Analysis of quantum phase transition in some different Curie-Weiss models: a unified approach
AAnalysis of quantum phase transition in somedifferent Curie-Weiss models: a unified approach ∗ Carla Maria Pontes Carneiro † Programa de P´os-gradua¸c˜ao em Modelagem Matem´atica e ComputacionalCEFET-MG - Centro Federal de Educa¸c˜ao Tecnol´ogica de Minas GeraisBelo Horizonte, MG, Brazil.Giancarlo Queiroz Pellegrino ‡ Departamento de Matem´aticaCEFET-MG - Centro Federal de Educa¸c˜ao Tecnol´ogica de Minas GeraisBelo Horizonte, MG, Brazil.July 19, 2018
PACS numbers: 05.70.Fh, 73.20.Mf, 03.65.SqKeywords: semiclassical limit, quantum phase transition
Abstract
A unified approach to the analysis of quantum phase transitionsin some different Curie-Weiss models is proposed such that they aretreated and analyzed under the same general scheme. This approachtakes three steps: balancing the quantum Hamiltonian by an appropri-ate factor, rewriting the Hamiltonian in terms of SU (2) operators only,and obtention of a classical Hamiltonian. SU (2) operators are obtainedfrom creation and annihilation operators as linear combinations in thecase of fermions and as an inverse Holstein-Primakoff transformationin the case of bosons. This scheme is successfully applied to Lipkin,pairing, Jaynes-Cummings, bilayer, and Heisenberg models. ∗ Submitted for publication on August 29, 2017. † [email protected] ‡ [email protected] (corresponding author) a r X i v : . [ qu a n t - ph ] D ec Introduction
Quantum phase transition (QPT) has been given much attention and scien-tific effort during the last years since it proved to be a collective phenomenonobserved in very different areas [1]. More recently, the concept — originallydefined as a property of the ground-state energy — was enlarged to encom-pass also the manifestations found in excited states (ESQPT) [2, 3]. Oneof the major interests in this field comes from the possibility of a betterunderstanding of the intricate behavior of many-body systems, observed invery different physical situations, although “direct comparison of treatmentsusing different complicated notations is difficult”, as described by Lipkin,Meshkov, and Glick in their famous 1965 paper [4].We propose here a unified approach to analyze the occurrence of QPTin a set of models coming from different areas: the Lipkin and pairing mod-els from nuclear physics [4, 5], the N -atom Jaynes-Cummings model fromquantum optics [6, 7, 8], the bilayer model from condensed matter [9, 10],and the Heisenberg model from magnetism [11]. Although they are very dif-ferent in their physical natures, these models share the common feature thatthe expected mean values of their observables admit an expansion in powersof 1 /N , where N is, generically speaking, the number of constituents of thesystem; in short, they are Curie-Weiss models [12]. This property is deeplyconnected with the possibility of a (semi)classical description based on aproper Hamiltonian function corresponding to the quantum Hamiltonianoperator for the model in question. Several quantum-classical connectionscan then be made, which lead to semiclassical characterization and analysisof QPT in these models. Such connections range from quantum spectra andclassical orbits to critical value of a parameter at QPT and the change in(in)stability in phase space.With this perspective, we propose the following scheme to analyze theabove-cited models in a unified approach.Firstly, the original quantum Hamiltonian is rewritten in terms of SU (2)operators only. This is achieved either by defining pseudo-spin or pseudo-angular momentum operators as suitable combinations of the original fermioniccreation-annihilation ones, or by performing an (inverse) Holstein-Primakofftransformation [13] on the original bosonic creation-annihilation operatorsto produce again SU (2) operators.Secondly, the quantum Hamiltonian, written as H = H + gH int , (1)is balanced by a factor N s [14], where N counts the (constant) number of2articles in the system and s = [ H int ] − [ H ] . (2)In this expression [ H a ] is the highest power of SU (2) operators in H a .In a final step, a classical Hamiltonian function is obtained from these SU (2)-based quantum Hamiltonians, by following a prescription set by Lieb[15]. With the quantum and classical Hamiltonians, it is possible to obtain,and compare, results associated with QPT both on quantum and classicallevels. These steps are summarized as H (cid:16) a, a † ; b, b † (cid:17) → H ( J z ; J ± ) → H balanced ( J z ; J ± ) → H ( p, q ) , (3)where a and a † are generic fermionic operators, b and b † are bosonic ones, J z and J ± are sets of SU (2) operators, and p and q are pairs of canonicalconjugate classical momenta and coordinates. H and H stand for quantumand classical Hamiltonians respectively.Moreover, since all the models we will treat can be put in the form H = H + gN s H int , (4)where H sums up the energy of the free constituents and H int takes intoaccount the contribution to the energy given by the interaction betweenthose