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Frustration-driven multi magnon condensates and their excitations Oleg Starykh, University of Utah, USA Current trends in frustrated magnetism, ICTP and Jawaharlal Nehru University, New Delhi, India, Feb 9-13, 2015 Collaborators Leon Balents, KITP, UCSB Andrey Chubukov, Univ of Minnesota not today but! closely related findings: spin-current state at the tip of 1/3 magnetization plateau, spontaneous generation of orbiting spin currents (ask me for details after the talk :)) Outline • Frustrated magnetism (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 in Sec. III. Inout, particular, the IsingIsing the BKT consistent with universality. Wetransitions also discuss the quantum phase diagram finiteof S. clearly indicated by lattice. aitsharp peak ofofthe specific heat pointed however, in Ref. 25and that these two continuous is divisible by theforsize the helix pitchsquare and liesIninthe the limit spinhave S !repeated 1, there is which a zero temperature Lifshit We a similar analysis for several(T) values of also 1 are considered in merge Secs. III A and III B, respectively. The transitions can into a single first-order one. These conJ =J and the complete phase diagram is shown in Fig. 5,0 is illustrated inlong-range Fig. 3 spiral [10].spinThis feature to be ordersharp for J3 > is range from 20 to numbers: 120. We apply the periodic (toric) boundary 3 at 1T ! 0 DOI: 10.1103/PhysRevLett.93.257206 PACS 75.10.Hk, 75.10.Jm 4 J1 . We present classical Monte h Z2 where we have plotted T versus J =J . We find that T neighborhood ofnot thedepend Lifshitz and theform phase diagram are c spin rotation 3 1 clusions do on point the particular of interactions in theory forthe T> 0 crossoversdisplayed near the Lifshitz point: symmetry isc r contrasted the broad maximum by the same conditions as well as the cylindrical onesto(i.e., with periodic f the vanishes linearly for J =J ! 1=4; a theory 1 behav2 3 1 there is a1 broken lattice reflection symmetry for 0 " T < Tc # $J3for%this the system soon the ground-state degeneracy remains the discussed in Sec.asIII C. as Section IV contains our conclusions. 0 4 J1 &S . Th condition along the b axis and the one along the a ior will now be presented. nded quantity for free J3 =J < , i.e., when the classical ground state 1 4 consistent with Ising universality. We also discuss the quantum phase diagram same. They are confirmed by numerical of the models also suggest that discrete lattice symmetries may play a Near the classical Lifshitz point, we can model quantum Frustrated antiferromagnets havestudies recently attracted axis). We have found thatdisplays both conditions to order. the same 11,13–17,23,24 ordinarylead Néel In particular, the maximum and thermal fluctuations by a continuum unit vector field0 mentioned above. role near other quantum critical points with spiral order [6]. much interest in connection with the possibility of stabilizDOI: 10.1103/PhysRevLett.93.257206 PACS n ansivalues of transition temperatures and indexes. In contrast, n"r; %#, where r ! "x; y# is spatial coordinate, % is imagiof the specific heat is consistent with a logarithmic depenNevertheless, the situation remains contradictory in 2D II. MODEL AND METHODS PHYSICAL REVIEW B 85, 174404 (2012) Broken discrete symmetries have also been discussed ing unconventional low-temperature (T) phases, with novel 2 occur nary time, and n ! 1 at all r, %. This field is proportional values of Binder’s cumulants andonthe chiral-order-parameter helimagnets belonging to the same Z ⊗ SO(2) class as dence system size (see the inset of Fig. corresponding week 2 x &y [7,8] in the context of the J types of ‘‘quantum order’’ [1]. A very promising candidate % J model, with firstand P H Y S I C A L R E V I E W L E T ENéel R 3) S order 1 (2004) 2 "%1#2004 n"rj ; %#. toTthe parameter with S^ j /ending sition We consider the model (1) of the classical XY magnet on a 17 DECEMBER PRL 93, 257206 distribution at J ≈ 0.309 depend on boundary conditions as 2 the FFXY model and the antiferromagnet on the triangular 4,5 for a spin-liquid phase is the J second-neighbor couplings on the square lattice. However, % J model Chiral spin liquid in two-dimensional XY helimagnets to a critical exponent " ! 