Dynamic Scaling of Growing Interfaces

Kardar, Mehran; Parisi, Giorgio; Zhang, Yi Cheng

doi:10.1103/physrevlett.56.889

Cited by 5,211 publications

(4,929 citation statements)

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“…Since the system is traversed in time t = L/v the integrated growth velocity is given by 2λu 2 ℓ/L which for a single pair of fixed size vanishes in the thermodynamic limit. On the other hand,the local growth velocity dh/dt is given by 2λu 2 = (λ/2)(∇h) 2 which is consistent with the averaged KPZ equation (1) in the stationary state.…”

Section: Statisticssupporting

confidence: 73%

“…In the height field this morphology corresponds to downward cusps connected by parabolic segments with superimposed linear modes [1,2,27,28]. In the noisy case the interface is driven into a stationary state, and anticipating the analysis in section 3, it turns out that the noise excites a left hand soliton of the shape…”

Section: The Noisy Burgers Equationmentioning

confidence: 76%

“…In one dimension the KPZ equation has the form (in a co-moving frame) [1,2] ∂h ∂t = ν∇ 2 h + λ 2 (∇h) 2 + η .…”

Section: Introductionmentioning

confidence: 99%

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Nonequilibrium dynamics of a growing interface

Fogedby

2002

J. Phys.: Condens. Matter

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Abstract. A growing interface subject to noise is described by the KardarParisi-Zhang equation or, equivalently, the noisy Burgers equation. In one dimension this equation is analyzed by means of a weak noise canonical phase space approach applied to the associated Fokker-Planck equation. The growth morphology is characterized by a gas of nonlinear soliton modes with superimposed linear diffusive modes. We also discuss the ensuing scaling properties.

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Section: Statisticssupporting

confidence: 73%

Section: The Noisy Burgers Equationmentioning

confidence: 76%

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Nonequilibrium dynamics of a growing interface

Fogedby

2002

J. Phys.: Condens. Matter

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“…For noisy systems in one spatial dimension, we argue that the critical exponents for entanglement growth are those of the Kardar-Parisi-Zhang (KPZ) equation, originally introduced to describe the stochastic growth of a surface with time t [45]. In the simplest setting, we find that the height of this surface at a point x in space is simply the von Neumann entanglement entropy Sðx; tÞ for a bipartition which splits the system in two at x.…”

Section: Introductionmentioning

confidence: 99%

“…A remarkable feature of the KPZ universality class is that it also embraces two classical problems that at first sight are very different from surface growth [45,46]. These connections lead us to powerful heuristic pictures for entanglement growth, in both 1D and higher dimensions.…”

Section: Introductionmentioning

confidence: 99%

Quantum Entanglement Growth under Random Unitary Dynamics

et al. 2017

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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 KardarParisi-Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like ðtimeÞ 1=3 and are spatially correlated over a distance ∝ ðtimeÞ 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 wellstudied problem of pinning of a membrane or domain wall by disorder.

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Surface roughness of sputter-deposited gold films: a combined x-ray technique and AFM study

Schug

Schempp

Lamparter

et al. 1999

Surf. Interface Anal.

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We applied grazing incidence x‐ray specular and non‐specular reflectivity (GIXR) and small‐angle x‐ray scattering (SAXS) to study the surface roughness of gold films on quartz glass substrates. Two model systems were prepared: one series of Au films was sputtered in Ar at 3×10−3mbar (Au1 films) and one in residual air at 3×10−1 mbar (Au2). The combined x‐ray experiments showed that the Au1 films are compact and smooth, with‐mean‐square (rms) roughness values too low to be quantified by means of atomic force microscopy (AFM) or SAXS. The surfaces of film and substrate exhibit a strong cross correlation that is gradually lost with increasing film thickness. The surface height–height correlation functions are well described by a self‐affine model. In contrast, as directly proved via AFM, the Au2 films consist of islands that are coarsening with increasing film thickness. The film surface showed no cross‐correlation with the substrate surface at all. The SAXS curves from the Au2 films were attributed to a volume scattering effect, by interpreting the films in terms of a two‐dimensional array of hard hemispheres. Copyright © 1999 John Wiley & Sons, Ltd.

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Dynamic Scaling of Growing Interfaces

Cited by 5,211 publications

References 20 publications

Nonequilibrium dynamics of a growing interface

Nonequilibrium dynamics of a growing interface

Quantum Entanglement Growth under Random Unitary Dynamics

Surface roughness of sputter-deposited gold films: a combined x-ray technique and AFM study

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