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arXiv:hep-th/0507209 · Journal of Statistical Mechanics Theory and Experiment

Spectral equivalences and symmetry breaking in integrableSUq(N) spin chains with boundaries

Alexander Nichols

Published Sep 26, 2005• 3 citations
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Abstract

We consider the SU q ( N ) invariant spin chain with diagonal and non-diagonal integrable boundary terms. The algebraic study of spin chains with different kinds of boundary terms is used to motivate a set of spectral equivalences between integrable chains with purely diagonal boundary terms and ones with an arbitrary non-diagonal term at one end. For each choice of diagonal boundary terms there is an iso-spectral one-boundary problem and vice versa. The cases N = 2, 3, 4 are used as examples throughout to illustrate the general structure. The quantum group SU q ( N ) symmetry is broken by the presence of a non-diagonal boundary term; however, one can use the spectral equivalence with the diagonal chain to easily understand the residual symmetries of the system. This is described in detail for the case of SU (3).

DOI
10.1088/1742-5468/2005/09/p09009

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We consider the SU q ( N ) invariant spin chain with diagonal and non-diagonal integrable boundary terms. The algebraic study of spin chains with different kinds of boundary terms is used to motivate a set of spectral equivalences between integrable chains with purely diagonal boundary terms and ones with an arbitrary non-diagonal term at one end. For each choice of diagonal boundary terms there is an iso-spectral one-boundary problem and vice versa. The cases N = 2, 3, 4 are used as examples throughout to illustrate the general structure. The quantum group SU q ( N ) symmetry is broken by the presence of a non-diagonal boundary term; however, one can use the spectral equivalence with the diagonal chain to easily understand the residual symmetries of the system. This is described in detail for the case of SU (3).
We construct an open quantum spin chain from the “twisted” [Formula: see text]R matrix in the fundamental representation which has the quantum algebra symmetry Uq[ su (2)]. This anisotropic spin-1 chain is different from the Uq[ su (2)]-invariant chain constructed from the “untwisted” [Formula: see text] spin-1 R matrix (namely, the spin-1 XXZ chain of Fateev-Zamolodchikov with boundary terms) but, nevertheless, is also completely integrable. We discuss the general case of an R matrix of the type g(k), where k∈{1, 2, 3}, and g is any simple Lie algebra.
The XXX spin- \frac{1}{2} 1 2 Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of U(1) U ( 1 ) symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model. For U(1) U ( 1 ) symmetric spin- \frac{1}{2} 1 2 chains such NLIEs involve two functions a(x) a ( x ) and \bar{a}(x) a ‾ ( x ) coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size N N . In the case considered here the U(1) U ( 1 ) symmetry is broken by the non-diagonal boundary fields and the equations involve a novel third function c(x) c ( x ) , which captures the inhomogeneous contributions to the T T - Q Q relation in the ODBA. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In \log(1+a(x)) log ( 1 + a ( x ) ) and \log(1+\bar a(x)) log ( 1 + a ‾ ( x ) )
Spin-sss QQQ-systems: Twist and open boundaries
In integrable spin chains, the spectral problem can be solved by the method of Bethe Ansatz, which transforms the problem of diagonalization of the Hamiltonian into the problem of solving a set of algebraic equations named Bethe equations. In this work, we systematically investigate the spin- s s XXX chain with twisted and open boundary conditions using the rational Q Q -system, which is a powerful tool to solve Bethe equations. We establish basic frameworks of the rational Q Q -system and confirm its completeness numerically in both cases. For twisted boundaries, we investigate the polynomiality conditions of the rational Q Q -system and derive physical conditions for singular solutions of Bethe equations. For open boundaries, we uncover novel phenomena such as hidden symmetries and boundary induced strings under specific boundary parameters. Hidden symmetries lead to the appearance of extra degeneracies in the Hilbert space, while the boundary induced string is a novel type of exact string configuration, whose length depends on the boundary magnetic fields. These findings, supported by both analytical and numerical evidences, offer new insights into the interplay between symmetries and boundary conditions.
The off-diagonal Bethe ansatz method is generalized to the high spin integrable systems associated with the su(2) algebra by employing the spin-s isotropic Heisenberg chain model with generic integrable boundaries as an example. With the fusion techniques, certain closed operator identities for constructing the functional T − Q relations and the Bethe ansatz equations are derived. It is found that a variety of inhomogeneous T − Q relations obeying the operator product identities can be constructed. Numerical results for two-site s = 1 case indicate that an arbitrary choice of the derived T − Q relations is enough to give the complete spectrum of the transfer matrix.