Thermal Hall effect in SrCu 2 (BO 3 ) 2

Transcription

1 Thermal Hall effect in SrCu (BO 3 ) Judit Romhányi R. Ganesh Karlo Penc Wigner Research Centre for Physics! Institute for Solid State Physics and Optics! Budapest Nat. Comm. 6, 685 (5) [arxiv:46.63] supported by Hungarian

2 The Shastry-Sutherland model exact eigenstate of a triangle: H 4 = S S + S S 3 + S S 3 = (S + S + S 3 ) 9 8 = 3 arbitrary spin state singlet! bond Majumdar-Ghosh model! a singlet coverings which is an exact ground states H = X H 4 = X i (S i S i+ + S i S i+ ) J =/ J=

3 The Shastry-Sutherland model exact eigenstate of a triangle: H 4 = S S + S S 3 + S S 3 = (S + S + S 3 ) 9 8 = 3 arbitrary spin state singlet! bond Majumdar-Ghosh model! a singlet coverings which is an exact ground states D extension: the dimer covering is an exact eigenstate Shastry and Sutherland Physica B 8, 69 (98)

4 Experimental realization: SrCu (BO 3 ) J J Smith & Keszler, J. Solid State Chem. 93, 43 (99) Kageyama et al., Phys. Rev. Lett. 8, 368 (999) J: AFM intradimer interaction J : AFM interdimer interaction Shastry-Sutherland model: singlet-dimer covering an exact ground state for J /J<.7

5 Magnetization energy levels of a single dimer in a field Many plateaus... E J 3J h crit triplets h singlet triplet excitations like quasi particles on the singlet background Kageyama H., et al Prog. Theor. Phys. Suppl. 45 But in this talk we will be interested what happens in low field...

6 Triplon dispersion from neutron spectra Kageyama et al., Phys. Rev. Lett. 84, 5876 () E gap J triplets 3J h crit singlet h Dispersionless band of triplets gap k Why are triplets localized?

7 Why are triplets localized in orthogonal dimer systems? The parity of the singlet and triplet bond is different: reflexion triplet even singlet odd minus sign enters hopping matrix elements not allowed by symmetry a triplon can be created next to an existing triplon

8 Why are triplets localized in orthogonal dimer systems? The parity of the singlet and triplet bond is different: reflexion triplet even singlet odd minus sign enters hopping matrix elements not allowed by symmetry a 6th order process in J /J gives a weak dispersion S. Miyahara, K. Ueda. (3)

9 Neutron spectra Kageyama et al., Phys. Rev. Lett. 84, 5876 () pair hopping k

10 Neutron spectra - better resolution Kageyama et al., Phys. Rev. Lett. 84, 5876 () Even though there is no external magnetic field, the triplet excitations split. Why? The reason: anisotropic Dzyaloshinskii-Moriya interaction +D (S i S j ) Gaulin et al. Phys. Rev. Lett. 93, 67 (4)

11 Effects of anisotropy - ESR spectra ESR: splitting of the triplets, gap does not closes in the filed E J 3J ESR triplets singlet h crit h H. Nojiri, H. Kageyama, Y. Ueda and M. Motokawa! J. Phys. Soc. Japan 7, 343 (3)

12 Effects of anisotropy - ESR, neutron, FIR spectra Gaulin et al. Phys. Rev. Lett. 93, 67 (4).8 mev 68 GHz! 3. mev 77 GHz T. Rõõm et al., PRB 7, 4447 (4). H. Nojiri, H. Kageyama, Y. Ueda and M. Motokawa! J. Phys. Soc. Japan 7, 343 (3)

13 Dzyaloshinskii s symmetry argument (958) lb i Certain antiferromagnets ( -Fe O 3, CoCO 3 ) show weak ferromagnetism (moment.-. of full moment), sensitive to crystal symmetry. The Heisenberg Hamiltonian is not enough. H Heis = JS S Dzyaloshinskii constructed the allowed spin invariants (Hamiltonian) taking into account the symmetries of the lattice. H DM = D (S S )

