In this chapter we will reveal another piece of the algebraic structure of the Hubbard model. As was first observed by Uglov and Korepin [462], the Hubbard Hamiltonian on the infinite line is invariant under the action of the direct sum of two so-called Yangian quantum groups, extending the rotational and the η-pairing su(2) symmetries we encountered earlier. Following [172] we shall address the issue in a more general context. We present two pairs of fermionic representations of the Y(su(2)) Yangian quantum group which commute with the trigonometric [162] and hyperbolic [40, 41] versions of a Hubbard Hamiltonian with non-nearest-neighbour hopping. In both cases the two representations are also mutually commuting, hence can be combined into a representation of Y(su(2))⊕Y(su(2)). The generators of the Yangian symmetry of the ordinary Hubbard model (with nearest-neighbour hopping) and of a number of other interesting models like the Haldane-Shastry spin-chain [194, 394] are obtained as special cases of our general result. Introduction Quantum groups were introduced by Drinfeld [107, 109]. His original intention was to put what we called the Yang-Baxter algebra into the mathematically more conventional context of Hopf algebras. The Yangians are special quantum groups. Their representation theory [80, 81] is intimately related to the classification of integrable quantum systems with rational R -matrices.
In this chapter we will reveal another piece of the algebraic structure of the Hubbard model. As was first observed by Uglov and Korepin [462], the Hubbard Hamiltonian on the infinite line is invariant under the action of the direct sum of two so-called Yangian quantum groups, extending the rotational and the η-pairing su(2) symmetries we encountered earlier. Following [172] we shall address the issue in a more general context. We present two pairs of fermionic representations of the Y(su(2)) Yangian quantum group which commute with the trigonometric [162] and hyperbolic [40, 41] versions of a Hubbard Hamiltonian with non-nearest-neighbour hopping. In both cases the two representations are also mutually commuting, hence can be combined into a representation of Y(su(2))⊕Y(su(2)). The generators of the Yangian symmetry of the ordinary Hubbard model (with nearest-neighbour hopping) and of a number of other interesting models like the Haldane-Shastry spin-chain [194, 394] are obtained as special cases of our general result. Introduction Quantum groups were introduced by Drinfeld [107, 109]. His original intention was to put what we called the Yang-Baxter algebra into the mathematically more conventional context of Hopf algebras. The Yangians are special quantum groups. Their representation theory [80, 81] is intimately related to the classification of integrable quantum systems with rational R -matrices.
We present two pairs of Y(sl2) Yangian symmetries for the trigonometric and hyperbolic versions of the Hubbard model with non-nearest-neighbour hopping. In both cases the Yangians are mutually commuting, hence can be combined into a Y(sl2)⊕Y(sl2) Yangian. Their mutual commutativity is of dynamical origin. The known Yangians of the Haldane-Shastry spin chain and the nearest neighbour Hubbard model are contained as limiting cases of our new representations. Yangian quantum groups were introduced by Drinfel’d [1] more then ten years ago. His original intention was to put the algebraic structure underlying the Yang-Baxter exchange relations with rational R-matrix, responsible for the integrability of the most prominent integrable systems, into the mathematically more conventional context of Hopf algebras. Yangian quantum groups and their representation theory are thus intimately connected with the classification of integrable quantum systems. ∗e-mail:
We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite interval at zero density. R-matrix and monodromy matrix are obtained as limits from their known counterparts on the finite interval. The R-matrix greatly simplifies in the considered limit. The new R-matrix contains a submatrix which turns into the rational R-matrix of the XXX-chain by an appropriate reparametrization. The corresponding submatrix of the monodromy matrix thus provides a representation of the Y(su(2)) Yangian. From its quantum determinant we obtain an infinite series of mutually commuting Yangian invariant operators which includes the Hamiltonian. Keywords Hubbard model, quantum inverse scattering, Yangian z E-mail:murakami@appi.t.u-tokyo.ac.jp x E-mail:frank@monet.phys.s.u-tokyo.ac.jp -- Address from Oct. 1996: Physikalisches Institut der Universitat Bayreuth, TP1, 95440 Bayreuth, Germany. 1 Introduction Among the exactly solvable 1d quantum systems the Hubbard...
In this thesis we theoretically investigate the close relation between spin chains and Yangians. First, we investigate the Haldane-Shastry model, which symmetry algebra is a representation of the Yangian. We show this in full detail for the spin-1/2 model. We also give an argument why a spin-1 chain with Yangian symmetry cannot exist. \nAfter that, we turn to integrable spin chains which are constructed using the quantum inverse scattering method. Using this method, an integrable spin chain can be constructed from each so called R-matrix, which is a solution of the Yang-Baxter equation. Our main result is the construction of an R-matrix that is invariant under the adjoint representation of SU(n), which is found using an intertwining operator for the corresponding Yangian representation. The integrable\nspin chain that corresponds to this R-matrix is non-Hermitian, making its physical interpretation unclear. These results have been published recently as a preprint under arXiv:1606.02516. \nFinally, integrable spin chains with defects are studied. We present the Hamiltonian for the Heisenberg model and the Babujian-Takhtajan model with defects, such that these models keep there integrability while staying Lie algebra invariant.
