Emergent Fine Structure Constant of Quantum Spin Ice Is Large

Condensed-matter systems provide alternative “vacua” exhibiting emergent low-energy properties drastically different from those of the standard model. A case in point is the emergent quantum electrodynamics (QED) in the fractionalized topological magnet known as quantum spin ice, whose magnetic monopoles set it apart from the familiar QED of the world we live in. Here, we show that the two greatly differ in their fine structure constant $\alpha$, which parametrizes how strongly matter couples to light: ${\alpha }_{\mathrm{QSI}}$ is more than an order of magnitude greater than ${\alpha }_{\mathrm{QED}}\approx 1/137$. Furthermore, ${\alpha }_{\mathrm{QSI}}$, the emergent speed of light, and all other parameters of the emergent QED, are tunable by engineering the microscopic Hamiltonian. We find that ${\alpha }_{\mathrm{QSI}}$ can be tuned all the way from zero up to what is believed to be the strongest possible coupling beyond which QED confines. In view of the small size of its constrained Hilbert space, this marks out quantum spin ice as an ideal platform for studying exotic quantum field theories and a target for quantum simulation. The large ${\alpha }_{\mathrm{QSI}}$ implies that experiments probing candidate condensed-matter realizations of quantum spin ice should expect to observe phenomena arising due to strong interactions.

Figure 1

(a) The pyrochlore lattice of quantum spin ice (QSI) is formed from corner sharing tetrahedra with spin $1/2\mathrm{s}$ residing at corners. The spins shown give an example of ice-rule violating tetrahedra that correspond to an electric charge-anticharge pair. (b) The emergent electric charges and photons can interact, just as electrons and photons do in QED, and their interaction strength is given by the emergent fine structure constant ${\alpha }_{\mathrm{QSI}}$. (c) The value of ${\alpha }_{\mathrm{QSI}}$ in the eQED phase of the microscopic QSI Hamiltonian [see Eq. (2)] shown as a function of $\mu$ (with $\zeta =0$) and $\zeta$ (with $\mu =0$). Error bars represent the standard deviation of ${\alpha }_{\mathrm{QSI}}$ among its shape-dependent variations at a fixed $(\mu ,\zeta )$. By varying the 3NN potential, $\alpha$ is tunable up to the maximum value ${\alpha }_c$ (dotted line) beyond which it is conjectured that any compact QED in $3+1\mathrm{D}$ confines [22, 23, 24].

Figure 2

(a) The emergent electric charge $e_{\mathrm{QSI}}$ as a function of RK ($\mu$,$\zeta =0$) and 3NN ($\mu =0$, $\zeta$) potential. A representative scatter plot of these data is shown in the inset (data corresponding to $\mu =\zeta =0$) with associated fit (red line). The dashed lines are fits giving $e_{\mathrm{QSI}}=0.20\sqrt{ag(1-\mu )}$ at $\zeta =0$ and $e_{\mathrm{QSI}}=0.38\sqrt{ag(0.28+\zeta )}$ at $\mu =0$. We note that the former dependence is predicted near the RK point at $\mu =1$ [7, 17], while the latter is a guide to the eye. (b) The emergent speed of light $c_{\mathrm{QSI}}$ as a function of RK and 3NN potential. Representative scatter plot of this dispersion is shown in the inset (at $\mu =\zeta =0$) with associated fit (red line). Dashed lines are fits giving $c_{\mathrm{QSI}}=0.51ag\sqrt{1-\mu }/\hslash$ and $c_{\mathrm{QSI}}=0.78ag\sqrt{0.41+\zeta }/\hslash$ along the $\zeta =0$ and $\mu =0$ axes, respectively. Again, we note that the dependence of $c$ on $\mu$ near the RK point is consistent with previous results [7, 18]. The error bars in both panels represent the standard deviation of $e_{\mathrm{QSI}}$ and $c_{\mathrm{QSI}}$ among its shape-dependent variations at a fixed $(\mu ,\zeta )$. Furthermore, in both insets, scatter points are brighter the denser their neighboring data points are.