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:

enter image description here

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


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.


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!:

enter image description here

enter image description here

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$.

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  • $\begingroup$ Nice answer. I think it would be better if you included a reference for those images, presumably wikipedia? $\endgroup$ – AccidentalFourierTransform Sep 16 '19 at 1:47
  • $\begingroup$ @AccidentalFourierTransform I just googled them, I'll try and find them again. $\endgroup$ – SuperCiocia Sep 16 '19 at 2:01
  • $\begingroup$ @SuperCiocia Thanks for the helpful answer on gauge theory. I have a quick question relating to your answer: it seems the answer is about the lattice gauge field which is commonly assumed to describe spin liquids. Can one define a spin liquid (SL) without the gauge field? Or does a gauge theory guarantee a SL? $\endgroup$ – Histoscienology Jun 6 at 14:00
  • 1
    $\begingroup$ @Histoscienology without a gauge field there are no interactions, so you’d have a high of non-interacting fermions. The quantum spin liquid phase is defined to be a phase formed by interacting quantum spins. So you need some sort of interaction to get a QSL. Interaction —> gauge field. U(1) is just one type. $\endgroup$ – SuperCiocia Jun 6 at 16:30

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