Imaging Anyons with Scanning Tunneling Microscopy

Anyons are exotic quasiparticles with fractional charge that can emerge as fundamental excitations of strongly interacting topological quantum phases of matter. Unlike ordinary fermions and bosons, they may obey non-Abelian statistics—a property that would help realize fault-tolerant quantum computation. Non-Abelian anyons have long been predicted to occur in the fractional quantum Hall (FQH) phases that form in two-dimensional electron gases in the presence of a large magnetic field, such as the $\nu =5/2$ FQH state. However, direct experimental evidence of anyons and tests that can distinguish between Abelian and non-Abelian quantum ground states with such excitations have remained elusive. Here, we propose a new experimental approach to directly visualize the structure of interacting electronic states of FQH states with the STM. Our theoretical calculations show how spectroscopy mapping with the STM near individual impurity defects can be used to image fractional statistics in FQH states, identifying unique signatures in such measurements that can distinguish different proposed ground states. The presence of locally trapped anyons should leave distinct signatures in STM spectroscopic maps, and enables a new approach to directly detect—and perhaps ultimately manipulate—these exotic quasiparticles.

Popular Summary

One of the most exotic consequences of both quantum mechanics and strong correlations is fractionalization, which allows an electron to lower its energy by splitting into several independent quasiparticles. The ability to detect and manipulate these quasiparticles could be used to store and process quantum information in a manner that is robust against many error sources. However, there is no reliable tool for pinpointing the location of these quasiparticles in a material. We propose an experimental approach for visualizing quasiparticles using a scanning tunneling microscope (STM).

A STM uses an atomically sharp tip to inject an electron into a material, allowing it to image the structure of the electronic wave function with atom-scale resolution. In a fractionalized material, the injected electron will split into several quasiparticles, which should leave some signature in STM images. We show, theoretically, that there is a telltale sign in the STM signal whenever the tip lies above a fractionalized quasiparticle.

In the same way that spectroscopy can be used to determine the composition of a molecule, STM spectroscopy can be used to determine the nature of the quasiparticle underneath the microscope tip. This ability could be used to image the location and type of fractionalized quasiparticles in the bulk of a material, a promising step towards their detection and control.

Figure 1

Experimental setup. A metallic gate (bottom) is separated by an insulating barrier of thickness $d_g$ from a 2DEG (top). An impurity of charge $Z$ is located inside the barrier at distance $d_i$ from the 2DEG. Tuning the gate voltage $V_{\mathrm{g}}$ to enter into a quantum Hall state, STM spectroscopy is used to measure the local density of states (LDOS), shown as a density plot. The LDOS will reveal a discrete set of ringlike resonances centered on the impurity, whose radius depends on the energy. By counting the number of resonances of a given radius, it is argued that the location and fractional exclusion statistics of the anyons can be determined.

Figure 2

The simulated occupied $\mathrm{LDOS}(E,\mathbf{r}=x,y)$ for the $\nu =1$ IQHE and $\nu =1/3$ Laughlin phase. The calculations assume a dipolar Coulomb interaction [Eq. (1)] and charged impurity located below the origin [Eq. (2)]. Each image is at a fixed $E$ annotated in the lower left in units of meV at $B=14\mathrm{T}$, and a scale bar indicates ${\ell }_B\sim 7\mathrm{nm}$. Away from the displayed $E$, there is no DOS. The white arcs denote the density maxima $r_m$ of the $m$th single-particle orbital. (a) For $\nu =1$, each orbital lights up once while sweeping through the energy, indicating there is a single hole excitation for each angular momentum $m$. The impurity has charge $e$ and is situated at $d_i=3.1{\ell }_B$ below the 2DEG. (b) For $\nu =1/3$, the $m=2$ ring lights up twice, once at energy $-30.77\mathrm{meV}$, and once again at $-29.78\mathrm{meV}$. This indicates there are multiple many-body excitations associated with removing an electron from orbital $m$, as a result of strong correlations. The system contains $N_{\varphi }=24$ flux quanta with impurity at $d_i=2.7{\ell }_B$.

Figure 3

The orbital-resolved $\mathrm{LDOS}(E,m)$ for various states. In all cases a phenomenological width of $7\mu \mathrm{eV}$ was added to the data. (a) The IQH state. There is one level for each $m$. (b) The $\nu =1/3$ FQH state, with no anyons localized on the impurity. There are now multiple energy levels for each $m$. The counting of the low-lying levels (i.e., the number of excitations at each $m$) is $1,1,2,3,4,5,\dots$. States above the brightest band form a nonuniversal continuum. (c) The $\nu =5/2$ Moore-Read state. The counting of the low-lying levels is $1,2,3,5,\dots$. In order to break particle-hole symmetry, a small component of three-body interaction $V_{3b}=0.1$ was added to the screened Coulomb Hamiltonian. The system contains $N_{\varphi }=25$ flux quanta with impurity at $d_i=2.3{\ell }_B$. (d) Same as (b) but in the presence of a charge-$e/3$ anyon bound to the impurity. The counting of the low-lying levels is now $0,1,1,2,3,\dots$. The system contains $N_{\varphi }=24$ flux quanta with impurity at $d_i=2.8{\ell }_B$.