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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 system supports solitons (domain quence, the symplanar' Q model on a triangular lattice reveal a wealth of interal antiferromagnetic walls) as additional elementary excitations. An orderto a transition LETTERS model REVIEW From this simple PHYSICAL critical phenomena. NUMBER 6 1.arise a zero-field ing VOLUME 52, in Fig. is shown example of a soliton ed, Kosand a critical point for with disordering, new mechanism spin the te of long-range Theorder, associated &3 order aparameter classithe class. new universality sociated possible where is given by pl, the comperiodicity ar (XY') with a W3 Breaking and Novel Critical Phenomena Discrete-Symmetry us && 6 FEBRUWRV 1984 in I as 70.Jk, 75. is 70. defined vector from an plex Ak Planar (XY) Model in Two Dimensions an 10. 64. 60. Antiferromagnetic numbers: Fh, 05. CS Cn, g05. ered phase -=— exp(-iq B,.)s, D, H. Iee, J. D. Joannopoul. os, and J. W. Negel. e eas the 03139 Institute Cambridge, Massachusetts of Technology, of Physics,of Massachusetts As a consestates. the tmo ground nal. spin systems cannot exhibitDepartmentordering partite solitons (domain the system supports and quence, comcontinuous der symHere a e ferro0) and (~~, q=(+~m, by breaking (~ are orthogonal ' Although parallel and perpendicular of ( chosen case for ponents walls) as additional elementary excitations. An ordertemperature. D. P. Landau H o 0. In terms of g,a fundamenfield when to the magnetic in Fig. 1. shown soliton is 6 FEBRUARY 1984 Kos- PHYSICAL of example LETTERS REVIEW itions are' VOLUME excluded, thereby 6 52, NUMBER Department x 1m(() of Physics, University of Georgia, Athens, Georgia 30602 the staggered helicity' is given by rt =Be(g) rustratiori. with1983) the &3 associated order parameter classi(KT)'= showed that thelattice ouless 14 November (Received ref l. ection The is a discrete pl' sing. Since ilson spin-disincreases, As the temperature (XY') is given analyses W3 periodicity onal. ferromagnetic planar by pl, andwhere calculations for the clasCarlocomMonte the heliandau-Ginzburg-Wilson can haveItwo symmetry the long-range arlo (MC) system symmetry, i of the 8 unbinding mechanisms compete: ordering antiferromagnetic as on a triangular lattice reveal a wealth of intersical defined vector g isplanar' transition Q model a unique phase plex without from Mermin-Wagner antiferroviolatingan the city order and and the both Re(ICI) vortex pairs the theorem. " ' Theinzero-field Im(() transition to a From this simple model arise a zero-field critical phenomena. esting low-temperature phase X Hy FTls& = 0 to a disordered ordered &, phase , &(s,. ,"s phase critical and between point correlation a for the Re(ICI) disordering, of mechanism phase new unlocking a spin of state long-range order, = ~i ⇥ S ~j is pairs. characterized thereforeWhereas atticeof and ( g) 0 and (rl) =+1 = S . )s, and discrete spin chirality ng vortex the bytakes Order parameters: continuous B, class. new exp(-iq universality with associated a possible through latter place The process and Im(ICI). so that although both Be(() and Im(p) have no longal behavonofasol. ic planarthe model. bipartite triangle transition occurs the PACS itons creation areand locked in phase 64. 60.Cn, 05.70.Fh, 05.70.Jk, 75.10.Ak range order, numbers: they (y). the 1.2 freeferrothe surface 0.4 orthogonal comto the Herewith energy associated equivalent 0) and (~~, (~ are q=(+~m, e lattice triangu- iswhen /J argu-of simple soliton vanishes. Using and ordering chosen perpendicular case stence l. , theofantiferromagnetic for stability ponents of the tmo ground states. As a conseTmo-dimensional. cannot parallel exhibit spin systems thatfundamenthe pair unbinding temperawe find is the ments" l. attics Two possibilities: cy the system In terms of g, supports solitons (domain fieldH~&when ordertobythe a continuous sym- H o 0.quence, magnetic l.ong-range breaking .g. ,in triangular) assois twice the temperature ture roughly T„ T, ' partite as additional walls) orderx 1m(()elementary excitations. An metry at finite temperature. is given by helicity' the with because of inherent frustratiori. rt =Be(g) staggeredAlthough 8 transition. Thisthesuggests ciated a soliton on two a soliton is shown in Fig. 1. example a of disorder transitions pl'aresing. thereby excluded, Kosdiscrete me • that helicity order will be lost first, = with a spon- Since Hylattice ref l. ection is order s. use For Landau-Ginzburg-Wilson the &3 transition: both The parameter associated with single terlitz and Thouless (KT)' showed that the classiAs verified of helisolitons. can have long-range and generation the system (XY') Monte Carlo by ry analyses symmetry, (MC) onsists of taneous spinstheand chiralities order, comW3 periodicity is given by pl, where cal iwo-dimensional. ferromagnetic planar solitons the shown the screen MC results below, 0' angles without the Mermin-Wagner antiferroorder violating the the o study phases of model exhibits acity unique phase transition from an plex vector g is defined as iminteraction and induce unbinding vortex pair " ' ipartite = "s zero-fieldphase The low-temperature phase ar (AFP) model, X theorem. a disordered ordered phase to &,algebraical. , &(s,. , forly spin disordering • l possibl. e mediately. This mechanism two separate transitions: =— . )s, =+1 via the from characterized unbinding isoftherefore vortex Whereas the by ( g) = 0 and (rl) exp(-iq and a and ar lattice pairs. triangul. square B, H, ' unvortex different is fundamentally pair flection Ising (chirality) transition a bipartite model. on both antiferromagnetic planar so behavin their differences critical Be(() and Im(p) have no longuninew although binding and leads to what seems to be athat Fig. 1, com-by the BKT (spins) lattice is equivalent to the ferroHere 0) and (~~, (~ are orthogonal (e.g. , square) is followed ar latphysics. l.ying versality class of critical phenomena.range order, they are locked in phaseq=(+~m, (y). case for model. antiferromagnetic ponents of chosen parallel and perpendicular , thestaggered other In the an in-plane field ofmagnetic magnetic by distinguishing the triangufeature states of opposite I. Two ground FIG.presence fundamen- 0. 