non-Hermitian Random Matrix Models Research Articles

Chiral random matrix theory for single-flavor spin-one Cooper pairing

We propose a new non-Hermitian chiral random matrix model that describes single-flavor spin-one Cooper pairing of quarks. For three colors the model shows spontaneous breaking of color $\mathrm{SU}(3)_C$ and spin $\mathrm{SO}(3)_J$ symmetries down to the diagonal $\mathrm{SO}(3)_{C+J}$ subgroup, in striking analogy to the color-spin locked phase of one-flavor QCD at high density. For two colors, color-singlet spin-one diquarks condense and trigger symmetry breaking $\mathrm{U}(1) \times \mathrm{SO}(3)_J \to \mathrm{SO}(2)_J$. In both cases the microscopic large-$N$ limit is rigorously taken and the effective theory of Nambu-Goldstone modes is derived.

Universal Spectra of Random Lindblad Operators.

To understand the typical dynamics of an open quantum system in continuous time, we introduce an ensemble of random Lindblad operators, which generate completely positive Markovian evolution in the space of the density matrices. The spectral properties of these operators, including the shape of the eigenvalue distribution in the complex plane, are evaluated by using methods of free probabilities and explained with non-Hermitian random matrix models. We also demonstrate the universality of the spectral features. The notion of an ensemble of random generators of Markovian quantum evolution constitutes a step towards categorization of dissipative quantum chaos.

On the spectrum of random anti-symmetric and tournament matrices

We consider a discrete, non-Hermitian random matrix model, which can be expressed as a shift of a rank-one perturbation of an anti-symmetric matrix. We show that, asymptotically almost surely, the real parts of the eigenvalues of the non-Hermitian matrix around any fixed index are interlaced with those of the anti-symmetric matrix. Along the way, we show that some tools recently developed to study the eigenvalue distributions of Hermitian matrices extend to the anti-symmetric setting.

Symmetries of the Feinberg–Zee random hopping matrix

We study the symmetries of the spectrum of the Feinberg–Zee Random Hopping Matrix introduced in [J. Feinberg and A. Zee, Spectral curves of non-Hermitian Hamiltonians, Nucl. Phys. B 552 (1999) 599–623] and studied in various papers thereafter (e.g. [S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, eigenvalue Problem meets Sierpinski triangle: Computing the spectrum of a non-self-adjoint random operator, Oper. Matrices 5 (2011) 633–648; S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators, Oper. Matrices 7 (2013) 739–775; S. N. Chandler-Wilde and E. B. Davies, Spectrum of a Feinberg–Zee sandom hopping matrix, J. Spectral Theory 2 (2012) 147–179; R. Hagger, On the spectrum and numerical range of tridiagonal random operators, preprint (2014), arXiv: 1407.5486; D. E. Holz, H. Orland and A. Zee, On the remarkable spectrum of a non-Hermitian random matrix model, J. Phys. A: Math. Gen. 36 (2003) 3385–3400]). In [J. Spectral Theory 2 (2012) 147–179], Chandler-Wilde and Davies proved that the spectrum of the Feinberg–Zee Random Hopping Matrix is invariant under taking square roots, which implied that the unit disk is contained in the spectrum (a result already obtained slightly earlier in [Oper. Matrices 5 (2011) 633–648]. In a similar approach we show that there is an infinite sequence of symmetries at least in the periodic part of the spectrum (which is conjectured to be dense). Using these symmetries and the result of [J. Spectral Theory 2 (2012) 147–179], we can exploit a considerably larger part of the spectrum than the unit disk. As a further consequence we find an infinite sequence of Julia sets contained in the spectrum. These facts may serve as a part of an explanation of the seemingly fractal-like behavior of the boundary.

Random matrix ensembles for PT-symmetric systems

Recently much effort has been made towards the introduction of non-Hermitian random matrix models respecting PT-symmetry. Here we show that there is a one-to-one correspondence between complex PT-symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian; and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. We conjecture that these ensembles represent universality classes for PT-symmetric matrices. For the case of 2 × 2 matrices we derive analytic expressions for the joint probability distributions of the eigenvalues, the one-level densities and the level spacings in the case of real eigenvalues.

Dysonian dynamics of the Ginibre ensemble.

