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Entanglement entropy in collective models Julien Vidal,1 Sébastien Dusuel,2 and Thomas Barthel3 1Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France 7 2Lycée Louis Thuillier, 70 Boulevard de Saint Quentin, 80098 Amiens Cedex 3, France 0 3Institute for Theoretical Physics C, RWTH Aachen, 52056 Aachen, Germany 0 Wediscussthebehavioroftheentanglemententropyofthegroundstateinvariouscollectivesys- 2 tems. Resultsforgeneralquadratictwo-modebosonmodelsaregiven,yieldingtherelationbetween n quantumphasetransitionsofthesystem(signaledbyadivergenceoftheentanglemententropy)and a theexcitationenergies. SuchsystemsnaturallyarisewhenexpandingcollectivespinHamiltoniansat J leadingorderviatheHolstein-Primakoffmapping. Inasecondstep,weanalyzeseveralsuchmodels 2 (the Dicke model, the two-level BCS model, the Lieb-Mattis model and the Lipkin-Meshkov-Glick 2 model) and investigate the properties of the entanglement entropy in the whole parameter range. Weshowthatwhenthesystemcontainsgaplessexcitationstheentanglemententropyoftheground ] statedivergeswith increasing systemsize. Wederiveandclassify thescaling behaviorsthat can be h met. c e m PACSnumbers: 03.65.Ud,03.67.Mn,21.10.Ev,73.43.Nq - t a I. INTRODUCTION also spin) systems [4, 5, 6, 7, 8, 9, 10, 11, 12]. Note that st in conformal field theory, the prefactor is determined by t. One of the most striking aspects of quantum mechan- the central charges [13, 14]. In higher-dimensional crit- a ics is certainly the superposition principle which has no ical systems, the scaling behavior is different for bosons m counterpart in classical physics. A direct consequence andfermions. Forbosons,fieldtheoreticalandnumerical - of this principle is the entanglement of physical states resultsshowthatthearealawprevailsalsointhecritical d n whose characterization has been a subject of intense re- case [1, 15, 16, 17, 18]. For fermions, the area law is in o search in the recent years. However, the interest for the this case corrected by a logarithmic factor [18, 19, 20]. c entropy in quantum systems has first emerged in black Note also that recently, noncritical fermionic systems [ holes physics [1] pre-empting the recent favour in con- with rapidly decaying interactions have been shown to 2 densed matter physics. The relationship between entan- violate this area law [21]. v glement and quantum phase transitions (QPTs) has es- All the above cited studies focus on low-dimensional 3 pecially drawn much attention since the original works systems. The aim of the present work is to investi- 3 on one-dimensional (1D) spin systems [2, 3, 4, 5]. In gate collective systems which, in a sense, can be seen 8 these studies, it has been shownthatentanglementmea- as infinite-dimensional. The physics of these models can 0 suresareverysensitive tothe presenceofacriticalpoint be easily analyzed by a classical approach but the en- 1 in the phase diagram. For example, in the 1D quantum tanglement entropy comes from the quantum fluctua- 6 0 Ising model in a transverse field, the entanglement en- tions around the classical ground state. Here, we map / tropy of the ground state has been shown to be finite those collective systems onto models with two interact- t a