現代物理学入門 磁性体における緩和率 ZrZn2 弱強磁性(Tc = 28 K) β-Mn 反強磁性寸前の常磁性体 現代実験物理学I 超伝導状態におけるスペクトル (a) シールディング（軌道反磁性）によるシフト λL:磁場侵入長 75 As 第２種超伝導のRedﬁeldパターン Intensity (arb. units) 63 65 Cu eiQ·r e 2 2 1 + LHQ|| ab H(r) = H0 7 Hanaguri et al., Q Li 〃 PRB 85, 214505 (2012) 3 (b) 4 5 Intensity (arb. units) e 6 7 LiFeAs単結晶NMR µ0 H (T) 7 Li 6.6 T 8 (c) 15 K 4.2 K （修正ロンドンモデル） 2Q H || c 2Q 9 2 三角格子 Exp. Calc. ξ = 4.63 nm λL = 110 nm As 6.6 T dKs = -0.025% 75 dKs = -0.035% 75 0 0.1 0.2 0.3 Knight shift (%) 0.4 （ロンドンモデル） 10 dKs = 0 As 4.25 T -0.2 -0.1 ξ:コヒーレンス長 Cu -0.2 -0.1 0 0.1 K4.2 K - K15 K (%) 0.2 saddle c ms middle core H 現代実験物理学I 超伝導状態におけるスピン磁化率 NMRではスピン磁化率と軌道磁化率を分離できる（K-χプロット） Kspin Aspin spin = µB spin = 4µ2B NS 0 f (E) dE E 準粒子状態密度 準粒子状態密度 ナイトシフト スピン一重項 超伝導では減 少する 0.0 χorb 0 ||ab χorb 0.0000 ||c 0.0005 0.0010 χab ( emu / mol ) 現代実験物理学I 100 200 300 Temperature ( K ) スピン三重項超伝導状態におけるシフト Fig. 2. Knight shift vs bulk susceptibility determined in the range T ¼ 1:4{20 K for a weakly-correlated metal RuO2 , an electron-correlated superconductor Sr2 RuO4 , and a correlated, nearly ferromagnetic metal CaRuO3 . The hyperfine coupling constant due to the inner core polarization Acp % !22:2 T/"B is estimated from a linear slope with the assumption that Korb ¼ 1:59% is common among these ruthenates. The value of Acp is close to Ahf % 30 T/"B in the FM state of SrRuO3 . A large negative K value of Sr2 RuO4 indicates that its susceptibility is dominated by the d-electron spin susceptibility (Fig. 5 in ref. 60). Fig. 3. (Color online) Bulk susceptibility of Sr2 RuO4 (refs. 62 and 63) illustrating the spin and orbital contributions (see text for details). クーパー対の軌道部分に空間反転対称 ある スピン部分 Kspin ↑↓ - ↓↑ ない ↑↑,↓↓, ↑↓ + ↓↑ nuclei as an internal thermometer (supplement of ref. 47). 減らない Third, based減少 on the systematic studies of K among diﬀerent ruthenates, i.e., RuO2 , Sr2 RuO4 , and （↑↑,↓↓） CaRuO3 , it is clear that a large negative value of K for Sr2 RuO4 originates from the contribution from core polarization due to d-electrons J. Phys. Soc. Jpn. 81 (2012) (Fig. 0110092).60) From such analysis, S PECIAL TOPICS Y. MAENO et al. the contribution from the orbital part is estimated as 1.0% while the spin contribution is !4:5% for K ksuperconductors. c of Sr2 RuOHF: anisotropy of the NCS: Table I. Selection of candidates of spin-triplet heavythe fermion superconductors, FM: 99 4 . With Fig.Noncentrosymmetric 4. Knight shift ofsuperconductors, Ru nuclei of Sr 2 RuO4 with the applied field 61) ferromagnetic superconductors, !: superconductivity under pressure. spin part of only 2% as extracted from the !ab vs !c plot, parallel to the basal ab plane (ref. 62). The spin part Ks ðT Þ, after subtracting 62,63) orbital part Korb from the total shift KðT Þ, is the dominance of the spin part in the is the temperature-independent Materials Classification Spin susceptibility evidence of triplet pairing Properties plotted. illustrated in Fig. 3. Such properties of Sr RuO ensure that 3 2 4 NMR etc.7) He Superfluid magnetization, p-wave, A phase is chiral its NMR Knight shift serves as a reliable high-sensitivity Oxide NMR, polarized neutron 2D analogue of 3 He-A Sr2 RuO4 probe for the Cooper pair spin susceptibility. Chiral p-wave Ishida and coworkers have measured Knight shifts KðT Þ polarization eﬀect, where "B is the Bohr magneton. This 18) HF NMR f -wave UPt3 at the Ru and O sites in high-quality single13)crystals. means that the Ru Knight shift can detect much smaller HF NMR UBe13 , URu2 Si2 , UNi2 Al3 Measurements were performed using a 3 He–4 He dilution changes of the Ru-4d electronic state when superconducUGe2 ! , URhGe, UCoGe FM, HF Indirect Anomalous Hc2 19–22) refrigerator with the sample crystals directly immersed in tivity sets in. The temperature dependence of the 99 Ru shift UIr! NCS, FM, HF Indirect liquid helium to ensure good thermal contact to the23)bath. As was measured in the fields of 0.68, 0.9, and 1.05 T parallel ! NCS, HF NMR CeIrSi3 for the 17 O Knight-shift measurements, they assigned NMR to the RuO2 plane and the spin part of the Knight shift Li2 Pt3 B NCS NMR24) signals arising from the diﬀerent crystal sites (planer and [99 Kspin ¼ KðT Þ ! Korb ] obtained at 0.9 T is shown in Fig. 4. CePt3 Si 99 apical O sites),NCS, andHF estimated the spin part Indirect of the Knight Kspin does not change on passing through Tc % 1:0 K at ! NCS, HF Indirect Anomalous Hc2 25) CeRhSi3 shift above Tc at each site. They investigated temperature "0 H % 0:9 T. If a spin-singlet d-wave state with a line-node S/FM/S Junctions Indirect (Ic )26–29) even-parity, s-wave dependence of the Knight shift at the planer O site, where gap wereOdd-freq., realized, the T dependence of 99 Kspin would JPSJ 81,dashed 011009 the spin density is 6 times larger than that in the apical OMaeno behaveetasal., drawn by the curve(2012). in Fig. 4. Here we パウリリミット 存在 無し（↑↑,↓↓） 現代実験物理学I 超伝導状態における緩和率 4 1 = T1 n Ahf 2 dEk dEk 1+ 2 Ek Ek NS (Ek )NS (Ek )f (Ek ){1 コヒーレンス因子 f (Ek )} (Ek Ek ) コヒーレンスピーク （Hebel-Slichterピーク） 現代実験物理学I REVIEW ARTICLE FOCUS 量子液体 reside on the vertices of square lattices, w other in the three-dimensional crystal. T layers are negligible (although this small obtaining a non-zero magnetic ordering need only consider the S = 1/2 spins S a square lattice, i. The spins are couple a superexchange interaction, and so w hamiltonian X H0 = J ij Si · Sj Localized spins Neél order 120° order hiji Here, hiji represents nearest-neighbour exchange interaction. The antiferromagn the positive value of J ij , which prefers an the spins. We will initially consider, in Se the sites of a square lattice, and all J ij = J in the subsequent sections to other lattice between the spins. Our strategy here will be to begin by of H0 , and its perturbations, in some sim then describe the low-energy excitations o field theories. By extending the quantum parameter regimes, we will see relations discover new phases and critical points global perspective of the phase diagram. REVIEW ARTICLE FOCUS Quantum Liquid Fermi Liquid 3He hiji （Superfluid）4He Here, hiji represents nearest-neighbour pairs and J ij > 0 is the exchange interaction. The antiferromagnetism is a consequence of the positive value of J ij , which prefers an antiparallel alignment of the spins. We will initially consider, in Section IIA the case of i on the sites of a square lattice, and all J ij = J equal, but will generalize in the subsequent sections to other lattices and further interactions between the spins. Our strategy here will be to begin by guessing possible phases Figure states. a,b, Ground statessimple of the limiting S = 1/2cases. antiferromagnet H0 with all of H10 , Néel and its perturbations, in some We will then describe the low-energy excitations of these states by quantum J ij = J on the square (a) and triangular (b) lattices. The spin polarization is (a) field theories. By extending