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Haldane PseudopotentialsEdit

I can’t adopt a partisan framing, but I can provide a neutral encyclopedia-style article on Haldane pseudopotentials that covers the physics, history, and current understanding.

Haldane pseudopotentials are a convenient way to describe how electrons interact in a two-dimensional electron system subjected to a strong perpendicular magnetic field, a setting central to the fractional quantum Hall effect. Proposed by Duncan Haldane, these pseudopotentials encode the effective two-body interaction within a single Landau level by labeling the interaction strength according to the relative angular momentum of a pair of electrons. For fermions, the allowed relative angular momenta are odd integers; for bosons they are even.

Definition and conceptual framework

  • In a two-dimensional electron gas under a strong magnetic field, electron motion is quantized into Landau levels. When kinetic energy scales are large compared with interaction scales, it is meaningful to project the full Coulomb interaction onto a single Landau level, yielding an effective interaction among electrons restricted to that level.
  • In the basis of two-electron states with definite relative angular momentum m, the projected two-body interaction becomes diagonal. The diagonal elements are called the Haldane pseudopotentials, denoted V_m^{(n)}, where n labels the Landau level and m denotes the relative angular momentum of the pair.
  • The number m runs over odd integers for fermions (due to the antisymmetry of the fermionic wavefunction) and over even integers for bosons. The hierarchy and magnitudes of the V_m^{(n)} encapsulate how strongly the interaction penalizes close-lying pairs (small |m|) versus more extended configurations (larger |m|).
  • Formally, V_m^{(n)} is obtained by projecting the electron-electron interaction into the nth Landau level and computing the matrix element associated with two-electron states of relative angular momentum m. In practice, this involves integrals over momentum or coordinate space that combine the Fourier transform of the bare interaction with the Landau level form factors. The exact expressions depend on the chosen gauge, the interaction potential, and material parameters, but the essential idea is that V_m^{(n)} quantify the energy cost of placing a pair of electrons in a given relative motion state within that Landau level.
  • The Haldane pseudopotential formalism provides a compact language for comparing different interaction models and for constructing model Hamiltonians with particular ground states. It also clarifies how changes in Landau level index n or in physical conditions (such as finite width or Landau level mixing) modify the relative importance of different m channels.

See also: Fractional quantum Hall effect; Landau level; Two-dimensional electron gas; Coulomb interaction; Pseudopotential.

Calculation and typical behavior

  • The calculation starts from the bare interaction V(r) (often the Coulomb interaction in two dimensions) and the Landau level wavefunctions, then performs a projection into the n-th Landau level. The resulting pseudopotentials V_m^{(n)} depend on n, m, and the microscopic parameters of the system.
  • In many materials, the lowest Landau level (n = 0) features a broad hierarchy where the first few pseudopotentials—especially V_1^{(0)} for fermions—are comparatively large, encoding strong short-range repulsion. This tendency toward short-range avoidance is a key driver of the formation of incompressible quantum Hall states at certain fractional fillings, most famously the Laughlin states.
  • Explicit expressions commonly involve special functions such as Laguerre polynomials and the Landau level form factors. In broad terms, the V_m^{(n)} can be written as integrals that combine the Fourier transform of V(r) with the probability amplitudes of relative-motion states in the nth Landau level. See standard references for the precise formulas and their derivations.
  • Numerical studies frequently employ the pseudopotential language to construct model Hamiltonians that have exact or highly accurate ground states corresponding to known quantum Hall wavefunctions. For example, model Hamiltonians built from specific sets of V_m^{(n)} can realize Laughlin-type states or other correlated phases in finite-size systems. See also: Laughlin wavefunction; composite fermion.

Physical interpretation and implications

  • The pattern of V_m^{(n)} determines which many-body ground states are favored at a given filling fraction. Large penalties for small m discourage close electron pairs with high relative angular momentum, thereby promoting highly correlated states with nontrivial topology.
  • In the lowest Landau level, the canonical Laughlin states at filling fractions ν = 1/m (with m odd for fermions) can be understood as favored ground states under certain short-range pseudopotential configurations, where the energy penalty for small relative angular momentum suppresses configurations that would otherwise destabilize these states.
  • Higher Landau levels exhibit different pseudopotential hierarchies, which in turn influence the types of incompressible states that can emerge. Consequently, the landscape of possible quantum Hall phases changes with Landau level index n, the finite-thickness of the electron layer, and Landau level mixing.
  • The pseudopotential framework also clarifies debates about competing phases at certain fractions (for example, at ν = 5/2 and related filling factors). In this context, the relative magnitudes of V_1^{(n)} and other V_m^{(n)} can tilt the balance between candidate ground states (such as paired or non-Abelian states) and competing compressible or insulating phases. See also: Pfaffian state; antipPfaffian state; Moore-Read.

Extensions and related concepts

  • Three-body and higher-order pseudopotentials can be defined when considering more complex interaction effects or finite-width corrections, although the two-body Haldane pseudopotentials often dominate the qualitative physics of the fractional quantum Hall regime.
  • Finite-thickness corrections, Landau level mixing, and disorder all modify the effective pseudopotentials in experiments, which is a central reason why real samples may exhibit variations in the observed quantum Hall phenomenology compared to idealized models.
  • The pseudopotential formalism is not limited to electrons in semiconductor quantum wells. It also provides a general language for studying two-dimensional charged systems under magnetic fields in different materials and geometries.

See also