Abstract
The lowenergy elementary excitations in frustrated quantum magnets have fascinated researchers for decades. In frustrated Ising magnets on a pyrochlore lattice possessing macroscopically degenerate spinice ground states, the excitations have been discussed in terms of classical magnetic monopoles, which do not contain quantum fluctuations. Here we report unusual behaviours of magnetothermal conductivity in the disordered spinliquid regime of pyrochlore Yb_{2}Ti_{2}O_{7}, which hosts frustrated spinice correlations with large quantum fluctuations owing to pseudospin1/2 of Yb ions. The analysis of the temperature and magnetic field dependencies shows the presence of gapped elementary excitations. We find that the gap energy is largely suppressed from that expected in classical monopoles. Moreover, these excitations propagate a long distance without being scattered, in contrast to the diffusive nature of classical monopoles. These results suggests the emergence of highly itinerant quantum magnetic monopole, which is a heavy quasiparticle that propagates coherently in threedimensional spin liquids.
Introduction
Rareearth pyrochlore oxides exhibit various exotic magnetic properties owing to their strong geometrical frustration experienced by coupled magnetic moments on the tetrahedral lattice (Fig. 1a)^{1}. The most explored materials are Ho_{2}Ti_{2}O_{7} and Dy_{2}Ti_{2}O_{7}, in which the magnetic moments can be regarded as classical spins with a strong easyaxis (Ising) anisotropy^{1,2}. The frustration of these moments results in a remarkable spin ice with macroscopically degenerate ground states, in which each tetrahedron has the two spins in and two spins out (2in2out) configuration. This spin structure is characterized by dipolar spin correlations with a power law decay, which is observable as the unusual pinchpoint shape of spin structure factor by neutron scattering^{3,4}. One of the most remarkable features of the spinice state is that it hosts emergent magnetic monopole excitations; the first excitation is a 3in1out configuration^{5,6}. This produces a bound pair of north and south poles, which can be fractionalized into two free magnetic monopoles. This classical monopole excitations are gapped and dispersionless (Fig. 1b). Therefore the propagation of monopoles occurs only diffusively and the monopole population decays exponentially at temperatures well below the gap. Of particular interest is how the spinice ground state is altered by the quantum fluctuations, which may lift the degeneracy of the spinice manifold, leading to a new ground state such as quantum spinice state^{7,8,9,10,11}. To clarify this issue, uncovering newly emergent elementary excitations in the presence of quantum fluctuations is crucially important. Although exotic excitations such as gapless photonlike mode have been proposed theoretically, the nature of excitations are poorly explored.
Among the magnetic pyrochlore materials, Yb_{2}Ti_{2}O_{7}, Er_{2}Ti_{2}O_{7}, Pr_{2}Sn_{2}O_{7} and possibly Tb_{2}Ti_{2}O_{7} host strong quantum fluctuations of magnetic dipoles owing to pseudospin1/2 of magnetic rareearth elements^{12,13,14,15}. In particular, Yb_{2}Ti_{2}O_{7} is a good model system to study the influence of the quantum effects on monopole excitations. This is because the lowtemperature physical properties are not influenced by the crystalline electric fieldexcited levels owing to the well separated excited levels from the ground state^{16}. In addition, the full set of Hamiltonian parameters has been determined by inelastic neutron scattering experiments^{12,16}. The Hamiltonian consists of three main interactions, J_{}, J_{⊥} and J_{z±}. Here J_{} (=2 K) is the Ising component of the nearest neighbour interaction, J_{⊥}(=0.58 K) is the XYcomponent and J_{z±}(=1.7 K) is the offdiagonal component. Finite J_{⊥} and J_{z±} produce quantum fluctuations (Fig. 1c). In Yb_{2}Ti_{2}O_{7}, J_{z±}, which is comparable to J_{}, gives rise to dispersive monopole excitations, that is, itinerant magnetic monopoles. Yb_{2}Ti_{2}O_{7} undergoes a firstorder ferromagnetic phase transition at T_{C}∼0.2 K (refs 17, 18). It is widely believed that quantum fluctuations keep spins from freezing and lead to a spinliquid state. Therefore, the pinchpoint structure observed in the ferromagnetic samples by neutron scattering above T_{C} indicates spinliquid phase with spinice correlations.
