arXiv subjects

K. Splittorff

Publications and source records attributed to K. Splittorff.

At least 55 records · Page 3Linked to original sources

arXiv2010

Degenerate distributions in complex Langevin dynamics: one-dimensional QCD at finite chemical potential

We demonstrate analytically that complex Langevin dynamics can solve the sign problem in one-dimensional QCD in the thermodynamic limit. In particular, it is shown that the contributions from the complex and highly oscillating spectral density of the Dirac operator to the chiral condensate are taken into account correctly. We find an infinite number of classical fixed points of the Langevin flow in the thermodynamic limit. The correct solution originates from a continuum of degenerate distributions in the complexified space.

hep-lat↗

arXiv2009

Lattice QCD and dense quark matter

This talk summarizes recent progress in lattice QCD for dense quark matter. The emphasis is on the insights obtained from analytical results derived within chiral perturbation theory.

hep-lat↗

arXiv2009

Fluctuations, correlations and the sign problem in QCD

We study the distribution of the phase angle and the magnitude of the fermion determinant as well as its correlation with the chiral condensate and the baryon number for QCD at non-zero quark chemical potential. Results are derived to one-loop order in Chiral Perturbation Theory (ChPT), as well as by analytical and numerical calculations in QCD in one Euclidean dimension. We find a qualitative change of the distribution of the phase of the fermion determinant when the quark mass enters the spectrum of the Dirac operator: it changes from a periodicized Gaussian distribution to a periodicized Lorentzian distribution. We also explore the possibility that some observables remain weakly correlated with the phase of the fermion determinant even though the sign problem is severe. We discuss the practical implications of our findings on lattice simulations of QCD at non-zero baryon chemical potential.

hep-lat↗

arXiv2009

The Fluctuations of the Quark Number and of the Chiral Condensate

The distributions of the quark number and chiral condensate over the gauge fields are computed for QCD in Euclidean space at nonzero quark chemical potential. As both operators are non-hermitian the distributions are in the complex plane. Moreover, because of the sign problem, the distributions are not real and positive. The computations are carried out within leading order chiral perturbation theory and give a direct insight into the delicate cancellations that take place in contributions to the total baryon number and the chiral condensate.

hep-lat↗

arXiv2009

Distributions of the Phase Angle of the Fermion Determinant in QCD

The distribution of the phase angle and the magnitude of the fermion determinant as well as its correlations with the baryon number and the chiral condensate are studied for QCD at non zero quark chemical potential. Results are derived to one-loop order in chiral perturbation theory. We find that the distribution of the phase angle is Gaussian for small chemical potential and a periodic Lorentzian when the quark mass is inside the support of the Dirac spectrum. The baryon number and chiral condensate are computed as a function of the phase of the fermion determinant and we discuss the severe cancellations which occur upon integration over the angle. We compute the distribution of the magnitude of the fermion determinant as well as the baryon number and chiral condensate at fixed magnitude. Finally, we consider QCD in one Euclidean dimension where it is shown analytically, starting from the fundamental QCD partition function, that the distribution of the phase of the fermion determinant is a periodic Lorentzian when the quark mass is inside the spectral density of the Dirac operator.

hep-lat↗

arXiv2008

Phase of the Fermion Determinant for QCD at Finite Chemical Potential

In this lecture we discuss various properties of the phase factor of the fermion determinant for QCD at nonzero chemical potential. Its effect on physical observables is elucidated by comparing the phase diagram of QCD and phase quenched QCD and by illustrating the failure of the Banks-Casher formula with the example of one-dimensional QCD. The average phase factor and the distribution of the phase are calculated to one-loop order in chiral perturbation theory. In quantitative agreement with lattice QCD results, we find that the distribution is Gaussian with a width $\sim μT \sqrt V$ (for $m_π\ll T \ll Λ_{\rm QCD}$). Finally, we introduce, so-called teflon plated observables which can be calculated accurately by Monte Carlo even though the sign problem is severe.

hep-lat↗

arXiv2008

Lessons from Random Matrix Theory for QCD at Finite Density

In this lecture we discuss various aspects of QCD at nonzero chemical potential, including its phase diagram and the Dirac spectrum, and summarize what chiral random matrix theory has contributed to this subject. To illustrate the importance of the phase of the fermion determinant, we particularly highlight the differences between QCD and phase quenched QCD.

hep-ph↗

arXiv2008

Statistical QCD with non-positive measure

In this talk we discuss the microscopic limit of QCD at nonzero chemical potential. In this domain, where the QCD partition function is under complete analytical control, we uncover an entirely new link between the spectral density of the Dirac operator and the chiral condensate: violent complex oscillations on the microscopic scale give rise to the discontinuity of the chiral condensate at zero quark mass. We first establish this relation exactly within a random matrix framework and then analyze the importance of the individual modes by Fourier analysis.

hep-lat↗

arXiv2008

Chiral Condensate at Nonzero Chemical Potential in the Microscopic Limit of QCD

The chiral condensate in QCD at zero temperature does not depend on the quark chemical potential (up to one third the nucleon mass), whereas the spectral density of the Dirac operator shows a strong dependence on the chemical potential. The cancellations which make this possible also occur on the microscopic scale, where they can be investigated by means of a random matrix model. We show that they can be understood in terms of orthogonality properties of orthogonal polynomials. In the strong non-Hermiticity limit they are related to integrability properties of the spectral density. As a by-product we find exact analytical expressions for the partially quenched chiral condensate in the microscopic domain at nonzero chemical potential.