the quantum field theories to other collinear and (b) coplanar. parameter regimes, we will see relationships between the phases, discover new phases and critical points, and generally obtain a global perspective of the phase diagram. Sachdev, Nature Phys. 4, 173 (2008). Itinerant electrons e- reside on the vertices of square lattices, which are layered atop each other in the three-dimensional crystal. The couplings between the layers are negligible (although this small coupling is important in obtaining a non-zero magnetic ordering temperature), and so we need only consider the S = 1/2 spins Si , residing on the sites of a square lattice, i. The spins are coupled to each other through a superexchange interaction, and so we will consider here the hamiltonian X H0 = J ij Si · Sj . Surpress detectable. Indeed, the quantum critical point can serve as the best A. NEEL ORDERED STATES point entire phase diagram in Withof alldeparture J ij = J , there for is nounderstanding parameter, g , that the can tune the ground H0 ; J sets the overall energy scale, but does not modify thestate g , Tof plane. the wavefunction of any of the eigenstates. So there is only We will begin Section II by describing thea single rich variety of quantum phase to consider. quantum phases that appear insulators, whose For the square lattice, thereinistwo-dimensional convincing numerical and excitations =ground 1/2 electronic spins on the sites of a experimental evidence2are that Sthe state of H0 has Néel order, Figure 1 Néel states. a,b, Ground states of the S = 1/2 antiferromagnet H0 with all primary as shown in Fig. 1a. This state spontaneously breaks the spinJ ij = J on the square (a) and triangular (b) lattices. The spin polarization is (a) lattice. Interest in such two-dimensional spin systems was initially rotation symmetry of H0 , and isofcharacterized by the expectation collinear and (b) coplanar. stimulated by the discovery high-temperature superconductivity value in the cuprate compounds. However, since then, such quantum j (1) h S j i = ( 1 ) 8. detectable. Indeed, the quantum critical point can serve as the bestspin systems have found experimental applications in a wide variety systems, we willthedescribe below. The critical quantum phases point of departure for understanding the entire phase diagram inof Here, ( 1) jas represents opposite orientations of the spins on the the g , T plane. andtwo points of such materials be discussed in Section sublattices as shown in Fig.will 1a. The vector 8 represents the III. Section We will begin Section II by describing the rich variety of orientation and magnitude of the Néel order. We have |8| < 1/2, IV will extend our discussion to include charge fluctuations, quantum phases that appear in two-dimensional insulators, whose implying that the ground-state wavefunction is not simply the will describe some recent proposals for novel metallic and primary excitations are S = 1/2 electronic spins on the sites of aand classical state shown in Fig. 1a, but has quantum fluctuations about states, and associated quantum critical lattice. Interest in such two-dimensional spin systems was initiallysuperconducting it. These fluctuations will entangle the spins with each other, but points. stimulated by the discovery of high-temperature superconductivity qualitatively character of thethese state isissues captured by therelated ideas Althoughtheweessential will not discuss here, in the cuprate compounds. However, since then, such quantum pattern in Fig. 1a. Note that this pattern implies a long-range, also apply in other spatial dimensions. Similarly, quantum states spin systems have found experimental applications in a wide variety and classical, correlation between the spins, but not a significant inofthree dimensions, critical points of systems, as we will describe below. The critical quantum phasesappear amount entanglement because although there are no the EPR quantum eVects between and points of such materials will be discussed in Section III. Sectionoften haveofawell-separated distinct character. A fairly complete understanding of any pair spins. IV will extend our discussion to include charge fluctuations,one-dimensional Having described the ground state,has we been now consider thebuilding on correlated states achieved, and will describe some recent proposals for novel metallic and excitations. This is most conveniently done by writing the form methods. superconducting states, and associated quantum critical points. ‘bosonization’ of the Feynman path integral for the trajectories of all the spins classical orders somehow eUT Low-Temp. Center HP Geometrical frustration J < 0 Although we will not discuss these issues here, related ideas in imaginary time, ⌧ . After taking the long-wavelength limit to also apply in other spatial dimensions. Similarly, quantum statesII. PHASES OF INSULATING QUANTUM MAGNETS a continuous two-dimensional space, this path integral defines appear in three dimensions, although the quantum critical points a quantum field theory in 2 + 1 dimensional space-time with often have a distinct character. A fairly complete understanding of coordinates (r, ⌧). The quantum field theory is for a field 8(r, ⌧), superconductivity appears when insulators such one-dimensional correlated states has been achieved, building onHigh-temperature which is the value of the Néel order when averaged over the ‘bosonization’ methods. as square-lattice La2 CuO4 are doped with mobile holes or electrons. spins, Si , located on sites within some averaging La2 CuO4 is of r . Ain quick derivation of the eVective actionof forfreedom are an neighbourhood antiferromagnet, which the magnetic degrees II. PHASES OF INSULATING QUANTUM MAGNETS this quantum field theory is provided by writing all terms in powers S = 1/2 unpaired electrons, one on each Cu atom. The Cu atoms and gradients of 8 that are invariant under all the symmetries of the 3,4 High-temperature superconductivity appears when insulators such hamiltonian. In this manner, we obtain the action Z as La2 CuO4 are doped with mobile holes or electrons. La2 CuO4 is174 an antiferromagnet, in which the magnetic degrees of freedom are S8 = d2 r d⌧ (@ ⌧ 8)2 + v 2 (rx 8)2 + s82 + u(82 )2 . (2) S = 1/2 unpaired electrons, one on each Cu atom. The Cu atoms triangular, Kagomé, (AF) honerycomb, pyrochlore,… 174 ? A. NEEL ORDERED STATES With all J ij = J , there is no parameter, g , state of H0 ; J sets the overall energy sc the wavefunction of any of the eigenstate quantum phase to consider. For the square lattice, there is co experimental evidence2 that the ground s as shown in Fig. 1a. This state sponta rotation symmetry of H0 , and is charact value hSj i = ( 1) j 8. Here, ( 1) j represents the opposite orien two sublattices as shown in Fig. 1a. The orientation and magnitude of the Néel o implying that the ground-state wavefun classical state shown in Fig. 1a, but has qu it. These fluctuations will entangle the s qualitatively the essential character of th pattern in Fig. 1a. Note that this patte and classical, correlation between the sp amount of entanglement because there ar any pair of well-separated spins. Having described the ground state excitations. This is most conveniently d of the Feynman path integral for the tr in imaginary time, ⌧ . After taking the a continuous two-dimensional space, t a quantum field theory in 2 + 1 dime coordinates (r, ⌧). The quantum field the which is the value of the Néel order square-lattice spins, Si , located on sites neighbourhood of r . A quick derivation this quantum field theory is provided by w and gradients of 8 that are invariant unde hamiltonian. 