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Quasi-one-dimensional spin nematic states and their excitations Oleg Starykh, University of Utah Leon Balents, KITP, UCSB APS March meeting, San Antonio, March 3, 2015 Outline • Very brief intro - emergence of composite orders from competing interactions • Nematic vs SDW in LiCuVO4 ✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave” • Volborthite kagome antiferromagnet - experimental status - magnetization plateau - Nematic, SDW and more - Field theory of the Lifshitz point • Conclusions Emergent nematic (Ising) order parameters ~1 · N ~ 2 = ±1 =N Outline • Very brief intro - emergence of composite orders from competing interactions • Nematic vs SDW in LiCuVO4 ✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave” • Volborthite kagome antiferromagnet - experimental status - magnetization plateau - Nematic, SDW and more - Field theory of the Lifshitz point • Conclusions LiCuVO4 : magnon superconductor? Letter rystal structure of LiCuVO4 . Cu-O chains separated by VO4 tetrahedra and estimates: he b direction. ∠ Cu-O-Cu ∼ 90◦ indicates the ferromagnetic interaction. J1 = - 1.6 meV J2 = 3.9 meV (subject of active debates) J5 = -0.4 meV tinger parameter.9) Recent numerical studies exhibit magnetization vs nd the quadrupole phase in fact persists down to rather low magnetic z z High-field analysis: condensate of bound magnon pairs + hS i = 0 + + hS S i = 6 0 Ferromagnetic J1 < 0 produces attraction in real space Chubukov 1991 Kecke et al 2007 Kuzian and Drechsler 2007 Hikihara et al 2008 Sudan et al 2009 Zhitomirsky and Tsunetsugu 2010 Magnon binding E-EFM = ε1 + h 1-magnon 2-magnon bound state E-EFM = ε2 + 2h E Sz=-2 Sz=-1 ε2 < 2 ε1 : “molecular” bound state h Formation of molecular fluid For d>1 at T=0 this is a molecular BEC = true spin nematic Hidden order No dipolar order + hSi i hSi+ Sj Nematic order nematic director =0 i⇠e + + hSi Si+a i |i j|/⇠ Sz=1 gap = 6 0 Magnetic quadrupole moment Symmetry breaking U(1) → Z2 can think of a fluctuating fan state LiCuVO4 experiment: collinear SDW along B Hagiwara, Svistov et al, 2011 Buttgen et al 2012, 2014 LiCuVO4 No spin-flip scattering above ~ 9 Tesla: longitudinal SDW state SF = spin flip, ΔS = 1" NSF = no spin flip, ΔS = 0 o 1d J1-J2 chain is only quasi-spin-nematic • • • No true condensation [ U(1) breaking] in d=1.! ! Inter-chain interaction is crucial for establishing! symmetry breaking in d=2.! ! Need to study weakly coupled “superconducting” chains J1< 0 (ferro) J2 >0, J’ > 0 (afm) in magnetic field Sato et al 2013 Starykh and Balents 2014 Inter-chain interaction Hinter chain XZ = y ~y · S ~y+1 ⇠ dx S XZ z dx Sy+ Sy+1 + Syz Sy+1 y Superconducting analogy: single-particle (magnon) tunneling between magnon superconductors is strongly suppressed at low energy (below the single-particle gap) ? Hinter = Z X dx J y 0 + hSy (x)Sy+1 (x + 1)inematic ground state !0 Superconducting analogy: fluctuations generate two-magnon (Josephson coupling) tunneling between chains. They are generically weak, ~ J1(J’/J1)2 << J’ , but responsible for a true two-dimensional nematic order Z 02 Hnem ⇠ (J /J1 ) X dx + [Ty (x)Ty+1 (x) + h.c.] Ty+ (x) ⇠ Sy (x)Sy (x + 1) y At the same time, density-density inter-chain interaction does not experience any suppression. It drives the system toward a two-dimensional collinear SDW order. p Syz z Hinter chain =M 2npair = M = Hsdw ⇠ J 0 X y z Syz Sy+1 Ã1 e ⇠J 0 i 2⇡ XZ '+ y (x) iksdw x dx cos[ e p 2⇡ ('+ y '+ y+1 )] y Away from the saturation, SDW is more relevant [and stronger, via J’ >> (J’)2/J1 ] than the nematic interaction: coupled 1d nematic chains order in a 2d SDW state. Simple scaling Hnem ⇠ (J 02 /J1 ) Z X dx [Ty+ (x)Ty+1 (x) + h.c.] y • describes kinetic energy of magnon pairs, linear in magnon pair density npair z Hinter chain = Hsdw ⇠ J 0 X z Syz Sy+1 y ⇠J 0 XZ dx cos[ p 2⇡ ('+ y '+ y+1 )] y • describes potential energy of interaction between