Kitaev Magnetism and Fractionalized Excitations in Double Perovskite Sm2ZnIrO6

The quest for Kitaev spin liquids in particular three dimensional solids is long sought goal in condensed matter physics, as these states may give rise to exotic new types of quasi-particle excitations carrying fractional quantum numbers namely Majorana Fermionic excitations. Here we report the experimental signature of this characteristic feature of the Kitaev spin liquid via Raman measurements. Sm2ZnIrO6 is a strongly spin orbit coupled Mott insulator, where Jeff = 1/2 controls the physics, which provide striking evidence for this characteristic feature of the Kitaev spin liquid. As the temperature is lowered, we find that the spin excitations form a continuum in contrast to the conventional sharp modes expected in ordered antiferromagnets. Our observation of a broad magnetic continuum and anomalous renormalization of the phonon self-energy parameters evidence the existence of Majorana fermions from spin fractionalization in double perovskites structure as theoretically conjectured in a Kitaev-Heisenberg geometrically frustrated double perovskite systems.

Quantum magnetic materials having partially filled 5d elements are subject of extensive studies as these are predicted to host exotic quantum phases. These materials open up new avenue in the field of condensed matter physics due to the presence of interesting competing interactions of similar energy scale such as on-site coulomb interaction, strong spin-orbit (SO) coupling, Hund's coupling and crystal-field splitting; involving interplay of spin, orbital, charge and lattice degrees of freedom [1][2][3][4][5]. Fierce competition between different degrees of freedom provides fertile ground to observe rich magnetic ground states such as Quantum Spin Liquid (QSL). The experimental realization of QSL, a state which precludes long range ordering of spins even at absolute zero temperature, is a long sought goal in physics, as they represent new states of matter. Systems with QSL as their ground state ( GS ) are anticipated to be topologically active and to host exotic low energy quasi-particle excitations such as Majorana fermions which results from spin fractionalization, and gauge fields.
The so-called Kitaev Hamiltonian, in which the nearest neighbour Ising type interactions are bonddirectional [6], is an exactly solvable model with QSL as the GS . The recent search is for systems which realize these exotic states. As one example, it was proposed [7] that the strong SO coupled Mott insulators with an edge-sharing octahedral environment may give rise to a QSL as ground state. To date very few systems such as 2D honeycomb based structure A2IrO3 (A = Na, Li) and RuCl3 [8][9][10] fulfils these criterion. However, the elusive Kitaev spin liquid state in these systems is pre-empted by the long range magnetic order at low temperature possibly resulting from the presence of interaction beyond the Kitaev model such as Heisenberg off-diagonal interactions.
Interestingly, in spite of long range magnetic ordering, the existence of spin fractionalization and Majorana fermionic type excitations in these systems was inferred from the light scattering experiments via the observation of a broad continuum in Raman and inelastic neutron scattering 3 [11][12][13] measurements, in contrary to the sharp magnon modes observed in a typical long range magnetically ordered systems [14][15] suggesting that the ordered GS may be proximate to a quantum phase transition into the Kitaev spin liquid GS . The hunt is on to find an experimental evidence of a true Kitaev spin liquid especially in 3D systems. Recently, it has been suggested theoretically that strong SO coupled Mott insulators controlled by Jeff = 1/2 physics such as found in Iridium-based double perovskites (DP) A2BIrO6 (A = La, B = Zn, Mg) support symmetry allowed nearest neighbour interactions such as Heisenberg, Kitaev and symmetric off-diagonal exchange [16][17]. It is advocated that these 3D geometrically frustrated DP systems support Kitaev [22][23]. Raman scattering is a very powerful probe to capture signatures of these exotic quasi-particle excitation as it can simultaneously probe charge, spin, lattice, orbital and electronic excitations [24][25][26][27]. The double perovskite Sm2ZnIrO6 crystallizes in monoclinic structure (P21/n; space group No. 14), and consists of corner sharing IrO6 octahedra [see Fig. 1a]; magnetic measurements suggest an AFM ordering at TN ~ 13 K and tiny magnetic moments [28]. To our knowledge there is so far no report on this 3D double perovskite system probing spin excitations and exploring the possibility of anticipated spin fractionalization driven by strong SO coupling, a necessary denominator of the Kitaev spin liquid GS .
