Axioms 2012, 1(2), 226-237; doi:10.3390/axioms1020226

Article
Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing
Jon Links
Centre for Mathematical Physics, School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia; Email: jrl@maths.uq.edu.au; Tel.: +61-7-3365-2400; Fax: +61-7-3365-1477
Received: 28 June 2012; in revised form: 13 August 2012 / Accepted: 4 September 2012 /
Published: 20 September 2012

Abstract

: 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).
Keywords:
Hopf algebra; Drinfel’d double construction; quantum integrability; Yang–Baxter equation

1. Introduction

Integrable quantum systems which admit exact solutions are central in advancing understanding of many-body systems. Classic examples are provided by the Heisenberg spin chain [1], the Bose [2] and Fermi [3] gases with delta-function interactions, the Bardeen–Cooper–Schrieffer pairing Hamiltonian with uniform scattering interactions [4], and the Hubbard model in one dimension [5]. With the development of the Quantum Inverse Scattering Method [6] as a systematic prescription for constructing integrable quantum systems through the Yang–Baxter equation [3,7,8], and solving them through the algebraic Bethe ansatz, it subsequently emerged that Hopf algebraic structures are fundamental in quantum integrability. The works of Jimbo [9] and Drinfel’d [10] were instrumental in formulating the notion of quantum algebras Axioms 01 00226 i001, deformations of the universal enveloping algebras of a Lie algebra g, which have the structure of a quasi-triangular Hopf algebra. The significance of the quasi-triangular structure is that it affords an algebraic solution of the Yang–Baxter equation. Matrix solutions of the Yang–Baxter equation are then generated through representations of these algebras. The simplest example of the two-dimensional loop representation of the untwisted affine quantum algebra Axioms 01 00226 i003 leads to the six-vertex model solution of the Yang–Baxter equation, which establishes integrability of the anisotropic (XXZ) Heisenberg chain. The precise form of six-vertex solution obtained depends on the choice of gradation for Axioms 01 00226 i003. The principal gradation leads to the symmetric solution, while the homogeneous gradation leads to an asymmetric solution [11]. Only in the latter case is the solution invariant with respect to the action of the non-affine subalgebra Axioms 01 00226 i004.

The work of Drinfel’d [10] also provides a means to construct a quasi-triangular Hopf algebra from any Hopf algebra and the dual algebra, through a procedure known as the double construction. The double construction applied to finite group algebras [12] yields a framework in which to develop anyonic models that lead to notions of topological quantum computation [13]. In a series of works [14,15,16], solutions of the Yang–Baxter associated with Drinfel’d doubles of dihedral group algebras, denoted D(Dn), have been studied. In particular, it was found that two-dimensional representations of these algebras belong to the aforementioned six-vertex model solution in the symmetric case. The symmetric solution was employed in [17] to construct an integrable anyonic pairing Hamiltonian, which generalises the pairing Hamiltonian with uniform scattering interactions solved by Richardson [4]. Below, this integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries given by D(Dn) for particular values of the coupling parameters.

2. The Integrable Hamiltonian for Anyonic Pairing

Consider a general anyonic pairing Hamiltonian of the reduced Bardeen–Cooper–Schrieffer form, which acts on a Hilbert space Axioms 01 00226 i006 of dimension 4L, given by

Axioms 01 00226 i008

Above, Axioms 01 00226 i009 represent single-particle energy levels (two-fold denegerate labelled by Axioms 01 00226 i010) and Axioms 01 00226 i011 are the pairing interaction coupling parameters of the model. For Axioms 01 00226 i012 the operators Axioms 01 00226 i013 satisfy the relations

Axioms 01 00226 i014

and those relations obtained by taking Hermitian conjugates. Throughout, I is used to denote an identity operator. These types of anyonic operators are considered as q-deformations of fermionic operators, with the usual fermionic commutation relations recovered in the limit Axioms 01 00226 i016. The anyonic creation and annihilation operators may be realised in terms of the canonical fermionic operators Axioms 01 00226 i017 through a generalised Jordan–Wigner transformation

Axioms 01 00226 i018

where Axioms 01 00226 i019.

