Thermal conductivity of quantum magnetic monopoles in the frustrated pyrochlore Yb2Ti2O7
Y. Tokiwa, T. Yamashita, M. Udagawa, S. Kittaka, T. Sakakibara, D. Terazawa, Y. Shimoyama, T. Terashima, Y. Yasui, T. Shibauchi, Y. Matsuda
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Thermal conductivity of quantum magnetic monopoles in the frustrated pyrochloreYb Ti O Y. Tokiwa , , T. Yamashita , M. Udagawa , S. Kittaka , T. Sakakibara , D.Terazawa , Y. Shimoyama , T. Terashima , Y. Yasui , T. Shibauchi , and Y. Matsuda Department of Physics, Kyoto University, Kyoto 606-8502, Japan Research Center for Low Temperature and Materials Science, Kyoto University, Kyoto 606-8501, Japan Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan Department of Physics, School of Science and Technology,Meiji University, Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan and Department of Advanced Materials Science, University of Tokyo, Chiba 277-8561, Japan
We report low-temperature thermal conductivity κ of pyrochlore Yb Ti O , which contains frus-trated spin-ice correlations with significant quantum fluctuations. In the disordered spin-liquidregime, κ ( H ) exhibits a nonmonotonic magnetic field dependence, which is well explained by thestrong spin-phonon scattering and quantum monopole excitations. We show that the excitation en-ergy of quantum monopoles is strongly suppressed from that of dispersionless classical monopoles.Moreover, in stark contrast to the diffusive classical monopoles, the quantum monopoles have a verylong mean free path. We infer that the quantum monopole is a novel heavy particle, presumablyboson, which is highly mobile in a three-dimensional spin liquid. Rare-earth pyrochlore oxides exhibit various exoticmagnetic properties owing to their strong geometricalfrustration experienced by coupled magnetic momentson the tetrahedral lattice (Fig. 1(a)) [1]. The most ex-plored materials are Ho Ti O and Dy Ti O , in whichthe magnetic moments can be regarded as classical spinswith a strong easy-axis (Ising) anisotropy [1, 2]. The frus-tration of these moments results in a remarkable spin icewith macroscopically degenerate ground states, in whicheach tetrahedron has the “two spins in, two spins out (2-in-2-out)” configuration. This spin structure is character-ized by dipolar spin correlations with a power-law decay,which is observable as the unusual pinch-point shape ofspin structure factor by neutron scattering [3, 4]. One ofthe most remarkable features of the spin-ice state is thatit hosts emergent magnetic monopole excitations; thefirst excitation is 3-in-1-out configuration [5, 6]. This pro-duces a bound pair of north and south poles, which canbe fractionalized into two free magnetic monopoles. Thisclassical monopole excitations are gapped and dispersion-less (Fig. 1(b)). Therefore the propagation of monopolesoccurs only diffusively and the monopole population de-cays exponentially at temperatures well below the gap.Of particular interest is how the spin-ice ground stateis altered by the quantum fluctuations, which may liftthe degeneracy of the spin-ice manifold, leading to a newground state such as quantum spin-ice state [7–11]. Toclarify this issue, uncovering newly emergent elementaryexcitations in the presence of quantum fluctuations iscrucially important. Although exotic excitations such asgapless photon-like mode have been proposed theoreti-cally, the nature of excitations are poorly explored.Among the magnetic pyrochlore materials, Yb Ti O ,Er Ti O , Pr Sn O and possibly Tb Ti O host strongtransverse quantum fluctuations of magnetic dipoles owing to speudospin-1/2 of magnetic rare earth ele-ments [12–15]. In particular, Yb Ti O with well sep-arated crystalline electric-field excited levels is a goodmodel system to study the influence of the quantumeffects on monopole excitations [16]. Yb Ti O under-goes a weakly first-order ferromagnetic phase transitionat T C ∼ . T C indicates the presence of aspin-liquid phase with spin-ice correlations [17]. The fullset of Hamiltonian parameters is determined by inelasticneutron scattering experiments, providing a prototypicalsystem described by an effective pseudospin-1/2 quantumspin-ice model [12]. The Hamiltonian consists of threemain interactions, J k , J ⊥ and J z ± . Here J k (= 2 K) isthe Ising component of the nearest neighbor interaction, J ⊥ (= 0 .
