Hae-Young Kee University of Toronto Novel Quantum States in Condensed Matter 2022, Yukawa Institute for Theoretical Physics, Japan (Nov. 22, 2022)

Theory
|J. Rau (Univ. of Windsor), |H.S. Kim (Kangwon Nat. Univ.), |A. Catuneanu (Dana/CanmetMaterials), | Q. Luo (Nanjiang Univ. of Aeronautics & Astro) P. Stavropoulos (Univ. of Minnesota) J. Gordon J. Cen, E. Sorensen (McMaster Univ.), W. Yang, A. Nocera, I. Affleck (Univ. British Columbia), Y.-B. Kim group (Univ. of Toronto), S. Trebst group (Univ. of Cologne), Matthias Gohlke (Okinawa Univ.)
Experiment
Y.-J. Kim group (Univ. of Toronto), K. Burch (Boston College), L. Yang (Tsinghua Univ.), D. Dessau (Univ. of Colorado Boulder), H. Takagi (Max Planck Institute), A. Seo (Kentucky Univ.) Review article on “Beyond Kitaev physics”, N. Perkins, I. Rousochatzakis, Q. Luo, HYK, in preparation
where  γ = x, y, z bond-dependent Ising interaction Kitaev spin liquid: emergent particles - Majorana fermion and vortices A. Kitaev, Annals of Physics 321, 2 (2006) Example: Ising Topological Order (Z2 vortex & Fermions)
vortex carrying unpaired MF
SU(2): U and single-orbital When orbital & spin are not conserved via SOC
: symmetry constrains matrix elements insufficient to understand why one term is larger over other
: Kitaev interaction is the largest interaction in full H Necessary (not sufficient) Requirements honeycomb Mott insulator : strong e-e interaction ● transition metals ● bond-dependent spin interaction for S=I/2 multi-orbital systems with Hund's coupling spin-orbit coupling G. Khaliullin on triangular lattice (2005); G. Jackeli, G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009)
nearest neighbour: ideal honeycomb H =
HY, another bond-dep. interaction H z  = [K z S z S z + | Γ z ( S z S y + S y S z )] |+ J S z · S z 〈ij〉Ez—bond + other interactions allowed by the symmetry H x = H z ( x → y → z → x ) J. Rau, E. Lee, HYK, PRL  I I 2, 077204 (2014) other ref. - V. Katukuri, et al, New J. Phys. 16, 013056 (2014); Y.Yamaji, et al, PRL 113, 107201 (2014)
A2IrO3 (A = Na, Li): Y. Singh, et al, PRB 82, 064412 (2010);... alpha-RuCl3    : K. Plumb, et al, PRB 90 041112(R) (2014); ... AlrO3 (A = Mg, Zn): Y. Haraguchi, et al, PRM 2, 054411 (2018),.... on, carnelleivistic SOC - play an squarand states and low-e llows:  Sec. eudospin-1/1 ospin Hami ige geometr ns written in J~ We regard and | ↓), corresp 
20 rty (conserving hopping;)Heisenberg 3d7: Jeff=I/2 honeycomb Cobaltates: R. Sano et al, PRB 97, 014408 (2018), H. Liu, G. Khaliulin, PRB 97, 014407 (2018), .... 3, 1う, 「〉 Debate over the strength of Kitaev interaction 312 ' Co²+ : 3d7
For arbitrary S, Wp = eiπ(Si +Si +Si +Si +Si +Si +Si ) ultra-short range correlations G. Baskaran, D. Sen, R. Shankar, PRB 78, 115116 (2008) Quantum spin liquid? Majorana excitations? half-integer vs. integer-S Kitaev?
S=1 Kitaev model: Plaquette operators that commute with the Hamiltonian [1] (b)1.4 May be a gapless spin liquid [2] Has incipient entropy plateau [2,3] 10 Q: How do we get S= | Kitaev interaction? [1] G. Baskaran, D. Sen, and R. Shankar, Phys. Rev. B 78, 115116 (2008). [2] A. Koga, H. Tomishige, and J. Nasu, Journal of the Physical Society of Japan 87, 063703 (2018). [3] J. Oitmaa, A. Koga, and R. R. P. Singh, Phys. Rev. B 98, 214404 (2018).
eg S=1 t2g
P. Peter Stavropoulos, D. Pereira, HYK PRL 123, 037203 (2019) ex: spin one (d8) Crystal field splitting   > Hund’s coupling   >>   spin-orbit coupling (SOC) No mixture of different orbitals & spins - No Kitaev?
perturbation theory: hopping between two M sites via heavy A sites Indirect K = −2Jind Direct Jd = 4t²/U L = 0 : up to 4th order H ij = K Sk Sj + JSj · Sj AF Kitaev J = − | Jind | + Jd ED calculation: S=I KJ model 12 & 18 sites P. Peter Stavropoulos, D. Pereira, HYK, PRL  123, 037203 (2019) Kitaev from heavy anion from ab-intio: C. Xu et al, npj Comput. Mater. 4, 57 (2018)
Single layer of Crl3 (S=3/2 honeycomb): Ferromagnetic insulator Kitaev interaction? μ 6 H (T) Single layer (2D) Ferromagnet B. Huang et al, Nature 546, 7657 (2017) trigonal distortion from SOC + distortion P. P. Stavropoulos, Liu, HYK, PRR3, 013216 (2021)
Angle dependent Ferromagnetic Resonance (FMR): Crl3            80
Ho (kOe) Kitaev dominant Inhee Lee, et al, Phys. Rev. Lett. 124, 017201 (2020) Inelastic neutron scattering J + DM interaction L. Chen et al, PRX II, 031047 (2021)
: Kitaev interaction is the largest interaction in full H Necessary (not sufficient) Requirements ●   honeycomb Mott insulator : strong e-e interaction Debate over the strength of Kitaev interaction ● bond-dependent spin interaction for S=I/2 multi-orbital systems with Hund's coupling spin-orbit coupling S >   I/2 : spin-orbit coupling in heavy ligand + Hund's coupling in transition metal Goal: how to determine the Kitaev strength? Use Symmetry of full Hamiltonian Jiefu Cen, HYK, Commun. Phys. 5, 119 (2022); Jiefu Cen, HYK, arXiv:2208.13807 (2022)
mirror plane C2b = 180 rotation about b-axis b is parallel to the bond; a/c-axis are perp. to the bond/plane
C2a= 180 rotation about a-axis C2a èc
S 0 V2 ê x Ey
 Èz b is parallel to the bond; a/c-axis are perp. to the bond/plane
H in abc coordinate

