Distribution of complex eigenvalues for symplectic ensembles of non-Hermitian matrices

A symplectic ensemble of disordered non-Hermitian Hamiltonians is studied. Starting from a model with an imaginary magnetic field, we derive a proper supermatrix σ-model. The zero-dimensional version of this model corresponds to a symplectic ensemble of weakly non-Hermitian matrices. We derive analytically an explicit expression for the density of complex eigenvalues. This function proves to differ qualitatively from those known for the unitary and orthogonal ensembles. In contrast to these cases, a depletion of the eigenvalues occurs near the real axis. The result about the depletion is in agreement with a previous numerical study performed for QCD models.     

Filling up complex spectral regions through non-Hermitian disordered chains

Eigenspectra that fill regions in the complex plane have been intriguing to many, inspiring research from random matrix theory to esoteric semi-infinite bounded non-Hermitian lattices. In this work, we propose a simple and robust ansatz for constructing models whose eigenspectra fill up generic prescribed regions. Our approach utilizes specially designed non-Hermitian random couplings that allow the co-existence of eigenstates with a continuum of localization lengths, mathematically emulating the effects of semi-infinite boundaries. While some of these couplings are necessarily long-ranged, they are still far more local than what is possible with known random matrix ensembles. Our ansatz can be feasibly implemented in physical platforms such as classical and quantum circuits, and harbors very high tolerance to imperfections due to its stochastic nature.     

The relevance of connectance on the Lyapunov characteristic exponents of products of symplectic random matrices

We study the effect of connectance on the Lyapunov characteristic exponents of products of symplectic random matrices, which mimic the chaotic behaviour of a large class of hamiltonian systems. It is shown that no significative modifications appear in the spectrum of the Lyapunov characteristic exponents when the number of interacting neighbours is increased.     

An ensemble of random particle-hole matrices with collective eigenstates

An ensemble of random particle-hole matrices is studied as a function of the signs of the matrix elements. When all matrix elements have the same sign, a theorem due to Perron ensures the existence of a coherent state: the collective particle-hole state. The example of Gamow-Teller transitions between²⁰⁸Bi(1+) and²⁰⁸Pb(0+) is considered and the strength of the Gamow-Teller giant resonance as well as the fluctuations in the ensemble are studied. When a fractionp of matrix elements are of opposite sign, we find that a giant resonance may still exist providedp is small enough. Its magnitude and position in the spectrum are studied. The method proposed is valid for any particle hole excitation. The two body matrix elements of several interactions are analysed and are shown to fit into a unique valuep=0.2 for the 1 + states.     

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

A complex symplectic model for the damped oscillator

The linearly damped harmonic oscillator is described by a hamiltonian complex manifold. The master equation for the complex eigenstates of the quantized hamiltonian is found.     

Statistics of real eigenvalues in Ginibre's ensemble of random real matrices

The integrable structure of Ginibre's orthogonal ensemble of random matrices is looked at through the prism of the probability p(n,k) to find exactly k real eigenvalues in the spectrum of an n x n real asymmetric Gaussian random matrix. The exact solution for the probability function p(n,k) is presented, and its remarkable connection to the theory of symmetric functions is revealed. An extension of the Dyson integration theorem is a key ingredient of the theory presented.     

Random matrix theory of singular values of rectangular complex matrices I: Exact formula of one-body distribution function in fixed-trace ensemble

The fixed-trace ensemble of random complex matrices is the fundamental model that excellently describes the entanglement in the quantum states realized in a coupled system by its strongly chaotic dynamical evolution [see H. Kubotani, S. Adachi, M. Toda, Phys. Rev. Lett. 100 (2008) 240501]. The fixed-trace ensemble fully takes into account the conservation of probability for quantum states. The present paper derives for the first time the exact analytical formula of the one-body distribution function of singular values of random complex matrices in the fixed-trace ensemble. The distribution function of singular values (i.e. Schmidt eigenvalues) of a quantum state is so important since it describes characteristics of the entanglement in the state. The derivation of the exact analytical formula utilizes two recent achievements in mathematics, which appeared in 1990s. The first is the Kaneko theory that extends the famous Selberg integral by inserting a hypergeometric type weight factor into the integrand to obtain an analytical formula for the extended integral. The second is the Petkovšek–Wilf–Zeilberger theory that calculates definite hypergeometric sums in a closed form.     

An optimum Hamiltonian for non-Hermitian quantum evolution and the complex Bloch sphere

For a quantum system governed by a non-Hermitian Hamiltonian, we studied the problem of obtaining an optimum Hamiltonian that generates nonunitary transformations of a given initial state into a certain final state in the smallest time τ. The analysis is based on the relationship between the states of the two-dimensional subspace of the Hilbert space spanned by the initial and final states and the points of the two-dimensional complex Bloch sphere.     

Reduced density matrices of a system of N coupled oscillators: 4. The eigenstructure of the p-particle matrices for the 2-temperature canonical ensemble

It is proved that the reduced p-particle matrix (p?N) corresponding to a 2-temperature equilibrium state of a system of N coupled oscillators (the two temperatures T_tr and T_re1 relating to the translational and relative motion of the latter) coincides with the density matrix of a two-temperature canonical ensemble of an effective system of p coupled oscillators, the effective temperature T?_re1 being equal to T_re1 while $\text{T}\text{?}{}_{\mathrm{tr}}\text{≠T}{}_{\mathrm{tr}}$, and in particular $\text{T}\text{?}{}_{\mathrm{tr}}\text{>0}$ for T_tr=0.     

Scaling laws of complex band random matrices

We study a model of complex band random matrices capable of describing the transitions between three different ensembles of Hermitian matrices: Gaussian orthogonal, Gaussian unitary and Poissonian. Analyzing numerical data we observe new scaling relations based on the generalized localization length of eigenvectors. We show that during transitions between canonical ensembles of random matrices the changes of statistical properties of eigenvalues and eigenvectors are correlated.     

Scaling laws of complex band random matrices

We study a model of complex band random matrices capable of describing the transitions between three different ensembles of Hermitian matrices: Gaussian orthogonal, Gaussian unitary and Poissonian. Analyzing numerical data we observe new scaling relations based on the generalized localization length of eigenvectors. We show that during transitions between canonical ensembles of random matrices the changes of statistical properties of eigenvalues and eigenvectors are correlated.     

非厄米哈密顿量的物理意义

分析了一个脉冲激光与原子相互作用的四能级系统，并考虑最上层能级的自电离过程，从而引入非厄米哈密顿量.在缀饰原子模型下，通过直接求解此哈密顿量的本征值与本征函数，得到系统布居的演化函数.与数值方法所得演化函数的对比表明二者相当符合，从而肯定了非厄米哈密顿量在量子力学框架中的地位，并得到其本征值虚部的物理意义.这将使传统量子力学中力学量的定义得以拓展. 关键词： 非厄米哈密顿量 缀饰原子模型