Phases of quantum dimers from ensembles of classical stochastic trajectories

TL;DR

This paper explores the deep connection between quantum dimer phases and classical stochastic dimer dynamics, revealing how classical trajectory statistics reflect quantum phase transitions, especially around the Rokhsar-Kivelson point.

Contribution

It demonstrates the correspondence between quantum phase behavior and classical trajectory ensembles, extending understanding beyond the RK point using large deviation statistics and trajectory sampling.

Findings

Quantum phases correspond to classical trajectory phases.

Classical trajectory analysis reveals quantum ground state order.

Trajectory phase transitions mirror quantum phase transitions.

Abstract

We study the connection between the phase behaviour of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the Hamiltonian corresponds to a non-stochastic "tilted" operator that encodes the statistics of time-integrated observables of the classical stochastic problem. This implies a direct relation between the phase behaviour of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. We make these ideas concrete by studying fully packed dimers on the square lattice. Using transition path sampling - supplemented by trajectory umbrella sampling - we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behaviour of the classical and quantum systems. The transition at the RK point between quantum phases of distinct order corresponds, in the classical case, to a trajectory phase transition between active and inactive dynamical phases. Furthermore, from the structure of stochastic trajectories in the active dynamical phase we infer that the ground state of quantum dimers has columnar order to one side of the RK point. We discuss how these results relate to those from quantum Monte Carlo, and how our approach may generalise to other problems.