Our measurements reveal a broad anomalous magnetic continuum, unlike sharp features which are hallmark of the long range magnetic ordering, that persists in the magnetically ordered state below 13 K. This anomalous evolution cannot be captured by the conventional two-magnon scattering, however, it seems consistent with the theoretical predictions for spin liquid phase. Figure 1b  and  are the phonon frequency and line-width, respectively, and q define the nature of asymmetry. Figure 2  continuum: a stronger coupling ( 0 q  ) causes the peak to be more asymmetric and in the weak coupling ( q ) limit Fano lineshape is reduced to a Lorentzian line shape. Interestingly, the 6 Fano asymmetry parameter (1/|q|) shows strong temperature dependence (see Figure 2 (c)), it has high value in the long-range ordered phase, and above TN (~13 K) continuously decreases till ~ 180 K, and thereafter remains nearly constant up to 330 K with a quite appreciable value till very high temperature (~ 180 K) suggesting the presence of active magnetic excitations far above TN.
We note that the Fano asymmetry parameter mapped nicely onto the dynamic spin susceptibility [discussed later, see Fig. 4b] implying that the Fano line shape is also an indicator of spin fractionalization and the increased value below 200 K may be translated to a growth of finite spin fractionalization below this temperature. We note that similar observation have been reported for the other 2D Kitaev materials RuCl3, Li2IrO3 [12,30].
We now focus on the temperature evolution of a broad magnetic continuum with spectral weight surviving till ~ 25 meV (see inset Fig. 1b, marked by yellow color shadings). In a pure Kitaev material, the underlying quasi-particle excitations in the form of continuum originates from Majorana fermion scattering [31][32][33] which survive even in the presence of Heisenberg exchange as a perturbative term [13,19]. In Fig 3a,b, we have shown the temperature evolution of the underlying continuum. To make a quantitative estimate we extracted the intensity of this continuum as a function of temperature (see Fig. 3c). We note that the continuum loses its intensity with decreasing temperature and becomes nearly constant below ~ 150 K till ~ 20 K; and increases on entering the long range magnetic ordered phase below ~ 20 K. Our observation suggest that there is a change in the magnetic scattering below ~ 150 K hinting on development of finite entanglement between spins, also the existence of this magnetic scattering much above TN is a clear signature of frustrated magnetism. Generally, a broad magnetic continuum evolves into sharp one/two magnon peaks in long range ordered AFM systems. However, in Sm2ZnIrO6, there is no signature of any sharp peak suggesting that broad continuum observed here is not a conventional 7 magnetic continuum but does have its origin in an intricate magnetic structure probably dominated by the Kitaev exchange interactions. We note that similar broad magnetic continua have been reported in other systems with QSL as their potential ground state such as herbertsmithite [34], Cs2CuCl4 [14], RuCl3 and Li2IrO3 [11][12]30]. In these systems as well, the continuum intensity gains spectral weight as spin liquid correlation start developing much above long range magnetic ordering temperature. Keeping that the magnetic continuum in quasi 2D system extends upto ~  , see solid green line in Fig 4b). Unlike a conventional antiferromagnet which show power law behaviour only below TN and saturate in the paramagnetic phase, dyn  display a power law behaviour much above the long range magnetic ordering temperature. The saturation of dyn  is a hallmark of spin gas configuration where spins do not talk to each other i.e. pure paramagnetic phase, however the power-law dependence of dyn  for double perovskite Sm2ZnIrO6 even much above the long range magnetic ordering temperature reflects the slowly decaying correlation inherent to spin liquid phase and the onset temperature (~ 160-180 K) triggered the fractionalization of spins into itinerant spinons [36][37]. This anomaly clearly suggest that the underlying magnetic continuum arises mainly from the fractionalized excitations. We note that our observation in Sm2ZnIrO6 is analogous to the observation in quasi Kitaev 2D, 3D honeycomb systems namely RuCl3, Li2IrO3 and similar characteristics of the underlying magnetic 9 continuum in Sm2ZnIrO6 and these systems indicate that in Sm2ZnIrO6, Kitaev magnetism may be realized.
