Extended Lipkin-Meshkov-Glick Hamiltonian
EExtended Lipkin-Meshkov-Glick Hamiltonian
R. Romano, X. Roca-Maza, ∗ G. Col`o, and Shihang Shen ( 申 时 行 ) Dipartimento di Fisica “Aldo Pontremoli”, Universit`a degli Studi di Milano, 20133 Milano, Italy andINFN, Sezione di Milano, 20133 Milano, Italy (Dated: September 9, 2020)The Lipkin-Meshkov-Glick (LMG) model was devised to test the validity of different approximateformalisms to treat many-particle systems. The model was constructed to be exactly solvable andyet non-trivial, in order to capture some of the main features of real physical systems. In the presentcontribution, we explicitly review the fact that different many-body approximations commonly usedin different fields in physics clearly fail to describe the exact LMG solution. With similar assumptionsas those adopted for the LMG model, we propose a new Hamiltonian based on a general two-bodyinteraction. The new model (Extended LMG) is not only more general than the original LMG modeland, therefore, with a potentially larger spectrum of applicability, but also the physics behind itsexact solution can be much better captured by common many-body approximations. At the basisof this improvement lies a new term in the Hamiltonian that depends on the number of constituentsof the system and polarizes it, producing an explicit symmetry breaking.
PACS numbers: 21.60.Fw, 11.15.Ex, 21.60.-n,21.60.Jz,71.70.Ej
A complete microscopic study of quantum many-bodysystems with realistic Hamiltonians is the central prob-lem of different fields in physics such as atomic physics,condensed matter, or nuclear physics among others [1–6].The systems under study are in general very complicatedto tackle and different many-body approximations havebeen proposed along the years in order to understand thediverse phenomenology: from pairing in superconductorsor superfluidity in He to collective states in nuclei. How-ever, to test, compare and better understand state-of-the-art formalisms may become unpractical, especially whenoriginating from different fields. One way forward is tosimplify the problem under study by proposing a moresimple albeit non-trivial Hamiltonian. That is, an Hamil-tonian that contains some of the relevant features of thephysical system under study. Then to test and learn fromthe very complex many-body techniques available in theliterature may become a manageable alternative [7–10].With this aim, the LMG model was proposed [7]. Suchmodel considers a system of N fermions distributed ontwo-levels, each of them M -fold degenerate and separatedby an energy ε . Each state is described by two quantumnumbers: σ specifies the level (+1 and − p specifies the particulardegenerate state within a given level. In this schematicmodel, fermions interact by a monopole interaction thatdoes not change the p quantum number. The interactionhas two channels. The first scatters pairs of particles inthe same level to the other level while the other scattersone particle to the upper level and, at the same time,another to the lower level. Since each particle has onlytwo possible states, the model can be also understood as asystem of spins. In absence of interaction, the model willpredict all spins aligned along the same direction whileother more complex configurations will be only favored ∗ [email protected] when the interaction is switched on. As it can be easilyunderstood from this analogy, the power of the LMGmodel arises from the fact that it mimics some features ofdifferent physical many-body systems and, in addition, itadmits an exact solution: using a quasi-spin formulation,the Hamiltonian can be written in terms of the operatorsthat generate the SU(2) algebra.To date the LMG model [7] has been applied and