Frustration-Driven Multi-Magnon Condensates

Frustration-driven multi-magnon condensates

Credit: Francis Pratt / ISIS / STFC

Oleg Starykh University of Utah !

RECENT PROGRESS IN LOW-DIMENSIONAL QUANTUM MAGNETISM LDQM2016 5-16 September 2016, EPFL, Lausanne Why do we do it?

Immense potential for practical applications Collaborators

Leon Balents, KITP, UCSB

Andrey Chubukov, FTPI, U Minnesota

Jason Alicea, Caltech General motivation: Exotic but ordered phases

ordered ! spin nematic! quantum! phases hidden order spin liquids spin nematic

composite order parameter Outline

• Experimental evidence: Frustrated ferromagnets and magnon binding

• Basic theory and some numerics

• Field theory of the Lifshitz point magnon “BCS” superconductor

• Spin-current state near the end-point of 1/3 magnetization plateau exotic superconductorof magnons RAPID COMMUNICATIONS

PHYSICAL REVIEW B 80, 140402͑R͒͑2009͒

Emergent multipolar spin correlations in a ﬂuctuating spiral: The frustrated ferromagnetic 1 spin-2 Heisenberg chain in a magnetic ﬁeld

Julien Sudan,1 Andreas Lüscher,1 and Andreas M. Läuchli2,* 1Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), CH-1015 Lausanne, Switzerland 2Max Planck Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany ͑Received 22 July 2008; published 7 October 2009͒ 1 We present the phase diagram of the frustrated ferromagnetic S= 2 Heisenberg J1 −J2 chain in a magnetic ﬁeld, obtained by large scale exact diagonalizations and density matrix renormalization group simulations. A vector chirally ordered state, metamagnetic behavior and a sequence of spin-multipolar Luttinger liquid phases up to hexadecupolar kind are found. We provide numerical evidence for a locking mechanism, which can drive spiral states toward spin-multipolar phases, such as quadrupolar or octupolar phases. Our results also shed light 1 on previously discovered spin-multipolar phases in two-dimensional S= 2 quantum magnets in a magnetic ﬁeld.

DOI: 10.1103/PhysRevB.80.140402 PACS number͑s͒: 75.10.Jm, 75.30.Kz, 75.40.Cx, 75.40.Mg

