Two-magnon instabilities and other surprises in magnetized quantum antiferromagnets

Credit: Francis Pratt / ISIS / STFC

Oleg Starykh University of Utah ! Andrey Chubukov, U Wisconsin

Conference on Field Theory Methods in Low-Dimensional Strongly Correlated Quantum Systems, August 25-29, 2014, ICTP, Trieste, Italy Outline

• Emergent Ising orders - a very brief history ! • UUD magnetization plateau and its instabilities ! • High-field phase diagram of a triangular antiferromagnet VOLUME 52, NUMBER 6 PHYSICAL REVIEW LETTERS 6 FEBRUWRV 1984

Discrete-Symmetry Breaking and Novel Critical Phenomena in an Antiferromagnetic Planar (XY) Model in Two Dimensions VOLUMED, H.52,Iee,NUMBERJ. D. Joannopoul.6 os, andPHYSICALJ. W. Negel. e REVIEW LETTERS 6 FEBRUWRV 1984 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 03139 Discrete-Symmetryand Breaking and Novel Critical Phenomena in anD. AntiferromagneticP. Landau Planar (XY) Model in Two Dimensions Department of Physics, University of Georgia, Athens, Georgia 30602 (Received 14 NovemberD, H.1983)Iee, J. D. Joannopoul. os, and J. W. Negel. e I andau-Ginzburg-WilsonDepartmentsymmetry ofanalysesPhysics,and MonteMassachusettsCarlo calculationsInstitutefor theof Technology,clas- Cambridge, Massachusetts 03139 sical antiferromagnetic planar' Q model on a triangular lattice reveal a wealth of inter- esting critical phenomena. From this simple model arise a zero-field transitionandto a state of long-range order, a new mechanism for spin disordering, and a critical point associated with a possible new universality class. D. P. Landau PACS numbers: 64.60.Cn, 05.70.Fh, 05.Department70.Jk, 75.10.ofAk Physics, University of Georgia, Athens, Georgia 30602 (Received 14 November 1983) of the tmo ground states. As a conse- Tmo-dimensional. spin systems cannotI andau-Ginzburg-Wilsonexhibit ordering symmetry analyses and Monte Carlo calculations for the clas- l. order breaking a continuous sym- quence, the system supports solitons (domain ong-range by sical antiferromagnetic planar' model on a triangular lattice reveal a wealth of inter- metry at finite temperature. ' Although order- walls) as additionalQ elementary excitations. An esting critical phenomena. FromVOLUMEthis simpleNUMBERmodel6 arise a zero-fieldPHYSICALtransitionREVIEWto a LETTERS 6 FEBRUWRV 1984 disorder transitions are thereby excluded, Kos- example of a soliton is shown52, in Fig. 1. a new mechanism for spin disordering, and a critical point terlitz and Thouless (KT)' showed thatstatethe ofclassi-long-rangeTheorder,order parameter associated with the &3 associated with &&a possible new universality class. cal iwo-dimensional. ferromagnetic planar (XY') W3 periodicity is given by I pl, whereDiscrete-Symmetrythe com- Breaking and Novel Critical Phenomena in model exhibits a unique phase transitionPACSfromnumbers:an plex64.60.vectorCn, g05.is 70.definedFh, 05.as 70.Jk, 75.an10.AntiferromagneticAk Planar (XY) Model in Two Dimensions algebraical. ly ordered phase to a disordered phase and Negel. e via the unbinding of vortex pairs. Whereas the =—- exp(-iq B,.)s, D, H. Iee, J. D. Joannopoul. os, J. W. g Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 03139 antiferromagnetic planar Tmo-dimensional.model. on a bipartite spin systems cannot exhibit ordering of the tmo ground states. As a conse- (e.g. , square) lattice isl.ong-rangeequivalent toorderthe ferro-by breakingHere aq=(+~m,continuous0) and (~~,sym-(~ are orthogonalquence, com-the system supports andsolitons (domain magnetic model. , the antiferromagneticmetry at finite casetemperature.for ponents' Althoughof ( chosenorder-parallel and perpendicularwalls) as additional elementary excitations. An l. fundamen- to the magnetic field when H o In terms of D. P. Landau a tripartite (e.g. , triangular) attics is 0. ofg,a soliton is shown in 6 FEBRUARY 1984 disorder transitions areVOLUMEthereby52, NUMBERexcluded,6 Kos- PHYSICALexampleDepartmentREVIEW LETTERSUniversity Fig. 1.Athens, Georgia 30602 tal. l.y different because of the inherent frustratiori. ' the staggered helicity' is given by rt =Be(g)x 1m(()of Physics, of Georgia, terlitz and Thouless (KT)'= showed that the classi- The order parameter(Receivedassociated14 Novemberwith1983)the &3 In this work, me use Landau-Ginzburg-Wilson I pl' sing. Since lattice ref l.ection is a discrete && the temperature two spin-dis- I (LGW) symmetry analysescal iwo-dimensional.and Monte Carlo (MC)ferromagneticAssymmetry, theplanarsystemincreases,(XY')can haveI andau-Ginzburg-Wilsonlong-rangeW3 periodicityheli- symmetryis given analysesby pl, andwhereMonte theCarlocom-calculations for the clas- mechanisms the unbinding of 8 i calculations to study themodelphasesexhibitsof the antiferro-a unique orderingphasecity ordertransitionwithout fromviolatingcompete:an thesicalMermin-Wagnerplexantiferromagneticvector g isplanar'definedQ modelas on a triangular lattice reveal a wealth of inter- = ." "s the vortex pairs' in both Re(ICI) andestingIm(()criticaland thephenomena. From this simple model arise a zero-field transition to a magnetic pl. anar (AFP)algebraical.model, X Jgly &,ordered,&(s, , phasetheorem.to a disorderedThe zero-fieldphaselow-temperature phase Hy FTls& = 0 unlocking of the phase correlationstatebetweenof long-rangeRe(ICI) order, a new mechanism for spin disordering, and a critical point + s,.'s,.'), on a triangul. ar and a square lattice and is therefore characterized by ( g) = 0 and (rl) =+1- . ~ ~ via the unbinding of vortexOrderpairs. parameters:Whereas continuousthe with =—newexp(-iquniversality B,class.)s, and discrete spin chirality = Si Sj find dramatic differences in their critical behav- andsoIm(ICI).that althoughThe latterbothprocessBe(() andtakesassociatedIm(p)placehavethroughnoa possiblelong- g ⇥ antiferromagnetic planarthe model.creationonofasol.bipartiteitons and the transition occurs triangle ior and underl. ying physics. range order, they are locked PACSin phasenumbers:(y). 64.60.Cn, 05.70.Fh, 05.70.Jk, 75.10.Ak I I X An essential. feature (e.distinguishingg. , square)thelatticetriangu- iswhenequivalentthe surfaceto thefreeferro-energy associatedHerewithq=(+~m,the 0) and (~~, (~ are0.4 orthogonal com-1.2 stability argu- /J lar and bipartite AFP modelsmagneticis themodel.existence, theofantiferromagneticsoliton vanishes. UsingcaseTmo-dimensional.simplefor ponentsspin systemsof ( chosencannot parallelexhibit and orderingperpendicularof the tmo ground states. As a conse- ments" we find that the pair unbinding tempera- discrete as mell as continuousa tripartitedegeneracy intriangular)the l.attics is fundamen-l.ong-range ordertobythebreakingmagnetica continuousfield whensym-H o 0.quence,In termsthe systemof supports solitons (domainTwo possibilities: (e.g. , twice the asso- H~& g, triangular AFP ground state. For a bipartite ture T„is roughly temperature T, ' walls) as additional elementary excitations. An tal. l.y different because of the inherent frustratiori.metry at finite' temperature.the staggeredAlthoughhelicity'order-is given by rt =Be(g)x 1m(() ! lattice, a ground state consists of spins on two ciated with a soliton transition. This suggests 8 disorder transitions= are thereby excluded,Hy Kos- example of a soliton is shown in Fig. 1. In this work, me use Landau-Ginzburg-Wilsonthat helicity order will be lost first, withI pl' asing.spon- Since lattice ref l.ection is a discrete • single transition: both sublattices aligned in opposite directions. For terlitz and Thouless (KT)' showed that the classi- The order parameter associated with the &3 l. analysestaneousand generationMonte Carloof solitons. As verified by the system can have long-range heli- the triangular attice, (LGW)a ground symmetrystate consists of (MC) symmetry, && cal iwo-dimensional. ferromagnetic planar (XY') W3 periodicity is given by I pl, wherespinsthe andcom- chiralities order, spins on three sublattiees forming + 120' angles MC results shown below, the solitons screen the calculations to study the phases of the antiferro-model exhibits acityuniqueorderphasewithouttransitionviolatingfrom anthe Mermin-Wagnerplex vector g is defined as leading to a W3&& v 3 periodicity. In a bipartite vortex interaction= and induce" "spair unbinding im-' ! magnetic pl. anar (AFP) model, X &,algebraical.