Eigenvalue Correlation Functions Research Articles

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.

Massive partition functions and complex eigenvalue correlations in matrix models with symplectic symmetry

We compute all massive partition functions or characteristic polynomials and their complex eigenvalue correlation functions for non-Hermitean extensions of the symplectic and chiral symplectic ensemble of random matrices. Our results are valid for general weight functions without degeneracies of the mass parameters. The expressions we derive are given in terms of the Pfaffian of skew orthogonal polynomials in the complex plane and their kernel. They are much simpler than the corresponding expressions for symplectic matrix models with real eigenvalues, and we explicitly show how to recover these in the Hermitean limit. This explains the appearance of three different kernels as quaternion matrix elements there in terms of derivatives of a single kernel here.

The complex Laguerre symplectic ensemble of non-Hermitian matrices

We solve the complex extension of the chiral Gaussian symplectic ensemble, defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion real matrices. This leads to the appearance of Laguerre polynomials in the complex plane and we prove their orthogonality. Alternatively, a complex eigenvalue representation of this ensemble is given for general weight functions. All k-point correlation functions of complex eigenvalues are given in terms of the corresponding skew orthogonal polynomials in the complex plane for finite-N, where N is the matrix size or number of eigenvalues, respectively. We also allow for an arbitrary number of complex conjugate pairs of characteristic polynomials in the weight function, corresponding to massive quark flavours in applications to field theory. Explicit expressions are given in the large-N limit at both weak and strong non-Hermiticity for the weight of the Gaussian two-matrix model. This model can be mapped to the complex Dirac operator spectrum with non-vanishing chemical potential. It belongs to the symmetry class of either the adjoint representation or two colours in the fundamental representation using staggered lattice fermions.

Distributions of Dirac operator eigenvalues

The distribution of individual Dirac eigenvalues is derived by relating them to the density and higher eigenvalue correlation functions. The relations are general and hold for any gauge theory coupled to fermions under certain conditions which are stated. As a special case, we give examples of the lowest-lying eigenvalue distributions for QCD-like gauge theories without making use of earlier results based on the relation to Random Matrix Theory.

Random matrices with external source and multiple orthogonal polynomials

We show that the average characteristic polynomial Pn(z) = E[det(zI−M)] of the random Hermitian matrix ensemble Z 1 n exp(−Tr(V (M) − AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue aj of A, there is a weight and Pn has nj orthogonality conditions with respect to this weight, if nj is the multiplicity of aj. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin. Here we give a different expression for the kernel. We derive a Christoffel-Darboux formula in case A has two distinct eigenvalues, which leads to a compact formula in terms of a Riemann-Hilbert problem that is satisfied by multiple orthogonal polynomials.

Microscopic correlations for non-Hermitian Dirac operators in three-dimensional QCD

In the presence of a nonvanishing chemical potential the eigenvalues of the Dirac operator become complex. We calculate spectral correlation functions of complex eigenvalues using a random matrix model approach. Our results apply to non-Hermitian Dirac operators in three-dimensional QCD with broken flavor symmetry and in four-dimensional QCD in the bulk of the spectrum. The derivation follows earlier results of Fyodorov, Khoruzhenko, and Sommers for complex spectra exploiting the existence of orthogonal polynomials in the complex plane. Explicit analytic expressions are given for all microscopic k-point correlation functions in the presence of an arbitrary even number of massive quarks, both in the limit of strong and weak non-Hermiticity. In the latter case the parameter governing the non-Hermiticity of the Dirac matrices is identified with the influence of the chemical potential.

Massive random matrix ensembles atβ=1and 4: QCD in three dimensions

The zero momentum sectors in effective theories of three dimensional QCD coupled to pseudoreal (two colors) and real (adjoint) quarks in a classically parity-invariant manner have alternative descriptions in terms of orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute finite-volume QCD partition functions and correlation functions of Dirac operator eigenvalues in a presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of quaternion determinants, are reduced to conventional results for the Gaussian ensembles in the quenched limit.

Massive chiral random matrix ensembles atβ=1and 4: Finite-volume QCD partition functions

In a deep-infrared (ergodic) regime, QCD coupled to massive pseudoreal and real quarks are described by chiral orthogonal and symplectic ensembles of random matrices. Using this correspondence, general expressions for the QCD partition functions are derived in terms of microscopically rescaled mass variables. In limited cases, correlation functions of Dirac eigenvalues and distributions of the smallest Dirac eigenvalue are given as ratios of these partition functions. When all masses are degenerate, our results reproduce the known expressions for the partition functions of zero-dimensional sigma models.

Massive chiral random matrix ensembles atβ=1and 4: QCD Dirac operator spectra

The zero momentum sectors in effective theories of quantum chromodynamics coupled to pseudoreal (two colors) and real (adjoint) quarks have alternative descriptions in terms of chiral orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute correlation functions of Dirac operator eigenvalues within a sector with an arbitrary topological charge in the presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of Pfaffians, interpolate between known results for the chiral and quenched limits as quark masses vary.

Fermionic mapping for eigenvalue correlation functions of weakly non-Hermitian symplectic ensemble

The eigenvalues of an arbitrary quaternionic matrix have a joint probability distribution function first derived by Ginibre. We derive the j.p.d. for the weakly non-Hermitian version of this problem and then show that there exists a mapping of this system onto a fermionic field theory. This mapping is used to integrate over the positions of the eigenvalues and obtain eigenvalue density as well as all higher correlation functions for both the strongly and weakly non-Hermitian cases.

Correlation functions of eigenvalues of multi-matrix models, and the limit of a time-dependent matrix

The universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems, and proved in some particular cases, as the Hermitian one-matrix model with polynomial potential. Here, we consider the more difficult case of a unidimensional chain of Hermitian matrices with first-neighbour couplings and polynomial potentials. An asymptotic expression of the orthogonal polynomials and a generalization of the Darboux-Christoffel theorem allow us to find new results for the correlations of eigenvalues of different matrices of the chain. Eventually, we consider the limit of the infinite chain of matrices, which can be interpreted as a time-dependent Hermitian one-matrix model, and give the correlation functions of eigenvalues at different times.

Consistency conditions for finite-volume partition functions

Using relations from random matrix theory, we derive exact expressions for all $n$-point spectral correlation functions of Dirac operator eigenvalues in terms of finite-volume partition functions. This is done for both chiral symplectic and chiral unitary random matrix ensembles, which correspond to $SU(N_c \geq 3)$ gauge theories with $N_f$ fermions in the adjoint and fundamental representations, respectively. In the latter case we infer from this an infinite sequence of consistency conditions that must be satisfied by the corresponding finite-volume partition functions.

Orthogonal polynomials and exact correlation functions for two cut random matrix models

Exact eigenvalue correlation functions are computed for large N hermitian one-matrix models with eigenvalues distributed in two symmetric cuts. An asymptotic form for orthogonal polynomials for arbitrary polynomial potentials that support a Z 2 symmetricdistribution is obtained. This results in an exact explicit expression for the kernel at large N which determines all eigenvalue correlators. The oscillating and smooth parts of the two-point correlator are extracted and the universality of local fine-grained and smoothed global correlators is established.

Laguerre ensembles of random matrices: Nonuniversal correlation functions

The eigenvalue correlation functions of the Laguerre ensembles (exponential ensembles) of real symmetric and self-dual quaternion random matrices are calculated. It is shown that in certain intervals of the spectrum the renormalized correlations are not universal. A physical example in the transmission spectrum of a disordered conductor where such nonuniversality occurs is given.

Elliptic Law

Elliptic Law