We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the
Z2 properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.
We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at ν = q i K −1 ij q j , where K ij is a 2 × 2 matrix and q i is a vector. Our model allows us to reproduce several well-known wavefunctions. We show that the wavefunction Ψ ( m , m , n ) constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor and in particular Ψ (3,3,1) at filling can be obtained from our vacuum configuration. The unpolarized Halperin wavefunction and especially that for the fractional quantum Hall state at filling factor can also be recovered from our approach. Generalization to more than two layers is straightforward.
We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at \\nu=q_iK_{ij}^{-1}q_j, where K_{ij} is a 2\\times 2 matrix and q_i is a vector. Our model allows us to reproduce several well-known wave functions. We show that the wave function \\Psi_{(m,m,n)} constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor {2\\over m+n} and in particular \\Psi_{(3,3,1)} at filling {1\\over 2} can be obtained from our vacuum configuration. The unpolarized Halperin wave function and especially that for the fractional quantum Hall state at filling factor {2\\over 5} can also be recovered from our approach. Generalization to more than 2 layers is straightforward.
It has long been known that Laughlin's wave functions, describing the fractional quantum Hall effect at filling fractions ν=1/(2k+1), can be obtained as correlation functions in conformal field theory. We show how to generalize this approach to construct explicit trial wave functions for all states in the quantum Hall hierarchy corresponding to quasiparticle (as opposed to quasihole) condensates, including the recently observed state at ν= 4/11. At the filling fractions ν= n/(2np+1) this construction exactly reproduces Jain's composite fermion wave functions. An explicit connection is made to Wen's topological classification of FQH states.
We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where
L1 is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd
q, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd
q, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as
L1 increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite
L1; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.