elements, it will be seen that the classical Hamiltonians can be put inthe form H = p n + λ H int ( p, q ) , (5)where n is 0, 1 or 2, depending on the model, and λ = λ ( g ) relates thequantum interaction parameter g to the corresponding classical one λ .It will then be seen that, in these models, QPT presents universal char-acteristics concerning quantum spectra and the behavior of mean values asfunctions of the interaction parameter g , and also concerning the structureand behavior of classical orbits with respect to the parameter λ .That scheme is illustrated in some detail with the Lipkin model andthen is applied comparatively to the other models. In each case, a briefdescription of the model is given, a quantum treatment characterizes QPTin terms of the energy spectra and of a given mean value, both as functionsof the interaction parameter g ; finally, a classical treatment analyzes theclassical phase space in order to characterize QPT in terms of the orbitsand (in)stability of the critical points.In the next section, the models are briefly presented, the above-describedscheme is applied to each model, and the results are shown. In section 3, the3esults are compared and discussed from the perspective of common featureseventually present. A short section 4 concludes this article. In this section, we take the Lipkin model as a prototype to illustrate our pro-cedure. The other models are treated more briefly within the same scheme.In all cases, sets of SU (2) operators are designed to accomplish for the in-teractions between the constituents of the model and to satisfy the usualcommutation relations[ J z , J ± ] = ± J ± [ J + , J − ] = 2 J z . (6)Also, we illustrate the quantum results in the same mathematical situation:the operators H are diagonalized within subspaces of the same dimension2 J + 1 with J = 100 and with the interaction parameter g varying in [0 , λ before and afterthe transition.We note here that the results presented below agree with those eventuallyfound in the literature and obtained by other methods, when this is the case. The Lipkin-Meshkov-Glick model appears for the first time in NuclearPhysics as a test for the validity of some many-body approximation methods[4]. Since then, it has become a treatable and very useful model in manybranches [16, 17, 18, 19, 20]. Particularly interesting is the fact that, beingexactly solvable, the Lipkin model can serve as a reliable laboratory to testideas and methods related to QPT [21, 22, 23].We consider a system of N fermions, each occupying one of two levelsseparated by an energy ε . This situation configures a N -fold degeneratetwo-level system with single-particle states given by quantum numbers k =1 , , . . . , N for the particular degenerate state, and σ = ± k . Taking a two-body interaction between thefermions, such that pairs of particles are scattered up or down from onelevel to the other without changing the values of k , this system can be We follow the established literature and refer to this model as the Lipkin model. H = 12 ε (cid:88) kσ σa † kσ a kσ + 12 V (cid:88) kk (cid:48) σ a † kσ a † k (cid:48) σ a k (cid:48) ( − σ ) a k ( − σ ) . (7)The first term just counts the difference between the number of particlesin the upper and lower levels, while the second term describes the change inenergy occurring when a pair of particles goes from the same level − σ (withdifferent k and k (cid:48) ) to the other level + σ .Defining SU (2) pseudo-spin operators as J + = (cid:88) k a † k (+1) a k ( − = J †− (8) J z = 12 (cid:88) kσ σa † kσ a kσ , (9)this quantum Hamiltonian becomes H = εJ z + 12 V (cid:16) J + J − (cid:17) . (10)In order to put in evidence the critical value of the interaction at QPT,we take the form [21, 14] H = J z + g N (cid:16) J + J − (cid:17) , (11)where the whole Hamiltonian (10) is scaled by the constant factor ε . Amongthe different values of J allowed by addition of angular momenta, the groundstate of Hamiltonian (7) is realized with the eigenvalues J = N/ m z = − J = − N/ J and J z .The next step is the obtention of a classical Hamiltonian with the usualdefinitions of variables [15] j k = lim J →∞ J k J ( k = + , − , z ) . (12)The relations j x = ( j + + j − ) = (cid:113) − j z cos φj y = i ( j + − j − ) = (cid:113) − j z sin φj z = cos θ (13)5ake the variables p = j z and q = φ canonical conjugate variables in thesense of the classical Hamilton equations of motion. In this limit, from thequantum Hamiltonian (11), we get the classical Hamiltonian H = p + λ (cid:16) − p (cid:17) cos (2 q ) (14)with − ≤ p ≤ (cid:104) J z (cid:105) for different values of the parameter g in eq.(11). Figures 1(c) and 1(d) show the classical phase space for two differentvalues of the parameter λ in eq. (14).We observe that the quantum spectra are symmetric with respect totheir centers, a property that appears correspondingly in the classical phasespaces. Inflection points, characteristic of QPT in these Curie-Weiss models,are present in the spectra of the Lipkin model for g ≥ . E = ± λ ≥ .
