Boom and bust cycles due to pseudospectra of matrices with unimodular spectra

TL;DR

The paper investigates dynamics of $f(t)=\langle w|A^t|v\rangle$ for non-normal matrices with unimodular spectra ($A^T=\mathbb{I}$), showing that $|f(t)|$ can grow exponentially due to pseudospectrum effects. Using an exactly solvable block-structured matrix, it shows the initial growth rate equals the largest pseudospectrum value $\lambda_{\mathrm{ps}}$, producing boom and bust within a period $T$; this is complemented by two additional examples: the Ehrenfest-urn transfer matrix with $A=e^{i\alpha H}$, and purity dynamics in staircase random quantum circuits, where large, non-normal pseudospectral features govern transient behavior. The results emphasize that the limit order between matrix size $N$ and time $t$ dramatically changes observed dynamics and that pseudospectra provide a robust mechanism for transient growth beyond what unimodular eigenvalues would suggest. The work thus connects transfer-matrix physics, quantum-circuit purity dynamics, and non-Hermitian spectral theory, with potential implications for open-system dynamics and other non-normal settings.

Abstract

We discuss dynamics obtained by increasing powers of non-normal matrices that are roots of the identity, and therefore have all eigenvalues on the unit circle. Naively, one would expect that the expectation value of such powers cannot grow as one increases the power. We demonstrate that, rather counterintuitively, a completely opposite behavior is possible. In the limit of infinitely large matrices one can have an exponential growth. For finite matrices this exponential growth is a part of repeating cycles of exponential growths followed by exponential decays. The effect can occur if the spectrum is different than the pseudospectrum, with the exponential growth rate being given by the pseudospectrum. We show that this effect appears in a class of transfer matrices appearing in studies of two-dimensional non-interacting systems, for a matrix describing the Ehrenfest urn, as well as in previously observed purity dynamics in a staircase random circuit.

Figures (5)

• Figure 1: Periodic exponential growth and exponential decay of $|f(t)|$, Eq. \ref{['eq:f']}. For $g\neq1$, we see a periodic behavior with exponential growth followed by an exponential decay within each period. $A$ is a $400\times 400$ matrix given by Eq. \ref{['eq:T']}. All eigenvalues of $A$ are unimodular, $|\lambda_j|=1$, while the largest value in the pseudospectrum is $\lambda_{\mathrm{ps}}=g$, see Sec. \ref{['sec:simple']} for more details. ${\langle w |}$ and ${| v \rangle}$ are normalized random vectors with real components.
• Figure 2: Time evolution of $f(t)$ (Eq. \ref{['eq:f']}) for special choices of the vectors ${| v \rangle}$ and ${\langle w |}$ with components $w_p=1$ and $v_p=\delta_{1,p}$, $p = 1,2,\dots,N$. The plot shows results for $g=2$ and two different matrix sizes $N=100$ and $N=200$. The behavior in this figure is similar to the behavior from Fig. \ref{['fig:example']} for random ${\langle w |}$ and ${| v \rangle}$. The exponential growth is given by $g^t$, where $g$ is the maximum value in the pseudospectrum of $A$. The exponential decay is instead given by $(1/g)^t$.
• Figure 3: The spectrum and pseudospectrum of the matrices $A$ (frame (a)) and $B$ (frame (b)), with $g=2$ and in the limit of infinite matrix size.
• Figure 4: Dynamics for the Ehrenfest urn matrix from Eq. \ref{['eq:H']}. Frame (a) shows periodic exponential growth and exponential decay of $f(t)$. The vectors ${| v \rangle}$ and ${\langle w |}$ are chosen to be normalized random vectors with real components. Frame (b) shows how $f(t)$ initially behaves for large system sizes. The initial behavior of $f(t)$ converges to $\mathrm{e}^{\pi t/2}$ as we increase the system size. Note that $\mathrm{e}^{\pi/2}$ corresponds to the largest value in the pseudospectrum of $A$, shown as a shaded area in the inline plot in frame (b).
• Figure 5: Time evolution of purity $I(t)$ in a staircase open boundary condition random quantum circuit with iSWAP gates. Frame (a) shows how purity of a qubit chain of $26$ and $20$ sites decays exponentially towards its asymptotic value $I(\infty)$. Frame (b) shows the spectrum of the transfer matrix $Z$ used to propagate $I(t)-I(\infty)$ for a chain of $10$ qubits. The dashed circle marks all eigenvalues with modulus $1/4$. Frame (c) shows the propagation of $I(t)-I(\infty)$ by the rescaled transfer matrix $\lambda_1^{-1} \cdot Z$. In this case, purity shows cycles of boom and bust similar to $f(t)$.