As with the more familiar fermionic pairing Hamiltonians, one of the notable features of Equation (1) is the blocking effect. For any unpaired anyon at level j, the action of the pairing interaction is zero since only paired anyons interact. This means that the Hilbert space can be decoupled into a product of paired and unpaired anyonic states in which the action of the Hamiltonian on the space for the unpaired anyons is automatically diagonal in the natural basis. In view of this property, the pair number operator

Axioms 01 00226 i020

commutes with Equation (1) and thus provides a good quantum number. Below, M will be used to denote the eigenvalues of the pair number operator.

In [17] it was shown that, for a suitable restriction on the coupling parameters, the Hamiltonian is integrable in the sense of the Quantum Inverse Scattering Method and admits an exact solution derived through the algebraic Bethe ansatz. To characterise the integrable manifold of the coupling parameter space, the set of parameters Axioms 01 00226 i022 are introduced with the following constraints imposed:

Axioms 01 00226 i023

Axioms 01 00226 i024

The conserved operators for this integrable model are obtained via the Quantum Inverse Scattering Method in a standard manner. Here, the key steps are noted. A transfer matrix Axioms 01 00226 i025 is constructed as

Axioms 01 00226 i026

where T(x) is the monodromy matrix and Axioms 01 00226 i028 is the partial trace over an auxiliary space labelled by a. The monodromy matrix is required to satisfy the relation

Axioms 01 00226 i030

which is an operator equation on Axioms 01 00226 i031, with the two auxiliary spaces labelled by a and b. Above,

Axioms 01 00226 i032

is the six-vertex solution (it is convenient for our purposes to express the deformation parameter as q2 rather than the more familiar q) of the Yang–Baxter equation [3,7,8]

Axioms 01 00226 i033

which acts on the three-fold space Axioms 01 00226 i034. The subscripts above refer to the spaces on which the operators act, e.g.,

Axioms 01 00226 i035

Two important properties of R(x), which will be called upon later, are

Axioms 01 00226 i037

Axioms 01 00226 i038

where t2 denotes partial transposition in the second space of the tensor product.

The monodromy matrix is

Axioms 01 00226 i039

where

Axioms 01 00226 i040

and Axioms 01 00226 i041. Bearing in mind the earlier comments regarding the blocking effect, we may write

Axioms 01 00226 i042

where

Axioms 01 00226 i043

Note that U is defined in a sector-dependent manner in terms of the eigenvalues M of N, which is legitimate since N is conserved.

A consequence of Equation (10), and the diagonal form of U, is that the transfer matrices form a commutative family; i.e.,

Axioms 01 00226 i044

The transfer matrices can be expanded in a Laurent series

Axioms 01 00226 i045

such that, because of Equation (12), the co-efficients commute

Axioms 01 00226 i046

Finally it can be verified that the Hamiltonian Equation (1), subject to the constraints of Equations (2,3), is expressible as (the corresponding expression in [17] contains typographical errors, which are corrected here)

Axioms 01 00226 i047

establishing that Axioms 01 00226 i048 provides a set of Abelian conserved operators for the system. In this sense the system is said to be integrable.

In the remainder of this work it will be shown that for certain further restrictions on the coupling parameters there are additional Hopf algebraic symmetries of the system. These non-Abelian symmetries are not related to a quantum algebra Axioms 01 00226 i004 structure, but are realised through the Drinfel’d doubles of dihedral group algebras.

3. Drinfel’d Doubles of Dihedral Group Algebras

The dihedral group Dn has two generators Axioms 01 00226 i049 satisfying:

Axioms 01 00226 i050

where e denotes the group identity. Considering Dn as a group algebra, the Drinfel’d double [10] of Dn, denoted D(Dn), has basis

Axioms 01 00226 i051

where g are the group elements and Axioms 01 00226 i052 are their dual elements. This gives an algebra of dimension 4n2. Multiplication of dual elements is defined by

Axioms 01 00226 i053

where Axioms 01 00226 i054 is the Kronecker delta function. The products Axioms 01 00226 i055 are computed using

Axioms 01 00226 i056

The algebra D(Dn) becomes a Hopf algebra by imposing the following coproduct, antipode and counit respectively:

Axioms 01 00226 i057

An important property of D(Dn) which will be called upon later is

Axioms 01 00226 i058

Defining Axioms 01 00226 i059 the universal R-matrix is given by

Axioms 01 00226 i061

This can be shown to satisfy the relations for a quasi-triangular Hopf algebra as defined in [10]:

Axioms 01 00226 i062

where Axioms 01 00226 i063 is the opposite coproduct

Axioms 01 00226 i064

When n is even, D(Dn) admits eight one-dimensional irreducible representations, Axioms 01 00226 i065 two-dimensional irreducible representations, and eight Axioms 01 00226 i066-dimensional irreducible representations. When n is odd, D(Dn) admits two one-dimensional irreducible representations, Axioms 01 00226 i067 two-dimensional irreducible representations, and two n-dimensional irreducible representations. The explicit irreducible representations are given in [14]. Our interest will be in the two-dimensional irreducible representations. To describe them, let Axioms 01 00226 i069. Then these representations have the form

Axioms 01 00226 i070

for Axioms 01 00226 i071 if n is even and Axioms 01 00226 i072 if n is odd,

Axioms 01 00226 i073

for Axioms 01 00226 i071 if n is even, and

Axioms 01 00226 i074

for Axioms 01 00226 i075 and where Axioms 01 00226 i076 if n is even, and Axioms 01 00226 i077 if n is odd.

For any of the above two-dimensional representations Axioms 01 00226 i078 the tensor product representation applied to the universal R-matrix Equation (18) yields the general form

Axioms 01 00226 i079

for some Axioms 01 00226 i080. Choosing Axioms 01 00226 i081 in Equation (6) we then find

Axioms 01 00226 i082

where

Axioms 01 00226 i083

is the permutation operator on the tensor product space. This shows that the Baxterisation of the D(Dn) R-matrix in two-dimensional representations leads to the symmetric six-vertex model at q a root of unity, which was previously reported in [14]. Baxterisation of the D(Dn) R -matrix in higher-dimensional representations lead to the Fateev–Zamolodchikov solution of the Yang–Baxter equation, as discussed in [15,16].

Having identified the relationship Equation (21) between the solution Equation (6) of the Yang–Baxter equation and representations of the universal R-matrix Equation (18) for D(Dn), we can now proceed to determine when D(Dn) is a symmetry algebra of the transfer matrix associated to the Hamiltonian Equation (1) subject to the constraints Equations 2 and 3.

4. Symmetries of the Transfer Matrix and Hamiltonian

First we define

Axioms 01 00226 i084

It follows from Equation (7) that

Axioms 01 00226 i085

Axioms 01 00226 i086

We then define a modified monodromy matrix

Axioms 01 00226 i087

Through use of Equations (7,22,23) it can be shown that this monodromy matrix satisfies a generalised version of Equation (5):

Axioms 01 00226 i088

The transfer matrix is again defined by Equation (4). From the results of [18] it is known that Equation (12) still holds by use of Equation (9).

The action of D(Dn) on an L-fold tensor product space is given through iterated use of the co-product action Equation (16):

Axioms 01 00226 i089

Below, for ease of notation, we will omit the representation symbols Axioms 01 00226 i090 when dealing with tensor product representations obtained through this action. Whenever we have

Axioms 01 00226 i091

the monodromy matrix Equation (24) commutes with the action of D(Dn) as a consequence of Equations (8–21). From the results of [19], the transfer matrix obtained from Equation (24) also commutes with the action of D(Dn) due to Equation (17).

Observing that we may write

Axioms 01 00226 i092

we may simplify Equation (24) as

Axioms 01 00226 i093

where

Axioms 01 00226 i094

Comparing Equations 11 and 26 and taking note of Equation (25), these matrices are made equal by choosing

Axioms 01 00226 i095

meaning that the transfer matrices obtained from the monodromy matrices Equations 10 and 24 are equal. Thus we have established that the transfer matrix associated to the integrable Hamiltonian Equation (1) subject to the constraints of Equations 2 and 3 commutes with action of the quasi-triangular Hopf algebra D(Dn) whenever Equations 25 and 27 hold.

