Is the Composite Fermion a Dirac Particle? (1502.03446v3)

Abstract: We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the $\nu=\frac12$ quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of $\pi$ around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, $\mathcal C\mathcal P$. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors $\frac n{2n+1}$ and $\frac{n+1}{2n+1}$, which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, $n$ and $n+1$, is now mapped to the same half-integer filling factor $n+\frac12$ of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as $d$-wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while $s$-wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a $\mathcal C\mathcal P$-breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.

• The paper proposes a particle-hole symmetric theory indicating that composite fermions act as massless Dirac particles with a π Berry phase.
• It establishes an effective field theory without a Chern-Simons term, offering fresh insights into quantum Hall states and d-wave paired phases.
• The study outlines testable predictions for transport properties, enabling experimental differentiation between Dirac and conventional composite fermion models.

An Analytical Perspective on Composite Fermion as Dirac Particles

The paper "Is the Composite Fermion a Dirac Particle?" authored by Dam Thanh Son offers a novel theoretical construct that attempts to reconcile the behavior of composite fermions (CF) in a half-filled Landau level with Dirac particle properties. This exploration is primarily focused on providing an explicitly particle-hole symmetric description of the Fermi-liquid ground state present at such conditions. The theory is applicable not only to the Dirac fermion in a magnetic field but extends to nonrelativistic fermions in the quantum Hall state with negligible inter-Landau level mixing.

Particle-hole Symmetry and Dirac Composite Fermions

The central thesis posited in the paper is the need for a particle-hole symmetric theory when dealing with half-filled Landau levels, leading to the conjecture that composite fermions could be massless Dirac fermions. This conceptualization is significant as it diverges from existing models where composite fermions lack such symmetry, leading to discrepancies such as different interpretations for the conjugate states in the Jain sequence. When symmetry is exact, the paper argues that these fermions bear a Berry phase of $\pi$ around the Fermi surface.

Effective Field Theory Framework

Son proposes an effective field theory reflective of these characteristics, which remains invariant under crucial discrete symmetries like $CP$ and $PT$ transformations. More uniquely, in this proposed framework, the composite fermions interact through a gauge field that does not include a Chern-Simons term, distinguishing it from traditional formulations. This model suggests that the Pfaffian and anti-Pfaffian phases could be understood as $d$-wave paired states of these Dirac composite fermions, a notable deviation from the conventional perspective where CFs are subjected to strong inter-Chern-Simons interactions.

Theoretical and Experimental Implications

The insights generated from this theoretical model have profound implications for both quantum Hall states and condensed matter systems that observe particle-hole symmetry, like graphene and topological insulators. This extends to broader applications involving the anomalous Hall effect and the dynamics of topological phases. One key assertion is that these Dirac composite fermions encapsulate a Fermi liquid state at half-filling, a notion supported by identifying the emergent Fermi momentum moved by particle density.

Numerical and Experimental Verification

Son's hypothesis provides a framework predicting observational signatures that differ from traditional CF theories, like specific features in the longitudinal and Hall conductivities measured across various scenarios, including finite frequency domains. The research invites verification through high-precision measurements of transport properties aimed at determining any Berry-phase influences on electron systems allowing experimentalists to distinguish between previously established CF models and this new Dirac interpretation.

Future Directions

Looking forward, the paper posits prospective experimental avenues, especially within systems exhibiting tunable parameters, such as bilayer graphene. It also lays the groundwork for seeing particle-hole symmetric phases like the PH-Pfaffian, a state with potential realizations not yet fully explored. Beyond condensed matter, the findings intersect with high-energy physics concepts, invoking mirror symmetry and vorticity considerations, evolving the theoretical dialogue surrounding Dirac systems in external fields.

The paper thus opens a dialogue in condensed matter physics probing more deeply the nature of quanta in strongly-correlated electron systems, providing a refreshed perspective on elemental symmetries and opening the floor for an enriched understanding of topological phases and emergent phenomena within quantum Hall effects.