Path Integral Representation Research Articles

Magnetic flux disorder and superconductor-insulator transition in nanohole thin films

We study the superconductor-insulator transition in nanohole ultrathin films in a transverse magnetic field by numerical simulation of a Josephson-junction array model. Geometrical disorder due to the random location of nanoholes in the film corresponds to random flux in the array model. Monte Carlo simulation in the path-integral representation is used to determine the critical behavior and the universal resistivity at the transition as a function of disorder and average number of flux quanta per cell, $f_o$. The resistivity increases with disorder for noninteger $ f_o$ while it decreases for integer $ f_o$, and reaches a common constant value in a vortex-glass regime above a critical value of the flux disorder $D_f^c$. The estimate of $D_f^c$ and the resistivity increase for noninteger $ f_o$ are consistent with recent experiments on ultrathin superconducting films with positional disordered nanoholes.

Dynamic density and spin responses of a superfluid Fermi gas in the BCS–BEC crossover: Path integral formulation and pair fluctuation theory

We present a standard field theoretical derivation of the dynamic density and spin linear response functions of a dilute superfluid Fermi gas in the BCS–BEC crossover in both three and two dimensions. The derivation of the response functions is based on the elegant functional path integral approach which allows us to calculate the density–density and spin–spin correlation functions by introducing the external sources for the density and the spin density. Since the generating functional cannot be evaluated exactly, we consider two gapless approximations which ensure a gapless collective mode (Goldstone mode) in the superfluid state: the BCS–Leggett mean-field theory and the Gaussian-pair-fluctuation (GPF) theory. In the mean-field theory, our results of the response functions agree with the known results from the random phase approximation. We further consider the pair fluctuation effects and establish a theoretical framework for the dynamic responses within the GPF theory. We show that the GPF response theory naturally recovers three kinds of famous diagrammatic contributions: the Self-Energy contribution, the Aslamazov–Lakin contribution, and the Maki–Thompson contribution. We also show that unlike the equilibrium state, in evaluating the response functions, the linear (first-order) terms in the external sources as well as the induced order parameter perturbations should be treated carefully. In the superfluid state, there is an additional order parameter contribution which ensures that in the static and long wavelength limit, the density response function recovers the result of the compressibility (compressibility sum rule). We expect that the f-sum rule is manifested by the full number equation which includes the contribution from the Gaussian pair fluctuations. The dynamic density and spin response functions in the normal phase (above the superfluid critical temperature) are also derived within the Nozières–Schmitt–Rink (NSR) theory.

Computer simulation of spin states of electrons in nanoscale cavities in the Feynman path integrals representation

A method is developed for the ab initio calculation of the distance between the energy levels of different spin states of electrons, as well as the spin of mixed quantum states subjected to thermal fluctuations. It is shown that theoretical investigations in the path integral representation can be carried out not only for mixed states, but also for pure quantum states. The approach proposed by us is free of the one-electron approximation and takes into account all spatial correlations between electrons. Demonstration computer calculations using the Monte Carlo method are carried out for the spin states of electrons in model ellipsoidal quantum dots. The inversion of levels corresponding to high and low spin states is detected in a system of three electrons. The calculations show that it is possible in principle to obtain spin states with arbitrarily close energies by varying the shape of a quantum dot. This fact is of fundamental importance for the prospects of constructing a new generation of storage and computing devices based on the spin qubits.

Coherent states and path integrals for model hamiltonians in quantum optics

A brief review of applications of the coherent states (CSes) in quantum optics is given. The CS representation of path integrals in the semi-classical approximation leads to the classical dynamics of CS parameters. Calculations of squeezing in the model of spontaneous parametric scattering are performed and the chaos in the collective dynamics of three-level atoms interacting with resonant photon modes is studied.

A probabilistic approach to the zero-mass limit problem for three magnetic relativistic Schrödinger heat semigroups

We consider three magnetic relativistic Schrödinger operators which correspond to the same classical symbol $\sqrt{(\xi - A(x))^2 + m^2} + V(x)$ and whose heat semigroups admit the Feynman-Kac-Itô type path integral representation $E[e^{ - S^m (x,\,t;\,X)} g(x + X(t))]$. Using these representations, we prove the convergence of these heat semigroups when the mass-parameter $m$ goes to zero. Its proof reduces to the convergence of $e^{- S^m (x,\,t;\,X)}$, which yields a limit theorem for exponentials of semimartingales as functionals of Lévy processes $X$.

Low-frequency pulse propagation over 510 km in the Philippine Sea: A comparison of observed and theoretical pulse spreading.

Observations of the spread of wander-corrected averaged pulses propagated over 510 km for 54 h in the Philippine Sea are compared to Monte Carlo predictions using a parabolic equation and path-integral predictions. Two simultaneous m-sequence signals are used, one centered at 200 Hz, the other at 300 Hz; both have a bandwidth of 50 Hz. The internal wave field is estimated at slightly less than unity Garrett-Munk strength. The observed spreads in all the early ray-like arrivals are very small, <1 ms (for pulse widths of 17 and 14 ms), which are on the order of the sampling period. Monte Carlo predictions show similar very small spreads. Pulse spread is one consequence of scattering, which is assumed to occur primarily at upper ocean depths where scattering processes are strongest and upward propagating rays refract downward. If scattering effects in early ray-like arrivals accumulate with increasing upper turning points, spread might show a similar dependence. Real and simulation results show no such dependence. Path-integral theory prediction of spread is accurate for the earliest ray-like arrivals, but appears to be increasingly biased high for later ray-like arrivals, which have more upper turning points.

The semiclassical propagator in Fock space: dynamical echo and many-body interference.

We present a semiclassical approach to many-body quantum propagation in terms of coherent sums over quantum amplitudes associated with the solutions of corresponding classical nonlinear wave equations. This approach adequately describes interference effects in the many-body space of interacting bosonic systems. The main quantity of interest, the transition amplitude between Fock states when the dynamics is driven by both single-particle contributions and many-body interactions of similar magnitude, is non-perturbatively constructed in the spirit of Gutzwiller's derivation of the van Vleck propagator from the path integral representation of the time evolution operator, but lifted to the space of symmetrized many-body states. Effects beyond mean-field, here representing the classical limit of the theory, are semiclassically described by means of interfering amplitudes where the action and stability of the classical solutions enter. In this way, a genuinely many-body echo phenomenon, coherent backscattering in Fock space, is presented arising due to coherent quantum interference between classical solutions related by time reversal.

Conversion of second-class constraints and resolving the zero-curvature conditions in the geometric quantization theory

In the approach to the geometric quantization, based on the conversion of second-class constraints, we resolve the respective non-linear zero curvature conditions for the extended symplectic potential. From the zero curvature conditions, we deduce new, linear, equations for the extended symplectic potential. Then we show that being the linear equations satisfied, their solution does certainly satisfy the non-linear zero curvature condition, as well. Finally, we give the functional resolution to the new linear equations, and then deduce the respective path integral representation. We do our consideration as to the general case of a phase superspace where both Boson and Fermion coordinates are present on equal footing.

Molecular thermodynamics of quantum square-well fluids using a path-integral perturbation theory

ABSTRACTWe present a thermodynamic perturbation theory for fluids composed of N quantum particles of diameter α and mass m contained within a volume V at temperature T, interacting via a pair square-well potential (SW) with energy depth ε and range λ. Our approach is based on the exact analogy between the discretised path- integral formalism of quantum mechanics and the partition function of a classical system composed of necklace molecules, introducing a Zwanzig expansion method, using a quantum hard-spheres fluid (QHS) as reference system in order to calculate the perturbation terms for the Helmholtz free energy AQSW of the SW system. These terms are obtained with path-integral Monte Carlo simulations in the canonical ensemble, and analytical results are provided as functions of the inverse temperature β = 1/kT, density ρ* = Nσ3/V, SW range λ and de Broglie thermal wavelength , where h and k are the Planck's and Boltzmann's constants, respectively. Accurate results are obtained for thermodynamic states comprised in the region ρ* ≤ 0.7, 1.2 ≤ λ ≤ 1.8, and λ*B ⩽ 0.9. Quantum effects are more noticeable for shorter SW ranges, although the main thermodynamic effects are given by the QHS free energy.

A note on the path integral representation for Majorana fermions

Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev–Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.

Superconductor-insulator transition of Josephson-junction arrays on a honeycomb lattice in a magnetic field

We study the superconductor to insulator transition at zero temperature in a Josephson-junction array model on a honeycomb lattice with $f$ flux quantum per plaquette. The path integral representation of the model corresponds to a (2+1)-dimensional classical model, which is used to investigate the critical behavior by extensive Monte Carlo simulations on large system sizes. In contrast to the model on a square lattice, the transition is found to be first order for $f=1/3$ and continuous for $f=1/2$ but in a different universality class. The correlation-length critical exponent is estimated from finite-size scaling of vortex correlations. The estimated universal conductivity at the transition is approximately four times its value for $f=0$. The results are compared with experimental observations on ultrathin superconducting films with a triangular lattice of nanoholes in a transverse magnetic field.

Can quantum transition state theory be defined as an exact t = 0+ limit?

The definition of the classical transition state theory (TST) as a t → 0+ limit of the flux-side time correlation function relies on the assumption that simultaneous measurement of population and flux is a well defined physical process. However, the noncommutativity of the two measurements in quantum mechanics makes the extension of such a concept to the quantum regime impossible. For this reason, quantum TST (QTST) has been generally accepted as any kind of quantum rate theory reproducing the TST in the classical limit, and there has been a broad consensus that no unique QTST retaining all the properties of TST can be defined. Contrary to this widely held view, Hele and Althorpe (HA) [J. Chem. Phys. 138, 084108 (2013)] recently suggested that a true QTST can be defined as the exact t → 0+ limit of a certain kind of quantum flux-side time correlation function and that it is equivalent to the ring polymer molecular dynamics (RPMD) TST. This work seeks to question and clarify certain assumptions underlying these suggestions and their implications. First, the time correlation function used by HA as a starting expression is not related to the kinetic rate constant by virtue of linear response theory, which is the first important step in relating a t = 0+ limit to a physically measurable rate. Second, a theoretical analysis calls into question a key step in HA's proof which appears not to rely on an exact quantum mechanical identity. The correction of this makes the true t = 0+ limit of HA's QTST different from the RPMD-TST rate expression, but rather equal to the well-known path integral quantum transition state theory rate expression for the case of centroid dividing surface. An alternative quantum rate expression is then formulated starting from the linear response theory and by applying a recently developed formalism of real time dynamics of imaginary time path integrals [S. Jang, A. V. Sinitskiy, and G. A. Voth, J. Chem. Phys. 140, 154103 (2014)]. It is shown that the t → 0+ limit of the new rate expression vanishes in the exact quantum limit.

Path Integral Representation of the Wigner Function in Canonical Ensemble

AbstractQuantum effects can affect the shape of the particle kinetic energy distribution function, as the interaction of a particle with its surroundings restricts the volume of configuration space, which, due to the uncertainty relation, results in an increase in the volume of the momentum space, i.e., in a rise in the fraction of particles with higher momenta. Allowing for quantum effects at calculations of the equilibrium rate constants of inelastic processes is important in consideration of such phenomena as the transition of combustion into detonation, flame propagation, vibrational relaxation, and even thermonuclear fusion at high pressure and low temperatures. Quantum effects are also important in treatment of transport properties of the strongly interacting systems of many particles. In this work the new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in harmonic approximation. New quantum Monte‐Carlo method for ab initio calculations of the average values of quantum operators, Wigner function, momentum and position distributions and wave functions of the ground state has been developed and tested. Obtained results are in a very good agreement with available analytical results and results of usual path‐integral Monte‐Carlo method. The developed approach allows simulation of thermodynamic and kinetic properties of quantum systems and calculation average values of quantum operators, when the usual path integral Monte Carlo methods in configurational space failed. (© 2016 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Temperature Dependence of NMR Parameters Calculated from Path Integral Molecular Dynamics Simulations

The influence of temperature on NMR chemical shifts and quadrupolar couplings in model molecular organic solids is explored using path integral molecular dynamics (PIMD) and density functional theory (DFT) calculations of shielding and electric field gradient (EFG) tensors. An approach based on convoluting calculated shielding or EFG tensor components with probability distributions of selected bond distances and valence angles obtained from DFT-PIMD simulations at several temperatures is used to calculate the temperature effects. The probability distributions obtained from the quantum PIMD simulations, which includes nuclear quantum effects, are significantly broader and less temperature dependent than those obtained with conventional DFT molecular dynamics or with 1D scans through the potential energy surface. Predicted NMR observables for the model systems were in excellent agreement with experimental data.

Publisher's Note: Local-time representation of path integrals [Phys. Rev. E 92, 062137 (2015)

Received 29 January 2016DOI:https://doi.org/10.1103/PhysRevE.93.029901©2016 American Physical Society

Ab initio molecular dynamics with nuclear quantum effects at classical cost: Ring polymer contraction for density functional theory.