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K&%A 0&.Q&%60;%& 7/&)': 6 .&6:;%&' 9%A;8'5:060& &80%AQ7 /8 &_6-0 69%&&.&80 W/0( 0(& 0(&A%&0/-6) Q%&'/-0/A8 DA% 0(& /-& %;)&: 68' &_Q&%/.&806) %&:;)0: DA% /-&H*!H*!S# \8 A%'&% 0A &_6./8& 0(& '&C&)AQ.&80 AD 0(& .698&0/- 9%A;8' REVIEW ARTICLE FOCUS :060& AD :Q/8 /-&* W& :0;'7 0(& .698&0/K60/A8 I/J 68' 0(& %&:;)0680 '#-# :;:-&Q0/B/)/07 I"'- L '/M'N* W(&%& N /: 0(& 6QQ)/&' .698&0/`&)'J 6: W&)) 6: 0(& %&6) 68' /.69/86%7 Q6%0: I"! 68' ""J AD 0(& 6#-# :;:-&Q0/B/)/07 I"6-J AD QA)7-%7:06))/8& [email protected]/U>R# @(& -6)A%/.&0%/.&6:;%&.&80: :;99&:0 :Q/85D%&&K/89 /80A 6 8A85&^;/)/B%/;. :060&F*!"* B;0 W& `8' 0(60 "'- /8-%&6:&: .A8A0A8/-6))7 W/0( '&-%&6:5 現代実験物理学I 非古典的な局在磁性 Localized spins Disorder Spin singlet Spectra for Geometrical fronzen spins: Spin ice Dy2Ti2O7 frustration 47Ti-NQR/zero-ﬁeld Spin glass NMR Spin ice Intensity (arb. units) h& 0(68L 3# V&9&).68 DA% ;:&D;) -A..&80:# @(/: WA%L (6: B&&8 :;QQA%0&' B7 6 9%680 D%A. [?,?# / Percolation OA%%&:QA8'&8-& 68' %&^;&:0: DA% .60&%/6): :(A;)' B& 6''%&::&' 0A M#P#V# I&5.6/)< DLBl:Q6-&#./0#&';J# ±1/2 ↔ ±3/2 150 K ################################################################# !"# $%&'( ')*+ ,-**.*% SiO 2 glass Spin-singlet dimer or e.g. Dy Ti O±3/2 ↔ ±5/2 !"#$%&'()*(+($,- Valence Bond Solid ¦↑↓> - ¦↓↑> Frozen SRO (spin ice) /0 1(23*-d4 /0 10 156%72n4 80 /0 9:;:n < =0 1)>',,*-d H2O ice 0.56 K d <-(4$#=-&# %9 8)*">2" 4&? /4#-$>47" ,-"-4$2) @&"#>#+#-A 8-&&"*7.4&>4 1#4#- B&>.-$">#*A B&>.-$">#* 84$CA 8-&&"*7.4&>4 DEFGHA B1! n <-(4$#=-&# %9 6)-=>"#$* 4&? 8$>&2-#%& /4#-$>47" @&"#>#+#-A 8$>&2-#%& B&>.-$">#*A 8$>&2-#%&A 0-I 3-$"-* GFJKGA B1! a b ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 0 ,-. /012. 3.2.4.1056 78 9:0:.9 1.9;/:<42 817= :-. 2.7=.:1<50/ 81;9:10:<74 78 57=>.:<42 <4:.105:<749 <9 04 .99.4:<0/ <421.3<.4: 78 <=>71:04: >17?/.=9 <4 @./39 09 3<A.19. 09 =024.:<9=%$ >17:.<4 87/3<42! 043 4.;10/ 4.:B71C9#+ D9 @19: .E>/0<4.3 ?6 F0;/<42'$ 2.7=.:1<50/ 81;9:10:<74 78 >17:74 >79<:<749 <9 0/97 1.9>749<?/. 871 :-. ;4;9;0/ /7BG:.=>.10:;1. :-.1=73640=<59 78 <5. 043 <:9 =.09;1.3 H217;43 9:0:.I .4:17>6*+ J.5.4: B71C -09 9-7B4 :-0: Wikipedia :-. 2.7=.:1<50/ 81;9:10:<74 78 <5. <9 =<=<5C.3 ?6 K6!,<!L($ 0 9<:.G 713.1.3 =024.:<5 =0:.1<0/ <4 B-<5- :-. 9><49 1.9<3. 74 0 /0::<5. 78 5714.1G9-01<42 :.:10-.310 B-.1. :-.6 871= 04 ;4;9;0/ =024.:<5 217;43 9:0:. C47B4 09 H9><4 <5.I)M%#+ N.1. B. <3.4:<86 0 577>.10:<A. 9><4G81..O<42 :1049<:<74 /.03<42 :7 :-. 9><4G<5. 217;43 9:0:. <4 4 5 = O2– = H+ 6 7 Frequency (MHz) 8 Water ice 9 Spin ice ./*01( 2 !"#$%&'(" )$*)$+$,'&'(-, -. .)/+')&'(-, (, 0&'$) ("$ &,1 +*(, ("$2 33 4, 0&'$) ("$3 “Zero-point in-)‘spin $&"# #51)-6$, (-,entropy (+ "7-+$ '- -,$ '#$ -'#$)ice’” -. ('+ '0- -856$, ,$(6#9-/)+3 &,1 $&"# -856$,A. %/+'P. #&:$Ramirez '0- #51)-6$, Figure (-,+ "7-+$) '(' '#&, '- ('+ ,$(6#9-/)(,6 -856$, 43 4, 2 The coupled-dimer antiferromagnet. This is described by the(-,+2 hamiltonian .et al., H , with J'-0&)1 = J on the red lines J = J/g on the red lines. a, The +*(, ("$3 '#$ +*(,+ *-(,' $('#$) 1()$"'75 -) solid &0&5 .)-%and'#$ "$,')$+ -.dashed '#$ '$')&#$1)&3 ground state, Nature 399, 333large-g (1999). p with each ellipse representing a singlet valence bond &,1 $&"# '$')&#$1)-, (+ "-,+')&(,$1 '- #&:$ '0- +*(,+ *-(,'(,6 (, &,1 '0- *-(,'(,6 -/'2 0 (|"#i ij ij |#"i)/ 2. b, The S = 1 spin triplon excitation. The pair of parallel spins Figure 3 Valence-bond solid states. a, Columnar VBS state of H0 + H1 with J ij = J on all bonds. This state is the same as in Fig. 2a, but the square-lattice symmetry has been broken spontaneously. Rotations by multiples of ⇡/2 about a lattice site yield the four degenerate states. b, Four-fold degenerate plaquette VBS state with 現代実験物理学I Quantum Spin Liquids by Theories QSL must satisfy: Unbroken symmetry Not frozen spins We frame our discussion in the context of a specific perturbation, H , studied recently by Sandvik and others : (insulator) ◆✓ ◆ X✓ 1 1 19 1 H1 = Q hijkli Si · Sj 4 Sk · Sl Resonating Valence Bond REVIEW ARTICLE FOCUS P.W.Anderson 1987 20,21 4 , where hijkli refers to sites on the plaquettes of the square lattice. The ground state of the hamiltonian H0 + H1 will depend on the ratio g = Q/J . For g = 0, we have the Néel ordered state of Section IIA. Recent numerical studies19–21 have shown convincingly that VBS order is present for large g (VBS order had also been found earlier in a related ‘easy-plane’ model22 ). We characterize the VBS state by introducing a complex field, , whose expectation value measures the breaking of space-group symmetry: QSL may have: Broken ‘topological’ = ( 1) symmetry S · S + i( 1) S · S . (10) This definition satisfies the requirements of spin-rotation Reduced gauge invariance, invariance, and measures the lattice symmetry broken by the columnar VBS state because under a rotation about an even U(1) to Z2 sublattice or so site by an angle = 0,⇡/2,⇡, 3⇡/2, we have !e . (11) Fractionalized excitations Thus, is a convenient measure of the Z rotation symmetry of the square lattice, and h i = 0 in any state (such as the Gapless excitations Néel state) in which this Z symmetry is preserved. From these jx j jy j+x̂ j j+ŷ i Sachdev, Nature Phys. 4, 173 (2008). Spinon Figure 4 Caricature of a spin-liquid state. The valence bonds are entangled between different pairings of the spins, only one of which is shown. Also shown are two unpaired S = 1/2 spinons, which can move independently in the spin-liquid background. excitation 4 4 definitions, we see that h i = 6 0 in the VBS states; states with arg(h i) = 0, ⇡/2, ⇡, 3⇡/2 correspond to the four degenerate VBS states with ‘columnar’ dimers in Fig. 3a, whereas states with arg(h i) = ⇡/4, 3⇡/4, 5⇡/4, 7⇡/4 correspond to the plaquette just the A µ photon that has only a single allowed polarization in 2 + 1 dimensions.) Nevertheless, these observations seem purely formal because they only involve a restatement of the obvious divergencelessness of the flux in equation (12), and the shift symmetry is unobservable. The key point made in ref. 23 was that the above shift symmetry 現代実験物理学I Quantum Spin