arXiv subjects

Tilo Wettig

Publications and source records attributed to Tilo Wettig.

At least 55 records · Page 3Linked to original sources

arXiv2014

Lattice QCD with Domain Decomposition on Intel Xeon Phi Co-Processors

The gap between the cost of moving data and the cost of computing continues to grow, making it ever harder to design iterative solvers on extreme-scale architectures. This problem can be alleviated by alternative algorithms that reduce the amount of data movement. We investigate this in the context of Lattice Quantum Chromodynamics and implement such an alternative solver algorithm, based on domain decomposition, on Intel Xeon Phi co-processor (KNC) clusters. We demonstrate close-to-linear on-chip scaling to all 60 cores of the KNC. With a mix of single- and half-precision the domain-decomposition method sustains 400-500 Gflop/s per chip. Compared to an optimized KNC implementation of a standard solver [1], our full multi-node domain-decomposition solver strong-scales to more nodes and reduces the time-to-solution by a factor of 5.

hep-lat↗

arXiv2014

Induced QCD with two auxiliary bosonic fields

Following a proposal of Budczies and Zirnbauer, we investigate an alternative lattice discretization of continuum ${\rm SU}(N_c)$ Yang-Mills theory in which the self-interactions of the gauge field are induced by a path integral over $N_b\ge N_c-1$ auxiliary bosonic fields which are coupled linearly to the gauge field. In two dimensions there exists an analytic proof that the new discretization reproduces Yang-Mills theory in its non-perturbative continuum limit. We provide numerical evidence that this is also the case in three and four dimensions and that, after a suitable matching of the free parameters, the results of the induced theory agree with results from the ordinary plaquette action up to lattice artifacts. The new discretization is ideally suited to change the order of integration in the QCD path integral to arrive at formulations in which the gauge fields have been integrated out. The resulting theories might be amenable to methods previously used in the infinite-coupling limit, and we briefly discuss possibilities to arrive at dual representations of lattice QCD.

hep-lat↗

arXiv2014

An Itzykson-Zuber-like Integral and Diffusion for Complex Ordinary and Supermatrices

We compute an analogue of the Itzykson-Zuber integral for the case of arbitrary complex matrices. The calculation is done for both ordinary and supermatrices by transferring the Itzykson-Zuber diffusion equation method to the space of arbitrary complex matrices. The integral is of interest for applications in Quantum Chromodynamics and the theory of two-dimensional Quantum Gravity.

hep-th↗

arXiv2014

Stressed Cooper pairing in QCD at high isospin density: effective Lagrangian and random matrix theory

We generalize QCD at asymptotically large isospin chemical potential to an arbitrary even number of flavors. We also allow for small quark chemical potentials, which stress the coincident Fermi surfaces of the paired quarks and lead to a sign problem in Monte Carlo simulations. We derive the corresponding low-energy effective theory in both $p$- and $ε$-expansion and quantify the severity of the sign problem. We construct the random matrix theory describing our physical situation and show that it can be mapped to a known random matrix theory at low baryon density so that new insights can be gained without additional calculations. In particular, we explain the Silver Blaze phenomenon at high isospin density. We also introduce stressed singular values of the Dirac operator and relate them to the pionic condensate. Finally we comment on extensions of our work to two-color QCD.

hep-ph↗

arXiv2014

Sign problem and subsets in one-dimensional QCD

We present a subset method that solves the sign problem for QCD at nonzero quark chemical potential in 0+1 dimensions. The subsets of gauge configurations are constructed using the center symmetry of the SU(3) group. These subsets completely solve the sign problem for up to five flavors. For a larger number of flavors the sign problem slowly reappears, and we propose an extension of the subsets that also solves the sign problem for these cases. The subset method allows for numerical simulations of the model at nonzero chemical potential. We also present some preliminary results on subsets for QCD in two, three, and four dimensions.

hep-lat↗

arXiv2014

Banks-Casher-type relations for complex Dirac spectra

For theories with a sign problem there is no analog of the Banks-Casher relation. This is true in particular for QCD at nonzero quark chemical potential. However, for QCD-like theories without a sign problem the Banks-Casher relation can be extended to the case of complex Dirac eigenvalues. We derive such extensions for the zero-temperature, high-density limits of two-color QCD, QCD at nonzero isospin chemical potential, and adjoint QCD. In all three cases the density of the complex Dirac eigenvalues at the origin is proportional to the BCS gap squared.

