Oleg Starykh University of Utah arXiv:1708.02980 E d i t o r - PowerPoint PPT Presentation

Chiral liquid phase of simple quantum antiferromagnets Oleg Starykh University of Utah arXiv:1708.02980 E d i t o r s ’ S u g g e s t i o n Andrey Chubukov (U Minnesota) Zhentao Wang (U Tennessee) Cristian Batista (U Tennessee) Andrian Feigun (Northeastern U) Wei Zhu (LANL) YITP workshop “Novel quantum states in condensed matter”, November 9, 2017

Outline Vector chirality ✴ Talks by Togawa, Nagaosa, Tokura 1/3 magnetization plateau and its instabilities: ✴ • spin-current phase Minimal s=1 XXZ model of spin-current phase ✴ Conclusions ✴

Exotic ordered phases, emergent (Ising) orders ordered spin nematics quantum phases composite order spin liquids spin nematic composite order parameter, bilinear in spins Annals of the Israel Physical Society, vol.2, p. 565 (1978) S 1 x S 2 S 2 S 1

Brief history T Ising = 0 . 513 J > T KT = 0 . 502 J T Ising = 0 . 5125 J > T KT = 0 . 5046 J

Emergent Ising order parameters (finite T) PHYSICAL REVIEW B 85 , 174404 (2012) Chiral spin liquid in two-dimensional XY helimagnets A. O. Sorokin 1,* and A. V. Syromyatnikov 1,2, † week ending P H Y S I C A L R E V I E W L E T T E R S PRL 93, 257206 (2004) 17 DECEMBER 2004 Low-Temperature Broken-Symmetry Phases of Spiral Antiferromagnets Luca Capriotti 1,2 and Subir Sachdev 2,3 � 1 � H = ( J 1 cos( ϕ x − ϕ x + a ) + J 2 cos( ϕ x − ϕ x + 2 a ) x S i � ^ ^ S i � ^ ^ ^ X X H � J 1 S j � J 3 S j ; − J b cos( ϕ x − ϕ x + b )) , h i;j i hh i;j ii 1.2 � a � � ^ S 1 � ^ S 3 � ^ S 2 � ^ S 4 � a ; T 1 T ising ξ spin ~ S / T 1/2 T c 0.8 0.3 T c 0.25 0.6 ξ spin ~ e c'S 2 / T ξ spin ~ e cS 2 / T Ising nematic order 0.2 T KT 0.15 0.4 0.1 J 3 J 1 / 4 0.05 Neel LRO Spiral LRO 0.2 Lifshitz point 0 0.25 0.3 0.35 0.4 at T=0 only 4 3 4 3 0 0 0.5 1 1.5 2 2.5 3 J /J 1 3 1 2 1 2 (Q,Q) (Q, − Q) FIG. 2. The two different minimum energy configurations with Vector spin chiral phase Q ? � � Q; � Q � with ~ ~ Q � � Q; Q � and magnetic wave vectors Q � 2 � = 3 , corresponding to J 3 =J 1 � 0 : 5 . is present, but the temperature interval is tiny . Can be enhanced by DM interaction + phonons, Onoda, Nagaosa PRL 2007

Vector chirality in 1d (T=0)

Today: Search for vector chirality without magnetic order in quantum 2d models ✴ Spin-current phase Cheshire Cat’s smile

Outline Vector chirality ✴ 1/3 magnetization plateau and its instabilities: ✴ • spin-current phase Minimal s=1 XXZ model of spin-current phase ✴ Conclusions ✴

Phase diagram of the Heisenberg (XXX) model in the field Seabra, Momoi, Sindzingre, Shannon 2011 Z 2 vortex (chirality ordering) transition Gvozdikova, Melchy, Zhitomirsky 2010

