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The Journal of Chemical Physics

Path integral molecular dynamics framework for spin-phonon systems

Ilija Srpak, Michael J. Willatt, Stuart C. Althorpe, Ali Alavi

Published Aug 19, 2026• 0 citations
Publisher

Abstract

We build on the framework for studying systems containing spin-phonon coupling using molecular dynamics as the method for sampling the positions of atoms carrying spins, whose interaction is described by the Heisenberg model, and which we presented in a previous paper Srpak et al. [Phys. Rev. B 112, 205132 (2025)]. In applying the real-space path integral framework to the spin-phonon problem, neighboring imaginary time slices become coupled via overlap of eigenvector matrices propagating along the path integral, leading to interactions that go beyond the pure harmonic interactions between neighboring beads of the ring-polymers in Born-Oppenheimer-based path integral molecular dynamics (PIMD) simulations of quantum nuclei. We show how the use of a spin-force, defined as the negative gradient of the spin free energy, accounts for this type of coupling, valid for systems in thermal equilibrium. The spin force is computed using either ring-polymer exact diagonalization (for small systems) or stochastic series expansion (for large systems). We provide the technical details for these implementations. We present a neural network approach for this type of PIMD simulation, where we compute the spin force based on training data obtained from the ring-polymer stochastic series expansion method. Utilizing these methods, we compare them with the literature and demonstrate how one can obtain phase diagrams clearly showing how the thermal and nuclear quantum effects influence the stability of the spin-Peierls dimerized phase. We also show how machine learning makes it possible to apply the method to two dimensional systems.

