Path Integrals — New Trends and Perspectives

The following sections are included:

• PREFACE

• CONFERENCE COMMITTEES

• Group photo

• CONTENTS

I offer some historical comments about the origins of Feynman's path-integral approach, as an alternative approach to standard quantum mechanics. Looking at the interaction between Einstein and Feynman, which was mediated by Feynman's thesis supervisor John Wheeler, it is argued that, contrary to what one might expect, the significance of the interaction between Einstein and Feynman pertained to a critique of classical field theory, rather than to a direct critique of quantum mechanics itself. Nevertheless, the critical perspective on classical field theory became a motivation and point of departure for Feynman's space-time approach to non-relativistic quantum mechanics.

The history of physics is short, compared with mathematics and astronomy, starting in about 1800. After the first century of electrodynamics and thermodynamics, come 60 years of quantum, relativity, and nuclei. In the last 50 years, physicists have profited from an incredible boom in their research. That is threatened now, because the applications of physics have become very complicated engineering. At the other end of the spectrum, a lot of esoteric physics is going nowhere. It is not the job of the universities to produce specialists. Young people have to learn and get a chance to use their intelligence and imagination. Professors have to show them the way, and industry has to have confidence in them.

Please refer to full text.

This paper discusses two basic issues of functional integration: domains of integration and volume elements adapted to a given domain of integration. Two examples of domain of integration are given explicitly in Sections 2 and 3 respectively: the domain of integration is a space of contractible paths and the domain of integration is a space of Poisson paths. A property of volume element, presented in Section 3, namely the Koszul formula, valid on totally different geometries (riemannian, symplectic, grassman) can be used for some infinite dimensional geometries.

In joint work with M. Roncadelli, I have shown that the quantum Hamilton-Jacobi equation, which at one time attracted considerable attention but was abandoned as intractable, could be solved through its relation to the propagator. (Of course solving for the propagator is also far from trivial.) This article reviews that work and discusses applications. In one application we find that quantum operator ordering reveals the classical density of paths.

Long periodic orbits of hyperbolic dynamics do not exist as independent individuals but rather come in closely packed bunches. Under weak resolution a bunch looks like a single orbit in configuration space, but close inspection reveals topological orbit-to-orbit differences. The construction principle of bunches involves close self-"encounters" of an orbit wherein two or more stretches stay close. A certain duality of encounters and the intervening "links" reveals an infinite hierarchical structure of orbit bunches. — The orbit-to-orbit action differences ∆S within a bunch can be arbitrarily small. Bunches with ∆S of the order of Planck's constant have constructively interfering Feynman amplitudes for quantum observables, and this is why the classical bunching phenomenon could yield the semiclassical explanation of universal fluctuations in quantum spectra and transport.

Path integration is carried out for the bound states of a particle in the combined field of a wedge disclination and a screw dislocation. The energy spectrum extracted from the Feynman kernel differs from that obtained by solving the Schrödinger equation.

The dynamics of strongly confined laser driven semiconductor quantum dots coupled to phonons is studied theoretically by calculating the time evolution of the reduced density matrix using the path integral method. We explore the cases of long pulses, strong dot-phonon and dot-laser coupling and high temperatures, which up to now have been inaccessible. We find that the decay rate of the Rabi oscillations is a non-monotonic function of the laser field leading to the decay and reappearance of the Rabi oscillations in the field dependence of the dot exciton population.

In a reasonably self-contained and explicit presentation we illustrate the efficiency of the Feynman–Kac formula by the rigorous derivation of three inequalities of interest in non-relativistic quantum mechanics.

The behavior of the electron density of states (DOS) for the Lifshitz tail states is studied in the limit of low energy using the Feynman path-integral method. This method was used to study the heavily doped semiconductors for the case of a Gaussian random potential. The main results obtained are that the tail states behave as DOS~exp (-B(E)), with B(E) = Eⁿ, for short-range interaction and n = 2 for long-range interaction. In this study it is shown that without the Gaussian approximation, the behavior of the Lifshitz tails for the Poisson distribution is obtained as DOS~exp (-B(E)) with B(E) = Eⁿ, . As in the case of heavily doped semiconductor, the method can be easily generalized to long-range interactions. A comparison with the method developed by Friedberg and Luttinger based on the reformulation of the problem in terms of Brownian motion is given.

