Vladimir Rittenberg

Supersymmetric Quantum Mechanics

Abstract We give a general construction for supersymmetric Hamiltonians in quantum mechanics. We find that N-extended supersymmetry imposes very strong constraints, and for N > 4 the Hamiltonian is integrable. We give a variety of examples, for one-particle and for many-particle systems, in different numbers of dimensions.

q-Deformations of the O(3) symmetric spin-1 Heisenberg chain

The authors present the general expression for the spin-1 Heisenberg chain invariant under the Uq(SO(3)) quantum algebra. Several physical and mathematical implications are discussed.

Reaction-Diffusion Processes, Critical Dynamics, and Quantum Chains

Abstract The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schrodinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and q -deformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions, and finite-size scaling are also discussed in detail.

Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries

We consider the one-dimensional partially asymmetric exclusion model with open boundaries. The model describes a system of hard-core particles that hop stochastically in both directions with different rates. At both boundaries particles are injected and extracted. By means of the method of Derrida et al the stationary probability measure can be expressed as a matrix-product state involving two matrices forming a Fock-like representation of a general quadratic algebra. We obtain the representations of this algebra, which were unknown in the mathematical literature and use the two-dimensional one to derive exact expressions for the density profile and correlation functions. Using the correspondence between the stochastic model and a quantum spin chain, we obtain exact correlation functions for a spin- Heisenberg XXZ chain with non-diagonal boundary terms. Generalizations to other reaction - diffusion models are discussed.

Bethe Ansatz Solution of the Fateev-zamolodchikov Quantum Spin Chain With Boundary Terms

Abstract We solve the Fateev-Zamolodchikov quantum spin chain (i.e., the spin-1 XXZ quantum Heisenberg chain) with a class of boundary terms by the quantum inverse scattering method. For a particular choice of boundary terms, the model has the quantum symmetry U q [SU (2)].

Magic in the spectra of the XXZ quantum chain with boundaries at Δ=0 and Δ=−1/2

Abstract We show that from the spectra of the U q ( sl ( 2 ) ) symmetric XXZ spin- 1 / 2 finite quantum chain at Δ = − 1 / 2 ( q = e π i / 3 ) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for Δ = 0 ( q = e π i / 2 ). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley–Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.

Reaction-diffusion processes as physical realizations of Hecke algebras

Abstract We show that the master equation governing the dynamics of simple diffusion and certain chemical reaction processes in one dimension gives time evolution operators (Hamiltonians) which are realizations of Hecke algebras. In the case of simple diffusion one obtains, after similarity transformations, reducible hermitian representations while in the other cases they are non-hermitian and correspond to supersymmetric quotients of Hecke algebras.

Temperley-Lieb stochastic processes

We discuss one-dimensional stochastic processes defined through the Temperley–Lieb algebra related to the Q = 1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmetry class of alternating sign matrices, objects that have received much attention in combinatorics.

Conformal invariance and finite one-dimensional quantum chains

Based on previous work of Cardy (1984), the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions. The method is illustrated in the case of the three- and four-state Potts models where the anomalous dimensions of the para-fermionic operators are found.

N-species stochastic models with boundaries and quadratic algebras

Stationary probability distributions for stochastic processes on linear chains with closed or open ends are obtained using the matrix product ansatz. The matrices are representations of some quadratic algebras. The algebras and the types of representations considered depend on the boundary conditions. In the language of quantum chains we obtain the ground state of N-state quantum chains with free boundary conditions or with non-diagonal boundary terms at one or both ends. In contrast to problems involving the Bethe ansatz, we do not have a general framework for arbitrary N, which when specialized, gives the known results for N = 2; in fact, the N = 2 and N>2 cases appear to be very different.