Multi-loop zeta function regularization and spectral cutoff in curved spacetime

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.     

Generalized spectral decomposition of the β-adic baker's transformation and intrinsic irreversibility

We construct a generalized spectral decomposition of the Frobenius-Perron operator of the β-adic baker's transformation using a general iterative operator method applicable in principle for any mixing dynamical system. The eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one. We explicitly define appropriate generalized function spaces, which provide mathematical meaning to the formally obtained spectral decomposition. The unitary Frobenius-Perron evolution of densities, when extended to the generalized function spaces, splits into two semigroups, one decaying in the future and the other in the past. This split, which reflects the asymptotic evolution of the forward and backward K-partitions, shows the instrinsic irreversibility of the baker's transformation.     

Direct measurement of group dispersion of optical components using white-light spectral interferometry

We present a simple white-light spectral interferometric technique employing a low-resolution spectrometer for a direct measurement of the group dispersion of optical components over a wide wavelength range. The technique utilizes an unbalanced Mach-Zehnder interferometer with a component under test inserted in one arm and the other arm with adjustable path length. We record a series of spectral interferograms to measure the equalization wavelength as a function of the path length difference. We measure the absolute group refractive index as a function of wavelength for a quartz crystal of known thickness and the relative one for optical fiber. In the latter case we use a microscope objective in front and a lens behind the fiber and subtract their group dispersion, which is measured by a technique of tandem interferometry including also a Michelson interferometer.     

Relative Oscillation Theory,Weighted Zeros of the Wronskian,and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm–Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein’s spectral shift function is established. Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.     

Renormalization of the Asymptotically Expanded Yang–Mills Spectral Action

We study renormalizability aspects of the spectral action for the Yang–Mills system on a flat 4-dimensional background manifold, focusing on its asymptotic expansion. Interpreting the latter as a higher-derivative gauge theory, a power-counting argument shows that it is superrenormalizable. We determine the counterterms at one-loop using zeta function regularization in a background field gauge and establish their gauge invariance. Consequently, the corresponding field theory can be renormalized by a simple shift of the spectral function appearing in the spectral action.     

The Spectral Shift Function and Spectral Flow

At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D +  V with D having compact resolvent belonging to a general semifinite von Neumann algebra and the perturbation . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples where D has compact resolvent belonging to and V is any bounded self-adjoint operator in . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].     

Quantum decoherence in the spectral functions of undoped lamno3

We study the one hole spectral function in a model for LaMnO3 including both the effects of orbital ordering and the quantum decoherence due to the antiferromagnetic coupling between the ferromagnetic layers. We find that the classical picture of a ferromagnetic polaron does not apply and free dispersion is replaced by rigid quasiparticles on the scale of magnetic excitations, while the spectra are dominated by the incoherent spectral weight at high energies. These results have important implications on the in-plane transport and angular resolved photoemission in the manganites.     

A holomorphic and background independent partition function for matrix models and topological strings

We study various properties of a nonperturbative partition function which can be associated with any spectral curve. When the spectral curve arises from a matrix model, this nonperturbative partition function is given by a sum of matrix integrals over all possible filling fractions, and includes all the multi-instanton corrections to the perturbative 1/N$1/N$ expansion. We show that the nonperturbative partition function, which is manifestly holomorphic, is also modular and background independent: it transforms as the partition function of a twisted fermion on the spectral curve. Therefore, modularity is restored by nonperturbative corrections. We also show that this nonperturbative partition function obeys the Hirota equation and provides a natural nonperturbative completion for topological string theory on local Calabi–Yau 3-folds.     

The truncation of the Mori continued fraction for the spectral function in Heisenberg and XY models

We study the truncation of the Mori continued fraction for the two-spin spectral function of the XY model in various schemes suggested by the 3-pole approximation. These schemes are (i) a fully consistent version of the 3-pole approximation; (ii) 5-pole approximation (not fully consistent); and (iii) 5-pole approximate (fully consistent). We establish the equilivalence between the fully consistent 3-pole approximation and the Bennett-Martin approximation. Finally, we calculate the time-dependent spin-spin correlation function for the classical Heisenberg model in both versions of the 3-pole approximation and compare it with the exact spin-spin correlation function obtained by direct numerical calculation in one dimension.     

Spectral Gap and Exponential Decay of Correlations

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.     

A simple algorithm for spectral line deconvolution

The objective of this work is to develop a numerical procedure to subtract the instrumental function from a measured spectral line profile. The measuring device (for example, a Fabry-Perot Interferometer) distorts the spectral line profile and the experimentally measured one is a convolution of this profile and the instrumental function. Restoring the spectral line profile is strongly affected by numerical instabilities and the problem has been overcome by using the Tikhonov regularization method. The approach is very simple and easy for programming and it is particularly useful for “noisy” experimental data.     