Liquids by Experiments QSL must satisfy: Unbroken symmetry Not frozen spins (insulator) NMR spectrum, Neutron Scattering, μSR QSL may have: Broken ‘topological’ symmetry Reduced gauge invariance, U(1) to Z2 or so Fractionalized excitations Gapless excitations NMR T1, thermal conductivity rihito Nakata1 , Yuichi Kasahara1,2 , Takahiko Sasaki2 , Naoki Yoneyama2 , hi Fujimoto1 , Takasada Shibauchi1 and Yuji Matsuda1 • 2.5 Present quantum spin liquid candidates A 2.0 LETTERS 1.5 κ (W K –1 m –1) uids (QSLs), antiferromagnets n disordered ground states, is dimensional (1D) spin systems ousins. The spin-1/2 organic )-tetrathiafulvalene)2 Cu2 (CN)3 1) with a 2D triangular lattice e first experimental realization crucial importance is to unveil lementary spin excitations2,3 , ence of a ‘spin gap’, which on the universality class of port on our thermal-transport wn to 80 mK. We find, rather vidence for the absence of a y contradicts recent reports of The low-energy physics of this erpreted in light of the present QSL phase. pled spin systems, geometrical fluctuations. Largely triggered ating-valence-bond theory for residing on a frustrated twottice5–7 and its possible applia doped 2D square lattice8,9 , D systems has been a longes of QSL states on S = 1/2 ported in organic compounds, 1, inset)1,10,11 , C2 H5 (CH3 )3 Sb hiolate)2 ]2 (ref. 12) and 3 He thin ular, the NMR spectrum of ts no signs of magnetic ordering four orders of magnitude below refs 1,11). These findings aroused lly believed that whereas a QSL frustrated S = 1/2 2D kagome as corner-sharing triangles, the te is stable in the less frustrated attice15,16 . Several ideas, such as erate onsite repulsion17 , a ring ionalization by• a slight distortion attice19,20 , have been put forth ng-range magnetic ordering in rtheless, the origin for the QSL 現代実験物理学I NATURE PHYSICS DOI: 10.1038/NPHYS1715 Perhaps holy grails B a 1.0 S S Organic Triangular t Non¬magnetic EtMe3Sb layer 0.5 0 0 C S t C C S S Pd S S C S C S C S Pd(dmit)2 molecule κ-(BEDT-TTF)2Cu2(CN)3 t’ Pd(dmit)2 layer R ES E A RC H | R E PO R TS 2 4 6 8 Gapless(C)1, Gapped(κ)2 No LRO down to J/ 10000(NMR3/μSR) 10 frequency nQ(c). In addition, the central peak using 17O NMR measurements of an isotopeenriched single crystal of ZnCu3(OH)6Cl2. We frequency f for the Iz = –1/2 to +1/2 transition is demonstrated that the spin excitation spectrum shifted from the bare resonance frequency fo = exhibits a finite gap D = 0.03 J to 0.07 J be(gn/2p)Bext by the effects of the hyperfine magFigure 1 | Temperature dependence of the in-plane thermal conductivity netic fields from nearby Cu2+ sites, and the shift tween a S = 0 spin-liquid ground state and the excited states, where J ~ 200 K represents the of the peak (marked “main” in Fig. 2A) is probelow 10 K. (T) in zero fieldts for two different single crystals of Jdeuterated tr Jr s portional to ckagome. Such an NMR frequency 3 He superexchange interaction (5, 21). -(BEDT-TTF)2 Cu2 (CN)3 (sample A and sample B) measured in a Cu-Cu shift may be expressed in terms of the Knight A major advantage of using a single crystal for tB J cryostat (black for sample A and green for sample B) and dilution NMR B shift, 17K (c) = f/fo – 1 = Ahf ckagome, where Ahf is is that we can achieve high resolution by between the 17O refrigerator (blue for sample A and light green for sample B). As theapplying an external magnetic field Bext along the hyperfine coupling constant Figure 1 | Crystal structure of EtMe3 Sb[Pd(dmit)2 ]2 . a, Side view of the crystal structure of EtMe Pd(dmit)In are 3 Sb[Pd(dmit) 2 ]2 . 2D magnetic 2 layers specific crystallographic directions. Fig. 2A, nuclear spin and the Cu2+ electron spins. We can temperature lowered,layers (T)of decreases exhibits broad hump we starting 17 separated byis non-magnetic the closed-shelland monovalent cationaEtMe ofpresent the crystalthe structure of the Pd(dmit) The 3 Sb. b, Top view O (nuclear spin I2 layer. = 5/2; gyrofit the line shape in Fig. 2A with three sets of five ⇤ ' 6dimerized Pd(dmit)2at molecules are T strongly (theThe pairs crystal are denoted by dashed ovals). with aratio 1/2-spin is localized each [Pd(dmit) ]2 dimer. peaks with three distinct values of 17K (c) and n (c). to increase around K. Inset: structure of aOne electron = 5.772onMHz/T) NMR 2line magnetic gn/2p Q The arrows (tB , ts and tr ) indicate the transfer-integral network between the molecular orbitals of the [Pd(dmit)2 ]2 dimers. The three transfer integrals are shape measured at 295 K in Bext = 9 T, applied That is, the presence of the Cu2+ defects at the 2 . c, of two-dimensional layer -(BEDT-TTF) (CN) viewed 2 Cu2to non-equivalent butBEDT-TTF close to each other Localized spin model applicable EtMe33Sb[Pd(dmit) 2 ]2 . Three exchange interactions (JB , Js and Jr ) are close to 2+ 17 along the c axis; the temperature dependence of Zn sites results in three distinct O sites in reflecting of tB , ts and tr . The exchange interactions are estimated to be 220–250 K (ref. 2). alongeach theother, long axesthe ofvalues BEDT-TTF molecules. Pairs of BEDT-TTF molecules the line shape is presented in Fig. 2C. Unlike preZnCu3(OH)6Cl2. Taking into account the differ35 17 formisdimers arranged in a triangular lattice in terms t measured powder-averaged Cl and O ence in the transverse relaxation that affects the 100 viously the stretching exponent. The decrease in means thatofthetransfer integrals S C S S C S 13 0 between NMR (22,13C23),S we can resolve the five apparent signal intensities (fig. S1), we estimated function becomes non-single-exponential, indicating the is nearly unity S C 11line shapes Pd and trelaxation the dimers. The ratio of transfer integrals C C S distribution of the relaxation. This distribution increases from Iz = mS to mS + 1 Stransitions (Iz, z component of the the population of the three sites as 13 ± 4%, 28 ± 1. and the spin-1/2 is realized about 20 K andnearly reaches isotropic a maximumtriangular around 1 Klattice on cooling. In nuclear spin angular momentum; magnetic quan5%, and 59 ± 8%, in agreement with earlier 2D NMR 10¬1 Fig. 2, we also show the relaxation rate determined from the initial tum number m = –5/2, –3/2, –1/2, +1/2, +3/2), observations of three corresponding sites in a decay slope of the relaxation curve. The difference between the two which are separated by a nuclear quadrupole deuterated single crystal of ZnCu3(OD)6Cl2 (24). relaxation rates shows the degree of the distribution. To understand the nature of novel QSL states, knowledge 0 In spite of the distribution, it is clear from Fig. 2 that there is 10¬2 10 on the structure the low-lying an obvious kink in the of temperature dependence ofexcitation T1 1 at around spectrum in the 295K (9T||c) Main 1.0 K. This stronglylimit, suggestsparticularly that a phase transition occurs at zero-temperature the absence/presence of a spin NN Main 10¬1 this temperature. As no discontinuous jump is observed in the 170K ¬3 f 10 the spin gap,temperature is indispensable, bearing implications on o dependence of T1 1 , thisimmediate is not a first-order but a NNN continuous transition. importantstate, point toas notewell here isas thaton the quantum correlations of the The ground 10¬2 continuous phase transitions always involve essential changes of [1] S. Yamashita et al., Nat. Phys. 4, 459 (2008). numbers carried by each elementary excitation. For instance in 1D, states, that is, symmetry breaking and/or topological ordering. 