Spiral or helical ground states are an old and well- ploy exact diagonalizations ͑EDs͒ on systems sizes up to understood concept in classical magnetism,1 and several ma- L=64 sites complemented by density matrix renormalization terials are successfully described based on theories of spiral group ͑DMRG͒͑Ref. 11͒ simulations on open systems of states. For low spin and dimensionality however quantum maximal length L=384, retaining up to 800 basis states. fluctuations become important and might destabilize the spi- The classical ground state of Hamiltonian ͑1͒ is ferromag- ral states. Given that spiral states generally arise due to com- netic for J2 Ͻ1/4 and a spiral with pitch angle ␸ peting interactions, fluctuations are expected to be particu- =arccos͑1/4J2͒෈͓0,␲/2͔ otherwise. The Lifshitz point is larly strong. located at J2 =1/4. In a magnetic field the spins develop a A prominent instability of spiral states is their intrinsic uniform component along the field, while the pitch angle in 2 twist ͗Si ϫSj͑͘vector chirality͒. It has been recognized that the plane transverse to the field axis is unaltered by the field. finite temperature3 or quantum4 fluctuations can disorder the The zero field quantum mechanical phase diagram for 1 spin moment ͗Si͘ of the spiral, while the twist remains finite. S= 2 is still unsettled. Field theoretical work predicts a finite, 5 12,13 Such a state is called p-type spin nematic. In the context of but tiny gap accompanied by dimerization for J2 Ͼ1/4, quantum fluctuations such a scenario has been confirmed re- which present numerical approaches are unable to resolve. 6 cently in a ring-exchange model, while possible experimen- The classical Lifshitz point J2 =1/4 is not renormalized for tal evidence for the thermal scenario has been presented in.7 1 S= 2 , and the transition point manifests itself on finite sys- The twist also gained attention in multiferroics, since it tems as a level crossing between the ferromagnetic multiplet couples directly to the ferroelectricity.8 and an exactly known singlet state.14 The theoretical phase In this Rapid Communication we provide evidence for the diagram at finite field has recently received considerable existence of yet a different instability of spiral states toward attention,15–17 triggered by experiments on quasi one- spin-multipolar phases. The basic idea is that many spin- dimensional cuprate helimagnets.18–20 One of the most pecu- multipolar order parameters are finite in the magnetically liar features of the finite size magnetization process is the NaCuMoO4 OH as a Candidate Frustrated J1–J2 Chain Quantum Magnet ( ) ordered spiral state, but that under a suitable amount of fluc- appearance of elementary magnetization steps of Kazuhiro Nawa1,∗ Yoshihiko Okamoto1,† Akiratuations Matsuo1, Koichi the Kindo primary1, Yoko Kitahara spin2, Syota order Yoshida2 is, lost, while a spin- z Shohei Ikeda2, Shigeo Hara3, Takahiro Sakurai3,SusumuOkubo4, Hitoshi Ohta4, and Zenji Hiroi1 ⌬S =2,3,4 in certain J2 and m regions. This has been attrib- 1Institute for Solid State Physics, Themultipolar University of Tokyo, orderKashiwa, parameter Chiba 277-8581, Japan survives. We demonstrate this 2Graduate School of Science, Kobe University, Nada, Kobe 657-8501, Japan uted to the formation of bound states of spin flips, leading to 3Center for Supports to Research and Educationmechanism Activities, Kobe University, based Nada, on the Kobe 657-8501, magnetic Japan field phase diagram of a 4MolecularNew Photoscience Research system: Center, Kobe University, Nada, Frustrated Kobe 657-8501, Japan dominant spin-multipolar correlations close to saturation. A 1 (Dated: September 5, 2014) prototypical model, the frustrated S= 2 Heisenberg chain detailed phase diagram is however still lacking. In a frustrated J1–J2 chain with the nearest-neighbor ferromagnetic interaction J1 and the next- nearest-neighbor antiferromagnetic interactionwithJ2,novelmagneticstatessuchasaspin-nematicstate ferromagnetic nearest-neighbor and antiferromagnetic We present our numerical phase diagram in the J2 /͉J1͉ vs. are theoretically expected. However,ferromagnet theynext have been nearest-neighbor rarely examined in experiments interactions. because of the Furthermore we show difficulty in obtaining suitable model compounds. We show herethatthequasi-one-dimensional m/msat plane in Fig. 1. At least five different phases are antiferromagnet NaCuMoO4(OH), which comprises edge-sharing CuO2 chains, is a good candidate J1–J2 chain1d antiferromagnet. S=1/2 The exchangechainthat interactions this instability are estimated as providesJ1 = −51 K a and naturalJ2 and unified understand- present. The low magnetization region consists of a single =36Kbycomparingthemagneticsusceptibility,heatcapacity, and magnetization data with the data obtained using calculations by theing exact diagonalizat of previouslyion method. High-field discovered magnetization two-dimensional spin- vector chiral phase ͑gray͒. Below the saturation magnetiza- measurements at 1.3 K show a saturation above 26 T with little evidence9, of10 a spin nematic state expected just below the saturation field, whichmultipolar is probably due phases. to smearing effectsJ1 caused<0 by thermalFM tion we confirm the presence of three different multipolar fluctuations and the polycrystalline nature of theTo sample. be specific, we determine numerically the phase dia- Luttinger liquid phases ͑red, green, and blue͒. The red phase 16 Low-dimensional quantum spin systems with geomet-gram of the following Hamiltonian: extends up to J2 ϱ, and its lower border approaches TABLE I. Candidate compounds for the J1–J2 chain system. rical frustration and/or competing magnetic interactions + → Listed are the nearest-neighbor intrachain interaction J1,the m=0 in that limit. All three multipolar liquids present a have attracted much attention in the field of magnetism. z J2>0 AF next-nearest-neighbor interaction J2,thebondanglesofCu- Low dimensionality, quantum fluctuations, and frustra- H = J1 Si · Si+1 + J2 Si · Si+2 − h Si , ͑1͒ O-Cu paths for J1͚,theantiferromagnetictransitiontemper-͚ ͚ crossover as a function of m/msat, where the dominant cor- tion are three ingredients that may effectively suppress ature at zero field TiN,andthesaturationfieldiHs. i conventional magnetic order and lead us to unconven- relations change from spin-multipolar close to saturation to tional magnetic order or exotic ground states such as a Compound J1, J2 ∠ Cu-O-Cu TN Hs quantum spin liquid[1, 2]. and we set J1 =−1, J2(K)Ն0 in(deg) the (K) following. (T) Si are spin-1/2 spin-density wave ͑SDW͒ character at lower magnetization. AfrustratedJ1–J2 chain of spin 1/2 defined as Li2ZrCuO4[12, 13] −151, 35 94.1 6.4 - operatorsRb2Cu at2Mo3 siteO12[14,i 15],− while138, 51 89.9,h 101.8denotes< 214 the uniform magnetic One also expects a tiny incommensurate p=2 phase close to z H = J1 ! sl · sl+1 + J2 ! sl · sl+2 − h ! sl (1) 91.9, 101.1 z 17 field. The magnetization− is defined as m 1 L S . We em- the p=3 phase, which we did not aim to localize in this l l l PbCuSO4(OH)2[16–18] 100, 36 91.2, 94.3 2.8 5.4 ª / ͚i i LiCuSbO4[19] −75, 34 89.8, 95.0 < 0.1 12 provides us with an interesting example: the competi- 92.0, 96.8 tion between the nearest-neighbor (NN) ferromagnetic LiCu2O2[20–22] −69, 43 92.2, 92.5 22.3 110 interaction J1 and the next-nearest-neighbor (NNN)1098-0121/2009/80 an- LiCuVO4[23–31]͑14͒−/14040219, 44 95.0͑4͒ 2.1 44.4 140402-1 ©2009 The American Physical Society − tiferromagnetic interaction J2 causes various quantum NaCuMoO4(OH) 51, 36 92.0, 103.6 0.59 26 states in magnetic fields h[3–7]. Realized in low fields is a long-range order of vector chirality defined as (sl × s l+n)z (n =1, 2). As the fieldTeVO increases,4 spin correlations K. Nawa et al, arXiv:1409.1310 change markedly because bound magnon pairs are stabi- such as Li2ZrCuO4[12, 13], Rb2Cu2Mo3O12[14, 15], lizedPregelj by ferromagnetic et al., Nat.Comm.2015J1. The bound magnon pairs form PbCu(SO4)(OH)2[16–18], LiCuSbO4[19], LiCu2O2[20– a spin density wave (SDW) in medium fields, whereas, in 22], and LiCuVO4[23–31], the key parameters of which high fields just below the saturation of magnetization, are listed in Table I. These compounds commonly they exhibit Bose–Einstein condensation into quantum have edge-sharing CuO2 chains made of CuO6 octahe- arXiv:1409.1310v1 [cond-mat.str-el] 4 Sep 2014 multipolar states[8–11]. One of the multipolar states ex- dra. NN Cu spins are magnetically coupled with each pected just below the saturation is a quadrupolar state other through two superexchange Cu–O–Cu paths with of magnon pairs called a spin nematic state, analogous approximately 90◦ bond angles, while NNN Cu spins are to nematic liquid crystals. coupled through two super-superexchange Cu–O–O–Cu To explore these quantum states theoretically pre- paths. Thus, according to the Goodenough–Kanamori dicted for the frustrated J1–J2 chain, many experimen- rule, J1 should be ferromagnetic while J2 can be antifer- tal studies have been performed on quasi-1D compounds romagnetic. This is in fact the case for these candidate compounds, which causes frustration in the J1–J2 chains. Among these compounds, the most often studied is ∗ [email protected] LiCuVO4 with J1 = −19 K and J2 = 44 K[25]. It has † Present address: Department of Applied Physics, Graduate been shown using large single crystals that LiCuVO4 ex- School of Engineering, Nagoya University, Chikusa, Nagoya 464- hibits an incommensurate helical order at low fields[25– 8603, Japan 29], which may be a 3D analogue of the vector chirality PRL 2016 multi-polar states with 9 p 2

