Mathematical Foundation for the Generalised Brillouin zone of m-banded Toeplitz operators

Abstract

We show that the spectrum of the open-boundary limit of banded Toeplitz matrices is real whenever the associated symbol function is real-valued along a closed polar curve. Building on this result, we develop both analytical and numerical methods to symmetrise a class of banded non-Hermitian Toeplitz matrices whose asymptotic spectra are real. Finally, we provide a rigorous mathematical foundation for the generalised Brillouin zone, a concept widely used in non-Hermitian physics, by proving that it coincides with the polar curve on which the symbol function takes real values.

Figures (12)

• Figure 2.1: Clearly the set $\mathbf{Z}(f_m)\subseteq f_m^{-1}(\mathbb{R})$ is parametrised by a polar curve. We emphasise that confluent roots are only situated on the real axis and by Remark \ref{['rem: real and imag confluent roots']} belong to $\mathbf{Z}(f_m)$. We see that there are no other confluent roots as the complex band in (A) is isolated. Computation performed for $m = 3$, $\kappa = 3.5$, $\rho = 6.5$.
• Figure 2.2: The set $\mathbf{Z}(f_m)\subseteq f_m^{-1}(\mathbb{R})$ is no longer parametrised by a polar curve. Equivalently the gap bands in $(A)$ cross each other. The open limit in this setting contains complex values. Computation performed for $m = 3$, $\kappa =1$, $\rho = 6.5$.
• Figure 2.3: Figure (A) illustrates the polar curve, which is not just a scaling of the torus but rather a homotopic deformation of the torus. Figure (B) illustrates that even for relatively small matrix sizes, i.e. $n = 100$, the empirical measure of $\mathbf{T}_n(f\circ p)$ approaches the density of states $\mathbf{T}_n(f)$. Computation performed for $\kappa = 3.5$, $\rho = 4.8$ and $m = 7$.
• Figure 2.4: The Hermitian matrix $\mathbf{T}_n(f\circ p)$ is less prone to numerical pollution compared to the non-Hermitian matrix $\mathbf{T}_n(f)$. This motivates the idea of considering the evaluation of the symbol function on a deformed torus as a numerical preconditioner. Computation performed for $\kappa = 3.5$, $\rho = 4.8$ and $m = 7$.
• Figure 2.5: Numerical experiments show that the $\ell^1$ distance behaves like $\mathcal{O}(n^{-1})$, for Toeplitz matrices that have a closed polar curve $p(\mathbb{T})$. In this computation the sampling size for the FFT scales with the matrix dimension. In this fashion, we achieve a robust numerical framework in which both the numerical errors and the analytical errors scale like $\mathcal{O}(n^{-1})$. For more details on this we refer the reader to Appendix \ref{['sec: Numerical analysis']}. Note that is numerically difficult to simulate matrices for sizes larger than $n = 100$, see Figure \ref{['Fig: Eigval conditionner']}.

Theorems & Definitions (26)

• Theorem 2.1: Gohberg
• Theorem 2.2: Schmidt-Spitzer
• Theorem 2.3: Ullman
• Definition 2.4
• Remark 2.6
• Definition 2.7
• Lemma 2.8
• proof
• Lemma 2.9
• proof