0, in agreement with Ising Spiral order will therefore appear as sinusoidal 1 3 er” square lattice. We set J = J 28 also suggest thatdependiscr Frustrated antiferromagnets recently attracted = (see 1 foralso simplicity, the value wemodel discuss below in detail. Standard Metropolis algorithm33havedence X 29) and 1 bX lattice. Garel and Doniach Ref. considered the of n on r. The action for n is the conventional this has only collinear, commensurate spin correlauniversality. gonal ^ ' S^ j ( J3 ^1,2,† role near much interest in connectionwithin with the possibility of sigma stabilizS^ The (1) H^ 1,* ! Ja1square O(3) nonlinear model, expanded to other includequantum quartic i 'extra j ; competing Low-Temperature Broken-Symmetry Phases of Spiral Antiferromagnets A. O. Sorokin A.SV.i J Syromyatnikov of the simplest extra exchange interaction is a variable. Lifshitz has and been The the thermalization was maintained helimagnet onand withSan tions, this used. makes both classical and quantum theory 2lattice eting Broken discrete ing unconventional low-temperature (T) phases, with novel This critical behavior can be directly related to the hi;ji Institute, Gatchina, hhi;jii St. Petersburg 188300, Russia gradient terms ( h ! ! k ! lattice spacing ! 1): S n sym ! tersburg Nuclear Physics Institute, NRC Kurchatov 5 B different fromCarlo that considered here. As will become exchange coupling is described by the quite point corresponds to J2along = 1/4one in axis this that notation. The system 4 × 10 Monte steps in each simulation. Averages have 1,2 R1=T 2,3 R 2 FIG. 4. Bot 2 elong Luca Capriotti and Subir Sachdev [7,8] in the context of types of ‘‘quantum order’’ [1]. A very promising candidate Department of Physics,^St. Petersburg State University, 198504 St. Petersburg, Russia d rLn with reflection symmetry by studying an appro0 d% 6 lattice clear below, the spiral order3.6 andbroken associated Lifshitz point 1within Hamiltonian where S are spin-S operators on a square lattice and has a collinear antiferromagnetic ground state at J < 1/4. To been calculated × 10 steps for ordinary points and i 2 Valuation Risk Group, Credit Suisse First Boston (Europe) Ltd., One Cabot Square, London E14 4QJ, United Kingdom Eq. (2)] for Received 7 November 2011; revised manuscript received 27 January 2012; published 3 May 2012) model fortheory a spin-liquid phase is the J1 % J3 model second-neighbor coupld 2structurepriate play a central role in the of our and in the T Ising nematic order parameter. From the symmetries 6 ! Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA J ; J ) 0 are the nearestand third-neighbor antiferro1 3 transition from the (quasi-)antiferromagnetic discuss the phase 6 × 10 for points close to the critical ones. We have used alsoNewXHaven, Connecticut 06520-8120, this X Box shows the dat The 3 model has only co carry out Monte Carlomagnetic simulations to discuss critical properties of a classical two-dimensional XY frustrated Department of Physics, Yale University, P.O. 208120, USA dependence of observables. H = (J cos(ϕ − ϕ ) + J cos(ϕ − ϕ ) ^ ^ ^ ^ ^ 1 x along x+a 2 coordinate x x+2a couplings the two axes. For S S ' S ( J ' S ; (1) H ! J of Fig. 2, we deduce that the order parameter is # ! 1 i j 3 i j = 0 and 0.1 (see phase to the paramagnetic one, we consider J (Received 23 September 2004; published 14 December 2004) tions, and this makes bo Ising expone spingnet on a square lattice. We find two successive phase transitions upon2the temperature decreasing: the first P x this model, early large N computations, [2] and recent hi;ji hhi;jii 1=M" # with and the second one is of of a discrete Z2 symmetry from thd We study Heisenberg with nearest- (J1 ) and third- (J3 ) neighbor exchange onquite the different Fig. with 1). breaking The ground has ϕa helical ordering attheJ2Berezinskii-Kosterlitz> 1/4. nassociated 3D a #aantiferromagnets temperature large scale density matrix renormalization Jb state cos(ϕ )), a narrow (1) diagram x− x+b with below, the spiral square lattice. In the limit^ of spin S ! 1, there is a zero temperature (T) Lifshitz point at J3 ! 14 J1 , clear ss (BKT) type at which the − SO(2) symmetry breaks. Thus, region exists on group the phase where Si atare operators on classical a square 0:5. The inse 1.2lattice and 1 (DMRG) for S ! 1=2 have the present Monte Carlo simulations and a long-range spiral spin order T !spin-S 0 for J3 > n lines of the Ising and the BKTcalculations transitions that corresponds to a [3] chiral spin suggested liquid. 