360 research outputs found
A Generalisation of Dyson's Integration Theorem for Determinants
Dyson's integration theorem is widely used in the computation of eigenvalue correlation functions in Random Matrix Theory. Here we focus on the variant of the theorem for determinants, relevant for the unitary ensembles with Dyson index beta = 2. We derive a formula reducing the (n-k)-fold integral of an n x n determinant of a kernel of two sets of arbitrary functions to a determinant of size k x k. Our generalisation allows for sets of functions that are not orthogonal or bi-orthogonal with respect to the integration measure. In the special case of orthogonal functions Dyson's theorem is recovered
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Individual complex Dirac eigenvalue distributions from random matrix theory and lattice QCD at nonzero chemical potential
We analyze how individual eigenvalues of the QCD Dirac operator at nonzero chemical potential are distributed in the complex plane. Exact and approximate analytical results for such distributions are derived from non-Hermitian random matrix theory. When comparing these to lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class
Distributions of individual Dirac eigenvalues for QCD at non-zero chemical potential: RMT predictions and lattice results
For QCD at non-zero chemical potential , 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 , where is the volume and is the familiar low-energy constant of chiral perturbation theory. For small , we use a Fredholm determinant expansion and observe that already the first few terms give an excellent approximation. For large , 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
Wigner surmise for Hermitian and non-Hermitian Chiral random matrices
We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral Random Matrix Theory, for both real and complex eigenvalues. Testing against known results
for zero and maximal non-Hermiticity in the microscopic large-N limit we find an excellent agreement, valid for a small number of exact zero-eigenvalues. New compact expressions are derived for real eigenvalues in the orthogonal and symplectic classes, and at intermediate non-Hermiticity for the unitary and symplectic classes. Such individual Dirac eigenvalue
distributions are a useful tool in Lattice Gauge Theory and we illustrate this by showing that our new results can describe data from two-colour QCD simulations with chemical potential in the symplectic class
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.Comment: 4 pages, 4 figures, minor changes, as published in Phys. Rev. Let
Gap probabilities in non-Hermitian random matrix theory
We compute the gap probability that a circle of
radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (beta=2) or quaternion real (beta=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is for rotationally invariant weights, the product of Fredholm eigenvalues for beta=4 follows from beta=2 by skipping every second factor, in contrast to the known relation for Hermitian ensembles. On additionally choosing Gaussian weights we give new explicit expressions for the Fredholm eigenvalues in the chiral case, in terms of Bessel-K and incomplete Bessel-I functions. This compares to known results for the Ginibre ensembles in terms of incomplete exponentials. Furthermore we present an asymptotic expansion of the logarithm of the gap probability for large argument r at large N in all four ensembles, up to including the third order linear term. We can provide strict upper and lower bounds and present numerical evidence for its conjectured values, depending on the number of exact zero eigenvalues in the chiral ensembles. For the Ginibre ensemble at beta=2 exact results were previously derived by Forrester
Optical Reflection Studies of Damage in Ion Implanted Silicon
Optical (3–6.5 eV) reflection spectra are presented for crystalline Si implanted at room temperature with 40 keV Sb ions to doses of less than 2×10^15/cm^2. These spectra, and their deviation from the reflection spectrum of crystalline Si, are discussed in terms of a model based on the average dielectric properties of the implanted region. For samples having a high ion dose (>10^15/cm^2) the observed spectra resemble the spectra of sputtered Si films. Anneal characteristics of the reflection spectra are found to be dose dependent. These observations are compared to, and found to substantiate, the results of other experimental techniques for studying lattice damage in Si
Sum Rules for the Dirac Spectrum of the Schwinger Model
The inverse eigenvalues of the Dirac operator in the Schwinger model satisfy
the same Leutwyler-Smilga sum rules as in the case of QCD with one flavor. In
this paper we give a microscopic derivation of these sum rules in the sector of
arbitrary topological charge. We show that the sum rules can be obtained from
the clustering property of the scalar correlation functions. This argument also
holds for other theories with a mass gap and broken chiral symmetry such as QCD
with one flavor. For QCD with several flavors a modified clustering property is
derived from the low energy chiral Lagrangian. We also obtain sum rules for a
fixed external gauge field and show their relation with the bosonized version
of the Schwinger model. In the sector of topological charge the sum rules
are consistent with a shift of the Dirac spectrum away from zero by
average level spacings. This shift is also required to obtain a nonzero chiral
condensate in the massless limit. Finally, we discuss the Dirac spectrum for a
closely related two-dimensional theory for which the gauge field action is
quadratic in the the gauge fields. This theory of so called random Dirac
fermions has been discussed extensively in the context of the quantum Hall
effect and d-wave super-conductors.Comment: 41 pages, Late
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