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Maciej A. Nowak

Publications and source records attributed to Maciej A. Nowak.

86 records · Page 5Linked to original sources
arXiv1998

Chiral Disorder and Diffusion of Light Quarks in the QCD Vacuum

We give a pedagogical introduction to the concept that light quarks diffuse in the QCD vacuum following the spontaneous breaking of chiral symmetry. By analogy with disordered electrons in metals, we show that the diffusion constant for light quarks in QCD is $D=2F_π^2/|\la\bar{q}q\to|$ which is about 0.22 fm. We comment on the correspondence between the diffusive phase and the chiral phase as described by chiral perturbation theory, as well as the cross-over to the ergodic phase as described by random matrix theory. The cross-over is identified with the Thouless energy $E_c=D/\sqrt{V_4}$ which is the inverse diffusion time in an Euclidean four-volume $V_4$.

hep-ph↗
arXiv1998

Chiral Disorder and QCD at Finite Chemical Potential

We investigate the effects of a finite chemical potential $μ$ in QCD viewed as a disordered medium. In the quenched approximation, $A_4=iμ$ induces a complex electric Aharonov-Bohm effect that causes the diagonal contribution to the quark return probability to vanish at $μ=m_π/2$ (half the pion mass). In two-color QCD, the weak-localization contribution to the quark return probability remains unaffected causing a mutation in the spectral statistics. In full QCD, the complex electric flux is screened and the light quarks are shown to diffuse asymmetrically with a substantial decrease in the conductivity along the `spatial' directions. Mean-field arguments suggest that a d=1 percolation transition may take place in the range $1.5ρ_0<ρ<3ρ_0$, where $ρ_0$ is nuclear matter density.

hep-ph↗
arXiv1998

Random Matrices and Chiral Symmetry in QCD

In this talk we review some recent results from random matrix models as applied to some non-perturbative issues in QCD. All of the issues we will discuss touched upon the important phenomenon related to the spontaneous breaking of chiral symmetry. The afore mentioned insights are: 1. Spontaneous breakdown of chiral symmetry and disorder. 2. Universal microscopic properties of the eigenvalues of the Dirac operator in the vacuum. 3. Universal microscopic properties of the eigenvalues of the Dirac operator in matter. 4. Structural changes of the Dirac spectrum - finite temperature. 5. Structural changes of the Dirac spectrum - finite baryonic density - ``phony vacua'' 6. Structural changes of the Dirac spectrum - finite baryonic density - ``true vacua'' . 7. Phase diagram. 8. Critical parameters. 9. Critical exponents. 10. $U(1)_A$ problem. 11. Screening of the pseudoscalar susceptibility. 12. Strong CP violation (finite $θ$).

hep-ph↗
arXiv1997

Various Shades of Blue's Functions

We discuss random matrix models in terms of elementary operations on Blue's functions (functional inverse of Green's functions). We show that such operations embody the essence of a number of physical phenomena whether at/or away from the critical points. We illustrate these assertions by borrowing on a number of recent results in effective QCD in vacuum and matter. We provide simple physical arguments in favor of the universality of the continuum QCD spectral oscillations, whether at zero virtuality, in the bulk of the spectrum or at the chiral critical points. We also discuss effective quantum systems of disorder with strong or weak dissipation (Hatano-Nelson localization).

hep-th↗
arXiv1997

New Developments In Non-Hermitian Random Matrix Models

In this talk we go over several new developments regarding the techniques for a large class of non-hermitian matrix models with unitary randomness (complex random numbers). In particular, we discuss: (a) - A diagrammatic approach based on a $1/N$ expansion (b) - A generalization of the addition theorem (R-transformation) (c) - A conformal transformation on the position of pertinent singularities (d) - A `phase' analysis using appropriate partition functions (e) - A number of two-point functions and the issue of universality.

hep-ph↗
arXiv1997

Nonhermitean Random Matrix Models

We introduce an extension of the diagrammatic rules in random matrix theory and apply it to nonhermitean random matrix models using the 1/N approximation. A number of one- and two-point functions are evaluated on their holomorphic and nonholomorphic supports to leading order in 1/N. The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic correlations. Generic form for the two-point functions are obtained, generalizing the concept of macroscopic universality to nonhermitean random matrices. We show that the holomorphic and nonholomorphic one- and two-point functions condition the behavior of pertinent partition functions to order $O(1/N)$. We derive explicit conditions for the location and distribution of their singularities. Most of our analytical results are found to be in good agreement with numerical calculations using large ensembles of complex matrices.

cond-mat↗
arXiv1997

Localization Transitions from Free Random Variables

We motivate and use the concept of free random variables for the studies of the de-pinning transition of flux lines in superconductors as recently discussed by Hatano and Nelson. We derive analytical conditions for the critical points of the eigenvalue distribution in the complex plane for all values of the transverse magnetic field in one-dimension, in agreement with numerical calculations. We suggest a relation between reduced nonhermitean quantum mechanics in D-dimensions and nonhermitean random matrix models with weak nonhermiticity.

cond-mat↗
arXiv1997

Macroscopic Universality : Why QCD in Matter is Subtle?

