Unusual ordered phases of magnetized frustrated antiferromagnets

Transcription

1 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

2 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

3 Why to do it? Immense potential for practical applications

4 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

5 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

6 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

7 LiCuVO4 : magnon superconductor? estimates: J1 = mev J2 = 3.9 mev J5 = -0.4 mev

8 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

9 LiCuVO4: NMR lineshape - collinear SDW along B Hagiwara et a, 2011 Buttgen et al 2012

10 LiCuVO4 No spin-flip scattering above ~ 9 Tesla: longitudinal SDW state SF = spin flip, ΔS = 1 NSF = no spin flip, ΔS = 0

11 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

12 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)

13 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 )]

14 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

15 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

16 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

17 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

18 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

19 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

20 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

21 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

22 } } 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]= 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

23 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

24 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

25 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!

26 arxiv: gapped single particles; spontaneous circulating currents Motivation: cold atoms in optical lattices