Quantum Spin Liquids

A quantum spin liquid is a magnetic quantum phase without conventional magnetic order that evades a short-range-entangled product-state description. Its spins remain strongly correlated, and its low-energy organization is typically expressed through fractionalized excitations, emergent gauge structure, or, for a gapped liquid in two dimensions, intrinsic topological order. Preservation of every discrete symmetry is not required: a chiral spin liquid breaks time reversal while retaining this nontrivial organization.

The name does not mean that individual spins fluctuate independently or that the phase is a thermal liquid. It refers to a many-body ground state and its low-temperature regime. Nor is absence of a magnetic Bragg peak enough: a trivial quantum paramagnet, a valence-bond solid, a spin glass, disorder-broadened excitations, a one-dimensional critical magnet, or a system whose ordering temperature lies below the measurement window can all look magnetically quiet.

Usage varies at the dimensional boundary. The antiferromagnetic spin- $1/2$ chain is often called a gapless spin liquid because it preserves spin rotation and has fractional spinons. Its criticality and fractionalization are nevertheless specifically one-dimensional. This page uses quantum spin liquid primarily for candidate deconfined phases in two- and three-dimensional materials and links to Heisenberg Chain for the exact one-dimensional benchmark.

Required background. Heisenberg Model supplies spin exchange and frustrated lattice benchmarks. Antiferromagnetism supplies conventional ordering and magnetic diffraction tests. Structure Factors supplies the momentum and energy resolution of spin correlations, while Topological Order Preview supplies the long-range-entanglement, ground-state-sector, and anyon diagnostics used for gapped spin liquids.

What a Spin Liquid Is Not

A missing order parameter is only an exclusion

Take the spin operators to be dimensionless, so exchange couplings have units of energy. For a candidate ordering wavevector $\mathbf Q$ , define the finite-size magnetic estimator

m_{\mathbf Q}^2(N) = \frac{1}{N^2} \sum_{i,j} e^{i\mathbf Q\cdot(\mathbf r_i-\mathbf r_j)} \left\langle \mathbf S_i\cdot\mathbf S_j \right\rangle.

Magnetic long-range order requires a nonzero thermodynamic limit of this quantity, with finite-size and domain effects treated consistently. Neutron diffraction, magnetic susceptibility, nuclear magnetic resonance, and muon spin rotation probe different consequences of order. Their null results become persuasive only when their time windows, volume fractions, moment sensitivity, and structural backgrounds agree.

The following alternatives must be tested separately.

Alternative	How it differs from a spin liquid
Trivial quantum paramagnet	Local singlets or a large single-ion term allow a gapped path to an unentangled state of unit cells.
Valence-bond solid	Translation or rotation breaks; look for bond anisotropy and symmetry-related patterns.
Spin glass	Moments freeze irregularly; look for aging, history dependence and a frozen local component.
One-dimensional critical magnet	Algebraic correlations and spinons arise from one-dimensional physics; interchain coupling can change the ground state.
Weakly ordered magnet	Small moments or a low transition scale can escape existing resolution.
Random-singlet regime	Broad exchange distributions produce low-energy singlets and inhomogeneous relaxation.

Glasses and Spin Glasses owns the freezing, aging, and disorder protocols. Antiferromagnetism owns the ordered benchmark, including small-moment and frustrated antiferromagnets.

A finite-temperature regime is not automatically a phase

Experiments operate at $T>0$ . A material with exchange scale $J$ and ordering temperature $T_N\ll J/k_{\mathrm B}$ can show a broad correlated-paramagnet regime,

T_N \ll T \ll J/k_{\mathrm B},

whose dynamics resemble a spin liquid even though its ground state orders. This proximate spin-liquid regime can be scientifically important, as in several Kitaev materials, but it should not be relabeled a zero-temperature spin-liquid phase.

