Topological Modes in Monitored Quantum Dynamics

Abstract

Dynamical quantum systems both driven by unitary evolutions and monitored through measurements have proved to be fertile ground for exploring new dynamical quantum matters. While the entanglement structure and symmetry properties of monitored systems have been intensively studied, the role of topology in monitored dynamics is much less explored. In this work, we investigate novel topological phenomena in the monitored dynamics through the lens of free-fermion systems. Free-fermion monitored dynamics were previously shown to be unified with the Anderson localization problem under the Altland-Zirnbauer symmetry classification. Guided by this unification, we identify the topological area-law-entangled phases in the former setting through the topological classification of disordered insulators and superconductors in the latter. As examples, we focus on 1+1D free-fermion monitored dynamics in two symmetry classes, DIII and A. We construct quantum circuit models to study different topological area-law phases and their domain walls in the respective symmetry classes. We find that the domain wall between topologically distinct area-law phases hosts dynamical topological modes whose entanglement is protected from being quenched by the measurements in the monitored dynamics. We demonstrate how to manipulate these topological modes by programming the domain-wall dynamics. In particular, for topological modes in class DIII, which behave as unmeasured Majorana modes, we devise a protocol to braid them and study the entanglement generated in the braiding process.

Figures (7)

• Figure 1: (a) Spacetime geometry of a symmetry-class-DIII monitored circuit on a Majorana chain of $L$ sites (black dots). This circuit has a staggered pattern with different measurement probabilities $p_{\rm odd/even}$ in different two-site gates. (b) Average steady-state mutual information $\overline{I_{A,B}}$ as a function of $p\equiv p_{\text{odd}}=1-p_{\text{even}}$ in the class-DIII monitored circuit shows two distinct area-law phases, separated by a critical phase. Inset shows the configuration of two antipodal regions, $A$ and $B$, on a ring geometry to compute $\overline{I_{A,B}}$.
• Figure 2: (a) Spacetime configuration $p(i,t)$ in a class-DIII monitored circuit acting on a 128-site Majorana chain. (b-c) Entanglement contour and total entanglement entropy (in binary logarithm) of the physical chain in a typical quantum trajectory. The peaks in the entanglement contour indicate the spacetime position of the dynamical topological domain-wall modes. (d) Schematics for the "ideal" entanglement structure of physical and reference systems at different times $t$. Each dot presents a Majorana mode, with the DTDMs colored red. The colored strip behind physical sites represents the configuration of $p(i,t)$ shown in (a). Each pair of modes connected by a wavy line is maximally entangled.
• Figure 3: Top: Schematics for the four stages of the DTDM braiding protocol, where $\alpha_1,\alpha_2$ and $\beta_1,\beta_2$ represent the DTDMs with finite correlation length. We numerically simulate the protocol for a T-junction of three 64-site chains. Bottom: Mutual information between the pair of chains $(\mathfrak{A}, \mathfrak{B})$, $(\mathfrak{B},\mathfrak{C})$, and $(\mathfrak{C},\mathfrak{A})$ in a typical quantum trajectory as a function of time $t$.
• Figure 4: (a) Spacetime geometry of a class-A monitored circuit on a 1D complex fermion chain with each unit cell (dashed box) containing two sublattices $\mathsf{A}$ and $\mathsf{B}$. Two-site (post-selected) measurements (purple and orange gates) and onsite unitary gates (gray gates) are defined in the main text. (b) Average steady-state mutual information $\overline{I_{A,B}}$ as a function of $\alpha_0$ with $\tanh^2{\alpha_0}+\tanh^2\alpha_1=1$. (c) Spacetime configuration $r(i,t)$ for a class-A monitored circuit on a 128-site chain, encoding DWs of different integer classes as labeled. (d-e) Entanglement contour and total entanglement entropy (in binary logarithm) of the physical chain in a typical quantum trajectory.
• Figure S1: Average steady-state mutual information $\overline{I_{A,B}}$ as a function of the measurement strength $\alpha_0$ in class-AIII monitored circuit, showing only one area-law phase.