Observing that distribution theory offers an extension of the concept of integration, we review a framework in which the Feynman integral becomes mathematically meaningful for large classes of interaction potentials. We present some examples and open problems.

We give an overview of a recently developed method which systematically improves the convergence of generic path integrals for transition amplitudes, partition functions, expectation values, and energy spectra. This was achieved by analytically constructing a hierarchy of discretized effective actions indexed by a natural number p and converging to the continuum limit as 1/N^p. We analyze and compare the ensuing increase in effciency of several orders of magnitude, and perform series of Monte Carlo simulations to verify the results.

We present an application of a recently developed method for accelerated Monte Carlo computations of path integrals to the problem of energy spectra calculation of generic many-particle systems. We calculate the energy spectra of a two-particle two dimensional system in a quartic potential using the hierarchy of discretized effective actions, and demonstrate agreement with analytical results governing the increase in efficiency of the new method.

All the geometric phases, adiabatic and non-adiabatic, are formulated in a unified manner in the second quantized path integral formulation. The exact hidden local symmetry inherent in the Schrödinger equation defines the holonomy. All the geometric phases are shown to be topologically trivial. The geometric phases are briefly compared to the chiral anomaly which is naturally formulated in the path integral.

This is a survey of our work in Ref. 1. We give a fairly general class of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of the phase space. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the Riemann integrals, the interchange of the order with a limit and the perturbation expansion formula hold in the phase space path integrals. The use of piecewise bicharacteristic paths naturally leads us to the semiclassical approximation on the phase space.

We propose a non-perturbative method for the evaluation of the functional integral with fourth order term in the action. We found the result in the form of an asymptotic series.

The coherent states for a quantum particle on a Möbius strip are constructed and their relation with the natural phase space for fermionic fields is shown. An explicit comparison of the obtained states with previous works, where the cylinder quantization was used and the spin 1/2 was introduced by hand, is worked out.

The functional integral has many triumphs in elucidating quantum theory. But incorporating charge fractionalization into that formalism remains a challenge.

Changes of field variables may lead to multivalued fields which do not satisfy the Schwarz integrability conditions. Their quantum field theory needs special care as is illustrated here in applications to superfluid and superconducting phase transitions. Extending the notions that first qantization governs fluctuating orbits while second quantization deals with fluctuating field, the theory of multivalued fields may be considered as a theory of third quantization. The lecture is an introduction to my new book on this subject.

A method named the Gaussian equivalent representation and developed to calculate path integrals over a Gaussian measure, is presented. As an example partition functions for simple liquids and proton plasma are calculated by this method. Free energy and a pair correlation function in the lowest and next approximations are obtained.

On the basis of the path-integral formulation of the Yang-Mills theory a gauge invariant infrared regularization is introduced. The regularized model includes higher derivatives, but in the limit when the regularization is removed, unphysical excitations decouple.

General response properties of interacting bosonic fields are investigated.

Recent developments in rigorous functional integration applied to the Nelson and Pauli-Fierz models are presented.

We discuss the properties of classical string-like topological solitons in the A3M model. This model describes gauge-invariant interaction of Maxwell and "easy-axis" 3-component spin field.

We summarize recent results for the Gribov-Zwanziger Lagrangian which includes the effect of restricting the path integral to the first Gribov region. These include the two loop and one loop MOM gap equations for the Gribov mass.

It is shown that the conventional mesons and the lowest glueball state can be reasonably described within a simple relativistic quantum-field model of interacting quarks and gluons under the analytic confinement by using a path-integral approach. The ladder Bethe-Salpeter equation is solved for the meson and glueball (gg) spectra. A minimal set of parameters (the quark masses m_f, the coupling constant α_s and the confinement scale Λ) is used to fit the latest experimental data. In spite of the simplicity, the model provides a reasonable framework to estimate the decay constants f_π and f_K as well as the non-exotic meson and glueball masses in a wide range of energy up to 10 GeV.