Calculated lower and upper two-particle energy boundaries for a one-dimensional cosine dispersion — A continuum comes from sharing a fixed total momentum in many ways. For the toy band $\varepsilon(k)=\Delta+2t[1-\cos(ka)]$ with $\Delta=t/2$ , the shaded region contains the pair energies $\varepsilon(k)+\varepsilon(q-k)$ . Its boundaries are calculated in Exercise 3. Shading marks allowed energies, not neutron intensity; matrix elements can suppress parts of the region. The same kinematics applies to ordinary particles, so this one-dimensional illustration is not a diagnosis of a material spin liquid.

Frustration and Quantum Fluctuations

Frustration creates competition, not a theorem

For antiferromagnetic exchange,

H_{\mathrm{spin}} = \sum_{i<j} J_{ij} \mathbf S_i\cdot\mathbf S_j +H_{\mathrm{anis}} +H_{\mathrm{ring}} +H_{\mathrm{dis}}.

On a triangle with all $J_{ij}>0$ , no collinear arrangement satisfies every bond. Kagome and pyrochlore lattices extend this geometric competition; further-neighbor or ring exchanges can generate frustration even on simpler lattices. Quantum fluctuations are strongest for small spin, low coordination, competing interactions, and low dimension.

Frustration suppresses simple order and produces many low-energy configurations, but it can also select a noncollinear magnet, a valence-bond crystal, a nematic state, or a glass. The ratio $\lvert\Theta_{\mathrm{CW}}\rvert/T_N$ is therefore a useful frustration index only when a Curie–Weiss fit is valid and $T_N$ is actually measured. A large ratio does not diagnose entanglement or fractionalization.

Resonating valence bonds

A singlet on sites $i,j$ is

\lvert s_{ij}\rangle = \frac{1}{\sqrt2} \left( \lvert\uparrow_i\downarrow_j\rangle - \lvert\downarrow_i\uparrow_j\rangle \right).

A resonating-valence-bond state superposes many singlet coverings instead of choosing one static pattern. The superposition may be short-range and gapped, algebraic and gapless, chiral, or unstable to valence-bond order. “RVB” names a construction principle, not one phase.

Lieb–Schultz–Mattis–Oshikawa–Hastings constraints

For a translation-invariant short-range spin system with an odd number of spin- $1/2$ moments per primitive cell and the needed spin symmetry, a unique, gapped, symmetric, short-range-entangled ground state is excluded. The infrared alternatives include:

spontaneous symmetry breaking;
gapless excitations;
a gapped phase with topological order.

This is a structural constraint, not proof that a particular Hamiltonian is a spin liquid. Real spin–orbit-coupled crystals may preserve less than full spin rotation, and nonsymmorphic or other crystalline symmetries can modify the filling constraint. The microscopic symmetry representation per primitive cell must be identified before invoking the theorem.

Spinons and Emergent Gauge Structure

Partons are a representation

A common fermionic parton representation is

\mathbf S_i = \frac12 f_{i\alpha}^{\dagger} \boldsymbol\sigma_{\alpha\beta} f_{i\beta}, \qquad \sum_{\alpha} f_{i\alpha}^{\dagger}f_{i\alpha} = 1.

The enlarged variables have a local redundancy. At minimum,

f_{i\alpha} \longrightarrow e^{i\theta_i} f_{i\alpha}

leaves the spin operator invariant. More complete formulations expose an SU(2) gauge redundancy before a mean-field ansatz reduces the invariant gauge group. The gauge field is not an extra microscopic force inserted by hand; it enforces the local redundancy and can acquire its own long-distance dynamics.

A mean-field spinon band is not yet a physical excitation. Gauge fluctuations can confine the partons, drive symmetry breaking, or qualitatively change their spectrum. Fractionalization is physical only when gauge-invariant observables and the stability of a deconfined phase support it.

What a local spin probe creates

When spin rotation is a symmetry, a local spin operator carries integer spin and can create a pair of spin- $1/2$ spinons with integer total spin. More generally, even without conserved spin, a local operator cannot create an isolated nontrivial topological charge in the bulk; the complete excitation must have trivial total topological charge. Sharing energy and momentum among excitations naturally produces continua, although matrix elements and bound states can also create sharp features.

