Dirac Spin Liquid Phase in Quantum Magnets

• The Dirac spin liquid phase is a quantum-disordered state in 2D frustrated magnets characterized by gapless Dirac spinons coupled to an emergent U(1) gauge field.
• Numerical and field-theory studies reveal algebraic decay of spin correlations and critical monopole scaling, offering distinct signatures on kagome, triangular, and square lattices.
• Theoretical models based on QED₃ and parton constructions show that DSL stability critically depends on lattice symmetries and quantum criticality, guiding experimental realizations.

The Dirac spin liquid (DSL) phase is a quantum-disordered state of two-dimensional magnetic systems in which elementary excitations are gapless, massless Dirac fermion “spinons” coupled to an emergent gauge field. This phase is characterized by the absence of conventional long-range magnetic or lattice order, algebraic (power-law) decay of correlation functions, and the emergence of exotic topological and symmetry properties. The DSL paradigm has been realized theoretically and numerically in a range of frustrated magnetic systems, notably on kagome, triangular, and honeycomb lattices, as well as in square-lattice and extended SU(N) models. Its field-theoretical description is anchored in compact quantum electrodynamics in 2+1 dimensions (QED₃) with N_f flavors of Dirac fermions minimally coupled to a U(1) gauge field.

1. Field-Theoretic Structure and Universal Properties

The DSL is encapsulated at low energies by the QED₃ action: $$\mathcal{L}_{\mathrm{DSL}} = \sum_{i=1}^{N_f} \bar{\psi}_i i \gamma^\mu (\partial_\mu - i a_\mu) \psi_i + \frac{1}{4e^2} f_{\mu\nu}^2$$ where $\psi_i$ are Dirac spinons, $a_\mu$ is an emergent gauge field, and $$f_{\mu\nu} = \partial_\mu a_\nu - \partial_\nu a_\mu$$. The flavor number $N_f$ depends on the lattice and spin symmetry; $N_f=4$ for spin-½ models on the square, triangular, and kagome lattices.

The DSL possesses several defining features:

• Gapless Dirac spectrum: Spinons exhibit Dirac-cone dispersions at symmetry-protected points in the Brillouin zone.
• Algebraic correlations: Physical correlators, such as spin–spin and dimer–dimer, decay as $1/r^{z + \eta}$, with exponents controlled by the underlying CFT and gauge fluctuations.
• Absence of quasi-particles: Correlation functions generically display non-trivial power-laws without sharp peaks, signaling the breakdown of Fermi-liquid behavior.
• Critical dynamic exponent: In fields or certain transitions, the dynamical exponent can deviate from $z=1$, and Lorentz invariance may be weakly or strongly broken depending on the microscopic realization (Shackleton et al., 2022).

2. Lattice Realizations and Numerical Signatures

Kagome Heisenberg Antiferromagnet

On the kagome lattice, the $S=1/2$ Heisenberg antiferromagnet hosts a U(1) DSL described by a projected fermionic wavefunction with zero flux through each triangle and $\pi$ flux through each hexagon. This construction yields Dirac nodes in the mean-field spectrum (Iqbal et al., 2012, Iqbal et al., 2013). Numerical variational Monte Carlo (VMC) and Green’s function Monte Carlo calculations, improved via Lanczos steps and zero-variance extrapolation, yield ground-state energies competitive—if not superior in some clusters—to DMRG estimates: $E^\mathrm{DSL}_\infty / J = -0.4365(2)$ Static spin structure factors $S(\mathbf{q})$ display weight at hexagon corners, while long-wavelength behavior ($q \to 0$) is dominated by the $1/r^4$ power law linked to Dirac spinons.

Triangular Lattice and Stability

On the triangular lattice, analogous parton constructions—often with a staggered $\pi$-flux pattern—support stable U(1) DSLs in a regime of high frustration (e.g., the $J_1$-$J_2$ model with $J_2/J_1 \approx 1/8$) (Budaraju et al., 24 Oct 2024, Calvera et al., 2020). These have been shown numerically to admit gapless Q = 1 monopole excitations (both singlet and triplet) in the thermodynamic limit; their variational excitation energies align well with large-$N_f$ field theory predictions, bolstering the identification with a robust DSL.

Square Lattice Quantum Antiferromagnets

For the $J_1$-$J_2$ Heisenberg model, machine-learning–based variational studies combining RBMs and pair-product states (RBM+PP) reveal a critical “nodal” DSL phase in a narrow window ($J_2/J_1 \in [0.49, 0.54]$) (Nomura et al., 2020), characterized by:

• Gapless singlet and triplet excitations with Dirac-like dispersions.
• Dual criticality: power-law decay in both spin and dimer correlations with distinct exponents, merging at a single critical symmetry point.

In extended models (e.g., SU(4) π-flux models (Liao et al., 2022)), the DSL emerges as a strong-coupling ground state, connected via Gross–Neveu quantum critical points to symmetry-broken (VBS) phases.

3. Quantum Criticality, Monopoles, and Phase Transitions

The stability and universality of the DSL are intimately tied to the properties of monopole operators in QED₃. The scaling dimension of the minimal monopole, calculated as $\Delta_{\mathcal{M}_1} \approx 1.02$ for $N_f=4$, is a critical quantity (He et al., 2021).

