1. Introduction
This communication is on a topical issue of the mixed valence compound SmB
6.[
1,
2,
3,
4,
5]—a narrow gap topological Kondo insulator (TKI) which, with a high-temperature metallic phase, transforms into a paramagnetic charge insulator below 45 K. It has been suggested [
6,
7], as well as it is increasingly apparent [
8,
9,
10] during the past several years, that SmB
6 is a non-trivial topological insulator. This has generated great deal of excitement in the condensed matter physics community and it still remains a matter of avid debate [
6]. Despite the supporting evidence for the TKI scenario [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10], there is no prognosis regarding the nature of the bulk and surface states of SmB
6 [
11,
12,
13]. In this paper our primary aim is to resolve this issue. We consider an extended periodic Anderson model (EPAM) [
14] for the compound SmB
6 for this purpose
. We introduce here the exchange interaction (
M) assuming the presence of the ferromagnetic magnetic impurities in the system. The slave boson (SB) mean-field-theoretic Anderson model of Legner [
15] refers to a simple cubic lattice with one spin-degenerate orbital per lattice site each for
electrons. We consider the low-energy version of this model together with the exchange interaction. Our minimalistic Hamiltonian, based on the slave boson (SB)mean field theory of ref. [
15]
, captures essential physics of TKI in the presence of the coulomb repulsion
( >>
between
f electrons on the same site, and the spin-orbit hybridization
V. The parameter
V is the harbinger of a topological dispensation. The terms (
) are the nearest neighbor hopping parameters for
electrons. There are three other parameters (
b, λ,
ξ ) of our theory [
14]. While the term
ξ enforces the fact that there are equal number of
d and
f fermions, the parameter
b represents a c-number slave-boson field. We note that the constraint
>>
imposes a non-holonomic constraint, viz. the exclusion of the double occupancy. The SB-protocol provides a platform to reformulate this nonholonomic constraint into a holonomic constraint that can be implemented with the Lagrange multiplier λ. We found that λ =
ξ =
. The admissible value of
is 1
−. Since the method to obtain them is explained clearly in ref. [
14] we will not reproduce the same here. We observe a Dirac cone like feature at
momentum in the surface state energy spectra for
M = 0 [
16], upon writing the SB Hamiltonian in the Dirac basis similar to the Bernevig–Hughes–Zhang (BHZ) model [
17]
. We obtain the Z
2 invariant (Z
2 =
) using the eigenvalues of the space inversion operator in the Fu-Kane framework [
18]
. This is the conclusive evidence of the fact that SmB
6 is a strong topological insulator for
M = 0. By calculating Berry curvature and the Chern number we have been able to show that
M corresponds to the quantum anomalous Hall state. It must be mentioned here that our low-energy model has been found adequate enough to capture important features of SmB
6, although the ground state of the compound SmB
6 has been shown to be is a quartet state [
1]
.
The exotic Floquet topological phases [
19,
20,
21,
22,
23,
24,
25,
26] with a high tunability could be realized using the circularly polarized optical field (CPOF). Analogous to the Bloch theory, here one can transform the time-dependent Hamiltonian problem to a time-independent one using the Floquet’s theorem [
27,
28,
29,
30]. The combinations of the Floquet theory with dynamical mean field theory [
31], and slave boson protocol [
32] were also formulated for strongly correlated systems. Our approach is in acquiescence to the latter. We use the Floquet theory[
27,
28,
29,
30,
33] in section 3 in the high-frequency limit to examine the Co
3 Sn
2 S
2 thin film system. Interestingly, the incidence of CPOF leads to the time reversal symmetry (TRS) broken phase despite
M = 0. The conclusive evidence of this phase being quantum spin Hall (QSH) phase is presented by calculating the spin Chern number [
34,
35]
.
The paper is organized as follows: In section 2, we calculate Z
2 invariant using the eigenvalues of the space inversion operator with in the Fu-Kane framework [
18]. In section 3, we are able to show the emergence of a novel phase with broken-TRS by the normal incidence of tunable CPOF despite M = 0. For this purpose, we make use of the Floquet theory in the high-frequency limit to investigate the system. The paper ends with discussion and brief concluding remarks in section 4.
