Path Integral Framework Research Articles

Effective action approach for preheating

We present a semiclassical non-perturbative approach for calculating the preheating process at the end of inflation. Our method involves integrating out the decayed particles within the path integral framework and subsequently determining world-line instanton solutions in the effective action. This enables us to obtain the effective action of the inflaton, with its imaginary part linked to the phenomenon of particle creation driven by coherent inflaton field oscillations. Additionally, we utilize the Bogoliubov transformation to investigate the evolution of particle density within the medium after multiple inflaton oscillations. We apply our approach to various final state particles, including scalar fields, tachyonic fields, and gauge fields. The non-perturbative approach provides analytical results for preheating that are in accord with previous methods.

Incorporation of Empirical Gain and Loss Mechanisms in Open Quantum Systems through Path Integral Lindblad Dynamics.

Path integrals offer a robust approach for simulating open quantum dynamics with advancements transcending initial system size limitations. However, accurately modeling systems governed by mechanisms that do not conserve the number of quantum particles, such as lossy cavity modes, remains a challenge. We present a method to incorporate such empirical source and drain mechanisms within a path integral framework using quantum master equations. This technique facilitates rigorous inclusion of bath degrees of freedom while accommodating empirical time scales via Lindbladian dynamics. Computational costs are primarily driven by the path integral method with minimal overhead from Lindbladian terms. We use it to study exciton transport in a four-site Fenna-Matthews-Olson model, examining the potential loss of the exciton to the reaction center. This path integral Lindblad method promises an enhanced ability to simulate dynamics and will be fundamental to simulation of spectra in diverse quantum processes in open systems.

Generalized free energy and dynamical state transition of the dyonic AdS black hole in the grand canonical ensemble

We study the generalized free energy of the dyonic AdS black hole in an ensemble with varying electric charge qE and fixed magnetic charge qM. When we adjust the temperature T and the electric potential ΦE of the ensemble, the Ricci scalar curvature R and electromagnetic potential Au usually diverge at the horizon. We regularize them and incorporate the off-shell corrections into the Einstein-Hilbert action. Alternatively, we find that the off-shell corrections can also be obtained by adding a boundary near the horizon to exclude the singularities. Ultimately, we derive the generalized free energy which is consistent with the definition of the thermodynamic relations. Based on the generalized free energy landscape, we can describe the dynamics of state transition as a stochastic process quantified by the Langevin equation. The path integral framework can be formulated to derive the time-dependent trajectory of the order parameter and the time evolution of the transition probability. By comparing the probability with the result of the classical master equation, we attribute the contribution to the probability of one pseudomolecule or antipseudomolecule (the instanton and anti-instanton pairs) to the rate of state transition. These results are consistent with the qualitative analysis of the free energy landscape.

Analysing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations.

Rigorous diagnostics of experiments with warm dense matter are notoriously difficult. A key method is X-ray Thomson scattering (XRTS), but the interpretation of XRTS measurements is usually based on theoretical models that entail various approximations. Recently, Dornheim et al. [Nat. Commun. 13, 7911 (2022)] introduced a new framework for temperature diagnostics of XRTS experiments that is based on imaginary-time correlation functions. On the one hand, switching from the frequency to the imaginary-time domain gives one direct access to a number of physical properties, which facilitates the extraction of the temperature of arbitrarily complex materials without relying on any models or approximations. On the other hand, the bulk of theoretical work in dynamic quantum many-body theory is devoted to the frequency domain, and, to the best of our knowledge, the manifestation of physics properties within the imaginary-time density-density correlation function (ITCF) remains poorly understood. In the present work, we aim to fill this gap by introducing a simple, semi-analytical model for the imaginary-time dependence of two-body correlations within the framework of imaginary-time path integrals. As a practical example, we compare our new model to extensive ab initio path integral Monte Carlo results for the ITCF of a uniform electron gas, and find excellent agreement over a broad range of wavenumbers, densities and temperatures. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

A General Path-Integral Monte Carlo Method for Exact Simulations of Chemical Reaction Networks.

