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The ground-state energies and the squared moduli of the ground-state wave functions are calculated for the {sup 3}H, {sup 3,4,6}He, {sup 6}Li, and {sup 9}Be nuclei by Feynman’s continual (path) integrals method. The results are in satisfactory agreement with experimental data.
In this paper we investigate the shape invariance property of a potential in one dimension. We show that a simple ansatz allows us to reconstruct all the known shape invariant potentials in one dimension. This ansatz can be easily extended to arrive at a large class of new shape invariant potentials in arbitrary dimensions. A reformulation of the shape invariance property and possible generalizations are proposed. These may lead to an important extension of the shape invariance property to Hamiltonians that are related to standard potential problems via space time transformations, which are found useful in path integral formulation of quantum mechanics.
The incorporation of the geometric phase in single-state adiabatic dynamics near a conical intersection (CI) seam has so far been restricted to molecular systems with high symmetry or simple model Hamiltonians. This is due to the fact that the ab initio determined derivative coupling (DC) in a multi-dimensional space is not curl-free, thus making its line integral path dependent. In a recent work [C. L. Malbon et al. , J. Chem. Phys. 145, 234111 (2016)], we proposed a new and general approach based on an ab initio determined diabatic representation consisting of only two electronic states, in which the DC is completely removable, so that its line integral is path independent in the simply connected domains that exclude the CI seam. Then with the CIs included, the line integral of the single-valued DC can be used to construct the complex geometry-dependent phase needed to exactly eliminate the double-valued character of the real-valued adiabatic electronic wavefunction. This geometry-dependent phase gives rise to a vector potential which, when included in the adiabatic representation, rigorously accounts for the geometric phase in a system with an arbitrary locus of the CI seam and an arbitrary number of internal coordinates. Here, we demonstrate this approach in a three-dimensional treatment of the tunneling facilitated dissociation of the S 1 state of phenol, which is affected by a C s symmetry allowed but otherwise accidental seam of CI. Here, since the space is three-dimensional rather than two-dimensional, the seam is a curve rather than a point. The nodal structure of the ground state vibronic wavefunction is shown to map out the seam of CI.
We show that universal transport coefficients of the fractional quantum Hall effect (FQHE) can be understood as a response to variations of spatial geometry. Some transport properties are essentially governed by the gravitational anomaly. We develop a general method to compute correlation functions of FQH states in a curved space, where local transformation properties of these states are examined through local geometric variations. We introduce the notion of a generating functional and relate it to geometric invariant functionals recently studied in geometry. We develop two complementary methods to study the geometry of the FQHE. One method is based on iterating a Ward identity, while the other is based on a field theoretical formulation of the FQHE through a path integral formalism.
In this work we investigate the ground state and the dissipative quantum dynamics of interacting charged particles in an external potential at finite temperature. The recently devised time-dependent quantum Monte Carlo (TDQMC) method allows a self-consistent treatment of the system of particles together with bath oscillators first for imaginary-time propagation of Schrödinger type of equations where both the system and the bath converge to their finite temperature ground state, and next for real time calculation where the dissipative dynamics is demonstrated. In that context the application of TDQMC appears as promising alternative to the path-integral related techniques where the real time propagation can be a challenge.
In this paper, we rigorously investigate the reduced density matrix (RDM) associated to the ideal Bose gas in harmonic traps. We present a method based on a sum-decomposition of the RDM allowing to treat not only the isotropic trap, but also general anisotropic traps. When focusing on the isotropic trap, the method is analogous to the loop-gas approach developed by Mullin [“The loop-gas approach to Bose-Einstein condensation for trapped particles,” Am. J. Phys. 68(2), 120 (2000)]. Turning to the case of anisotropic traps, we examine the RDM for some anisotropic trap models corresponding to some quasi-1D and quasi-2D regimes. For such models, we bring out an additional contribution in the local density of particles which arises from the mesoscopic loops. The close connection with the occurrence of generalized-Bose-Einstein condensation is discussed. Our loop-gas-like approach provides relevant information which can help guide numerical investigations on highly anisotropic systems based on the Path Integral Monte Carlo method.