ex-tended in a variety of fields and for the study of dif-ferent phenomena. An introduction to the LMG modelhas been given in the context of the many-body nuclearproblem in Ref. [8]. The LMG has been studied atfinite temperature within the mean-field approximation[11], applied to the study of quantum phase transitions[12, 13], spontaneous symmetry breaking [14–21], finitesize effects [22–25], long-range interacting spin-chains[26], quantum metrology [27], optical cavity QED [28],Bose-Einstein condensation [29], quantum spin squeez-ing [30, 31] or quantum entanglement [32, 33], amongothers [34, 35]. It has been extended to three levelsystems [36, 37] and used to test different many-bodyapproaches such as the Random Phase Approximation(RPA) [38, 39], the coupled-cluster method [40] or den-sity functional theory [41]. Nevertheless, a modificationof the Hamiltonian has not been previously studied.For the two level system described above, the mostgeneral Hamiltonian that does not change the quantumnumber p , written in second quantization, reads H = 12 ε (cid:88) pσ σa † pσ a pσ + 12 (cid:88) σ ,σ σ ,σ (cid:88) p,p (cid:48) V σ σ ,σ σ a † pσ a † p (cid:48) σ a p (cid:48) σ a pσ (1)where, for convenience, we have introduced a short-hand notation for the matrix elements V σ σ ,σ σ ≡(cid:104) pσ , p (cid:48) σ |V| pσ , p (cid:48) σ (cid:105) . Expanding the sums in Eq. (1)one arrives at the following expression for the Hamilto- a r X i v : . [ nu c l - t h ] S e p nian: H = εJ z − V J + J − ) − W J + J − + J − J + ) − G ( J + + J − )( N −
1) + W N − F N . (2)In Eq. (2) we have defined the quasi-spin operator J andthe particle number operator N as follows, J z = 12 (cid:88) pσ σa † pσ a pσ , N = (cid:88) pσ a † pσ a pσ ,J + = (cid:88) p a † p + a p − , J − = (cid:88) p a † p − a p + . (3)The components of the quasi-spin operator J followthe usual SU(2) algebra, namely [ J + , J − ] = 2 J z and[ J z , J ± ] = ± J ± . The particle number operator com-mutes with all of them. Hence, the Hamiltonian (2) canbe solved exactly using the angular momentum represen-tation. The coupling constants V, W, G and F in Eq. (2)are defined in terms of the matrix elements in Eq. (1) asfollows, − V ≡ V ++ , −− − F ≡ V + − , + − = V ++ , ++ = V −− , −− − W ≡ V + − , − + − G ≡ V ++ , − + = V −− , + − . (4)Matrix elements should be hermitian and symmetric withrespect to the exchange of particles, σ ↔ σ and σ ↔ σ . The equalities (4) between the matrix elements defin-ing F and G in our model are representative of physicalcases in which V ++ , ++ and V −− , −− as well as V ++ , − + and V −− , + − , can be expected to be of the same order. Hence,one may also assume V + − , + − = ( V ++ , ++ + V −− , −− ) / N , the lasttwo terms in Eq. (2) produce just a constant shift in theenergy and, thus, can be renormalized by a change in thevalue of ε without losing generality. Under this condition,and neglecting also the term in G in the Hamiltonian (2),one can recover the original LMG model proposed in Ref.[7]. In the present work we propose instead to keep themore general formulation of the Hamiltonian within thetwo-level assumption of the LMG model, that is, H = εJ z − V J + J − ) − W J + J − + J − J + ) − G ( J + + J − )( N − . (5)The new one-body term proportional to G ( N −
1) scattersone particle upward or downward. It is important to notethat:i) the strength of this term increases with the numberof particles, hence dominating for large N ;ii) the new term produces a fundamental differencein the energy dependence on the particle number,when compared to the LMG model; iii) the term in G ( N −