2 to the0.magnetic when H o 0. In terms of g, 4 T a tripartite 0.field 6 (e. g.The ,&3triangular) &v3 perithe still mmetry statesexistence of l. attics is the helicity domain wall. helicity ground by a preserve is separated the of e AFP models ' the staggered helicity' /J is given by rt =Be(g) x 1m(() tal. l.y ground-state different because of the inherent frustratiori. is indicated. each triangle hei. icity magnetizaThe sublattice odicity. = pl'for ell as continuous in the degeneracy Landau-Ginzburg-Wilson In this sing. me all usei and FIG. 2. Phase diagrams AFP Since models lattice on the ref l. ection is a discrete . l = 1 for tions m„m» m, satisfy m,work, and triangular square (bottom) the lattices. (top)(MC) can have long-range helia bipartite state. ForSociety P The Carlo systemComground symmetry, (LGW) symmetry analyses and 433Monte 84 American Physical of the order parameters ponents are violating zero are the Mermin-Wagner without the antiferroto study the phases city order which of spins calculations on two nd state consists (2) of indicated m,. =H/3Z. The circle illustrate explicitly. diagrams the " ' )=1 . ,"s theorem. The zero-field low-temperature phase &, , &(s, directions.magnetic For pl. anar (AFP) model, X = spin configurations gned in opposite in the ordered phases. The dots +though is therefore characterized a square and arhasandnoleads a triangul.model by ( g) = 0 and (rl) =+1 s,. 's,. '),the around circles the number represent of additionaland exceedingly simple tolattice athesurprising richness of phases critical behavior. ! stateeven l. attice, aSurprisingly, consists ofonHamiltonian ground “An and so that both behavhave longno in their critical although dramatic differences find there Im(p) Be(() exist continuous symmetry, continuously distinct and degenerate spin configurations obtained sublattiees forming + 120'iorangles arewhich locked range order, and underl. they in physics. ying The manner the in phase (y). permuting the sublattices. can solutions to it be Moreover, by degenerate The (2). triangular lattice and the underlying associated degeneracy a crucial role in this physics.”! play 3&& v 3 periodicity. a In bipartite boundaries merge at & is not determined thephase trianguAn essential.as feature distinguishingthree P is increased shown that at zero temperature in this e (2) change spin rotation AFP models is theprecisely existence of work. larpossibl. and bipartite abovegenerates from to H, =3J, theallsolutions discrete (corresponding as mell as continuous while an two extra l. attice reflection manifolds to two degeneracy in the disconnected [unfortunately, incorrect identification of spin configurations] triangular ground state.to For a bipartite hei. icity states) to (corresponding lattice. In one the triangular Fig.manifold 1, AFP a ground of =[0. on two is associated with spontaneous genlattice, spins495(5)]Z Above 9J all consists a state the helicity). II, = state ar latof of thezero triangul. round states in opposite directions. paramagnetic eading to thealigned sol. itons as described earlier. spins are aligned, l.sublattices eration ofFor g I && I -=—g I I ( ' ! I && I ! Jg - g ( I l g Jg I Classical isotropic triangular AFM in magnetic field • Zero field: co-planar spiral (120 degree) state! • Magnetic field: accidental degeneracy! ! =J H ! ! ⃗i · S ⃗j − S i,j ! ⃗h · S ⃗i i ⃗h #2 1 !"! ⃗ Si − H! = J 2 3J ! △ i∈△ • ⃗i1 + S ⃗i2 + S ⃗i3 all states with S • Accidental degeneracy! 120o state co-planar supersolid non-coplanar superfluid ⃗h form the lowest-energy manifold! = 3J – 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) 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). 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 J. Phys. C: Solid State Phys. 19 (1986) 5927-5935. Printed in Great Britain Phase diagram of the antiferromagneticXY 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, Kosygina 2,117940 Moscow, USSR Antiferromagnetic XY model with triangular lattice 5933 Received 16 December 1985 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. Three of them are characterised by different configurations of mean magnetic moments for the three sublattices. The existence of one more non-trivial phase with an algebraic decay of the correlation functions is very probable. 