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Register a new userKardar-Parisi-Zhang physics in integrable rotationally symmetric dynamics on discrete space-time lattice
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We introduce a deterministic SO(3) invariant dynamics of classical spins on a discrete space-time lattice and prove its complete integrability by explicitly finding a related non-constant (baxterized) solution of the set-theoretic quantum Yang-Baxter equation over the 2-sphere. Equipping the algebraic structure with the corresponding Lax operator we derive an infinite sequence of conserved quantities with local densities. The dynamics depend on a single continuous spectral parameter and reduce to a (lattice) Landau-Lifshitz model in the limit of a small parameter which corresponds to the continuous time limit. Using quasi-exact numerical simulations of deterministic dynamics and Monte Carlo sampling of initial conditions corresponding to a maximum entropy equilibrium state we determine spin-spin spatio-temporal (dynamical) correlation functions with relative accuracy of three orders of magnitude. We demonstrate that in the equilibrium state with a vanishing total magnetization the correlation function precisely follow Kardar-Parisi-Zhang scaling hence the spin transport belongs to the universality class with dynamical exponent z=3/2, in accordance to recent related simulations in discrete and continuous time quantum Heisenberg spin 1/2 chains.
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We study the dynamics of vortices in a two-dimensional, non-equilibrium system, described by the compact Kardar-Parisi-Zhang equation, after a sudden quench across the critical region. Our exact numerical solution of the phase-ordering kinetics shows that the unique interplay between non-equilibrium and the variable degree of spatial anisotropy leads to different critical regimes. We provide an analytical expression for the vortex evolution, based on scaling arguments, which is in agreement with the numerical results, and confirms the form of the interaction potential between vortices in this system.
We study atypically large fluctuations of height $H$ in the 1+1-dimensional Kardar-Parisi-Zhang (KPZ) equation at long times $t$, when starting from a droplet initial condition. We derive exact large deviation function of height for $lambda H<0$, where $lambda$ is the nonlinearity coefficient of the KPZ equation. This large deviation function describes a crossover from the Tracy-Widom distribution tail at small $|H|/t$, which scales as $|H|^3/t$, to a different tail at large $|H|/t$, which scales as $|H|^{5/2}/t^{1/2}$. The latter tail exists at all times $t>0$. It was previously obtained in the framework of the optimal fluctuation method. It was also obtained at short times from exact representation of the complete height statistics. The crossover between the two tails, at long times, occurs at $|H|sim t$ as previously conjectured. Our analytical findings are supported by numerical evaluations using exact representation of the complete height statistics.
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.
We study the scaling regimes for the Kardar-Parisi-Zhang equation with noise correlator R(q) ~ (1 + w q^{-2 rho}) in Fourier space, as a function of rho and the spatial dimension d. By means of a stochastic Cole-Hopf transformation, the critical and correction-to-scaling exponents at the roughening transition are determined to all orders in a (d - d_c) expansion. We also argue that there is a intriguing possibility that the rough phases above and below the lower critical dimension d_c = 2 (1 + rho) are genuinely different which could lead to a re-interpretation of results in the literature.
Circular KPZ interfaces spreading radially in the plane have GUE Tracy-Widom (TW) height distribution (HD) and Airy$_2$ spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a cup, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as $langle L(t) rangle = L_0+omega t^{gamma}$, while their mean height $langle h rangle$ increases as usual [$langle h ranglesim t$]. We show that the competition between the $L$ enlargement and the correlation length ($xi simeq c t^{1/z}$) plays a key role in the asymptotic statistics of the interfaces. While systems with $gamma>1/z$ have HDs given by GUE and the interface width increasing as $w sim t^{beta}$, for $gamma<1/z$ the HDs are Gaussian, in a correlated regime where $w sim t^{alpha gamma}$. For the special case $gamma=1/z$, a continuous class of distributions exists, which interpolate between Gaussian (for small $omega/c$) and GUE (for $omega/c gg 1$). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for $omega/c approx 10$. Despite the GUE HDs for $gamma>1/z$, the spatial covariances present a strong dependence on the parameters $omega$ and $gamma$, agreeing with Airy$_2$ only for $omega gg 1$, for a given $gamma$, or when $gamma=1$, for a fixed $omega$. These results considerably generalize our knowledge on the 1D KPZ systems, unveiling the importance of the background space in their statistics.
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