Spin Liquid Pr₂Ir₂O₇: Quantum Criticality & Chiral Order

• Spin Liquid Pr₂Ir₂O₇ is a metallic quantum spin liquid arising from geometrical frustration, weak Kondo coupling, and non-Kramers doublet physics.
• It exhibits chiral order with a spontaneous Hall effect below 1.5 K, reflecting a novel interplay of local moment and itinerant electron physics.
• Quantum critical scaling and emergent gauge field dynamics in Pr₂Ir₂O₇ provide experimental benchmarks for theories of frustrated Kondo lattices and quantum spin liquids.

Spin liquid Pr₂Ir₂O₇ is a pyrochlore iridate in which strong geometrical frustration of the Pr³⁺ local moments, weak Kondo coupling to itinerant Ir 5d electrons, and non-Kramers doublet crystal-field physics combine to stabilize a metallic quantum spin liquid ground state featuring chiral order and quantum critical scaling. Pr₂Ir₂O₇ exhibits a spontaneous Hall effect below 1.5 K in zero magnetic field, with no conventional magnetic order, marking it as a paradigm of quantum-critical metallic spin liquid behavior in frustrated Kondo lattices.

1. Pyrochlore Lattice Structure and Local Moment Physics

Pr₂Ir₂O₇ crystallizes in the Fd–3m space group, forming a three-dimensional pyrochlore lattice of corner-sharing tetrahedra. Pr³⁺ ions populate the sites of this network, subject to strong local crystalline electric fields that split the J=4 manifold into a non-Kramers doublet ground state (Ising-like, with easy axes along local ⟨111⟩ directions) and an excited singlet at 162 K (Tokiwa et al., 2014). Each Pr³⁺ moment thus behaves as a classical spin-ice variable at low temperature, realizing “two-in/two-out” configurations per tetrahedron.

Ir ions (5d⁵) form an interpenetrating sublattice; conduction electrons reside primarily on the Ir network. The electronic structure is characterized by a small carrier density, weak correlation, and strong spin–orbit coupling, putting the material near a Mott instability and endowing itinerant electrons with substantial magneto-chiral susceptibility (Flint et al., 2013).

2. Frustrated Kondo Lattice and Emergence of the Metallic Spin Liquid

The minimal model is a frustrated Kondo-lattice Hamiltonian (Tokiwa et al., 2014):

$$H = J_{\text{ff}} \sum_{\langle ij\rangle} S_i^z S_j^z + J_K\sum_{i} S_i\cdot c_i^\dagger \sigma c_i + H_{\text{itin}}$$

with:

• $J_{\text{ff}}>0$: nearest-neighbor ferromagnetic Ising exchange between Pr moments (favoring spin-ice manifold)
• $J_K$: weak Kondo/RKKY coupling
• $H_{\text{itin}}$: Ir conduction electron kinetic energy

This combination leads to a spin-ice degenerate manifold at high T with strong frustration. Kondo screening is insufficient to fully suppress local moment formation, but it promotes novel interactions—including a “chiral RKKY” mechanism where conduction-electron chirality fluctuations mediate ferro-chiral coupling between Pr moments (Flint et al., 2013).

The slave-rotor mean-field calculation yields a ferro-chiral coupling $|J_\kappa|\sim10~\text{K}$, with ordering suppressed by spin-ice entropy, setting the transition temperature $T_H \sim 1.5~\text{K}$ (Flint et al., 2013).

3. Chiral Spin Liquid, Time-Reversal Breaking, and Anomalous Hall Effect

The low-T phase (1.5 K > T > 0.4 K) hosts a chiral spin liquid (CSL*) marked by finite scalar spin chirality on triangular loops:

$$\chi_{ijk} = \langle \vec{S}_i\cdot (\vec{S}_j \times \vec{S}_k) \rangle$$

A uniform expectation of $\langle\chi\rangle\neq0$ breaks time-reversal symmetry without generating macroscopic magnetization. This state is indicated by the onset of a spontaneous Hall conductivity $\sigma_{xy}$ in zero field, with measured values $\sim 7~\Omega^{-1}\text{cm}^{-1}$ along [111] comparable to theoretical predictions (Rau et al., 2013, Tokiwa et al., 2014).