51 t Figure 3. Configuration of spins of the three sublattices with the minimal spin wave free energy: ( a ) , 0 < h < hcl;( b ) ,h = hCl;( c ) ,h,, < h < hc2;( d ) , in the case of the opposite sign of the anisotropic part of the free energy. sublattices are non-equivalent. On one of them the spins are antiparallel to the field, and on the two others they have the perpendicular-to-field components of opposite signs P (figure 3(a)). This state is six-fold degenerate in accordance with a number of possible In the exchange approximation a planar antiferromagnet can be described with the permutations of non-equivalent sublattices. Hamiltonian: With increasing h the angle between the spins, which are not antiparallel to the field, diminishes, and at h = hcl vanishes (figure 3(b)). Two of the sublattices become H =J m i mi!= J cos(qj - qj,) (1) (11') (ii') equivalent, the spins on them being parallel to the field. The degeneracy multiplicity of state 4 qj, sin qj)are unit planar vectors defined on lattice sitesthis where mi= (cos and theis equal to three. For hcl < h < hc2the equivalence of two of the sublattices is retained, but the symsummation is performed over pairs of nearest neighbours. In the case of a flat triangular in the direction perpendicular to the field becomes broken (figure 3(c)). That lattice (and this is the case we are interested in) the ground state consistsmetry of three 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 ,I anharmonicities being taken into account, the magnetic moment in this state is not angles 120" with respect to each other: parallel to the field (at finite temperatures). Thus(2) we have considered which states are preferred by the spin wave free energy at @* = @ I ? 120" = @ I 7 120". the lowest temperatures. Here Figure q$ ( I = 4. 1 ,Phase 2 , 3 ) denote theofcommon values of q iininan each of themagnetic three sublattices. In diagram the AF XY(t) model external field. In phases a, b addition continuous degeneracy caused the invariance of form the Hamiltonian with andtoc the mean magnetic moments of by different sublattices configurations similar to respect to homogeneous rotation of all the ground state also dpossesses two-fold order those shown in figures 3(a),3(b) andspins 3(c) respectively. In phase only the long-range The order parameter degeneracy discrete degeneracy (upper and lower signs in p(2)). with respect to helicities is retained. Phase is paramagnetic. space R is, accordingly, a pair of circumferences: 1. Introduction hlJ Order of helicities (chiralities) only - /id\ R = Z 2 x SI. 1977). So the curve convenient BA (or attoleast partthisofadditional it) should split into two curves joining in the It proves describe (discrete) degeneracy by introducing Phase diagram of the Heisenberg (XXX) model in the field Seabra, Momoi, Sindzingre, Shannon 2011 Z2 vortex (chirality ordering) transition Gvozdikova, Melchy, Zhitomirsky 2010 can merge into with a single first-order one. These con-the quantum in2 Sec.transitions III. In particular, the Ising and the BKT consistent Ising universality. Wetransitions also discuss phase diagram finiteof S. clearly indicated by lattice. aitsharp peak ofofthe specific heat is divisible by theforsize the helix pitchsquare and liesIninthe the th Z limit spinhave S !repeated 1, there is which a zero temperature Lifshit We a similar analysis for several(T) values of clusions do not depend on the particular form of interactions in 1 are considered in Secs. III A and III B, respectively. The J =J and the complete phase diagram is shown in Fig. 5,0 is illustrated inlong-range Fig. 3 spiral [10].spinThis feature to be of the 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 the system as soon as the ground-state degeneracy remains the where we have plotted T versus J =J . We find that T neighborhood of the Lifshitz point and the phase diagram are c spin rotation 3 1 theory forthe T> 0 crossoversdisplayed near the Lifshitz point: symmetry isc r anded contrasted the broad maximum by the same conditions as well as the cylindrical onesto(i.e., with periodic vanishes linearly for J =J ! 1=4; a theory 1 behav2 same. They are confirmed by numerical studies of the models 3 1 there is a1 broken lattice reflection symmetry for 0 " T < Tc # $J3for%this discussed in Sec. III C.11,13–17,23,24 Section IV contains our conclusions. 0 4 J1 &S . Th condition along the b axis and the free one along the a ior will now be presented. quantity for J =J < , i.e., when the classical ground state 3 1 mentioned above. PHYSICAL REVIEW B 85, 174404 (2012) 4 consistent with Ising universality. We also discuss the quantum phase diagram ransiNear the classical Lifshitz point, we can model quantum also suggest symmetries may play a to order. Frustrated the antiferromagnets have contradictory recently attracted axis). We that havediscrete foundlattice thatdisplays both conditions lead the same Nevertheless, situation remains in 2D ordinary Néel In particular, the maximum and thermal fluctuations by a continuum unit vector field0 occur 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 values of transition temperatures and indexes. In contrast, helimagnets belonging to the same Z ⊗ SO(2) class as Chiral spin liquid in two-dimensional XY helimagnets n"r; %#, where r ! "x; y# is spatial coordinate, % is imagi2 of the specific heat is consistent with a logarithmic depenII. MODELlow-temperature AND METHODS sition Broken discrete symmetries have also been discussed ing unconventional (T) phases, with novel 2 nary time, and n ! 