&(s,. orderedtheorem.phase toThea disorderedzero-fieldphaselow-temperature phase mediately. ThisJgmechanism, , forly spin disordering • lattice global spin rotation generates all possibl. e = - . two separate transitions: + s,.'s,.'), on a triangul. ar and a square latticevia the andunbinding isofthereforevortex pairs.characterizedWhereasH, ' the by ( g) 0 and (rl)=—=+1exp(-iq B,)s, ground states, while an extra l. attice reflection is fundamentally different from vortex pair un- g antiferromagnetic planar model. on a bipartite Ising (chirality) transition is needed for the triangularfind dramaticlattice. In differencesFig. 1, bindingin theirand leadscriticalto whatbehav-seems to beso athatnew althoughuni- both Be(() and Im(p) have no long- (e. square) lattice is equivalent to the ferro- Here q=(+~m, 0) and are orthogonal com- me show two ground statesior andof theunderl.triangul.yingar lat-physics.versality class of criticalg. , phenomena.range order, they are locked in phase (y). (~~, (~ is followed by the BKT (spins) model. the antiferromagnetic case ponents of chosen parallel and perpendicular magnetic , for ( I tice which are not obtainableAn essential.from each otherfeatureby distinguishingIn theFIG.presenceI. Two groundtheof antriangu-statesin-planeof oppositemagneticstaggeredfield a tripartite triangular) l.attics is fundamen- 0.2 to the0.magnetic4 0.field6 when H o 0. In terms of global spin rotation. This additional. symmetry thehelicityground separatedstates stillby a preservedomain wall.(e.theg.The,&3 &v3helicityperi-of T g, lar and bipartite AFP models is the existencel. of ' helicity'/J is given rt x breaking is manifested by the opposite hei. icity odicity.each triangleThe sublatticeis indicated.tal. y ground-statedifferent becausemagnetiza-of the inherent frustratiori. the staggered by =Be(g) 1m(() the = discrete as mell as continuous degeneracyIn thisin work,. = me use Landau-Ginzburg-WilsonFIG. 2. Phase diagramsI pl'forsing.AFP Sincemodels latticeon the ref l.ection is a discrete tions m„m» m, satisfy l m, l 1 for all i and triangular AFP1984 ThegroundAmericanstate.PhysicalForSocietya bipartite(LGW) symmetry analyses and433MontesquareCarlo(top)(MC)and triangularsymmetry,(bottom) thelattices.systemCom-can have long-range heli- ponents of the order parameters which are zero are lattice, a ground state consistsg m,. =H/3Z.of spins calculationson two to study the phases(2) of the antiferro- city order without violating the Mermin-Wagner )=1 anar X =indicated explicitly.." "s Thetheorem.circle 'diagramsThe zero-fieldillustrate low-temperaturethe phase sublattices aligned in opposite directions.magneticFor pl. (AFP) model, Jg &, ,&(s, , spin configurations in the ordered phases. The dots = even +thoughs,.'s,.'),theonHamiltoniana triangul. arhasandno a square lattice and is therefore characterized by ( g) 0 and (rl) =+1 the triangular l.attice, aSurprisingly,ground “Anstate exceedinglyconsists of simple model leadsaround to athe surprisingcircles represent richnessthe number of phasesof additional and critical behavior. ! find dramatic differences in their critical behav- so that although both Be(() and Im(p) have no long- spins on three sublattieescontinuousformingsymmetry,+ 120' anglesthere exist continuously distinct and degenerate spin configurations obtained degenerate solutionsiortoand(2).underl.Moreover,ying physics.it can be by permuting the sublattices.range order,The mannerthey inarewhichlockedthe in phase (y). leading to W3&& v 3 TheIn underlying triangular lattice and the associated degeneracy play a crucial role in this physics.”! a periodicity.shown that at zeroa bipartitetemperatureAn essential.asfeatureP is increaseddistinguishingthreethephasetriangu-boundaries merge at & is not determined lattice global spin rotationabovegeneratesH, =3J, theallsolutionslarpossibl.and bipartitetoe (2) changeAFP modelsfrom is thepreciselyexistencein thisof work. ground states, while an twoextradisconnectedl. attice reflectionmanifoldsdiscrete (correspondingas mell[unfortunately,as continuousto two degeneracyincorrect inidentificationthe of spin configurations] is needed for the triangularhei. icitylattice.states) toIn onetriangularFig.manifold1, AFP(correspondingground state.to For a bipartite = me show two ground statesa stateof ofthezerotriangul.helicity).lattice,ar lat-Abovea groundII, state9J allconsiststhe of =[0.spins495(5)]Zon two is associated with spontaneous gen- spins are aligned, l.sublatticeseading to thealignedparamagneticin opposite directions.eration ofForsol.itons as described earlier. tice which are not obtainable from each other by phase. the triangular l.attice,FIG.aI.groundTwo groundstateSinceconsistsstatesMC resultsofof oppositeindicatestaggeredthat all the phase additional. global spin rotation. ThisThe entire II- Tsymmetryphasespins diagramson three forsublattieeshelicityboth squareseparatedforming boundaries+by120'a domainanglesshownwall.in Fig.The 2helicityare continuousof transi- breaking is manifested byand thetriangul.oppositear AFPhei.leadingmodelsicity todetermineda W3&&eachv 3 byperiodicity.triangleMC is indicated.Intions,a bipartiteLQW symmetry anal. yses were performed calculations are shownlatticein Fig.global2. spin(Therotationsimula- generatesto understandall possibl. theire nature. All. the invariants of tions are performed1984groundwithThe useAmericanstates,of standardwhilePhysicalantech-extraSocietyl. atticethe reflectionLGW Hamiltonian (to infinite order)433can be niques' for L &L latticesis neededwith forperiodicthe triangularboundary lattice.builtIn upFig.from1, twelve elements. The first few conditions, with L meup toshow72. twoDatagroundwere obtainedstates of the triangul.terms inarthislat-expansion are with use of between tice2000whichand 6000are notMC obtainablesteps for from each other by FIG. I. Two ground states of opposite staggered +LGW +(Ill ) + +(4x) +I ((Ily 4z) y l.ocating phase boundariesglobal spinand asrotation.many asThis4&&104additional. symmetry helicity separated by a domain wall. The helicity of MC steps for studyingbreakingcriticalis behavior.manifested) Theby the oppositewherehei. icity each triangle is indicated. phase diagram for the square lattice (top panel) is easily understood. At II =0, there is a conven- 1984+(IIIThe) +2(IIIAmerican) +aRe(IllPhysical)+ Society 433 tional KT transition at T„= 90(2)]A At H & [0. 0, = only two degenerate ground states remain, lead- K(ICI, ) K,'((,)+a'Re(ICI, ')+.. ., (3) ing to an Ising order-disorder phase boundary. = &((II, 0, ) &Re(ICI ')+ cRe[(g, ICI, *)'] For the triangular lattice (bottom panel) the phase „$, diagram is far richer including four phase bound- + dim[((„g~*)']+... . aries and three mul. ticritical points T„P» and A. The zero-field critical temperature T, Here , and +,' define isotropie two-component 434 Classical isotropic triangular AFM in magnetic field

• Zero field: co-planar spiral (120 degree) state! • Magnetic field: accidental degeneracy! = ⃗ · ⃗ − ⃗ · ⃗ H! J ! Si Sj ! h Si 120o state co-planar supersolid non-coplanar! ! i,j i ⃗ 2 superfluid 1 ⃗ h H! = J Si − 2 ! " ! 3J # ! △ i∈△ ⃗ ⃗ ⃗ ⃗ h • all states with S i + S i + S i = form the lowest-energy manifold! 1 2 3 3J • Accidental degeneracy! – O(2) spins: 3 angles, 2 equations => 1 continuous angle undetermined! – O(3) spins: 6 angles, 3 equations => 2 continuous angles (upto global U(1) rotation about h) Antiferromagnetic XY model with triangular lattice 5931