3. Floquet Theory
The polarized periodic optical field provides a potent modus operandi to carry out theoretical proposition and experimental realization, manipulation, and detection of diverse unconventional/novel optical and electronic properties of materials, such as the realization of novel quantum phases without static counterparts like light induced quantum anomalous phase (QAH) phase [
19], the topological phase transitions in semi-metals [
21,
22,
23,
24]
, the Floquet engin-eering of magnetism in topological insulator thin films [
41,
42], and so on. The exotic Floquet topological phases with a high tunability could be realized using the polarized periodic optical field. In fact, there has been an upsurge on experimental front in the search for topological states, in solid state [
42], cold-atom [
43] and optical systems [
44], which are driven periodically. The circularly polarized optical field
(CPOF) is described by a time-periodic (time period =
T = 2π/ ω where ω is the frequency of light) gauge field. Upon using the Peierls substitution, lattice electrons couple to the electromagnetic gauge field. In the presence of COPF, the thin film Hamiltonian
, apart from breaking the time reversal invariance (TRS), becomes periodic in time. One can now transform the time-dependent Hamiltonian problem to a time-independent one using the Floquet’s theorem [
25,
26,
27,
28]. Analogous to the Bloch theory, a solution for the time-dependent Schrodinger equation of the system is obtained here involving the Floquet quasi-energy and the time-periodic Floquet state with the periodicity
T. The Floquet state could be expanded in a Fourier series which makes us arrive at an infinite dimensional eigenvalue equation in the Sambe space [
27]
.The circularly polarized optical field incident upon the film may be described by a time-varying gauge field
through the relation:
,
Here
The optical field. In particular, when the phase
ψ = 0 or π, the optical field is linearly polarized. When
ψ = + π/2 (
ψ = −π/2), the optical field is left-handed ( right-handed) circularly polarized. Once we have included a gauge field, it is necessary that we make the Peierls substitution
. The dimensionless quantity
corresponds to the frequency of the incident light.
We assume the normal incidence of CPOF on the surface SmB
6 with the thickness
d = 30
nm. Suppose the angular frequency of the optical field incident on the film is ω
and wavelength
Therefore the ratio
d/
Upon taking the field into consideration our Hamiltonian becomes time dependent. As stated above, the Floquet theory can be applied to our time-periodic Hamiltonian
with the period
= 2π/ω. Analogous to the Bloch theory involving crystal quasi-momentum, a solution |
involving the Floquet quasi-energy
could be written down for the time-dependent Schrodinger equation of the system. The Floquet state satisfies
and, therefore, could be expanded in a Fourier series
where r is an integer. Then the wave function, in terms of the quasi-energy
has the form |
=
This makes us arrive at an infinite dimensional eigenvalue equation in the Sambe space (the extended Hilbert space)[
27,
28]:
The matrix element of the Floquet state surface Hamiltonian
where (
) are integers. In view of the Floquet theory [
29,
30,
31,
32], in the high-frequency limit, a thin film system, irradiated by the circularly polarized radiation, can be described by an effective, static Hamiltonian. in the off-resonant regime using the Floquet-Magnus (high-frequency) expansion [
30]
:
where
with n
For M <<
, we can write
From the action of the time reversal operator on the wave function we see, that it leads to a complex conjugation of the wave function. Thus, in the case of spin-less wave functions as
Θ =
K, where
K is the operator for complex conjugation. More generally, we can write
Θ = UK where
U is a unitary operator. Furthermore, for a spin-1/2 particle, flipping the spin coincides with the time-reversal. This means
Θ is the vector of Pauli matrices. In view of these, one may also choose
Θ =
Upon making use of the results
ΘΘ −1 =
ΘΘ −1 =
and so on, where
……., we find that
where
ΘΘ −1 =
only when
when the optical field is linearly polarized. In this case, the time reversal symmetry (TRS) is not broken. However, when
, TRS is broken. We now consider the particular cases where ψ = + π/2 and ψ = −π/2. For the former the optical field is left-handed circularly polarized, whereas for the latter it is right-handed. Thus, the (previously not known) consequence is that the incidence of the CPOF on the SmB
6 surface will be able to create a novel state with the broken TRS.
