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New path integral simulation algorithms and their application to creep in the quantum sine-Gordon chain

Florian R. Krajewski

Published Dec 31, 2002• 0 citations
Publisher

Abstract

A path integral simulation algorithm which includes a higher-order Trotter approximation (HOA)is analyzed and compared to an approach which includes the correct quantum mechanical pair interaction (effective Propagator (EPr)). It is found that the HOA algorithmconverges to the quantum limit with increasing Trotter number P as P^{-4}, while the EPr algorithm converges as P^{-2}.The convergence rate of the HOA algorithm is analyzed for various physical systemssuch as a harmonic chain,a particle in a double-well potential, gaseous argon, gaseous helium and crystalline argon. A new expression for the estimator for the pair correlation function in the HOA algorithm is derived. A new path integral algorithm, the hybrid algorithm, is developed.It combines an exact treatment of the quadratic part of the Hamiltonian and thehigher-order Trotter expansion techniques.For the discrete quantum sine-Gordon chain (DQSGC), it is shown that this algorithm works more efficiently than all other improved path integral algorithms discussed in this work. The new simulation techniques developed in this work allow the analysis of theDQSGC and disordered model systems in the highly quantum mechanical regime using path integral molecular dynamics (PIMD)and adiabatic centroid path integral molecular dynamics (ACPIMD).The ground state phonon dispersion relation is calculated for the DQSGC by the ACPIMD method.It is found that the excitation gap at zero wave vector is reduced by quantum fluctuations. Two different phases exist: One phase with a finite excitation gap at zero wave vector, and a gapless phase where the excitation gap vanishes.The reaction of the DQSGC to an external driving force is analyzed at T=0.In the gapless phase the system creeps if a small force is applied, and in the phase with a gap the system is pinned. At a critical force, the systems undergo a depinning transition in both phases and flow is induced. The analysis of the DQSGC is extended to models with disordered substrate potentials. Three different cases are analyzed: Disordered substrate potentials with roughness exponent H=0, H=1/2,and a model with disordered bond length. For all models, the ground state phonon dispersion relation is calculated.

