Unfolding the complexity of quasi-particle physics in disordered materials

The concept of quasi-particles forms the theoretical basis of our microscopic understanding of emergent phenomena associated with quantum mechanical many-body interactions. However, quasi-particle theory in disordered materials has proven difficult, resulting in the predominance of mean-field solutions. Here we report first-principles phonon calculations and inelastic x-ray and neutron scattering measurements on equiatomic alloys (NiCo, NiFe, AgPd, and NiFeCo) with force constant dominant disorder - confronting a key 50-year-old assumption in the Hamiltonian of all mean-field quasi-particle solutions for off-diagonal disorder. Our results have revealed the presence of a large, and heretofore unrecognized, impact of local chemical environments on the distribution of the species-pair-resolved force constant disorder that can dominate phonon scattering. This discovery not only identifies a critical analysis issue that has broad implications for other elementary excitations such as magnons and skyrmions in magnetic alloys, but also provides an important tool for the design of materials with ultra-low thermal conductivity.

Quasi-particle elementary excitations including electron quasi-particles, phonons, magnons, plasmons, excitons, etc., along with skyrmions 1,2 , Majorana fermions 3 and their mutual interactions, represent remarkably successful theoretical descriptions of emergent phenomena associated with quantum mechanical many-body interactions. For example, the seminal discovery of the quantized thermal Hall effect in the spin liquid candidate RuCl3 4,5 can be explained by the coupling of quasi-particles (phonons and chiral Majorana edge modes) in Kitaev's non-Abelian spin liquid 6,7 . As such, quasi-particle physics has provided a microscopic understanding of the underlying science of phenomena ranging from the novel properties of quantum materials to the electronic and vibrational based physics in solids. In contrast to fully ordered crystals that can be described within the Bloch theorem, the broken translational symmetry in alloys associated with substitutional chemical disorder has long challenged the development of analogous robust quasi-particle theories of configurationally averaged observables that inherently lead to the finite quasi-particle lifetimes measured experimentallyeven at zero temperature. Therefore, describing both the spectral dispersion and lifetimes of quasi-particles in disordered materials is fundamentally important for understanding the underlying science of condensed matter and the design of technological materials with targeted properties.
Historically, a landmark advance in treating compositional disorder was the coherent potential approximation (CPA) [8][9][10] . As an analytic "single-site" theory of the configurationally averaged observables that restores the translational invariance of a self-consistently determined effective medium, the CPA formalism was initially designed to treat on-site (site-diagonal) fluctuations associated with the different alloying elements -different on-site orbital energies for the case of electrons, different atomic masses for the case of phonons. Effective as the original CPA was, it was quickly realized that extending the single-site CPA to include fluctuations in two-site quantities -inter-site hopping integrals for electrons; pairwise force constants for phonons -is nontrivial [11][12][13][14] . This problem is particularly acute for phonons where force constant (off-diagonal) disorder is crucial and entangled with the mass (diagonal) disorder, as illustrated in Fig. 1 (a). For electron quasi-particles it was possible to account for both diagonal and off-diagonal fluctuations by reformulating CPA within the language of multiple scattering theory 15 and ab initio density functional theory [16][17][18] . For phonon quasi-particles, no similar "simple" transformation of the underlying theory has been possible.
For phonons, the formulation of itinerant-CPA (ICPA) by Ghosh et al. 19 provided a substantial advance in analytic mean-field theories of phonon quasi-particle physics in disordered materials based on the augmented space formalism of Mookerjee 20 . The theory was demonstrated initially for the cases of NiPd and NiPt that contain large mass disorder as well as force-constant disorder 19 , and has been applied subsequently to multiple binary alloys with similarly dominant mass disorder [21][22][23][24] . Remarkably, with the exception of a limited study of face-centered cubic (fcc) Ni88Cr12 and bcc Fe53Cr47 using an augmented space recursion (ASR) approach 25, 26  The Hamiltonian for the ICPA and all previous mean-field theories (including the CPA) for offdiagonal as well as diagonal disorder make use of the simplifying assumption that force constant disorder between individual AA, BB, and AB type pairs can be approximated by their global average in AB binary alloys 13, 27 . Similar to lattice vibrations (phonons), the magnetic excitations (e.g. magnons) of disordered alloys are also challenging to describe due to site-diagonal (local moment) and off-diagonal (exchange interactions) disorder, as depicted in Fig. 1 (b). The crucial exchange interactions between individual AA, BB, and AB type pairs are usually approximated by their ensemble averages as calculated from linear response theory 28 . Unaccountably, although this 50-year old assumption within the Hamiltonian is at the core of the analytic mean-field formalisms, it has so far not been questioned.
In this work we address this knowledge gap for the first time through a combined firstprinciples theory and experimental measurement investigation of the phonon quasiparticle physics (dispersions and linewidths) of concentrated disordered alloys, NiCo, NiFe, AgPd, and NiFeCo with strong force constant disorder but minimal mass disorder. Remarkably, we have discovered from ab-initio supercell phonon unfolding (SPU) simulations [29][30][31] and their comparison with ICPA and experimental measurements, that force constant fluctuations considering each individual AA, BB, and AB type species-pairs far surpasses that of the usual global average force constant fluctuations. Moreover, we have shown that the source of the enhanced fluctuations is the inherent random variations in local chemical environments surrounding individual AA, BB, and AB type pairs. Accordingly, we have demonstrated that the longstanding approximation of replacing individual-pair force-constant fluctuations with their global averages in the Hamiltonian of quasiparticle mean-field theories for disordered materials must be reconsidered.

