Emergent Ising orders of frustrated magnets

Transcription

1 Emergent Ising orders of frustrated magnets Oleg Starykh University of Utah Jason Alicea (Caltech) Andrey Chubukov (U Minnesota) Leon Balents (KITP) Zhentao Wang (U Tennessee) Cristian Batista (U Tennessee) SPIN COHERENCE, CONDENSATION, AND SUPERFLUIDITY University of California Gump Station, Moorea, French Polynesia (February 16, 2017)

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5 Quantum magnetism vs Spintronics Spin is (almost) conserved No dipolar coupling (small magnetic moments) Notable exception > spin ice Basically isolated system of spins No coupling to phonons, etc.! Notable exception -> hybridization of magnons and phonons in non-collinear spin structures Spin transport > mostly thermal 1) heat transport in chains/ladders 2) thermal transport in organic spin liquid candidate materials 3) magnon Hall effect (due to DM interactions) 4) Spin Seebeck effect in ordered and critical magnetic insulators

6 Interacting magnons Repulsive Attractive superfluid collapse Anything else? -> emergence of composite order = nematic!

7 Exotic ordered phases, emergent (Ising) orders ordered! phases spin nematic! composite order spin nematic composite order parameter quantum! spin liquids

8 PHYSICAL REVIEW B 85,174404(2012) Emergent Ising order parameters Chiral spin liquid in two-dimensional XY helimagnets A. O. Sorokin 1,* and A. V. Syromyatnikov 1,2, g Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, St. Petersburg 188 PRL 93, (2004) P H Y S I C A L R E V I E W L E T T E R S week ending 17 DECEMBER 2004 H = (J 1 cos(ϕ x ϕ x+a ) + J 2 cos(ϕ x ϕ x+2a ) x J b cos(ϕ x ϕ x+b )), = Tising TKT T 1 ξ spin ~ e c's2 / T Neel LRO Low-Temperature Broken-Symmetry Phases of Spiral Antiferromagnets ξ spin ~ S / T 1/2 J 1 / 4 Lifshitz point ξ spin ~ e cs2 / T Spiral LRO at T=0 only Luca Capriotti 1,2 and Subir Sachdev 2,3 J 3 X ^H J 1 ^S i ^S X j J 3 ^S i ^S j ; T c hi;ji hhi;jii a ^S 1 ^S 3 ^S 2 ^S 4 a ; Ising nematic order 4 3 T c J /J (Q,Q) (Q, Q) FIG. 2. The two different minimum energy configurations with magnetic wave vectors ~Q Q; Q and ~Q? Q; Q with Q 2 =3, corresponding to J 3 =J 1 0:5.

9 Outline Magnetization process of triangular lattice antiferromagnet - accidental degeneracy and selection of the magnetization plateau at M=1/3 by quantum (thermal) fluctuations! Instabilities of the plateau - moving down/up in magnetic field - spatial anisotropy of exchange interactions! Two-magnon pairing and condensation - Spin-current (chiral Mott) phase - Spontaneous generation of DM interaction!! Generic features: (at least) two-fold spectrum degeneracy strong interactions (in the vicinity of single-particle condensation)! Mott (PM) to Superfluid (cone) transition proceeds via intermediate chiral Mott phase U(1) x Z 2!!

10 Phase diagram of the Heisenberg (XXX) model in the field Seabra, Momoi, Sindzingre, Shannon 2011 Z2 vortex (chirality ordering) transition Gvozdikova, Melchy, Zhitomirsky 2010

11 Quantum fluctuations, S >> 1, T=0. J = J: Quantum fluctuations select co-planar and collinear phases UUD plateau is due to interactions between spin waves h c2 - h c1 = (0.6/2S) h sat up-up-down collinear state

12 Exp: M=1/3 magnetization plateau in Cs2CuBr4 Observed in Cs 2 CuBr 4 (Ono 2004, Tsuji 2007) J S=1/2 J first observation of up-up-down state in spin-1/2 triangular lattice antiferromagnet! total of 9 phases -- instead of 3 expected! Important: the lattice is strongly anisotropic

