Haldane Halperin HierarchyEdit
The Haldane–Halperin hierarchy is a theoretical framework within the study of the fractional quantum Hall effect. It describes a systematic way to generate new incompressible quantum fluids by considering the quasiparticles of a parent fractional quantum Hall state as their own two-dimensional system capable of forming a correlated, liquid-like state. This approach extends the idea of Laughlin states and provides a structured way to understand a broad family of observed filling factors in two-dimensional electron systems under high magnetic fields. For the broader physical context, see Fractional Quantum Hall Effect and Laughlin state.
The hierarchy was developed by F. D. M. Haldane and B. I. Halperin in the 1980s as part of the effort to map the rich landscape of collective phenomena that emerge from strong interactions in a partially filled Landau level. Their idea rests on the observation that the excitations of an incompressible liquid—quasiparticles carrying fractional charge and anyonic statistics—can themselves organize into an additional, correlated quantum Hall state. In this sense, a cascade of generations of states can arise: a parent state at a given filling factor can give rise to daughter states at new filling factors through the condensation of its quasiparticles, and those daughter states can, in turn, host their own quasiparticle condensations. See Haldane–Halperin hierarchy and Laughlin wavefunction for foundational connections.
Overview
The basic setting is a two-dimensional electron gas in a strong perpendicular magnetic field. Electrons occupy Landau levels, and interactions between electrons drive the system into a variety of correlated phases. The most famous example is the Laughlin state at ν = 1/m (with m an odd integer), which provides a simple, highly successful description of several experimentally observed fractions. See Landau level and Laughlin state.
The Haldane–Halperin scheme generalizes this by treating the quasiparticles of a parent Laughlin state as a new “species” that can form its own quantum Hall liquid. When these quasiparticles condense, they yield a new incompressible state at a different filling factor. This process can, in principle, be repeated to generate further generations of states, producing a hierarchical structure of fractions. See quasiparticle and Topological order.
In practice, the hierarchy illuminates why many fractional fillings appear in experiments and how they relate to each other. Early fractions such as 2/5 and 3/7 are often cited as characteristic descendants of the primary Laughlin states. The framework also connects to broader ideas about topological order and long-range quantum entanglement in condensed matter systems. See Fractional Quantum Hall Effect and Topological order.
Generating mechanism
Starting from a parent FQHE state (for example, a Laughlin state at ν = 1/m), the system supports quasiparticle excitations that carry fractional charge and obey anyonic statistics. If the density and interactions permit, these quasiparticles can themselves form an incompressible liquid, effectively creating a new two-dimensional electron-like system within the original. This yields a new filling factor and a new set of excitations. See quasiparticle and Anyons.
The process can be iterated: each new descendant state may have its own quasiparticles, which could condense into their own hierarchical state. In this way the Haldane–Halperin picture envisions a nested family of quantum Hall liquids, each linked to its parent by the physics of fractional charge, statistics, and strong correlations. See Haldane–Halperin hierarchy.
The scheme is often discussed in relation to, and in dialogue with, alternative explanations of the FQHE, most notably the composite fermion picture. The composite fermion approach reinterprets many observed fractions as integer quantum Hall states of bound electron–vortex objects, while the HH hierarchy emphasizes a condensation sequence of quasiparticles. See Composite fermion.
Experimental context and interpretation
The fractional quantum Hall effect has been observed in high-midelity two-dimensional electron systems, particularly in GaAs-based heterostructures, where extremely clean samples enable precise Hall plateau quantization at fractional values. The HH hierarchy provides one language to categorize and relate many of these observed fractions. See GaAs and Fractional Quantum Hall Effect.
In practice, experimental observations have both supported and challenged aspects of the hierarchical picture. Some fractions align well with HH predictions, while others are more naturally understood within the composite fermion framework or other theoretical constructions. The ongoing dialogue among theories reflects the richness of electron interactions in partially filled Landau levels and the subtleties of real-world materials. See Read–Rezayi states for a broader family of topological phases that enrich this discussion.
Beyond static filling factors, researchers study quasiparticle charge measurements, interferometry experiments aimed at exposing anyonic statistics, and energy gaps that characterize the stability of these liquids. These investigations inform how faithfully the HH hierarchy captures the underlying physics and where its description intersects with or diverges from alternative models. See Quasiparticle and Anyons.
Variants and related theories
The Haldane–Halperin hierarchy is one among several frameworks for organizing the FQHE landscape. While the original hierarchy emphasized successive quasiparticle condensations, other approaches, such as the composite fermion model, map observed fractions to simpler effective problems, offering complementary intuition and predictive power. See Haldane–Halperin hierarchy and Composite fermion.
More recent developments explore non-Abelian quantum Hall states and more intricate topological orders, such as Read–Rezayi states, which extend the concept of hierarchy into richer statistics and degeneracy patterns. See Read–Rezayi states.
The broader context includes topological phases of matter and the role of long-range entanglement in stabilizing exotic excitations. See Topological order.