14 Moriya's perturbational calculation (96) PH YSICAL REVIEW VOLUME, NUM BER OCTOBER, 96 Anisotropic Superexchange Interaction and Weak Ferromagnetism TQRU MORIYA Bell Telephone Laboratories, 3IIurray Hill, Eem Jersey (Received May 5, 96) A theory of anisotropic superexchange interaction is developed by extending the Anderson theory of superexchange to include spin-orbit coupling. The antisymmetric spin coupling suggested by Dzialoshinski from purely symmetry grounds and the symmetric pseudodipolar interaction are derived. Their orders of magnitudes are estimated to be (Ag/g) and (ng/g) times the isotropic superexchange energy, respectively. Higher order spin couplings are also discussed. As an example of antisymmetric spin coupling the case of CuC H~O is illustrated. In CuCl~ H, a spin arrangement which is different from one accepted so far is proposed. This antisymmetric interaction is shown to be responsible for weak ferromagnetism in o.-fe&3, MnCO3, and CrF3. The paramagnetic susceptibility perpendicular to the trigonal axis is expected to increase very sharply near the Neel temperature as the temperature is lowered, as was actually observed in CrF3.

15 Moriya's perturbational calculation (96) action between the spins at R and Eg, g = &'& Jg, a &'&(S(R). S(R')) antisymmetric +D.&& LS(R)XS(R')] +S(R) Fg.a S(R') &'& (.3) symmetric where the scalar, vector and tensor quantities: Jzz (", Dg, g &~, and F~a &'& are given in the case of one electron per ion as follows: Jg, g &'& = ib (R R') i'/u, (.4a) Dg, s, &"= (4i/U) Lb (R R') C ~ (R. ' R) C (R R')b (R' R)], (.4b) rgg &'&= (4/U)LC (R R')C (R' R) ~ +C (R' R) C..(R R') isotropic Heisenberg (C. (R R') C (R' R))]. (.4c) J» D (hg/w) J, F (Ag/g)'J.

16 Symmetry point group of SrCu (BO 3 ) p4g high symmetry wallpaper group (above T s =395K) Wikipedia: C 4h paintings from China fly screen from US cmm low symmetry wallpaper group (below T s =395K) C v painting from an egyptian tomb S 4 C v

17 DM - interactions allowed by the symmetry H = J S i S j + J S i S j h S j nn nnn j D A A +D nn (S i S j )+D nnn (S i S j ) y D D D B x B Low symmetry phase (T< 395K): intradimer: interdimer: D D? D kns D ks e.g. K Kodama et al., J. Phys.: Condensed Matt 7, L6 (5)

18 Shastry-Sutherland model + DM interaction (a) D H = J S i S j + J n.n. n.n.n. S i S j h z i + D ij (S i S j )+ D ij (S i S j ) n.n. n.n.n. S z i Ground state: = s A s B D J J D E = H s A = s A + t y A s B = s B t x B minimization gives = D J JR et al PRB () the new triplet basis: t x i A = t x i A t x i B = si B + t x i B t y i A = si A + t y i A t y i B = t y i B t z i A = t z i A t z i B = t z i B correspond to the 6 different triplet -like excitation t x i = i p ( ""i ##i) t y i = p ( ""i + ##i) t z i = i p ( "#i + #"i)

19 The variational approach and bond-wave spectra Step : we minimize the energy with a dimer-product variational wave function (entanglement within dimers taken care of): = bonds ( s s + t + t + t )( s s + t + t + t ) Step : starting from the variational solution, we construct a bond-wave hamiltonian analogous to the spin-wave approach: S j, = i t,j s j s j t i,j,, t,j t,j, i S j, = t,j s j s j t i,j,, t,j t,j. s and t bosons are like Schwinger-bosons, and condensing one boson (variational solution) we get the Holstein-Primakoff representation and a /M expansion: s,s M µ t µt µ M M µ t µt µ quadratic Hamiltonian which can be Bogoliubov diagolized: H () = X kbz t k t k! T M N N M t k t k!

