From coupled-wire construction of quantum Hall states to wave functions and hydrodynamics

In this paper we use a close connection between the coupled-wire construction (CWC) of Abelian quantum Hall states and the theory of composite bosons to extract the Laughlin wave function and the hydrodynamic effective theory in the bulk, including the Wen-Zee topological action, directly from the CWC. We show how rotational invariance can be recovered by fine tuning the interactions. A simple recipe is also given to construct general Abelian quantum Hall states described by the multicomponent Wen-Zee action.

Figure 1

(a) An array of coupled wires in a perpendicular magnetic field. (b) The energy band structure of the coupled wires. In the limit of decoupled wires, a perpendicular magnetic field in the Landau gauge shifts the quadratic dispersion relations of the individual wires (labeled by $j)$ in a $j$-dependent manner: $k\to k+bj$ (thick curves). When $2k_{\text{F}}=\left|b\right|\left(\nu =1\right)$, the single-particle hopping between the adjacent wires opens the gaps at Fermi points and the anisotropic Landau levels are formed (thin blue curves).

Figure 2

(a) Original wires (located at $j)$ on which the original bosons $\{{\mathrm{\Phi }}_j,{\mathrm{\Theta }}_j\}$ are defined and (b) virtual wires (with positions specified by $j^{*})$ corresponding to the on-strip fields $\{{\phi }_{j^{*}},{\vartheta }_{j^{*}}\}$. Dashed lines denote the new set of chiral bosons ${\tilde{\varphi }}_{j^{*}}^{\text{L/R}}$, with which interwire interaction (9) looks like an ordinary backscattering $cos\left(2{\vartheta }_{j^{*}}\right)$.

Figure 3

Loop used to calculate the phase of the Laughlin wave function. Particles on wires 1 to $M-1$ are fully encircled and each gives the phase $2m\pi$. Exchanging the positions with particles on wires 1 or $M$ amounts to replacing the straight lines with half-circles, each one giving rise to a phase $m\pi$.

Figure 4

Multilayer construction that reproduces the Abelian Chern-Simons gauge theory (61).

Figure 5

Points $({\mathrm{\Phi }}_j/\pi ,{\mathrm{\Theta }}_j/\pi )$ identified by redundancy for fermionic (left) and bosonic (right) cases. In the fermionic case, the entire lattice splits into two sublattices corresponding to fermionic (red) and bosonic (black) sectors.