4 J1 . We ^ ^ ^ ^ play a central role intotht # ! " S ( S & S ( S # ; (2) ent.11 J ; J ) 0 are the nearestand third-neighbor antiferrotheory for T > 0 crossovers near the Lifshitz point: spin rotation symmetry is restored at any T > 0, but a 1 3 2 4 a whereexistence the sum of runs over sites x = (xastate ,xb ) with of the lattice, a = sponding a gapped spin-liquid exponentially 1 3 1 2 T # $J % J &S . The transition at T ! T is there is a broken lattice reflection symmetry for 0 " T < T dependence of observa 1 axes. For c 3 c 0.1103/PhysRevB.85.174404 PACS number(s): 64.60.De, ature 4 1 magnetic couplings along the two coordinate spin correlations and no translation sym- 75.30.Kz (1,0) decaying and b = (0,1) are unit vectors of broken the lattice, the coupling consistent with Ising universality. We also discuss the quantum phase diagram for finite S. first large N computations, [2] and recent metryJ1,2 in the of strong frustration 0:5).they are regime positive. Usingising arguments(Jof3 =J Ref. constants 1 ’ 30, ξspin~ S / T 1/2 this model, early T DOI: 10.1103/PhysRevLett.93.257206 PACS0.8numbers:group 75.10.Hk, 75.10.Jm 28 c large scale density matrix renormalization s I.the This Letter properties of the above model 257206-2 16 concluded that atwill lowdescribe temperatures vertices bound INTRODUCTION model, thethe Coulomb gasare system of half-integer charges,17 0.3 18 S. Our 13,19,20 (DMRG) calculations for S ! 1=2 [3] have suggested the for large S and discuss consequences for general n by two the coupled models, and Ising-XY and the by strings, which would BKTXY transition make model, 0.25 0.6 ets have attracted much attention in recentinhibit cS2 /of T a gapped spin-liquid state with exponentially 21 c'S2 / T results, obtained by classical Monte Carlo simulations and existence chiral generalized fully frustrated XY model. And surely, the most 0.2 ξ ~ e ξspin~ e Frustrated antiferromagnets thewhich Ising have transition occurinfirst with the temperature increasing. spin Ising nematic order liquid phases, been found also suggest that discrete lattice symmetriesTmay play a have recently attracted famous of them is the fully frustrated XY model (FFXY ) a theory described below, are summarized in Fig. 1 for the decaying spin correlations nonear broken translation sym- 0.15 31 hofthe special Kolezhuk interest.1 A noticed chiral spin-liquid role other quantum much interest in connection with the possibility of stabiliz- and that those arguments are not valid for a 0.4 critical points with spiral order [6]. 22 introduced by Villain. This model is of great interest because 0.1 limit S ! 1. There is a T ! 0 state with long-range spiral metry in the regime of strong frustration (J =J ’ 0:5). Broken discussed ing unconventional (T)Jphases, with novel etems of such an helimagnet, exotic state of matter in whichthat the KT P H Ydiscrete S I C3Asymmetries L1 R E V I Ehave W also L E Tbeen TER S and showed Ising transition temperature J1 /low-temperature 4 1 it describes a superconducting array of Josephson junctions 3 PRL 93, 257206 (2004) 0.05 spin order for J > J . We establish that at 0 < T < T # [7,8] in the context of the J types of ‘‘quantum order’’ [1]. A very promising candidate % J model, with first- and This Letter will describe properties of the above model uasi-long-range nor long-range magnetic 3 1 c 1 2 Neel LRO 4 one at least near the Lifshitz point y (see 0.2 23 Spiral LRO is larger than the BKT an external transverse magnetic field. It was foundfor a Lifshitz point 0 spin-liquid phase is the second-neighbor couplings S. on the However, % J3 model 12 ×14SJj1⟩&Sis2 ,nonzero. above this under state there is a phase with broken $J⟨S 0.25 0.3 0.35 0.4 forJ1large S and discuss3 consequences for3 general Oursquare lattice. 3 i% Jl 1order -J2 parameter that the temperature of the Ising transition T is 1%–3% larger 4 4 J = J /4. It was found by Monte Carlo simulations in the X X I this model has only collinear, commensurate spin correlaat T=0 only 2 1 in context of onea phase is discussed c'S2 / T discrete symmetry of lattice reflections about the x andfory most of above-named Monte Carlo simulations and1 and1.510quantum S^ i ' S^ j ; by classical ' S^ j ( J3 obtained (1) H^ ! J1 S^ i results, 0 ξ ~ e than that of the BKT transition 2 tions, and this makes both the classical theory ^ 0 0.5 2 2.5 3 spin FIG. 1. Phase diagram of H in the limit S ! 