Exceptional degeneracies in interacting Kitaev spin liquids: realistic parameter regimes, self-energy and flux-sector corrections, and experimental detection

Yang, Varjas, Bergholtz, Morampudi, and Wilczek (Phys. Rev. Research 4, L042025 (2022)) showed that phonons and quenched disorder in Kitaev honeycomb quantum spin liquids generate a non-Hermitian self-energy whose anti-Hermitian components produce exceptional rings (time reversal preserved on average) or exceptional points connected by Fermi arcs (phonons), with distinctive signatures in spectral functions, the dynamical spin structure factor, the density of states, and the low-temperature specific heat. The result rests on deliberately idealized approximations: a lowest-order self-consistent Born treatment with dimensionless couplings, neglect of the frequency-linear self-energy term responsible for wavefunction renormalization, restriction to the zero-flux sector, and only sketched detection strategies. The open problem is the quantitative, material-realistic completion of this program: determine the experimentally realistic ranges of couplings, phonon gaps, disorder variances, and temperatures in which anti-Hermitian level attraction dominates Hermitian gap opening; treat the linear-in-$\omega$ self-energy systematically; assess how flux disorder and thermally excited vortices modify the exceptional signatures; and develop concrete, validated detection protocols for candidate compounds. Each of these four components is left open by the source and its Supplemental Material.

Background

Quantum spin liquids are low-temperature magnetic phases in which quantum fluctuations prevent long-range spin order; their distinctive features are fractionalized excitations (spinons) and emergent gauge fields. The Kitaev honeycomb model (A. Kitaev, Ann. Phys. 321, 2 (2006)) is a rare exactly solvable example: spins-1/2 on a honeycomb lattice interact via bond-direction-dependent Ising couplings $-\sum_{\langle jk\rangle_\alpha}J_\alpha\sigma^\alpha_j\sigma^\alpha_k$, where $\alpha=x,y,z$ labels the three bond orientations. In its Majorana representation (each spin fractionalizes into four Majorana fermions, particles that are their own antiparticles, three of which freeze into a static $\mathbb{Z}_2$ gauge field), the low-energy sector reduces to free Majoranas with a Dirac-cone dispersion; the elementary gauge excitations are fluxes (vortices) attached to lattice plaquettes, separated from the flux-free ground-state sector by the flux gap. Candidate materials such as $\alpha$-RuCl$_3$ realize proximate Kitaev physics, and phonons (quantized lattice vibrations) couple appreciably to the spins there (Metavitsiadis, Natori, Knolle, and Brenig, Phys. Rev. B 105, 165151 (2022)). Dynamical response is described by the Green's function $G^{R}(\omega)$, which obeys the Dyson equation $[\omega-H_0-\Sigma^{R}(\omega)]G^{R}(\omega)=1$, with the self-energy $\Sigma$ encoding interactions and disorder. Unlike the bare Hamiltonian, the retarded self-energy is generally non-Hermitian: it decomposes into a Hermitian part that shifts energies and can open gaps, and an anti-Hermitian part that gives quasiparticles finite lifetimes and, when off-diagonal, generates level attraction between bands. Expanding $\Sigma^{R}(\omega)=\Sigma^{R}(0)+\omega\Sigma^{(1)}+\cdots$ and discarding the term linear in $\omega$ yields the frequency-independent effective Hamiltonian $H_{\mathrm{eff}}=H_0+\Sigma^{R}(0)$, which is non-Hermitian and can host exceptional points: degeneracies at which both eigenvalues and eigenvectors coalesce because the Hamiltonian becomes defective (non-diagonalizable), with a characteristic square-root band splitting; in two dimensions such degeneracies can also organize into closed exceptional rings (see Bergholtz, Budich, and Kunst, Rev. Mod. Phys. 93, 015005 (2021) for a review of non-Hermitian degeneracies and their topology). Building on their earlier study of spin liquids coupled to environmental modes (Yang, Morampudi, and Bergholtz, Phys. Rev. Lett. 126, 077201 (2021)), the source Letter analyzed how $\Sigma$ transforms under the particle-hole, inversion, and time-reversal symmetries of the Majorana problem and found: (a) particle-hole symmetry plus inversion symmetry forces the anti-Hermitian part to be trivial (mere resonance broadening), so inversion must be broken; (b) random magnetic fields with zero mean that preserve time-reversal symmetry on average but break inversion (different variances on the two sublattices, e.g., via proximity to a paramagnetic substrate) produce exceptional rings, computed within the self-consistent Born approximation (SCBA, a self-consistent lowest-order disorder treatment) and verified by kernel-polynomial-method numerics (a Chebyshev-expansion technique for spectral functions of large disordered systems); the resulting density of states shows a dip rather than a peak at zero energy, with a corresponding anomaly in the low-temperature specific heat; (c) coupling to gapped optical phonons through symmetry-allowed vertices that also break time reversal produces exceptional points connected by Fermi arcs (constant-real-energy surfaces terminating on the exceptional points), with a strongly anisotropic dynamical spin structure factor of the kind measured in inelastic neutron scattering. The Letter is deliberately idealized, and its own text marks the gaps: the self-energy is obtained at SCBA order with dimensionless model couplings rather than material-specific parameters, so the experimentally realistic ranges of coupling strengths, phonon gaps, disorder variances, and temperatures where anti-Hermitian level attraction wins over Hermitian gap opening are unknown; the term $\omega\Sigma^{(1)}$, responsible for wavefunction renormalization, is neglected even though the Supplemental Material shows it can suppress a non-negligible part of the anti-Hermitian self-energy for strong disorder and explicitly defers a generalized-eigenvalue treatment to future study; the random-field perturbation theory is restricted to the zero-flux sector, while flux disorder or thermally excited vortices may obscure the signatures — the Letter itself notes that in $\alpha$-RuCl$_3$ the optical phonon energy lies well above the flux gap, so exceptional-point signatures may be obscured by effective flux disorder; and the proposed detection strategies (substrate-induced inversion breaking, proximity-induced random fields, nonlinear spectroscopy of the kind analyzed by Kanega, Ikeda, and Sato, Phys. Rev. Research 3, L032024 (2021)) are sketched but not developed into concrete, material-specific protocols.

