Spinon Fermi Surface in Quantum Spin Liquids

• Spinon Fermi Surface is a quantum phase where local moments fractionalize into neutral spinon excitations forming a gapless Fermi sea with topological order.
• It displays clear experimental signatures such as broad, gapless two-spinon continua and distinct 2k_F anomalies observable in inelastic neutron scattering and STM.
• The state is modeled using slave-particle techniques under gauge constraints, uniting analytical methods, numerical simulations, and experimental observations in frustrated magnets.

A spinon Fermi surface (SFS) is a fundamentally nontrivial quantum many-body state in strongly correlated insulators, in which local moments fractionalize into emergent fermionic excitations—spinons—that form a gapless Fermi sea. This state is prototypical for certain two-dimensional U(1) quantum spin liquids and constitutes an example of non-Landau metallicity in Mott insulators, distinguished from conventional metals by the neutrality of their Fermi-surface excitations, topological order, and strong gauge fluctuations. The SFS paradigm unifies theoretical constructions, numerical simulations, and a growing array of experimental observations in frustrated magnets, rare-earth quantum crystals, and intrinsically two-dimensional Mott materials.

1. Theoretical Construction of the Spinon Fermi Surface

In the modern framework of parton or slave-particle theory, a spinon Fermi surface emerges from fractionalizing local spin operators into partonic fermions subject to a gauge constraint. For an $S=1$ system such as Ba$_3$NiSb$_2$O$_9$, the $S=1$ moment at site $i$ is written as $$S_i^\alpha = \sum_{\mu\nu} f_{i,\mu}^\dagger (S^\alpha)_{\mu\nu} f_{i,\nu}$$, with $f_{i,\mu}$ ($\mu=1,2,3$ labeling three flavors) fermions and on-site single occupancy $\sum_\mu f_{i,\mu}^\dagger f_{i,\mu}=1$ (Fak et al., 2016). For $S=1/2$ moments, one typically uses $f_{i,\alpha}$ with $\alpha = \uparrow,\downarrow$, and the constraint $\sum_\alpha f_{i,\alpha}^\dagger f_{i,\alpha}=1$ (Shen et al., 2016).

The canonical mean-field Hamiltonian for the three-flavor ($S=1$) case is

$$H_0 = \sum_{k,\alpha} \epsilon_k f_{k,\alpha}^\dagger f_{k,\alpha} - \mu \sum_{i,\alpha} f_{i,\alpha}^\dagger f_{i,\alpha}$$

where, on the triangular lattice with lattice vectors $a_1$, $a_2$,

$$\epsilon_k = -2t[\cos(k \cdot a_1) + \cos(k \cdot a_2) + \cos(k \cdot (a_1+a_2))]$$

and the chemical potential $\mu$ is fixed by the occupancy constraint (e.g., $1/3$ per flavor for $S=1$). Each spinon species forms a Fermi surface enclosing the appropriate fraction of the Brillouin zone (Fak et al., 2016).

Physical observables require projection to the constraint space (see Gutzwiller projection), generating strong quantum fluctuations and coupling the mean-field spinons to emergent U(1) gauge fields. This gives rise to "algebraic" (non-Fermi-liquid) corrections to low-energy properties due to gauge fluctuations (Freire, 2014).

2. Characteristic Physical Signatures and Experimental Observations

A defining property of a spinon Fermi surface state is the presence of broad, gapless two-spinon continua in momentum- and energy-resolved spin response functions. The dynamical spin structure factor,

$$S(\mathbf{q},\omega) = \frac{1}{N} \sum_{k,\alpha,\beta} |M_{\alpha\beta}(k,\mathbf{q})|^2\, \delta[\omega - (\epsilon_{k+\mathbf{q},\beta} - \epsilon_{k,\alpha})]$$

exhibits:

• A gapless continuum for all $\mathbf{q}$ within particle–hole phase space, with intensity features ("2$k_F$" singularities) at $|\mathbf{q}| \approx 2k_F$ (Fak et al., 2016, Shen et al., 2016, Li et al., 2017).
• Non-dispersive rods of scattering at specific momentum shells corresponding to $2k_F$ values in powder-averaged inelastic neutron scattering (INS) (Fak et al., 2016).
• Absence of sharp magnon modes and of energy gaps down to meV or sub-meV resolution (Fak et al., 2016, Shen et al., 2016, Dai et al., 2020).

