Application of Berry phases and Wannier functions in solids
Creators: name: Ren, Shang
Publisher: name: No Publisher Supplied
Year: 2024

description: The Berry phase is the geometrical phase acquired by a quantum state after evolving along a loop in parameter space. Wannier functions, which are Fourier transforms of Bloch functions, have centers related to the Berry phases of Bloch functions acquired along a loop winding through the Brillouin zone. The application of Berry phases and Wannier functions in solids is the main topic of this thesis, and we present two case studies to illustrate these applications. In the first case study, we demonstrate that Berry phase plays an essential role for phonons in the absence of time-reversal symmetry (TRS). The conventional approach to phonons treats them as harmonic oscillators, which does not fully account for the effects of broken TRS in magnetic systems. A recent attempt to rectify this involves considering the force induced by the velocity of atoms, which appears naturally after implementing the Born-Oppenheimer approximation, with the coefficient being the nuclear Berry curvature. We have calculated the nuclear Berry curvature under the assumption that all other degrees of freedom are fully relaxed with respect to the nuclear coordinates. The results show that the main contribution comes from spin precessions during the loop of nuclear coordinates. However, the quasiparticles associated with spin precessions, magnons, have energy scales similar to those of phonons. This contradicts the aforementioned assumption, indicating that phonons and magnons need to be treated on an equal footing. This is achieved by utilizing adiabatic theory, which will be discussed in detail in this thesis. Our case studies reveal that degenerate phonons can have different energies in CrI$_3$ with atoms rotating, thus forming chiral phonons, and Raman-active even-parity phonons in Cr$_2$O$_3$ can acquire some infrared activity. In the second case study, we use Berry phases and Wannier functions to predict the corner charges for two-dimensional (2D) finite flakes. In the one-dimensional (1D) case, the bulk-boundary correspondence tells us that the end charge of a finite 1D chain can be predicted by the bulk polarization (modulo $e$). The bulk polarization is related to the Berry phase of Bloch functions calculated along a loop winding through the Brillouin zone, which is equivalent to calculating the polarization using the centers of Wannier functions. For a 2D finite flake, the corner charge can be predicted by the sum of edge polarizations for the two edges intersecting at the corner and the interior quadrupole moment. Similar to the bulk polarization, edge polarizations and quadrupole moments can also be calculated using Wannier functions. Edge polarizations can be calculated using 1D ribbon geometries, where one direction is periodic and the other is finite, containing multiple unit cells. The quadrupole moment can be calculated using a 2D bulk. We show that although these quantities are individually gauge-dependent, their gauge dependence cancels out when summed, indicating that we must work under the same Wannier gauge when calculating them. The direct calculation of corner charges for the 2D finite flake validates our theory. Besides the two cases about Berry phase applications, we will also discuss a collaborative study with experimentalists on Humble defects, a kind of planar defect observed in some semiconductors. Although there are several types of Humble defects, only the (a)-type was observed in GeSi alloy. Our density functional theory calculations show that this type has the lowest energy among the various types. We also simulate the core-level electron energy loss spectra for the Si atoms in both the defect and bulk regions, and our simulations agree well with experimental results.