Path Integral Representation Research Articles

Resurgence of a de Sitter Glauber‐Sudarshan State: Nodal Diagrams and Borel Resummation

AbstractIt is shown in this article that an explicit construction of a four‐dimensional de Sitter space may be performed using a diagrammatic approach via nodal diagrams emanating from the path integral representation of the Glauber‐Sudarshan state. Sum of these diagrams typically leads to an asymptotic series of Gevrey kind which can then be Borel resummed, thus reproducing the non‐perturbative structure of the system. The analysis shows that four‐dimensional de Sitter space is not only possible in string theory overcoming the no‐go and the swampland criteria—albeit as a Glauber‐Sudarshan state—but it may also be non‐perturbatively stable within a controlled temporal domain. Somewhat consistently, the Borel resummation of the Gevrey series provides strong hint towards the positivity of the cosmological constant.

Structural and Thermodynamic Properties of Magnesium-Rich Liquids at Ultrahigh Pressure

We explore the structural properties of Mg, MgO, and MgSiO3 liquids from ab initio computer simulations at conditions that are relevant for the interiors of giant planets, stars, shock compression measurements, and inertial confinement fusion experiments. Using path-integral Monte Carlo and density functional theory molecular dynamics, we derive the equation of state of magnesium-rich liquids in the regime of condensed and warm dense matter, with densities ranging from 0.32 to 86.11 g cm−3 and temperatures from 20,000 K to 5 × 108 K. We study the electronic structure of magnesium as a function of density and temperature and the correlations of the atomic motion, finding an unexpected local maximum in the pair correlation functions that emerges at high densities which decreases the coordination number of elemental magnesium and reveals a higher packing. This phenomenon is not observed in other magnesium liquids, which maintain a rather constant coordination number.

Most probable path of active Ornstein-Uhlenbeck particles.

Using the path integral representation of the nonequilibrium dynamics, we compute the most probable path between arbitrary starting and final points that is followed by an active particle driven by persistent noise. We focus our attention on the case of active particles immersed in harmonic potentials, where the trajectory can be computed analytically. Once we consider the extended Markovian dynamics where the self-propulsive drive evolves according to an Ornstein-Uhlenbeck process, we can compute the trajectory analytically with arbitrary conditions on position and self-propulsion velocity. We test the analytical predictions against numerical simulations and we compare the analytical results with those obtained within approximated equilibriumlike dynamics.

K-space theory and convective gains of stimulated Raman side scattering.

We have developed a k-space theory for stimulated Raman scattering. The theory is used to calculate the convective gain of stimulated Raman side scattering (SRSS) in order to clarify the discrepancies found between previous gain formulas proposed in the literature. The gains are dramatically modified by the eigenvalue of SRSS, and the maximum gain occurs not at the perfect wave-number matching condition but instead at a wave number having a small deviation that is associated with the eigenvalue. Numerical solutions of the k-space theory equationsare compared with, and used to verify, analytically derived gains. We demonstrate connections to the existing path integral theories, and we derive a similar path integral formula in k space.

Toverline{T} + Λ2 from a 2d gravity path integral

We develop a two-dimensional gravity path integral formulation of the Toverline{T} + Λ2 deformation of quantum field theory. This provides an exactly solvable generalization of the pure Toverline{T} deformation that is relevant for de Sitter and flat space holography. The path integral sheds light on quantum aspects of these flows in curved space, most notably the Weyl anomaly, the operator relation for the trace of the energy-momentum tensor, and the renormalization of the composite Toverline{T} operator. It also applies to both the Hagedorn and the holographic signs of such flows. We present explicit calculations for the torus and sphere partition functions that reproduce previous results in the literature, now in path integral language. Finally, we use the path integral representation in order to establish an explicit map with 3d gravity, and obtain new predictions for flat space holography.