hep-lat↗

arXiv2013

Subset method for one-dimensional QCD

We present a subset method which solves the sign problem for QCD at nonzero quark chemical potential in 0+1 dimensions. The subsets gather gauge configurations based on the center symmetry of the SU(3) group. We show that the sign problem is solved for one to five quark flavors and that it slowly reappears for a larger number of flavors. We formulate an extension of the center subsets that solves the sign problem for a larger number of flavors as well. We also derive some new analytical results for this toy model.

hep-lat↗

arXiv2013

iDataCool: HPC with Hot-Water Cooling and Energy Reuse

iDataCool is an HPC architecture jointly developed by the University of Regensburg and the IBM Research and Development Lab Böblingen. It is based on IBM's iDataPlex platform, whose air-cooling solution was replaced by a custom water-cooling solution that allows for cooling water temperatures of 70C/158F. The system is coupled to an adsorption chiller by InvenSor that operates efficiently at these temperatures. Thus a significant portion of the energy spent on HPC can be recovered in the form of chilled water, which can then be used to cool other parts of the computing center. We describe the architecture of iDataCool and present benchmarks of the cooling performance and the energy (reuse) efficiency.

cs.DC↗

arXiv2013

Banks-Casher-type relation for the BCS gap at high density

We derive a new Banks-Casher-type relation which relates the density of complex Dirac eigenvalues at the origin to the BCS gap of quarks at high density. Our relation is applicable to QCD and QCD-like theories without a sign problem, such as two-color QCD and adjoint QCD with baryon chemical potential, and QCD with isospin chemical potential. It provides us with a method to measure the BCS gap through the Dirac spectrum on the lattice.

hep-lat↗

arXiv2012

Singular values of the Dirac operator at nonzero density

At nonzero density the eigenvalues of the Dirac operator move into the complex plane, while its singular values remain real and nonnegative. In QCD-like theories, the singular-value spectrum carries information on the diquark (or pionic) condensate. We have constructed low-energy effective theories in different density regimes and derived a number of exact results for the Dirac singular values, including Banks-Casher-type relations for the diquark (or pionic) condensate, Smilga-Stern-type relations for the slope of the singular-value density, and Leutwyler-Smilga-type sum rules for the inverse singular values. We also present a rigorous index theorem for non-Hermitian Dirac operators.

hep-lat↗

arXiv2012

Wigner surmise for mixed symmetry classes in random matrix theory

We consider the nearest-neighbor spacing distributions of mixed random matrix ensembles interpolating between different symmetry classes, or between integrable and non-integrable systems. We derive analytical formulas for the spacing distributions of 2x2 or 4x4 matrices and show numerically that they provide very good approximations for those of random matrices with large dimension. This generalizes the Wigner surmise, which is valid for pure ensembles that are recovered as limits of the mixed ensembles. We show how the coupling parameters of small and large matrices must be matched depending on the local eigenvalue density.

math-ph↗

arXiv2011

Singular values of the Dirac operator in dense QCD-like theories

We study the singular values of the Dirac operator in dense QCD-like theories at zero temperature. The Dirac singular values are real and nonnegative at any nonzero quark density. The scale of their spectrum is set by the diquark condensate, in contrast to the complex Dirac eigenvalues whose scale is set by the chiral condensate at low density and by the BCS gap at high density. We identify three different low-energy effective theories with diquark sources applicable at low, intermediate, and high density, together with their overlapping domains of validity. We derive a number of exact formulas for the Dirac singular values, including Banks-Casher-type relations for the diquark condensate, Smilga-Stern-type relations for the slope of the singular value density, and Leutwyler-Smilga-type sum rules for the inverse singular values. We construct random matrix theories and determine the form of the microscopic spectral correlation functions of the singular values for all nonzero quark densities. We also derive a rigorous index theorem for non-Hermitian Dirac operators. Our results can in principle be tested in lattice simulations.

