Quantum phases and transitions of spatially anisotropic triangular antiferromagnets. Leon Balents, KITP, Santa Barbara, CA

Transcription

1 Quantum phases and transitions of spatially anisotropic triangular antiferromagnets Leon Balents, KITP, Santa Barbara, CA Hanoi, July 14, 2010

2 Collaborators Oleg Starykh, University of Utah Masanori Kohno, NIMS, Tsukuba, Japan Hosho Katsura, KITP

3 Outline Introduction to quantum magnetism on triangular and anisotropic triangular lattices Magnetic order, spin liquid,... Cs2 CuCl 4, dimensional reduction, and low temperature phases of anisotropic triangular lattice antiferromagnets

4 Frustration and triangles Simplest idea: pairwise exchange interactions cannot be simultaneously satisfied geometric frustration But this is a bit simplistic, and overstates the problem

5 Triangular Ising model Wannier (1950): thermodynamic ground state degeneracy Ω = e S/k B S 0.34Nk B 1 frustrated bond per triangle Classical Ising model does not order at any T>0 but very sensitive, and no known examples of this

6 Heisenberg Systems H = J ij S i S j Unlike the Ising model, this is not intrinsically classical However, it turns out that the semi-classical spin wave expansion tends to work well even for small S (unfortunately!) We may still ask about the degeneracy in the classical limit

7 Classical ground state Spins must sum to zero

8 Classical ground state Degrees of freedom: 2

9 Classical ground state Degrees of freedom: 2+1

10 Classical ground state Degrees of freedom: 2+1 order parameter is SO(3) matrix S i =Re ψ(ˆn 1 + iˆn 2 )e iq r i

11 Quantum Spin Liquid (QSL) Anderson (1973) proposed Resonating Valence Bond (RVB) state of S=1/2 Heisenberg magnets Fazekas + Anderson (1974) made first calculations suggesting this for the triangular lattice in particular Ψ = Valence bond = spin singlet VB = 1 2 ( )

12 Reality: 120 state Semi-classical ordered state has been verified for S=1/2 spins by exact diagonalization and other numerics. best estimate Ms = ±0.015 compare full moment Ms =0.5, spin wave theory gives M s =0.24

13 Hubbard Model If charge fluctuations are strong, reduction to the Heisenberg model is not accurate H = t ij c iα c jα + U i n i (n i 1) H = J ij S i S j + K 4 P P 1234 Calculations (Imada, Motrunich, Lee2 ) suggest QSL state at intermediate t/u (K/J > 0.15)

14 κ-(bedt-ttf) 2 Cu 2 (CN) 3 and EtMe 3 Sb[Pd(dmit) 2 ] 2 (1/T 1 T) max Crossover Organics proximate to a Mott transition Mott insulator (dr/dt) max R = R 0 + AT 2 T 1 T = const. Non-activated transport (Spin liquid) onset T C Metal (Fermi liquid) Optical pseudogap Superconductor Pressure (10-1 GPa)

15 Specific Heat Linear specific heat as expected for free fermions - approximately as expected for QSL - but very strange for an insulator C p T 1 (mj K 2 mol 1 ) T 1 T 4 T 8 T κ -(d 8 :BEDT-TTF) 2 Cu[N(CN) 2 ]Br κ -(BEDT-TTF) 2 Cu[N(CN) 2 ]CI β'-(bedt-ttf) 2 ICI T 2 (K 2 ) S. Yamashita et al,

16 Challenges Thermal conductivity shows a gap Not consistent with uniform RVB state κ /T (W K 2 m 1 ) 0.10 Clean limit Dirty Consistency with specific heat? 0.01 Different behavior seen for other organic very recently T 2 (K 2 ) M. Yamashita et al, 2008