1 at all r, %. This field is proportional the FFXY model and the antiferromagnet on the triangular values of Binder’s cumulants andonthe chiral-order-parameter dence system size (see the inset of Fig. corresponding week der”4,5 x &y 1,* order’’ 1,2,† [7,8] in the context of the J types of ‘‘quantum [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 28V. Syromyatnikov 1 (2004) 2 "%1#2004 n"rj ; %#. toTthe parameter with S^ j /ending A. O. Sorokin and A. 17 DECEMBER PRL 93, 257206 lattice. Garel and Doniach (see also Ref. 29) considered the We consider the model (1) of the classical XY magnet on a distribution at J ≈ 0.309 depend on boundary conditions as 2 gonal for a spin-liquid phase is the J second-neighbor couplings on the square lattice. However, % J model to a critical exponent " ! 0, in agreement with Ising Spiral order will therefore appear as sinusoidal tersburg Nuclear Physics Institute, NRC Kurchatov Institute, St. Petersburg 188300, Russia 1 Gatchina, 3 also suggest thatdependiscr Frustrated antiferromagnets recently attracted simplest helimagnet square lattice with anPetersburg, extra competing square lattice. We set J1 =onJaState =University, 1 for simplicity, and the value this 2 wemodel discuss below in detail. Standard Metropolis algorithm33havedence X St. bX of n on r. The action for n is the conventional has only collinear, commensurate spin correlaDepartment of Physics, St. Petersburg 198504 Russia peting universality. ^ i ' S^axis ^ !along role near much interest in connectionwithin with the possibility of sigma stabilizS S^ i described (aJanuary Jthat ' S^published (1) H J1 received exchange one is the O(3) nonlinear model, expanded to other includequantum quartic 3 j; Low-Temperature Broken-Symmetry Phases of Spiral Antiferromagnets Received 7 November 2011;coupling revisedinteraction manuscript 27 2012; 3 by May 2012) tions, of the extra exchange J2 jis variable. The Lifshitz has and been The the thermalization was maintained this used. makes both classical and quantum theory elong Broken discrete ing unconventional low-temperature (T) phases, with novel This critical behavior can be directly related to the hi;ji hhi;jii Hamiltonian gradient terms ( h ! ! k ! lattice spacing ! 1): S n sym ! 5 B quite fromCarlo that considered here. As will become corresponds J2 =critical 1/4 properties in thisofnotation. The system 4 ×different 10 Monte steps in each simulation. Averages have model 1,2 R1=T 2,3 R 2 carrypoint out Monte Carlo simulationsto to discuss a classical two-dimensional XY frustrated FIG. 4. Bot Luca Capriotti and Subir Sachdev [7,8] in the context of types of ‘‘quantum order’’ [1]. A very promising candidate ! d rLn with reflection symmetry by studying an appro9gnet on a square lattice. 0 d% 6 lattice ^ itwo clear below, the spiral order3.6 andbroken associated Lifshitz point 1within Weantiferromagnetic find successive phase transitions upon the temperature decreasing: the first where S are spin-S operators on a square lattice and The has a collinear ground state at J < 1/4. To been calculated × 10 steps for ordinary points and 2 Valuation Risk Group, Credit Suisse First Boston (Europe) Ltd., One Cabot Square, London E14 4QJ, United Kingdom Eq. (2)] for H of = a discrete (J1 Z cos(ϕ ) +second J2 cos(ϕ − ϕ Berezinskii-Kosterlitz) fortheory a spin-liquid phase is the J1 % J3 model second-neighbor coupld x − ϕand x+athe 2structurepriate play a central role in the of our and in the T one isxof the x+2a associated with breaking Ising nematic order parameter. From the symmetries 6 2 symmetry Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA J ; J ) 0 are the nearestand third-neighbor antiferrospin1 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 3 model has only co x symmetry breaks. Thus, a narrow region exists on the phase diagram ss (BKT) type at which the SO(2) Department of Physics, Yale University, P.O. 208120, USA dependence of observables. ^ ^ ^ ^ ^ magnetic couplings along the two coordinate axes. For S S ' S ( J ' S ; (1) H ! J of Fig. 2, we deduce that the order parameter is # ! in 3D 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 2 spin liquid. n lines of the Ising and the BKT that to a chiral P − Jtransitions − corresponds ϕx+b (1) b cos(ϕ this model, early xlarge N )), computations, [2] and recent hi;ji hhi;jii 1=M" # with from thd We study Heisenberg with nearest- (J1 ) and third- (J3 ) neighbor exchange onquite the different Fig. 1). The ground has a helical at J2 >group 1/4. a #aantiferromagnets temperature 11 large scalestate density matrix ordering renormalization 1 0.1103/PhysRevB.85.174404 PACS number(s): 64.60.De, 75.30.Kz ent. ! J , with square lattice. In the limit of spin S ! 1, there is a zero temperature (T) Lifshitz point at J 3 1 where the sum runs over sites x = (x ,x ) of the lattice, a = 4 clear below, where S^ i atare operators on classical a square 0:5. the Thespiral inse 1.2lattice and 1 (DMRG) calculations for S ! 1=2 a[3]bhave suggested the present Monte Carlo simulations and a long-range spiral spin order T !spin-S 0 for J3 > 4 J1 . We rature ^ ^ ^ ^ play a central role in (1,0)existence and b = of (0,1) are unit vectors of the lattice, the coupling # ! " S ( S & S ( S # ; (2) 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 sponding totht a gapped spin-liquid state with exponentially 1 3 1 2 T 16 17 eI.first INTRODUCTION # $J % J &S . The transition at T ! T is there is a broken lattice reflection symmetry for 0 " T < T dependence of observa are positive. Using arguments of Ref. 30, they constants J model, Coulomb gas system of half-integer charges, 1 axes. For c 3 c 1,2 spin correlations 4 1 magnetic couplings along the two coordinate decaying and nothe broken translation sym18 13,19,20 consistent with Ising universality. We