Nature Physics  Letter
Topological excitations and the dynamic structure factor of spin liquids on the kagome lattice
 Matthias Punk^{1, 2}^{, }
 Debanjan Chowdhury^{1}^{, }
 Subir Sachdev^{1}^{, }
 Journal name:
 Nature Physics
 Volume:
 10,
 Pages:
 289–293
 Year published:
 DOI:
 doi:10.1038/nphys2887
Recent neutron scattering experiments on the spin1/2 kagome lattice antiferromagnet ZnCu_{3}(OH)_{6}Cl_{2} (Herbertsmithite) provide the first evidence of fractionalized excitations in a quantum spin liquid state in two spatial dimensions^{1}. In contrast to existing theoretical models of both gapped and gapless spin liquids^{2, 3, 4, 5, 6, 7, 8}, which give rise to sharp dispersing features in the dynamic structure factor^{9, 10}, the measured dynamic structure factor reveals an excitation continuum that is remarkably flat as a function of frequency. Here we show that many experimentally observed features can be explained by the presence of topological vison excitations in a Z_{2} spin liquid^{11}. These visons form flat bands on the kagome lattice, and thus act as a momentum sink for spincarrying excitations that are probed by neutron scattering. We compute the dynamic structure factor for two different Z_{2} spin liquids^{2} and find that our results for one of them are in qualitative agreement with the neutron scattering experiments above a very low energy cutoff, below which the structure factor is probably dominated by impurities.
Subject terms:
At a glance
Figures
Main
Herbertsmithite, a layered spin1/2 kagome lattice antiferromagnet^{12}, is one of the strongest contenders for an experimental realization of a spin liquid state^{13}. 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 larger^{14}. Neutron scattering experiments^{1} on single crystals of this material are consistent with a continuum of fractionalized spinon excitations as expected in a quantum spin liquid state. However, meanfield theories predict a vanishing structure factor below the onset of the twospinon 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 momentumdependent 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 Z_{2} spin liquids^{2, 3, 4}. Besides spinons, these states exhibit gapped vortex excitations^{15, 16} of an emergent Z_{2} gauge field^{17, 18}, socalled visons^{11}, which indeed have a lowest energy band that is nearly flat^{19, 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 twospinon 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 calculations^{21, 22, 23}, which indicate that a Z_{2} spin liquid groundstate 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 Z_{2} spin liquids that have been discussed in detail in ref. 2. We start from the standard bosonic spin liquid meanfield theory of the spin1/2 antiferromagnetic Heisenberg model on the kagome lattice. Using a Schwingerboson representation of the spin1/2 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 meanfield decoupling in the spinsinglet 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 meanfield variables Q_{ij} in our theory. The Z_{2} spin liquid corresponds to the Higgs phase of the resulting emergent gauge theory, where the phase fluctuations are described by an Ising bond variable σ_{ij}^{z}. 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 σ_{ij}^{z}. K and h are phenomenological parameters that set the energy scale for fluctuations of the Z_{2} gauge field. Vison excitations are vortices of this emergent Z_{2} gauge field—that is, excitations where the product ∏ σ_{ij}^{z} on a plaquette changes sign. For practical calculations it is more convenient to switch to a dual description of the Z_{2} gauge field in terms of its vortex excitations^{24}, where the pure gauge field terms in the second line of equation (1) take the form of a fullyfrustrated 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 nearestneighbour 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 longrange statistical interaction (a spinon picks up a Berry’s phase of π when encircling a vison^{20}), 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 nondispersing comes to the rescue here. Because these excitations are localized and can only be created in pairs, the longrange 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 longrange statistical part. Accordingly we choose the simplest, gaugeinvariant Hamiltonian of bosonic spinons on the kagome lattice coupled to a single, nondispersing vison mode on the dual Dice lattice
Here, the real field ϕ_{i} describes visons living on the dice lattice sites i, g_{0} denotes the spinon–vison coupling strength and Δ_{v} is the vison gap. The sum in the interaction term runs only over the threecoordinated 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 Z_{2} spin liquid state that has been identified in ref. 2. For the nearestneighbour kagome antiferromagnet there are two independent bond expectation values Q_{ij} ∈ {Q_{1}, Q_{2}} and the two distinct, locally stable meanfield solutions have Q_{1}=Q_{2} or Q_{1}=−Q_{2}. The Q_{1}=Q_{2} state has flux π in the elementary hexagons, whereas the Q_{1}=−Q_{2} state is a zeroflux state. During the remainder of this article we focus only on the Q_{1}=Q_{2} 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 Z_{2} states have been identified on the kagome lattice^{3}, 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 R_{i} 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 S_{i}⋅S_{j} in terms of Schwinger bosons and diagonalizing the free spinon Hamiltonian with a Bogoliubov transformation, the oneloop expression for the dynamic spin susceptibility shown in Fig. 2, χ(k, iω_{n}), 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 Q_{1}=Q_{2} state for different spinon–vison interaction strengths g_{0}. In the region around and between the highsymmetry 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).
