Deformed Richardson-Gaudin model

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1 Deformed Richardson-Gaudin model P Kulish 1, A Stolin and L H Johannesson 1 St. Petersburg Department of Stelov Mathematical Institute, Russian Academy of Sciences Fontana 7, St. Petersburg, Russia University of Gothenburg, Box 100, SE Gothenburg, Sweden Abstract. The Richardson-Gaudin model describes strong pairing correlations of fermions confined to a finite chain. The integrability of the Hamiltonian allows the algebraic construction of its eigenstates. In this wor we show that the quantum group theory provides a possibility to deform the Hamiltonian 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 require further investigation. The Richardson-Gaudin model 1, ] is an integrable spin- 1 periodic chain with Hamiltonian H = ɛ j Sj z + g j, S j S+ (1 g is a coupling constant and S ± l = Sl x ± is y l, with N copies of the Lie algebra su( generators Sl α, S α l, Sβ l ] = iε α β γ S γ δ l l, α, β = x, y, z As shown by Cambiaggio et al 3], by introducing the fermion operators c lm and c lm related to the sl( generators by S z l = 1/ m c lm c lm 1/, S + l = 1 c lm c l m = (S l m the Richardson-Gaudin model in Eq. (1 gets mapped onto the pairing model Hamiltonian H P = l ɛ lˆn l + g/ l,l A l A l ( Here c lm (c lm creates (annihilates a fermion in the state lm (with l m in the time reversed state of lm and n l = m c lm c lm, A l = (A l = m c lm c l m Content from this wor may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this wor must maintain attribution to the author(s and the title of the wor, journal citation and DOI. Published under licence by Ltd 1

2 are the corresponding number- and pair-creation operators. The pairing strengths g ll are here approximated by a single constant g, with ɛ l the single-particle level corresponding to the m-fold degenerate states lm. As it is well-nown, the pairing model in Eq. ( 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 the small number of fermions prohibits the use of conventional BCS theory 4]. Moreover, its (pseudospin representation in the guise of the Richardson-Gaudin model, Eq. (1, provides a striing lin between quantum magnetism and pairing phenomena, both 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-matrix r(λ, µ = 4 S α S α 1 ( (3 α s= h(λ, X + (λ, X (λ are the generators of the loop algebra L(sl( as the L-matrix is ( h(λ X L(λ = (λ X + (λ h(λ The commutation relations (CR of loop algebra generators are given in compact matrix form L 1 (λ, L (µ] = r 1 (λ, µ, L 1 (λ + L (µ] L 1 (λ = L(λ I, L (µ = I L(µ and r(λ, µ is the 4 4 c-number matrix in Eq. (3. A consequence of this form is the commutativity of transfer matrices, t(λ = 1 tr 0(L (λ L(sl(, t(λ, t(µ] = 0 (4 The corresponding mutually commuting operators extracted from the decomposition of t(λ define a Gaudin model, 7]. However, to get Richardson Hamiltonian a mild change of the L-operator is necessary L(λ L(λ; c := c h 0 + L(λ h 0 = σ0 z in auxiliary space C 0 of spin 1/. This transformation does not change the CR of matrix elements of this matrix L(λ; c due to the symmetry of the r-matrix (3: Y I + I Y, r(λ, µ] = 0, Y sl( The resulting transfer matrix obtains some extra terms t(λ; c = 1 tr 0 (L(λ; c = c 1 + c h(λ + h (λ + ( X + (λx (λ + X (λx + (λ Let us consider a spin- 1 representation on auxiliary space V 0 C and spin l representations on quantum spaces V C l +1 with extra parameters ɛ corresponding to site = 1,,..., N.

3 The whole space of quantum states is H = N 1 V and the highest weight vector (highest spin, ferromagnetic state Ω + satisfies X + (λ Ω + = 0, h(λ Ω + = ρ(λ Ω + (5 ρ(λ = l /(λ ɛ It is useful to introduce notation for global operators of sl(-representation Y gl := N Y. To find the eigenvectors and spectrum of t(λ on H one requires that vectors of the form µ 1,..., µ M = M X (µ j Ω + are eigenvectors of t(λ, t(λ {µ j } M = Λ ( λ; {µ j } M {µj } M provided that the parameters µ j satisfy the Bethe equations: c + M l /(µ i ɛ /(µ i µ j = 0, i = 1,..., M (6 j i The realization of the loop algebra generators on the space H taes the form h(λ = h λ ɛ, X (λ = X λ ɛ, X + (λ = X + λ ɛ (7 The coupling constant g of (1 is connected with parameter c = 1/g while the Hamiltonian (1 is obtained as operator coefficient of term 1/λ in the expansion of t(λ; c at λ. The quantum group theory provides a possibility to deform a Hamiltonian preserving integrability 8, 9]. Specifically, we use the so-called Jordanian r-matrix to quantum deform the Hamiltonian of Richardson-Gaudin model (1. We add to sl( symmetric r-matrix (3 the Jordanian part r J (λ, µ = C + ξ ( h X + X + h with Casimir element C C = h h + ( X + X + X X + in the tensor product of two copies of sl(, After Jordanian twist the r-matrix (14 is commuting with the generator X 0 + ] X 0 + I + I X+ 0, r(j (λ, µ = 0 only 3