magnon pairs on! neighboring chains, quadratic in magnon pair density npair (J 0 )2 ⇤ 0 2 0 0 n ⇠ J npair ⇠ J npair , hence npair,c ⇠ J /J/J 1 1 pair J1 • Competition • Hence:! - Spin Nematic near saturation, for n - SDW for n > n pair *pair pair < n*pair! T=0 schematic phase diagram of weakly coupled nematic spin chains Spin Nematic SDW 1/2 - O(J’/J) cf: Sato, Hikihara, Momoi 2013 1/2 Fully Polarized BEC physics M Cautionary remark: ! maybe impurity effect Intermediate Summary • Interesting magnetically ordered states: SDW and Spin Nematic - Gapped ΔS=1 excitations (no usual spin waves!) - 2d Nematic very near the full saturation 2d SDW from nematic chains, occupies most of the phase diagram toy problem of “magnon high Tc” Outline • Very brief intro - emergence of composite orders from competing interactions • Nematic vs SDW in LiCuVO4 ✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave” • Volborthite kagome antiferromagnet - experimental status - magnetization plateau - Nematic, SDW and more - Field theory of the Lifshitz point • Conclusions n an g the f latconds to nteration. nical spin peritions nical model constate that ually y. As cy of Volborthite NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1875 a H Cu2 V O5 c b a b c z2 − r2 when onds, ds are oxyween s partions the Cu2 + xis is b Cu2 2, c a Cu1 Cu1 quantum spin liquid?! J " Si • S j impurity ordering at low T? magnetization steps? 0.8 6 8 0.4 CD /T ● 0T ○ 1T mol-Cu–1) K– 2 150 60 40 1 4 3/2 1 6 9/2 150 K 30 magnetization plateau C/T (mJ 4 T (K) 40 1 4 7/2 magnetic order ! C/T (mJ K– 2 mol-Cu–1) 2 50 I 2/a (a × b 2014 T* 0.0 0 60 2012 0.1 0.6 70 Intensity (counts) 2009 M/H (10– 3 cm3 mol-Cu –1) . Thus, Tp corresponds approximately to J / 4.24) Another marked indication from the χ data is the absence of any spin-gap behavior. Although χ would tend to zero as T approaches zero, if a gap is opened, as illustrated in Fig. 1(a) for the theoretical case of Δ = J / 4,8) the χ of volborthite can remain large and finite at ~3 × 10-3 cm3 mol-Cu-1, implying the absence of a gap or the presence of a very small gap. Furthermore, we have extended our χ measurements down to 60 mK, as shown in Fig. 1(b), and observed an almost T-independent behavior with neither an anomaly nor any indication of a downturn. Therefore, the spin gap can be no more than J / 1500, which is muchNATURE smallerCOMMUNICATIONS than theoreti- | DOI: 10.1038/ncomms1875 cally predicted values.8, 9) This strongly suggests that the ground state of volborthite is nearly gapless and probably a spin liquid. 4 1.20 Spin glass transitions are observed even in our clean sample at Tg = 1.1 and 0.32 K at magnetic fields of 0.1 and 1 T, 1.10 respectively (Fig. 1(b)). It has been pointed out, however, 3 based on the previous NMR results, that this spin glass can 1.00 be associated with domains based around impurity spins Ts having local staggered moments and, therefore, is not intrinsic.19, 24) Fortunately, because the impurity-induced spin 320 280 300 2 T (K) glass disappears with increasing field, we can study the intrinsic properties of the kagome lattice at high magnetic fields, above 2 T. 1 Hare = 1 freT Microscopic probes, such as µSR and NMR, quently used to investigate the dynamics of spins. Polycrystal The Single previous µSR study21, 22) revealed a significant increase in crystal relaxation rate λ at low temperatures below 3 0K, towards T ~ 200 300 1 K (Fig. 2(a)), due to the slowing down of the0spin fluctua-100 T (K) tions, which remain dynamic with a correlation time of 20 ns down to 50 mK.27) On the other hand, we observed a sudden broadening of the 51V NMR line below 1 K, as M/H (10– 3 cm3 mol-Cu–1) 2001 Cmag T -1 / mJ K-2 Cu-mol-1 of es its re as on or his ty” m. ds ee As be on temperature was 320 K. Thus determined lattice contribution CD/T is plotted with the broken line in Fig. 3. A magnetic contribution is determined as Cm = C -CD and is