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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 _{q}_{n}

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 [

The work of Drinfel’d [_{n}_{n}

Consider a general anyonic pairing Hamiltonian of the reduced Bardeen–Cooper–Schrieffer form, which acts on a Hilbert space ^{L}

Above,

and those relations obtained by taking Hermitian conjugates. Throughout,

where

As with the more familiar fermionic pairing Hamiltonians, one of the notable features of Equation (1) is the

commutes with Equation (1) and thus provides a good quantum number. Below,

In [

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

where

which is an operator equation on

is the six-vertex solution (it is convenient for our purposes to express the deformation parameter as ^{2} rather than the more familiar

which acts on the three-fold space

Two important properties of

where _{2} denotes partial transposition in the second space of the tensor product.

The monodromy matrix is

where

and

where

Note that

A consequence of Equation (10), and the diagonal form of

The transfer matrices can be expanded in a Laurent series

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

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 [

establishing that

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

The dihedral group _{n} has two generators

where _{n} as a group algebra, the Drinfel’d double [_{n}, denoted _{n}), has basis

where ^{2}. Multiplication of dual elements is defined by

where

The algebra _{n}) becomes a Hopf algebra by imposing the following coproduct, antipode and counit respectively:

An important property of _{n}) which will be called upon later is

Defining

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

where

When _{n}) admits eight one-dimensional irreducible representations, _{n}) admits two one-dimensional irreducible representations,

for

for

for

For any of the above two-dimensional representations

for some

where

is the permutation operator on the tensor product space. This shows that the Baxterisation of the _{n}) _{n})

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

First we define

It follows from Equation (7) that

We then define a modified monodromy matrix

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

The transfer matrix is again defined by Equation (4). From the results of [

The action of _{n}) on an

Below, for ease of notation, we will omit the representation symbols

the monodromy matrix Equation (24) commutes with the action of _{n}) as a consequence of Equations (8–21). From the results of [_{n}) due to Equation (17).

Observing that we may write

we may simplify Equation (24) as

where

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

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 _{n}) 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 _{n}) action does not preserve sectors, and specifically

whereas

These relations follow from the above two-dimensional matrix representations for which it is seen that representations of

Recall that the Hamiltonian is defined through the transfer matrix by Equation (13). Consequently, while _{n}) is a symmetry of the transfer matrix obtained from Equation (24) in the conventional sense, the interpretation of _{n}) as a symmetry of the Hamiltonian is more subtle as the choice Equation (27) is sector-dependent and thus

Using Equation (13) we obtain

From the trigonometric identity

this then leads to the following

This relation shows how the spectrum of the Hamiltonian maps under a particle-hole transformation

Thus as a result of Equations 29 and 30, the action of

Finally, if the above procedure is followed using the asymmetric

a transfer matrix is obtained which commutes with the co-product action of

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 _{n}) 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 _{n}). These results were utilised in _{n}) 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 _{n}) as a symmetry algebra. However the interpretation of _{n}) 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 _{n}) generator that induces a particle-hole transformation.

This work was supported by the Australian Research Council through Discovery Project