# What are $U(n)$ or $\mathbb{Z}_2$ quantum spin liquids?

Quantum spin liquid is a state of matter in which spins are correlated and fluctuate even at zero temperature.

My question is about these terms in general. When we say that a state or a quasi-particle is $$U(n)$$, $$SU(n)$$ or $$\mathbb{Z}_2$$, what do we physically mean?

They specify which gauge symmetry the Quantum Spin Liquids is subject to, when treated via a Lattice Gauge Theory. The gauge field (some sort of interaction) is defined over a discretised space(-time).

For example imagine atoms are situated on the vertices on this square lattice:

where the lines between vertices are links, and the area defined by $$4$$ links is a plaquette.

# --- $$U(1)$$ ---

Gauge variables are unphysical and not directly observable.
Let's start with a $$U(1)$$ gauge theory, corresponding to usual electromagnetism.

The gauge fields live on links like $$nm$$. Physical (observable) variables are electric fluxes on links $$E_{nm}$$ and magnetic fluxes through plaquettes $$\Phi_{mnpq}$$: $$\mathbf{E} = -\frac{\partial \mathbf{A}}{\partial t} \quad\Rightarrow \quad E_{mn} = -\dot{A}_{mn},$$ $$\Phi = \int \mathbf{B}\cdot \mathrm{d}^2\mathbf{r} = \oint_{\mathrm{plaquette}} \mathbf{A}\cdot \mathrm{d}\mathbf{r} \quad \Rightarrow \quad \Phi_{mnpq} = A_{mn}+A_{nq}+A_{pq}+A_{qn}.$$

$$E$$ is quantised, because it is related to electric charges. In some units, then, $$E = 0, \pm 1, \pm 2, \dots$$.

Normally $$A$$ can take any value from $$-\infty$$ to $$\infty$$. Usually, however, one considers a compact U(1) gauge theory by requiring $$0 \leq A \leq 2\pi, \quad V(A + 2\pi) = V(A) \Rightarrow V(A) = f(e^{\mathrm{i}A}),$$ where $$V$$ would be the potential energy in the Hamiltonian $$H$$: $$H = \sum_{\mathrm{links}} \frac{E^2_{mn}}{2I} - \sum_{\mathrm{plaquette}} \lambda \cos \Phi_{mnpq}.$$

The $$E^2$$ term makes sense as it is the energy density of the electric field. The $$\cos$$ is introduced to preserve the compactness and periodicity of the magnetic potential $$A$$. For small fluxes $$\Phi$$, this reduces to $$\propto 1 + B^2$$ which, again, makes sense as it is the energy density of the magnetic field.

# --- $$\mathbb{Z}_2$$ ---

For $$\mathbb{Z}_2$$, you switch from integer arithmetics $$(\mathbb{Z})$$ to binary arithmetics $$(\mathbb{Z}_2)$$ for the electric flux along the links. They can be either "up" or "down", and not taking any value $$\in \mathbb{R}$$ between $$0$$ and $$2\pi$$.

The simplest way to try and do this is to exponentiate all your old variables $$E$$, $$A$$ and $$\Phi$$:

$$\begin{array}{|c|c|} \hline \mathrm{(compact)}\, U(1) & \mathbb{Z}_2 \\ \hline E = 0, \pm 1, \pm 2, \dots & (-1)^E \leftrightarrow \sigma^x = \pm 1 \\ 0 \leq A \leq 2\pi & e^{\pm \mathrm{i}A} \leftrightarrow \sigma^z = \pm 1 \\ \Phi = \sum_{\mathrm{plaquette}}A & e^{\mathrm{i}\Phi} \leftrightarrow \Pi_{\mathrm{plaquettes}} \sigma^z = \pm 1 \\ H = \sum_{\mathrm{links}} \frac{E^2_{mn}}{2I} - \lambda\sum_{\mathrm{plaquette}} \cos \Phi_{mnpq} & H = -\Gamma\sum_{\mathrm{links}}\sigma^x_{mn} - \lambda \sum_{\mathrm{plaquettes}} \sigma^z_{mn}\dots \sigma^z_{qm}\\ \hline \end{array}$$

An example of a $$\mathbb{Z}_2$$ theory, with $$\sigma_x$$ and $$\sigma_z$$, is the Heisenberg model. This can employed, for instance, to model the Kagome lattice -- which is one of those expected to display quantum spin liquids, and which hopefully someone will soon investigate experimentally!:

EDIT: reference for figures: here and here.

# --- $$U(N)$$, $$SU(N$$) etc. ---

If you want to have $$SU(2)$$, instead of $$U(1)$$, you need a state which has $$2$$ internal degrees of freedom (internal states). So you don't just have a scalar wavefunction $$\psi \rightarrow \underbrace{e^\phi}_{\in U(1)}\psi$$, but a multi-component one $$\psi = \left ( \begin{array}{c} \psi_1\\ \psi_2 \end{array} \right ) \rightarrow \underbrace{\left ( \begin{array}{cc} a & b \\ c & d \end{array} \right )}_{\in SU(2)} \left ( \begin{array}{c} \psi_1\\ \psi_2 \end{array} \right )$$, and so on for $$N \geq 2$$.

• Nice answer. I think it would be better if you included a reference for those images, presumably wikipedia? – AccidentalFourierTransform Sep 16 at 1:47
• @AccidentalFourierTransform I just googled them, I'll try and find them again. – SuperCiocia Sep 16 at 2:01