# Rotor representation of creation and annihilation operators?

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I read the paper "Edge–Entanglement correspondence for gapped topological phases with symmetry"

Where a lattice $U(1)$ model is introduced with bosons both on the sites and links of the lattice. The site boson are created by $b_s^{\dagger}$ and the paper uses the "rotor representation" where $b_s^{\dagger}=e^{i\theta_s}$ and $[\theta_s,n_s]=i$. The link bosons $b_{ss'}$  are hard-core bosons whose number is $0$ or $1$.

I have some questions about it:

1. It is implied that $b_s=e^{-i\theta_s}$ so $$b_s^{\dagger}b_s=b_sb_s^{\dagger}=1=n_s$$ This is weird since the number of bosons can be any integer. Also, the creation and annihilation operators should not commute.
2. How is $\theta_s$ defined mathematically? Is it a number? is it an Hermitian operator? Does it have only integer eigenvalues?
3. How does the bosons transform under the $U(1)$ phase symmetry? If $b_s^{\dagger}\rightarrow e^{i\alpha_s}b_s^{\dagger}$ It would changer the number of bosons, possibly to a non-integer number!
4. The Hamiltonian has a term $Q_s=2n_s+\sum_{s'}n_{ss'}$, its states with $q_s$  are said to have fraction $\frac{1}{2}$ $U(1)$  charge, How is this reflected in the $U(1)$
transformation of the site bosons?

asked Jan 5

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