Phase Diagrams of Three-Component Attractive Ultracold Fermions in One-Dimension
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Phase Diagrams of Three-Component AttractiveUltracold Fermions in One-Dimension
C.C.N. Kuhn and A. Foerster
Instituto de Fisica da UFRGS, Av. Bento Gon¸calves, 9500, Porto Alegre,91501-970,BrazilE-mail: [email protected] , [email protected] Abstract.
We investigate trions, paired states and quantum phase transitions inone-dimensional SU (3) attractive fermions in external fields by means of the Betheansatz formalism. Analytical results for the ground state energy, critical fields andcomplete phase diagrams are obtained for the weak coupling regime. Higher ordercorrections for these physical quantities are presented in the strong attractive regime.Numerical solutions of the dressed energy equations allow us to examine how thedifferent phase boundaries modify by varying the inter-component coupling throughoutthe whole attractive regime. The pure trionic phase existing in the strong couplingregime reduces smoothly by decreasing this coupling until the weak limit is reached.In this weak regime, a pure BCS-like paired phase can be sustained under certainnonlinear Zeeman splittings.PACS numbers: 02.30.Ik, 03.75.Ss, 03.75.Hh, 64.70.Tg Submitted to:
New J. Phys. hase Diagrams of Three-Component Attractive Fermions
1. Introduction
Recent experiments on ultracold atomic systems confined to one dimension (1D) [1–4]have attracted renewed interest in Bethe ansatz integrable models of interacting bosonsand multi-component fermions. The most recent experimental breakthrough is therealization of a 1D spin-imbalanced Fermi gas of Li atoms under the degeneratetemperature [5]. This study demonstrates how ultracold atomic gases in 1D may be usedto create non-trivial new phases of matter, and also paves the way for direct observationand further study of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states [6, 7].Three-component fermions exhibit a rich scenario, revealing more exotic phases[8–15]. Notably, strongly attractive three-component ultracold atomic fermions canform three-body bound states called trions . Consequently, a phase transition betweenBardeen-Cooper-Schrieffer (BCS) like pairing superfluid and trionic states is expectedto occur in the strong attractive regime [10, 13, 16–22]. So far, most of the theoreticalanalysis has focused on the attractive strong coupling limit. A pertinent discussion inthis context is what happens at intermediate and weak attractive coupling regimesIn this paper we consider one-dimensional three-component ultracold fermionswith δ -function interaction in external magnetic fields. From a mathematical pointof view, this model was solved long ago by Sutherland [23] and Takahashi [24] throughBethe ansatz techniques. Recently, integrable models of three-component interactingfermions [16, 17, 25] have received renewed interest in connection with ultracold atomicgases. Advanced experimental techniques newly developed allow to explore three-component Fermi gas with different phases of trions, dimers and free atoms [26–29].Remarkably, direct observation of a trimer state consisting of fermionic Li atoms inthe three energetically lowest substates has been just reported in [30]. This opens upan opportunity to experimentally study such novel quantum phases of trions and pairsin 1D three-component Fermi gases, providing a physical ground and stimulus for theinvestigation of their theoretical aspects.Our aim here is to expand on the theoretic knowledge of 1D integrable model ofthree-component fermions by undertaking a detailed analysis of how the different phasesmodify as the inter-component interaction decreases, ranging from a strong to a weakregime. We obtain analytical expressions for the critical fields and construct the fullphase diagrams in the weak coupling regime by solving the Bethe ansatz equations. Weextend previous work on this model [16, 19] to derive higher order corrections for thesephysical quantities in the strong coupling regime. Numerical solutions of the dressedenergy equations show that the pure trionic phase existing in the strong coupling regimereduces as the coupling decreases and it disappears in the presence of external fieldswhen the weak regime is approached. In contrast to the two-component interactingfermions [31], nonlinear Zeeman splitting may sustain a BCS-like paired phase in thethree-component attractive fermions for weak coupling regime. hase Diagrams of Three-Component Attractive Fermions
