Frustration-driven multi magnon condensates and their excitations - PowerPoint PPT Presentation

Frustration-driven multi magnon condensates and their excitations Oleg Starykh, University of Utah, USA Current trends in frustrated magnetism , ICTP and Jawaharlal Nehru University, New Delhi, India, Feb 9-13, 2015

Collaborators Leon Balents, Andrey Chubukov, KITP, UCSB Univ of Minnesota not today but � closely related findings: spin-current state at the tip of 1/3 magnetization plateau, spontaneous generation of orbiting spin currents (ask me for details after the talk :))

Condensed Matter Physics at the University of Utah Scanning Probe Microscopy Nano-optics Low Temperature Transport Exotic Matter and High Pressure Spin electronics Organic Semiconductors NMR and MRI � Strongly Correlated Electron Physics Topological insulators Frustrated magnetism Superconductivity

Life at (and near) University of Utah

Outline • Frustrated magnetism (brief intro) - emergence of composite orders from competing interactions • Nematic vs SDW in LiCuVO 4 ✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave” • Volborthite kagome antiferromagnet - experimental status - magnetization plateau - Nematic, SDW and more - Field theory of the Lifshitz point • Conclusions

Emergent Ising order parameters PHYSICAL REVIEW B 85 , 174404 (2012) 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 Chiral spin liquid in two-dimensional XY helimagnets A. O. Sorokin 1,* and A. V. Syromyatnikov 1,2, † 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 ) ^ S i � ^ ^ S i � ^ ^ X X H � J 1 S j � J 3 S j ; x h i;j i hh i;j ii − J b cos( ϕ x − ϕ x + b )) , 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 0.15 T KT 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 Q ? � � Q; � Q � with ~ ~ Q � � Q; Q � and magnetic wave vectors Q � 2 � = 3 , corresponding to J 3 =J 1 � 0 : 5 . Ising order: spin chirality X S i × ~ ~ � = S j triangle

Ising nematic in collinear spin system � = ~ N 1 · ~ N 2 = ± 1

Outline • Frustrated magnetism (brief intro) - emergence of composite orders from competing interactions • Nematic vs SDW in LiCuVO 4 ✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave” • Volborthite kagome antiferromagnet - experimental status - magnetization plateau - Nematic, SDW and more - Field theory of the Lifshitz point • Conclusions

LiCuVO 4 : magnon superconductor? estimates: J 1 = - 1.6 meV J 2 = 3.9 meV (subject of active debates) J 5 = -0.4 meV

High-field analysis: condensate of bound magnon pairs h S + i = 0 h S + S + i 6 = 0 Ferromagnetic J 1 < 0 produces attraction in real space Chubukov 1991 Kecke et al 2007 Kuzian and Drechsler 2007 Hikihara et al 2008 Sudan et al 2009 Zhitomirsky and Tsunetsugu 2010

Magnon binding E-E FM = ε 1 + h 1-magnon 2-magnon E-E FM = ε 2 + 2h bound state S z =-2 E ε 2 < 2 ε 1 : “molecular” S z =-1 bound state h Formation of molecular fluid For d>1 at T=0 this is a molecular BEC = true spin nematic

Hidden order No dipolar order h S + i i = 0 j i ⇠ e − | i − j | / ξ S z =1 gap h S + i S − Nematic order h S + i S + i + a i 6 = 0 Magnetic quadrupole moment Symmetry breaking U(1) → Z 2 nematic director can think of a fluctuating fan state

LiCuVO 4 : NMR lineshape - collinear SDW along B Hagiwara, Svistov et al, 2011 Buttgen et al 2012

LiCuVO 4 No spin-flip scattering above ~ 9 Tesla: longitudinal SDW state SF = spin flip, Δ S = 1 � NSF = no spin flip, Δ S = 0

Geometry (motivated by LiCuVO 4 ) o J 1 < 0 (ferro) J 2 >0, J’ > 0 (afm) in magnetic field No true condensation [ U(1) breaking] in d=1. � • � Inter-chain interaction is crucial for establishing � • Sato et al 2013 symmetry breaking in d=2. � Starykh and Balents 2014 � • Need to study weakly coupled “superconducting” chains

