Eugene Kanzieper

research topics
 

[ research topics | selected publications | recent talks | curriculum vitae | teaching | joint physics seminar | contact me ]
[ options: replicas | random matrices | mesoscopics | arXived papers | citebase | google scholar | postdoctoral position ]
     

Research Highlights in Brief

     

 

 

Order and Chaos through the eyes of Escher

   

 

My major research interests include: (i) Mathematical and Physical Aspects of the Random Matrix Theory (RMT); (ii) Field Theory Applications to Condensed Matter Problems (Supersymmetry, Replica, and Keldysh Nonlinear Sigma Models); (iii) Disordered and Quantum Chaotic Systems; (iv) Quantum Transport and Nonequilibrium Mesoscopics.

The "magic world of random matrices" (expression by Freeman J. Dyson), related either directly or indirectly to almost all of the above topics, is central to my research. (Dyson, referring to a series of his most influential papers, writes: "I found it very difficult to keep my mind... from wandering into the magic world of random matrices"). Indeed, the Random Matrix Theory is exceptional in a way it combines mathematical richness and physical ubiquity, with a boundary between physics and mathematics being hard to draw. 

    

   Group Members (past and present):

 

The latest preprint(s)/paper(s): 

  • New E. Kanzieper and N. Singh, Non-Hermitean Wishart random matrices (I); e-print: arXiv:1006.3096, Journal of Mathematical Physics 51, 103510 (2010).

  • New V. Al. Osipov and E. Kanzieper, Correlations of RMT characteristic polynomials: Hermitean matrices; e-print: arXiv:1003.0757, Annals of Physics 325, 2251 (2010).

  • E. Kanzieper, Replica approach in random matrix theory; e-print: arXiv:0909.3198. Invited chapter for The Oxford Handbook on Random Matrix Theory, Oxford University Press (2010).  

  • V. Al. Osipov and E. Kanzieper, Statistics of thermal to shot noise crossover in chaotic cavities; e-print: arXiv:0902.3069, Journal of Physics A 42, 475101 (2009).

  • V. Al. Osipov and E. Kanzieper, Integrable theory of quantum transport in chaotic cavities; e-print: arXiv:0806.2784, Physical Review Letters 101, 176804 (2008).

  • V. Al. Osipov and E. Kanzieper, Are bosonic replicas faulty? e-print: arXiv:0704.2968, Physical Review Letters 99, 050602 (2007).

  • G. Akemann and E. Kanzieper, Integrable structure of Ginibre's ensemble of real random matrices and a Pfaffian integration theorem, e-print: math-ph/0703019, Journal of Statistical Physics 129, 1159 (2007).

  • A. Borodin and E. Kanzieper, A note on the Pfaffian integration theorem, e-print: arXiv:0707.2784, Journal of Physics A 40, F849 (2007).
 

Voronoi Tesselation can be modelled by non-Hermitean random matrix models

Density of complex eigenvalues in Ginibre's orthogonal ensemble

Powerpoint presentations available (some symbols can be displayed incorrectly): 

 

   
  Replica Field Theories in 0 Dimensions  
 

     
 

 

This is my most recent, and also currently run, activity. It addresses a long standing problem of legitimacy of replica field theories formulated by Wegner (1979), Schäfer and Wegner (1980), Efetov, Larkin and Khmelnitskii (1980), and Finkelstein (1983). These are also known as nonlinear replica sigma models.

 

  
     

Replica field theories, in either form, rest on the replica "trick" which substitutes a disordered system by its n identical copies, or replicas, making a nonperturbative averaging of a physical observable of interest over various disorder realisations viable. As a result, an observable can formally be represented as an n ® 0 limit of a matrix integral running over an n×n matrix field Q; the number of replicas n is a parameter of the field theory. Not unexpectedly, the limiting procedure n ® 0 ruins a classic notion of matrix and raises both conceptual problems regarding mathematical foundations of replica field theories (Parisi, 2002) and operational problems of dealing with weird objects where theoretical physicist intuition often refuses to work.
   