2. The model
We consider a δ -function (contact potential) interacting system of N fermions with equalmass m , which can occupy three possible hyperfine levels ( | i , | i and | i ) with particlenumber N , N and N , respectively. They are constrained to a line of length L withperiodic boundary conditions. The Hamiltonian reads [23] H = − ¯ h m N X i =1 ∂ ∂x i + g X ≤ i
0. Here we will focus in the attractive case. For simplicity, we choose thedimensionless units of ¯ h = 2 m = 1 and use the dimensionless coupling constant γ = c/n with linear density n = N/L .The Hamiltonian (1) exhibits spin SU (3) symmetry and was solved long ago bymeans of the nested Bethe ansatz [23, 24]. In this approach its spin content wasincorporated via the symmetry of the wavefunction. The energy eigenspectrum is givenin terms of the quasimomenta { k i } of the fermions through E = N X j =1 k j , (2)satisfying the following set of coupled Bethe ansatz equations (BAE) [23, 24] exp (i k j L ) = M Y ℓ =1 k j − Λ ℓ + i c/ k j − Λ ℓ − i c/ , N Y ℓ =1 Λ α − k ℓ + i c/ α − k ℓ − i c/ − M Y β =1 Λ α − Λ β + i c Λ α − Λ β − i c M Y ℓ =1 Λ α − λ ℓ − i c/ α − λ ℓ + i c/ , M Y ℓ =1 λ µ − Λ ℓ + i c/ λ µ − Λ ℓ − i c/ − M Y ℓ =1 λ µ − λ ℓ + i cλ µ − λ ℓ − i c , (3) hase Diagrams of Three-Component Attractive Fermions α and λ µ . Above j = 1 , . . . , N , α = 1 , . . . , M , µ = 1 , . . . , M , with quantum numbers M = N + 2 N and M = N . For the irreducible representation h N N N i , a three-column Young tableau encodes the numbers of unpaired fermions ( N = N − N ),bound pairs ( N = N − N ) and trions ( N = N ).
3. Ground states
For attractive interaction, the BAE allow charge bound states and spin strings. Inparticular, the SU (3) symmetry carries two kinds of charge bound states: trions andpairs. In principle, different numbers of unpaired fermions, pairs and trions can bechosen to populate the ground state by carefully tuning H and H .In the strong coupling regime L | c | ≫
1, the imaginary parts of the bound statesbecome equal-spaced, i.e., a trionic state has the form { k j = Λ j ± i | c | , λ j } and for thebound pair { k r = Λ r ± i | c | / } . Substituting these root patterns into the BAE (3), wefind their real parts, from which the ground state energy in the strongly attractiveregime can be obtained [16] EL ≈ π n n + 4 n | c | + 12 c (2 n + n ) ! − n c π n n + 6 n + 16 n | c | + 13 c (6 n + 3 n + 8 n ) ! − n c + π n n + 32 n + 18 n | c | + 127 c (6 n + 16 n + 9 n ) ! . (4)Here n a = N a /L ( a = 1 , ,
3) is the density for unpaired fermions, pairs and trions,respectively. This state can be considered as a mixture of unpaired fermions, pairs andtrionic fermions, behaving basically like particles with different statistical signatures [34].For strong attractive interaction, trions are stable compared to the BCS-like pairingand unpaired states. From (4) we can obtain the binding energy for a trion, given by ε t = ¯ h c /m and the pair binding energy, which is ε b = ¯ h c / m .In the weak coupling regime L | c | ≪ y of the charge boundstates are the roots of Hermite polynomials H k of degree k . Specifically, H k ( q L | c | y ) = 0,with k = 2 , c = 0 case. Withthis root configuration, the ground state energy in the weak attractive regime can beobtained EL ≈ π n + 2 n + 3 n )+ π ( n ( n + n )( n + n + n ) + 2 n n ( n + n )) − | c | ( n n + 2 n n + 4 n n + n + 3 n ) . (5)The ground state energy (5) is dominated by the kinetic energy of compositeparticles and unpaired fermions and has an interaction energy consisting of density hase Diagrams of Three-Component Attractive Fermions ε t = 3¯ h | c | /mL and the pair binding energy, which is ε b = ¯ h | c | /mL . For weakattractive interaction, the trionic state is unstable against thermal and spin fluctuations.This becomes apparent in the weak coupling phase diagrams presented in Fig. 1(d), Fig.2(c) and Fig. 3(c) below.
4. Dressed energy formalism
In the thermodynamic limit, i.e.