Results
Phonon properties of NiCo and NiFe: Phonon dispersion and linewidth measurements for NiCo and NiFe samples were performed at room temperature using high resolution inelastic x-ray scattering (IXS) and inelastic neutron scattering (INS) measurements along the [001] and [011] reciprocal lattice directions. In addition, density functional theory (DFT) was employed to calculate the force constants of NiCo, NiFe, AgPd, and NiFeCo alloys with chemical disorder modeled by the supercell method. Phonon spectral functions and corresponding linewidths for equiatomic NiCo, NiFe, AgPd and NiFeCo alloys were extracted using both the ICPA 19 and the SPU approach 31 . We note that the analytical mean-field ICPA approach employs the symmetryaveraged force constant matrix for a pair of atoms, i.e., one single matrix for each of Φ AA , Φ BB and Φ AB in binary A-B alloys, while the nonanalytical SPU approach uses the full force constant matrix of the supercell. The SPU maps phonon bands within the Brillouin zone (BZ) of large supercells to those of the primitive BZ. While the phonon dispersion curves for an N-atom supercell form 3N continuous (sharp) phonon dispersion bands, when unfolded into the primitive BZ they map into three effective acoustic bands with finite-width distributions of discrete eigenvalues. This is illustrated in Fig. 1 (c, d). More experimental and calculation details can be found in the Methods section.  -contour overview of the phonon dispersions and phonon linewidths   as obtained by SPU and ICPA calculations, along with corresponding IXS and INS measurements for disordered solid-solution NiCo and NiFe, respectively. These plots present results as a function of the wavevector q= [zx,zy,zz] in reciprocal lattice units (rlu) of 2p/a0 (a0 = cubic lattice parameter) for longitudinal (LA) and transverse (TA) acoustic phonons. The color-contours in Fig. 2 give the simulated phonon spectral functions overlaid by discrete phonon dispersion measurements (black solid circles). We note general agreement between first-principles phonon dispersions (SPU and ICPA) and the corresponding measurements for both NiCo and NiFe, albeit small discrepancies occur near the [0,0,z] BZ boundary X-point and for TA1 phonons along the [0,z,z] direction.
More important for disordered materials, though, are the phonon linewidths which provide a direct probe of force constant disorder, when mass disorder is minimal. Figure 3  We observe first of all, significant phonon broadening for both NiCo and NiFe in Fig. 3, especially for q>0.7 rlu. This broadening implies that force constant disorder alone (in the absence of mass disorder) can cause significant phonon scattering and, hence, lead to shorter phonon lifetimes corresponding to substantial linewidth broadening. Furthermore, the measured linewidths are not only larger for NiFe than for NiCo, but dramatically so for q <~0.5 rlu. The linewidths for NiCo tend to fall below ~0.5 meV for q < 0.5 rlu while the linewidths for NiFe range from 1.5-2 meV down to q as low as ~0.2 rlu, the importance of which will be discussed below.
Focusing on the simulations, we observe in Fig. 3 (a) that (within the measurement uncertainties) both the SPU and ICPA simulations agree with the measured linewidths for NiCo, albeit ICPA tends to overestimate those of LA phonons in the [0,0,z] direction near the BZ boundary, and that the SPU simulations tend to underestimate the linewidths in the same region.
In Fig. 3 (b), it is demonstrated that the linewidth simulations for NiFe using SPU are in excellent agreement with the measured linewidths for all wavevectors sampled. Conversely, the ICPA linewidths significantly underestimate the measured linewidths for NiFe -lower by more than a factor of two for the LA phonons and lower by factors of five to ten for the TA phonons. Comparison between the SPU calculated and the measured linewidths for NiFe and NiCo provides a direct validation of the SPU approach for predicting vibrational properties in disordered binary alloys where force constant disorder dominates. This success, now raises the question as to why the ICPA so significantly underestimates the linewidths in NiFe, given that the ICPA approach 19 was specifically designed to incorporate force constant disorder within an analytic mean-field theory.