everywhere except at the critical field where it diverges. ing bosonic modes, each of them corresponding to one m This entropy, which is the central topic of this paper is part of a bipartite splitting of the original system. We - defined as follows. Suppose that we split the degrees of show that the scaling of the entanglement entropy de- d freedom of the system under consideration into two sub- pends mainly on two factors: the number of Goldstone n systems and . The entanglement entropy of any modes (0, 1 or 2) or, when approaching a critical point, o A B E pure quantum state ρ = ψ ψ with respect to this bi- the number of vanishing excitation gaps (1 or 2). After c | ih | : partition is defined as the von Neumann entropy of the giving the general results for quadratic two-mode boson v reduced density matrix Hamiltonians,wediscussindetailseveralcollectivemod- i X els covering different behavior of entropy scaling. = Tr (ρ lnρ )= Tr (ρ lnρ ), (1) r E − A A A − B B B The structure of this paper is the following. In Sec. a where ρ = Tr ρ. Several fundamental questions II, we derive the expression of the ground state entan- A,B B,A arise concerning this entropy. In particular when, why glement entropy for general quadratic two-mode boson and how does it diverge? In d-dimensional non-critical Hamiltonians and extract its explicit dependence on the (gapped) systems, the entanglement entropy is generi- excitationenergies. SectionIIIisdevotedtothestudyof callyproportionaltothesurfaceareaLd−1 oftheconsid- theDickemodelforwhichtheentropyisshowntodiverge ered subsystem with linear size L (area law). In critical as 1lnN at the critical point where N is the number of 6 systems, logarithmic corrections to this scaling may oc- two-level systems. In Sec. IV, we analyze the two-level cur. Inparticular,alogarithmicscaling lnLisfound BCS model in which the entropy diverges as 1lnN in in critical one-dimensional boson and feErm∝ion (and thus the broken phase and as 1lnN at the critical p2oint. In 3 2 the antiferromagnetic long-range Lieb-Mattis model dis- particle energies are necessarily positive. This is given, cussed in Sec. V, we exactly compute the ground state if we restrict to ω >0. After a Bogoliubovtransforma- ± entanglement entropy which behaves as lnN. Section tion,H isdiagonalizedintermsofnewbosonicoperators VI focuses on the Lipkin-Meshkov-Glick (LMG) model b˜ [22], for which we analyze the dependence of the entropy on several parameters (anisotropy, system and subsystem H =(b˜†)T ∆b˜+ 1Tr(∆ V), (5) sizes). In each case, we compare our analytical predic- 2 − tions based on the scaling hypothesis (see Appendix A) with numerics. and the Green’s function matrices read G++ =R†(ϕ)ω1/2R†(ψ)∆−1R(ψ)ω1/2R(ϕ), (6a) II. GENERAL QUADRATIC TWO-MODE BOSON MODEL G−− = R†(ϕ)ω−1/2R†(ψ)∆R(ψ)ω−1/2R(ϕ). (6b) − Let us consider the most general quadratic two-mode Theentanglemententropybetweenmodes1and2isthen boson Hamiltonian given by [18] 1 H =(b†)TVb+ (b†)TWb†+h.c. , (2) µ+1 µ+1 µ 1 µ 1 2 = ln − ln − , (7a) h i E 2 2 − 2 2 whereb=(b ,b ). Werestrictheretothecasewherethe 1 2 µ= G++G−− = G++G−−. (7b) couplingmatrices V andW arerealso thatthere aresix − 1,1 1,1 − 2,2 2,2 free parameters. The aim is to find a simple expression Now, let us expqress