L, N → ∞ with
N/L finite, the grand partitionfunction Z = tr (e −H / T ) = e − G/T is given in terms of the Gibbs free energy G = E + E Z − µN − T S , written in terms of the Zeeman energy E Z , chemical potential µ , temperature T and entropy S [16, 36–38]. The Gibbs free energy can be expressedin terms of the densities of particles and holes for unpaired fermions, bound pairs andtrions, as well as spin degrees of freedom, which are determined from the BAE (3). Thusthe equilibrium state is established by minimizing the Gibbs free energy with respect tothese densities. This procedure leads to a set of coupled nonlinear integral equations,from which the dressed energy equations are obtained in the limit T → ǫ (3) ( λ ) = 3 λ − c − µ − a ∗ ǫ (1) ( λ ) − [ a + a ] ∗ ǫ (2) ( λ ) − [ a + a ] ∗ ǫ (3) ( λ ) ǫ (2) (Λ) = 2Λ − µ − c − H − a ∗ ǫ (1) (Λ) − a ∗ ǫ (Λ) − [ a + a ] ∗ ǫ (3) (Λ) (6) ǫ (1) ( k ) = k − µ − H − a ∗ ǫ (2) ( k ) − a ∗ ǫ (3) ( k ) . Here ǫ ( a ) , a = 1 , , a j ( x ) = π j | c | ( jc/ + x . The symbol “ ∗ ” denotes the convolution a j ∗ ǫ ( a ) ( x ) = R + Q a − Q a a j ( x − y ) ǫ ( a ) ( y ) dy with the integration boundaries Q a given by ǫ ( a ) ( ± Q a ) = 0. The Gibbs free energy per unit length at zero temperature can bewritten in terms of the dressed energies as G = P a =1 a π R + Q a − Q a ǫ ( a ) ( x ) dx .The dressed energy equations (6) can be analytically solved just in some speciallimits. In particular, they were solved in [16] for strongly attractive interaction througha lengthy iteration method. Here we numerically solve these equations to determine thefull phase diagram of the model for any value of the coupling. This allows to examinehow the different phase boundaries deform by varying the coupling from strong to weakregime. The numerical solution is also employed to confirm the analytical expressionsfor the physical quantities and the resulting phase diagrams of the model in the weakcoupling limit (see [19] for a similar discussion in the strong regime). hase Diagrams of Three-Component Attractive Fermions
5. Full phase diagrams
Basically, there are two possible Bethe ansatz schemes to construct the phase diagramof the system. One possibility is to handle with the dressed energy equations (6). Thisapproach was discussed for the strong attractive regime in [16], where expressions forthe fields in terms of the densities were obtained up to order of 1 / | c | . Alternatively, onecan handle directly with its discrete version (equations (2) and (3)) by solving the BAE.We adopt this second strategy here. In order to obtain the explicit forms for the fieldsin terms of the polarizations we consider the energy for arbitrary population imbalances E/L = µn + G/L + n H + n H , (7)which coincides with the ground state energy (2) obtained by solving the BAE (3). Thenthe fields H and H are determined through the relations H = ∂E/L∂n , H = ∂E/L∂n (8)together with the constraint n = n + 2 n + 3 n . (9)In the strong coupling regime, using the ground state energy (4) we find H = π n − n | c | + 8 n | c | + 4 n | c | + 12 c (2 n + n ) − n c (2 n + n ) ! − π n n | c | + 32 n | c | + 4 n | c | + (6 n + 16 n + 9 n ) c − n (6 n + 16 n + 9 n )9 c ! + 10 π n | c | n + 3 n + 8 n | c | ! + 2 c .H = π n n | c | + 40 n | c | + 16 n | c | + (6 n + 3 n + 8 n ) c − n (6 n + 3 n + 8 n )27 c ! − π n n | c | + 32 n | c | + 8 n | c | + (6 n + 16 n + 9 n ) c − n (6 n + 16 n + 9 n )27 c ! + 16 π n | c | n + n ) | c | ! + 5 c . (10)These equations provide higher order corrections to those derived in [16] using thedressed energy equations. For determining the full phase boundaries, we also need theenergy-field transfer relation between the paired and unpaired phases H − H /
2, whichcan be extracted from the underlying two-component system with SU (2) symmetry[31, 34]. These equations determine the full phase diagram and the critical fieldsactivated by the fields H and H .Fig. 1(a) shows the ground state energy versus Zeeman splitting parameters H and H determined from equation (4) with the densities n and n obtained from (10).There are three pure phases: an unpaired phase A , a pairing phase B and a trion hase Diagrams of Three-Component Attractive Fermions Figure 1.
