The measurement-induced phase transition in strongly disordered spin chains

Abstract

We investigate the dynamics of strongly disordered spin chains in the presence of random local measurements. By studying the transverse-field Ising model with a site-dependent random longitudinal field and an effective $l$-bit many-body localized Hamiltonian, we show that the prethermal and MBL regimes are unstable to local measurements along any direction. Any non-zero measurement density induces a volume-law entangled phase with a subsequent phase transition into an area-law state as the measurement rate is further increased. The critical measurement rate $p_c$, where the transition occurs, is exponentially small in the strength of disorder $W$ and the average overlap between the measurement operator and the local integrals of motion $O$ as $p_c \sim \exp[-αW/(1-O^2)]$. In the measurement-induced volume-law phase, the saturation time scales as $t_s \sim L $, contrasting the exponentially slow saturation $t_s \sim e^{aL}$ in the prethermal and MBL regimes at $p = 0$.

Figures (8)

• Figure 1: (a) The top panel is a schematic of the typical support of an $l$-bit operator, localized at site $i$ in the microscopic model, with each qubit measured randomly with $M_i(\theta)$ in the $x-z$ plane. The bottom panel shows a schematic of the $l$-bit operator in the $l$-bit model, with each qubit measured randomly in the $x$ direction. (b) Illustration of measurement protocol with unitary Hamiltonian dynamics $U(\Delta t)$ followed by random measurements $M_i(\theta)$ with rate $p$. (c) Illustration of the phase diagram for the MBL system with measurement in the space of disorder strength $W$ and measurement rate $p_{\theta}$ at angle $\theta$. For any $\theta$, the critical measurement rate $p_c\sim \exp\{- \frac{\alpha W}{1-O^2}\}$ for large $W$ labels the boundary between volume and area-law entangled phases with $O$ being the overlap between the measurement operator (defined eplicitly for in Eq. \ref{['Overlap']}).
• Figure 2: Tripartite mutual information $I_3(t)$ for the evolution under Hamiltonian \ref{['Hmbl']} and random measurements for measurement angle $\theta = \frac{\pi}{2}$ and various system sizes $L$. The disorder strength is $W=5$ and the measurement probability is $p=0.01$ in (a) and and $p=0.09$ in (b) corresponding to the volume-law and the area-law entangled phases, respectively. The inset in (b) depicts the geometry used to compute $I_3(t)$. (c) Tripartite mutual information in the steady state $I_3^s$ as a function of $p$ with a critical value $p_c\simeq 0.039$. The inset shows the data collapse for the microscopic model using the critical exponent $\nu\simeq 1.6$ extrapolated from the $l$-bit model.
• Figure 3: (a) The $W$- and $\theta$-dependence of the critical measurement rate $p_c(W,\theta)$ defines the phase boundary between volume-law entanglement, $p<p_c$, and area-law entanglement, $p>p_c$, for the model \ref{['Hmbl']}. The inset shows the data collapse onto the curve given by Eq. \ref{['eq_pc']}. The hollow markers indicate data points where $p_c$ shows a significant finite-size effect. (b) Overlaps between the local integral of motion and the measurement operator $O(\theta,W)$ as defined in Eq. \ref{['Overlap']} for Hamiltonian \ref{['Hmbl']}
• Figure 4: Entanglement dynamics for the $l$-bit model \ref{['Hlbit']}. (a) Plots of the steady state mutual information $I_3^s$ vs. the measurement rate $p$ for different system sizes $L$ crossing at $p=p_c$. The inset shows the data collapse with the scaling $(p-p_c)L^{1/\nu}$ as defined in Eq. \ref{['Eq.collapse']}, where $\nu\simeq 1.6$. (b) Saturation time $t_s$ ($t_s/L$ in the inset) as a function of $p$. The $t_s/L$ plots collapse consistent with $t_s\sim\mathcal{O}(L)$ when $p<p_c=0.18$ and the system is in the volume-law entanglement phase, while the $t_s$ plots collapse for $p>p_c$ when the system is in the area-law entangled phase where $t_s\sim\mathcal{O}(1)$. Legend is shared with (a). (c) Half-cut entanglement entropy for different measurement rates $p$ demonstrating that the entanglement entropy can grow faster with measurements than without.
• Figure 5: We evolve the state $\ket{\psi_0}=\otimes_{i=1}^L \ket{\uparrow}$ with the Hamiltonian \ref{['Hmbl']} for system size $L=8$ and disorder strength $W=5$. (a) Norm of $\ket{\psi_\text{KPM}}$ as a function of time for different truncation orders $\Lambda$. (b) Fidelity of the KPM wavefunction $\ket{\psi_\text{KPM}}$ for different $\Lambda$ as compared with the exact time evolved state $\ket{\psi_\text{ex}}$.