An hbar-expansion of the Toda hierarchy: a recursive construction of solutions

A construction of general solutions of the \hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and \bar W. The dressing operators are set in an exponential form as W = e^{X/\hbar} and \bar W = e^{\phi/\hbar}e^{\bar X/\hbar}, and the auxiliary operators X,\bar X and the function \phi are assumed to have \hbar-expansions X = X_0 + \hbar X_1 + ..., \bar X = \bar X_0 + \hbar\bar X_1 + ... and \phi = \phi_0 + \hbar\phi_1 + .... The coefficients of these expansions turn out to satisfy a set of recursion relations. X,\bar X and \phi are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form \Psi = e^{S/\hbar} and \bar\Psi = e^{\bar S/\hbar}, which leads to an \hbar-expansion of the logarithm of the tau function.Comment: 37 pages, no figures. arXiv admin note: substantial text overlap with arXiv:0912.486

DMRG and periodic boundary conditions: a quantum information perspective

We introduce a picture to analyze the density matrix renormalization group (DMRG) numerical method from a quantum information perspective. This leads us to introduce some modifications for problems with periodic boundary conditions in which the results are dramatically improved. The picture also explains some features of the method in terms of entanglement and teleportation.Comment: 4 page

Critical Point of a Symmetric Vertex Model

We study a symmetric vertex model, that allows 10 vertex configurations, by use of the corner transfer matrix renormalization group (CTMRG), a variant of DMRG. The model has a critical point that belongs to the Ising universality class.Comment: 2 pages, 6 figures, short not

qq-analogue of modified KP hierarchy and its quasi-classical limit

A $q$-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear $q$-difference equations. These bilinear equations are translated to the language of wave functions, which turn out to satisfy a system of linear $q$-difference equations. These linear $q$-difference equations are used to formulate the Lax formalism and the description of quasi-classical limit. These results can be generalized to a $q$-analogue of the Toda hierarchy. The results on the $q$-analogue of the Toda hierarchy might have an application to the random partition calculus in gauge theories and topological strings.Comment: latex2e, a4 paper 15 pages, no figure; (v2) a few references are adde

Toda Lattice Hierarchy and Generalized String Equations

String equations of the $p$-th generalized Kontsevich model and the compactified $c = 1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c \le 1$ string models is discussed in this new framework.Comment: 35pages, LaTeX Errors are corrected in Eqs. (2.21), (2.36), (2.33), (3.3), (5.10), (6.1), sentences after (3.19) and theorem 5. A few references are update