Quantum field theory based quantum information: Measurements and correlations

This is the first in a series of papers aiming to develop a relativistic quantum information theory in terms of unequal-time correlation functions in quantum field theory. In this work, we highlight two formalisms which together can provide a useful theoretical platform suitable for further developments: (1) Quantum field measurements using the Quantum Temporal Probabilities (QTP) method; (2) Closed-Time-Path (CTP) formalism for causal time evolutions. QTP incorporates the detector into the quantum description, while emphasizing that the records of measurement are macroscopic and can be expressed in terms of classical spacetime coordinates. We first present a new, elementary derivation of the QTP formulas for the probabilities of n measurement events. We then demonstrate the relation of QTP with the Closed-Time-Path formalism, by writing an explicit formula that relates the associated generating functionals. We exploit the path integral representation of the CTP formalism, in order to express the measured probabilities in terms of path integrals. After this, we provide some simple applications of the QTP formalism. In particular, we show how Unruh–DeWitt detector models and Glauber’s photodetection theory appear as limiting cases. Finally, with quantum correlation being the pivotal notion in relativistic quantum information and measurements, we highlight the role played by the CTP two-particle irreducible effective action which enables one to tap into the resources of non-equilibrium quantum field theory for our stated purpose.

Hong-Ou-Mandel interference of longitudinal spatially coherent beams.

The Hong-Ou-Mandel (HOM) interference of two light beams having different longitudinal spatial coherence properties is investigated theoretically and experimentally. The normalized second-order correlation function (g (2)) is determined for the interfering photons from two sources of different angular widths using Feynman's path integral theory. We find that the difference in angular width of the sources has an explicit impact on the HOM interference pattern, which can be quantified through the visibility and full width at half maxima of the HOM dip. The effect of distinguishability of the interfering longitudinally spatially coherent beams on the HOM dip is verified experimentally and is analogous to non-classical HOM interference. The magnitude of the angular width of beams manifested through the difference in angular width has a significant impact. Along with the difference of two sources, this HOM scheme is sensitive to the angular spectrum width of each source. The enhanced sensitivity can be useful in the remote sensing of objects and beams in metrological applications. This work can play a significant role in fundamental and applied physics.

Research progress of measurement of propagators in path integrals

The propagator plays a central role in path integral theory and therefore has significant value in various fields of modern quantum physics, where path integral representations can be used. However, owing to the fact that it has not been directly measured in experiment, progress of experimental studies of quantum systems based on path integral representations has been seriously limited. Recently, we proposed a propagator measurement scheme based on the direct measurement of the wave function and successfully performed the first experimental measurement of the propagator by using a single photon experiment. Furthermore, in this study, the quantum principle of least action is demonstrated for the first time. This research successfully addresses the technical challenges of path integral experimental studies. In this work, we review the research progress in this field, including a brief introduction to the basic concepts and research progress of direct wave function measurement, and a detailed description of the theoretical model, experimental design, and experimental results of propagator measurement. Finally, we introduce an important application example, which can serve as the experimental demonstration of the quantum principle of least action through propagator measurement. The research progress of propagator measurement reviewed in this work will provide important references for future experimental studies by using this method.

Diffusion of a tracer in a dense mixture of soft particles connected to different thermostats.

We study the dynamics of a tracer in a dense mixture of particles connected to different thermostats. Starting from the overdamped Langevin equationsthat describe the evolution of the system, we derive the expression of the self-diffusion coefficient of a tagged particle in the suspension, in the limit of soft interactions between the particles. Our derivation, which relies on the linearization of the Dean-Kawasaki equationsobeyed by the density fields and on a path-integral representation of the dynamics of the tracer, extends previous derivations that held for tracers in contact with a single bath. Our analytical result is confronted to results from Brownian dynamics simulations. The agreement with numerical simulations is very good even for high densities. We show how the diffusivity of tracers can be affected by the activity of a dense environment of soft particles that may represent polymer coils-a result that could be of relevance in the interpretation of measurements of diffusivity in biological media. Finally, our analytical result is general and can be applied to the diffusion of tracers coupled to different types of fluctuating environments, provided that their evolution equationsare linear and that the coupling between the tracer and the bath is weak.