I give a brief introduction into quantum-gravitational path integrals and discuss some recent developments and applications.

We show how the four-sphere appears as a background geometry with superimposed quantum fluctuations in the background independent approach to quantum gravity which uses causal dynamical triangulations as a regularization.

A sketch of a recent approach to quantum gravity is presented which involves several unconventional aspects. The basic ingredients include: (1) Affine kinematical variables; (2) Affine coherent states; (3) Projection operator approach for quantum constraints; (4) Continuous-time regularized functional integral representation without/with constraints; and (5) Hard core picture of nonrenormalizability. Emphasis is given to the functional integral expressions.

The role of the Thomas-Fermi approach in Neutron Star matter cores is presented and discussed with special attention to solutions which are globally neutral and do not fulfil the traditional condition of local charge neutrality. A new stable and energetically favorable configuration is found. This new solution can be of relevance in understanding unsolved issues of gravitational collapse processes and their energetics.

We discuss how to extract information about the cosmological constant from the Wheeler-DeWitt equation, considered as an eigenvalue of a Sturm-Liouville problem. A generalization to a f(R) theory is taken under examination. The equation is approximated to one loop with the help of a variational approach with Gaussian trial wave functionals. We use a zeta function regularization to handle with divergences. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation.

It is shown that the problem of the cosmological constant is connected with the problem of emergence of quantum mechanics. Both of them are principal aspects of physics at the Planck scale. Probability amplitudes describe evolution of the harmonic oscillator non-equilibrium distributions in a thermal bath. The Planck constant ħ, the Fock space and the Schrödinger equation appear in the natural way. For massless fields it leads, in particular, to appearance of masses, and, consequently, of the cosmological constant in the gravitational equations. The path integral for the relativistic particle propagator is presented.

In the Euclidean formulation of functional integration we discuss a dimensional reduction of quantum field theory near the horizon in terms of Green functions. We show that a massless scalar quantum field in D dimensions can be approximated near the bifurcate Killing horizon by a massless two-dimensional conformal field.

The semi-classical approximation to black hole partition functions is not well-defined, because the classical action is unbounded and the first variation of the uncorrected action does not vanish for all variations preserving the boundary conditions. Both problems can be solved by adding a Hamilton-Jacobi counterterm. I show that the same problem and solution arises in quantum mechanics for half-binding potentials.

By using path integral methods we obtain effective Lagrangians for noncommutative Quantum Mechanics. The starting point is a relatively simple modification of standard phase-space path integrals, which leads in configuration space to Lagrangians depending also on the accelerations. We comment on the subtleties involved.

I shall briefly review what I consider as some of the important physics results in which the use of path or field integrals has played an essential role. This is by now a long history.^1,2

We study the finite-size critical behavior of the anisotropic φ⁴ lattice model with periodic boundary conditions in a d-dimensional hypercubic geometry above, at, and below T_c. Our perturbation approach at fixed d = 3 yields excellent agreement with the Monte Carlo (MC) data for the finite-size amplitude of the free energy of the three-dimensional Ising model at T_c by Mon [Phys. Rev. Lett. 54, 2671 (1985)]. Below T_c a minimum of the scaling function of the excess free energy is found. We predict a measurable dependence of this minimum on the anisotropy parameters. Our theory agrees quantitatively with the non-monotonic dependence of the Binder cumulant on the ferromagnetic next-nearest neighbor (NNN) coupling of the two-dimensional Ising model found by MC simulations of Selke and Shchur [J. Phys. A 38, L739 (2005)]. Our theory also predicts a non-monotonic dependence for small values of the anti-ferromagnetic NNN coupling and the existence of a Lifshitz point at a larger value of this coupling. The tails of the large-L behavior at T ≠ T_c violate both finite-size scaling and universality even for isotropic systems as they depend on the bare four-point coupling of the φ⁴ theory, on the cutoff procedure, and on subleading long-range interactions.

An overview is given of recent results concerning systems described by a set of at least two slow dynamical variables. The simplest model contains a relaxing order parameter coupled to the energy density (model C). The effects induced by randomness in such a model are discussed. At the superconducting transition the gauge dependence of the critical dynamics is considered for a model of two coupled relaxation equations.

