Dissociation of bulk and entanglement phase transitions in the Haldane phase

We revisit the momentum-resolved entanglement spectrum (ES) of the spin-1/2 ladder in the Haldane phase, long believed to exhibit a des Cloizeaux-Pearson (dCP)-type $\sin|k|$ dispersion. Using exact diagonalization up to 40 spins, we resolve two distinct branches at $k=0$ and $k=π$, which were previously interpreted as a single smooth mode due to SU(2) degeneracy and limited resolution. Breaking SU(2) symmetry via XXZ anisotropy opens a spin or neutral gap at $k=π$, depending on the anisotropy direction, triggering an entanglement quantum phase transition that is disconnected from the bulk critical point. In the easy-plane regime, the entanglement ground state is unique in the $S_A^z=0$ sector but becomes quasi-degenerate across $S_A^z$ sectors in the thermodynamic limit, consistent with spontaneous U(1) symmetry breaking. This separation between bulk and entanglement transitions demonstrates that the Li-Haldane correspondence can fail in topological ladders, even at the level of low-energy dispersions. These results revise the prevailing picture of ES in spin ladders under extensive bipartitioning, and suggest that the entanglement Hamiltonian, being nonlocal, can circumvent conventional constraints such as the Lieb-Schultz-Mattis and Mermin-Wagner theorems.