21aAK-5 Singular eigenstates in the even(odd) length Heisenberg spin chain

A. Benyoussef, Abdelrhani Boubekri, Hamid Ez-zahraouy A

Pulak Ranjan Giri, Tetsuo Deguchi

We study the implications of the regularization for the singular solutions on the even(odd) length spin- 1 / 2 XXX chains in some specific down-spin sectors. In particular, the analytic expressions of the Bethe eigenstates for three down-spin sector have been obtained along with their numerical forms in some fixed length chains. For an even-length chain if the singular solutions { λ α } are invariant under the sign changes of their rapidities { λ α } = { − λ α } , then the Bethe ansatz equations are reduced to a system of ( M − 2 ) / 2 ( ( M − 3 ) / 2 ) equations in an even (odd) down-spin sector. For an odd N length chain in the three down-spin sector, it has been analytically shown that there exist singular solutions in any finite length of the spin chain of the form N = 3 ( 2 k + 1 ) with k = 1 , 2 , 3 , ⋯ . It is also shown that there exist no singular solutions in the four down-spin sector for some odd-length spin- 1 / 2 XXX chains.

Adriana Moreo

Yoshihiro Nishiyama, Keisuke Totsuka, Naomichi Hatano +1 more

Tetsuo Deguchi, Pulak Ranjan Giri

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 ( N 2 − 6 N + 8)/8 singular solutions for M = 6, M = 7 down-spins in an even-length chain with N ⩾ 2 M . Correspondence of the non self-conjugate string solutions and the singular string solutions to the rigged configurations has also been shown.

Jae Dong Noh, Deok‐Sun Lee, Doochul Kim +1 more

We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω=±1 with Ω the eigenvalue of T. We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω, ϒ)=(±1, ∓1) where ϒ is the eigenvalue of V. Therefore, the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω, ϒ)=(±1, ∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.