The Hamiltonian to describe this broken TRS system, in the basis (
T, could be written as
where
. The functions
and
are defined below:
The eigenvalues (
) of the matrix (15) is given by the quartic
(α= 1,2,3,4 ) where the coefficients
b) (β = 1,2,3,4) are given in
Appendix A (see Eqs.(A.21)—(A.23)). It may be noted that to denote these eigenvalues we have used the symbol var epsilon which is distinct from that in Eq. (A.1). Once again, in view of the Ferrari’s solution of a quartic equation, we find the roots as
where α = 1,2,3,4,
is the spin index and
is the band-index. The spin-down (
conduction band (
and the spin-up (down) (
valence bands (
, denoted respectively by
,
somewhat peculiar as will be shown below. The functions appearing in Eq. (18) are given by
The eigenvectors corresponding to
could be calculated in a manner given in the
Appendix A. The value of
around 0.8 which is good enough for the
field of frequency
under consideration. Moreover,
= +1 (
= − 1 sign) corresponds to the left-handed (right-handed) circularly polarized radiation above. We notice from above that CPOF not only renormalizes
d and
f electron hopping integrals but also does the renormalization of the hybridization parameter(HP)
We have the renormalized hybridization parameters(HP) as
and
. As
[
e find that the renormalized HP
>
(
<
for the left-handed (right-handed) CPOF. However, the renormalized HP
<
(
>
for the left-handed(right-
note that, in principle, when a renor-malized parameter is less than
it is possible that there is a critical intensity of the radiation
at a given frequency at which the RHP in question will be zero. This, however, may affect the topological nature of the material. Now the nearest neighbor hopping elements
are related to the band masses by
If one takes for the band masses
where
then the corresponding values of the hopping matrix elements are
≈ 150 meV and
≈ 4.5 meV. This yields the critical intensity of the radiation
which is roughly three times the intensity value assumed in the graphical representations in
Figure 3.
Figure 3.
The plots of energy eigenvalues in Eq.(31) as a function of ak for intensity of incident radiation = (0.80, 0.50). The Figures 4(a) and 4(b), respectively, corresponds to the plots for the left handed and the right-handed CPOF. The same is true for the Figures 4(c) and 4(b). The numerical values of the parameters used in the plots in (a) and (b) are = 1, = 0.8, = 0.01, = 0.01, = 0.001, = 0.001, 0.02, V = 0.1, b =0.98,μ = The numerical values corresponding to (c) and (d) are μ = ; the rest of the values are the same as in (a) and (b).The horizontal lines represent the Fermi energy.
Figure 3.
The plots of energy eigenvalues in Eq.(31) as a function of ak for intensity of incident radiation = (0.80, 0.50). The Figures 4(a) and 4(b), respectively, corresponds to the plots for the left handed and the right-handed CPOF. The same is true for the Figures 4(c) and 4(b). The numerical values of the parameters used in the plots in (a) and (b) are = 1, = 0.8, = 0.01, = 0.01, = 0.001, = 0.001, 0.02, V = 0.1, b =0.98,μ = The numerical values corresponding to (c) and (d) are μ = ; the rest of the values are the same as in (a) and (b).The horizontal lines represent the Fermi energy.
In
Figure 3 (a),3(b), 3(c), and 3(d) we have plotted the energy eigenvalues
as a function of
ak for
= (0.80,0.50) for the circularly polarized light. Whereas Figures 3(a) and 3(c) correspond to left-handed CPOF, 3(b) and 3(d) to the right-handed CPOF. The numerical values of the parameters used in the plots are
= 1,
=
0.8,
= 0.01,
= 0.01,
= 0.001,
= 0.001,
0.02, V = 0.1, b =0.98,μ = (
The horizontal lines represent the Fermi energy. The conduction and valence bands denoted by
,
represent-ed by differently colored curves, apart from the band-inversion, exhibit some peculiarities by way of the multiple avoided crossings and the near absence of a surface Dirac cone at
k = 0 in 3(a) and 3(b) unlike that in
Figure 1(a).This non-trivial feature could be ascribed to the interaction of the system with the incident radiation. The figures show that when TRS is broken despite
M = 0, the fledgling novel phase of the system is very robust. The reason being in both the figures the Fermi energy intersects the band
only in the same BZ an odd pair number of times. This pair of surface state crossings (SSC)corresponds to the momenta
(
or (
in
Figure 3(a) and
Figure 3(b). However, in
Figure 3(c) and
Figure 3(d) the same happens at the momenta
(
or (
These momenta satisfy the condition
where the reciprocal lattice vector
G is (
or (
in
Figure 3(a) and
Figure 3(b) and (
or (
in
Figure 3(c) and
Figure 3(d).
ur graphical representation lead to the fact that, due to the light-matter interaction, the emergent unconventional phase possibly corresponds to QSH. However, as stated in section 1, the conclusive evidence of this TRS-broken phase being QSH/QAH phase will be obtained once we calculate the spin Chern number[
34] and the Z
2 invariant which are future tasks.
