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T. Wettig

Publications and source records attributed to T. Wettig.

At least 19 recordsLinked to original sources
arXiv2018

Dirac spectrum and chiral condensate for QCD at fixed $θ$-angle

We analyze the mass dependence of the chiral condensate for QCD at nonzero $θ$-angle and find that in general the discontinuity of the chiral condensate is not on the support of the Dirac spectrum. To understand this behavior we decompose the spectral density and the chiral condensate into contributions from the zero modes, the quenched part, and a remainder which is sensitive to the fermion determinant and is referred to as the dynamical part. We obtain general formulas for the contributions of the zero modes. Expressions for the quenched part, valid for an arbitrary number of flavors, and for the dynamical part, valid for one and two flavors, are derived in the microscopic domain of QCD. We find that at nonzero $θ$-angle the quenched and dynamical part of the Dirac spectral density are strongly oscillating with an amplitude that increases exponentially with the volume $V$ and a period of order of $1/V$. The quenched part of the chiral condensate becomes exponentially large at $θ\ne0$, but this divergence is canceled by the contribution from the zero modes. The oscillatory behavior of the dynamical part of the density is essential for moving the discontinuity of the chiral condensate away from the support of the Dirac spectrum. As important by-products of this work we obtain analytical expressions for the microscopic spectral density of the Dirac operator at nonzero $θ$-angle for both one- and two-flavor QCD with nonzero quark masses.

hep-th↗
arXiv2014

Dirac spectrum of one-flavor QCD at θ=0 and continuity of the chiral condensate

We derive exact analytical expressions for the spectral density of the Dirac operator at fixed θ-angle in the microscopic domain of one-flavor QCD. These results are obtained by performing the sum over topological sectors using novel identities involving sums of products of Bessel functions. Because the fermion determinant is not positive definite for negative quark mass, the usual Banks-Casher relation is not valid and has to be replaced by a different mechanism first observed for QCD at nonzero chemical potential. Using the exact results for the spectral density we explain how this mechanism results in a chiral condensate that remains constant when the quark mass changes sign.

hep-th↗
arXiv2011

Random matrix theory of unquenched two-colour QCD with nonzero chemical potential

We solve a random two-matrix model with two real asymmetric matrices whose primary purpose is to describe certain aspects of quantum chromodynamics with two colours and dynamical fermions at nonzero quark chemical potential mu. In this symmetry class the determinant of the Dirac operator is real but not necessarily positive. Despite this sign problem the unquenched matrix model remains completely solvable and provides detailed predictions for the Dirac operator spectrum in two different physical scenarios/limits: (i) the epsilon-regime of chiral perturbation theory at small mu, where mu^2 multiplied by the volume remains fixed in the infinite-volume limit and (ii) the high-density regime where a BCS gap is formed and mu is unscaled. We give explicit examples for the complex, real, and imaginary eigenvalue densities including Nf=2 non-degenerate flavours. Whilst the limit of two degenerate masses has no sign problem and can be tested with standard lattice techniques, we analyse the severity of the sign problem for non-degenerate masses as a function of the mass split and of mu. On the mathematical side our new results include an analytical formula for the spectral density of real Wishart eigenvalues in the limit (i) of weak non-Hermiticity, thus completing the previous solution of the corresponding quenched model of two real asymmetric Wishart matrices.

hep-lat↗
arXiv2011

Lattice QCD Applications on QPACE

QPACE is a novel massively parallel architecture optimized for lattice QCD simulations. A single QPACE node is based on the IBM PowerXCell 8i processor. The nodes are interconnected by a custom 3-dimensional torus network implemented on an FPGA. The compute power of the processor is provided by 8 Synergistic Processing Units. Making efficient use of these accelerator cores in scientific applications is challenging. In this paper we describe our strategies for porting applications to the QPACE architecture and report on performance numbers.

