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arXiv:2501.17606 · Physical Review A

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

Paranjoy Chaki, Protyush Nandi, Ujjwal Sen, Subinay Dasgupta

Published Mar 9, 2026• 1 citation
arXivHTMLPDF

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 $τ$, 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 $τ$ 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 $τ_c \sim 0.2$ 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 $τ_c \sim 1/\sqrt{L}$. The transition at finite \(τ_c\) for \(L \sim 28\) seems therefore to recede to \(τ_c = 0\) 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.

DOI
10.1103/4mjh-767c

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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 τττ, followed by a measurement that provides a yes/no'' answer to the question, Are all spins up?''. The survival probability after nnn time-steps is defined as the probability that the answer is ``no'' in all the nnn time-steps. For various τττ values, we compute the survival probability, entanglement in bipartition, and the generalized geometric measure, a genuine multiparty entanglement, for a chain of size L∼26L \sim 26L∼26, and identify a transition at τc∼0.2τ_c \sim 0.2τc​∼0.2 for field strength h=1/2h=1/2h=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 τc∼1/Lτ_c \sim 1/\sqrt{L}τc​∼1/L​. The transition at finite (τ_c) for (L \sim 28) seems therefore to recede to (τ_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.
We find that the measurement-induced phase transition generated by deterministic global measurements, previously observed in the integrable transverse-field Ising model (TFIM), persists in non-integrable variants of the same. To address this question, we consider the TFIM with longitudinal field and the axial next-nearest-neighbor Ising (ANNNI) model. We show that both the survival probability and the bipartite entanglement entropy consistently capture a transition between area-law and volume-law entangled phases for two distinct initial states: a product state with all spins polarized along the transverse direction and a Greenberger-Horne-Zeilinger (GHZ) state. Finite-size scaling reveals a pronounced initial state dependence: for the polarized product state, the transition point follows an inverse-square-root scaling with system size in both non-integrable models, consistent with the integrable TFIM, whereas for the GHZ initial state, it deviates from this scaling and approaches zero considerably more slowly in the non-integrable models than in the integrable TFIM. These results establish the robustness of measurement-induced transitions under deterministic measurements against integrability breaking while highlighting the crucial role of the initial state in governing their scaling behavior.
Non-equilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce different non-equilibrium regimes and an abrupt change in the scaling-law of the bipartite entanglement entropy. However, our understanding of how these regimes appear{, how they affect the statistics of local quantities and}, finally whether they survive in the thermodynamic limit, is much less established. Here we investigate measurement-induced phase transitions in the Quantum Ising chain coupled to a monitoring environment. In particular we show that local projective measurements induce a quantitative modification of the out-of-equilibrium probability distribution function of the local magnetization. Starting from a GHZ state, the relaxation of the paramagnetic and the ferromagnetic order is analysed. In particular we describe how the probability distributions associated to them show different behaviour depending on the measurement rate.
Suppose a quantum system starts to evolve under a Hamiltonian from some initial state. When, for the first time, will an observable attain a preassigned value? To answer this question, one method often adopted is to make instantaneous measurements periodically and note down the serial number for which the desired result is obtained for the first time. We apply this protocol to an interacting spin system at zero temperature and show analytically that the response of this system shows a nonanalyticity as a function of the parameter of the Hamiltonian and the time interval of measurement. In contrast to quantum phase transitions, this type of phase transition is not a property of the ground state and arises from the Hamiltonian dynamics and quantum-mechanical nature of the measurement. The specific system studied is the transverse Ising chain, and the measurement performed is whether the total transverse magnetic moment (per site) is not equal to 1. The results for some other types of measurement are also discussed.
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 lll-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 pcp_cpc​, where the transition occurs, is exponentially small in the strength of disorder WWW and the average overlap between the measurement operator and the local integrals of motion OOO as pc∼exp⁡[−αW/(1−O2)]p_c \sim \exp[-αW/(1-O^2)]pc​∼exp[−αW/(1−O2)]. In the measurement-induced volume-law phase, the saturation time scales as $t_s \sim L $, contrasting the exponentially slow saturation ts∼eaLt_s \sim e^{aL}ts​∼eaL in the prethermal and MBL regimes at p=0p = 0p=0.
Nonequilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce an abrupt change in the scaling law of the bipartite entanglement entropy, thus suggesting the existence of different nonequilibrium regimes. However, our understanding of how these regimes appear and whether they survive in the thermodynamic limit is much less established. Here we investigate these questions on a one-dimensional quadratic fermionic model: this allows us to reach system sizes relevant in the thermodynamic sense. We show that local projective measurements induce a qualitative modification of the time growth of the entanglement entropy which changes from linear to logarithmic. However, in the stationary regime, the logarithmic behavior of the entanglement entropy does not survive in the thermodynamic limit and, for any finite value of the measurement rate, we numerically show the existence of a single area-law phase for the entanglement entropy. Finally, exploiting the quasiparticle picture, we further support our results by analyzing the fluctuations of the stationary entanglement entropy and its scaling behavior.
Measurement-induced phase transitions (MIPT) give rise to novel dynamical states of quantum matter realized by balancing unitary evolution and measurements. We present large-scale numerical simulations of a trapped-ion native MIPT, argued to belong to the universality class described by the Haar non-unitary conformal field theory. First, through a finite-size analysis we obtained the critical measurement rate, and correlation length exponent, which falls close to the percolation value. Second, by leveraging a monotone computable via semi-definite programming, we uncover robust algebraic decay of genuine multiparty entanglement (GME) versus separation for 2, 3, and 4 parties. The corresponding critical exponents are lower-bounded by those of the multiparty mutual information, which we determine up to 4 parties, and conjecture to be (k+2) for k parties. Additionally, we derive lower bounds for both GME and multiparty mutual information.