Here, to study the elementary excitations in the spinliquid state of Yb_{2}Ti_{2}O_{7}, we measured the thermal conductivity, which is a powerful probe for lowenergy excitations at low temperatures, providing a sensitive measurement of a flow of entropy conducted by magnetic excitations and phonons. The thermal conductivity has been reported in the classical spinice state of Dy_{2}Ti_{2}O_{7} recently^{19,20}. However, the interpretation of the thermal conductivity of Dy_{2}Ti_{2}O_{7} appears to be complicated owing to the strongly suppressed phonon thermal conductivity by unknown additional scatterings (Supplementary Fig. 1; Supplementary Note 1). In fact, suggested heat transport by classical monopole is at odds with the diffusive motion of the dispersionless classical monopoles. We show that the thermal conductivity of Yb_{2}Ti_{2}O_{7} is rather simple. The monopole thermal conductivity can be well separated from the phonon contribution, which obeys magnetic field/temperature (H/T) scaling. Our analysis shows the evidence of the substantial heat transport by quantum magnetic monoples, whose excitation energy is significantly suppressed from that of classical monopoles. The quantum magnetic monopoles become itinerant due to quantum fluctuations, in stark contrast to the localized and diffusive nature of classical monopoles.
Results
Thermal conductivity and specific heat at zero magnetic field
Figure 2a shows the temperature dependence of thermal conductivity divided by temperature κ/T in zero field and at μ_{0}H=12 T measured on a single crystal of Yb_{2}Ti_{2}O_{7}, where μ_{0} is the vacuum permeability. Distinct jump in κ/T at zero field is observed at T_{C}. As shown in Fig. 2b, the specific heat C of the single crystal taken from the same batch shows a sharp and large jump at the same T_{C} (ref. 17). We note that there are also some experiments reporting the absence of longrange order even below the critical temperature^{21,22,23,24,25}. However, in the previous studies, a sharp single jump in C/T had been reported only in the powdered samples, but not in single crystals^{21,26}. In contrast, in the recent highquality single crystals having a sharp C/T jump, the longrange ferromagnetic ordering below T_{C} and the pinchpoint features in neutron scattering have been clearly observed^{17,18}. As shown in Fig. 2c, zero field κ/T above T_{C} follows a Tlinear dependence with negligibly small intercept at T=0 K. The absence of residual in the spinliquid state with spinice correlations will be discussed later.
Magnetothermal conductivity
As clearly seen in Fig. 2a, magnetic field strongly enhances the thermal conductivity. Figure 3a–d show the field dependence of κ(H)/T for different field directions. As illustrated in the inset of Fig. 3e, there are three characteristic regimes; lowfield regime where κ(H)/T decreases with H, intermediatefield regime where κ(H)/T increases, and highfield regime where κ(H)/T exhibits a saturation.
In the present system, heat is transferred by phonons and magnetic excitations: κ=κ_{p}+κ_{m}. We point out that the field dependence of κ(H)/T in the intermediate and highfield regimes are dominated by the phonon contribution κ_{p} determined by spinphonon scattering, which contains elastic and inelastic processes. The elastic scattering is enhanced with increasing disorder of spins and thus this scattering process should be monotonically suppressed by the alignment of spins with increasing magnetic field. A recent calculations of magnetoresistance in a fluctuating spinice state indicates that the electronspin elastic scattering rate decreases with increasing magnetization^{27}, which supports this trend. The inelastic scattering is directly related to the quantum dynamics of spin. In this inelastic scattering, the leading spinflip process accompanies a hopping of a monopole to the neighbouring tetrahedron (which is related to J_{⊥}), because this process requires much lower energies than creation or annihilation of monopoles. This scattering is suppressed with field by the formation of Zeeman gap. (see Supplementary Note 2 for discussion in more detail.) Therefore an external magnetic field suppresses both elastic and inelastic scatterings, leading to the enhancement of the phonon thermal conductivity κ_{p}.