The phase-ordering kinetics of the ferromagnetic two-dimensional Ising model with uniform disorder is characterised by a dynamical exponent z = 2 + ε/T which depends continuously on the disorder and on temperature. This allows for a detailed test of local scale invariance for several distinct values of z.

Finite-size effects are investigated in the mean spherical model in film geometry with nonperiodic boundary conditions above and below bulk T_c. We have obtained exact results for the excess free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed boundary conditions in 2 < d ≤ 3 dimensions. Analytic results are presented in 2 < d < 3 dimensions for Dirichlet boundary conditions and for d = 3 for Neumann-Dirichlet boundary conditions. We find an unexpected leading size dependence ∝ C_± t/L² of the Casimir force, with dfferent amplitudes C₊ and C_- above and below T_c for large L at fixed t ≡ (T - T_c)/T_c ≠ 0 for other than periodic boundary conditions.

The theory presented is based on a simple Hamiltonian for a vortex lattice in a weak impurity background which includes linear elasticity and plasticity, the latter in the form of integer valued fields accounting for defects. By using the variational approach of Mézard and Parisi established for random manifolds, we obtain the phase diagram including glass transition lines for superconductors with a melting line near H_c2 like YBCO and also for superconductors with a melting line in the deep H_c2 region like BSCCO.

We study the critical behavior of the classical φ⁴-model approaching criticality from the broken symmetry phase using the functional renormalization group (RG). We derive and solve RG flow equations for the flowing order parameter and the coupling constant. We also calculate the scaling function for the momentum dependent self-energy at the critical temperature T_c.

Higher-order perturbative calculations in Quantum (Field) Theory suffer from the factorial increase of the number of individual diagrams. Here I describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature. This method (originally proposed by Bogolyubov and Plechko) is applied to the calculation of higher-order terms for the ground-state energy of the polaron. Using state-of-the-art multidimensional integration routines two new coefficients are obtained corresponding to a 4- and 5-loop calculation.

Recent experiments by Kim and Chan on solid ⁴He have been interpreted as discovery of a supersolid phase of matter. Arguments based on wavefunctions have shown that such a phase exists, but do not necessarily apply to solid ⁴He. Imaginary time path integrals, implemented using Monte Carlo methods, provide a definitive answer; a clean system of solid ⁴He should be a normal quantum solid, not one with superfluid properties. The Kim-Chan phenomena must be due to defects introduced when the solid is formed.

The diffusion quantum Monte Carlo method is extended to solve the old theoretical physics problem of many-electron atoms and ions in intense magnetic fields. The feature of our approach is the use of adiabatic approximation wave functions augmented by a Jastrow factor as guiding functions to initialize the quantum Monte Carlo prodecure. We calculate the ground state energies of atoms and ions with nuclear charges from Z = 2, 3, 4,…, 26 for magnetic field strengths relevant for neutron stars.

Hard and soft disks in external periodic (light-) fields show rich phase diagrams including freezing and melting transitions when the density of the system is varied. Monte Carlo simulations for detailed finite-size scaling analysis of various thermodynamic quantities like the order parameter, its cumulants, etc. have been used in order to map the phase diagram of the system for various values of the density and the amplitude of the external potential.

Interpreting hard disks as the simplest model of an atomic fluid, quantum effects on the phase diagram are investigated by path integral Monte Carlo simulations.

By Hamiltonian path-integration a purely-quantum, self-consistent, spin-wave approximation can be developed for spin models on a lattice, that finally allows to map the original quantum problem to a classical one ruled by an effective classical spin Hamiltonian. Such approach has revealed especially valuable to investigate systems with S > 1/2 which cannot be easily addressed by other methods. This has made possible to quantitatively interpret experimental data for intermediate-spin compounds and to study how different observables reach the classical limit by increasing S. Here, we focus on the spin-flop phase of a quantum 2D antiferromagnet frustrated by an applied magnetic field that acts as an effective easy-plane anisotropy and determines Berezinskii-Kosterlitz-Thouless (BKT) behavior. By acting on the field one can tune the BKT transition temperature, giving a unique opportunity to observe the otherwise elusive BKT critical behavior in real magnetic systems. The calculated data are shown to well concur with the experimental findings for the S = 5/2 compound manganese-formate-dihydrate.

