Virtues and limitations of the truncated Holstein-Primakoff description of quantum rotors
Jorge G. Hirsch, Octavio Castanos, Ramon Lopez-Pena, Eduardo Nahmad-Achar
VVirtues and limitations of the truncatedHolstein-Primakoff description of quantum rotors
Jorge G. Hirsch, Octavio Casta˜nos, Ram´on L´opez-Pe˜na, andEduardo Nahmad-Achar
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exicoApdo. Postal 70-543, Mexico D. F., C.P. 04510E-mail: [email protected]
Abstract.
A Hamiltonian describing the collective behaviour of N interacting spinscan be mapped to a bosonic one employing the Holstein-Primakoff realisation, at theexpense of having an infinite series in powers of the boson creation and annihilationoperators. Truncating this series up to quadratic terms allows for the obtentionof analytic solutions through a Bogoliubov transformation, which becomes exact inthe limit N → ∞ . The Hamiltonian exhibits a phase transition from single spinexcitations to a collective mode. In a vicinity of this phase transition the truncatedsolutions predict the existence of singularities for finite number of spins, which haveno counterpart in the exact diagonalization. Renormalisation allows to extract fromthese divergences the exact behaviour of relevant observables with the number of spinsaround the phase transition, and relate it with the class of universality to which themodel belongs. In the present work a detailed analysis of these aspects is presentedfor the Lipkin model.PACS numbers: 42.50.Ct, 03.65.Fd, 64.70.Tg Keywords : quantum optics, coherent states, phase transitions, Holstein-Primakoff.
1. Introduction
The dynamics of N two-level systems is well described by the Lipkin Hamiltonian[1]. Born in nuclear physics, it has found extensive use in quantum optics, in thegeneration of squeezed states [2], multipartite entanglement [3], two-mode Bose-Einsteincondensates [4], and monomolecular magnets [5]. It represents an approximation toferromagnetic Ising models [6], exhibiting a second-order phase transition in the limit oflarge number of particles which is well described by mean field techniques [7, 8, 9]. Inmost cases a Holstein-Primakoff realisation is employed which, when truncated, providesanalytical solutions in the thermodynamic limit [10, 11]. For a finite number of atoms,observables like the ground state energy, the energy gap and the number of excitedatoms exhibit a singular behaviour at the phase transition, going to zero or to infinity,while numerical calculations show that they should remain finite. a r X i v : . [ qu a n t - ph ] M a r he truncated Holstein-Primakoff description of quantum rotors
2. The Lipkin Hamiltonian
The Lipkin Hamiltonian [1] describes the collective behaviour of N spins or two-levelatoms, with energy separation (cid:15) , which interact by scattering pairs of particles betweenthe two levels. In the quasi-spin formalism it has the form H = (cid:15)J z + γ x N J x + γ y N J y , (1)where J x , J y , J z are the three components of the angular momentum operator, with theusual commutation relations [ J j , J k ] = iε jkl J l , and γ x , γ y are the coupling strenghts. he truncated Holstein-Primakoff description of quantum rotors In the literature it is customary to employ the Holstein-Primakoff representation of theangular momentum operators [22] J + = √ N b † (cid:115) − b † bN , J − = √ N (cid:115) − b † bN b, J z = b † b − N , (2)where J + = J x + iJ y , J − = J x − iJ y , the Bose operators b † , b obey the commutationrelation [ b, b † ] = 1 , and the vacuum | (cid:105) of the bosons satisfies b | (cid:105) = 0. Making thesesubstitutions into H , Eq. (1), the bosonic Hamiltonian is built. Care must be takenin using the commutation relations to move the creation operators (outside the squareroots) to the right, and the annihilation ones to the left.The mean field description of this Hamiltonian is easily obtained employing asa trial state the Heinsenberg-Weyl coherent state | α (cid:105) , which is the eigenstate of thebosonic annihilation operator b | α (cid:105) = α | α (cid:105) , where α = ρ e iφ is a complex numbers, with ρ ≥ ≤ φ < π .The expectation value of the Hamiltonian for this coherent state provides the energysurface . Employing the approximation (cid:113) − b † bN | β (cid:105) ≈ (cid:113) − ρ N | β (cid:105) [23], which becomesexact in the thermodynamic limit, when the number of atoms goes to infinity, it takesthe simple form [24] (cid:104) α | H | α (cid:105) = (cid:15) (cid:18) ρ − N (cid:19) + γ x + γ y (cid:32) − ρ N (cid:33) (cid:16) ρ (cid:17) (3)+ γ x − γ y (cid:32) − ρ N (cid:33) ρ cos[2 φ ]For a given