Experiments in Ba$_3$NiSb$_2$O$_9$ ($S=1$) (Fak et al., 2016), YbMgGaO$_4$ (Shen et al., 2016, Li et al., 2016, Li et al., 2017), and NaYbSe$_2$ (Dai et al., 2020) have identified precisely such broad, gapless, and non-dispersive spectral weight extending to the lowest energies accessible, peaking at the characteristic $2k_F$ wavevectors, and robust against temperature up to $O(10$–$50$ K$)$.

In real space, scanning tunneling spectroscopy (STS) can image long-wavelength charge or spinon density oscillations at $2k_F$. In monolayer 1T-TaSe$_2$, STM/STS reveals incommensurate supermodulation in the local density of states at $q_\mathrm{ICS}=2k_F$, enabling direct extraction of the spinon Fermi wavevector in excellent match with tight-binding theory (Ruan et al., 2020).

The interplay of gauge fluctuations and SFS geometry can be probed by quantum noise measurements using nitrogen-vacancy (NV) centers, which detect the universal transverse magnetic noise scaling with the perimeter of the Fermi surface, reduced by a gauge-screening factor (Khoo et al., 2022, Khoo et al., 2021).

3. Thermodynamic and Transport Properties

The SFS state manifests metallic-like thermodynamics (finite density of states at the chemical potential) without charge transport. Key signatures include:

• A linear-in-$T$ contribution to the magnetic specific heat, $C_{\text{mag}}(T)\sim \gamma T$, with $\gamma$ proportional to the spinon density of states (Dai et al., 2020).
• A nearly temperature-independent Pauli spin susceptibility, directly reflecting the finite density of spinons at the Fermi energy (Dai et al., 2020).
• Power-law ($\sim T^\gamma$) thermal conductivity, where $\gamma<1$ arises from non-Fermi liquid corrections due to U(1) gauge field fluctuations ($\gamma=1/2+5\epsilon/4$ for Landau-damped gauge field with dynamical exponent $z_b=2+\epsilon$) (Freire, 2014).
• An unquantized, field- and temperature-dependent thermal Hall conductivity $\kappa_{xy}/T$ arising from the Berry curvature of spinon bands in the presence of broken time-reversal symmetry, in stark contrast to the quantized plateau of chiral topological phases (Teng et al., 2020).

These thermodynamic and transport fingerprints distinguish the SFS state sharply from both conventional ordered magnets and gapped $\mathbb{Z}_2$ or Dirac spin liquids.

4. Field Response and Experimental Probes

External magnetic fields induce distinctive modifications to the spinon spectrum. The Zeeman term splits the spinon bands, resulting in:

• A sharp spectral-weight enhancement at the $\Gamma$ point at energy $\omega=\Delta=g\mu_B B$, i.e., a field-linear peak ("Zeeman peak").
• An "X"-shaped crossing in the $S(\mathbf{q},\omega)$ map, with upper and lower excitation edges at $\omega^\pm(\mathbf{q})=\Delta\pm v_F |\mathbf{q}|$, uniquely characteristic of spinon Fermi surfaces as opposed to conventional magnon bands (Li et al., 2017).
• Field-dependent shift and splitting of $2k_F$ singularities, testable by high-resolution cold-neutron INS.

These features have been predicted and detailed for YbMgGaO$_4$ and Ba$_3$NiSb$_2$O$_9$, and constitute a "smoking-gun" for fractionalized Fermi-surface states (Li et al., 2017).

