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Ivan Hip

Publications and source records attributed to Ivan Hip.

17 recordsLinked to original sources
arXiv2023

A pion decay constant in the multi-flavor Schwinger model

The pion decay constant $F_π$ plays an important role in QCD and in Chiral Perturbation Theory. It is hardly known, however, that a corresponding constant exists in the Schwinger model with $N_{\rm f} \geq 2$ degenerate fermion flavors. In this case, the ``pion'' does not decay and $F_π$is dimensionless. Still, $F_π$ can be defined by 2d analogies to the Gell-Mann--Oakes--Renner relation, the Witten--Veneziano formula and the residual ``pion'' mass in the $δ$-regime. With suitable assumptions, and by inserting simulation data, these QCD-inspired relations are all compatible with $F_π \simeq 1/\sqrt{2π}$ at zero fermion mass, as we observe for $N_{\rm f} = 2, \dots , 6$. We conclude that this is a meaningful constant in the multi-flavor Schwinger model.

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arXiv2023

An analogue to the pion decay constant in the multi-flavor Schwinger model

We study the Schwinger model with $N_{\rm f} \geq 2$ degenerate fermion flavors, by means of lattice simulations. We use dynamical Wilson fermions for $N_{\rm f} = 2$, and re-weighted quenched configurations for overlap-hypercube fermions with $N_{\rm f} \leq 6$. In this framework, we explore an analogue of the QCD pion decay constant $F_π$, which is dimensionless in $d=2$, and which has hardly been considered in the literature. We determine $F_π$ by three independent methods, with numerical and analytical ingredients. First, we consider the 2-dimensional version of the Gell-Mann--Oakes--Renner relation, where we insert both theoretical and numerical values for the quantities involved. Next we refer to the $δ$-regime, {\it i.e.\ a small spatial volume, where we assume formulae from Chiral Perturbation Theory to apply even in the absence of Nambu-Goldstone bosons. We further postulate an effective relation between $N_{\rm f}$ and the number of relevant, light bosons, which we denote as "pions". Thus $F_π$ is obtained from the residual "pion" mass in the chiral limit, which is a finite-size effect. Finally, we address to the 2-dimensional Witten--Veneziano formula: it yields a value for $F_η$, which we identify with $F_π$, as in large-$N_{\rm c}$ QCD. All three approaches consistently lead to $F_π \simeq 1/\sqrt{2 π}$ at fermion mass $m=0$, which implies that this quantity is meaningful.

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arXiv2022

New insight in the 2-flavor Schwinger model based on lattice simulations

We consider the Schwinger model with two degenerate, light fermion flavors by means of lattice simulations. At finite temperature, we probe the viability of a bosonization method by Hosotani et al. Next we explore an analogue to the pion decay constant, which agrees for independent formulations based on the Gell-Mann--Oakes--Renner relation, the 2-dimensional Witten--Veneziano formula and the $δ$-regime. Finally we confront several conjectures about the chiral condensate with lattice results.

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arXiv2021

Finite temperature and delta-regime in the 2-flavor Schwinger model

The Schwinger model is often used as a testbed for conceptual and numerical approaches in lattice field theory. Still, some of its rich physical properties in anisotropic volumes have not yet been explored. For the multi-flavor finite temperature Schwinger model there is an approximate solution by Hosotani et al. based on bosonization. We perform lattice simulations and check the validity of this approximation in the case of two flavors. Next we exchange the rôle of the coordinates to enter the $δ$-regime, and measure the dependence of the residual "pion" mass on the spatial size, at zero temperature. Our results show that universal features, which were derived by Leutwyler, Hasenfratz and Niedermayer referring to quasi-spontaneous symmetry breaking in $d>2$, extend even to $d=2$. This enables the computation of the Schwinger model counterpart of the pion decay constant $F_π$. It is consistent with an earlier determination by Harada et al. who considered the divergence of the axial current in a light-cone formulation, and with analytical results that we conjecture from 2d versions of the Witten--Veneziano formula and the Gell-Mann--Oakes--Renner relation, which suggest $F_π = 1/ \sqrt{2π}$.

