Discretized Path Integral Research Articles

Operator Formulation of Feynman Path Centroid Dynamics for Rotations.

An operator formulation of centroid molecular dynamics (CMD) for rotational degrees of freedom is presented. The quasi-density operator concept was introduced by Jang and Voth [J. Chem. Phys 111, 2357 (1999)] and is used to obtain a phase-space mapping without the need for discretized path integrals. The approach allows the calculation of approximate Kubo-transformed time correlation functions. The particle on a ring is chosen as an illustrative example. Numerical results demonstrate that the proposed approach leads to accurate results when compared with exact diagonalization calculations for linear operators. At very low temperatures, it is found that rotational CMD yields results that are in exact agreement with the quantum dynamics of a spin-1 system.

Topological error correcting processes from fixed-point path integrals

We propose a unifying paradigm for analyzing and constructing topological quantum error correcting codes as dynamical circuits of geometrically local channels and measurements. To this end, we relate such circuits to discrete fixed-point path integrals in Euclidean spacetime, which describe the underlying topological order: If we fix a history of measurement outcomes, we obtain a fixed-point path integral carrying a pattern of topological defects. As an example, we show that the stabilizer toric code, subsystem toric code, and CSS Floquet code can be viewed as one and the same code on different spacetime lattices, and the honeycomb Floquet code is equivalent to the CSS Floquet code under a change of basis. We also use our formalism to derive two new error-correcting codes, namely a Floquet version of the 3+1-dimensional toric code using only 2-body measurements, as well as a dynamic code based on the double-semion string-net path integral.

Penentuan Density Matrix Sistem Kuantum Satu Partikel Dimensi Satu dengan Metode Discretized Path Integral (DPI)

A mixed quantum system can be represented by a density matrix. By knowing the density matrix of the system, other thermodynamic properties of the system such as Helmholtz free energy and entropy can be determined. Computation of the density matrix of a one-dimensional single-particle quantum system using the discretized path integral(DPI) method is presented. The results of the DPI method were validated using the finite difference time domain (FDTD) method for a particle in an infinite square well and harmonic oscillator potentials. The results of density matrix, Helmholtz’s free energy and entropy have shown that the DPI method produces correct numerical values compared to the FDTD results. Then, the DPI method is used in the double-well potential by giving variations in barrier potential and temperature of the system. The results show that the particle will be tunneling when the barrier potential energy is more than the kinetic energy of the particle. The particle also can tunnel easily if the system has the higher temperature.

Towards topological fixed-point models beyond gappable boundaries

We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the mathematical community as state-sum constructions or lattice topological quantum field theories. All of the established ansatzes for fixed-point models imply the existence of a gapped boundary as well as a commuting-projector Hamiltonian. Thus, they fail to capture topological phases without a gapped boundary or commuting-projector Hamiltonian, most notably chiral topological phases in $2+1$ dimensions. In this work, we present a more general fixed-point ansatz not affected by the aforementioned restrictions. Thus, our formalism opens up a possible way forward towards a microscopic fixed-point description of chiral phases and we present several strategies that may lead to concrete examples. Furthermore, we argue that our more general ansatz constitutes a universal form of topological fixed-point models, whereas established ansatzes are universal only for fixed-points of phases which admit topological boundaries.

A unified diagrammatic approach to topological fixed point models

We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework is reminiscent of state-sum models and lattice topological quantum field theories, but is formalised and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism represent discrete path integrals in Euclidean space-time. This language is more directly related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We introduce our formalism by simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.

The Ponzano–Regge cylinder and propagator for 3d quantum gravity

We investigate the propagator of 3d quantum gravity, formulated as a discrete topological path integral. We define it as the Ponzano–Regge amplitude of the solid cylinder swept by a 2d disk evolving in time. Quantum states for a 2d disk live in the tensor products of N spins, where N is the number of holonomy insertions connecting to the disk boundary. We formulate the cylindric amplitude in terms of a transfer matrix and identify its eigen-modes in terms of spin recoupling. We show that the propagator distinguishes subspaces with different total recoupled spin. This may select the vanishing overall spin sector at late time depending on the chosen cylinder boundary data, leading to an emergent symmetry scenario in the continuum limit. We discuss applications to quantum circuits and the possibility of experimental simulations of this 3d quantum gravity propagator.

Symmetry-resolved entanglement in symmetry-protected topological phases

Symmetry protected topological phases (SPTs) have universal degeneracies in the entanglement spectrum in one dimension (1D). Here, we formulate this phenomenon in the framework of symmetry-resolved entanglement (SRE) using cohomology theory. We develop a general approach to compute entanglement measures of SPTs in any dimension and specifically SRE via a discrete path integral on multi-sheet Riemann surfaces with generalized defects. The resulting path integral is expressed in terms of group cocycles describing the topological actions of SPTs. Their cohomology classification allows to identify universal entanglement properties. Specifically, we demonstrate an equi-block decomposition of the reduced density matrix into symmetry sectors, for all 1D topological phases protected by finite Abelian unitary symmetries.