also discuss the quantum phase diagram for finite S. 28 s the two coupledthe XY vertices models, are and the concluded atrecent low oftemperatures bound21model, large N computations, [2] and recent ets have attracted much in metryattention in that the regime strong frustration (J3 =J1 ’ Ising-XY 0:5). ising ξspin~ S / T 1/2 this model, early generalized fully frustrated XY model. And surely, the most en by T DOI: 10.1103/PhysRevLett.93.257206 PACS0.8numbers:group 75.10.Hk, 75.10.Jm bywhich strings, which would inhibit the BKT transition and make liquid phases, have been found in c large scale density matrix renormalization This Letter will describe properties ofisthe above model XY model (FFXY ) 257206-2 famous of them the fully frustrated 0.3 1 chiral of special interest. A chiral spin-liquid the Ising transition firstconsequences with the temperature increasing. (DMRG) calculations for S ! 1=2 [3] have suggested the 22 for large S andoccur discuss for general S. Our introduced by Villain. This model is of great interest because 0.25 0.6 ehof the such an exotic state ofnoticed matter in 31which cS2 /of T a gapped spin-liquid state with exponentially c'S2 / T Kolezhuk that those arguments aresimulations not validarray for ofa Josephson junctions results, obtained by classical Monte aCarlo and existence it describes superconducting 0.2 ξ ~ e ξspin~ e Frustrated antiferromagnets spin Ising nematic order uasi-long-range nor long-range magnetic also suggest that discrete lattice symmetriesTmay play a have recently attracted 23 stems a theory described below, are summarized in Fig. 1 for the an external transverse magnetic field. It was foundmuch interest in connection decaying spin correlations nonear broken translation sym- 0.15 helimagnet, theKT Ising transition temperature role other quantum with the possibility of stabiliz- and ⟩ is showed nonzero. thatunder l order parameter ⟨Si × Sjand 0.4 critical points with spiral order [6]. ya (see that the temperature of the Ising transition T is 1%–3% larger 0.1 limit S ! 1. There is a T ! 0 state with long-range spiral I metry in the regime of strong frustration (J =J ’ 0:5). is larger than the BKT one at least near the Lifshitz point Broken discussed ing unconventional (T)Jphases, with novel phase is discussed in context of oneP H Ydiscrete S I C3Asymmetries L1 R E V I Ehave W also L E Tbeen TER S J1 /low-temperature 4 1 3 PRL 93, 257206 (2004) than that of the BKT transition for most of above-named 0.05 2 J -J 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 2 This Letter will describe properties of the above model 3 1 c ted1 quantum systems, and it is Jmagnetic = J /4. It was found by Monte Carlo simulations in the 1 2 Neel LRO 4 2 1 0.2 11,13,23,24 Spiral LRO point 0 for a Lifshitz spin-liquid phase is the second-neighbor couplings S. on the However, % J3 model 12 ly in Ref. 3. state there is a phase $J3 % 14 J1 &S2 , above this systems. 0.25 0.3 0.35 0.4 forJ1large S and discuss3 consequences for3 general Oursquare lattice. 25 with broken Korshunov argued that a phase transition, driven by un4 4 X X this model has only collinear, commensurate spin correlaat T=0 only sions, one of thediscrete systems symmetry in which the c'S2 / T of lattice reflections about the xonand ydomain walls associated Monte Carlo simulations and1 and1.510quantum S^ i ' S^ j ; by classical ' S^ j ( J3 obtained (1) H^ ! J1 S^ i results, 0 ξspin 174404-1 ©2012 American Physical Society binding of kink-antikink pairs the tions, and this makes both the theory ^ 0 0.5classical 2 2.5 ~ e3 FIG. 1. Phase diagram of H in the limit S ! 1. The shaded hase can be found at finitewhile temperature a hi;ji axes, spin isrotation invariance is preserved. Thisin models similar to 2D a theoryhhi;jii described below, are quite summarized in Fig. 1 for the here.J3As /J 1 will become with the Z2 symmetry, can take place different from that considered region has a broken symmetry of lattice reflections about the x 8 Y ) helimagnet with Z2 ⊗ SO(2) symmetry phase has ‘‘Ising nematic’’ order. We present strong nubelow, spiral order spiral and associated Lifshitz point FFXY one at temperatures appreciably smaller TBKT (seewhere limit Son!atransition 1. There T ! 0clear state withthelong-range S^ i are spin-Sorder. operators square lattice and than y axes, leading to Ising nematic The Ising is1is aand cal structure results from a competition FIG. 5. in Critical temperature a 6function theT frustration play a central the structure of ourastheory and inofthe Ref. 26).atSuch could to athe decoupling ofJ1 ; JT merical spins. evidence that thealso transition Tc aistransition indeed in the lead at 2 spin 0 $J are %the1 Jnearestandorder third-neighbor for J31>antiferrothatrole 3 c)# temperature ctions between localized Critical Neel LRO 1 ratio 2Jat=J0. < T < Tc # 4 J12. We establish 3 4 1 &S . The spin correlation dependence of observables. 3 1 phase coherence across domain boundaries, producing in this Lif 1 magnetic couplings along the two coordinate axes. For 2 Ising universality class. Such Ising nematic order[4] was stems from this class is described by two %dependencies J1 &S , above is a phase with broken 4 length, !spin the3 T as this state there 27 , is finite for all T > 0, with $J 4 (Q,Q) (Q,−Q) way two separate bulk transitions with T < T . It was this model, early large N computations, [2] and recent BKT I originallymagnetization proposed in Ref. [2] for S ! 1=2 in a T ! 