Symmetry-protected topological polarons

Significance

When electrons move through a solid, they can distort the surrounding lattice and become trapped, forming composite particles known as polarons. These distortions were long assumed to be trivial, consisting of simple contractions or expansions of the lattice around the electron. In this work, we show that topologically nontrivial real-space textures, characterized by integer-valued topological indices, are a general feature of polarons in a broad range of materials. The finding of universal topological quantization in polarons opens opportunities for using these quasiparticles as information carriers in post-Moore electronics and quantum information science.

Abstract

Emergent quasiparticles in solids often exhibit unique topological properties as a result of the complex interplay between charge, orbital, spin, and lattice degrees of freedom. Among these quasiparticles, the polaron occupies a special place as the first known manifestation of the interaction between a fermion and a boson field. While polarons have been investigated for almost a century, whether these quasiparticles exhibit topological properties and why remain open questions. Here, we establish the universal symmetry principles governing the topology of polar textures in large polarons. Using a group-theoretic analysis, we identify four distinct classes of polar textures in time-reversal-invariant systems, and we show that they carry integer topological charges. We validate this classification by performing state-of-the-art first-principles calculations of materials representative of each class. For these materials, we compute the fingerprints of polaron topology in Huang diffuse scattering and propose ultrafast electron and X-ray scattering experiments to detect these quasiparticles.

Topological invariants of emergent quasiparticles, as encoded in the real-space spin textures of skyrmions (1, 2) and the momentum-space Berry curvature of Dirac, Weyl, and Majorana fermions (3–6), often give rise to exotic physical phenomena and constitute promising candidates for information carriers in next-generation quantum devices. Despite extensive research on topological invariants, their potential role in the physics of one of the most fundamental quasiparticles, the polaron, is only beginning to emerge (7–9). Polarons are electronic excitations dressed by a phonon field (10, 11); they are ubiquitous in materials (12–14) and play key roles in artificial photosynthesis (15–17), neuromorphic computing (18, 19), and emerging photovoltaics (20–24). A significant body of knowledge exists on polaron energetics and dynamics (14, 25, 26), spanning model Hamiltonian approaches (27–32) and first-principles calculations (33–37); however, very little is known about the real-space polar textures emerging from electron–phonon couplings, whether and under which conditions they exhibit nontrivial topology, and which physical observables may be associated with such topological properties.

Very recently, topologically nontrivial polar textures accompanying polarons have been identified in the halide double perovskite Cs2AgBiBr6 (9) and in the quantum paraelectric SrTiO₃ (38) via first-principles calculations. In addition, several experimental observations of nontrivial polar textures have been reported in engineered ferroelectric oxides like Pb(Zr,Ti)O₃ and BaTiO₃ (39, 40), and their moirés (41, 42), including polar analogs of magnetic skyrmions (43, 44), merons (45, 46), and hopfions (47). These topological objects attracted considerable interest as they could offer novel pathways for storing and manipulating quantum information via all-electrical means. For example, single-polaron write/move/erase operations using STM tips have recently been demonstrated (48, 49).

Here, we set out to systematically identify and analyze topologically nontrivial textures of polarons. For other quasiparticles, topological invariants are fundamentally dictated by symmetry, such as time-reversal, particle–hole, and chiral symmetries, as well as crystalline point-group and space-group symmetries (50–54). By analogy, we hypothesize that symmetry principles offer the most natural starting point for investigating polaron topology.

The symmetry of polarons in coupled electron–phonon systems remains an open question. For noninteracting electrons and phonons, the relevant symmetry is the little group of operations that preserves their wavevectors. However, when electrons and phonons couple to form small polarons, the resulting electron wavefunction and polar distortion are coherent superpositions of states from the entire Brillouin zone (9, 55–57), incorporating low-symmetry wavevectors with trivial little groups. This complexity makes a symmetry-based analysis impractical. In contrast, large polarons primarily involve low-energy valley electrons and long-wavelength phonons, which imposes symmetry constraints on the effective interaction Hamiltonian. These constraints enable a general symmetry analysis of large polarons, as demonstrated for cubic crystals in ref. 36, and provide a framework to investigate their topological properties.

Results

Symmetry-Based Classification of Polarons.

