Dissociation of bulk and entanglement phase transitions in the Haldane phase

Abstract

Quantum entanglement provides a sensitive probe of topological phases and strong correlations in quantum many-body systems. We revisit the momentum-resolved entanglement spectrum (ES) of the spin-$\frac12$ XXZ ladder in the Haldane phase, whose SU(2)-symmetric spectrum has for over fifteen years been interpreted as a single des Cloizeaux--Pearson mode. Using momentum-resolved entanglement spectroscopy based on exact diagonalization of ladders with up to 40 spins, we instead resolve two distinct modes crossing at $k=\pi/2$ and forming a sharp cusp rather than a single $\sin|k|$ branch, revising the SU(2)-symmetric ES. We then introduce explicit SU(2) symmetry breaking through an XXZ anisotropy. The ES undergoes a phase transition at the isotropic point, while the physical ladder remains in the Haldane phase until a distinct bulk transition at larger anisotropy, establishing a dissociation between bulk and entanglement phase transitions. Between the entanglement and bulk critical points, no gapless entanglement branch remains that can be interpreted as a Haldane-edge excitation, signalling breakdown of the Li--Haldane correspondence. In the easy-plane regime, the ES develops an Anderson tower of states, and the entanglement ground state exhibits long-ranged spin correlations on the finite chains studied, consistent with emergent U(1)-broken order and incompatible with conventional short-range one-dimensional Hamiltonians satisfying the assumptions underlying the Lieb--Schultz--Mattis and Mermin--Wagner--Hohenberg--Coleman theorems. Together, these results establish entanglement-only quantum criticality as a distinct manifestation of critical behaviour encoded in the entanglement description.