Description
The entanglement Hamiltonian provides a powerful way of characterizing the structure of a quantum many-body state, but a fundamental question remains: what kind of many-body Hamiltonian does it actually define? I will address this question through two contrasting examples in quantum spin ladders. I will first discuss a non-Hermitian spin ladder in the rung-singlet regime, where the reduced density matrix can be perturbatively mapped onto a thermal state of a renormalized single-chain Hamiltonian. This provides a controlled setting in which the entanglement Hamiltonian admits a simple interpretation in terms of an effective temperature. I will then turn to a Hermitian spin ladder in the Haldane phase, where a very different picture emerges. While conventional bulk diagnostics indicate that the physical system remains within the same gapped Haldane phase, the entanglement sector undergoes a distinct quantum phase transition, exhibiting critical scaling and long-range order. To understand the microscopic origin of this behavior, we reconstruct the entanglement Hamiltonian by projecting it onto few-body operators. We find that long-range two-body interactions alone fail to reproduce key features of the low-energy entanglement spectrum, whereas the inclusion of finite-range four-body interactions recovers its characteristic structure. These results suggest that the physics of an entanglement Hamiltonian can depend crucially not only on interaction range, but also on its many-body operator content. I will conclude by discussing the implications for locality and Lieb–Schultz–Mattis-type constraints on entanglement Hamiltonians, and the broader question of which properties of a physical Hamiltonian must—or need not—be inherited by its entanglement Hamiltonian. Ref: [1]Pei-Yun Yang, YCT, SciPost Phys. Core 7, 074 (2024) [2]YCT, Gunnar Möller, arXiv:2509.03588