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a nontrivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of a new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the nonanalytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics that we observe for the Ginibre ensemble is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.

A diagrammatic equation for oriented planar graphs

In this paper we derive a diagrammatic equation for the planar sector of square non-Hermitian random matrix models. Our fundamental equation is first obtained by a graph counting argument (inspired by the Polchinski equation in quantum field theory) and subsequently derived independently by a precise saddle point analysis of the corresponding random matrix integral. We solve the equation perturbatively for a generic model and conclude by exhibiting two duality properties of the perturbative solution.

Complex Hermite polynomials: from the semi-circular law to the circular law

We study asymptotics of orthogonal polynomial measures of the form |HN| 2 d∞ where HN are real or complex Hermite polynomials with re- spect to the Gaussian measure ∞. By means of dierential equations on Laplace transforms, interpolation between the (real) arcsine law and the (complex) uniform distribution on the circle is emphasized. Suitable aver- ages by an independent uniform law give rise to the limiting semi-circular and circular laws of Hermitian and non-Hermitian Gaussian random matrix models. The intermediate regime between strong and weak non-Hermiticity is clearly identified on the limiting dierential equation by means of an addi- tional normal variable in the vertical direction.

Universal Results from an Alternate Random-Matrix Model for QCD with a Baryon Chemical Potential

We introduce a new non-Hermitian random-matrix model for QCD with a baryon chemical potential. This model is a direct chiral extension of a previously studied model that interpolates between the Wigner-Dyson and Ginibre ensembles. We present exact results for all eigenvalue correlations for any number of quark flavors using the orthogonal polynomial method. We also find that the parameters of the model can be scaled to remove the effects of the chemical potential from all thermodynamic quantities until the finite density phase transition is reached. This makes the model and its extensions well suited for studying the phase diagram of QCD.

QCD dirac operator at nonzero chemical potential: lattice data and matrix model.

Recently, a non-Hermitian chiral random matrix model was proposed to describe the eigenvalues of the QCD Dirac operator at nonzero chemical potential. This matrix model can be constructed from QCD by mapping it to an equivalent matrix model which has the same symmetries as QCD with chemical potential. Its microscopic spectral correlations are conjectured to be identical to those of the QCD Dirac operator. We investigate this conjecture by comparing large ensembles of Dirac eigenvalues in quenched SU(3) lattice QCD at a nonzero chemical potential to the analytical predictions of the matrix model. Excellent agreement is found in the two regimes of weak and strong non-Hermiticity, for several different lattice volumes.

On the remarkable spectrum of a non-Hermitian random matrix model

A non-Hermitian random matrix model proposed a few years ago has a remarkably intricate spectrum. Various attempts have been made to understand the spectrum, but even its dimension is not known. Using the Dyson?Schmidt equation, we show that the spectrum consists of a non-denumerable set of lines in the complex plane. Each line is the support of the spectrum of a periodic Hamiltonian, obtained by the infinite repetition of any finite sequence of the disorder variables. Our approach is based on the 'theory of words'. We make a complete study of all four-letter words. The spectrum is complicated because our matrix contains everything that will ever be written in the history of the universe, including this particular paper.

Green's functions in non-hermitian random matrix models

We review some recent techniques for dealing with non-hermitian random matrix models based on generalized Green's functions. We introduce the diagrammatic methods in the hermitian case and generalize them to the non-hermitian case. The results are illustrated in terms of the eigenvalue distribution, eigenvector statistics and addition laws.

Correlations of eigenvectors for non-Hermitian random-matrix models.

We establish a general relation between the diagonal correlator of eigenvectors and the spectral Green's function for non-Hermitian random-matrix models in the large-N limit. We apply this result to a number of non-Hermitian random-matrix models and show that the outcome is in good agreement with numerical results.

Non-hermitian random matrix models

We introduce an extension of the diagrammatic rules in random matrix theory and apply it to non-hermitian random matrix models using the 1/ N approximation. A number of one-and two-point functions are evaluated on their holomorphic and non-holomorphic supports to leading order in 1/ N. The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic cotrelations. The generic form for the two-point functions is obtained, generalizing the concept of macroscopic universality to non-hermitian random matrices. We show that the holomorphic and non-holomorphic 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.