↵ = J / J 2,e↵ | 1,e↵ | Huge 1/3 magnetization plateau ! Phase diagram

small plateau’s onset ﬁeld of 27 Tesla, relative to J ~ 100 K, 1/3 T suggest the presence of ferromagnetic plateau exchange interactions N? SDW

1/3 1K SDW plateau

? spin nematic? 1T 26T B

FIG. 2 (color online). (a) 51V NMR spectra measured on a single-maydomain piece be of a crystala spin in nematic?? magneticH. Ishikawa fields between et al 15, PRL and 30 2015 T applied perpendicular to the ab plane at T = 0.4 K. (b) Magnetization curve of single crystals (top, black line) and its field derivative (bottom, red line) in B

ab at 1.4 K after the subtraction of the Van Vleck paramagnetic magnetization (MVV). Magnetization deduced from the center of the gravity of the NMR spectra is also plotted (top, blue circles). Expected spin structures in phases II and III are schematically depicted in the inset.

But the model keeps changing…

1/3 plateau N? SDW

FIG. 2 (color online). (a) 51V NMR spectra measured on a single-domain piece of a crystal in magneticH. Ishikawa fields between et al 15, PRL and 30 2015 T applied perpendicular to the ab plane at T = 0.4 K. (b) Magnetization curve of single crystals (top, black line) and its field derivative (bottom, red line) in B

Outline

• Experimental evidence: Frustrated ferromagnets and magnon binding

• Basic theory and some numerics

• Field theory of the Lifshitz point

• Spin-current state near the end-point of 1/3 magnetization plateau RAPID COMMUNICATIONS

PHYSICAL REVIEW B 80, 140402͑R͒͑2009͒

Emergent multipolar spin correlations in a ﬂuctuating spiral: The frustrated ferromagnetic 1 spin-2 Heisenberg chain in a magnetic ﬁeld

DOI: 10.1103/PhysRevB.80.140402 PACS number͑s͒: 75.10.Jm, 75.30.Kz, 75.40.Cx, 75.40.Mg

0 1/5 1 J2/(|J1|+J2) RAPID COMMUNICATIONS

PHYSICAL REVIEW B 80, 140402͑R͒͑2009͒

Emergent multipolar spin correlations in a ﬂuctuating spiral: The frustrated ferromagnetic 1 spin-2 Heisenberg chain in a magnetic ﬁeld

DOI: 10.1103/PhysRevB.80.140402 PACS number͑s͒: 75.10.Jm, 75.30.Kz, 75.40.Cx, 75.40.Mg

0 1/5 1 J2/(|J1|+J2)

z=2 z=4 z=1 (spin-wave) dispersion 2 4 3/2 ! k ! k ! (4J2 J1 ) k Ki ⇠ ⇠ ⇠ | | | | Approaching from fully polarized state — Multipolar phases

