A NEW EXOTIC STATE IN AN OLD MATERIAL: A TALE OF SmB
M. Dzeroa, V. Galitskih*
" Department of Physics, Kent State University, Kent, OH 44242, USA
b Condensed Matter Theory Center and Department of Physics, University of Maryland, College Park, MD 20742, USA
Received April 26, 2013
Dedicated to the memory of Professor Anatoly Larkin
We review current theoretical and experimental efforts to identify a novel class of intermetallic 4/ and 5/ orbital materials in which strong interactions between itinerant and predominately localized degrees of freedom gives rise to a bulk insulating state at low temperatures, while the surface remains metallic. This effect arises due to inversion of even-parity conduction bands and odd-parity very narrow /-electron bands. The number of band inversions is mainly determined by the crystal symmetry of a material and the corresponding degeneracy of the hybridized /-electron bands. For an odd number of band inversions, the metallic surface states are chiral and therefore remain robust against disorder and time-reversal invariant perturbations. We discuss a number of unresolved theoretical issues specific to topological Kondo insulators and outline experimental challenges in probing the chiral surface states in these materials.
DOI: 10.7868/S0044451013090101
1. INTRODUCTION
In the past toil years, researchers have been fascinated with a peculiar kind of materials: topological insulators fl 6]. These materials host spin-momentuni-lockod (i.e., chiral) metallic surface states, which allow them to remain robust to time-reversal invariant perturbations [7 13]. In addition, these materials, when brought into contact with s-wavo superconductors, support Ma.jorana formions that are their own antiparticlos [14, 15]. The combination of these properties makes topological insulators promising platforms for spintronics and quantum computing applications. At the same time, these materials, which have been proved to possess topologically protected metallic surface states, have significant bulk conductivity [16 19]. In this sense they are therefore not ideal topological insulators.
One promising route to the discovery of ideal topological insulators is to examine materials with strong electron electron interactions. First, the electron correlations may fully suppress the bulk conductivity. Second, electronic interactions may significantly enhance
E-mail: galitski'fi'phvsics.umd.edu
the spin orbit coupling, which is responsible for the inversion of the bands with opposite parity. In weakly-correlated Bi-basod topological insulators, spin orbit coupling inverts the ,s- and /»-bands. In correlated topological insulators, we expect bands with higher orbital numbers, either />- and ii-orbitals or ,s-, d-, and /-orbi-tals, to invert. For example, a topological Mott insulating state has been theoretically predicted for ii-orbital pyrochloro irridatos within the extended Hubbard model on a honeycomb lattice [20 24], while inversion between Os ii-bands and Co /-bands leads to a topological insulator in filled skutterudites [25] and the general two-dimensional Kondo system where topological insulating state is hidden inside the ferromagnetic metallic state [26]. It is also worth mentioning the theoretical realization of various interaction-driven topological phases in ultracold atom systems and graphene [27 29].
In this article, we focus on recent theoretical and experimental breakthroughs in the search for the ideal topological insulator in higher orbital systems. The special attention is given to the already existing /-orbital materials [30], such as CeNiSii, Ce3Bi4Pt3, YbBi2, and SmB6. These materials, which are called Kondo insulators, have all the necessary features needed for realizing topological behavior: strong spin orbit coupling, strong electron electron interactions, and orbitals with
Table. Strength of the Hubbard interaction U and spin-orbit coupling A depending on the orbital state type
4d 5 d 4/ 5/
U, oV A, oV 1.5 0.1,... ,0.2 1 0.4,... ,0.6 1.7 0.7,... ,1 2.1 1,... ,2
opposite parity (see Table). The strong spin orbit coupling is inherent in /-electron systems and guarantees the inversion of the bands at the high-symmetry points in the Brillouin zone. The predominantly localized character of the /-electrons furnishes a strong Coulomb repulsion between them, while hybridization between the even-parity conduction electrons and /-electrons leads to the emergence of the hybridization gap. Interestingly enough, the onset of the hybridization gap, observed by Raman spectroscopy in some Ivondo insulators, has clear features of a second-order phase transition. In any case, the opening of the hybridization gap does not guarantee an insulating gap, of course. But in Ivondo insulators, the total number of conduction and /-electrons per unit cell is even and, consequently, it immediately follows that Ivondo insulators are strongly-correlated analogues of band insulators.