In the presence of large polarons, the host crystal can be considered as a continuous medium, and the symmetry of the undistorted crystal is preserved on average, since atomic displacements are small. Under these approximations, the response of the lattice to an excess electron induces atomic displacement patterns that can be described via a cell-averaged, slowly varying vector field $\mathbf{u}$. This vector field describes the change of the atomic positions from the pristine crystal structure without polarons to the distorted configuration in the presence of polarons, in analogy with the definition of macroscopic polarization in the modern theory of polarization (58–60). This field is constrained by the crystal point-group symmetry:

$$\begin{array}{c}\hfill \mathbf{u}(\widehat{S}\mathbf{r})=\widehat{S}\mathbf{u}(\mathbf{r}),\end{array}$$ [1]

where $\widehat{S}$ represents the point operation of any symmetry belonging to the crystal space group (61, 62), and $\mathbf{r}$ is the position relative to the polaron center. The above equation is proven in SI Appendix, Supplementary Note 1, starting from the ab initio polaron equations of ref. 63 and including the leading terms in the electron–phonon couplings at long wavelength, namely Fröhlich (64–67) and piezoelectric couplings (68–72).

Eq. 1 dictates the possible forms that $\mathbf{u}$ can take in a given crystal. To illustrate this point, we consider two examples, leaving detailed derivations to SI Appendix, Supplementary Note 2. i) In the simplest case of cubic rock-salt crystals, which belong to the $m\overline{3}m$ point group, symmetry requires the displacement field $(u_x,u_y,u_z)$ near the polaron center to transform like $(x,y,z)$ plus terms of third order and higher in $x,y,z$. As a result, atomic displacements form a monopole-like field with a characteristic hedgehog pattern, as shown in Fig. 1A. This is precisely the behavior observed in recent ab initio calculations of large polarons in LiF (36, 55). ii) If inversion symmetry is removed from the $m\overline{3}m$ group, the resulting subgroup is $\overline{4}3m$, which describes, for example, cubic zinc-blende crystals (see SI Appendix, Fig. S1 for group–subgroup relations). In this case, symmetry dictates a displacement field of the type $a(x,y,z)+b(yz,zx,xy)$ plus terms of third order and higher, with $a$ and $b$ being material-specific constants. Therefore, in addition to the hedgehog texture, in this case, we also have a three-dimensional antivortex field, as shown in Fig. 1 D and H Here, the lack of inversion symmetry is precisely what enables nontrivial textures, because vortex-like fields involving $\mathit{xy}$, $\mathit{xz}$, $\mathit{yz}$ are parity-forbidden. This is analogous to the case of magnetic skyrmions driven by the Dzyaloshinskii–Moriya interaction, which is allowed only in noncentrosymmetric crystals (2, 73).

Fig. 1.

Symmetry-based classification of topological polarons. (A) Monopole-like hedgehog polaron with topological charge $|Q|=1$, vorticity $v=1$, and helicity $\gamma =0$. The arrows show the atomic displacement field on a sphere enclosing the polaron center. (B) Pruned graph of group–subgroup relations between crystal point groups without inversion symmetry, extracted from SI Appendix, Fig. S1. Color-coded groups host symmetry-protected topological polarons (green: antivortex; red: vortex; yellow: double-antivortex; blue: vertical flow). (C) The point groups in the colored regions admit only one type of polaron texture, while all other groups admit combinations of two, three, or four textures that are not protected by symmetry. (D) Antivortex polaron on a sphere, with topological charge $|Q|=3$, vorticity $v=-1$, and helicity $|\gamma |={90}^{°}$. The color code is a visual aid. (E) Vortex polaron, with topological charge $|Q|=1$, vorticity $v=1$, and helicity $|\gamma |={90}^{°}$. (F) Double antivortex polaron, with topological charge $|Q|=2$, vorticity $v=-2$, and helicity $|\gamma |={90}^{°}$. (G) Vertical flow polaron, with topological charge $|Q|=0$ (vorticity and helicity are not defined in this case). (H–K) and (L–O): Volumetric plots and planar cuts of the polaron textures shown in (D–G), respectively. The spheres are the same as those shown in (D–G). A detailed analysis of each of these textures and their topological invariants is provided in SI Appendix, Fig. S2 and Supplementary Note 3.

