Majorana Landau-level Raman spectroscopy

The unambiguous experimental detection of quantum spin liquids and, in particular, of the long-sought Kitaev quantum spin liquid (KQSL) with its Majorana fermion excitations remains an outstanding challenge. One of the major obstacles is the absence of signatures that definitively characterize this phase. Here, we propose the Landau levels known to form in the Majorana excitation spectrum of the KQSL when certain strain fields are applied as a direct signature of Majorana fermions with Dirac-like dispersion. In particular, we show that the Majorana Landau level quantization of strained films of the KQSL can be directly probed by Raman spectroscopy. Such experiments are feasible in thin films of $\alpha -{\mathrm{RuCl}}_3$, which are a promising place to search for the KQSL.

Figure 1

Schematic plot of a nanobubble of an $\alpha -{\mathrm{RuCl}}_3$ thin film is shown in (a). The resulting lattice distortion induces a strain pattern that acts as a pseudo-orbital magnetic field for the Majorana fermions which emerge from spin fractionalization in the QSL of the Kitaev model. The resulting LL quantization can be directly probed by Raman scattering. The evolution of the response for finite ($s=0.04$) and zero strain ($s=0$) is shown in (b).

Figure 2

The DOS and all possible Raman correlation functions for finite strain ($s=0.04$) (solid lines) and zero strain ($s=0$) (dotted lines) with $r=50$ corresponding to $6*{(50+1)}^2=15606$ sites. DOS (a) shows peaks at $\sqrt{n}{\omega }_c$ (gray vertical lines) while $I_{\text{nonres}}$ (b) features them at $\left(\sqrt{n}+\sqrt{n+1}\right){\omega }_c$ and $I_{\text{res}}$ (c) at $2\sqrt{n}{\omega }_c$. $I_{\text{res}}$ is the resonant channel, which is antisymmetric in polarization, as discussed in Ref. [33].

Figure 3

A set of density plots showing the progression of Landau level peaks as a function of (a) strain $s$, (b) system size $r$, and (c) temperature $T$. Where not specified, we used the parameters $s=0.04$ and $T=0$ with the system sizes (a) $r=30$ and (c) $r=15$. The temperature of the crossover in (c) agrees well with the expected spinon-confinement transition for noninteracting fluxes at $T^{*}\approx 0.02J$ ($s=0.04, r=15$).

Figure 4

The nature of the crossover out of the KQSL phase is demonstrated for (a) an unstrained flake and (b) a strained flake with $s=0.04$. For uniform noninteracting fluxes the flux density in (1) is expected to follow a Fermi-Dirac distribution at energy $(ln2)\mathrm{\Delta }$, where $\mathrm{\Delta }$ is the flux gap [39]. (2) shows the flux density resolved over the radial coordinate as a function of temperature. (3) plots the single-flux gap as a function of position on the honeycomb flake [2]. (4) give the expectation value of the flux operator as a function of position for one particular run at temperatures $T=0.057J$ in (a)(4) and $T=0.020J$ in (b)(4), both corresponding to the critical flux density of the strained flake ${\rho }_c\approx 0.12$. Although there is significant noise in (4) we see that the flux distribution is largely random without strain (a)(4) and it becomes focused away from the center under stain. This effect is responsible for the appreciable critical flux density ${\rho }_c\approx 0.12$ for Landau levels in the strained flake, for which Landau orbits can form within the “clean” center region.

Figure 5

Plots of unstrained Raman and DOS to compare with Fig. 3(a) of Ref. [27] and Fig. 3(a) of Ref. [31], respectively. The comparisons appear quite favorable, although the present study did not involve the superlattice treatment that allowed Ref. [31] to suppress the finite-size effects that are large in our plot of the DOS. The system sizes used were $r=11$ and $r=7$, chosen to most closely mimic the ones used in those references.