Unusual ordered phases of magnetized frustrated antiferromagnets Credit: Francis Pratt / ISIS / STFC Oleg Starykh University of Utah Leon Balents and Andrey Chubukov Novel states in correlated condensed matter from model systems to real materials, April 8-10, 2014
Outline Frustrated magnetism - competing orders - symmetry breaking vs topological order Nematic vs SDW in LiCuVO4 spin nematic: magnon superconductor collinear SDW: magnon charge density wave Instability of the 1/3 plateau: spin-current nematic phase Conclusions
Why to do it? Immense potential for practical applications
Different kinds of orders Fermi liquids charge density waves superconductors spin liquids quantum Hall effect topological spin liquids topological insulators entanglement... S. Trebst, U Cologne
To break (the symmetry) or not to break? (symmetry order parameter) (top. order) This talk: Break the symmetry in an interesting way spin nematic: magnon superconductor collinear SDW: magnon charge density wave spontaneous generation of spin currents
Outline Frustrated magnetism - competing orders - symmetry breaking vs topological order Nematic vs SDW in LiCuVO4 spin nematic: magnon superconductor collinear SDW: magnon charge density wave Instability of the 1/3 plateau: spin-current nematic phase Conclusions
LiCuVO4 : magnon superconductor? estimates: J1 = - 1.6 mev J2 = 3.9 mev J5 = -0.4 mev
High-field analysis: condensate of magnon pairs hs + i =0 hs + S + i6=0 Ferromagnetic J1 < 0 produces attraction in real space Chubukov, PRB 1991 Zhitomirsky, Tsunetsugu EPL 2010
LiCuVO4: NMR lineshape - collinear SDW along B Hagiwara et a, 2011 Buttgen et al 2012
LiCuVO4 No spin-flip scattering above ~ 9 Tesla: longitudinal SDW state SF = spin flip, ΔS = 1 NSF = no spin flip, ΔS = 0
o Geometry (motivated by LiCuVO4) today: J1< 0 (ferro) J2 >0, J > 0 (afm) in magnetic field Chubukov 1991; Kecke et al 2007; Hikihara et al 2008; Zhitomirsky and Tsunetsugu 2010 Sato et al 2013 Starykh and Balents 2014
Nematic chain S z -S z (SDW) channel: in-chain J1< 0 gaps out relative mode H intra chain = X y Z dxj 1 sin[ M] cos[p 8 'y ] ' y =(' y,odd ' y,even )/ p 2 y, even o J1-J2 chain y, odd S + y (x) ( 1) x A 3 e i p 2 + y (x) e i( 1)x p 2 y (x) hs + i =0 quantum-disordered, decays exponentially: S z = 1 excitations are gapped Standard (in 1d) T y + = S y + (x)s y + (x + 1) e ip 2 + y (x) power-law decay: critical nematic spin correlations ht + i = hs + S + i6=0 Physical picture: 1d magnon superconductor Kolezhuk, Vekua (2005); Hikihara et al. (2008); Sato, Hikihara, Momoi (2013)
Inter-chain interaction H inter chain = X y Z dx ~ S y ~S y+1 X y Z dx S + y S y+1 + S z ys z y+1 Superconducting analogy: single-particle (magnon) tunneling between magnon superconductors is strongly suppressed at low energy (below the single-particle gap) H? inter = X y Z dx J 0 hs + y (x)s y+1 (x + 1)i nematic ground state! 0 Superconducting analogy: fluctuations generate two-magnon (Josephson) tunneling between chains. They are generically weak, ~ J1(J /J1) 2 << J1, but responsible for a true two-dimensional nematic order. H nem (J 02 /J 1 ) X y Z dx [T + y (x)t y+1 (x)+h.c.] 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. H z inter S z y M 2n pair = M Ã1e chain = H sdw J 0 X y S z ys z y+1 J 0 X y p 2 ' + y (x) Away from the saturation, SDW is more relevant [and stronger, via J1 >> J1(J /J1) 2 ] than the nematic interaction: coupled 1d nematic chains order in a 2d SDW state. Z dx cos[ p 2 (' + y ' + y+1 )]
T=0 schematic phase diagram of weakly coupled nematic spin chains M 1/2 - O(J /J) 1/2 Fully Polarized Spin Nematic BEC physics SDW
Excitations (via spin-spin correlation functions) 2d SDW hs z (r)i = M +Re e ik sdw r 1. preserves U(1) [with respect to magnetic field] -> no transverse spin waves 2. breaks translational symmetry - longitudinal phason mode at ksdw = π(1-2μ) and k=0 phason OS, Balents PRB 2014
Excitations (via spin-spin correlation functions) 2d Spin Nematic hs + (r)s + (r 0 )i = e ik nm r cm 1. breaks U(1) but ΔS=1 excitations are gapped (magnon superconductor) hs + (r)i =0 2. gapless density fluctuations at k=0 OS, Balents PRB 2014
Conclusions I Interesting magnetically ordered states: SDW and Spin Nematic - Gapped ΔS=1 excitations - Linearly-dispersing phason mode with ΔS=0 in SDW - Linearly-dispersing magnon density waves in SN - useful analogy with superconductor/charge density wave competition
Outline Frustrated magnetism - competing orders - symmetry breaking vs topological order Nematic vs SDW in LiCuVO4 spin nematic: magnon superconductor collinear SDW: magnon charge density wave Instability of the 1/3 plateau: spin-current nematic phase Conclusions
Exp: M=1/3 magnetization plateau in Cs2CuBr4 Observed in Cs 2 CuBr 4 (Ono 2004, Tsuji 2007) J 0 /J 0.75 year 2007! 0.4 year 2014 S=1/2 J J up-up-down state is commensurate collinear SDW Important: the lattice is strongly anisotropic
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
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. Hence, very stable. -k1 +k1 Alicea, Chubukov, OS PRL 2009
} } Bosonization of 2d interacting 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]= 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,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
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 mode @ 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
h c2 Two-magnon condensate = Spin-current nematic state distorted umbrella uud spincurrent J J > 0 < 0 h c1 distorted umbrella δ cr 4 δ no transverse magnetic order J hs x,y r i =0 domain wall hs r S r 0i is not affected Finite scalar (and vector) chiralities. Sign of Υ 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
Conclusions Nematic vs SDW in real material LiCuVO4 spin nematic: magnon superconductor collinear SDW: magnon charge density wave Two-magnon instability of the 1/3 plateau: spin-current nematic phase breaks spatial inversion (Z2) spontaneous generation of DM interaction Ordered states can be quite interesting!
arxiv:1308.3237 gapped single particles; spontaneous circulating currents Motivation: cold atoms in optical lattices