Chemical Reaction Networks (CRNs) are stochastic many-body systems used to model real-world chemical systems through a differential Master Equation (ME); analytical solutions to these equations are only known for the simplest systems. In this paper, we construct a path-integral inspirited framework for studying CRNs. Under this scheme, the time-evolution of a reaction network can be encoded in a Hamiltonian-like operator. This operator yields a probability distribution which can be sampled, using Monte Carlo Methods, to generate exact numerical simulations of a reaction network. We recover the grand probability function used in the Gillespie Algorithm as an approximation to our probability distribution, which motivates the addition of a leapfrog correction step. To assess the utility of our method in forecasting real-world phenomena, and to contrast it with the Gillespie Algorithm, we simulated a COVID-19 epidemiological model using parameters from the United States for the Original Strain and the Alpha, Delta and Omicron Variants. By comparing the results of these simulations with official data, we found that our model closely agrees with the measured population dynamics, and given the generality of this framework it can also be applied to study the spread dynamics of other contagious diseases.

Theoretical Simulations of Kinetic Isotope Effects on Decarboxylation of 3-Carboxybenzisoxazole.

Kinetic isotope effect values on the decarboxylation of 3-carboxybenzisoxazole have been computed using the second-order Kleinert's variational perturbation theory in the framework of Feynman's path integrals along with the potential energy surface obtained at the MP2/6-31+G(d) level. Good agreement with the experimental data was obtained, demonstrating that this novel computational approach for computing KIE values of organic reaction is a viable alternative to the traditional method employing the Bigeleisen equation and harmonic vibrational frequencies. Compared with the experimental measurements, consideration of anharmonicity and tunneling effects can significantly improve the calculated KIE values, reducing the root-mean-square deviation from 1.19 % for traditional method to 0.20 % for path-integral method.

Efficient estimation of boundary integrals for path-space differentiable rendering

Boundary integrals are unique to physics-based differentiable rendering and crucial for differentiating with respect to object geometry. Under the differential path integral framework---which has enabled the development of sophisticated differentiable rendering algorithms---the boundary components are themselves path integrals. Previously, although the mathematical formulation of boundary path integrals have been established, efficient estimation of these integrals remains challenging. In this paper, we introduce a new technique to efficiently estimate boundary path integrals. A key component of our technique is a primary-sample-space guiding step for importance sampling of boundary segments. Additionally, we show multiple importance sampling can be used to combine multiple guided samplings. Lastly, we introduce an optional edge sorting step to further improve the runtime performance. We evaluate the effectiveness of our method using several differentiable-rendering and inverse-rendering examples and provide comparisons with existing methods for reconstruction as well as gradient quality.

Most probable transition paths in piecewise-smooth stochastic differential equations

We develop a path integral framework for determining most probable paths for a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin–Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider an n-dimensional system with a switching manifold in the drift that forms an (n−1)−dimensional hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. To do this, we mollify the drift and use Γ−convergence to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin–Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of the derived functional through two case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.

Path integral and instantons for the dynamical process and phase transition rate of Reissner-Nordström-AdS black holes

We propose a new approach to study the dynamical phase transition of RNAdS black holes on the underlying free energy landscape. By formulating a path integral framework, we can quantify the kinetic paths representing the history from the initial state to the end state, which provides us a visualized yet quantified picture about how the phase transition proceeds. Based on these paths, we derive the analytical formulas for the time evolution of the transition probability and provide a physical interpretation of the contribution to the probability from one "pseudomolecule" ("anti-pseudomolecule"), which is actually the phase transition rate from the small(large) to the large(small) black hole state. These numerical results show a good consistency with the underlying free energy landscape topography.

Path Integral Framework for Characterizing and Controlling Decoherence Induced by Nonstationary Environments on a Quantum Probe

A framework to characterize and control the influence of out-of-equilibrium environments on a quantum device is presented: A methodology inspired by field theory with relevance to quantum technologies.

