Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach (original) (raw)
Related papers
Theoretical and Mathematical Physics, 2011
We solve the loop equations of the β-ensemble model analogously to the solution found for the Hermitian matrices β = 1. For β = 1, the solution was expressed using the algebraic spectral curve of equation y 2 = U (x). For arbitrary β, the spectral curve converts into a Schrödinger equation (( ∂) 2 −U (x))ψ(x) = 0 with ∝ ( √ β−1/ √ β)/N . This paper is similar to the sister paper I, in particular, all the main ingredients specific for the algebraic solution of the problem remain the same, but here we present the second approach to finding a solution of loop equations using sectorwise definition of resolvents. Being technically more involved, it allows to define consistently the B-cycle structure of the obtained quantum algebraic curve (a D-module of the form y 2 −U (x), where [y, x] = ) and to construct explicitly the correlation functions and the corresponding symplectic invariants F h , or the terms of the free energy, in 1/N 2 -expansion at arbitrary . The set of "flat" coordinates comprises the potential times t k and the occupation numbers ǫ α . We define and investigate the properties of the Aand B-cycles, forms of 1st, 2nd and 3rd kind, and the Riemann bilinear identities. We use these identities to find explicitly the singular part of F 0 that depends exclusively on ǫ α .
Loop equations and topological recursion for the arbitrary-β two-matrix model
2012
We write the loop equations for the β two-matrix model, and we propose a topological recursion algorithm to solve them, order by order in a small parameter. We find that to leading order, the spectral curve is a "quantum" spectral curve, i.e. it is given by a differential operator (instead of an algebraic equation for the hermitian case). Here, we study the case where that quantum spectral curve is completely degenerate, it satisfies a Bethe ansatz, and the spectral curve is the Baxter TQ relation.
Topological expansion of the Bethe ansatz, and quantum algebraic geometry
Arxiv preprint arXiv:0911.1664, 2009
Abstract: In this article, we solve the loop equations of the β-random matrix model, in a way similar to what was found for the case of hermitian matrices β = 1. For ... 4.2 Third kind differential: kernel G(x, z) . . . . . . . . . . . . . . . . . . . 25 ... 4.3 The Bergman kernel B(x, z) . . . . . . . . . . . . . . . . . . . . . ...
Topological expansion and exponential asymptotics in 1D quantum mechanics
Journal of Physics A: Mathematical and General, 2000
Borel summable semiclassical expansions in 1D quantum mechanics are considered. These are the Borel summable expansions of fundamental solutions and of quantities constructed with their help. An expansion, called topological, is constructed for the corresponding Borel functions. Its main property is to order the singularity structure of the Borel plane in a hierarchical way by an increasing complexity of this structure starting from the analytic one. This allows us to study the Borel plane singularity structure in a systematic way. Examples of such structures are considered for linear, harmonic and anharmonic potentials. Together with the best approximation provided by the semiclassical series the exponentially small contribution completing the approximation are considered. A natural method of constructing such an exponential asymptotics relied on the Borel plane singularity structures provided by the topological expansion is developed. The method is used to form the semiclassical series including exponential contributions for the energy levels of the anharmonic oscillator.
