We investigate the existence and properties of a double asymptotic expansion in $1/N^{2}$ and $1/\sqrt{D}$ in $\mathrm{U}(N)\times\mathrm{O}(D)$ invariant Hermitian multi-matrix models, where the $N\times N$ matrices transform in the vector representation of $\mathrm{O}(D)$. The crucial point is to prove the existence of an upper bound $η(h)$ on the maximum power $D^{1+η(h)}$ of $D$ that can appear for the contribution at a given order $N^{2-2h}$ in the large $N$ expansion. We conjecture that $η(h)=h$ in a large class of models. In the case of traceless Hermitian matrices with the quartic tetrahedral interaction, we are able to prove that $η(h)\leq 2h$; the sharper bound $η(h)=h$ is proven for a complex bipartite version of the model, with no need to impose a tracelessness condition. We also prove that $η(h)=h$ for the Hermitian model with the sextic wheel interaction, again with no need to impose a tracelessness condition.
This paper provides the constructive loop vertex expansion for stable matrix models with (single trace) interactions of arbitrarily high even order in the Hermitian and real symmetric cases. It relies on a new and simpler method which can also be applied in the previously treated complex case. We prove analyticity in the coupling constant of the free energy for such models in a domain uniform in the size of the matrix
This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and $W$-representation. We apply it to prove the formula for the Jack functions averages - appropriate analogue of characters for the $β$-deformed Hermitian Gaussian matrix model. We also sketch the construction of $W$-operators from Calogero-Ruijsenaars H…
Virasoro constraint is the operator algebra version of one-loop equation for a Hermitian one-matrix model, and it plays an important role in solving the model. We construct the realization of the Virasoro constraint from the Conformal Field Theory (CFT) method. From multi-loop equations of the one-matrix model, we get a more general constraint. It can be expressed in terms of the operator algebras, which is the Virasoro subalgebra with extra parameters. In this sense, we named as generalized Virasoro constraint. We enlarge this algebra with central extension, this is a new kind of algebra, and…
A Novel Approach to Non-Hermitian Random Matrix Models
本文提出了一种研究非埃尔米特随机矩阵系综谱性质的新方法。类似于复格林函数通过不连续性对厄米特系综的实谱进行编码，这里提出的格林函数的四元数扩展在非厄米特系综的情况下直接导致复谱，并对特征向量的一些谱特性进行编码。四元数的标准二乘二矩阵表示导致了最近在图解方法的背景下提出的所谓矩阵值预解式的推广[1-6]。我们证明了四元数格林函数服从自由变量微积分[7，8]。特别地，矩阵格林函数的四元数函数逆，在[9]之后称为Blue函数，服从简单加法定律，正如不久前观察到的那样[1，3]。利用这一定律，我们推导出新的、通用的、算法性的和有效的方法来寻找所有形式为H+ iH '的非埃尔米特系综的非全纯格林函数，其中系综H和H'是独立的（在Voiculescu [7]的意义上是自由的）埃尔米特系综任意测量。我们证明了一个简单的重新推导的非厄米随机矩阵模型的几个例子的谱特性的方法的力量。
Using techniques of supersymmetric quantum mechanics, scattering properties of Hermitian Hamiltonians, which are related to non-Hermitian ones by similarity transformations, are studied. We have found that the scattering matrix of the Hermitian Hamiltonian coincides with the phase factor of the non-unitary scattering matrix of the non-Hermitian Hamiltonian. The possible presence of a spectral singularity in a non-Hermitian Hamiltonian translates into a pronounced resonance in the scattering cross section of its Hermitian counterpart. This opens a way for detecting spectral singularities in sca…
We present the $W_{1+\infty}$ constraints for the Gaussian Hermitian matrix model, where the constructed constraint operators yield the $W_{1+\infty}$ $n$-algebra. For the Virasoro constraints, we note that the constraint operators give the null 3-algebra. With the help of our Virasoro constraints, we derive a new effective formula for correlators in the Gaussian Hermitian matrix model.