17 Challenges very small or no optical gap (pseudogap) non-activated resistivity (or gap <15 mev) Kézsmárki et al, 2006

18 Challenges Dramatic dielectric anomalies observed at T<60K Points to molecular dipoles in individual organic dimers - not taken into account by RVB theory!!! " " B6 '0C($?5 %& $ # "! %& *" %& *( %& *# %& *> %& *$ %& *= %& *%& 0-5 0D5!*01&23*334567! '''':'0;<5 &'() %'&) ('&) %&) (&) &'%* &'(* %'&* :'0;<5 %& #! "#$ %&'%! &'(('$)"##*+,-./' %& ( %& " '''3 $ '0A5 '& %& 9 '# '= %&!!& 9& "& (& 3?-E '0A5 &!& "& #& $& 3/?+/)-@7)/'0A5 M. Abdel-Jawad et al, 2010 FIG. 2: (Color online) (a) The dielectric constant of a single crystal of κ-(bedt-ttf) 2 Cu 2 (CN) 3 along the a-axis (crossplane) direction at various frequencies as a function of tem-

19 S=1 Systems Spin S=1 antiferromagnets allow biquadratic exchange K J S i S j K( S i S j ) 2 H = ij j j This can induce quadrupolar order, or spin nematic state (a) π/2 S i =0 S µ i Sν i = qn µ i nν j 3π/4 π AFQ FM AFM FQ SU(3) ϑ=0 Läuchli et al, 2006 J SU(3) π/2 π/4 K

20 NiGa 2 S 4 Very 2d isotropic S=1 material S. Nakatsuji et al, 2005 Considered a candidate for spin-nematic state Tsunetsugu et al, 2006; Bhattacharjee et al, 2006; Läuchli et al, 2006; Stoudenmire, LB, 2009

21 T>0 Heisenberg Systems Mermin-Wagner theorem implies that 2d Heisenberg magnet cannot have magnetic order at T> L= fitting L=48 L=96 L= L=384 L=768 L=1536 L=48 L=96 L=192 L=384 L= (a) Eq. (2) Regular part 1.8 L=48 L= L=192 L= L=768 L= (b) L= Eq. (4) Regular part L=48 L=96 L=192 L=384 L=768 L= lor online) Temperature dependence of the finite-size spin correlation length, together figures from classical MC simulations by Kawamura et al, 2010 Non-linear sigma model: ξ ~ exp (c J/T), stiffness=0 e bulk one. The inset is a scaling plot, i.e., ξ 2L /ξ L plotted vs. ξ L /L. ite large, ξ Our fitting analysis also gives an estimate of the crossover smooth crossovers seen in measured quantities e T and the crossover length l 380. It should be noticed, however, ue of ξ at T = T v (at T = T )isanon-universalpropertydependingonthedetails em. previous studies, as an order parameter characterizing the Z 2 -vortex transition, lson-loop (vorticity function) 12) or a vorticity modulus 20) has been proposed. Here, e following Refs. 14, 21) the latter quantity, the vorticity modulus v, definedasthe

22 T>0 Heisenberg Systems Mermin-Wagner theorem implies that 2d Heisenberg magnet cannot have magnetic order at T> L= fitting L=48 L=96 L= L=384 L=768 L=1536 L=48 L=96 L=192 L=384 L= Kawamura + Miyashita (1984) proposed a topological transition due to Z 2 vortices, point defects of the SO(3) order parameter ue of ξ at T = T v (at T = T )isanon-universalpropertydependingonthedetails em L=48 L= L=192 L= L=768 L= existence of phase transition is controversial (a) Eq. (2) Regular part lor online) Temperature dependence of the finite-size spin correlation length, together e bulk one. The inset is a scaling plot, i.e., ξ 2L /ξ L plotted vs. ξ L /L. ite large, ξ Our fitting analysis also gives an estimate of the crossover e T and the crossover length l 380. It should be noticed, however, previous studies, as an order parameter characterizing the Z 2 -vortex transition, lson-loop (vorticity function) 12) or a vorticity modulus 20) has been proposed. Here, e following Refs. 14, 21) the latter quantity, the vorticity modulus v, definedasthe (b) L= Eq. (4) Regular part L=48 L=96 L=192 L=384 L=768 L= figures from classical MC simulations by Kawamura et al, 2010

23 Spatially anisotropic case J J J chains Issues: Is there a QSL state? Magnetization plateaux in strong magnetic field?