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

Spatially anisotropic model Need to understand end-points PRB 2013 J J 0

Spatially anisotropic model: classical vs quantum J J 0 z H J S S h S ∑ ∑ = ⋅ − ij i j i ij i 〈 〉 Umbrella state: S = ∞ favored classically; h h 0 energy gain (J-J’) 2 /J sat J ′ ̸ = J Planar states: favored by S = 1 quantum fluctuations; h h 2 0 sat energy gain J/S 1/3-plateau The competition is controlled by δ = S ( J − J ′ ) 2 /J 2 dimensionless parameter Alicea, Chubukov, OS PRL 2009

Emergent Ising order near the end-point of the 1/3 magnetization plateau h sat H D X J ij ~ S i · ~ H = S j U(1) * U(1) U(1) * U(1) * Z 2 h i,j i U(1) * Z 2 U(1) * Z 3 b h c2 G Z 3* Z 2 B C 2 Z 3 Cone C U(1) * Z 3* Z 2 C 1 1/3 plateau A h c1 F a U(1) * Z 3 U(1) * U(1) * Z 2 U(1) * Z 2 E U(1) * U(1) ⇣ J − J 0 δ = 40 ⌘ 2 O 3 S J δ cr δ 0 1 3 4 OAS, Reports on Progress in Physics 78, 052502 (2015), OAS, Wen Jin, Chubukov, Phys. Rev. Lett. 113, 087204 (2014)

UUD-to-cone phase transition Z 3 → U (1) × Z 2 or Z 3 → smth else → U (1) × Z 2 ? h sat H D X J ij ~ S i · ~ H = S j U(1) * U(1) U(1) * U(1) * Z 2 h i,j i U(1) * Z 3 b h c2 G Z 3* Z 2 B C 2 Z 3 (UUD) U(1) * Z 2 (cone) C C 1 A h c1 F a U(1) * Z 3 U(1) * U(1) * Z 2 U(1) * Z 2 E U(1) * U(1) ⇣ J − J 0 δ = 40 ⌘ 2 O 3 S J δ cr δ 0 1 3 4

Low-energy excitation spectra ✏ d 2 = h c 2 − h + 9 Jk 2 4 Magnetization plateau is for δ < 3 collinear phase: preserves O(2) rotations about magnetic field -- no gapless spin waves. Breaks only discrete Z 3 . -k 2 d 2 Hence, very stable . vacuum of d 1,2 h c 2 − h c 1 = 0 . 6 2 S h sat = 0 . 6 2 S (9 JS ) δ = 40 S (1 − J 0 /J ) 2 3 Bose-Einstein condensation of d 1 (d 2 ) mode at k =0 leads to d 1 ✏ d 1 = h − h c 1 + 3 Jk 2 lower (upper) co-planar phase 4 for δ < 1 Alicea, Chubukov, OS PRL 2009

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

Interaction between low-energy magnons d 1 d 2 = 3 ⇣ ⌘ H (4) X d † 1 , k 0 + p d † 2 , − k 0 − p d 1 , − k 0 + q d 2 , k 0 − q − d † 1 , k 0 + p d † 2 , − k 0 − p d † 1 , − k 0 + q d † Φ ( p, q ) + h.c. 2 , k 0 − q N p,q } } Ψ † 1 ,p Ψ † Φ ( p, q ) ∼ ( − 3 J ) k 2 Ψ † 0 1 ,p Ψ 2 ,q 2 ,q | p || q | } singular magnon interaction Ψ 1 ,p = d 1 ,k 0 + p d 2 , − k 0 − p magnon pair Out[25]= 1 2 operators Ψ 2 ,p = d 1 , − k 0 + p d 2 ,k 0 − p 2 1 Obey canonical Bose commutation relations in the UUD ground state ⇣ ⌘ 1 + d † 1 , k 0 + p d 1 , k 0 + p + d † [ Ψ 1 , p , Ψ 2 , q ] = δ 1 , 2 δ p , q 2 , k 0 + p d 2 , k 0 + p → δ 1 , 2 δ p , q h d † 1 d 1 i uud = h d † In the UUD ground state 2 d 2 i uud = 0 ★ Interacting magnon Hamiltonian in terms of d 1,2 bosons = non-interacting Hamiltonian in terms of Ψ 1,2 magnon pairs Chubukov, OS PRL 2013