In the highfield regime, the Zeeman splitting energy gμ_{B}H well exceeds both of the magnetic interactions and thermal energy, gμ_{B}HJ_{}, J_{⊥}, J_{z±} and k_{B}T, where g is the gfactor, μ_{B} is the Bohr magneton and k_{B} is the Boltzmann constant. In this situation, where all spins are fully polarized and the magnetic (spinwave) excitations are gapped with a gap gμ_{B}H, thermal conductivity is almost entirely dominated by the pure phonon contribution without spin scattering because of the following reasons. First, elastic spinphonon scattering is absent due to the perfect alignment of spins. Second, inelastic scattering is also absent due to the formation of the large Zeeman gap. Third, spins do not carry the heat due to the Zeeman gap. As purely phononic thermal conductivity is insensitive to magnetic field, κ(H)/T in the highfield regime is nearly independent of H. In the intermediatefield regime, the phonon mean free path is significantly reduced by the spinphonon scattering due to the spins thermally excited across the Zeeman gap. In fact, as shown in Fig. 3e which plots κ/T as a function of μ_{B}H/k_{B}T, all data collapse into a single curve except for the low μ_{B}H/k_{B}T regime. The fact that data for both field directions stabilizing different spin configurations (3in1out for H  [1, 1, 1] and 2in2out for H  [0, 0, 1]) follow the same curve implies that the elastic spinphonon scattering dominates over the inelastic scattering in this regime. It is intriguing that the H/T scaling curve appears to follow the Brillouin function (the dashed line in Fig. 3e). This coincidence with the Brillouin function calls for further theoretical investigations.
A particularly important information for the elementary excitations is provided by κ(H)/T in the lowfield regime, where κ(H)/T decreases with H (Fig. 3c,d) and exhibits clear deviations from the H/T scaling curve (Fig. 3e). This lowfield behaviour of κ(H)/T arises from the magnetic excitations because the initial reduction with H cannot be explained by spinphonon scattering, which always increases κ(H) with H as discussed above. We point out that the magnetic monopoles are most likely to be responsible for this excitations because of the following reasons. First, the deviations in the lowfield regime appear below a characteristic temperature T*∼4 K, where the pinchpoint features in neutron scattering appears^{17}. In addition, T* is close to the temperature 2J_{}/k_{B}, above which monopole excitation disappears. Second, as shown in Fig. 3c (Supplementary Figs 2, 3 and 4; Supplementary Notes 3 and 4), the initial reduction of κ(H)/T disappears below T_{C}, which is consistent with the monopole scenario because ferromagnetic ordering prevents the monopole formation. Third, the effect of ferromagnetic fluctuations as the origin of the anomalous lowfield behaviour is safely excluded, as we discuss below. Since classical localized monopoles do not transport the heat, these results suggest the emergence of itinerant quantum magnetic monopoles illustrated in Fig. 1c,d.
The appearance of the quantum monopoles are supported by the field dependence of κ(H)/T shown in Fig. 3c,d. The decrease of κ(H)/T with H implies that the number of monopoles is reduced with H at low fields. This initial reduction is expected in the dispersionless classical monopoles with gap 2J_{} (Supplementary Fig. 5; Supplementary Note 5). However, in the classical case, the number of monopoles will decay exponentially with decreasing temperature below T*∼2J_{}/k_{B}. Therefore the observed quite substantial reduction of κ(H)/T even at low temperatures well below T* is inconsistent with the classical monopoles. The results indicate that the monopole excitation gap is largely suppressed from the classical monopole, which is consistent with the dispersive quantum monopoles (Fig. 1c). We also note that the substantial reduction of monopole density by the low field will result in a reduction of the inelastic spinphonon scattering process related to the monopole hopping discussed above, which further emphasizes the significant role of the quantum monopoles themselves as a heat conducting carrier at low fields.