all quantitiqes in terms of the initial for the entanglement entropy between both modes. As coupling matrices. Therefore, let us set: shown in Ref. [18], the reduced density matrix obtained bytracingoveroneofthetwomodescanbe expressedin d d terms of the Green function matrices defined as G+ij+ = V −W = d0 d1 , (8) (b† +b )(b† +b ) and G−− = (b† b )(b† b ) . It (cid:20) (cid:21) h i i j j i ij h i − i j − j i isthereforeconvenienttoreparametrizethe Hamiltonian which leads to (2) as 1 V W =R†(ϕ)ωR(ϕ), (3a) ω± = d0+d1 ǫ 4d2+(d0 d1)2 , (9) − 2 ± − V +W =R†(ϕ)ω−12 R†(ψ)∆2R(ψ)ω−12R(ϕ), (3b) h p i where ǫ=sign(d d ). 0 1 − where Similarly, setting: cosϕ sinϕ R(ϕ)= , (4) s s sinϕ cosϕ V +W = 0 , (10) (cid:20)− (cid:21) s s 1 (cid:20) (cid:21) ω =diag(ω ,ω ) and ∆=diag(∆ ,∆ ). The matrices + − + − V W need to be positive definite, because the single- one obtains the excitation energies: ± 1 ∆ = 2ds+d s +d s ǫ′ (2ds+d s +d s )2 4(d d d2)(s s s2) , (11) ± 0 0 1 1 0 0 1 1 0 1 0 1 2 ± − − − r h p i with ǫ2d[s(d +d )+d(s +s )]+(d d )(d s d s ) ǫ′ =sign 0 1 0 1 0− 1 0 0− 1 1 . (12) { 4d2+(d d )2 } 0 1 − Of course,∆± haveto be positive,giving the paramepterrangefor whichthis transformationis well-defined. It is also useful to introduce the two parameters t=tan(2ϕ)=2d/(d d ), (13) 0 1 − 2(d d d2)1/2[s(d d ) d(s s )] 0 1 0 1 0 1 u=tan(2ψ)= − − − − . (14) 2d[s(d +d )+d(s +s )]+(d d )(d s d s ) 0 1 0 1 0 1 0 0 1 1 − − 3 The quantity µ defined in Eq. (7b) is then given by Holstein-Primakoff(HP) bosonrepresentationofthe an- gular momentum [30] reads: µ= 1+(X κ+X κ−1)2, (15) − + S = b†N1/2(1 b†b/N)1/2 =(S )† , (19) p + − − with N S = b†b , (20) z t(1+u2)1/2(ν ν−1) [2u+t(ν+ν−1)] − 2 X = − ± , (16) ± 4(1+t2)1/2(1+u2)1/2 with [b,b†]=1, so that we now have to deal with a two- bosonproblem. Inthethermodynamicallimitandinthe and normal phase, one has b†b /N 1 and we can expand h i ≪ the square root in (19) to obtain the following form of ω ∆ ν = +, κ= + . (17) the Hamiltonian: ω− s∆− r N H = ω +ω b†b+ωa†a+λ a†+a b†+b 0 0 Note that these expressions are still valid for the special − 2 case d =d provided one sets ǫ=sign(d). + (1/N) . (cid:0) (cid:1)(cid:0) ((cid:1)21) 0 1 O This simple form allows one to investigate the various behaviors of the entropy that can be met in two-mode At this order, one thus has to deal with a quadratic boson systems. Indeed, Eq. (15) clearly states that the Hamiltonianand we can makeuse of the results givenin entropydivergeswhenanyofthequantitiesω ,∆ van- Sec. II. AsfoundinRef. [26],thetwoexcitationenergies ± ± ishes or diverges and several cases must then be distin- are given by guished. In the following, we will only focus on systems for ω2+ω2 16λ2ωω +(ω2 ω2)2 which gaps vanish, signalling the presence of a quantum 0± 0 − 0 ∆ = r .(22) phase transition. In the next section, we consider the ± q √2 Dicke model [23] in which one gap vanishes at the critical point but the other remains finite. In Sec. IV, we Note that ∆±, as defined in Eq. (11), depends on consider the two-level BCS model for which both gaps ε = sign(ω2 ω2) which is assumed here to be pos- − 0 vanish simultaneously at the transition. We show how itive without loss of generality. At the critical point these different behaviors lead to different finite-size