83 research outputs found
Random Matrix Models for Dirac Operators at finite Lattice Spacing
We study discretization effects of the Wilson and staggered Dirac operator
with using chiral random matrix theory (chRMT). We obtain
analytical results for the joint probability density of Wilson-chRMT in terms
of a determinantal expression over complex pairs of eigenvalues, and real
eigenvalues corresponding to eigenvectors of positive or negative chirality as
well as for the eigenvalue densities. The explicit dependence on the lattice
spacing can be readily read off from our results which are compared to
numerical simulations of Wilson-chRMT. For the staggered Dirac operator we have
studied random matrices modeling the transition from non-degenerate eigenvalues
at non-zero lattice spacing to degenerate ones in the continuum limit.Comment: 7 pages, 6 figures, Proceedings for the XXIX International Symposium
on Lattice Field Theory, July 10 -- 16 2011, Squaw Valley, Lake Tahoe,
California, PACS: 12.38.Gc, 05.50.+q, 02.10.Yn, 11.15.H
On the Efetov-Wegner terms by diagonalizing a Hermitian supermatrix
The diagonalization of Hermitian supermatrices is studied. Such a change of
coordinates is inevitable to find certain structures in random matrix theory.
However it still poses serious problems since up to now the calculation of all
Rothstein contributions known as Efetov-Wegner terms in physics was quite
cumbersome. We derive the supermatrix Bessel function with all Efetov-Wegner
terms for an arbitrary rotation invariant probability density function. As
applications we consider representations of generating functions for Hermitian
random matrices with and without an external field as integrals over
eigenvalues of Hermitian supermatrices. All results are obtained with all
Efetov-Wegner terms which were unknown before in such an explicit and compact
representation.Comment: 23 pages, PACS: 02.30.Cj, 02.30.Fn, 02.30.Px, 05.30.Ch, 05.30.-d,
05.45.M
Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials
The Realization of the Sharpe-Singleton Scenario
The microscopic spectral density of the Wilson Dirac operator for two flavor
lattice QCD is analyzed. The computation includes the leading order a^2
corrections of the chiral Lagrangian in the microscopic limit. The result is
used to demonstrate how the Sharpe-Singleton first order scenario is realized
in terms of the eigenvalues of the Wilson Dirac operator. We show that the
Sharpe-Singleton scenario only takes place in the theory with dynamical
fermions whereas the Aoki phase can be realized in the quenched as well as the
unquenched theory. Moreover, we give constraints imposed by gamma_5-Hermiticity
on the additional low energy constants of Wilson chiral perturbation theory.Comment: 33 pages, 4 figures, version to appear in PR
Closed-form performance analysis of linear MIMO receivers in general fading scenarios
Linear precoding and post-processing schemes are ubiquitous in wireless
multi-input-multi-output (MIMO) settings, due to their reduced complexity with
respect to optimal strategies. Despite their popularity, the performance
analysis of linear MIMO receivers is mostly not available in closed form, apart
for the canonical (uncorrelated Rayleigh fading) case, while for more general
fading conditions only bounds are provided. This lack of results is motivated
by the complex dependence of the output signal-to-interference and noise ratio
(SINR) at each branch of the receiving filter on both the squared singular
values as well as the (typically right) singular vectors of the channel matrix.
While the explicit knowledge of the statistics of the SINR can be circumvented
for some fading types in the analysis of the linear Minimum Mean-Squared Error
(MMSE) receiver, this does not apply to the less complex and widely adopted
Zero-Forcing (ZF) scheme. This work provides the first-to-date closed-form
expression of the probability density function (pdf) of the output ZF and MMSE
SINR, for a wide range of fading laws, encompassing, in particular,
correlations and multiple scattering effects typical of practically relevant
channel models.Comment: 16 pages, 2 figures, contents submitted to IEEE/VDE WSA 201
Dirac spectrum and chiral condensate for QCD at fixed angle
Kieburg M, Verbaarschot JJM, Wettig T. Dirac spectrum and chiral condensate for QCD at fixed angle. PHYSICAL REVIEW D. 2019;99(7): 074515.We analyze the mass dependence of the chiral condensate for QCD at nonzero theta 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 theta angle the quenched and dynamical parts 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 theta not equal 0, 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 theta angle for both one- and two-flavor QCD with nonzero quark masses
New term in effective field theory at fixed topology
A random matrix model for lattice QCD which takes into account the positive
definite nature of the Wilson term is introduced. The corresponding effective
theory for fixed index of the Wilson Dirac operator is derived to next to
leading order. It reveals a new term proportional to the topological index of
the Wilson Dirac operator and the lattice spacing. The new term appears
naturally in a fixed index spurion analysis. The spurion approach reveals that
the term is the first in a new family of such terms and that equivalent terms
are relevant for the effective theory of continuum QCD.Comment: 22 pages, 2 figures, version to appear in PR
Phase Diagram of Wilson and Twisted Mass Fermions at finite isospin chemical potential
Wilson Fermions with untwisted and twisted mass are widely used in lattice
simulations. Therefore one important question is whether the twist angle and
the lattice spacing affect the phase diagram. We briefly report on the study of
the phase diagram of QCD in the parameter space of the degenerate quark masses,
isospin chemical potential, lattice spacing, and twist angle by employing
chiral perturbation theory. Moreover we calculate the pion masses and their
dependence on these four parameters.Comment: 7 pages, 1 figure, conference proceeding of LATTICE201
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