Path integrals for fractional Brownian motion and fractional Gaussian noise.

Wiener's path integral plays a central role in the study of Brownian motion. Here we derive exact path-integral representations for the more general fractional Brownian motion (FBM) and for its time derivative process, fractional Gaussian noise (FGN). These paradigmatic non-Markovian stochastic processes, introduced by Kolmogorov, Mandelbrot, and van Ness, found numerous applications across the disciplines, ranging from anomalous diffusion in cellular environments to mathematical finance. Their exact path-integral representations were previously unknown. Our formalism exploits the Gaussianity of the FBM and FGN, relies on the theory of singular integral equations, and overcomes some technical difficulties by representing the action functional for the FBM in terms of the FGN for the subdiffusive FBM and in terms of the derivative of the FGN for the super-diffusive FBM. We also extend the formalism to include external forcing. The exact and explicit path-integral representations make inroads in the study of the FBM and FGN.

Efficient and scalable path integral Monte Carlo simulations with worm-type updates for Bose-Hubbard and XXZ models

We present a novel and open-source implementation of the worm algorithm, which is an algorithm to simulate Bose-Hubbard and sign-positive spin models using a path-integral representation of the partition function. The code can deal with arbitrary lattice structures and assumes spin-exchange terms, or bosonic hopping amplitudes, between nearest-neighbor sites, and local or nearest-neighbor interactions of the density-density type. We explicitly demonstrate the near-linear scaling of the algorithm with respect to the system volume and the inverse temperature and analyze the autocorrelation times in the vicinity of a U(1)U(1) second order phase transition. The code is written in such a way that extensions to other lattice models as well as closely-related sign-positive models can be done straightforwardly on top of the provided framework.

Codebase release 1.0 Worm

We present a novel and open-source implementation of the worm algorithm, which is an algorithm to simulate Bose-Hubbard and sign-positive spin models using a path-integral representation of the partition function. The code can deal with arbitrary lattice structures and assumes spin-exchange terms, or bosonic hopping amplitudes, between nearest-neighbor sites, and local or nearest-neighbor interactions of the density-density type. We explicitly demonstrate the near-linear scaling of the algorithm with respect to the system volume and the inverse temperature and analyze the autocorrelation times in the vicinity of a U(1)U(1) second order phase transition. The code is written in such a way that extensions to other lattice models as well as closely-related sign-positive models can be done straightforwardly on top of the provided framework.

Fermionic matrix models and bosonization

We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a $U(1)\times SU(N)\times SU(L)$ symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (non-singlet) bosonic fields. The bosonic path integral representations in the vector limits $(N,1)$ and $(1,L)$ are matched to fermionic Fock space Hamiltonians expressed in terms of quadratic Casimirs and some additional terms involving the Cartan subalgebra, which makes explicit the symmetries preserved by scalar and matrix bosonizations at the quantum level. For the case of non-singlet bosonization we find an equivalence between the vector model and the Polychronakos+Frahm spin model. Using this relation we compute the free energy. Finally, we compute the eigenvalue distribution in the large $N,L$-limit with $ \alpha = \frac{L}{N}$ fixed. The model displays a third order phase transition as we vary the temperature which, in the $\alpha\gg1$ limit, can be characterized analytically. We conclude finding the critical curve in the parameter space were the eigenvalue distribution transitions from single to double cut.

Optimal stochastic modeling in random media propagation: Dynamically orthogonal parabolic equations?