5. Theory–Numerics–Experiment Synthesis and SFS Stability

Conclusive signatures of the SFS state arise from the confluence of:

• Analytical mean-field spinon band theory, supported by projective symmetry group (PSG) classification that identifies the unique U(1) state consistent with large Fermi surfaces and observed spectral features (Li et al., 2016).
• Exact numerical methods (DMRG, variational Monte Carlo, machine learning on quantum snapshot data) showing that "gapless" QSL phases exhibit $2k_F$ oscillations (Friedel oscillations), central-charge scaling consistent with a Fermi surface, and ring-shaped maxima in momentum-space correlation functions (He et al., 2018, Zhang et al., 2023).
• Time-domain and spatially resolved probes (time-resolved photoemission in cold-atom simulators, STM imaging in real-space) that reveal direct signatures of the SFS geometry, including occupation beyond the mean-field Fermi surface and collective spinon–spinon interactions (Schuckert et al., 2021, Zhang et al., 2023, Ruan et al., 2020).
• Magnetic noise and plasma mode features that are analytically identical to metallic Fermi surfaces in the appropriate regime, suggesting new directions in non-contact probes (Khoo et al., 2022, Khoo et al., 2021, Ma et al., 2014).

The stability of the SFS against symmetry-breaking tendencies remains nontrivial. While many variational and Gutzwiller-projected numerics have favored SFS ground states in models close to the Mott transition or with strong ring exchange, recent systematic studies in the $J_1$-$J_4$ triangular model show that, for realistic parameters, lattice symmetry–broken phases or valence-bond solids can preempt the SFS, challenging its global stability (Yang et al., 2023).

6. Material Realizations and Open Questions

Materials where signatures consistent with spinon Fermi surfaces have been observed include:

• $S=1$ triangular lattice: Ba$_3$NiSb$_2$O$_9$ (Fak et al., 2016).
• $S=1/2$ rare-earth triangular: YbMgGaO$_4$ (Shen et al., 2016, Li et al., 2016, Li et al., 2017), NaYbSe$_2$ (Dai et al., 2020).
• Layered cluster Mott insulators: 1T-TaS$_2$, 1T-TaSe$_2$ (Ruan et al., 2020, He et al., 2018, Chen et al., 2020).
• Organic $\kappa$-phase and dmit compounds.
• Honeycomb-lattice Kitaev–Heisenberg models under field, numerically identified via machine learning and snapshot analysis (Zhang et al., 2023).

Outstanding issues include the precise conditions for SFS stability, impact of disorder and strong correlations, the detailed role of gauge fluctuations in transport and dynamics beyond mean-field, and the proximity and crossover to heavy-fermion or Fermi-liquid metal states as coupling to itinerant electrons is introduced (Chen et al., 2020, Han et al., 2016).

7. Summary Table: Key Experimental and Theoretical Indicators

Signature	SFS State Prediction	Distinctive vs. Conventional States
INS continuum shape	Broad, gapless, non-dispersive, extends to lowest $\omega$	Discrete magnons, gapped continua
$2k_F$ features (momentum)	Intensity at $|\mathbf{q}|=2k_F$	Absent for magnons or Dirac nodes
Pauli susceptibility, $C(T)/T$	Finite, metallic-like	Vanishes (gapped, spin-gapped)
Thermal conductivity	$\kappa \sim T^\gamma<1$ or metallic	Activated or rapidly vanishing
STM/STS modulations	$2k_F$ Friedel oscillations	Trivial or absent
Field-induced Zeeman peak/cross	X-shape crossing in $S(\mathbf{q},\omega)$ at $\omega= \Delta \pm v_F |\mathbf{q}|$	Absent or single shift
Magnetic noise scaling (NV)	Proportional to FS perimeter $P_F$	Differs for insulators, metals

The SFS state thus stands as a robust, highly entangled phase with clear theoretical construction, sharp experimental phenomenology, and a nontrivial position in the broader taxonomy of quantum spin liquids and strongly correlated quantum matter (Fak et al., 2016, Shen et al., 2016, Li et al., 2016, Khoo et al., 2022, Dai et al., 2020, Yang et al., 2023, Ruan et al., 2020, Zhang et al., 2023).