set of Hamiltonian parameters (cid:15), γ x , γ y , the values ρ c , φ c which minimisethis expression provide the mean field wave function. They are obtained by solving theequations for the energy surface critical points ∂ (cid:104) α | H | α (cid:105) ∂ρ = 0 , ∂ (cid:104) α | H | α (cid:105) ∂φ = 0 . (4)The solutions of these equations associated with the minima of the energy surface are[8, 9] ρ c = 0 , φ c undetermined, if γ x ≥ γ c and γ y ≥ γ c ,ρ c = (cid:114) N (cid:16) − γ c γ x (cid:17) , φ c = 0 , π, if γ x < γ c and γ x ≤ γ y ,ρ c = (cid:114) N (cid:16) − γ c γ y (cid:17) , φ c = 0 , π, if γ y < γ c and γ x > γ y , (5)with γ c ≡ − (cid:15) . The first case, γ x ≥ γ c and γ y ≥ γ c , defines the normal region (I), wherein the ground state all atoms are in their lowest energy state. In the first deformed region (II), where γ x < γ c and γ x ≤ γ y , it is energetically favoured to collectively exciteall atoms, and the ground state is doubly degenerate, as there are two critical phases φ c = 0 , π . The second deformed region (III), where γ y < γ c and γ x > γ y , is symmetricto the first one: it is obtained by interchanging J x ↔ J y . In what follows we will restrictour analysis to regions I and II. he truncated Holstein-Primakoff description of quantum rotors ε gs = (cid:104) α c | H | α c (cid:105) /N , and thefraction of excited atoms n e = 2 ρ c /N : ε gs = γ c γ x + γ y N , n e = 0 , region I ε gs = γ c + γ x γ x + ( γ x + γ c )( γ y + γ x )8 N γ x , n e = 1 − γ c γ x , region II . (6)Although they were obtained using different techniques, these mean fieldexpressions, shown as dashed red lines in Fig. 1, exactly coincide with those presented in[3, 8]. They reproduce reasonably well the numerical results even for a relatively smallnumber of particles, as can be seen in Fig. 1, where the numerical diagonalization ofthe Hamiltonian matrix H is shown as a continuous green line, for N= 10 and 40 atoms,for γ c = − γ y = 1. At this point it is worth to emphasize that the mean field (cid:45) (cid:45) (cid:45) Γ x (cid:45) (cid:45) (cid:45) Ε gs (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Γ x n e Figure 1. (Color online) Left: Ground state energy per atom, calculated throughnumerical diagonalization of the matrix Hamiltonian H for N= 10 (continuous greenline), and from Eq. (6) (dashed red line). Right: Fraction of excited atoms calculatedthrough numerical diagonalization of the matrix Hamiltonian H for N= 40 (continuousgreen line) and from Eq. (6) (dashed red line). Both plots were calculated using γ c = − γ y = 1. Peaked blue lines show the spurious results obtained with thetruncated Hamiltonian (see next section). description becomes exact in the thermodynamic limit, where only the first terms in theenergy per atom remain finite. The peak in the continuous blue lines at γ x = −
1, whichgoes to infinity for the fraction of excited atoms, is similar to the one shown in Fig.1 ofRef. [15]. It is an exact result coming from the truncated version of the Hamiltonianwhich has no counterpart in the numerical diagonalization of the Lipkin Hamiltonian(1) for any finite number of atoms, as explained in the next section.
3. Beyond mean field
The first step in going beyond the mean field description is the introduction of thedisplaced boson operators, as in [13, 3], c † = b † − ρ c , c = b − ρ c , (7) he truncated Holstein-Primakoff description of quantum rotors c | α c (cid:105) = 0. While in the normal phase, where ρ c = 0, there is not displacement, using c † , c allows for a general treatment of the Lipkinmodel beyond mean field.If the ground state is well described by the coherent state (the vacuum, in the caseof the normal phase), it is valid to make the approximation c † cN → (cid:104) c † c (cid:105) N (cid:28) . (8)It must be stressed that this expansion becomes exact only in the thermodynamic limit,where N → ∞ . For any finite N it will have problems, which become particularlyrelevant close to the phase transition.Far from the phase transition region it is possible to expand the square roots inEq. (2) in powers of N series, conserving terms of order N , N and N , and neglectingall negative powers of N . Care must be taken because ρ c is of order N . The truncatedversion H ( t ) of the Hamiltonian H , which is quadratic in the new bosons, is H ( t ) = A + B c † c + C (cid:16) c † + c (cid:17) (9)with A = − γ c (cid:18) ρ c − N (cid:19) + γ x N (cid:16) N − ρ c + 4 ρ c (cid:16) N − ρ c (cid:17)(cid:17) + γ y N (cid:16) N − ρ c (cid:17) ,B = − γ c + ( N − ρ c ) γ x N + ( N − ρ c ) γ y N , (10) C = ( N − ρ c ) γ x N − ( N − ρ c ) γ y N .