Here we comment on the field direction dependence of κ(H)/T. As shown in Fig. 3a,b,e, κ(H)/T is nearly isotropic with respect to the Hdirection at high fields. Anisotropic κ(H)/T may be expected at high fields, because with increasing H, the monopole density increases for H  [1, 1, 1], while it decreases to zero for H  [0, 0, 1] (Supplementary Fig. 5; Supplementary Table 1). However, monopoles tend to localize at high fields because the energy of spinflip, which is necessary for the monopole propagation, increases linearly with H. Thus monopole propagation does not contribute to the thermal conductivity at high fields, which is consistent with the observed isotropic κ(H)/T.
As shown in the inset of Fig. 4, κ(H) decreases as κ(H)=κ(0)−αH^{2} (α>0) at very low fields. As the thermal conduction by magnetic excitations is determined by the number of lowenergy itinerant quasiparticles, this α is a measure of the suppression rate of magnetic monopoles at low fields. Figure 4 depicts the temperature dependence of α for H  [0, 0, 1] and [1, 1, 1]. As the temperature is lowered, α first increases, decreases after showing a maximum at =0.3–0.5 K and suddenly vanishes at T_{C}. Here, we stress that the initial reduction of κ(H) is not caused by ferromagnetic fluctuations, since monotonic increase of ferromagnetic fluctuations with approaching T_{C} is inconsistent with the nonmonotonic temperature dependence of α. The difference in the magnitude of α in the two field directions may be related to the expected difference in the density of 3in1out configuration at high fields, but the trends of α(T) are similar in both cases. The enhancement of α with decreasing T below T* can be accounted for by the reduction of thermal smearing of the monopole excitations. At low temperatures below the gap energy of monopole excitation, the monopole density decreases rapidly with decreasing temperature, leading to a suppression of α. As a result, α exhibits nonmonotonic temperature dependence with a maximum. The calculation shows that the initial suppression of the monopole density reaches a maximum at around T_{max}≈Δ/2.5k_{B}, where Δ is the monopole excitation gap (Supplementary Fig. 6; Supplementary Note 5). In Yb_{2}Ti_{2}O_{7}, assuming that monopole band minimum is lowered by ∼J_{z±} due to its quantum motion, Δ/k_{B}=(2J_{}−J_{z±})/k_{B} is estimated to be ∼2 K, which yields T_{max}∼0.8 K. The fact that compares with T_{max} suggests that the maximum of α appears as a result of gap, which is largely suppressed from the classical one (2J_{}/k_{B}∼4 K). We point out that the further reduction of from T_{max} may be due to the influence of the quantum fluctuations on the velocity and mean free path included in the thermal conductivity. The present results lead us to conclude that the thermally excited quantum monopoles carry substantial portion of the heat particularly in the lowfield regime. This is reinforced by the fact that κ/T at zero field shows a distinct decrease below T_{C} (Fig. 2a), where the phonon contribution κ_{p} is expected to be enhanced owing to the ferromagnetic spin alignment.
Estimation of mean free path
Next we demonstrate that quantum monopoles are highly itinerant in the crystal lattice. Assuming the kinetic approximation, the monopole contribution to the thermal conductivity κ_{m} is written as , where C_{m} is the monopole contribution in the specific heat, v is the velocity and is the mean free path of the monopoles. We estimate at 0.6 K simply by assuming that the amount of initial reduction of κ(H)/T shown by red doubleheaded arrow in Fig. 3d is attributed to the monopole contribution. The total specific heat C≈1 J/Ybmol K at 0.6 K (Fig. 2b) and v, which is roughly determined by v∼aJ_{z±}/2πħ∼15 m s^{−1}, where a(=0.43 nm) is the distance between neighbouring tetrahedra, yield nm, or equivalently the scattering time ns. We stress that this long is still underestimated, since the total specific heat and the initial reduction of thermal conductivity give only an overestimate and underestimate, respectively, for the monopole contribution. This indicates that the excitations are mobile to a very long distance, , without being scattered. We note that is much longer than the intermonopole distance, which is estimated to be at most 5a, assuming monopole density of 1% of total number of tetrahedra. This corresponds to a very large coherent volume including more than ∼10^{7} tetrahedra, demonstrating highly itinerant transport of this longlived particle, whose effective mass is as heavy as ∼2,000 times the bare electron mass^{28}. This very small scattering rate may be due to the quantum feature, which prohibits the simple monopole–antimonopole pair annihilation that violates energy conservation.