13 Magnetic phase transitions are detected electrically!

14 this talk: Emergent Ising order near the end-point of the 1/3 magnetization plateau hsat H D H = X hi,ji J ij ~ Si ~S j U(1)*U(1) U(1)*U(1)*Z2 U(1)*Z3 U(1)*Z2 hc2 G B b Z3*Z2 Z3 U(1)*Z3*Z2 C2 C1 C = 40 J J S J hc1 F O 0 U(1)*Z3 U(1)*U(1) A U(1)*U(1)*Z2 E δcr OAS, Reports on Progress in Physics 78, (2015), OAS, Wen Jin, Chubukov, Phys. Rev. Lett. 113, (2014) a U(1)*Z2 δ

15 Spatially anisotropic model: classical vs quantum H = J S S h ij i j ij i S z i J J S = 1 J J 0 h h sat Umbrella state: favored classically; energy gain (J-J ) 2 /J S = h sat h 1/3-plateau The competition is controlled by dimensionless parameter Planar states: favored by quantum fluctuations; energy gain J/S δ = S(J J ) 2 /J 2

16 Emergent Ising orders in quantum two-dimensional triangular antiferromagnet at T=0 h H = X hi,ji J ij ~ Si ~S j 0 sat hsat h planar hc2 hc1 H G F O 0 U(1)*Z3 Z3 A U(1)*Z3 U(1)*U(1) E D U(1)*U(1) b? B a C2 C1 δcr C U(1)*Z2 = 40 J J S J h h cone

17 Emergent Ising order near the end-point of the 1/3 magnetization plateau the result hsat H D H = X hi,ji J ij ~ Si ~S j U(1)*U(1) U(1)*U(1)*Z2 U(1)*Z3 U(1)*Z2 hc2 G B b Z3*Z2 Z3 U(1)*Z3*Z2 C2 C1 C = 40 J J S J hc1 F O 0 U(1)*Z3 U(1)*U(1) A U(1)*U(1)*Z2 E δcr OAS, Reports on Progress in Physics 78, (2015), OAS, Wen Jin, Chubukov, Phys. Rev. Lett. 113, (2014) a U(1)*Z2 δ

18 UUD-to-cone phase transition Z 3! U(1) Z 2 or Z 3! smth else! U(1) Z 2? hsat H D H = X hi,ji J ij ~ Si ~S j U(1)*U(1) U(1)*U(1)*Z2 U(1)*Z3 hc2 G B b Z3*Z2 Z3 (UUD) C2 C1 C U(1)*Z2 (cone) hc1 F U(1)*Z3 A U(1)*U(1)*Z2 a U(1)*Z2 = 40 J J S J O 0 U(1)*U(1) E δcr δ

19 Low-energy excitation spectra d2 d2 = h c2 h + 9Jk2 4 for δ < 3 -k2 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 h c2 h c1 = 0.6 2S h sat = 0.6 2S (9JS) = 40 S 3 (1 J 0 /J) 2 d1 d1 = h h c1 + 3Jk2 4 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

20 Low-energy excitation spectra near the plateau s end-point = 40 S 3 (1 J 0 /J) 2 parameterizes anisotropy J /J Out[24]= -k2 +k2 extended symmetry: 4 gapless modes at the plateau s end-point d2 vacuum of d1,2 δ=4 Out[25]= d1 k1 = k2 = k0 -k0 +k0 k 0 = r 3 10S Out[19]= S>>1 = 40 S 3 (1 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

