A wave function subject to unitary time evolution and exposed to measurements undergoes pure state dynamics, with deterministic unitary and probabilistic measurement-induced state updates, defining a quantum trajectory. For many-particle systems, the competition of these different elements of dynamics can give rise to a scenario similar to quantum phase transitions. To access this competition despite the randomness of single quantum trajectories, we construct an $n$-replica Keldysh field theory for the ensemble average of the $n$th moment of the trajectory projector. A key finding is that this field theory decouples into one set of degrees of freedom that heats up indefinitely, while $n-1$ others can be cast into the form of pure state evolutions generated by an effective non-Hermitian Hamiltonian. This decoupling is exact for free theories, and useful for interacting ones. In particular, we study locally measured Dirac fermions in ($1+1$) dimensions, which can be bosonized to a monitored interacting Luttinger liquid at long wavelengths. For this model, the non-Hermitian Hamiltonian corresponds to a quantum sine-Gordon model with complex coefficients. A renormalization group analysis reveals a gapless critical phase with logarithmic entanglement entropy growth, and a gapped area law phase, separated by a Berezinskii-Kosterlitz-Thouless transition. The physical picture emerging here is a measurement-induced pinning of the trajectory wave function into eigenstates of the measurement operators, which succeeds upon increasing the monitoring rate across a critical threshold.

中文翻译：

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狄拉克费米子测量诱导相变的有效理论

受到幺正时间演化并暴露于测量的波函数经历纯状态动力学，具有确定性的幺正和概率测量引起的状态更新，定义了量子轨迹。对于多粒子系统，这些不同动力学元素的竞争可以产生类似于量子相变的场景。尽管单量子轨迹具有随机性，但为了参加这种竞争，我们构建了一个$n$- 集合平均的复制 Keldysh 场理论 $n$轨迹投影仪的时刻。一个关键发现是，该场论解耦为一组无限加热的自由度，而$n-1$其他的可以转化为由有效的非厄米哈密顿量生成的纯状态演化的形式。这种解耦对于自由理论是准确的，对于相互作用的理论很有用。特别是，我们研究了 ($1+1$) 维度，可以将其玻色化为受监测的长波长相互作用的 Luttinger 液体。对于该模型，非厄米哈密顿量对应于具有复系数的量子正弦-戈登模型。重整化组分析揭示了具有对数纠缠熵增长的无间隙临界相和有间隙的面积定律相，由 Berezinskii-Kosterlitz-Thouless 转换分隔。这里出现的物理图像是测量引起的轨迹波函数钉扎到测量算子的本征态中，这成功地将监测率提高到临界阈值。