Symmetry and Topology of Monitored Quantum Dynamics

TL;DR

The paper addresses symmetry and topology in monitored (nonunitary) quantum dynamics of free fermions, establishing a tenfold classification for Kraus operators and non-Hermitian generators in $(d+1)$-dimensional spacetime. It builds a framework connecting spacetime symmetries (time-reversal, particle-hole, and chiral) to classifying spaces, and shows how topology governs measurement-induced phase transitions through topological terms in nonlinear sigma models. A bulk-boundary correspondence is demonstrated: nontrivial spacetime topology yields topological steady states and anomalous Lyapunov-boundary modes, such as Lyapunov zero modes and chiral edge modes, leading to a topologically protected slowdown of dynamical purification. The work provides an open-quantum analogue of the periodic table for monitored systems, offers concrete 1+1 and 2+1 dimensional examples, and lays groundwork for incorporating interactions and exploring experimental implications.

Abstract

The interplay between unitary dynamics and quantum measurements induces diverse phenomena in open quantum systems with no counterparts in closed quantum systems at equilibrium. Here, we generally classify Kraus operators and their effective non-Hermitian dynamical generators, thereby establishing the tenfold classification for symmetry and topology of monitored free fermions. Our classification elucidates the role of topology in measurement-induced phase transitions and identifies potential topological terms in the corresponding nonlinear sigma models. Furthermore, we establish the bulk-boundary correspondence in monitored quantum dynamics: nontrivial topology in spacetime manifests itself as topologically nontrivial steady states and gapless boundary states in Lyapunov spectra, such as Lyapunov zero modes and chiral edge modes, leading to the topologically protected slowdown of dynamical purification.

Figures (2)

• Figure 1: (a) Monitored dynamics of a Majorana chain generated by repeating the operations inside the black (blue) dashed lines for class BDI (D). (b) Monitored dynamics of the double Majorana chain generated by the operations in subfigure (a) on individual chains and the unitary gates coupling the two chains. (c) $\mathbb{Z}$ and (d) $\mathbb{Z}_2$ topological markers for the steady states in classes BDI and D, respectively. The fluctuations are due to the spatial randomness in $K_{[0, t]}$. (e), (f) Lyapunov spectra of the monitored single (left) and double (right) Majorana chains in classes (e) BDI and (f) D. Insets: smallest (or two smallest) positive Lyapunov exponent(s) as a function of the system size. Due to particle-hole symmetry, Lyapunov exponents appear in opposite-sign pairs $(\eta_i, -\eta_i)$ and shown only for $\eta_i \geq 0$. Details of the models can be found in Ref. supplemental.
• Figure 2: (a) $\left( 2+1 \right)$-dimensional quantum circuit on a lattice of size $L_x \times L_y$. Each site $\bm r$ incorporates one fermion $c_{\bm r}^{\dag}$. At each time step, measurements and unitary gates are applied to the bonds of a specific color, based on the sequence shown at the bottom. (b) Lyapunov spectra of the dynamics with the homogeneous measurement strength under periodic boundary conditions (PBC) along the $x$ direction, and PBC or open boundary conditions (OBC) along the $y$ direction. (c), (d) Monitored dynamics with inhomogeneous measurement strength: (c) Local Chern marker of the steady state. (d) Lyapunov spectra under PBC and OBC. Details of the models can be found in Ref. supplemental.