Problem Statement

Determine, at a quantitative and material-realistic level, whether the exceptional degeneracies (exceptional rings and exceptional points with Fermi arcs) predicted for the Kitaev honeycomb spin liquid coupled to phonons and quenched disorder survive under experimentally realistic conditions, and how they can be unambiguously detected. This single objective is made concrete by four linked components, reflecting the limitations left open by Yang, Varjas, Bergholtz, Morampudi, and Wilczek (Phys. Rev. Research 4, L042025 (2022)) and its Supplemental Material: (1) Parameter regimes — derive, by microscopic modeling beyond the self-consistent Born approximation (including higher-order or vertex corrections where needed, with material-specific couplings), the ranges of coupling strengths, phonon gaps, disorder variances, and temperatures for which the anti-Hermitian components of the self-energy dominate the Hermitian gap-opening components, so that exceptional degeneracies actually occur; equivalently, map the boundary between the trivially broadened phase, the Hermitian-gapped phase, and the exceptional phase in realistic parameter space. (2) Linear-in-$\omega$ self-energy — treat the wavefunction-renormalization term $\omega\Sigma^{(1)}$ systematically, and quantify how it modifies the size, shape, and observability of the exceptional degeneracies (the Supplemental Material shows this term can suppress a non-negligible part of the anti-Hermitian self-energy at strong disorder). (3) Flux sector — go beyond the zero-flux restriction of the random-field perturbation theory and determine how quenched flux disorder and thermally excited vortices modify or obscure the exceptional signatures in bulk observables (single-particle spectral function, dynamical spin structure factor, density of states, low-temperature specific heat). (4) Detection — translate components (1)-(3) into concrete, quantitatively validated (demonstrated in model calculations for a named compound) protocols for candidate compounds: material-specific modeling of substrate-engineered inversion breaking and proximity-induced random fields, and spectroscopic discriminants (neutron scattering, nonlinear THz spectroscopy) that separate exceptional-degeneracy signatures such as square-root dispersions, asymmetric Fermi arcs, and directional anisotropy of the structure factor from conventional disorder broadening. Components (2) and (3) are the theoretical corrections that make the regime map (1) reliable, and (4) is its observable translation, so the four components are jointly posed. Relative terms such as 'experimentally realistic' are to be read against the conventions of the source and the field: couplings in units of the Kitaev exchange $J$, phonon gaps and temperatures relative to the flux gap, and disorder variances comparable to those induced by the proposed substrate geometries.

References

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