Helical Edge States and Quantum Phase Transitions in Tetralayer Graphene

Helical conductors with spin-momentum locking are promising platforms for Majorana fermions. Here we report observation of two topologically distinct phases supporting helical edge states in charge neutral Bernal-stacked tetralayer graphene in Hall bar and Corbino geometries. As the magnetic field $B_{\perp }$ and out-of-plane displacement field $D$ are varied, we observe a phase diagram consisting of an insulating phase and two metallic phases, with 0, 1, and 2 helical edge states, respectively. These phases are accounted for by a theoretical model that relates their conductance to spin-polarization plateaus. Transitions between them arise from a competition among interlayer hopping, electrostatic and exchange interaction energies. Our work highlights the complex competing symmetries and the rich quantum phases in few-layer graphene.

Figure 1

(a) Low energy band structure $E(k_x)$ of 4LG in the absence of external fields. Red and blue lines denote BLG-like bands with light ($m$) and heavy ($M$) effective masses, respectively. (b)–(c) Schematics of Hall bar and Corbino devices, respectively, and propagation of 1 pair of helical edges therein. (d) Landau fan $R_{xx}(n,B_{\perp })$ for a Hall bar device (H1) at $D=0$. The unit is $\text{k}\mathrm{\Omega }$, in logarithmic scale. Blue numbers indicate filling factors. (e) $R_{xx}(n)$ at $D=0$ and $B_{\perp }=29$ and 32 T, respectively.

Figure 2

(a) Electronic phase diagram $R_{xx}(D,B_{\perp })$ at $n=D=0$, different phases are labeled I, II, and III. The unit is $\mathrm{k}\mathrm{\Omega }$. Dotted box indicates the region shown in Figs. 3. (b) $R_{xx}(B_{\perp })$ at $n=D=0$ at selected temperatures. (c)–(d) Temperature dependence of $R_{xx}(D)$ of the $\nu =0$ state, at $B_{\perp }=25$ and 34 T, respectively. Inset in (d): Enlarged plot of $R_{xx}(D)$ at $T=0.3\mathrm{K}$, showing near quantization of phase I and II to $4e^2/h$ and $2e^2/h$, respectively.

Figure 3

(a) Conductance in unit of $e^2/h$ for a Corbino device (C1) as a function of $D$ and $B_{\perp }$, respectively. (b) Conductance contributed by edge states. (c) Phase diagram $R_{xx}(D,B_{\perp })$ measured in tilt magnetic field at an angle $\theta =49°$, in units of $\mathrm{k}\mathrm{\Omega }$. The red dotted curves outline the phase boundaries in Figs. 2 at $\theta =0$ and $B_{\parallel }=0$. (d) $R_{xx}(D,B_{\perp })$ at $n=D=0$ measured at different tilt angles from device H2. Inset: location of the quantum critical point in the plane of perpendicular ($B_{\perp }$) and total ($B_t$) magnetic field from 4 different devices. Solid line is a linear fit to the data.

Figure 4

(a) Spin- and valley-degenerate LL spectrum of 4LG at $D=0$. Red and blue lines denote electron and holelike LLs, respectively. Numbers between the LLs indicate filling factors. LLs are labeled by heavy ($M$) or light ($m$) mass BLG-like band, orbital index, and spin polarization. (b) Similar to (a), except for $25<B_{\perp }<45\mathrm{T}$ and each LL is spin split by a $g$ factor of 2. (c)–(d) LL energy vs interlayer potential $\mathrm{\Delta }$ at $B_{\perp }=25$ and 35 T, respectively. Circles and triangles indicate crossing points of electron and holelike LLs that form the $\nu =0$ state. (e) Single particle phase diagram in $\mathrm{\Delta }\text{−}{B}_{\perp }$ plane, using the same symbols for LL crossing points in (c)–(d). (f) Theoretical phase diagram by taking electronic interactions into account. PP1 and PP2 refer to two distinct ground states with partial spin polarization that support no conducting edges, with spin canting next-layer coherence and nearest-neighbor interlayer coherence, respectively [24].