We discuss how the Phase Integration Method (PIM), recently developed to compute symmetrized time correlation functions [M. Monteferrante, S. Bonella, and G. Ciccotti, Mol. Phys. 109, 3015 (2011)], can be adapted to sampling/generating the thermal Wigner density, a key ingredient, for example, in many approximate schemes for simulating quantum time dependent properties. PIM combines a path integral representation of the density with a cumulant expansion to represent the Wigner function in a form calculable via existing Monte Carlo algorithms for sampling noisy probability densities. The method is able to capture highly non-classical effects such as correlation among the momenta and coordinates parts of the density, or correlations among the momenta themselves. By using alternatives to cumulants, it can also indicate the presence of negative parts of the Wigner density. Both properties are demonstrated by comparing PIM results to those of reference quantum calculations on a set of model problems.
The Lagrangian of the Liouville theory with topological defects is analyzed in detail and general solution of the corresponding defect equations of motion is found. We study the heavy and light semiclassical limits of the defect two-point function found before via the bootstrap program. We show that the heavy asymptotic limit is given by the exponential of the Liouville action with defects, evaluated on the solutions with two singular points. We demonstrate that the light asymptotic limit is given by the finite-dimensional path integral over solutions of the defect equations of motion with a vanishing energy–momentum tensor.
The thermodynamic and superfluid properties of the dilute two-dimensional binary Bose mixture at low temperatures are discussed. We also considered the problem of the emergence of the long-range order in these systems. All calculations are performed by means of the celebrated Popov path-integral approach to the Bose gas with a short-range interparticle potential.
We study the Förster resonant energy transfer rate, absorption and emission spectra in multichromophoric systems. The multichromophoric Förster theory (MCFT) is determined from an overlap integral of generalized matrices related to the donor’s emission and acceptor’s absorption spectra, which are obtained via a full 2nd-order cumulant expansion technique developed in this work. We calculate the spectra and MCFT rate for both localized and delocalized systems, and calibrate the analytical 2nd-order cumulant expansion with the exact stochastic path integral method. We present three essential findings: (i) The role of the initial entanglement between the donor and its bath is found to be crucial in both the emission spectrum and the MCFT rate. (ii) The absorption spectra obtained by the cumulant expansion method are nearly identical to the exact spectra for both localized and delocalized systems, even when the system-bath coupling is far from the perturbative regime. (iii) For the emission spectra, the cumulant expansion can give reliable results for localized systems, but fail to provide reliable spectra of the high-lying excited states of a delocalized system, when the system-bath coupling is large and the thermal energy is small. This paper also provides a simple golden-rule derivation of the MCFT, reviews existing methods, and motivates further developments in the subsequent papers.
We study the 2+1 dimensional boson–fermion duality in the presence of background curvature and electromagnetic fields. The main players are, on the one hand, a massive complex |ϕ|{sup 4} scalar field coupled to a U(1) Maxwell–Chern–Simons gauge field at level 1, representing a relativistic composite boson with one unit of attached flux, and on the other hand, a massive Dirac fermion. We show that, in a curved background and at the level of the partition function, the relativistic composite boson, in the infinite coupling limit, is dual to a short-range interacting Dirac fermion. The coupling to the gravitational spin connection arises naturally from the spin factors of the Wilson loop in the Chern–Simons theory. A non-minimal coupling to the scalar curvature is included on the bosonic side in order to obtain agreement between partition functions. Although an explicit Lagrangian expression for the fermionic interactions is not obtained, their short-range nature constrains them to be irrelevant, which protects the duality in its strong interpretation as an exact mapping at the IR fixed point between a Wilson–Fisher–Chern–Simons complex scalar and a free Dirac fermion. We also show that, even away from the IR, keeping the |ϕ|{sup 4} term is of key importance as it provides the short-range bosonic interactions necessary to prevent intersections of worldlines in the path integral, thus forbidding unknotting of knots and ensuring preservation of the worldline topologies.