A crucial point to bear in mind is that the transfer matrices were defined in a sector-dependent manner, where each sector is associated with a fixed number of Cooper pairs. However the D(Dn) action does not preserve sectors, and specifically Axioms 01 00226 i096 acts as a particle-hole transformation:

Axioms 01 00226 i097

whereas

Axioms 01 00226 i098

Axioms 01 00226 i099

These relations follow from the above two-dimensional matrix representations for which it is seen that representations of Axioms 01 00226 i100 and Axioms 01 00226 i101 are always diagonal in the basis in which the action of N is diagonal. In the same basis, representations of Axioms 01 00226 i096 are orthogonal matrices with non-zero off-diagonal entries.

Recall that the Hamiltonian is defined through the transfer matrix by Equation (13). Consequently, while D(Dn) is a symmetry of the transfer matrix obtained from Equation (24) in the conventional sense, the interpretation of D(Dn) as a symmetry of the Hamiltonian is more subtle as the choice Equation (27) is sector-dependent and thus α needs to be treated as an operator-valued quantity. From Equation (28) we have for α given by Equation (27) that for each sector where N has eigenvalue M

Axioms 01 00226 i103

Using Equation (13) we obtain

Axioms 01 00226 i104

From the trigonometric identity

Axioms 01 00226 i105

this then leads to the following anti-symmetry relation for the integrable Hamiltonian Equations (1–3) whenever Equations 25 and 27 hold

Axioms 01 00226 i106

This relation shows how the spectrum of the Hamiltonian maps under a particle-hole transformation Axioms 01 00226 i107 induced by Equation (28). On the other hand,

Axioms 01 00226 i108

Thus as a result of Equations 29 and 30, the action of Axioms 01 00226 i100 and Axioms 01 00226 i101 leaves the spectrum of the Hamiltonian invariant in each sector with fixed M.

Finally, if the above procedure is followed using the asymmetric R-matrix

Axioms 01 00226 i109

a transfer matrix is obtained which commutes with the co-product action of Axioms 01 00226 i004 [20,21]. However in this setting the corresponding conserved operator Axioms 01 00226 i110 contains additional interaction terms. As a result, an expression analogous to Equation (13) does not yield an operator in the form of Equation (1).

5. Conclusions

An analysis of an integrable Hamiltonian for anyonic pairing, as given by Equation (1) subject to Equations 2 and 3, was undertaken. Values of the coupling parameters were identified for which the model admits Hopf algebraic symmetries. In Section 2 the construction of the integrable model was outlined in terms of the Quantum Inverse Scattering Method. This was achieved through the symmetric, six-vertex solution of the Yang–Baxter equation. The Hamiltonian was identified through a conserved operator associated to the corresponding transfer matrix. In Section 3 a description of the quasi-triangular Hopf algebra D(Dn) was presented, including explicit expressions for all irreducible, two-dimensional representations. Through these representations it was established that the symmetric, six-vertex solution of the Yang–Baxter equation is related to representations of the universal R-matrix for D(Dn). These results were utilised in Section 4 to construct a transfer matrix which preserved the D(Dn) symmetry. From this transfer matrix, values of the coupling parameters were identified for which the Hamiltonian Equation (1) subject to Equations 2 and 3 has D(Dn) as a symmetry algebra. However the interpretation of D(Dn) as a symmetry algebra for the Hamiltonian is somewhat unconventional in that both commuting and anti-commuting actions for the generators were found. The anti-commuting action is associated with a particular D(Dn) generator that induces a particle-hole transformation.

Acknowledgments

This work was supported by the Australian Research Council through Discovery Project Topological properties of exactly solvable, two-dimensional quantum systems (DP110101414).