In conclusion, our comprehensive Raman scattering study on the double perovskite Sm2ZnIrO6 evince the signature of fractionalized excitations. Anomalous evolution of the broad magnetic continuum and Fano asymmetry suggest thermal fractionalization of the spins into fermionic excitations and suggest that anticipated three dimensional geometrically frustrated nonhoneycomb Iridium based double perovskite systems also realize the spin liquid state. Our results broaden the idea of fractionalized quasi-particle to a non-honeycomb based 3D geometrically frustrated Kitaev system with Heisenberg exchange as perturbation.   The first principle calculations were performed in the framework of density functional theory (DFT) via employing the plane wave method implemented in the QUANTUM ESPRESSO package [3]. We adopted generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) as an exchange-correlation function. The dynamical matrix and phonon frequencies were calculated using density functional perturbation theory [4]. A dense 4 × 4 × 4 Monkhorst-Pack grid was used for numerical integration of the Brillouin zone (BZ) [5].  According to our calculations phonon modes S1-S18 are attributed to be first order phonon mode, however high frequency modes S19-S23 are second ordered modes. Figure S2 Table-S2. 18 C. Temperature dependence of the phonon modes Figure S4 and S5 illustrate the mode frequency and line-width of the prominent first as well as second order phonon modes as a function of temperature. The following observation can be made: (i) the phonon modes are observed to show anomalous softening ~ 1.5 % in low frequency modes ( S1-S3 ) and ~ 0.5 % in high frequency modes (S7, S9, S10 and S13) in the low temperature region with respect to its value at 16 K. (ii) Furthermore, we see the substantial line broadening (increase in phonon life-time) of phonon mode, ~ 10 % and ~ 5 % for low frequency modes ( S1and S3) and high frequency modes ( S9, S10 ) below ~ 16 K with decrease in temperature, respectively. (iii) Interestingly, we note clear and sharp discontinuity at ~ 180 K, phonon hardening ~ 2 cm -1 in the S1, S3 and S7 modes with decrease in temperature. This discontinuity in phonon mode frequencies may occur due to local structural changes within the crystal or spin reorientations around this temperature.
To quantify the temperature dependence of phonon frequencies and damping constant, we have  a a a a a a  a a a a a a a Figure S4 are fit of the frequency and line-width to the anharmonic interaction model illustrating that in the temperature range of 20-330 K. The temperature variation of phonon modes in this range can be well explained within the cubic and quartic anharmonicities. The value obtained from fitting are summarized in Table-S2. It is worth to note that from the fitted values the cubic anharmonicity decay is primarily responsible for temperature dependences of phonon modes. The deviation of mode frequencies and line-width from the curve estimated by anharmonic interaction model is observed in low temperature regime below ~ 20 K. The phonon renormalization in the low temperature could be due to additional decay channels, namely, the interaction of phonons with other quasi-particle excitations. As at low temperature the effect of phonon-phonon interaction on self-energy parameters is minimal. Therefore, the renormalization of phonon frequency below magnetic ordering temperature can be understood due to the coupling of lattice with magnetic degrees of freedom through spin-phonon coupling. The spin-phonon coupling Hamiltonian can be written as   S7, S9, S10 and S13 modes. Solid red lines are fitted curve as described in the text and yellow lines are guide to eye. Inset of S1, S3 and S7 shows zoomed view of frequency in temperature range of 60 K to 300 K and solid red color vertical line is guide to eye.