0. We refer to these orbits asopen and closed orbits, respectively. The appearance of the closed orbitsaround equilibrium points (maximum at q = 0 and minimum at q = ± π/ (cid:104) J z (cid:105) indeed shows a sudden rise in the collective excitation. We note,furthermore, that the inflection points which are seen in the upward half ofthe spectra predict a corresponding collective de-excitation for the highest-energy state. The pairing model was conceived after a suggestive analogy between thespectra of nuclei and those of superconducting metallic states, where inter-acting pairs of particles with equal and opposite momenta are the majorinterest [24]. We take here the form studied by Krieger and Goeke [25], inwhich two N -fold degenerate levels have an energy separation ε , just like inthe Lipkin model. These could be two nuclear j -shells j and j , with stateslabeled as kσ , where σ = − j and σ = +1 for j identify the shellsand where k specifies the z -component of an angular momentum. In thepairing model, one is interested in the scattering of pairs of particles whichare coupled with opposite values k and − k .In order to take into account both the two levels and these pairing in-teractions, we write the quantum Hamiltonian H = 12 ε (cid:88) kσ σa † kσ a kσ − g (cid:88) kσ a † kσ a † ¯ kσ (cid:88) k (cid:48) σ (cid:48) a ¯ k (cid:48) σ (cid:48) a k (cid:48) σ (cid:48) . (15)6igure 1: Lipkin model : (a) quantum spectra for interaction parametervalues g = 0 , . , . . . ,
3; (b) quantum mean value (cid:104) J z (cid:105) ; (c) classical phasespace for λ = 0 .
5; (d) classical phase space for λ = 1 .
5. The phase transitionoccurs at g = 1 . λ = 1 .
0. 7ere a † and a are fermion creation and annihilation operators, a † ¯ k (+1) =( − j − k a † ( − k )(+1) and a † ¯ k ( − = ( − j − k a † ( − k )( − .Differently from the Lipkin model, in this pairing model the interactionbetween the particles may change the value of k , if a coupled pair in states( k, − k ) is scattered to a new state ( k (cid:48) , − k (cid:48) ). Due to this coupling, we definetwo sets of pseudo-spin operators, one for the upper level and another forthe lower one, as J = (cid:88) k> a † k ( − a † ¯ k ( − = J † − (16) J z = 12 (cid:88) k a † k ( − a k ( − − N J = (cid:88) k> a † k (+1) a † ¯ k (+1) = J † − (18) J z = 12 (cid:88) k a † k (+1) a k (+1) − N . (19)These are SU (2) sets of operators and [ J α , J β ] = 0 for any α, β = − , + , z .Applying these definitions to Hamiltonian (15), we get H = ε ( J z − J z ) − gN ( J + J ) ( J − + J − ) . (20)Moreover, the operators J , J and ( J z + J z ) commute with theHamiltonian and their mean values are constants of motion. As in theLipkin model, the ground state here is realized for eigenvalues J = J = N ,and m z = − N for both J z and J z . Since J z refers to