Reliable underwater acoustic propagation is challenging due to complex ocean dynamics such as internal-waves and to the uncertain larger-scale ocean physics, acoustics, bathymetry, and seabed fields. For accurate acoustic propagation, capturing the important environmental uncertainties and variabilities and predicting the probability distributions of the acoustic pressure field is then what matters. Prior works towards addressing this goal include (i) wave propagation in random media techniques such as perturbation methods, path integral theory, and coupled-mode transport theory, and (ii) probabilistic modeling techniques such as Monte Carlo sampling and Polynomial Chaos expansions. Recently, we developed a novel technique called the Dynamically Orthogonal Parabolic Equations (DO-ParEq) which represent the sound speed, density, bathymetry, and acoustic pressure fields using optimal dynamic Karhunen-Loeve decompositions. The DO-ParEq are range-evolving partial and stochastic differential equations preserving acoustic nonlinearities and non-Gaussian properties. In this presentation, we showcase the theoretical and computational advantages of the DO-ParEq framework compared to the state-of-the-art techniques in the Pekeris waveguide and wedge benchmark problems, in addition to a realistic ocean example in the New York Bight region.

Renormalization of excitonic properties by polar phonons.

We employ quasiparticle path integral molecular dynamics to study how the excitonic properties of model semiconductors are altered by electron-phonon coupling. We describe ways within a path integral representation of the system to evaluate the renormalized mass, binding energy, and radiative recombination rate of excitons in the presence of a fluctuating lattice. To illustrate this approach, we consider Fröhlich-type electron-phonon interactions and employ an imaginary time influence functional to incorporate phonon-induced effects nonperturbatively. The effective mass and binding energies are compared with perturbative and variational approaches, which provide qualitatively consistent trends. We evaluate electron-hole recombination rates as mediated through both trap-assisted and bimolecular processes, developing a consistent statistical mechanical approach valid in the reaction limited regime. These calculations demonstrate how phonons screen electron-hole interactions, generically reducing exciton binding energies and increasing their radiative lifetimes.

Coarse-Graining of Imaginary Time Feynman Path Integrals: Inclusion of Intramolecular Interactions and Bottom-up Force-Matching.

Feynman's imaginary time path integral formalism of quantum statistical mechanics and the corresponding quantum-classical isomorphism provide a tangible way of incorporating nuclear quantum effect (NQE) in the simulation of condensed matter systems using well-developed classical simulation techniques. Our previous work has presented the many-body coarse-graining of path integral (CG-PI) theory that builds an isomorphism between the quantum partition function of N distinguishable particles and the classical partition function of 2N pseudoparticles. In this present work, we develop a generalized version of the many-body CG-PI theory that incorporates many-body interactions in the force field. Based on the new derivation, we provide a numerical CG-PI (n-CG-PI) modeling strategy parametrized from the underlying path integral molecular dynamics (PIMD) trajectories using force matching and Boltzmann inversion. The n-CG-PI models for two liquid systems are shown to capture well both the intramolecular and intermolecular structural correlations of the reference PIMD simulations. The generalized derivation of the many-body CG-PI theory and the n-CG-PI model presented in this work extend the scope of the CG-PI formalism by generalizing the previously limited theory to incorporate force fields of realistic molecular systems.

Spin and exchange in mixed quantum states from first principles in Feynman path integral quantum statistical mechanics representation

The exchange and space correlations in the system of indistinguishable electrons against the background of thermal fluctuations are studied from first principles. A complete set of mutually orthogonal spin-coordinate basis functions symmetric with respect to permutations in coordinate and spin variables is constructed by the action of Young symmetry operators. As an illustration of a general approach, Path Integral Monte Carlo (PIMC) computer simulation of mixed quantum states of four electrons in a model ellipsoidal nanoscopic trap with soft walls is performed. The equilibrium value of the spin is calculated as well as the mechanism of its formation depending on the temperature and shape of the trap in connection with the evolution of the spatial organization of electrons is analyzed. The Quantum-statistical calculation, executed with a fundamentally accurate description of the permutation symmetry of electrons and the spin variable, includes in explicit form the Coulomb and exchange correlations of all orders and the energy exchange with a thermal bath. The equilibrium values ​​of the spin in dependence of the shape of the potential well and temperature as well as statistical distributions over the eigenfunctions of the spin squared operator are obtained. A general, fundamentally accurate and numerically implementable on computers available today computers approach for describing the spin states of systems of different numbers of indistinguishable fermions with spin 1/2 in the representation of path integrals is formulated.The regularities in dependence of spin on the shape of the system and on the temperature are studied. The variations in the geometry of the trap in direction of elongated shapes are accompanied by the transfer of a part of the electron cloud first from the equator to the poles of the ellipsoidal potential well, and then in the opposite direction with the subsequent collapse of the ring-shaped form into a compact clot in the center and its decomposition into two fragments localized on the axis of rotation. The spin of the mixed quantum state of four electrons at room temperature varies depending on the geometry of the system in the opposite direction, compared with the system of two and three electrons, reaching its maximum at a spherical shape. Spin states of electron ensembles several nanometers in size are unstable to thermal perturbations already in the region of room temperatures. Compression leads to a stabilization of spin states. The calculation results indicate a strong dependence of the energy of collective mixed spin states of electrons on a detailed account of exchange and Coulomb high-orders spatial correlations.