Ground state energy vs Zeeman splitting for different coupling values (a)strong interaction | γ | = 10, (b) | γ | = 5, (c) | γ | = 1 and (d) weak interaction | γ | = 0 . C , presentin the strong coupling regime, reduces smoothly as | γ | decreases and is suppressed inthe weak limit. phase C and four different mixtures of these states. For small H , a transition from atrionic state into a mixture of trions and pairs occurs as H exceeds the lower criticalvalue H c . When H is greater than the upper critical value H c , a pure pairing phasetakes place. Trions and BCS-like pairs coexist when H c < H < H c . These criticalfields, derived from equation (10), are given by H c ≈ n (cid:16) γ − π (1 + | γ | − γ ) (cid:17) hase Diagrams of Three-Component Attractive Fermions H c ≈ n (cid:16) γ + π (1 + | γ | − γ ) (cid:17) . The phase transitions from B → A + B → A induced by increasing H are reminiscent of those in the two-component systems [34,41].Basically in this region the highest level is far away from the other two levels, sothe system reduces to the spin-1/2 fermion case. The mixed phase containing BCS-like pairs and unpaired fermions can be called a FFLO phase. We mention that adiscussion about the pairing nature of 1D many-body systems can be found, for instance,in [39, 40]. For small H , a phase transition from a trionic into a mixture of trionsand unpaired fermions occur. Using equation (10), we find that the trionic state withzero polarization n /n = 0 forms the ground state when the field H < H c , where H c ≈ n (cid:16) γ − π (1 + | γ | + γ ) (cid:17) . When H is greater than the upper critical value H c ≈ n (cid:16) γ + π (1 − | γ | ) (cid:17) , all trions are broken and the state becomes a normalFermi liquid.At intermediate coupling regimes, it is not possible to construct the full phasediagrams analytically. However, they can be determined by numerically solving thedressed energy equations (6), as illustrated in Figs. 1(b), 1(c); 2(a), 2(b) and 3(a), 3(b),for the intermediary values of the coupling | c | = 5 and | c | = 1, respectively. The differentphase boundaries modify slightly by varying the inter-component coupling through thewhole attractive regime. In particular, the pure trionic phase existing in the strongcoupling regime reduces smoothly by decreasing this coupling until it is completelysuppressed. A careful numerical analysis of the phase diagrams for n = 1 and differentvalues of | c | between | c | = 1 and | c | = 0 . c c ≈ .
6. Other mixed phases involvingtrions, specially the phase ( B + C ) also reduce by decreasing | c | .In the weak coupling regime we obtain the expressions between the fields and thepolarizations using equations (5), (8) and (9) H = π n + n + 4 n n + 4 n n + 2 n n )+ 2 | c | n + n ) .H = π n + 2 n + 2 n n + 2 n n + 4 n n )+ 2 | c | n + n ) . (11)These equations together with the energy-field transfer relation H − H / H and H . We observe that the density of trions n does no appear independently in eqs.(11),in contrast to the corresponding equations in the strong regime (10). Fig. 1(d) presentsthe ground state energy versus the fields H , H while Figs. 2(c) and 3(c) show thepolarizations n /n and n /n in terms of Zeeman splitting, respectively. Now in theweak coupling regime there are just six different phases in the H – H plane: we observethe disappearance of the pure trionic phase C in the presence of the fields, i.e., the trionic hase Diagrams of Three-Component Attractive Fermions Figure 2.
