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Register a new userThe Toeplitz matrix $e^{- kappa |i-j|}$ and its application to a layered electron gas
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We present an explicit solution of the eigen-spectrum Toeplitz matrix $C_{ij}= e^{- kappa |i-j|}$ with $0leq i,j leq N$ and apply it to find analytically the plasma modes of a layered assembly of 2-dimensional electron gas. The solution is found by elementary means that bypass the Wiener-Hopf technique usually used for this class of problems. It rests on the observation that the inverse of $C_{ij}$ is effectively a nearest neighbor hopping model with a specific onsite energies which can in turn be diagonalized easily. Extensions to a combination of a Toeplitz and Hankel matrix, and to a generalization of $C_{ij}$, are discussed at the end of the paper.
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We calculated the optical properties of an $N$-layer graphene by formulating the dynamical conductivity of each layer. This is the conductivity when an electromagnetic field is localized at a particular layer and differs from the standard conductivity calculated assuming a uniform field throughout all layers. By combining these conductivities with a transfer matrix method, we took into account the spatial variation of the electromagnetic field caused by internal reflections. The results obtained from the two conductivities show that similar peak structures originating from the interlayer electronic interaction appear in reflectance of an $N$-layer graphene at any $N$. The peak is inherent to the AB stacking and is not seen for the AA stacking, and the peak corresponding to a sufficiently large $N$ is considered to the one observed for natural graphite. We also gave physical explanations of the existing experimental results on highly oriented pyrolytic graphite and natural graphite under high pressure. Although a layered conductivity underestimates the reflectance of graphite at photon energies below the peak, we will show that the disagreement is attributed to a nonlocal conductivity caused by interlayer interaction. The calculations with layered conductivity are useful in knowing the local response to light and may be further validated by an observation of a correction by interlayer electronic interaction to the universal layer number that we have discovered recently.
Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of $|mathbf{x}|^2$ diverges. Intermediate regimes are forbidden. Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane (Phys. Rev. Lett. 61, 2017 (1988)). We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in (Commun. Math. Phys. 359, 61-100 (2018)), and finally we announce further generalizations to non-periodic models.
The purpose of this article is to study the eigenvalues $u_1^{, t}=e^{ittheta_1},dots,u_N^{,t}=e^{ittheta_N}$ of $U^t$ where $U$ is a large $Ntimes N$ random unitary matrix and $t>0$. In particular we are interested in the typical times $t$ for which all the eigenvalues are simultaneously close to $1$ in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first orders of the large $N$ asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widoms formula. Eventually we explain why the first return time is expected to converge towards an exponential distribution when $N$ is large. Numeric simulations are provided along the paper to illustrate the results.
We carry out diagonalization of a $3times3$ Hermitian matrix of which Real component and Imaginary part are commutative and apply it to Majorana neutrino mass matrix $M=M_ u M_ u^dagger$ which satisfies the same condition. It is shown in a model-independent way for the kind of matrix M of which Real component and Imaginary part are commutative that $delta = pmpi/2$ which implies the maximal strength of CP violation in neutrino oscillations. And we obtain the prediction $cos (2theta_{23})=0$ for this kind of M. It is shown that the kind of Hermitian Majorana neutrino mass matrix M has only five real parameters and furthermore, only one free real parameter (D or A) if using the measured values of three mixing angles and mass differences as input.
We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assumption that the Landauer-Butticker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of Random Matrix Theory (RMT). The starting points are the finite-n formulae that we recently discovered (Mezzadri and Simm, J. Math. Phys. 52 (2011), 103511). Our analysis includes all the symmetry classes beta=1,2,4; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer (J. Math. Phys. 37 (1996), 4986-5018) and Altland and Zirnbauer (Phys. Rev. B. 55 (1997), 1142-1161). Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed by Berkolaiko et al. (J. Phys. A.: Math. Theor. 41 (2008), 365102) and Berkolaiko and Kuipers (J. Phys. A: Math. Theor. 43 (2010), 035101 and New J. Phys. 13 (2011), 063020). Our approach also applies to the Selberg-like integrals. We calculate the first two terms in their asymptotic expansion explicitly.
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