0 Aside from the conventional shown, with c=2 large ! c0 !scale 8"jJ3density % 14 J1 j; matrix the crossovers betweenofgroup discrete symmetry lattice reflections about the x and y renormalization pointed out, however, in Ref. 25 that these two continuous 2 257 FIG. 1. Phase diagram phase also described by a Z2 gauge theory [5]. Thus etry, one has to spin-liquid take into account the different behaviors of ! are at the dashed lines at T # FIG. 2. The two different minimum energy configurations with axes, while spin rotation invariance is preserved. This (DMRG) calculations for S ! 1=2 [3] have suggested the spin transitions can merge into a single first-order one. These con′ rameter FIG. that is1.an(Color Ising variable Z2 diagram region has a broken sym 0 1 the same Isingwith nematic order canofappear when spiral spinis form 23. (Color online) Phase thedepend model (1)particular that FIG. online) Distribution of the in We "Q; &Q# strong with magnetic wave vectors ! "Q;order. Q# and Q~ ? !present ofsymmetry a gapped spin-liquid state with jJof rotation is phase broken only atvalue Texponentially ! 0EaQ~defined 0 0.1 0.2 0.3 0.4 0.5 clusions do not on the interactions inexistence has ‘‘Ising nematic’’ nu3% 4 J1 jS . Spin T rameter characterizes the direction of the and y axes, leadingT/Jto Isin Q !broken 2!=3, corresponding to J3 =J1 ! 0:5. order is destroyed either the by system thermal fluctuations (as in the degeneracy decaying spin correlations and no translation sym= 0.5, T = 0.67 < T , and different L. found in this paper. Eq. (10) for J where ! ! 1. There is no Lifshitz point at finite S because it as soon as the ground-state remains the 1 2 I spin merical evidence that the transition at Tc is indeed in the1/2 at the temperature T # stinguishes left-handed andLetter; right-handed in the regime of strong frustration (J3 =J c present see Fig. 1)same. or byThey quantum fluctuations (as in studies 1 ’ 0:5). are confirmed by numerical of the models ξspin~ Swas /T is preempted [13]metry by quantum effects within the dotted semiIsing universality class. Such Ising states nematic order[4] Tc ,ofisthe FIG. 3. T dependence specific This Letter will describe properties of the above model 11,13–17,23,24 length, ! finite for We begin by recalling [9] the ground of H at S ! spin Ref. [2]). Our large S results are therefore mentioned above. consonant with circle: here there is a T ! 0 spin gap ! # S exp$%~ cS&proposed and spin in Ref. [2] for S ! 1=2 in a T ! onal (3D) helimagnets, the phase transiDifferent 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 ! Nevertheless, the situation remains contradictory in 2D a spin-liquid phase at S ! 1=2 as derotation symmetryresults, is preserved. semicircular region extends between L ! 24 cS and / T L ! 120. Data for J tic and the chiral the orderpossibility parameters of occur c'S /1T 1Z gauge obtainedThis by classical Monte Carlo simulations and ~ spin-liquid phase described by a theory [5]. Thus ξ ~ e ξ ~ e vector Q ! "!; !# for J =J $ . For J =J > , the 1 2 spin 3 1 3 1 spin to the same ⊗ SO(2) class as 1 jS # !. Further details on the physics within the different behaviors o 4 4 scribed Refs. [2,3]; wehelimagnets will discussbelonging the quantum finite S Z2 over T # jJ for comparison (full dots and dashed line) 3 % 4J was found numerically thatinthe transition a theory described below, the are summarized Fig. 1 forincommensurate the groundIsing stateinhas planar antiferromagnetic 1 specific same nematic order can appear when spiral spin 2 . Spin the FFXY model and the antiferromagnet on the triangular the maximum of the heat. 4,5 jJ % J jS rotati phase diagram further towards the end of the Letter. We this region appearlimit at the the isLetter. 3 order or of the “almost-second-order” S !end 1. of There a T ! 0 state with long-range spiral 4 1 J1 /the 4 order at a wave vector Q~ !by "Q;thermal Q#, with Q decreasing from lattice. Garel and Doniach28 (see also Ref. 29) considered thespin order for J > 1 J . We order is destroyed either fluctuations (as in 3 ! 1. There i where !Jspin establish that at 0 < T < T # erromagnets on a body-centered tetragonal 3 c 1 Neel LRO 4 1 ! as J =J > and approaching Q ! !=2 monotonically Spiral LRO simplest helimagnet on a square lattice with an extra competing 3 1 Lifshitz point 1 2 4with present Fig. 1) or by quantum fluctuations (as in there Letter; is a phasesee broken $J3 % 4 J1 &S , above this state mple cubic lattice with an extra competing where a labels each plaquette of qu th is preempted [13] by 1. The spiral order is incommensurate for 14 < for J3 =J1 ! exchange coupling along one axis that is described by the 257206-1 © 2004 The American Physical Society 7 0031-9007=04=93(25)=257206(4)$22.50 Ref. [2]). Our large S results are therefore consonant with "1; 2; 3; 4# are its corners. The variabl along one axis. These systems belong discrete symmetry of lattice reflections about the x and y circle: here there is a T ! where Q !diagram 2!=3, corJ3 =J1 < 1, except at J3 =J1 ! 0:5 1. Phase of H^ in the limit S ! 1. The shaded Hamiltonian Néel antiferromagnet, while they axes, while spin rotationthe invariance is preserved. This % FIG.phase do)universality class as, e.g., the model possibility of a spin-liquid at S ! 1=2 as derotation symmetry is assu pres hasspins a broken about the x responding to an angle of 120 region between (seesymmetry Fig. 2). of lattice reflections ! 