scal- λc =√ωω0/2,thesystemundergoesasecond-orderQPT ing for the entropy at the critical point. and∆ vanishesas(λ λ)1/2 whereas∆ = ω2+ω2 − c− + 0 remains finite. p In this model, the parameter µ is given by: III. THE DICKE MODEL 4ωω λ2 ∆ ∆ µ(0) = 1+ 0 + + − 2 , whLicehtudsecsocrnisbiedsertthheeisnitnegrlaec-mtioondeoDf iNcketwHoa-mleivlteolnsiyasnte[2m3s] s 16λ2ωω0+(ω2−ω02)2(cid:18)∆− ∆+ − (cid:19) with a single bosonic mode wherethesuperscript”(0)”referstotheorderinthe1/N expansion. The entanglement entropy between modes a λ H =ω S +ωa†a+ a†+a (S +S ) , (18) and b is then given by Eq. (7a). In the vicinity of the 0 z + − √N critical point, the entropy thus diverges as: (cid:0) (cid:1) wherea† anda arebosoniccreationandannihilationop- 1/4 eratorssatifying[a,a†]=1. Thecollectivespinoperators (0) =1+ln λ5c + [(λ λ)1/2]. are defined as Sα = Ni=1σαi/2 where the σα’s are the E "2(λc−λ)(ω2+ω02)2# O c− Pauli matrices, and S =S iS . (23) ± x y P ± We refer the reader to Ref. [24] for a description of We note that this expression is in agreement with those the phase diagram as well as the super-radiance physics obtainedinRef. [26]withadifferentapproach. However, in this model. Here, we focus on the entanglement en- the constant term in this latter study is wrong and we tropy studied in Refs. [25, 26] and discuss the linear en- give here the correct value. tropyrecentlyanalyzedinRef. [27]. Letus alsomention To extract the behavior of the entropy at the critical that other entanglement measures such as the concur- point,onesimplyinvokesthescalinghypothesisdiscussed rence have been studied in this model [25, 28, 29]. in Ref. [29] for this model and detailed in Appendix A. Our goal here is not to reproduce the results of Lam- In the present context, one indeed has, near the critical bert et al. [25] but to show how to extractthe finite-size point, (0) 1ln(λ λ) which yields 1lnN. E ∼ −4 c − E ∼ 6 behavior of the entropy at the critical point. Thus, we Note that this exponent 1/6 is close to the numerical only consider the normal (symmetric) phase (λ < λ = result(0.14 0.01)obtained fromexact diagonalizations c √ωω0/2) characterized by limN→∞ a†a /N = 0. The [26]. ± h i 4 Similarly, one can easily compute the linear entropy withSA,B =SA,B iSA,B and[a,a†]=[b,b†]=1. Inthe ± x ± y which reads : thermodynamicallimitandinthe normalphase,onehas a†a /N 1 and b†b /N 1 so that the Hamiltonian 1 (0) =1 , (24) chan bie wr≪itten as:h i ≪ Elin. − µ(0) Nh 1 1 1 as shown in Appendix B. In the vicinity of λc, one has H = + h (a†a+b†b) (a†b†+ab) − 2 − 2 − 2 − 2 El(i0n). =1−"(λc−λ)8(cid:0)λω5c2+ω02(cid:1)2#1/4+O[(λc−λ)3/(42]5.) ++N11(cid:16)aa†ab†+b2b+(cid:16)†ba+†ba††22(cid:17)ba+2+a†ba†22bb+2(cid:17)a†2ab† 2N Consequently, using the scaling hypothesis, and since + (1(cid:16)/N2). (cid:17) (30) (0) 1 (λ λ)1/4 one expects 1 N−1/6 at O Elin.− ∼ c− Elin.− ∼ the transition as predicted in Ref. [27] in the adiabatic At order N0, the Hamiltonian is easily diagonalized and regime (ω/ω 1). Here again, we wish to emphasize 0 onethenfinds that, inthis phase,bothgapsareequalto thattheexpres≪siongivenbyLibertiet al. isonlyvalidin ∆ = h(h 1). Thus, at the critical point h = 1, both this regime. Thus, their expression of the linear entropy − gaps vanish (κ=1). is only correct at first order in ω/ω as can be easily p 0 The parameter µ can however still be expressed in checked by expanding (25) in powers of this parameter terms of the gap ω/ω . 