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 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:
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 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.
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.
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...
Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-vertex solution of the Yang–Baxter equation, the Hopf algebra symmetries are not in terms of the quantum algebra Uq(sl(2)). Rather, they are associated with the Drinfel’d doubles of dihedral group algebras D(Dn).
In this chapter we carry out the thermodynamic limit on the level of the monodromy matrix introduced in Chapter 12. This means to change the strategy as compared to the Bethe ansatz solutions put forward in Chapters 3 and 12 which depended crucially on the use of periodic boundary conditions. The discreteness of the quasi momenta k j in the Bethe ansatz wave function was due to the finite length L of the system. For infinite L there will be no Lieb-Wu equations which were our main tool for studying the Hubbard model in this book. Instead the commutation relations for the elements of the infinite interval monodromy matrix will utilized in the calculations shown below. We basically follow the articles [335, 336], where the quantum inverse scattering method, in the way as originally designed in [131, 404, 454], was applied to the Hubbard model. Our account will be restricted to the case of zero electron density. Quite generally, the quantum inverse scattering method, as originally conceived in the spirit of the ‘inverse scattering theory’ for classical non-linear evolution equations, is restricted to uncorrelated vacua (ground states) which limits the applicability of the method. Nevertheless, applying it to the empty band ground state of the Hubbard model we shall obtain valuable additional insights into its structure. We shall reveal the connection between Shastry's R -matrix and the Yangian symmetry discussed in Chapter 14.