in-plane vs. out-of-plane anisotropy due to Gamma J. Chaloupka, G. Khaliullin, PRB 92, 24413 (2015); Pyrochlore lattice, S. Onoda, J. Phys.:Conf. Ser. 320, 012065 (2011); K. Ross, et al, PRX 1, 021002 (2011)
Consequence of broken C2a
+ single-ion anisotropy, DM on 2nd n.n. with d//c,  further n.n. Heisenberg + ... All interactions are invariant under C2a  Except Jac →一Jac Jac = − K − − − − . •   A magnetic field can detect the broken C 2 α symmetry and be used to isolate the interaction that breaks C2a・
Two different directions of field related by C2a: different excitation energies only if Jac finite En(d) — En(—θ) × Jac field is in the ac-plane needs to isolate both Gamma and K
Recipe to isolate Kitaev
I.apply  B in the ac-plane with  θ 2. measure the spin excitations 3. rotate B with angle of — θ 4. measure the spin excitations 5. energy difference between 2 & 4:
6. In-plane and out-of-plane energy excitation = Gamma
7. subtract Gamma from 5, then single out the Kitaev interaction! ☎ For S=1/2
Angle-dependent Ferromagnetic Resonance, NMR, Optical spectroscopy R δω A ≡ ω(θ = 0) — ω(θ = 90°) ∞ Jxy = Jz = L
Inelastic Neutron Scattering
θ =土30° Can we use a “fixed” magnetic field direction? — Two different momenta under a certain magnetic field
Symmetry of H

Jacギ0 Jac = 0 bc-plane: no mirror symmetry Jieful Cen, HYK, arXiv:2208.13807 (2022) Example J = -1, K=0.5,  h =1
Current debate Angle dependent Ferromagnetic Resonance (FMR): Crl3 Ho (kOe) Kitaev dominant Inhee Lee, et al, Phys. Rev. Lett. 124, 017201 (2020) Inelastic neutron scattering J + DM interaction L. Chen et al, PRX II, 031047 (2021)
| Interaction | value (meV) | Interaction | value (meV)
20÷ Future experiments — search for candidate materials : +45° 10
(バッハJiefu Cen, HYK, arXiv:2208.13807 (2022)
●   Higher-spin S Kitaev interaction: combination of Hund’s at transition metal + SOC in ligand • Proposal to estimate the Kitaev interaction for general S: apply magnetic field in the bc-plane; measure spin excitations at two momenta (MI and M3) - broken mirror symmetry difference is due to Kitaev interaction (Gamma, Gamma’ ~ 0 for spin > 1/2)