The attentive reader can get an impression that at finite temperatures performing a quadric expansion on the background of the fixed ground state produces a result that is inconsistent due to the absence of a rigorous long-range order in rp. We want to stress that when the anisotropy found is self-consistently taken into account a gap appears in the gapless mode. The long-range order becomes rigorous. It seems worth remembering that in two-dimensional systems with a continuous Abelian symmetry an arbitrary small anisotropy proves to be relevant if the temperature is small in comparison with the constant in the gradient energy term (Pokrovsky and Uimin 1973a, b, JosC et a1 1977). J. Phys. C: Solid State Phys. 19 (1986) 5927-5935. Printed in Great Britain Let us now consider what kind of states have the minimal free energy of the spin waves. The configurations of spins in different sublattices maximising S( q52, r#J3) with constraints (5) being taken into account are shown in figure 3. For h < hcl the three

Phase diagram of the antiferromagnetic XY model with a triangular lattice in an external magnetic field

S E Korshunov L D Landau Institute for Theoretical Physics, Academy of Sciences of the USSR, AntiferromagneticKosygina 2,117940 XY Moscow, model USSR with triangular lattice 5933

Received 16 December 1985 t

Abstract. The ordered states of a planar antiferromagnet with a triangular lattice are investigated in the presence of a magnetic field. The spin wave free energy is taken into account and proves to be important for determining the properties of the system. The phase diagram is constructed. It contains four different phases with rigorous long-range order. Figure 3. Configuration of spins of the three sublattices with the minimal spin wave free Three of them are characterised by different configurations of mean magnetic moments for energy: (a),0 < h < hcl;(b), h = hCl;(c), h,, < h < hc2;(d), in the case of the opposite sign the three sublattices. The existence of one more non-trivial phase with an algebraic decay of the anisotropic part of the free energy. of the correlation functions is very probable.

sublattices are non-equivalent. On one of them the spins are antiparallel to the field, 1. IntroductionhlJ P and on the two others they have the perpendicular-to-field components of opposite signs Order of(figure helicities 3(a)). This state(chiralities) is six-fold degenerate only in accordance with a number of possible In the exchange approximation a planar antiferromagnet can be described with the 51 permutations of non-equivalent sublattices. Hamiltonian: With increasing h the angle between the spins, which are not antiparallel to the H = J mi- mi!= J cos(qj - qj,) field, diminishes,(1) and at h = hcl vanishes (figure 3(b)).Two of the sublattices become (11') (ii') equivalent, the spins on them being parallel to the field. The degeneracy multiplicity of where mi= (cos4 qj, sin qj)are unit planar vectors defined on lattice sitesthis and state the is equal to three. summation is performed over pairs of nearest neighbours. In the case of a flat triangularFor hcl < h < hc2the equivalence of two of the sublattices is retained, but the sym- lattice (and this is the case we are interested in) the ground state consistsmetry of three in the direction perpendicular to the field becomes broken (figure 3(c)). That leads to the degeneracy multiplicity being increased up to six, as for 0 < h < hcl.The sublattices. The magnetic moments (spins) mibelonging to different sublattices form the , anharmonicities being taken into account, the magnetic moment in this state is not angles 120" with respect to eachI other: /id\ parallel to the field (at finite temperatures). @* = @I ? 120" = @I 7 120". Thus(2) we have considered which states are preferred by the spin wave free energy at the lowest temperatures. Here Figureq$ (I = 4.1,2,3) Phasedenote diagram the of common the AF XY(t) values model of qi inin an each external of the magnetic three sublattices. field. In phases In a, b additionand to c thecontinuous mean magnetic degeneracy moments caused of bydifferent the invariance sublattices of formthe Hamiltonian configurations with similar to respectthose to homogeneous shown in figures rotation 3(a),3(b) of alland spins 3(c) respectively.the ground state In phase also dpossesses only the long-rangetwo-fold order discretewith degeneracy respect to helicities(upper and is retained. lower signs Phase in p(2)). is paramagnetic. The order parameter degeneracy space R is, accordingly, a pair of circumferences: R = Z2 x SI. 1977). So theIt provescurve convenientBA (or at to least describe part this of additional it) should (discrete) split into degeneracy two curves by introducing joining in the point A. the helicity (vorticity) U, of each triangular plaquette r,: Let us now consider the form of the phase diagram for the low fields. The numerical U, = (1/2n) {Vi- qy - n}. (3) simulation of Miyashita and Shiba(ii') E r (1984) shows that in zero field the BKT transition precedes that of the Ising type. This Ising-type transition restores the equivalence between 0022-3719/86/295927the states differing + in 09 signs $02.50 of @ the 1986 helicities The Institute in each of Physics elementary cell and5927 therefore is essentially the same transition as the transition on the line DC which, so it seems, should terminate at the point C2 of the Ising-type transition in zero field. For small fields the influence of the spin wave free energy is equivalent to that of the three-fold anisotropy field for a continuous variable. So the point C1 of the BKT transition in zero field should not be an isolated singular point but it should serve as an end-point for a line of phase transitions between the six-fold-degenerate and two-fold- degenerate states (cf JosC et a1 1977). This makes us conclude that one more ordered phase should exist (phase d in figure 4). In this phase the mean values of magnetic moments are equal for all the sublattices, but the ‘antiferromagnetic’ order with respect to the helicities is retained. The point of coexistence of four phases (phases a, b, d and paramagnetic phase p) is likely to split into two triple points (points C and C’ in figure 4). The line C’C2is the line of the Ising-type transitions, and the line C‘C1that of the three-state Potts model transitions. The phase transitions on the line CC’ are likely to be of first order, just as in the six-state Potts model (Baxter 1973) or in a six-state cubic model (Nienhuis et aZ1983). When the interaction of more distant neighbours is taken into account the sequence and types of the transitions in zero field can change (Korshunov and Uimin 1986). The Phase diagram of the Heisenberg (XXX) model in the field

Seabra, Momoi, Sindzingre, Shannon 2011

Z2 vortex (chirality ordering) transition

Gvozdikova, Melchy, Zhitomirsky 2010 PHYSICAL REVIEW LETTERS week ending PRL 93, 257206 (2004) 17 DECEMBER 2004