The converse fails. Two magnons, damped paramagnons, phonons, random singlets, and spatially broadened crystal-field transitions also form continua. A convincing fractionalization claim predicts continuum boundaries, polarization, momentum dependence, temperature transfer of spectral weight, and field evolution from one microscopic Hamiltonian.

Gapped and Gapless Families

Phase family	Excitations and stability questions
Gapped $\mathbb Z_2$ liquid	Spinons and gauge fluxes (visons), with four ideal torus sectors; defects can obscure intrinsic gaps.
Dirac spin liquid	Nodal spinons and a gauge field; allowed masses or monopoles can destabilize the phase.
Spinon Fermi surface	Neutral fermions on a momentum-space surface coupled to U(1) fluctuations; heat response is affected by interactions and disorder.
Quantum spin ice	A three-dimensional U(1) photon and gapped electric/magnetic gauge charges; additional interactions can select order.
Chiral spin liquid	Anyons and a neutral chiral edge with broken time reversal; phonon Hall response can obscure edge transport.

Gapped liquids

In a gapped two-dimensional spin liquid, local correlations can decay exponentially even though the state is long-range entangled. Topology-dependent ground sectors, deconfined anyons, and a universal entanglement correction distinguish it from a trivial dimer paramagnet. These universal data are developed in Topological Order; a material experiment normally sees only projections of them.

A spin gap extracted from Knight shift, relaxation, heat capacity, or neutron scattering must identify which excitation and momentum is gapped. Defect spins can add low-energy spectral weight without closing the intrinsic kagome-layer gap. Conversely, an apparent activation law can arise from a finite field, a bottleneck, or limited resolution.

Gapless liquids

Gapless spin liquids need not possess intrinsic topological order in exactly the same fully gapped sense, but they can still have stable emergent gauge structure and fractionalized excitations. Their power laws depend on dimension, symmetry, and gauge dynamics. A linear heat-capacity term in an electrical insulator is suggestive of neutral gapless matter, yet nuclear Schottky terms, disorder, two-level systems, and phonons must be removed before assigning a spinon density of states.

The low-temperature longitudinal thermal conductivity is often extrapolated as

\frac{\kappa_{xx}}{T} = a_0 +bT^{p-1}.

Here $a_0=\lim_{T\to0}\kappa_{xx}/T$ is the extrapolated intercept and $p>1$ describes a fitted phonon power. A nonzero $a_0$ can support mobile gapless neutral carriers, but boundary-limited phonons and sample cracks make the extrapolation delicate. A vanishing intercept does not rule out gapless spinons if they are strongly scattered or localized.

The Kitaev Spin-Liquid Benchmark

Bond-directional exchange

For spin $1/2$ , the honeycomb Kitaev Hamiltonian couples one spin component on each bond type,

H_K = \sum_{\langle ij\rangle_x} K_xS_i^xS_j^x + \sum_{\langle ij\rangle_y} K_yS_i^yS_j^y + \sum_{\langle ij\rangle_z} K_zS_i^zS_j^z.

Despite strong interactions, the model is exactly solvable. Writing each spin with four Majoranas,

S_i^{\gamma} = \frac{i}{2} b_i^{\gamma}c_i, \qquad b_i^xb_i^yb_i^zc_i = 1,

turns the Hamiltonian into itinerant $c$ Majoranas moving in a static $\mathbb Z_2$ gauge background. In the enlarged Majorana Hilbert space, the oriented-bond variable $u_{ij}=ib_i^{\gamma}b_j^{\gamma}=\pm1$ is conserved but gauge dependent. The corresponding plaquette flux is gauge invariant; projection imposes the displayed local constraint on physical states.

