Enlarged symmetry algebras of spin chains, loop models, and S-matrices

Abstract

The symmetry algebras of certain families of quantum spin chains are considered in detail. The simplest examples possess m states per site ( m ⩾ 2 ), with nearest-neighbor interactions with U ( m ) symmetry, under which the sites transform alternately along the chain in the fundamental m and its conjugate representation m ¯ . We find that these spin chains, even with arbitrary coefficients of these interactions, have a symmetry algebra A m much larger than U ( m ) , which implies that the energy eigenstates fall into sectors that for open chains (i.e., free boundary conditions) can be labeled by j = 0 , 1 , … , L , for the 2 L-site chain such that the degeneracies of all eigenvalues in the jth sector are generically the same and increase rapidly with j. For large j, these degeneracies are much larger than those that would be expected from the U ( m ) symmetry alone. The enlarged symmetry algebra A m ( 2 L ) consists of operators that commute in this space of states with the Temperley–Lieb algebra that is generated by the set of nearest-neighbor interaction terms; A m ( 2 L ) is not a Yangian. There are similar results for supersymmetric chains with gl ( m + n | n ) symmetry of nearest-neighbor interactions, and a richer representation structure for closed chains (i.e., periodic boundary conditions). The symmetries also apply to the loop models that can be obtained from the spin chains in a spacetime or transfer matrix picture. In the loop language, the symmetries arise because the loops cannot cross. We further define tensor products of representations (for the open chains) by joining chains end to end. The fusion rules for decomposing the tensor product of representations labeled j 1 and j 2 take the same form as the Clebsch–Gordan series for SU(2). This and other structures turn the symmetry algebra A m into a ribbon Hopf algebra, and we show that this is “Morita equivalent” to the quantum group U q ( sl 2 ) for m = q + q −1 . The open-chain results are extended to the cases | m | < 2 for which the algebras are no longer semisimple; these possess continuum limits that are critical (conformal) field theories, or massive perturbations thereof. Such models, for open and closed boundary conditions, arise in connection with disordered fermions, percolation, and polymers (self-avoiding walks), and certain non-linear sigma models, all in two dimensions. A product operation is defined in a related way for the Temperley–Lieb representations also, and the fusion rules for this are related to those for A m or U q ( sl 2 ) representations; this is useful for the continuum limits also, as we discuss in a companion paper.