the lower particlelevels, with energies − ε , it is a maximum for this ground state, while J z isa minimum. In this way, one needs only to diagonalize H in the subspacewhere J z + J z = 0.Also, using the same definitions (12) and (13), we get from the quantumHamiltonian (20) the classical Hamiltonian H = ( p − p ) − Λ (cid:20) − (cid:16) p + p (cid:17) + 2 (cid:113) − p (cid:113) − p cos ( q − q ) (cid:21) . (21)The following canonical transformations p a = p + p q a = q + q p b = p − p q b = q − q rewrites (21) as H = 2 p b − Λ (cid:110) − (cid:104) ( p a + p b ) + ( p a − p b ) (cid:105) +2 (cid:113) − ( p a + p b ) (cid:113) − ( p a − p b ) cos ( q b ) (cid:27) . (23)Now, the restriction J z + J z = 0 takes the classical form p + p = 0 = p a , so H becomes H = p + λ (cid:16) − p (cid:17) cos (cid:18) q (cid:19) , (24)in appropriate energy unity, and where the index b is suppressed as it isno longer necessary. Note that, here, λ assumes negative values in order tofollow the interaction assumed in (15).Again, Figures 2(a) and 2(b) show respectively quantum results for en-ergy spectra and the mean value (cid:104) J z (cid:105) for different values of the parameter g in eq. (20). Figures 2(c) and 2(d) show the classical phase space for twodifferent values of the parameter λ in eq. (24).It is seen that the same general structure found in the Lipkin modelreappears with the pairing model. In this case, the spectra are not symmetricbut still repeat the lower half of Lipkin spectra, with the inflections pointsassociated with the quantum phase transition for g ≥ .
0. Also, there arethe closed orbits (librations) around an equilibrium point, for λ ≤ − . (cid:104) J z (cid:105) measures the number of elements in the shell j andshows a collective population of this shell for g above the critical value. N -atom Jaynes-Cummings model The Jaynes-Cummings model [26] with N atoms — also named in the lit-erature as Dicke model [6] or Tavis-Cummings model [27] — considers theinteraction of N two-level atoms with a single mode radiation field of fre-quency ω (see also [7] and [8]). For each atom, the two levels are separatedby an energy ε . Since its beginning, the model takes the N atoms (ormolecules) as a single spin system described by SU (2) operators J , J − , J z with N = 2 J , and the radiation field described by bosonic creation andannihilation operators b , b † , and b † b . The quantum Hamiltonian for thisspin-boson system, with ¯ hω = ε , is written as H = ε b † b + εJ z + g (cid:16) bJ + b † J − (cid:17) + g (cid:48) (cid:16) b † J + bJ − (cid:17) , (25)9igure 2: Pairing model : (a) quantum spectra for interaction parametervalues g = 0 , . , . . . ,
3; (b) quantum mean value (cid:104) J z (cid:105) ; (c) classical phasespace for λ = − .
5; (d) classical phase space for λ = − .