12

43 43 15 10

10 8 max C

C 6 5 1 2 12 (Q,Q) (Q,−Q) 4 1 1.5 2 2.5 log(L) 2 FIG. 2. The two different minimum energy configurations with PHYSICAL REVIEW B 85,174404(2012) magnetic wave vectors Q~ Q; Q and Q~ ? Q; Q with 0 ˆ † ˆ ÿ † 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Q 2=3, corresponding to J3=J1 0:5. Chiral spin liquid in two-dimensional XY helimagnets ˆ ˆ T/J1 A. O. SOROKIN AND A. V. SYROMYATNIKOV PHYSICAL REVIEW B 85,174404(2012) 1,* 1,2, We begin by recalling [9] the ground states of at S FIG. 3. T dependence of the specific heat for J3=J1 0:5. A. O. Sorokin and A. V. Syromyatnikov † H ˆ ˆ 1Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, St. Petersburg 188300, Russia . There is conventional Ne´el order with magnetic wave Different symbols refer to different clusters with liner size recent paper32 that T >T at J 0.3andJ J 1 2Department of Physics, St. Petersburg State University, 198504BKT St. Petersburg,I Russia2 1 b 1 ~ 1 1 between L 24 and L 120. Data for J3=J1 0:1 are shown = = = vector Q ;  for J3=J1 . For J3=J1 > , the ˆ ˆ ˆ (Received 7 November 2011; revised manuscript(i.e., received very near 27 January the Lifshitz 2012; published point) in 3 May accordance 2012) with Ref. 28 ˆ †  4 4 for comparison (full dots and dashed line). Inset: size scaling of and in contrast to Ref. 31. ground state has planar incommensurate antiferromagnetic the maximum of the specific heat. We carry out Monte Carlo simulations to discuss criticalTo properties account of for a classical the discordance two-dimensional betweenXY frustrated results for helical order at a wave vector Q~ Q; Q , with Q decreasing from helimagnet on a square lattice. We find two successivemagnets phase transitions and the upon general the temperature arguments decreasing: for the firstSO(2) class, 1 ˆ † Z2  as J3=J1 > 4 and approaching Q =2 monotonically one is associated with breaking of a discrete Z2 symmetry and the second one is of the Berezinskii-Kosterlitz-⊗ ˆ where a labels each plaquette of the square lattice and we perform extensive Monte Carlo simulations of the model for J =J . The spiral order is incommensurate for 1 < Thouless (BKT) type at which the SO(2) symmetry breaks. Thus, a narrow region exists on the phase diagram 3 1 !1 4 1; 2; 3; 4 are its corners. The variables  are zero for a between lines of the Ising and the BKT transitions(1) that forcorresponds different to a values chiral spin of liquid.J2.Weobtainreliableresultsat J =J < , except at J =J 0:5 where Q 2=3, cor- a 3 1 3 1 Ne ´el antiferromagnet,† while they assume opposite signs on J2 > 0.4J1 which show that TBKT gas system of half-integer charges,17 ˆ ÿ † h ai temperature from the chiral-order-parameter18 distribution13,19,20 and (for Q Þ ). This state cannot be obtained from the one Our numerical results contain strong evidence for a Frustrated magnets have attracted much attention in recent two coupled XY models, Ising-XY model, and the find thatgeneralizedTBKT phase transition between a low T phase years. Exotic spin-liquid phases, which have been foundtime, in we find in accordance with results of Ref. 32 that the 1 famous of them is the fully frustrated XY model (FFXY) two configurations are connected by a global rotation with  Þ 0, and a homogeneous high T phase with some of them, are of special interest. Achiralspin-liquid 22 h i specificintroduced heat and by susceptibilities Villain. This model have is of subsidiaryPHYSICAL great interest peaks because REVIEW at LETTERS week ending PHYSICAL 0. REVIEW The divergence LETTERS in the specific heatweek ending (Fig. 3) is phase is an example of such an exotic state of matter in which PRL 93, 257206 (2004) combined17 DECEMBER with a 2004 reflection aboutPRL the93,x257206or y (2004)axes. The 17 DECEMBER 2004 low TTI , T to the0 maximum, while of con- the susceptibility, which is 10 ‡ ‰
† ‡
† Š 1= 2 In larger dimensions, one of the systems in which theof the chiral order parameter. We find no such features in the  @ n1 @ n (3) 2 The turn angle θ between two neighboring spins along the a expected to scale as Tmax L Tc aLÿ , where Tc is 2 x y binding of kink-antikinkKavli Institute pairs for on Theoretical the domain Physics, walls University associated of California, Santa Barbara,0 California 93106-4030,tinuous O(3) USA symmetryLow-Temperature is immediately Broken-Symmetry restored at †ˆ any Phases‡ of Spiral Antiferromagnets 8 ‡ 0.315
†


chiral spin-liquid phase can be found at finite temperature isspecific a heat and susceptibilities3Department farof Physics, from the Yale Lifshitz University, point P.O. Box(at 208120, New Haven, Connecticut 06520-8120, USA the thermodynamic critical temperature; Fig. 4 (upper in- axis is given by cos θ J /4J at zero temperature. σ where the ellipses denote a finite number of additional  with the Z2 symmetry, can take place in models similar to 2D 0 1 2 set) shows  1, as expected. The exponent is also in max 0.31 i classical planar (XY)helimagnetwithZ2 SO(2) symmetryJ2 > 0.4J1). As a result, we obtain the(Received phase diagram 23 September shown 2004; published 14 December 2004) = −nonzero T. We established this by extensiveˆ Monte1,2 Carlo 2,3 6 couplings involvingT /J four powers of n and four spatial ⊗ FFXY one at temperatures appreciably smaller than TBKT (see Todiscussthenumberandthesequenceofphasetransitions agreementLuca with Capriotti Ising universality.and Subir This Sachdev can be extracted in which the helical structure results from a competition 1 derivatives invariant under spin rotations and lattice sym- in Fig. 1. We study Heisenberg antiferromagnets with nearest- (J ) and third- (J ) neighbor exchangesimulation on theValuation using Risk a combination Group, Credit Suisse of Metropolis First Boston (Europe) and over- Ltd., = One Cabot Square, London E14 4QJ, United0.305 Kingdom also Ref. 26). Such a transition could lead to a decoupling of from1 the (quasi-)helical3 phase to the paramagnetic one, we from the scaling law  Lÿ f x , where f x is the 4 metries. In the limit S , we have 1= 8J ,  of exchange interactions between localized spins. Critical 1 2 j j ˆ 1= † † 1 The rest of this papersquare is organized lattice. In the as limit follows. of spin WeS discuss, there is in a zero temperature (T) Lifshitz point at Jrelaxed3 4 J1, with algorithmKavli Institute for periodic for Theoretical clustersscaling Physics, of function size University and upx to ofM California,L with  SantaT Barbara,Tc ; this CaliforniaJ 4J 93106-4030,S2,  16!1 USAJ J S2=12, ?ˆ 0 , and† allˆ phase coherence across domain boundaries, producing!1 in1 this consider J2 0.309, 0.5, andˆ 1.76 corresponding3 at T 0 ˆ ˆ j ÿ j 2 1 3 1 3 0 1 0.02 2 0.04 behavior of spin systems from this class is described by two long-range spiral spin order at T 0 for27J3 > J1. We present classical Monte Carlo simulations and a Department of Physics, Yalescaling University, is shown P.O. in Box Fig. 208120, 4,ˆ upper New inset, Haven, with ConnecticutTc ÿ 06520-8120,† ˆ USA ÿ † ˆ Sec. IIwaythe model two separate(1) in more bulk transitions detail and with introduceT 0 crossovers nearBKT theˆ LifshitzI point: spinto rotation angles symmetry of commensurate is restored at any T> helices0, but θ0 4π/5, 2π/3, and(Received0:303 2313 estimated September1 from 2004; the positionpublished of the14 Decembermaximum of 2004) ˆ order parameters. Aside from the conventional magnetization  † 0 and so can be regarded as the tuning parameter;  0 be foundpointed in our out, calculations. however,there is in a brokenNumerical Ref. 25 latticethat reflectionresults these aretwo symmetry discussed continuous for 0 T is phaseJ to. We diagram be present is shown classical in Fig. Monte 5,0.3 Carlo simulations and a DOI: 10.1103/PhysRevLett.93.257206range from 20 PACS to numbers: 120. We 75.10.Hk, apply 75.10.Jm the periodic (toric) boundary 3 4 1 A convenient analysis of the properties of n is provided

