On the generalized eigenvalue method for energies and matrix elements in lattice field theory

We discuss the generalized eigenvalue problem for computing energies and matrix elements in lattice gauge theory, including effective theories such as HQET. It is analyzed how the extracted effective energies and matrix elements converge when the time separations are made large. This suggests a particularly efficient application of the method for which we can prove that corrections vanish asymptotically as exp(-(E(N+1) - E(n))t). The gap E(N+1) - E(n) can be made large by increasing the number N of interpolating fields in the correlation matrix. We also show how excited state matrix elements can be extracted such that contaminations from all other states disappear exponentially in time. As a demonstration we present numerical results for the extraction of ground state and excited B-meson masses and decay constants in static approximation and to order 1/m(b) in HQET.

Chaotic thermalization in Yang-Mills-Higgs theory on a spacial lattice

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

THEORY OF ELASTICITY OF THE ABRIKOSOV FLUX-LINE LATTICE FOR UNIAXIAL SUPERCONDUCTORS - PARALLEL FLUX LINES

In this paper, we consider the extension of the Brandt theory of elasticity of the Abrikosov flux-line lattice for a uniaxial superconductor for the case of parallel flux lines. The results show that the effect of the anisotropy is to rescale the components of the wave vector k and the magnetic field and order-parameter wave vector cut off by a geometrical parameter previously introduced by Kogan.

Analytic perturbation theory for bound states in the transfer matrix spectrum of weakly correlated lattice ferromagnetic spin systems

We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region ('beta' << 1). Each model is characterized by a single site apriori spin distribution taken to be even. We also take the parameter 'alfa' = ('S POT.4') - 3 '(S POT.2') POT.2' > 0, i.e. in the region which we call Gaussian subjugation, where ('S POT.K') denotes the kth moment of the apriori distribution. Associated with the model is a lattice quantum field theory known to contain a particle of asymptotic mass -ln 'beta' and a bound state below the two-particle threshold. We develop a 'beta' analytic perturbation theory for the binding energy of this bound state. As a key ingredient in obtaining our result we show that the Fourier transform of the two-point function is a meromorphic function, with a simple pole, in a suitable complex spectral parameter and the coefficients of its Laurent expansion are analytic in 'beta'.

Quantum simulation of (1+1)D QED via a Zn lattice Gauge theory

La simulazione di un sistema quantistico complesso rappresenta ancora oggi una sfida estremamente impegnativa a causa degli elevati costi computazionali. La dimensione dello spazio di Hilbert cresce solitamente in modo esponenziale all'aumentare della taglia, rendendo di fatto impossibile una implementazione esatta anche sui più potenti calcolatori. Nel tentativo di superare queste difficoltà, sono stati sviluppati metodi stocastici classici, i quali tuttavia non garantiscono precisione per sistemi fermionici fortemente interagenti o teorie di campo in regimi di densità finita. Di qui, la necessità di un nuovo metodo di simulazione, ovvero la simulazione quantistica. L'idea di base è molto semplice: utilizzare un sistema completamente controllabile, chiamato simulatore quantistico, per analizzarne un altro meno accessibile. Seguendo tale idea, in questo lavoro di tesi si è utilizzata una teoria di gauge discreta con simmetria Zn per una simulazione dell'elettrodinamica quantistica in (1+1)D, studiando alcuni fenomeni di attivo interesse di ricerca, come il diagramma di fase o la dinamica di string-breaking, che generalmente non sono accessibili mediante simulazioni classiche. Si propone un diagramma di fase del modello caratterizzato dalla presenza di una fase confinata, in cui emergono eccitazioni mesoniche ed antimesoniche, cioè stati legati particella-antiparticella, ed una fase deconfinata.

Light-cone Wilson loop in classical lattice gauge theory

The transverse broadening of an energetic jet passing through a non-Abelian plasma is believed to be described by the thermal expectation value of a light-cone Wilson loop. In this exploratory study, we measure the light-cone Wilson loop with classical lattice gauge theory simulations. We observe, as suggested by previous studies, that there are strong interactions already at short transverse distances, which may lead to more efficient jet quenching than in leading-order perturbation theory. We also verify that the asymptotics of the Wilson loop do not change qualitatively when crossing the light cone, which supports arguments in the literature that infrared contributions to jet quenching can be studied with dimensionally reduced simulations in the space-like domain. Finally we speculate on possibilities for full four-dimensional lattice studies of the same observable, perhaps by employing shifted boundary conditions in order to simulate ensembles boosted by an imaginary velocity.

Compactified N = 1 supersymmetric Yang-Mills theory on the lattice: continuity and the disappearance of the deconfinement transition

Fermion boundary conditions play a relevant role in revealing the confinement mechanism of N=1 supersymmetric Yang-Mills theory with one compactified space-time dimension. A deconfinement phase transition occurs for a sufficiently small compactification radius, equivalent to a high temperature in the thermal theory where antiperiodic fermion boundary conditions are applied. Periodic fermion boundary conditions, on the other hand, are related to the Witten index and confinement is expected to persist independently of the length of the compactified dimension. We study this aspect with lattice Monte Carlo simulations for different values of the fermion mass parameter that breaks supersymmetry softly. We find a deconfined region that shrinks when the fermion mass is lowered. Deconfinement takes place between two confined regions at large and small compactification radii, that would correspond to low and high temperatures in the thermal theory. At the smallest fermion masses we find no indication of a deconfinement transition. These results are a first signal for the predicted continuity in the compactification of supersymmetric Yang-Mills theory.

