Main

Herbertsmithite, a layered spin-1/2 kagome lattice antiferromagnet12, is one of the strongest contenders for an experimental realization of a spin liquid state13. Indeed, no sign of magnetic ordering is observed down to temperatures around 50 mK, whereas the natural energy scale set by the magnetic exchange coupling J ~200 K is four orders of magnitude larger14. Neutron scattering experiments1 on single crystals of this material are consistent with a continuum of fractionalized spinon excitations as expected in a quantum spin liquid state. However, mean-field theories predict a vanishing structure factor below the onset of the two-spinon continuum, which is at a finite energy even for gapless spin liquids, apart from the small set of crystal momenta where the spinon gap closes. This is in stark contrast to experiments, where the measured structure factor is finite and almost constant as a function of frequency down to energies of the order of ~J/10 (ref. 1).

Here we propose an explanation for the lack of a momentum-dependent spinon continuum threshold via the interaction of spinons with another set of excitations which form a (nearly) flat band. Such localized excitations act as a momentum sink for the spinons, thereby flattening the dynamic structure factor. So far, the only theoretical model for a spin liquid state on the kagome lattice which naturally gives rise to a flat excitation band at low energies consists of the Z2 spin liquids2,3,4. Besides spinons, these states exhibit gapped vortex excitations15,16 of an emergent Z2 gauge field17,18, so-called visons11, which indeed have a lowest energy band that is nearly flat19,20. Because the visons carry neither charge nor spin, they do not couple directly to neutrons. They interact with spinons, however, and we show that this coupling is responsible for flattening the dynamic structure factor and removing the sharp onset at the two-spinon continuum, in accordance with experimental results. Note that the vison gap has to be small for this mechanism to work. This assumption is justified by numerical density matrix renormalization group calculations21,22,23, which indicate that a Z2 spin liquid ground-state on the kagome lattice is proximate to a valence bond solid (VBS) transition, at which the vison gap vanishes.

Model

Our aim is to compute the dynamic structure factor for two Z2 spin liquids that have been discussed in detail in ref. 2. We start from the standard bosonic spin liquid mean-field theory of the spin-1/2 antiferromagnetic Heisenberg model on the kagome lattice. Using a Schwinger-boson representation of the spin-1/2 operators where σ denotes the vector of Pauli matrices and is the creation operator of a boson with spin α on lattice site i, and performing a mean-field decoupling in the spin-singlet channel, the Heisenberg Hamiltonian can be written as

with is the fully antisymmetric tensor of SU(2), h.c. is the hermitian conjugate term and λ denotes the Lagrange multiplier that fixes the constraint of one Schwinger boson per lattice site. Sums over Greek indices are implicit. To study the effect of vison excitations on the spinons, we have to include phase fluctuations of the mean-field variables Qij in our theory. The Z2 spin liquid corresponds to the Higgs phase of the resulting emergent gauge theory, where the phase fluctuations are described by an Ising bond variable σijz. The Hamiltonian describing bosonic spinons and their coupling to the Ising gauge field takes the form

where the terms on the second line are responsible for the dynamics of the gauge field σijz. K and h are phenomenological parameters that set the energy scale for fluctuations of the Z2 gauge field. Vison excitations are vortices of this emergent Z2 gauge field—that is, excitations where the product ∏ σijz on a plaquette changes sign. For practical calculations it is more convenient to switch to a dual description of the Z2 gauge field in terms of its vortex excitations24, where the pure gauge field terms in the second line of equation (1) take the form of a fully-frustrated Ising model on the dice lattice. This model has been studied in detail in refs 19 and 20 and gives rise to three flat vison bands if restricted to nearest-neighbour vison hopping. As only the gap to the lowest vison band is small, we neglect the effects of the other two bands in the following.

