Thermoelectric Transport Signatures of Dirac Composite Fermions in the Half-Filled Landau Level

The half-filled Landau level is expected to be approximately particle-hole symmetric, which requires an extension of the Halperin-Lee-Read (HLR) theory of the compressible state observed at this filling. Recent work indicates that, when particle-hole symmetry is preserved, the composite fermions experience a quantized $\pi$-Berry phase upon winding around the composite Fermi surface, analogous to Dirac fermions at the surface of a 3D topological insulator. In contrast, the effective low-energy theory of the composite fermion liquid originally proposed by HLR lacks particle-hole symmetry and has vanishing Berry phase. In this paper, we explain how thermoelectric transport measurements can be used to test the Dirac nature of the composite fermions by quantitatively extracting this Berry phase. First, we point out that longitudinal thermopower (Seebeck effect) is nonvanishing because of the unusual nature of particle-hole symmetry in this context and is not sensitive to the Berry phase. In contrast, we find that off-diagonal thermopower (Nernst effect) is directly related to the topological structure of the composite Fermi surface, vanishing for zero Berry phase and taking its maximal value for $\pi$ Berry phase. In contrast, in purely electrical transport signatures, the Berry phase contributions appear as small corrections to a large background signal, making the Nernst effect a promising diagnostic of the Dirac nature of composite fermions.

Popular Summary

A composite fermion liquid is a remarkable phase of matter that forms when low-density electrons are subjected to an intense magnetic field. In this situation, only half of a Landau level is filled. In contrast with fractional quantum Hall states that occur at nearby fillings, such as $1/3$, a $1/2$-filled Landau level displays the metallic behavior of an entirely new type of object—a composite fermion (i.e., a bound state of a pair of vortices and an electron). Recent theoretical developments have argued that composite fermions possess attributes typically associated with Dirac fermions, such as a Berry phase. Such attributes would imply an entirely new avenue to explore Dirac physics, similar to graphene and topological insulator surfaces, except that the fermions in this case are not electrons but instead composite fermions.

A major open question in quantum mechanics pertains to the experimentally testable differences between the new Dirac theory of composite fermions and the preexisting Halperin-Lee-Read theory. Unfortunately, the Seebeck coefficients of Dirac composite fermions and the Halperin-Lee-Read state are similar and nonzero, which makes it impossible to differentiate between the two states. Here, we show that a readily accessible thermoelectric measurement—the Nernst effect—may represent an ideal way to distinguish between the two scenarios and, accordingly, probe the particle-hole asymmetry. To demonstrate the Nernst effect, one applies a heat current along one direction and measures the voltage in the perpendicular direction. We show that the Nernst effect is present for Dirac composite fermions but absent otherwise. This effect is therefore a robust diagnostic of Dirac physics in composite fermions.

While challenging, the measurements suggested by our work appear accessible with current capabilities and pave the way for experimental tests of the Dirac nature of composite fermions.

Figure 1

Schematic of charge vortex duality for transport. (a) An electron (blue dot) is a source of electric field $\mathbf{E}$ (blue line) and $-4\pi$ flux of the emergent gauge field $b$ (circulating dashed red line indicates winding of phase for Dirac composite fermion). The Dirac composite fermion object (red dot) is a dual object that is a source of emergent electric field lines $\mathbf{e}$ (dashed red lines) and $4\pi$ flux of the electron phase (circulating blue line). Panel (b) shows the electron phase ${\varphi }_e$, winding by $4\pi$ across the $y$ direction for each composite fermion propagating along $x$. Panels (c,d) schematically depict the composite fermion dispersion and Fermi-surface pseudospin texture for the Dirac ($\pi$ Berry phase) and HLR (0 Berry phase), respectively.

Figure 2

Field dependence of $S^{xy}$. In addition to having a zero or nonzero value at half-filling ($B=B_{1/2}$), the asymmetric field dependence of the transverse thermopower (Nernst) signal distinguishes between the HLR state with zero Berry phase ($\gamma =0$) and the particle-hole symmetric Dirac state with $\pi$ Berry phase ($\gamma =1$).