# Quantum Monte Carlo Loop Algorithm for quantum spin: why is the freezing graph present in ferromagnetic Ising model?

I study the loop algorithm (Evertz et al). I cannot understand, why the freezing graph type where we have to flip all 4 spins together is not present for the quantum-XY model and the anti-/ferromagnetic quantum Heisenberg model but for the Ising Model?

Edit: dash "-" denotes a graph that is not possible for a configuration because of spin conservation and has to be zero. some references: http://edu.itp.phys.ethz.ch/fs11/cqp/script05.pdf https://arxiv.org/abs/cond-mat/9711022

• isn't the notation here $-$ for not allowed and $0$ for allowed but not present? It might be allowed, just not needed to arrive at the total weight – user2723984 Jul 31 at 18:18
• You are right. Thank you. I changed it. But why is it not present? Actually I cannot made the link between the Hamiltonian of these models and the table. How can I see, what kind of graphs can be assigned to a plaquette configuration? – Suslik Jul 31 at 18:26
• IIRC a graph is allowed if flipping the spins in the a cluster does not violate conservation laws of the Hamiltonian in the time direction, more explicitely the time evolution at first order is equal to the Hamiltonian, so for example because of the term $S^+S^-+S^-S^+$ the XXZ model is allowed to flip a spin from one timestep to the other if it also flips the neighbor, and they are opposite. If it's allowed, whether you put it in or not only depends on arriving at the correct weight, I don't know if there's a general rule. But I'm not an expert on this stuff. – user2723984 Jul 31 at 18:50
• Thank you, this was a helpful hint. – Suslik Jul 31 at 19:37