Hartle-Hawking state in the real-time formalism

We study self-interacting massive scalar field theory in static spacetimes with a bifurcate Killing horizon and a wedge reflection. In this theory the Hartle-Hawking state is defined to have the $N$-point correlation functions obtained by analytically continuing those in the Euclidean theory, whereas the double Kubo-Martin-Schwinger (KMS) state is the pure state invariant under the Killing flow and the wedge reflection which is regular on the bifurcate Killing horizon and reduces to the thermal state at the Hawking temperature in each of the two static regions. We demonstrate in the Schwinger-Keldysh operator formalism of perturbation theory the equivalence between the Hartle-Hawking state and the double KMS state with the Hawking temperature, which was shown before by Jacobson in the path-integral framework.

A multisite decomposition of the tensor network path integrals.

Tensor network decompositions of path integrals for simulating open quantum systems have recently been proven to be useful. However, these methods scale exponentially with the system size. This makes it challenging to simulate the non-equilibrium dynamics of extended quantum systems coupled with local dissipative environments. In this work, we extend the tensor network path integral (TNPI) framework to efficiently simulate such extended systems. The Feynman-Vernon influence functional is a popular approach used to account for the effect of environments on the dynamics of the system. In order to facilitate the incorporation of the influence functional into a multisite framework (MS-TNPI), we combine a matrix product state (MPS) decomposition of the reduced density tensor of the system along the sites with a corresponding tensor network representation of the time axis to construct an efficient 2D tensor network. The 2D MS-TNPI network, when contracted, yields the time-dependent reduced density tensor of the extended system as an MPS. The algorithm presented is independent of the system Hamiltonian. We outline an iteration scheme to take the simulation beyond the non-Markovian memory introduced by solvents. Applications to spin chains coupled to local harmonic baths are presented; we consider the Ising, XXZ, and Heisenberg models, demonstrating that the presence of local environments can often dissipate the entanglement between the sites. We discuss three factors causing the system to transition from a coherent oscillatory dynamics to a fully incoherent dynamics. The MS-TNPI method is useful for studying a variety of extended quantum systems coupled with solvents.

Constrained stochastic optimal control with learned importance sampling: A path integral approach

Modern robotic systems are expected to operate robustly in partially unknown environments. This article proposes an algorithm capable of controlling a wide range of high-dimensional robotic systems in such challenging scenarios. Our method is based on the path integral formulation of stochastic optimal control, which we extend with constraint-handling capabilities. Under our control law, the optimal input is inferred from a set of stochastic rollouts of the system dynamics. These rollouts are simulated by a physics engine, placing minimal restrictions on the types of systems and environments that can be modeled. Although sampling-based algorithms are typically not suitable for online control, we demonstrate in this work how importance sampling and constraints can be used to effectively curb the sampling complexity and enable real-time control applications. Furthermore, the path integral framework provides a natural way of incorporating existing control architectures as ancillary controllers for shaping the sampling distribution. Our results reveal that even in cases where the ancillary controller would fail, our stochastic control algorithm provides an additional safety and robustness layer. Moreover, in the absence of an existing ancillary controller, our method can be used to train a parametrized importance sampling policy using data from the stochastic rollouts. The algorithm may thereby bootstrap itself by learning an importance sampling policy offline and then refining it to unseen environments during online control. We validate our results on three robotic systems, including hardware experiments on a quadrupedal robot.

Dualities and models in various dimensions

Working within the path-integral framework we first establish a duality between the partition functions of two U(1) gauge theories with a theta term in d=4 space-time dimensions. Then, after a dimensional reduction to d=3 dimensions we arrive to the partition function of a U(1) gauge theory coupled to a scalar field with an action that exhibits a Dirac monopole solution. A subsequent reduction to d=2 dimensions leads to the partition function of a theory in which the gauge field decouples from two scalars which have non-trivial vortex-like solutions. Finally this d=2 partition function can be related to the bosonized version of the two-dimensional QED2 (Schwinger) model.

Path integral for new models of time-dependent harmonic oscillators

We have presented a particular direct method to solve time-dependent quantum problems within the framework of path integrals by using explicitly time-dependent transformations. We have applied it to the case of harmonic oscillator with time-dependent mass and frequency. Some examples have illustrated the method used. From their exact propagators, normalized solutions have been obtained.