hep-lat↗
arXiv2009

QPACE -- a QCD parallel computer based on Cell processors

QPACE is a novel parallel computer which has been developed to be primarily used for lattice QCD simulations. The compute power is provided by the IBM PowerXCell 8i processor, an enhanced version of the Cell processor that is used in the Playstation 3. The QPACE nodes are interconnected by a custom, application optimized 3-dimensional torus network implemented on an FPGA. To achieve the very high packaging density of 26 TFlops per rack a new water cooling concept has been developed and successfully realized. In this paper we give an overview of the architecture and highlight some important technical details of the system. Furthermore, we provide initial performance results and report on the installation of 8 QPACE racks providing an aggregate peak performance of 200 TFlops.

hep-lat↗
arXiv2009

Topology and chiral random matrix theory at nonzero imaginary chemical potential

We study the effect of topology for a random matrix model of QCD at nonzero imaginary chemical potential or nonzero temperature. Non-universal fluctuations of Dirac eigenvalues lead to normalization factors that contribute to the $θ$-dependence of the partition function. These normalization factors have to be canceled in order to reproduce the $θ$-dependence of the QCD partition function. The reason for this behavior is that the topological domain of the Dirac spectrum (the region of the Dirac spectrum that is sensitive to the topological charge) extends beyond the microscopic domain at nonzero imaginary chemical potential or temperature. Such behavior could persist in certain lattice formulations of QCD.

hep-th↗
arXiv2008

Status of the QPACE Project

We give an overview of the QPACE project, which is pursuing the development of a massively parallel, scalable supercomputer for LQCD. The machine is a three-dimensional torus of identical processing nodes, based on the PowerXCell 8i processor. The nodes are connected by an FPGA-based, application-optimized network processor attached to the PowerXCell 8i processor. We present a performance analysis of lattice QCD codes on QPACE and corresponding hardware benchmarks.

hep-lat↗
arXiv2008

Character expansion method for supergroups and extended superversions of the Leutwyler-Smilga and Berezin-Karpelevich integrals

We introduce an extension of the character expansion method to the case of supergroups. This method allows us to calculate a superversion of the Leutwyler-Smilga integral which, to the best of our knowledge, has not been calculated before. We also use the method to generalize a previously calculated superversion of the Berezin-Karpelevich integral. Our character expansion method should also allow for the calculation of other supergroup integrals.

math-ph↗
arXiv2008

Individual complex Dirac eigenvalue distributions from random matrix theory and comparison to quenched lattice QCD with a quark chemical potential

We analyze how individual eigenvalues of the QCD Dirac operator at nonzero quark chemical potential are distributed in the complex plane. Exact and approximate analytical results for both quenched and unquenched distributions are derived from non-Hermitian random matrix theory. When comparing these to quenched lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the quark chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class.

hep-lat↗
arXiv2007

An iterative method to compute the sign function of a non-Hermitian matrix and its application to the overlap Dirac operator at nonzero chemical potential

The overlap Dirac operator in lattice QCD requires the computation of the sign function of a matrix. While this matrix is usually Hermitian, it becomes non-Hermitian in the presence of a quark chemical potential. We show how the action of the sign function of a non-Hermitian matrix on an arbitrary vector can be computed efficiently on large lattices by an iterative method. A Krylov subspace approximation based on the Arnoldi algorithm is described for the evaluation of a generic matrix function. The efficiency of the method is spoiled when the matrix has eigenvalues close to a function discontinuity. This is cured by adding a small number of critical eigenvectors to the Krylov subspace, for which we propose two different deflation schemes. The ensuing modified Arnoldi method is then applied to the sign function, which has a discontinuity along the imaginary axis. The numerical results clearly show the improved efficiency of the method. Our modification is particularly effective when the action of the sign function of the same matrix has to be computed many times on different vectors, e.g., if the overlap Dirac operator is inverted using an iterative method.

hep-lat↗
arXiv2007

QCD on the Cell Broadband Engine

We evaluate IBM's Enhanced Cell Broadband Engine (BE) as a possible building block of a new generation of lattice QCD machines. The Enhanced Cell BE will provide full support of double-precision floating-point arithmetics, including IEEE-compliant rounding. We have developed a performance model and applied it to relevant lattice QCD kernels. The performance estimates are supported by micro- and application-benchmarks that have been obtained on currently available Cell BE-based computers, such as IBM QS20 blades and PlayStation 3. The results are encouraging and show that this processor is an interesting option for lattice QCD applications. For a massively parallel machine on the basis of the Cell BE, an application-optimized network needs to be developed.