58 K) is the XY-component and J z ± (= 1 . J ⊥ and J z ± producequantum fluctuations (Fig. 1(c)).Here, to study the elementary excitations in the spinliquid state of Yb Ti O , we measured the thermal con-ductivity, which is a powerful probe for low energy exci-tations at low temperatures, providing a sensitive mea-surement of a flow of entropy conducted by magnetic ex-citations and phonons. The thermal conductivity hasbeen reported in the classical spin-ice state of Dy Ti O recently. However, the interpretation of the thermal con-ductivity of Dy Ti O appears to be complicated owingto the strongly suppressed phonon thermal conductiv-ity by unknown additional scatterings (see SupplementalMaterial [21]). In fact, suggested heat transport by clas-sical monopole is at odds with the diffusive motion ofthe dispersionless classical monopoles. We show that thethermal conductivity of Yb Ti O is rather simple: the J || classical spin ice classical monopoles quantum spin ice photonquantum monopoles ~2 J z+ ~ J /J || (b)(c)(a) FIG. 1. (color online). (a) Spin-ice structure on frustrated py-rochlore lattice. (b) Magnetic monopole excitations in classi-cal and quantum spin ice. In classical spin ice the gap energyis twice the Ising interaction of magnetic moments 2 J k . In thequantum spin ice the off-diagonal interaction J z ± gives riseto a dispersive monopole excitation. The photon excitationsbased on the XY -component J ⊥ lift the ice degeneracy inthe ground state. (c) Collective motion of quantum magneticmonopoles. phonon term shows a B/T scaling and the monopole con-tribution vanishes below T C as expected. Our analysisshows the evidence of the substantial heat transport byquantum magnetic monoples, whose excitation energy issignificantly suppressed from that of classical monopoles.The quantum magnetic monopoles are highly mobile dueto quantum fluctuations, in stark contrast to the local-ized and diffusive nature of classical monopoles.High quality single crystals of Yb Ti O were grownby the floating zone method. Thermal conductivity wasmeasured along [1,-1,0] direction by the standard steady-state method in a dilution refrigerator. Magnetic fieldwas applied along [1,1,1] and [0,0,1], perpendicular to theheat current. Specific heat was determined by the quasi-adiabatic heat pulse method in a dilution refrigerator.Figure 2(a) shows the temperature dependence of ther-mal conductivity divided by temperature κ/T in zerofield and at µ H = 12 T measured on a single crystalof Yb Ti O . Distinct jump in κ/T at zero field is ob-served at T C . As shown in Fig. 2(b), the specific heat C of the single crystal taken from the same batch showsa sharp and large jump at T C [17]. We note that inthe previous studies, such a sharp single jump in C/T had been reported only in the powered samples [19, 22],demonstrating the high quality of the present crystal.We also note that the pinch point features in neutronscattering has been clearly observed in the single crystalwhich shows a similar specific heat jump [17]. As shownin Fig. 2(c), zero-field κ/T above T C follows a T -lineardependence with negligibly small intercept at T = 0 K.The absence of residual κ/T | T → in the spin-liquidstate with spin-ice correlations will be discussed later.As clearly seen in Fig. 2(a), magnetic field strongly en-hances the thermal conductivity. Figures 3(a) and 3(b)and their insets show the field dependence of κ ( H ) /T for κ / T ( W / K m ) T (K) H =12T //[111] H =0T C / T ( J / m o l K ) T (K)0 g µ B H H heaterheat bath thermometer ∆ TQ //[1,-1,0] T c T c T (K) κ / T ( W / K m ) (c)(a) (b) FIG. 2. (color online). (a) κ/T at zero and µ H = 12 Tapplied along [1,1,1] direction is plotted against temperature.The heat current is applied along [1,-1,0]. At T C , κ (0) /T exhibits a jump, indicated by an arrow. Inset illustrates themeasurement configuration of the thermal conductivity. (b)Specific heat divided by temperature C/T at zero field. (c) κ/T at zero field in the spin liquid state above 0.2 K. Greyline is a fit to a T -linear dependence κ / T = AT with A =0 .
