Nonperturbative Approximation Research Articles

First-order transition and critical end point in vortex liquids in layered superconductors

We calculate various thermodynamic quantities of vortex liquids in a layered superconductor by using the nonperturbative parquet approximation method, which was previously used to study the effect of thermal fluctuations in two-dimensional vortex systems. We find there is a first-order transition between two vortex liquid phases which differ in the magnitude of their correlation lengths. As the coupling between the layers increases,the first-order transition line ends at a critical point. We discuss the possible relation between this critical end-point and the disappearance of the first-order transition which is observed in experiments on high temperature superconductors at low magnetic fields.

Dual quantum electrodynamics: Dyon-dyon and charge-monopole scattering in a high-energy approximation

We develop the quantum field theory of electron-point magnetic monopole interactions and more generally, dyon-dyon interactions, based on the original string-dependent ``nonlocal'' action of Dirac and Schwinger. We demonstrate that a viable nonperturbative quantum field theoretic formulation can be constructed that results in a string {\em independent} cross section for monopole-electron and dyon-dyon scattering. Such calculations can be done only by using nonperturbative approximations such as the eikonal and not by some mutilation of lowest-order perturbation theory.

Analytical ground state for the three-band Hubbard model

For the calculation of charge excitations as those observed in, e.g., photo-emission spectroscopy or in electron-energy loss spectroscopy, a correct description of ground-state charge properties is essential. In strongly correlated systems like the undoped cuprates this is a highly non-trivial problem. In this paper we derive a non-perturbative analytical approximation for the ground state of the three-band Hubbard model on an infinite, half filled CuO_2 plane. By comparison with Projector Quantum Monte Carlo calculations it is shown that the resulting expressions correctly describe the charge properties of the ground state. Relations to other approaches are discussed. The analytical ground state preserves size consistency and can be generalized for other geometries, while still being both easy to interpret and to evaluate.

Dynamical ansatz for path integrals and nonperturbative trace formulas.

It is shown that a recently discovered representation of the Green's function is equivalent to a certain "dynamical ansatz" for the corresponding path integral, which brings about a convenient method of nonperturbative approximations. Based on this observation, a set of nonperturbative approximations to the trace of the Green's function is established.

QUANTUM EVOLUTION OF INHOMOGENEITIES IN CURVED SPACE

We obtain the renormalized equations of motion for matter and semiclassical gravity in an inhomogeneous space–time. We use the functional Schrödinger picture and a simple Gaussian approximation to analyze the time evolution of the λϕ4 model, and we establish the renormalizability of this nonperturbative approximation. We also show that the energy–momentum tensor in this approximation is finite once we consider the usual mass and coupling constant renormalizations, without the need of further geometrical counterterms.

Flow equations for Yang-Mills theories in general axial gauges

We present a formulation of non-Abelian gauge theories in general axial gauges using a Wilsonian (or 'Exact') Renormalisation Group. No 'spurious' propagator divergencies are encountered in contrast to standard perturbation theory. Modified Ward identities, compatible with the flow equation, ensure gauge invariance of physical Green functions. The axial gauge nA=0 is shown to be a fixed point under the flow equation. Possible non-perturbative approximation schemes and further applications are outlined.

Minimal adiabatic approximation of molecular energies

The minimal necessary and sufficient condition for Hamiltonians to yield so-called adiabatic energies of molecules is determined and used to formulate the non-perturbative non-variational Minimal Adiabatic Approximation for obtaining them. No finite electronic energy eigenvalues which result from this theory can become accidentally degenerate at any localized configuration of their nuclei, thus establishing a general ‘non-crossing’ rule of overall molecular potentials, which obviates the Jahn-Teller effect entirely on an adiabatic basis. The resulting electronic energy eigenfunctions are single-valued continuous bounded appropriately differentiable functions of their parametric nuclear coordinates and cannot have any geometric phase factor ascribed to them. The ground-state electronic energy eigenfunctions cannot have any phase factors which depend on nuclear configuration affixed to them and still retain their ground-state status. The ground-state electronic energy eigenvalues are bounded from below by those of the Born-Oppenheimer approximation and from above by those of the BornHuang approximation, as are their corresponding ground-state total energies.