! (k2 Q2)2 (h h) ⇠ sat

-Q Q

H/(|J1|+J2) phases with bound FM complexes made out of p=2 p magnons , 2008 , 2009 3 et al et al 4 VC (p=1) 0 1 Sudan 1/5 Hikihara Lifshitz point J2/(|J1|+J2) J = J /4 2 | 1| Magnon BEC

! (k2 Q2)2 (h h) ⇠ sat

1-magnon Emin = ✏1(k)+h

-Q Q

Emin Condensation: z + p S =-1 ak = N Q k,Q h i iQx S e n h n i⇠ Q hsat h Magnon (quasi-)BEC 2 S(4J2 J1 ) hsat = | | 4J2 Today: condensation of magnon pairs

1-magnon Emin = ✏1(k)+h

2-magnon bound state Emin = ⌦2(k)+2h Sz=-2 Emin z S =-1 “molecular” bound state Min ⌦ =2✏ (Q) { 2} 1 bound hsat,2 = hsat + bound/2 hsat hsat,2 h Formation of molecular fluid For d>1 at T=0 this is a molecular BEC = true spin nematic Hidden order θ

No dipolar order S =0 φ h n i + i j /⇠ z S S e| | S =1 gap h i j i⇠

Nematic order SS = =0 h n n+ai 6 x x y y x y y x 2 SS = S S S S i(S S + S S ) sin ✓(cos 2' i sin 2') h n mi h n m n m n m n m i⇠ nematic Magnetic quadrupole moment → director Symmetry breaking U(1) Z2 -θ θ think of a fluctuating fan state: φ is constant, while θ fluctuates

in the interval (θ0, - θ0) Condensate of bound magnon pairs

S =0 SS = =0 h n i h n n+ai 6

Ferromagnetic J1 < 0 produces attraction in real space quasi-1d 2d

Chubukov 1991 Kecke et al 2007 Kuzian and Drechsler 2007 Hikihara et al 2008 Sudan et al 2009 Zhitomirsky and Tsunetsugu 2010 Shannon, Momoi, Sindzingre PRL 2006 RAPID COMMUNICATIONS

PHYSICAL REVIEW B 80, 140402͑R͒͑2009͒

Emergent multipolar spin correlations in a ﬂuctuating spiral: The frustrated ferromagnetic 1 spin-2 Heisenberg chain in a magnetic ﬁeld

DOI: 10.1103/PhysRevB.80.140402 PACS number͑s͒: 75.10.Jm, 75.30.Kz, 75.40.Cx, 75.40.Mg

0 1/5 1 J2/(|J1|+J2) s=1/2 s=1/2 s=1,3/2,2 Numerics: nematicity and st RAPID COMMUNICATIONS 1 orderSUDAN, seem LÜSCHER, AND LÄUCHLI connected? PHYSICAL REVIEW B 80, 140402͑R͒͑2009͒ 0.03 0.03 J2/J1 -0.25-0.275 -0.3 -0.35 -0.4 -0.5 -0.75 -1 nnn Bond 1 ? 0.025 0.025 p=4 p=3 p=2 0.02 0.02 0.8 VC VC VC octupolar quadrupolar 2

κ 0.015 0.015 p=4 p=3 p=2

0.6 hexadecupolar SDW (p=2) sat 1d frustrated chain SDW Sudan0.01 et al,metamagnetic 2009 metamagnetic 0.01 SDW (p=3) 1 nn Bond

