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I'll state one version of the theorem, valid for classical systems. I'll not give the most general framework, as things become messy, but this should still give you an idea of how general the result is. We need the following ingredients: Spins: to each vertex of the lattice $\mathbb{Z}^2$, we attach a spin $\phi_x$ taking values in some compact ...

3

Unfortunately, the global spin rotation symmetry is not essential for spin liquid in the general sense. The defining property of spin liquid is the intrinsic topological order. The Kitaev spin liquid possesses the $\mathbb{Z}_2$ topological order, which makes it a spin liquid, although the spin rotation symmetry is explicitly broken even on the model level. ...

3

I think I got the answer now. The main idea is this: When we gauge continuous symmetries we identify all the states $$A^\mu=A^\mu+\partial^\mu\chi$$ (which are continuously many) as a unique physical state. When we gauge a discrete symmetry (let's assume it's generated by $\theta$) we identify all the states $$|\Psi\rangle=\theta^n|\Psi\rangle$$ where ...

1

Focus on the integral $$I_{ij}(k) = \int k_i k_j\ \mathrm{d}\Omega_k.$$ This is a rank 2 symmetric tensor which can only depend on $\vec{k}$ through its magnitude $k^2$, since the direction has been integrated over. So the only possibility is that $I_{ij}$ is proportional to the unit tensor (Kronecker delta): $$I_{ij}(k) = f(k^2) \delta_{ij},$$ where ...

1

Let there be given a physical system. What charges are Noetherian and what charges are topological often depend on the precise action formulation and field content of the physical system, see also this Phys.SE post. To simplify the discussion, let us assume that the action formulation is fixed, and below definitions will then refer to this fixed ...

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The metric is spherically symmetric. This means that angular momentum of the system is conserved (you can show this directly using the metric by computing the three killing vectors associated with spatial rotation and their corresponding conserved quantities) and therefore that the motion is contained to lie in a plane. If the motion is in a given plane, ...

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