Building on this insight, we perform a classification of nontrivial polaron textures by inspecting the group–subgroup relations of crystal point groups. From the full graph shown in SI Appendix, Fig. S1, we obtain the pruned graph in Fig. 1B by eliminating all groups that possess inversion symmetry. By applying the symmetry operations of each group to Eq. 1, we obtain four distinct nontrivial vector fields $\mathbf{u}(\mathbf{r})$: an antivortex with texture given by $(yz,zx,xy)$; a vortex with texture $(-yz,xz,0)$; a double antivortex $(2xy,x^2-y^2,0)$; and a vertical flow $(0,0,z^2)$. The rationale for this nomenclature will become apparent shortly. These vector fields are shown on a sphere in Fig. 1D–G, respectively, as well as 3D plots and 2D cuts in panels (H–K) and (L–O) of the same figure, respectively. Groups admitting these vector fields are highlighted in color in Fig. 1B and include $\overline{4}3m$, $4mm$, $422$, $622$, $6mm$, $\overline{6}m2$, as well as two subgroups, $\overline{6}$ and $23$, which inherit the textures of their respective parent groups (see SI Appendix, Table S1 for a summary). Fig. 1C shows how all other groups admit combinations of two, three, or four of the above elementary vector fields; for materials in these other groups, polaron textures are not symmetry-protected, therefore we do not consider them further.

Topological Invariants and Topological Protection.

A complete classification of polaron textures requires identifying their associated topological invariants. To this end, we focus on the skyrmion number or topological charge $Q$ of a vector field, which is also referred to as the “degree of mapping” in ref. 74. This quantity is defined as the flux of the topological density $\mathbf{\Omega }$ through a unit sphere enclosing its center: $Q=(1/4\pi )\int \mathbf{\Omega }·d\mathbf{S}$, with ${\mathbf{\Omega }}_{\alpha }={ϵ}_{\alpha \beta \gamma }\widehat{\mathbf{u}}·({\partial }_{\beta }\widehat{\mathbf{u}}\times {\partial }_{\gamma }\widehat{\mathbf{u}})/2$ (1, 9). Here, $\widehat{\mathbf{u}}(\mathbf{r})$ is the normalized atomic displacement field, Greek subscripts denote Cartesian directions, ${ϵ}_{\alpha \beta \gamma }$ is the Levi-Civita symbol, ${\partial }_{\alpha }$ is the spatial derivative with respect to the direction $\alpha$, $d\mathbf{S}$ is the surface element, and summation over repeated indices is implied. The density $\mathbf{\Omega }$ measures the local winding of the field, and the charge $Q$ counts how many times the displacement field wraps a closed surface enclosing the polaron center.

In SI Appendix, Supplementary Note 3, we evaluate the topological charge for the hedgehog, antivortex, vortex, double antivortex, and vertical flow fields identified above. We find integer charges $Q=0,\pm 1,\pm 2,\pm 3$ across these vector fields, with each texture carrying a uniquely defined charge. The sign of the topological charge is determined by the constant $a$ and is a material-specific property. SI Appendix, Fig. S2 and Table S2 report the complete assignment of topological charges for these textures.

Different textures can uniquely be identified by their topological charge, except for the hedgehog and the vortex which share the same charge $Q=\pm 1$. To further distinguish these patterns and motivate our nomenclature, we also consider the vorticity $v$ and the helicity $\gamma$ of the vector field, which are well defined for any 2D slice at constant elevation $z$, cf. Fig. 1L–O (1). The vorticity is the winding of the in-plane component of the vector field along a closed loop, $v=(1/2\pi )\oint d\phi$ with $\phi ={tan}^{-1}(u_y/u_x)$; the helicity is the average angle between the field and the in-plane radial direction (SI Appendix, Fig. S2). The polaron textures identified here correspond to $v=1$ for the hedgehog (no helicity, $\gamma =0$) and the vortex ($|\gamma |={90}^{°}$); $v=-1$ for the antivortex; and $v=-2$ for the double antivortex. The latter texture winds twice around the polaron center, hence its name.

Beyond the topological charge, point group symmetries also dictate the vectorial character of these textures. Indeed, following ref. 75, all time-reversal-even vectors can be classified into four categories: neutral, polar, chiral, and axial. Among the textures identified here, the hedgehog field is invariant under inversion symmetry and thus belongs to the neutral category. The antivortex, double antivortex, and vertical flow fields reverse sign under inversion; therefore, they are polar vectors. The vortex field lacks both inversion and mirror symmetries and is classified as a truly chiral field, similar to the chiral phonons in $\alpha$-HgS (76–79). The only missing element in our list is the axial texture, which is associated with ferrotoroidic order (80, 81); this polaron class has recently been identified in halide perovskites (9), but is absent here because we are focusing on noncentrosymmetric point groups. For completeness, we discuss the polar textures of ref. 9 in SI Appendix, Supplementary Note 4. The neutral, polar, chiral, and axial classes exhaust all possible symmetry-protected topological polaron textures in time-reversal-even crystals. In SI Appendix, Table S3, we also discuss the relation between the dimensionality and the topological stability of these polarons.

Validation via First-Principles Calculations.

To validate our topological classification of polarons, we proceed to direct ab initio calculations using the method of ref. 63 (Materials and Methods). Based on the map in Fig. 1C, we consider one representative compound per class: i) zinc-blende BeO (Fig. 2A), a theoretically predicted ultra-wide-band-gap semiconductor (82–84) with point group $\overline{4}3m$, for which our theory predicts a 3D antivortex pattern; ii) $\gamma$-LiAlO₂ (Fig. 2B), a representative oxide for nuclear fusion applications and battery cathodes with point group $422$ (85), for which we predict a vortex-type polaron; iii) hexagonal BN (86), a common insulator in 2D electronics, with point group $\overline{6}m2$, for which we predict a double antivortex polaron; for ease of visualization, in Fig. 2C we consider instead an h-BN monolayer (87), which shares the same in-plane symmetry; iv) the tetragonal PbTiO₃ perovskite (Fig. 2D), a prototypical displacive ferroelectric with point group $4mm$ (88); for this compound, our theory predicts a vertical flow polaron texture.

Fig. 2.

First-principles calculations of symmetry-protected topological polarons. (A–D) Ball-stick models of the conventional unit cells of zb-BeO ($F\overline{4}3m$ space group), $\gamma$-LiAlO₂ ($P{4}_1{2}_{1}2$), 2D h-BN ($P3m1$), and PbTiO₃ ($P4mm$), respectively. (E–H) The polaron displacement field computed from first principles for each of the systems in the first row, in the same order. The color code is a visual aid. The orange ellipsoids visible in the center represent the envelope function of the electron wavefunction for zb-BeO, $\gamma$-LiAlO₂, and PbTiO₃, and of the hole wavefunction for 2D h-BN. (I–L) Displacement field on a sphere centered at the polaron center with radius $2{\sigma }_{\mathrm{p}}$, where ${\sigma }_{\mathrm{p}}$ is the SD obtained from the ab initio polaron wavefunctions in SI Appendix, Fig. S6: ${\sigma }_{\mathrm{p}}=$10.4 Å, 14.4 Å, and 10.8 Å for zb-BeO, $\gamma$-LiAlO₂, and 2D h-BN, respectively; ${\sigma }_{\mathit{xy}}=8.2$ Å and ${\sigma }_z=1.8$ Å for PbTiO₃. Here, we recognize the antivortex, vortex, double antivortex, and vertical flow fields, respectively, in the same order as in Fig. 1. In SI Appendix, Fig. S8, we perform a detailed comparison between these ab initio results and the symmetry-based polaron textures shown in Fig. 1. For ease of visualization, in each panel the displacement field is rendered for a single atomic species: O for BeO and PbTiO₃, Li for $\gamma$-LiAlO₂, and B for 2D h-BN. The displacements of the other species follow the same patterns. Note that, for h-BN, we use the 2D monolayer for ease of visualization; the group–subgroup relations for 2D rosette groups are shown in SI Appendix, Fig. S1.

For each of these crystals, we solved the ab initio polaron equations (63) and we found large polarons with formation energies in the range 10 to 50 meV (SI Appendix, Fig. S3). In all cases, the formation of these polarons is driven by long-wavelength phonons (SI Appendix, Fig. S4), and for these phonons we verified that the electron–phonon coupling matrix elements correctly satisfy crystal point-group symmetries, see SI Appendix, Fig. S5. In all cases, the electron or hole charge density resembles a Gaussian envelope with a SD ranging between 0.8 nm (PbTiO₃) and 1.4 nm ($\gamma$-LiAlO₂); these solutions are shown in SI Appendix, Fig. S6.