Nonperturbative Flow Equations in QCD

We review the formalism of the effective average action in quantum field theory which corresponds to a coarse grained free energy in statistical mechanics. The associated exact renormalization group equation and possible nonperturbative approximations for its solution are discussed. This is applied to QCD where one observes the consecutive emergence of mesonic bound states and spontaneous chiral symmetry breaking as the coarse graining scale is lowered. We finally present a study of the chiral phase transition in two flavor QCD. A precision estimate of the universal critical equation of state for the three-dimensional 0( 4) Heisenberg model is presented. We explicitly connect the 0( 4) universal behavior near the critical temperature and zero quark mass with the physics at zero temperature and a realistic pion mass. For realistic quark masses the pion correlation length near Tc turns out to be smaller than its zero temperature value. Quantum chromodynamics (QCD) describes qualitatively different physics at different length scales. At short distances the relevant degrees of freedom are quarks and gluons which can be treated perturbatively. At long distances we observe hadrons, and an essential part of the dynamics can be encoded in the masses and interactions of mesons. Any attempt to deal with this situation analytically and to predict the meson properties from the short distance physics (as functions of the strong gauge coupling o: 8 and the current quark masses mq) has to bridge the gap between two qualitatively different effective descriptions. Two basic problems have to be mastered for an extrapolation from short distance QCD to mesonic length scales: • The effective couplings change with scale. This does not only concern the running gauge coupling, but also the coefficients of non-renormalizable opera­ tors as, for example, four quark operators. Typically, these non-renormalizable terms become important in the momentum range where 0:8 is strong and deviate substantially from their perturbative values. Consider the four-point function which results after integrating out the gluons. For heavy quarks it contains the information about the shape of the heavy quark potential whereas for light quarks the complicated spectrum of light mesons and chiral symmetry breaking are encoded in it. At distance scales around 1 fm one expects that the effec­ tive action resembles very little the form of the classical QCD action which is relevant at short distances. • Not only the couplings, but even the relevant variables or degrees of freedom are different for long distance and short distance QCD. It seems forbiddingly

Out of equilibrium dynamics of an inflationary phase transition

We study the nonlinear dynamics of an inflationary phase transition in a quartically self-coupled inflaton model within the framework of a de Sitter background. Large $N$ and Hartree nonperturbative approximations combined with nonequilibrium field theory methods are used to study the self-consistent time evolution including back reaction effects. We find that when the system cools down from an initial temperature ${T}_{i}>{T}_{c}$ to below ${T}_{c}$ with the initial value of the zero mode of the inflaton $\ensuremath{\varphi}(0)\ensuremath{\ll}m{\ensuremath{\lambda}}^{\ensuremath{-}1/4}$, the dynamics is determined by the growth of long-wavelength quantum fluctuations. For $\ensuremath{\varphi}(0)\ensuremath{\gg}m{\ensuremath{\lambda}}^{\ensuremath{-}1/4}$ the dynamics is determined by the evolution of the classical zero mode. In the regime where spinodal quantum fluctuations give the most important contribution to the nonequilibrium dynamics, we find that they modify the equation of state providing a graceful exit from the inflationary stage. Inflation ends through this new mechanism at a time scale ${t}_{s}>~[{H/m}^{2}]\mathrm{ln}[{\ensuremath{\lambda}}^{\ensuremath{-}1}]$ which for $H>~m$ and very weak coupling allows over one hundred $e$-folds during the de Sitter phase. Spatially correlated domains grow to be of horizon size and quantum fluctuations ``freeze-out'' for times $t>{t}_{s}.$

Nonlinear operators and their propagators

Mathematical physicists are familiar with a large set of tools designed for dealing with linear operators, which are so common in both the classical and quantum theories; but many of those tools are useless with nonlinear equations of motion. In this work a general algebra and calculus is developed for working with nonlinear operators: The basic new tool being the “slash product,” defined by A(1+εB) =A+εA/B+O(ε2). For a generic time development equation, the propagator is constructed and then there follows the formal version of time dependent perturbation theory, in remarkable similarity to the linear situation. A nonperturbative approximation scheme capable of producing high accuracy computations, previously developed for linear operators, is shown to be applicable as well in the nonlinear domain. A number of auxiliary mathematical properties and examples are given.