the system as soon as the ground-state degeneracy remains the ˆ L neighborhood of the Lifshitz point and the phase diagram are theory for T>0 crossoverswhere near we the have Lifshitz plotted point:Tc versus spin rotationJ3=J1. We symmetry find that isTc restored at any T>0, but S helix twist and distinguishes left-handed and right-handed conditions as well as the cylindricalcontrasted ones to (i.e., the with broad the maximum periodic displayed by the same by a direct generalization 0.1 of the 1=N expansion of same. They are confirmed by numerical studies of the models vanishes linearly for J3=J1 1=4;atheoryforthisbehav-1 2 σ there is a1 broken lattice reflection symmetry for 0 T 1/4. temperature dependence of the susceptibility!1 of  for J3=J1 DOI: 10.1103/PhysRevB.85.174404large scale density matrix renormalizationPACS number(s): 64 group.60.De, 75.30.Kz square lattice. In the limit of† spin S , there is a zero temperature (T) Lifshitz point at J 1 J , with !n 0 term in Eq. (4) [12]. A solution for m exists for all In two dimensions (2D), the situation is rather different. where the sum runs over sites x (x ,x )ofthelattice,a P ^ 3 4 1 clear below, the spiral order and associated Lifshitz point ˆ (DMRG) calculations for S 1=2a[3]b have suggested the where Si are spin-!1S operators1 on a square1.2 lattice andˆ 0:5. The inset showsT>ˆ0, and the leads size to scalingthe crossovers of the in the temperature spin correlation corre- = = long-range spiral spin order at T 0 for J3 > 4 J1. We present classical Monte Carlo simulations and a Two successive transitions were observed with the temperature (1,0) and b (0,1) are unit vectorsˆ of the lattice, the coupling ˆ ^ ^ ^ ^ play a central role in the structure of our theory and in the T existence of a gapped spin-liquid state with exponentially theory for T>J01crossovers;J3 0 nearare the thea Lifshitz nearest-S point:1 spinS and3 rotation third-neighborS2 symmetryS4 a is; restored antiferro- at any T>(2)0, but sponding to thelength maximumspin shown of inthe Fig. susceptibility. 1. The value of spin, and the decreasing. The chiral order appears as a result of the first = 16 T 17 there is a broken lattice reflection symmetryˆ for 0
Tstate of matter in31 which 2 0.6 spin-liquid phase arises between these transitions with the Kolezhukresults, noticed obtainedthat by classical those arguments Monte Carlo are simulations not valid and for a c'S2 / T existencecS / ofT a gapped spin-liquidˆ state with exponentially it describes a superconducting array of Josephson junctionsξspin~ e ξspin~ e 0.2 there are neither quasi-long-range nor long-range magnetic 23 Frustrated antiferromagnets have recentlyIsing attracted nematicalso suggest order that discrete lattice symmetriesT may play a  n @x@yn @xn @yn: (6) chiral order and without a magnetic one. Various 2D systems helimagnet,a theory and described showed below, thatunder are the summarized Ising an external transition transverse in Fig. temperature 1 magnetic for the field. It was found decaying spin correlations and no broken translation sym- 0.15 ˆ
ÿ
orders, but a chiral order parameter Si Sj is nonzero. TKT much interest in connection with the possibility of stabiliz- role near other quantum0.4 critical points with spiral order [6]. 0.1 from the class Z2 SO(2) were investigated numerically (see is largerlimit⟨ × thanS ⟩ the. There BKT is one a Tthat at the0 leaststate temperature near with the long-range of the Lifshitz Ising spiral transition point TI is 1%–3% largering unconventional low-temperaturemetry in (T the) phases, regime with of novel strong frustrationBroken discrete (J3=J symmetries1 0:5). have also been discussed Theweek ending Ising1/2 susceptibility, , is then Existence of such a phase is discussed in context of one- J1 / 4 PHYSICAL REVIEW LETTERS ξ ~ S /1 =TT ⊗ 14,15 !1 1 thanˆ that of the BKT transition for most of above-named PRL 93,J257206 (2004) ’ 0.05 17spin DECEMBER 2004 2 Ref. 13 for review): triangular antiferromagnet, J1-J2 spin order for2 J > J . We establish that at 0