The spectrum of the Sutherland model for fermions with spin has a degeneracy. The orthogonal set of the basis of the Fock space of degenerate states is usually related to a Lie algebra such as su 2 and su 3 . In our case, however, the degeneracy extends to states with different values of total spin, and the ordinary Lie algebra is insufficient. The relevant algebra is called Yangian [43, 44], and is the algebra controlling those operators satisfying the Yang–Baxter relation [118, 178]. In this chapter we describe how the Yangian naturally emerges in the many-particle states described by Jack polynomials, and constructs an orthogonal basis of degenerate states. We concentrate here on the simplest case of the gl 2 Yangian, corresponding to spin 1/2 particles. More general cases will be discussed in Chapter 9. We introduce a variant of the R-matrix in Section 8.1, and its standard form in Section 8.2. We then introduce a monodromy matrix and discuss its algebraic properties, such as the Yang–Baxter relation and the intertwining property. In Section 8.4, we consider the action of the monodromy matrix on the set of states constituting the degenerate eigenstates. We will see that an element of the monodromy matrix plays the role of a creation or annihilation operator of orthogonal eigenfunctions. This means that all eigenfunctions can be generated algebraically from the highest-weight state, for example, by successive action of an element of the monodromy matrix. Thus, the degeneracy beyond the Lie algebra is accounted for by the symmetry of the monodromy matrix, which is nothing but the Yangian. A demonstration of this construction will be given in Section 8.5 for the two-particle and three-particle systems.
We now describe the properties of the SU( >K ) chain in more algebraic terms. The relevant symmetry is the SU( K ) Yangian, which is usually referred to as Y ( sl K ). On the other hand, the Yangian symmetry of the Sutherland model with spin 1/2 is given by Y ( gl 2 ). The difference between Y ( sl K ) and Y ( gl K ) is that the charge degrees of freedom U(1) is absent in Y ( sl K ), while the spin degrees of freedom is included in both Y ( sl K ) and Y ( sl K ). In Section 4.8.3, we have removed the U(1) component by means of the freezing trick, i.e., by taking the infinitely large repulsion parameter. In this chapter we take a more algebraic approach to study Y ( sl K ), which makes the relation between Y ( gl K ) and Y ( sl K ) clearer. In Section 9.1, we begin with the construction of the SU( K ) monodromy matrix as the product form. The commuting property of the Cherednik–Dunkl operators plays a crucial role here. The monodromy matrix is the basic building block of the Yangian algbebra, as discussed in Section 8.2. Because the Hamiltonian of the SU( K ) spin chain and the monodromy matrix commute, each eigenstate of the spin chain constitutes a basis for representations of the SU( K ) Yangian. In Section 9.2 and 9.3, we discuss the quantum determinant by comparing it with the ordinary determinant. It is demonstrated in Section 9.4 that the quantum determinant has a scalar nature that leads to a simple evaluation. Because of the scalar nature, however, the quantum determinant cannot distinguish different SU( K ) Yangian states.
We proceed to study the symmetries of integrable open boundaries in the one dimensional Hubbard model, the Heisenberg XXX spin chain and Inozemtsev's hyperbolic spin chain. \nFor the Hubbard model, we show that when placed on the left half-line, the known integrable open boundaries (a magnetic field and chemical potential) break the bulk Yangian symmetry to a twisted Yangian corresponding to the (sl2; u1) symmetric pair. Furthermore, we consider two additional boundaries, corresponding to the symmetric pairs (so4; sl2) and (sl2; sl2) and construct their twisted Yangian symmetries. This provides a step forward in the classification of integrable boundaries of the open Hubbard model. We conclude our study of this model by examining the symmetries of its bulk and open SU(n) generalisation. \n \nFor the Heisenberg XXX spin chain and Inozemtsev's hyperbolic spin chain we construct a procedure to, given the integrable bulk models, systematically obtain \ntheir integrable boundaries and corresponding Yangian symmetries for the symmetric pairs (sl2; u1); (so4; sl2) and (sl2; sl2). We call this method `folding', and it is \nmotivated by the wish to study integrable boundaries for long-range spin chains. We test this procedure by first applying it on the Heisenberg XXX spin chain and confirming it reproduces well known results. We then apply the folding to Inozemtsev's hyperbolic spin chain and classify its integrable open boundaries and their twisted Yangian symmetries.