1. The shaded ted quantum magnetic and itrotation is 11,13,23,24is preserved. This hi;ji axes,systems, while spin invariance areflections theoryhhi;jii described below, are quite summarized in Fig. 1 for the here.J3As systems. /J 1 will become different from that considered region has a broken symmetry of lattice about the x ly in174404-1 Ref. 3. 8 ©2012 American Physical Society 25 phase has ‘‘Ising nematic’’ order. We present strong nuKorshunov argued that a phase transition, un-where below, spiral order spiral and associated Lifshitz point limit Son!atransition 1. There T ! 0clear state withthelong-range S^ i are spin-Sorder. operators square lattice and ydriven axes, by leading to Ising nematic The Ising is1is aand sions, one of the systems in which the FIG. 5. in Critical temperature a 6function theT frustration play a central the structure of ourastheory and inofthe of kink-antikink pairs on associatedJ1 ; J3 ) 0 are the1 nearestmerical evidence that transition at Tc is indeed inthe thedomain andorder third-neighbor for J31>antiferrothatrole hase can be found at finite temperature is a thebinding The spin correlation at walls the temperature Tc # $J3 % 4 J1 &S2 . spin Neel LRO 1 ratio 2Jat=J0. < T < Tc # 4 J12. We establish dependence of observables. 3 1 with the Z symmetry, can take place in models similar to 2D Lif 1 magnetic couplings along the two coordinate axes. For 2 2 class. Such Ising nematic order[4] was Y ) helimagnet withIsing Z2 ⊗ universality SO(2) symmetry %dependencies above is a phase with broken 4 length, !Tspin , is(see finite for all T > 0, with $J the3 T as this state there 4 J1 &S , [2] (Q,Q) (Q,−Q) FFXY one at temperatures appreciably smaller than this model, early large N computations, and recent BKT originally proposed in Ref. [2] for S ! 1=2 in a T ! 0 cal structure results from a competition shown, with c=2oflarge ! c0 !scale 8"jJ3density % 14 J1 j; matrix the crossovers betweenofgroup discrete symmetry lattice reflections about the x and y also Ref. 26). Such a transition could lead to a decoupling renormalization 2 257 FIG. 1. Phase diagram ctions between localized spins. Critical spin-liquid phase described by a Z2 gauge theory [5]. Thus the differentinbehaviors of calculations !spin are atfor theaxes, dashed attwo Tdifferent # FIG. 2.lines Thespin minimum energy configurations with This while rotation invariance is preserved. (DMRG) S! 1=2 [3] have suggested the phase coherence across domain boundaries, producing this ′ stems from described two diagram region has a broken sym 0 1 27 23. (Color online) Distribution the issame Ising by nematic order canof appear whentransitions spiral spin FIG.this 1.class (Color online) Phase the model (1) that is TBKT of the in We "Q; &Q# strong with magnetic wave vectors ! "Q;order. Q# and Q~ ? !present existence a gapped spin-liquid state with jJ3 % Spin rotationofsymmetry is phase broken only atvalue Texponentially ! 0EaQ~defined 0 0.1 0.2 0.3 0.4 0.5 way two separate bulk with < FIG. has ‘‘Ising nematic’’ nuI .1 jS It. was 4TJ T Aside from the conventional magnetization and y axes, leadingT/Jto Isin Q !broken 2!=3, corresponding to J3 =J1 ! 0:5. order is destroyed either pointed by thermal fluctuations (as 25 in the decaying spin correlations and no translation sym= 0.5, T = 0.67 < T , and different L. found in this paper. Eq. (10) for J out, however, in Ref. that these two continuous where ! ! 1. There is no Lifshitz point at finite S because it 1 2 I spin merical evidence that the transition at Tc is indeed in the1/2 at the temperature T # etry, one has to present take intoLetter; account in the regime of strong frustration (J3 =J c seealso Fig. 1)transitions or by quantum fluctuations (asfirst-order in 1 ’ 0:5). can merge into a single These conξspin~ Swas /T is one. preempted [13]metry by quantum effects within the dotted semiIsing universality class. Such Ising states nematic order[4] rameter that is an Ising variable with Z2 Tc ,ofisthe FIG. 3. T dependence specific This Letter will describe properties of the above model length, ! finite for We begin by recalling [9] the ground of H at S ! spin Ref. [2]). Our large S results aredo therefore consonant with form clusions not depend on the particular of interactions circle: here thereinis a T ! 0 spin gap ! # S exp$%~ cS&proposed and spin in Ref. [2] for S ! 