Analysis of the singular band structure occurring in one-dimensional topological normal and superfluid fermionic systems: A pedagogical description

Abstract

Topological properties of solid-state materials arise when crossings occur in their band-structure eigenvalues, which give rise to discontinuities in the associated Bloch-function eigenvectors once these are mapped over the whole Brillouin zone. These nonanalytic properties have direct consequences on the spatial decay of the corresponding Wannier functions, leading to what is nowadays referred to as the "obstruction to finding symmetric Wannier functions" for a given set of bands, as well as on the need for shifting the Wannier functions to interstitial positions, related to what is nowadays known as the "bulk-boundary correspondence." The importance of nonanalytic points of Bloch eigenfunctions and their consequences for the spatial decay of Wannier functions were historically anticipated back in 1978 [G. Strinati, Phys. Rev. B 18, 4104-4119 (1978)], somewhat before the work of Berry on what came to be referred to as the "Berry phase" [M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)]. In particular, the former paper identified key precursors and physical insights that are now understood, in hindsight, to be closely related to the later developments mentioned above. Here, we recap the essential features of these key issues in a rather pedagogical way, by considering in full details two instructing examples for which the origin of the discontinuities in the eigenvectors can be readily traced and mapped out, and the rate of the spatial falloff of the associated Wannier functions can be fully determined. For this analysis to be as complete as possible, two cases, one for noninteracting and one for interacting fermions, are considered on equal footing.

Figures (10)

• Figure 1: The non-interacting dispersion relation $\lambda(k) = - 2 t \cos (k)$ is plotted in the BZ $- \pi \le k \le + \pi$. Two different values of the chemical potential are considered, such that $\mu_{1} > 2t$ and $\mu_{2} < 2t$. In the second case, there is a region of the BZ (dashed segment) where $\xi(k) = \lambda(k) - \mu_{2} > 0$.
• Figure 2: The functions $|u(k)|$ (blue lines) and $|v(k)|$ (red lines) from Eqs. (\ref{['u-notation']}) and (\ref{['v-notation']}) are plotted in the top panels over the whole BZ, for several values of $\mu$ across the QCP at $\mu=1$ (with the horizontal dashed line marking the unit value). The corresponding eigenvalues $E(k) = \pm \epsilon(k)$ from Eq. (\ref{['epsilon']}) are plotted in the bottom panels. In (the pairs of) panels from (a) to (c) corresponding to the trivial phase with $\mu>1$, $|u(k)|$ and $|v(k)|$ do not cross each other and the eigenvalues $\pm\epsilon(k)$ have a gap $\Delta E = 2 \, \epsilon(k=\pm \pi)$. In (the pairs of) panels (d), where the QCP occurs at $\mu=1$, $|u(k)|$ and $|v(k)|$ touch each other at $k=\pm\pi$ and the energy gap closes. In this case, $|u(k=\pm\pi)|=|v(k=\pm\pi)| = 1/\sqrt{2}$ are both nonzero. In (the pairs of) panels from (e) to (f) corresponding to the non-trivial phase with $\mu < 1$ past the QCP, $|u(k)|$ and $|v(k)|$ cross each other at some value of $k$ inside the BZ and the energy gap opens again.
• Figure 3: The eigenvectors (\ref{['+eigenvector-final']}) for the positive eigenvalue $+ \epsilon(k)$ are shown over the whole BZ, for two typical cases corresponding to (a)-(b)$\mu > 1$ and (c)-(d)$\mu < 1$. The corresponding values of the overall phase $\varphi$ of the wave-function components are also shown in both halves of the BZ.
• Figure 4: Dependence on the lattice sites $n$ of the magnitude of the Wannier-like functions (\ref{['Wannier-like-function-1']}) (panels from (a) to (c)) and (\ref{['Wannier-like-function-2']}) (panels from (d) to (f)). Data from the numerical evaluation of the integrals in Eqs. (\ref{['Wannier-like-function-1']}) and (\ref{['Wannier-like-function-2']}) are shown as blue dots and green crosses, respectively, while the expected asymptotic behavior obtained analytically is shown by a red solid line in all cases. Pairs of panels correspond to three values of $\mu$ across the QCP at $\mu=1$. To highlight the exponential decay for large $n$ of the Wannier-like functions when $\mu>1$, in panels (a) and (d) a log-linear scale is used. To highlight instead the power-law decay for large $n$ of the Wannier-like functions when $\mu\leq1$, in the remaining panels from (b) to (f) a log-log scale is used. In all cases, the analytic behavior for large-$n$ (obtained as explained in the text) is reported in the boxes.
• Figure 5: Dependence on the lattice site $n$ of the pair wave function $|g(n)|$ from Eq. \ref{['definition-g']} (panels from (a) to (c)) and of the correlation function $|f(n)|$ from Eq. \ref{['definition-f']} (panels from (d) to (f)). Data from the numerical evaluation of the integrals in Eq. \ref{['definition-g']} and \ref{['definition-f']} are shown by green dots and cyan open circles, respectively, while the expected asymptotic behavior obtained analytically is shown by red solid lines in all cases. Like in Fig. \ref{['Figure-4']}, pairs of panels correspond to three values of $\mu$ across the QCP at $\mu=1$. Data for $|g(n)|$ are reported in a linear scale, showing an alternating behavior for even and odd $n$ when $\mu>1$ in panel (a), and the reaching of a single uniform value for $\mu\leq1$ in panels (b) and (c). Data for $|f(n)|$ are reported in a log-linear scale in panel (d) for $\mu>1$ and in panel (f) for $\mu<1$ to highlight an exponential falloff for large $n$, while data for $\mu=1$ are reported in panel (e) in a log-log scale to highlight a power-law falloff. In addition, the asymptotic falloff of $|f(n)|$ for large-$n$ is given in the boxes as explained in the text (while the asymptotic falloff of $|g(n)|$ for large-$n$ will be discussed in Fig. \ref{['Figure-6']}).