Fig. 2E–H show how, in each of these representative compounds, polarons exhibit precisely the displacement texture predicted by our theory: a 3D antivortex for BeO, a vortex for $\gamma$-LiAlO₂, a double antivortex for 2D h-BN, and a vertical flow for PbTiO₃. We note that the vertical flow pattern in PbTiO₃ is aligned with the ferroelectric polarization in this compound; a detailed analysis is presented in SI Appendix, Fig. S7. Panels (I–L) in the same figure show the calculated displacement fields on a sphere for each of these textures. A side-by-side comparison of these patterns with the symmetry-protected fields of Fig. 1 is shown in SI Appendix, Fig. S8. The close agreement between our theoretical predictions and our explicit ab initio calculations demonstrates the reliability of our symmetry-based approach.

To further validate the theory, we computed the topological charge $Q$ for each of these polarons (Materials and Methods), and we found that each charge is indeed quantized and matches exactly what we predicted from symmetry analysis. We also verified that these polar textures fulfill the Poincaré-Hopf theorem (89), whereby the sum of the topological charges of a smooth vector field on a compact manifold equals its Euler characteristic, $Q_{\mathrm{tot}}=\chi$. Since the Born-von Kármán supercells used in our calculations are topologically equivalent to hypertori ($\chi =0$), the theorem requires the total charge in the supercell to vanish. Focusing on 2D h-BN for illustration purposes, close examination of the displacement field in SI Appendix, Fig. S9 reveals two topological sources located in the interstitial regions between periodic images of the polaron. Each of these topological defects carries a charge $Q=+1$; furthermore, the polaron carries a charge $Q=-2$. Together, these contributions yield a total topological charge $Q_{\mathrm{tot}}=+1+1-2=0$, in agreement with the Poincaré-Hopf theorem. Similar considerations apply to the other systems considered here. The present findings confirm the existence of symmetry-protected, topologically nontrivial polaron textures in materials, at least at the level of atomistic first-principles calculations.

Analytical Model of Topological Polaron Textures.

The nontrivial polaron textures found here can be rationalized with a remarkably simple analytical model. Since the calculated polaron wavefunctions are relatively featureless (SI Appendix, Fig. S6), in a first approximation, they can be described via Gaussian envelopes. The simplest polaron model that yields Gaussian wavefunctions is the Landau-Pekar model (25). In this model, the Gaussian charge density of the excess electron or hole generates an electric field, which polarizes in turn the ionic lattice, establishing the atomic displacement texture. The simplest coupling term between electric field and atomic displacements, the Fröhlich coupling, only allows for hedgehog-type polar textures (SI Appendix, Supplementary Note 5). To enable more complex patterns, we must consider the next-to-leading order term, which is provided by piezoelectric couplings.

To investigate the effect of piezoelectric couplings, we make the following observations: i) the electric field $\mathbf{E}$ generates a local strain field ${\epsilon }_{\alpha \beta }$ via the converse piezoelectric strain coefficients $d_{\alpha \beta \gamma }$, ${\epsilon }_{\alpha \beta }=d_{\alpha \beta \gamma }{E}_{\gamma }$ (61); ii) the strain tensor is related to the displacement field via the standard relation ${\epsilon }_{\alpha \beta }=(\partial u_{\alpha }/\partial r_{\beta }+\partial u_{\beta }/\partial r_{\alpha })/2$; iii) the electric field generated by a Gaussian charge density centered at $\mathbf{r}=0$ is linear in $\mathbf{r}$ near the origin; since topological properties are insensitive to continuous deformations, this field can be taken to be isotropic. By combining these three relations, the displacement field can be expressed in terms of the converse piezoelectric tensor, as we show in SI Appendix, Supplementary Note 5:

$$\begin{array}{c}\hfill \mathbf{u}=(3d_{\alpha \beta \gamma }-d_{\beta \gamma \alpha }-d_{\gamma \alpha \beta }){\widehat{\mathbf{r}}}_{\alpha }{r}_{\beta }{r}_{\gamma }.\end{array}$$ [2]

Here, ${\widehat{\mathbf{r}}}_{\alpha }$ denotes the unit vector along the direction $\alpha$. For instance, in point groups 422 and 622, the only nonvanishing tensor components are $d_{312}=-d_{321}$ (90, 91); by using these components in Eq. 2, we obtain the displacement field $(yz,-zx,0)$. The corresponding displacement patterns for all other groups are reported in SI Appendix, Table S3 and are illustrated schematically in Fig. 3; a comparison between the displacement field and the strain field is shown in SI Appendix, Fig. S10 for 2D h-BN. These patterns are fully consistent with our ab initio calculations, indicating that the piezoelectric effect plays a key role in shaping these textures.

