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

The 1983 paper by F. D. M. Haldane that introduced the hierarchy of incompressible quantum fluid states for the fractional quantum Hall effect explicitly left open a microscopic question: for a given two-body interaction (paradigmatically the Coulomb interaction projected into the lowest Landau level), which of the hierarchical incompressible fluid states actually exist in the thermodynamic limit, with what excitation gaps, and which of them survive against competing phases such as the Wigner crystal. Later literature substantially built the requested machinery — exact diagonalization with finite-size scaling on the Haldane sphere, pseudopotential analysis, and composite-fermion and parton constructions (partons: trial states built by splitting each electron into charged constituents) — and settled the principal fractions (1/3, 2/5, 3/7, ...). However, the determination for higher-generation hierarchy states (such as the $\nu=4/11$ state), the validity of the original quasiparticle-condensation assumption at higher hierarchy levels, and which states survive against the competing Wigner crystal for a specified interaction remain open; direct evidence shows the original assumption fails in at least one studied case.

Background

When a two-dimensional electron gas is placed in a strong perpendicular magnetic field, its kinetic energy is quantized into highly degenerate Landau levels, and the fractional quantum Hall effect (FQHE) occurs when the highest occupied Landau level is only fractionally filled, at a filling factor $\nu$ (the ratio of electron density to the density the level can hold). The integer effect ( $\nu$ an integer) is explained by single-particle level filling, but the fractional effect requires many-body correlations. Laughlin (1983) proposed variational many-body wave functions at fillings $\nu=1/m$ with odd $m$ describing an 'incompressible quantum fluid': a homogeneous liquid whose lowest excitation costs a finite energy gap, which is the condition for a quantized Hall plateau with vanishing longitudinal resistance. Its fractionally charged quasiparticle excitations carry charge $\pm e/m$ .

Haldane (1983) reformulated this physics on the 'Haldane sphere': $N$ electrons confined to the surface of a sphere threaded by $2S$ units of magnetic flux (a Dirac monopole at the center), where the single-particle states are monopole harmonics (the spherical analogues of the planar Landau-level orbitals) forming a finite-dimensional lowest-Landau-level (LLL) Hilbert space. This geometry is rotationally invariant (unlike a droplet, which breaks translational symmetry at its edge) and makes the LLL interaction problem, for finite $N$ , a finite matrix that can be diagonalized numerically. On the sphere the interaction enters through pseudopotentials $V_J^{(S)}$ : the interaction energy of a pair of electrons with total pair angular momentum $J$ ; Laughlin correlations mean avoiding pair states with the largest $J$ (smallest separations).

Haldane then constructed the hierarchy. If a Laughlin fluid at $\nu=1/m$ is slightly over- or under-filled, the excess or deficit appears as dilute quasiparticles (particles or holes of fractional charge $\pm e/m$ ) that, on the sphere, behave as bosons (their exchange statistics is trivial because the two-particle and two-hole excitation states are symmetric in the excitation coordinates). If the interaction energy of these quasiparticles is small compared with the gap separating them from other excitations, their ground-state problem is analogous to the original electron problem with Fermi statistics replaced by Bose statistics, so they can condense into a Laughlin fluid of their own at quasiparticle filling $1/p$ with even $p$ . Iterating this construction yields incompressible daughter states at continued fractions $\nu=1/(m\pm1/(p_1\pm\dots))$ , covering all odd-denominator filling factors, on the sphere at flux $S(N;m,p_1,\dots,p_n)$ with divisibility conditions on $N$ .

The closing discussion of the source states the unresolved issue explicitly: 'What is so far missing is a calculational scheme for the direct determination of whether a given fluid state exists for a given interaction potential, e.g., the Coulomb interaction.' The paper proposes the needed ingredients — evaluation of the pseudopotential coefficients $V_J^{(S)}$ and the pair-angular-momentum weights of trial wave functions for variational correlation energies, and direct numerical diagonalization of the low-lying levels at increasing $N$ with finite-size scaling of the gap (for example the charge gap, the energy to create a widely separated quasiparticle-quasihole pair, or the neutral collective-excitation gaps) toward the planar thermodynamic limit — and notes the stability competition against the Wigner crystal (a periodic electron solid, expected at low filling, which is gapless and therefore cannot quantize the Hall resistance). Whether the weakly-interacting-bosonic-quasiparticle assumption itself holds at higher hierarchy levels was flagged as an assumption, not a result. The question is the microscopic phase diagram problem of the FQHE: which incompressible fluids does a given interaction actually support?

