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arXiv:2608.14296 · Chinese Physics C

Entanglement spectrum ordering and flavor polarization in the two-flavor Schwinger model at vacuum angle $θ= π$

Boliang Yu, Ruixin Zhou, Meisen Gao

hep-latcond-mat.stat-mechquant-phSubmitted Aug 14, 2026• Announced Aug 17, 2026• 0 citations
v1 · Aug 14, 2026v2 · Aug 17, 2026
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Abstract

Entanglement spectra in gauge theories can encode both symmetry breaking and the organization of gauge sectors. In the two-flavor Schwinger model at vacuum angle $θ=π$, we find that the joint Schmidt distribution $p_{Q_A,F_A}$, labeled by the subsystem gauge charge $Q_A$ and flavor imbalance $F_A$, reveals a cut-dependent gauge-sector hierarchy and a mass-induced flavor asymmetry: changing the staggered cut reorganizes the gauge-charge distribution $p_{Q_A}$, while mass imbalance breaks the $F_A\leftrightarrow-F_A$ symmetry of the conditional flavor distribution $p_{F_A|Q_A}$. Defining the combined weight $W_F^{(q)}=p_{q,+1}+p_{q,-1}$ and conditional polarization $\mathcal P_F^{(q)}=(p_{q,+1}-p_{q,-1})/W_F^{(q)}$, we find that changing the cut reverses the weight hierarchy, $W_F^{(-1)}>W_F^{(+1)}$ at unit-cell boundaries but $W_F^{(+1)}>W_F^{(-1)}$ at intra-cell cuts, while mass imbalance drives $\mathcal P_F^{(q)}$ away from zero with a $q$-dependent cut response. Symmetry resolution therefore separates gauge-sector ordering, sector weight, and flavor polarization that are mixed in the globally ordered entanglement spectrum.

Comments
21 pages, 6 figures. Submitted to $\textit{Chinese Physics C}$

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The entanglement asymmetry has emerged in recent years as a practical quantity to study phases of matter. We present the first study of entanglement asymmetry in gauge theories by considering the chiral anomaly of the analytically solvable massless Schwinger model at both zero and finite temperatures. At zero temperature, we find the asymmetry exhibits logarithmic growth with system size. At finite temperature, we show that it is parametrically more sensitive to chiral symmetry-breaking than the corresponding local order parameter: while the chiral condensate decays exponentially, the asymmetry decreases only logarithmically. This establishes the entanglement asymmetry as a promising tool to probe (finite-temperature) phase transitions in gauge theories.
We compute the entanglement entropy and the entanglement spectrum of the vacuum state in the massive Schwinger model at a finite θθθ angle. By computing two-point correlation functions we show that the correlation length diverges in the critical region. We clarify the physical origin of entanglement entropy enhancement at θ=πθ=πθ=π by relating it to the competition between distinct electric-flux vacuum branches. We show that the peak near θ=πθ=πθ=π persists across the range of masses studied and corresponds to the point of maximal competition between distinct vacuum branches with opposite electric-field orientation, where quantum fluctuations due to fermion pair creation are maximized. While this entropy enhancement is generic, a pronounced compression of the low-lying entanglement spectrum at θ=πθ=πθ=π distinguishes the near-critical case (m/g≃0.33m/g\simeq0.33m/g≃0.33). Using the Bisognano--Wichmann (BW) theorem, we construct a lattice BW entanglement Hamiltonian and compare it with the exact modular Hamiltonian obtained from the reduced density matrix. We observe agreement in the infrared sector, indicating that the entanglement Hamiltonian is well approximated by a spatially weighted Hamiltonian. Finally, we discuss possible applications to entanglement in topological insulators and quantum wires. In this work, we used the chirally rotated formulation of the Schwinger model Hamiltonian. We identify an additive vacuum-energy artifact induced by the discretization of the chirally rotated lattice and derive its exact free-lattice form to subtract it in the continuum limit.
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