C R phase⊗ has ‘‘Ising nematic’’FFXY order.one at We temperatures present strong appreciably nu- smaller than T (see ^ limit S . There is a T 0clearstate below, with the long-range spiral order spiral and associated Lifshitz point length rescaling analysis of this theory in which the i and in which the helical structure results from a competition and y axes,BKT leadingwhere to IsingSi are nematic spin-S order.operators The Ising on a transition square lattice is and J / 4 C 1 also Ref. 26). Such a transition could lead to a decoupling of !1 1 ˆ play a centralFIG. role 5. inCritical the structure temperature of our as theory a 6 function and in of the theT frustration 5 i are fixed, we see that bothJ3T and  scale as inverse merical evidence that the transition at Tc is indeed in the J1;J3 0 are the1 nearest-2 spin and order third-neighbor for J3 > antiferro-J1. We establish that at 0 0, with theJ T dependenciesJ S , above as this state there is a phase with broken 4 1 1.5 2 2.5 way two separate bulk transitions with T for J03=J, with1 0:5 the. T dependencies as Ref. [2]). Our large S resultsmentioned are therefore above. consonant with circle: here there is a T 0 spin gap
S exp cS~ and spin spin ˆ In three-dimensional (3D) helimagnets, the phase transi- for large S and discuss consequencesoriginally. There forproposed is general conventionalS in. Our Ref. Ne´el [2] order for withS magnetic1H=2 in waveˆ a T Different0 symbols refer to different clusters with1 liner size Nevertheless, the situation remains174404-2 contradictory in 2D ˆ  ÿ † shown, with c=2 c0 8 J3 J1 ; the crossovers between the possibility of a spin-liquid phase at S 1=2 as de- rotation symmetry is preserved. This semicircular1 region extends ˆ 2 ˆbetween L 24 2and L 120. Data for J =J 0:1 4are shown tions on the magnetic and the chiral order parameters occur results, obtained by classicalspin-liquid Monte Carlo~ phase simulations described and by a1Z gauge~ e theoryc'S /1 T [5]. Thus ~ ecS / T ˆ ˆ 3 1j ÿ j helimagnets belongingˆ to the same Z2 SO(2) class1 as vector Q ;  for J3=J1 4 .2 ForξspinJ3=J1 > 4 , the ξspintheˆ differentˆ behaviors of  ˆ are at the dashed lines at T simultaneously. It was found numericallyscribed that in the Refs. transition [2,3]; we will discuss the quantum finite S over T J3 4 Ja1 theoryS
described. Further below, details are on summarized the physicsˆ in within Fig.† 1 for the  for comparison (full dots and dashed line). Inset:spin size scaling of the FFXY model and the antiferromagnet⊗ on the triangularj ÿ j  the sameground Ising state has nematic planar incommensurate order can appear antiferromagnetic when spiral spin 1 2  phase diagram further4,5 towards the end of the Letter. We this region appear at the end of the Letter. the maximumJ3 of theJ1 specificS . Spin heat. rotation symmetry is broken only at T 0 is of the weak first order or of the “almost-second-order” 28 limit S . There is a T 0 state with long-range spiral~ 4 lattice. Garel and Doniach (see also Ref. 29) considered the !1 1 orderˆ order is destroyed at a wave vector eitherQ byQ; thermal Q , with Q fluctuationsdecreasing from (as inJ1 /the 4 j ÿ J3j ˆ type in helical antiferromagnets on a body-centered tetragonal spin order for J3 > J1. We establish that at 01 and approaching Q Neel=2 LROmonotonically Spiral LRO 6 simplest helimagnet on a square lattice with an extra competing 1 2 present Letter;3 1 see4 Fig. 1) or by quantum fluctuationsLifshitz point (as in ˆ 1 lattice and on a simple cubic lattice with an extra competing J3 4 J1 S , above this state there is a phase with broken ˆ 1 where a labelsis preempted each plaquette [13] by of quantum the square effects lattice andwithin the dotted semi- 7 0031-9007=04=93(25)=257206(4)$22.50exchange coupling along one axis 257206-1 that is described by the ©ÿ2004† The American Physicalfor J3=J1 Society. The spiral order is incommensurate for 4 < exchange coupling along one axis. These systems belong discrete symmetry of latticeRef. reflections [2]). Our about!1 large the xSandresultsy are therefore consonant with1; 2; 3; 4 circle:are its corners.here there The is variables a T 0spina are gap zero
for aS exp cS~ and spin Hamiltonian J3=J1 < , except at J3=J1 0FIG.:5 where 1. PhaseQ diagram2=3, cor- of H^ in the limit S† . The shaded to the same (pseudo)universality class as, e.g., the model axes, while spin rotationthe invariance possibility is1 preserved. of a spin-liquid This ˆ phase atˆ S 1=2 asNe de-´el antiferromagnet,rotation!1 symmetry while they is assumepreserved.ˆ opposite This signs semicircular on ÿ region† extends 8 9 responding to an angle of 120  regionbetween has spins a broken (see symmetry Fig. 2). of lattice reflections about the x on a stacked-triangular lattice and V2,2 Stiefel model. The phase has ‘‘Ising nematic’’ order. We present strong nu- ˆ the two degenerate ground1 states in the spiral phase. H (J1 cos(ϕx ϕx a) J2 cos(ϕx ϕx 2a) scribedInterestingly, in Refs. for [2,3]; each spiralwe will stateand discuss withy axes,Q~ leadingtheQ; quantum Q tothere Ising nematicis finite order.S Theover IsingT transitionJ3 is 4 J1 S
. Further details on the physics within possibility of existence and stabilization of the chiral spin- = − + + − + merical evidence that the transition at Tc is indeed in the at the temperatureˆ † T J 1Consequently,J S2. The spin a phase correlation j withÿ Isingj nematic order is signaled x phasea distinct diagram but further equivalent towards configuration the end at Q~ ? of thecQ; Letter. Q 3 4 We1 this region appear at the end of the Letter. liquid phase by, e.g., Dzyaloshinsky-Moria interaction in 3D ! Ising universality class. Such Ising nematic order[4] was  ÿ by† a a Þ 0. J ϕ ϕ , length, spin, is finiteˆ ÿ for all †T>0, withh thei T dependencies as b cos( x x b)) (1)originally proposed in Ref. [2](for forQSÞ 1).=2 Thisin a stateT cannot0 be obtained from the one 1 Our numerical results contain strong evidence for a helimagnets, is discussed recently in Ref. 10. − − + shown, with c=2 c0 8 J J ; the crossovers between ˆ ~ ˆ ˆ ˆ j 3 ÿ 4 1j 11 spin-liquid phase described by awithZ2 gauge wave vectortheoryQ [5].by Thus a globalthe spin different rotation. behaviors Instead, of the arecontinuous at the dashed Ising lines phase at T transition between a low T phase In two dimensions (2D), the situation is rather different. where the sum runs over sites x (xa,xb)ofthelattice,a spin the same Ising nematic order0031-9007 cantwo appear configurations=04 when=93(25) spiral are spin=257206(4)$22.50 connected by1 a2 global rotation with 257206-1 Þ 0, and a homogeneous © 2004 high TheT phase American with Physical Society Two successive transitions were observed with the temperature = = J3 J1 S . Spin rotation symmetry is broken only at T 0 (1,0) and b (0,1) are unit vectors of the lattice, the couplingorder is destroyed either by thermalcombined fluctuations with a (as reflection in the aboutj ÿ the4 jx or y axes. The  h0.i The divergenceˆ in the specific heat (Fig. 3) is decreasing. The chiral order appears as a result of the first = where spin . There is no Lifshitz point at finite S because it constants J1,2 are positive. Using arguments of Ref. 30,theypresent Letter; see Fig. 1) or byglobal quantum symmetry fluctuations of the (as inclassical groundˆ state1 is O 3 accompaniedh i ˆ by a divergence in the susceptibility of the transition that is of the Ising type. Another one is the concluded28 that at low temperatures the vertices are bound is preempted [13] by quantum † effects within the dotted semi- Ref. [2]). Our large S results areZ2 therefore,withanadditionaltwofolddegeneracybeyondthatof consonant with circle: here there is a T 0 spin gap
S exp cS~ and spin Berezinskii-Kosterlitz-Thouless (BKT) transition driven by by strings, which would inhibit the BKT transition and make ˆ  ÿ † 12 the possibility of a spin-liquidthe phase Ne´el at case.S 1=2 as de- rotation symmetry is preserved. This semicircular region extends the unbinding of vortex-antivortex pairs. Then, the chiral ˆ 1 12 the Ising transition occur first with the temperature increasing.scribed in Refs. [2,3]; we will discussOne the of the quantum main finiteclaimsS in Fig.over 1T is thatJ3 theJ broken1 S
.Z Further2 details on the physics within 31  j ÿ 4 j  0.32 spin-liquid phase arises between these transitions with the Kolezhuk noticed that those arguments are not valid for aphase diagram further towardssymmetry the end of survives the Letter. for a We finite rangethis region of T> appear0, while at the con- end of the Letter. 10 1 chiral order and without a magnetic one. Various 2D systems helimagnet, and showed that the Ising transition temperature tinuous O(3) symmetry is immediately restored at any 8 0.315 σ from the class Z SO(2) were investigated numerically (see max 0.31 2 is larger than the BKT one at least near the Lifshitz point nonzero T. We established this by extensive Monte Carlo 6 T /J ⊗ 14,15 0031-9007=04=93(25)=257206(4)$22.50 257206-1 © 2004 The American Physical Society Ref. 13 for review): triangular antiferromagnet, J1-J2 J2 J1/4. It was found by Monte Carlo simulations in the simulation using a combination of Metropolis and over- 4 0.305 = relaxed algorithm for periodic clusters of size up to M ˆ 2 0 0.02 0.04 1098-0121/2012/85(17)/174404(10)174404-1 ©2012 American Physical Society 120 120, and for several values of J3=J1 between 0.25 1/L and 4. Indeed, the presence of a finite T phase transition is 0 0.4 clearly indicated by a sharp peak of the specific heat which 1 is illustrated in Fig. 3 [10]. This sharp feature is to be 0.3 L contrasted to the broad maximum displayed by the same 0.1 σ quantity for J =J < 1 , i.e., when the classical ground state σ 0.2 β 1/8 3 1 4 Tc /J 1 0.303 displays ordinary Ne´el order. In particular, the maximum 0.01 0.1 0.1 1 10 of the specific heat is consistent with a logarithmic depen- dence on system size (see the inset of Fig. 3) corresponding 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 to a critical exponent 0, in agreement with Ising ˆ T/J universality. 1 This critical behavior can be directly related to the broken lattice reflection symmetry by studying an appro- FIG. 4. Bottom: T dependence of the order parameter, , [see Eq. (2)] for different cluster sizes and J3=J1 0:5. The inset priate Ising nematic order parameter. From the symmetries shows the data collapse according to the scalingˆ hypothesis with of Fig. 2, we deduce that the order parameter is  Ising exponents 1=8 and  1, and T 0:303. Top: 1=M  with ˆ ˆ ˆ c ˆ a a† temperature dependence of the susceptibility of  for J3=J1 P 0:5. The inset shows the size scaling of the temperature corre-ˆ  S^ S^ S^ S^ ; (2) a ˆ 1
3 ÿ 2
4†a sponding to the maximum of the susceptibility.

257206-2 Ising nematic in collinear spin system

= N~ N~ = 1 1 · 2 ± this talk: Emergent Ising orders in quantum two-dimensional triangular antiferromagnet at T=0

h sat H D

H = JijS~i S~j · U(1)*U(1) U(1)*U(1)*Z2 i,j hXi

U(1)*Z3 U(1)*Z2 b hc2 G B Z3*Z2 C2 Z3 U(1)*Z3*Z2 C C1

A hc1 F U(1)*Z3 a U(1)*U(1)*Z2 U(1)*Z2

2 40 J J 0 U(1)*U(1) E = S O 3 J ⇣ ⌘ 0 1 3 δcr 4 δ Spatially anisotropic model: classical vs quantum J J’ H J S S h Sz = ∑ ij i ⋅ j − ∑ i 〈ij〉 i

Umbrella state: ! S = 1 favored classically;! 0 hsat h energy gain (J-J’)2/J J ′ ̸= J 1 Planar states: favored by ! S = quantum fluctuations;! 2 0 hsat h 1/3-plateau energy gain J/S The competition is controlled by − ′ 2 2 dimensionless parameter δ = S(J J ) /J Emergent Ising orders in quantum two-dimensional triangular antiferromagnet at T=0

h

h H = J S~ S~ ij i · j i,j hXi h h h sat H D

sat

sat

h

h

U(1)*U(1) sat

sat

h

h

U(1)*Z3

planar b hc2 G B C2

C cone Z3 U(1)*Z2 ? C1 A hc1 F U(1)*Z3 a

1/3-plateau

1/3-plateau

U(1)*U(1) E

0

O 0

0

0 2 δ 40 J J 0 0 1 3 cr 4 δ = S 3 J ⇣ ⌘ UUD-to-cone phase transition

Z U(1) Z or Z smth else U(1) Z ? 3 ! ⇥ 2 3 ! ! ⇥ 2 h sat H D

H = JijS~i S~j · U(1)*U(1) U(1)*U(1)*Z2 i,j hXi

U(1)*Z3 b hc2 G B Z3*Z2 C2 Z3 (UUD) U(1)*Z2 (cone) C1 C

A hc1 F U(1)*Z3 a U(1)*U(1)*Z2 U(1)*Z2

2 40 J J 0 U(1)*U(1) E = S O 3 J ⇣ ⌘ 0 1 3 δcr 4 δ Low-energy excitation spectra

9Jk2 ✏d2 = hc2 h + 4 Magnetization plateau is for δ < 3 collinear phase: preserves O(2) rotations about magnetic field -- no gapless spin waves.