1) introduces an explicit sym-metry breaking.In order to clarify the last point, we briefly recall thesymmetries of the LMG and ELMG Hamiltonians. Theunperturbed term depends on J z , the term on W is pro-portional to J − J z while the term in V is proportionalto J x − J y . Both Hamiltonians commute with J whilethe terms in V and G do not commute with J z and in-troduce an explicit symmetry breaking. The term in V proportional to J x − J y limits the continuous rotationalsymmetry of the LMG Hamiltonian to a discrete sym-metry [7]. The term in G , that can be also written as2 G ( N − J x , explicitly breaks the rotational symmetry.In Fig. 1, we show some exact and approximate resultsfor the LMG model (left panels) and for the extendedmodel presented here ELMG (right panels). For detailson the exact and some approximate solutions of the LMGmodel, we refer the reader to Refs. [7, 38]. The method-ology to solve the ELMG model is fully analogous to thatof the LMG model. In this figure, we explore the behaviorof the ground as well as the lowest excited state energieswith respect to the model parameters, comparing exactresults for a system with N = 20 particles with threedifferent approximations of common use in many-bodyphysics [8]: the Hartree-Fock (HF), the 1-particle 1-holeRandom Phase Approximation (RPA) and the 2-particle2-hole RPA or second RPA (SRPA). In the figure, wehave redefined the coupling constants as v ≡ V ( N − /ε , w ≡ W ( N − /ε and g ≡ G ( N − /ε as well as the totalenergy as e ≡ E/ε where E is the total energy solu-tion of the Hamiltonian in Eq. (5). We have chosen toshow the results for some specific values of the parame-ters but the general features displayed in Fig. 1 lead tosimilar conclusions for other choices. In the left panels,the results for the LMG model are shown. Two regionsare highlighted since the LMG model predicts the ex-istence of two different HF ground states depending onthe value of the parameters v and w : v + w < v + w > v + w ≈ v + w < G pro- v w=v g= e r e l a ti v e [ % ] e v w=v g=v -505 x e r e l a ti v e [ % ] ExactHFRPASRPA e Ground StateGround State 1 st Exc. State2 nd Exc. State1 st Exc. State2 nd Exc. State
N=20 v+w >1 v+w <1 FIG. 1. Relative energy of the ground state with respect tothe exact result [%], and energy e ≡ E/ε of the first andsecond excited states, for a system with N = 20 particles asa function of the model parameters v ≡ V ( N − /ε , w ≡ W ( N − /ε and g ≡ G ( N − /ε . The exact solution iscompared to HF, RPA and SRPA (this latter only for excitedstates). The left panel corresponds to the LMG model for v = w and the right panel to the new ELMG model for v = w = g . duces a fundamental difference in the ground state ofthe system when solved within the simplest many-bodyapproximation, that is, within the HF approach. TheHF ground state can be used as a basis for higher-orderapproximations, like RPA and SRPA, in a conceptuallysimilar way in which it serves as a basis for second order–and higher orders– many-body perturbation theory. Onthis regard, we have noticed by performing SRPA calcu-lations that the LMG model lacks contributions form thecoupling of 1 p -1 h to 2 p -2 h states, which is a consequenceof the simplicity of the Hamiltonian. As a consequence,many-body methods beyond the RPA should not be ex-pected to be accurate in the LMG model. This problemis overcome by the new ELMG Hamiltonian. In short,we can stress that this new Hamiltonian is more generalthan the original LMG Hamiltonian and, therefore, hasa potentially larger spectrum of applicability; moreover,common many-body approximations capture much bet-ter the physics behind the exact solution of the ELMGmodel.Let us now inspect in some detail the HF ground statein the two models. The HF wave function is a Slater de-terminant that has the general form | HF (cid:105) = Π p b † p, − | (cid:105) .The b † p,σ must be related with a † p,σ by a unitary transfor-mation, (cid:32) b † p, + b † p, − (cid:33) = (cid:18) cos α − sin α sin α cos α (cid:19) (cid:32) a † p, + a † p, − (cid:33) , (6)and for α (cid:54) = 0 each state will be represented as a super-position of bare particles in the upper and lower levels.This is an example of quasi -particles (see for example Eq. -3 -2 -1 0 1 2 3 α -15-10-50510 e H F v + w = 0.5 v + w = 1.0 v + w = 1.5 v + w = 2.0 -3 -2 -1 0 1 2 3 α -30-20-1001020 e H F v=w g=0 v=w=g N=20
FIG. 2. Hartree-Fock energy e HF ≡ E HF /ε as a function ofthe variational parameter α for different values of the couplingconstants, as predicted by the LMG model (left panel) andthe ELMG model (right panel). (7.1) of Ref. [8]). As a function of the variational param-eter α , the ground state within the HF approximation(cf. left panel of Fig. 2) is: α = 0 for v + w < α = 1 / ( v + w ) for w + v >
1. In-stead, for the case of the ELMG model (cf. right panelof Fig. 2), the minimum of the ground state energy, E HF ε = − N (cid:18) cos α + wN − v + w α + 2 g sin α (cid:19) , (7)evolves as a function of the parameters in a continuousway. The value of the variational parameter which corre-sponds to the minimum of the energy is always positiveprovided g (cid:54) = 0. The ELMG model shows two differenttypes of transitions : one with g = 0 (LMG model) be-tween a non-degenerate ground state with v + w < v + w >
1; the other, be-tween a degenerate ground state with g = 0 and v + w > g (cid:54) = 0 and ∀ v , w .From our discussion here, one realizes that the HF ap-proximation for the ground state ( α corresponding to theminimum of E HF ) works much better for deformed states( α (cid:54) = 0) in both models, that is, for quasi-particles ratherthan particles. This can be seen in the lower panels ofFig. 1: the better the exact solution of the LMG modelis described in HF, the larger the value of the param-eters ( v and w ) is, which corresponds to a larger valueof α at the minima of E HF . Specifically, the HF solu-tion of the LMG model is accurate within a 0.1% forvalues of v + w >
2; and the HF solution of the ELMGmodel is accurate for all values of the parameters withina 0.05%, yet improving also its accuracy as the value ofthe parameters increases, that also corresponds to largervalues of α . We can conclude that our findings provide aground for the idea that HF works better in the case of deformed solutions. We should note that there is an ex-ception to this general trend. In the perturbative limit,the non-deformed HF solution ( α = 0) for the case ofthe LMG model, and the slightly deformed HF solutionfor the ELMG model ( α →
0) are also very accurate.Regarding the description of the lowest excited statesstudied here, it is evident that an accurate HF groundstate together with the extended Hamiltonian containingthe G term is of paramount importance for approximatemethods to satisfactorily describe the exact result.In summary, we have presented a new exactly solv-able model (ELMG) inspired by the LMG model thathas been shown to be useful for applications in differentfields in physics. The spectrum of applicability of thenew model is broader than that of the LMG model. Anew term in the ELMG Hamiltonian is responsible for anexplicit symmetry breaking leading, in general, to a non-degenerate HF ground state energy with a variational pa-rameter ( α ) that favors the description of the system as a superposition of bare particles in the upper and lowerlevels. The new term in the Hamiltonian also enablesother fundamental differences: the coupling between 1 p -1 h and 2 p -2 h is non zero as it happens for the LMG inSRPA calculations; and displays an explicit dependenceon the number of