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arXiv:2605.10459

Renormalization of Quantum Operations: Parity-Time Transition and Chaotic Flows

Atsushi Oyaizu, Hongchao Li, Masaya Nakagawa, Masahito Ueda

cond-mat.stat-mechquant-phSubmitted May 11, 2026• Announced May 12, 2026• 0 citations
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Abstract

The renormalization group (RG) in statistical physics focuses on ground-state properties of equilibrium systems. However, it is unclear how it should be generalized to nonunitary quantum dynamics caused by dissipation and measurement backaction, in which the notion of conserved energy is absent. Here, we extend the RG to cover nonunitary quantum dynamics governed by quantum operations. By performing coarse-graining in real time, we find that the competition between decoherence and coherent dynamics plays a decisive role in the behavior of the RG flow. In particular, we find that chaotic behavior without fixed points emerges in the RG flow when coherent dynamics is dominant, with the parity-time transition serving as a prototypical example. The measurement-induced parity-time transition belongs to the universality class of the one-dimensional Yang-Lee edge singularity, which serves as a guide for experimentally realizing imaginary fields in lattice spin systems with a quantum system.

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17 pages, 4 figures

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The renormalization group (RG) in statistical physics focuses on ground-state properties of equilibrium systems. However, it is unclear how it should be generalized to nonunitary quantum dynamics caused by dissipation and measurement backaction, in which the notion of conserved energy is absent. Here, we extend the RG to cover nonunitary quantum dynamics governed by quantum operations. By performing coarse-graining in real time, we find that the competition between decoherence and coherent dynamics plays a decisive role in the behavior of the RG flow. In particular, we find that chaotic behavior without fixed points emerges in the RG flow when coherent dynamics is dominant, with the parity-time transition serving as a prototypical example. The measurement-induced parity-time transition belongs to the universality class of the one-dimensional Yang-Lee edge singularity, which serves as a guide for experimentally realizing imaginary fields in lattice spin systems with a quantum system.
This paper proposes a quantum renormalization group (RG)-based theoretical framework aimed at cooperatively emerging spatial geometry and temporal causal structurefrom a background-independent quantum gravity ground state. Traditional renormalization methods primarily focus on coarse-graining spatial degrees of freedom, while time istreated as an external parameter. This study introduces spatio-temporal joint renormalization transformations, unifying temporal evolution with spatial scaling behavior, andconstructs RG transformations based on the spatio-temporal tensor network category,where the transformations are implemented by a series of local quantum operations thatpreserve causal structure. The research proves that at specific fixed points of the RGflow, the system spontaneously emerges with a clear arrow of time, manifested as thebreaking of scale invariance of causal correlation functions. Through numerical simulations of a (1+1)-dimensional quantum lattice model, the cooperative scaling behavior ofspacetime structure is verified. This paper provides a concrete dynamical mechanism for“time emerging from quantum processes” and offers a new path for quantum gravity andholographic duality research.
We study renormalization group (RG) flows in far-from-equilibrium states. The study is made tractable by focusing on states that are spatially homogeneous, time independent, and scale invariant. Such states, in which mode k has occupation numbers n k ∼ k − γ , are well known in nonlinear physics. RG flow in such states is qualitatively different from that in the vacuum—a positive γ decreases the dimension of an operator, turning marginal interactions into relevant interactions. We compute one-loop beta functions. Depending on the sign of the beta function, backreaction may either cause a minor shift of the state in the IR or completely change the nature of the state. Focusing on nearly marginal interactions, we construct an analog of the epsilon expansion and IR fixed points, with epsilon now set by the scaling of the interaction rather than the spacetime dimension. In the language of RG flow, critical balance scaling—which has applications in fields as varied as astrophysics and ocean waves—corresponds to the state dynamically adjusting itself along the RG flow until the interaction becomes marginal.
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Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases - Part I: The O(N)O(N)O(N) model
The functional renormalization group (FRG) approach is a powerful tool for studies of a large variety of systems, ranging from statistical physics over the theory of the strong interaction to gravity. The practical application of this approach relies on the derivation of so-called flow equations, which describe the change of the quantum effective action under the variation of a coarse-graining parameter. In the present work, we discuss in detail a novel approach to solve such flow equations. This approach relies on the fact that RG equations can be rewritten such that they exhibit similarities with the conservation laws of fluid dynamics. This observation can be exploited in different ways. First of all, we show that this allows to employ powerful numerical techniques developed in the context of fluid dynamics to solve RG equations. In particular, it allows to reliably treat the emergence of non-analytic behavior in the RG flow of the effective action as it is expected to occur in studies of, e.g., spontaneous symmetry breaking. Second, the analogy between RG equations and fluid dynamics offers the opportunity to gain novel insights into RG flows and their interpretation in general, including the irreversibility of RG flows. We work out this connection in practice by applying it to zero-dimensional quantum-field theoretical models. The generalization to higher-dimensional models is also discussed. Our findings are expected to help improving future FRG studies of quantum field theories in higher dimensions both on a qualitative and quantitative level.
Fliss [.] extended the exact renormalization group (ERG) to arbitrary wave functionals in quantum field theory. Applying this formalism, we show that a Lindblad master equation gives the ERG flow of density matrices. The Lindbladian consists of a “Hamiltonian” term, which is the sum of a scaling and a coarse-graining (disentangling) operator, and a dissipative term with absorption and emission rates for each momentum mode. We consider as examples the flow of Gaussian states and the perturbative ground state of λ ϕ 4 theory and highlight the role of the dissipative terms in generating accurate contributions to the flow of couplings at nontrivial order in λ . Integrating the Lindblad master equation, we find that a finite ERG flow of density matrices is described by a quantum channel. It follows from the data processing inequality that any distinguishability measure of states is an ERG monotone.
Yang-Lee edge singularities (YLES) are the edges of the partition function zeros of an interacting spin model in the space of complex control parameters. They play an important role in understanding non-Hermitian phase transitions in many-body physics, as well as characterizing the corresponding nonunitary criticality. Even though such partition function zeroes have been measured in dynamical experiments where time acts as the imaginary control field, experimentally demonstrating such YLES criticality with a physical imaginary field has remained elusive due to the difficulty of physically realizing non-Hermitian many-body models. We provide a protocol for observing the YLES by detecting kinked dynamical magnetization responses due to broken PT symmetry, thus enabling the physical probing of nonunitary phase transitions in nonequilibrium settings. In particular, scaling analyses based on our nonunitary time evolution circuit with matrix product states accurately recover the exponents uniquely associated with the corresponding nonunitary CFT. We provide an explicit proposal for observing YLES criticality in Floquet quenched Rydberg atomic arrays with laser-induced loss, which paves the way towards a universal platform for simulating non-Hermitian many-body dynamical phenomena.