Measurement-Induced Dark State Phase Transitions in Long-Ranged Fermion Systems

We identify an unconventional algebraic scaling phase in the quantum dynamics of long-range hopping, free fermions, which are exposed to continuous local measurements. The algebraic phase occurs for hopping decay exponents $1<p\lesssim 3/2$, and features an algebraic entanglement entropy growth, and a slow algebraic decay of the density-density correlation function, both with a fractional exponent. It is separated from a critical phase with logarithmic entanglement growth at small, and an area law phase with constant entanglement entropy at large monitoring rates. A perturbative renormalization group analysis predicts that the transitions to the long-range phase correspond to an unconventional, modified sine-Gordon theory. Exact numerical simulations of the monitored wave functions are in excellent agreement with an analytical replica field theory approach, which confirms the view of the measurement-induced phase transition as a quantum phase transition in the dark state of an effective, non-Hermitian Hamiltonian.

Figure 1

(a) Free fermions on a chain with algebraic hopping ($1<p<\infty$) and continuous monitoring with rate $\gamma$. (b) Schematic phase diagram: we detect and characterize three qualitatively different phases. (c) Phase diagram as determined from the effective central charge $c$ for $L=600$. Circles indicate the points discussed in Fig. 2. The color-coded vertical lines are studied in (d),(e): comparison of the numerically determined scaling exponents at $\gamma =0.3$ (blue), $\gamma =1$ (black) and $\gamma =1.5$ (red) with the dark state (solid black line). (d) Correlation function exponent $a$. Dashed lines yield an estimate for the point where the algebraic scaling terminates and is replaced by exponential scaling. (e) Entanglement entropy growth exponent $b$. A vanishing exponent corresponds to logarithmic scaling or area law. The critical points found by this procedure are indicated as crosses in (c).

Figure 2

Phase characterization of conformally invariant ($\gamma =0.3$, $p=1.25$, orange), area-law ($\gamma =2$, $p=5$, light blue), and algebraic scaling phase ($\gamma =0.3$, $p=5$, purple). (a) Half-system entanglement entropy and best fit using the ansatz $S=(c/3)\log (L/\pi )+s_0+B{L}^b$. The entropy always grows slower than a volume law ($\sim L$, dashed line). (b) The scaling exponent of the correlation function at opposite ends of the system is determined by fitting to $C=1/[A{L}^a+DL]$ (dashed line $\sim L^{-2}$).

Figure 3

(a) Effective central charge $c$ determined from a fit at $l\in [L/4,3L/4]$ for system sizes $L=200$, 400, 600, 800. For $1/p\lesssim 0.6$, $c$ saturates, while it diverges with $L$ for $1>1/p\gtrsim 0.6$, indicating faster than log entanglement growth. (b) Entanglement entropy and density-density correlation function deep in the algebraic scaling phase ($\gamma =0.3$, $p=1.25$) for $L=1600$. At intermediate distances, where finite-size effects are negligible, the algebraic scaling matches the analytical prediction $b=2-a$.