15 W/K m. different field directions. As illustrated in the inset ofFig. 3(c), there are three characteristic regimes; low-fieldregime (i) where κ ( H ) /T decreases with H , intermediatefield regime (ii) where κ ( H ) /T increases, and high-fieldregime where κ ( H ) /T exhibits a saturation.In the present system, heat is transferred by phononsand magnetic excitations: κ = κ p + κ m . We point outthat the field dependence of κ ( H ) /T in the (ii) and (iii)regimes are dominated by the phonon contribution κ p de-termined by spin-phonon scattering, which contains elas-tic and inelastic processes. The elastic scattering (de-termined by J k ) is enhanced with increasing disorder ofspins and thus this scattering process should be mono-tonically suppressed by the alignment of spins with in-creasing magnetic field. A recent calculations of mag-netoresistance in a fluctuating spin-ice state indicatesthat the electron-spin elastic scattering rate decreaseswith increasing magnetization [25], which supports thistrend. The inelastic scattering is directly related to thequantum dynamics of spin. In this inelastic scattering,the leading spin-flip process accompanies a hopping ofa monopole to the neighboring tetrahedron (which is re-lated to J ⊥ ), because this process requires much lower en-ergies than creation or annihilation of monopoles. Thisscattering is suppressed with field by the formation ofZeeman gap. Therefore an external magnetic field sup-presses both elastic and inelastic scatterings, leading tothe enhancement of the phonon thermal conductivity κ p .In the regime (iii), the Zeeman splitting energy gµ B H well exceeds both of the magnetic interactions and ther-mal energy, gµ B H ≫ J k , J ⊥ , J z ± and k B T . In this situa-tion, where all spins are fully polarized and the magnetic(spin-wave) excitations are gapped with a gap gµ B H ,thermal conductivity is almost entirely dominated bythe pure phonon contribution without spin scattering be-cause of the following reasons. First, elastic spin-phononscattering is absent due to the perfect alignment of spins.Second, inelastic scattering is also absent due to the for-mation of the large Zeeman gap. Third, spins do notcarry the heat due to the Zeeman gap. Since purelyphononic thermal conductivity is insensitive to magneticfield, κ ( H ) /T in the regime (iii) is nearly independentof H . In the regime (ii), the phonon mean free path issignificantly reduced by the spin-phonon scattering dueto the spins thermally excited across the Zeeman gap.In fact, as shown in Fig. 3(c) which plots κ/T as a func-tion of µ B H/k B T , all data collapse into a single curveexcept for the low µ B H/k B T regime. The fact that datafor both field directions stabilizing different spin config-urations (3-in-1-out for H k [1 , ,
1] and 2-in-2-out for H k [0 , , H/T scaling curve appears to follow the Brillouin function (thedashed line in Fig. 3(c)). Here we fitted the data with g -factor of 0.79, which is comparable to Land´e g -factor(8/7) of Yb. This coincidence with the Brillouin functioncalls for further theoretical investigations.A particularly important information for the elemen-tary excitations is provided by κ ( H ) /T in the regime (i),where κ ( H ) /T decreases with H (the insets of Figs. 3(a)and (b)) and exhibits striking deviations from the H/T -scaling curve (Fig. 3(c)). This low-field behavior of κ ( H ) /T is most likely due to the monopole contribution κ m because of the following reasons. First, the initialreduction with H cannot be explained by spin-phononscattering, which always increases κ ( H ) with H as dis-cussed above. Second, the deviations in regime (i) appearbelow T ∗ ∼ T ∗ is close tothe temperature 2 J k /k B , above which monopole excita-tion disappears. Third, as shown in the inset of Fig. 3(a)(see Supplemental Material [21] for more detail), the ini-tial reduction of κ ( H ) /T disappears below T C , which isconsistent with the monopole scenario because ferromag-netic ordering prevents the monopole formation.The decrease of κ ( H ) /T with H implies that the num-ber of monopoles is reduced with H at low fields. Thisreduction is expected even in the dispersionless classi-cal monopoles with gap 2 J k (see Supplemental Mate-rial [21]). However, in the classical case, the numberof monopoles will decay exponentially with decreasingtemperature below T ∗ ∼ J k /k B . Therefore the ob-served quite substantial reduction of κ ( H ) /T even at lowtemperatures well below T ∗ is totally inconsistent withthe classical monopoles. The results provide strong evi-dence that the monopole excitation gap is dramatically κ / κ s a t H //[1,1,1]5K2K1.5K1.2K1K H //[1,0,0]2K1.5K1K µ B H /k B T (i) (iii)(ii) H (arb. unit) κ ( a r b . un i t ) µ H (T) H //[0,0,1]2K1K0.6K0.3K κ / T ( W / K m ) H //[1,1,1]5K2K1.5K1K0.5K0.3K κ / T ( W / K m ) (b) (a) (c) µ H (T ) κ / T ( W / K m ) κ / T ( W / K m ) µ H (T)0.010 FIG. 3. (color online). (a) Field dependence of κ/T ofYb Ti O for H k [1,1,1] with the heat current along [1,-1,0]. The inset shows κ ( H ) /T at low field. Solid red andblack lines are κ/T ( H ) at 0.18 and 1 K, respectively. Data areshifted vertically for clarity. (b) The same plot for H k [0,0,1].Double-headed red arrow in the inset of (a) indicates the ini-tial reduction of κ ( H ) /T for H k [0,0,1] at T =0.6 K, which isestimated to be 0.03 W/K m, giving a lower-bound estimateof the monopole contribution. (c) Normalized thermal con-ductivity κ/κ sat plotted against µ B H/k B T , where κ sat is thesaturated thermal conductivity at high fields. κ sat at hightemperatures is determined so as to fit the scaling curve. Thedashed line represents the Brillouin function with spin=1/2,assuming g =0.79. The inset illustrates the typical behavior of κ ( H ) /T . There are three characteristic field regimes, (i),(ii)and (iii) indicated by different colors. suppressed from the classical monopole, suggesting theemergence of dispersive quantum magnetic monopoles il-lustrated in Figs. 1(c) and (d). We also note that thesubstantial reduction of monopole density by the lowfield will result in a reduction of the inelastic spin-phononscattering process related to the monopole hopping dis-cussed above, which further emphasizes the significantrole of the quantum monopoles themselves as a heat con-ducting carrier at low fields.As shown in the inset of Fig. 4, κ ( H ) decreases as κ ( H ) = κ (0) − αH ( α >
0) at very low fields. Asthe thermal conduction by magnetic excitations is de-termined by the number of low-energy itinerant quasi-particles, this α is a measure of the suppression rate ofmagnetic monopoles at low fields. Figure 4 depicts thetemperature dependence of α for H k [0,0,1] and [1,1,1].As the temperature is lowered, α first increases, decreasesafter showing maximum at T max = 0 . T C . The difference in the magnitudeof α in the two field directions may be related to theexpected difference in the density of 3-in-1-out configu- α ( W / T K m ) T (K) H //[111] H //[100] H //[111]1.5K1K0.5K H //[100]1.5K1K0.5K κ µ H (T ) FIG. 4. (color online). Temperature dependence of the initialslope of κ ( H ) determined by fitting κ ( H ) = κ (0) − αH , for H k [0,0,1] and [1,1,1]. Inset shows κ ( H ) plotted as a functionof H at very low field. ration at high fields, but