There is an extra term proportional to c † + c whose coefficient vanishes exactly whenemploying the values of ρ c given in Eq. (5). The Hamiltonian H ( t ) can be diagonalizedthrough the Bogoliubov transformation [10] c † = cosh (cid:20) Θ2 (cid:21) a † + sinh (cid:20) Θ2 (cid:21) a, c = cosh (cid:20) Θ2 (cid:21) a + sinh (cid:20) Θ2 (cid:21) a † , (11)in terms of the new bosons a † , a . When replaced in Eq. (9), it reads H ( t ) = A + B sinh (cid:20) Θ2 (cid:21) + C sinh[Θ] + ( B cosh[Θ] + 2 C sinh[Θ]) a † a + (cid:18) B C cosh[Θ] (cid:19) (cid:16) a † + a (cid:17) . (12)The last term cancels out by selecting tanh[Θ] = − CB . The truncated Hamitonian hasthe final diagonal form [10] H ( t ) = A + 12 (cid:16) √ B − C − B (cid:17) + √ B − C a † a = N ε ( t ) gs + ∆ a † a. (13)The ground state of the truncated Hamiltonian is the new vacuum | (cid:105) , which satisfies a | (cid:105) = 0. The coefficient of the last term is the gap ∆, the energy separation betweenthe ground and the first excited state in the normal phase, and between the ground andthe second excited state in the deformed region, where the first excited state becomes he truncated Holstein-Primakoff description of quantum rotors ≡ √ B − C = (cid:113) ( γ x − γ c )( − γ c + γ y ) region I , (cid:113) ( γ x − γ c ) ( γ x − γ y ) /γ x region II . (14) (cid:45) (cid:45) (cid:45) Γ x (cid:68) Figure 2. (Color online) The gap, Eq. (14), as function of γ x , shown as a dashedblue line for γ c = − γ y = 1, and the first and second excitation energies, obtainednumerically for N= 40, displayed as continuos green lines. Figure 2 plots the gap as a function of γ x as dashed blue line. It becomes null at thephase transition γ x = γ c . The excitation energies of the first and second excited states,obtained through exact diagonalization for N=40, are also displayed. The truncatedHamiltonian allows a good description of the gap, which becomes exact when N → ∞ .The minimum of the excitation energy is a precursor of the phase transition at finite N,which takes place at a different value of γ x for each N [25].The ground state energy per atom ε ( t ) gs is given by the constant terms in H ( t ) ,Eq. (13), divided by N ε ( t ) gs = γ c + γ c N + N (cid:113) ( γ x − γ c )( γ y − γ c ) region I , γ c + γ x γ x + γ x N + N (cid:113) ( γ x − γ c ) ( γ x − γ y ) /γ x region II . (15)It is plotted as a continuous blue line in Fig. 1 (left), and displays a spike at γ x = γ c which is an artifact of the truncation, and vanishes as N → ∞ .The fraction of excited atoms is obtained by expressing J z in terms of the newbosons ( a † , a ), and evaluating n e = (cid:104) J z (cid:105) N + 1 in the new vacuum. n ( t ) e = N γ x + γ y − γ c √ ( γ x − γ c )( γ y − γ c ) − N region I , γ x − γ c γ x + N √ γ x − γ c γ c γ y + γ x (3 γ c − γ x + γ y ) √ γ x ( γ c + γ x )( γ x − γ y ) − N region II . (16)The curve n ( t ) e vs. γ x is displayed as a continuous blue line in Fig. 1 (right). Itfollows closely the mean field prediction, except in a vicinity around γ x = γ c , where it he truncated Holstein-Primakoff description of quantum rotors
4. Renormalisation and critical exponents
The renormalisation procedure postulates that it is possible to extract the correctfunctional form O ( γ, N ), smooth and finite for any finite N, of any observable O ( t ) ( γ, N )which becomes singular at the phase transition [26] when described using the truncatedHamiltonian. If O ( t ) ( γ, N ) is singular as N β ( γ − γ c ) − α , employing a scaling function[ N ν ( γ − γ c )] α the regular function is built as [10] O ( γ, N ) = O ( t ) ( γ, N ) [ N ν ( γ − γ c )] α → N β ( γ − γ c ) − α [ N ν ( γ − γ c )] α = N β + α ν , (17)where the power α in the second term was selected to cancel out the singularity at γ c ,and ν defines the class of universality to which the model belongs. In the Lipkin modelnumerical analysis points to ν = [6, 25, 20]. We will show that this number can bededuced analytically using the fidelity susceptibility. Also we will exhibit the singularbehaviour of the point ( γ x , γ y ) = ( γ c , γ c ) in the parameter region, where ν has a differentvalue when approached along the lines γ y = γ c or γ x = γ c .The singular term in the energy per atom, which gives rise to the spurious spikewhen γ x → γ c , behaves like √ γ x − γ c /N , as shown in Eq. (15). Renormalizing it weobtain ε rengs ( γ x , N ) → ( γ x − γ c ) N − (cid:104) N ( γ x − γ c ) (cid:105) − = N − . (18)The singular term can be calculated numerically, substracting from the exact energyper atom the regular terms in Eq. (15). It goes to zero as function of N as predicted[10]. The divergence in the fraction of excited atoms can be manipulated in the sameway. Taking the divergent term from Eq. (16), it gives n rene ( γ x , N ) → ( γ x − γ c ) − N − (cid:104) N ( γ x − γ c ) (cid:105) = N − . (19)In Ref. [10] this dependence on N was also confirmed numerically. In classical information theory the fidelity measures the accuracy of a transmission [27].It also provides a powerful tool to study quantum phase transitions [21, 20]. For a purestate | ψ ( λ ) (cid:105) which varies as a function of a control parameter λ , the fidelity is definedas F ( λ, λ + δλ ) ≡ |(cid:104) ψ ( λ ) | ψ ( λ + δλ ) (cid:105)| . (20) he truncated Holstein-Primakoff description of quantum rotors H = H + λH I , which makes explicit thedependence on the control parameter, the fidelity susceptibility can be calculated as[20, 25] χ F = (cid:88) k (cid:54) =0 |(cid:104) ψ k ( λ ) | H I | ψ ( λ ) (cid:105)| ( E k ( λ ) − E ( λ )) , (21)where | ψ k ( λ ) (cid:105) denotes the k eigenstate of H with energy E k ( λ ), and k = 0 refers to theground state.In our case, we select H I = J x /N and study the dependence on γ x . The way toproceed is to express it in terms of the bosons a † , a , as J x = j + j a † a + j (cid:16) a † + a (cid:17) + j (cid:16) a † + a (cid:17) . (22)The first two terms do not contribute to χ F . The third one connects the ground statewith a one boson state, with energy E = ∆, the last one with a two boson state withenergy E = 2∆. In the normal phase, region I, we find j n = 0 , j n = N (cid:115) γ y − γ c γ x − γ c , (23)and in the deformed phase, region II,( j d ) = − N γ c γ x (cid:115) ( γ x − γ c ) ( γ x − γ y ) γ x , (24) j d = − N ( γ c (5 γ x − γ y ) − γ c γ x ( γ y + 3 γ x ) + 2 γ x γ y )8 γ x (cid:113) γ x ( γ x − γ c ) ( γ x − γ y ) . (25)Substituting in Eq. (21) we obtain χ nF = 132 ( γ x − γ c ) , (26) χ dF = − N (cid:113) ( γ x − γ c ) γ c γ x (cid:115) γ x ( γ c + γ x ) ( γ x − γ y )+ 1( γ c − γ x ) [ γ c γ x ( − γ c + 3 γ x ) + (3 γ c − γ x ) ( γ c + γ x ) γ y ] γ x ( γ c + γ x ) ( γ x − γ y ) . (27)Note that in the deformed region II γ x < γ c = − (cid:15) < γ x ≤ γ y , making the argumentin the square roots, ( j d ) and χ dF positive. In the deformed region the first term is oforder N and the second of order N . For any value of γ x (cid:54) = γ c the first term is the oneto be considered in the thermodynamic limit [20, 28].The fidelity susceptibility χ F exhibits very interesting features. It is divergent inboth phases as γ x → γ c , but with different powers of ( γ x − γ c ) and of N in each phase[20, 28]. Renormalising the fidelity susceptibility the following exponents are found he truncated Holstein-Primakoff description of quantum rotors α β β + α νγ x ≥ γ c νγ x < γ c ν The Lipkin model shows distinct critical exponents for the fidelity susceptibilityaround the critical point. The exact numerical calculations fully confirm that the criticalexponents are different on both sides of the critical point. It can be clearly seen in theplots of the rescaled fidelity susceptibility against N ( γ x − γ c ), which for any values of Nfalls exactly on the same line, with a noticeably asymmetry on both sides of the criticalpoint [20, 28, 25].It implies that the renormalized fidelity susceptibility χ F scales as N ν in thenormal region, and as N ν in the deformed region. As χ F is continuos at thephase transition for any finite N, the two exponents should be equal, implying that ν = .In this way we have shown analytically that the Lipkin Hamiltonian belongs to thisclass of universality. It follows that χ F diverges as N at the phase transition. Thisdependence of χ F at γ c is confirmed numerically [25]. It is shown in the left of Fig.3, where the maximum value of χ F is plotted against N in a log -log plot. Thepoints are clearly along a straight line, with slope 1 . ≈ /
3. The numerical fit is χ F = 0 . N . . The values of N range from 2 = 1024 to 2 = 65536 . To the bestof our knowledge, this is the first time that the discontinuity in the critical exponentsof the fidelity susceptibility is employed to obtain analytically the class of universalityassociated with the Lipkin model.