A simple microcanonical strategy for the simulation of first-order phase transitions is presented. The method does not require iterative parameters optimization, nor long waits for tunneling between the ordered and the disordered phases. It is illustrated in the Q-states Potts model in two dimensions for which several exact results are known, and where a cluster method nicely works.

Random graphs are widely used for modeling complex networks. Instead of considering many different models, to study dynamical phenomena on networks, it is desirable to design a general algorithm which produces random graphs with a variety of properties. Here we present a Monte Carlo method based on a random walk in the space of graphs. By ascribing to each graph a statistical weight we can generate networks of different types by tuning the weight function. The algorithm allows in particular to perform multicanonical simulations known, e.g., from spin models.

We investigate the behaviour of stochastic differential equations, especially Burgers' equation, by means of Monte Carlo techniques. By analysis of the produced configurations, we show that direct and often intuitive insight into the fundamentals of the solutions to the underlying equation, like shock wave formation, intermittency and chaotic dynamics, can be obtained. We also demonstrate that very natural constraints for the lattice parameters are sufficient to ensure stable calculations for unlimited numbers of Monte Carlo steps.

I discuss the use of path integrals to study strong-interaction physics from first principles. The underlying theory is cast into path integrals which are evaluated numerically using Monte Carlo methods on a space-time lattice. Examples are given in progress made in nuclear physics.

The investigation of superfluid atomic Fermi gases in the regime of strong interactions is conveniently investigated with the path-integral method at temperature zero, or at the critical temperature where the gap vanishes, by taking particle-pair or hole-pair fluctuations into account. Here, we also take the particle-hole excitations into account, which is important to investigate intermediate temperatures. The additional terms in the fluctuation propagator are identified, and a contour integral representation is used to calculate the contribution of these terms to the free energy and to the density of noncondensed fermions.

We construct an effective field theory for a condensate of cold Fermi atoms whose scattering is controlled by a Feshbach resonance, with particular emphasis on the speed of sound and its hydrodynamic description.

The appearance of coherence and order is studied for ultracold bosonic atoms in the presence of additional disorder potentials. These arise either naturally like in current carrying wire traps, or artificially and controllably like in laser speckle fields. The description of such disordered bosons within a suitably generalized Bogoliubov theory, first given by Huang and Meng, is rederived here within a functional integral approach for replicated bosonic fields. The superfluidity in homogeneous Bose systems with condensates depleted by weak interactions and disorder can thereby be discussed.

A mixture of spin-1/2 fermionic atoms and molecules of paired fermionic atoms is studied in an optical lattice. The molecules are formed by an attractive nearest-neighbor interaction. A functional integral is constructed for this many-body system and analyzed in terms of a mean-field approximation and Gaussian fluctuations. This provides a phase diagram with the two merging Mott insulators and an intermediate superfluid. The Gaussian fluctuations give rise to an induced repulsive dimer-dimer interaction mediated by the unpaired fermions. The effect of an unbalanced distribution of spin-up and spin-down fermions is also discussed.

A mixture of two types of fermions with different mass in an optical lattice is considered. The light fermions are subject to thermal and quantum fluctuations, the heavy fermions only to thermal fluctuations. We derive the distribution of heavy fermions and study numerically the localization length of the light fermions in a two-dimensional lattice. Depending on the temperature of the system, a transition from extended states at low temperatures to localized states at high temperatures is found.

The dilute Bose gas is studied in the large-N limit using functional integration.

We set up a recursion relation for the partition function of a fixed number of harmonically confined bosons. For an ideal Bose gas this leads to the well-known results for the temperature dependence of the specific heat and the ground-state occupancy. Due to the diluteness of the gas, we include both the isotropic contact interaction and the anisotropic dipole-dipole interaction by an infinite-bubble sum of the lowest-order perturbative results. Due to the anisotropy of the dipole-dipole interaction, the thermodynamic quantities of interest crucially depend on the trap configuration.