4. Discussion and concluding remarks
The strong correlation effects and diverse surface conditions make SmB
6 extremely complicated and almost a Gordian knot. Despite this, as we have seen above, our low-energy model was able to show that the compound is a strong TI. Our low-energy model was also able to capture the fact that there should be a surface Dirac cone at
k=0(as in ref.[
16]
) in
Figure 1(a) and 1(b) for
M = 0. For
MFigure 1(d)), since TRS is broken, there is no Kramers degeneracy. By calculating BC and the Chern number we have been able to show that the
Figure 1(d) corresponds to QAH state. In the case of the light-matter interaction in section 3, however, we need to show that the novel TRS-broken phase (despite M = 0) corresponds to QSH state. The problem needs an extensive investigation introducing an additional term
which is the Rashba spin-orbit coupling (RSOC) between the
d-electrons, in Eq. (1). Here
These are highlights of the present report.
The Rashba coupling can arise in the present system due to proximity of material lacking in the structural inversion symmetry. In view of the spin-polarized ARPES measurements which appear to confirm the surface helical spin texture [
45,
46]
, it would be interesting to see how does surface state react to Rashba splitting as there is evidence of for a massive surface state at the surface Brillouin zone center which can exhibit Rashba splitting[
47]. The Rashba SOC is of particular importance as it is a crucial ingredient for several spintronics and topological phenomena [
48].
There are many other complications to bring home the point that the system needs concerted investigations. In fact, we are presently working on three of the several issues to be discussed below in brief:
(i) Since we have considered surface state quite extensively, the next step forward is an investigation on the Kondo break down[
49,
50].
(ii)In YbB
12, a finite residual temperature-linear term in the thermal conductivity κ/T(T → 0) is observed demonstrating the presence of gapless and itinerant neutral fermions [
51]. On the other hand, κ/T (T → 0) in SmB
6 has been controversial [
52,
53]. While κ/T(T → 0) of SmB
6 has been reported to be very small but finite [
54,
55,
56], the absence of κ /T(T → 0) has been reported in references [
55,
57]. It is worth mentioning that in ref.[
58] κ/T(T → 0) has been shown to be finite. In view of the fact that TKI are found to be exceptionally sensitive to impurities
, the issue needs to be looked into.
(iii) Finally, in the quantum oscillation (QO) experiments of Li et al. [
59]
, the signatures of two-dimensional Fermi surfaces supporting the presence of topological surface states were obtained. Various theoretical models were put forward that invoke novel itinerant low-energy neutral excitations [
60] within the charge gap. These excitations were proposed to produce magneto quantum oscillation (MQO) signals. The theoretical models which entered the fray are based on magnetoexcitons [
61], scalar Majorana fermions [
62], emergent fractionalized quasiparticles [
63] and non-Hermitian states [
64]. As has been reported earlier [
61], in SmB
6, the QOs are observed only in the magnetization (de Haas-van Alphen(dHvA)effect). The dHvA oscillations strongly deviate below 1 K from the Lifshitz-Kosevich theory (LKT) possibly due to the presence of magnetic impurities [
65]. It must be mentioned that the QOs in YbB
12 [
66] are observed in both magnetization (the de Haas-van Alphen, dHvA, effect) and resistivity (the Shubnikov-de Haas, SdH, effect) at applied magnetic fields
H where the hybridization gap is still finite. The temperature-dependence of the oscillation amplitude complies with the expectations of Fermi-liquid theory [
66]. It is hoped that the details of the problematic issues given above, related/unrelated to the present communication, will motivate the condensed matter physics community to delve deeper into this problem.
In conclusion, looking at the controversies and the possibilities, it is anybody’s guess that there are many unsettled issues. Unless other TKI candidates are discovered and thoroughly studied, it is perhaps difficult to achieve enhancement in the current understanding of strongly correlated topological insulators. In this backdrop, it is pertinent to make an attempt to investigate thoroughly what exactly are the physical explanation of the issues involved. In a future communication, as we already stated, we undertake a part of this demanding task.