1=2 in a T ! rameter characterizes the direction of the Different symbols refer to different0 clu originally 0 for large S and discuss consequences foris general S. OurNéel order with magnetic wave 174404-2 1. There conventional shown, with c=2 ! c ! the system as soon as the ground-state degeneracy remains the the possibility of a spin-liquid phase at S ! 1=2 as derotation symmetryresults, is preserved. semicircular region extends between L ! 24 stinguishes left-handed and right-handed cS and / T L ! 120. Data for J c'S /1T 1Z gauge obtainedThis by classical Monte Carlo simulations and ~phase spin-liquid described by a theory [5]. Thus X ξ ~ e ξ ~ e vector Q ! "!; !# for J =J $ . For J =J > , the 1 2 spin same. They are confirmed by numerical studies of the models 3 1 3 1 spin the different behaviors o 4 4 scribed in Refs. [2,3]; we will discuss the11,13–17,23,24 quantum finite S jS # !. Furtherbelow, detailsare onsummarized the physicsinwithin over T # jJ3 % 4 Ja1theory for comparison (full dots and dashed line) described Fig. 1 forincommensurate the ~ ~ groundIsing state has planar antiferromagnetic 1 specific the same nematic order can appear when spiral spin 2 . Spin mentioned above. the maximum of the heat. = S ⇥ S jJ % J jS rotati phase the diagram further towards this region appearlimit at the the isLetter. i the end j of the Letter. We onal (3D) helimagnets, phase transi3 S !end 1. of There a T ! 0 state with long-range spiral 4 1 J1 /the 4 order at a wave vector Q~ ! "Q; Q#, with Q decreasing from Nevertheless, the situation remains contradictory in 2D order is destroyed fluctuations (as in 3 ! 1. There i where !Jspin tic and the chiral order parameters occur spin order for J3 > 14 J1 . We establish that at 0 1< Teither < Tc #by thermalNeel LRO ! as J =J > and approaching Q ! !=2 monotonically Spiral LRO triangle helimagnets belonging to the same Z ⊗ SO(2) class as 3 1 Lifshitz point 2 1 2 4with present Fig. 1) or by quantum fluctuations (as in was found numerically that the transition there Letter; is a phasesee broken $J3 % 4 J1 &S , above this state where a labels each plaquette of qu th is preempted [13] by 1. The spiral order is incommensurate for 14 < for J3 =J1 ! the FFXY model and the antiferromagnet on the triangular © 2004 The American 4,5 0031-9007=04=93(25)=257206(4)$22.50 257206-1 Physical Society Ref. [2]). Our large S results are therefore consonant with "1; 2; 3; 4# are its corners. The variabl order or of the “almost-second-order” discrete symmetry of lattice reflections about the x and y circle: here there is a T ! where Q !diagram 2!=3, corJ3 =J1 < 1, except at J3 =J1 ! 0:5 lattice. Garel and Doniach28 (see also Ref. 29) considered theaxes, while spin rotation invariance 1. Phase of H^ in the limit S ! 1. The shaded Néel antiferromagnet, while they is preserved. This % FIG.phase erromagnets on a body-centered tetragonal the possibility of a spin-liquid at S ! 1=2 as derotation symmetry is assu pres hasspins a broken about the x responding to an angle of 120 region between (seesymmetry Fig. 2). of lattice reflections simplest helimagnet on a square lattice with an extra competing the two degenerate ground states i phase has ‘‘Ising nematic’’ order. We present strong numple cubic lattice with an extra competing 1 and y axes, leading to there Ising nematic order. IsingTtransition is 4 J1 jS # ! scribed in Refs.for [2,3]; we will discuss the quantum finite S Theover # jJ3 % ~! Interestingly, each spiral state with Q "Q; Q# is exchange coupling along one axis that is described by the 7 Consequently, a phase with Ising nem merical evidence that the transition at Tc is indeed in the 2 along one axis. These systems belong $J % 1 JWe spin correlation at the Tc # 1 &S . The phase diagram further towards the temperature end Letter. Hamiltonian a distinct but order[4] equivalent configuration at Q~ ?of!the "&Q; Q#3 4 by a h#a i this ! 0.region appear at the Ising universality class. Such Ising nematic was do)universality class as, e.g., the model 1 3 σ Emergent Ising order parameters j j Tc T T C C max 15 10 5 1 Ising order: spin chirality ular lattice8 and V2,2 Stiefel model.9 The ence and stabilization of the chiral spin., Dzyaloshinsky-Moria interaction in 3D ussed recently in Ref. 10. ns (2D), the situation is rather different.11 2 ! H = (J1 cos(ϕx − ϕx+a ) + J2 cos(ϕx − ϕx+2a ) x − Jb cos(ϕx − ϕx+b )), length, ! 2 , is finite for all T > 0, with the T dependencies as spin be obtained from the one numerical results contain str originally proposed in Ref. [2](for forQS!!!). 