5. The phasetransition occurs at g = 1 . λ = − . SU (2) operators, we apply an inverse Holstein-Primakoff transformation[13, 28] on the bosonic operators b , b † , and b † b , so as to spinorize them. Wedefine b † b = J + J z b † = J √ J − J z b = 1 √ J − J z J − , (26)and take | m (cid:105) = | ψ n (cid:105) , where | ψ n (cid:105) are the eigenstates of b † b with eigenvalues n = J + m . Note that | ψ (cid:105) = |− J (cid:105) and b † b counts the number of excitations,starting at the lowest eigenvalue of J z .Applying the transformation (26) to Hamiltonian (25), we get H = J + ( J z + J z )+ g √ N (cid:18) √ J − J z J − J + J √ J − J z J − (cid:19) (27)+ g (cid:48) √ N (cid:18) J √ J − J z J + 1 √ J − J z J − J − (cid:19) , where, again, we use ε as the energy unity.In the rotating wave approximation, with g (cid:48) = 0, the observable (cid:16) b † b + J z (cid:17) commutes with H and is a constant of motion; so ( J z + J z ) also does.In the counter-rotating wave approximation, with g = 0, the observable (cid:16) b † b − J z (cid:17) commutes with H , as well as ( J z − J z ). In diagonalizingHamiltonian (27), we take advantage of these constants to reduce the sizeof the matrix.Again, taking definitions (12) and (13), followed by the canonical trans-formations (22), the classical Hamiltonian reads H = A + p a + λ (cid:113) p a + p b ) (cid:113) − ( p a − p b ) cos ( q b ) (28)+ λ (cid:48) (cid:113) p a + p b ) (cid:113) − ( p a − p b ) cos ( q a ) . In this expression, A is a constant with no contribution to dynamics, and λ , λ (cid:48) are appropriate parameters derived from g and g (cid:48) respectively.For the Jaynes-Cummings model, we separate two cases of interest. Inthe rotating wave approximation, λ (cid:48) = 0 and the chosen constant J z + J z =0 leads to p + p = 0 = p a . In this case, we have H λ (cid:48) =0 = λ (1 + p ) (cid:112) − p cos ( q ) . (29)11t is seen that the parameter λ is a global scale factor, so the phase spaceremains unaltered for any value λ > λ = 0, we take J z − J z =0 to get p − p = 0 = p b and H λ =0 = p + λ (cid:48) (1 + p ) (cid:112) − p cos ( q ) . (30)In the sequel, we present two sets of figures for these two cases (Figs. 3and 4). Both sets bring spectra and mean value of an observable as quantumresults, and phase space pictures as classical results.In the rotating-wave case, due to the commutation of J z + J z with theHamiltonian, the interaction parameter g is a global scale factor, and theclassical phase space is invariant with respect to λ . The mean value (cid:104) J z (cid:105) does not show any change for g >
0. One can observe that there is, in fact,an inflection point at the center of the spectra, but there is no accompanyingchange in stability in the classical phase space. Indeed, the mean value (cid:104) J z (cid:105) ,calculated for the energy eigenstate at the center of the spectrum has thesame constant behavior as that shown in Fig. 3(b).In the counter-rotating-wave case, however, the whole phase-transitionstructure observed in Lipkin and pairing models is again present. One ob-serves the inflection points in the lower part of the spectra, the collectiveexcitation of the atoms for g (cid:48) > . The bilayer model, in which two separate layers are each occupied by a two-dimensional fermionic gas, has been used to test Einsteins’ prediction thatat low temperatures all the bosons in the system should condense into thesame quantum state. These bilayer models provide the necessary bosonsin the form of excitons given by bound states of electron-hole pairs [29].In 2004, an experiment with a bilayer system under strong magnetic fielddescribed exciton condensation in electron-electron parallel layers [30]. Amodel Hamiltonian was proposed to mimic the dynamics of creation andannihilation of excitons formed from the N/ H = δb † b + g N/ (cid:88) k =1 (cid:16) a † k a † k b + b † a k a k (cid:17) , (31)12igure 3: Rotating-wave Jaynes-Cummings model : (a) quantum spec-tra for interaction parameter values g = 0 , . , . . . ,
3; (b) quantum meanvalue (cid:104) J z (cid:105) ; (c) classical phase space for any λ > Counter-rotating-wave Jaynes-Cummings model : (a) quan-tum spectra for interaction parameter values g (cid:48) = 0 , . , . . . ,
3; (b) quantummean value (cid:104) J z (cid:105) ; (c) classical phase space for λ (cid:48) = 0 .
7; (d) classical phasespace for λ (cid:48) = 0 .