References

  1. Bethe, H. Zur Theorie der Metalle: I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Zeitschrift für. Physik 1931, 71, 205–226, doi:10.1007/BF01341708.
  2. Lieb, E.H.; Liniger, W. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 1963, 130, 1605–1616, doi:10.1103/PhysRev.130.1605.
  3. Yang, C.N. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 1967, 19, 1312–1315, doi:10.1103/PhysRevLett.19.1312.
  4. Richardson, R.W. A restricted class of exact eigenstates of the pairing-force Hamiltonian. Phys. Lett. 1963, 3, 277–279, doi:10.1016/0031-9163(63)90259-2.
  5. Lieb, E.H.; Wu, F.Y. Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension. Phys. Rev. Lett. 1968, 20, 1445–1448, doi:10.1103/PhysRevLett.20.1445.
  6. Takhtadzhan, L.A.; Faddeev, L.D. The quantum method of the inverse problem and the Heisenberg XYZ model. Russ. Math. Surveys 1979, 34, 11–68, doi:10.1070/RM1979v034n05ABEH003909.
  7. McGuire, J.B. Study of exactly soluble one-dimensional N-body problems. J. Math. Phys. 1964, 5, 622–636, doi:10.1063/1.1704156.
  8. Baxter, R.J. Partition function of the eight-vertex lattice model. Ann. Phys. 1972, 70, 193–228, doi:10.1016/0003-4916(72)90335-1.
  9. Jimbo, M. A q-difference analog of U(g) and the Yang–Baxter equation. Lett. Math. Phys. 1985, 10, 63–69, doi:10.1007/BF00704588.
  10. Drinfel’d, V.G. Quantum Groups. In Proceedings of the International Congress of Mathematicians; Gleason, A.M., Ed.; American Mathematical Society: Providence, Rhode Island, 1986; pp. 798–820.
  11. Bracken, A.J.; Delius, G.W.; Gould, M.D.; Zhang, Y.-Z. Infinite families of gauge equivalent R-matrices and gradations of quantized affine algebras. Int. J. Mod. Phys. B 1994, 8, 3679–3691, doi:10.1142/S0217979294001585.
  12. Gould, M.D. Quantum double finite group algebras and their representations. Bull. Aust. Math. Soc. 1993, 48, 275–301, doi:10.1017/S0004972700015707.
  13. Kitaev, A.Y. Fault-Tolerant quantum computation by anyons. Ann. Phys. 2003, 303, 2–30, doi:10.1016/S0003-4916(02)00018-0.
  14. Dancer, K.A.; Isaac, P.; Links, J. Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang–Baxter equation. J. Math. Phys. 2006, 47, 1–18.
  15. Finch, P.E.; Dancer, K.A.; Isaac, P.; Links, J. Solutions of the Yang–Baxter equation: Descendents of the six-vertex model from the Drinfeld doubles of dihedral group algebras. Nucl. Phys. B 2011, 847, 387–412, doi:10.1016/j.nuclphysb.2011.01.034.
  16. Finch, P.E. Integrable Hamiltonians with D(Dn) symmetry from the Fateev-Zamolodchikov model. J. Stat. Mech. Theory Exp. 2011, doi:10.1088/1742-5468/2011/04/P04012.
  17. Dunning, C.; Ibañez, M.; Links, J.; Sierra, G.; Zhao, S.-Y. Exact solution of the p + ip pairing Hamiltonian and a hierarchy of integrable models. J. Stat. Mech. Theory Exp. 2010, doi:10.1088/1742-5468/2010/08/P08025.
  18. Links, J.; Foerster, A. On the construction of integrable closed chains with quantum supersymmetry. J. Phys. A Math. Gen. 1997, 30, 2483–2487, doi:10.1088/0305-4470/30/7/026.
  19. Links, J.R.; Gould, M.D. Casimir invariants for Hopf algebras. Rep. Math. Phys. 1992, 31, 91–111, doi:10.1016/0034-4877(92)90007-N.
  20. Grosse, H.; Pallua, S.; Prester, P.; Raschhofer, E. On a quantum group invariant spin chain with non-local boundary conditions. J. Phys. A Math. Gen. 1994, 27, 4761–4771, doi:10.1088/0305-4470/27/14/007.
  21. Karowski, M.; Zapletal, A. Quantum group invariant integrable n-state vertex models with periodic boundary conditions. Nucl. Phys. 1994, 419, 567–588, doi:10.1016/0550-3213(94)90345-X.
Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert