arXiv subjects

Jacques Bloch

Publications and source records attributed to Jacques Bloch.

36 records · Page 2Linked to original sources

arXiv2015

Complex Langevin in low-dimensional QCD: the good and the not-so-good

We present our latest results on the application of the complex Langevin method to one- and two-dimensional QCD. Although the method is stable, it unfortunately converges to an incorrect result when applied as such. After applying additional gauge cooling steps, the results agree with the known analytical results in the one-dimensional case. However, in the two-dimensional case the disagreement subsists, even with gauge cooling, when the sign problem is sufficiently large.

hep-lat↗

arXiv2015

QPACE 2 and Domain Decomposition on the Intel Xeon Phi

We give an overview of QPACE 2, which is a custom-designed supercomputer based on Intel Xeon Phi processors, developed in a collaboration of Regensburg University and Eurotech. We give some general recommendations for how to write high-performance code for the Xeon Phi and then discuss our implementation of a domain-decomposition-based solver and present a number of benchmarks.

cs.DC↗

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↗

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↗

arXiv2012

Subsets and the canonical partition functions

We explain the physical nature of the subset solution to the sign problem in chiral random matrix theory: The subset sum is shown to project out the canonical determinant with zero quark charge from a given configuration. As the grand canonical chiral random matrix partition function is independent of the chemical potential, the zero quark charge sector provides the full result.

hep-lat↗

arXiv2012

A subset solution to the sign problem in random matrix simulations

We present a solution to the sign problem in dynamical random matrix simulations of a two-matrix model at nonzero chemical potential. The sign problem, caused by the complex fermion determinants, is solved by gathering the matrices into subsets, whose sums of determinants are real and positive even though their cardinality only grows linearly with the matrix size. A detailed proof of this positivity theorem is given for an arbitrary number of fermion flavors. We performed importance sampling Monte Carlo simulations to compute the chiral condensate and the quark number density for varying chemical potential and volume. The statistical errors on the results only show a mild dependence on the matrix size and chemical potential, which confirms the absence of sign problem in the subset method. This strongly contrasts with the exponential growth of the statistical error in standard reweighting methods, which was also analyzed quantitatively using the subset method. Finally, we show how the method elegantly resolves the Silver Blaze puzzle in the microscopic limit of the matrix model, where it is equivalent to QCD.

hep-lat↗

arXiv2012

Level spacings for weakly asymmetric real random matrices and application to two-color QCD with chemical potential

We consider antisymmetric perturbations of real symmetric matrices in the context of random matrix theory and two-color quantum chromodynamics. We investigate the level spacing distributions of eigenvalues that remain real or become complex conjugate pairs under the perturbation. We work out analytical surmises from small matrices and show that they describe the level spacings of large random matrices. As expected from symmetry arguments, these level spacings also apply to the overlap Dirac operator for two-color QCD with chemical potential.

hep-lat↗

arXiv2011

Evading the sign problem in random matrix simulations

We show how the sign problem occurring in dynamical simulations of random matrices at nonzero chemical potential can be avoided by judiciously combining matrices into subsets. For each subset the sum of fermionic determinants is real and positive such that importance sampling can be used in Monte Carlo simulations. The number of matrices per subset is proportional to the matrix dimension. We measure the chiral condensate and observe that the statistical error is independent of the chemical potential and grows linearly with the matrix dimension, which contrasts strongly with its exponential growth in reweighting methods.

hep-lat↗

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↗

arXiv2010

Random matrix analysis of the QCD sign problem

The severity of the sign problem in lattice QCD at nonzero baryon density is measured by the average phase of the fermion determinant. Motivated by the equivalence of chiral random matrix theory and QCD to leading order in the epsilon regime, we compute the phase of the fermion determinant for general topology in random matrix theory as a function of the quark chemical potential and the quark mass. We find that the sign problem becomes milder with increasing topological charge. The analytic predictions are verified by detailed numerical random matrix simulations.

hep-lat↗

arXiv2009

Random matrix analysis of the QCD sign problem for general topology

Motivated by the important role played by the phase of the fermion determinant in the investigation of the sign problem in lattice QCD at nonzero baryon density, we derive an analytical formula for the average phase factor of the fermion determinant for general topology in the microscopic limit of chiral random matrix theory at nonzero chemical potential, for both the quenched and the unquenched case. The formula is a nontrivial extension of the expression for zero topology derived earlier by Splittorff and Verbaarschot. Our analytical predictions are verified by detailed numerical random matrix simulations of the quenched theory.

hep-lat↗

arXiv2008

Comparing iterative methods to compute the overlap Dirac operator at nonzero chemical potential

The overlap Dirac operator at nonzero quark chemical potential involves the computation of the sign function of a non-Hermitian matrix. In this talk we present iterative Krylov subspace approximations, with deflation of critical eigenvalues, which we developed to compute the operator on large lattices. We compare the accuracy and efficiency of two alternative approximations based on the Arnoldi and on the two-sided Lanczos method. The short recurrences used in the latter method make it faster and more effective for realistic lattice simulations.

hep-lat↗

arXiv2007

Distributions of individual Dirac eigenvalues for QCD at non-zero chemical potential: RMT predictions and lattice results

For QCD at non-zero chemical potential $μ$, the Dirac eigenvalues are scattered in the complex plane. We define a notion of ordering for individual eigenvalues in this case and derive the distributions of individual eigenvalues from random matrix theory (RMT). We distinguish two cases depending on the parameter $α=μ^2 F^2 V$, where $V$ is the volume and $F$ is the familiar low-energy constant of chiral perturbation theory. For small $α$, we use a Fredholm determinant expansion and observe that already the first few terms give an excellent approximation. For large $α$, all spectral correlations are rotationally invariant, and exact results can be derived. We compare the RMT predictions to lattice data and in both cases find excellent agreement in the topological sectors $ν=0,1,2$.

hep-lat↗

arXiv2007

An iterative method to compute the overlap Dirac operator at nonzero chemical potential

The overlap Dirac operator at nonzero quark chemical potential involves the computation of the sign function of a non-Hermitian matrix. In this talk we present an iterative method, first proposed by us in Ref. [1], which allows for an efficient computation of the operator, even on large lattices. The starting point is a Krylov subspace approximation, based on the Arnoldi algorithm, for the evaluation of a generic matrix function. The efficiency of this method is spoiled when the matrix has eigenvalues close to a function discontinuity. To cure this, a small number of critical eigenvectors are added to the Krylov subspace, and two different deflation schemes are proposed in this augmented subspace. The ensuing method is then applied to the sign function of the overlap Dirac operator, for two different lattice sizes. The sign function has a discontinuity along the imaginary axis, and the numerical results show how deflation dramatically improves the efficiency of the method.

hep-lat↗

arXiv2007

Domain-wall and overlap fermions at nonzero quark chemical potential

We have recently given a construction of the overlap Dirac operator at nonzero quark chemical potential. Here, we introduce a quark chemical potential in the domain-wall fermion formalism and show that our earlier result is reproduced if the extent of the fifth dimension is taken to infinity and its lattice spacing is taken to zero. We also extend this result to include a bare quark mass, consider its continuum limit, and prove a number of properties of the overlap operator at nonzero quark chemical potential. In particular, we show that the relation between the anomaly and the index of the overlap operator remains valid.

hep-lat↗

arXiv2006

Nonzero chemical potential in the overlap Dirac operator and comparison to random matrix theory

In this talk we present the results published recently in Ref. [1], where we showed how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer gamma_5-Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions of non-Hermitian chiral random matrix theory for both trivial and nontrivial topology. We also explain an observed change in the number of zero modes as a function of chemical potential.

hep-lat↗

arXiv2006

Overlap Dirac operator at nonzero chemical potential and random matrix theory

We show how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer gamma_5-hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions of nonhermitian chiral random matrix theory for both trivial and nontrivial topology.

hep-lat↗