The Kondo hybridization mechanism forces projective symmetry incompatibility between Pr spinons and Ir electrons, breaking all but C₃ + inversion symmetries, and leading to a chiral–nematic metallic phase with Berry curvature and anomalous Hall effect. Magnetic/quadrupolar moments are exceedingly small ($<10^{-3}\mu_B$) (Rau et al., 2013).

4. Quantum Criticality and Thermodynamic Scaling

Below $T_m \approx 0.4~\text{K}$, quantum fluctuations melt the ice-rule manifold and the system enters a quantum-critical spin-liquid regime. The magnetic Grüneisen ratio $\Gamma_H=(1/T)(dT/dH)_S=-\partial M/\partial T/C$ diverges as $\Gamma_H(T)\sim T^{-3/2}$ at zero field, and all data collapse under the scaling $\Gamma_HH = F(T/H^{4/3})$, yielding a critical exponent $\nu z = 2/3$ (Tokiwa et al., 2014). This is distinct from conventional Hertz–Millis quantum criticality, reflecting the local-moment Kondo physics and frustrated metallic environment.

Thermal conductivity measurements reveal a strong suppression above $T_s\sim0.12~\text{K}$ attributed to scattering of phonons by fluctuating spins; the absence of mobile fermionic spinons and conventional electron behavior (Wiedemann–Franz law holds at the QCP) further constrain the models for the quantum spin liquid (Ni et al., 2018).

5. Gauge Field Structure and Spin-Liquid Transitions

The effective low-energy model for non-Kramers pyrochlores like Pr₂Ir₂O₇ is the so-called XYZ pseudospin Hamiltonian (Lozano-Gómez et al., 2023, An et al., 23 May 2025):

$$\mathcal{H} = \sum_{\langle ij\rangle}[J_{zz}S^z_iS^z_j - J_{\pm}(S^+_iS^-_j + S^-_iS^+_j) + J_{\pm\pm}(\gamma_{ij}S^+_iS^+_j + \gamma_{ij}^*S^-_iS^-_j)]$$

with both dipolar (Ising) and quadrupolar (transverse) exchanges. Near the “dipolar-quadrupolar-quadrupolar” (DQQ) point, three irrep modes coexist, giving rise to emergent vector (rank-1) and tensor (rank-2) gauge fields. The intermediate-temperature regime ($T^*\lesssim T \lesssim T_{gl}\sim J$) features simultaneous vector and tensor gauge signatures in S(q):

• Vector field: $\nabla\cdot B^{\text{Ice}} = 0$ → two-fold pinch points
• Tensor gauge: $\partial_\alpha \mathcal{M}_{\alpha\beta}^{xy}=0$ → four-fold pinch points/pinch-lines

Entropic selection at low T depopulates tensor gauge degrees of freedom, resulting in a Coulomb phase (pure spin ice) (Lozano-Gómez et al., 2023). Quantum fluctuations provide a stability window for such coexisting gauge fields down to T = 0, making Pr₂Ir₂O₇ a prime candidate for multiscale emergent electromagnetism in magnetic solids.

Schwinger-boson mean-field theory and projective symmetry group classification identify multiple Z₂ QSL phases in the parameter space, including candidates with dominant transverse exchanges, modeled to produce broad rods in neutron scattering and robust inelastic continua (An et al., 23 May 2025).

6. Chiral Spin-Texture Dynamics, Jellyfish Excitations, and Topological Memory

Short-range defect-defect interactions supplement long-range entropic Coulomb forces in the spin-ice manifold, enabling exotic non-equilibrium dynamics (Udagawa et al., 2016). At special coupling ($J=1/4$), “jellyfish” clusters—hexagonal rings of same-sign monopoles with circulating toroidal magnetization—proliferate, providing persistent local time-reversal breaking without macroscopic moment. These topological textures can freeze a bias in chirality following a field quench, offering a mechanism for the observed spontaneous Hall conductivity even after magnetization relaxes to zero (Udagawa et al., 2016).

