Ruelle-Pollicott resonances of diffusive U(1)-invariant qubit circuits

Abstract

We study Ruelle-Pollicott resonances of translationally invariant magnetization-conserving qubit circuits via the spectrum of the quasi-momentum-resolved truncated propagator of extensive observables. Diffusive transport of the conserved magnetization is reflected in the Gaussian quasi-momentum $k$ dependence of the leading eigenvalue (Ruelle-Pollicott resonance) of the truncated propagator for small $k$. This, in particular, allows us to extract the diffusion constant. For large $k$, the leading Ruelle-Pollicott resonance is not related to transport and governs the exponential decay of correlation functions. Additionally, we conjecture the existence of a continuum of eigenvalues below the leading diffusive resonance, which governs non-exponential decay, for instance, power-law hydrodynamic tails. We expect our conclusions to hold for generic systems with exactly one U(1) conserved quantity.

Figures (12)

• Figure 1: Diagram of the $k$-dependent RP resonance spectrum in homogeneous systems without (a), and with one conserved quantity (b). Labels in figure (b) show which parts of the spectrum are responsible for which physical behavior of the system. The shaded region in figure (b) denotes the location of conjectured continuums of RP resonances, which we cannot observe numerically with the methods used. We show only positive $k$ since $|\lambda(k)| = \left\lvert\lambda(-k)\right\rvert$, for details see Sec. \ref{['sec:trunc']}.
• Figure 2: Circuits with no conserved quantities. (a) Diagram of operator spreading in a brickwall circuit with the same gate $V$ acting between all nearest neighbors (the shown circuit has $s = 2$, $\delta r=2$). (b) The leading eigenvalue of the propagator truncated to the space of extensive observables with support $r$ for two realizations of a circuit with a different gate $V$ chosen randomly according to the unitary Haar measure. (c) Infinite temperature autocorrelation functions of extensive observables $A$\ref{['eq:a_ext_simple']} in the circuit with gate i from (b). The local density $a$ is either $\sigma^z$, or chosen randomly according to the Gaussian unitary ensemble (GUE) mehtaRandomMatrices2004 with support $r = 2$. Dashed lines show the RP prediction $\propto \left\lvert\lambda_1\right\rvert^t$, while full curves are an exact calculation in a circuit with $N = 32$ qubits; the $\sigma^z, k = 0$ case is multiplied by $15$ for better presentation.
• Figure 3: Circuits with conserved magnetization. (a) Circuit diagram with magnetization-conserving 3-qubit gate $V$ (the shown circuit has $s = 3$ and $\delta r=6$), whose block structure is shown on the left. (b) The leading RP eigenvalue of the truncated propagator with support $r$ for 3 choices of $V$ (chosen according to the Haar measure). The same gates will be used throughout the paper.
• Figure 4: Transport of magnetization in the circuit with gate 1 (a) and gate 2 (b). In both instances we show the leading RP resonance ($r=11$) and the fit to ${\rm e}^{-Dk^z}$ (insets i), and the domain-wall evolution (TEBD, $N=354$ with $\mu = 10^{-3}$ and $\chi = 256$) at times with equal spacing $\Delta t = 55$ up to $t = 500$ (main plots) with the associated transferred magnetization and the diffusion constant fit (insets ii).
• Figure 5: Leading RP resonance and correlation function decay in the $k = \pi$ sector for magnetization-conserving circuits. (a) The whole spectrum of the truncated propagator in $k = 0, \pi$ sectors for the circuit with gate 3 and $r = 7$. (b) Convergence of the leading RP resonances in the $k = \pi$ sector with $r$, with dashed lines denoting an extrapolation to the $r \to \infty$ limit. (c) Infinite temperature autocorrelation functions of extensive observables $A$ defined in Eq. \ref{['eq:a_ext']} with $k = \pi$. The local densities are either $\sigma^x$ (i.e., $x$ magnetization) or chosen randomly according to the Gaussian unitary ensemble (GUE) mehtaRandomMatrices2004 with support $r = 2$. The correlation functions are calculated in a finite circuit with $N = 30$ sites. Dashed lines show the RP predictions $\propto \left\lvert\lambda_1\right\rvert^t$.