the trends of α ( T ) are similarin both cases. The enhancement of α with decreasing T below T ∗ can be accounted for by the reduction ofthermal smearing of the monopole excitations. The re-duction of α at lower temperatures is expected when thethermal energy scale k B T max becomes comparable to afraction of the monopole excitation gap (see Supplemen-tal Material [21]). In addition, possible ferromagneticfluctuations, because of the weakly first-order nature ofthe transition at T C , would also suppress α near T C . Inany case, the observed low energy scale of k B T max indi-cates that the excitation gap is strongly reduced than theclassical monopole case (2 J k = 4 K). The present resultslead us to conclude that the thermally excited quantummonopoles carry substantial portion of the heat partic-ularly in the regime (i). This is reinforced by the factthat κ/T at zero field shows a distinct decrease below T C (Fig. 2), where the phonon contribution κ p is expected tobe enhanced owing to the ferromagnetic spin alignment.Next we demonstrate that quantum monopoles arehighly mobile in the crystal lattice. Assuming the ki-netic approximation, the monopole contribution to thethermal conductivity κ m is written as κ m = C m vℓ/ C m is the monopole contribution in the specificheat, v is the velocity and ℓ is the mean free path of themonopoles. We estimate ℓ at 0.6 K simply by assumingthat the amount of initial reduction of κ ( H ) /T shown reddouble-headed arrow in the inset of Fig. 3(b) is attributedto the monopole contribution. The total specific heat C ≈ v ,which is roughly determined by v ∼ aJ z ± / π ~ ∼
15 m/s,where a (= 0 .
43 nm) is the distance between neighbor-ing tetrahedra, yield ℓ ∼
100 nm, or equivalently thescattering time τ ∼ . ℓ is still underestimated, since the total specific heat andthe initial reduction of thermal conductivity give onlyan overestimate and underestimate, respectively, for themonopole contribution. This indicates that the excita-tions are mobile to a very long distance, ℓ > a , with-out being scattered. We stress that ℓ is much longerthan the inter-monopole distance, which is estimated tobe at most 5 a , assuming monopole density of 1% of totalnumber of tetrahedra. This corresponds to a very largecoherent volume including more than ∼ tetrahedra,demonstrating highly mobile transport of this long-livedparticle, whose effective mass is as heavy as ∼ E photon ≈ J ⊥ /J k ∼ .
05 K is oneorder of magnitude smaller than the present tempera-ture range, and hence the strongly temperature depen-dent α is incompatible with the photon excitations. Thehighly mobile heavy quantum monopoles in the spin liq-uid state is the most salient feature of the elementary ex-citations in frustrated magnetic pyrochlore systems withstrong quantum fluctuations. Nearly ballistic propaga-tion phenomena of fractionalized magnetic excitationsin spin-liquid states have been reported in spin-1/2 1DHeisenberg chain [27, 28] and 2D triangular lattice withantiferromagnetic interactions [29]. In the former ele-mentary excitation is spinon which obeys semion statis-tics [30] and in the latter excitation has been discussed interms of spinon which obeys fermionic statistics [31–36].In the present 3D system elementary excitation in thespin liquid state is quantum monopole, which is anotherfractionalized spinon. The residual κ/T | T → , whichis distinctly present in the 2D case [29, 36], is absentin Yb Ti O (Fig. 2(c)), implying that this 3D spinonis unlikely to be fermionic. In fact, bosonic spinon hasbeen presumed theoretically in 3D pyrochlore lattice [37].In 1D Heisenberg system, the mean free path is