24 Theories (zero field) Spin wave theory? spiral J /J DMRG disordered spiral variational QMC 1d SL 2d SL spiral series expansion spiral 0 1 coupled cluster method collinear spiral

25 Materials Cs2 CuCl 4 : J /J = 0.34 Cs2 CuBr 4 : J /J = spiral AF spiral AF κ-(et)2 X; X=Cu 2 (CN) 3, Cu[N (CN) 2 ]Cl : J /J 1 AF and spin liquid X[Pd(dmit)2 ] 2 ; X= EtMe 3 Sb, EtMe 3 P, Me 4 P... : J /J? AFs, spin liquid, dimerized NaTiO2? little studied 3 He on graphite? spin liquid at 4/7 coverage

26 Materials Cs2 CuCl 4 : J /J = 0.34 Cs2 CuBr 4 : J /J = Organics close to the Mott transition. spiral AF Hamiltonian must include charge fluctuations, and spiral is AF not well understood. κ-(et)2 X; X=Cu 2 (CN) 3, Cu[N (CN) 2 ]Cl : J /J 1 AF and spin liquid X[Pd(dmit)2 ] 2 ; X= EtMe 3 Sb, EtMe 3 P, Me 4 P... : J /J? AFs, spin liquid, dimerized NaTiO2? little studied 3 He on graphite? spin liquid at 4/7 coverage

27 Materials Cs2 CuCl 4 : J /J = 0.34 Cs2 CuBr 4 : J /J = spiral AF spiral AF κ-(et)2 X; X=Cu 2 (CN) 3, Cu[N (CN) 2 ]Cl : J /J 1 X[Pd(dmit)2 ] 2 ; X= EtMe 3 Sb, EtMe 3 P, Me 4 P... : J /J? AF and spin liquid Orbital quasi-degeneracy probably is important. AFs, spin liquid, dimerized NaTiO2? little studied 3 He on graphite? spin liquid at 4/7 coverage

28 Materials Cs2 CuCl 4 : J /J = 0.34 Cs2 CuBr 4 : J /J = spiral AF spiral AF κ-(et)2 X; X=Cu 2 (CN) 3, Cu[N AF and spin liquid (CN) 2 ]Cl : J /J 1 Experimental probes are X[Pd(dmit)2 ] 2 ; X= EtMe 3 Sb, limited, and there are EtMe 3 P, Me 4 P... : J /J? competing models. AFs, spin Likely liquid, dimerized significant multi-spin ring NaTiO2? exchange is little important studied 3 He on graphite? spin liquid at 4/7 coverage

29 Materials Cs2 CuCl 4 : J /J = 0.34 Cs2 CuBr 4 : J /J = spiral AF spiral AF κ-(et)2 X; X=Cu 2 (CN) 3, Cu[N (CN) 2 ]Cl : J /J 1 AF and spin liquid X[Pd(dmit)2 ] 2 ; X= EtMe 3 Sb, EtMe 3 P, Me 4 P... : J /J? AFs, spin liquid, dimerized NaTiO2? little studied 3 He on graphite? spin liquid at 4/7 coverage

30 Cs 2 CuCl 4 Cu 2+ spin-1/2 spins Couplings measured by measuring singlemagnon spectrum in polarizing magnetic field (9T) J=0.37meV J =0.34J J =0.045J D=0.053J chains diagonals inter-layer Dzyaloshinskii-Moriya (a) (021) (010) (020) (021) (011) (001) [0,ζ,ζ-1] [0,2,l] Energy (mev) (b) Counts (/mon~10 mins) Intensity (arb. units) (c) FIG. 2. B =12T a T <0.2K k=(0,1.447,0) B=12 T a [0,k,0] Energy (mev) (d) 2.0 (011)+Q (011)-Q (020)-Q 1.5 [0,k,1] (002)+Q R. Coldea et al, (013)-Q _ Intensity ratio I + /I (a) Magnetic dispersion relations in the saturated phase

31 Dimensional reduction? Frustration of interchain coupling makes it less relevant First order energy correction vanishes Interchain correlations are established only at O[(J )4 /J 3 ]! Other smaller interchain interactions can dominate Elementary excitations of 1d chains - spinons - form a good basis for excitations of the coupled system

32 Dimensional reduction? Numerics show that correlations are weak for J /J < 0.7 Weng et al, 2006 Very different from spin wave theory Very weak inter-chain correlations

33 Broad lineshape: free spinons Power law fits well to free spinon result Fit determines normalization J (k)=0 here

34 Bound state Compare spectra at J (k)<0 and J (k)>0: Curves: 2-spinon 4-spinon theory RPA w/ w/ experimental resolution