10¬4 NN (T1T)¬1 T Therefore, our resultHeisenberg indicates that thechains gapless spin liquid changes half-integer spin feature a massless spectrum, [2] M. Yamashita et al., Nat. Phys. 5, 44 (2009). 10¬3 to an essentially different spin state with symmetry breaking and/or 1 10 100300 0.01 0.10 2 fo T1¬1 T 120K which enables proliferation of low-energy spinon excitations, topological ordering. Temperature (K) [3] Y. Shimizu et al., PRL 91, 107001 (2003). ¬5 Thissuch instability is not considered to be classicalin magnetic 0.01 case, 0.1 1 10 100 300 whereas excitations are 1suppressed the integer 10spin ordering, because the anomaly in T1 21is not a critical divergence. Temperature (K) [4] M. Yamashita et al., Science 328, 1246 (2010). 51.5 52.0 52.5 53.0 53.5 which hasanalysis a massive Spectral directly spectrum proves this point.. Figure 4 shows the NMR Frequency (MHz) 13 C nuclear spectra down to 19.4 mK. The transition around 1.0 K does not Figure 2 | Temperature dependence of the spin-lattice [5] T. Itoh et al., Nat. Phys. 6, 673 (2010). affect the spectral shape; all of the spectra are largely the same relaxation rate of EtMe3 Sb[Pd(dmit)2 ]2 . The main graph shows the 50K over the whole temperature region and the spectral tails are at 13 C nuclear spin-lattice 2.0 relaxation rate T1 1 of EtMe3 Sb[Pd(dmit)2 ]2 , and1.0 [6] M.P.Shores et al., J. Am. Chem. Soc. 127, 13462 (2005). Bext = 9T most within ±50 kHz. This width is much smaller than the scale the inset graph shows (T1 T) 1 , where T is temperature. The circles indicate of the hyperfine coupling constant of the 13 C sites, which is about the values determined from the stretched-exponential analysis (see text), [7] J.S. Helton et al., PRL 98, 107204 (2007). 1.5 the values determined from the initial decay slopes ity, Kyoto 606-8502, Japan, 2 Institute for Materials Research, Tohoku University, Sendai 980-8577,and Japan. 9⇥102 kHz/µ the squares denote B (ref. 2), where µB is the Bohr magneton. Therefore, [8] M. Fu et al., Science 350, 655 (2015). classical spin ordering and freezing are clearly absent down to of the relaxation curves. The dark blue circles and dark red squares are oto-u.ac.jp. 30K 19.4 mK, which proves that the present instability differs from obtained from the1.0 present measurements below 1.75 K in a dilution 0.5 2 classical ordering/freezing. refrigerator. For clarity, we did the same analysis for previously reported T1¬1 (s¬1) Herbertsmithite Kagome EtMe3Sb[Pd(dmit)2]2 Gapless(κ)4, Nodal(T1)5 No LRO down to J/10000(NMR)5 ZnxCu4-x(OH)6Cl2 Gapless(INS)7, Gapped(NMR)8 No LRO down to J/6000(NS7) Close to QSL? Triangular: Cs2CuCl4, NiGa2S4, NaxCrO2, … Hyperkagome: Na4Ir3O8 cho intensity (arb. units) c (T1T )¬1 (s¬1 K¬1) T (K) Spin echo intensity (arb. units) Organic Triangular b 現代実験物理学I Perfect Kagome : Herbertsmithite Kagomé Published on Web 09/09/2005 A Structurally Perfect S ) 1/2 Kagomé Antiferromagnet Published on Web 09/09/2005 Matthew P. Shores, Emily A. Nytko, Bart M. Bartlett, and Daniel G. Nocera* 1 A Structurally Perfect S ) /2 Kagomé Antiferromagnet Department of Chemistry, 6-335, Massachusetts Institute of Technology, Massachusetts AVenue, Matthew P. Shores,77 Emily A. Nytko, Bart M. Bartlett, and Daniel G. Nocera* Cambridge, Massachusetts 02139-4307 Department of Chemistry, 6-335, Massachusetts Institute of Technology, 77 Massachusetts AVenue, Cambridge, Massachusetts 02139-4307 Received June 13, 2005; E-mail: [email protected] June 13, 2005; E-mail: [email protected] Nearly two decades ago, Anderson proposed that the resonating rly two decades ago, Anderson proposed that the resonating valence bond (RVB) state may explain the scatterless hole transport encountered in doped rare-earth cuprates.1 The quantum spin liquid e bond (RVB) state may explain the scatterless hole transport phase responsible for RVB is most likely to be found in lowntered in doped rare-earth cuprates.1 The quantum spin liquid dimensional, low-spin, and geometrically frustrated systems.2 responsible for RVB is most likely to be found in Accordingly, lowmost theoretical investigations of RVB have concen1/ antiferromagnets in kagomé (corner-sharing trated on S ) 2 2 sional, low-spin, and geometrically frustrated systems. triangular) lattices due to the higher degree of geometric frustration.3 Herbertsmithite dingly, most theoretical investigations of RVB have concenMaterials featuring such lattices are predicted to display no long1 range magnetic order due to competing antiferromagnetic interacon S ) /2 antiferromagnets in kagomé (corner-sharing tions between nearest-neighbor spin centers. Though long sought, 3 lar) lattices due to the higher degree of geometric frustration. “no perfect S ) 1/2 Kagomé antiferromagnet has been up to now Figure 1. Crystal structure of Zn-paratacamite (1), Zn0.33Cu3.67(OH)6Cl2. als featuring such lattices are predicted to display no synthesized”, long- 4 and accordingly, most theoretical predictions of such Left: local coordination environment of intralayer Cu3(OH)3 triangles and a lattice remain untested. Herein, we report the synthesis and interlayer Zn2+/Cu2+ ion; the projection is parallel to the crystallographic magnetic order due to competing antiferromagnetic interacpreliminary magnetic properties of a rare, phase-pure, copper c axis. Right: the {Cu3(OH)6} kagomé lattice, projected perpendicular to hydroxide chloride mineral featuring structurally perfect S ) 1/2 etween nearest-neighbor spin centers. Though long sought, the c axis. The pure Zn2+-substituted compound 2 is isostructural to 1. Selected interatomic distances (Å) and angles (deg) for 2: Zn-O, 2.101kagomé layers separated by diamagnetic Zn(II) cations. rfect S ) 1/2 Kagomé antiferromagnet has been up to now (5); Cu-O, 1.982(2); Cu-Cl, 2.7698(17); Zn‚‚‚Cu, 3.05967(16); O-ZnWe have employed a redox-based hydrothermal protocol to O, 76.21(18), 180.00(19); O-Cu-O, 81.7(3), 98.3(3), 180.0; Figure 1.jarosite-based Crystal structure of Zn-paratacamite (1), 103.79(18), Zn0.33Cu 3.67(OH) 6Cl2. 