21 } } Interaction between low-energy magnons H (4) d 1 d 2 = 3 N X p,q (p, q) d 1,k 0 +p d 2, k 0 p d 1, k 0 +qd 2,k0 q d 1,k 0 +p d 2, k 0 p d 1, k 0 +q d 2,k 0 q +h.c. (p, q) ( 3J)k2 0 p q singular magnon interaction } 1,p 2,q 1,p 2,q magnon pair operators 1,p = d 1,k0 +pd 2, k0 p 2,p = d 1, k0 +pd 2,k0 p Out[25]= Obey canonical Bose commutation relations in the UUD ground state [ 1,p, 2,q] = 1,2 p,q 1+d 1,k 0 +p d 1,k 0 +p + d 2,k 0 +p d 2,k 0 +p! 1,2 p,q In the UUD ground state hd 1 d 1i uud = hd 2 d 2i uud =0 Interacting magnon Hamiltonian in terms of d1,2 bosons = non-interacting Hamiltonian in terms of Ψ1,2 magnon pairs Chubukov, OS PRL 2013

22 Two-magnon instability Magnon pairs Ψ1,2 condense before single magnons d1,2 Equations of motion for Ψ - Hamiltonian h h 1,p 1,pi = 6Jf2 p 3 p N 2,p 2,pi = 6Jf2 p 3 p N X fq 2 h q X fq 2 h q 2,q 2,qi 1,q 1,qi `Superconducting solution with imaginary order parameter h 1,pi = h 2,pi i p 2 Instability = softening of twomagnon δcr = 4 - O(1/S 2 ) 1= 1 S 1 N X p k 0 p p 2 +(1 /4)k 2 0 no single particle condensate hd 1 i = hd 2 i =0 Chubukov, OS PRL 2013

23 Two-magnon condensate = Spin-current nematic state h c2 distorted! umbrella > 0 < 0 uud spin-! current J J h c1 distorted! umbrella J δ cr 4 δ no transverse magnetic order hs x,y r i =0 domain wall hs r S r 0i is not affected Finite scalar (and vector) chiralities. Sign of determines sense of spin-current circulation hẑ S A S C i = hẑ S C S B i = hẑ S B S A 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

24 Side remark: J1-J2 spin-s chain The end-plateau 2-magnon instability occurs at! Should be checked by numerics! 1/ p S>>1/S 2 Wen Jin, OAS

25 Spontaneous generation of Dzyaloshinskii-Moriya interaction H (4) d 1 d 2 / 1 N X ik p 0 (d k2+k 0 (k k0 ) 2 +(1 /4)k0 2 1,k d 2, k d 1,k d 2, k ) X ik p 0 (d p2 k 0 (p + k0 ) 2 +(1 /4)k0 2 1,p d 2, p d 1,p d 2, p ) a1 a2 continuum limit of DM in triangular lattice X ẑ S r (S r+a1 + S r+a2 ) r Mean-field approximation: B z H (4) d 1 d 2! D X k ( k 0 k k 0 + k 0 k + k 0 )(d 1,k d 2, k d 1,k d 2, k ) D spin currents appear due to spontaneously generated DM! similar to Lauchli et al (PRL 2005) for Heisenberg+ring exchange model; also chiral Mott insulator, Dhar et al, PRB 2013; Zaletel et al, 2013

26 Continuous transitions: plateau ->spin-current > cone! Z 3! Z 3 Z 2! U(1) Z 2 hsat H D H = X hi,ji J ij ~ Si ~S j U(1)*U(1) U(1)*U(1)*Z2 U(1)*Z3 hc2 G B b Z3*Z2 Z3 (UUD) C2 C1 C U(1)*Z2 (cone) hc1 F U(1)*Z3 A U(1)*U(1)*Z2 a U(1)*Z2 = 40 J J S J O 0 U(1)*U(1) E δcr δ

27 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 Paramekanti 2,3,6,7 1 Indian Institute of Astrophysics, Bangalore , India Spin-current phase = chiral Mott insulator gapped single particles; but spontaneously broken time-reversal = spontaneous circulating currents

