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arXiv:2010.15161 · Physical review. E

Efimov effect at the Kardar-Parisi-Zhang roughening transition

Yu Nakayama, Yusuke Nishida

Published Jan 19, 2021• 7 citations
arXivHTMLPDF

Abstract

Surface growth governed by the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than two undergoes a roughening transition from smooth to rough phases with increasing the nonlinearity. It is also known that the KPZ equation can be mapped onto quantum mechanics of attractive bosons with a contact interaction, where the roughening transition corresponds to a binding transition of two bosons with increasing the attraction. Such critical bosons in three dimensions actually exhibit the Efimov effect, where a three-boson coupling turns out to be relevant under the renormalization group so as to break the scale invariance down to a discrete one. On the basis of these facts linking the two distinct subjects in physics, we predict that the KPZ roughening transition in three dimensions shows either the discrete scale invariance or no intrinsic scale invariance.

DOI
10.1103/physreve.103.012117

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Surface growth governed by the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than two undergoes a roughening transition from smooth to rough phases with increasing the nonlinearity. It is also known that the KPZ equation can be mapped onto quantum mechanics of attractive bosons with a contact interaction, where the roughening transition corresponds to a binding transition of two bosons with increasing the attraction. Such critical bosons in three dimensions actually exhibit the Efimov effect, where a three-boson coupling turns out to be relevant under the renormalization group so as to break the scale invariance down to a discrete one. On the basis of these facts linking the two distinct subjects in physics, we predict that the KPZ roughening transition in three dimensions shows either the discrete scale invariance or no intrinsic scale invariance.
The one-dimensional Kardar-Parisi-Zhang (KPZ) equation is becoming an overarching paradigm for the scaling of nonequilibrium, spatially extended, classical and quantum systems with strong correlations. Recent analytical solutions have uncovered a rich structure regarding its scaling exponents and fluctuation statistics. However, the zero surface tension or zero viscosity case eludes such analytical solutions and has remained ill-understood. Using numerical simulations, we elucidate a well-defined universality class for this case that differs from that of the viscous case, featuring intrinsically anomalous kinetic roughening (despite previous expectations for systems with local interactions and time-dependent noise) and ballistic dynamics. The latter may be relevant to recent quantum spin chain experiments which measure KPZ and ballistic relaxation under different conditions. We identify the ensuing set of scaling exponents in previous discrete interface growth models related with isotropic percolation, and show it to describe the fluctuations of additional continuum systems related with the noisy Korteweg-de Vries equation. Along this process, we additionally elucidate the universality class of the related inviscid stochastic Burgers equation.
We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate $\ensuremath{\gamma}$ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a nontrivial effect on the probability distribution of the random walker. At small $\ensuremath{\gamma}$, we show analytically that the time evolution of the latter can be mapped to the stochastic heat equation. In this limit, the width of the log-probability thus follows a Family-Vicsek scaling law, ${N}^{\ensuremath{\alpha}}f(t/{N}^{\ensuremath{\alpha}/\ensuremath{\beta}})$, with roughness and growth exponents corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e., ${\ensuremath{\alpha}}_{\mathrm{KPZ}}^{1\mathrm{D}}=1/2$ and ${\ensuremath{\beta}}_{\mathrm{KPZ}}^{1\mathrm{D}}=1/3$, respectively. When $\ensuremath{\gamma}$ is increased outside this regime, we find numerically in 1D a crossover from the KPZ class to a new universality class characterized by exponents ${\ensuremath{\alpha}}_{M}^{1\mathrm{D}}\ensuremath{\approx}1$ and ${\ensuremath{\beta}}_{M}^{1\mathrm{D}}\ensuremath{\approx}1.4$. In 3D, varying $\ensuremath{\gamma}$ beyond a critical value ${\ensuremath{\gamma}}_{M}^{c}$ leads to a phase transition from a smooth phase that we identify as the Edwards-Wilkinson class to a new universality class with ${\ensuremath{\alpha}}_{M}^{3\mathrm{D}}\ensuremath{\approx}1$.
The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise δ-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.
Abstract We revisit the Lieb–Liniger model for n bosons in one dimension with attractive delta interaction in a half-space with diagonal boundary conditions. This model is integrable for the arbitrary value of , the interaction parameter with the boundary. We show that its spectrum exhibits a sequence of transitions, as b is decreased from the hard-wall case , with successive appearance of boundary bound states (or boundary modes) which we fully characterize. We apply these results to study the Kardar–Parisi–Zhang equation for the growth of a one-dimensional interface of height , on the half-space with boundary condition and droplet initial condition at the wall. We obtain explicit expressions, valid at all time t and arbitrary b , for the integer exponential (one-point) moments of the KPZ height field . From these moments we extract the large time limit of the probability distribution function (PDF) of the scaled KPZ height function. It exhibits a phase transition, related to the unbinding to the wall of the equivalent directed polymer problem, with two phases: (i) unbound for where the PDF is given by the GSE Tracy–Widom distribution (ii) bound for , where the PDF is a Gaussian. At the critical point , the PDF is given by the GOE Tracy–Widom distribution.
Abstract In its original version, the KPZ equation models the dynamics of an interface bordering a stable phase against a metastable one. Over the past few years the corresponding two-dimensional field theory has been applied to models with different physics. Out of a wide choice, the spin–spin time correlations for the Heisenberg chain will be discussed at some length, also the equilibrium time correlations of the conserved fields for 1D fluids. An interesting recent theoretical advance is the construction of the scale-invariant asymptotic theory, the so-called KPZ fixed point.
We discuss the universal behavior linked to the Goldstone mode associated with the spontaneous breaking of time-translation symmetry in many-body systems, in which the order parameter traces out a limit cycle. We show that this universal behavior is closely tied to Kardar-Parisi-Zhang physics, which can strongly affect the scaling properties in all dimensions. Our Letter predicts and rationalizes the emergence of Kardar-Parisi-Zhang in numerous systems such as nonreciprocal phases in active matter, active magnets, driven-dissipative quantum systems, and synchronization of oscillators.