11 12 13 14 15 16 Log N12141618Log Χ max
11 12 13 14 15 16 Log N20222426283032Log Χ max Figure 3. (Color online) Log-log plots of the maximum of the fidelity susceptibilityas function of the number of particles N in the system, for γ c = − γ y = 1 . γ y = − . The above deduction is valid for γ y > γ c in the normal phase, region I, and for γ y < γ x in the deformed phase, region II. There is a very singular behaviour along theline γ y = γ c in the deformed region II. In this case the fidelity susceptibility has thefunctional form χ dF ( γ y = γ c ) = − N ( γ x − γ c ) γ c γ x (cid:115) γ x γ x + γ c . (28) he truncated Holstein-Primakoff description of quantum rotors ν ( γ y = γ c ) = 1, and χ F → N as it approachesthe phase transition. This behaviour is plotted in the right of Fig 3, with slope 2 . χ F = 0 . N along the line γ y = γ c , γ x → γ c − .
5. Conclusions
We have studied the Lipkin model of collective spins employing the Holstein-Primakoffmapping of the quantum rotor to a bosonic field. We have shown how the differentphases of the system are found by using the coherent Heisenberg-Weyl state, which isan eigenstate of this boson. The possibility of going beyond the mean field descriptionwas reviewed. It was obtained by introducing a second set of displaced bosonic operatorswhose vacuum is the coherent state, expanding the square roots appearing in themapping of the angular momentum operators as an infinite series, replacing it in theHamiltonian, and truncating by keeping terms of order N , N / and N . In this waya truncated Hamiltonian is built, which is quadratic in the new bosons, and can bediagonalized exactly by means of a Bogoliubov transformation. It provides a harmonicdescription of the Hamiltonian, with an excitation energy, the gap, which vanishes atthe phase transition. It was shown that this description is consistent with the exactbehaviour of the system at finite number of spins N, obtained through a numericaldiagonalization of the full Hamiltonian.On the other hand, it was also exhibited that the predictions for the ground stateenergy per particle and for the fraction of excited atoms exhibit a spike around the phasetransition which is not present in the exact numerical calculations for a finite numberof atoms. The size of this spike goes, in the case of the energy per particle, to zero inthe thermodynamic limit, while for the fraction of excited atoms is infinite, and remainsinfinite even in the limit N → ∞ . These singularities are a remainder that resultsobtained employing the truncated Hamiltonian for a finite of atoms close to the phasetransition do not correspond to the exact ones, but carry anyway useful information.Through renormalisation, it was obtained a recipe to build smooth and well behavedfunctions from singular ones by multiplying them with an appropriate function of thecontrol parameter and the number of particles.The behaviour for large N of the singular term in the energy per particle and of thenumber of excited atoms was obtained in this way. It was also shown that the fidelitysusceptibility, an observable widely used in quantum optics, provides not only a veryefficient tool to detect the precursor of the phase transition at finite N, but allows oneto obtain analytically the class of universality to which the Lipkin model belongs: theone which is renormalized with a general factor N / ( γ − γ c ) α . The exponent 2 / γ y = γ c and γ x < γ c , in which the renormalizing function is different, i.e., the system behavesin a different way along this line. Numerical calculations confirm the N dependence of he truncated Holstein-Primakoff description of quantum rotors References [1] Lipkin H J, Meshkov N and Glick A J 1965
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