Gapless Spin Liquids and Dirac Spinons

A gapless spin liquid has no symmetry-breaking order yet supports fractionalized low-energy degrees of freedom. In a Dirac spin liquid, projected spinons form Dirac nodes and couple to a compact emergent U(1) gauge field. The mean-field cones identify a candidate QED3 theory; stability requires every symmetry-allowed gauge-invariant mass, pairing/Higgs channel, and monopole perturbation to be irrelevant.

Required background. Parton constraints supplies projection and gauge redundancy; compact U(1) gauge fields supplies the monopole test.

Helpful background. Critical Fermi-surface patch theories supplies the contrasting spinon-Fermi-surface case.

Projected Dirac ansatz

A symmetry-compatible hopping pattern can produce nodes at momenta $\mathbf K_a$. Linearizing and restoring gauge fluctuations gives the Euclidean theory

$$\mathcal L_{\rm QED_3}= \sum_{a=1}^{N_f}\bar\psi_a\gamma^\mu(\partial_\mu-ia_\mu)\psi_a +\frac{1}{4e^2}f_{\mu\nu}f_{\mu\nu} +\sum_q\lambda_q(\mathcal M_q+\mathcal M_q^\dagger)+\cdots.$$

Repeated spacetime indices are contracted with the Euclidean metric. For common spin-$1/2$ ansätze, spin and node multiplicities often give $N_f=4$ two-component Dirac fermions, but the value and projective symmetry action are lattice dependent. The microscopic gauge field is compact; $\mathcal M_q$ inserts $2\pi q$ flux. Omitting it converts a stability question into an assumption.

The physical spin operator is gauge neutral. Its long-distance expansion contains fermion bilinears and, at momenta fixed by the projective symmetry group, monopole operators. Consequently, a spin structure factor probes scaling dimensions of these composites, not a single-spinon pole Hermele et al. 2004.

Stability tests

A candidate algebraic spin liquid survives only if:

• all symmetry-allowed gauge-invariant fermion masses and interactions in pairing/Higgs channels are irrelevant or forbidden;
• the lowest symmetry-allowed monopole has $\Delta_{\mathcal M}>3$ in $2+1$ dimensions;
• velocity anisotropy and four-fermion interactions flow to the proposed fixed point;
• projection preserves the intended symmetry and does not generate order;
• finite-size spectra and physical correlations converge to one operator dictionary.

Large-$N_f$ QED3 suppresses monopoles and provides a controlled expansion. Extrapolation to $N_f=4$ is nontrivial; a relevant monopole can confine and its quantum numbers then select valence-bond or magnetic order. A bare spinon pair is gauge charged and is not itself an allowed local perturbation of the gauge-invariant action. Condensation of a gauge-charge-two Higgs field, or an equivalent pairing instability generated by a gauge-invariant interaction, instead Higgses U(1) to Z2 and may yield a gapless Z2 Dirac liquid or eventually a gapped phase.

Current model status

Numerical studies continue to distinguish U(1), Z2, chiral, and ordered states in closely competing Hamiltonians. A 2025 projected-state-guided DMRG study of the square-lattice $J_1$–$J_2$ model reported a Z2 Dirac spin-liquid candidate and compared nearby chiral orders Jin, Tu, and Zhang 2025. This is model- and method-specific evidence, not a general demonstration that physical $N_f=4$ compact QED3 is a stable phase.

The mutable numerical record was checked through 10 August 2026. Formal QED3 operator and stability criteria are durable; claims about particular Hamiltonians require current size, bond-dimension, and cross-method evidence. See Quantum Matter and Emergence Research for dated updates.

Exercise

A lattice symmetry permits only quadrupled monopoles, and the charge-four monopole has $\Delta_4=3.6$. What is its linear RG status?

Solution

Its fugacity has eigenvalue $y_4=3-\Delta_4=-0.6$ and is irrelevant at linear order. One must still check lower-charge monopoles are genuinely forbidden, all other perturbations, and possible dangerously irrelevant effects away from the fixed point.

References

• Michael Hermele, T. Senthil, Matthew P. A. Fisher, Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen, “Stability of U(1) Spin Liquids in Two Dimensions,” Physical Review B 70 (2004) 214437, doi:10.1103/PhysRevB.70.214437.
• Hui-Ke Jin, Hong-Hao Tu, and Ya-Hui Zhang, “Dirac and Chiral Spin Liquids on the Spin-1/2 Square-Lattice Heisenberg Antiferromagnet,” Physical Review B 112 (2025) 035159, doi:10.1103/4yrt-nsth.