Rayleigh-Ritz variational approximation and symmetry nonrestoration

The investigation of symmetry nonrestoration scenarios has led to a controversy, with certain nonperturbative approximation schemes giving indications in sharp disagreement with those found within conventional perturbation theory. A Rayleigh-Ritz variational approach to the problem, which might be useful in bridging the gap between perturbative and nonperturbative viewpoints, is here proposed. As a first application, this approach is used in the investigation of a Z 2 × Z 2-invariant thermal field theory with two scalar fields, placing particular emphasis on the region of parameter space that has been claimed to support symmetry nonrestoration.

Gauge invariance, the quantum action principle, and the renormalization group

If the Wilsonian renormalization group (RG) is formulated with a cutoff that breaks gauge invariance, then gauge invariance may be recovered only once the cutoff is removed and only once a set of effective Ward identities is imposed. We show that an effective quantum action principle can be formulated in perturbation theory which enables the effective Ward identities to be solved order by order, even if the theory requires nonvanishing subtraction points. The difficulties encountered with nonperturbative approximations are briefly discussed.

Ground-state magnetization of 209Bi in a dynamic-correlation model.

The nuclear magnetization distribution and the magnetic moment of the ground state of $^{209}\mathrm{Bi}$ are calculated within a dynamic-correlation model. The obtained ground-state magnetization reproduces successfully the hyperfine structure splitting (HFS) of mesonic and electronic $^{209}\mathrm{Bi}$. The evaluated nuclear radius and the electric charge distribution are also in very good agreement with experimental data. The dynamic model is based on the introduction of configuration mixing wave functions (CMWF), generated by breaking the particle-hole symmetry of the $^{208}\mathrm{Pb}$ closed-shell core via the two-body interaction. The 2-particle--1-hole states in this calculation are restricted to a 2\ensuremath{\Elzxh}\ensuremath{\omega} configuration space in the protons and in the neutrons. The particle-hole excitations, introduced by this nonperturbative approximation are of collective character and therefore can also be associated with the creation of virtual bosons in the dynamic nuclear structure calculation. In the framework of this dynamic nuclear model it is possible to set a realistic limit for the nuclear contributions for calculations of the $^{209}\mathrm{Bi}^{82+}$ hyperfine structure splitting, and therefore to allow, for the first time, a test of QED corrections in the strong magnetic field of high-Z atoms.

Non-perturbative approximations of path integrals with some applications to quantum statistics

Some methods for constructing uniform non-perturbative approximations of path integrals over a conditional Wiener measure are examined. The relation of these methods and the results obtained with their help to the ones known in the literature is established. The concrete analytical procedures and the formulae for the corresponding approximations are constructed and some applications in quantum statistical mechanics are considered.

THE EXACT RENORMALIZATION GROUP AND APPROXIMATE SOLUTIONS

We investigate the structure of Polchinski’s formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff Green’s functions are given. A promising nonperturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in “irrelevancy” of operators. We illustrate with two simple models of four-dimensional λφ4 theory: the cactus approximation, and a model incorporating the first irrelevant correction to the renormalized coupling. The qualitative and quantitative behaviour give confidence in a fuller use of this method for obtaining accurate results.

CUSP ANNIHILATION ON ORDINARY COSMIC STRINGS

The order of magnitude of energy emission from cusps to light bosons on ordinary cosmic string is calculated perturbatively. The analysis is applicable to both closed string loops and long cosmic strings. The perturbative result obtained here is much less than what is found by nonperturbative approximations.

Virtual particles versus superconductive vacuum polarizations in interacting systems.