Inter-chain interaction Z Z X dx ~ S y · ~ X dx S + H inter − chain = S y +1 ∼ y S − y +1 + S z y S z y +1 y y Superconducting analogy: single-particle (magnon) tunneling between magnon superconductors is strongly suppressed at low energy (below the single-particle gap) Z X y ( x ) S � H ? dx J 0 h S + inter = y +1 ( x + 1) i nematic ground state ! 0 y Superconducting analogy: fluctuations generate two-magnon ( Josephson coupling ) tunneling between chains. They are generically weak, ~ J 1 (J’/J 1 ) 2 << J’ , but responsible for a true two-dimensional nematic order Z X H nem ∼ ( J 0 2 /J 1 ) dx [ T + y ( x ) T � y +1 ( x ) + h . c . ] T + y ( x ) ∼ S − y ( x ) S − y ( x + 1) y At the same time, density-density inter-chain interaction does not experience any suppression. It drives the system toward a two-dimensional collinear SDW order . √ 2 π ϕ + y ( x ) e ik sdw x y = M − 2 n pair = M − ˜ A 1 e i S z β √ 2 π Z inter � chain = H sdw ∼ J 0 X y +1 ∼ J 0 X y − ϕ + ( ϕ + dx cos[ y +1 )] H z S z y S z β y y Away from the saturation, SDW is more relevant [and stronger, via J’ >> (J’) 2 /J 1 ] than the nematic interaction : coupled 1d nematic chains order in a 2d SDW state.

Simple scaling Z X H nem ∼ ( J 0 2 /J 1 ) dx [ T + y ( x ) T � y +1 ( x ) + h . c . ] y • describes kinetic energy of magnon pairs, linear in magnon pair density n pair √ 2 π Z inter � chain = H sdw ∼ J 0 X y +1 ∼ J 0 X y − ϕ + ( ϕ + dx cos[ y +1 )] H z S z y S z β y y • describes potential energy of interaction between magnon pairs on � neighboring chains, quadratic in magnon pair density n pair ( J 0 ) 2 n ⇤ pair ∼ J 0 /J 1 • Competition n pair ∼ J 0 n 2 pair , hence n pair , c ∼ J 0 /J 1 J 1 • Hence: � - Spin Nematic near saturation , for n pair < n *pair � - SDW for n pair > n *pair

T=0 schematic phase diagram of weakly coupled nematic spin chains M Cautionary remark: � maybe impurity effect BEC physics Fully Polarized 1/2 - O(J’/J) 1/2 Spin Nematic SDW cf: Sato, Hikihara, Momoi 2013 0

Excitations (via spin-spin correlation functions) • 2d SDW ⇣ Φ e i k sdw · r ⌘ h S z ( r ) i = M + Re • preserves U(1) [with respect to magnetic field] -> hence NO transverse spin waves • breaks translational symmetry -> longitudinal phason mode at k sdw = π (1-2 Μ ) and k=0 (solitons (kinks) of massive sine-Gordon model � which describes 2d ordered state) phason OS, Balents PRB 2014

Excitations (via spin-spin correlation functions) � • 2d Spin Nematic h S + ( r ) S + ( r 0 ) i ⇠ Ψ 6 = 0 • breaks U(1) but Δ S=1 excitations are gapped h S + ( r ) i = 0 (magnon superconductor) • gapless density fluctuations at k=0 ( - sector: solitons of � massive sine-Gordon � model describing � ( + sector: solitons of � 1d zig-zag chain. ) � J 1 massive sine-Gordon � Energy scale model which describes � 2d ordered state.) � ( J 0 ) 2 /J 1 Energy scale OS, Balents PRB 2014

Intermediate Summary • Interesting magnetically ordered states: SDW and Spin Nematic - Gapped Δ S= 1 excitations (no usual spin waves!) - Linearly-dispersing phason mode with Δ S= 0 in 2d SDW - SDW naturally sensitive to structural disorder - Linearly-dispersing magnon density waves in 2d Spin Nematic - analogy with superconductor/charge density wave competition

Outline • Frustrated magnetism (brief intro) - emergence of composite orders from competing interactions • Nematic vs SDW in LiCuVO 4 ✓ spin nematic: “magnon superconductor” ✓ collinear SDW: “magnon charge density wave” • Volborthite kagome antiferromagnet - experimental status - magnetization plateau - Nematic, SDW and more - Field theory of the Lifshitz point • Conclusions

�� � Volborthite n a e t- - o - H Cu1 n. Cu2 l in V i- O5 s c l del b a - e t b c y s f en s, e - en - s b Cu2 e Cu1 a c is e

� � � � � Volborthite’s timeline 2001 2009 2012 2014 impurity ordering at low T? magnetic order ! magnetization quantum spin liquid?! magnetization steps? plateau 0.8 0.1 C / T (mJ K – 2 mol-Cu –1 ) 120 T * 150 60 0.6 C mag T -1 / mJ K -2 Cu-mol -1 80 C / T (mJ K – 2 mol-Cu –1 ) C/T (J/K 2 mol Cu) 0.0 40 0 2 4 6 8 T (K) 40 100 0 � 0.4 0 0.5 1.0 1.5 2.0 ● 0 T T 2 (K 2 ) ○ 1 T C D / T T * △ 3 T 20 ▼ 5 T 0.2 50 □ 7 T C m / T r � * Polycrystal Single crystal 0 0.0 0 1 2 3 0 10 20 30 40 50 60 70 0 0 1 2 3 4 T / K T (K) T (K) time = material quality � � �