At the isotropic point the zero-field Majorana spectrum is gapless. Sufficient exchange anisotropy produces a gapped Abelian phase. Starting in the gapless phase, a weak time-reversal-breaking perturbation with all three field components gaps the cones and yields a non-Abelian Ising topological phase with $\lvert c_-\rvert=1/2$ . The sign depends on the perturbation’s handedness and the stated edge orientation. Its ideal two-dimensional edge therefore carries half of the thermal conductance quantum in magnitude. The general $c_-$ relation and its equilibration caveats live in Topological Order.

A local spin flip changes the neighboring gauge fluxes as well as exciting the itinerant Majorana sector. At zero temperature, the neutron response can therefore have a nonzero threshold set by flux creation even when the Majorana band is gapless (Knolle and collaborators, 2014). A measured spin-response gap is not automatically the gap of every excitation; the probe’s operator matters.

From exact model to crystal

Spin–orbit-entangled ions in edge-sharing octahedra can generate dominant bond-directional exchange, making honeycomb iridates and $\alpha$ -RuCl $_3$ important Kitaev-material platforms. Real Hamiltonians also contain Heisenberg exchange, symmetric off-diagonal $\Gamma$ terms, further-neighbor coupling, interlayer exchange, phonons, and disorder.

$\alpha$ -RuCl $_3$ orders in a zigzag pattern at zero field, so it is not a zero-field spin liquid. Its high-energy continuum and field-suppressed order support proximity to Kitaev physics and fractionalization-compatible dynamics. Reports of thermal Hall response near the half-quantized value in selected field and temperature windows are important, but sample dependence, a sizable phonon Hall background, nonquantized datasets, and alternative magnon or phonon descriptions keep the field-induced topological-phase assignment active rather than settled.

How Experiments Distinguish the Possibilities

Start with the magnetic degrees of freedom: valence, crystal-field levels, charge gap, effective moment and exchange tensor. A pseudospin doublet can differ substantially from a bare electron spin. Compare diffraction and local probes over their respective time windows, and test freezing where disorder is appreciable.

The strongest positive tests depend on the proposed phase: a spinon continuum with predicted momentum and polarization dependence, separate flux and matter energy scales, a neutral heat channel, or the photon response of a three-dimensional spin ice. Compare each with the same microscopic model used for thermodynamics and its evolution in field or pressure. Neutron Scattering explains instrument and normalization issues; Magnetic Susceptibility explains Curie tails and freezing tests.

Probe	Useful signal and ambiguity
Diffraction	Tests static periodic order; tiny moments or small volume fractions can escape detection.
$\mu$ SR or NMR	Tests local fields and dynamics; time windows, probe perturbation and signal loss matter.
Susceptibility	Reveals correlated scales and freezing; defect Curie tails can dominate.
Heat capacity	Constrains entropy and excitation gaps; separate phonons, nuclear terms and glassy defects.
Inelastic neutrons	Resolves continuum boundaries and weight; test multimagnon, disorder and crystal-field alternatives.
Raman or terahertz	Resolves polarization and field dependence; phonons and ordinary multiparticle states contribute.
Longitudinal heat transport	Can reveal mobile neutral excitations; phonons, cracks and localization complicate extrapolation.
Thermal Hall response	Can test a chiral edge; quantify phonon/magnon backgrounds, layer normalization and equilibration.

Material Examples

Herbertsmithite

Herbertsmithite realizes well-separated kagome planes of Cu $^{2+}$ moments and shows no conventional magnetic order down to temperatures tiny compared with its exchange scale. Single-crystal neutron scattering reveals an extended continuum, while site-resolved NMR has been used to separate intrinsic kagome susceptibility from interlayer defects.

The central unresolved issue is the intrinsic infrared theory. Cu on nominally nonmagnetic Zn sites supplies orphan-like interlayer spins, and possible Zn substitution in kagome layers changes the local network. Different probes and analyses have supported gapped and gapless interpretations. The robust statement is that herbertsmithite is a leading, strongly constrained kagome spin-liquid candidate with fractionalization-compatible dynamics, not that its exact anyon theory has been experimentally measured.

