Real space finite difference method for conductance calculations

We present a general method for calculating coherent electronic transport in quantum wires and tunnel junctions. It is based upon a real space high order finite difference representation of the single particle Hamiltonian and wave functions. Landauer's formula is used to express the conductance as a scattering problem. Dividing space into a scattering region and left and right ideal electrode regions, this problem is solved by wave function matching (WFM) in the boundary zones connecting these regions. The method is tested on a model tunnel junction and applied to sodium atomic wires. In particular, we show that using a high order finite difference approximation of the kinetic energy operator leads to a high accuracy at moderate computational costs.Comment: 13 pages, 10 figure

Maximizing sum rate and minimizing MSE on multiuser downlink: Optimality, fast algorithms and equivalence via max-min SIR

Maximizing the minimum weighted SIR, minimizing the weighted sum MSE and maximizing the weighted sum rate in a multiuser downlink system are three important performance objectives in joint transceiver and power optimization, where all the users have a total power constraint. We show that, through connections with the nonlinear Perron-Frobenius theory, jointly optimizing power and beamformers in the max-min weighted SIR problem can be solved optimally in a distributed fashion. Then, connecting these three performance objectives through the arithmetic-geometric mean inequality and nonnegative matrix theory, we solve the weighted sum MSE minimization and weighted sum rate maximization in the low to moderate interference regimes using fast algorithms

Tightness of the maximum likelihood semidefinite relaxation for angular synchronization

Maximum likelihood estimation problems are, in general, intractable optimization problems. As a result, it is common to approximate the maximum likelihood estimator (MLE) using convex relaxations. In some cases, the relaxation is tight: it recovers the true MLE. Most tightness proofs only apply to situations where the MLE exactly recovers a planted solution (known to the analyst). It is then sufficient to establish that the optimality conditions hold at the planted signal. In this paper, we study an estimation problem (angular synchronization) for which the MLE is not a simple function of the planted solution, yet for which the convex relaxation is tight. To establish tightness in this context, the proof is less direct because the point at which to verify optimality conditions is not known explicitly. Angular synchronization consists in estimating a collection of $n$ phases, given noisy measurements of the pairwise relative phases. The MLE for angular synchronization is the solution of a (hard) non-bipartite Grothendieck problem over the complex numbers. We consider a stochastic model for the data: a planted signal (that is, a ground truth set of phases) is corrupted with non-adversarial random noise. Even though the MLE does not coincide with the planted signal, we show that the classical semidefinite relaxation for it is tight, with high probability. This holds even for high levels of noise.Comment: 2 figure

Planckian Axions in String Theory

We argue that super-Planckian diameters of axion fundamental domains can naturally arise in Calabi-Yau compactifications of string theory. In a theory with $N$ axions $\theta^i$, the fundamental domain is a polytope defined by the periodicities of the axions, via constraints of the form $-\pi<Q^{i}_{j} \theta^j<\pi$. We compute the diameter of the fundamental domain in terms of the eigenvalues $f_1^2\le\...\le f_N^2$ of the metric on field space, and also, crucially, the largest eigenvalue of $(QQ^{\top})^{-1}$. At large $N$, $QQ^{\top}$ approaches a Wishart matrix, due to universality, and we show that the diameter is at least $N f_{N}$, exceeding the naive Pythagorean range by a factor $>\sqrt{N}$. This result is robust in the presence of $P>N$ constraints, while for $P=N$ the diameter is further enhanced by eigenvector delocalization to $N^{3/2}f_N$. We directly verify our results in explicit Calabi-Yau compactifications of type IIB string theory. In the classic example with $h^{1,1}=51$ where parametrically controlled moduli stabilization was demonstrated by Denef et al. in [1], the largest metric eigenvalue obeys $f_N \approx 0.013 M_{pl}$. The random matrix analysis then predicts, and we exhibit, axion diameters $>M_{pl}$ for the precise vacuum parameters found in [1]. Our results provide a framework for achieving large-field axion inflation in well-understood flux vacua.Comment: 42 pages, 4 figure

Universal Cubic Eigenvalue Repulsion for Random Normal Matrices

Random matrix models consisting of normal matrices, defined by the sole constraint $[N^{\dag},N]=0$, will be explored. It is shown that cubic eigenvalue repulsion in the complex plane is universal with respect to the probability distribution of matrices. The density of eigenvalues, all correlation functions, and level spacing statistics are calculated. Normal matrix models offer more probability distributions amenable to analytical analysis than complex matrix models where only a model wth a Gaussian distribution are solvable. The statistics of numerically generated eigenvalues from gaussian distributed normal matrices are compared to the analytical results obtained and agreement is seen.Comment: 15 pages, 2 eps figures. to appar in Physical Review

Entanglement Perturbation Theory for Antiferromagnetic Heisenberg Spin Chains

A recently developed numerical method, entanglement perturbation theory (EPT), is used to study the antiferromagnetic Heisenberg spin chains with z-axis anisotropy $\lambda$ and magnetic field B. To demonstrate the accuracy, we first apply EPT to the isotropic spin-1/2 antiferromagnetic Heisenberg model, and find that EPT successfully reproduces the exact Bethe Ansatz results for the ground state energy, the local magnetization, and the spin correlation functions (Bethe ansatz result is available for the first 7 lattice separations). In particular, EPT confirms for the first time the asymptotic behavior of the spin correlation functions predicted by the conformal field theory, which realizes only for lattice separations larger than 1000. Next, turning on the z-axis anisotropy and the magnetic field, the 2-spin and 4-spin correlation functions are calculated, and the results are compared with those obtained by Bosonization and density matrix renormalization group methods. Finally, for the spin-1 antiferromagnetic Heisenberg model, the ground state phase diagram in $\lambda$ space is determined with help of the Roomany-Wyld RG finite-size-scaling. The results are in good agreement with those obtained by the level-spectroscopy method.Comment: 12 pages, 14 figure