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arXiv:hep-th/9604079 · Journal of Physics A Mathematical and General

The structure of conserved charges in open spin chains

Marek Grabowski, Pierre Mathieu

Published Dec 7, 1996• 21 citations
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Abstract

We study the local conserved charges in integrable spin chains of the XYZ type with non-trivial boundary conditions. The general structure of these charges consists of a bulk part, whose density is identical to that of a periodic chain, and a boundary part. In contrast with the periodic case, only charges corresponding to interactions of even number of spins exist for the open chain. Hence, there are half as many charges in the open case as in the closed case. For the open spin- XY chain, we derive the explicit expressions of all the charges. For the open spin- XXX chain, several lowest-order charges are presented and a general method of obtaining the boundary terms is indicated. In contrast with the closed case, the XXX charges cannot be described in terms of a Catalan tree pattern.

DOI
10.1088/0305-4470/29/23/024

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We study the local conserved charges in integrable spin chains of the XYZ type with non-trivial boundary conditions. The general structure of these charges consists of a bulk part, whose density is identical to that of a periodic chain, and a boundary part. In contrast with the periodic case, only charges corresponding to interactions of even number of spins exist for the open chain. Hence, there are half as many charges in the open case as in the closed case. For the open spin- XY chain, we derive the explicit expressions of all the charges. For the open spin- XXX chain, several lowest-order charges are presented and a general method of obtaining the boundary terms is indicated. In contrast with the closed case, the XXX charges cannot be described in terms of a Catalan tree pattern.
Abstract We revisit the conserved quantities of the spin- 1 2 XY model with open boundary conditions. In the absence of a transverse field, we find new families of local charges and show that half of the seeming conservation laws are conserved only if the number of sites is odd. In even chains the set of noninteracting charges is abelian, like in the periodic case when the number of sites is odd. In odd chains the set is doubled and becomes non-abelian, like in even periodic chains. The dependence of the charges on the parity of the chain’s size undermines the common belief that the thermodynamic limit of diagonal ensembles exists. We consider also the transverse-field Ising chain, where the situation is more ordinary. The generalization to the XY model in a transverse field is not straightforward and we propose a general framework to carry out similar calculations. We conjecture the form of the bulk part of the local charges and discuss the emergence of quasilocal conserved quantities. We provide evidence that in a region of the parameter space there is a reduction of the number of quasilocal conservation laws invariant under chain inversion. As a by-product, we study a class of block-Toeplitz-plus-Hankel operators and identify the conditions that their symbols satisfy in order to commute with a given block-Toeplitz.
Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstrate that the Fredkin spin chain, under both periodic and open boundary conditions, lacks local conserved charges, thereby confirming its nonintegrable nature. Furthermore, we find that when one or a portion of the Hamiltonian terms are removed (referred to as the truncated Fredkin spin chain), local conserved charges are still absent. Our findings suggest that in models involving three-site interactions, integrable models are generally rare.
A class of quantum chains possessing a family of local conserved charges with a Catalan tree pattern is studied. Recently, we have identified such a structure in the integrable SU(N)-invariant chains. In the present work we find sufficient conditions for the existence of a family of charges with this structure in terms of the underlying algebra. Two additional systems with a Catalan tree structure of conserved charges are found. One is the spin 1/2 XXZ model with Δ=−1. The other is a new octonionic isotropic chain, generalizing the Heisenberg model. This system provides an interesting example of an infinite family of noncommuting local conserved quantities.
It is known that integrable models associated to rational R matrices give rise to certain non-abelian symmetries known as Yangians. Analogously ‘boundary ’ symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay and Short from the field theory point of view, in the context of the principal chiral model on the half line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton non-preserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary non-local charges directly at the quantum level. The non-local charges are simply coproduct realizations of particular boundary quantum algebras called ‘boundary ’ or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the symmetry of the open spin chain, namely we show that a number of the boundary non-local charges are in fact conserved quantities.
It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.
The N site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine–Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non-local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary one of the non-local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian.