Non self-conjugate strings, singular strings and rigged configurations in the Heisenberg model

Abstract

We observe a different type of complex solutions in the isotropic spin-1/2 Heisenberg chain starting from N = 12, where the central rapidity of some of the odd-length strings becomes complex so that not all the strings self-conjugate individually. We show that there are at most (N − 2)/2 singular solutions for M = 4, M = 5 down-spins and at most (N2 − 6N + 8)/8 singular solutions for M = 6, M = 7 down-spins in an even-length chain with N ⩾ 2M. Correspondence of the non self-conjugate string solutions and the singular string solutions to the rigged configurations has also been shown.