hep-lat↗
arXiv2006

QCD Dirac operator at nonzero chemical potential: lattice data and matrix model

Recently, a non-Hermitian chiral random matrix model was proposed to describe the eigenvalues of the QCD Dirac operator at nonzero chemical potential. This matrix model can be constructed from QCD by mapping it to an equivalent matrix model which has the same symmetries as QCD with chemical potential. Its microscopic spectral correlations are conjectured to be identical to those of the QCD Dirac operator. We investigate this conjecture by comparing large ensembles of Dirac eigenvalues in quenched SU(3) lattice QCD at nonzero chemical potential to the analytical predictions of the matrix model. Excellent agreement is found in the two regimes of weak and strong non-Hermiticity, for several different lattice volumes.

hep-lat↗
arXiv2005

Random Matrices

We review elementary properties of random matrices and discuss widely used mathematical methods for both hermitian and nonhermitian random matrix ensembles. Applications to a wide range of physics problems are summarized. This paper originally appeared as an article in the Wiley Encyclopedia of Electrical and Electronics Engineering.

hep-ph↗
arXiv2003

Hardware and software status of QCDOC

QCDOC is a massively parallel supercomputer whose processing nodes are based on an application-specific integrated circuit (ASIC). This ASIC was custom-designed so that crucial lattice QCD kernels achieve an overall sustained performance of 50% on machines with several 10,000 nodes. This strong scalability, together with low power consumption and a price/performance ratio of $1 per sustained MFlops, enable QCDOC to attack the most demanding lattice QCD problems. The first ASICs became available in June of 2003, and the testing performed so far has shown all systems functioning according to specification. We review the hardware and software status of QCDOC and present performance figures obtained in real hardware as well as in simulation.

hep-lat↗
arXiv2003

Comparing matrix models and QCD lattice data with chemical potential

We present a quantitative analysis of the microscopic Dirac spectrum which is complex in the presence of a non-vanishing quark chemical potential. Data from quenched SU(3) lattice simulations for different volumes V and small values of the chemical potential are compared to analytical predictions from matrix models. We confirm the existence of two distinct limits for weakly and strongly nonhermitian Dirac operators. Good agreement is found in both limits, confirming the different scaling of chemical potential and eigenvalues with the volume.

hep-lat↗
arXiv2003

The QCDOC supercomputer: hardware, software, and performance

An overview is given of the QCDOC architecture, a massively parallel and highly scalable computer optimized for lattice QCD using system-on-a-chip technology. The heart of a single node is the PowerPC-based QCDOC ASIC, developed in collaboration with IBM Research, with a peak speed of 1 GFlop/s. The nodes communicate via high-speed serial links in a 6-dimensional mesh with nearest-neighbor connections. We find that highly optimized four-dimensional QCD code obtains over 50% efficiency in cycle accurate simulations of QCDOC, even for problems of fixed computational difficulty run on tens of thousands of nodes. We also provide an overview of the QCDOC operating system, which manages and runs QCDOC applications on partitions of variable dimensionality. Finally, the SciDAC activity for QCDOC and the message-passing interface QMP specified as a part of the SciDAC effort are discussed for QCDOC. We explain how to make optimal use of QMP routines on QCDOC in conjunction with existing C and C++ lattice QCD codes, including the publicly available MILC codes.

hep-lat↗
arXiv2003

Matrix model correlation functions and lattice data for the QCD Dirac operator with chemical potential

We apply a complex chiral random matrix model as an effective model to QCD with a small chemical potential at zero temperature. In our model the correlation functions of complex eigenvalues can be determined analytically in two different limits, at weak and strong non-Hermiticity. We compare them to the distribution of the smallest Dirac operator eigenvalues from quenched QCD lattice data for small values of the chemical potential, appropriately rescaled with the volume. This confirms the existence of two different scaling regimes from lattice data.

hep-lat↗