Exact integrability in quantum field theory and statistical systems

Algebraic solution of the Hubbard model on the infinite interval

“…We have to require the amplitude of the incident wave to be unity. In previously studied cases [26,27,49] it turned out that such kind of normalization was obtainable by introducing the operator analog of the reflection coefficient of the corresponding classical problem. In other words, the creation operators were normalized by multiplying with the inverse of certain generators of conserved quantities.…”

“…For this reason the method [26,27,23,28] is so far restricted to In the following we will restrict ourselves to the empty band vacuum |0 which is defined by…”

“…We take the thermodynamic limit first and construct the eigenstates afterwards. This is the original idea of the quantum inverse scattering method [26,27,28], which was designed in close analogy to the inverse scattering theory for the solution of classical integrable systems [29]. The disadvantage of this method is that it is so far restricted to uncorrelated vacua (ground states).…”

Yang-Baxter equation and reflection equations in integrable models

“…where the standard QISM notations [1][2][3][4][5][6][7] are used T 1 = T ⊗ I, T 2 = I ⊗ T to embed matrices in C n into C n ⊗ C n . From the structure of the L-operator one can conclude that the n elements of the row C(λ) act on the vacuum as some creation operators.…”

“…Another valuable characteristic is related to the combination of different mathematical methods required to solve some specific model thus giving rise to mutual interrelations among formally separated fields of mathematics. The start of the recent activity in the quantum integrable models (see reviews [1,2,3,4,5,6,7,8] and Refs. therein) was definitely related with the development of the soliton theory, although the influence of the preceding pioneering contributions is out of discussion (see [10,11,12,13] and Refs.…”