Thermal Transport Signatures of Broken-Symmetry Phases in Graphene

In the half filled zero-energy Landau level of bilayer graphene, competing phases with spontaneously broken symmetries and an intriguing quantum critical behavior have been predicted. Here we investigate signatures of these broken-symmetry phases in thermal transport measurements. To this end, we calculate the spectrum of spin and valley waves in the $\nu =0$ quantum Hall state of bilayer graphene. The presence of Goldstone modes enables heat transport even at low temperatures, which can serve as compelling evidence for spontaneous symmetry breaking. By varying external electric and magnetic fields, it is possible to determine the nature of the symmetry breaking. Temperature-dependent measurements may yield additional information about gapped modes.

Figure 1

(a) Experimental setup: bilayer graphene in perpendicular magnetic and electric fields between two metallic leads. (b) Diagrammatic representation of the vertex function ${\mathrm{\Gamma }}_{A,ab}$ comprising the Fock contribution of the long-range Coulomb interaction (wavy line) and Hartree and Fock contributions of the anisotropic short-range interaction (dotted line). (c) Thermal conductance from particle-hole excitations at $T=1\mathrm{K}$ as a function of the Zeeman energy and displacement field [32]. Inset: Phase diagram of the $\nu =0$ state in bilayer graphene. The axes are the same as in the main plot.

Figure 2

(a) Excitation spectrum in the CAF phase for ${\epsilon }_V=3\mathrm{K}$. The two low-energy modes (red solid and dashed curves) are spin waves at low wave vectors, while the higher-energy modes (black dotted curve) correspond to isospin fluctuations. Inset: Full spectrum with energies up to $\sim U_c$. (b) Excitation spectrum at ${\epsilon }_V=19.2\pm 0.1\mathrm{K}$ on the two sides of the CAF-PLP transition. We use the same color coding as in (a) for the spectrum on the CAF side and green solid lines on the PLP side.

Figure 3

(a) Thermal conductance normalized by $T^2$ inside the three phases CAF, PLP, and FLP at ${\epsilon }_V=15$, 25, and 31 K (solid lines) as well as near the phase transitions CAF-PLP and PLP-FLP at ${\epsilon }_V=19.21$ and 27 K (dashed lines). (b) Thermal conductance vs ${\epsilon }_V$ for various temperatures. Curves have been offset for clarity (the dashed lines indicate zero conductance), and the vertical gray lines mark the phase transitions.