35 Low energy phases - zoology? R. Coldea et al, 2002 Energy (mev) S c (arb units) Incommensuration ε (2π/b) B (T) a (a) (b) (c) T <0.2K T <0.1K B (T) a S a (a.u.) Cone (e) (020) (010) (0,.447,1) + (0,.447,1) - B a c b Ferromagnetic (0,1.5-ε,0) (0,0.5-ε,1) T <0.1K B a: only one phase B a: many phases Y. Tokiwa et al, 2006

36 Strategy Quantitative understanding of high energy neutron scattering implies that a description in terms of weakly correlated spin chains is appropriate Use the low energy field theory of the decoupled spin chains to study the low energy physics when they are coupled This field theory is critical (i.e. the system is gapless), with conformal invariance Use renormalization group methods to study the coupling

37 Operators For example, in zero magnetic field: N: Néel vector N ε: staggered dimerization ε>0 ε<0

38 Inter-chain exchange Translates into operator couplings e.g. rectangular lattice J H J dx N y N y+1 Triangular lattice J J J H J dx N y x N y+1 reflection symmetry

39 Power counting Operators scale like L -Δ, with scaling dimension Δ e.g. Δ(N)=1/2. Dimensionless coupling, measured with respect to v k ~ v/l: e.g. rectangular lattice H J dx N y N y+1 g (L) J L L 2 v/l J v L2 2 J v L relevant interaction: generates Néel order at kt~v/l~j

40 Power counting Operators scale like L -Δ, with scaling dimension Δ e.g. Δ(N)=1/2. Dimensionless coupling, measured with respect to v k ~ v/l: triangular lattice H J dx N y x N y+1 g (L) J L L 2 1 v/l J v marginal interaction: exponentially weak effects?

41 Fluctuations Relevant couplings between second neighbor chains are allowed H g 2 dx N y N y+2 Interactions of this type are generated by fluctuations a nasty calculation shows g2 ~ (J ) 4 /v 3 leads to commensurate, collinear AF order with ktc ~ g 2 O. Starykh + LB, 2007

42 Theories (zero field) Spin wave theory? spiral J /J DMRG disordered spiral variational QMC 1d SL 2d SL spiral series expansion spiral 0 1 coupled cluster method collinear spiral

43 DM Cs2 CuCl 4 has a spiral ground state! This is due to Dzyaloshinskii-Moriya interactions H D = Dẑ S x,y H D D S x 1 2,y+1 S x+ 1 2,y+1 dx ẑ N y N y+1 This leads to spiral order with kt ~ D ~ 0.05J >> (J )4 /v 3 ~ (0.3) 4 J ~ 0.008J

44 Behavior in a field The approach is readily generalized to arbitrary magnetic fields Need to understand the operator content of the Heisenberg chain in a field Crucial fact: anisotropic correlations N = N x N y N z N ± S z π±2δ ± < 1/2 XY-like correlations transverse to field z > 1/2 incommensurate SDW correlations parallel to field 2δ=2πM

45 CAF Ideal 2d model: J-J only Competition between longitudinal (less frustrated) and transverse (less fluctuating) couplings T SDW plateau SDW cone (J ) 4 v 3 M = 1 3 M sat ~0.7H sat H sat H

46 CAF Ideal 2d model: J-J only Competition between longitudinal (less frustrated) and transverse (less fluctuating) couplings T up, up, down state SDW plateau SDW cone (J ) 4 v 3 M = 1 3 M sat ~0.7H sat H sat H

47 Cs 2 CuBr 4 Isostructural to Cs2 CuCl 4, but with larger J /J= Shows a well-formed magnetization plateau at Msat /3 (b) M s 2M s / 3 H c3 H c4 M [µ B /Cu 2+ ] M s / 3 H c 0 H H c2 c1 0.2 H b H a Cs 2 CuBr 4 T = 400 mk H [T] T. Ono et al, 2004 Fortune et al, 2009