To avoid operations with "matrices" of non-integer dimensionality, one evaluates a matrix integral over  Q for all n integers in a hope to analytically continue the integer-n result into a vicinity of n=0 to make taking the replica limit  well-defined and safe. Analytic continuation away from n positive integers turns out to be the problem.

 

In the context of Random Matrix Theory (RMT), subtleties of the replica "trick" were discussed at length by Verbaarschot and Zirnbauer (1985) who attributed the failure of replica field theories to correctly account for all nonperturbative contributions to a physical observable to a non-uniqueness of the analytic continuation away from n integers. This standpoint formed the prevailing opinion in the literature that the replica method may at best be considered as a perturbative tool not being able to reproduce truly nonperturbative results accessible by, e.g., an alternative Efetov's supersymmetric field theory. This, in turn, raised the question whether or not the nonperturbative sector of replica field theories is reliable at all. The issue is of conceptual importance yet is not pure academic because the replica field theories are among a very few means available to address the problems involving both the disorder and interactions, about which the supersymmetry approach – a prime tool in studying noninteracting disordered systems – has nothing to say.

 

The paper that challenged the opinion about inner deficiency of replica field theories in the RMT context and triggered their further reassessment was that of Kamenev and Mezárd (1999). Adopting the idea of replica symmetry breaking originally devised in the theory of spin glasses, the authors have shown that an approximate, saddle-point like treatment of replica field theories brings seemingly nonperturbative results in a certain region of parameter space. Unfortunately, representations derived by Kamenev and Mezárd do not exist in the vicinity of n=0, the domain crucially important for a correct evaluation of the replica limit n ® 0. In fact, approximate evaluation of matrix integrals over the n×n field Q is the major drawback of replicas à la Kamenev and Mezárd.

 

My contribution to the ongoing discussion on legitimacy replica field theories consists in claiming exact integrability of nonlinear replica sigma models derived in the context of Random Matrix Theory. (i) The claim is based on the observation that replica partition functions for U(N) invariant ensembles of random matrices form an integrable hierarchy described by a semi-infinite Toda Lattice equation. (ii) The latter can exactly be reduced to one of the six Painlevé equations. Operationally, the Toda-to-Painlevé reduction rests on the Hamiltonian theory of Painlevé transcendents. (iii) Since the resulting Painlevé equations contain the number of replicas n as a single parameter, they serve as a proper staring point for building a consistent analytic continuation of nonperturbative replica partition functions to a vicinity of n=0 that would allow for a safe implementation of the replica limit. (iv) This method was shown to bring exact nonperturbative results for quantum correlation functions in Hermitean random matrix models belonging to A and AIII symmetry classes (in Cartan classification) and in Ginibre's ensemble of complex non-Hermitean random matrices.

 

More details are reported in:

This research is supported by the Israel Science Foundation through the grant No 286/04.
 
 

 

Random Matrices: General Formalism and Applications
        

 

Since my PhD studies I have been involved into exciting studies of integrable structure possessed by invariant ensembles of generic non-Gaussian distributed random matrices. A particular emphasis was placed on a quantitative description of eigenvalue correlations in spectra of large random matrices.

Shohat's Method and Universality in Random Matrix Theory

Suppose we are given a unitary invariant random matrix model described by a general non-Gaussian probability density function P parameterised by the confinement potential V[H] such that Tr V[H] ~ – log P[H]. For one, the confinement potential may contain singular points which locally affect a portion of matrix eigenvalues, or it may globally deform matrix spectrum leading to opening of one or more forbidden gaps in the spectrum. What can be said about the density of eigenlevels and the eigenvalue correlations in such a non-Gaussian matrix model? This problem posed in the very early days of the random matrix theory was not answered till the late nineties of the past century. Yet, basic mathematical ideas to deal with the problem, were formulated well before the RMT was conceived.

