Breaking the Spin Waves: Spinons in Cs2cucl4 and Elsewhere

Breaking the spin waves: spinons in !!!Cs2CuCl4 and elsewhere

Oleg Starykh, University of Utah

In collaboration with: K. Povarov, A. Smirnov, S. Petrov, Kapitza Institute for Physical Problems, Moscow, Russia, A. Shapiro, Shubnikov Institute for Crystallography, Moscow, Russia

Leon Balents (KITP), H. Katsura (Gakushuin U, Tokyo), M. Kohno (NIMS, Tsukuba) Jianmin Sun (U Indiana), Suhas Gangadharaiah (U Basel)

arxiv:1101.5275 Phys. Rev. B 82, 014421 (2010) Phys. Rev. B 78, 054436 (2008) Spin Waves 2011, St. Petersburg Nature Physics 3, 790 (2007) Phys. Rev. Lett. 98, 077205 (2007)

Wednesday, April 18, 12 Outline

• Spin waves and spinons • Experimental observations of spinons

➡ neutron scattering, thermal conductivity, 2kF oscillations, ESR

• Cs2CuCl4: spinon continuum and ESR • ESR in the presence of uniform DM interaction - ESR in 2d electron gas with Rashba SOI - ESR in spin liquids with spinon Fermi surface • Conclusions

Wednesday, April 18, 12 Spin wave or magnon = propagating disturbance in magnetically ordered state (ferromagnet, antiferromagnet, ferrimagnet...)

+ i k.r Σr S r e |0> sharp ω(k)

La2CuO4

Carries Sz = 1. Observed via inelastic neutron scattering as a sharp single-particle excitation.

magnon is a boson (neglecting ﬁnite dimension of the Hilbert space for ﬁnite S) Coldea et al PRL 2001

Wednesday, April 18, 12 But the history is more complicated Regarding statistics of spin excitations:

“The experimental facts available suggest that the magnons are submitted to the Fermi statistics; namely, when T << TCW the susceptibility tends to a constant limit, which is of 5 the order of const/TCW ( ) [for T > TCW, χ=const/(T + TCW)]. Evidently we have here to deal with the Pauli paramagnetism which can be directly obtained from the Fermi distribution. Therefore, we shall assume the Fermi statistics for the magnons.”

Wednesday, April 18, 12 But the history is more complicated Regarding statistics of spin excitations:

(5) A. Perrier and Kamerlingh Onnes, Leiden Comm. No.139 (1914)

I. Pomeranchuk, JETP 1940

Wednesday, April 18, 12 But the history is more complicated Regarding statistics of spin excitations:

(5) A. Perrier and Kamerlingh Onnes, Leiden Comm. No.139 (1914)

solid oxygen

I. Pomeranchuk, JETP 1940

Wednesday, April 18, 12 30+ years later P. W. Anderson, Resonating valence bonds: a new kind of insulator? Mat. Res. Bul. (1973) P. Fazekas and P. W. Anderson, Philos. Magazine (1974)

Science 235, 1196 (1987)

and the arguments are still evolving ...

Wednesday, April 18, 12 Spinons are natural in d=1

Bethe’s solution: 1933 identiﬁcation of spinons: 1981

S=1 spin wave breaks into two domain walls / spinons: hence each is carrying S=1/2

Wednesday, April 18, 12 Two-spinon continuum of spin-1/2 chain

Spinon energy de Cloizeaux-Peason dispersion, 1962

S=1 excitation

Upper boundary ε

Energy Variables:

kx1 and kx2 Qx or Lower Low-energy sector QAFM=π ε and Qx boundary

Wednesday, April 18, 12 Highly 1d antiferromagnet CuPzN

Stone et al, PRL 2003 J’/J < 10-4

Wednesday, April 18, 12 J J’

Along the chain

Cs2CuCl4

J’/J=0.34 J to chain transverse J’ k Very unusual response: broad and strong continuum; spectral intensity varies strongly with 2d momentum (kx, ky)

Wednesday, April 18, 12 J J’