Two-magnon instability Magnon pairs Ψ 1,2 condense before single magnons d 1,2 1 , p � Ψ 1 , p i = 6 Jf 2 3 Equations of motion for Ψ - Hamiltonian h Ψ † q h Ψ † X f 2 p 2 , q � Ψ 2 , q i Ω p N q 2 , p � Ψ 2 , p i = 6 Jf 2 3 h Ψ † X q h Ψ † f 2 p 1 , q � Ψ 1 , q i Ω p N q h Ψ 1 ,p i = h Ψ 2 ,p i ⇠ i Υ `Superconducting’ solution with imaginary order parameter p 2 1 = 1 1 k 0 Instability = softening of two- X magnon mode @ δ cr = 4 - O(1/S 2 ) | p | 2 + (1 − δ / 4) k 2 p S N 0 p h d 1 i = h d 2 i = 0 no single particle condensate Chubukov, OS PRL 2013

Spin-current nematic state near the end-point of the 1/3 magnetization plateau (large-S analysis) � < 0 � > 0 distorted umbrella h c2 Z 3* Z 2 uud J’ spin- J current J’ h c1 distorted umbrella domain wall δ δ cr 4 h S r · S r 0 i is not affected h S x,y i = 0 no transverse magnetic order r Finite vector chirality h ˆ z · S A ⇥ S C i = h ˆ z · S C ⇥ S B i = h ˆ z · S B ⇥ S A i / Υ Spontaneously broken Z 2 -- spatial inversion [in addition to broken Z 3 inherited from the UUD state] Chubukov, OS PRL 2013

Spin current visualization B C A Precessing spins on sub lattices A, B, C are phase shifted by 2 π /3: S A = (cos[ ω t ] , sin[ ω t ] , M A ) , S B = (cos[ ω t ± 4 π 3 ] , sin[ ω t ± 4 π 3 ] , M B ) , S C = (cos[ ω t ± 2 π 3 ] , sin[ ω t ± 2 π 3 ] , M C ) Then no dipolar transverse order: h S A · S C i = h S C · S B i = h S B · S A i = cos[2 π h S x,y i = 0 and 3 ] r But finite chirality , determined by the sign of 2 π /3 shift between the sublattices: h S A ⇥ S C i = h S C ⇥ S B i = h S B ⇥ S A i = ± sin[2 π 3 ]

End-point of the plateau on kagome lattice PRB 2017 Kagome geometry Spin-current pattern 1/3 plateau

Outline Vector chirality ✴ 1/3 magnetization plateau and its instabilities: ✴ • spin-current phase Minimal s=1 XXZ model of spin-current phase ✴ Conclusions ✴

The minimal 2d quantum spin model • Spin-1 model with featureless Mott ground state at large D > 0 [ ] S z r = 0 Triangular lattice: two-fold degenerate spectrum, at + Q and - Q • X X r ) 2 r S y H = J ( S x r 0 + S y r 0 + ∆ S z r 0 ) + D ( S z r S x r S z h r,r 0 i r Out[24]= -Q +Q ⟨ S r ⟩ ̸ = 0 ⟨ S r ⟩ = 0 ⟨ S r ⟩ = 0 ⟨ S r × S r + e ν ⟩ ̸ = 0 ⟨ S r × S r + e ν ⟩ ̸ = 0 ⟨ S r × S r + e ν ⟩ = 0 e 2 e 1 e 3 U (1) × U (1) � U (1) � Z 2 × Z 2 × Z 2 � CL XY PM g b 0 g c g c

The minimal 2d quantum spin model • Spin-1 model with featureless Mott ground state at large D > 0 [ ] S z r = 0 Triangular lattice: two-fold degenerate spectrum, at + Q and - Q • X X r ) 2 r S y H = J ( S x r 0 + S y r 0 + ∆ S z r 0 ) + D ( S z r S x r S z h r,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.]