Pis’ma v ZhETF, vol. 112, iss. 10, pp. 686 - 687

Can the highly symmetric SU (4) spin-orbital model be realized

in α-ZrCl₃?

A. V. Ushakov⁺, I. V. Solovyev^+∗×, S. V. Streltsov^+∗1)

⁺Institute of Metal Physics, Russian Academy of Sciences, 620041 Ekaterinburg, Russia

^∗Ural Federal University, 620002 Ekaterinburg, Russia

^×International Center for Materials Nanoarchitectonics, National Institute for Materials Science,

1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan

Submitted 1 October 2020

Resubmitted 12 October 2020

Accepted 18

October 2020

DOI: 10.31857/S1234567820220097

Highly symmetric models play a special role not only

α-ZrCl₃ at temperatures ∼ 500 K in the same way they

in the condensed matter physics, but in a whole physics.

do in Li₂RuO₃ [18].

A special efforts were put into studying of highly sym-

We found that α-ZrCl₃ appears to be an insulator

metric spin and spin-orbital models, since they are im-

even at the GGA level in contrast to metallic α-RuCl₃.

portant for description of magnetic materials. In partic-

The lowest in energy t_2g orbitals looking towards each

ular it was shown that in case of the common-face geom-

other in edge-sharing geometry form molecular orbitals

etry the Kugel-Khomskii spin-orbital Hamiltonian has

and this results in strong bonding-antibonding splitting

unexpectedly high symmetry [1, 2]. Another example is

seen in the density of states plot (Fig.1). Two electrons

the Kitaev model, which naturally appears in layered

of the dimer occupy the bonding state leading to the in

materials with the honeycomb lattice and heavy transi-

the non-magnetic ground state, while α-RuCl₃ is mag-

tion metal ions, such as Ir⁴⁺ or Ru³⁺ [3-7] with a pos-

netic.

sibility of spin-liquid ground state realization. Recently

Yamada and co-authors [8] noticed that α-ZrCl₃ with

one electron residing in the relativistic j_eff = 3/2 man-

ifold can be a physical realization of SU(4) symmetric

spin-orbital model.

In the present paper we performed ab initio study to

check the hypothesis about realization of this model in

α-ZrCl₃. We used the generalized gradient approxima-

tion (GGA) [9] and projector augmented-wave (PAW)

method as realized in the VASP code [10] for the calcu-

lations.

We used data of α-RuCl₃ [6] for the structural opti-

mization of α-ZrCl₃ as a starting point and relaxed all

possible parameters in magnetic GGA. As a result α-

ZrCl₃ dimerizes (Zr-Zr distance turns out to be smaller

than in Zr metal [11]). The dimers are parallel to each

other. Similar dimerization has been observed in α-

RuCl₃ under pressure [12], TiCl₃ [13] and many other ti-

tanites [14-17]. While the lowest in energy configuration

corresponds to parallel dimers, the other one with arm-

chair geometry is rather close in energy and one might

expect that dimers might start to flow over the lattice in

Fig. 1. (Color online) The partial densities of states of α-

ZrCl3 calculated in the GGA and GGA + U + SOC ap-

proximations for the dimerized structure with parallel

¹⁾e-mail: streltsov.s@gmail.com

dimers

686

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Can the highly symmetric SU(4) spin-orbital model be realized in α-ZrCl₃?

687

Further we construct an effective five-orbital

S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den

Hubbard-type model for Zr 4d bands using the Wannier

Brink, Y. Singh, P. Gegenwart, and R. Valenti, J. Phys.

Cond. Matt. 29, 493002 (2017).

functions technique and dimerized crystal structure,

obtained from the optimization in the frameworks of

H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and

S. E. Nagler, Nat. Rev. Phys. 1, 264 (2019).

GGA [19]. Without spin-orbit coupling, the t_2g levels

R. D. Johnson, S. C. Williams, A. A. Haghighirad,

are split by 8 and 186 meV, separating the lowest-middle

J. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li,

and middle-highest levels, respectively. The spin-orbit

H. O. Jeschke, R. Valenti, and R. Coldea, Phys. Rev.

coupling constant has been estimated to be about

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70 meV. It additionally splits the lowest t_2g levels.

J. Zheng, K. Ran, T. Li, J. Wang, P. Wang, B. Liu,

Thus, the crystal field, though not particularly strong,

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lifts the orbital degeneracy, substantially modifies the

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j_eff = 3/2 character of the lowest energy states and kills

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SU(4) invariance of the spin-orbital model.

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j have been calculated to be tooij = -1.262 eV and

10.

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11.

S. V. Streltsov and D. I. Khomskii, Physics-Uspekhi 60,

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1121 (2017).

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12.

G. Bastien, G. Garbarino, R. Yadav, F. J. Martinez-

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Heisenberg model as rather weak J ∼ 0.26 meV. The

S. P. Limandri, R. Ray, P. Lampen-Kelley, D.G. Man-

constrained random phase approximation (cRPA) [19]

drus, S. E. Nagler, M. Roslova, A. Isaeva, T. Doert,

yields U = 1.53 and J_H = 0.58 eV. These values were

L. Hozoi, A. U. B. Wolter, B. Büchner, J. Geck, and

used in the subsequent GGA + U + SOC calculations.

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Basically U renormalizes GGA energy differences be-

13.

S. Ogawa, J. Phys. Soc. Japan 15, 1901 (1960).

tween different solutions discussed above, but it does

14.

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One may notice, that formation of molecular or-

15.

S. V. Streltsov and D. I. Khomskii, Phys. Rev. B 77,

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(∼ 10^-3µ_B) for α-ZrCl₃. However, in some dimerized or

16.

A. Seidel, C. A. Marianetti, F. C. Chou, G. Ceder, and

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17.

M. Schmidt, W. Ratcliff, P. G. Radaelli, K. Refson,

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sultations on the crystal optimization of α-RuCl₃ and

18.

S. A. J. Kimber, I. I. Mazin, J. Shen, H. O. Jeschke,

especially to G.Jackeli and D.I.Khomskii for various

S. V. Streltsov, D.N. Argyriou, R. Valenti, and

D. I. Khomskii, Phys. Rev. B 89, 081408 (2014).

stimulating discussions on spin-orbital physics.

19.

I. V. Solovyev, J. Phys.: Condens. Matter 20, 293201

This work was supported by the Russian Science

(2008).

Foundation through RSF 20-62-46047 research grant.

20.

J. Terzic, J. C. Wang, F. Ye, W. H. Song, S. J. Yuan,

Full text of the paper is published in JETP Letters

S. Aswartham, L. E. DeLong, S. V. Streltsov,

journal. DOI: 10.1134/S002136402022004X

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2020