• If $\Delta_{\mathcal{M}_1} < 3$, the monopole is a relevant perturbation: on the square lattice, this destabilizes the “parent” DSL and drives either Néel or VBS order depending on subleading anisotropies. The DSL is thus strictly speaking not a stable phase, but an “unnecessary” quantum critical point within a symmetry-broken phase (Zhang et al., 17 Apr 2024), with scaling exponent flow and order parameter vanishing characterized by the DSL CFT.
• On the triangular/kagome lattices, lattice symmetries can forbid the lowest-charge monopoles or render them irrelevant at the fixed point, thus stabilizing a robust deconfined DSL phase (He et al., 2021, Budaraju et al., 24 Oct 2024).

The condensation of monopoles with particular symmetry quantum numbers (e.g., maximal total spin) provides a symmetry-based “selection rule,” dictating the nature of the ordered phase emerging from DSL confinement (Dupuis et al., 2021).

The transition from U(1) DSL to a gapless $\mathbb{Z}_2$ spin liquid (Z2Azz13) can proceed via Higgs condensation of a charge-2 complex scalar field (Shackleton et al., 2022). The resultant $\mathbb{Z}_2$ state retains gapless Dirac spinons, but with modified topological order and gauge structure. The critical theory exhibits velocity anisotropy, a dynamical exponent $z \neq 1$, and distinct angular scaling of Néel and VBS correlations due to weak breaking of emergent SO(5) symmetry (Shackleton et al., 2021, Shackleton et al., 2022).

4. Signatures and Experimental Relevance

A suite of signatures—static and dynamical—can be associated with the DSL:

• Algebraic spin correlations: $C(r) \sim 1/r^{z+\eta}$, with $z+\eta \approx 1.4$–$1.5$ in numerically observed QSL windows (Nomura et al., 2020, Maity et al., 30 Dec 2024).
• Spin excitation spectra: Dirac cones manifest as linearly vanishing two-spinon continua at symmetry-related momenta, often observed as broad continua or isomorphic features to $d$-wave superconducting nodes in cuprates.
• Thermal and transport properties: In certain materials, a finite $\kappa_0/T$ at low temperature ("metallic" thermal conductivity) is attributed to mobile, gapless spinons from the parent DSL (Bose et al., 2022).
• Response to magnetic fields: The field-induced evolution from zigzag order to a gapless DSL (and ultimately to a chiral QSL) in the honeycomb Kitaev–$\Gamma$ model explains experimental thermal and spectroscopic data on $\alpha$-RuCl$_3$ (Liu et al., 2017).
• Dynamical structure factors: Calculations show “double-lobe” intensity features in $S(\mathbf{q}, \omega)$ near incommensurate $\mathbf{q}$, consistent with Dirac cone locations (Maity et al., 30 Dec 2024).

5. Stability and Material Realizations

The stability of the DSL is precarious. On bipartite lattices (square, honeycomb), relevant monopole perturbations or certain disorders destabilize the phase, confining spinons and driving conventional order (Dey, 2020, Calvera et al., 2021). On non-bipartite lattices with suitable symmetry constraints, the DSL can be robust—the critical scaling dimension of the lowest monopole is bounded from below (e.g., $\Delta_{2\pi} > 1.046$ on triangular and $>1.105$ on kagome (He et al., 2021)) and matches numerically estimated values.

Disorder or coupling to lattice distortions (phonons) can promote ordering via monopole condensation or spin-Peierls instabilities, but a regime where the DSL remains robust exists if the perturbations (phonon gaps, disorder strength) are sufficiently small (Seifert et al., 2023).

Experimental indications for DSL behavior include the observed QSL signatures in kagome and triangular magnets (e.g., Herbertsmithite, organic triangular-lattice salts), honeycomb cobaltates (e.g., BaCo$_2$(AsO$_4$)$_2$), and SU(N) cold atom systems (Bose et al., 2022, Bose et al., 5 Dec 2024). The interpretation is further sharpened in cases where broad spin excitation continua and metallic thermal conductivities are measured.

6. Symmetry, Gauge Structure, and Topological Aspects

DSL phases are classified by their projective symmetry group (PSG) structure, determining the allowed pattern of hopping, pairing and corresponding gauge structure (U(1), SU(N), $\mathbb{Z}_2$, $\mathbb{Z}_4$). Transitions between various DSL descendants—such as from U(1) to $\mathbb{Z}_2$ via Higgsing, or to chiral states via time-reversal breaking—furnish a rich landscape of topological quantum matter, with distinct anyonic excitations (e.g., semions, Fibonacci anyons) and possible experimental fingerprints (Calvera et al., 2020, Calvera et al., 2021).

7. Outlook and Open Challenges

Current research directions focus on:

• Precisely pinning the role of monopole operators and their scaling dimensions via conformal bootstrap, quantum Monte Carlo, and field-theory techniques (He et al., 2021, Zerf et al., 2019).
• Understanding disorder effects, lattice coupling, and deconfined criticality in lattice realizations.
• Probing dynamical signatures, topological responses (e.g., quantized thermal Hall effect), and emergent gauge fluctuations in experiments.
• Extending parton and PSG constructions to higher-spin and SU(N) systems relevant to cold atom and Moiré materials.

The Dirac spin liquid phase provides a unifying theoretical framework and quantitative target for experimental and numerical searches for exotic quantum ground states beyond conventional order in correlated quantum magnets, but its realization is ultimately determined by a fine interplay of lattice symmetries, topology, gauge fluctuations, and external perturbations.