Fingerprinting fractons with pump-probe spectroscopy

Abstract

We demonstrate how pump-probe techniques enable specific diagnostics of fracton phases of matter by exploring how lineon-planon braiding in the paradigmatic X-cube phase may be probed spectroscopically. Our discussion builds on works explaining how to probe anyonic exchange statistics spectroscopically in traditional spin liquids. However, the extension to fracton phases reveals qualitatively new features coming from the existence of multi-anyon bound states, which alter the long-time asymptotic behavior of the signal. In particular, the signal we examine is sensitive to (i) the existence of nontrivial braiding statistics in three dimensions, (ii) the fact that some of the fractionalized excitations can form bound states, and (iii) that some of the fractionalized excitations are lineonic in nature (i.e., mobile only in one dimension). Thus, one can spectroscopically detect not only the existence of anyonic braiding statistics in fracton phases, but can crisply distinguish it from anyons in traditional (non-fractonic) spin liquids.

Figures (3)

• Figure 1: Different pairings of four adjacent fractons with a single $Z$ operator acting on a $y$-oriented link. (a) Planons separated in the $x$-direction can move in the $xy$-plane. This configuration is labeled by the relative coordinate as ${\boldsymbol{r}}_{xy}=(1,0)$. (b) Planons separated in the $z$-direction can move in the $yz$-plane. This configuration is labeled as ${\boldsymbol{r}}_{yz}=(0,1)$. (a) and (b) correspond to the same underlying wave function, thereby gluing together the $xy$- and $yz$-planes.
• Figure 2: Spacetime diagram of a pump-probe protocol for braiding statistics in the X-cube model. A pair of lineons is created at $t=0$, which move in one dimension along the trajectories ${\boldsymbol{x}}=\pm {\boldsymbol{v}}t + {\boldsymbol{x}}_i$, where ${\boldsymbol{v}} =v\hat{x}$. A pair of planons is created and annihilated at times $t_1$ and $t_1+t_2$, respectively, with trajectories labeled by ${\boldsymbol{r}}_1(t)$ and ${\boldsymbol{r}}_2(t)$ that together form a loop. The red shaded region indicates the two-dimensional area spanned by the initial lineon positions ${\boldsymbol{x}}_i$ for which one of the lineon world-lines intersects the planon world-loop. The braiding probability is therefore proportional to this area, which can be estimated as $A[{\boldsymbol{r}}_{1,2}(t); v] \sim \Delta x_{\parallel} \Delta x_{\perp}$, where $\Delta x_{\parallel}$ and $\Delta x_{\perp}$ denote the widths parallel and perpendicular to the lineon velocity.
• Figure 3: The inverse Green's functions $|g_s(E)|^{-1}$ and $|g_d(E)|^{-1}$ denote the $s$- and $d$- wave channel in the presence of hardcore constraint, while $|g^0_s(E)|^{-1}$ is the $s$-wave Green's function without the hardcore constraint. The edge of the two-planon continuum locates at $E/h_z =-8$. The red curve shows the energy-dependent effective attractive potential $|U(E)|$ obtained by integrating out the extra degrees of freedom. The bound state solution is given as $|U(E)|=|g_\alpha(E)|^{-1}$.