Spin Superfluidity in the $\nu =0$ Quantum Hall State of Graphene

Strong electron interactions can lead to a variety of broken-symmetry phases in monolayer graphene. In the quantum Hall regime, the interaction effect are enhanced by the formation of highly degenerate Landau levels, catalyzing the emergence of such phases. Recent magnetotransport studies show evidence that the $\nu =0$ quantum Hall state of graphene is in an insulating canted antiferromagnetic phase with the Néel vector lying within the graphene plane. Here, we show that this Néel order can be detected via two-terminal spin transport. We find that a dynamic and inhomogeneous texture of the Néel vector can mediate nearly dissipationless (superfluid) transport of spin angular momentum polarized along the $z$ axis, which could serve as a strong support for the antiferromagnetic scenario. The injection and detection of spin current in the $\nu =0$ region can be achieved using the two spin-polarized edge channels of the $|\nu |=2$ quantum Hall state. Measurements of the dependence of the spin current on the length of the $\nu =0$ region would provide direct evidence for spin superfluidity.

Figure 1

Proposed setup for realizing and detecting superfluid spin transport through the $\nu =0$ QH state of graphene. (a) Top view of the graphene Hall bar. The yellow regions are top gates and the gray regions denote Ohmic contacts held at their respective voltages. Two independently biased spin-polarized edge channels on opposite sides of the $\nu =0$ region are used to inject and detect spin current flowing through the CAF. The spin states of the $\nu =-2$ edge channels are polarized collinearly to the $z$ axis outside of the injection and detection regions. (b) A cartoon energy diagram at a $\nu =0$ to $\nu =-2$ transition region (across the bold red line). The spin axes are viewed from the side along the $y$ direction. In the $\nu =-2$ region, the energies of the two spin states, oppositely polarized along the $z$ axis, are drawn; the Zeeman effect gives an energetic advantage to the spin-down state. In the $\nu =0$ region, the two occupied branches of the CAF spectrum are shown. There, an external field in the positive $z$ direction results in a ferromagnetic canting of spins in the negative $z$ direction inside the antiferromagnet. Spin orientations of the chiral edge modes are intermediate between the up and down spin eigenstates within the $\nu =-2$ region (left side) and the canted spins within the CAF (right side). The black lines are merely a rough guide for the energies of the spin states in the transition region. The above illustration does not contain two other branches of the spectrum that are a part of the zeroth Landau level but not essential for the edge physics in the transition region.

Figure 2

A cartoon of the CAF in a dynamic superfluid state. The Néel vector rotates within the graphene plane about the $z$ axis with a global precession frequency $\mathrm{\Omega }$. The static contribution to spin current ${\overline{I}}_s\propto V_{-}=V_{\uparrow }-V_{\downarrow }$ is injected into the CAF while the dynamic (spin-pumping) contribution $i_s\propto \mathrm{\Omega }$ pumps spin current back out into the edge. Two analogous contributions exist on the detection side.

Figure 3

The injection region. Charge currents entering vertices $a$ and $b$ redistribute according to the scattering probability matrix $\widehat{S}$. The interchannel scattering inside the line junction is quantified by an effective conductance $g(y)$ per unit length.

Figure 4

Effective spin conductance $G_{\mathrm{eff}}^s\equiv I_s^{\prime }/e{V}_{-}$ as a function of the aspect ratio $w=W/\ell$. The solid, dashed, and dotted curves are, respectively, for full ($t=0.5$), partial ($t=0.75$), and no ($t=1$) interchannel mixing at the vertices.