The dynamic structure factor of a homogeneous, weakly interacting BOSE gas has been calculated for finite temperatures within RPA and compared to corresponding results for a Fermi gas. Approaching T_c from high temperatures, the boson S(q,ω) exhibits a noteworthy BEC precursor. Temperature–dependent renormalization of the Bogoliubov frequency and the Landau damping of density excitations in the BOSE gas are discussed in detail. The isothermal compressibility of a Bose gas in the condensed phase turns out to vary only slightly with temperature.

We introduce a stochastic field equation based on the P-representation of the grand canonical density operator of a Bose gas which is free from ultraviolet problems. Numerical simulations for a harmonic trap potential are presented. Although strictly valid for an ideal gas only, we argue that the behavior of weakly interacting Bose gases at finite temperatures may also be described.

We investigate a trapped Bose-Einstein condensate underlying besides a short-range contact interaction especially a long-range 1/r interaction. The latter one can artificially be created by the radiation field of a certain laser configuration via induced electric dipoles. For this system we calculate the leading shift of the critical temperature with respect to the ideal gas.

We consider a dilute Bose gas moving in a harmonic trap with a superimposed frozen random potential which arises in experiments either naturally in wire traps or artificially and controllably with laser speckles. The critical temperature, which characterises the onset of Bose-Einstein condensation, depends on the disorder realization within the ensemble. Therefore, we introduce an effective grand-canonical potential from which we determine perturbatively the disorder averages of both the first and the second moment of the critical temperature in leading order. We discuss our results for a finite number of particles by assuming a Gaussian spatial correlation for the quenched disorder potential.

We analyze in detail recent experiments on ultracold dilute ⁸⁷Rb–⁴⁰K mixtures in Hamburg and in Florence within a mean-field theory. To this end we determine how the stationary bosonic and fermionic density profiles in this mixture depend in the Thomas-Fermi limit on the respective particle numbers. Furthermore, we investigate how the observed stability of the Bose-Fermi mixture with respect to collapse is crucially related to the value of the interspecies s-wave scattering length.

At first, we consider an ideal gas of harmonically trapped fermions with total angular momentum F = 3/2 and calculate, e.g., the temperature dependence of the heat capacity for a fixed magnetization. Afterwards, the isotropic short-range contact-interaction is treated perturbatively and its influence on the ground-state energy is worked out. Such spinor Fermi gases are important, for example, in the context of a ⁵²Cr-⁵³Cr boson-fermion mixture.¹

Electrical noise produced when charges flow through a mesoscopic conductor is non-Gaussian, where higher order cumulants carry valuable information about the microscopic transport process. In these notes we briefly discuss the experimental situation and show that theoretical descriptions for the detection of electrical noise with Josephson junctions lead to generalizations of classical and quantum theories, respectively, for decay rates out of metastable states.

Some time ago, DeWitt-Morette¹ et al. discussed the problem of the propagation of radiation or particles in the presence of a wedge. Their treatment includes path integral solutions of the wedge problem with Dirichlet or Neumann boundary conditions for various situations. Recently, the superconducting phase in a wedge has gained increasing interest. The linearized Ginzburg-Landau equation for the order parameter is formally similar to the Schrödinger equation. But the conditions of no normal current at the boundary pose very specific problems, which are discussed in the present paper.

It has recently been shown (K. Park, Phys. Rev. Lett. 95, 027001 (2005)) that the ground state of the t-J model at half filling is entirely equivalent to the ground state of the Gutzwiller-projected Bardeen-Cooper-Schrieffer (BCS) Hamiltonian with strong pairing. Here we extend this result to finite doping. We show that in the immediate vicinity of half filling the projected 2D BCS Hamiltonian with strong pairing still develops the antiferromagnetically (AF) ordered ground state.

We study Landau-Zener transitions of a two-level system that is coupled to a quantum heat bath at zero temperature. In particular, we reveal that for a whole class of models, the probability for a nonadiabatic transition is bath-independent.