We discuss the effective infrared theory governing a light scalar's long wavelength dynamics in de Sitter spacetime. We show how the separation of scales around the physical curvature radius k / a ∼ H can be performed consistently with a window function and how short wavelengths can be integrated out in the Schwinger-Keldysh path integral formalism. At leading order, and for time scales Δ t >> H {sup −1}, this results in the well-known Starobinsky stochastic evolution. However, our approach allows for the computation of quantum UV corrections, generating an effective potential on which the stochastic dynamics takes place. The long wavelength stochastic dynamical equations are now second order in time, incorporating temporal scales Δ t ∼ H {sup −1} and resulting in a Kramers equation for the probability distribution—more precisely the Wigner function—in contrast to the more usual Fokker-Planck equation. This feature allows us to non-perturbatively evaluate, within the stochastic formalism, not only expectation values of field correlators, but also the stress-energy tensor of φ.
Abstract 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.
We present an integral-based technique (IBT) algorithm to accelerate supernova (SN) radiative transfer calculations. The algorithm utilizes “integral packets,” which are calculated by the path integral of the Monte Carlo (MC) energy packets, to synthesize the observed spectropolarimetric signal at a given viewing direction in a 3D time-dependent radiative transfer program. Compared to the event-based technique (EBT) proposed by M. Bulla et al., our algorithm significantly reduces the computation time and increases the MC signal-to-noise ratio (S/N). Using a 1D spherical symmetric Type Ia SN ejecta model DDC10 and its derived 3D model, the IBT algorithm has successfully passed the verification of spherical symmetry and cross comparison on a 3D SN model with the direct-counting technique and EBT. Notably, with our algorithm implemented in the 3D MC radiative transfer code SEDONA, the computation time is faster than EBT by a factor of 10−30, and the S/N is better by a factor of 1.5−3, with the same number of MC quanta.
Visible Cherenkov radiation can offer a method of the measurement of the velocity of charged particles. The angular width of the radiation is important since it determines the resolution of the velocity measurement. In this article, the angular width of Cherenkov radiation with inclusion of multiple scattering is calculated through the path-integral method, and the analytical expressions are presented. The condition that multiple scattering processes dominate the angular distribution is obtained.
The development and implementation of increasingly accurate methods for electronic structure calculations mean that, for many atomistic simulation problems, treating light nuclei as classical particles is now one of the most serious approximations. Even though recent developments have significantly reduced the overhead for modeling the quantum nature of the nuclei, the cost is still prohibitive when combined with advanced electronic structure methods. Here we present how multiple time step integrators can be combined with ring-polymer contraction techniques (effectively, multiple time stepping in imaginary time) to reduce virtually to zero the overhead of modelling nuclear quantum effects, while describing inter-atomic forces at high levels of electronic structure theory. This is demonstrated for a combination of MP2 and semi-local DFT applied to the Zundel cation. The approach can be seamlessly combined with other methods to reduce the computational cost of path integral calculations, such as high-order factorizations of the Boltzmann operator or generalized Langevin equation thermostats.
In Jackiw-Teitelboim (JT) gravity, which is dual to a random matrix ensemble, the annealed entropy differs from the quenched entropy at low temperatures and goes negative. However, computing the quenched entropy in JT gravity requires a replica limit that is poorly understood. To circumvent this, we define an intermediate quantity called the semiquenched entropy, which has the positivity properties of the quenched entropy, while requiring a much simpler replica trick. We compute this in JT gravity in different regimes using (i) a bulk calculation involving wormholes corresponding to the Airy limit of the dual matrix integral and (ii) a boundary calculation involving one-eigenvalue instanton saddles proposed by Hernández-Cuenca, demonstrating consistency between these two calculations in their common regime of validity. We also clarify why similar one-eigenvalue instanton saddles cannot be used to compute the quenched entropy due to a breakdown of the saddle-point approximation for the one-eigenvalue instanton in the replica limit. Our results show how to use the gravitational path integral to prove that black holes in JT gravity have isolated ground states and to study their properties.
It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin–Kasteleyn–Ginibre property in this model and the Ghirlanda–Guerra identities in artificial models in a path integral representation based on the Lie–Trotter–Suzuki formula enable us to extend Chatterjee’s proof for the random field Ising model to the quantum model.