Scale invariant dynamics of surface growth

Keldysh field theory for driven open quantum systems

Recent experimental developments in diverse areas-ranging from cold atomic gases to light-driven semiconductors to microcavity arrays-move systems into the focus which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in equilibrium condensed matter physics. This concerns both their non-thermal stationary states and their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macroscopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.

Strong-coupling phases of the anisotropic Kardar-Parisi-Zhang equation

We study the anisotropic Kardar-Parisi-Zhang equation using nonperturbative renormalization group methods. In contrast to a previous analysis in the weak-coupling regime, we find the strong-coupling fixed point corresponding to the isotropic rough phase to be always locally stable and unaffected by the anisotropy even at noninteger dimensions. Apart from the well-known weak-coupling and the now well-established isotropic strong-coupling behavior, we find an anisotropic strong-coupling fixed point for nonlinear couplings of opposite signs at noninteger dimensions.

Kardar-Parisi-Zhang equation with spatially correlated noise: a unified picture from nonperturbative renormalization group

We investigate the scaling regimes of the Kardar-Parisi-Zhang (KPZ) equation in the presence of spatially correlated noise with power-law decay D(p) ∼ p(-2ρ) in Fourier space, using a nonperturbative renormalization group approach. We determine the full phase diagram of the system as a function of ρ and the dimension d. In addition to the weak-coupling part of the diagram, which agrees with the results from Europhys. Lett. 47, 14 (1999) and Eur. Phys. J. B 9, 491 (1999), we find the two fixed points describing the short-range- (SR) and long-range- (LR) dominated strong-coupling phases. In contrast with a suggestion in the references cited above, we show that, for all values of ρ, there exists a unique strong-coupling SR fixed point that can be continuously followed as a function of d. We show in particular that the existence and the behavior of the LR fixed point do not provide any hint for 4 being the upper critical dimension of the KPZ equation with SR noise.

Multisurface coding simulations of the restricted solid-on-solid model in four dimensions

We study the restricted solid-on-solid model for surface growth in spatial dimension d=4 by means of a multisurface coding technique that allows us to analyze samples of size up to 256(4) in the steady-state regime. For such large systems we are able to achieve a controlled asymptotic regime where the typical scale of the fluctuations are larger than the lattice spacing used in the simulations. A careful finite-size scaling analysis of the critical exponents clearly indicate that d=4 is not the upper critical dimension of the model.

Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1, and 3+1 dimensions

We investigate the strong-coupling regime of the stationary Kardar-Parisi-Zhang equation for interfaces growing on a substrate of dimension d = 1, 2, and 3 using a nonperturbative renormalization group (NPRG) approach. We compute critical exponents, correlation and response functions, extract the related scaling functions, and calculate universal amplitude ratios. We work with a simplified implementation of the second-order (in the response field) approximation proposed in a previous work [Phys. Rev. E 84, 061150 (2011) and Phys. Rev. E 86, 019904(E) (2012)], which greatly simplifies the frequency sector of the NPRG flow equations, while keeping a nontrivial frequency dependence for the two-point functions. The one-dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second-order NPRG equations and with the exact scaling function. Furthermore, the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in d = 2 and d = 3. We argue that our ansatz is reliable up to d [Symbol: see text] 3.5.

Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation

We present a simple approximation of the nonperturbative renormalization group designed for the Kardar-Parisi-Zhang equation and show that it yields the correct phase diagram, including the strong-coupling phase with reasonable scaling exponent values in physical dimensions. We find indications of a possible qualitative change of behavior around d=4. We discuss how our approach can be systematically improved.

Anisotropic model of kinetic roughening: the strong-coupling regime

We study the strong coupling (SC) limit of the anisotropic Kardar-Parisi-Zhang (KPZ) model. A systematic mapping of the continuum model to its lattice equivalent shows that in the SC limit, anisotropic perturbations destroy all spatial correlations but retain a temporal scaling which shows a remarkable crossover along one of the two spatial directions, the choice of direction depending on the relative strength of anisotropicity. The results agree with exact numerics and are expected to settle the long-standing SC problem of a KPZ model in the infinite range limit.

