Angular‐Momentum Transfer Mediated by a Vibronic‐Bound‐State

Abstract The notion that phonons can carry pseudo‐angular momentum has many major consequences, including topologically protected phonon chirality, Berry curvature of phonon band structure, and the phonon Hall effect. When a phonon is resonantly coupled to an orbital state split by its crystal field environment, a so‐called vibronic bound state forms. Here, a vibronic bound state is observed in NaYbSe2, a quantum spin liquid candidate. In addition, field and polarization dependent Raman microscopy is used to probe an angular momentum transfer of ΔJ z = ±ℏ between phonons and the crystalline electric field mediated by the vibronic bound stat. This angular momentum transfer between electronic and lattice subsystems provides new pathways for selective optical addressability of phononic angular momentum via electronic ancillary states.

Density functional theory (DFT) calculations were performed based on the projector augmented wave method (PAW) as implemented in the Vienna Ab-initio Simulation Package (VASP) [1][2][3][4] to obtain the phonon dispersion relationship in NaYbSe 2 , .The generalized gradient approximation, parameterized by Perdew, Burke, and Ernzerhof (PBE) [5] was used for exchange-correlations.A 520 eV kinetic energy cutoff in the plane-wave expansion and energy convergence criteria of 10 −6 eV were employed.Ionic relaxations were performed until Hellmann-Feynman forces converged to 10 −4 meV/ Å.The structure was relaxed with a Γcentered 15×15×3 k-mesh.The harmonic interatomic force constants (IFCs) were calculated using the finite displacement method implemented in the phonopy package [6] in a 3×3×1 supercell with Γ-centered 5×5×3 k-meshes.The DFT+U method [7] was used to include the Coulomb correlations with U eff =6 eV [7] for Yb atoms.Van der Waals interactions were taken into account via DFT-D3 method [8].  Figure S5 shows the peak positions extracted from Figure S2 using Bayesian inference [10].In each subplot, the trace is the median of the posterior distribution of the peak position or peak width, and the two shades of different opacities are corresponding to 1σ (more opaque) and 2σ (more transparent) bands of the posterior distribution.Figure S7 shows the original magneto-Raman spectra for NaYbSe 2 and Figure S8 for CsYbSe 2 .The magnetic field dependence given polarization configuration for the CEF modes are similar for NaYbSe 2 and CsYbSe 2 , i.e., (i) with (σ incident , σ scatter ) = (σ + , σ − ) configuration, the CEF1 and CEF2 shift toward higher energy in positive field, and CEF3 toward lower energy and (ii) the trends are inverted with inverted polarization configurations or inverted field.However, the ω is observed only in a subset of spatial locations [9], and with much weaker intensity, likely due to larger detuning between the E 2 2g and CEF1 in CsYbSe 2 .Small residual peaks that violate the proposed selection rules are present in both datasets: for instance, a small ω peak is present in Figure S7 (c) and Figure S7 (e) due to less than ideal polarization contrast in the optics train.For CsYbSe 2 , the extinction ratio of the selection rules are not as high as that of NaYbSe 2 , suggesting that the selection rules (and hence the assignment of the states) proposed in the text may only apply to NaYbSe 2 .

V. ANALYSIS OF TRANSITIONS FOR THE VBS MODE
The VBS ω is a hybridized state between the CEF1 and the E g mode.As discussed in the main text, CEF1 can arise from four possible transitions: , while E g can be E g,+ and E g,-depending on the sign of the angular momentum.Therefore, the ω could originate from eight possible transitions: According to the discrete angular momentum conservation rule |∆J photon +∆J ω | modulo3 = 0, the eight transitions are allowed in either cross-circular or co-circular polarization, which is inconsistent with the experimental observation that the ω can only be observed in cocircular polarization.We note that the above discussions have a basic assumption that the total angular momentum between the CEF1 and the E g mode follows the same rule for the conventional coupling between orbital angular momentum and spin angular momentum.
However, the electron-phonon coupling may be more complicated according to a recent work [11]: Coupling = −αL phonon • J electron , where α is a coupling constant in units of the inverse of the moment of inertia.Based on this concept, we hypothesize that the coupling between the CEF1 and the E g mode is not the same as the conventional coupling between orbital and spin angular momenta, and there is a coupling constant before the derived ∆J ω

Figure S2 (
Figure S2 (raw spectra) and Figure S3 (contour) shows temperature-dependent unpolarized Raman spectra taken from T = 3.3 K to T = 270 K.The spectra were taken with Semrock dichroic and longpass filters with cutoff at 90 cm −1 instead of a set of volume Bragg gratings.Similar to CsYbSe 2 [9], a multitude of possible combination modes (e.g., possible CEF1 + CEF2) -behavior of resonant Raman excitation -is present.
FIG. S3.Contour plot for Raman spectra as function of temperature from T = 3.3 K to T = 270 K. Same data as in Figure S2.