Problem Statement

For a specified microscopic two-body interaction — paradigmatically the Coulomb interaction projected into the lowest Landau level, with the interaction convention stated (the pair-distance convention on the sphere, chord or geodesic; the transverse width of the confinement layer; and whether Landau-level mixing is included or projected away, i.e., whether single-particle states outside the lowest Landau level are admixed) — determine directly, in the planar thermodynamic limit, which hierarchy incompressible quantum fluid states actually exist and are stable. This is the source's open request: 'a calculational scheme for the direct determination of whether a given fluid state exists for a given interaction potential, e.g., the Coulomb interaction', and thereby which hierarchy incompressible fluid states the specified interaction supports, and with what excitation gaps. Two conditions belong to the existence question itself: the hierarchy construction rests on the conditional assumption that the low-energy quasiparticles are sufficiently weakly interacting and effectively bosonic to condense into Laughlin-like daughter fluids, so the validity of that assumption at a given hierarchy level enters whether the corresponding daughter state exists; and stability must be assessed against the competing gapless Wigner solid (a periodic electron solid) that the source identifies at low filling, which cannot quantize the Hall resistance. For a single interaction convention declared under the conditions above, determine which of the hierarchy states $\nu=1/3$ , $2/5$ , $3/7$ , and $4/11$ exist and are stable, what their charge gaps are (the energy to create a widely separated quasiparticle-quasihole pair), whether they survive against the competing Wigner solid, and whether the quasiparticle-condensation assumption holds at each hierarchy level.

References

F. D. M. Haldane, 'Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States', Physical Review Letters 51, 605-608 (1983). DOI: 10.1103/PhysRevLett.51.605, https://doi.org/10.1103/PhysRevLett.51.605 (source paper)
R. B. Laughlin, 'Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations', Physical Review Letters 50, 1395-1398 (1983). DOI: 10.1103/PhysRevLett.50.1395, https://doi.org/10.1103/PhysRevLett.50.1395
B. I. Halperin, 'Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States', Physical Review Letters 52, 1583-1586 (1984). DOI: 10.1103/PhysRevLett.52.1583, https://doi.org/10.1103/PhysRevLett.52.1583
F. D. M. Haldane and E. H. Rezayi, 'Finite-Size Studies of the Incompressible State of the Fractionally Quantized Hall Effect and its Excitations', Physical Review Letters 54, 237-240 (1985). DOI: 10.1103/PhysRevLett.54.237, https://doi.org/10.1103/PhysRevLett.54.237
J. K. Jain, 'Composite-Fermion Approach for the Fractional Quantum Hall Effect', Physical Review Letters 63, 199-202 (1989). DOI: 10.1103/PhysRevLett.63.199, https://doi.org/10.1103/PhysRevLett.63.199
Arkadiusz Wójs, Kyung-Soo Yi, and John J. Quinn, 'Fractional quantum Hall states of clustered composite fermions', Physical Review B 69, 205322 (2004). DOI: 10.1103/PhysRevB.69.205322, https://doi.org/10.1103/PhysRevB.69.205322 (arXiv:cond-mat/0312290, https://arxiv.org/abs/cond-mat/0312290)
Arkadiusz Wójs, George Simion, and John J. Quinn, 'Spin phase diagram of the $\nu_e=4/11$ composite fermion liquid', Physical Review B 75, 155318 (2007). DOI: 10.1103/PhysRevB.75.155318, https://doi.org/10.1103/PhysRevB.75.155318 (arXiv:cond-mat/0612610, https://arxiv.org/abs/cond-mat/0612610)
John J. Quinn, Arkadiusz Wójs, Kyung-Soo Yi, and George Simion, 'The hierarchy of incompressible fractional quantum Hall states', Physics Reports 481, 29-81 (2009). DOI: 10.1016/j.physrep.2009.06.002, https://doi.org/10.1016/j.physrep.2009.06.002 (arXiv:0906.5182, https://arxiv.org/abs/0906.5182)
Koyena Bose and Ajit C. Balram, 'Prediction of non-Abelian fractional quantum Hall effect at $\nu=2+4/11$ ', Physical Review B 107, 235111 (2023). DOI: 10.1103/PhysRevB.107.235111, https://doi.org/10.1103/PhysRevB.107.235111 (preprint arXiv:2302.02478, https://arxiv.org/abs/2302.02478)