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Non-perturbative Quantum Effects

Valence bound ground states in quantum antiferromagnets and quadratic algebras

V. Rittenberg

Published Sep 25, 2000• 0 citations
Publisher

Abstract

The wave functions corresponding to the zero-energy eigenvalue of a one-dimensional quantum chain Hamiltonian can be written in a simple way using quadratic algebras.We describe here two applications.The non-hermitian Hamiltonians related to one-dimensional stochastic processes have always zero-energy eigenvalues corresponding to the stationary states.A class of quadratic algebras corresponding to diffusion processes was classified.As a second application, for a class of q-deformed O(N ) symmetric antiferromagnetic quantum chains, we give the zero-energy wavefunctions for periodic boundary conditions corresponding to momentum zero and for zone boundary states.We also consider free and various non-diagonal boundary conditions.All correlation lengths are derived.

DOI
10.22323/1.006.0048

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The wave functions corresponding to the zero-energy eigenvalue of a one-dimensional quantum chain Hamiltonian can be written in a simple way using quadratic algebras.We describe here two applications.The non-hermitian Hamiltonians related to one-dimensional stochastic processes have always zero-energy eigenvalues corresponding to the stationary states.A class of quadratic algebras corresponding to diffusion processes was classified.As a second application, for a class of q-deformed O(N ) symmetric antiferromagnetic quantum chains, we give the zero-energy wavefunctions for periodic boundary conditions corresponding to momentum zero and for zone boundary states.We also consider free and various non-diagonal boundary conditions.All correlation lengths are derived.
The wavefunctions corresponding to the zero-energy eigenvalue of a one-dimensional quantum chain Hamiltonian can be written in a simple way using quadratic algebras. Hamiltonians describing stochastic processes have stationary states given by such wavefunctions and various quadratic algebras have been found and applied to several diffusion processes. We show that similar methods can also be applied for equilibrium processes. As an example, for a class of q -deformed O( N ) symmetric antiferromagnetic quantum chains, we give the zero-energy wavefunctions for periodic boundary conditions corresponding to momenta zero and π. We also consider free and various non-diagonal boundary conditions and give the corresponding wavefunctions. All correlation lengths are derived.
The wave functions corresponding to the zero energy eigenvalue of a onedimensional quantum chain Hamiltonian can be written in a simple way using quadratic algebras. Hamiltonians describing stochastic processes have stationary states given by such wave functions and various quadratic algebras were found and applied to several diffusions processes. We show that similar methods can also be applied for equilibrium processes. As an example, for a class of q-deformed O(N) symmetric antiferromagnetic quantum chains, we give the zero energy wave functions for periodic boundary conditions corresponding to momenta zero and π. We also consider free and various non-diagonal boundary conditions and give the corresponding wave functions. All correlation lengths are derived.
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.
The possibility of a matrix product representation for eigenstates with energy and momentum zero of a general m -state quantum spin Hamiltonian with nearest-neighbour interaction and periodic boundary condition is considered. For this representation we use the quadratic algebra defined in Krebs and Sandow ( 1997 J. Phys. A: Math. Gen. 30 3165 ), which is generated by 2 m operators which fulfil m 2 quadratic relations, endowed with a trace-like function reflecting periodic boundary conditions. It is shown that not every eigenstate with energy and momentum zero can be written in this way. An explicit counter-example is given. This is in contrast to the case of open boundary conditions where every zero-energy eigenstate can be written as a matrix product state using a Fock-like representation of the same quadratic algebra.
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.