The Resurrection of LSM Theorem in Open Quantum Systems

You know how the LSM theorem works in closed systems, right? It's like a rule that says certain spin chains can't have a single, non-degenerate ground state. But what happens when these systems aren't closed anymore? They interact with their environment, and things get a bit messy. Imagine you're looking at an open quantum system. It's like a party where the guests (spin chains) are dancing with the environment (bath). The LSM theorem seems to lose its power here because energy isn't conserved anymore. But, hey, we found something interesting! When the system starts to interact with the environment and becomes short-range correlated, the LSM theorem can make a comeback. This time, it's in the entanglement Hamiltonian. You see, the entanglement spectrum can't have just one minimum state, isolated by a gap. We've done some math and found that the topological constraints and something called UV data are crucial in shaping entanglement in these open systems. We've even checked this with some numerical examples. We took a spin-