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arXiv:cond-mat/9910452 · Journal of Physics A Mathematical and General

On matrix product states for periodic boundary conditions

Klaus Krebs

Published Mar 31, 2000• 9 citations
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Abstract

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.

DOI
10.1088/0305-4470/33/14/102

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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.
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. The quadratic algebra used for this representation is generated by 2m operators which fulfil m 2 quadratic relations and is endowed with a trace. It is shown that not every eigenstate with energy and momentum zero can be written as matrix product state. 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.
Recently, it has been shown that the zero-energy eigenstate – corresponding to the stationary state – of a stochastic Hamiltonian with nearest-neighbour interaction in the bulk and single-site boundary terms, can generically be written in the form of a so-called matrix-product state. We generalize this result to stochastic Hamiltonians with arbitrary, but finite, interaction range. As an application two different particle-hopping models with three-site bulk interaction are studied. For these models which can be interpreted as cellular automata for traffic flow, we present exact solutions for periodic boundary conditions and some suitably chosen boundary interactions.
Using the algebraic Bethe Ansatz, we derive a matrix product representation of the exact Bethe-Ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-1/2) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation, and we further show this by providing an additional example of the same matrix product state construction for a well-known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries, and thus they suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.
Abstract This article concerns the matrix product operators for one-dimensional spin Hamiltonians under open and periodic boundary conditions. A novel approach called ‘the dynamics of blocks’ is proposed to obtain the matrix product operators in the case of Hamiltonians with periodic boundary conditions. This approach works universally for any Hamiltonian with two-particle as well as single-particle interactions. The main significance of the proposed approach is that, the dimensions of the matrix product operators obtained are the same for both open and periodic boundary conditions. This method gives us a generic way of constructing MPOs for any Hamiltonian with arbitrary neighboring interactions, immaterial of how the interaction strength falls. A general numerical implementation of the ‘dynamics of blocks’ method is also carried out. The DMRG algorithm was implemented, and the ground state eigenvalue was obtained using these MPOs to numerically validate the method.
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.