We introduce a model of effective conformal quantum field theory in dimension d = 1 + 1 coupled to stochastic noise, where Kardar-Parisi-Zhang (KPZ) class fluctuations can be observed. The analysis of the quantum dynamics of the scaling operators reduces to the study of random trajectories in a random environment, modeled by Brownian vector fields. We use recent results on random walks in random environments to calculate the time-dependent entanglement entropy of a subsystem interval, starting from a factorized state. We find that the fluctuations of the entropy in the large deviation regime are governed by the universal Tracy-Widom distribution. This enlarges the KPZ class, previously observed in random circuit models, to a family of interacting many-body quantum systems.
We introduce a model of effective conformal quantum field theory in dimension d = 1 + 1 coupled to stochastic noise, where Kardar-Parisi-Zhang (KPZ) class fluctuations can be observed. The analysis of the quantum dynamics of the scaling operators reduces to the study of random trajectories in a random environment, modeled by Brownian vector fields. We use recent results on random walks in random environments to calculate the time-dependent entanglement entropy of a subsystem interval, starting from a factorized state. We find that the fluctuations of the entropy in the large deviation regime are governed by the universal Tracy-Widom distribution. This enlarges the KPZ class, previously observed in random circuit models, to a family of interacting many-body quantum systems.
Characterizing how entanglement grows with time in a many-body system, for example, after a quantum quench, is a key problem in nonequilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time-dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the “entanglement tsunami” in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar-Parisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like (time)[superscript 1/3] and are spatially correlated over a distance ∝(time)[superscript 2/3]. We derive KPZ universal behavior in three complementary ways, by mapping random entanglement growth to (i) a stochastic model of a growing surface, (ii) a “minimal cut” picture, reminiscent of the Ryu-Takayanagi formula in holography, and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading-order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple coarse grained minimal cut picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the “velocity” of entanglement growth in the 1D entanglement tsunami. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder. Subject Areas: Condensed Matter Physics, Quantum Information, Statistical Physics
We study theoretically entanglement and operator growth in a spin system coupled to an environment, which is modeled with classical dephasing noise. Using exact numerical simulations we show that the entanglement growth and its fluctuations are described by the Kardar-Parisi-Zhang equation. Moreover, we find that the wavefront in the out-of-time ordered correlator (OTOC), which is a measure for the operator growth, propagates linearly with the butterfly velocity, and broadens diffusively with a diffusion constant that is larger than the one of spin transport. The obtained entanglement velocity is smaller than the butterfly velocity for finite noise strength, yet both of them are strongly suppressed by the noise. We calculate perturbatively how the effective timescales depend on the noise strength, both for uncorrelated Markovian and for correlated non-Markovian noise.
This paper provides an in-depth analysis of entanglement entropy (EE) in quantum field theory (QFT), with a particular focus on its computation using the replica trick and its applications to both conformal and non-conformal systems. Beginning with an introduction to the basics of QFT, the study explains how entanglement entropy quantifies the quantum correlations between subsystems in a pure state, represented by the von Neumann entropy of the reduced density matrix. The replica trick is employed to derive the entanglement entropy, involving path integrals over n-sheeted Riemann surfaces. The paper demonstrates the method’s utility in 1+1-dimensional conformal field theory (CFT), where the central charge governs universal properties of entropy. It further extends the analysis to massive field theories, finite systems, and topological phases, exploring how deviations from conformal symmetry affect entanglement. The work also delves into the AdS/CFT correspondence, showcasing how holographic techniques facilitate entanglement entropy calculations in higher-dimensional systems via the Ryu-Takayanagi formula. In addition, the study investigates entanglement entropy in quantum lattice systems, considering the effects of spatial and thermal fluctuations. Numerical methods are used to compute the scaling of entanglement entropy with interval length for varying system masses and temperatures, highlighting the influence of these parameters on entropy behavior. Fractal lattice structures are explored to uncover unique entropy scaling laws and self-similar entanglement patterns. This study advances the under-standing of EE in diverse quantum systems and establishes a foundation for exploring its role in quantum criticality and fractal geometries. The paper also explores multiscale entanglement entropy (MSE), offering a comprehensive framework for future research in quantum critical phenomena and non-equilibrium systems, shedding light on the complex relationship between quantum entanglement and thermal effects. This work serves as a foundational reference for future theoretical and computational studies in quantum entanglement
Recent breakthroughs in many-body quantum dynamics have unexpectedly revealed that the growth of entanglement entropy and quantum decoherence strictly follow the Kardar-Parisi-Zhang (KPZ) universality class, an equation originally formulated to describe macroscopic stochastic surface growth. In this paper, we theoretically bridge this empirical observation with the Dynamic Metric Topology (DMT) framework. We demonstrate that the macroscopic "roughness" governed by the KPZ equation in quantum spin systems is not driven by fundamentally stochastic noise, but by a deterministic topological friction between the spinorial phase and the scalar field pseudo-Riemannian manifold. By explicitly mapping the KPZ non-linear growth term to the DMT scalar impedance threshold at H˜ = 114.12 GeV via the fine structure constant (α), we provide a unified tensorial master equation. This integration redefines quantum noise as an uncompensated metric drag, providing a definitive proof through the deterministic evolution of Bell’s inequality parameters.
We study the growth of entanglement in quantum systems with a conserved quantity exhibiting diffusive transport, focusing on how initial inhomogeneities are imprinted on the entropy. We propose a simple effective model, which generalizes the minimal cut picture of Jonay, Huse, and Nahum [arXiv:1803.00089] in such a way that the ``line tension'' of the cut depends on the local entropy density. In the case of noisy dynamics, this is described by the Kardar-Parisi-Zhang (KPZ) equation coupled to a diffusing field. We investigate the resulting dynamics and find that initial inhomogeneities of the conserved charge give rise to features in the entanglement profile, whose width and height both grow in time as $\ensuremath{\propto}\sqrt{t}$. In particular, for a domain wall quench, diffusion restricts entanglement growth to be ${S}_{\text{vN}}\ensuremath{\lesssim}\sqrt{t}$. We find that for charge density wave initial states, these features in the entanglement profile are present even after the charge density has equilibrated. Our conclusions are supported by numerical results on random circuits and deterministic spin chains.
We give a detailed theory for the leading coarse-grained dynamics of entanglement entropy of states and of operators in generic short-range interacting quantum many-body systems. This includes operators spreading under Heisenberg time evolution, which we find are much less entangled than "typical" operators of the same spatial support. Extending previous conjectures based on random circuit dynamics, we provide evidence that the leading-order entanglement dynamics of a given chaotic system are determined by a function E(v), which is model-dependent, but which we argue satisfies certain general constraints. In a minimal membrane picture, E(v) is the "surface tension" of the membrane and is a function of the membrane's orientation v in spacetime. For one-dimensional (1D) systems this surface tension is related by a Legendre transformation to an entanglement entropy growth rate Γ(∂S/∂x) which depends on the spatial "gradient" of the entanglement entropy S(x,t) across the cut at position x. We show how to extract the entanglement growth functions numerically in 1D at infinite temperature using the concept of the operator entanglement of the time evolution operator, and we discuss possible universality of E at low temperatures. Our theoretical ideas are tested against and informed by numerical results for a quantum-chaotic 1D spin Hamiltonian. These results are relevant to the broad class of chaotic many-particle systems or field theories with spatially local interactions, both in 1D and above.