The coupling between spinons and visons is a long-range statistical interaction (a spinon picks up a Berry’s phase of π when encircling a vison20), which cannot be expressed in the form of a simple local Hamiltonian in the vortex representation. However, the fact that visons on the dice lattice are non-dispersing comes to the rescue here. Because these excitations are localized and can only be created in pairs, the long-range statistical interaction is effectively cancelled. Indeed, if a spinon is carried around a pair of visons, it does not pick up a Berry’s phase. This is in precise analogy to an electron carried around a pair of superconducting Abrikosov vortices, where the total encircled flux is 2π and thus no phase is accumulated. The vison pairs are excited locally by a spinon, and thus it is reasonable to model the spinon–vison interaction by a local energy–energy coupling, neglecting the long-range statistical part. Accordingly we choose the simplest, gauge-invariant Hamiltonian of bosonic spinons on the kagome lattice coupled to a single, non-dispersing vison mode on the dual Dice lattice

Here, the real field ϕi describes visons living on the dice lattice sites i, g0 denotes the spinon–vison coupling strength and Δv is the vison gap. The sum in the interaction term runs only over the three-coordinated Dice lattice sites i and couples the spinon bond energy on the triangular kagome plaquettes to the local vison gap at the plaquette centre. Further terms, where spinons on the hexagonal kagome plaquettes interact with visons at the centre of the hexagons are allowed, but neglected for simplicity.

A more detailed discussion of this interaction term can be found in the Supplementary Methods. We are going to compute the dynamic structure factor S(k, ω) using the model equation (2) for a particular Z2 spin liquid state that has been identified in ref. 2. For the nearest-neighbour kagome antiferromagnet there are two independent bond expectation values Qij ∈ {Q1, Q2} and the two distinct, locally stable mean-field solutions have Q1=Q2 or Q1=−Q2. The Q1=Q2 state has flux π in the elementary hexagons, whereas the Q1=−Q2 state is a zero-flux state. During the remainder of this article we focus only on the Q1=Q2 state, as it gives rise to a little peak in S(k, ω) at small frequencies at the M point of the extended Brillouin zone, in accordance with experimental results. Results for the other state are discussed in the Supplementary Methods. Two other bosonic Z2 states have been identified on the kagome lattice3, but we refrain from computing the structure factor for these states, because both have a doubled unit cell, which complicates the calculations considerably.

Dynamic structure factor

Neutron scattering experiments measure the dynamic structure factor

which we are going to compute for the model presented in equation (2). Here Ri denotes the position of lattice site i. Note that S(k, ω) is periodic in the extended Brillouin zone depicted in (Fig. 1e). After expressing Si⋅Sj in terms of Schwinger bosons and diagonalizing the free spinon Hamiltonian with a Bogoliubov transformation, the one-loop expression for the dynamic spin susceptibility shown in Fig. 2, χ(k, iωn), can be derived straightforwardly (Methods). The dynamic structure factor can then be obtained from the susceptibility via

Results of this calculation at zero temperature are shown in Figs 1 and 3 for the Q1=Q2 state for different spinon–vison interaction strengths g0. In the region around and between the high-symmetry points M and K the lowest order vertex correction shown in Fig. 2 gives only a relatively small contribution to S(k, ω) and thus has been neglected in the data shown in these figures (see Supplementary Methods for a discussion).

Figure 1: Density plots of the the dynamic spin-structure factor S(k, ω) for the Q1 = Q2 spin liquid state.
figure 1

a–c, Plots of S(k,ω) at zero temperature for different spinon–vison interaction strengths as a function of frequency and momentum along the high-symmetry directions between the Γ, M and K points of the extended Brillouin zone, indicated by the blue arrows in e. a, Non-interacting spinons. Note that in the Q1=Q2 state two of the three spinon bands are degenerate, whereas the third, highest energy spinon band is flat. This flat spinon band gives rise to the horizontal feature at ω≃0.75J. b, Spinon–vison interaction g0=0.2. c, Spinon–vison interaction g0=0.6. d,e, S(k,ω) for non-interacting spinons at fixed frequency ω/J=0.4 (d) and ω/J=0.85 (e). The elementary Brillouin zone of the kagome lattice is indicated by a dashed hexagon in e. Note the sharp onset of the two-spinon continuum for non-interacting spinons in a and d, which is washed out when interactions with visons are accounted for. All data in this figure were calculated for |Q1|=0.4 and the spinon gap was fixed at Δs≃0.05J. The vison gap is set to Δv=0.025J in b and c.