48 But Cs 2 CuCl 4 looks different R. Coldea et al, 2002 Energy (mev) S c (arb units) Incommensuration ε (2π/b) B (T) a (a) (b) (c) T <0.2K T <0.1K B (T) a S a (a.u.) Cone (e) (020) (010) (0,.447,1) + (0,.447,1) - B a c b Ferromagnetic (0,1.5-ε,0) (0,0.5-ε,1) T <0.1K B a: only one phase B a: many phases Y. Tokiwa et al, 2006

49 But Cs 2 CuCl 4 looks different R. Coldea et al, 2002 Energy (mev) S c (arb units) Incommensuration ε (2π/b) B (T) a (a) (b) (c) T <0.2K T <0.1K B (T) a S a (a.u.) Cone (e) (020) (010) (0,.447,1) + (0,.447,1) - B a c b Ferromagnetic (0,1.5-ε,0) (0,0.5-ε,1) T <0.1K B a: only one phase H D D Field and DM share the same z axis dx N + y N y+1 +h.c. The DM-induced order is strengthened by the field, because Δ decreases length N ± decreases approaching saturation

50 In-plane field DM and field compete to establish the z axis When g μb B >> D, the field establishes the axis: H D D dx N y + S2δ,y+1e z i2δx +h.c. Due to the oscillation, D is now strongly irrelevant Expect to see ideal J-J behavior?

51 No! Reason: J exchange between triangular planes is unfrustrated and not suppressed by the field another strongly relevant perturbation Y. Tokiwa et al, 2006 It is only slightly weaker than D, and takes over when D is washed out by the field B a: many phases

52 Correlated planes? Due to small J << J, J, it is natural to view Cs2 CuCl 4 in terms of correlated b-c planes of spins a c b

53 Correlated planes? But actually, when the magnetic field is in the b-c plane, the spins are more correlated in a-b planes perpendicular to the triangular layers! a c b

54 Extreme sensitivity Order in the a-b planes further amplifies the effects of very weak but unfrustrated interactions (of 1% of J or so) Second neighbor exchange J2 Fluctuation-generated 4-spin interactions Dzyaloshinskii-Moriya coupling Dc on the chain bonds

55 Results vs. Experiment (b) h b T (c) DM spiral ~ h c T D AF Cone hsat FP h Y. Tokiwa et al, 2006 DM spiral ~ D IC AF Cone hsat FP h B a: many phases

56 Results vs. Experiment (b) h b T a commensurate state stabilized by very weak (fluctuation-generated) 4- spin interactions (and possibly J 2 ) (c) DM spiral ~ h c T D AF Cone hsat FP h AF Y. Tokiwa et al, 2006 DM spiral ~ D IC AF Cone hsat FP h AF B a: many phases

57 AF state The AF state has a period of 4 chains in the triangular plane consistent with neutron scattering for B c which observed a commensurate state with this periodicity (R. Coldea) Commensurate state also observed by NMR in this field range (M. Takigawa)

58 Results vs. Experiment (b) h b T (c) DM spiral ~ h c T D AF Cone FP h hsat an incommensurate state, in which the AF order is deformed by D c AF Y. Tokiwa et al, 2006 DM spiral ~ D IC AF Cone hsat FP h IC AF B a: many phases

59 IC State Incommensurate state deformed from AF one by spiraling spins in opposite directions on even and odd chains Evidence for incommensurate state seen in NMR (Takigawa) Neutron data looks very similar to AF state. However, wavevector of incommensurability is expected to be very small (<2%), which may be too small to resolve.

60 Summary Frustration has strong effects on S=1/2 spatially anisotropic triangular antiferromagnets It greatly enhances the regime of quasi-one-dimensionality It leads to tremendous sensitivity to weak perturbations, producing a very complex phase diagram Some additional source of fluctuations (other interactions, charge fluctuations...) is needed to obtain a true QSL state

61 Summary M. Kohno, O. Starykh, L.B., Nature Physics 11, 790 (2007) O. Starykh, L.B., Phys. Rev. Lett. 98, (2007) O. Starykh, H. Katsura, L.B., arxiv: (2010) (a) h a (b) h b T T Cone hsat FP h DM spiral ~ D AF Cone hsat FP h (c) h c (d) Ideal 2d model T T DM spiral ~ D IC AF Cone hsat FP h CAF ( J ' ) ~ J 2 3 SDW ~ h 2 3 Cone sat hsat FP h