Disappearance of measurement-induced phase transition in a quantum spin system for large sizes

TL;DR

This study investigates measurement-induced phase transitions in a quantum spin system with global measurements, revealing that the transition diminishes as system size increases, suggesting it vanishes in the thermodynamic limit.

Contribution

It introduces a model with global measurements at each step, derives a recursion relation for survival probability, and shows the transition scales down with system size, indicating no finite transition in large systems.

Findings

Transition at finite  observed for small sizes, but scales as   with system size.

Survival probability decays logarithmically only when the ground state is paramagnetic.

Transition recedes to zero in the thermodynamic limit, implying no true phase transition for large systems.

Abstract

Measurement-induced phase transitions are often studied in random quantum circuits, with local measurements performed with a certain probability. We present here a model where a global measurement is performed with certainty at every time-step of the measurement protocol. Each time step, therefore, consists of evolution under the transverse Ising Hamiltonian for a time $\tau$, followed by a measurement that provides a ``yes/no'' answer to the question, ``Are all spins up?''. The survival probability after $n$ time-steps is defined as the probability that the answer is ``no'' in all the $n$ time-steps. For various $\tau$ values, we compute the survival probability, entanglement in bipartition, and the generalized geometric measure, a genuine multiparty entanglement, for a chain of size $L \sim 26$, and identify a transition at $\tau_c \sim 0.2$ for field strength $h=1/2$. We then analytically derive a recursion relation that enables us to calculate the survival probability for system sizes up to 1000, which provides evidence of a scaling $\tau_c \sim 1/\sqrt{L}$. The transition at finite \(\tau_c\) for \(L \sim 28\) seems therefore to recede to \(\tau_c = 0\) in the thermodynamic limit. Additionally, at large time-steps, survival probability decays logarithmically only when the ground state of the Hamiltonian is paramagnetic. Such decay is not present when the ground state is ferromagnetic.