Importance of off-diagonal randomness: To examine off-diagonal disorder, we have used SPU calculations to test the fundamental approximation within the present ICPA Hamiltonian for AB binary alloys 19 , which includes only the inter-species disorder between the "global averaged" force constants associated with individual species-pairs (<F AA >, <F AB >, <F BB >). That is, ICPA averages over the actual distribution of force constants for individual species-pairs, i.e. intraspecies force constant disorder. While this approximation, (in general use for mean-field quasiparticle theories) greatly simplifies the ICPA formalism and has underpinned attempts to generalize the original CPA to include off-diagonal randomness 13,27 , it tacitly assumes that the local atomic environment surrounding interacting pairs does not impact the force constants for individual atomic pairs significantly. In contrast, however, we show in Figs The thick gray SPU-A curves in Fig. 3 explicitly demonstrate the consequence of using averaged species-pair force constants. SPU-A replaces the individual AA, AB, and BB speciespair force constants of the full SPU with their averages (<F AA >, <F AB >, <F BB >), as detailed in supplementary note 1.2 33 . Therefore, SPU-A is a supercell phonon unfolding calculation that mimics the force constant averaging of the ICPA Hamiltonian. We note that the SPU-A, the full SPU and the ICPA linewidth simulations are quite similar for NiCo. While the force-constantaveraged SPU-A calculations for NiFe depart strongly from those of SPU, they reproduce almost exactly the (anomalously low) ICPA linewidth results. These observations demonstrate conclusively that it is indeed the intra-species force constant averaging associated with the present ICPA Hamiltonian that is responsible for the breakdown in the ICPA linewidth predictions for NiFe. Conversely, the agreement between the SPU-A and the ICPA results indicates directly that the present ICPA properly accounts for the effects of off-diagonal disorder properly when it is limited to inter-species (averaged) randomness. It is important to note that the underestimated ICPA linewidths are due solely to the oversimplified Hamiltonian rather than the ICPA methodology itself.
This limitation was not apparent in previous ICPA alloy studies 19,21-24,34 because the impact of the large (100-150%) mass disorder on phonon dispersion curves tended to mask the impact of force constant disorder on phonon linewidths. In any case, the results presented above indicate that the incorporation of (individual) intra-species force constant fluctuations in the ICPA phonon Hamiltonian would lead to an accurate description of phonons in disordered materials. While outside the scope of the present study, in principle the inclusion of individual-pair force constant disorder in ICPA could be achieved by discretizing intra-species force constant distributions through the introduction of additional, appropriately-weighted, ICPA components. This is similar to the discretization of the distribution of the relevant random variables (atomic displacements, spin disorder) by introducing more CPA components for configurational averaging in the electronic structure methodology 35 .
To demonstrate that the importance of retaining full force constant distributions in NiFe is not an isolated case, we performed SPU and SPU-A phonon linewidth calculations for the equiatomic 4d binary AgPd and the 3d ternary NiFeCo alloys, both of which have negligible mass disorder (few percent). Analyzing further, the magnitude and the wavevector, q, dependence of the simulated and measured linewidths for NiCo and NiFe (see Fig. 3) provide direct quantitative insight into the force constant disorder induced phonon linewidths D averaged over a length scale l = 2p/q. In particular, linewidths of D = 1 meV correspond to phonon lifetimes of 0.66 ps, and q = 0.25 rlu corresponds to l ~15 Å. Accordingly, at low q = 0.25 rlu, the phonon broadening is expected to be small (such as observed for LA phonons for NiCo in Fig. 3) since longer wavelength phonons are expected to be less sensitive to local force constant disorder. Surprisingly, however, for NiFe at q = [0,0,0.25] rlu -corresponding to l ~15 Å -the broadening of the LA mode is substantial and more than five times that of NiCo. This result is consistent with the observations in

