Angular quantization and the density matrix renormalization group

Path integral techniques for the density matrix of a one-dimensional statistical system near a boundary previously employed in black-hole physics are applied to providing a new interpretation of the density matrix renormalization group: its efficacy is due to the concentration of quantum states near the boundary. There has recently been a revival of interest in the study of statistical properties of quantum field theories in the presence of a boundary. This subject has a long history: it goes back (at least) to the idea of the entropy of black holes. This entropy, associated to Hawking radiation, was studied with quantum field theory in the corresponding curved space-time [1]. It has been proposed that black-hole entropy arises solely from the presence of a horizon and the consequent ignorance of its interior, being an entanglement entropy and having little to do with the curvature of the space-time [2]. Actually, this is indicated by the classical calculation of the density matrix in Rindler space [1], which will be important in the following. An interesting connection arose from the study of integrable one-dimensional lattice models, where R. Baxter introduced the corner transfer matrix. In the continuum limit of these models, it is natural to use rapidity variables, in which the integrability conditions adopt a simpler form and, moreover, their symmetry with respect to phase shifts can be understood as a consequence of Lorentz invariance [3]. Substantiating this connection, the type of quantization used in the Euclidean version of Rindler space, namely, angular quantization, has been invoked as a suitable computational formalism in 1+1 integrable quantum field theory [4]. The corner transfer matrix method is only appropriate for integrable models, but T. Nishino and collaborators realized that it is related to an approximate method to solve quantum systems, namely, the density matrix renormalization group (DMRG) [5]. Also at Instituto de Matemáticas y F́ısica Fundamental, CSIC, Serrano 123, 28006 Madrid, Spain Associated to NASA Astrobiology Institute