Discussion
The present results indicate the significant heat conduction by magnetic excitations, which are most likely magnetic monopoles. This implies that the monopole excitation becomes dispersive due to the offdiagonal term J_{z±} (Fig. 1c). This is consistent with the strongly suppressed monopole excitation gap, indicated by our analysis. We note that the observed nearly gapless excitations are not relevant to the photon excitations predicted by refs 7, 8, 9, 10, 11. This is because the characteristic photon energy is one order of magnitude smaller than the present temperature range, and hence the strongly temperature dependent α is incompatible with the photon excitations.
The itinerant heavy quantum monopoles in the spinliquid state appear to be a characteristic feature of the elementary excitations in frustrated magnetic pyrochlore systems with strong quantum fluctuations. Nearly ballistic propagation phenomena of fractionalized magnetic excitations in spinliquid states have been reported in spin1/2 onedimensional Heisenberg chain^{29,30} and twodimensional (2D) triangular lattice with antiferromagnetic interactions^{31}. In the former elementary excitation is spinon which obeys semion statistics^{32} and in the latter excitation has been discussed in terms of spinon which obeys fermionic statistics^{33,34,35,36,37,38}. In the present threedimensional (3D) system elementary excitation in the spinliquid state is quantum monopole, which is another fractionalized spinon. The residual , which is distinctly present in the 2D case^{31,38}, is absent in Yb_{2}Ti_{2}O_{7} (Fig. 2c), implying that this 3D spinon is unlikely to be fermionic. In fact, bosonic spinon has been presumed theoretically in 3D pyrochlore lattice^{39,40}. In onedimensional Heisenberg system, the mean free path is infinite at nonzero temperature due to the integrability of the Hamiltonian. The highly itinerant fermionic spinons in 2D and bosonic quantum monopoles in 3D may be a key feature of the elementary excitations in highly frustrated quantum magnets and its origin is an open question.
Methods
Singlecrystal growth
Highquality single crystals of Yb_{2}Ti_{2}O_{7} were grown by the floating zone method. Stoichiometric amount of Yb_{2}O_{3} and TiO_{2} powder were mixed, pressed into rods and sintered at 1,150 °C for 24 h. Single crystals were grown from the rods in air at a rate of 1.5 mm h^{−1}. The crystal ingot has a typical diameter of ∼6 mm and a length of ∼20 mm. Powder Xray diffraction experiments on pulverized single crystal show no appreciable amount of impurity phase.
Thermal conductivity and specific heat measurements
Thermal conductivity was measured along [1, −1, 0] direction by the standard steadystate method in a dilution refrigerator. Magnetic field was applied along [1, 1, 1] and [0, 0, 1], perpendicular to the heat current. As shown in the inset of Fig. 2a, the temperature difference within the sample, ΔT, due to the heat current from the heater to heat bath was measured by two Ruthenium oxide thermometers. Sample temperature was measured with high accuracy with use of alternating current resistance bridges. About 1 kΩ chip resistor was used for a heater. A single crystalline sample was well thermally coupled to the thermometers, heater and heat bath by thermally connecting with 50 μm silver wires and silver paint as a glue. Specific heat was determined by the standard quasiadiabatic heat pulse method in a dilution refrigerator. Sample temperature was measured by ruthenium oxide thermometer and heat pulse was produced by Joule heating of resistive strain gauge attached to the sample.
Additional information
How to cite this article: Tokiwa, Y. et al. Possible observation of highly itinerant quantum magnetic monopoles in the frustrated pyrochlore Yb_{2}Ti_{2}O_{7}. Nat. Commun. 7:10807 doi: 10.1038/ncomms10807 (2016).
References
 1
Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materials. Science 294, 1495–1501 (2001).
 2
Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. Zeropoint entropy in spin ice. Nature 399, 333–335 (1999).
 3
Bramwell, S. T. et al. Spin correlations in Ho2Ti2O7: a dipolar spin ice system. Phys. Rev. Lett. 87, 047205 (2001).