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Figure 2: Feynman diagrams for the spinon self energy and spin susceptibility for the theory in equation (2).
figure 2

Spinon self energy (left), one-loop contribution to the spin susceptibility (middle) and corresponding lowest order vertex correction (right). Double lines are dressed spinon propagators and dashed lines are bare vison propagators.

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Figure 3: Qualitative comparison between experimental measurements1 and our theoretical results for the dynamic structure factor S(k, ω).
figure 3

a,b, Experimental data at fixed frequency are shown for ω=0.75 meV (a) and ω=6 meV (b). c,d, Theoretical results for the Q1=Q2 spin liquid at fixed frequency are plotted for ω=0.37J (c) and ω=0.6J (d). The extended Brillouin zone is indicated by the dashed hexagons. Note that the peak at the M point at low frequencies, as well as the flatness of S(k,ω) between the M and K points at higher frequencies is captured by our theory. e,f, Cuts of our theoretical results for S(k,ω) along high-symmetry directions at different frequencies are plotted between the M and K point (e), as well as between the Γ and M point (f), again showing the peak at the M point at low frequencies. g, Details of the calculated structure factor as a function of frequency for various momenta between the M (bottom curve) and K point (top curve). Note that all curves in g are shifted by 0.12 J with respect to each other for better visibility. All theoretical data shown was computed for the Q1=Q2 state with a spinon–vison interaction strength g0=0.6 and other parameters as in Fig. 1.

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Discussion

Fig. 1 shows the two-spinon contribution to the dynamic structure factor for the Q1=Q2 state (results for the Q1=−Q2 state can be found in the Supplementary Methods). The onset of the two-spinon continuum, which has a minimum at the M point, is clearly visible in Fig. 1a as the line of frequencies below which the dynamic structure factor vanishes. Moreover, several sharp peaks appear inside the spinon continuum. We note that such features in the two-spinon contribution to S(k, ω) are generic and are present also for gapless Dirac spin liquids.

Fig. 1b, c show the dynamic structure factor along the same high-symmetry directions as in Fig. 1a, but now including the effect of spinon-induced vison pair production for two different interaction strengths g0. The non-dispersing visons act as a powerful momentum sink for the spinons and lead to a considerable shift of spectral weight below the two-spinon continuum. The computed structure factor is considerably flattened at intermediate energies. Our results for the Q1=Q2 state also capture the small low-frequency peak in S(k, ω) at the M point, which has been seen in experiment. This peak is a remnant of a minimum in the threshold of the two-spinon continuum at the M point, and we conjecture that it might be an indication that this particular Z2 spin liquid state is realized in Herbertsmithite. In Fig. 3 we show plots of S(k, ω) at constant energy, where this peak is clearly visible, and compare our results qualitatively to the experimental data. Note that we did not choose the parameters to fit the experimental data, instead we tried to use reasonable values for the spinon gap Δs≃0.05J and the vison gap Δv=0.025J to make features related to the momentum-independent onset of the dynamic structure factor better visible. Also the spinon bandwith was adjusted to be on the order of J.

In Fig. 1c, 3g one can barely see small oscillations of S(k, ω) at low frequencies. These oscillations originate from the self-consistent computation of the spinon self-energy Σ (k, ω) and are related to resonances in the self-energy at energies corresponding to the creation of two, four and higher even numbers of vison excitations.

The experimental results show a strong increase of the dynamic structure factor at energies below 1meV away from the M point. We attribute this feature to impurity spins, which are not accounted for in our approach. In Herbertsmithite excess copper substitutes for zinc in the interlayer sites. These spin-1/2 impurities are only weakly coupled to the kagome layers, with an exchange constant that is on the order of one kelvin25. Although it is unlikely that these impurities contribute considerably to a flattening of the dynamical structure factor as discussed in this paper, we believe they are responsible for the above-mentioned low-energy contribution. This is in accordance with recent low-energy neutron scattering measurements on powder samples of Herbertsmithite26, but a detailed calculation remains an open problem for future study. Also note that such a low-energy contribution would hide the momentum-independent onset of the dynamic structure factor, which is at the energy ωonset=2Δv+2Δs in the scenario discussed here.