1=2This in astate T !cannot 0 shown, with c=2 ! c0 ! 8"jJ3 % 14 JOur 1 j; the crossovers between ~ continuous Isinglines phase betw wave vector a globalthe spin rotation. Instead,ofthe spin-liquid phase described by awith Z2 gauge theory Q [5].byThus different behaviors !spin are at the dashed at Ttransition # 0031-9007=04=93(25)=257206(4)$22.50 257206-1 © 20 with h#i ! 0, and a homogeneous 1 twoappear configurations arespin connected a 2 global rotation the same Ising nematic order can when spiral jJ3 % by 4 J1 jS . Spin rotation symmetry is broken only at T ! 0 (1)order is destroyed either by thermal h#i ! 0. The divergence in the spec combined with a (as reflection or1.y There axes. isThe fluctuations in the about wherethe !spinx ! no Lifshitz point at finite S because it accompanied a divergence in the symmetry of the classical ground state ' effects present Letter; see Fig. 1) or by global quantum fluctuations (as in is preempted [13] isbyO"3# quantum within thebydotted semi- where the sum runs over sites x = (xa ,xb ) of the lattice, a = Ising nematic in collinear spin system ~1 · N ~ 2 = ±1 =N Outline • Frustrated magnetism (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: NMR lineshape - collinear SDW along B Hagiwara, Svistov et al, 2011 Buttgen et al 2012 LiCuVO4 No spin-flip scattering above ~ 9 Tesla: longitudinal SDW state SF = spin flip, ΔS = 1" NSF = no spin flip, ΔS = 0 o Geometry (motivated by LiCuVO4) • • • 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 Excitations (via spin-spin correlation functions) • 2d SDW z hS (r)i = M + Re ⇣ e iksdw ·r ⌘ • preserves U(1) [with respect to magnetic field] -> hence NO transverse spin waves • breaks translational symmetry -> longitudinal phason mode at ksdw = π(1-2Μ) and k=0 (solitons (kinks) of massive sine-Gordon model! which describes 2d ordered state) phason OS, Balents PRB 2014 Excitations (via spin-spin correlation functions) ! • + + 0 2d Spin Nematic hS (r)S (r )i ⇠ 6= 0 • breaks U(1) but ΔS=1 excitations are gapped + hS (r)i = 0 (magnon superconductor) • gapless density fluctuations at k=0 - ( sector: solitons of ! massive sine-Gordon ! model describing! 1d zig-zag chain.)! Energy scale J1 (+ sector: solitons of ! massive sine-Gordon ! model which describes ! 2d ordered state.)! Energy scale (J 0 )2 /J1 OS, Balents PRB 2014 Intermediate Summary • Interesting magnetically ordered states: SDW and Spin Nematic - Gapped ΔS=1 excitations (no usual spin waves!) - SDW naturally sensitive to structural disorder - analogy with superconductor/charge density wave competition Linearly-dispersing phason mode with ΔS=0 in 2d SDW Linearly-dispersing magnon density waves in 2d Spin Nematic Outline • Frustrated magnetism (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 1/3 plateau ? 26T B our interpretation 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?! Phase diagram 1K ? SDW 1T spin nematic? T 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 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) m/msat 0.8 0.6 0.4 0.2 0 -4 SDW -3.5 p=4 p=3 octupolar 144404-4 -3 -0.4 ? SDW (p=3) J1/J2 -2.5 -0.5 (p=4) -2 -0.75 -1.5 -1 p=2 quadrupolar 0.025 Ψ “dominant” 0.02 SDW (p=2) 1 0.8 0.6 0.4 0.2 0 0 0 Vector Chiral Order -1 FIG. 2. !Color online" Squared vector chirality order parameter " $Eq. !2"% in the low magnetization phase for different values of 2 VC 0.015 0.01 0.2 m/msat 0.4 VC ɸ “dominant” (a) (b) 0.6 0 0.1 m/msat 0.2 0.3 0 VC 0.2 metamagnetic + 2 (S ) m/msat p=2 -0.35 p=3 -0.3 p=4 SUDAN, LÜSCHER, AND LÄUCHLI Ψ~ : spin-nematic ɸ ~ Sz ei q x : SDW J2/J1 metamagnetic %p,k;&r1, . . . ,r p−1'# = " p n n eikl /psl− %FM#, ) * (" l=1 n=1 1 1d J1-J2 chain is only quasi-spin-nematic power-law correlations $9! FIG. 4. Schematic picture of antiferronematic quasi-long-range order in the nematic phase. Ellipses represent directors of the nematic order on each bond. -0.275 2 1 h/hsat where -0.25 κ ure of the vector chiral order. The arrows g of the parity symmetry by the vector z $1! n! , which obeys the relation J1"# # hexadecupolar tion of the sz spin current, shown by the ng, and there is no net spin current flow. ) κ(2) of the phase diagram by examining polarized state. To that end, 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 Quasi-1d nematic RAPID COMMUNICATIONS PHYSICAL REVIEW B 80, 140402!R" !2009" 0.03 0.03 nnn Bond 0.025 0.02 nn Bond 0.005 