The bound state S-matrix of the deformed Hubbard chain

Abstract

In this work we use the q-oscillator formalism to construct the atypical (short) supersymmetric representations of the centrally extended U q (su(2|2)) algebra. We then determine the S-matrix describing the scattering of arbitrary bound states. The crucial ingredient in this derivation is the affine extension of the aforementioned algebra. IntroductionIntegrable systems constitute a special class of models in mathematics and physics. Their properties allow them to be solved exactly and thus they appear to be a very useful playground for studying various systems. One common feature shared by these models is that they are closely related to some underlying algebraic structures. Thus for most of the quantum integrable systems there is some sort of large and powerful symmetry hidden in the origins of it, for example a Yangian or a quantum affine algebra. A particularly interesting example is the Hubbard model.-1 -The Hubbard model, which was named after John Hubbard, is the simplest model of interacting particles on a lattice, with only two terms in the Hamiltonian, the hopping term (kinetic energy) and the Coulomb potential [1]. The model describes an ensemble of particles in a periodic potential at sufficiently low temperatures such that all the particles may be considered to be in the lowest Bloch band and also any long-range interactions between the particles are considered to be weak enough and thus are ignored. It is based on the tight-binding approximation of the superconducting systems and the motion of electrons between the atoms of a crystalline solid. Despite its apparent simplicity, it is very rich applications and generalizations describing including phase shifts and a plethora of interesting phenomena. In the case when interactions between particles on different sites of the lattice can not be neglected and are included, the model is often referred to as the Extended Hubbard model. The particles can either be fermions, as in Hubbard's original work, or bosons, and the model is then referred as either the BoseHubbard model or the boson Hubbard model that can be used to study systems such as bosonic atoms on an optical lattice (for a decent overview of various generalizations see reprint volumes [2][3][4] and also a more recent book [5]).A very specific class of models are those that share features of the one-dimensional Hubbard model and the supersymmetric t-J model [6]. The very interesting case being the Alcaraz and Bariev model [7] having an extra spin-spin interaction term in the Hamiltonian and showing some characteristics of superconductivity. This model can be viewed as a quantum deformation of the Hubbard model in much the same way as the Heisenberg XXZ model is a quantum deformation of the XXX model. This model has a specific R-matrix which can not be written as a function of the difference of two associated spectral parameters. This paradigm is related to the very interesting but at the same time complicated algebraic properties of the model.In recent years there has been renewed in...

S-matrix for strings on η-deformed AdS5 × S5

“…It appears that this centrally extended psu(2|2), or more precisely its universal enveloping algebra, admits a natural deformation psu q (2|2) in the sense of quantum groups [6,7]. This algebraic structure is the starting point for the construction of a psu q (2|2) ⊕ psu q (2|2)-invariant S-matrix, giving a quantum deformation of the AdS 5 × S 5 world-sheet S-matrix [6,8,9]. The deformation parameter q can be an arbitrary complex number, but in physical applications is typically taken to be either real or a root of unity.…”

Integrable boundaries in AdS/CFT: revisiting the Z=0 giant graviton and D7-brane

“…This can be done in a similar way as in [34], where the bound-state S-matrix for the algebra Q was found. However these calculations are rather complicated and thus we will reduce our goal to finding the analytic expressions of the reflection matrices with the total bound-state number M ≤ 3.…”

“…We will start by briefly reclling the construction of the quantum affine coideal subalgebras [28] (see [40] for explicit details on the non-affine coideal subalgebras) and the boundstate representation of the quantum affine algebra Q [33,34]. We will then construct the corresponding boundary algebras using the same approach as for the q-deformed model of the reflection from the Y = 0 giant graviton [39].…”

“…We shall be using the q-oscillator representation (for any complex q not a root of unity) constructed in [34]. The bound-state representation is defined on vectors and describes an excitation with momentum p defined by the relation U 2 = e ip .…”

“…A quantum affine algebra Q leading to a q-deformed S-matrix which in the q → 1 limit specializes to the AdS/CFT worldsheet S-matrix was constructed in [33] and the corresponding q-deformed bound-state S-matrices were found in [34]. However, finding fundamental scattering matrices does not require the full quantum affine algebra, thus the fundamental q-deformed S-matrix was found earlier in [35].…”