Hermitian vs. Anti-Hermitian 1-Matrix Models and Their Hierarchies
基于\v C. Crnković，M. Douglas和G. Moore，多临界多割单矩阵模型及其相关的$sl(2,C)$可积层次的研究，进一步追求。具有不同标度变换的厄米矩阵模型的双标度极限导致KdV族、修正KdV族和部分非线性薛定谔族。相反，反厄米矩阵模型，在两个弧扇区，在Zakharov-Shabat层次，其中包含KdV和mKdV作为减少的结果。对于所有的层次结构，它被发现，维拉索罗约束作用于相关的τ-函数。而它是已知的ZS和KdV模型导致维拉索罗约束的$sl(2,C)$真空，我们发现，mKdV模型导致维拉索罗约束的最高权状态与任意的共形尺寸。
We introduce Random Matrix Models for the Hermitian Wilson-Dirac operator of QCD-like theories. We show that they are equivalent to the $ε$-limit of the chiral Lagrangian for Wilson chiral perturbation theory. Results are obtained for two-color QCD with quarks in the fundamental representation of the color group as well as any-color QCD with quarks in the adjoint representation. For $N_c=2$ we also have obtained the lattice spacing dependence of the quenched average spectral density for a fixed value of the index of the Dirac operator. Comparisons with direct numerical simulations of the rando…
A formal expansion for the Green's functions of an interacting quantum field theory in a parameter that somehow encodes its "distance" from the corresponding non-interacting one was introduced more than thirty years ago, and has been recently reconsidered in connection with its possible application to the renormalization of non-hermitian theories. Besides this new and interesting application, this expansion has special properties already when applied to ordinary (i.e. hermitian) theories, and in order to disentangle the peculiarities of the expansion itself from those of non-hermitian theories…
We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the c…
We construct the multi-variable realizations of the $W_{1+\infty}$ algebra such that they lead to the $W_{1+\infty}$ $n$-algebra. Based on our realizations of the $W_{1+\infty}$ algebra, we derive the $W_{1+\infty}$ constraints for the hermitian one-matrix model. The constraint operators yield not only the $W_{1+\infty}$ algebra but also the closed $W_{1+\infty}$ $n$-algebra.
J. Harer and D. Zagier have found a strikingly simple generating function for exact (all-genera) 1-point correlators in the Gaussian Hermitian matrix model. In this paper we generalize their result to 2-point correlators, using Toda integrability of the model. Remarkably, this exact 2-point correlation function turns out to be an elementary function - arctangent. Relation to the standard 2-point resolvents is pointed out. Some attempts of generalization to 3-point and higher functions are described.
Given a diagonalizable $N\times N$ matrix $H$, whose non-degenerate spectrum consists of $p$ pairs of complex conjugate eigenvalues and additional $N-2p$ real eigenvalues, we determine all metrics $M$, of all possible signatures, with respect to which $H$ is pseudo-hermitian. In particular, we show that any compatible $M$ must have $p$ pairs of opposite eigenvalues in its spectrum so that $p$ is the minimal number of both positive and negative eigenvalues of $M$. We provide explicit parametrization of the space of all admissible metrics and show that it is topologically a $p$-dimensional torus…
Since the ($β$-deformed) hermitian one-matrix models can be represented as the integrated conformal field theory (CFT) expectation values, we construct the operators in terms of the generators of the Heisenberg algebra such that the constraints can be derived by inserting the constructed operators into the integrated expectation values. We also obtain the second order total derivative operators associating with the derived constraint operators and analyze their properties. We explore the intrinsic connection between the derived constraint operators and $W$-representations of some matrix models…
Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite dimensional subspace. As such, they can be used, in the usual spirit of random matrix theory, to model chaotic or disordered PT-symmetric quantum systems, or their gain-loss-balanced classical analogs, in the phase of broken PT-symmetry. The eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. In this paper we introduc…
We analyze two types of hermitian matrix models with asymmetric solutions. One type breaks the symmetry explicitly with an asymmetric quartic potential. We give the phase diagram of this model with two different phase transitions between the one cut and two cut solutions. The second type, describing real scalar field theory on fuzzy spaces, breaks the symmetry spontaneously with multitrace terms. We present two methods to study this model, one direct and one using a connection with the first type of models. We analyze the model for the fuzzy sphere and obtain a phase diagram with the location …
Real critical exponents from the $\varepsilon$-expansion in an interacting $U(1)$ model with non-Hermitian $Z_4$ anisotropy
Institute for Theoretical Solid State Physics, IFW Dresden（德累斯顿亥姆霍兹中心IFW固体物理研究所）；Bogolyubov Institute for Theoretical Physics（博戈留博夫理论物理研究所）；Institute for Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, TU Dresden（德累斯顿工业大学理论物理研究所与维尔茨堡-德累斯顿卓越集群ct.qmat）
We develop relativistic wave equations in the framework of the new non-hermitian ${\cal PT}$ quantum mechanics. The familiar Hermitian Dirac equation emerges as an exact result of imposing the Dirac algebra, the criteria of ${\cal PT}$-symmetric quantum mechanics, and relativistic invariance. However, relaxing the constraint that in particular the mass matrix be Hermitian also allows for models that have no counterpart in conventional quantum mechanics. For example it is well-known that a quartet of Weyl spinors coupled by a Hermitian mass matrix reduces to two independent Dirac fermions; here…
