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Annals of Physics

Spin hamiltonian for which quantum hall wavefunction is exact

R. B. Laughlin

Published Apr 1, 1989• 59 citations
Publisher

Abstract

A Hamiltonian is introduced for which the spin-12 quantum liquid wavefunction proposed by Kalmeyer and Laughlin is the exact ground state. An explicit proof is given that this state is a doubly degenerate spin singlet. The excitation spectrum of the Hamiltonian is calculated variationally and a case is made for the presence of an energy gap. The spin-1 collective mode, calculated in the single-mode approximation, is shown to have a large energy that is minimized at the Brillouin zone corner. The spin-0 collective mode behaves similarly but has a lower overall energy. Wavefunctions for the neutral spin-12 excitations are shown to form an exact spin doublet. The energy of a pair of such excitations increases or decreases logarithmically with separation depending on the total spin. When widely separated, the excitations possess an internal 2-fold degree of freedom, a Brillouin zone that is half the linear dimension of the full zone, and a mass comparable to that of the collective modes.

DOI
10.1016/0003-4916(89)90339-4

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A Hamiltonian is introduced for which the spin-12 quantum liquid wavefunction proposed by Kalmeyer and Laughlin is the exact ground state. An explicit proof is given that this state is a doubly degenerate spin singlet. The excitation spectrum of the Hamiltonian is calculated variationally and a case is made for the presence of an energy gap. The spin-1 collective mode, calculated in the single-mode approximation, is shown to have a large energy that is minimized at the Brillouin zone corner. The spin-0 collective mode behaves similarly but has a lower overall energy. Wavefunctions for the neutral spin-12 excitations are shown to form an exact spin doublet. The energy of a pair of such excitations increases or decreases logarithmically with separation depending on the total spin. When widely separated, the excitations possess an internal 2-fold degree of freedom, a Brillouin zone that is half the linear dimension of the full zone, and a mass comparable to that of the collective modes.
We propose that the disordered state of a two-dimensional spin-1/2 Heisenberg antiferromagnet is physically equivalent to the incompressible liquid state of the fractional quantum Hall system. The fractional quantum Hall state for bosons is shown to be an exact spin singlet and to possess a low variational energy for the near-neighbor Heisenberg model on a triangular lattice. Variational wave functions for neutral spin-1/2 excitations are constructed and shown to form an exact spin doublet. Variational energies of these states are calculated, and their spin density profiles are determined. We find that a localized spin-1/2 quaisparticle has a size comparable to a lattice bond length and an excitation energy \ensuremath{\Delta}=1.3J. The energy-momentum dispersion of quasiparticles and spin-1 collective modes, obtained variationally, supports the hypothesis that the spin liquid state has a finite energy gap. The 1/2 fractional statistics exhibited by the quasiparticle excitations is explicitly demonstrated.
It is shown that one of the class of variational spin-liquid wave functions recently derived by Wen, Wilczek, and Zee [Phys. Rev. B 39, 11413 (1989)] is identical to the fractional-quantum-Hall analog state proposed by Kalmeyer and Laughlin [Phys. Rev. Lett. 59, 2095 (1987); Phys. Rev. B 39, 11 879 (1989)] and Laughlin [Ann. Phys. (N.Y.) 191, 163 (1989)], and that the neutral spin-1/2 excitations of the two states are also the same. The 1/2 fractional statistics obeyed by these particles is demonstrated explicitly. The spin-spin correlation function and quasiparticle profile for the Wen-Wilczek-Zee states are calculated both numerically and by a hypernetted-chain procedure. Both are consistent with the idea that spontaneous breaking of time-reversal symmetry is an essential feature of any spin state lacking magnetic order. A version of this state for a three-dimensional spin system is reported and shown to have similar properties. A case is made that the neutral spin-1/2 excitations of the three-dimensional spin liquid behave like anisotropic monopoles.
The properties of a class of wave functions describing charged spinless excitations of the Kalmeyer-Laughlin spin-liquid state are studied within the context of the t-J model and the case made that they are legitimate elementary excitations of such a state. Using these wave functions and those previously proposed for the neutral spin-1/2 excitations, a variational calculation of the hole propagator is performed. The spectral density at a given momentum is found to be a broad continuum with resonant structures that move as the momentum is varied, and to be consistent with exact-diagonalization studies.
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.
The one dimensional t-J Hamiltonian is diagonalized exactly for the supersymmetric case 2t=\ifmmode\pm\else\textpm\fi{}J using the Bethe ansatz. In this limit it is identical with models previously considered by Lai, Sutherland, and Schlottmann. In the present paper we discuss the ground-state properties and excitation spectrum in zero magnetic field. The ground state is a liquid of singlet bonds with varying spatial separation. Its most remarkable feature is the presence of bonds connecting particles at arbitrarily large distances. The ground-state energy is an analytic function of the band filling. There is no difference in the chemical potential for adding one or two particles and no evidence for the binding of holes. The low-lying part of the spectrum consists of two types of gapless excitations (charge and spin) with effective Fermi surfaces at 2kF{\mathit{k}}_{\mathit{F}}kF​ and kF{\mathit{k}}_{\mathit{F}}kF​, respectively. An interpretation of the energy spectrum in terms of spinons and holons is appropriate at low energies.
Collective modes capture the dynamical aspects of fractional quantum Hall (FQH) fluids. Depending on the active degrees of freedom, different types of collective modes can arise in a FQH state. In this work, we consider spinful FQH states in the lowest Landau level (LLL) along the Jain sequence of fillings ν=n/(2n±1)ν{=}n/(2n{\pm}1)ν=n/(2n±1) and compute the Coulomb dispersion of their spin-flip and spin-conserving collective modes in the spherical geometry. We use the LLL-projected density-wave and composite fermion (CF) exciton states as trial wave functions for these modes. To evaluate the dispersion of density-wave states, we derive the commutation algebra of spinful LLL-projected density operators on the sphere, which enables us to extract the gap of the density-wave excitations from the numerically computed density-density correlation function, i.e., the static structure factor, of the FQH ground state. We find that the CF excitons provide an accurate description of the collective modes at all wavelengths, while the density-wave states fail to do so. Specifically, the spin-flip density wave reliably captures the spin-flip collective mode only for the Laughlin and Halperin states, and that too only in the long-wavelength limit. Interestingly, for spin-singlet primary Jain states, the spin-conserving density mode is inaccurate even in the long-wavelength regime. We show that this discrepancy stems from the presence of an additional high-energy spin-conserving parton mode, similar to that found in fully polarized secondary Jain states at ν=n/(4n±1)ν{=}n/(4n{\pm}1)ν=n/(4n±1). We propose an ansatz for this parton mode and compute its Coulomb dispersion in the singlet state at ν=2/5ν{=}2/5ν=2/5. The predicted parton mode can be observed in circularly polarized inelastic light scattering experiments.