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arXiv2014

Features of a 2d Gauge Theory with Vanishing Chiral Condensate

The Schwinger model with $N_f \geq 2$ flavors is a simple example for a fermionic model with zero chiral condensate Sigma (in the chiral limit). We consider numerical data for two light flavors, based on simulations with dynamical chiral lattice fermions. We test properties and predictions that were put forward in the recent literature for models with Sigma = 0, which include IR conformal theories. In particular we probe the decorrelation of low lying Dirac eigenvalues, and we discuss the mass anomalous dimension and its IR extrapolation. Here we encounter subtleties, which may urge caution with analogous efforts in other models, such as multi-flavor QCD.

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arXiv2013

Microscopic Dirac Spectrum in a 2d Gauge Theory with Zero Chiral Condensate

Fermionic theories with a vanishing chiral condensate (in the chiral limit) have recently attracted considerable interest; in particular variants of multi-flavour QCD are candidates for this behaviour. Here we consider the 2-flavour Schwinger model as a simple theory with this property. Based on simulations with light dynamical overlap fermions, we test the hypothesis that in such models the low lying Dirac eigenvalues could be decorrelated. That has been observed in 4d Yang-Mills theories at high temperature, but it cannot be confirmed for the 2-flavour Schwinger model. We also discuss subtleties in the evaluation of the mass anomalous dimension and its IR extrapolation.

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arXiv2012

A Numerical Study of the 2-Flavour Schwinger Model with Dynamical Overlap Hypercube Fermions

We present numerical results for the 2-flavour Schwinger model with dynamical chiral lattice fermions. We insert an approximately chiral hypercube Dirac operator into the overlap formula to construct the overlap hypercube operator. This is an exact solution to the Ginsparg-Wilson relation, with an excellent level of locality and scaling. Due to its similarity with the hypercubic kernel, a low polynomial in this kernel provides a numerically efficient Hybrid Monte Carlo force. We measure the microscopic Dirac spectrum and discuss the corresponding scale-invariant parameter, which takes a surprising form. This is an interesting case, since Random Matrix Theory is unexplored for this setting, where the chiral condensate Σ vanishes in the chiral limit. We also measure Σ and the "pion" mass, in distinct topological sectors. In this context we discuss and probe the topological summation of observables by various methods, as well as the evaluation of the topological susceptibility. The feasibility of this summation is essential for the prospects of dynamical overlap fermions in QCD.

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arXiv2012

Topological Summation in Lattice Gauge Theory

In gauge theories the field configurations often occur in distinct topological sectors. In a lattice regularised system with chiral fermions, these sectors can be defined by referring to the Atiyah-Singer Index Theorem. However, if such a model is simulated with local updates of the lattice gauge configuration, the Monte Carlo history tends to get stuck in one sector for many steps, in particular on fine lattices. Then expectation values can be measured only within specific sectors. Here we present a pilot study in the 2-flavour Schwinger model which explores methods of approximating the complete result for an observable - corresponding to a suitable sum over all sectors - based on numerical measurements in a few specific topological sectors. We also probe various procedures for an indirect evaluation of the topological susceptibility, starting from such topologically restricted measurements.

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arXiv2009

Spectral study of a chiral limit without chiral condensate

Random Matrix Theory (RMT) has elaborated successful predictions for Dirac spectra in field theoretical models. However, a generic assumption by RMT has been a non-vanishing chiral condensate $Σ$ in the chiral limit. Here we consider the 2-flavour Schwinger model, where this assumption does not hold. We simulated this model with dynamical overlap hypercube fermions, and entered terra incognita by analysing this Dirac spectrum. The usual RMT prediction for the unfolded level spacing distribution in a unitary ensemble is precisely confirmed. The microscopic spectrum does not perform a Banks-Casher plateau. Instead the obvious expectation is a density of the lowest eigenvalue $λ_{1}$ which increases $\propto λ_{1}^{1/3}$. That would correspond to a scale-invariant parameter $\propto λV^{3/4}$, which is, however, incompatible with our data. Instead we observe to high precision a scale-invariant parameter $z \propto λV^{5/8}$. This surprising result implies a microscopic spectral density $\propto λ_1^{3/5}$, which still remains to be understood in the light of RMT.