8. The phase transition occurs at g (cid:48) = 1 . λ (cid:48) = 1 / √ b , b † stand for boson and a , a † stand for fermion creation and an-nihilation operators. This Hamiltonian can be written in terms of SU (2)operators only with the aid of definitions similar to (16-17) for the fermionicoperators J = N/ (cid:88) k =1 a † k a † k = J † − (32) J z = 12 N/ (cid:88) k =1 (cid:16) a † k a k + a † k a k (cid:17) − N , (33)and of the inverse Holstein-Primakoff introduced for the Jaynes-Cummingsmodel (26) for the bosonic ones b † b = J + J z b † = J √ J − J z b = 1 √ J − J z J − . (34)With these new operators, and taking δ = 1 as energy unity, the bilayerHamiltonian reads H = ( J + J z ) + g √ N (cid:18) J √ J − J z J − + J √ J − J z J − (cid:19) . (35)In this model, the total number N of fermions is constant and is givenby the operator N op = N/ (cid:88) k =1 (cid:16) a † k a k + a † k a k (cid:17) + 2 b † b, (36)therefore, N op = 4 J + 4( J z + J z ) (37)and N = 4 J with ( J z + J z ) also commuting with Hamiltonian (35). Wework in the subspace J z + J z = 0 in this model.Once again, definitions (12) and (13), followed by the canonical trans-formations (22), put the classical Hamiltonian in the form H = (1 + p a ) + p b + λ (cid:113) p a + p b ) (cid:113) − ( p a − p b ) cos ( q b ) . (38)Since we choose J z + J z = 0, we get p a = 0 and, neglecting the constantterm, we arrive at H = p + λ (1 + p ) (cid:112) − p cos ( q ) (39)15igure 5: Bilayer model : (a) quantum spectra for interaction parametervalues g = 0 , . , . . . ,
3; (b) quantum mean value (cid:104) J z (cid:105) ; (c) classical phasespace for λ = 0 .
7; (d) classical phase space for λ = 0 .
8. The phase transitionoccurs at g = 1 / √ λ = 1 / √
2. 16s the classical Hamiltonian.We show below quantum and classical results for the bilayer model. Fig-ure 5 follows the same displaying of the precedent cases.It is seen that the bilayer model presents results with the same structureas does the N -atom Jaynes-Cummings model in the counter-rotating case.For the bilayer model, at the critical value g = 1 / √
2, there happens a collec-tive formation of excitons. The difference between the quantum parametersis due to the fact that in the bilayer model one has N = 4 J , whereas in theJaynes-Cummings model N = 2 J . Some care, however, must be exercisedin comparing these models. The interaction part of the quantum bilayerHamiltonian is, in fact, equal to the interaction part of the rotating-waveHamiltonian for the Jaynes-Cummings model. The Heisenberg model is a spin model, usually studied in statistical me-chanics to show magnetic phase transitions [11]. However, it is an attractivemodel also in the few-body domain, where possible integrable and chaoticversions were considered [31, 32, 33, 34].Here we look at the anisotropic two-spin Heisenberg model, often writtenas H = F ( S · S + σS z S z ) (40)or, in our scheme, H = J z J z + g J J − + J − J ) . (41)In this model, ( J z + J z ) commutes with H and provides a constant ofmotion, which we take as J z + J z = 0. Relations (12) and (13), followed bythe same canonical transformations (22), produce the classical Hamiltonian H = ( p a − p b ) ( p a + p b ) + λ (cid:113) − ( p a − p b ) (cid:113) − ( p a + p b ) cos ( q b ) . (42)From J z + J z = 0, we have p a = 0, so we end with H = p + λ (cid:16) − p (cid:17) cos ( q ) , (43)where λ and g have opposite signs.Figure 6, showing quantum and classical results for the Heisenberg model,appears below. Observe that the quantum mean-value curve is symmetric,due to the quadratic term J z J z in Hamiltonian (41).17igure 6: Heisenberg model : (a) quantum spectra for interaction pa-rameter values g = − , − . , . . . , − .
5; (b) quantum mean value (cid:104) J z (cid:105) ; (c)classical phase space for λ = 0 .
5; (d) classical phase space for λ = 1 .
5. Thephase transition occurs at g = − . λ = 1 . (cid:104) J z (cid:105) (Fig. 6(b)) — and another one at energy zero for g > − .
0. Accordingly, there are two types of critical points in the classi-cal phase space, but only one of them reflects QPT, with a correspondingchange in stability. One type, at q = ± π , is always present and its sur-rounding librations are confined by a separatrix orbit — as seen in Fig. 6(c)— while the other one, at q = 0, appears as λ goes above 1 .
0, with closedorbits around it and the — now rectangular — separatrix between the crit-ical points. Therefore, Fig. 6(c) is associated with the mean value (cid:104) J z (cid:105) for g > − .
0, and Fig. 6(d) with that part with g < − .
