Computational Complexity of Determining the Barriers to Interface Motion in Random Systems

TL;DR

This paper proves that calculating energy barriers in interface motion within random systems is NP-complete, indicating a fundamental computational difficulty unlike the polynomial-time solvable ground-state energy determination.

Contribution

It establishes the NP-completeness of computing energy barriers in interface models, highlighting a key computational complexity distinction in disordered systems.

Findings

Ground-state energy can be computed in polynomial time.

Energy barrier determination is NP-complete.

Exact barrier computation is significantly more complex than ground-state calculation.

Abstract

The low-temperature driven or thermally activated motion of several condensed matter systems is often modeled by the dynamics of interfaces (co-dimension-1 elastic manifolds) subject to a random potential. Two characteristic quantitative features of the energy landscape of such a many-degree-of-freedom system are the ground-state energy and the magnitude of the energy barriers between given configurations. While the numerical determination of the former can be accomplished in time polynomial in the system size, it is shown here that the problem of determining the latter quantity is NP-complete. Exact computation of barriers is therefore (almost certainly) much more difficult than determining the exact ground states of interfaces.