The Path Integral Formulation Of Quantum Mechanics & Its Topological Applications
In this report, we deliver a detailed introduction to the methods of path integration in the focus of quantum mechanics. After an overview of the techniques of integration and the relationship to the familiar results of quantum mechanics such as the Schroedinger equation, we study some of the applications to mechanical systems with non-trivial degrees of freedom and discuss the advantages that path integration has as a formulation in studying these systems. To this end, in chapter 3, we introduce the topological concept of homotopy classes, and apply this to derive the spin statistics theorem, and discuss the phenomenon of parastatistics for systems constrained to dimension d<3. Following on from this, we calculate a general formula for the quantum mechanical propagator in terms of path integrals over different homotopy classes. The final chapter of this report studies the so called instanton solution as a way of describing quantum mechanical vacuum tunneling effects as a semi-classical problem. We conclude with discussing some applications in gauge field theory and describe the topological quality of the classical vacua of SU(2) Yang-Mills gauge theory; apply the path integral method to arrive at the theta-vacuum, and briefly say a few words about its far reaching consequences and applications.\\The content is aimed predominantly at a mathematical audience with a physical interest, moreover, we assume that the reader has a good grounding in topology and in both the classical and quantum mechanical theories. For completeness, essentials of these topics are reviewed in the appendices. (NOTE: Previous version uploaded was an earlier draft accidentally uploaded. This is the correct final version)
DOI: 10.1088/1742-5468/2012/01/P01011 One-cut solution of the β-ensembles in the Zhukovsky
2013
In this article, we study in detail the modified topological recursion of the one matrix model for arbitrary β in the one cut case. We show that for polynomial potentials, the recursion can be computed as a sum of residues. However the main difference with the hermitian matrix model is that the residues cannot be set at the branchpoints of the spectral curve but require the knowledge of the whole curve. In order to establish non-ambiguous formulas, we place ourselves in the context of the globalizing parametrization which is specific to the one cut case (also known as Zhukovsky parametrization). This situation is particularly interesting for applications since in most cases the potentials of the matrix models only have one cut in string theory. Finally, the article exhibits some numeric simulations of histograms of limiting density of eigenvalues for different values of the parameter β. Contents 1
Topological quantum theories and integrable models
Physical Review D, 1991
The path-integral generalization of the Duistermaat-Heckman integration formula is investigated for integrable models. It is shown that for models with periodic classical trajectories the path integral reduces to a form similar to the finite-dimensional Duistermaat-Heckman integration formula. This provides a relation between exactness of the stationary-phase approximation and Morse theory. It is also argued that certain integrable models can be related to topological quantum theories. Finally, it is found that in general the stationary-phase approximation presumes that the initial and final configurations are in difFerent polarizations. This is exemplified by the quantization of the SU(2) coadjoint orbit.
Journal of Statistical Mechanics: Theory and Experiment, 2019
We consider a parameter dependent ensemble of two real random matrices with Gaussian distribution. It describes the transition between the symmetry class of the chiral Gaussian orthogonal ensemble (Cartan class B|DI) and the ensemble of antisymmetric Hermitian random matrices (Cartan class B|D). It enjoys the special feature that, depending on the matrix dimension N , it has exactly ν = 0 (1) zero-mode for N even (odd), throughout the symmetry transition. This "topological protection" is reminiscent of properties of topological insulators. We show that our ensemble represents a Pfaffian point process which is typical for such transition ensembles. On a technical level, our results follow from the applicability of the Harish-Chandra integral over the orthogonal group. The matrixvalued kernel determining all eigenvalue correlation functions is explicitly constructed in terms of skeworthogonal polynomials, depending on the topological index ν = 0, 1. These polynomials interpolate between Laguerre and even (odd) Hermite polynomials for ν = 0 (1), in terms of which the two limiting symmetry classes can be solved. Numerical simulations illustrate our analytical results for the spectral density and an expansion for the distribution of the smallest eigenvalue at finite N .
Free energy topological expansion for the 2-matrix model
Journal of High Energy Physics, 2006
We compute the complete topological expansion of the formal hermitian twomatrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1 N expansion of the nonmixed correlation functions and give a new formulation of the spectral curve. We extend these rules obtaining a closed formula for correlation functions in all orders of topological expansion. We then integrate it to obtain the free energy in terms of residues on the associated Riemann surface. *
The quantum canonical ensemble
Journal of Mathematical Physics, 1998
The phase space Γ of quantum mechanics can be viewed as the complex projective space CP n endowed with a Kählerian structure given by the Fubini-Study metric and an associated symplectic form. We can then interpret the Schrödinger equation as generating a Hamiltonian dynamics on Γ. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting density matrix for the canonical Γ-ensemble differs from density matrix of the conventional approach. As an illustration, the results are applied to the case of a spin one-half particle in a heat bath with an applied magnetic field.