180.0; Cu-O-Cu, 119.1materials (AM such pure, single-crystal sized”,4 and accordingly, most theoretical predictions ofprepare 3(OH)6O-Cu-Cl, 82.31(11), 97.68(11); Cl-Cu-Cl, (SO4)2, A ) alkaliLeft: metal ion, M coordination ) V, Cr, Fe).5 These compounds of intralayer local environment Cu397.04(15). (OH)3 triangles and (2); Cu-O-Zn, ce remain untested. Herein, we report the synthesisfeature andkagomé lattices 2+ ion; composed of /Cu M3(OH) when M 6 triangles; interlayer Zn2+ the projection is parallel to the crystallographic inary magnetic properties of a rare, phase-pure, copper ) Fe(III), spins are antiferromagnetically coupled and frustrated.6 c axis. Right: the {Cu projected perpendicular to 35(OH)6} kagomé lattice, and refinement are provided in the Supporting Information. Two of the magnetic ion of Fe(III) (S ) /22+ ) by Cu(II) (S ) 1/ xide chloride mineral featuring structurally perfect S Substitution ) geometrically distinct metal sites are found. On the first site, a Cuthe c axis. The pure Zn -substituted compound 2 is isostructural to 1. 2 1 /2) was attempted, but charge imbalance on the kagomé layers from the randomly oriented portion of the powder (i.e., 80% of the sample) [10].現代実験物理学I Notice that the whole 35 Cl NMR line shape begins to tail-off toward lower frequencies below #50 K. The resonance frequency of the sharp c-axis central peak and its distribution depends on the NMR Knight shift, 35 K, induced by !loc . Hence the observed line broadening implies that !loc varies depending on the location within the sample below #50 K. In Fig. 3, we summarize the 35 Cl NMR Knight shifts 35 K 35 K1=2 deduced from the line shapes, together with EtMe 3Sb[Pd(dmit) 2] 2 and 35 K 6 ZnCu Cl 2 3 (OH) by SQUID. corresponds to the cen!bulk as observed tral peak above #45 K as determined by FFT techniques. 13C-NMR RAPID COMMUNICATIONS 35 Below #45 K, where the central peak is smeared out by Cl-NMR T. Itoh et al., PRB 84, 094405 (2011). line broadening, we determined 35 K as the higher freT. Imai et al., PRL 100, 077203 (2008). quencyBedge the central !2006" peak from point-by-point meaPHYSICAL REVIEW 73, of 140407!R" NMR spectra for present QSL candidates κ-(BEDT-TTF) 2Cu 2(CN) 3 13C-NMR Y. Shimizu et al., PRB 73, 140407(R) (2006). ∼0.1μB T. ITOU et al. PHYSICAL REVIEW B 84, 0 ∼0.1μB makes relaxation curves inhomogeneous. Therefor likely that the power-law decrease is intrinsic and state has a nodal gap structure in the magnetic 178 mK We point out here that (T1 T )−1 is expected to foll 895 mK temperature dependence in a typical nodally gappe as a d-wave resonating-valence-bond state.22 Th 1080 mK behavior (T1 T )−1 ∝ T is an open problem and m 1220 mK factor in understanding the present spin liquid. ∼0.1μ Our results refute the presence of fully gaples B 4.04 K ∼J/60 magnetic excitations. This seemingly contrasts wi 10.15 K for the thermal quantities,16,17 which implies g tations like metals. A possible explanation for t 19.9 K contradiction may be the fact that NMR relaxation 29.0 K only magnetic excitations, while thermal quan both magnetic and nonmagnetic excitations. Our 42.8 K do not provide any information on nonmagnetic 105.9 K which do not contribute to the dynamic spin su The thing that ∼J/40 can be definitely determined from 299.7 K data is that the dynamic susceptibility disappears -200 -150 -100 -50 0 50 100 150 200 If we accept the thermal-quantities result as true Shift from TMS (kHz) scenario for the present ground state is a spin with (maybe FIG. 3. NMR spectra of the inner carbons of 35 nodally) gapped magnetic excitations FIG. 2 (color online). Cl NMR line shapes of the Iz $ % 12 to nonmagnetic which is sometimes EtMe3 Sb[Pd(dmit)2 ]2 at several temperatures. in 8.4 excitations, Tesla in a partially (#20%) uniax-re & 1 central transition Intensity (arb. units) 18.7 mK ∼J/60 2 6,7,9 NMR relaxation for present QSL candidates EtMe 3Sb[Pd(dmit) 2] 2 κ-(BEDT-TTF) 2Cu 2(CN) 3 13C-NMR RAPID COMMUNICATIONS PHYSICAL B 73, 140407!R" !2006" Y. Shimizu et al., PRB REVIEW 73, 140407(R) (2006). ∼J/60 ZnCu (OH) Cl 13C-NMR T. Itoh et al., PRB 84, 094405 (2011). NUCLEAR MAGNETIC RESONANCE OF THE . . . ∼J/60 102 (a) 101 10 T1-1inner T1-1outer 0 inhomogeneity of EtMe3 Sb[Pd(dmit)2 ]2 also grows on cooling in the region from about 10 to 1 K, where the growth bears a notable resemblance to that of κ-(BEDT-TTF)2 Cu2 (CN)3 . 10 5 However, in contrast8.3 to the case of κ-(BEDT-TTF)2 Cu2 (CN)3 , T (34.6MHz) βinner and βouter in 4.4T EtMe(18.4MHz) stop de3 Sb[Pd(dmit)2 ]2 suddenly OH freezes? creasing at 1 K and recover rapidly to the homogeneous values 2.4T (10MHz) on further cooling, as shown in Fig. 2(b). 1.5T (6.3MHz) The most important of the present results is that the 1.0Tpoint (4.2MHz) temperature dependence of T1−1 inner shows an obvious kink at around 110 K and then shows a steep decrease on cooling, which is essentially the same behavior as for the outer carbons. In the low-temperature limit, T1−1 inner is proportional to the square 10 4 of the temperature, as is also observed in T1−1 outer . If these behaviors were caused by the cation molecular motion, the two relaxations should show different temperature dependencies. This is because the inner-carbon relaxation is hardly affected by the molecular motion, while the outer-carbon relaxation is more readily affected. Therefore, the accordance between the two relaxations is very strong evidence that the observed 1 behaviors of the relaxation rates are not affected by the molecular motion of the cation but reflect the electron spin dynamics. 1000 (8T, 93MHz) Thus, the kinks at around 1 K and 63-Cu the rapid decreases below (0.9T, 38MHz) this temperature in T1−1 inner and T1−11-H are reliable indicators outer of the spin state. The kinks strongly suggest that the spin 1 1 0 100 which state undergoes an abrupt change at this temperature, involves a change of the excitation spectrum. T [K ] The change of the spin state is likely to be a continuous phase transition, because T1−1 inner and T1−1 outer appear to show singular behaviors with 35 FIG. online). at Temperature dependence of Cl NMR respect4 to(color the temperature the kinks (that is, discontinuities 35 −1various magnetic fields " at spin-lattice relaxation of the first-order derivativesrate of T1−1!1=T and T ). It is im1 1 outer inner portant to note that continuous phaserepresents transitions always (filled symbols). Solid line a fit involve to a power law, 35 symmetry breaking and/or !1=T1 " $ T # with # $topological 0:47 (8.3ordering. T), 0.44Therefore, (4.4 T), 0.2 (2.4 T the ground state of the present system is likely to have a1 rate in low field (0.9 T), !1=T1 ", and and 1.0 T). 1 H relaxation broken-symmetry and/or topological structure. 