Discussion
Fig. 1 shows the twospinon contribution to the dynamic structure factor for the Q_{1}=Q_{2} state (results for the Q_{1}=−Q_{2} state can be found in the Supplementary Methods). The onset of the twospinon 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 twospinon 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 highsymmetry directions as in Fig. 1a, but now including the effect of spinoninduced vison pair production for two different interaction strengths g_{0}. The nondispersing visons act as a powerful momentum sink for the spinons and lead to a considerable shift of spectral weight below the twospinon continuum. The computed structure factor is considerably flattened at intermediate energies. Our results for the Q_{1}=Q_{2} state also capture the small lowfrequency 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 twospinon continuum at the M point, and we conjecture that it might be an indication that this particular Z_{2} 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 Δ_{s}≃0.05J and the vison gap Δ_{v}=0.025J to make features related to the momentumindependent 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 selfconsistent computation of the spinon selfenergy Σ (k, ω) and are related to resonances in the selfenergy 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 spin1/2 impurities are only weakly coupled to the kagome layers, with an exchange constant that is on the order of one kelvin^{25}. 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 abovementioned lowenergy contribution. This is in accordance with recent lowenergy neutron scattering measurements on powder samples of Herbertsmithite^{26}, but a detailed calculation remains an open problem for future study. Also note that such a lowenergy contribution would hide the momentumindependent onset of the dynamic structure factor, which is at the energy ω_{onset}=2Δ_{v}+2Δ_{s} in the scenario discussed here.
Dzyaloshinskii–Moriya (DM) interactions as well as an easyaxis 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 expansion^{9, 27}, where the Q_{1}=Q_{2} state is favoured over the Q_{1}=−Q_{2} 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 rule^{1}. 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 Q_{1}=0.4 and a spinon gap Δ_{s}=0.05) the structure factor for noninteracting 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 g_{0}=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 noninteracting spinons. We note that similarly large bandwidths have been found in exact diagonalization studies^{28}.
Methods
The oneloop expression for the dynamic spin susceptibility, χ(k, iω_{n}), 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 spinsusceptibility χ(k, iω_{n}) and the spinon propagator G(q, iΩ_{n}) are expressed as functions of the bosonic Matsubara frequencies iΩ_{n} and iω_{n}. The summation over the sublattice indices i, j, ℓ, m∈{1,2,3} is implicit here and the 3×3 matrices U_{ij} and V _{ij} form the Bogoliubov rotation matrix
as defined in ref. 2, which diagonalizes the meanfield spinon Hamiltonian. G_{ℓ} (q, iΩ_{n}) denotes the dressed spinon Green’s function with bandindex ℓ
where ε_{ℓ} (q) is the bare spinon dispersion. The spinon selfenergy (Fig. 2), which we compute selfconsistently, 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) spinsymmetry. Furthermore, note that the flat vison band is not renormalized at arbitrary order in the spinon–vison coupling.
We emphasize here that a selfconsistent computation of the spinon selfenergy is necessary, because the real part of Σ (k, ω) is large and broadens the spinon bands. A nonselfconsistent 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) nonselfconsistently 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 Q_{1} and λ variationally. Instead, we use them to fix the spinon gap as well as the spinon bandwidth. Q_{1} is restricted to values between 0 and and quantifies antiferromagnetic correlations of nearestneighbour spins ( if nearestneighbour spins form a singlet). All data shown in this paper was computed for Q_{1}=0.4, and λ has been adjusted such that the spinon gap takes the value Δ_{s}/J ≃0.05. As mentioned in the introduction, we assume that the vison gap Δ_{v} is small owing to evidence of proximity to a VBS state, and we chose Δ_{v}/J =0.025 for all data shown in this Article—namely, the vison gap is roughly half the spinon gap.
Change history
 Corrected online 22 April 2016
 In the version of this Letter originally published square root symbols were mistakenly included in the xaxis tick labels in Figure 3ad. This has now been corrected in the online versions of the Letter.
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Acknowledgements
We acknowledge illuminating discussions with M. Babadi, S. Gopalakrishnan, M. Lawler, J. D. Sau and especially Y. S. Lee. Furthermore, we thank TH. Han and Y. S. Lee for providing the neutron scattering data shown in Fig. 3. This research was supported by the US NSF under Grant DMR1103860 and by the John Templeton Foundation. This research was also supported in part by the Perimeter Institute for Theoretical Physics; research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. M.P. is supported by the Erwin Schrödinger Fellowship J 3077N16 of the Austrian Science Fund (FWF). The computations were performed in part on the Odyssey cluster supported by the FAS Science Division Research Computing Group at Harvard University.
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Affiliations

Department of Physics, Harvard University, Cambridge, Massachussetts 02138, USA
 Matthias Punk,
 Debanjan Chowdhury &
 Subir Sachdev

Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
 Matthias Punk
Contributions
M.P. performed the numerical computations. M.P., D.C. and S.S. contributed to the theoretical research described in the paper and the writing of the manuscript.
Competing financial interests
The authors declare no competing financial interests.
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