Computational:
To explore the effect of force constant disorder on the phonon properties of considered equiatomic alloys, we employed the projector augmented wave method (PAW) 41  Finally, the SPU-A calculations were performed for a single 256-atom supercell to capture aspects of the configurational averaging inherent to ICPA, whereas each of the full SPU phonon spectral function calculations presented (except for AgPd) represent averages over several (i.e. six for 64 and three for 108 atom) supercells. The SPU phonon spectral function for AgPd were obtained from a single 108-atom supercell.

Connection between ICPA and SPU:
The connection of the SPU with the ICPA can be established through the postulated ansatz that configurational averaging for an infinite random system can be approximated by manually averaging the observables (PDOS and spectral functions) over many finite SQS cells. A particular advantage of the SPU approach, however, is that it couples straightforwardly with density functional theory (DFT) computed force constants without the need for any special averaging procedure. Hence, SPU has the required capability to provide complete information on the impact of force constant disorder, i.e., the fluctuations between different atomic pairs as well as their full environment dependence. However, we note that so far, neither ICPA nor SPU has been tested for random alloys dominated by force constant disorder.

Data availability
The authors declare that the data supporting this study are available from the corresponding author upon request.

Force constants versus bond lengths
The bond stiffness, or the force constants in materials are directly associated with the bond length.
In the bond proportion model (BPM) (see Ref. 1    indicating that the force constant is more sensitive to the chemical environment than the bond length. On the other hand, the force constants within the 4d-transition metal AgPd alloy (with only 0.7% mass disorder) obey the BPM with little force constant variations at particular bond lengths (see Fig. 1 and 2). This suggests that the bond length plays a direct and dominant role on the force constants. Further, freezing all atoms in their ideal positions, the resulting species-resolved force constants have much weaker fluctuations, as seen in Fig. 2.

