Path Integral Monte Carlo in the Angular Momentum Basis for a Chain of Planar Rotors

TL;DR

The paper develops a Path Integral Monte Carlo framework in the angular momentum basis to study a chain of planar dipolar rotors, enabling direct computation of momentum-related observables without path-breaking. Central to the approach are the Bond-Hamiltonian decomposition of the propagator and cluster-loop moves that restore ergodicity under parity constraints, coupled with PIGS to project to the ground state. The method yields ground-state energies and angular-momentum properties in good agreement with DMRG benchmarks away from the quantum phase transition, and uses the derivative of the kinetic energy with respect to the interaction strength as an effective order parameter for detecting the QPT. The work provides a pathway to extend discrete-path-sum QMC to rotor systems and highlights symmetry-driven sampling challenges and potential non-local update strategies for enhanced performance near criticality.

Abstract

We introduce a Path Integral Monte Carlo (PIMC) approach that uses the angular momentum representation for the description of interacting rotor systems. Such a choice of representation allows the calculation of momentum properties without having to break the paths. The discrete nature of the momentum basis also allows the use of rejection-free Gibbs sampling techniques. To illustrate the method, we study the collective behavior of $N$ confined planar rotors with dipole-dipole interactions, a system known to exhibit a quantum phase transition from a disordered to an ordered state at zero temperature. Ground state properties are obtained using the Path Integral Ground State (PIGS) method. We propose a Bond-Hamiltonian decomposition for the high temperature density matrix factorization of the imaginary time propagator. We show that \textit{cluster-loop} type moves are necessary to overcome ergodicity issues and to achieve efficient Markov Chain updates. Ground state energies and angular momentum properties are computed and compared with Density Matrix Renormalization Group (DMRG) benchmark results. In particular, the derivative of the kinetic energy with respect to the interaction strength estimator is presented as a successful order parameter for the detection of the quantum phase transition.

Figures (12)

• Figure 1: Illustration of the $N$ planar rotor chain in a co-planar arrangement.Serwatka2024
• Figure 2: Graphical representation of the "checker-board" grid defined in Eq. \ref{['eq:part_function_H_bond_odd_even']} for a system of $N=5$ planar rotors and $L=4$ ($P\equiv 2L+1=9$ total beads). The black circles represent the particles (solid for the $m=0$ state and unfilled for a generic state $m$) and the black solid rectangles represent the 2-body density matrix $\rho^{\text{2-B}}_{\text{bond}}(\tau)$. The blue circles indicate the middle bead for $P=L+1=5$. The red solid lines stand for the kinetic energy density matrix operators $e^{- \frac{\tau}{2} \hat{K}_{1}}$ and $e^{- \frac{\tau}{2} \hat{K}_{5}}$ acting on the $1$st and $5$th particle due to the OBC.
• Figure 3: Diagrammatic representation of: (a) The matrix element (cluster) of the 2-body density matrix $\hat{\rho}^{\text{2-B}}_{\text{bond}}(\tau)$ for some coordinate particle-bead $(i,p)$ of the grid; (b) The interaction of two clusters where the target particle $\ket{m_t}$ (light green) interacts with the six remaining ones $\{\ket{m_k} : k \in [1,6]\}$ (light purple).
• Figure 4: Graphical representation of the process of updating a cluster by changing the particles in two different manners: (i) in pairs; and (ii) individually and sequentially.
• Figure 5: Representation of the two simplest possible closed loops (in light green color) connecting pairs of particles that share the same cluster in the PIGS grid.