also plotted with solid circles in the figure. Integrating Cm/T between 1.8 K and 60 K, we find a value of 4.1 J/mol K which is about 30 % smaller than the total magnetic entropy (Rln2) for S = 1/2. The discrepancy is most likely due to crudeness of the estimation of the lattice contribution particularly at high temperature. To be noted here is that Cm/T seems to show a broad maximum at 20-25 K and then rather steep decrease below 3 K. Alternatively, one can say that there is a second peak or shoulder around 3 K. The first maximum must be ascribed to shortrange AF ordering, because its temperature coincides 2 or mé is 2a is he ly he ue ce he es ed 1). re all in J/ ro. re of of ly. ite S= he um ns. he me ns improvements in crystallinity and particle size. Magnetization at moderately high magnetic fields was measured in a Quantum Design MPMS equipment between 2 and 350 K and in a Faraday-force capacitive magnetometer down to!60 sensitive to a phase transition than magnetic 25) susceptibility especially in the case of quantummK. AFMs. High-field magnetization measurements were carThe specific heat of Volborthite exhibited no using a pulsed magnet up to 55 T at T = 1.4 and 4.2 ried out anomalies down to 1.8 K, which evidences absence of K. Specific heat was measured in a Quantum Design LRO above this temperature (Fig. 3). The anomaly PPMS equipment down to 0.5 K. seen at 9 K is an experimental artifact. It was notmagnetic susceptibility χ shown in Fig. 1(a) in a wide The easy to extract a magnetic contribution Cm from the temperature range exhibits a Curie-Weiss increase on coolmeasured data, because a nonmagnetic isomorph is ing from high temperature, followed by a broad maximum at not available at present from which the lattice Tp ~ 22 K without any anomaly indicating LRO. From contribution could be estimated. We fitted the data fitting to the theoretical model for the S-1/2 KAFM8) above at 50 - 70 K to the simple Debye model assuming 150 K, the average antiferromagnetic interaction26) is deternegligible magnetic contributions at this hight e m p e r a t u r e r a n g e . T h e e s t i m a t e d mined D e b y eto be J = 86 K on the basis of the spin Hamiltonian C/T (J/K mol Cu) ue des he by nt 50 Volborthite’s timeline 100 120 80 40 0 0 0.5 1.0 1.5 2.0 T 2 (K2) T* △ 3T χ Fig. 1. Temperature dependence of the magnetic susceptibility 20 ▼ 5T of volborthite measured using a high-quality polycrystalline 0.2 50 7T a Cm /T sample. (a) χ for the wide temperature range measured□with Polycrystal * Quantum Design MPMS at µ0H = 0.1 T on heating, after cooling at zero field. The solid curve above 150 K represents a fit Single crystal to the theoretical model for0the S-1/2 kagome antiferromag8) 0.0 0 0 10 20 30 40 50 60 net,70 which yields J = 86 K. 0The dotted 1 curve2is obtained 3 0 1 2 3 4 from theoretical calculations on finite clusters for a spin gap of T (K) T/K T (K) Δ = J / 4 to open.8) The inset schematically shows a snapshot of a long-range resonating-valence-bond state on a kagome lattice Fig. 2. (a) Relaxation rates λ (triangles) previous µSR and heat capacity of single crystals of Fig.4 Figurefrom 2 | Magnetic properties 51 made of Cu atoms shown by balls, which consists of various measurements21) at µ0H = 0.01 T andvolborthite. 1/T1 from the present V dependence of magnetic susceptibilities (a) Temperature Fig. 3. Specific heat of Volborthite. The open circlesranges show of spin-singlet pairs, as indicated by broken ovals. (b) χ NMR experiments at µ0H = 1 (circles)measured and 4 (squares) T. The using an assembly of randomly oriented single crystals in a the raw data, while the magnetic contribution estimated measured by the Faraday method with a dilution refrigerator on * -1 field inset shows the temperature evolution of the NMRof spectra ta- 2 and 350 K upon cooling and heating. The magnetic is shown with closed circles. The estimated, Debye-type Fig. 4. Magnetic heat capacity