Polymer density functional approach to efficient evaluation of path integrals

We discuss a novel solution for the sign problem which prevents first principle Monte-Carlo computations of QCD at finite chemical potential (especially important for both the search for the critical point and neutron star physics) as well as real time quantities such as transport coefficients. The solution is based on deforming the region of integration in the path integral into a complex manifold where the sign problem can be mitigated substantially. We explain the new Monte-Carlo algorithm based on this idea and give examples of interacting quantum field theories (bosonic and fermionic) with nonzero chemical potential as well as real time dynamics where this method successfully solves the sign problem. This approach generalizes the “Lefschetz thimble” method that received much attention lately. We also compare/contrast with the complex Langevin method.

A new approach to the solution of sign problems in Monte-Carlo methods, based on cluster algorithms, has emerged recently. The underlying idea is to identify clusters that help match configurations of equal weight but opposite signs and hence do not contribute to the path integral. Avoiding these clusters, called merons, alleviates the sign problem and in certain cases solves it completely. Among the variety of problems that can be solved using these techniques there also exist some severe fermion sign problems. A pedagogical discussion of the above ideas is presented here with an emphasis on applications to fermionic systems.

There has been renewed interest in the path-integral computation of the partition function of AdS3 gravity, both in the metric and Chern-Simons formulations. The one-loop partition function around Euclidean AdS3 turns out to be given by the vacuum character of Virasoro group. This stems from the work of Brown and Henneaux (BH) who showed that, in AdS3 gravity with sensible asymptotic boundary conditions, an infinite group of (improper) diffeomorphisms arises which acts canonically on phase space as two independent Virasoro symmetries. The gauge group turns out to be composed of so-called “proper” diffeomorphisms which approach the identity at infinity fast enough. However, it is sometimes far from evident to identify where BH boundary conditions enter in the path integral, and much more difficult to see how the improper diffeomorphisms are left out of the gauge group. In particular, in the metric formulation, Giombi, Maloney and Yin obtained the one-loop partition function around thermal AdS3 resorting to the heat kernel method to compute the determinants coming from the path integral. Here we identify how BH boundary conditions follow naturally from the usual requirement of square-integrability of the metric perturbations. Also, and equally relevant, we clarify how the quotient by only proper diffeomorphisms is implemented, promoting the improper diffeomorphisms to symmetries in the path integral. Our strategy is general enough to apply to other approaches where square integrability is assumed. Finally, we show that square integrability implies that the asymptotic symmetries in higher dimensional AdS gravity are just isometries.

In the present paper on the one hand we apply the central limit theorem to the solution of the sign problem of a path integral of two-interacting particles in potential and give an expression for the sign solved propagator (SSP) derived from that solution and on the other hand we perform the angular decomposition of the path integrals of the 2D and 3D Helium atoms. Finally, we combine those two results and derive the SSPs of the 2D and 3D Helium atoms.

The Markov chain Monte Carlo (MCMC) method is used to evaluate the imaginary-time path integral of a quantum oscillator with a potential that includes a quadratic term and a quartic term whose coupling is varied by several orders of magnitude. This path integral is discretized on a time lattice on which calculations for the energy and probability density of the ground state and energies of the first few excited states are carried out on lattices with decreasing spacing to estimate these quantities in the continuum limit. The variation of the quartic coupling constant produces corresponding variations in the optimum simulation parameters for the MCMC method and in the statistical uncertainty for a fixed number of paths used for measurement. The energies and probability densities are in excellent agreement with those obtained from numerical solutions of Schrödinger’s equation. The theoretical and computational framework presented here introduces undergraduates to the path integral formulations of quantum mechanics in real time and the partition function in statistical mechanics in imaginary time. The example of the anharmonic oscillator helps to build an intuition about the MCMC method of evaluating the partition function, which can then be used to solve other problems in physics and beyond.

In this paper rigorous path integral treatments are presented of free motion on the Poincare disc, the Poincare upper half-plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincare upper half-plane to the existing one. All the calculations are mainly based on Fourier-expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ2, Liouville quantum mechanics and the modified Poschl-Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave-functions and the energy-spectrum are derived. The energy-spectra are the same in all three spaces and read E = (1/2m) (p2 + 1/4) (p -momentum). The “zero-point” energy E0 = 1/8m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ2, D, U and S are also discussed.

