Deformed Richardson-Gaudin model

The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a finite chain. The integrability of the Hamiltonian allows for its eigenstates to be constructed algebraically. In this work we show that quantum group theory provides a possibility to deform the Hamiltonian while preserving integrability. More precisely, we use the so-called Jordanian r-matrix to deform the Hamiltonian of the Richardson-Gaudin model. In order to preserve its integrability, we need to insert a special nilpotent term into the auxiliary L-operator which generates integrals of motion of the system. Moreover, the quantum inverse scattering method enables us to construct the exact eigenstates of the deformed Hamiltonian. These states have a highly complex entanglement structure which requires further investigation.

As shown by Cambiaggio et al. [3], by introducing fermion operators c † lm and c lm related to the sl(2) generators by the Richardson-Gaudin model in Eq. (1) gets mapped onto the pairing model Hamiltonian Here c † lm (c lm ) creates (annihilates) a fermion in the state | lm (with | lm the time reversed state of | lm ), and n l = m c † lm c lm and A † l = (A l ) † = m c † lm c † lm are the corresponding number-and pair-creation operators. The pairing strengths g ll ′ are here approximated by a single constant g, with ǫ l the single-particle level corresponding to the m-fold degenerate states | lm .
As is well-known, the pairing model in Eq. (4) is central in the theory of superconductivity. Richardson's exact solution of the model [1], exploiting its integrability, has been important for applications in mesoscopic and nuclear physics where the small number of fermions prohibits the use of conventional BCS theory [4]. Moreover, its (pseudo)spin representation in the guise of the Richardson-Gaudin model, Eq. (1), provides a striking link between quantum magnetism and pairing phenomena, two central concepts in the physics of quantum matter.
The eigenstates of the Richardson-Gaudin Hamiltonian, Eq. (1), can be constructed algebraically using the quantum inverse scattering method (QISM) [5,6]. The main objects of this method are the classical r-matrices and the L-matrix of the loop algebra L(sl(2)) generators h(λ), X + (λ), X − (λ) The commutation relations (CR) of loop algebra generators are given in compact matrix form where L 1 (λ) = L(λ) ⊗ I, L 2 (µ) = I ⊗ L(µ) and r(λ, µ) is the 4 × 4 c-number matrix in Eq. (5). A consequence of this form is the commutativity of transfer matrices, The corresponding mutually commuting operators extracted from the decomposition of t(λ) define a Gaudin model [2,7]. However, to get the Richardson Hamiltonian a mild change of the L-operator is necessary, where h 0 = σ z 0 in auxiliary space C 2 0 of spin 1/2. This transformation does not change the CR of matrix elements of this matrix L(λ; c) due to the symmetry of the r-matrix (5): The resulting transfer matrix obtains some extra terms Let us consider a spin-1/2 representation on the auxiliary space V 0 ≃ C 2 and spin ℓ k representations on quantum spaces V k ≃ C ℓ k +1 with extra parameters ǫ k corresponding to site k = 1, 2, . . . , N. The whole space of quantum states is H = ⊗ N 1 V k and the highest weight vector (highest spin, "ferromagnetic state") | Ω + satisfies where It is useful to introduce the notation Y gl for global operators of the sl(2)-representation: To find the eigenvectors and spectrum of t(λ) on H one requires that vectors of the form are eigenvectors of t(λ), provided that the parameters µ j satisfy the Bethe equations: The realization of the loop algebra generators on the space H takes the form The coupling constant g of (1) is connected with the parameter c = 1/g, while the Hamiltonian (1) is obtained as the operator coefficient of the term 1/λ 2 in the expansion of t(λ; c) at λ → ∞.
Quantum group theory gives the possibility to deform a Hamiltonian preserving integrability [8,9]. Specifically, we can use the so-called Jordanian r-matrix to quantum deform the Hamiltonian of Richardson-Gaudin model (1). We add to the sl(2) symmetric r-matrix (5) the Jordanian part with Casimir element C ⊗ 2 in the tensor product of two copies of sl (2), After the Jordanian twist the r-matrix (14) is commuting with the generator X + 0 only, Hence, one can add the term cX + 0 + L(λ, ξ) to the L-operator. This yields the twisted transfermatrix t (J) (λ) = 1 2 tr 0 (cX + 0 + L(λ, ξ)) 2 , The corresponding commutation relations between the generators of the twisted loop algebra are explicitly given by The realization of the Jordanian twisted loop algebra L J (sl (2)) with CR (19) is given similar to (15) with extra terms proportional to the deformation parameter ξ, To construct eigenstates for the twisted model one has to use operators of the form [9, 10] and acting by these operators on the ferromagnetic state | Ω + . The deformed Richardson-Gaudin model Hamiltonian can now be extracted from the transfermatrix t (J) (λ) as the operator coefficient in its expansion λ → ∞.
According to Eqs. (8) and (18) one can also extract quantum integrals of motion J k using the realization (20), reading off the expressions for J k from the expansion The corresponding quantum deformed Hamiltonian reads It is instructive to write down a simplified case without the Jordanian twist: ξ = 0. One thus obtains The case ξ = 0 can also be obtained by taking off from the inhomogeneous XXX spin chain. The model can be described by a 2 × 2 monodromy matrix [5] T satisfying the quadratic relations If we multiply T (λ) by a constant 2 × 2 matrix M the resulting matrix T (λ) = M · T (λ) will satisfy the same relation (25). Choosing a triangular matrix the entries of monodromy matrices become simply related: This choice of M(ǫ) (of the same type as considered in Ref. [11]) permits us to use the same reference state | Ω + ∈ H (11) and B as a creation operator of the algebraic Bethe ansatz [5]. Bethe states are given by the same action of product operators B(µ j ) = B(µ j ) + εD(µ j ) although operators B(µ j ) do not commute with D(µ j ): where Similar formulas are valid for M-magnon states. Hence, acting on the ferromagnet state | Ω + we obtain filtration of states with eigenvalues of S z : N 2 , N 2 − 1, N 2 − 2, N 2 − 3. More complicated deformations of the Richardson-Gaudin model can be obtained using rmatrices related to the higher rank Lie algebras [12]. The structure of the eigenstates of the transfer matrix and their entanglement properties [13] are under investigation.