Fig. 3.

Rationalizing polaron textures in terms of local strain. (A) Schematic of a cube enclosing the polaron excess charge. In a first approximation, this charge induces electric fields perpendicular to the cube faces, which generate local strains through the converse piezoelectric tensor (SI Appendix, Table S4). With reference to the top face, and considering point group $\overline{4}m3$, the only nonvanishing component of the converse piezoelectric tensor for $E_z<0$ is $d_{\mathit{xyz}}>0$, leading to the strain ${\epsilon }_{\mathit{xy}}<0$. This strain corresponds to a rhombohedral distortion. The cumulative distortions of all cube faces produce the pattern shown in (B), which matches the antivortex polaron of zb-BeO in Fig. 2I. In this panel, the unstrained cube is shown in dark blue, the strained cube is in blue, and arrows denote the displacements of each vertex. (C) By repeating the same reasoning for point group $422$, the combination of the distortions of each face causes a counterrotation of the top and bottom faces about the $z$-axis. This chiral distortion pattern matches the vortex polaron of $\gamma$-LiAlO₂ in Fig. 2J. (D and E) Strain-induced distortion patterns allowed within the $\overline{6}m2$ and $4mm$ point groups, respectively. These patterns match the double antivortex polaron of 2D h-BN in Fig. 2K and the vertical flow polaron of PbTiO₃ in Fig. 2L, respectively.

Huang Diffuse Scattering of Topological Polarons.

Experimental observation of the topological textures predicted here should be possible via Huang diffuse scattering (92). When a crystal is illuminated by X-ray or electron beams, diffraction by the periodic arrangement of atoms leads to the standard Bragg peaks. Thermal fluctuations broaden these peaks by inducing dynamic distortions of the crystal lattice (93, 94). In the presence of nonthermal distortions, such as the strain fields arising from point defects (95), extended defects (96), alloying (97), quasicrystalline order (98), and small polarons (99), diffraction lineshapes exhibit additional anisotropic components known as Huang scattering. These features carry the fingerprints of the underlying nonperiodic crystal structures (Materials and Methods). Huang diffuse scattering has recently been employed in combination with ultrafast X-ray diffraction (UXRD) (23) and ultrafast electron diffraction (UED) (100–102) to investigate the structure and dynamics of polarons in several classes of materials.

For an isolated polaron, the amplitude of Huang diffuse scattering is governed by the Fourier transform of its displacement field. As shown in Fig. 4A, a hedgehog-type polaron gives rise to a dipole-like pattern. This characteristic shape is well established in the crystallography of point defects (103) and is commonly known as the “double-drop” pattern. Such a pattern was recently observed with high resolution in UED experiments on photoexcited GeS (104).

Fig. 4.

Huang diffuse scattering of topological polarons. (A) Huang diffuse scattering intensity calculated for a model hedgehog-type polaron in a simple cubic lattice, for the displacement pattern $(x,y,z)$ modulated by the Gaussian profile $\text{exp}(-r^2/2{\sigma }^2)$; $\sigma$ = 6 Å from the electron polaron in LiF (63). The double-drop structure is aligned with the Bragg vector [001], as shown at the Bottom. Orange/blue indicates normalized positive/negative intensity. The solid square represents the first Brillouin zone, while the dashed square represents the region around the Bragg peaks shown in the other panels, extending from $-$0.1 Å⁻¹ to 0.1 Å⁻¹ along each direction. The contours are guides to the eye. (B–E) Huang diffuse scattering intensities calculated for the model antivortex [$(yz,zx,xy)$], vortex [$(yz,-xz,0)$], double antivortex [$(2xy,x^2-y^2,0)$], and vertical flow polaron [$(0,0,z^2)$], respectively. The patterns are modulated by Gaussian profiles with $\sigma$ = 10 Å comparable to the ab initio calculations in Fig. 2. Contour lines correspond to 40%, 60%, and 80% of the maximum value in each panel. These idealized scattering intensities do not include thermal disorder and correspond to the displacements of the acoustic phonons, which dominate at long timescales. The effect of thermal disorder and phonon contributions at short timescales is analyzed in SI Appendix, Fig. S11.