Accelerating equilibrium isotope effect calculations. II. Stochastic implementation of direct estimators.

Path integral calculations of equilibrium isotope effects and isotopic fractionation are expensive due to the presence of path integral discretization errors, statistical errors, and thermodynamic integration errors. Whereas the discretization errors can be reduced by high-order factorization of the path integral and statistical errors by using centroid virial estimators, two recent papers proposed alternative ways to completely remove the thermodynamic integration errors: Cheng and Ceriotti [J. Chem. Phys. 141, 244112 (2015)] employed a variant of free-energy perturbation called "direct estimators," while Karandashev and Vaníček [J. Chem. Phys. 143, 194104 (2017)] combined the thermodynamic integration with a stochastic change of mass and piecewise-linear umbrella biasing potential. Here, we combine the former approach with the stochastic change in mass in order to decrease its statistical errors when applied to larger isotope effects and perform a thorough comparison of different methods by computing isotope effects first on a harmonic model and then on methane and methanium, where we evaluate all isotope effects of the form CH4-xDx/CH4 and CH5-xDx +/CH5 +, respectively. We discuss the reasons for a surprising behavior of the original method of direct estimators, which performed well for a much larger range of isotope effects than what had been expected previously, as well as some implications of our work for the more general problem of free energy difference calculations.

Discrete worldline instantons

The semiclassical approximation of the worldline path integral is a powerful tool to study nonperturbative electron-positron pair creation in spacetime-dependent background fields. Finding solutions of the classical equations of motion, i.e., worldline instantons, is possible analytically only in special cases, and a numerical treatment is nontrivial as well. We introduce a completely general numerical approach based on an approximate evaluation of the discretized path integral that easily and robustly gives the full semiclassical pair production rate in nontrivial multidimensional fields, and apply it to some example cases.

An open-chain imaginary-time path-integral sampling approach to the calculation of approximate symmetrized quantum time correlation functions.

We introduce a scheme for approximating quantum time correlation functions numerically within the Feynman path integral formulation. Starting with the symmetrized version of the correlation function expressed as a discretized path integral, we introduce a change of integration variables often used in the derivation of trajectory-based semiclassical methods. In particular, we transform to sum and difference variables between forward and backward complex-time propagation paths. Once the transformation is performed, the potential energy is expanded in powers of the difference variables, which allows us to perform the integrals over these variables analytically. The manner in which this procedure is carried out results in an open-chain path integral (in the remaining sum variables) with a modified potential that is evaluated using imaginary-time path-integral sampling rather than requiring the generation of a large ensemble of trajectories. Consequently, any number of path integral sampling schemes can be employed to compute the remaining path integral, including Monte Carlo, path-integral molecular dynamics, or enhanced path-integral molecular dynamics. We believe that this approach constitutes a different perspective in semiclassical-type approximations to quantum time correlation functions. Importantly, we argue that our approximation can be systematically improved within a cumulant expansion formalism. We test this approximation on a set of one-dimensional problems that are commonly used to benchmark approximate quantum dynamical schemes. We show that the method is at least as accurate as the popular ring-polymer molecular dynamics technique and linearized semiclassical initial value representation for correlation functions of linear operators in most of these examples and improves the accuracy of correlation functions of nonlinear operators.

Group field theory for quantum gravity minimally coupled to a scalar field

We construct a group field theory model for quantum gravity minimally coupled to relativistic scalar fields, defining as well a corresponding discrete gravity path integral (and, implicitly, a coupled spin foam model) in its Feynman expansion. We also analyze a number of variations of the same model, the corresponding discrete gravity path integrals, its generalization to the coupling of multiple scalar fields and discuss its possible applications to the extraction of effective cosmological dynamics from the full quantum gravity formalism, in the context of group field theory condensate cosmology.

Primal-dual approach to optical tomography with discretized path integral with efficient formulations.

.We propose an efficient optical tomography with discretized path integral. We first introduce the primal-dual approach to solve the inverse problem formulated as a constraint optimization problem. Next, we develop efficient formulations for computing Jacobian and Hessian of the cost function of the constraint nonlinear optimization problem. Numerical experiments show that the proposed formulation is faster () than the previous work with the log-barrier interior point method () for the Shepp–Logan phantom with a grid size of , while keeping the quality of the estimation results (root-mean-square error increasing by up to 12%).

Accelerating equilibrium isotope effect calculations. I. Stochastic thermodynamic integration with respect to mass

Accurate path integral Monte Carlo or molecular dynamics calculations of isotope effects have until recently been expensive because of the necessity to reduce three types of errors present in such calculations: statistical errors due to sampling, path integral discretization errors, and thermodynamic integration errors. While the statistical errors can be reduced with virial estimators and path integral discretization errors with high-order factorization of the Boltzmann operator, here we propose a method for accelerating isotope effect calculations by eliminating the integration error. We show that the integration error can be removed entirely by changing particle masses stochastically during the calculation and by using a piecewise linear umbrella biasing potential. Moreover, we demonstrate numerically that this approach does not increase the statistical error. The resulting acceleration of isotope effect calculations is demonstrated on a model harmonic system and on deuterated species of methane.

