Symmetry-protected topological polarons

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 classfication 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.

Figures (4)

• Figure 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 Supplemental Fig. 1. 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 $|\gamma|=90^\circ$. The color code is a visual aid. e Vortex polaron, with topological charge $|Q|=1$, vorticity $v=1$, and helicity $|\gamma|=90^\circ$. f Double antivortex polaron, with topological charge $|Q|=2$, vorticity $v=-2$, and helicity $|\gamma|=90^\circ$. 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 Supplemental Fig. 2 and Supplemental Note 3.
• Figure 2: First-principles calculations of symmetry-protected topological polarons. a-d Ball-stick models of the conventional unit cells of zb-BeO ($F\bar{4}3m$ space group), $\gamma$-LiAlO2 ($P4_{1}2_{1}2$), 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 $2\,\sigma_{\rm p}$, where $\sigma_{\rm p}$ is the standard deviation obtained from the ab initio polaron wavefunctions in Supplemental Fig. 6: $\sigma_{\rm p}=$10.4 Å, 14.4 Å, and 10.8 Å for zb-BeO, $\gamma$-LiAlO2, and 2D h-BN, respectively; $\sigma_{xy}=8.2$ 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 Supplemental Fig. 8, 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 Supplemental Fig. 1.
• Figure 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 (see Supplemental Table 4). With reference to the top face, and considering point group $\bar{4}m3$, the only nonvanishing component of the converse piezoelectric tensor for $E_z<0$ is $d_{xyz}>0$, leading to the strain $\varepsilon_{xy}<0$. 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. 2(i). 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 cause a counter-rotation of the top and bottom faces about the $z$-axis. This chiral distortion pattern matches the vortex polaron of $\gamma$-LiAlO2 in Fig. 2(j). d, e Strain-induced distortion patterns allowed within the $\bar{6}m2$ and $4mm$ point groups, respectively. These patterns match the double antivortex polaron of 2D h-BN in Fig. 2(k) and the vertical flow polaron of PbTiO3 in Fig. 2(l), respectively.
• Figure 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 $\exp(-r^2/2\sigma^2)$; $\sigma$ = 6 Å from the electron polaron in LiF Sio2019b. The double-drop structure is aligned with the Bragg vector [001], as shown at the bottom. Orange/blue indicate 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 are analyzed in Supplemental Fig. 11.