28 Other systems Ferrimagnetic state of kagome (anti)ferromagnet 1/3 magnetization plateau PHYSICAL REVIEW B 82, Coupled frustrated quantum spin- 1 2 chains with orbital order in volborthite Cu 3V 2 O 7 (OH) 2 2H 2 O O. Janson, 1, * J. Richter, 2 P. Sindzingre, 3 and H. Rosner 1, 1 Max-Planck-Institut für Chemische Physik fester Stoffe, D Dresden, Germany 2 Institut für Theoretische Physik, Universität Magdeburg, D Magdeburg, Germany 3 Laboratoire de Physique Théorique de la Matière Condensée, Univ. P. & M. Curie, Paris, France Received 9 August 2010; published 30 September 2010 We present a microscopic magnetic model for the spin-liquid candidate volborthite Cu V O OH 2H O. J 1 < 0,J 2 > 0,J 0 > 0 J 1 FM, J 2 AF J AF polarized chains?!

29 Frustrated ferromagnetism PHYSICAL REVIEW B 82, Coupled frustrated quantum spin- 1 2 chains with orbital order in volborthite Cu 3V 2 O 7 (OH) 2 2H 2 O O. Janson, 1, * J. Richter, 2 P. Sindzingre, 3 and H. Rosner 1, 1 Max-Planck-Institut für Chemische Physik fester Stoffe, D Dresden, Germany 2 Institut für Theoretische Physik, Universität Magdeburg, D Magdeburg, Germany 3 Laboratoire de Physique Théorique de la Matière Condensée, Univ. P. & M. Curie, Paris, France Received 9 August 2010; published 30 September 2010 m/m s 0.2 We present a microscopic magnetic model for the spin-liquid candidate volborthite Cu V O OH 2H O T (K) kagome 0.8 J 2 / J 1 = J 2 / J 1 = 1.4 J 2 / J 1 = J ic / J 1 = h (T) c b a DFT gets it right! J 1 < 0,J 2 > 0,J 0 > 0 b a JJ 0 ic J 1 J 2 FM

30 Other systems arxiv: Kagome geometry Spin-current pattern 1/3 plateau

31 Is this a generic feature of the Mott-to-Superfluid transition? Zhentao Wang (U Tenn), Adrian Feiguin (Northeastern U), Cristian Batista (U Tenn), OAS Spin-1 model with featureless Mott ground state at large D > 0 [ Sr z =0] Triangular lattice: two-fold degenerate spectrum, at +Q and -Q X H = J(Sr x Sr x 0 + Sr y S y r 0 + Sr z S z X r 0 )+D (Sr z ) 2 hr,r 0 i r 1. Toy problem of two-spin exciton. Derive Schrodinger eqn for the pair wave function ψ + charge, - charge Solution which is odd under inversion is the first instability when approaching from large-d limit. Indicates chiral Mott phase. [Single-particle condensation occurs at D=3J.]

32 Is this a generic feature of the Mott-to-Superfluid transition? Zhentao Wang (U Tenn), Adrian Feiguin (Northeastern U), Cristian Batista (U Tenn), OAS Spin-1 model with featureless Mott ground state at large D > 0 Triangular lattice H = X hr,r 0 i J(S x r S x r 0 + S y r S y r 0 + S z r S z r 0 )+D X r (S z r ) 2 2. Solve Bethe-Salpeter eqn. 3 2 Preliminary result Δ 1 0

33 Conclusions Mott -> superfluid transition in frustrated lattice requires U(1) x Z2 breaking.! This proceeds via intermediate spin-current (chiral Mott) phase (breaking Z2 only). Spontaneously breaks inversion/time reversal. All single particle excitations are gapped.!! Thank you!

34 h/j Side remark: spin-1/2 triangular lattice AFM is different 4 fully polarized plateau width no plateau end-point in the s=1/2 model! 3 C planar IC planar R 2 1 C planar 1/3 IC planar SDW SDW quasi-collinear cone plateau for all J /J (crystal of spindowns; end-point at J =0) R=1-J /J S=1/2 vs S=1 difference for the ordered 2d phase!!!