Fig. 4B–E show how different polarons imprint distinct diffuse scattering signatures, each carrying a unique nodal structure that is qualitatively different from the classic double-drop associated with point defects. Thermal broadening tends to smear out the fine structure of these fingerprints, but the essential features remain, as seen in SI Appendix, Fig. S11. These unique diffuse scattering patterns should be detectable with state-of-the-art time-resolved UXRD or UED at long timescales (22, 23, 105, 106), and thus offer a potential pathway to directly observe topological polarons in materials.

Discussion

The present finding of several classes of symmetry-protected topological polarons establishes connections between electron–phonon physics, crystal symmetry, and topology (107). These topologically nontrivial real-space textures emerge clearly in first-principles calculations, and exhibit characteristic fingerprints that should be accessible via ultrafast Huang diffuse scattering. A potential consequence of topological polarons is that the accompanying strain fields should act as sources of pseudomagnetic fields and Berry curvature, in analogy with the quantum Hall physics of strained graphene and related Dirac materials (108, 109). Some of these possibilities are investigated in SI Appendix, Supplementary Note 6. Extension of our theory from a single polaron to multipolaron scenarios may result in topological polar lattices in the form of polaron Wigner crystals (110, 111). This possibility is investigated in SI Appendix, Fig. S12.

More generally, the present classification captures all four vectorial types of polaron textures allowed in time-reversal-invariant systems, including the recently identified ferrotoroidic fields (9). Further extending this framework to systems with broken time-reversal symmetry could reveal an even richer taxonomy, potentially bridging electron–phonon physics with ongoing efforts on electric Dzyaloshinskii–Moriya interactions (112), magnetic skyrmions (2, 73), and other magnetoelectric phenomena (113). In this work, we have not addressed cases where the electronic bands and phonon dispersions underlying the polaron also exhibit nontrivial topology. These cases may appear, for example, in the study of polarons in topological insulators (3–5) and could offer opportunities to investigate the interplay between the reciprocal-space topology of bands, phonons, or electron–phonon matrix elements and that of the real-space atomic displacement patterns. A promising class of materials where such effects may be investigated is that of correlated topological insulators (114, 115); in these materials, relatively narrow bands and heavy effective masses could favor polaron formation, while at the same time harboring quantum Hall phases where edge states interact with topological displacements patterns. Another interesting direction of future investigation could be to promote the classical displacement fields considered in this work to quantum fields, so as to investigate the interplay between fermionic and bosonic degrees of freedom and topology within a field-theoretic framework. This could be achieved, for example, by mapping the present displacement fields into the amplitude of coherent phonons (116). Similarly, it would be interesting to investigate whether the topology of the displacement field carries any observable imprints on the electronic wavefunctions; answering this question will require going beyond the adiabatic Born–Oppenheimer approximation, and consider coupled electron-nuclear dynamics, e.g., via exact factorization (117). The present framework could also prove fruitful to investigate phonon vorticity around shallow impurities in semiconductors, as recently observed via atomic-resolution vibrational energy-loss spectroscopy (118). Looking ahead, these far-reaching connections could present opportunities for engineering emergent gauge fields and electronic Berry phases through electron–phonon couplings and thus open pathways for controlling charge, spin, and lattice degrees of freedom in quantum materials.

Materials and Methods

All ab initio calculations are performed using the Quantum ESPRESSO package (119) (electronic structure and lattice vibrational properties), the Wannier90 code (120) (maximally localized Wannier functions), the EPW code (121) (interpolation of electron–phonon matrix elements and polaron calculations), and the ABINIT package (122) (quadrupole tensors). We describe all the materials using the local density approximation (LDA) (123), optimized norm-conserving Vanderbilt (ONCV) pseudopotentials (124, 125), and a plane-wave basis with a kinetic energy cutoff of 100 Ry. Phonon frequencies and electron–phonon matrix elements are computed within density functional perturbation theory (126). Identical, unshifted uniform coarse grids of $\mathbf{k}$- and $\mathbf{q}$-points are employed for these calculations and are interpolated to dense grids by means of Wannier-Fourier interpolation (127, 128). The methods presented in refs. 66, 67, 71, 72, 129, and 130 are used to handle the long-range contributions of the electron–phonon vertex, namely dipolar and quadrupolar effects. Complete computational details are available in SI Appendix.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Raw data files and post-processing scripts are available in the Materials Cloud Archive: https://doi.org/10.24435/materialscloud:y1-js (131). All study data are included in the article and/or SI Appendix.