[Paper Review] Electrons and composite Dirac fermions in the lowest Landau level

Kartik Prabhu, Matthew M. Roberts|arXiv (Cornell University)|Sep 8, 2017

Quantum and electron transport phenomena27 references3 citations

TL;DR

This paper constructs a symmetry-consistent effective action for composite Dirac fermions in the lowest Landau level, showing that such fermions must couple to a nontrivial geometric connection even in flat space. The theory reveals that composite fermions acquire an electric dipole moment orthogonal to their wavevector, and the stress tensor and current operators are derived in curved spacetime, enforcing constraints from particle-hole symmetry and Galilean invariance in the massless limit.

ABSTRACT

We construct an action for the composite Dirac fermion consistent with symmetries of electrons projected to the lowest Landau level. First we construct a generalization of the $g=2$ electron that gives a smooth massless limit on any curved background. Using the symmetries of the microscopic electron theory in this massless limit we find a number of constraints on any low-energy effective theory. We find that any low-energy description must couple to a geometry which exhibits nontrivial curvature even on flat space-times. Any composite fermion must have an electric dipole moment proportional and orthogonal to the composite fermion's wavevector. We construct the effective action for the composite Dirac fermion and calculate the physical stress tensor and current operators for this theory.

Motivation & Objective

• To construct a low-energy effective field theory for composite Dirac fermions that respects the symmetries of electrons projected into the lowest Landau level.
• To identify geometric constraints arising from the massless limit of the $g=2$ electron theory on curved backgrounds.
• To derive the physical stress tensor and current operators for the composite Dirac fermion in a curved, non-relativistic spacetime.
• To resolve inconsistencies in previous models like Halperin-Lee-Read and Son's theory by enforcing particle-hole symmetry and proper gauge structure.
• To show that any low-energy description must couple to a nontrivial connection, even in flat space, due to emergent curvature in the effective geometry.

Proposed method

• Generalize the $g=2$ electron action to allow a smooth massless limit on arbitrary curved backgrounds, preserving key symmetries.
• Use the symmetries of the massless electron theory to constrain possible low-energy effective actions, particularly regarding coupling to geometry.
• Introduce a new connection (distinct from the spin connection) to which the composite fermion must couple, due to symmetry constraints.
• Construct an effective action for the composite Dirac fermion coupled to a dynamical gauge field $a_\mu$, a background gauge field $A_\mu$, and the geometric connection.
• Derive the stress tensor and current operators via variation of the action with respect to the coframe $e^a$, spin connection $\tilde{\omega}$, and gauge fields $A$, $a$, using a covariant variational procedure.
• Apply an 'improvement' procedure to obtain a symmetric Cauchy stress-mass tensor and consistent physical currents.

Experimental results

Research questions

• RQ1 How can a consistent effective field theory for composite Dirac fermions be constructed that respects the symmetries of electrons in the lowest Landau level on curved backgrounds?
• RQ2 What geometric structures must emerge in the low-energy description of composite fermions, even in flat space, due to symmetry constraints?
• RQ3 Why does the standard spin connection fail to couple directly to neutral composite fermions, and what alternative geometric connection is required?
• RQ4 How do the stress tensor and current operators transform under diffeomorphisms and gauge symmetries in this framework?
• RQ5 What is the role of particle-hole symmetry in constraining the form of the effective action and the resulting transport properties?

Key findings

• The effective theory requires coupling to a nontrivial geometric connection, even in flat space, due to symmetry constraints in the massless limit of the electron theory.
• Composite fermions must possess an electric dipole moment that is proportional and orthogonal to their wavevector, arising from the interplay of gauge and geometric symmetries.
• The physical stress tensor and current operators are derived explicitly, with the current operator containing contributions from both the gauge field and the geometric connection.
• The theory enforces a nontrivial curvature in the effective geometry, even when the spacetime is classically flat, due to the coupling of the composite fermion to the modified connection.
• The action correctly reproduces the Jain sequence of fractional quantum Hall states and the topological shift $N+2$ or $1-N$ on the sphere, confirming consistency with particle-hole symmetry.
• The construction resolves inconsistencies in prior models by ensuring proper gauge structure and symmetry realization, particularly through the use of a field redefinition and Lagrange multiplier fields.