0.4 0.015 0.01 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 'Sz = 2 uted to domin detaile We m / msa presen vector tion w Luttin extend + m = 0 H = J1 % Si · Si+1 + J2 % Si · Si+2 − h % Szi , !1" crosso VECTOR CHIRAL AND MULTIPOLAR PHYSICAL REVIEW B 78, 144 i i ORDERS IN THE… i relatio 0.8 and we set J = −1, J " 0 in the following. spin-1/2 bosons whichSiareare actually two-magnon spin-d bound s 1 2 total momentum k = ". The boson creationOne operato h/J N site i, while h denotes the uniform F a1 operators at magnetic − − † sponds to s s and the boson density n = b b # l 0.6 l l+1 l l 2 z thea fini p= em- costs field. The magnetization is defined as m ª 1a /two-magnon L%iSi . We breaking bound-state Hikihara et al, 2008 Multipolar phases 2 IN 0.4 SDW2 1098-0121/2009/80!14"/140402!4" T Q (a) 0 −4 Is it an infinite progression? SDW3 0.2 VC −3 energy, the transverse-spin correlation #s+0 s−l $ is sh where s+0 = sx0 + is0y . Being a TL liquid, the ground s its power-law decaying correlations of the si 140402-1 − $, and the density fl propagator, #b0b†l $ # #s+0 s+1 s−l sl+1 z z z z #n0nl$ − #n0$#nl$ # #s0sl $ − #s0$#sl $. When the boson decays slower than the density-density correlatio propriate to call this phase the !spin" nematic ph opposite case when the latter incommensurate den lation is dominant, we call this phase the spin-de −2 J1 / J2 −1 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 . 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 Z 0 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 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, Berry m̂ @ m̂ m̂ @ m̂ A = . phase(3) 1 + m̂ important point for us is that Aterm contains a vative of imaginary time ⌧ . 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 hm̂z To find the actual saddle point,symmetry we make an assumptwo tunes tion that it is of the form of an umbrella state (I tried also to look for a planar state, but it seemed to be allowed interactions QCP energetically unfavorable). To avoid having to rescale, 4 Let m̂ = we work in the original variables of Eq. (1). p (' cos qx, ' sin qx, 1 '2 ). Then the action is just the integral of the energy density p 2 2 4 2 4 4 "= q ' + Kq ' + uq ' h( 1 '2 1), (6) at O(q ) 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 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 v derives from quantum fluctuations By a spin wave analysis, one finds v ~ -3/(2S) < 0 h first order hc = 8K p |v|(1 1<v< local instability of FM state (1-magnon condensation) FM IC cone 0 2 spiral 1 4 p |v|) Phase diagram VECTORchain CHIRAL AND MULTIPOLAR ORDERS Frustrated ferromagnetic First order metamagnetic transition near Lifshitz point 0.8 h / J2 N F 0.6 IN 0.4 Higher dimensions? SDW2 T SDW3 0.2 Q Hikihara et al, 2008 (a) 0 −4 VC −3 −2 J1 / J2 −1 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 Phase diagram ECTOR CHIRAL AND MULTIPOLAR ORDERS IN THE… 0.8 h / J2 N F 0.6 IN 0.4 SDW2 T SDW3 0.2 Q (a) 0 −4 VC −3 −2 J1 / J2 −1 bosons w multipolar phases total mom from QCP? sponds to breaking energy, th + where s0 its powe propagato #n0nl$ − #n decays sl propriate opposite lation is d Origin of multipolar phases VECTOR CHIRAL AND MULTIPOLAR ORDERS IN THE… 0.8 h / J2 N F 0.6 IN 0.4 SDW2 T SDW3 0.2 Q (a) 0 −4 VC −3 −2 J1 / J2 −1 M 0.4 0.2 (b) PHYSICAL REVIEW B 78, 144404 !2008" bosons which are actually two-magnon bound states with total momentum k = ". The boson creation operator b†l corre− sponds to s−l sl+1 and the boson density nl = b†l bl # 21 − szl . Since breaking a two-magnon bound-state costs a finite binding energy, the transverse-spin correlation #s+0 s−l $ is short ranged, where s+0 = sx0 + is0y . Being a TL liquid, the ground state exhibits power-law decaying correlations of the single-boson − $, and the density fluctuations, propagator, #b0b†l $ # #s+0 s+1 s−l sl+1 #n0nl$ − #n0$#nl$ # #sz0szl $ − #sz0$#szl $. When the boson propagator decays slower than the density-density correlation, it is appropriate to call this phase the !spin" nematic phase. In the opposite case when the latter incommensurate density correlation is dominant, we call this phase the spin-density-wave !SDW2" phase. The SDW2 phase is extended to the antiferromagnetic side J1 $ 0 across the decoupled-chain limit J1 = 0; it is called even-odd phase in Ref. 25. The boundary between the SDW2 phase and the nematic phase is shown by a dotted line in Fig. 1. In the semiclassical picture we can write s−l = e−i%l, where %l is the angle of the two-dimensional vector !sxl , sly" measured from the positive x direction, 0 & %l ' 2". The product − = e−i!