Spontaneous Symmetry Breaking in Chiral Current-Carrying High-Energy Eigenstates

Typical finite-energy-density eigenstates of nonintegrable systems are expected to obey the eigenstate thermalization hypothesis and to reproduce thermal local observables. In integrable systems, typical states are instead finite-entropy Bethe macrostates, or generalized Gibbs ensembles, with smooth quasiparticle occupations. Here we show that the spin-$\tfrac12$ XXX Heisenberg chain contains an exactly solvable exception. By biasing mutually commuting conserved charges, we use the ground state of a selector Hamiltonian to construct rare eigenstates of the undeformed XXX Hamiltonian. These states are atypical ordered, chiral, current-carrying, critical, zero-entropy Bethe macrostates at tunable XXX energy density. They lie outside the finite-entropy manifold dominating generalized Gibbs ensembles, yet remain exact XXX eigenstates with sharp Bethe occupations, finite scalar chirality, nonthermal local observables, and gapless Luttinger-liquid correlations. A finite-interval thermodynamic Bethe ansatz reveals an asymmetric chiral Bethe sea whose zero-field limit selects an extensive spin sector through $SU(2)$ symmetry breaking. A commuting exchange bias moves the same ordered macrostate through the full thermodynamic XXX energy band, including the Hilbert-space trace center. Finite-size DMRG and exact-diagonalization benchmarks indicate that the magnetochiral signatures persist under weak integrability breaking over prethermal time scales. The construction provides a controlled integrable realization of ETH-violating, scar-like large-deviation eigenstates and a route to ordered critical matter deep inside a many-body spectrum.