m/m (p=4) 0.8

0.4 0.6 sat 0.005 0.005

0.4 h/h 0.2 (a) (b) (c) 0 0 0 0.2 0 0.2 0.4 0.6 0 0.1 0.2 0.3 0 0.2 0.4 0.6

m/msat m/msat m/msat Vector Chiral Order 0 -4 -3.5 -3 -2.5 -2 -1.5 -1 FIG. 2. ͑Color online͒ Squared vector chirality order parameter ␬2 ͓Eq. ͑2͔͒ in the low magnetization phase for different values of J1/J2 J2 /͉J1͉ as a function of m/msat: ͑a͒ J2 /͉J1͉=0.27, ͑b͒ J2 /͉J1͉=0.32, 1stFIG. order 1. ͑Color online͒ Phase2nd diagram order of the frustrated ferromag- and ͑c͒ J2 /͉J1͉=0.38. The order parameter vanishes in the multipo- netic chain ͑1͒ in the J1 /J2 vs m/msat plane. The gray low-m region lar Luttinger liquids. exhibits vector chiral long range order. The colored regions denote spin-multipolar Luttinger liquids of bound states of p=2,3,4 spin multipolar phases at larger m seems to occur generically via flips. Close to saturation the dominant correlations are multipolar, metamagnetic behavior ͑cf. left and right panels of Fig. 2͒. while below the dashed crossover lines, the dominant correlations For the parameter set in the middle panel we expect the same 2d frustrated are of SDW p type. The tiny cyan colored region corresponds to an ͑ ͒ behavior, but it can’t be resolved based on the system sizes incommensurate p=2 phase. The white region denotes a metamag- square lattice used. netic jump. Finally the scribbled region close to the transition Hamiltonian ͑1͒ presents unusual elementary step sizes J1 /J2 −4 has not been studied here, but consists most likely of a z z low field→ vector chiral phase, followed by a metamagnetic region ⌬S Ͼ1 in some extended J2 /͉J1͉ and m domains, where ⌬S 15 extending up to saturation magnetization. The inset shows the same is independent of the system size. This phenomenon has been explained based on the formation of bound states of diagram in the J1 /J2 vs h/hsat plane. p=⌬Sz magnons in the completely saturated state, and at finite m/m a description in terms of a single component study. Finally the multipolar Luttinger liquids are separated sat Shannon, Momoi, Sindzingre PRL 2006 Luttinger liquid of bound states has been put forward.16,17 from the vector chiral phase by a metamagnetic transition, We have determined the extension of the ⌬Sz =2,3,4 regions which occupies a larger and larger fraction of m as in Fig. 1, based on exact diagonalizations on systems sizes J 1 4+, leading to an absence of multipolar liquids com- 2 / up to 32 sites and DMRG simulations on systems up to 192 posed→ of five or more spin flips. In the following we will sites. The boundaries are in very good agreement with pre- characterize these phases in more detail and put forward an vious results15 where available. The ⌬Sz =3 and 4 domains explanation for the occurrence and locations of the spin- form lobes which are widest at m and whose tips do not multipolar phases. sat extend down to zero magnetization. The higher lobes are For mϾ0 we reveal a contiguous phase sustaining long successively narrower in the J direction. We have also range vector chiral order21 breaking discrete parity symme- 2 ͑ searched for ⌬Sz =5 and higher regions, but found them to be try , similar to phases recently discovered for J ,J Ͼ0.22,23 ͒ 1 2 unstable against a direct metamagnetic transition from the Direct evidence for the presence of this phase is obtained vector chiral phase to full saturation. Individual bound states from measurements of the squared vector chiral order param- of pՆ5 magnons do exist ͑see below͒, but they experience a eter, too strong mutual attraction to be thermodynamically stable. 2 z z An exciting property of the Luttinger liquids of p bound ␬ ͑r,d͒ ª ͓͗S0 ϫ Sd͔ ͓Sr ϫ Sr+d͔ ͘. ͑2͒ magnon states17 is that the transverse spin correlations are In Fig. 2 we display DMRG results for long distance corre- exponentially decaying as a function of distance due to the lations between J1 bonds ͑d=1, black symbols͒ and J2 bonds binding, while p-multipolar spin correlations ͑d=2, red symbols͒ obtained on a L=192 system. The three chosen values of J reflect positions underneath each of the p−1 p−1 2 1 1/K three spin-multipolar Luttinger liquids shown in Fig. 1. The + − r ͟ S0+n͟ Sr+n ϳ͑− 1͒ ͑3͒ non-monotonic behavior of the correlations at very small m ͳ n=0 n=0 ʹ ͩ r ͪ is probably a finite size artifact or convergence issue. Beyond the long range order in the vector chirality, the system be- are critical with wave vector ␲ ͑multipolar correlations with haves as a single channel Luttinger liquid ͑with central pЈϽp also decay exponentially͒. p=2,3,4 correspond to charge c=1, confirmed by our DMRG based entanglement quadrupolar, octupolar, and hexadecupolar correlations, re- entropy analysis24͒ with critical incommensurate transverse spectively. Therefore they can be considered as one- spin correlation functions.22 The transition to the spin- dimensional analogs of spin multipolar ordered phases found

140402-2 Outline

• Experimental evidence: Frustrated ferromagnets and magnon binding

• Basic theory and some numerics

• Field theory of the Lifshitz point

• Spin-current state near the end-point of 1/3 magnetization plateau RAPID COMMUNICATIONS

PHYSICAL REVIEW B 80, 140402͑R͒͑2009͒

Emergent multipolar spin correlations in a ﬂuctuating spiral: The frustrated ferromagnetic 1 spin-2 Heisenberg chain in a magnetic ﬁeld

DOI: 10.1103/PhysRevB.80.140402 PACS number͑s͒: 75.10.Jm, 75.30.Kz, 75.40.Cx, 75.40.Mg

0 1/5 1 J2/(|J1|+J2) Notes on Lifshitz point etc.

Leon Balents1 1Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: August 23, 2014)