Phase diagram showing the polarization n /n versus the fields H and H for different coupling values (a) | γ | = 5, (b) | γ | = 1 and (c) weak interaction | γ | = 0 . state is unstable against thermal and spin fluctuations. This behaviour is in contrast tothe strong coupling regime, where the phase C is robust and trion states populate theground state for a considerable interval of the fields. In addition, the phase where trionsand pairs coexist ( B + C ) reduces significantly compared to the strong coupling regime.Interestingly, in contrast to the weak attractive spin-1/2 fermion system, a pure pairedphase can be sustained under certain Zeeman splittings. For certain tuning H and H ,the two lowest levels are almost degenerate. Therefore, the paired phase naturally occursand is stable. The persistence of this phase is relevant for the investigation of phasetransition between BCS-like pairs and FFLO states. All these boundary modificationsoccur smoothly, as shown by a numerical analysis of the phase diagrams for differentvalues of the coupling across all regimes. This indicates that all phase transitions in thevicinities of critical points are second order. This conclusion is consistent with previousanalytical results [34, 41]. We also mention that quantum phase transitions betweendifferent superfluid phases have been discussed in [42].We perform a similar analysis as in the previous strong case to extract the criticalfields. Since the trionic phase C disappears for non-vanishing fields, less critical fields arefound compared to the strong coupling case. For small H a transition from a mixtureof trions and pairs into a pure paired phase occurs as H exceeds the critical value H c ≈ n (cid:16) γ + π (cid:17) . The transition from a mixture of trions and unpaired fermions intoa normal Fermi liquid phase occurs as H exceeds the critical value H c ≈ n (cid:16) γ + π (cid:17) . The phase transitions B → A + B → A are reminiscent of those in spin-1/2 fermionsystems. However, in the weak attractive two-component case the pure BCS-like pairedphase is suppressed [31] and consequently it is not possible to investigate the phase hase Diagrams of Three-Component Attractive Fermions Figure 3.
Phase diagram showing the polarization n /n versus the fields H and H for different coupling values (a) | γ | = 5, (b) | γ | = 1 and (c) weak interaction | γ | = 0 . separation between a BCS-like paired phase and a FFLO state, in contrast to the three-component case, where this study is still possible in a weak regime.
6. Conclusion
We have studied the three-component attractive 1D Fermi gas in external fields throughthe Bethe ansatz formalism. New results for the critical fields and complete zerotemperature phase diagrams have been presented for the weak coupling regime. Previouswork on this model has been extended to derive higher order corrections to these physicalquantities in the strong regime. We have further confirmed that the system exhibitsexotic phases of trions, bound pairs, a normal Fermi liquid and mixture of these phasesin the strongly attractive limit. We have also shown how the different phase boundariesdeform by varying the inter-component coupling across the whole attractive regime. Inparticular, the trionic phase that may occur in the strong coupling regime for certainvalues of the Zeeman splittings reduces smoothly by decreasing the coupling until theweak limit is approached, when the trionic phase is suppressed. Interestingly, in theweak regime, a pure paired phase can be maintained under certain nonlinear Zeemansplittings, in contrast to the two-component attractive 1D Fermi gas. Our high precisionof critical phase boundaries pave the way to further investigate quantum criticality inthree-component interacting Fermi gas through the finite temperature Bethe ansatz. hase Diagrams of Three-Component Attractive Fermions
7. Acknowledgments
This work has been supported by CNPq (Conselho Nacional de DesenvolvimentoCientifico e Tecnol´ogico). The authors would like to thank X.W. Guan for helpfuldiscussions and comments.
8. References [1] H. Moritz, T. Stoferle, K. Gunter, M. Kohl and T. Esslinger, Phys. Rev. Lett. , 210401 (2005).[2] T. Kinoshita, T. Wenger and D.S. Weiss, Science , 1125 (2004).[3] T. Kinoshita, T. Wenger and D.S. Weiss, Phys. Rev. Lett. , 190406 (2005).[4] E. Haller, M. Gustavsson, M. J. Mark, J. G. Danzl, R. Hart, G. Pupillo, H.-C. N¨agerl, Science , 1224 (2009).[5] Y. Liao, A. Rittner, T. Paprotta, W. Li, G. Patridge, R. Hulet, S. Baur and E. Mueller, Nature , 567 (2010).[6] P. Fulde and R. A. Ferrell, Phys. Rev. , A550 (1964); A. I. Larkin and Yu. N. Ovchinnikov,Sov. Phys.: JETP , 762 (1965).[7] W. V. Liu and F. Wilczek, Phys. Rev. Lett. , 047002 (2003).[8] ´A. Rapp, G. Zar´and, C. Honerkamp and W. Hofstetter, Phys. Rev. Lett. , 160405 (2007); C.Honerkamp and W. Hofstetter, Phys. Rev. Lett. , 170403 (2004).[9] R.W. Cherng, G. Refael and E. Demler, Phys. Rev. Lett. , 130406 (2007).[10] P. Lecheminant, E. Boulat and P. Azaria, Phys. Rev. Lett. , 240402 (2005); S. Capponi, G.Roux, P. Lecheminant, P. Azaria, E. Boulat, and S. R. White, Phys. Rev. A , 013624 (2008).[11] F. Wilczek, Nature Phys. , 375 (2007).[12] T. Paananen, J.-P. Martikainen and P. T¨orm¨a, Phys. Rev. A , 053606 (2006).[13] ´A. Rapp, W. Hofstetter and G. Zar´and, .Phys. Rev.B , 144520 (2008).[14] H. Zhai, Phys. Rev. A , 031603(R) (2007).[15] T. N. de Silva, Phys. Rev. A , 013620 (2009)[16] X. W. Guan, M.T. Batchelor, C. Lee, and H.-Q. Zhou, Phys. Rev. Lett. , 200401 (2008)[17] X.-J. Liu, H. Hu and P. D. Drummond, Phys. Rev. A , 013622 (2008).[18] Peng He, X. Yin , X.-W. Guan , M. T. Batchelor and Y. Wang, Phys. Rev. A , 053633, (2010).[19] M. T. Batchelor, A. Foerster, X.-W. Guan and C.C.N. Kuhn, J. Stat. Mech. P12014 (2010).[20] X.-W. Guan, J.-Y. Lee, M. T. Batchelor, X. G. Yin and S. Chen, Phys. Rev. A , 021606 (2010).[21] P. F. Bedaque, J. P. D´Incao, Ann. Phys. , 1763 (2009).[22] K. Inaba and S. Suga, Phys. Rev. A , 041602 (2009).[23] B. Sutherland, Phys. Rev. Lett. , 98 (1968).[24] M. Takahashi, Prog. Theor. Phys. , 899 (1970).[25] B. Errea, J. Dukelsky and G. Ortiz, Phys. Rev. A , 203202(2008); J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Stites, and K. M. O ´Hara, Phys.Rev. Lett. , 165302 (2009); J. R. Williams, E. L. Hazlett, J. H. Huckans, R. W. Stites, Y.Zhang, and K. M. O ´Hara, Phys. Rev. Lett. , 130404 (2009).[27] T. Lompe, T. B. Ottenstein, F. Serwane, K. Viering, A. N. Wenz, G. Z¨urn and S. Jochim, Phys.Rev. Lett. , 103201 (2010);[28] S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon, M. Ueda, Phys. Rev. Lett. , 023201(2010); S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon and M. Ueda, Phys. Rev. Lett. , 143201 (2011);[29] S.Knoop, F. Ferlaino, M.Mark, M.Berninger, H.Sch¨obel, H.-C.N¨agerl and R.Grimm, NaturePhysics , 227 (2009).[30] T. Lompe, T. B. Ottenstein, F. Serwane, A. N. Wenz, G. Z¨urn, S. Jochim, Science , 940 (2010). hase Diagrams of Three-Component Attractive Fermions [31] J.S. He, A. Foerster, X.-W. Guan and M. T. Batchelor, New Journal of Physics , 073009-1(2009).[32] M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, S. Jochim, C. Chin, J. Hecker Denschlag, andR. Grimm, Phys. Rev. Lett. , 103201 (2005).[33] T. Bergeman, M. G. Moore and M. Olshanii, Phys. Rev. Lett. , 163201 (2003).[34] X.-W. Guan, M. T. Batchelor, C. Lee and M. Bortz, Phys. Rev. B , 085120 (2007).[35] X. W. Guan, M. T. Batchelor, C. Lee and J. Y. Lee, Europhys. Lett. , 50003 (2009).[36] M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press,Cambridge, 1999).[37] P. Schlottmann, Int. J. Mod. Phys. B , 355 (1997).[38] M. T. Batchelor, X.-W. Guan, N. Oelkers and Z. Tsuboi, Adv. Phys. , 465 (2007).[39] C. H. Gu and C. N. Yang, Commun. Math. Phys. , 105 (1989).[40] J.-Y. Lee and X.-W. Guan, Nucl. Phys. B , 125 (2011).[41] F. Woynarovich and K. Penc, Z. Phys. B, , 269 (1991).[42] G. Catelani and E. A. Yuzbashyan, Phys. Rev. A78