Organic triangular salts

The organic salts place spin- $1/2$ molecular dimers on anisotropic triangular networks near a pressure-driven Mott transition. NMR and bulk probes found no ordinary magnetic order despite sizable exchange. Heat capacity and early thermal transport motivated gapless spinon-Fermi-surface interpretations.

Later work exposed low-temperature lattice anomalies, sample and cooling-history dependence, and conflicting residual thermal-conductivity intercepts. These materials remain valuable because pressure connects the insulating state to metal and superconductivity, but that proximity also makes charge fluctuations, disorder, and lattice coupling part of the minimal problem.

Spin–orbit and Kitaev platforms

Rare-earth triangular magnets and honeycomb Kitaev candidates broaden the Hamiltonian beyond isotropic Heisenberg exchange. Their anisotropy is an opportunity and a burden: it can stabilize new liquids, but it also permits conventional ordered states and makes defects act as random exchange tensors.

For $\alpha$ -RuCl $_3$ , any thermal Hall claim must report field orientation, layer spacing, longitudinal conductivity, contact geometry, antisymmetrization, and sample history. Agreement with a half-quantized value over one window is stronger when accompanied by a plateau, a bulk gap, predicted sign changes, thickness or edge tests, and reproducibility across growths. The cited literature contains competing experimental and theoretical interpretations for Majorana, magnon, and phonon contributions.

Numerical Evidence

A lattice model can establish a spin-liquid phase more directly than a bulk material because the Hamiltonian and wavefunction are available, but only after convergence checks. Useful diagnostics include:

order parameters extrapolated in both system size and bond dimension;
singlet and triplet gaps with boundary and topological-sector control;
correlation lengths and transfer-matrix spectra;
flux insertion and anyon-sector identification;
topological entanglement entropy or modular data for a gapped state;
algebraic exponents and operator content for a gapless state;
energy competition against valence-bond and magnetically ordered ansatzes.

The nearest-neighbor spin- $1/2$ kagome Heisenberg antiferromagnet is a landmark case. Density-matrix renormalization group studies produced strong evidence for a gapped $\mathbb Z_2$ liquid, while later methods and geometries have supported competing gapless descriptions, including the tensor-network study of Liao and collaborators (2017). Small energy differences and long correlation lengths make algorithmic convergence part of the physics claim. A numerical phase assignment should not be transferred to herbertsmithite without adding Dzyaloshinskii–Moriya coupling, further-neighbor exchange, and defects.

Exercises

1. A silent magnet is not yet a liquid

A spin- $1$ dimer crystal has one dimer per primitive cell and remains uniquely gapped as the coupling between dimers is turned off. Its final state is a product of intradimer singlets. It preserves spin rotation and translation. Is it a quantum spin liquid under the definition used here?

Solution

No. The specified gapped path explicitly connects the state to independent unit-cell singlets. Spins within each dimer are entangled, but the state has no long-range entanglement between cells. A gap and short correlations alone would not have established this path; here it is part of the premise. A gapped spin liquid would require long-range entanglement or equivalent evidence such as topology-dependent ground sectors and deconfined fractional excitations.

2. Apply the filling constraint

Consider a two-dimensional translation-invariant magnet with three spin- $1/2$ moments in each primitive cell, short-range interactions, and unbroken spin rotation and translation. Numerics find a nonzero gap and no local symmetry breaking. What additional structure is required in the thermodynamic limit?

Solution

Three spin- $1/2$ moments give a half-integer spin representation per primitive cell. Under the stated symmetries, the Lieb–Schultz–Mattis–Oshikawa–Hastings constraint excludes a unique gapped symmetric short-range-entangled ground state. If the numerical gap and symmetry conclusions survive extrapolation, the state must have topological order and the associated topology-dependent low-energy sectors. A finite cylinder can hide those sectors, so flux insertion and sector-resolved calculations are appropriate checks.