Universality classes of the Kardar-Parisi-Zhang equation

We reexamine mode-coupling theory for the Kardar-Parisi-Zhang equation in the strong-coupling limit and show that there exist two branches of solutions. One branch (or universality class) exists only for dimensionalities d<dc=2 and is similar to that found by a variety of analytic approaches, including replica symmetry breaking and Flory-Imry-Ma arguments. The second branch exists up to dc=4 and gives values for the dynamical exponent z similar to those of numerical studies for d>or=2.

Kardar-Parisi-Zhang equation in the weak noise limit: pattern formation and upper critical dimension

We extend the previously developed nonperturbative weak noise scheme, applied to the noisy Burgers equation in one dimension, to the Kardar-Parisi-Zhang equation for a growing interface in arbitrary dimensions. By means of the Cole-Hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes. In the Cole-Hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear Schrödinger equation, subsequently boosted to finite velocity by a Galilei transformation. We discuss the dynamics of the pattern formation and, briefly, the superimposed linear modes. Implementing the stochastic interpretation we discuss kinetic transitions and in particular the preliminary scaling properties pertaining to the pair mode or dipole sector. In the dipole sector we obtain the Hurst exponent H=(3-d)/(4-d) or dynamic exponent Zdip(4-d)/(3-d) for the random walk of growth modes. Below d=3 the dipole growth modes show anomalous diffusion, above d=3 the dipole growth modes freeze. Finally, applying Derrick's theorem based on constrained minimization we show that the upper critical dimension is d=4 in the sense that growth modes cease to exist above this dimension.

Stochastic model in the Kardar-Parisi-Zhang universality class with minimal finite size effects

We introduce a solid-on-solid lattice model for growth with conditional evaporation. A measure of finite size effects is obtained by observing the time invariance of distribution of local height fluctuations. The model parameters are chosen so that the change in the distribution in time is minimum. On a one-dimensional substrate the results obtained from the model for the roughness exponent from three different methods are same as predicted for the Kardar-Parisi-Zhang equation. One of the unique features of the model is that as obtained from the structure factor S(k,t) for the one-dimensional substrate growth exactly matches the predicted value of 0.5 within statistical errors. The model can be defined in any dimensions. We have obtained results for this model on two- and three-dimensional substrates.

Width distributions and the upper critical dimension of Kardar-Parisi-Zhang interfaces

Simulations of restricted solid-on-solid growth models are used to build the width distributions of d=2-5 dimensional Kardar-Parisi-Zhang (KPZ) interfaces. We find that the universal scaling function associated with the steady-state width distribution changes smoothly as d is increased, thus strongly suggesting that d=4 is not an upper critical dimension for the KPZ equation. The dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity.

Upper critical dimension, dynamic exponent, and scaling functions in the mode-coupling theory for the Kardar-Parisi-Zhang equation

We study the mode-coupling approximation for the Kardar-Parisi-Zhang equation in the strong-coupling regime. By constructing an ansatz consistent with the asymptotic forms of the correlation and response functions we determine the upper critical dimension d(c) = 4 and the expansion z = 2-(d-4)/4+O((4-d)2) around dc. We find the exact z = 3/2 value in d = 1, and estimate the values z approximately 1.62, z approximately 1.78 in d = 2, 3. The result dc = 4 and the expansion around dc are very robust and can be derived just from a mild assumption on the relative scale on which the response and correlation functions vary as z approaches 2.

Renormalization-group study of one-dimensional systems with roughening transitions

A recently introduced real-space renormalization-group technique, developed for the analysis of processes in the Kardar-Parisi-Zhang universality class, is generalized and tested by applying it to a different family of surface-growth processes. In particular, we consider a growth model exhibiting a rich phenomenology even in one dimension. It has four different phases and a directed percolation-related roughening transition. The renormalization method reproduces extremely well all of the phase diagram, the roughness exponents in all the phases, and the separatrix among them. This proves the versatility of the method and elucidates interesting physical mechanisms.