-k2 Breaks only discrete Z3. d2 Hence, very stable.

vacuum of d1,2 0.6 0.6 h h = h = (9JS) c2 c1 2S sat 2S

40 S 2 = (1 J 0/J) 3 Bose-Einstein condensation d1 3Jk2 of d1 (d2) mode at k =0 leads to ✏ = h h + d1 c1 4 lower (upper) co-planar phase for δ < 1 Alicea, Chubukov, OS PRL 2009 Low-energy excitation spectra near the plateau’s end-point

40 S 2 = (1 J 0/J) parameterizes anisotropy J’/J 3

Out[24]= extended symmetry: 4 gapless modes at the plateau’s end-point -k-k22 +k2 d2

Out[25]= vacuum of d1,2 δ=4 k1 = k2 = k0 d1 3 -k0 +k0 k0 = r10S S>>1 Out[19]=

40 S 2 = (1 J 0/J) 3 -k1 +k1 Magnetization plateau is collinear phase: preserves O(2) rotations about magnetic field -- no gapless spin waves. Breaks only discrete Z . 3 Alicea, Chubukov, OS PRL 2009 Bosonization of 2d interacting magnons

(4) 3 † † † † † † d d = (p, q) d1,k +pd2, k pd1, k0+qd2,k0 q d1,k +pd2, k pd1, k +qd2,k q +h.c. H 1 2 N 0 0 0 0 0 0 }

p,q } X ⇣ ⌘

( 3J)k2 (p, q) 0 † † ⇠ p q 1†,p 2,q 1,p 2,q | || | singular magnon interaction

= d d Out[25]= magnon pair 1,p 1,k0+p 2, k0 p 1 2 operators 2,p = d1, k +pd2,k p } 0 0 2 1

Obey canonical Bose commutation relations in the UUD ground state

[ 1,p, 2,q]=1,2p,q 1+d† d1,k +p + d† d2,k +p 1,2p,q 1,k0+p 0 2,k0+p 0 ! ⇣ ⌘ In the UUD ground state d†d = d†d =0 h 1 1iuud h 2 2iuud

★ Interacting magnon Hamiltonian in terms of d1,2 bosons = non-interacting Hamiltonian in terms of Ψ1,2 magnon pairs Chubukov, OS PRL 2013 Two-magnon instability

Magnon pairs Ψ1,2 condense before single magnons d1,2

2 6Jfp 3 2 Equations of motion for Ψ - Hamiltonian † = f † h 1,p 1,pi ⌦ N q h 2,q 2,qi p q X 2 6Jfp 3 2 † = f † h 2,p 2,pi ⌦ N q h 1,q 1,qi p q X

⌥ `Superconducting’ solution with = i imaginary order parameter h 1,pi h 2,pi⇠ p2

Instability = softening of two- 1 1 k0 2 1= magnon mode @ δcr = 4 - O(1/S ) S N 2 2 p p +(1 /4)k0 X | | p no single particle condensate d = d =0 h 1i h 2i

Chubukov, OS PRL 2013 Two-magnon condensate = Spin-current nematic state distorted ! umbrella Υ > 0 Υ < 0 hc2 uud spin-! J’ current J h J’ c1 distorted ! umbrella domain wall δcr 4 δ x,y no transverse magnetic order S =0 Sr Sr is not affected h r i h · 0 i Finite scalar (and vector) chiralities. Sign of ⌥ determines sense of spin-current circulation zˆ S S = zˆ S S = zˆ S S ⌥ h · A ⇥ C i h · C ⇥ Bi h · B ⇥ Ai/

Spontaneously broken Z2 -- spatial inversion [in addition to broken Z3 inherited from the UUD state] ! Leads to spontaneous generation of Dzyaloshisnkii-Moriya interaction

Chubukov, OS PRL 2013 Spontaneous generation of Dzyaloshinskii-Moriya interaction

(4) 1 ik0 ik0 d d (d1†,kd2†, k d1,kd2, k) (d1†,pd2†, p d1,pd2, p) H 1 2 / N 2 2 2 2 k +k (k k0) +(1 /4)k0 p k (p + k0) +(1 /4)k0 2X 0 2X 0 p p

continuum limit of DM in triangular lattice a1 zˆ S (S + S ) · r ⇥ r+a1 r+a2 B || z r a2 X Mean-field approximation:

(4) k0 k0 D ( + )(d† d† d d ) d1d2 1,k 2, k 1,k 2, k H ! k k0 k + k0 Xk | | | | D ⌥ ⇠

spin currents appear due to spontaneously generated DM (similar to Lauchli et al (PRL 2005) for Heis.+ring exchange model; also ‘chiral Mott insulator’, Dhar et al, PRB 2013; Zaletel et al, 2013 ) PHYSICAL REVIEW B 87,174501(2013)

Chiral Mott insulator with staggered loop currents in the fully frustrated Bose-Hubbard model

Arya Dhar,1 Tapan Mishra,2 Maheswar Maji,3 R. V. Pai,4 Subroto Mukerjee,3,5 and Arun Paramekanti2,3,6,7 1Indian Institute of Astrophysics, Bangalore 560 034, India 2International Center for Theoretical Sciences (ICTS), Bangalore 560 012, India 3Department of Physics, Indian Institute of Science, Bangalore 560 012, India 4Department of Physics, Goa University, Taleigao Plateau, Goa 403 206, India 5Centre for Quantum Information and Quantum Computing, Indian Institute of Science, Bangalore 560 012, India 6Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 7Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8 (Received 9 July 2012; revised manuscript received 15 March 2013; published 3 May 2013) Motivated by experiments on Josephson junction arrays in a magnetic field and ultracold interacting atoms in an optical lattice in the presence of a “synthetic” orbital magnetic field, we study the “fully frustrated” Bose-Hubbard model and quantum XY model with half a flux quantum per lattice plaquette. Using Monte Carlo simulations and the density matrix renormalization group method, we show that these kinetically frustrated boson models admit three phases at integer filling: a weakly interacting chiral superfluid phase with staggered loop currents which spontaneously break time-reversal symmetry, a conventional Mott insulator at strong coupling, and a remarkable “chiral Mott insulator” (CMI) with staggered loop currents sandwiched between them at intermediate correlation. We discuss how the CMI state may be viewed as an exciton condensate or a vortex supersolid, study a Jastrow variational wave function which captures its correlations, present results for the boson momentum distribution gapped single particles; across the phase diagram, and consider various experimental implications of our phase diagram. Finally, we consider generalizations to a staggered flux Bose-Hubbard model and a two-dimensional (2D) version of the but CMI in weakly coupled ladders.

DOI: 10.1103/PhysRevB.87.174501 PACS number(s): 67.85.Hj, 03.75.Lm, 75.10spontaneously.Jm broken time-reversal

