Field Theory of Charge Sharpening in Symmetric Monitored Quantum Circuits

Monitored quantum circuits (MRCs) exhibit a measurement-induced phase transition between area-law and volume-law entanglement scaling. MRCs with a conserved charge additionally exhibit two distinct volume-law entangled phases that cannot be characterized by equilibrium notions of symmetry-breaking or topological order, but rather by the nonequilibrium dynamics and steady-state distribution of charge fluctuations. These include a charge-fuzzy phase in which charge information is rapidly scrambled leading to slowly decaying spatial fluctuations of charge in the steady state, and a charge-sharp phase in which measurements collapse quantum fluctuations of charge without destroying the volume-law entanglement of neutral degrees of freedom. By taking a continuous-time, weak-measurement limit, we construct a controlled replica field theory description of these phases and their intervening charge-sharpening transition in one spatial dimension. We find that the charge fuzzy phase is a critical phase with continuously evolving critical exponents that terminates in a modified Kosterlitz-Thouless transition to the short-range correlated charge-sharp phase. We numerically corroborate these scaling predictions also hold for discrete-time projective-measurement circuit models using large-scale matrix-product state simulations, and discuss generalizations to higher dimensions.

Figure 1

Model and phase diagram.—(a) Monitored random circuit (MRC) with charged qubits (solid lines) and neutral large- $d$ qudits (dashed lines) interacting with Haar-random gates (blue boxes) and randomly placed measurements (red dots). (b) Statistical mechanics (stat-mech) model for replicated MRC consists of replica permutation “spins” (arrows) interacting with random-walking charge world lines (orange lines). (c) The phase diagram, with phases labeled in the MRC (bottom) and stat-mech (top) language, respectively. In addition to the entanglement transition at $p_c$, which corresponds to a (dis)ordering transition of the permutation “spins,” there is a charge-sharpening transition in the volume law phase, corresponding to an interreplica superfluid (IRSF) to Mott-insulator (MI) transition in the statistical mechanics language. In $1+1d$, this charge sharpening transition has a modified Kosterlitz-Thouless universality class, and the IRSF is a critical Goldstone phase with scaling exponents that vary continuously with $p$.

Figure 2

TEBD data.—(a) Charge fluctuations $C_z(x)=\mathbb{E}[⟨{\sigma }_x^z{\sigma }_0^z⟩-⟨{\sigma }_x^z⟩⟨{\sigma }_0^z⟩]$, scaling as $C_z(x)\sim x^{-\alpha }$ in the fuzzy phase. Inset: charge variance of an interval of size $x$, ${\mathrm{Var}}_q(x)={\sum }_{0<i,j<x}\mathbb{E}[⟨{\sigma }_i^z{\sigma }_j^z{⟩}_c]$, predicted to scale as $\sim (8{\rho }_s/\pi )\log x$. (b) Dual string disorder parameter $C_W(x)=\mathbb{E}[⟨W_{[0,x]}{⟩}^2]$ with $W_{[0,x]}={\prod }_{0<i<x}{\sigma }_i^z$, showing power-law decay $C_W(x)\sim x^{-2\pi {\rho }_s}$ in the charge-fuzzy phase. (c) Continuously evolving superfluid density ${\rho }_s$ as a function of $p$, extracted from the local charge variance (blue) and the dual correlator $C_W(x)$ (orange). The dashed horizontal line indicates the critical threshold ${({\rho }_s)}_{\#}={\pi }^{-1}$. For $p<p_{\#}\sim 0.2$, the charge correlator $C_z(x)$ decays with an exponent $\alpha =2$ (green).