0 To conclude this section, we mention that the same 2h 1 µ(0) = − , (31) calculationcanbe performedinthe brokenphaseλ>λc 2∆ (see Ref. [26]). Further, the exact correction of order 1/N could, in principle, be easily calculated but its ex- where the superscript ”(0)” again refers to the order in pressionwouldbetoolongtobegivenforthismodel. We the 1/N expansion. The entanglement entropy between shall compute it for the systems considered thereafter. sets and , in the thermodynamical limit and in the A B symmetric phase is then givenby Eq. (7a). In the vicinity of the critical point, the entropy thus behaves as: IV. THE TWO-LEVEL BCS MODEL (0) =1 2ln2 ln(h 1)1/2+ (h 1). (32) E − − − O − We now turn to another problem which can also be The scaling hypothesis (see Appendix A) immediately mapped onto a two-mode boson system, but where both implies 1lnN at the transition point. gaps vanish at the critical point. The two-level BCS E ∼ 3 Before comparing with numerics and discussing the modeloriginatesfromsuperconductivity[31,32]butcan broken phase, we also give the next order correction be translated into a simple spin model (see for instance which can be computed exactly in this model. At or- Ref. [33]). Its entanglement properties have recently der 1/N, the Hamiltonian can be diagonalized using the been discussed but only from the concurrence point of canonical transformations method [29]. This allows one view [34, 35]. Here, we shall show that its entropy cap- to evaluate the correction of order 1/N to µ tures some nontrivialfeatures due to the vanishing gaps. The two-level BCS Hamiltonian reads h h 1 1 µ(1) = + . (33) H = S2+S2 h(SA SB). (26) 2 − ∆4 ∆3 −N x y − z − z (cid:18) (cid:19) Here, the indices (cid:0)and re(cid:1)fer to two distinct sets of As explained in Ref. [36], µ(1) is sufficient to determine N/2 spins 1/2 subAjected tBo antiparallel magnetic fields. thefirstcorrectiontotheentropy(seeAppendixC). The Here, we have introduced the collective spin operators entropy at order 1/N is indeed obtained by Taylor ex- SA,B = σi/2, and S = SA +SB (where α = pandingexpression(7a)whereµisalsocomputedatthis α i∈A,B α α α α order as above. One then obtains: x,y,z). We further restrict our study to the total mag- P netization S = 0 which is reminiscent of the fermion z µ(1) µ(0)+1 number conservation in the initial BCS problem. Let us (1) = ln . (34) E 2 µ(0) 1 first discuss the symmetric phase h>1 characterizedby (cid:18) − (cid:19) lim 4 SA /N = lim 4 SB /N =1. As previ- N→∞ h z i − N→∞ h zi This expressionallows to check explicitly the scaling hy- ously, we perform a 1/N expansion using the HP boson pothesis(seeAppendixA)since,atthisorder,andinthe representation of the spin operators: vicinity of the critical point, one has SA = N/4 a†a, SB =b†b N/4, (27) SzA = (SA)−† =(N/2)1z/2 1 2−a†a/N 1/2a, (28) eE = eE(0)+N1E(1), (35) + − − 1 S+B = (S−B)† =(N/2)1/2b(cid:0)† 1−2b†b/N(cid:1) 1/2 , (29) ∼ (h−1)−1/2(cid:20)1− N(h−1)3/2(cid:21). (36) (cid:0) (cid:1) 5 33333 55555 h=0 44444 h=1 2 22222 h=1 33333 EEEEE EEEEE 22222 11111lllllnnnnnNNNNN 11111 