Emergent particle-hole symmetry in the half-filled Landau level

We provide an effective description of a particle-hole symmetric state of electrons in a half-filled Landau level, starting from the traditional approach pioneered by Halperin, Lee, and Read [Phys. Rev. B 47, 7312 (1993)]. Specifically, we study a system consisting of alternating quasi-one-dimensional strips of composite Fermi liquid (CFL) and composite hole liquid (CHL), both of which break particle-hole symmetry. When the CFL and CHL strips are identical in size, the resulting state is manifestly invariant under the combined action of a particle-hole transformation with respect to a single Landau level (which interchanges the CFL and CHL) and translation by one unit, equal to the strip width, in the direction transverse to the strips. At distances long compared to the strip width, we demonstrate that the system is described by a Dirac fermion coupled to an emergent gauge field, with an antiunitary particle-hole symmetry, as recently proposed by Son [Phys. Rev. X 5, 031027 (2015)].

Figure 1

The left side of the figure depicts the CFL in the lower half-plane and the topologically trivial vacuum in the upper half-plane. The chiral field ${\varphi }_f$ is indicated by the short dashed line and described by the Lagrangian in Eq. (2.5). The right side displays the CHL in the lower half-plane and the $\nu =1$ integer quantum Hall state in the upper half-plane. The chiral field ${\varphi }_h$ is indicated by the long dashed line and described by the Lagrangian in Eq. (2.7). A particle-hole transformation maps the left and right configurations into one another and transforms the part of the Lagrangians describing ${\varphi }_f$ and ${\varphi }_h$ into one another.

Figure 2

The left side of the figure depicts the CFL in the lower half-plane and the CHL in the upper half-plane. The right side displays the CFL in the lower half-plane and the CHL in the upper half-plane. The chiral field ${\varphi }_f$ is indicated by the short dashed line, the ${\varphi }_h$ field by the long dashed line, and the chiral mode ${\varphi }_{\mathrm{LL}}$ of the filled Landau level by the full line. A particle-hole transformation maps the left and right configurations into one another.

Figure 3

Schematic picture of the two-dimensional spatial plane patterned with alternating strips of the composite Fermi liquid, shown in red, and the composite hole liquid, shown in blue. The strips have width equal to $\delta$ and run along the $x$ direction.

Figure 4

Wires hosting composite fermions are depicted by short-dashed lines along $y/\delta \in 2\mathbb{Z}-\frac{1}{2}$ within the (red) composite Fermi liquid regions, while wires hosting composite holes are indicated by long-dashed lines along $y/\delta \in 2\mathbb{Z}+\frac{1}{2}$ within the (blue) composite hole liquid regions. The solid lines along $y/\delta \in \mathbb{Z}$ denote line interfaces between the composite Fermi and composite hole liquids with excitations described by the action $S_{\mathrm{interfaces}}$. The solid lines host the copropagting left-moving (right-moving) fields ${\varphi }_f$ and ${\varphi }_h$ and the counterpropagating right-moving (left-moving) filled Landau level mode ${\varphi }_{\mathrm{LL}}$ for $y/\delta \in 2\mathbb{Z}$ ($y/\delta \in 2\mathbb{Z}+1$).