Quantum Hall hierarchy from coupled wires

The coupled-wire construction provides a useful way to obtain microscopic Hamiltonians for various two-dimensional topological phases, among which fractional quantum Hall states are paradigmatic examples. Using the recently introduced flux attachment and vortex duality transformations for coupled wires, we show that this construction is remarkably versatile to encapsulate phenomenologies of hierarchical quantum Hall states: the Jain-type hierarchy states of composite fermions filling Landau levels and the Haldane-Halperin hierarchy states of quasiparticle condensation. The particle-hole conjugate transformation for coupled-wire models is also given as a special case of the hierarchy construction. We also propose coupled-wire models for the composite Fermi liquid, which turn out to be compatible with a sort of the particle-hole symmetry implemented in a nonlocal way at $\nu =\frac{1}{2}$. Furthermore, our approach shows explicitly the connection between the Moore-Read Pfaffian state and a chiral $p$-wave pairing of the composite fermions. This composite-fermion picture is also generalized to a family of the Pfaffian state, including the anti-Pfaffian state and Bonderson-Slingerland hierarchy states.

Figure 1

Tunneling Hamiltonian for the $\nu =1/q$ Laughlin state in terms of (a) the original particles, (b) composite fermions, (c) composite bosons, and (d) vortices from the composite bosons. The horizontal lines represent wires where the bosonic fields are defined. The red straight arrows represent single-particle hopping involving the ${\phi }_j$ field. The red curly arrows represents charge density fluctuations involving ${\theta }_j$ with the wave number $k_F$ multiplied by the factor indicated in red. For the composite fermions, the black horizontal arrows and the red curved arrows, respectively, represent chiral fermion modes ${\psi }_{R/L}^{\mathrm{CF}}$ and single-fermion hopping terms. The associated Chern-Simons couplings are also shown.

Figure 2

Tunneling Hamiltonian for the $\nu =\frac{2}{5}$ state in terms of (a) the electrons, (b) composite fermions, (c) composite bosons, and (d) vortices of the composite bosons. The same notation as in Fig. 1 is used.

Figure 3

Tunneling Hamiltonian for a second-level hierarchy state at $\nu =\frac{3}{7}$. Successive applications of the flux attachment and vortex duality transformations finally yield the Hamiltonian describing the Mott insulator of vortices. The resulting $K$ matrix is also given above.

Figure 4

PH transformation to the Laughlin $\nu =1/\tilde{q}=1/(2\tilde{p}+1)$ state. The PH conjugate of the Laughlin state involves $\tilde{p}$ electron hopping and obviously leaves the chiral edge states corresponding to the $\nu =1$ IQH state.

Figure 5

Tunneling Hamiltonian for the CFL at $\nu =1/M$ in terms of (a) the original particles and (b) composite fermions.

Figure 6

The Fermi surface of the composite fermion from Eq. (123). For $g_R=g_L=0$, the composite fermion has linear dispersions at $k_x=\pm k_F$ while a flat dispersion along $k_y$. Nonzero $g_R=g_L$ develop an open Fermi surface with the shape of cosine.

Figure 7

Tunneling Hamiltonian for the CFL at $\nu =\frac{1}{2}$. After $4\pi$-flux attachment to electrons, the Hamiltonian is written in terms of composite fermions. Starting from the hole picture, we can also write the Hamiltonian in terms of composite holes by attaching $-4\pi$ flux to the holes.

Figure 8

Tunneling Hamiltonian for the Pfaffian state at $\nu =1/M$. (a) Quantum wires are placed in a magnetic field of the average flux $b=2M{k}_F$ with modulation $\delta b=2k_F$. The Hamiltonian is given in terms of the bosonic fields corresponding to (a) the original particles and (b) composite fermions. In the latter case, two adjacent wires can be decomposed into a bosonic charge mode ${\mathrm{\Phi }}_l^c$ (solid line) and neutral chiral Majorana modes ${\xi }_{p,l}^{1,2}$ (dashed arrow).

Figure 9

Tunneling Hamiltonian for the Bonderson-Slingerland hierarchy state at $\nu =2/(2M-1)$ in terms of (a) the original particles and (b) composite fermions. The same notation as in Fig. 8 is used.

Figure 10

Tunneling Hamiltonian for the anti-Pfaffian state at $\nu =\frac{1}{2}$ in terms of (a) the original particles and (b) composite fermions. A similar notation to Fig. 8 is used.

Figure 11

Tunneling Hamiltonians for the $\ell =1$ paired state of the composite fermions at $\nu =\frac{1}{2}$, which has the same topological order as the PH-Pfaffian state but with the explicitly broken PH symmetry. When only the coupling constants $t_{00}$ and $t_{11}$ are relevant, the models describe (a) the Laughlin $\nu =\frac{1}{8}$ state of charge-2 bosons or (b) the Halperin 113 state.