of volborthite at low temperature around T = 1 K in a C T vs T 1 T between mag heating from 60 mK and cooling to 100 mK at µ0H = 0.1 and 1 ken at µ0H = 1 T at frequencies between 8 and 14.5 MHz. sample (b) data for a polycrystalline measured under the same conditions are lattice contribution is shown with the broken line. The time = material quality ** 20 10 0 200 [101] * [110] 250 T (K Figure 3 | First-order structural phase tran dependence of the intensity of superlattice indices − 1 − 4 − 7/2, 1 − 4 − 3/2 and − 1 6 The inset shows a CCD image obtained at marked by * and ** show major superlattice 1/2 and 5 10 1/2, respectively. Further single-crystal structural a 150 and 323 K to investigate the struc phase has a monoclinic structure, wit tice parameters of a = 10.657(3) Å, b = 5 B = 95.035(8)°. Structural refinements crystal structure as that reported by La ity factors R [I > 2S(I)] = 4.13% and wR of the LT phase was determined to b parameters are a = 10.6418(1) Å, b = 5 and B = 95.443(1)° at 150 K, and the [I > 2S(I)] = 2.93% and wR2 = 7.67%. Th are provided in Supplementary Data 1 plementary Data 2 for the LT phase. Th could not be determined in the presen Of primary interest is to understa transition modifies the Cu kagomé around the Cu ions. The structures of HT and LT phases are depicted in Fig result of the structural transition, a mi at the Cu1 site are lost with respect to t causes a large change in the coordina site, including the O3 site splitting int 2014: huge plateau! H. Ishikawa…M.Takigawa…Z.Hiroi, unpublished, 2014 High-field magnetization more different MH curves in a pile of 50 large “thick” arrowhead-shaped crystals 30 days growth 0.5 0.3 ~2/5 Van Vleck M( a pile of thin crystals B ab 0.2 0.1 T = 1.4 K 0.0 0 10 20 30 40 B (T) 50 Huge 1/3 plateau! further optical meas. @ Takeyama lab It survives over 120 T! B / Cu) 0.4 polycrystals a pile of ~50 thick crystals B ab B // ab 60 70 Kagome plateau or ferrimagnetic state? coupled to lattice, but already distorted high-field mag. meas. @ Tokunaga & Kindo labs Phase diagram T 1K ? SDW 1T spin nematic? small plateau’s onset field of 27 Tesla, relative to J ~ 100 K, suggest the presence of ferromagnetic exchange interactions 1/3 plateau 26T B may be a spin nematic?? FIG. 2 (color H. online). (a) 51V NMR Ishikawa et al,spectra measured on a single-domain piece of a crystal in magnetic fields between 15 and 30 T applied perpendicular to the ab plane at T = 0.4 K. (b) unpublished PHYSICAL REVIEW B 82, 104434 I. INTRODUCTION Cu-O !2010" bonds; “2 + 4”" while Cu!2" resides in a plaquette formed by four short bonds !Fig. 1, top". Recently, densityfunctional theory !DFT" studies of CuSb2O6, implying the 2 + 4 local environment of Cu atoms, revealed that orbital ordering !OO" drastically changes the nature of the magnetic coupling from three-dimensional to one-dimensional !1D".18 The search for new magnetic ground states !GSs" is a major subject in solid-state physics. Magnetic monopoles in experiment the spin ice system Dy2Ti2O7 !Refs. 1–3", the metal-insulator 3.5 kagome transition in the spin-Peierls compound TiOCl !Ref. 4" and !Refs.lattice 5 and 1 6" the quantum critical behavior in Li2ZrCuON4=18 N=24 are among thelattice power 1 of 3 recent discoveries that demonstrate =18 lattice 2 PHYSICAL REVIEWtechniques B 82, 104434N !2010" combining precise experimental with modern N=24 latticeex2 theory. However, for a rather large number of problems 1 Coupled frustrated quantum spin- 2 chains with orbital order in volborthite Cu3V2O7(OH)2 · 2H2O periment 2.5and theory do not keep abreast, since it is often O. Janson, * J. Richter, P. Sindzingre, and H. Rosner tricky to find Max-Planck-Institut a real material realization well-studied für Chemische Physik fester Stoffe, for D-01187aDresden, Germany Institut für Theoretische Physik, Universität Magdeburg, D-39016 Magdeburg, Germany theoretical Laboratoire model.de Physique