We exactly evaluate the partition function (index) of $$ \mathcal{N} $$ = 4 supersymmetric quiver quantum mechanics in the Higgs phase by using the localization techniques. We show that the path integral is localized at the fixed points, which are obtained by solving the BRST equations, and D-term and F-term conditions. We turn on background gauge fields of R-symmetries for the chiral multiplets corresponding to the arrows between quiver nodes, but the partition function does not depend on these R-charges. We give explicit examples of the quiver theory including a non-coprime dimension vector. The partition functions completely agree with the mathematical formulæ of the Poincaré polynomials (χ y -genus) and the wall crossing for the quiver moduli spaces. We also discuss exact computation of the expectation values of supersymmetric (Q-closed) Wilson loops in the quiver theory.

A definition of the Feynman path integral which does not rest on a limiting procedure based on time-slicing has been given by DeWitt-Morette. We present in this paper a discussion of real Gaussian measures and formulate expressions for the quantum statistical partition function directly in terms of measures of integration on the topological vector space o0 of continuous functions defined on the time intervalT = (ta,tb), such thatx(ta,tb)=0 for allx ɛ o0. We give a definition of a measure for the space o0 equivalent to the path integral based on the Uhlenbeck-Ornstein probability distribution. We give expressions for the partition function using the Wiener-Feynman measure and the Uhlenbeck-Ornstein measure. As an exercise in the use of the new techniques, we present calculations of moments of potential functions. The techniques will enable one to solve in a rigorous manner practical problems in quantum statistical mechanics.

The theory of dilute magnetic alloys is studied using Anderson's model. The Coulomb interaction is represented by the fluctuating potential acting on single electrons at the impurity site, and the partition function is rigorously formulated as a path integral over all possible time histories of this potential. For any particular path, the response of the electron gas is calculated using a method introduced by Nozi\`eres and De Dominicis, which is exact in the limit that the potential fluctuations are slow. From this, the contribution of any particular path to the partition function is obtained as an explicit and rather simple functional. When the Coulomb interaction is large compared to the width of the virtual bound state, a particular group of paths are singled out on the basis that they make the largest contributions. The functional is evaluated for this set of paths, and gives an expression which can be interpreted as the grand partition function for a one-dimensional gas of classical particles interacting through a logarithmic pair potential. This is identical to the result of a recent study of the $s\ensuremath{-}d$ exchange model by Anderson and Yuval. An analysis of this result has been given earlier, and it yields a satisfactory description of the Kondo effect. The resistivity is estimated, and found to approach the unitarity limit below the Kondo temperature and the Hartree-Fock value above the Kondo temperature. The correspondence between the Anderson and $s\ensuremath{-}d$ exchange models is shown to break down when the former is only weakly magnetic.

The electronic and liquid structures of water and its thermodynamic properties are studied over a wide range of temperature (0°–600 °C) and density (0.6–1.4 g/cm3) based on the ab initio molecular orbital theory combined with the integral equation method of liquid. Unlike standard treatments of water by means of the classical statistical mechanics including molecular simulations, the effective charges on oxygen and hydrogen atoms in water molecules are not “input parameters,” but naturally predicted from the theory in the course of self-consistent determination of the electronic structure and the density pair correlation functions in liquids. It is found that the molecular dipole moments and electronic polarization energies decrease with increasing temperature and/or density. The theoretical results for dipole moments are in quantitative accord with the experimental data, which has been determined based on the NMR chemical shift coupled with the molecular dynamics simulation [N. Matsubayashi, C. Wakai, and M. Nakahara, J. Chem. Phys. 110, 8000 (1999)]. The state dependence of the electronic structure is discussed in terms of the thermal activation of molecules and intermolecular interactions including the hydrogen bonds. The liquid structure of water is discussed in the wide range of thermodynamic states in terms of S(r), an average response of the pair correlation functions to temperature change which is introduced in the present study in order to make structural characteristics of water more distinctive. It is concluded from the behavior of the function that the short range structure of water retains the characteristics to ice, or the tetrahedral configuration, over relatively wide range of temperature in the normal density (1.0 g/cm3). The ice-like characteristics of water disappear to large extent both at high (1.4 g/cm3) and low (0.6 g/cm3) densities for different reasons: in the high density due to the packing effect, while in the low density due to essentially entropic cause, or increased configuration space available to a molecule. The distance between the nearest-neighbor molecules in water are insensitive to the density change compared with those corresponding to the Lennard-Jones fluid. The difference in the behaviors between the two fluids is explained in terms of the intermolecular interactions and liquid structures. The number of hydrogen bonds is calculated from the oxygen–hydrogen pair correlation function using a new definition based on S(r), which enables us to distinguish the hydrogen-bonded OH pairs from those just in contact due to packing effect. The temperature and density dependence of the quantity is discussed in terms of the liquid structure of water.

