Tricritical directed percolation with long-range interaction in one and two dimensions

TL;DR

This paper extends the tricritical directed percolation model to include long-range interactions in one and two dimensions, analyzing critical behavior and universality classes through numerical methods.

Contribution

It introduces a long-range interaction version of the TDP model and characterizes its critical exponents and phase diagram in various dimensions.

Findings

Critical exponents for LTDP class are numerically obtained.

The interval of the long-range parameter $\sigma$ for the LTDP class is determined.

A universality class diagram in ($d$, $\sigma$) space is constructed.

Abstract

Recently, the quantum contact process, in which branching and coagulation processes occur both coherently and incoherently, was theoretically and experimentally investigated in driven open quantum spin systems. In the semi-classical approach, the quantum coherence effect was regarded as a process in which two consecutive atoms are involved in the excitation of a neighboring atom from the inactive (ground) state to the active state (excited $s$ state). In this case, both second-order and first-order transitions occur. Therefore, a tricritical point exists at which the transition belongs to the tricritical directed percolation (TDP) class. On the other hand, when an atom is excited to the $d$ state, long-range interaction is induced. Here, to account for this long-range interaction, we extend the TDP model to one with long-range interaction in the form of $\sim 1/r^{d+\sigma}$ (denoted as LTDP), where $r$ is the separation, $d$ is the spatial dimension, and $\sigma$ is a control parameter. In particular, we investigate the properties of the LTDP class below the upper critical dimension $d_c=$ min$(3,\,1.5\sigma)$. We numerically obtain a set of critical exponents in the LTDP class and determine the interval of $\sigma$ for the LTDP class. Finally, we construct a diagram of universality classes in the space ($d$, $\sigma$).