We use a chiral random matrix model with 2N_f flavors to mock up the QCD Dirac spectrum at finite chemical potential. We show that the 1/N approximation breaks down in the quenched state with spontaneously broken chiral symmetry. The breakdown condition is set by the divergence of a two-point function, that is shown to follow the general lore of macroscopic universality. In this state, the fermionic fluctuations are not suppressed in the large N limit.

hep-ph↗
arXiv1996

The U(1) Problem in Chiral Random Matrix Models

We show that conventional asymmetric chiral random matrix models (ChRMM), with a gaussian distribution in the asymmetry, provide for a screening of the topological charge and a resolution of the $U(1)$ problem in the unquenched approximation. Our exact results to order $1/N$ are in agreement with numerical estimates using large ensembles of asymmetric ChRMM with gaussian distributions.

hep-lat↗
arXiv1996

Nonhermitean Random Matrix Models : a Free Random Variable Approach

Using the standard concepts of free random variables, we show that for a large class of nonhermitean random matrix models, the support of the eigenvalue distribution follows from their hermitean analogs using a conformal transformation. We also extend the concepts of free random variables to the class of nonhermitean matrices, and apply them to the models discussed by Ginibre-Girko (elliptic ensemble) and Mahaux-Weidenmüller (chaotic resonance scattering).

hep-ph↗
arXiv1996

Dirac spectrum in QCD and Quark Masses

We use a chiral random matrix model to investigate the effects of massive quarks on the distribution of eigenvalues of QCD inspired Dirac operators. Kalkreuter's lattice analysis of the spectrum of the massive (hermitean) Dirac operator for two colors and Wilson fermions is shown to follow from a cubic equation in the quenched approximation. The quenched spectrum shows a Mott-transition from a (delocalized) Goldstone phase softly broken by the current mass, to a (localized) heavy quark phase, with quarks localized over their Compton wavelength. Both phases are distinguishable by the quark density of states at zero virtuality, with a critical quark mass of the order of 100-200 MeV. At the critical point, the quark density of states is given by $ν_Q (λ) \sim |λ|^{1/3}$. Using Grassmannian techniques, we derive an integral representation for the resolvent of the massive Dirac operator with one-flavor in the unquenched approximation, and show that near zero virtuality, the distribution of eigenvalues is quantitatively changed by a non-zero quark mass. The generalization of our construction to arbitrary flavors is also discussed. Some recommendations for lattice simulations are suggested.

hep-ph↗
arXiv1996

QCD-Inspired Spectra from Blue's Functions

We use the law of addition in random matrix theory to analyze the spectral distributions of a variety of chiral random matrix models as inspired from QCD whether through symmetries or models. In terms of the Blue's functions recently discussed by Zee, we show that most of the spectral distributions in the macroscopic limit and the quenched approximation, follow algebraically from the discontinuity of a pertinent solution to a cubic (Cardano) or a quartic (Ferrari) equation. We use the end-point equation of the energy spectra in chiral random matrix models to argue for novel phase structures, in which the Dirac density of states plays the role of an order parameter.

hep-ph↗
arXiv1996

Lattice QCD Spectra at Finite Temperature: a Random Matrix Approach

We suggest that the lattice Dirac spectra in QCD at finite temperature may be understood using a gaussian unitary ensemble for Wilson fermions, and a chiral gaussian unitary ensemble for Kogut-Susskind fermions. For Kogut-Susskind fermions, the lattice results by the Columbia group are in good agreement with the spectral distribution following from a cubic equation, both for the valence quark distribution and the anomalous symmetry breaking. We explicitly construct a number of Dirac spectra for Wilson fermions at finite temperature, and use the end-point singularities to derive analytically the pertinent critical lines. For the physical current masses, the matrix model shows a transition from a delocalized phase at low temperature, to a localized phase at high temperature. The localization is over the thermal wavelength of the quark modes. For heavier masses, the spectral distribution reflects on localized states with competitive effects between the quark Compton wavelength and the thermal wavelength. Some further suggestions for lattice simulations are made.

hep-ph↗