infiniteat nonzero temperature due to the integrability of theHamiltonian. The highly mobile fermionic spinons in 2Dand bosonic quantum monopoles in 3D may be a keyfeature of the elementary excitations in highly frustratedquantum magnets and its origin is an open question.We thank L. Balents, K. Behnia, S. Fujimoto,H. Kawamura, S. Onoda, and K. Totsuka for useful dis-cussions. Financial support for this work was providedby Grants-in-Aid for Scientific Research from the JapanSociety for the Promotion of Science (JSPS). [1] S. T. Bramwell, and M. J. P. Gingras, Science , 1495(2001).[2] A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan,and B. S. Shastry, Nature , 333 (1999).[3] S. T. 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Supplemental Materials for ”Thermalconductivity of quantum magneticmonopoles in the frustrated pyrochloreYb Ti O ” ADDITIONAL PHONON SCATTERINGS INDy Ti O κ/T of Y Ti O , Dy Ti O at zero field and Yb Ti O at H =12 T are compared in Fig.S1. κ/T of Dy Ti O athigh temperature is strongly suppressed from Y Ti O .It has been reported that κ/T of Dy Ti O decreasesmonotonically with H , indicating even smaller phonon κ/T [S1]. This implies the presence of unknown phononscatterings, which are not likely caused by crystal fieldexcitations because of the large energy gap of first excitedstate, ∼
380 K [S2]. As discussed in the main text, κ/T of Yb Ti O at H =12 T represents the purely phononic κ/T at low temperatures, where k B T << µ B H . κ/T at2 K is much larger than Dy Ti O and close to Y Ti O ,suggesting the absence of such unknown scatterings. κ / T ( W / K m ) T (K) Yb Ti O µ H =12T Y Ti O H =0Dy Ti O H =0 FIG. S1. (color online). κ/T of Y Ti O , Dy Ti O atzero field and Yb Ti O at H =12 T applied parallel to [1,1,1]with the same heat current direction along [1,-1,0]. Data ofY Ti O , Dy Ti O are taken from Ref. [S1]. DISAPPEARANCE OF INITIAL REDUCTIONOF κ ( H ) /T WITH H BELOW T C =0.19 K In a temperature range of 0.2 K ≤ T ≤ κ ( H ) /T for H k [1,1,1] at the transitionbetween spin liquid and ferromagnetic states, indicatedby arrows in Fig. S2. The kink is shifted to lower fieldwith decreasing temperature and vanishes below T C inthe ferromagnetic state. The position of the kink agreeswith the field dependence of T C ( H ) determined by spe-cific heat measurements, which increases with H at low H //[1,1,1] κ / T ( W / K m ) µ H (T) FIG. S2. (color online). Field dependence of κ/T ofYb Ti O for H k [1,1,1] with the heat current along [1,-1,0].Arrows indicate the field-induced transition from paramag-netic to ferromagnetic states. κ / T ( W / K m ) H (T) H //[0,0,1] FIG. S3. (color online). Field dependence of κ/T ofYb Ti O for H k [0,0,1] with the heat current along [1,-1,0] field region [S3].As discussed in the main text, the initial reductionof κ ( H ) /T with H observed above T C indicates thermalconduction of magnetic quantum monopoles. The initialreduction in the temperature range 0.2 K ≤ T ≤ κ ( H ) /T shows a characteristic enhancement withthe field in the ferromagnetic state. The enhancementis understood by the suppression of elastic scattering ofphonon due to the ordering of magnetic moments. Asthe kink disappears in the ferromagnetic state, the ini-tial reduction, which is the signature of monopole heatconduction, also disappears, in consistent with the sup- || [111] h || [001]Energy Energy Energy J J J J -2 J -4/3 h -2 J +4/3 h -2 h h -2/3 h h -2 J -2 J (cid:127) (cid:127) -4 h /√3-2 J (cid:127) (cid:127) +4 h /√3 (cid:127) (cid:127) -2 h /√3√3 2 h / TABLE S1. (color online). The energy of spin configurationsin a single tetrahedron pression of spin-ice correlations below T C reported byneutron scattering experiments [S4]. The initial reduc-tion disappears below T C also for H k [0,0,1](Fig. S3). INITIAL REDUCTION OF CLASSICALMONOPOLE DENSITY WITH MAGNETICFIELD
