Recently, it was discussed that the $\nu=1/3$ fractional quantum Hall state can be expressed by a one-dimensional lattice model with an exact matrix-prduct ground state, when toroidal boundary conditions are assumed for a narrow strip [Phys. Rev. Lett. {\bf 109} (2012) 016401]. In this article, we discuss how the excitation spectra of this system are calculated analytically based on the matrix-product formalism. We introduce variational wave functions for various momenta and optimize them. The obtained spectra of the charge neutral excitations show the magneto-roton behavior.
Recently, it was discussed that the ν=1/3 fractional quantum Hall state can be expressed by a one-dimensional lattice model with an exact matrix-prduct ground state, when toroidal boundary conditions are assumed for a narrow strip [Phys. Rev. Lett. {\bf 109} (2012) 016401]. In this article, we discuss how the excitation spectra of this system are calculated analytically based on the matrix-product formalism. We introduce variational wave functions for various momenta and optimize them. The obtained spectra of the charge neutral excitations show the magneto-roton behavior.
Recently, it was discussed that the ν=1/3 fractional quantum Hall state can be expressed by a one-dimensional lattice model with an exact matrix-prduct ground state, when toroidal boundary conditions are assumed for a narrow strip [Phys. Rev. Lett. {\bf 109} (2012) 016401]. In this article, we discuss how the excitation spectra of this system are calculated analytically based on the matrix-product formalism. We introduce variational wave functions for various momenta and optimize them. The obtained spectra of the charge neutral excitations show the magneto-roton behavior.
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 introduce an exactly solvable fermion chain that describes a ν=1/3 fractional quantum Hall (FQH) state beyond the thin-torus limit. The ground state of our model is shown to be unique for each center-of-mass sector, and it has a matrix product representation that enables us to exactly calculate order parameters, correlation functions, and entanglement spectra. The ground state of our model shows striking similarities with the BCS wave functions and quantum spin-1 chains. Using the variational method with matrix product ansatz, we analytically calculate excitation gaps and vanishing of the compressibility expected in the FQH state. We also show that the above results can be related to a ν=1/2 bosonic FQH state.
he understanding and simulation of quantum many-body states in one space dimension has experienced revolutionary progress with the advent of the density matrix renormalization g roup. In modern language, this method can be viewed as a variational optimization over the set of matrix product states (MPS). Due to their perimeter law entanglement, 2-D system s such as the fractional quantum Hall effect are harder to simulate by MPS. Two recent articles[1,2] have proposed an exact MPS representation of a large series of fractional quantum Hall states in various geometries. The model states include all paired k=2 Jack polynomials, such as the Moore-Read and Gaffnian states, as well as the Read-Rezayi k=3 state. We will discuss how a controlled truncation can be performed on this r epresentation and we will give a natural interpretation from the entanglement spectrum perspective. We will discuss the numerical accuracy of this truncated prescription for vari ous geometries. [1] M.P. Zaletel and R.S. K. Mong, Phys. Rev. B 86, 245305 (2012). [2] B. Estienne, Z. Papic, N. Regnault, B. A. Bernevig, arXiv:1211.3353.
We show model wavefunctions for neutral collective modes in fractional quantum Hall (FQH) states have simple analytic forms obtained from judicially reducing the powers of selected pairs in the ground state Jastrow factor. This scheme of "pair excitations" works for the magneto-roton modes of single-component Abelian and non-Abeliean FQH states, as well as neutral fermion mode for the Moore-Read (MR) state. The analytic wavefunctions enable computations in the thermodynamic limit previous inaccessible to numerics, and the long wavelength limit of the neutral energy gap of the magneto-roton modes can be interpreted as fusion of charges in two-dimensional plasma picture, extending the plasma analogy to neutral excitations. A lattice diagrammatic method of representing these many-body wavefunctions and FQH elementary excitations is also presented.
Ground-state of fractional and integral quantum Hall systems at ν≤1 and it excitations
Many-body variational ground-state wave function of two-dimensional electron system (2DES), localized in the main strip (MS)Lx□×Ly of the finite width Lx□=2πmℓ0 (and the periodic boundary condition (PBC) imposed along x−direction), is presented at the fractional and the integral filling factors ν=1/m for two different ion backgrounds: microscopical uniform ion background (UIB) and classical ion jellium background (IJB); ℓ0 is the magnetic length, m=2ℓ+1 and ℓ=0,1,2,.... It is shown that the ground-state and the lowest excited-state can correspond to partial crystal-like correlation order among N electrons of the main region (MR) Lx×Ly; then the study of 2DES of N electrons within MR is exactly reduced to the treatment of 2DES of N~=NLx□/Lx electrons localized within MS, with PBC along x. The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the y− direction, with the period Lx□/m, and independent of x. For IJB, at m=3,5, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state (the Laughlin model uses IJB); the same holds at m=1. Obtained compound exciton and compound spin-exciton states show finite excitation gaps. The excited compound electron (hole) is composed, within MS, from m strongly correlated quasielectrons (quasiholes) of the charge e/m (−e/m. Quantized Hall conductance σH=e2/(2mπℏ) is obtained. The theory is in good agreement with experiments.