8 9 the two degenerate ground states i phase has ‘‘Ising nematic’’ order. We present strong nu1 ular lattice and V2,2 Stiefel model. The and y axes, leading to there Ising nematic order. IsingTtransition is 4 J1 jS # ! scribed in Refs.for [2,3]; we will discuss the quantum finite S Theover # jJ3 % ~! H = (J1 cos(ϕx − ϕx+a ) + J2 cos(ϕx − ϕx+2a ) Interestingly, each spiral state with Q "Q; Q# is Consequently, a phase with Ising nem merical evidence that the transition at Tc is indeed in the 2 ence and stabilization of the chiral spin$J % 1 JWe spin correlation at the Tc # 1 &S . The x phase diagram further towards the temperature end Letter. 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 ., Dzyaloshinsky-Moria interaction in 3D ussed recently in Ref. 10. ns (2D), the situation is rather different.11 sitions were observed with the temperature iral order appears as a result of the first of the Ising type. Another one is the 1 3 σ Emergent Ising order parameters j j Tc T T C C max 15 10 5 1 2 − Jb cos(ϕx − ϕx+b )), 2 length, !spin , is finite for all T > 0, with the T dependencies as (1)originally proposed in Ref. [2](for be obtained from the one numerical results contain str forQS!!!). 1=2This in astate T !cannot 0 shown, with c=2 ! c0 ! 8"jJ3 % 1 JOur 1 j; the crossovers between ~ byThus wave vector a global spin rotation. Instead, the spin-liquid phase described by awith Z gauge theory Q [5]. 4 continuous Ising phase transition betw 2 the different behaviors of !spin are at the dashed lines at T # where the sum runs over sites x = (xa ,xb ) of the lattice, a =the same Ising nematic order 0031-9007=04=93(25)=257206(4)$22.50 257206-1 © 20 with h#i ! 0, and a homogeneous 1 twoappear configurations arespin connected by a 2 global rotation can when spiral jJ 3 % 4 J1 jS . 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 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 constants J1,2 are positive. Using arguments of Ref. 30, theypresent Letter; see Fig. 1) or by global accompanied by a divergence in the symmetry of the classical ground state ' effects within the dotted semiquantum fluctuations (as in is preempted [13] isbyO"3# quantum concluded28 that at low temperatures the vertices are bound Ising nematic in collinear spin system ~1 · N ~ 2 = ±1 =N this talk: Emergent Ising orders in quantum two-dimensional triangular antiferromagnet at T=0 hsat H H= X hi,ji D ~i · S ~j Jij S U(1)*U(1) U(1)*U(1)*Z2 U(1)*Z2 U(1)*Z3 b hc2 G Z3 hc1 F 40 ⇣ J J 0 ⌘2 = S 3 J Z3*Z2 B U(1)*Z3*Z2 C2 C C1 A a U(1)*Z3 U(1)*U(1) O 0 U(1)*Z2 U(1)*U(1)*Z2 1 E 3 δcr 4 δ Spatially anisotropic model: classical vs quantum J H = ∑ J ijSi ⋅ S j − h ∑ S 〈 ij 〉 S=1 ′ J ̸= J 1 S= 2 0 1/3-plateau J’ i hsat 0 z i hsat h h Umbrella state: ! favored classically;! energy gain (J-J’)2/J Planar states: favored by ! quantum fluctuations;! energy gain J/S The competition is controlled by ′ 2 2 δ = S(J − J ) /J dimensionless parameter h X hi,ji hsat hsat H ~i · S ~j Jij S h H= D hsat U(1)*U(1) planar hc1 F ? b B Z3 A U(1)*U(1) 0 C C1 U(1)*Z2 a U(1)*Z3 O C2 1 E 3 δcr 4 40 ⇣ J J 0 ⌘2 δ= S 3 J 0 1/3-plateau hc2 G cone U(1)*Z3 0 h hsat 0 Emergent Ising orders in quantum two-dimensional triangular antiferromagnet at T=0 UUD-to-cone phase transition Z3 ! U (1) ⇥ Z2 or Z3 ! smth else ! U (1) ⇥ Z2 ? hsat H H= X hi,ji D ~i · S ~j Jij S U(1)*U(1) U(1)*U(1)*Z2 U(1)*Z3 b hc2 G C2 Z3 (UUD) hc1 F 40 ⇣ J J 0 ⌘2 = S 3 J Z3*Z2 B C C1 U(1)*Z2 (cone) A a U(1)*Z3 U(1)*U(1) O 0 U(1)*Z2 U(1)*U(1)*Z2 1 E 3 δcr 4 δ Low-energy excitation spectra ✏d2 9Jk 2 = hc2 h + 4 for δ < 3 -k2 d2 Magnetization plateau is collinear phase: preserves O(2) rotations about magnetic field -no gapless spin waves. Breaks only discrete Z3. Hence, very stable. vacuum of d1,2 hc2 40 S = (1 3 d1 ✏d1 = h 3Jk 2 hc1 + 4 hc1 0.6 0.6 = hsat = (9JS) 2S 2S J 0 /J)2 Bose-Einstein condensation of d1 (d2) mode at k =0 leads to 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 = (1 3 J 0 /J)2 parameterizes anisotropy J’/J extended symmetry: 4 gapless modes at the plateau’s end-point Out[24]= d2 -k -k22 +k2 Out[25]= δ=4 k1 = k2 = k0 vacuum of d1,2 d1 -k0 +k0 k0 = r 3 10S S>>1 Out[19]= = 40 S (1 3 J 0 /J)2 Magnetization plateau is collinear phase: preserves O(2) rotations about magnetic field -no gapless spin waves. Breaks only discrete Z3. -k1 +k1 Alicea, Chubukov, OS PRL 2009 Bosonization of 2d interacting magnons d†1,k0 +p d†2, k0 p d†1, k0 +q d†2,k0 q k0 p d1, k0 +q d2,k0 q } ⇣ 3 X = (p, q) d†1,k0 +p d†2, N p,q } (4) Hd1 d2 ( 3J)k02 (p, q) ⇠ |p||q| } † 1,p † 1,p 2,q ⌘ + h.c. † 2,q singular magnon interaction magnon pair operators 1,p = d1,k0 +p d2, k0 2,p = d1, k0 +p d2,k0 Out[25]= p p 1 2 2 1 Obey canonical Bose commutation relations in the UUD ground state [ 1,p , 2,q ] = 1,2 p,q ⇣ ⌘ 1 + d†1,k0 +p d1,k0 +p + d†2,k0 +p d2,k0 +p ! † † In the UUD ground state hd1 d1 iuud = hd2 d2 iuud = 0 1,2 p,q ★ 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 Equations of motion for Ψ - Hamiltonian h † 1,p h † 2,p `Superconducting’ solution with imaginary order parameter Instability = softening of twomagnon mode @ δcr = 4 - O(1/S2) no single particle condensate 6Jfp2 3 X 2 f h 1,p i = ⌦p N q q 6Jfp2 3 X 2 f h 2,p i = ⌦p N q q † 2,q 2,q i † 1,q 1,q i ⌥ h 1,p i = h 2,p i ⇠ i 2 p 1 1 X k0 p 1= SN p |p|2 + (1 /4)k02 hd1 i = hd2 i = 0 Chubukov, OS PRL 2013 hc2 Two-magnon condensate = Spin-current nematic state distorted ! Υ<0 Υ>0 umbrella uud hc1 spin-! current J J’ distorted ! umbrella δcr 4 J’ domain wall δ no transverse magnetic order 0i hS · S hSx,y i = 0 r r r is not affected