In this article we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang-Baxter and boundary Yang-Baxter equations depending on the choice of boundary conditions. The relation between the aforementioned equations and the braid group is briefly discussed. A short review on quantum groups as well as the quantum inverse scattering method (algebraic Bethe ansatz) is also presented.
The main purpose of this article is to offer a review on the basic ideas of quantum integrability as well as familiarize the reader, who has not necessarily a background on the subject, with the fundamental concepts.
Quantum integrability in 1+1 dimensions has been a very rich research subject, especially after the seminal works of the St. Petersburg group (see e.g. [1]- [5]) on the quantum inverse scattering method (QISM). We refer the interested reader to a number of lecture notes and review articles on algebraic Bethe ansatz, special topics on integrable models, or articles with emphasis on statistical and thermodynamic properties or applications to condensed matter physics (see e.g [6]- [11]). In these notes we are basically focusing on the algebraic content of quantum integrable systems giving particular emphasis on the quantum algebras and their connections to braid groups and Hecke algebras. We also review the quantum inverse scattering method and briefly discuss lattice integrable models with open boundary conditions.
The outline of the article is as follows: in the next section we introduce the basic notation on tensor products of matrices and vectors and we briefly review the su 2 algebra as well as its representations. We then introduce the Heisenberg model [12], describing first neighbors spin-spin interaction. In section 3 we present in more detail the XXX (isotropic) and XXZ (anisotropic Heisenberg) models. In particular, we give a first flavor on the corresponding spectra and eigenstates for small a number of sites. We also discuss the zero temperature phase diagram. The next section is basically devoted to the Yang-Baxter [13] equation and its solution, the so called R matrix. This is the fundamental equation within the QISM context. We introduce the equation and also provide systematic means for solving it via its structural similarity with the braid group. The braid group and certain quotients, such as the Hecke and Temperley-Lieb algebras [14,15,16], are also discussed.
In section 5 we introduce the quantum Lax operator, and the fundamental algebraic equation governing the underlying quantum algebras (Yangians and q deformed Lie algebras) [17,18]. We then construct tensorial representations of the underlying algebras, and eventually build the closed (periodic) transfer matrix of a spin chain-like system. We show the integrability of the system, and also extract the corresponding local Hamiltonian. In the next section we discuss in more detail the non-trivial co products arising in quantum algebras and we show how one can exploit them in order to investigate the symmetry of the associated R matrix. In section 7 we present representations of the U q (sl 2 ) algebra and discuss in detail the algebraic Bethe ansatz technique for diagonalizing the generalized XXZ spin chain. In the last section we discuss integrable lattice models with generic integrable boundary conditions [19]. The corresponding fundamental algebraic relation i.e. the reflection equation [20] is introduced and solutions (reflection matrices) are obtained with the help of the B-type braid group and its quotients [21,22,23,24].
Tensorial representations of the reflection algebra are constructed and the open transfer matrix is introduced. Finally, the U q (sl 2 ) invariant open XXZ spin chain [25] is discussed and the corresponding quadratic Casimir is extracted from the open transfer matrix.
Before we proceed with the presentation of the fundamental notions of quantum integrability it is necessary to introduce some basic notation.
Consider the tensor vector space V ⊗ V then define
(2.1)
We then attach subscripts on the various elements to define the respective vector space on which they act non-trivially. For example, suppose that A, B ∈ EndV . In the described notation the tensor product between them can be written as
In general, consider the tensor sequence of N vector spaces V ⊗ V ⊗ . . . ⊗ V then define:
A n = I ⊗ . . . ⊗ I ⊗ A n ⊗I ⊗ . . . ⊗ I, n ∈ {1, 2, . . . , N}.
(2.3)
We shall extensively use such notation subsequently when constructing one dimensional integrable quantum spin chains, which is one of the primary objectives of this review.
Some basic properties of the tensor product are listed below1 :
There is a simple rule that gives the tensor product of two matrices. Consider for sim- In general for two n × n matrices A, B the corresponding tensor product is a n 2 × n 2 matrix and the rule generalizes in a straightforward manner: (A ⊗ B) ij,kl = a ij b kl .
The tensor product of two vectors a, b ∈ C 2 is derived as
and in general for n column vectors a, b ∈ C n we obtain an n 2 column vector: (a ⊗ b) i,j = a i b j .
The su 2 algebra is defined by the generators J ± , J z and the exchange relations
(2.7)
The spin 1 2 representation of su 2 maps the three generators of the algebra to the three Pauli matrices. Indeed consider the spin 1 2 representation π : su 2 ֒→ End(C 2 ) such that:
and σ ± , σ
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