R. B. Laughlin's 1988 Science article argued that the spin-liquid state hypothesized for doped cuprate Mott insulators exhibits the same principles of fractional quantization as the fractional quantum Hall effect — fractionalized quasiparticles (neutral spin-1/21/2 spinons and, upon doping, spinless holons with ν=1/2\nu=1/2 anyonic statistics), an excitation gap, and long-range gauge forces — and that the statistical gauge attraction of charge-2e2e holon pairs provides a non-BCS mechanism of high-temperature superconductivity with TcT_c set by the spin gap. The article itself flagged the unresolved issues: existence of a spin liquid in dimensions higher than one, its fractional quantization, its occurrence in real high-TcT_c materials, the three-dimensional analog of fractional statistics (with possible confinement), the role of disorder and gaplessness, and the absence of quantitative predictions. Later work settled parts of this agenda: exactly solvable two- and three-dimensional spin-liquid models with anyonic or loop-braiding excitations exist, the anyon-superconductivity mechanism has been realized in idealized models, and fractional statistics has been directly observed — but only in fractional quantum Hall semiconductor devices. What remains open is the original real-matter core: no real more-than-one-dimensional material has been conclusively shown to host a spin liquid with fractionalized excitations; fractional statistics and gauge forces have been observed in no magnet or superconductor; and no superconductor is known to derive its pairing from fractionalization. The problem is to experimentally establish or exclude this fractional-quantization mechanism in real Mott materials, with quantitative predictions of gaps, dispersions, and doping dependence.

Background

A Mott insulator is a material in which strong electron–electron repulsion localizes the electrons even though band theory would predict a metal; its low-energy degrees of freedom are localized spins rather than mobile charges. Most Mott insulators develop static magnetic order at low temperature. P. W. Anderson proposed instead, in 1973 and again in 1987, that certain frustrated magnets realize a resonating valence bond (RVB) or spin-liquid state: a quantum-disordered ground state of spin-1/21/2 moments that keeps fluctuating among singlet pairings and never orders magnetically. Interest sharpened when the parent compounds of the high-temperature (high-TcT_c) superconducting cuprates — layered Mott insulators such as La2_2CuO4_4 — turned out to become superconducting upon chemical doping: if the undoped parent were a spin liquid, superconductivity would emerge from doping that exotic state rather than from a conventional metal.

In 1983 Laughlin explained the fractional quantum Hall effect (FQHE) — the quantization of the Hall conductance of a two-dimensional electron gas in a strong magnetic field to rational fractions of e2/he^2/h — as an incompressible quantum liquid whose elementary excitations carry a fraction of the electron charge rather than the electron charge itself. In a 1988 Science article, Laughlin argued that the hypothesized cuprate spin liquid realizes the same principles of fractional quantization. The construction has four ingredients: (i) a featureless liquid ground state; (ii) elementary excitations carrying a fraction of a quantum number — neutral spin-1/21/2 spinons, obtained conceptually by splitting a spin-1 spin flip in two, and, upon doping, spinless charged holons; (iii) an energy gap in the spin-wave (collective magnetic excitation) spectrum, analogous to the magnetoroton gap of the FQHE (the minimum energy of its neutral collective modes), which measures how 'liquid' the spins are; and (iv) long-range gauge forces acting between the fractionalized objects.

In two spatial dimensions, quantum statistics is not restricted to bosons and fermions: exchanging two identical particles may multiply the wave function by any phase eiπνe^{i\pi\nu} with 0<ν<10<\nu<1 (fractional statistics; such particles are called anyons), and the resulting interactions can be represented as a coupling to an emergent gauge field. This is impossible in one dimension, where exchange is meaningless, and in three dimensions point particles are restricted to bosons or fermions, although loop-like excitations can still braid. Laughlin proposed that the holons of the doped spin liquid are ν=1/2\nu=1/2 anyons (semions) and showed, in a Chern–Simons mean-field treatment — an approximation that replaces the statistics by an average effective magnetic field and whose control was never established — that such a gas forms a superfluid of charge-2e2e pairs. This gave a mechanism of superconductivity distinct from the phonon-mediated Bardeen–Cooper–Schrieffer (BCS) mechanism, with the transition temperature TcT_c set by the collapse of the spin gap, Δs≈kBTc\Delta_s \approx k_B T_c, and with the physically observable excitations (spin waves, charged currents, electrons) being confined bound pairs of the fractionalized objects.

The article's own closing concerns define the problem posed here: the existence of a spin-liquid state in more than one dimension was unproved — the only exactly solved paradigm was the one-dimensional Heisenberg chain, whose spin-1/21/2 spinons are permitted by the peculiar statistics of one dimension, where bosons and fermions are not sharply distinct; the fractional statistics and the gap were asserted by analogy with the FQHE rather than derived; whether these phenomena occur in real high-TcT_c superconductors was left open; the three-dimensional analog of fractional statistics was unclear, with the worry that three-dimensional gauge forces might become confining; the roles of disorder and gaplessness were unresolved; and quantitative predictions — the spin-wave dispersion, the gap sizes, and the doping dependence — were beyond the calculational tools of the time.

For the modern reader, two further conventions are needed. 'Fractionalization' now denotes the general phenomenon that emergent quasiparticles of a many-body ground state carry quantum numbers that are fractions of those of the microscopic degrees of freedom, and 'topological order' denotes the associated long-range entanglement that symmetry breaking cannot characterize. Since 1988 the theoretical landscape has changed drastically — exactly solvable two- and three-dimensional spin-liquid models now exist — so the open residue of the 1988 agenda is concentrated in real materials, which is where the problem below is posed.

Problem Statement

The problem is the experimentally open core of Laughlin's 1988 agenda, stated for real matter — in the source's venue, the doped cuprate high-TcT_c Mott insulators, and real more-than-one-dimensional Mott materials generally (the model-Hamiltonian existence part having been resolved in the intervening literature).

Establish — or conclusively exclude — that the form of fractional quantization identified in the fractional quantum Hall effect occurs in real more-than-one-dimensional Mott materials, and determine whether it accounts for superconductivity emerging from them. The following specific open facets of this single question are identified from the source's open concerns and the subsequent literature:

(1) Demonstrate, in a named real material, a spin-liquid ground state with fractionalized excitations — persisting to the lowest measured temperatures and free of magnetic order — and quantitatively determine its excitation gap, spinon dispersion, and their dependence on doping. The source's concern about the absence of quantitative predictions, and its disorder and gaplessness questions — whether disorder degrades the quantization or gaplessness masks the predicted gaps — are part of this determination; a controlled microscopic or effective calculation against which these quantities could be checked does not currently exist for doped two-dimensional Mott insulators.

(2) Detect, in such a material, the fractional statistics and the emergent gauge forces of its excitations — the analog, for neutral or charge-carrying fractionalized quasiparticles, of the fractional-charge and anyonic-braiding demonstrations achieved in quantum Hall devices.

(3) Settle the three-dimensional analog: whether a three-dimensional quantum spin liquid with loop-like fractionalization and an emergent gauge field exists in a real material (the classical spin-ice realization does not settle the quantum question), or whether ordering or confinement always wins in practice.

(4) Determine whether superconductivity that appears upon doping a spin-liquid insulator arises from the fractionalization mechanism — statistical gauge attraction of fractionalized charge carriers, with TcT_c tied to the spin gap — in the real doped Mott materials of the source's question, in particular the cuprates, discriminating it quantitatively from conventional BCS pairing; or establish that it does not.

Facets (1), (2), and (4) jointly address the overarching question of whether fractional quantization, as identified in the fractional quantum Hall effect, occurs in real more-than-one-dimensional Mott materials and accounts for the superconductivity that emerges from them; facet (3) is its separate three-dimensional extension. The class at issue is the real more-than-one-dimensional Mott materials, in particular the doped cuprates of the source's venue.

References

  1. R. B. Laughlin (1988). The Relationship Between High-Temperature Superconductivity and the Fractional Quantum Hall Effect. Science 242, 525–533. DOI: 10.1126/science.242.4878.525. https://doi.org/10.1126/science.242.4878.525
  2. P. W. Anderson (1973). Resonating valence bonds: A new kind of insulator? Materials Research Bulletin 8, 153–160. DOI: 10.1016/0025-5408(73)90167-0. https://doi.org/10.1016/0025-5408(73)90167-0
  3. P. W. Anderson (1987). The Resonating Valence Bond State in La2CuO4 and Superconductivity. Science 235, 1196–1198. DOI: 10.1126/science.235.4793.1196. https://doi.org/10.1126/science.235.4793.1196
  4. R. B. Laughlin (1983). Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations. Physical Review Letters 50, 1395–1398. DOI: 10.1103/PhysRevLett.50.1395. https://doi.org/10.1103/PhysRevLett.50.1395
  5. Yi-Hong Chen, Frank Wilczek, Edward Witten, Bertrand I. Halperin (1989). On anyon superconductivity. International Journal of Modern Physics B 3, 1001–1067. DOI: 10.1142/S0217979289000725. https://doi.org/10.1142/S0217979289000725
  6. C. E. Gough, M. S. Colclough, E. M. Forgan, R. G. Jordan, M. Keene, C. M. Muirhead, A. I. M. Rae, N. Thomas, J. S. Abell, S. Sutton (1987). Flux quantization in a high-Tc superconductor. Nature 326, 855. DOI: 10.1038/326855a0. https://doi.org/10.1038/326855a0
  7. K. B. Lyons, J. Kwo, J. F. Dillon, G. P. Espinosa, M. McGlashan-Powell, A. P. Ramirez, L. F. Schneemeyer (1990). Search for circular dichroism in high-Tc superconductors. Physical Review Letters 64, 2949–2952. DOI: 10.1103/PhysRevLett.64.2949. https://doi.org/10.1103/PhysRevLett.64.2949
  8. S. Spielman, Beth Parks, J. Orenstein, D. T. Nemeth, Frank Ludwig, John Clarke, Paul Merchant, D. J. Lew (1994). Quasiparticle Hall effect in superconducting YBa2Cu3O7−δ. Physica C: Superconductivity 235–240, 2021–2022. DOI: 10.1016/0921-4534(94)92232-2. https://doi.org/10.1016/0921-4534(94)92232-2
  9. Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, G. Saito (2003). Spin Liquid State in an Organic Mott Insulator with a Triangular Lattice. Physical Review Letters 91, 107001. DOI: 10.1103/PhysRevLett.91.107001. https://doi.org/10.1103/PhysRevLett.91.107001
  10. A. Yu. Kitaev (2003). Fault-tolerant quantum computation by anyons. Annals of Physics 303, 2–30. DOI: 10.1016/S0003-4916(02)00018-0. https://doi.org/10.1016/S0003-4916(02)00018-0
  11. Alexei Kitaev (2006). Anyons in an exactly solved model and beyond. Annals of Physics 321, 2–111. DOI: 10.1016/j.aop.2005.10.005. https://doi.org/10.1016/j.aop.2005.10.005
  12. Michael A. Levin, Xiao-Gang Wen (2005). String-net condensation: A physical mechanism for topological phases. Physical Review B 71, 045110. DOI: 10.1103/PhysRevB.71.045110. https://doi.org/10.1103/PhysRevB.71.045110
  13. Michael Hermele, Matthew P. A. Fisher, Leon Balents (2004). Pyrochlore photons: The U(1) spin liquid in a S=1/2 three-dimensional frustrated magnet. Physical Review B 69, 064404. DOI: 10.1103/PhysRevB.69.064404. https://doi.org/10.1103/PhysRevB.69.064404
  14. Chenjie Wang, Michael Levin (2014). Braiding Statistics of Loop Excitations in Three Dimensions. Physical Review Letters 113, 080403. DOI: 10.1103/PhysRevLett.113.080403. https://doi.org/10.1103/PhysRevLett.113.080403
  15. C. Castelnovo, R. Moessner, S. L. Sondhi (2008). Magnetic monopoles in spin ice. Nature 451, 42–45. DOI: 10.1038/nature06433. https://doi.org/10.1038/nature06433
  16. T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prabhakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow, S. T. Bramwell (2009). Magnetic Coulomb Phase in the Spin Ice Ho2Ti2O7. Science 326, 415–417. DOI: 10.1126/science.1177582. https://doi.org/10.1126/science.1177582
  17. Leon Balents (2010). Spin liquids in frustrated magnets. Nature 464, 199–208. DOI: 10.1038/nature08917. https://doi.org/10.1038/nature08917
  18. J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M. Bartlett, Y. Qiu, D. G. Nocera, Y. S. Lee (2010). Dynamic Scaling in the Susceptibility of the Spin-1/2 Kagome Lattice Antiferromagnet Herbertsmithite. Physical Review Letters 104, 147201. DOI: 10.1103/PhysRevLett.104.147201. https://doi.org/10.1103/PhysRevLett.104.147201
  19. Simeng Yan, David A. Huse, Steven R. White (2011). Spin-Liquid Ground State of the S=1/2 Kagome Heisenberg Antiferromagnet. Science 332, 1173–1176. DOI: 10.1126/science.1201080. https://doi.org/10.1126/science.1201080
  20. Tian-Heng Han, Joel S. Helton, Shaoyan Chu, Daniel G. Nocera, Jose A. Rodriguez-Rivera, Collin Broholm, Young S. Lee (2012). Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature 492, 406–410. DOI: 10.1038/nature11659. https://doi.org/10.1038/nature11659
  21. Mingxuan Fu, Takashi Imai, Tian-Heng Han, Young S. Lee (2015). Evidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnet. Science 350, 655–658. DOI: 10.1126/science.aab2120. https://doi.org/10.1126/science.aab2120
  22. Lucile Savary, Leon Balents (2016). Quantum spin liquids: a review. Reports on Progress in Physics 80, 016502. DOI: 10.1088/0034-4885/80/1/016502. https://doi.org/10.1088/0034-4885/80/1/016502
  23. R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu (1997). Direct observation of a fractional charge. Nature 389, 162–164. DOI: 10.1038/38241. https://doi.org/10.1038/38241
  24. H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J.-M. Berroir, E. Bocquillon, B. Plaçais, A. Cavanna, Q. Dong, U. Gennser, Y. Jin, G. Fève (2020). Fractional statistics in anyon collisions. Science 368, 173–177. DOI: 10.1126/science.aaz5601. https://doi.org/10.1126/science.aaz5601
  25. Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, Sixiao Ma, K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Motome, T. Shibauchi, Y. Matsuda (2018). Majorana quantization and half-integer thermal quantum Hall effect in a Kitaev spin liquid. Nature 559, 227–231. DOI: 10.1038/s41586-018-0274-0. https://doi.org/10.1038/s41586-018-0274-0
  26. J. A. N. Bruin, R. R. Claus, Y. Matsumoto, N. Kurita, H. Tanaka, H. Takagi (2022). Robustness of the thermal Hall effect close to half-quantization in α-RuCl3. Nature Physics 18, 401–405. DOI: 10.1038/s41567-021-01501-y. https://doi.org/10.1038/s41567-021-01501-y
  27. Aaron T. Breidenbach, Arthur C. Campello, Jiajia Wen, Hong-Chen Jiang, Daniel M. Pajerowski, Rebecca W. Smaha, Young S. Lee (2025). Identifying universal spin excitations in candidate spin-1/2 kagome quantum spin liquid materials. Nature Physics 21, 1957–1964. DOI: 10.1038/s41567-025-03069-3. https://doi.org/10.1038/s41567-025-03069-3
  28. Stefan Divic, Valentin Crépel, Tomohiro Soejima, Xue-Yang Song, Andrew J. Millis, Michael P. Zaletel, Ashvin Vishwanath (2025). Anyon superconductivity from topological criticality in a Hofstadter–Hubbard model. Proceedings of the National Academy of Sciences 122 (33), e2426680122. DOI: 10.1073/pnas.2426680122. https://doi.org/10.1073/pnas.2426680122
  29. Cheong-Eung Ahn, Donghae Seo, Gyeoul Lee, Youngwook Kim, Gil Young Cho (2026). Superconductivity proximate to non-Abelian fractional spin Hall insulator in twisted bilayer MoTe2. Nature Communications 17, 7005. DOI: 10.1038/s41467-026-73777-8. https://doi.org/10.1038/s41467-026-73777-8
  30. Avi Shragai, Ezekiel Horsley, Subin Kim, Young-June Kim, B. J. Ramshaw (2026). Phonon Hall viscosity and the intrinsic thermal Hall effect of α-RuCl3. Nature 652, 1166–1172. DOI: 10.1038/s41586-026-10420-y. https://doi.org/10.1038/s41586-026-10420-y