Finite scalar (and vector) chiralities. Sign of ⌥determines sense of spin-current circulation hẑ · SA ⇥ SC i = hẑ · SC ⇥ SB i = hẑ · SB ⇥ SA i / ⌥ 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) Hd1 d2 1 X p / N (k k2+k0 ik0 k0 )2 + (1 † † (d d 1,k 2, k /4)k02 d1,k d2, k) X p2 k0 p ik0 † † (d d 1,p 2, 2 (p + k0 )2 + (1 /4)k0 continuum limit of DM in triangular lattice X ẑ · Sr ⇥ (Sr+a1 + Sr+a2 ) a1 p d1,p d2, B || z r a2 Mean-field approximation: (4) Hd1 d2 !D X k k0 k0 † † ( + )(d1,k d2, |k k0 | |k + k0 | k d1,k d2, 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 ) k) p) 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 1 Indian Institute of Astrophysics, Bangalore 560 034, India International Center for Theoretical Sciences (ICTS), Bangalore 560 012, India 3 Department of Physics, Indian Institute of Science, Bangalore 560 012, India 4 Department of Physics, Goa University, Taleigao Plateau, Goa 403 206, India 5 Centre for Quantum Information and Quantum Computing, Indian Institute of Science, Bangalore 560 012, India 6 Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 7 Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8 (Received 9 July 2012; revised manuscript received 15 March 2013; published 3 May 2013) 2 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 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 CMI in weakly coupled ladders. DOI: 10.1103/PhysRevB.87.174501 I. INTRODUCTION The effect of frustration in generating unusual states of matter such as fractional quantum Hall fluids or quantum spin liquids is an important and recurring theme in the physics of condensed matter systems.1,2 Recently, research in the field of ultracold atomic gases has begun to explore this area, gapped single particles; but spontaneously broken time-reversal PACS number(s): 67.85.Hj, 03.75.Lm, 75.10.Jm = spontaneous circulating fluctuations which can “sense” the local flux on a plaquette. In a recent paper, we have found numerical evidence for the currents existence of such a remarkable intermediate state in frustrated two-leg ladders of bosons for the so-called fully frustrated Bose-Hubbard (FFBH) model which has half a flux quantum per plaquette. We call this state a “chiral Mott insulator” (CMI) since it is fully gapped due to boson-boson interactions, exactly Phases of a triangular-lattice antiferromagnet near saturation hsat H H= X hi,ji ~i · S ~j Jij S D U(1)*U(1)*Z2 U(1)*U(1) U(1)*Z2 U(1)*Z3 b hc2 G C2 Z3 hc1 F 40 ⇣ J J 0 ⌘2 = S 3 J Z3*Z2 B C C1 A a U(1)*Z3 U(1)*U(1) O 0 U(1)*Z2 U(1)*U(1)*Z2 1 E 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 1 B < 2 U(1)*U(1) U(1)*Z3 ~ ~ hsat incommensurate planar U(1)*Z2 cone D commensurate planar (V) U(1)*U(1)*Z2 double cone C plateau, Z3 0 J 0 /J 1 Low-density expansion E0 /N = = (1) 1| µ(| 2 1 2 +| = 2| 2 )+ (0) 1 2 + 1 X ⇣ (J0 + 5Jk )2 = 16S J0 Jk k2BZ 1 (| 1 | 4 (1) +| 2| 4 )+ 2| 1| 9( J)2 = J 2 | 2| 2 1.6J S (J0 4JQ+k )2 ⌘ 3J + ⇡ JQ+k JQ 8S 1.6J S Phases of a triangular-lattice antiferromagnet near saturation U (1) ⇥ Z3 ! U (1) ⇥ U (1) Q = (4⇡/3, 0) ! Qi = incommensurate hsat H H= X hi,ji ~i · S ~j Jij S D U(1)*U(1)*Z2 U(1)*U(1) U(1)*Z2 U(1)*Z3 b hc2 G C2 Z3 hc1 F 40 ⇣ J J 0 ⌘2 = S 3 J Z3*Z2 B C C1 A a U(1)*Z3 U(1)*U(1) O 0 U(1)*Z2 U(1)*U(1)*Z2 1 E 3 δcr 4 δ OS, Jin Wen, Andrey Chubukov, PRL 2014 High-field phases: commensurate-incommensurate transition h A Q = (4⇡/3, 0) ! Qi = incommensurate fully polarized state B hsat U(1)*U(1) U(1)*Z3 ~ ~ incommensurate planar U(1)*Z2 cone D commensurate planar (V) plateau, Z3 B U(1)*U(1)*Z2 δ1 double cone C 1 δ3 A J0 δ2 A 0 J C J 0 /J Low-density expansion -> classical sine-Gordon model of the (relative) phase fluctuations e i3Q·r =1 first calculation of 3 3 X ⇣ (5Jk + J0 )(5JQ+k + J0 )JQ = 32S 2 (J0 Jk )(J0 JQ+k ) k2BZ k (5Jk + J0 )(Jk + J0 ) ⌘ 3J0 + ⇡ 2(J0 Jk ) 64S 2 0.69J S2 Phases of a triangular-lattice antiferromagnet near saturation U (1) ⇥ U (1) ! U (1) ⇥ Z2 or U (1) ⇥ U (1) ! smth else ! U (1) ⇥ Z2 ? hsat H H= X hi,ji ~i · S ~j Jij S D U(1)*U(1)*Z2 U(1)*U(1) U(1)*Z2 U(1)*Z3 b hc2 G C2 Z3 hc1 F 40 ⇣ J J 0 ⌘2 = S 3 J Z3*Z2 B C C1 A a U(1)*Z3 U(1)*U(1) O 0 U(1)*Z2 U(1)*U(1)*Z2 1 E 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 1 B < 2 U(1)*U(1) U(1)*Z3 ~ ~ hsat incommensurate planar U(1)*Z2 cone D commensurate planar (V) U(1)*U(1)*Z2 double cone plateau, Z3 C 0 1 J 0 /J Elementary excitation spectrum of the cone phase: Goldstone mode at k = +Q, gapped at Transition at BD: softening of the mode at -Q’ -Q’ -Q +Q 1.45(hsat h) p Q =Q+ hsat S 0 Q0 6= Q High-field phases, XXZ model: commensuratecommensurate transitions HXXZ = X x x J(Si Sj + y y Si Sj ) + z z J z Si Sj h fully polarized state B C A ~ ~ U(1)*Z3 V state U(1)*Z3 state co-planar c2 = 0.45/S = 0.53/S 3 3J = 2S hsat U(1)*Z2 cone =1 c2 c1 c1 z Sj j hi,ji h X 0.69J S2 commensurate ! non-coplanar 3JS 2 E✓ = (@x ✓)2 + S 4hsat Jz /J? µ3 3 3 cos[6✓] 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 fully polarized 4 plateau width 0.6 0.4 0.2 3 0.0 0.0 IC planar C planar 2 0.4 C planar R 0.6 0.8 1.0 cone SDW 1/3 1 0.2 plateau for all J’/J (crystal of spindowns; end-point at J’=0) SDW IC planar 0 0.0 quasi-collinear 0.2 0.4 0.6 R=1-J’/J 0.8 1.0 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]= h A fully polarized state B 2 1 -Q Q hsat U(1)*U(1) U(1)*Z3 ~ ~ incommensurate planar cone D commensurate planar (V) plateau, Z3 0 U(1)*Z2 B U(1)*U(1)*Z2 δ1 double cone C A 1 J δ3 A J0 δ2 C J 0 /J OS, Jin Wen, Andrey Chubukov, PRL 2014