The authors investigate here the dynamics of free fermions in a random potential system under continuous monitoring. Repeated measurements are known to induce an entanglement phase transition from a logarithmically entangled critical state to area law, while disorder leads to Anderson localization. They show that their interplay results in a stable critical phase at finite disorder strength and dissipation, which drive a transition consistent with Berezinskii-Kosterlitz-Thouless universality. This work is important for testing such phase transitions in quantum dot arrays and nanowires.
We analytically study interacting Dirac fermions, described by the Thirring model, under weak local particle number measurements with monitoring rate $\ensuremath{\gamma}$. This system maps to a bosonic replica field theory, analyzed via the renormalization group. For a nonzero attractive interaction, a phase transition occurs at a critical measurement strength ${\ensuremath{\gamma}}_{c}$. When $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}$, the system enters a localized phase characterized by exponentially decaying density-density correlations beyond a finite correlation length; for $\ensuremath{\gamma}<{\ensuremath{\gamma}}_{c}$, the correlations decay algebraically. The transition is of Berezinskii-Kosterlitz-Thouless (BKT) type, reflected by a characteristic scaling of the correlation length. In the noninteracting limit, ${\ensuremath{\gamma}}_{c}\ensuremath{\rightarrow}0$ shifts to zero, reducing the algebraic phase to a single point in parameter space. This identifies weak measurements in the free case as an implicit double fine tuning to the critical end point of the BKT phase transition. Along the noninteracting line, we compute the entanglement entropy from density-density correlation functions and find no entanglement transition at nonzero measurement strength in the thermodynamic limit.
We study measurement-induced phases of free fermion systems with U ( 1 ) symmetry. Following a recent approach developed for Majorana chains, we derive a field theory description for the purity and bipartite entanglement at large space and timescales. We focus on a multiflavor one-dimensional chain with random complex hoppings and continuous monitoring of the local fermion density. By means of the replica trick, and using the number of flavors as a large parameter controlling our approximations, we derive an effective field theory made up of a SU(N) nonlinear sigma model ( NL σ M ) coupled to fluctuating hydrodynamics. Contrary to the case of noninteracting Majorana fermions, displaying no U ( 1 ) symmetry, we find that the bipartite entanglement entropy satisfies an area law for all monitoring rates but with a nontrivial scaling of entanglement when the correlation length is large. We provide numerical evidence supporting our claims. We briefly show how imposing a reality condition on the hoppings can change the NL σ M and also discuss higher-dimensional generalizations. Published by the American Physical Society 2024
Monitored quantum circuits exhibit entanglement transitions at certain measurement rates. Such a transition separates phases characterized by how much information an observer can learn from the measurement outcomes. We study SU(2)-symmetric monitored quantum circuits, using exact numerics and a mapping onto an effective statistical-mechanics model. Due to the symmetry's non-Abelian nature, measuring qubit pairs allows for nontrivial entanglement scaling even in the measurement-only limit. We find a transition between a volume-law entangled phase and a critical phase whose diffusive purification dynamics emerge from the non-Abelian symmetry. Additionally, we identify a ``spin-sharpening transition.'' Across the transition, the rate at which measurements reveal information about the total spin quantum number changes parametrically with system size.
On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations (chains up to
L=96 and up to 128 Born-rule trajectories per parameter point), whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation
FA=∑kνk(1−νk) and a participation-style effective number of entangling modes,
MA=exp[−∑kwklnwk], with
wk∝νk(1−νk). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values,
M is nearly a deterministic function of
F: a pooled curve explains 99.7\% of its variance across all rates, so
M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of
∼2 smaller than that of the entropy or
F under strong monitoring. Subsystem-scaling comparisons at
L=48--
96 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus,
F remains the natural experimentally motivated companion to the entropy, whereas
M is best viewed as a variance-reduced numerical summary of essentially the same information.
Global symmetries that define the number of low-energy degrees of freedom have profound consequences on universal properties near topological quantum critical points and in other gapless or nearly gapless states of emergent fermions. We take a
Z2 global symmetry (such as time-reversal) as an example to study its effect on thermodynamic and transport properties. Although the thermal entropy density of
Z2 symmetric systems is simply twice that of their counterparts without any global symmetries or the
Z1 class, the temperature dependence of thermal conductivity $\ensuremath{\kappa}$ is distinctly and drastically different for different symmetries. For systems with dynamic exponent
z=1, in the
Z2 symmetric class, we have $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}(d\ensuremath{-}1)}$ in the quantum critical regime near weakly interacting fixed points, while for systems with no global symmetries (i.e., the
Z1 class), we have $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}(d+3)}$, with
d being the spatial dimension. Only near strong-coupling fixed points, both cases with or without
Z2 global symmetries follow the same scaling function, $\ensuremath{\kappa}\ensuremath{\propto}{T}^{d\ensuremath{-}1}$. These distinct scalings of thermal conductivity can also appear in gapless surface Majorana states.