This article is organized as follows. In the next section, we review the theoretical models that lead to the original prediction [31] that Ivondo insulators with tetragonal or orthorhombic crystalline symmetries can naturally become a host to topologically protected metallic surface states. Section 3 is devoted to the review of recent experimental and theoretical efforts toward the understanding of the physics of cubic topological Ivondo insulators and, specifically, SmB6. In Sec. 4, we discuss open questions, the answers to which will deepen our understanding of topological Ivondo insulators. We summarize the current status of the field and present our conclusions in Sec. 5.
2. THEORIES OF TOPOLOGICAL KONDO INSULATORS
In this section, we review the recent theories of topological Ivondo insulators. We consider the case where the /-ion is in tetragonal crystalline field environment and review the theoretical results obtained for this case first. We then proceed with the discussion of the theories for cubic topological Ivondo insulators, which are relevant for SmB6, YbBi2, and Ce3Bi4Pt3 materials.
2.1. Tetragonal topological Kondo insulators
The minimal model of Kondo insulators must involve conduction and strongly correlated /-electrons as well as hybridization between them. Before we write the corresponding periodic Anderson model, we need to specify the /-electron states. Since most of the tetragonal systems that are insulating or semi-metallic contain Co, wo consider the model for the Co ion in a state with the total angular momentum J = 5/2. The six-fold degenerate multiplet is then split into the three Kramers doublets with the eigenvectors written conveniently in terms of the eigenvectors of the angular momentum projection operator J- as [32]
|// = ±1) = | ± 1/2),
I//. = ±2) = cosa:| ± 5/2) — sina:| =F 3/2), (1)
I//. = ±3) = sin a| ± 5/2) + cosa:| =F 3/2),
where the angle a: determines the degree of mixing between the corresponding orbitals. These eigenvectors can be conveniently expressed in terms of the spin part of the electron wave function \,T as
5/2 3
1/')= E E (2)
m=-l5/2 m=-3
where AffntT are the Clebsch Gordan coefficients and B^m are some known constants determined from the crystalline electric field (CEF) potential.
The minimal model Hamiltonian for the tetragonal Kondo insulator then takes the form [31 33]
kfT
+ E ¿"fU*» + E 'c.JU^+h.C.) +
k fjL k/LiCT
-¿<7/ E flUfl-U'- (3)
Vrfl^fl1
where cj^ creates an electron in the conduction band in a plane-wave state with the momentum k, spin a =f, 4-and energy (relative to the chemical potential of the conduction band), while /^ creates an /-electron in a state with momentum k and the multiplet component fi in (1) and the energy e\[\ We note that the bandwidth of the /-electrons is much smaller than the one for the conduction electrons. The third term in (3) describes the momentum-dependent hybridization between the conduction and /-electrons, while the last terms accounts for strong local correlations between the /-electrons on site i. This last terms is important because it leads to the local moment formation.
It is very intuitivo that the momentum dependence of the hybridization amplitude determines the anisotropy of the hybridization gap, which becomes an insulating gap if the total number of electrons per unit cell is even. Formally, the momentum dependence of Vk/i<T can be written in terms of the spherical harmonic functions [32, 341:
\
5/2
k,Jtr =
D
Il M v4tTV m
£ ^
(4)
m=
-5/2
where \).i are the matrix elements, which can be expressed in terms of the corresponding Slater Ivoster matrix elements [35]. We note that the values of the momentum in (4) are defined everywhere in the Bril-louin zone and
Yrw y'w^)e
ik-R
Z
(5)
R^O
is a tight-binding generalization of the spherical Harmonics that preserves the translational symmetry of the hybridization, I k = Vk+G, where G is a reciprocal lattice vector [31, 34]. Here, R are the positions of the Z nearest-neighbor sites around the magnetic ion.
The low-energy properties of the model in (3) can be analyzed by the using the following conjecture: the effect of the local correlations between the /-electrons leads to the renormalization of the hybridization amplitude and a shift of the /-energy level:
.(/) , _-(/) _ 7i
k fi
IV
k ßcr
IV
(6)
k/iíT I
''k/ifT I
where the renormalization factor Zk;j is determined by the /-electron self-energy part S/J/J(k,u;):
Zkfl —
duJ
(7)
u)=0
Then the low-energy model can be diagonalized, yielding a band structure that consists of two doubly degenerate bands separated by the momentum-dependent energy gap A^ given by
= ^Tr[FkHk].
(8)
where we suppressed the spin and orbital indices for brevity. Since the total number of electrons per unit cell is even, the lowest two bands are guaranteed to be fully occupied, and we have a band insulator if Ak does not have nodes anywhere in the Brillouin zone
apart from the high-symmetry point
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