Evaluation of Euler number of complex Grassmann manifold G(k,N) via Mathai-Quillen formalism

In this paper, we provide a recipe for computing Euler number of Grassmann manifold G(k,N) by using Mathai-Quillen formalism (MQ formalism) [9] and Atiyah-Jeffrey construction [1]. Especially, we construct path-integral representation of Euler number of G(k,N). Our model corresponds to a finite dimensional toy-model of topological Yang-Mills theory which motivated Atiyah-Jeffrey construction. As a by-product, we construct free fermion realization of cohomology ring of G(k,N).

Artificial neural network-based path integral simulations of hydrogen isotope diffusion in palladium

The contribution of nuclear quantum effects (NQEs) to the kinetics and dynamics of interstitial H isotopes in face-centered cubic Pd was intensively investigated using several path-integral techniques, along with a newly developed machine-learning interatomic potential based on artificial neural networks for Pd–H alloys. The diffusion coefficients (D) of protium, deuterium, and tritium in Pd were predicted over a wide temperature range (50–1500 K) based on quantum transition-state theory (QTST) combined with path-integral molecular-dynamics simulations. The importance of NQEs even at high temperatures was illustrated in terms of the characteristic temperature dependence of the activation free energies for H-isotope migration in Pd. This illuminates the overall picture of anomalous D crossovers among the three H isotopes in Pd. In addition, the D of protium in Pd was directly computed using two approximate quantum-dynamics methods based on Feynman’s path-integral theory, i.e. centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD), in the temperature range 370–1500 K. The D values obtained from the CMD and RPMD simulations were very similar and agreed better with the reported experimental values than the QTST results in this temperature range. Our machine learning-based path-integral calculations elucidate the underlying quantum nature of the ‘reversed S’-type nonlinear behavior of D for the three H isotopes in Pd on the Arrhenius plots.

Evidence for Dissociation and Ionization in Shock Compressed Nitrogen to 800GPa.

Triple bonding in the nitrogen molecule (N_{2}) is among the strongest chemical bonds with a dissociation enthalpy of 9.8 eV/molecule. Nitrogen is therefore an excellent test bed for theoretical and numerical methods aimed at understanding how bonding evolves under the influence of the extreme pressures and temperatures of the warm dense matter regime. Here, we report laser-driven shock experiments on fluid molecular nitrogen up to 800GPa and 4.0 g/cm^{3}. Line-imaging velocimetry measurements and impedance matching method with a quartz reference yield shock equation of state data of initially precompressed nitrogen. Comparison with numerical simulations using path integral MonteCarlo and density functional theory molecular dynamics reveals clear signatures of chemical dissociation and the onset of L-shell ionization. Combining data along multiple shock Hugoniot curves starting from densities between 0.76 and 1.29 g/cm^{3}, our study documents how pressure and density affect these changes in chemical bonding and provides benchmarks for future theoretical developments in this regime, with applications for planetary interior modeling, high energy density science, and inertial confinement fusion research.