By calculating the classical monopole density ( ρ , 3-in-1-out and 1-in-3-out configurations) in magnetic field,we show that ρ decreases with H at zero-field limit,regardless of the field direction.Hamiltonian of a nearest-neighbor spin ice model iswritten as H = J X σ zi σ zj − gµ B H · X j S j where J , which corresponds to J k in the main text, isthe nearest neighbor Ising interaction, the spin S j is anIsing spin: S j = σ zj d j , σ zj = ±
1, with the anisotropyaxes, d = [1 , , / √ d = [1 , − , − / √ d =[ − , , − / √ d = [ − , − , / √
3. With h = gµ B H ,the energy of all the spin configurations are shown inTable S1.For H k [1,1,1], ρ is derived as, ρ = 8 cosh (2 h J / t ) N [1 , , N [1 , , = 6 exp (2 /t ) cosh (4 h J / t )+ 8 cosh (2 h J / t ) + 2 exp ( − /t )Here, h J = h/J and t = k B T /J are normalized field andtemperature, respectively.For H k [0,0,1], ρ = 8 cosh (2 h J / √ t ) N [0 , , ρ g µ B H / J k B T / J =5k B T / J =2k B T / J =1=0.5 H //[0,0,1] H //[1,1,1] FIG. S4. (color online). Classical monopole density ρ is plotted against normalized magnetic field gµ B H/J for H k [1,1,1] and [0,0,1]. Dotted vertical line indicates the field gµ B H/J =3, at which level crossing of 3-in-1-out and 2-in-2-out occurs when H is applied along [1,1,1]. α k B T / J FIG. S5. (color online). h J coefficient of ρ , α , is plottedagainst normalized temperature, k B T /J . N [0 , , = 2 exp (2 /t )[2 + cosh (4 h J / √ t )]+ 8 cosh (2 h J / √ t ) + 2 exp ( − /t )The resulting field dependencies at different tempera-tures are plotted in Fig.S4. For H k [0,0,1], the energy ofone 2-in-2-out configuration decreases the most by Zee-man effect, leading to monotonic increase of 2-in-2-outdensity. As a result, ρ decreases monotonically. For H k [1,1,1], the energy of one 3-in-1-out configuration de-creases the most and crosses with the lowest energy of2-in-2-out configuration at gµ B H/J =3. As this crossingoccurs, ρ rapidly increases with H . On the other hand,in low field region, ρ decreases with field isotropically.We verify this by expanding ρ with h J around h J =0.For both the two field directions, h J -linear term vanishesand h J term is identical. The isotropic field dependenceof ρ at zero-field limit is then, ρ ( h J ) = ρ (0) − α h J α = 8(exp (2 /t ) − exp ( − /t ))3 t (exp ( − /t ) + 3 exp (2 /t ) + 4) The h J coefficient α is plotted against t = k B T /J in Fig.S5. It exhibits a maximum at k B T max,α /J =0.8.The relation between T max,α and the monopole excita-tion energy ∆ is then ∆ =2 J =2.5 k B T max,α . In themain text, the initial H decrease of κ/T is ascribed todecreasing number of monopoles and the H coefficient α in the field dependence of κ/T exhibits a maximumat T max =0.3-0.5 K. If T max is related to the monopole gap energy, it corresponds to 0.75-1.25K of monopole ex-citation energy, which is strongly suppressed from theclassical one, 2 J k =4 K [S5]. It should be noted, however,that this estimation is of purely classical monopole. [S1] G. Kolland , M. Valldor, M. Hiertz, J. Frielingsdorf, andT. Lorenz, Phys. Rev. B , 054406 (2013).[S2] S. Rosenkranz, A. P. Ramirez, A. Hayashi, R. J. Cava, R.Siddharthan and B. S. Shastry, J. Appl. Phys. , 5914(2000)[S3] Y. Yasui et al., in preparation.[S4] L.-J. Chang S. Onoda, Y. Su, Y. -J. Kao, K. -D. Tsuei,Y. Yasui, K. Kakurai and M. R. Lees, Nat. Commun. ,992 (2012).[S5] K. A. Ross, L. Savary, B. D. Gaulin, and L. Balents,Phys. Rev. X1