The structure of the reference vacuum state plays a very important role in a theoretical dynamic description of interacting particles. This structure is generated by the residual interaction acting between the valence particles and, in systems under extreme high temperature and consequently high pressure, cannot be treated in the framework of the perturbation theory. In this paper we elaborate a nonperturbative approximation to include the vacuum-polarization effects of superconductive type in the calculation of the many-body dynamics. In this proposed model, the wave functions of the system are characterized by the strong coupling of the valence particles with the intrinsic vacuum states. These are associated with (a) superconductive vacuum polarizations, and (b) superconductive virtual particles formed coupling the vacuum excitations to the non-normal parity states of the system. The coupling of the valence particles with the vacuum excitations (a) and (b), in this paper, is generated within the equations of motion methods which, in the limit of factorization methods defined to compute the matrix elements of the nuclear Hamiltonian in the resulting complex states, and with the use of energy-dependent linearization approximations, introduced to generate the superconductive collective modes, are of easy application in the low-energy domain of the interacting systems. The energy-dependent linearization approximations define collective modes which are loose and freely moving in this low-energy domain, while strongly interacting with increasing energy.

Phase transitions out of equilibrium: Domain formation and growth.

We study the dynamics of phase transitions out of equilibrium in weakly coupled scalar field theories. We consider the case in which there is a rapid supercooling from an initial symmetric phase in thermal equilibrium at temperature ${T}_{i}>{T}_{c}$ to a final state at low temperature ${T}_{f}\ensuremath{\approx}0$. In particular we study the formation and growth of correlated domains out of equilibrium. It is shown that the dynamics of the process of domain formation and growth (spinodal decomposition) cannot be studied in perturbation theory, and a nonperturbative self-consistent Hartree approximation is used to study the long time evolution. We find in weakly coupled theories that the size of domains grows at long times as ${\ensuremath{\xi}}_{D}(t)\ensuremath{\approx}\sqrt{t\ensuremath{\xi}(0)}$. The size of the domains and the amplitude of the fluctuations grow up to a maximum time ${t}_{s}$ which in weakly coupled theories is estimated to be ${t}_{s}\ensuremath{\approx}\ensuremath{-}\ensuremath{\xi}(0)\mathrm{ln}\left[{\left(\frac{3\ensuremath{\lambda}}{4{\ensuremath{\pi}}^{3}}\right)}^{\frac{1}{2}}\left(\frac{{(\frac{{T}_{i}}{2{T}_{c}})}^{3}}{[\frac{{T}_{i}^{2}}{{T}_{c}^{2}}\ensuremath{-}1]}\right)\right]$ with $\ensuremath{\xi}(0)$ the zero-temperature correlation length. For very weakly coupled theories, their final size is several times the zero-temperature correlation length. For strongly coupled theories the final size of the domains is comparable to the zero-temperature correlation length and the transition proceeds faster.

An accurate short-time small-wave vector approximation for the dynamic structure factor of a Rouse polymer

A nonperturbative approximation and a fortuitous cancellation of errors lead to an accurate, simple approximation for the dynamic single-chain structure factor—or coherent intermediate scattering function—of a Rouse polymer fluid in the short to intermediate time and wave vector region. With this approximation the breakdown at small wave vector of the wave vector to the fourth scaling of the ‘‘Rouse frequency’’ is illustrated, and compared to recent neutron spin-echo measurements on polydimethylsiloxane. Experimental study of this breakdown might provide information about the relation between static structure and dynamic behavior in polymer melts.

Vacuum structures in Hamiltonian light-front dynamics

Hamiltonian light-front dynamics of quantum fields may provide a useful approach to systematic nonperturbative approximations to quantum field theories. We investigate inequivalent Hilbert-space representations of the light-front field algebra in which the stability group of the light front is implemented by unitary transformations. The Hilbert space representation of states is generated by the operator algebra from the vacuum state. There is a large class of vacuum states besides the Fock vacuum which meet all the invariance requirements. The light-front Hamiltonian must annihilate the vacuum and have a positive spectrum. We exhibit relations of the Hamiltonian to the nontrivial vacuum structure.