Luttinger liquids with boundaries: Power-laws and energy scales

We present a study of the one-particle spectral properties for a variety of models of Luttinger liquids with open boundaries. We first consider the Tomonaga-Luttinger model using bosonization. For weak interactions the boundary exponent of the power-law suppression of the spectral weight close to the chemical potential is dominated by a term linear in the interaction. This motivates us to study the spectral properties also within the Hartree-Fock approximation. It already gives power-law behavior and qualitative agreement with the exact spectral function. For the lattice model of spinless fermions and the Hubbard model we present numerically exact results obtained using the density-matrix renormalization-group algorithm. We show that many aspects of the behavior of the spectral function close to the boundary can again be understood within the Hartree-Fock approximation. For the repulsive Hubbard model with interaction U the spectral weight is enhanced in a large energy range around the chemical potential. At smaller energies a power-law suppression, as predicted by bosonization, sets in. We present an analytical discussion of the crossover and show that for small U it occurs at energies exponentially (in -1/U) close to the chemical potential, i.e. that bosonization only holds on exponentially small energy scales. We show that such a crossover can also be found in other models. Received 8 February 2000 and Received in final form 25 April 2000     

Forward scattering approximation and bosonization in integer quantum Hall systems

In this work, we present a model and a method to study integer quantum Hall (IQH) systems. Making use of the Landau levels structure we divide these two-dimensional systems into a set of interacting one-dimensional gases, one for each guiding center. We show that the so-called strong field approximation, used by Kallin and Halperin and by MacDonald, is equivalent, in first order, to a forward scattering approximation and analyze the IQH systems within this approximation. Using an appropriate variation of the Landau level bosonization method we obtain the dispersion relations for the collective excitations and the single-particle spectral functions. For the bulk states, these results evidence a behavior typical of non-normal strongly correlated systems, including the spin-charge splitting of the single-particle spectral function. We discuss the origin of this behavior in the light of the Tomonaga-Luttinger model and the bosonization of two-dimensional electron gases.     

Effective dimensionality approach to the RKKY interaction in multilayer systems

We consider RKKY interaction in a quasi 2D system with nonparabolic dispersion. In our paper we calculate the RKKY range function assuming the in-layer confinement via effective dimensionality approach. We show, that indirect magnetic exchange in our system can be modelled by the effective spectral dimension which equals one.     

Quark spectral properties above <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript> from Dyson–Schwinger equations

We report on an analysis of the quark spectral representation at finite temperatures based on the quark propagator determined from its Dyson–Schwinger equation in Landau gauge. In Euclidean space we achieve nice agreement with recent results from quenched lattice QCD. We find different analytical properties of the quark propagator below and above the deconfinement transition. Using a variety of ansätze for the spectral function we then analyze the possible quasiparticle spectrum, in particular its quark mass and momentum dependence in the high temperature phase. This analysis is completed by an application of the Maximum Entropy Method, in principle allowing for any positive semi-definite spectral function. Our results motivate a more direct determination of the spectral function in the framework of Dyson–Schwinger equations.     

Measuring the spectrum of colored noise by dynamical decoupling

Decoherence is one of the most important obstacles that must be overcome in quantum information processing. It depends on the qubit-environment coupling strength, but also on the spectral composition of the noise generated by the environment. If the spectral density is known, fighting the effect of decoherence can be made more effective. Applying sequences of inversion pulses to the qubit system, we developed a method for dynamical decoupling noise spectroscopy. We generate effective filter functions that probe the environmental spectral density without requiring assumptions about its shape. Comparing different pulse sequences, we recover the complete spectral density function and distinguish different contributions to the overall decoherence.     

Dispersion error of a beam splitter cube in white-light spectral interferometry

We revealed that the phase function of a thin-film structure measured by a white-light spectral interferometric technique depends on the path length difference adjusted in a Michelson interferometer. This phenomenon is due to a dispersion error of a beam splitter cube, the effective thickness of which varies with the adjusted path length difference. A technique for eliminating the effect in measurement of the phase function is described. In a first step, the Michelson interferometer with same metallic mirrors is used to measure the effective thickness of the beam splitter cube as a function of the path length difference. In a second step, one of the mirrors of the interferometer is replaced by a thin-film structure and its phase function is measured for the same path length differences as those adjusted in the first step. In both steps, the phase is retrieved from the recorded spectral interferograms by using a windowed Fourier transform applied in the wavelength domain.     

Spectroscopy of the Hubbard dimer: the spectral potential

The spectral potential is the dynamical generalization of the Kohn–Sham potential. It targets, in principle exactly, the spectral function in addition to the electronic density. Here we examine the spectral potential in one of the simplest solvable models exhibiting a non-trivial interplay between electron-electron interaction and inhomogeneity, namely the asymmetric Hubbard dimer. We discuss a general strategy to introduce approximations, which consists in calculating the spectral potential in the homogeneous limit (here represented by the symmetric Hubbard dimer) and importing it in the real inhomogeneous system through a suitable “connector”. The comparison of different levels of approximation to the spectral potential with the exact solution of the asymmetric Hubbard dimer gives insights about the advantages and the difficulties of this connector strategy for applications in real materials.     

Specification of spectral line shape and multiplex dispersion by self-imaging and moiré technique

Due to finite width of a spectral line, the visibility of the moiré fringes formed by a grating and the self-image of another similar grating reduces by the increase of the self-image order. This effect is exploited to specify the spectral line shape by evaluating the Fourier transform of a function related to the visibility. Even, by using in-expensive optics, the technique can provide the spectral line shapes of rather broad widths—of the order of nanometer and more—by precisions that are comparable by those obtained by expensive Fourier transform spectrometers.Besides, it is shown that by comparing the line shapes obtained with and without a dispersive medium between the gratings, one can specify the dispersion function of the medium in the wavelength range covered by the spectrum.     

Spectral Extension of the Quantum Group Cotangent Bundle

The structure of a cotangent bundle is investigated for quantum linear groups GL _q (n) and SL _q (n). Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on SL _q (n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators—the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SL _q (n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of the q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. The relation between the two operators is given by a modular functional equation for the Riemann theta function.