The most example is the Théorique remarkable de la Matière Condensée, Univ. P. & M. Curie, Paris, conFrance !Received 9 August 2010; published 307September 2010" 2 J /|J1| = 1.1 Jic/|Jbond” | = 2—a magnetic GS cept of a “resonating valence 1 We present 2 a microscopic magnetic model for the spin-liquid candidate volborthite Cu V O !OH" · 2H O. formed The byessentials pairsof thisofdensity-functional-theory-based coupled spin-singlets longand model are !i" thelacking orbital orderingthe of Cu!1" 3d , !ii" three relevant couplings J , J , and J , !iii" the ferromagnetic nature of J , and !iv" Cu!2" 3d range magnetic order !LRO". studies revealed a 150 0 governed 50 Subsequent 100 implies magnetism of frustration by the next-nearest-neighbor exchange interaction J . Our model T (K) 8,9 frustrated coupled chains in contrast to the previously proposed anisotropic kagome model. Exact diagonalfascinating variety of disordered GS, commonly called ization 1 studies reveal agreement with experiments. “spin liquids” in order to emphasize their dynamic nature DOI: 10.1103/PhysRevB.82.104434 PACS number!s": 75.10.Jm, 75.25.Dk, 71.20.Ps, 91.60.Pn kagome c and even raised the discussion of their possible b 0.8 I.10INTRODUCTION J /| J | = 1.1 a Cu-O bonds; “2 + 4”" while Cu!2" resides in a plaquette 2 1 applications. formed by four short bonds !Fig. 1, top". Recently, densityThe search for new the magnetic is that a Jground /|Jstates | belief =!GSs" 1.4 functional theory !DFT" studies CuSb O , implying the Following common the spin-liquid GSof may 1 monopoles major subject in solid-state physics. 2 Magnetic in 0.6 2 + 4 local environment of Cu atoms, revealed that orbital O !Refs. the metal-insulator the spin ice system Dy Tithe emerge from interplay low dimensionality, quantum ordering !OO" drastically changes the nature of the magnetic J1–3", /| J | =of1.6 transition in the spin-Peierls compound TiOCl 2 1 !Ref. 4" and coupling from three-dimensional to one-dimensional !1D". fluctuations, frustration, considerable effort has 5 and 6" the quantum critical behavior in Li ZrCuO !Refs. 0.4 and magnetic are among recent discoveries that demonstrate the power of1 been spent the search for combining precise on experimental techniques withspinmodern2 Heisenberg magnets with Jicof/|J1herbertsmithite |=2 theory. However, for a rather large number of problems exkagome geometry. The periment and theory do not keep abreast, since it synthesis is often 0.2 tricky to find a real material11realization for a well-studied Cu the example first is inorganic spin- 21 system with 3Zn!OH" 6Cl 2,remarkable theoretical model. The most the concept of a “resonating valence bond” —a magnetic GS ideal kagome 0 geometry and subsequent studies revealed beformed by pairs of coupled spin-singlets lacking the long0 !LRO". Subsequent 50 studies 150 200 range magnetic order revealed a(T) LRO sides the desired absence of magnetic !Ref. 12" !i" inh fascinating variety of disordered GS, commonly called “spin liquids” in order structural to emphasize theirdisorder dynamic nature trinsic Cu/Zn and !ii" the c presence of anand even raised the discussion of their possible 13 The b aphysics. isotropic interactions complicating the spin applications. Following the common belief that the spin-liquid GS may 14 was G. 4. !Color online" Top:quantum fits to predicted the experimental !!T" !Ref. recently synthesized kapellasite to imply emerge from the interplay of low dimensionality, b fluctuations, and magnetic frustration, considerable effort has kagome physics due to an additional relevant Jic he modified solution of the J -J -J model yields an improved descripbeen spent on the search for spin- Heisenberg 1 2magnetsicwith 15 a kagome geometry. The synthesis of herbertsmithite coupling. 