20 Comparison with experiments: neutron spectra at H= bond-wave result: see also: Y. F. Cheng, PRB 75, 444 (7). q = q = J ± J q J ± q J D D J,s ( cos a cos b) + 6D cos a b cos J. Romhányi, K. Totsuka, K. Penc, Phys. Rev. B 83, 443 (). / B. Gaulin et al.,! PRL 93, 67 (4). ω/j.9.8 (,) (π,π) (π,) (,) (q a,q b ) The triplet dispersion due to J (6th order process) are missing from our treatment.

21 (a) Quantitative comparison with measured ESR & IR spectra: gap in the field T o - T e - z m J. Romhányi, K. Totsuka, K. Penc, Phys. Rev. B 83, 443 ()..5.5 (a) ν (GHz) T e T o T e T o 9 8 (b) T e ω (b) Miyahara et al four bosons two bosons 7.4. D=. J D z=. J h z (T) h z..4 H. Nojiri, H. Kageyama, Y. Ueda and M. Motokawa! J. Phys. Soc. Japan 7, 343 (3) T. Rõõm et al., PRB 7, 4447 (4).

22 Hamiltonian describing the hopping of triplets H = assuming that t x,b,k t y,b,k t z,b,k t x,a,k t y,a,k t z,a,k T h, D, D J, J J ih z D ih z J D idj J 3 idj J J idj D 3 J J ih z idj D J ih z J idj J J idj 3 J idj 3 J idj J t x,b,k t y,b,k t z,b,k t x,a,k t y,a,k t z,a,k analytical solution:, = J 3,4 = J 4 dispersive bands 5,6 = J + ± flat bands geometrical factors: = cos k x cos k y =sin k x + k y 3 =sin k x k y with ± = (h z ± D ) + D J full bond-wave theory gives the same results, linear in D 4J ( + 3 ) y D D x D A D A B B

23 The dispersion: D A A y D D D B x B analytical solution:, = J 3,4 = J 4 dispersive bands 5,6 = J + ± flat bands with ± = (h z ± D ) + D J 4J ( + 3 ) kx ky Note: the dispersions are degenerate at the Brillouin zone boundary!

24 The 3x3 matrix for the triplets Unfold 6x6 into 3x3 H = t x,k t y,k t z,k C A T DJ J 3 J ih z +id DJ ih z id J DJ J 3 J DJ J J C B t x,k t y,k t z,k C A t x,k = t x,a,k or i t x,b,k t y,k = t y,a,k or i t y,b,k t z,k = t z,a,k or i t z,b,k

25 Band touching transition of triplons in magnetic field (c) h c =D h z =h c / C n = i BZ d kf xy n (k). + Berry curvature F xy n (k) =@ kx ky ky kx n(k)i n(k) is the normalized wave function which belongs to n(k) X Γ M

26 Band touching transition of triplons in magnetic field h c =D (a) (b) (c) (d) (e) h z = h z =h c / h z =h c h z =3h c / J+D J+D + ω J J D J D X Γ M X Γ M X Γ M X Γ M C n = i BZ Berry curvature d kf xy n (k). F xy n (k) =@ kx ky ky kx n(k)i n(k) is the normalized wave function which belongs to n(k)

27 Band touching transition of triplons in magnetic field h c =D (a) (b) (c) (d) (e) h z = h z =h c / h z =h c h z =3h c / J+D J+D + ω J J D J D X Γ M X Γ M X Γ M X Γ M Note: the ground state is trivial all the time! h = topologically nontrivial excitations, finite Chern numbers h = h c band touching transition topologically trivial

28 How do we understand the Chern numbers? H = t x,k t y,k t z,k C A T DJ J 3 J ih z +id DJ ih z id J DJ J 3 J DJ J J C B t x,k t y,k t z,k C A