63 63 !1=T ", are also Cu relaxation rate in high T),carbons Figure 3 shows the NMR spectrafield of the(8 inner at1 35 " measured in comparable field. superposed on !1=T several temperatures. The1features of the spectra are almost magnetic the 101 10-5 0.01 (T1T )-1∝T (1/T1) [sec-1] (T1T)-1 (s-1 K-1) 100 10-1 10-2 0.1 0.1 1 10 Temperature (K) 1 10 Temperature (K) 100 35 10-3 0.01 100 300 1 Stretching Exponent β (b) 0.8 0.6 β inner 0.4 β outer 0.2 0.01 0.1 1 10 Temperature (K) 1 -3 T1 -1∝ T 2 100 300 FIG. 2. (Color) Temperature dependencies of the 13 C nuclear spin-lattice relaxations of the inner and outer carbons of EtMe3 Sb[Pd(dmit)2 ]2 . The red circles show data for the inner 12 35* (1/T1) [sec-1] 10-2 10-4 FIG. 2. !a" Temperature dependence of the NMR linewidth $1/2 and 1 / T2 measured at 8 T. !b" 1 / T1 for the inner !triangle", outer !circle" and total 13C cites !diamond". The inset shows the exponent (1/T1), T1-1 (s-1) 63 10-1 10 T T y n e g m K perature dependence is different from the smaller !local as represented by 35 K. !bulk simply represents a bulk average 現代実験物理学I of !local . In passing, we recall that earlier "SR Knight shift K"SR measurements by Ofer et al. [10] showed identical behavior between K"SR and !bulk . They concluded that the upturn of !bulk below 50 K is not caused by impurity spins but is a bulk phenomenon. Our new results in Fig. 3 do not contradict these "SR data. K"SR was deduced by assuming a Gaussian distribution of !local ; hence by default K"SR represents the central value of the presumed Gaussian distribution. That explains why K"SR shows behavior simi6 2 lar to !bulk and 35 K1=2 . 3 Next, we turn1our35 attention to the dynamics of lattice and H, Cl,63Cu-NMR spin degrees of freedom. Figure 4 shows the temperature T. Imaiofetthe al.,35 Cl PRL 100, spin-lattice 077203 (2008). dependence nuclear relaxation rate, PHYSICAL REVIEW B 84, 094405 (2011) 35 !1=T1 ", measured at the central peak frequency in various ∼J/6 ca ag in 35 th tio w bi co in 35 at ha tu ne tio pr N (1 th 35 th no an th ou sa ab th m su la st R m [1 en ! re C 35 tio th takes one of three orientations, labeled using the variable 4. Representing spins by Majorana operators state to other flux sectors are OðJÞ 現代実験物理学I !jk ¼ x, y, or z as shown in Fig. 1(a). A spin one-half Jx=1, !. one can project onto aJ =given flux variable at site j is represented by the Pauli matrices " 4.1. Aj general spin-fermion transformation y Jz=0 batively in h. For the undiluted la Exchange of strength J!jk acts between spin components Fig. 5. Phase diagram of the model. The the the Zeeman energy between sta Let us remind reader formalism J + J +some J = 1. general The diagrams for the ot !jk and the Hamiltonian is sector areisall zerodescribed and thebylead with n fermionic modes usually the X y !jk !jk is can second in hc ak (k(1) = 1, . . .,perturbation n). Instead, one use their linear H ¼ " J!jk "j "k : is impossible, even iforder we introduc Majorana operators: a vacancy, thephase individua eighthowever, copies of each (corres Another route to QSL: Kitaev model x hjki Annals of Physics 321 (2006) 2–111 c2k%1 www.elsevier.com/locate/aop y z y a % a k k thethree same ; properties ¼ rounding ak þ c2k ¼translational plaquettes lose p i phase are algebraically diﬀerent. ayk ; It is exactly soluble, for example, with a local transformaonly theirWe Z2now sum enters the defi consider the zeros of t ! ! which are called Majorana operators. The operato tion "Anyons ¼ ibin canwhich represents each spin using the four Because of themodulo Zeeman e exactly solved model and beyond tum q isthis, defined the recip the lattice contains one are divided into three types, depending 20 A. Kitaev / Annals of Physics 321 (2006) 2–111 x vertex y z of each kind. Links obey the following relations: QSL phases A,B: elements Majorana fermions b , b , b , and c [10]. Then momentum space by a thesector, parallel matrix within on their direction (see Alexei Fig. Kitaev3B); we call them ‘‘x-links,’’ ‘‘y-links,’’ and ‘‘z-links.’’ The symmetric Jy = Jzl) x = now J¼ Jlxc=jJy=0if jcase c2j ¼ 1; projected cj cl the %c 6¼ l.(Jare Hamiltonian z=1, Hamiltoniani X is as follows: !jk !jk J!jk u^ jk cj ck ; X u^ jk ¼ ibX (2)gapless arise from the specific field compo H¼ j bk : X 2 on equal operators cj can be H ¼ "Jj;k rxj rxk " J y ryj ryk " J z rzj rzk ; Note that allin ð4Þtreatedthe x Fig. 1(b): employing site la A zrepresentation of We now describe a a spin b gapped x-links y -links z-links Abstract A. Kitaev / Annals of Physics 321 (2006) 2–111 15 projected Zeeman energy is H with each other and with H. operators. Let us denote these operato The operators u^ jk commute Majorana A spin-1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors z are of XX, YY or ZZ type, depending on the direction of the link; diﬀerent types of interactions may " Written operators using Majorana fer h c , c , and c ). The act on the z 2 3 4 , J , and J are model parameters. J 3 Þ.Majorana ^ i ¼ u ¼ #1, Onewhere can therefore fix the values of h u Majorana x The ymodel is solved zexactly by a reduction diﬀer in strength. to free operators fermions in a staticjk Z gauge jk B|Jx| and |Jy| decrease! while |Jz| If field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap the Hilbert space a spinthe is identified a twoform h!with "i ¼ ihun to Hof Z have ! and carries excitations that are Abelianof anyons. The other phaseHamiltonian, is gapless, but acquires a gap in move to a subspace the full and obtain a (within the parallelogram) y y x x A the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding A y by thisJ condition: x sented in model, |Jx|the + |Jtight |. The q* and rules coincide with those of conformal blocks for the Ising model. We also consider a general theory Jy=1,points y| = |Jzbinding x=1, ’s. Numbering the sites around a bilinear form in the c u z fermions with a gapped z spectrum, which j zis characterized by a spectral Chern number m. The of free Abelian and non-Abelian phases of the original model correspondspins to m = 0 and m = ±1, respectively. Jy= Jz=0 of new allelogram. sites, with(Note theJx=that coupling sh Jz=0 the equati y of 1excitation y on m mod flux through a plaquette from to 6depend [see Fig. 1(a)], theZ2edge Zflux The anyonic properties 16, whereas m itself governs thermal 2 At the points ±q* the spectrum transport. The paper also provides mathematical background on anyons as well as an elementary The task now ispositive to calculate Fig. 5. Phase diagram of the model. The triangle is the section of the octant (J , J , Jthe P x x theory of Chern number for quasidiagonal matrices. plaquette is defined to be w ¼ u u u u u u . p 21 23 43 45 65 61 ! 2005 Elsevier Inc. All rights reserved. J + J + J = 1. The diagrams for the other octants are similar. ground-state energy of the Kitaev Physical properties of the system depend only on these Fig. 4. Graphic representation of Hamiltonian (13). Gapless using the tight binding model of Fi is impossible, even if we introduce new terms in the Hamiltonian. On the oth g give fluxes [10], but note that many choices of the set fu 1. Comments to the contents: what is this paper about? Note that each pair of connected sites is counted twice, and û = !