Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface

TL;DR

This paper develops a Landau theory to analyze short-time dynamical phase transitions in the KPZ interface, revealing how spatial separation and height difference influence transition types and distribution tails.

Contribution

It introduces an effective Landau theory for the KPZ interface's dynamical phase transition, connecting spatial and height parameters to phase behavior and deriving analytical distribution limits.

Findings

Identifies a second-order phase transition at zero separation.

Discovers a first-order transition when the separation changes sign.

Derives asymptotic forms of the height difference distribution tails.

Abstract

We study the short-time distribution $\mathcal{P}\left(H,L,t\right)$ of the two-point two-time height difference $H=h(L,t)-h(0,0)$ of a stationary Kardar-Parisi-Zhang (KPZ) interface in 1+1 dimension. Employing the optimal-fluctuation method, we develop an effective Landau theory for the second-order dynamical phase transition found previously for $L=0$ at a critical value $H=H_c$. We show that $|H|$ and $L$ play the roles of inverse temperature and external magnetic field, respectively. In particular, we find a first-order dynamical phase transition when $L$ changes sign, at supercritical $H$. We also determine analytically $\mathcal{P}\left(H,L,t\right)$ in several limits away from the second-order transition. Typical fluctuations of $H$ are Gaussian, but the distribution tails are highly asymmetric. The tails $-\ln\mathcal{P}\sim\left|H\right|^{3/2} \! /\sqrt{t}$ and $-\ln\mathcal{P}\sim\left|H\right|^{5/2} \! /\sqrt{t}$, previously found for $L=0$, are enhanced for $L \ne 0$. At very large $|L|$ the whole height-difference distribution $\mathcal{P}\left(H,L,t\right)$ is time-independent and Gaussian in $H$, $-\ln\mathcal{P}\sim\left|H\right|^{2} \! /|L|$, describing the probability of creating a ramp-like height profile at $t=0$.