29 How do we understand the Chern numbers? H = t x,k t y,k t z,k C A T DJ J 3 J ih z +id DJ ih z id J DJ J 3 J DJ J J C B t x,k t y,k t z,k C A Q x = Q y = Q @ i i A A A A more appealing form: d(k) = H(k) DJ J DJ J d(k) Q kx ky sin kx+ky sin h z +D? cos kx ky cos A Q,Q = i Q Q : generators of an SU() algebra, T= pseudospin d(k) : a fictitious magnetic field, Zeeman splits the T= pseudospins

30 Chern number and skyrmions d(k) = H(k) DJ J DJ J kx ky sin kx+ky sin h z +D? cos kx d(k) Q ky cos A Texture of the d(k) fictitious magnetic field in the Brillouin zone. It makes a skyrmion. N s = 4 Z dk x dk y y ˆd Chern numbers (a) (b) Skyrmion numbers C= Ns= (c) (d) C= Ns=

31 Band touching and skyrmions (a) (b) the d surface of the d(k) energy dispersion (c) (d) d(k) DJ J DJ J kx ky sin kx+ky sin h z +D? cos kx ky cos A H(k) =d(k) Q

32 Chern number and skyrmions : arbitrary spin Berry curvature F xy n (k) =@ x y y x n(k)i =i X m6=n Im hn (@ xh) mihm (@ y H) ni (E n E m ). Hamiltonian (Zeeman levels) H(k) =J d(k) Q H(k) ni =[J nd(k)] ni Q,Q = i" Q SU() algebra F xy n (k) =i X, =i x d (k)@ y d (k) d x d (k)@ y d (k) d (k) X m6=n Im hn Q mihm Q ni (n m) Im hn Q n+ihn+ Q ni + hn Q n ihn Q ni. = xˆd(k) the Berry curvature is proportional to the skyrmion density skyrmion number N s = Z dk x dk 4 C n = i Z dk x dk y F xy n = nn s The Chern number of the n-th band is n times the number of skyrmions -> n edge states

33 Band touching and Berry curvature Berry curvature color coded on the energy surface energy dispersion Fn xy (k) =i X Im hn (@ xh) mihm (@ y H) ni (E n E m ). m6=n H(k) =d(k) Q d(k) DJ J DJ J kx ky sin kx+ky sin h z +D? cos kx ky cos A

34 Band touching transition in magnetic field vs ESR (a) triplon density of states the blue and red lines are at the point.5 ω/j.95.5 C = C = C =+ C = ω/j (b) C = C = h z /J C = C = negative/positive Berry curvature

35 Band touching transition in magnetic field vs ESR A Dirac cone in the neutron inelastic spectrum at h c : h c =D

36 Edge states and local density of triplons (a) J+3D J+D h z = h / c C=- (b) J+D ω J J D J D C=+ J 3D π π/ π/ π k x number of edge states is given by C Chern numbers edge state carry energy and spin current

37 Thermal Hall Conductivity Chern bands in electrons quantum Hall effect! B transverse heat current T Z apple xy = X n n = e! n c ( ) = Z BZ d k c ( n ) F n xy (k) i dt ln ( + t ) thermal Hall effect in bosons: linear response (Kubo formula) formalism! Katsura et al., PRL 4, 6643 (),! Matsumoto et al PRL 6 97, ()

38 Observation of the Magnon Hall Effect SCIENCE VOL 39 6 JULY Y. Onose,, * T. Ideue, H. Katsura, 3 Y. Shiomi,,4 N. Nagaosa,,4 Y. Tokura,,4 LuVO7 is a FM insulator Magnetic field variation of the thermal Hall conductivity of LuVO7 at various temperatures. The magnetic field is applied along the [] direction. The solid lines are guides to the eye. The crystal structure of Lu V O 7 and the magnon Hall effect. (A) The V sublattice of Lu V O 7, which is composed of corner-sharing tetrahedra. (B) The direction of the Dzyaloshinskii-Moriya vector on each bond of the tetrahedron. The Dzyaloshinskii-Moriya interaction acts between the i and j sites. (C) The magnon Hall effect. A wave packet of magnon (a quantum of spin precession) moving from the hot to the cold side is deflected by the Dzyaloshinskii-Moriya interaction playing the role of a vector potential.