Path integral molecular dynamics simulations, combined with an ab initio evaluation of interactions using electronic structure theory, incorporate the quantum mechanical nature of both the electrons and nuclei, which are essential to accurately describe systems containing light nuclei. However, path integral simulations have traditionally required a computational cost around two orders of magnitude greater than treating the nuclei classically, making them prohibitively costly for most applications. Here we show that the cost of path integral simulations can be dramatically reduced by extending our ring polymer contraction approach to ab initio molecular dynamics simulations. By using density functional tight binding as a reference system, we show that our ring polymer contraction scheme gives rapid and systematic convergence to the full path integral density functional theory result. We demonstrate the efficiency of this approach in ab initio simulations of liquid water and the reactive protonated and deprotonated water dimer systems. We find that the vast majority of the nuclear quantum effects are accurately captured using contraction to just the ring polymer centroid, which requires the same number of density functional theory calculations as a classical simulation. Combined with a multiple time step scheme using the same reference system, which allows the time step to be increased, this approach is as fast as a typical classical ab initio molecular dynamics simulation and 35× faster than a full path integral calculation, while still exactly including the quantum sampling of nuclei. This development thus offers a route to routinely include nuclear quantum effects in ab initio molecular dynamics simulations at negligible computational cost.

Adaptive Importance Sampling for Control and Inference

Path integral (PI) control problems are a restricted class of non-linear control problems that can be solved formally as a Feyman-Kac path integral and can be estimated using Monte Carlo sampling. In this contribution we review path integral control theory in the finite horizon case. We subsequently focus on the problem how to compute and represent control solutions. Within the PI theory, the question of how to compute becomes the question of importance sampling. Efficient importance samplers are state feedback controllers and the use of these requires an efficient representation. Learning and representing effective state-feedback controllers for non-linear stochastic control problems is a very challenging, and largely unsolved, problem. We show how to learn and represent such controllers using ideas from the cross entropy method. We derive a gradient descent method that allows to learn feed-back controllers using an arbitrary parametrisation. We refer to this method as the Path Integral Cross Entropy method or PICE. We illustrate this method for some simple examples. The path integral control methods can be used to estimate the posterior distribution in latent state models. In neuroscience these problems arise when estimating connectivity from neural recording data using EM. We demonstrate the path integral control method as an accurate alternative to particle filtering.

Second-order interference of two independent and tunable single-mode continuous-wave lasers**Project supported by the National Natural Science Foundation of China (Grant No. 11404255) and the Doctor Foundation of Education Ministry of China (Grant No. 20130201120013).

The second-order temporal interference of two independent single-mode continuous-wave lasers is discussed by employing two-photon interference in Feynman’s path integral theory. It is concluded that whether the second-order temporal interference pattern can or cannot be retrieved via two-photon coincidence counting rate is dependent on the resolution time of the detection system and the frequency difference between these two lasers. Two identical and tunable single-mode continuous-wave diode lasers are employed to verify the predictions. These studies are helpful to understand the physics of two-photon interference with photons of different spectra.

Deriving the exact nonadiabatic quantum propagator in the mapping variable representation.

We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting Liouvillian is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact Liouvillian lead to existing approximate semiclassical and mixed quantum-classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact Liouvillian, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.

Auxiliary field loop expansion of the effective action for a class of stochastic partial differential equations

We present an alternative to the perturbative (in coupling constant) diagrammatic approach for studying stochastic dynamics of a class of reaction diffusion systems. Our approach is based on an auxiliary field loop expansion for the path integral representation for the generating functional of the noise induced correlation functions of the fields describing these systems. The systems we consider include Langevin systems describable by the set of self interacting classical fields ϕi(x,t) in the presence of external noise ηi(x,t), namely (∂t−ν∇2)ϕ−F[ϕ]=η, as well as chemical reaction annihilation processes obtained by applying the many-body approach of Doi–Peliti to the Master Equation formulation of these problems. We consider two different effective actions, one based on the Onsager–Machlup (OM) approach, and the other due to Janssen–deGenneris based on the Martin–Siggia–Rose (MSR) response function approach. For the simple models we consider, we determine an analytic expression for the Energy landscape (effective potential) in both formalisms and show how to obtain the more physical effective potential of the Onsager–Machlup approach from the MSR effective potential in leading order in the auxiliary field loop expansion. For the KPZ equation we find that our approximation, which is non-perturbative and obeys broken symmetry Ward identities, does not lead to the appearance of a fluctuation induced symmetry breakdown. This contradicts the results of earlier studies.