I. NLSM Can we see this formally somehow? Let us try rescaling to bring out the behavior for small .Weletx K/x ! and ⌧ K ⌧, where the second rescaling follows from A. Classical limit 2 p the linear! derivative nature of the Berry phase term. The magnetization itself does not rescale asm ˆ is a unit vector. Let us consider the Non-Linear sigma Model (NLsM) Carrying out this rescaling, we find which should describe the behavior near the Lifshitz point of the J1 J2 chain. The action in 1+1 dimen- Lifshitz K Point 2 2 2 sions is S = dxd⌧ is B[ˆm] @xmˆ + @ mˆ A | | | x | r Z Balents, Starykh PRL 2016 4 S = dxd⌧ is [ˆm] @ mˆ 2 + K @2mˆ 2 v @xmˆ hmˆ z , (5) AB | x | | x | | | Z 4 2 +u @xmˆ hmˆ z . (1) where we defined v = u/K and h = hK/ .Wesee | | Unusual QCP:that order-to-order when /K 1, the action istransition large in dimensionless • ⌧ Here s is the spin and B is the Berry phase term de- terms, and we expect a saddle point approximation to scribing those spins. ItA can be written in various ways, apply. This is precisely the classical limit! Note that this for example is valid whenσu/K is fixed, and also h 2/K,whichm • Effective actionfixed - the NL overallM field scalefor of unit the problem. vector⇠ 1 B = du mˆ @⌧ mˆ @um,ˆ (2) A 0 · ⇥ Z 2 2 2 4 where we introduceS a= fictitiousdxd auxiliary⌧ is coordinateB[ˆmu]+ @xmˆ + B.K Saddle@xmˆ point+ u @xmˆ hmˆ z such thatm ˆ (u = 0) =z ˆ andm ˆ (u = 1) =Am ˆ is the | | | | | | physical value, or equivalently,Z Berry To find the actual saddle point, we make an assump- tiontunes that it is of the formtwo of an umbrellasymmetry state (I tried mˆ 1@⌧ mˆ 2 mˆ 2@⌧ mˆ 1 B = . (3) also to look for a planar state, but it seemed to be A 1+m ˆ allowed interactions 3 phase energeticallyQCP unfavorable). To avoid having to rescale, we work in the original variables of Eq. (1).4 Letm ˆ = The main important point for us is that B contains a at O(q ) single derivative of imaginary time ⌧. Aterm (' cosJqx,1 ' sin qx,4J21 '2). Then the action is just the /integral| of| the energy density The action in Eq. (1) needs a condition for stability p against large gradients ofm ˆ . To get it, we note that by 2 2 4 2 4 4 di↵erentiation twice ofm ˆ mˆ = 1 we obtain " = q ' + Kq ' + uq ' h( 1 '2 1), (6) All· properties near Lifshitz point obey “one parameter 2 2 2 where we chose to add a constant h factorp so that " =0 @xmˆ = mˆ @xmˆ @xmˆ , (4) | | · universality”| | dependentwhen ' = 0. This is easilyupon minimized u/K over ratio wavevector where the final inequality is obvious. This in turn implies 2 2 4 2 that @xmˆ > @xmˆ , which is enough to show stability q = , (7) is present| | so long| as| u + K>0. This means we may 2(K + u'2) consider negative u so long as u> K. whence Note that for < 0, or suciently large h, the NLsM has an exact ferromagnetic ground state, described by 2'2 " = h( 1 '2 1). (8) just constantm ˆ , or a wavefunction of fully polarized 4(K + u'2) spins. This is a fully classical state. When > 0 and p h is not too large, however, there will be an incommen- We can see by direct expansion in a Taylor series in ' surate ground state, and thereby quantum fluctuations that a second order transition is possible for u> K/4. will occur. In this regime the NLsM is non-trivial. Nev- For more negative u we can find the first order point ertheless, we can expect that near the Lifshitz point, at by standard means. There are two conditions. First, a least on scales that are not too long, a classical descrip- minimum exists @'" = 0, and second, the minimum has tion should be correct (we expect that the L and the same energy as the trivial one, "(') = 0. This gives classical limits may not commute, but at least!1 the clas- two conditions which determine the order parameter ' sical analysis should lead us to a first understanding). at the transition and the field h at this point. According Lifshitz Point S = dxd⌧ is [ˆm]+ @ mˆ 2 + K @2mˆ 2 + u @ mˆ 4 hmˆ AB | x | | x | | x | z Z • Intuition: behavior near the Lifshitz point should be semi-classical, since “close” to FM state which is classical

K K x x ⌧ ⌧ ! ! 2 s| | K S = dxd⌧ is [ˆm] + sgn() @ mˆ 2 + @2mˆ 2 + v @ mˆ 4 hmˆ AB | x | | x | | x | z r Z u hK Large parameter: v = h = K 2 saddle point! Lifshitz point K S = dxd⌧ is [ˆm] + sgn() @ mˆ 2 + @2mˆ 2 + v @ mˆ 4 hmˆ AB | x | | x | | x | z r Z v derives from quantum fluctuations

Need it be positive? 2 2 mˆ mˆ =1 @xmˆ @xmˆ = mˆ @ mˆ @ mˆ · · · x  | x | Theory is stable for v>-1 In fact, v<0 3 v = • Semiclassical large s limit: 2s 9 vs=1/2 = 0.42 • s=1/2 estimate: (2 + p7)2 ⇡ Saddle point K S = dxd⌧ is [ˆm] + sgn() @ mˆ 2 + @2mˆ 2 + v @ mˆ 4 hmˆ AB | x | | x | | x | z r Z Solution: cos(qx + ) | | mˆ = sin(qx + ) 0 ±| | 1 1 2 | | @ A cone state p 1 Obtain v = q, versus h, v, many physical