39 Thermal Hall conductivity in the frustrated pyrochlore magnet Tb Ti O 7! M. Hirschberger, J. W. Krizan, R. J. Cava, and N. P. Ong! arxiv 5.6 FIG. 3: Curves of the thermal Hall conductivity κxy/t vs. H in TbTiO7 (Sample ). From 4 to 5 K, κxy/t, is H- linear (Panel A). Below 45 K, it develops pronounced curvature at large H, reaching its largest value near K. The sign is always hole-like. Panel B shows the curves below 5 K. A prominent feature is that the weak-field slope [κxy/tb] is nearly T independent below 5 K. Below 3 K, the field profile changes qualitatively, showing additional features that become prominent as T, namely the sharp peak near T and the broad maximum at 6 T.

40 Thermal Hall Conductivity Chern bands in electrons quantum Hall effect! Triplons do not carry charge currents, only heat (energy) current B transverse heat current T Z apple xy = X n n = e! n c ( ) = Z BZ d k c ( n ) F n xy (k) i dt ln ( + t ) thermal Hall effect in bosons: linear response (Kubo formula) formalism! Katsura et al., PRL 4, 6643 (),! Matsumoto et al PRL 6 97, ()

41 Thermal Hall Conductivity Chern bands in electrons quantum Hall effect! Triplons do not carry charge currents, only heat (energy) current B transverse heat current apple xy ( )=R(! )apple xy T R(x) = apple xy = Z x 4 sinh x BZ d k @k y thermal Hall effect in bosons: linear response (Kubo formula) formalism! Katsura et al., PRL 4, 6643 (),! Matsumoto et al PRL 6 97, ()

42 Thermal Hall Conductivity Chern bands in electrons quantum Hall effect! Triplons do not carry charge currents, only heat (energy) current B transverse heat current T κ xy ( W/Km) (a) h z (T) T=5 K T= K T=5 K T= K (b) (c) ν = ν = ν = h z =h c / T (K) κ xy ( W/Km) filling fraction thermal Hall effect in bosons: linear response (Kubo formula) formalism! Katsura et al., PRL 4, 6643 (),! Matsumoto et al PRL 6 97, ()

43 ..5 effect of diagonal nd neighbor hopping (,) (π,) (π/,π/) (,) K =.4 b K = (k x,k y ) ω/ b.95.9 (,) (π,π) (π,) (,) (q x,q y ) The triplet dispersion due to J (6th order process) are missing from our treatment. B. Gaulin et al.,! PRL 93, 67 (4) H(k) =[J K 4 (k)] + d(k) Q

44 Effect of second neighbor triplon hopping..5 (,) (π,) (π/,π/) (,) K =.4 b K = (k x,k y ) J+3D J+D J+D Edge states ω/ b ω J J D.95 J D.9 (,) (π,π) (π,) (,) (q x,q y ) apple xy = The thermal Hall response! Z BZ d k (! (k) ) sinh (! (k) ) d d @k x does not change for small K hopping, as the wave functions unaffected H(k) =[J K 4 (k)] + d(k) Q π π/ π/ π. k x

45 Summary DM interactions lead to Dirac cone in the excitation spectrum! Chern numbers ± topological triplon bands! Protected edge states! Observable thermal Hall signature! Topological effects in plateaux phases of SrCu (BO 3 ) or other dimer/paramagnetic systems?! How to apply general symmetry arguments?! Look for the level crossings in the ESR spectra!

46 Thank you for your attention