0. Once they exist,these critical points are static with respect to variations in the interactionparameter. This is a different behavior in comparison with the other models,where the critical point moves up or down as a function of λ . As a finalremark, for values g > − .
0, the inflection at energy zero has precisely theenergy of the separatrix seen in Fig. 6(c). However, like the inflection pointfound in the rotating-wave case of the Jaynes-Cummings model, it is notaccompanied by a change in stability in phase space.
A number of universal characteristics have been pointed out in the literature,concerning quantum phase transitions (QPT) in these Curie-Weiss models[22, 35, 36, 37, 38, 9, 10, 33]. In the following, we comment on some of them.QPT is signaled by the presence of an inflection point in the spectrum.As long as the interaction parameter g is varied, this inflection point movesupwards in the spectrum and a critical parameter for QPT is usually taken asthe lowest energy level becomes the inflection point, which then disappears.This situation is seen in all models here, except for the Jaynes-Cummingsmodel in the rotating wave approximation. Also, as the lower excited energylevels become the inflection point, the corresponding quantum states man-ifest QPT. This is a clear illustration of a more general definition of QPT,known as excited state quantum phase transition (ESQPT) [2, 3].At QPT, there happens a conspicuous increase in the mean value of anobservable associated with the number of one type of the elements in thesystems. Again, this is seen in all but one of our cases.Finally, the energy of the inflection point in the quantum spectrum is the19nergy of a separatrix orbit in the classical phase space, and the appearanceof this separatrix, as the classical interaction parameter λ crosses the criticalvalue, marks the transition.In addition to these characteristics, the unified approach presented hereallows a number of new observations and comparisons.Starting with the obvious resemblance in the curves for the mean values(part (b) in the figures), one realizes that the lower half of the spectra havethe same structure. This resemblance shows up also in the structure of theclassical phase space. In fact, since the open orbits are clearly associatedwith the free term H of the quantum Hamiltonian, and the closed ones withthe interaction term H int , the competition between these terms responds forthe mechanism under which QPT takes place. In this way, the appearance ofclosed orbits around an equilibrium point — a sudden change in the global(in)stability of phase space [39, 36] — signals that a new quantum regimeis accessible to the system as a whole, the collective formation of excitonsin the bilayer model or the collective excitation of fermions in the Lipkinmodel, as examples.Moreover, a unified approach may bring up interesting questions, con-cerning physical effects in different systems with similar behavior in thegeneral scheme. Counter-rotating Jaynes-Cummings model (eq. (30)) andbilayer model (eq. (39)) have both the same classical expression. Do thephysical effects seen in one system have analogs in the other? This samequestion arises for the pair formed by Lipkin model (14) and pairing model(24). In the Heisenberg model, QPT assumes a somewhat different classicalmanifestation. The (dis)appearance of closed orbits around a critical pointdoes not follow the movement of the equilibrium point with respect to theborder of the phase space. This is an otherwise static critical point, withrespect to the interaction parameter. Static critical points are present alsoin the rotating-wave Jaynes-Cummings model but, in this case, there is nochange in classical phase space neither in the quantum mean value. Arethere physical effects differentiating the quantum phase transitions in thisHeisenberg model and the other ones? In this work, we propose a unified approach to the analysis of quantumphase transitions in some different Curie-Weiss models. Two points are dis-tinctive in the proposal: the balancing of free and interaction Hamiltoniansby a proper factor, and the transformation of fermionic and bosonic creation20nd annihilation operators into SU (2) operators. The first point avoids aneventual QPT from occurring for an interaction parameter value too close tozero, which would make difficult its observation. The second point puts allthese models under the same general scheme, which permits treating differ-ent physical situations, making comparisons, and raising analogous questionsfor different areas.We do not present here a detailed analysis of the physical results for thesemodels, mainly of the possible translations of physical questions usuallypresent in one model but not in a similar other, as pointed out by theirgeneral form in this scheme. Work along this line is in progress and willappear in due future. Acknowledgements
The authors wish to express their warm gratitude to Jos´e Geraldo Peixoto deFaria and Helen Barreto Lara for many illuminating discussions. C.M.P.C.thanks to CAPES-Brazil.
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