We review recent advances in the full counting statistics (FCS) of transport of charge through an impurity in a 1D quantum wire. The model also applies to a coherent conductor in a resistive environment and to a resistively shunted Josephson device. Starting out from the path integral Coulomb gas representation both the weak and strong tunneling series representations for the cumulant generating function can be found in analytic form in particular regions of the parameter space. The zero temperature case is discussed in some detail.

In contrast to standard critical phenomena, disordered systems need to be treated via the Functional Renormalization Group. The latter leads to a coarse grained disorder landscape, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We review recent progress on how the non-analytic effective action can be measured both in simulations and experiments, and confront theory with numerical work.

We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^d-σ. Using FRG we obtained the phase diagram in (d, σ, N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4 + σ, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.

The critical behavior of the d-dimensional uniaxial ferromagnetic systems with weak quenched disorder is studied. In the critical region this model is known to be described by both the conventional Ginzburg-Landau Hamiltonian and the two-dimensional fermionic Gross-Neveu model in the n = 0 limit. Renormalization group calculations are used to obtain the temperature dependences near the critical point of some thermodynamic quantities, the large distance behavior of the two-spin correlation function and the equation of state at criticality. The ϵ expansion of the critical exponents is also discussed as well as the Kramers-Wannier duality of two-dimensional dilute systems. Most important questions of the theory of critical phenomena are the problem of universality of the critical behavior of random systems and the role of Griffiths singularities.

The functional measure of the three-dimensional complex |ψ|⁴ theory allows for line-like topological excitations which can be related to vortex lines in super-fluid helium by universality arguments. Upon approaching the λ point, these lines proliferate and destroy the superfluidity. To study the phase transition from this geometrical point of view, we investigated the statistical properties of the emerging vortex-loop network in the vicinity of the critical point by means of high-precision Monte Carlo simulations. For comparison the standard magnetic properties of the system were considered as well. Using sophisticated embedded cluster update techniques we examined if both of them exhibit the same critical behaviour leading to the same critical exponents and therefore to a consistent description of the phase transition. We find that different definitions for constructing the vortex-loop network lead to slightly (but statistically significantly) different results in the thermodynamic limit, and that the percolation thresholds are close to but do not really coincide with the thermodynamic phase transition point.

A magnetic system is usually described in terms of the exchange coupling between neighboring spins lying on the sites of a given lattice. Our goal here is to account for the unavoidable quantum effects due to the further coupling with the vibrations of the ions constituting the lattice. A Caldeira-Leggett scheme allows one to treat such effects through the analysis of the associated influence action, obtained after tracing-out the phonons. In a physically sound model, it turns out that one must deal with an environmental coupling which is nonlinear in the system's variables. The corresponding path integral can be dealt with by suitably extending the pure-quantum self-consistent harmonic approximation. In this way one can obtain extended phase diagrams for magnetic phase transitions, accounting for the environmental interaction.

The path integral approach is used for the calculation of the correlation functions of the XY Heisenberg chain. The obtained answers for the two-point correlators of the XX magnet are of the determinantal form and are interpreted in terms of the generating functions for the random turns vicious walkers.

With continuous time quantum Monte Carlo simulations we investigate a continuous quantum phase transition in the mixed quantum spin chain with spin arrangement - S^a - S^a - S^b - S^b -, with S^a = 1/2 and S^b = 1. By finite-size scaling analysis we calculate estimates of the critical control parameter as well as estimates of critical exponents.

The issue of the number, nature and sequence of phase transitions in the fully frustrated XY (FFXY) model is a highly non trivial one due to the complex interplay between its continuous and discrete degrees of freedom. In this contribution we attack such a problem by means of a twisted conformal field theory (CFT) approach¹ and show how it gives rise to the U(1) ⊗ Z₂ symmetry and to the whole spectrum of excitations of the FFXY model.²

We performed Monte Carlo simulations of the two-dimensional Ising model in the low-temperature phase at T = 1.5 at constant magnetisation (Kawasaki dynamics) and measured the size of the largest minority droplet, i.e., the largest cluster and all overturned spins within. The measured values are compared to theoretical predictions by Biskup et al., which can explain a jump of the droplet size in the vicinity of the spontaneous magnetisation.

