Scientists / Open questions

F. Duncan M. Haldane

Quantum many-body physics; topological phases

2016 Nobel Prize in Physics

16 questions

Nonperturbative low-energy theory of Haldane's boson-coupled Anderson impurity model for mixed valence: the valence-fluctuation scale and long-time dynamics
Rigorous status of the conjectured upper bound on the sixth-order coefficient $\bar S_6$ of the guiding-center structure factor of chiral fractional quantum Hall states
Microscopic determination of which hierarchy incompressible quantum fluid states exist for a given two-body interaction in the fractional quantum Hall effect
Completing Haldane's exact solution of the inverse-square spin chain: the discrete-anisotropy spectrum beyond the Jastrow states, and exact resonating-valence-bond states in two dimensions
Exact solvability of the one-dimensional Hubbard-type lattice gas with arbitrary permutation symmetry at finite interaction and finite density (the Choy-Haldane problem)
Umklapp and finite-spin quantum corrections to the semiclassical soliton theory of ferromagnetic Heisenberg chains
A controlled nonperturbative theory of the strong-coupling regime of the O(3) nonlinear sigma model with hedgehog Berry phases for two-dimensional quantum antiferromagnets
Haldane's open questions on the asymmetric Anderson impurity model: universal mixed-valence crossover functions, the crossover constants, and crossover-region physics beyond the scaling equations
Spectral statistics of the circle billiard with a point scatterer: exact spacing law for affected levels, simultaneous Bessel zeros, rationality effects, and genericity of smooth location dependence
Status of Haldane's Luttinger relation $v_S=(v_Nv_J)^{1/2}$ for Bethe-ansatz models with phase-shift kernels not of difference form
Relation between Haldane's counting definition of fractional statistics and braid (anyon) statistics: equivalence conditions, the nonorthogonality mechanism, and microscopic realizations
Controlled microscopic foundation of the dipolar composite-fermion Fermi liquid at the half-filled Landau level ( $\nu=1/2$ )
Integrability and multisoliton solutions of the massive easy-axis O(3) sigma model, Lorentz-breaking anisotropy effects on the soliton sector, and nonperturbative soliton energies in the one-dimensional antiferromagnet
Model-dependent content and range of validity of the harmonic-fluid (Luttinger-liquid) description of one-dimensional quantum fluids
Integrability structure and spinon-Bethe-rapidity correspondence of the Inozemtsev interpolation between the Haldane-Shastry and Heisenberg spin-1/2 chains
Microscopic foundation of Haldane's geometric description of fractional quantum Hall states: intrinsic metric elasticity, geometry across the compressible transition, and quasiparticles as curvature defects

Back to open problems

Nonperturbative low-energy theory of Haldane's boson-coupled Anderson impurity model for mixed valence: the valence-fluctuation scale and long-time dynamics

Rigorous status of the conjectured upper bound on the sixth-order coefficient Sˉ6\bar S_6Sˉ6​ of the guiding-center structure factor of chiral fractional quantum Hall states

Microscopic determination of which hierarchy incompressible quantum fluid states exist for a given two-body interaction in the fractional quantum Hall effect

Completing Haldane's exact solution of the inverse-square spin chain: the discrete-anisotropy spectrum beyond the Jastrow states, and exact resonating-valence-bond states in two dimensions

Exact solvability of the one-dimensional Hubbard-type lattice gas with arbitrary permutation symmetry at finite interaction and finite density (the Choy-Haldane problem)

Umklapp and finite-spin quantum corrections to the semiclassical soliton theory of ferromagnetic Heisenberg chains

A controlled nonperturbative theory of the strong-coupling regime of the O(3) nonlinear sigma model with hedgehog Berry phases for two-dimensional quantum antiferromagnets

Haldane's open questions on the asymmetric Anderson impurity model: universal mixed-valence crossover functions, the crossover constants, and crossover-region physics beyond the scaling equations

Spectral statistics of the circle billiard with a point scatterer: exact spacing law for affected levels, simultaneous Bessel zeros, rationality effects, and genericity of smooth location dependence

Status of Haldane's Luttinger relation vS=(vNvJ)1/2v_S=(v_Nv_J)^{1/2}vS​=(vN​vJ​)1/2 for Bethe-ansatz models with phase-shift kernels not of difference form

Relation between Haldane's counting definition of fractional statistics and braid (anyon) statistics: equivalence conditions, the nonorthogonality mechanism, and microscopic realizations

Controlled microscopic foundation of the dipolar composite-fermion Fermi liquid at the half-filled Landau level (ν=1/2\nu=1/2ν=1/2)

Integrability and multisoliton solutions of the massive easy-axis O(3) sigma model, Lorentz-breaking anisotropy effects on the soliton sector, and nonperturbative soliton energies in the one-dimensional antiferromagnet

Model-dependent content and range of validity of the harmonic-fluid (Luttinger-liquid) description of one-dimensional quantum fluids

Integrability structure and spinon-Bethe-rapidity correspondence of the Inozemtsev interpolation between the Haldane-Shastry and Heisenberg spin-1/2 chains

Microscopic foundation of Haldane's geometric description of fractional quantum Hall states: intrinsic metric elasticity, geometry across the compressible transition, and quasiparticles as curvature defects

Rigorous status of the conjectured upper bound on the sixth-order coefficient $\bar S_6$ of the guiding-center structure factor of chiral fractional quantum Hall states

Status of Haldane's Luttinger relation $v_S=(v_Nv_J)^{1/2}$ for Bethe-ansatz models with phase-shift kernels not of difference form

Controlled microscopic foundation of the dipolar composite-fermion Fermi liquid at the half-filled Landau level ( $\nu=1/2$ )