 4
Fennell, T. et al. Magnetic Coulomb phase in the spin ice Ho2Ti2O7 . Science 326, 415–417 (2009).
 5
Castelnovo, C., Moessner, R. & Sondhi, S. Magnetic monopoles in spin ice. Nature 451, 42–45 (2008).
 6
Morris, D. J. P. et al. Dirac strings and magnetic monopoles in the spin ice Dy2Ti2O7 . Science 326, 411–414 (2009).
 7
Shannon, N., Sikora, O., Pollmann, F., Penc, K. & Fulde, P. Quantum ice: a quantum Monte Carlo study. Phys. Rev. Lett. 108, 067204 (2012).
 8
Benton, O., Sikora, O. & Shannon, N. Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice. Phys. Rev. B 86, 075154 (2012).
 9
Gingras, M. J. P. & McClarty, P. A. Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets. Rep. Prog. Phys. 77, 056501 (2014).
 10
Savary, L. & Balents, L. Coulombic quantum liquids in spin1/2 pyrochlores. Phys. Rev. Lett. 108, 037202 (2012).
 11
Hermele, M., Fisher, M. P. A. & Balents, L. Pyrochlore photons: The U(1) spin liquid in a S=1/2 threedimensional frustrated magnet. Phys. Rev. B 69, 064404 (2004).
 12
Ross, K. A., Savary, L., Gaulin, B. D. & Balents, L. Quantum excitations in quantum spin ice. Phys. Rev. X 1, 021002 (2011).
 13
Applegate, R. et al. Vindication of Yb2Ti2O7 as a model exchange quantum spin ice. Phys. Rev. Lett. 109, 097205 (2012).
 14
Hirschberger, M., Krizan, J. W., Cava, R. J. & Ong, N. P. Large thermal Hall conductivity of neutral spin excitations in a frustrated quantum magnet. Science 348, 106–109 (2015).
 15
Gardner, J. S., Gingras, M. J. P. & Greedan, J. E. Magnetic pyrochlore oxides. Rev. Mod. Phys. 82, 53 (2010).
 16
Hodges, J. A. et al. The crystal field and exchange interactions in Yb2Ti2O7 . J. Phys. Cond. Matter 13, 9301 (2001).
 17
Chang, L.J. et al. Higgs transition from a magnetic Coulomb liquid to a ferromagnet in Yb2Ti2O7 . Nat. Commun. 3, 992 (2012).
 18
Yasui, Y. et al. Ferromagnetic transition of pyrochlore compound Yb2Ti2O7 . J. Phys. Soc. Jpn 72, 3014–3015 (2003).
 19
Kolland, G. et al. Thermal conductivity and specific heat of the spinice compound Dy2Ti2O7: experimental evidence for monopole heat transport. Phys. Rev. B 86, 060402 (2012).
 20
Kolland, G., Valldor, M., Hiertz, M., Frielingsdorf, J. & Lorenz, T. Anisotropic heat transport via monopoles in the spinice compound Dy2Ti2O7 . Phys. Rev. B 88, 054406 (2013).
 21
Ross, K. A. et al. Dimensional evolution of spin correlations in the magnetic pyrochlore Yb2Ti2O7 . Phys. Rev. B 84, 174442 (2011).
 22
Hodges, J. A. et al. Firstorder transition in the spin dynamics of geometrically frustrated Yb2Ti2O7 . Phys. Rev. Lett. 88, 077204 (2002).
 23
D’Ortenzio, R. M. et al. Unconventional magnetic ground state in Yb2Ti2O7 . Phys. Rev. B 88, 134428 (2013).
 24
Ross, K. A. et al. Twodimensional kagome correlations and field induced order in the ferromagnetic XY pyrochlore Yb2Ti2O7 . Phys. Rev. Lett. 103, 227202 (2009).
 25
Gardner, J. S., Ehlers, G., Rosov, N., Erwin, R. W. & Petrovic, C. Spinspin correlations in Yb2Ti2O7: a polarized neutron scattering study. Phys. Rev. B 70, 180404 (2004).