I. INTRODUCTION fluctuations which can “sense” the local flux on a plaquette.= spontaneous circulating In a recent paper, we have found numerical evidence for the The effect of frustration in generating unusual states of existence of such a remarkable intermediate state in frustrated currents matter such as fractional quantum Hall fluids or quantum spin two-leg ladders of bosons for the so-called fully frustrated liquids is an important and recurring theme in the physics of Bose-Hubbard (FFBH) model which has half a flux quantum condensed matter systems.1,2 Recently, research in the field per plaquette. We call this state a “chiral Mott insulator” (CMI) of ultracold atomic gases has begun to explore this area, since it is fully gapped due to boson-boson interactions, exactly spurred on by the creation of artificial gauge fields using like an ordinary Mott insulator, and in addition possesses chiral Raman transitions in systems of cold atoms.3,4 These gauge order associated with the spontaneously broken time-reversal fields can be used to thread fluxes through the plaquettes of symmetry arising from resolving the kinetic frustration. The optical lattices giving rise to “kinetic frustration” by producing superfluid state of this system also possesses this chiral order multiple minima in the band dispersion and frustrating simple and we thus dub it a chiral superfluid (CSF).13 Other recent Bose condensation into a single nondegenerate minimum. studies have also focused on various such exotic bosonic states Similarly, time-dependent shaking of the optical lattice5,6 or driven by “ring-exchange” interactions,14–19 which again arise populating higher bands of an optical lattice7 can be used due to virtual charge fluctuations in a Mott insulator. to control the sign of the hopping amplitude in an optical In this paper, we discuss further details of our work on this lattice, again leading to such “kinetic frustration.” For bosonic FFBH ladder model and its close cousin, the fully frustrated atoms with weak repulsion, such kinetic frustration gets quantum XY model, to which it reduces at high filling factors. resolved in a manner such that the resulting superfluid state can have a broken symmetry corresponding to picking out We also discuss how one might stabilize such a Mott insulator aparticularlinearcombinationofthedifferentminima.7–11 in higher dimensions. Such a CMI may also be viewed as a bosonic Mott-insulating version of the staggered loop current Increasing the strength of the interactions at commensurate 20–22 filling can be expected to eventually yield a Mott insulator states studied in the context of high-temperature cuprate (MI) with the motion of the bosons quenched, which thus superconductivity. renders the kinetic frustration ineffective. In a synthetic flux Classical analogs of the CMI and CSF states have been 9–11 and at strong coupling, the fully gapped MI is identical to the studied in the past. The simplest classical model displaying one expected for the same lattice without a frustrating flux analogous phases is the fully frustrated XY model in two 23,24 per plaquette;12 this simply means that at strong coupling, dimensions. At small but finite temperature, this model we can adiabatically remove the flux without encountering a has a phase with algebraic U(1) order for the spins along with quantum phase transition. However, there could exist a state astaggeredpatternofvorticityassociatedwitheachplaquette intermediate to the superfluid and the MI described above for corresponding to broken Z2 symmetry. This is the analog of which charge motion has been suppressed enough to open the CSF phase. As the temperature is increased, the U(1) up a gap but not restore the broken symmetry associated symmetry is restored while the Z2 symmetry continues to be with frustration. Such a state is stabilized by virtual boson broken in a state that is the analog of the CMI. Upon further

1098-0121/2013/87(17)/174501(13)174501-1 ©2013 American Physical Society Phases of a triangular-lattice antiferromagnet near saturation

h sat H D U(1)*U(1)*Z2

H = JijS~i S~j U(1)*U(1) · U(1)*Z2 i,j hXi U(1)*Z3

b hc2 G B Z3*Z2 C2 Z3 C1 C

A hc1 F U(1)*Z3 a U(1)*U(1)*Z2 U(1)*Z2

2 40 J J 0 U(1)*U(1) E = S O 3 J ⇣ ⌘ 0 1 3 δcr 4 δ OS, Jin Wen, Andrey Chubukov, PRL 2014 High-field phases: from cone to incommensurate planar at h = hsat

fully polarized state h A 1 > 2 B 1 < 2

U(1)*U(1) hsat incommensurate U(1)*Z2 U(1)*Z3 planar cone D commensurate U(1) U(1) Z ~ planar (V) * * 2 ~ double cone C plateau, Z3

0 1 J 0/J 1 E /N = µ( 2 + 2)+ ( 4 + 4)+ 2 2 Low-density expansion 0 | 1| | 2| 2 1 | 1| | 2| 2| 1| | 2| 9(J)2 1.6J = = (0) + (1) = 2 1 J S 1 (J +5J )2 (J 4J )2 3J 1.6J (1) = 0 k 0 Q+k + 16S J0 Jk JQ+k JQ 8S ⇡ S k BZ X2 ⇣ ⌘ Phases of a triangular-lattice antiferromagnet near saturation U(1) Z U(1) U(1) ⇥ 3 ! ⇥ Q =(4⇡/3, 0) Q = incommensurate ! i h sat H D U(1)*U(1)*Z2

H = JijS~i S~j U(1)*U(1) · U(1)*Z2 i,j hXi U(1)*Z3

b hc2 G B Z3*Z2 C2 Z3 C1 C

A hc1 F U(1)*Z3 a U(1)*U(1)*Z2 U(1)*Z2

2 40 J J 0 U(1)*U(1) E = S O 3 J ⇣ ⌘ 0 1 3 δcr 4 δ OS, Jin Wen, Andrey Chubukov, PRL 2014 High-field phases: commensurate-incommensurate transition Q =(4⇡/3, 0) Q = incommensurate ! i h A fully polarized state B hsat U(1)*U(1) incommensurate U(1)*Z2 U(1)*Z3 planar cone D B J A commensurate U(1) U(1) Z ~ planar (V) * * 2 δ1 δ3 ~ double cone J 0 δ2 C A plateau, Z3 C

0 1 J 0/J

Low-density expansion -> classical sine-Gordon model of the (relative) phase fluctuations

i3Q r 3 (5Jk + J0)(5JQ+k + J0)JQ k (5Jk + J0)(Jk + J0) 3J0 0.69J e · =1 first calculation of 3 = 2 + 2 2 32S (J0 Jk)(J0 JQ+k) 2(J0 Jk) 64S ⇡ S k BZ X2 ⇣ ⌘ Phases of a triangular-lattice antiferromagnet near saturation

U(1) U(1) U(1) Z or U(1) U(1) smth else U(1) Z ? ⇥ ! ⇥ 2 ⇥ ! ! ⇥ 2

h sat H D U(1)*U(1)*Z2

H = JijS~i S~j U(1)*U(1) · U(1)*Z2 i,j hXi U(1)*Z3

b hc2 G B Z3*Z2 C2 Z3 C1 C

A hc1 F U(1)*Z3 a U(1)*U(1)*Z2 U(1)*Z2

2 40 J J 0 U(1)*U(1) E = S O 3 J ⇣ ⌘ 0 1 3 δcr 4 δ OS, Jin Wen, Andrey Chubukov, PRL 2014 High-field phases: from cone to incommensurate planar at finite density

fully polarized state h A 1 > 2 B 1 < 2

U(1)*U(1) hsat incommensurate U(1)*Z2 U(1)*Z3 planar cone D commensurate U(1) U(1) Z ~ planar (V) * * 2 ~ double cone C plateau, Z3

0 1 J 0/J Elementary excitation spectrum of the cone phase: Goldstone mode at k = +Q, gapped at Q0 = Q 6 Transition at BD: softening of the mode at -Q’

1.45(hsat h) Q0 = Q + -Q’ -Q +Q hsatpS High-field phases, XXZ model: commensurate- commensurate transitions

H = J(SxSx + SySy)+J SzSz h Sz XXZ i j i j z i j j i,j j hXi X fully polarized state h A B C hsat U(1)*Z3 U(1)*Z3 U(1)*Z2 V state state cone ~ c1 c2 =1 Jz/J ? co-planar commensurate ! non-coplanar c1 =0.45/S 3J 0.69J 3JS2 µ3 = (@ ✓)2 + S cos[6✓] 3 = 2 ✓ x 3 3 c2 =0.53/S 2S S E 4hsat hsat

Solid-Solid transition due to the sign change of Γ OS, Jin Wen, Andrey Chubukov, PRL 2014 Conclusions! ! Emergent Ising orders ! Two-dimensional chiral spin-current phase Z3*Z2 ! High-field phases of triangular antiferromagnet U(1)*U(1)*Z2

“An exceedingly simple model leads to a surprising richness of phases and critical behavior. ! The underlying triangular lattice and the associated degeneracy play a crucial role in this physics.”!

HB2U, Alyosha! two-dimensional

Schematic phase diagram for spin-1/2 triangular lattice AFM h/J 0.8 plateau 4 fully polarized 0.6 width

0.4

0.2 3 0.0 0.0 0.2 0.4 R 0.6 0.8 1.0 IC planar C planar

2 SDW cone 1/3 plateau for 1 C planar all J’/J SDW (crystal of spin- IC planar downs; end-point at J’=0) 0 quasi-collinear 0.0 0.2 0.4 0.6 0.8 1.0

R=1-J’/J spin-current state does not apply directly to s=1/2 model Compare with: Cluster mean-field theory:! direct 1st order transition! between the UUD and the cone phases.! ! Can there be an intermediate! spin-current (chiral Mott) phase! with broken Z3*Z2 ? High-field phases, J-J’ model

Out[24]=

2 1 h A fully polarized state B -Q Q hsat U(1)*U(1) incommensurate U(1)*Z2 U(1)*Z3 planar cone D B J A commensurate U(1) U(1) Z ~ planar (V) * * 2 δ1 δ3 ~ double cone J 0 δ2 C A plateau, Z3 C

0 1 J 0/J

OS, Jin Wen, Andrey Chubukov, PRL 2014