434 research outputs found
Non-Commutativity of the Zero Chemical Potential Limit and the Thermodynamic Limit in Finite Density Systems
Monte Carlo simulations of finite density systems are often plagued by the
complex action problem. We point out that there exists certain
non-commutativity in the zero chemical potential limit and the thermodynamic
limit when one tries to study such systems by reweighting techniques. This is
demonstrated by explicit calculations in a Random Matrix Theory, which is
thought to be a simple qualitative model for finite density QCD. The
factorization method allows us to understand how the non-commutativity, which
appears at the intermediate steps, cancels in the end results for physical
observables.Comment: 7 pages, 9 figure
Non-Hermitian Random Matrix Theory and Lattice QCD with Chemical Potential
In quantum chromodynamics (QCD) at nonzero chemical potential, the
eigenvalues of the Dirac operator are scattered in the complex plane. Can the
fluctuation properties of the Dirac spectrum be described by universal
predictions of non-Hermitian random matrix theory? We introduce an unfolding
procedure for complex eigenvalues and apply it to data from lattice QCD at
finite chemical potential to construct the nearest-neighbor spacing
distribution of adjacent eigenvalues in the complex plane. For intermediate
values of , we find agreement with predictions of the Ginibre ensemble of
random matrix theory, both in the confinement and in the deconfinement phase.Comment: 4 pages, 3 figures, to appear in Phys. Rev. Let
The Fractal Geometry of Critical Systems
We investigate the geometry of a critical system undergoing a second order
thermal phase transition. Using a local description for the dynamics
characterizing the system at the critical point T=Tc, we reveal the formation
of clusters with fractal geometry, where the term cluster is used to describe
regions with a nonvanishing value of the order parameter. We show that,
treating the cluster as an open subsystem of the entire system, new
instanton-like configurations dominate the statistical mechanics of the
cluster. We study the dependence of the resulting fractal dimension on the
embedding dimension and the scaling properties (isothermal critical exponent)
of the system. Taking into account the finite size effects we are able to
calculate the size of the critical cluster in terms of the total size of the
system, the critical temperature and the effective coupling of the long
wavelength interaction at the critical point. We also show that the size of the
cluster has to be identified with the correlation length at criticality.
Finally, within the framework of the mean field approximation, we extend our
local considerations to obtain a global description of the system.Comment: 1 LaTeX file, 4 figures in ps-files. Accepted for publication in
Physical Review
Fermion determinants in matrix models of QCD at nonzero chemical potential
The presence of a chemical potential completely changes the analytical
structure of the QCD partition function. In particular, the eigenvalues of the
Dirac operator are distributed over a finite area in the complex plane, whereas
the zeros of the partition function in the complex mass plane remain on a
curve. In this paper we study the effects of the fermion determinant at nonzero
chemical potential on the Dirac spectrum by means of the resolvent, G(z), of
the QCD Dirac operator. The resolvent is studied both in a one-dimensional U(1)
model (Gibbs model) and in a random matrix model with the global symmetries of
the QCD partition function. In both cases we find that, if the argument z of
the resolvent is not equal to the mass m in the fermion determinant, the
resolvent diverges in the thermodynamic limit. However, for z =m the resolvent
in both models is well defined. In particular, the nature of the limit is illuminated in the Gibbs model. The phase structure of the
random matrix model in the complex m and \mu-planes is investigated both by a
saddle point approximation and via the distribution of Yang-Lee zeros. Both
methods are in complete agreement and lead to a well-defined chiral condensate
and quark number density.Comment: 27 pages, 6 figures, Late
Three-dimensional QCD in the adjoint representation and random matrix theory
In this paper we complete the derivations of finite volume partition
functions for QCD using random matrix theories by calculating the effective
low-energy partition function for three-dimensional QCD in the adjoint
representation from a random matrix theory with the same global symmetries. As
expected, this case corresponds to Dyson index , that is, the Dirac
operator can be written in terms of real quaternions. After discussing the
issue of defining Majorana fermions in Euclidean space, the actual matrix model
calculation turns out to be simple. We find that the symmetry breaking pattern
is , as expected from the correspondence
between symmetric (super)spaces and random matrix universality classes found by
Zirnbauer. We also derive the first Leutwyler--Smilga sum rule.Comment: LaTeX, 19 pages. Minor corrections, added comments, to appear on
Phys. Rev.
Hatano-Nelson model with a periodic potential
We study a generalisation of the Hatano-Nelson Hamiltonian in which a
periodic modulation of the site energies is present in addition to the usual
random distribution. The system can then become localized by disorder or
develop a band gap, and the eigenspectrum shows a wide variety of topologies.
We determine the phase diagram, and perform a finite size scaling analysis of
the localization transition.Comment: 7 pages, 10 figure
Chiral Phase Transition within Effective Models with Constituent Quarks
We investigate the chiral phase transition at nonzero temperature and
baryon-chemical potential within the framework of the linear sigma
model and the Nambu-Jona-Lasinio model. For small bare quark masses we find in
both models a smooth crossover transition for nonzero and and a
first order transition for T=0 and nonzero . We calculate explicitly the
first order phase transition line and spinodal lines in the plane.
As expected they all end in a critical point. We find that, in the linear sigma
model, the sigma mass goes to zero at the critical point. This is in contrast
to the NJL model, where the sigma mass, as defined in the random phase
approximation, does not vanish. We also compute the adiabatic lines in the
plane. Within the models studied here, the critical point does not
serve as a ``focusing'' point in the adiabatic expansion.Comment: 22 pages, 18 figure
Statistics of S-matrix poles for chaotic systems with broken time reversal invariance: a conjecture
In the framework of a random matrix description of chaotic quantum scattering
the positions of matrix poles are given by complex eigenvalues of an
effective non-Hermitian random-matrix Hamiltonian. We put forward a conjecture
on statistics of for systems with broken time-reversal invariance and
verify that it allows to reproduce statistical characteristics of Wigner time
delays known from independent calculations. We analyze the ensuing two-point
statistical measures as e.g. spectral form factor and the number variance. In
addition we find the density of complex eigenvalues of real asymmetric matrices
generalizing the recent result by Efetov\cite{Efnh}.Comment: 4 page
Correlation functions of the BC Calogero-Sutherland model
The BC-type Calogero-Sutherland model (CSM) is an integrable extension of the
ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is
related to the chiral classes of random matrix ensembles (RMEs) in exactly the
same way as the A-CSM is related to the Dyson classes. We first develop the
fermionic replica sigma-model formalism suitable to treat all chiral RMEs. By
exploiting ''generalized color-flavor transformation'' we then extend the
method to find the exact asymptotics of the BC-CSM density profile. Consistency
of our result with the c=1 Gaussian conformal field theory description is
verified. The emerging Friedel oscillations structure and sum rules are
discussed in details. We also compute the distribution of the particle nearest
to the reflection point.Comment: 12 pages, no figure, REVTeX4. sect.V updated, references added (v3
Random Matrices close to Hermitian or unitary: overview of methods and results
The paper discusses progress in understanding statistical properties of
complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and
non-Hermitian random matrices. Ensembles of this type emerge in various
physical contexts, most importantly in random matrix description of quantum
chaotic scattering as well as in the context of QCD-inspired random matrix
models.Comment: Published version, with a few more misprints correcte
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