Entanglement and correlation functions of the quantum Motzkin spin-chain

We present exact results on the exactly solvable spin chain of Bravyi et al [Phys. Rev. Lett. 109, 207202 (2012)]. This model is a spin one chain and has a Hamiltonian that is local and translationally invariant in the bulk. It has a unique (frustration free) ground state with an energy gap that is polynomially small in the system's size ($2n$). The half-chain entanglement entropy of the ground state is $\frac{1}{2}\log n+const.$. Here we first write the Hamiltonian in the standard spin-basis representation. We prove that at zero temperature, the magnetization is along the $z-$direction i.e., $\langle s^{x}\rangle=\langle s^{y}\rangle=0$ (everywhere on the chain). We then analytically calculate $\langle s^{z}\rangle$ and the two-point correlation functions of $s^{z}$. By analytically diagonalizing the reduced density matrices, we calculate the Schmidt rank, von Neumann and R\'enyi entanglement entropies for: 1. Any partition of the chain into two pieces (not necessarily in the middle) and 2. $L$ consecutive spins centered in the middle. Further, we identify entanglement Hamiltonians (Eqs. 49 and 59). We prove a small lemma (Lemma1) on the combinatorics of lattice paths using the reflection principle to relate and calculate the Motzkin walk 'height' to spin expected values. We also calculate the, closely related, (scaled) correlation functions of Brownian excursions. The known features of this model are summarized in a table in Sec.I.Comment: 25 pages, 5 figure

Eigenvalue Attraction

We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix $M$. If $M$ is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when $M$ is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.Comment: v1:15 pages, 12 figures, 1 Matlab code. v2: very minor changes, fixed a reference. v3: 25 pages, 17 figures and one Matlab code. The results have been extended and generalized in various ways v4: 26 pages, 10 figures and a Matlab Code. Journal Reference Added. http://link.springer.com/article/10.1007%2Fs10955-015-1424-

The q-analogue of the wild fundamental group and the inverse problem of the Galois theory of q-difference equations

In previous papers, we defined $q$-analogues of alien derivations for linear analytic $q$-difference equations with integral slopes and proved a density theorem (in the Galois group) and a freeness theorem. In this paper, we completely describe the wild fundamental group and apply this result to the inverse problem in $q$-difference Galois theory. The new version contains an appendix on pronilpotent completion and the main result on the direct problem is made more precise. (Submitted for publication

The Last Degrees Conferred by the University of Solsona (1701-1715)

En este artículo se estudian los últimos grados conferidos por la Universidad de Solsona (1701-1715). Con ello se pretende conocer mejor el perfil de los graduados por esta universidad irregular, que colacionó títulos de todas las Facultades a estudiantes catalanes y baleares, y que fue denunciada por la Universidad de Barcelona por práctica fraudulenta.This article studies the last degrees conferred by the University of Solsona (1701-1715). Our aim is to draw a picture of the type of student who graduated from this unorthodox university which conferred degrees from all faculties on students from Catalonia and the Balearic Islands, and which was denounced by the University of Barcelona for fraudulent practice