field Ising chain, comparing local magnetization with Rényi-2 entanglement asymmetry (EA) in ground states and after switching on the fields. Two ancilla spins restore a quadratic-fermion representation, while selecting their sectors recovers the physical boundary fields. For boundary-connected subsystems, the parity-broken reduced state yields the formula $\Delta S_2=-\ln[(I-\Gamma^2)^{-1}_{kk}]$, where $\Gamma$ is the restricted Majorana covariance matrix and $k$ labels the relevant ancilla Majorana. Derived bounds are $-\ln(1-s/2)\leq\Delta S_2\leq\ln(1+s)$, with $s=\sum_j|\Gamma_{kj}|^2$, and a further lower bound replaces $s$ by the maximum squared subsystem magnetization. Ground-state calculations show an EA plateau in the disordered phase, approach to $\ln2$ at criticality, and rapid near-saturation in the ordered phase. Critical scaling is compared qualitatively with related BCFT predictions; the fitted magnetization exponent is $0.100$, versus $1/8$ asymptotically. After boundary quenches, magnetization fronts follow $v_{\max}=2\min(J,h)$, whereas subsystem-integrated EA can be dominated by boundary contributions. A proposed asymmetry contour averages left- and right-subsystem EA finite differences and resolves propagation structures alongside nonlocal contributions. Dynamics use $L=50$; the fermionic contour calculation is checked against exact diagonalization at $L=10$. The contour need not be nonnegative and can become finite ahead of the quasiparticle front. Critical comparisons remain limited by finite size, boundary crossover scales, and differing geometries. A quantitative quasiparticle description and extensions beyond this setting remain open.