22222 NNNNN 11111 −−−−− 11111 11111lllllnnnnnNNNNN 33333 00000 00000 22222 44444 66666 88888 1111100000 00000 00000.....55555 11111 11111.....55555 22222 lllllnnnnnNNNNN hhhhh FIG.3: (Coloronline)Entanglemententropyasafunctionof FIG.1: (Coloronline)Entanglemententropyinthetwo-level lnN inthetwo-levelBCSmodelforh=0,1/2,1andanalytic BCS model as a function of h for N = 32,64,128,256 (from predictions: slope 1/2 (green line) for h<1, slope 1/3 (blue numerics [black lines]) and ( (0) [red line]). Arrows indi- ∞ E line) for h=1. cate the behavior of the finite-size correction in various regions. whose entropy is known [37, 38] to diverge as 1lnN. 00000 2 Sincenodrasticphysicalchangesoccurwhenvaryingthe field below h < 1, it is reasonable to expect a similar -----22222 behavior in the entire broken phase. We have checked this handwaving argument with numerics and, indeed, -----44444 (((((11111))))) the ground state entropy diverges as 12lnN in the entire EEEEE broken phase (see Fig. 3). -----66666 To complete this study, it is interesting to analyze a similar model for which the classical analysis predicts -----88888 two-Goldstone modes instead of one. The Lieb-Mattis Hamiltonian [39] is a natural candidate. -----1111100000 11111 11111.....22222 11111.....44444 11111.....66666 11111.....88888 22222 hhhhh V. THE LIEB-MATTIS MODEL FIG.2: (Color online) Behaviorof Nˆ num (0)ãs afunc- E −E tion of h in thetwo-level BCS model for N =32,64,128,256 (from numerics [black lines]) and ( (1) [red line]). The Lieb-Mattis Hamiltonian [39] is given by ∞ E H = 1 SA SB = 1 S2 SA 2 SB 2 . (39) N · 2N − − Ascanbe seeninFig. 1,anexcellentagreementisfound h (cid:0) (cid:1) (cid:0) (cid:1) i between the entanglement entropy for increasing N and As for the two-level BCS model, the indices and A B the thermodynamic limit. The leading finite N correc- refer to two distinct sets of N/2 spins 1/2. The (non- tionisalsoextractedfromnumericsandcomparedtothe degenerate) ground state ψ is readily found to be 0 | i analytical result (see Fig. 2) given above confirming the the eigenstate of S2, SA 2, SB 2 , with eigenvalues validity of the scaling hypothesis. Let us now turn to the broken phase h < 1. In this 0,N N+1 ,N Nn+1(cid:0) . H(cid:1)ow(cid:0)eve(cid:1)r,otheclassicalground 4 4 4 4 phase,atorderN0,oneofthetwogapsvanisheswhereas sntate i(cid:0)s foun(cid:1)d to(cid:0)have(cid:1)towo Goldstone modes stemming the other remains finite. More precisely, one has fromtheisotropyofthe Hamiltonian(39). This suggests that the scaling behavior of the entanglement entropy ∆+ = 1 h2, (37) here should differ from those of the models considered − ∆− = p0. (38) previously. AsinthebrokenphaseoftheBCSmodel,the diagonalizationofthe quadraticHamiltonianthat would We do not give here the details of this analysis since it derive from a HP expansion does not give the correct is similar to the isotropic LMG model discussed in the groundstatebecauseofthe Goldstonemodes. Neverthe- next section. Unfortunately, one can not use the scaling less, the entanglement entropy of the ground state can hypothesis to determine the behavior of the entropy due be evaluated exactly since: to the presenceof the Goldstone mode in this parameter range(∆− =0). However,onecanexactlydeterminethe 