quantities 0 0.5 1 1.5 Saddle point K S = dxd⌧ is [ˆm] + sgn() @ mˆ 2 + @2mˆ 2 + v @ mˆ 4 hmˆ AB | x | | x | | x | z r Z Solution: cos(qx + ) | | mˆ = sin(qx + ) 0 ±| | 1 1 2 | | @ A cone state p 1 Obtain v = q, versus h, v, many physical metamagnetism

quantities 0 0.5 1 1.5 Saddle point K S = dxd⌧ is [ˆm] + sgn() @ mˆ 2 + @2mˆ 2 + v @ mˆ 4 hmˆ AB | x | | x | | x | z r Z h h first order K 2 K hc = 8K v (1 v ) | | | | second order p p FM FM 2 h = c 2K IC cone IC cone spiral spiral 0 0 1 1/4 Note: at saddle point level there is no scale for δ Saddle point predicts 1st order transition for S < 6 ! = h EFM Econe K FM ✏1 =0

3 v = ﬁnd Scr=6 2s 0 saddle point Missing? = h EFM Econe H/(|J1|+J2) K FM FM ✏1 =0 2

3 4 VC 0 dimerized 1 0 J2/(|J1|+J2)

J1-J2 saddle point misses metamagnetic endpoint and multipolar phases Metamagnetic endpoint? = h EFM Econe K FM ✏1 =0 a2 EFM Econe ⇠

0 2 Quantum corrections /K penalize Econe but not EFM ⇠ E ⇥ p =+f(v)5/2 Econe 2

 and , quartic in derivatives, which is crucial in the follow- h>hc, the solution is simply the ferromagnetic one, with ing. The term has been ignored in previous field theoretic ' =0. On reducing the field, there are two possible behav- approaches[16, 17]. iors. For > /4 (v<1/4), a continuous transition occurs 2 The action (2) needs a condition for stability against large at the critical field hc = h0 = /(2). The “order param- gradients of mˆ . Starting from constraint mˆ mˆ =1, it is easy eter” ', which represents the local moment transverse to the · to obtain @2mˆ 2 > @ mˆ 4, which is enough to show stability magnetic field, increases smoothly from zero below h . This | x | | x | 0 is present so long as +  > 0. This means negative in (2) corresponds to the point of local instability of the FM phase to is allowed so long as > . single magnons, which Bose condense when their energy van- The action describes several distinct dynamical regimes. ishes at h . For < /4 (v>1/4), the transition occurs 0 For < 0, the excitations above the ground states are quadrat- discontinuously at hc >h0, at which point the ferromagnetic ically dispersing spin waves, ! kz, characterized by the state is still locally stable. The order parameter jumps to a ⇠ dynamical critical exponent z =2, which is easily seen by non-zero value ' for h = h 0+. This is a metamagnetic c c equating the linear ⌧ derivative in with the second spatial transition, described by AB derivative in the term. For =0, the dynamics changes to 2 z =4. For > 0, the theory is more non-trivial, and there is 2 2pv 1 2 ' = ,hc = ,q = , even a z =1regime (see below). c v 8pv(1 pv) c 4(1 pv) Asymptotic solubility: Physically, the absence of fluctua- (6) tions in the FM state suggests a saddle point approximation which hold for 1/4

⌘⌘¯ ⌘¯ + ⌘ ⌘¯ ⌘ ⌘⌘¯ mˆ = 2 [ eˆ1 + i eˆ2]+(1 )ê3, (7) r s 2ps 2ps s effectivewhere Bogoliubov the rotating dreibein Hamiltonianeˆj(x) are chosen as follows: eˆ eˆ =ê mˆ . The fields ⌘¯, ⌘ describe magnons, trans- 1 ⇥ 2 3 ⌘ sp FIG. 1. Saddle point result for the magnetization m(h) for different verse fluctuations of the magnetization. To quadratic order3 the S = Ssp + dx d⌧ ⌘@⌧ ⌘ + H(⌘, ⌘) +O(⌘ ) values of interaction parameter v, which is shown next to each curve. action in Eq. (2) becomes{ S = d⌧ dx} ⌘@¯ ⌧ ⌘ + Hfluct , which showsZ that ⌘¯, ⌘ are canonical Bose operators, and R ⇥R ⇤ The saddle point of Eq. (2) with minimum action describesdiagonalizationHfluc(¯⌘, ⌘) is gives a Hamiltonian. correction Fourier to GS transforming energy it into a cone (umbrella) state: momentum space shows that Hfluc contains both normal and anomalous terms: mˆ =(' cos qx, ' sin qx, 1 '2), (5) sp Hfluc = 2Ak⌘¯k⌘k + Bk(⌘k⌘ k +¯⌘k⌘¯ k). (8) with 0 ' 1 and q functions of thep parameters of the ac-   k tion. Solutions with both sign of q are degenerate, which re- X flects spontaneous breaking of reflection symmetry and chiral Here coefficients Ak,Bk are functions of momentum k and order: zˆ mˆ @ mˆ = '2q =0. For sufficient large field, depend on parameters , ,v,hand ' of the saddle point ac- · sp ⇥ x sp 6 Text