Proteins are the "work horses" in biological systems. In almost all functions specific proteins are involved. They control molecular transport processes, stabilize the cell structure, enzymatically catalyze chemical reactions; others act as molecular motors in the complex machinery of molecular synthetization processes. Due to their significance, misfolds and malfunctions of proteins typically entail disastrous diseases, such as Alzheimer's disease and bovine spongiform encephalopathy (BSE). Therefore, the understanding of the trinity of amino acid composition, geometric structure, and biological function is one of the most essential challenges for the natural sciences. Here, we glance at conformational transitions accompanying the structure formation in protein folding processes.

We give an overview of the glassy wormlike chain model, which describes the dynamics of a semiflexible polymer interacting with a surrounding sticky solution. The model is then generalized to the Rouse chain and we compare the macroscopic shear modulus.

We investigate the effect of hydrodynamic interactions on the non-equilibrium dynamics of an ideal flexible polymer pulled by a constant force applied at one polymer end using the perturbation theory. For moderate force, if the polymer elongation is small, the hydrodynamic interactions are not screened and the velocity and the longitudinal elongation of the polymer are computed using the renormalization group method. For large chain lengths and a finite force the hydrodynamic interactions are only partially screened, which in three dimensions results in unusual logarithmic corrections to the velocity and the longitudinal elongation.

We analyze the impact of a porous medium (structural disorder) on the scaling of the partition function of a star polymer immersed in a good solvent. We show that corresponding scaling exponents change if the disorder is long-range-correlated and calculate the exponents in the new universality class. A notable finding is that star and chain polymers react in qualitatively different manner on the presence of disorder: the corresponding scaling exponents increase for chains and decrease for stars. We discuss the physical consequences of this difference.

In this work the problem of describing the dynamics of a continuous chain with rigid constraints is treated using a path integral approach.

In this work we discuss the dynamics of a three-dimensional chain. It turns out that the generalized sigma model presented in Ref. 1 may be easily generalized to three dimensions. The formula of the probability distribution of two topologically entangled chains is provided. The interesting case of a chain which can form only discrete angles with respect to the z–axis is also presented.

Biopolymer conformations are investigated using the white noise path integral approach. Analytical evaluation of the path integral exhibits various features such as chirality, overwinding of biopolymers when stretched, and the helix-turn-helix structure.

The Feynman path integral for the quantum mechanical propagator is interpreted as the T-transform (infinite dimensional generalized Fourier transform in the Hida calculus) of a suitable functional in the space of Hida white noise distributions. Essential features of the approach are given and applications in the evaluation of various path integrals are noted.

The function of biopolymers depends to a large part on their shape statistics, on length scales ranging from one to thousands of monomers. I present a continuous model in which the equilibrium ensemble of macromolecular conformations is generated by a random walk with values in the Euclidean group. It includes local bending, twisting, stretching and shearing modes. The model exhibits helical structure on an intermediate length scale, while in the limit of long chains, the well-known worm-like chain behavior is recovered.

In recent years the Continuous Time Random Walk (CTRW) has been used to model anomalous diffusion in a variety of complex systems. Since this process is non-Markovian, the knowledge of single time probability distributions is not sufficient to characterize the CTRW. Using the method of subordination we construct an extension of the Wiener path integral for Brownian motion to CTRW processes. This contribution is a step towards a path integral formulation of CTRWs.

The scaling behavior of linear polymers in disordered media, modelled by self-avoiding walks (SAWs) on the backbone of percolation clusters in two, three and four dimensions is studied by numerical simulations. We apply the pruned-enriched Rosenbluth chain-growth method (PERM). Our numerical results bring about the estimates of critical exponents, governing the scaling laws of disorder averages of the configurational properties of SAWs.

It is shown that superpositions of path integrals with arbitrary Hamiltonians and different scaling parameters υ ("variances") obey the Chapman-Kolmogorov relation for Markovian processes if and only if the corresponding smearing distributions for υ have a specific functional form. Ensuing "smearing" distributions substantially simplify the coupled system of Fokker-Planck equations for smeared and unsmeared conditional probabilities. Simple application in financial models with stochastic volatility is presented.

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