constituents of the system.Importantly, we have shown that many-body approx-imations of common use such as the HF, RPA or SRPAdescribe remarkably well the exact ELMG solution forthe ground state and first two excited states, providingin that way the possibility of a simple yet reliable study ofcomplicated real physical systems whenever the ELMGmodel can be applied.The authors thank H. Sagawa for useful discussions.Funding from the European Union’s Horizon 2020 re-search and innovation programme under grant agreementNo 654002 is acknowledged. [1] R. B. Wiringa, Rev. Mod. Phys. , 231 (1993).[2] J. Sapirstein, Rev. Mod. Phys. , 55 (1998).[3] G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. , 601 (2002).[4] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[5] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev.Mod. Phys. , 517 (2008).[6] M. I. Katsnelson, V. Y. Irkhin, L. Chioncel, A. I. Licht-enstein, and R. A. de Groot, Rev. Mod. Phys. , 315(2008).[7] H. J. Lipkin, N. Meshkov, and A. J. Glick, NuclearPhysics , 188 (1965).[8] P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1980).[9] W. P. L. Jr. and D. H. Kobe, American Journal of Physics (1973), https://doi.org/10.1119/1.1987540.[10] M. Moshinsky, American Journal of Physics , 763(1968).[11] A. H. Blin, B. Hiller, and L. Junqing, Journal of PhysicsA: Mathematical and General , 3993 (1996).[12] H. T. Quan, Z. D. Wang, and C. P. Sun, Phys. Rev. A , 012104 (2007).[13] P. Solinas, P. Ribeiro, and R. Mosseri,Physical Review A (2008),https://doi.org/10.1103/PhysRevA.78.052329.[14] H. A. Jahn and E. Teller, Proceedings of the Royal Soci-ety A , 220 (1937).[15] I. Bersuker, Jahn Teller Effect (Cambridge UniversityPress, Cambridge, England, 2006).[16] P.-G. Reinhard and E. Otten, Nuclear Physics A ,173 (1984).[17] F. S. Ham, Phys. Rev. , A1727 (1965).[18] A. J. Millis, B. I. Shraiman, and R. Mueller, Phys. Rev.Lett. , 175 (1996).[19] J. K¨ugel, P.-J. Hsu, M. B¨ohme, K. Schneider, J. Senkpiel,D. Serrate, M. Bode, and N. Lorente, Phys. Rev. Lett. , 226402 (2018).[20] O. Vendrell, Phys. Rev. Lett. , 253001 (2018).[21] H. Liu and G. Khaliullin, Phys. Rev. Lett. , 057203(2019). [22] S. Dusuel and J. Vidal, Phys. Rev. Lett. , 237204(2004).[23] S. Dusuel and J. Vidal, Phys. Rev. B , 224420 (2005).[24] P. Ribeiro, J. Vidal, and R. Mosseri, Phys. Rev. E ,021106 (2008).[25] Y. Huang, T. Li, and Z.-q. Yin, Phys. Rev. A , 012115(2018).[26] A. Lerose, B. Zunkovic, J. Marino, A. Gam-bassi, and A. Silva, Physical Review B (2019),https://doi.org/10.1103/PhysRevB.99.045128.[27] G. Salvatori, A. Mandarino, and M. G. A.Paris, Physical Review A (2014),https://doi.org/10.1103/PhysRevA.90.022111.[28] S. Morrison and A. S. Parkins, Phys. Rev. Lett. ,040403 (2008).[29] T. Zibold, E. Nicklas, C. Gross, and M. K. Oberthaler,Phys. Rev. Lett. , 204101 (2010).[30] J. Ma, X. Wang, C. Sun, and F. Nori, Physics Reports , 89 (2011).[31] Y.-C. Zhang, X.-F. Zhou, X. Zhou, G.-C. Guo, and Z.-W. Zhou, Phys. Rev. Lett. , 083604 (2017).[32] R. Or´us, S. Dusuel, and J. Vidal, Phys. Rev. Lett. ,025701 (2008).[33] X. xing Zhang and F. li Li, Physics Letters A , 1053(2013).[34] Y. Ma, T. M. Hoang, M. Gong, T. Li, and Z.-q. Yin,Phys. Rev. A , 023827 (2017).[35] Y.-H. Ma, S.-H. Su, and C.-P. Sun, Phys. Rev. E ,022143 (2017).[36] K. Hagino and G. F. Bertsch, Physical Review C (2000), https://doi.org/10.1103/PhysRevC.61.024307.[37] M. Grasso and F. Catara, Phys. Rev. C , 014317(2000).[38] G. Co’ and S. D. Leo, Modern Physics Letter A (2015), 10.1142/S0217732315501965.[39] E. I. Duzzioni and J. R. Marinelli (2004) pp. 322–327.[40] N. I. Robinson, R. F. Bishop, and J. Ar-ponen, Physical Review A (1989),https://doi.org/10.1103/PhysRevA.40.4256.[41] D. Lacroix, Physical Review C79