A representation basis for the quantum integrable spin chain associated with the su(3) algebra

Abstract

An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N = 2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.

New construction of eigenstates and separation of variables for SU(N) quantum spin chains

“…While the SoV program has been thoroughly studied for SU (2)-type models and to a lesser degree for the SU (3) case (see e.g. [22,50]), its extension to the case of higher rank groups presents a challenge. One of the key required ingredients is the operator B(u) providing the separated variables.…”

“…21 See the beginning of section 4.3 for a brief summary of our numerical strategy. 22 Note that for L = 4 we already have to deal with rather large matrices having 256 × 256 = 65536 elements. 23 Accordingly, it also does not generate the two-magnon eigenstates of the transfer matrix already for L = 2.…”

“…The most transparent method to obtain the states available in the literature is the nested algebraic Bethe ansatz approach, in which the eigenvectors are built recursively based on the solution of a lower rank spin chain [13][14][15]. Other known constructions include an explicit representation via sums over partitions of Bethe roots [16], as well as the rather sophisticated trace formulas of [17] and the Drinfeld current construction [18,19] (see also [20][21][22][23]). These methods have been explored in-depth and have facilitated numerous calculations of various observables such as form factors and scalar products [19,[24][25][26][27][28][29][30][31][32][33].…”

A convenient basis for the Izergin–Korepin model

“…We have introduced a convenient basis (3.11) and (3.12) of the Hilbert space for the IK model with the periodic boundary condition, which is the quantum spin chain associated with the A It is well-known [17] that taking the rational limit (i.e.,η → 0) the IK model becomes the su(3)-invariant spin chain. It is easy to show that in this limit the resulting basis of (3.11) and (3.12) is exactly the rational version of the basis given recently in [26] which coincides with the F-basis [7] of the su(3)-invariant closed chain.…”

“…For the su(2) case, the corresponding states are the SoV states for the XXZ spin chain, and was shown in [25] that it coincides with the so-called F-basis [3]. For the su(n) case, the corresponding states are the nested generalization of the SoV states [26] for the trigonometric su(n) spin chain and coincide with the associated F-basis [7,8,9,10,11].…”