0 J1 Zn!OH" Cl , K the compared first inorganic spin- system with ownCuidealto 50 to the kagome model !bold gray line". Since the and search system kagome geometry subsequent studies a revealed be1 for 2 representing the pure J2 sides the desired absence of magnetic LRO !Ref. 12" !i" inkagome model is far from being completed, it is natural to m: magnetization trinsic Cu/Zn structural disorder andcurves !ii" the presence !N of an- = 36 sites" for different solutions isotropic interactions complicating the spin physics. The consider systems with where the distortion recently synthesized kapellasite was lower predicted tosymmetry imply b J1-J -J model in comparison to the kagome model.FIG. 1. !Color online" Top: Cu!1"O2 dumbbells !yellow/gray", 16 modified J 2 kagome ic physics due to an additional relevant -3 χ (10 emu / mol) Frustrated ferromagnetism 1, 2 3 1,† 1 2 3 3 2 7 2 2 3z2−r2 x2−y 2 ic 1 2 1 2 m/ms 2 2 2 6 7 18 2 4 7 8,9 DFT gets it right! 10 1 2 3 6 2 11 J < 0, J > 0, J > 0 1 2 13 14 FM Ferrimagnetic state PHYSICAL REVIEW B 82, 104434 !2010" 1 Coupled frustrated quantum spin- 2 chains with orbital order in volborthite Cu3V2O7(OH)2 · 2H2O O. Janson,1,* J. Richter,2 P. Sindzingre,3 and H. Rosner1,† 1Max-Planck-Institut für Chemische Physik fester Stoffe, D-01187 Dresden, Germany für Theoretische Physik, Universität Magdeburg, D-39016 Magdeburg, Germany 3Laboratoire de Physique Théorique de la Matière Condensée, Univ. P. & M. Curie, Paris, France !Received 9 August 2010; published 30 September 2010" 2Institut 0 J1 < 0, J2 > 0, J > 0 We present a microscopic magnetic model for the spin-liquid candidate volborthite Cu3V2O7!OH"2 · 2H2O. The essentials of this density-functional-theory-based model are !i" the orbital ordering of Cu!1" 3d3z2−r2 and Cu!2" 3dx2−y2, !ii" three relevant couplings Jic, J1, and J2, !iii" the ferromagnetic nature of J1, and !iv" frustration governed by the next-nearest-neighbor exchange interaction J2. Our model implies magnetism of frustrated coupled chains in contrast to the previously proposed anisotropic kagome model. Exact diagonalization studies reveal agreement with experiments. DOI: 10.1103/PhysRevB.82.104434 PACS number!s": 75.10.Jm, 75.25.Dk, 71.20.Ps, 91.60.Pn I. INTRODUCTION The search for new magnetic ground states !GSs" is a major subject in solid-state physics. Magnetic monopoles in the spin ice system Dy2Ti2O7 !Refs. 1–3", the metal-insulator transition in the spin-Peierls compound TiOCl !Ref. 4" and the quantum critical behavior in Li2ZrCuO4 !Refs. 5 and 6" are among recent discoveries that demonstrate the power of combining precise experimental techniques with modern theory. However, for a rather large number of problems experiment and theory do not keep abreast, since it is often tricky to find a real material realization for a well-studied theoretical model. The most remarkable example is the concept of a “resonating valence bond”7—a magnetic GS formed by pairs of coupled spin-singlets lacking the longrange magnetic order !LRO". Subsequent studies revealed a fascinating variety of disordered GS,8,9 commonly called “spin liquids” in order to emphasize their dynamic nature and even raised the discussion of their possible applications.10 Following the common belief that the spin-liquid GS may emerge from the interplay of low dimensionality, quantum fluctuations, and magnetic frustration, considerable effort has been spent on the search for spin- 21 Heisenberg magnets with kagome geometry. The synthesis of herbertsmithite Cu3Zn!OH"6Cl2,11 the first inorganic spin- 21 system with ideal kagome geometry and subsequent studies revealed besides the desired absence of magnetic LRO !Ref. 12" !i" intrinsic Cu/Zn structural disorder and !ii" the presence of anisotropic interactions complicating the spin physics.13 The recently synthesized kapellasite14 was predicted to imply modified kagome physics due to an additional relevant J1 FM, J2 AF Cu-O bonds; “2 + 4”" while Cu!2" resides in a plaquette formed by four short bonds !Fig. 1, top". Recently, densityfunctional theory !DFT" studies of