Various metastable regimes emerge, such as fragmented spin liquids and magnetization plateaux, stabilized by kinetic barriers and defect interactions. Neutron-scattering signatures include half-moon patterns and persistent dynamical “memory,” characteristic of topological bottlenecks.

7. STM Spectroscopy and Atomic-Scale Correlations

Atomic-resolution STM measurements reveal a kagome-terminated (111) Pr₃Ir surface, exposing the frustrated Pr network and subsurface Ir centers (Song et al., 22 Nov 2025). A Fano-shaped Kondo-lattice resonance appears just below E_F (E₀ ≈ –10 meV, Γ ≈ 4.5 meV), yielding a spectroscopic Kondo temperature $T_K\approx 37~\text{K}$. The spatial amplitude of the resonance modulates with the kagome lattice and exhibits three-fold symmetry.

Magnetic field induces significant Zeeman splitting of the resonance at Ir sites (g_eff ≈ 1.8), while Pr sites show negligible splitting, signifying strong coupling of the conduction electrons to the chiral spin-liquid background. This atomic-scale evidence corroborates a scenario in which Kondo hybridization, geometrical frustration, and chiral spin-texture intertwine to generate the many-body physics observed in macroscopic transport and thermodynamic probes.

8. Quantum Spin Ice, Monopole Condensation, and Critical Theory

U(1) quantum spin-ice theory, using compact QED formalism, describes Pr₂Ir₂O₇ near the critical region (Chen, 2016). The system resides close to the quantum phase transition from deconfined spin-liquid to confined Ising order, governed by condensation of magnetic monopoles. The proximate ordered state is the antiferromagnetic “two-in/two-out” pattern at Q=2π(001), accessible via a continuous transition, while the ferromagnetic state at Q=(000) requires a strongly first-order transition.

Critical scaling laws for susceptibility ($\chi(Q)\sim\ln(1/T)$), heat capacity (T³ contributions from photons and monopoles), and field response (metamagnetic jump in B∥[111]) are predicted, providing quantitative benchmarks for comparison to experiment.

Table: Key Temperature and Energy Scales in Pr₂Ir₂O₇

Phenomenon	Energy/Temperature Scale	Description
Kondo resonance	$T_K \sim 37–45~\text{K}$	Spectroscopic, macroscopic, transport
Ising exchange	$J_{ff}/k_B \approx 1.4~\text{K}$	Ferromagnetic Pr–Pr interaction
Chiral CSL* ordering	$T_H \approx 1.5~\text{K}$	Onset of spontaneous Hall effect
Quantum criticality	$T_m \approx 0.4~\text{K}$	Melting of ice manifold, scaling
Spin–phonon freezing	$T_s \approx 0.12~\text{K}$	Thermal conductivity suppression
Chiral RKKY coupling	$|J_\kappa| \approx 5–15~\text{K}$	Ferro-chiral Pr–Pr interaction

Conclusion

Pr₂Ir₂O₇ exemplifies a quantum-critical metallic spin liquid resulting from the interplay between geometrical frustration, non-Kramers doublet physics, and weak Kondo screening. The chiral spin liquid phase breaks time-reversal symmetry without conventional order and is accessed via the chiral RKKY mechanism and Kondo hybridization. Quantum criticality persists to very low temperatures, manifesting universal thermodynamic scaling, critical spin fluctuations, and unusual transport phenomena including a spontaneous Hall effect. Atomic-scale STM and spectroscopy provide direct evidence of the underlying Kondo-lattice resonance, geometric frustration, and spin-liquid order. The temperature-tunable gauge hierarchy and topological chiral textures further enrich the dynamical landscape, positioning Pr₂Ir₂O₇ as a central platform for experimentally testing concepts in quantum spin liquid theory and strongly correlated electron systems (Tokiwa et al., 2014, Flint et al., 2013, Rau et al., 2013, Ni et al., 2018, Lozano-Gómez et al., 2023, Udagawa et al., 2016, Song et al., 22 Nov 2025, An et al., 23 May 2025, Chen, 2016).