Metamagnetic endpoint? = h EFM Econe K FM ✏1 =0 a2 EFM Econe ⇠ f(v)5/2

0 2 Corrected first order curve bends slightly downward to intersect second order line Text

Metamagnetic endpoint? = h EFM Econe ✏1 =0 K 2 FM FM cone a E E ⇠ f(v)5/2

2 2 0 c

v = 1/4 " Control? 3 2 2 5/2 FM cone " " E E |✏1=0 ⇠ "2 1 c ⇠ ⌧ Text

Instabilities

• Choose EFM=0 FM h =0 Econe K ✏1 =0

0 2

What about multi-particle instabilities? Low density limit

1/2 mˆ x + imˆ y = 2 mˆ z =1 iqx iqx Low energy 1e + 2e ⇠ 2 (@ + h 4@2) L ⇠ a ⌧ 2K x a 2 2 +1[( 1 1) +( 2 2) ]+2 1 1 2 2

2 2 = (1 + 4v) 2 = (5 + 4v) 1 4K K 2 "2 < 0 + ⇠ ⇠ Low density limit

@2 H = 4 +2 (x x ) @x2 1 i j i i i

H/(|J1|+J2) FM 2 3 4 VC 0 dimerized 1 J2/(|J1|+J2) expect bubbles of multi-magnon condensates to proliferate near the metamagnetic end-point A guess • n magnon bound state 1 ` n ⇠ n ⇠ n" | 1| • Matching to de Broglie wave length 1 1 `0 ⇠ q0 ⇠ p • Suggests maximum bound state

1 1 nmax ⇠ ✏p ⇠ ✏2 (at this scale, 3-body interactions enter) Text

Instabilities

• Choose EFM=0

FM ✏4 =0 h ✏3 =0 cone =0 ✏ =0 K E 2 ✏1 =0

0 2 At low density level it appears higher bound state instabilities dominate Text

Instabilities

• Choose EFM=0 FM h cone =0 K E

0 2 Expect that n-boson bound states bend with increasing n to approach continuum line Text

Instabilities

• Choose EFM=0 FM h 2 4 3 =0 5 K Econe

0 2 Expect that n-boson bound states bend with increasing n to approach continuum line Text

2-magnon check of the proposed scenario

•Compute exact 2-magnon energy in QFT FM h K 2-magnon =0 Econe

2 2 2 2 2 .07s ✏ c2 .2s ✏ 0 c ⇠ ⇠

Separation of metamagnetism and multipole formation d>1

d 1 2 2 2 2 4 S = dxd yd⌧ is [ˆm]+ @ mˆ + c @ mˆ + K @ mˆ + u @ mˆ hmˆ AB | x | | y | | x | | x | z Z • Rescaling: K K pcK x x ⌧ ⌧ y y ! ! 2 s| | !

p d d 1 K c d 1 2 2 2 2 4 S = dxd yd⌧ is [ˆm] + sgn() @ mˆ + @ mˆ + @ mˆ + v @ mˆ hmˆ d 1/2 AB | x | | x | | y | | x | z Z ∴ Similar theory applies in d>1, and very similar conclusions apply Summary

Lifshitz point is a “parent” of many phases

S = dxd⌧ is [ˆm]+ @ mˆ 2 + K @2mˆ 2 + u @ mˆ 4 hmˆ AB | x | | x | | x | z Z Outline

• Experimental evidence: Frustrated ferromagnets and magnon binding

• Basic theory and some numerics

• Field theory of the Lifshitz point

• Spin-current state near the end-point of 1/3 magnetization plateau Question

• Is magnon pairing possible in a system with purely repulsive (antiferromagnetic) interactions?

Nematic — superconductor analogy suggests positive answer: Magnon analogue of Kohn-Luttinger mechanism (e.g. pairing due to repulsive interactions) this talk: Emergent Ising order near the end-point of the 1/3 magnetization plateau

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 OAS, Reports on Progress in Physics 78, 052502 (2015), OAS, Wen Jin, Chubukov, Phys. Rev. Lett. 113, 087204 (2014) 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 PRB 2013 Side remark: spin-1/2 triangular lattice AFM is different h/J 0.8 plateau 4 fully polarized 0.6 width

0.4 no plateau end-point in 0.2 the s=1/2 model! 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 this talk: Emergent Ising order near the end-point of the 1/3 magnetization plateau

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 OAS, Reports on Progress in Physics 78, 052502 (2015), OAS, Wen Jin, Chubukov, Phys. Rev. Lett. 113, 087204 (2014) Emergent Ising orders in quantum two-dimensional triangular antiferromagnet at T=0

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

sat

U(1)*U(1) sat

sat

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

U(1)*U(1) E

O 0 0 2 0 40 J J 0 δcr = S 0 1 3 4 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 Continuous transition: plateau —> spin-current —> cone !

Z Z Z U(1) Z 3 ! 3 ⇥ 2 ! ⇥ 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 PHYSICAL REVIEW B 87,174501(2013) Spin-current phase = 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 chiral Mott insulator Indian 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

Magnon pairing is a fascinating problem ! Route to multipolar orders of frustrated ferromagnets extention to d=2 problems? ! Spin-current/Chiral Mott insulators