hep-ph↗

arXiv2011

Geometry dependence of RMT-based methods to extract the low-energy constants Sigma and F

The lowest-order low-energy constants $Σ$ and $F$ of chiral pertubation theory can be extracted from lattice data using methods based on the equivalence of random matrix theory (RMT) and QCD in the epsilon regime. We discuss how the choice of the lattice geometry affects such methods. In particular, we show how to minimize systematic deviations from RMT by an optimal choice of the lattice geometry in the case of two light quark flavors. We illustrate our findings by determining $Σ$ and $F$ from lattice configurations with two dynamical overlap fermions generated by JLQCD, using two different lattice geometries.

hep-lat↗

arXiv2011

The QCD sign problem and dynamical simulations of random matrices

At nonzero quark chemical potential dynamical lattice simulations of QCD are hindered by the sign problem caused by the complex fermion determinant. The severity of the sign problem can be assessed by the average phase of the fermion determinant. In an earlier paper we derived a formula for the microscopic limit of the average phase for general topology using chiral random matrix theory. In the current paper we present an alternative derivation of the same quantity, leading to a simpler expression which is also calculable for finite-sized matrices, away from the microscopic limit. We explicitly prove the equivalence of the old and new results in the microscopic limit. The results for finite-sized matrices illustrate the convergence towards the microscopic limit. We compare the analytical results with dynamical random matrix simulations, where various reweighting methods are used to circumvent the sign problem. We discuss the pros and cons of these reweighting methods.

hep-lat↗

arXiv2011

Exact results for two-color QCD at low and high density

We discuss a random matrix theory that was originally constructed to describe two-color QCD at low density in the phase with a nonzero chiral condensate. With a particular choice of a parameter, the same random matrix theory also describes the high-density phase of two-color QCD. In this phase a BCS superfluid of diquark pairs is formed, and the pattern of chiral symmetry breaking is very different from that at low density. Analytical results for the spectral density obtained from this random matrix theory allow for the extraction of the BCS gap from lattice data.

hep-lat↗

arXiv2010

Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential

The overlap operator in lattice QCD requires the computation of the sign function of a matrix, which is non-Hermitian in the presence of a quark chemical potential. In previous work we introduced an Arnoldi-based Krylov subspace approximation, which uses long recurrences. Even after the deflation of critical eigenvalues, the low efficiency of the method restricts its application to small lattices. Here we propose new short-recurrence methods which strongly enhance the efficiency of the computational method. Using rational approximations to the sign function we introduce two variants, based on the restarted Arnoldi process and on the two-sided Lanczos method, respectively, which become very efficient when combined with multishift solvers. Alternatively, in the variant based on the two-sided Lanczos method the sign function can be evaluated directly. We present numerical results which compare the efficiencies of a restarted Arnoldi-based method and the direct two-sided Lanczos approximation for various lattice sizes. We also show that our new methods gain substantially when combined with deflation.

hep-lat↗

arXiv2010

The epsilon expansion at next-to-next-to-leading order with small imaginary chemical potential

We discuss chiral perturbation theory for two and three quark flavors in the epsilon expansion at next-to-next-to-leading order (NNLO) including a small imaginary chemical potential. We calculate finite-volume corrections to the low-energy constants $Σ$ and $F$ and determine the non-universal modifications of the theory, i.e., modifications that cannot be mapped to random matrix theory (RMT). In the special case of two quark flavors in an asymmetric box we discuss how to minimize the finite-volume corrections and non-universal modifications by an optimal choice of the lattice geometry. Furthermore we provide a detailed calculation of a special version of the massless sunset diagram at finite volume.

hep-lat↗

arXiv2010

Chiral random matrix theory for two-color QCD at high density

We identify a non-Hermitian chiral random matrix theory that corresponds to two-color QCD at high density. We show that the partition function of the random matrix theory coincides with the partition function of the finite-volume effective theory at high density, and that the Leutwyler-Smilga-type spectral sum rules of the random matrix theory are identical to those derived from the effective theory. The microscopic Dirac spectrum of the theory is governed by the BCS gap, rather than the conventional chiral condensate. We also show that with a different choice of a parameter the random matrix theory yields the effective partition function at low density.

hep-ph↗