%l+%l+1" can be represented by the vector Nl+1/2 s−l sl+1 = !cos (l,2 , sin (l,2" with (l,2 = −!%l + %l+1" / 2. We now realize that we need to identify Nl+1/2 with −Nl+1/2 because of the physical identification !%l , %l+1" % !%l + 2" , %l+1" % !%l , %l+1 + 2"". We can thus consider Nl+1/2 as a director representing the nematic order. We will show in Sec. VI that the nematic phase has antiferronematic quasi-long-range order of the director, as shown schematically in Fig. 4. The ground state is not dimerized in this phase as opposed to the initial proposal of Chubukov.1 Incommensurate nematic phase. The incommensurate First order transition: partially polarized state coexists with plateau one With enough quantum fluctuations, “bubbles” of partially polarized phase may become many-magnon bound states and form multipolar phases 0 −4 :N : SDW2 : IN :T : SDW3 :Q : VC −3 −2 J1 / J2 −1 FIG. 1. !Color online" Magnetic phase diagram of the spin-1/2 zigzag chain with ferromagnetic J1 and antiferromagnetic J2 !a" in the J1 / J2 versus h / J2 plane and !b" in the J1 / J2 versus M plane. Crosses show the transition and crossover points obtained from the magnetization curves and correlation functions. In !a", symbols VC, N, IN, T, Q, and F indicate the vector chiral !*Sztot = 1", nematic !*Sztot = 2", incommensurate nematic !*Sztot = 2", triatic !*Sztot = 3", Origin of multipolar phases VECTOR CHIRAL AND MULTIPOLAR ORDERS IN THE… 0.8 h / J2 N F 0.6 IN 0.4 SDW2 T SDW3 0.2 Q (a) 0 −4 VC −3 −2 J1 / J2 −1 M 0.4 0.2 (b) PHYSICAL REVIEW B 78, 144404 !2008" bosons which are actually two-magnon bound states with total momentum k = ". The boson creation operator b†l corre− sponds to s−l sl+1 and the boson density nl = b†l bl # 21 − szl . Since breaking a two-magnon bound-state costs a finite binding energy, the transverse-spin correlation #s+0 s−l $ is short ranged, where s+0 = sx0 + is0y . Being a TL liquid, the ground state exhibits power-law decaying correlations of the single-boson − $, and the density fluctuations, propagator, #b0b†l $ # #s+0 s+1 s−l sl+1 #n0nl$ − #n0$#nl$ # #sz0szl $ − #sz0$#szl $. When the boson propagator decays slower than the density-density correlation, it is appropriate to call this phase the !spin" nematic phase. In the opposite case when the latter incommensurate density correlation is dominant, we call this phase the spin-density-wave !SDW2" phase. The SDW2 phase is extended to the antiferromagnetic side J1 $ 0 across the decoupled-chain limit J1 = 0; it is called even-odd phase in Ref. 25. The boundary between the SDW2 phase and the nematic phase is shown by a dotted line in Fig. 1. In the semiclassical picture we can write s−l = e−i%l, where %l is the angle of the two-dimensional vector !sxl , sly" measured from the positive x direction, 0 & %l ' 2". The product − = e−i!%l+%l+1" can be represented by the vector Nl+1/2 s−l sl+1 = !cos (l,2 , sin (l,2" with (l,2 = −!%l + %l+1" / 2. We now realize that we need to identify Nl+1/2 with −Nl+1/2 because of the physical identification !%l , %l+1" % !%l + 2" , %l+1" % !%l , %l+1 + 2"". We can thus consider Nl+1/2 as a director representing the nematic order. We will show in Sec. VI that the nematic phase has antiferronematic quasi-long-range order of the director, as shown schematically in Fig. 4. The ground state is not dimerized in this phase as opposed to the initial proposal of Chubukov.1 Incommensurate nematic phase. The incommensurate First order transition: partially polarized state coexists with plateau one With enough quantum fluctuations, “bubbles” of partially polarized phase may become many-magnon bound states and form multipolar phases 0 −4 :N : SDW2 : IN :T : SDW3 :Q : VC −3 −2 J1 / J2 −1 FIG. 1. !Color online" Magnetic phase diagram of the spin-1/2 zigzag chain with ferromagnetic J1 and antiferromagnetic J2 !a" in the J1 / J2 versus h / J2 plane and !b" in the J1 / J2 versus M plane. Crosses show the transition and crossover points obtained from the magnetization curves and correlation functions. In !a", symbols VC, N, IN, T, Q, and F indicate the vector chiral !*Sztot = 1", nematic !*Sztot = 2", incommensurate nematic !*Sztot = 2", triatic !*Sztot = 3", 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