Quantum Monte Carlo simulation with a black hole

We perform quantum Monte Carlo simulations in the background of a classical black hole. The lattice discretized path integral is numerically calculated in the Schwarzschild metric and in its approximated metric. We study spontaneous symmetry breaking of a real scalar field theory. We observe inhomogeneous symmetry breaking induced by inhomogeneous gravitational field.

One-Shot Decoupling and Page Curves from a Dynamical Model for Black Hole Evaporation.

One-shot decoupling is a powerful primitive in quantum information theory and was hypothesized to play a role in the black hole information paradox. We study black hole dynamics modeled by a trilinear Hamiltonian whose semiclassical limit gives rise to Hawking radiation. An explicit numerical calculation of the discretized path integral of the S matrix shows that decoupling is exact in the continuous limit, implying that quantum information is perfectly transferred from the black hole to radiation. A striking consequence of decoupling is the emergence of an output radiation entropy profile that follows Page's prediction. We argue that information transfer and the emergence of Page curves is a robust feature of any multilinear interaction Hamiltonian with a bounded spectrum.

Optical tomography with discretized path integral.

We present a framework for optical tomography based on a path integral. Instead of directly solving the radiative transport equations, which have been widely used in optical tomography, we use a path integral that has been developed for rendering participating media based on the volume rendering equation in computer graphics. For a discretized two-dimensional layered grid, we develop an algorithm to estimate the extinction coefficients of each voxel with an interior point method. Numerical simulation results are shown to demonstrate that the proposed method works well.

Isotope effects of ammonia umbrella flip using semiclassical instanton calculations based on discretized path integrals

We have recently developed a new algorithm to calculate tunneling splittings for molecular systems using the discretized path integral instanton method Kawatsu and Miura (2014) [29]. Compared with the standard method, the new algorithm enables us to reduce the computational cost without decreasing the numerical accuracy. The algorithm has been combined with ab initio electronic structure calculations on the fly. In this Letter, we apply our method to ammonia umbrella flips with five different isotopologues: ND3, NT3, 15NH3, 15ND3, NH2D and NHD2. The calculated tunneling splittings are found to be in quantitative agreement with experimental data in all of these calculations as well as for NH3. The method provides a useful tool for evaluating tunneling splittings for such rigid molecular systems.

Quantum partition functions of composite particles in a hydrogen-helium plasma via path integral Monte Carlo.

We compute two- and three-body cluster functions that describe contributions of composite entities, like hydrogen atoms, ions H(-), H2(+), and helium atoms, and also charge-charge and atom-charge interactions, to the equation of state of a hydrogen-helium mixture at low density. A cluster function has the structure of a truncated virial coefficient and behaves, at low temperatures, like a usual partition function for the composite entity. Our path integral Monte Carlo calculations use importance sampling to sample efficiently the cluster partition functions even at low temperatures where bound state contributions dominate. We also employ a new and efficient adaptive discretization scheme that allows one not only to eliminate Coulomb divergencies in discretized path integrals, but also to direct the computational effort where particles are close and thus strongly interacting. The numerical results for the two-body function agree with the analytically known quantum second virial coefficient. The three-body cluster functions are compared at low temperatures with familiar partition functions for composite entities.

Path-integral solution of the one-dimensional Dirac quantum cellular automaton

Quantum cellular automata have been recently considered as a fundamental approach to quantum field theory, resorting to a precise automaton, linear in the field, for the Dirac equation in one dimension. In such linear case a quantum automaton is isomorphic to a quantum walk, and a convenient formulation can be given in terms of transition matrices, leading to a new kind of discrete path integral that we solve analytically in terms of Jacobi polynomials versus the arbitrary mass parameter.

Efficient algorithms for semiclassical instanton calculations based on discretized path integrals.

Path integral instanton method is a promising way to calculate the tunneling splitting of energies for degenerated two state systems. In order to calculate the tunneling splitting, we need to take the zero temperature limit, or the limit of infinite imaginary time duration. In the method developed by Richardson and Althorpe [J. Chem. Phys. 134, 054109 (2011)], the limit is simply replaced by the sufficiently long imaginary time. In the present study, we have developed a new formula of the tunneling splitting based on the discretized path integrals to take the limit analytically. We have applied our new formula to model systems, and found that this approach can significantly reduce the computational cost and gain the numerical accuracy. We then developed the method combined with the electronic structure calculations to obtain the accurate interatomic potential on the fly. We present an application of our ab initio instanton method to the ammonia umbrella flip motion.