SU(4) spin waves in the $\nu =\pm 1$ quantum Hall ferromagnet in graphene

We study generalized spin waves in graphene under a strong magnetic field when the Landau-level filling factor is $\nu =\pm 1$. In this case, the ground state is a particular SU(4) quantum Hall ferromagnet, in which not only the physical spin is fully polarized but also the pseudospin associated with the valley degree of freedom. The nature of the ground state and the spin-valley polarization depend on explicit symmetry-breaking terms that are also reflected in the generalized spin-wave spectrum. In addition to pure spin waves, one encounters valley-pseudospin waves as well as more exotic entanglement waves that have a mixed spin-valley character. Most saliently, the SU(4) symmetry-breaking terms not only yield gaps in the spectra, but also under certain circumstances, namely, in the case of residual ground-state symmetries, render the originally quadratic (in the wave vector) spin-wave dispersion linear.

Figure 1

(a) Phase diagram of the QHFM ground state composed of four phases: charge density wave (CDW), Kekulé distortion (KD), antiferrimagnetic (AFI), and canted antiferromagnetic (CAF). (b)–(e) Spin magnetization on the A and B sublattices of the different phases: (b) CDW, (c) KD, (d) AFI, and (e) CAF.

Figure 2

Phase diagram of the QHFM ground state with the valley Zeeman term ${\mathrm{\Delta }}_P$ such that ${\mathrm{\Delta }}_P={\mathrm{\Delta }}_Z$. The KD and CAF phases are modified compared to the case without the valley Zeeman term and are turned to a canted KD phase (CDW) and a different CAF phase (${\mathrm{CAF}}^{\prime }$).

Figure 3

Four sub-LLs of the $n=0$ LL and the three associated spin wave modes corresponding to the mixing of the filled sub-LL described by the spinor $|F\rangle$ with each of the three empty sub-LLs described by the spinors $|C_i\rangle$.

Figure 4

Dispersion relation of the three modes in the CDW phase for $u_z=-{\mathrm{\Delta }}_Z$ and $u_{\perp }=2{\mathrm{\Delta }}_Z$. The three modes are gapped and quadratically dispersing.

Figure 5

Bloch spheres corresponding to each modes in the CDW phase. (a) Spin Bloch sphere of the pure spin mode 1. (b) Pseudospin Bloch sphere of the pseudospin mode 2. (c) Entanglement Bloch sphere corresponding to the entanglement mode 3. The black arrow indicates the ground-state polarization, while the red arrow corresponds to the magnetization at a point in space in the presence of a spin wave. The red arrow rotates periodically around the ground-state polarization according to Eq. (88).

Figure 6

Three modes of the CDW phase. (a) “Snapshot” of the pure spin wave mode $a=1$ seen from the top with wave vector $\mathbf{k}$ along the axis $y$. We observe the precession of the spins around the $z$ axis of the spins in the A sublattice. (b) Sublattice polarization of the pseudospin wave mode $a=2$. We observe a small electronic density on sublattice B. The dynamic part of the field is encoded in the relative phase of the superposition between valleys $K$ and $K^{\prime }$. The spin magnetization is proportional to the sublattice density and points along ${\mathbf{s}}_z$. (c) Spin magnetization on the A and B sublattices of the entanglement mode $a=3$; there is a small spin magnetization on the B sublattice with opposite direction as on sublattice A.

Figure 7

Size of the gap of (a) the pseudospin and (b) the entanglement waves as a function of $u_{\perp }$ and $u_z$ in the CDW region. We observe that the pseudospin gap ${\mathrm{\Delta }}_2$ vanishes at the boundary with the KD phase, and the entanglement gap ${\mathrm{\Delta }}_3$ vanishes at the boundary with the AFI entangled phase. (c) “Phase diagram” of the spin mode with the lowest gap. We can see that the pseudospin and entanglement modes have the lowest energy near the phase boundaries, whereas the spin mode dominates elsewhere.

Figure 8

Bloch spheres corresponding to each modes in the KD phase in the same way as Fig. 5. (a) Spin Bloch sphere of the spin mode 1. (b) Pseudospin Bloch sphere of the pseudospin mode 2. The ground state has a U(1) symmetry for rotations around the $z$ axis. (c) Entanglement Bloch sphere corresponding to the entanglement mode 3. For the spin and entanglement modes, the red arrow rotates periodically around the ground-state polarization according to Eq. (88). At low energy, the pseudospin mode is restricted to the equator of the pseudospin Bloch sphere, which costs no anisotropic energy, while at higher energy, it acquires an element along the $z$ direction.

Figure 9

Dispersion relation of the three modes in the KD phase for $u_z=2{\mathrm{\Delta }}_Z$ and $u_{\perp }=-2{\mathrm{\Delta }}_Z$. We observe that the pseudospin mode is gapless, linear at low momentum $\left|\mathbf{k}\right|\ll k_0$ and becomes quadratic at higher momentum. The two other modes are gapped and quadratic.

Figure 10

Dispersion relation of the three modes in the AFI phase for $u_z=2{\mathrm{\Delta }}_Z$ and $u_{\perp }=6{\mathrm{\Delta }}_Z$. The modes $a=1$ and $a=2$ are coupled and form the $\alpha =1$ and $\alpha =2$, which are quadratically dispersing, while the entanglement mode is linear and gapless.

Figure 11

Entanglement Bloch spheres corresponding to the entanglement mode in (a) the AFI phase and (b) the CAF phase. The spinor $|F\rangle$ indicated by the black arrow corresponds to the ground state, while the spinor $|C_3\rangle$ is located at opposite direction of the Bloch sphere. The ground states possess a U(1) symmetry associated with the angle $\beta$ corresponding to the latitude indicated by the circle at the tip of the black and red arrows. At low energy, the entanglement wave corresponds to a small deviation at equilatitude indicated by the red arrow.

Figure 12

Dispersion relation of the three modes in the CAF region for $u_z=12{\mathrm{\Delta }}_Z$ and $u_{\perp }=2{\mathrm{\Delta }}_Z$. We observe two gapless modes: the entanglement mode and the mode $\alpha =2$, which originates from the gapless mode of the KD region.

Figure 13

Summary of the low-energy dispersion relation in the four phases. The indices 1, 2, and 3 refer to the spin, pseudospin, and entanglement modes, respectively, except in the CAF and AFI regions, where the spin and pseudospin modes are coupled. In the schematic expression of the dispersion relations, we have set ${\rho }_s/{\rho }_0=2\pi {\rho }_s{l}_B^2\equiv 1$. In the CDW region, the three modes are gapped. In the KD region, there are two gapped modes and one gapless linear mode, the pseudospin mode. In the AFI region, the entanglement mode is gapless, while the two other modes are gapped. Finally, in the CAF region, there are two gapless modes, the entanglement and the coupled mode $\alpha =2$, the descendant of the pseudospin mode.