CuSb2O6, implying the 2 + 4 local environment of Cu atoms, revealed that orbital ordering !OO" drastically changes the nature of the magnetic coupling from three-dimensional to one-dimensional !1D".18 J’ AF c b a b J polarized chains?! tuations the primary spin order is lost, while a spinmultipolar order parameter survives. We demonstrate this mechanism based on the magnetic field phase diagram of a 1 prototypical model, the frustrated S = 2 Heisenberg chain with ferromagnetic nearest-neighbor and antiferromagnetic next nearest-neighbor interactions. Furthermore we show thatferromagnetic this instability provides a natural and unified understandFrustrated chain ing of previously discovered two-dimensional spinmultipolar phases.9,10 J1 FM To be specific, we determine numerically the phase diaof the following Hamiltonian: Jgram 2 AF Spin chain redux H = J1 % Si · Si+1 + J2 % Si · Si+2 − h % Szi , i i !1" i H/(|J1|+J2and ) we set J1 = −1, J2 " 0 in the following. Si are spin-1/2 operators at site i, while h denotes the uniform magnetic z S field. The magnetization is defined as m ª 1 / L% i i . We emFM 1098-0121/2009/80!14"/140402!4" quasi-spin-nematic 0 1/5 'Sz = 2 uted to domin detaile We m / msa presen vector tion w Luttin extend m = 0+ crosso relatio spin-d One a the p = 140402-1 1 J2/(|J1|+J2) 0.8 0.6 0 -4 SDW SDW (p=3) (p=4) 0.2 -3.5 -3 octupolar J1/J2 -2.5 -2 quadrupolar 1 0.8 0.6 0.4 0.2 0 -1.5 SDW (p=2) Vector Chiral Order -1 0.8 0.025 0.02 0 h / J2 Ψ “dominant” VC 0.015 0.01 (a) ɸ “dominant” 0 0.2 0 −4 −3 −2 −1 J / J FIG. 2. !Color online" Squared vector chirality order parameter 1 2 (a) "2 $Eq. !2"% in the low magnetization phase for different values of J2 / 'J1' as a function of m / msat: !a" J2 / 'J1' = 0.27, !b" J2 / 'J1' = 0.32, m/msat 0.4 0.6 0.005 0.2 0.6 0 F VC Q0.1 VC 0.4 IN T (b)SDW3 m/msat 0.2 0.3 0 VC m/msat 0.2 p=2 + 2 (S ) metamagnetic -1 p=3 -0.75 p=4 p=2 144404-4 -0.5 metamagnetic SUDAN, LÜSCHER, AND LÄUCHLI %p,k;&r1, . . . ,r p−1'# = " p n n eikl /psl− %FM#, ) * (" l=1 n=1 1 $9! 1d J1-J2 chain is only quasi-spin-nematic power-law correlations FIG. 4. Schematic picture of antiferronematic quasi-long-range order in the nematic phase. Ellipses represent directors of the nematic order on each bond. ? 2 p=3 -0.4 κ -0.35 J2/J1 h/hsat -0.3 ure of the vector chiral order. The arrows g of the parity symmetry by the vector z $1! n! , which obeys the relation J1"# # tion of the sz spin current, shown by the ng, and there is no net spin current flow. p=4 where -0.275 ) 0.4 κ(2) 1 hexadecupolar of the phase diagram by examining m/m polarized state. To that end, sat we nurgy dispersion of low-energy excitaber of magnons $down spins!. The Hikihara et al, 2008 Sudan et al, 2009 AGNON INSTABILITY bound states is the soft mode. We calculate energy of p-magnon excitations using the method we introduced in Ref. 6. The number of magnons p and the total momentum k are good quantum numbers of Hamiltonian $1!. We thus expand eigenstates in the sector of p magnons with the basis -0.25 Quasi-1d nematic Ψ~ : spin-nematic AND MULTIPOLAR ORDERS IN THE… q x : CHIRAL ɸ ~ Sz eiVECTOR SDW RAPID COMMUNICATIONS PHYSICAL REVIEW B 80, 140402!R" !2009" 0.03 0.03 nnn Bond 0.025 N 0.02 nn Bond 0.4 0.015 0.01 SDW 2 0.005 (c) 0 0.6 tuations the primary spin order is lost, while a spinmultipolar order parameter survives. We demonstrate this mechanism based on the magnetic field phase diagram of a 1 prototypical model, the frustrated S = 2 Heisenberg chain with ferromagnetic nearest-neighbor and antiferromagnetic next nearest-neighbor interactions. Furthermore we show thatferromagnetic this instability provides a natural and unified understandFrustrated chain ing of previously discovered two-dimensional spinmultipolar phases.9,10 J1 FM To be specific, we determine numerically the phase diaof the following Hamiltonian: Jgram 2 AF A QCP parent? H = J1 % Si · Si+1 + J2 % Si · Si+2 − h % Szi , i i !1" i