hep-th↗

arXiv2008

Phase Diagram of the Dirac Spectrum at Nonzero Chemical Potential

The Dirac spectrum of QCD with dynamical fermions at nonzero chemical potential is characterized by three regions, a region with a constant eigenvalue density, a region where the eigenvalue density shows oscillations that grow exponentially with the volume and the remainder of the complex plane where the eigenvalue density is zero. In this paper we derive the phase diagram of the Dirac spectrum from a chiral Lagrangian. We show that the constant eigenvalue density corresponds to a pion condensed phase while the strongly oscillating region is given by a kaon condensed phase. The normal phase with nonzero chiral condensate but vanishing Bose condensates coincides with the region of the complex plane where there are no eigenvalues.

hep-lat↗

arXiv2008

Nonhermitian Supersymmetric Partition Functions: the case of one bosonic flavor

We discuss the supersymmetric formulation of the nonhermitian $β= 2$ random matrix partition function with one bosonic flavor. This partition function is regularized by adding one conjugate boson and fermion each. A supersymmetric nonlinear $σ$-model for the resulting Goldstone degrees of freedom is obtained using symmetry arguments only. For a Gaussian probability distribution the same results are derived using superbosonization and the complex orthogonal polynomial method. The symmetry arguments apply to any model with the same symmetries and a mass gap, and demonstrate the universality of the nonlinear $σ$-model.

hep-th↗

arXiv2008

A new Chiral Two-Matrix Theory for Dirac Spectra with Imaginary Chemical Potential

We solve a new chiral Random Two-Matrix Theory by means of biorthogonal polynomials for any matrix size $N$. By deriving the relevant kernels we find explicit formulas for all $(n,k)$-point spectral (mixed or unmixed) correlation functions. In the microscopic limit we find the corresponding scaling functions, and thus derive all spectral correlators in this limit as well. We extend these results to the ordinary (non-chiral) ensembles, and also there provide explicit solutions for any finite size $N$, and in the microscopic scaling limit. Our results give the general analytical expressions for the microscopic correlation functions of the Dirac operator eigenvalues in theories with imaginary baryon and isospin chemical potential, and can be used to extract the tree-level pion decay constant from lattice gauge theory configurations. We find exact agreement with previous computations based on the low-energy effective field theory in the two special cases where comparisons are possible.

hep-th↗

arXiv2007

Triage of the Sign Problem

We discuss the sign problem in QCD at nonzero chemical potential and its relation with chiral symmetry breaking and the spectrum of the Dirac operator using the framework of chiral random matrix theory. We show that the Banks-Casher formula is not valid for theories with a sign problem and has to be replaced by an alternative mechanism that is worked out in detail for QCD in one dimension at nonzero chemical potential.

hep-th↗

arXiv2007

The Approach to the Thermodynamic Limit in Lattice QCD at μ\neq0

The expectation value of the complex phase factor of the fermion determinant is computed to leading order in the $p$-expansion of the chiral Lagrangian. The computation is valid for $μ<m_π/2$ and determines the dependence of the sign problem on the volume and on the geometric shape of the volume. In the thermodynamic limit with $ L_i \to \infty $ at fixed temperature $1/L_0$, the average phase factor vanishes. In the low temperature limit where $L_i/L_0$ is fixed as $L_i$ becomes large the average phase factor approaches one. The results for a finite volume compare well with lattice results obtained by Allton {\it et al}.. After taking appropriate limits, we reproduce previously derived results for the $ε$-regime and for 1-dimensional QCD. The distribution of the phase itself is also computed.

hep-lat↗

arXiv2007

The Sign Problem via Imaginary Chemical Potential

We calculate an analogue of the average phase factor of the staggered fermion determinant at imaginary chemical potential. Our results from the lattice agree well with the analytical predictions in the microscopic regime for both quenched and phase-quenched QCD. We demonstrate that the average phase factor in the microscopic domain is dominated by the lowest-lying Dirac eigenvalues.

hep-lat↗

arXiv2007

Random Matrix Theory at Nonzero $μ$ and $T$

We review applications of random matrix theory to QCD at nonzero temperature and chemical potential. The chiral phase transition of QCD and QCD-like theories is discussed in terms of eigenvalues of the Dirac operator. We show that for QCD at $μ\ne 0$, which has a sign problem, the discontinuity in the chiral condensate is due to an alternative to the Banks-Casher relation. The severity of the sign problem is analyzed in the microscopic domain of QCD.

hep-ph↗

arXiv2007

The QCD Sign Problem for Small Chemical Potential

The expectation value of the complex phase factor of the fermion determinant is computed in the microscopic domain of QCD at nonzero chemical potential. We find that the average phase factor is non-vanishing below a critical value of the chemical potential equal to the half the pion mass and vanishes exponentially in the volume for larger values of the chemical potential. This holds for QCD with dynamical quarks as well as for quenched and phase quenched QCD. The average phase factor has an essential singularity for zero chemical potential and cannot be obtained by analytic continuation from imaginary chemical potential or by means of a Taylor expansion. The leading order correction in the $p$-expansion of the chiral Lagrangian is calculated as well.

hep-lat↗

arXiv2006

The sign problem in the $ε$-regime of QCD

QCD in the $ε$-regime at nonzero baryon chemical potential $μ$ is reviewed. The focus is on aspects of the sign problem which are relevant for lattice QCD. It is discussed how spontaneous chiral symmetry breaking and the sign problem are related through the spectrum of the Dirac operator. The strength of the sign problem is linked to the quark mass and the chemical potential. Specific implications for lattice QCD are discussed.

hep-lat↗