DOI
10.1063/5.0337420

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We build on the framework for studying systems containing spin-phonon coupling using molecular dynamics as the method for sampling the positions of atoms carrying spins, whose interaction is described by the Heisenberg model, and which we presented in a previous paper Srpak et al. [Phys. Rev. B 112, 205132 (2025)]. In applying the real-space path integral framework to the spin-phonon problem, neighboring imaginary time slices become coupled via overlap of eigenvector matrices propagating along the path integral, leading to interactions that go beyond the pure harmonic interactions between neighboring beads of the ring-polymers in Born-Oppenheimer-based path integral molecular dynamics (PIMD) simulations of quantum nuclei. We show how the use of a spin-force, defined as the negative gradient of the spin free energy, accounts for this type of coupling, valid for systems in thermal equilibrium. The spin force is computed using either ring-polymer exact diagonalization (for small systems) or stochastic series expansion (for large systems). We provide the technical details for these implementations. We present a neural network approach for this type of PIMD simulation, where we compute the spin force based on training data obtained from the ring-polymer stochastic series expansion method. Utilizing these methods, we compare them with the literature and demonstrate how one can obtain phase diagrams clearly showing how the thermal and nuclear quantum effects influence the stability of the spin-Peierls dimerized phase. We also show how machine learning makes it possible to apply the method to two dimensional systems.
We build on the framework for studying systems containing spin-phonon coupling using molecular dynamics (MD) as the method for sampling the positions of atoms carrying spins, whose interaction is described by the Heisenberg model, and which we presented in a previous paper Srpak et al. [Phys. Rev. B 112, 205132 (2025)]. In applying the real-space path integral (PI) framework to the spin-phonon problem, neighboring imaginary time slices become coupled via overlap of eigenvector matrices propagating along the path integral, leading to interactions that go beyond the pure harmonic interactions between neighboring beads of the ring-polymers in Born-Oppenheimer-based PIMD simulations of quantum nuclei. We show how the use of a spin-force, defined as the negative gradient of the spin free energy, accounts for this type of coupling, valid for systems in thermal equilibrium. The spin force is computed using either ring-polymer exact diagonalization (for small systems) or stochastic series expansion (for large systems). We provide the technical details for these implementations. We present a neural network approach for this type of PIMD simulation, where we compute the spin force based on training data obtained from the ring-polymer stochastic series expansion method. Utilizing these methods, we compare with the literature and demonstrate how one can obtain phase diagrams clearly showing how the thermal and nuclear quantum effects influence the stability of spin-Peierls dimerized phase.We also show how machine learning makes it possible to apply the method to two dimensional systems.
To study the spin-Peierls transition at finite temperatures, we use path integral molecular dynamics (PIMD) in combination with stochastic series expansion or exact diagonalization to sample the quantum Boltzmann distribution of the spin-phonon-coupled one-dimensional isotropic Heisenberg chains. The use of PIMD in this context has apparently been overlooked up to now and offers several advantages over other approaches, including no restriction on the functional forms of the phonon potential or the exchange coupling. This approach allows us to study the spin-spin correlation in such systems in detail. Looking at S = 1 / 2 , we find that it exhibits algebraic decay in the undimerized phase and exponential decay in the dimerized phase, but with many nuanced trends present. The decay exponentials in all cases roughly obey the scaling of ( g 2 / ( m ω π 2 ) ) γ , where g is the spin-phonon coupling constant, m is mass, and ω π is the frequency of the k = π phonon.
This thesis describes several novel approaches in quantum molecular dynamics for obtaining properties of molecular systems in different regimes. We investigate ground state properties of chains of linear rotors with dipole–dipole interactions via the density matrix renormalization group (DMRG), by deriving the appropriate form of the interaction operator and implementing it in ITensor. This provides us with further evidence of a quantum phase transition in this system. We also improve the sampling of Gaussian mixture distributions for finite temperature path integral Monte Carlo (PIMC) of vibronic Hamiltonians. To do this, we replace random sampling by quasi-random sampling, and improve sampling distributions by optimizing their parameters. Finally, we introduce estimators and integrators for constrained free energy simulations in path integral molecular dynamics (PIMD). This method is applied to the study of a water dimer, for which we obtain a quantum potential of mean force.
Trapped bosons exhibit fundamental physical phenomena and are at the core of emerging quantum technologies. We present a method for simulating bosons using path integral molecular dynamics. The main difficulty in performing such simulations is enumerating all ring-polymer configurations, which arise due to permutations of identical particles. We show that the potential and forces at each time step can be evaluated by using a recurrence relation which avoids enumerating all permutations, while providing the correct thermal expectation values. The resulting algorithm scales cubically with system size. The method is tested and applied to bosons in a 2-dimensional (2D) trap and agrees with analytical results and numerical diagonalization of the many-body Hamiltonian. An analysis of the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations, is also presented.
Using imaginary time, quantum statistical mechanics of a finite-temperature system in thermodynamical equilibrium turns into a problem of classical ring polymers made of particle paths. We need to consider all possible ring polymers at a given temperature, which again is where Monte Carlo methods are effective, especially using a so-called worm algorithm. We show how paths can be sampled using Monte Carlo, and how the physical properties, such as energy and radial distribution, are computed from the paths. Identical particles can have their paths exchanged at will, and we see how boson superfluidity comes out of this picture. Finally, the stochastic series expansion method is introduced to show finite temperature Hubbard model can be simulated.
We introduce a novel approach for a fully quantum description of coupled electron–ion systems from first principles. It combines the variational quantum Monte Carlo solution of the electronic part with the path integral formalism for the quantum nuclear dynamics. On the one hand, the path integral molecular dynamics includes nuclear quantum effects by adding a set of fictitious classical particles (beads) aimed at reproducing nuclear quantum fluctuations via a harmonic kinetic term. On the other hand, variational quantum Monte Carlo can provide Born–Oppenheimer potential energy surfaces with a precision comparable to the most-advanced post-Hartree–Fock approaches, and with a favorable scaling with the system size. In order to cope with the intrinsic noise due to the stochastic nature of quantum Monte Carlo methods, we generalize the path integral molecular dynamics using a Langevin thermostat correlated according to the covariance matrix of quantum Monte Carlo nuclear forces. The variational parameters of the quantum Monte Carlo wave function are evolved during the nuclear dynamics, such that the Born–Oppenheimer potential energy surface is unbiased. Statistical errors on the wave function parameters are reduced by resorting to bead grouping average, which we show to be accurate and well-controlled. Our general algorithm relies on a Trotter breakup between the dynamics driven by ionic forces and the one set by the harmonic interbead couplings. The latter is exactly integrated, even in the presence of the Langevin thermostat, thanks to the mapping onto an Ornstein–Uhlenbeck process. This framework turns out to be also very efficient in the case of noiseless (deterministic) ionic forces. The new implementation is validated on the Zundel ion (H 5 O 2 + ) by direct comparison with standard path integral Langevin dynamics calculations made with a coupled cluster potential energy surface. Nuclear quantum effects are confirmed to be dominant over thermal effects well beyond room temperature, giving the excess proton an increased mobility by quantum tunneling.