Density of complex eigenvalues in Ginibre's unitary ensemble

 

A regular formalism for quantitative description of spectral fluctuations in spectra of non-Gaussian U(N) invariant ensembles of random matrices was developed in my PhD Theses. It makes use of the Gaudin-Mehta approach of orthogonal polynomials, and utilises a simple and elegant (yet well forgotten!) idea due to J. Shohat (1930, 1939), the French mathematician of Jewish descent, who had shown how a three-term recurrence equation for generic orthogonal polynomials can be mapped onto a second order differential equation. A technique based on the asymptotic analysis of the latter turned out to be surprisingly fruitful in calculation of both local and global spectral characteristics of random matrices. Yet, it provided an overall look at the celebrated phenomenon of spectral universality in Random Matrix Theory. In particular, I used this technique (i) to give a first universality proof of so-called Airy correlations and also (ii) to find asymptotics of so-called universal multicritical correlations near the spectrum edge, of which the Airy correlations are a particular case. 

More details are reported in:

  • E. Kanzieper and V. Freilikher, Novel universal correlations in invariant random matrix models, e-print: chao-dyn/9701006; published in: Physical Review Letters 78, 3806 (1997).

  • E. Kanzieper and V. Freilikher, Spectra of large random matrices: A method of study, e-print: cond-mat/9809365; published in: NATO ASI Series C: Math. and Phys. Sciences, vol. 531, 165 (Kluwer Press, Dordrecht, 1999).

Parametric Level Statistics in Random Matrix Theory

A parameter-dependent perturbation can be added to above random matrix Hamiltonian (think of a magnetic flux, for one). What can be said about the parametric level statistics in non–Gaussian ensembles of Hermitean random matrices? It turns out that, for a Gaussian perturbation, the parametric fluctuations of eigenlevels can still be determined exactly within the orthogonal polynomials technique, no matter whether the confinement potential V[H] of an unperturbed ensemble is strong or soft. An exact solution takes the form of a double integral transformation that relates the parametric "density–density" correlation function to the two–point scalar kernel of an unperturbed matrix ensemble and to the level confinement V[H]. In random matrix ensembles with strong level confinement, this integral transformation becomes particularly simple in the thermodynamic limit leading to emergence of a connection relation between the parametric level statistics and the scalar two–point kernel of an unperturbed ensemble. Within this picture, the universality of parametric correlations is a simple consequence of the universality of an unperturbed random matrix ensemble.

More details are reported in:

Random Matrix Theory Applications to Quantum Chromodynamics Problems

 

Quantum Chromodynamics (QCD) is one of the fields where the RMT can bring parameter free predictions. Spectral fluctuations of QCD Dirac operators are of particular interest as they can be confronted against lattice simulations. In my publications with G. Akemann, we have shown that integrable structure of chiral random matrix models incorporating global symmetries of QCD Dirac operators (labeled by the Dyson index b = 1,2, and 4) is such that a connection relation between the spectral statistics of massive and massless Dirac operators can be established. For so called b-fold degenerate massive fermions, the link established was used to explicitly derive (and prove the random matrix universality of) statistics of low-lying eigenvalues of QCD Dirac operators in the presence of SU(2) massive fermions in the fundamental representation (b=1) and SU(Nc ³ 2) massive adjoint fermions (b=4). Comparison of our parameter free predictions with available lattice data for SU(2) dynamical staggered fermions has revealed a good agreement.
 

More details are reported in:
   
 

Mesoscopic Physics

  
   

During my stay at the Abdus Salam ICTP (1997-2000), Trieste, Italy I was interested in application of the supersymmetric field theory to description of quasi-one-dimensional conductors, as well as in nonequilibrium mesosocopics:
     
 

Department of Applied Mathematics, Faculty of Sciences, H.I.T. - Holon Institute of Technology, Holon 58102, Israel

  
     

Eugene Kanzieper (c) 2004-2011