Dzyaloshinskii–Moriya (DM) interactions as well as an easy-axis anisotropy on the order of ~J/10 are known to exist in Herbertsmithite, but have been neglected in our analysis for simplicity. The effect of DM interactions has been studied within a 1/N expansion9,27, where the Q1=Q2 state is favoured over the Q1=−Q2 state if the DM interactions are sufficiently strong.

Last, neutron scattering experiments explored energies up to ω≃0.65J and concluded that the integrated weight accounts for roughly 20% of the total moment sum rule1. Consequently it is reasonable to expect that the dynamic structure factor is finite up to energies of a few J. For the parameters chosen in our calculation (that is Q1=0.4 and a spinon gap Δs=0.05) the structure factor for non-interacting spinons has a sharp cutoff at an energy around ω≃1.3J, corresponding to roughly twice the spinon bandwidth. However, if interactions with visons are included, this upper cutoff is shifted to considerably larger energies. For a spinon–vison coupling g0=0.6, the structure factor has a smooth upper cutoff at an energy around ω≃3J. Such large bandwidths are hardly achievable in theories with non-interacting spinons. We note that similarly large bandwidths have been found in exact diagonalization studies28.

Methods

The one-loop expression for the dynamic spin susceptibility, χ(k, iωn), is given by

where the dots represent similar terms that give a contribution at negative frequencies after analytic continuation and thus play no role in calculating S(k, ω) at zero temperature. Note that we are working in a Matsubara representation here, where the spin-susceptibility χ(k, iωn) and the spinon propagator G(q, iΩn) are expressed as functions of the bosonic Matsubara frequencies iΩn and iωn. The summation over the sublattice indices i, j, ℓ, m∈{1,2,3} is implicit here and the 3×3 matrices Uij and V ij form the Bogoliubov rotation matrix

as defined in ref. 2, which diagonalizes the mean-field spinon Hamiltonian. Gℓ (q, iΩn) denotes the dressed spinon Green’s function with band-index ℓ

where εℓ (q) is the bare spinon dispersion. The spinon self-energy (Fig. 2), which we compute self-consistently, is determined by the equation

Here the 6×6 matrix λ(p, q) denotes the bare spinon–vison interaction vertex, with p (q) the momentum of the outgoing (incoming) spinon. Note that the six spinon bands come in three degenerate pairs owing to the SU(2) spin-symmetry. Furthermore, note that the flat vison band is not renormalized at arbitrary order in the spinon–vison coupling.

We emphasize here that a self-consistent computation of the spinon self-energy is necessary, because the real part of Σ (k, ω) is large and broadens the spinon bands. A non-self-consistent computation thus leads to sharp spinon excitations above the bare spinon band, which are unphysical as they would decay immediately via vison pair production. A different approximation, which circumvents this problem, would be to calculate Σ (k, ω) non-self-consistently and neglect the real part completely. This approximation violates sum rules however, as the integrated spectral weight of the spinon is no longer unity (for a detailed discussion, see the Supplementary Methods).

Note that we do not determine the parameters |Q1| and λ variationally. Instead, we use them to fix the spinon gap as well as the spinon bandwidth. |Q1| is restricted to values between 0 and and quantifies antiferromagnetic correlations of nearest-neighbour spins ( if nearest-neighbour spins form a singlet). All data shown in this paper was computed for |Q1|=0.4, and λ has been adjusted such that the spinon gap takes the value Δs/J ≃0.05. As mentioned in the introduction, we assume that the vison gap Δv is small owing to evidence of proximity to a VBS state, and we chose Δv/J =0.025 for all data shown in this Article—namely, the vison gap is roughly half the spinon gap.