Hamiltonian symmetries of quantum spin systems
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One of the basic questions in physics is to ask what symmetries are present in a system and what does this tell us about the systems observables?
In this note, I review Pauli operator algebra and how using such relations can be used to check/find symmetries of quantum spin Hamiltonians.
A given Hamiltonian \(H\) has a symmetry if it commutes with some symmetry transformation \(S\). We can start with a simple system of 3 spins described by the following Hamiltonian:
\[H = -h \sum_{i \in {1, 2, 3}} \sigma_x + -J \sum_{\langle ij\rangle} \sigma_z^{(i)} \sigma_z^{(j)}\]We can show that this Hamiltonian is symmetric under the transformation: $\hat{P}_x = \sigma_x^{(1)} \sigma_x^{(2)} \sigma_x^{(3)}$, which flips the spin of all 3 sites.
Note: \(\sigma_x\) is a spin flip because: \(\sigma_x = \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\) thus \(\sigma_x |\uparrow \rangle = |\downarrow \rangle\) and \(\sigma_x |\downarrow \rangle = |\uparrow \rangle\)
Let’s first consider \(P_x\) for a single site. We want to show that \([P_x, H] = 0\).
Useful references: