Quantum spin liquids and superconductivity: models and evidence

Scope

A quantum spin liquid is a quantum magnetic state with correlations and excitations beyond a conventional ordered magnet. Absence of static magnetic order alone is not a definition that identifies it experimentally: a trivial singlet state, disorder or a finite measurement window can also suppress the expected signal. Candidate materials require positive evidence and tests of alternatives. Chiral spin liquids can break time-reversal symmetry, so preserving every conventional symmetry is not a universal requirement. R1 R2 R6

The connection to superconductivity is a research route, not an automatic consequence. Anderson's RVB proposal motivates doping a correlated singlet state, but undoped La2CuO4 has antiferromagnetic order. A spin-only Hamiltonian has no mobile electrical charge or superconducting stiffness. Emergent neutral Majorana fermions in such a model do not make its material a topological superconductor. A specific charge sector and gauge-invariant superconducting response must be established separately. R1 R3 R11 R12

Mathematical assumptions

An explicit counterexample clarifies the diagnostic problem. A product of disconnected spin singlets has $\langle\mathbf S_i\rangle=0$ and vanishing correlations between different dimers, yet is not thereby a long-range-entangled spin liquid. For two spin-one-half singlets, direct matrix evaluation gives $\langle\mathbf S_1\cdot\mathbf S_2\rangle=-3/4$ and $\langle\mathbf S_1\cdot\mathbf S_3\rangle=0$. These finite-state checks disprove the inference from zero local moments alone; they do not classify an interacting bulk material. R2 R6

Use the honeycomb convention $H=-\sum_{\langle ij\rangle_\alpha}J_\alpha\sigma_i^\alpha\sigma_j^\alpha$, with Pauli matrices $\sigma=2S$. A Hamiltonian written with spin operators requires couplings $K_\alpha=4J_\alpha$ for the same energy. The vortex-free zero-field matter spectrum has zeros when the three $|J_\alpha|$ satisfy triangle inequalities. Kitaev calls this the gapless B region, including isotropy; the dominant-coupling A regions are gapped. The source draft had these labels reversed. R1

Near isotropy, a weak field with all three spin-axis components nonzero generates a time-reversal-breaking term scaling as $h_xh_yh_z/J^2$ at third order. This perturbative result is not exact solvability at arbitrary field. For an ideal equilibrated chiral Majorana edge above a gapped bulk, $\kappa_{xy}^{2D}/T=c_-\pi^2k_B^2/(3h)$ with $c_-=\pm1/2$. This is thermal conductance per layer, not three-dimensional conductivity or electrical Hall conductance. Independent checks verify the phase-boundary algebra, spin normalization and positive-energy thermal integral. R1

For a gapped two-dimensional topologically ordered ground state and a sufficiently large smooth region, $S=\alpha L-\gamma+\cdots$ with $\gamma=\ln\mathcal D$ and $\mathcal D=\sqrt{\sum_a d_a^2}$. The Abelian Z2 example gives $\gamma=\ln2$. This expression is not a universal entropy formula for every gapless spin liquid, and a single entropy constant does not fully specify anyon statistics. Finite geometry, boundaries and correlation length matter before identifying a universal term. R2

Experimental criteria

Shimizu's organic candidate kappa-(BEDT-TTF)2Cu2(CN)3 shows no proton-NMR signature of long-range magnetic order down to 32 mK. The single-crystal experiment applied 2.2 T perpendicular to the conducting planes; the susceptibility-derived exchange scale around 250 K came from a separate fit. Neither number is a superconducting transition temperature, and the experiment does not directly observe the zero-field, zero-temperature ground state. Kurosaki's pressure phase diagram separately connects the correlated insulator with metallic and superconducting regions; it does not identify a unique pairing mechanism. R4 R5

Pustogow's 2023 strain-transport study instead favors a low-entropy, spin-gapped valence-bond-solid interpretation of the organic material's 6 K anomaly. Its hydrostatic-equivalent pressure is inferred from comparison with transport data, rather than directly applied hydrostatic pressure. The paper retains a gapped chiral-spin-liquid alternative and discusses mounting-related microcracks. Thus a revised interpretation must preserve both the measured anomaly and the remaining discrimination problem. R6

Han's neutron study used deuterated herbertsmithite, ZnCu3(OD)6Cl2, at 1.6 K. Broad momentum-energy scattering supports the authors' fractionalization interpretation, but the measured low-energy signal can include interlayer Cu defects. The apparent absence of a gap down to the measured energy limit does not establish a universally gapless intrinsic kagome state. A later primary study explicitly analyzes correlated impurity contributions. Preserve isotope, background subtraction, resolution and momentum window before comparing these results with a theoretical continuum. R7 R8

In alpha-RuCl3, zero-field magnetic order and candidate field-induced behavior must be separated. Xing's 2025 ultraclean-crystal study reports a narrower near-half-quantized thermal Hall plateau and weakened oscillation-like features. It discusses sample-dependent magnetic boundaries, possible stacking faults and phonon-edge coupling. The 5.0 K Hall measurement used heat current along b and field along a; the comparison crystal was measured at 4.3 K. These are not identical conditions or a universal material threshold. R9

Shragai's 2026 ultrasound study measures acoustic Faraday rotation and infers a phonon Hall-viscosity contribution. Its conversion to thermal transport requires assumptions about frequency, phonon branches and viscosity components; the directly measured single-component estimate does not independently measure every thermal carrier. One displayed comparison also combines 2 K, 55-degree-tilted ultrasound with 4.9 K, 45-degree-tilted thermal data. This supplies substantive phonon evidence without making every field geometry or claimed quantized plateau equivalent. R10

Limits and disputes

A continuum, a low-temperature heat-capacity term, absent NMR splitting and a thermal Hall signal test different observables. Their interpretation depends on defects, phonons, magnetic order, probe time scales and sample history. Do not turn one positive signal into a sufficient spin-liquid criterion, or one disputed material interpretation into a claim about every possible quantum spin liquid. Broad statements that all materials are settled or that none can ever be established would exceed this review. R4 R6 R7 R9 R10

Complete scientific acceptance remains open. The present check covers selected primary passages, figures and restricted independent models. Several references retain abstract-only or earlier manuscript scopes; complete final-version comparisons, defect subtraction, thermal calibration and raw-data reproduction remain unfinished. The exact-model results and experimental interpretations are therefore presented at different evidence levels. R1 R2 R4 R5 R7 R8 R9 R10

Discovery implications

Track the insulating magnetic state, tuning parameter and superconducting response independently. For a pressure or doping trajectory, distinguish structural changes and disorder from changed bandwidth or filling. Compare several probes on characterized samples and ask what would exclude a trivial singlet, ordered or disordered alternative. A successful spin-liquid model provides testable mechanisms and excitations; establishing superconductivity additionally requires mobile charge, phase coherence and the corresponding experimental criteria. R5 R6 R11 R12

Related topics

Spin-liquid routes to superconductivity, RVB pairing, Organic superconductors, and Mottness.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.