DOI
10.25358/openscience-2459

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A path integral simulation algorithm which includes a higher-order Trotter approximation (HOA)is analyzed and compared to an approach which includes the correct quantum mechanical pair interaction (effective Propagator (EPr)). It is found that the HOA algorithmconverges to the quantum limit with increasing Trotter number P as P^{-4}, while the EPr algorithm converges as P^{-2}.The convergence rate of the HOA algorithm is analyzed for various physical systemssuch as a harmonic chain,a particle in a double-well potential, gaseous argon, gaseous helium and crystalline argon. A new expression for the estimator for the pair correlation function in the HOA algorithm is derived. A new path integral algorithm, the hybrid algorithm, is developed.It combines an exact treatment of the quadratic part of the Hamiltonian and thehigher-order Trotter expansion techniques.For the discrete quantum sine-Gordon chain (DQSGC), it is shown that this algorithm works more efficiently than all other improved path integral algorithms discussed in this work. The new simulation techniques developed in this work allow the analysis of theDQSGC and disordered model systems in the highly quantum mechanical regime using path integral molecular dynamics (PIMD)and adiabatic centroid path integral molecular dynamics (ACPIMD).The ground state phonon dispersion relation is calculated for the DQSGC by the ACPIMD method.It is found that the excitation gap at zero wave vector is reduced by quantum fluctuations. Two different phases exist: One phase with a finite excitation gap at zero wave vector, and a gapless phase where the excitation gap vanishes.The reaction of the DQSGC to an external driving force is analyzed at T=0.In the gapless phase the system creeps if a small force is applied, and in the phase with a gap the system is pinned. At a critical force, the systems undergo a depinning transition in both phases and flow is induced. The analysis of the DQSGC is extended to models with disordered substrate potentials. Three different cases are analyzed: Disordered substrate potentials with roughness exponent H=0, H=1/2,and a model with disordered bond length. For all models, the ground state phonon dispersion relation is calculated.
Numerical algorithms are developed for the centroid molecular dynamics (centroid MD) method to calculate dynamical time correlation functions for general many-body quantum systems. Approaches based on the normal mode path integral molecular dynamics and staging path integral Monte Carlo methods are described to carry out a direct calculation of the force on the centroid variables in the centroid MD algorithm. A more efficient, but approximate, scheme to compute the centroid force is devised which is based on the locally optimized harmonic approximation for the centroid potential. The centroid MD equations in the latter method can be solved with the help of an iterative procedure or through extended Lagrangian dynamics. A third algorithm introduces an effective centroid pseudopotential to approximate the full many-body centroid mean force potential by effective pairwise centroid interactions. Numerical simulations for both prototype models and more realistic many-body systems are performed to explore the feasibility and limitations of each algorithm.
The commensurate Frenkel Kontorova (FK) model is studied using path-integral molecular dynamics (PIMD). We focus on the highly discrete case, in which the embedding potential has a much greater maximum curvature than the harmonic potential connecting two particles in the FK chain. When efficient sampling methods are used, the dynamical interpretation of adiabatic PIMD appears to represent quite accurately the true time correlation functions of this highly correlated many-body system. We have found that the discrete, quantum FK model shows different behavior than its continuum version. The spectral density does not show the characteristic omega-2Theta(omega-omegac) cusp of the continuum solution in the pinned phase (m>m(c)). We also identify a dynamical quantum hysteresis in addition to the regular classical hysteresis when an external force is applied to the FK chain. In the unpinned phase (m<or=m(c)), we find a linear response damping coefficient which is finite and only weakly dependent on temperature T at small values of T.
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.
The discrete time path integral Monte Carlo with a one-particle density matrix approximation is applied to study the quantum phase transition in the coupled double-well chain. To improve the convergence properties, the exact action for a single particle in a double-well potential is used to construct the many-particle action. The algorithm is applied to the interacting quantum double-well chain for which the zero-temperature phase diagram is determined. The quantum phase transition is studied via finite-size scaling, and the critical exponents are shown to be compatible with the classical two-dimensional Ising universality class--not only in the order-disorder limit (deep potential wells), but also in the displacive regime (shallow potential wells).
Quantum mechanical effects play an important role in dynamics of condensed phases. It is also well known that the difficulty of solving the full Schrödinger equation grows exponentially with time. One approach to cope up with this difficulty is to restrict quantum treatment to a few particles (system) and treat the dynamics of the rest of particles (bath) using classical trajectories. Path integral methods due to their trajectory like nature provide excellent tool to develop quantum-classical methods, but to capture long time dynamics, the number of classical trajectories grow exponentially. A part of this work focuses on improving Quantum-Classical Path Integral (QCPI) treatments that allow for larger path integral time step by building “smarter” propagators. These improved solvent-driven reference propagators is developed by incorporating physically motivated approximations to the solvent. When used in QCPI expression they allow convergence with larger time steps leading to an exponential reduction of the number of trajectories required. Further advantages of these propagators include improvements in path filtering techniques, smaller number of path integral steps for achieving the memory decoherence time, and smoothing of the integrand which leads to convergence with fewer Monte Carlo sample points. These ideas have been validated on the spin boson model - a prototypical model to study condensed phase dynamics, which consists of a two-level system coupled to a harmonic bath. The new approach of building the reference propagators combined with their iterative evaluation and filtering is validated using parameters that mimic the first electron transfer in wild-type photosynthetic reaction centres. A real-world dynamical simulation contains various anharmonic effects, Ferrocene (donor) - Ferrocenium (acceptor) system in liquid Benzene was studied where the anharmonic effects of the bath of liquid Benzene on the dynamics of the system were treated using Ensemble Averaged Classical Path. The process of electron or proton transfer occurs between two species having different charges and fluctuations in the solvent plays a key role in determining the rates of such reaction dynamics. To correctly account for the effects of initial distribution of the solvent on the dynamics of the quantum system, the concept of equilibrating the bath to the donor state of the system is used. Two different approaches have been used within the influence functional framework.
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.