4 Hence, one can add the term cx L(λ, ξ to the L-operator. This yields the twisted transfermatrix t (J (λ = 1 tr 0(cX L(λ, ξ = cx + (λ + h(λ h (λ + (X (λ + ξx + (λ (8 The corresponding commutation relations between the generators of the twisted loop algebra are explicitly given by h(λ, h(µ] = ξ ( X + (λ X + (µ, X (λ, X (µ ] = ξ ( X (λ X (µ X + (λ, X (µ ] h(λ h(µ = + ξx + (λ, h(λ, X (µ ] = X (λ X (µ + ξh(µ, X + (λ, X + (µ ] = 0 (9 h(λ, X + (µ ] = X+ (λ X + (µ The realization of the Jordanian twisted loop algebra L J (sl( with CR (9 is given similar to (7 with extra terms proportional to the deformation parameter ξ h(λ = ( h +ξx + λ ɛ, X (λ = ( X ξ λ ɛ h, X + (λ = X + λ ɛ (10 To construct eigenstates for the twisted model one has to use operators of the form 9, 10] B M (µ 1,..., µ M = X (µ 1 ( X (µ + ξ... ( X (µ M + ξ(m 1 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 (4 and (8 one can also extract quantum integrals of motion J using the realization (10. It would yield rather cumbersome expressions for J : t (J (λ = J λ J λ J +... The corresponding quantum deformed Hamiltonian reads H J = c ɛ j X j + + ξ ɛ j h j X + gl h gl ɛ j X j + ( + hgl + h gl + 4X gl X+ gl It is instructive to write down a simplified case without the Jordanian twist: ξ = 0. One thus obtains J 0 = 0, J 1 = X + gl, J ( ɛ X + + g/ hgl + h gl + 4X gl X+ gl The case ξ = 0 can also be obtained by taen off from the inhomogeneous XXX spin chain. The model can be described by a monodromy matrix 5] ( A(λ B(λ T (λ = C(λ D(λ 4

5 and entries of this matrix satisfy quadratic relations R(λ, µt (λ T (µ = (I T (µ (T (λ I R(λ, µ (11 If we multiplay T (λ by a constant matrix M(ε the resulting matrix T (λ = M(ε T (λ will satisfy the same relation (11. Choosing a triangular matrix M(ε = ( ε the entries of monodromy matrices become simply related: Ã = A + εc, B = B + εd, C = C, D = D. This choice of M(ε (of the same type as considered in 11] permits us to use the same reference state Ω + H (5 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 : D(λB(µ = α(λ, µb(µd(λ + β(λ, µb(λd(µ α(λ, µ = ( + η/(, β(λ, µ = η/( For a 3 magnon state one gets due to B-D ordering 3 B(µ j = B(µ j + ε α(µ, µ s α(µ s, µ l B(µ B(µ l D(µ s s=1 3 + ε α(µ, µ s α(µ l, µ s B(µ s D(µ D(µ l + ε 3 s=1 D(µ j Similar formula is valid for M-magnon state. Hence, acting on ferromagnet state Ω +, we obtain filtration of states with eigenvalues of S z : N, N 1, N, N 3. More complicated deformations of the Richardson-Gaudin model can be obtained using r- matrices related to the higher ran Lie algebras 1]. The structure of the eigenstates of the transfer matrix and their entanglement properties 13] are under investigation. Acnowledgments This wor was supported by RFBR grants a, a, ofi- M (P.K., and by STINT grant IG (A.S. and H.J.. We would lie to than E. Damasinsy for useful discussion. References 1] Richardson R W 1963 J. Math. Phys ] Gaudin M 1983 La fonction d onde de Bethe (Paris: Masson chapter 13 3] Cambiaggio M C, Rivas A M F and Saraceno M 1997 Nucl. Phys. A ] Duelsy J, Pittel S and Sierra G 004 Rev. Mod. Phys ] Faddeev L D 1998 How algebraic Bethe ansatz wors for integrable models Quantum symmetries, Proceedings of the Les Houches summer school, session LXIV eds A Connes, K Gawedzi and J Zinn-Justi (North- Holland p 149 6] Kulish P P and Slyanin E K 198 Quantum spectral transform method. Recent developments Lecture Notes in Phys. 151 (Springer-Verlag p 61 7] Slyanin E K 1999 Lett. Math. Phys ] Kulish P P and Stolin A A 1997 Czech. J. Phys ] Kulish P P 00 Twisted sl( Gaudin model Preprint PDMI 08/00 10] Antonio N C and Manojlovic N 005 J. Math. Phys ] Muhin E, Tarasov V and Varcheno A 010 Contemp. Math ] Stolin A A 1991 Math. Scand ] Dunning C, Lins J and Zhou H-Q 005 Phys. Rev. Lett