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Breaking the Spin Waves: Spinons in Cs2cucl4 and Elsewhere

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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: “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.” (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: “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.” (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 transverseto chain 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 transverseto chain 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 =  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) -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 Another (more frequent) geometry: staggered DM field-induced shift field-induced gap! Oshikawa, Affleck ( 1)xD S S Essler, Tsvelik  − · 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.
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