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arXiv2008

Topological Summation of Observables Measured with Dynamical Overlap Fermions

HMC histories for light dynamical overlap fermions tend to stay in a fixed topological sector for many trajectories, so that the different sectors are not sampled properly. Therefore the suitable summation of observables, which have been measured in separate sectors, is a major challenge. We explore several techniques for this issue, based on data for the chiral condensate and the (analogue of the) pion mass in the 2-flavour Schwinger model with dynamical overlap-hypercube fermions.

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arXiv2007

Interactive Visualization Package for 4D Lattice Field Theories

Recent interest in exploring local vacuum structure of QCD through the properties of the eigenmodes of the lattice Dirac operators rises again the challenge to visualize four-dimensional objects and structures which appear in lattice field theories. In spite of complex and powerful commercial visualization software packages on the market, there are reasons to develop Interactive Visualization Package (IVP). We believe that an apprehension of the complex structures is possible only through the interactive approach, with the user being able to manipulate data representations and slices through the lattice in real-time. Further insight should also be gained by an interactive parallel examination of different physical quantities, e.g. eigenmode density with topological charge or action densities. Finally, thanks to constantly falling hardware prices, IVP makes it possible to use almost any Linux PC as a visualization tool for research in lattice field theory.

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arXiv2000

Approximate Ginsparg-Wilson fermions: A first test

We construct a 4-d lattice Dirac operator D using a systematical expansion in terms of simple operators on the lattice. The Ginsparg-Wilson equation turns into a system of coupled equations for the expansion coefficients of D. We solve these equations for a finite parametrization of D and find an approximate solution of the Ginsparg-Wilson equation. We analyze the spectral properties of our D for various ensembles of quenched SU(3) configurations. Improving the gauge field action considerably improves the spectral properties of our D.

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arXiv2000

New approximate solutions of the Ginsparg-Wilson equation - tests in 2-d

A new method for finding approximate solutions of the Ginsparg-Wilson equation is tested in 2-d. The Dirac operator is first constructed and then used in a dynamical simulation of the 2-flavor Schwinger model. We find a very small mass of the pi-particle implying almost chirally symmetric fermions. The generalization of our method to 4-d is straightforward.

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arXiv1998

Clover improvement, spectrum and Atiyah-Singer index theorem for the Dirac operator on the lattice

We study the role of the O(a)-improving clover term for the spectrum of the lattice Dirac operator using cooled and thermalized SU(2) gauge field configurations. For cooled configurations we observe improvement of the spectral properties when adding the clover term. For the thermalized case (12^4, beta = 2.4) without clover term we find a rather bad separation of physical and doubler branches making a probabilistic interpretation of the Atiyah-Singer index theorem on the lattice questionable for this beta and lattice size. Adding the clover term leads to the creation of additional real eigenvalues which come in pairs of opposite chirality thus further worsening the situation for the index theorem.

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arXiv1998

Analyzing the spectrum of general, non-hermitian Dirac operators

We discuss the computational problems when analyzing general, non-hermitian matrices and in particular the un-modified Wilson lattice Dirac operator. We report on our experiences with the Implicitly Restarted Arnoldi Method. The eigenstates of the Wilson-Dirac operator which have real eigenvalues and correspond to zero modes in the continuum are analyzed by correlating the size of the eigenvalues with the chirality of the eigenstates.

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arXiv1998

On the spectrum of the Wilson-Dirac lattice operator in topologically non-trivial background configurations

We study characteristic features of the eigenvalues of the Wilson-Dirac operator in topologically non-trivial gauge field configurations by examining complete spectra of the fermion matrix. In particular we discuss the role of eigenvectors with real eigenvalues as the lattice equivalents of the continuum zero-modes. We demonstrate, that those properties of the spectrum which correspond to non-trivial topology are stable under adding fluctuations to the gauge fields. The behavior of the spectrum in a fully quantized theory is discussed using QED_2 as an example.

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