H/(|J1|+J2and ) we set J1 = −1, J2 " 0 in the following. Si are spin-1/2 operators at site i, while h denotes the uniform magnetic z S field. The magnetization is defined as m ª 1 / L% i i . We emFM “Lifshitz” 1098-0121/2009/80!14"/140402!4" quasi-spin-nematic QCP 0 1/5 'Sz = 2 uted to domin detaile We m / msa presen vector tion w Luttin extend m = 0+ crosso relatio spin-d One a the p = 140402-1 1 J2/(|J1|+J2) I. A. NLSM Classical limit Lifshitz Point onsider the Non-Linear sigma Model (NLsM) uld describe the behavior near the Lifshitz he J1 J2 chain. The action in 1+1 dimenZ 2 |@x m̂| + dxd⌧ isAB [m̂] • • +u|@x m̂|4 hm̂z . Can we see this formally somehow? Let us try rescaling p to bring out the behavior for small . We let x ! K/ x and ⌧ ! K2 ⌧ , where the second rescaling follows from the linear derivative nature of the Berry phase term. The magnetization itself does not rescale as m̂ is a unit vector. Carrying out this rescaling, we find r Z K S = dxd⌧ isAB [m̂] |@x m̂|2 + |@x2 m̂|2 v|@x m̂|4 K|@x2 m̂|2 hm̂z , (5) Unusual QCP: order-to-order transition (1) the spin and AB is the Berry phase term dehose spins. It can be written in various ways, e Z 1 AB = du m̂ · @⌧ m̂ ⇥ @u m̂, (2) where we defined v = u/K and h = hK/ 2 . We see that when /K ⌧ 1, the action is large in dimensionless terms, and we expect a saddle point approximation to apply. This is precisely the classical limit! Note that this is valid when u/K is fixed, and also h ⇠ 2 /K, which fixed the overall field scale of the problem. Effective action - NLσM for unit vector m Z 0 2 2 2 4 S = dxd⌧ isA [ m̂] + |@ m̂| + K|@ m̂| + u|@ m̂| B. Saddle introduce a fictitious auxiliary coordinate u B x x x point m̂(u = 0) = ẑ and m̂(u = 1) = m̂ is the alue, or equivalently, hm̂z To find the actual saddle point,symmetry we make an assumpBerry two tunes tion that it is of the form of an umbrella state (I tried m̂ @ m̂ m̂ @ m̂ A = . phase(3) also to look for a planar state, but it seemed to be allowed interactions 1 + m̂ QCP energetically unfavorable). To avoid having to rescale, 4 Let m̂ = we work in the original variables of Eq. (1). important point for us is that Aterm contains a at O(q ) p / |Jqx, 4J21 ' ). Then the action is just the (' cos vative of imaginary time ⌧ . 1 |' sin qx, B 1 ⌧ 2 2 ⌧ 1 3 B ion in Eq. (1) needs a condition for stability ge gradients of m̂. To get it, we note that by tion twice of m̂ · m̂ = 1 we obtain |@x m̂|2 2 integral of the energy density p 1), (6) All properties near Lifshitz point obey “one parameter where we chose to add a constant h factor so that " = 0 = m̂ · @ m̂ |@ m̂|, (4) universality” dependent upon u/Kover ratio when ' = 0. This is easily minimized wavevector 2 x 2 x "= 2 2 4 2 4 q ' + Kq ' + uq ' 4 h( 1 '2 S= Z • Lifshitz Point 2 dxd⌧ isAB [m̂] + |@x m̂| + S= + u|@x m̂| 4 hm̂z Intuition: behavior near the Lifshitz point should be semi-classical, since “close” to FM state which is classical x! r 2 2 K|@x m̂| K Z s K x | | ⌧! K ⌧ 2 dxd⌧ isAB [m̂] + sgn( )|@x m̂|2 + |@x2 m̂|2 + v|@x m̂|4 Large parameter: saddle point! u v= K hm̂z h= hK 2 S= r K Z Saddle point dxd⌧ isAB [m̂] + sgn( )|@x m̂|2 + |@x2 m̂|2 + v|@x m̂|4 hm̂z -1 < v < -1/4 derives from quantum fluctuations Large S >>1: v ~ -3/(2S) < 0 S=1/2: v = -5/8 h order parameter discontinuity p ( 4|v| 1)/|v| ⌧ 1 for v ⇡ 1/4 FM first order spiral hc = 8K p |v|(1 1<v< local instability of FM state (1-magnon condensation) IC cone 0 2 1 4 p |v|) A natural speculation Summary • Spin chains keep showing up in unexpected places ✓ Nematic physics of frustrated ferromagnets ✓ Explored Lifshitz point as a “parent” for multipolar states and metamagnetism S= Z d>1 dxdd • 1 yd⌧ isAB [m̂] + |@x m̂|2 + c|@y m̂|2 + K|@x2 m̂|2 + u|@x m̂|4 Rescaling: x! S= p K d cd 1 d 1/2 Z dxdd s 1 K x | | ⌧! K ⌧ 2 y! p cK hm̂z y yd⌧ isAB [m̂] + sgn( )|@x m̂|2 + |@x2 m̂|2 + |@y m̂|2 + v|@x m̂|4 ∴ Similar theory applies in d>1, and very similar conclusions apply hm̂z