From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

We regularize compact and non-compact Abelian Chern–Simons–Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group R, each local Hilbert space is analogous to the one of a charged “particle” moving in the link-pair group space R2 in a constant “magnetic” background field. In the compact case, the link-pair group space is a torus U(1)2 threaded by k units of quantized “magnetic” flux, with k being the level of the Chern–Simons theory. The holonomies of the torus U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from U(1) to Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern–Simons limit of a large “photon” mass, this results in a Z(k)-symmetric variant of Kitaev’s toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle . Non-Abelian U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.

Proof theory for lattice-ordered groups

Proof-theoretic methods are developed and exploited to establish properties of the variety of lattice-ordered groups. In particular, a hypersequent calculus with a cut rule is used to provide an alternative syntactic proof of the generation of the variety by the lattice-ordered group of automorphisms of the real number chain. Completeness is also established for an analytic (cut-free) hypersequent calculus using cut elimination and it is proved that the equational theory of the variety is co-NP complete.

Performance analysis of adaptive lattice filters for FM signals and alpha-stable processes

The performance of an adaptive filter may be studied through the behaviour of the optimal and adaptive coefficients in a given environment. This thesis investigates the performance of finite impulse response adaptive lattice filters for two classes of input signals: (a) frequency modulated signals with polynomial phases of order p in complex Gaussian white noise (as nonstationary signals), and (b) the impulsive autoregressive processes with alpha-stable distributions (as non-Gaussian signals). Initially, an overview is given for linear prediction and adaptive filtering. The convergence and tracking properties of the stochastic gradient algorithms are discussed for stationary and nonstationary input signals. It is explained that the stochastic gradient lattice algorithm has many advantages over the least-mean square algorithm. Some of these advantages are having a modular structure, easy-guaranteed stability, less sensitivity to the eigenvalue spread of the input autocorrelation matrix, and easy quantization of filter coefficients (normally called reflection coefficients). We then characterize the performance of the stochastic gradient lattice algorithm for the frequency modulated signals through the optimal and adaptive lattice reflection coefficients. This is a difficult task due to the nonlinear dependence of the adaptive reflection coefficients on the preceding stages and the input signal. To ease the derivations, we assume that reflection coefficients of each stage are independent of the inputs to that stage. Then the optimal lattice filter is derived for the frequency modulated signals. This is performed by computing the optimal values of residual errors, reflection coefficients, and recovery errors. Next, we show the tracking behaviour of adaptive reflection coefficients for frequency modulated signals. This is carried out by computing the tracking model of these coefficients for the stochastic gradient lattice algorithm in average. The second-order convergence of the adaptive coefficients is investigated by modeling the theoretical asymptotic variance of the gradient noise at each stage. The accuracy of the analytical results is verified by computer simulations. Using the previous analytical results, we show a new property, the polynomial order reducing property of adaptive lattice filters. This property may be used to reduce the order of the polynomial phase of input frequency modulated signals. Considering two examples, we show how this property may be used in processing frequency modulated signals. In the first example, a detection procedure in carried out on a frequency modulated signal with a second-order polynomial phase in complex Gaussian white noise. We showed that using this technique a better probability of detection is obtained for the reduced-order phase signals compared to that of the traditional energy detector. Also, it is empirically shown that the distribution of the gradient noise in the first adaptive reflection coefficients approximates the Gaussian law. In the second example, the instantaneous frequency of the same observed signal is estimated. We show that by using this technique a lower mean square error is achieved for the estimated frequencies at high signal-to-noise ratios in comparison to that of the adaptive line enhancer. The performance of adaptive lattice filters is then investigated for the second type of input signals, i.e., impulsive autoregressive processes with alpha-stable distributions . The concept of alpha-stable distributions is first introduced. We discuss that the stochastic gradient algorithm which performs desirable results for finite variance input signals (like frequency modulated signals in noise) does not perform a fast convergence for infinite variance stable processes (due to using the minimum mean-square error criterion). To deal with such problems, the concept of minimum dispersion criterion, fractional lower order moments, and recently-developed algorithms for stable processes are introduced. We then study the possibility of using the lattice structure for impulsive stable processes. Accordingly, two new algorithms including the least-mean P-norm lattice algorithm and its normalized version are proposed for lattice filters based on the fractional lower order moments. Simulation results show that using the proposed algorithms, faster convergence speeds are achieved for parameters estimation of autoregressive stable processes with low to moderate degrees of impulsiveness in comparison to many other algorithms. Also, we discuss the effect of impulsiveness of stable processes on generating some misalignment between the estimated parameters and the true values. Due to the infinite variance of stable processes, the performance of the proposed algorithms is only investigated using extensive computer simulations.

Van der Waals-corrected density functional theory : benchmarking for hydrogen–nanotube and nanotube–nanotube interactions

Density functional theory (DFT) is a powerful approach to electronic structure calculations in extended systems, but suffers currently from inadequate incorporation of long-range dispersion, or Van der Waals (VdW) interactions. VdW-corrected DFT is tested for interactions involving molecular hydrogen, graphite, single-walled carbon nanotubes (SWCNTs), and SWCNT bundles. The energy correction, based on an empirical London dispersion term with a damping function at short range, allows a reasonable physisorption energy and equilibrium distance to be obtained for H2 on a model graphite surface. The VdW-corrected DFT calculation for an (8, 8) nanotube bundle reproduces accurately the experimental lattice constant. For H2 inside or outside an (8, 8) SWCNT, we find the binding energies are respectively higher and lower than that on a graphite surface, correctly predicting the well known curvature effect. We conclude that the VdW correction is a very effective method for implementing DFT calculations, allowing a reliable description of both short-range chemical bonding and long-range dispersive interactions. The method will find powerful applications in areas of SWCNT research where empirical potential functions either have not been developed, or do not capture the necessary range of both dispersion and bonding interactions.