Along the chain

Cs2CuCl4

J’/J=0.34 J to chain transverse J’ k Very unusual response: broad and strong continuum; spectral intensity varies strongly with 2d momentum (kx, ky)

Wednesday, April 18, 12 Efective Schrödinger equation in two-spinon basis Kohno, OS, Balents: Nat. Phys. 2007

• Study two spinon subspace

z (two spinons on chain y with S tot =+1) – Momentum conservation: 1d Schrödinger equation in ε space (k = (kx, ky))

• Crucial matrix elements known exactly

Bougourzi et al, 1996 • Calculate dynamic spin structure factor

Wednesday, April 18, 12 Dimensional reduction due to frustration: spinons in Cs2CuCl4 are of 1d origin, propagate between chains by forming S=1 triplon pairs

• Comparison:

Kohno, Starykh, Balents, Nature Physics 2007

’ ’  Vertical green lines: J’(k)=4J’ cos[k x/2] cos[k y/2] = 0.

Wednesday, April 18, 12 How else can we probe/see spinons?

mean-free path ~1 micron !

Wednesday, April 18, 12 Friedel oscillations due to spinon Fermi surface

RKKY like coupling mediated by spinons

• Charge Friedel oscillations in a Mott insulator, Mross and Senthil, arxiv:1007.2413

Wednesday, April 18, 12 ESR - electron spin resonance

• Simple (in principle) and sensitive probe of magnetic anisotropies (and, also, q=0 probe: S = Σr Sr) Zeeman H along z, w iwt + I(h,w)= dte [S (t),S] microwave radiation h L h i 4 Z polarized perpendicular to it. • For SU(2) invariant chain in paramagnetic phase I(H,ω) ~ δ(ω-H) m(H) [Kohn’s Th] Oshikawa, Afﬂeck PRB 2002 ﬁnite H along z, H=0 transverse structure factor Sxx and Syy

gμBH

Wednesday, April 18, 12 Main anisotropy in Cs2CuCl4: asymmetric Dzyaloshinskii-Moriya interaction ~D ~S ~S ij· i ⇥ j • Is known from inelastic neutron scattering data (Coldea et al. 2001-03) • 3D ordered state - determined by minute residual interactions - interplane and Dzyaloshinskii-Moriya (DM) OS, Katsura, Balents 2010

y z Dy,z =( 1) Dccˆ+( 1) Daaˆ (a) 4 3 J”

2 a 5 different c 1 b focus on the J” DM terms (b) 4 in-chain DM 3 allowed

2 a c 1 b

Wednesday, April 18, 12 Theory I: H along DM axis

H = Â JSx,y,z Sx+1,y,z Dy,z Sx,y,z Sx+1,y,z gµBH Sx,y,z x,y,z · · ⇥ · chain uniform DM along the chain magnetic ﬁeld • Unitary rotation S+(x) -> S+(x) ei D x/J removes DM term from the Hamiltonian (to D2 order) ‣ This boosts momentum to D/(J a0) q = 0 q = D/(Ja ) 2ph¯n = gµ H pD/2 ! 0 ) R/L B ±

rotated basis: q=0 original basis: q=D/J D=0 picture Oshikawa, Afﬂeck 2002

Chiral probe: ESR probes right- and left- moving modes (spinons) independently

Wednesday, April 18, 12 Theory II: arbitrary orientation

Relevant spin degrees of freedom

• Spin-1/2 AFM chain = half-filled (1 electron per site, kF=π/2a ) fermion chain

. q=0 fluctuations: right (R) and left (L) spin currents ~s M~ = Y† ss0 Y R/L R/L,s 2 R/L,s0 . 2k (= /a) fluctuations: charge density wave , spin density wave N F π ε Susceptibility

Staggered 1/q Spin flip ΔS=1 -kF kF Magnetization N 1/q

Staggered 1/q Dimerization -kF kF ΔS=0 x ε = (-1) Sx Sx+a

• Must be careful: often spin-charge separation must be enforced by hand

Wednesday, April 18, 12 Theory II: arbitrary orientation (cont’d)

2pv 2 2 vD d d z z [(M~ R) +(M~ L) ] [M M ] gµBH[M + M ] H = 3 J R L R L unperturbed chain uniform DM along the chain magnetic ﬁeld

BL H BR

Uniform DM produces internal momentum-dependent magnetic field along d-axis -D +D • Total field acting on right/left movers gµ H~ hv¯ ~D/J B ± • Hence ESR signals at 2ph¯n = gµ H~ hv¯ ~D/J R/L | B ± | • Polarization: for H=0 maximal absorption when microwave field hmw is perpendicular to the internal (DM) one. Hence hmw || b is most effective.

Gangadharaiah, Sun, OS, PRB 2008

Wednesday, April 18, 12 Temperature regimes ESR Paramagnetic J S individual spins single line

universal quasi-classical correlated spins regime (high T ﬁeld theory)

-2πS spin-correlated T0 ~ J e (spin liquid) Haldane scale does exist for S=1/2 chains: different response for S=1 and S=1/2 chains well-developed below this temperature two lines correlations along chains; spinons but little correlations (low T ﬁeld theory) between chains

ordered phase TN = 0.6 K strongly coupled chains; spin waves AFMR 2d (or 3d) description (at low energy) Povarov et al, 2011 C. Buragohain, S. Sachdev PRB 59 (1999)

Wednesday, April 18, 12 Temperature regimes ESR Paramagnetic J S individual spins single line

universal quasi-classical correlated spins regime (high T ﬁeld theory)

ordered phase TN = 0.6 K strongly coupled chains; spin waves AFMR 2d (or 3d) description (at low energy) Povarov et al, 2011 C. Buragohain, S. Sachdev PRB 59 (1999)

Wednesday, April 18, 12 Another (more frequent) geometry: staggered DM

ﬁeld-induced shift

ﬁeld-induced gap!

Oshikawa, Afﬂeck ( 1)xD S S Essler, Tsvelik Â · x ⇥ x+1 x but: single ESR line!

Wednesday, April 18, 12 Uniform vs staggered DM

~D ~S ~S ( 1)n~D ~S ~S · n ⇥ n+1 · n ⇥ n+1 h=0: free spinons free spinons

ﬁnite h: free spinons conﬁned spinons

but subject to varying generate strongly relevant ~ ~ n D h with momentum transverse magnetic ﬁeld ( 1) ⇥ ~Sn magnetic ﬁeld that binds spinons together 2J ·

ESR: generically two lines single line ✓ splitting ✓ shift ✓ shift ✓ width ✓width (?)

Wednesday, April 18, 12 Analogy with Rashba S-O interaction in 2d electron gas

Original suggestion: Kalevich, Korenev JETP Lett 52, 230 (1990)

aR(pxsy pysx) aR jx sy !

Wednesday, April 18, 12 Analogy with Rashba S-O interaction in 2d electron gas

• 2d gas: dc current produces static Rashba ﬁeld BR (due to drift velocity) which adds to external B0 • 1d AFM: uniform DM interaction produces k- dependent internal magnetic ﬁeld which adds to external B0 • But note that in 1d antiferromagnet the spinons are neutral fermions ! • Key question: can this approach be extended to two-dimensional spin liquids with spinon Fermi surface? (uniform RVB state of P. W. Anderson 1987)

Wednesday, April 18, 12 Tentative sketch for 2d (using Rashba model as an example)

ESR signal no DM in-plane h q=0 transitions ω = gμh at single ω = gμh

a (p s p s ) DM R x y y x ωmin

in-plane h

ωmax

2 2 splitting of Fermi surfaces DSO + DZ + 2DSODZ sinf Raikh, Chen 1999 q ESR signal the width ~ h line shape? effect of gauge fluctuations?... ωmin ωmax h

Wednesday, April 18, 12 Conclusions

• Spinons vs. spin waves • ESR as a novel probe of critical spinons (neutral fermions) - small momentum ~ D/J - allows to extract parameters of DM interaction • Higher-dimensional extensions?!

“old” ESR probe can access “new” deconfined spinons with a help from DM interactions

Wednesday, April 18, 12