Topology-guided path integral approach for stochastic optimal control in cluttered environment

This paper addresses planning and control of robot motion under uncertainty that is formulated as a continuous-time, continuous-space stochastic optimal control problem, by developing a topology-guided path integral control method. The path integral control framework, which forms the backbone of the proposed method, re-writes the Hamilton–Jacobi–Bellman equation as a statistical inference problem; the resulting inference problem is solved by a sampling procedure that computes the distribution of controlled trajectories around the trajectory by the passive dynamics. For motion control of robots in a highly cluttered environment, however, this sampling can easily be trapped in a local minimum unless the sample size is very large, since the global optimality of local minima depends on the degree of uncertainty. Thus, a homology-embedded sampling-based planner that identifies many (potentially) local-minimum trajectories in different homology classes is developed to aid the sampling process. In combination with a receding-horizon fashion of the optimal control the proposed method produces a dynamically feasible and collision-free motion plans without being trapped in a local minimum. Numerical examples on a synthetic toy problem and on quadrotor control in a complex obstacle field demonstrate the validity of the proposed method.

Duality between the massive sine-Gordon and the massive Schwinger models at finite temperature

The massive Schwinger and the massive sine-Gordon models are proved to be equivalent at finite temperature, using the path-integral framework. The well known relations among the parameters of these models to establish the duality at $T=0$, also remain valid at non zero temperature.

Vacuum expectation value of twist fields

Twist fields emerge in a number of physical applications ranging from entanglement entropy to scattering amplitudes in four-dimensional gauge theories. In this work, their vacuum expectation values are studied in the path integral framework. By performing a gauge transformation, their correlation functions are reduced to field theory of matter fields in external Aharonov-Bohm vortices. The resulting functional determinants are then analyzed within the zeta function regularization for the spectrum of Bessel zeros and concise formulas are derived.

Parity effect in a mesoscopic Fermi gas

We develop a quantitative analytic theory that accurately describes the odd-even effect observed experimentally in a one-dimensional, trapped Fermi gas with a small number of particles [G. Z\"urn et al., Phys. Rev. Lett. 111, 175302 (2013)]. We find that the underlying physics is similar to the parity effect known to exist in ultrasmall mesoscopic superconducting grains and atomic nuclei. However, in contrast to superconducting nanograins, the density (Hartree) correction dominates over the superconducting pairing fluctuations and leads to a much more pronounced odd-even effect in the mesoscopic, trapped Fermi gas. We calculate the corresponding parity parameter and separation energy using both perturbation theory and a path integral framework in the mesoscopic limit, generalized to account for the effects of the trap, pairing fluctuations, and Hartree corrections. Our results are in an excellent quantitative agreement with experimental data and exact diagonalization. Finally, we discuss a few-to-many particle crossover between the perturbative mesoscopic regime and non-perturbative many-body physics that the system approaches in the thermodynamic limit.

Nuclear quantum effects on the thermal expansion coefficient of hexagonal boron nitride monolayer

This work examines the importance of vibrational delocalization on a basic thermomechanical property of a hexagonal boron nitride monolayer, namely its thermal expansion coefficient (TEC). Using a recently parametrized bond-order potential of the Tersoff type, the TEC was theoretically obtained from the thermal variations of the lattice parameter a(T) calculated using three different methods: (i) the quasiharmonic approximation; (ii) its anharmonic improvement based on self-consistent phonons; (iii) fully anharmonic Monte Carlo simulations possibly enhanced within the path-integral framework to account for nuclear quantum effects. The results obtained with the three methods are generally consistent with one another and with other recently published data, and indicate that the TEC is negative at least up to ca. 700 K, quantum mechanical effects leading to a significant expansion by about 50% relative to the classical result. Comparison with experimental data on bulk hexagonal BN suggests significant differences, which originate from possible inaccuracies in the model that tend to underestimate the lattice parameter itself, and most likely from the 2D nature of the monolayer and the key contribution of out-of-plane modes. The effects of isotopic purity in the natural abundances of boron are found to be insignificant.