 26
Yaouanc, A., Dalmas de Réotier, P., Marin, C. & Glazkov, V. Singlecrystal versus polycrystalline samples of magnetically frustrated Yb2Ti2O7: specific heat results. Phys. Rev. B 84, 172408 (2011).
 27
Udagawa, M. Magnetic response of itinerant spin ice. Spin 5, 1540004 (2015).
 28
Pan, L. et al. A measure of monopole inertia in the quantum spin ice Yb2Ti2O7. Preprint at http://arxiv.org/abs/1501.05638 (2015).
 29
Sologubenko, A. V., Giannó, K., Ott, H. R., Ammerahl, U. & Revcolevschi, A. Thermal conductivity of the holedoped spin ladder system Sr14−xCaxCu24O41 . Phys. Rev. Lett. 84, 2714 (2000).
 30
Kudo, K. et al. Spin gap and hole pairing in the spinladder cuprate Sr14−xAxCu24O41 (A=Ca and La) studied by the thermal conductivity. J. Phys. Soc. Jpn 70, 437–444 (2001).
 31
Yamashita, M. et al. Highly mobile gapless excitations in a twodimensional candidate quantum spin liquid. Science 328, 1246–1248 (2010).
 32
Haldane, F. D. M. “Fractional statistics” in arbitrary dimensions: a generalization of the Pauli principle. Phys. Rev. Lett. 67, 937–940 (1991).
 33
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
 34
Lee, S.S. & Lee, P. A. U(1) Gauge theory of the Hubbard model: spin liquid states and possible application to κ(BEDTTTF)2Cu2(CN)3 . Phys. Rev. Lett. 95, 036403 (2005).
 35
Lee, S.S., Lee, P. A. & Senthil, T. Amperean pairing instability in the U(1) spin liquid state with Fermi surface and application to κ(BEDTTTF)2Cu2(CN)3 . Phys. Rev. Lett. 98, 067006 (2007).
 36
Block, M. S., Sheng, D. N., Motrunich, O. I. & Fisher, M. P. A. Spin Bosemetal and valence bond solid phases in a spin1/2 model with ring exchanges on a fourleg triangular ladder. Phys. Rev. Lett. 106, 157202 (2011).
 37
Barkeshli, M., Yao, H. & Kivelson, S. A. Gapless spin liquids: Stability and possible experimental relevance. Phys. Rev. B 87, 140402 (2013).
 38
Watanabe, D. et al. Novel Pauliparamagnetic quantum phase in a Mott insulator. Nat. Commun. 3, 1090 (2012).
 39
Hao, Z., Day, A. G. R. & Gingras, M. J. P. Bosonic manybody theory of quantum spin ice. Phys. Rev. B 90, 214430 (2014).
 40
Wang, C. & Senthil, T. Timereversal symmetric U (1) quantum spin liquids. Preprint at http://arxiv.org/abs/1505.03520 (2015).
Acknowledgements
We thank L. Balents, K. Behnia, S. Fujimoto, H. Kawamura, S. Onoda, and K. Totsuka for useful discussions. Financial support for this work was provided by GrantsinAid for Scientific Research (Nos. 26400339, 24340076, 15K13533 and 15H05852) from the Japan Society for the Promotion of Science (JSPS).
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Y.T., T.Y., T.S. and Y.M. conceived and designed the study. Y.T., T.Y., D.T. and Y.S. performed the thermal conductivity measurements. S.K., T.S. and Y.Y. performed the specific heat measurements. Y.Y. synthesized the highquality single crystalline samples. Y.T., T.Y., M.U., T.T., T.S. and Y.M. discussed the results. Y.T., M.U., T.S. and Y.M. prepared the manuscript.
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Supplementary Figures 16, Supplementary Table 1, Supplementary Notes 15 and Supplementary References (PDF 260 kb)
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Tokiwa, Y., Yamashita, T., Udagawa, M. et al. Possible observation of highly itinerant quantum magnetic monopoles in the frustrated pyrochlore Yb_{2}Ti_{2}O_{7}. Nat Commun 7, 10807 (2016). https://doi.org/10.1038/ncomms10807
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