3. Two-spinon kinematics

Suppose noninteracting spinons have dispersion $\varepsilon(\mathbf k)$ . A local spin operator creates a pair with total momentum $\mathbf q$ . Write the lower continuum boundary and explain why a broad neutron signal alone does not determine $\varepsilon$ uniquely. Then find both boundaries for the one-dimensional toy dispersion $\varepsilon(k)=\Delta+2t[1-\cos(ka)]$ , with $\Delta,t>0$ .

Solution

The lower boundary is

\hbar\omega_{\min}(\mathbf q) = \min_{\mathbf k} \left[ \varepsilon(\mathbf k) + \varepsilon(\mathbf q-\mathbf k) \right].

For the toy band, use the cosine sum identity:

\varepsilon(k)+\varepsilon(q-k) =2\Delta+4t-4t\cos(qa/2)\cos(ka-qa/2).

Allowing the last cosine to range from $-1$ to $1$ gives

E_\pm(q)=2\Delta+4t\pm4t\lvert\cos(qa/2)\rvert.

The collapse of this ideal band’s continuum width at $qa=\pi$ is a special property of nearest-neighbor cosine dispersion, not a general spin-liquid prediction. The observed intensity also contains matrix elements, spinon interactions, gauge fluctuations, instrumental resolution, and possible disorder averaging. Different dispersions can produce overlapping continua, while conventional two-magnon states can have similar kinematics. Momentum-resolved boundaries and polarization are more discriminating than breadth alone.

4. Defects and an apparent gap

An intrinsic kagome layer with gap energy $\Delta$ has susceptibility $\chi_{\mathrm{int}}\propto e^{-\Delta/(k_{\mathrm B}T)}$ , while a fraction $x$ of defect spins contributes $xC/(T+\theta_d)$ , with $\theta_d\ge0$ in the fitted temperature interval. What happens to the bulk low-temperature susceptibility, and which probe strategy can recover the intrinsic behavior?

Solution

The measured susceptibility is approximately

\chi_{\mathrm{bulk}}(T) = A e^{-\Delta/(k_{\mathrm B}T)} + \frac{xC}{T+\theta_d}.

Within this model, even small $x$ makes the Curie-like defect term dominate as the intrinsic term becomes exponentially small. Bulk susceptibility can therefore look gapless despite a gapped host. Site-resolved NMR, momentum dependence, impurity-field scaling, or a joint fit constrained by structural site occupancy can separate the components.

5. Half a thermal quantum

The field-gapped Kitaev phase has one chiral Majorana edge mode. What chiral central charge does it carry, and what two material checks are required before interpreting a measured thermal Hall value near the ideal prediction as proof?

Solution

A chiral Majorana mode carries $c_-=\pm1/2$ , with the sign fixed by its handedness and the chosen orientation, so its ideal layer conductance has half the thermal conductance quantum in magnitude. Material checks include demonstrating a plateau over field and temperature rather than an isolated crossing, and bounding phonon or magnon Hall backgrounds. One should also verify layer normalization, field-angle sign structure, a compatible bulk gap, edge equilibration, and reproducibility across samples.

6. Rank a candidate claim

A triangular insulator has no magnetic Bragg peaks down to $k_{\mathrm B}T/J=10^{-3}$ , a broad neutron continuum, and a frequency-dependent susceptibility cusp with sample-dependent onset. Which explanation must be tested first, and what measurements discriminate it from a clean spin liquid?

Solution

The cusp and sample dependence make glassy or random-singlet physics an immediate alternative. Test field-cooled and zero-field-cooled histories, aging and memory, frequency scaling of the cusp, local relaxation-rate distributions, structural disorder, and sample-to-sample variation. A clean spin liquid should give phase-consistent intrinsic spectra and thermodynamics after impurity contributions are controlled; a glass develops a frozen component and protocol dependence even without Bragg order.

Further Connections

Fractionalization compares spinons with charge fractionalization and one-dimensional spin–charge separation. Emergent Gauge Fields derives the constraint, gauge dynamics and confinement tests. Quantum Matter Frontiers and Open Problems discusses unresolved material identifications and measurements that can distinguish them.

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