Comments on the averaged force constant model
Although the point group symmetry on each site of disordered alloys is broken due to random chemical environments and the resulting local atomic relaxations, we employed the symmetry operations for the ideal face-centered cubic (fcc) lattice to correlate the same type of speciesdependent force constants within a particular neighboring shell. For example, all the nearest neighbor force constants are rotated so that all nearest neighbor bonds are along the (0.5,0.5,0) a0 direction (a0 is the lattice constant) with the help of the ideal symmetry operations, regardless of the local environments. In a perfect fcc lattice or a fully disordered solid solution, the averaged nearest neighbor force constants between any species pair for the first nearest neighbor bond (0.5, 0.5, 0) a0, denoted as Φ for short, should sustain the following symmetry with only three independent components with Φ (( = Φ )) and Φ () = Φ )( In this symmetric force constant matrix, both Φ (( and Φ () are large components. Furthermore, 1/2(Φ (( + Φ () ) gives the longitudinal force constants or central force constants (bond-stretching), while 1/2(Φ (( − Φ () ) and Φ --gives the transverse force constants, which are much weaker.
Note that this symmetry does not hold for individual force constants in alloys, but roughly holds after averaging over all the equivalent pairwise interactions, denoted as 〈Φ "# $% 〉 for i, j component

Bond stiffness in NiFeCo, NiCo and NiFe alloys
In NiFeCo, 〈Φ 6767 〉 > 〈Φ 9"9" 〉 > 〈Φ and therefore Fe has a larger band width and stronger electron hybridization. Accordingly, the Fe-Fe force constant is larger than the Ni-Ni force constant in NiFe.

Phonon density of states (PDOS)
By projecting the eigenvectors of the phonon modes to different species, species-resolved partial PDOS have been calculated for NiCo, NiFe, NiFeCo and AgPd alloys, as seen in Fig. 5. In the fcc lattice, there are two peaks indicating the resonance due to atomic motion; the highest frequency peak in the PDOS plot corresponds to the LA mode where the phonon band becomes flat, and the lower frequency peak relates to the top of the lower-lying TA modes. By analyzing the peak positions, the peak values, and the chemical contributions, conclusions can be drawn regarding the type of atomic motions in the energy spectrum. For example, the 8 THz resonance in NiCo is dominated by Ni atoms due to the larger intensity of the Nickel partial DOS. In addition, a 0.2 THz downward shift of the Co peak relative to the Ni peak can be seen from Fig. 5. This is due only to the relatively weaker force constants (see Table 1).
In NiFe, the relatively higher 8.2 THz resonance is dominated by the motion of Fe atoms, and the slightly lower-lying 7.8 THz resonance peak is mainly Ni-mediated. A weak shoulder remains at Mixing Ni, Fe, Co together, the partial DOS of NiFeCo (see Fig. 5c) sustains all the features in both NiFe and NiCo, with slight peak shifts. In the AgPd alloy, the partial DOSs of Ag and Pd ((see Fig. 5d)) are almost overlapping, indicating that Ag and Pd species in this binary alloy are almost indistinguishable.

Supplementary Note 2: Effect of pure force disorder in NiFe
In NiCo, the mass disorder is negligibly small (~0.4%) so the phonon broadening comes primarily from the force constant disorder. In contrast, despite the relatively small mass disorder in NiFe (~4.8%), it is still sizable compared with the mass disorder in NiCo. To isolate the pure force disorder effect in NiFe, we employ the full set of force constants of NiFe, obtained from standard ab-initio calculations based on a 108-atom SQS, but then calculate the phonon spectral functions by replacing all Fe masses with Co masses. The spectral functions, dispersion curves, and line widths for the actual NiFe and this "artificial NiCo" are compared in Fig. 6. As can be seen, the dispersion of the "artificial NiCo" shifts towards the low-energy region due to the slightly larger averaged mass. The phonon line widths are reduced as well due to the smaller mass disorder, but the reduction is quite weak so it has little bearing on our conclusions.

Supplementary Note 3: Local relaxation effect on phonon properties
Due to the small atomic size mismatch and the similar electronegativities of the alloying elements in the alloys studied here (NiCo, NiFe, NiFeCo, AgPd), the atomic displacements away from the ideal sites are small. However, NiFe displays greater local atomic displacements than those in

Tables
Supplementary Table 1 The averaged force constants in NiCo, extracted from supercell calculations. The force constants are given in eV/Å 2 . Values in parentheses are the standard deviations (eV/Å 2 ) of the corresponding species-pair force constant distribution.