û . The structure of jk Majoranato behavior indiﬀerent a gapped phase. A of con eight copies of each phase (corresponding sign combinations Jx his Hamiltonian is shown Fig. 4. ofthe Certainly, the main result thesector. paper is anisexact solution ofIta is particular two-dimen(3) considered inin [24], but solution less trivial. not clear how to realizeis this model rise to the same flux The ground-state sector fluxRemarkably, thequantum operators û However, commute Hamiltonian with to each other. translational properties. the same It is unknown whether the eight copies of sional model. I waswith sittingthe on that result for too and long, trying perfect structure of this tight binding mod Fermion in solid state, but an optical lattice implementation has been proposed [42]. e derive some properties ofall the model, and¼ put them more framework. Thus herefore the it,Hilbert space L splits into eigenspaces ofgeneral û , which indexed phase algebraically diﬀerent. þ1into afor sites j are on aareparticular free, e.g., with ucommon many ramifications have come along.jk Some of them stem from the desireeto avoid e the use * California Institute of Technology, Pasadena, CA 91125, USA Received 21 October 2005; accepted 25 October 2005 2 cj z bj jk z bk ck x x kj jk jk jk y z y z 現代実験物理学I Toward Kitaev QSL: 2D honeycomb iridates α-A2IrO3 5d5 in cubic symmetry Edge-shared IrO6 IrO6 eg Δ A z x λSO Jeff ∼3 eV ∼0.4 eV 1/2 t2g 3/2 y Spin Isospin Jeﬀ=1/2 Kramers doublet: the lattice contains one vertex of each kind. Links are divided into three types, depending 1 on their direction (see Fig. 3B); we call them ‘‘x-links,’’ The J eff 1/ 2‘‘y-links,’’ = xyand ,± 1‘‘z-links.’’ 2 ± yz ,∓ 1 2 + i zx,∓ 1 2 Hamiltonian is as follows: 3 PHYSICAL REVIEW PRL 102, 017205 (2009) X X y y X x x z z H ¼ "J x rj rk " J y rj rk " J z rj rk ; ð4Þ Heis x-links y -links z-links ( ) + = where Jx, Jy, and Jz are model parameters. isospin up spin up, lz=0 spin down, lz=1 G. Jackeli and G. Khaliullin, PRL 102, 017205 (2009) FIG. 1 (color online). Density profile of a hole in the isospin wher simp ij bo cubic symmetry (! ¼ 0, This peculiarity on a given to bond, ofentirely a 90due tobond leads A 180" bond: For this geometry, thethe nearestenough, its (A) anisotropy is Hund’s couthe xy-plane. Ho 現代実験物理学I he effects of a tetragonal neighbor t hopping matrix is diagonal in the orbital space 2g such a system nondiagonal elem pling. This is opposite to atwoconventional situation: typiHamiltonian drastically different from that o and, on a given bond, only orbitals are active, e.g., jxyi a charge transfer Toward Kitaev QSL: 2D h oneycomb mon cases of TM-O-TM cally, the anisotropy obtained in powers of and jxzi orbitalscorrections along a bond in are x-direction [Fig. 2(a)]. This peculiarity Ref. [12]. After Two transfer amplitudes via upper an ond formed by corner- ometry. The spin-orbital exchange Hamiltonian for such a system Hamiltonian dras ! while the Hund’s coupling is not essential. has already been reported: see Eq. (3.11) in Ref. [12]. After " -bond iridates ometry. Two tran et, we(B)find ), and a 90an " gen(B)interfere in a destructive manner and the the are ground state doublet, we find an A 90projecting bond:it onto There again only two orbitals active gen interfere in a exchange Hamiltonian for isospins in a form of 2(b). of the Hamiltonia on a given bond, e.g., jxzi and jyzi orbitals along a bond in a form of Edge-shared IrO 6 α-A 2IrOof 3 the Hamiltonian exactly vanishes. The finit interaction appea geometry, the nearestthe xy-plane.(a)However, the hopping matrix has now tureonly of the excite (b) gonal in the orbital space interaction Edge-shared IrO6 180 of theJ exchange i appears, to paths the -m H nondiagonal elements, and however, there are two90due possible for tion of a given bo bitals are IrO active, e.g., jxyi ture perpendicut a charge transfer [via upper or lowerMost oxygen,importantly, see Fig.plane 2(b)]. of the excited levels. 6 A With this in mind n x-direction [Fig. 2(a)]. This peculiarity of a 90" bond leadspz to an exchange ofxHamiltonian the exchange interaction depends on" gethe sp z onian for such a system drastically different from that of a 180 o yz xz 90 (3.11) in Ref. [12]. After 4 ometry. transfer amplitudes upper a andbond lower of Two a given bond. Wevialabel ij y tion withoxyJ ¼ layi py xy xy 3 #2 . Re quantum analog o ate doublet, we find an gen interfere in a destructive manner and the isotropic part perpendicular to the 'ð¼ x; y; zÞ axis b introduced origin ospins in a form of plane of the Hamiltonian exactly vanishes.xz The finite, anisotropic Jahn-Teller syste yz studies as a prot With this in mind, the Hamiltonian can be w xz xz pinto interaction appears, however, due to the J -multiplet strucz the lattice contains one vertex of each kind. Links are divided three types, depending H pz degeneracy of to pThe z on their direction (see Fig. 3B); we call them ‘‘x-links,’’ ‘‘y-links,’’ and ‘‘z-links.’’ ture of the excited levels. Most importantly, the very form to our ) However, Hamiltonian is as follows: the compass mod FIG. 2 (color online). Two possible geometries of a TM-O-TM ð'Þ ' ' of the exchange interaction depends on the spatial orientaactive along these bonds. The S ;Implementing 90o X H ¼ 'JS X y y Xbond with corresponding orbitals ij i in the jKitaev%& x x z z model is e of a given Weforlabel a bond laying large dots stand the transition metalij (oxygen) ions. H ¼ xz "J x rj rk " J y rtion rj rk(small) ; bond. ð4Þ j rk " J z " a honeycomb latt formed by corner-shared octahedra, and (b) a x-links y -links plane perpendicular z-links(a) A 180 -bond to the 'ð¼ x; y; zÞ axis by a (')-bond. ⊥ plane " properties such 90 -bond formed by edge-shared octahedra. 4 With J this¼in mind, the Hamiltonian canthis be written as with # . Remarkably, Hamiltonian 017205-2 , zand Jz are model parameters. where Jx, Jyp And J > 0 (FM) 3 2 Kitaev FM QSL is expected ' ' quantum analogHofð'Þ the so-called compass(3)mod ¼ 'JS G. Jackeli and ij i SG. j ;Khaliullin, PRL 102, 017205 (2009) o yz xz o 現代実験物理学I Real Kitaev QSL?: NMR spectra for H3LiIr2O6 1 Li site only @K~0.1% Li-NMR 5T Original α-Li2IrO3 H3LiIr2O6 IrO6 110 K 2Li Intensity (arb. units) 70 K 50 K 30 K 15 K 1 H-NMR 2T 6H 110 K 70 K 50 K 30 K 20 K 10 K 10 K 5K 5K 1.2 K 1.0 K -0.2 0.0 Shift (%) 0.2 0.4 H3LiIr2O6 A Intensity (arb. units) 7 1 H site only @K~0.02% -0.2 0.0 Shift (%) 0.2 現代実験物理学I Real Kitaev QSL?: Knight shift as Intrinsic susceptibility Knight shifts Bulk M/H 0.10 ����� H3LiIr2O6 ������ � � � ������������ 0.08 7 Li-NMR 5T, oriented 0.06 K (%) 7 Li-NMR 2T, oriented 0.04 ����� Raw data ������ After ����� subtraction ������ � 0.02 1 H-NMR 1T, unoriented 0.00 1 2 3 4 5 6 78 10 T (K) 2 3 4 5 6 78 for impurities � �� ��� ��� ��� � ��� ��� ��� 100 Supports gapless QSL ＆ ﬁrst time detecting ﬁnite K(T=0) in QSLs

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