Symmetry characterization of the collective modes of the phase diagram of the $\nu =0$ quantum Hall state in graphene: Mean-field phase diagram and spontaneously broken symmetries

We devote this work to the study of the mean-field phase diagram of the $\nu =0$ quantum Hall state in bilayer graphene and the computation of the corresponding neutral collective modes, extending the results of recent works in the literature. Specifically, we provide a detailed classification of the complete orbital-valley-spin structure of the collective modes and show that phase transitions are characterized by singlet modes in orbital pseudospin, which are independent of the Coulomb strength and suffer strong many-body corrections from short-range interactions at low momentum. We describe the symmetry breaking mechanism for phase transitions in terms of the valley-spin structure of the Goldstone modes. For the remaining phase boundaries, we prove that the associated exact $\text{SO}\left(5\right)$ symmetry existing at zero Zeeman energy and interlayer voltage survives as a weaker mean-field symmetry of the Hartree-Fock equations. We extend the previous results for bilayer graphene to the monolayer scenario. Finally, we show that taking into account Landau level mixing through screening does not modify the physical picture explained above.

Figure 1

Phase diagram of the $\nu =0$ QH state. (Left) Phase diagram in the parameter space $u_{\perp },u_z$. The point $V$ is the intersection of all phase boundaries. (Right) Expected phase diagram for fixed $u_{\perp },u_z$, with $u_z>-u_{\perp }>0$, as a function of the Zeeman and voltage energies, ${\epsilon }_Z,{\epsilon }_V$.

Figure 2

Dispersion relation of the collective modes for the different phases of the $\nu =0$ QH state in bilayer graphene. We take the values $F_C=0.5\hslash {\omega }_B, u_z=0.1\hslash {\omega }_B$, and $u_{\perp }=-0.05\hslash {\omega }_B$ for the Coulomb and short-range energies, respectively. We remark that, when several branches are represented in the same plot, the modes with $N=1,2,\text{and}3$ are so close to each other that is difficult to distinguish them. (Upper left corner) Dispersion relations of the F phase with ${\epsilon }_V=0.02\hslash {\omega }_B$ and ${\epsilon }_Z=0.2\hslash {\omega }_B$, with the left plot devoted to the valley-triplet mode ${\omega }_N^{11}\left(k\right)$ and the right plot devoted to the modes with $L_z=0, {\omega }_N^{\mathrm{L}0}\left(k\right), L=0,1$ being depicted in solid (dashed) line. (Upper right corner) Dispersion relations of the FLP phase with ${\epsilon }_V=0.2\hslash {\omega }_B$ and ${\epsilon }_Z=0.02\hslash {\omega }_B$. The left plot displays the spin-triplet mode ${\omega }_N^{11}\left(k\right)$ and right plot the spin-singlet mode ${\omega }_N^{00}\left(k\right)$. (Lower left corner) Dispersion relations of the CAF phase with ${\epsilon }_V=0.02\hslash {\omega }_B$ and ${\epsilon }_Z=0.02\hslash {\omega }_B$. Left plot corresponds to the valley-triplet mode ${\omega }_N^{11}\left(k\right)$ and right plot to the modes with $L_z=0$, the $+(-)$ branches being depicted in solid (dashed) line. (Lower right corner) Dispersion relations of the PLP phase with ${\epsilon }_V=0.1\hslash {\omega }_B$ and ${\epsilon }_Z=0.02\hslash {\omega }_B$. Left plot corresponds to the spin-triplet mode ${\omega }^{11}\left(k\right)$ and right plot the spin-singlet mode ${\omega }^{00}\left(k\right)$.

Figure 3

Dispersion relation for complex-frequency modes, where the real (imaginary) part of the complex-frequency mode is plotted in solid (dashed) line. (Left) Plot of ${\omega }^{11}\left(k\right)$ for a CAF state with parameters $F_C=0.5\hslash {\omega }_B, u_z=0.05\hslash {\omega }_B, u_{\perp }=-0.1\hslash {\omega }_B, {\epsilon }_V=0.02\hslash {\omega }_B$, and ${\epsilon }_Z=0.02\hslash {\omega }_B$. (Right) Plot of ${\omega }^{11}\left(k\right)$ for a PLP state with parameters $F_C=0.5\hslash {\omega }_B, u_z=0.1\hslash {\omega }_B, u_{\perp }=-0.05\hslash {\omega }_B, {\epsilon }_V=0.02\hslash {\omega }_B$, and ${\epsilon }_Z=0.02\hslash {\omega }_B$.

Figure 4

Plot of the velocity of the Goldstone modes as a function of the parameter ${\delta }_c$ for the CAF phase (solid blue line) and for the PLP phase (dashed black line), with the rest of parameters keeping the same values as in Fig. 2.

Figure 5

Dispersion relation of the collective modes for the different phases of the $\nu =0$ QH state in monolayer graphene. In all the plots, the strength of the Coulomb interaction is set to $F_C=0.5\hslash {\omega }_B$. (Upper left) Dispersion relation of the F phase for $u_z=0.05\hslash {\omega }_B, u_{\perp }=-0.02\hslash {\omega }_B$, and ${\epsilon }_Z=0.045\hslash {\omega }_B$. (Upper right) Dispersion relation of the CDW phase for $u_z=-0.05\hslash {\omega }_B, u_{\perp }=-0.02\hslash {\omega }_B$, and ${\epsilon }_Z=0.01\hslash {\omega }_B$. (Lower left) Dispersion relation of the CAF phase for $u_z=0.05\hslash {\omega }_B, u_{\perp }=-0.02\hslash {\omega }_B$, and ${\epsilon }_Z=0.01\hslash {\omega }_B$. (Lower right) Dispersion relation of the KD phase for $u_z=0.01\hslash {\omega }_B, u_{\perp }=-0.02\hslash {\omega }_B$, and ${\epsilon }_Z=0.01\hslash {\omega }_B$.

Figure 6

Plot of the function $f\left(x\right)$ defined in Eq. (75), where we have imposed a finite cutoff in the series $N_C=150$ for the computation.

Figure 7

Dispersion relation of the collective modes for the different phases of the $\nu =0$ QH state in bilayer graphene but now computed using the screened potential (76) with $F_C=4\hslash {\omega }_B$, while the rest of the parameters keep the same values of Fig. 2.

Figure 8

Dispersion relation of the complex-frequency mode ${\omega }^{11}\left(k\right)$ for a PLP state, computed using the screened potential (76) with $F_C=4\hslash {\omega }_B$ and the rest of the parameters with the same values of right Fig. 3. The real (imaginary) part of the complex-frequency mode is plotted in solid (dashed) line.

Figure 9

Phase diagram of the $\nu =0$ QH state in parameter space $u_{\perp },u_z$ (left) and ${\epsilon }_Z,{\epsilon }_V$ (right), including the boundaries of the dynamically stable regions, represented as dashed lines.

Figure 10

(Top) Diagrammatic representation of the HF equations. The double (single) lines represent the dressed (bare) Green's function. The wiggly line represents the Coulomb interaction and the dashed line represents the short-range interactions. The direct (Hartree) term of the Coulomb interaction is suppressed by the uniform positive charge background. (Middle) Diagrammatic representation of the total potential and the correlation function ${\mathrm{\Pi }}_A$, Eq. (B49). (Bottom) Diagrammatic representation of the vertex equation in the TDHFA, Eq. (B53).