If the diffeomorphism symmetry of general relativity is fully implemented into a path integral quantum theory, the path integral leads to a partition function which is an invariant of smooth manifolds. We comment on the physical implications of results on the classification of smooth and piecewise-linear 4-manifolds which show that the partition function can already be computed from a triangulation of space–time. Such a triangulation characterizes the topology and the differentiable structure, but is completely unrelated to any physical cut-off. It can be arbitrarily refined without affecting the physical predictions and without increasing the number of degrees of freedom proportionally to the volume. Only refinements at the boundary have a physical significance as long as the experimenters who observe through this boundary, can increase the resolution of their measurements. All these are consequences of the symmetries. The Planck scale cut-off expected in quantum gravity is rather a dynamical effect.

In this paper, an importance sampling method based on the Generalized Feynman–Kac (GFK) method has been used to calculate the mean values of quantum observables from quantum correlation functions for many-body systems with the Born–Oppenheimer approximation in the nonrelativistic limit both at zero and finite temperature. Specifically, the expectation values [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for the ground state of the lithium and beryllium and the density matrix, the partition function, the internal energy and the specific heat of a system of quantum harmonic oscillators are computed, in good agreement with the best nonrelativistic values for these quantities. Although the initial results are encouraging, more experimentation will be needed to improve the other existing numerical results beyond chemical accuracies specially for the last two properties for lithium and beryllium. Also more work needs to be done to improve the trial functions for finite temperature calculations. Although these results look promising, more work needs to be done to achieve the spectroscopic accuracy at zero temperature and to estimate the finite temperature effects from the non-Born–Oppenheimer calculations. Also more experimentation will be needed to study the convergence criteria for the inverse properties for atoms at zero temperature.

Entanglement entropy for a spatial partition of a quantum system is studied in theories which admit a dual description in terms of the anti-de Sitter (AdS) gravity one dimension higher. A general proof of the holographic formula which relates the entropy to the area of a codimension 2 minimal hypersurface embedded in the bulk AdS space is given. The entanglement entropy is determined by a partition function which is defined as a path integral over Riemannian AdS geometries with non-trivial boundary conditions. The topology of the Riemannian spaces puts restrictions on the choice of the minimal hypersurface for a given boundary conditions. The entanglement entropy is also considered in Randall-Sundrum braneworld models where its asymptotic expansion is derived when the curvature radius of the brane is much larger than the AdS radius. Special attention is payed to the geometrical structure of anomalous terms in the entropy in four dimensions. Modification of the holographic formula by the higher curvature terms in the bulk is briefly discussed.

This paper systematically studies and improves the statistical theory of the kink-phonon interactive mixture in a one-dimensional model for the displacive phase transition. By taking applicable forms of the path integral for both the partition functions of kink and phonon and altering the method in calculating the path integral of phonon, a more reasonable and general expression of the partition function of the Grand ensemble is obtained. Under the lowest eigenstate approximation, the Grand.partition function may be reduced to a result similar to that of reference[6]. Under the classical approximation, the average density of the kink calculated from the Grand partition function is in agreement with the result of the computer simulation better than those obtained in earlier publications.

The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are universal SPT invariants, fully characterizing SPTs. In this work, we use gauge fields to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows us to express the SPT invariants in terms of continuum field theory. We show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gauge-gravity actions describe the beyond-group-cohomology SPTs. We find new examples of mixed gauge-gravity actions for U(1) SPTs in (4+1)D via the gravitational Chern-Simons term. Field theory representations of SPT invariants not only serve as tools for classifying SPTs, but also guide us in designing physical probes for them. In addition, our field theory representations are independently powerful for studying group cohomology within the mathematical context.