1 N/4 entropyinzerofield. There,the groundstate is givenby ψ0 = ( 1)−M+N/4 M, M , (40) the Dicke state corresponding to S = N/2 and Sz = 0 | i N2 +1M=X−N/4 − | − i q 6 where the state M, M denotes the eigenstate of where S = σi/2, and the σ ’s are the Pauli matri- | − i α i α α SA 2,SA, SB 2,SB witheigenvalues N N+1 ,M, ces. It describes a system of N spin 1/2 mutually inter- z z 4 4 actinginafiePldh,transversetotheinteractiondirections nN4(cid:0) N4 +(cid:1) 1 ,M(cid:0) . T(cid:1)hereoduceddensitymatrnixρ(cid:0)A obta(cid:1)ined (XY). Here, we restrict our study to the most interest- biny(cid:0)tthreaceiing(cid:1)genobvaoesrisseotfB iSsAth2u,sSeAasily computed and reads iansgsufmerero0m6agγne<ti1ccaansdehan>d,0w.iTthhoeuitsolotsrsopoifcgceanseeraγli=ty,1wies z discussed separately at the end of the section. n(cid:0) (cid:1) o 1 As was early discussed in the original work of Lipkin, ρ =Tr ψ ψ = M M . (41) A B| 0ih 0| N +1| ih | MeshkovandGlick,thissystemundergoesasecond-order 2 QPT at h = 1, between a symmetric (h > 1) and a The entanglement entropy is thus given by broken(h<1) phase. In both regimes,the groundstate liesinthemaximumspinsectorS =N/2. Toanalyzethe E =−Trρlnρ=ln N2 +1 , (42) entanglemententropy,wepartitionthesetofN spinsinto two blocks and , of sizes L and (N L) respectively and thus behaves as lnN at large(cid:0)N. (cid:1) A B − [51]. We alsointroducethe correspondingspinoperators To conclude this brief review of entanglement entropy SA,B = σi/2. in collective systems, we shall now consider a collective α i∈A,B α spin model with an arbitrary bipartition. Let usPfirst focus on the symmetric phase h > 1 in which the classical ground state is fully polarized in the z-direction. As previously,itis convenienttouse theHP VI. THE LIPKIN-MESHKOV-GLICK MODEL representation of the spin operators InitiallyproposedfourtyyearsagobyLipkin,Meshkov andGlick[40,41,42]todescribephasetransitionsinnu- SA = L/2 a†a, (44) clei, the LMG model has, since then, been widely used z − to describe many physicalsystems suchas Bose-Einstein S−A = L1/2a†(1−a†a/L)1/2 =(S+A)†, (45) condensates [43], Josephson junctions or long-range in- SB = (N L)/2 b†b, (46) teracting spin systems [44, 45]. Here, we shall consider z − − SB = (N L)1/2b†(1 b†b/(N L)1/2 =(SB)†,(47) the latter formulation which is especially well-suited to − − − − + our problematics. The entanglement properties of the LMGmodelhavedrawnmuchattentioninthe lastyears [28, 46, 47, 48, 49, 50] but the entropy has been analyt- withSA,B =SA,B iSA,B. Thistransformationmapsthe ically investigated only very recently [51] following the ± x ± y LMG Hamiltonian onto a system of two bosonic modes. numerical work of Latorre et al. [38]. At fixed τ = L/N, the Hamiltonian can be expanded in The Hamiltonian of the LMG model is given by powers of 1/N since, in this phase, one has a†a /L 1 and b†b /(N L) 1. At order (1/N) anhd foir h >≪1, H = 1 S2+γS2 hS , (43) one thhenigets− ≪ −N x y − z (cid:0) (cid:1) H = Nh 1+γ + 2h−γ−1 a†a+b†b + γ−1 τ a†2+a2 +(1 τ) b†2+b2 +2 τ(1 τ) a†b†+ab + − 2 − 4 2 4 − − 1 1−γ (1 τ) a†a(cid:16)2b+a†2a(cid:17)b† +τ ahb†b(cid:16)2+a†b†2(cid:17)b + 1+γ(cid:16) a†a+b(cid:17)†b+ap†2a2+b†2(cid:16)b2+2a†ab(cid:17)†ib + N 4 τ(1 τ) − 2 (cid:26) − h (cid:16) (cid:17) (cid:16) (cid:17)i (cid:16) (cid:17) 1−γpa†2+a2+b†2+b2+2 a†a3+a†3a+b†b3+b†3b + (1/N2). (48) 8 O h (cid:16) (cid:17)i(cid:27) AtorderN0, this Hamiltonianis easilydiagonalizedand mains finite except for γ = 1 (see discussion at the end the excitation energies are given by: of the section). The parameter µ is then given by: 2h γ 1 ∆ = − − , (49) + 2 h 1 1/4 h γ 1/4 2 µ(0) = 1+τ(1 τ) − − , ∆− = (h−1)(h−γ). (50) vuu − "(cid:18)h−γ(cid:19) −(cid:18)h−1(cid:19) # Thus, at the critical popint, ∆− vanishes whereas ∆+ re- t (51) 7 111111......555555 00000 111111llllllnnnnnnNNNNNN γγγγγγ====== 111111 666666 444444 000...777555 ττττττ ====== 111111 -----11111 444444 111111 EEEEEEllllllnnnnnn222222 lllnnn222000 000...555 111 EEEEE(((((11111)))))-----22222 τττττγγγγγ==========111114444411111 44444 NNNNNN 111111 000000......555555 −−−−−− -----33333 NNNNNN 111111 −−−−−− -----44444 11111 11111.....22222 11111.....44444 11111.....66666 11111.....88888 22222 000000 000000 000000......555555 111111 111111......555555 222222 hhhhh hhhhhh FIG.5: (Color online) Behavior ofNˆ num (0)ãs afunc- E −E FIG.4: (Coloronline)EntropyasafunctionofhintheLMG tion of h in the LMG model for N = 32,64,128,256 (from model for γ = 1/4, τ = 1/4 and N = 32,64,128,256 (from numerics [black lines]) and ( (1) red line). ∞ E numerics[blacklines])and ( (0) redline). Arrowsindicate ∞ E the behavior of the finite-size correction in various regions. For h < 1, we plotted (0) +ln2 to take into account the to the entropy then reads E parity-broken nature of the classical states taken as starting point of the 1/N expansion [26, 51]. The inset is a zoom µ(1) µ(0)+1 (1) = ln . (54) around h= √γ where (0) = 0, for N = 64 (black line) and E 2 µ(0) 1 (red line). E (cid:18) − (cid:19) ∞ We emphasize that this is the exact correction of order 1/N whereas in Ref. [51], only its expansion near h = 1 which is equivalent to the form presented in Ref. [51]. was given. We display in Fig. 5, a comparison between The entropy is then given by Eq. (7a). Figure 4 shows the numericalcorrectionoforder1/N andthe analytical a comparison between the entropy at finite N obtained expression obtained from Eqs. (53)-(54). from numerical diagonalization of the Hamiltonian and Inthevicinityofthecriticalpointh=1,onethushas, the entropy (0) obtained from Eqs. (7a) and (51). As at this order: E canbeseen,theentropyisfiniteforallh=1anddiverges 6 eE = eE(0)+N1E(1), (55) at the transition point. Near the critical point h=1+, it behaves as [τ(1 τ)]1/2(1−γ)1/4 1 3(1−γ)1/2 . ∼ − (h 1)1/4 − 8N(h 1)3/2 − (cid:20) − (cid:21) 1 1 1 (0) = ln(h 1)+ ln[τ(1 τ)]+ ln(1 γ)+ This suggests that the scaling variable is indeed a func- E −4 − 2 − 4 − tion of γ but has no dependence on τ. Following Ref. 1 ln2+ [(h 1)1/2], (52) − O − [51],wereformulatethescalinghypothesisforthismodel by assuming that in the vicinity of the critical point, a wherethesuperscript”(0)”referstotheorderinthe1/N physical observable Φ can be written as the sum of a expansion. AsintheDickemodelstudiedinSec. III,the regular and a singular contribution, scaling hypothesis (see Appendix A) yields 1lnN. As can be seen in Eq. (52), the entropy aElso∼di6verges Φ (h,γ)=Φreg(h,γ)+Φsing(h,γ). (56) N N N when γ goes to 1 and when τ reaches its extremal val- ues 0 or 1. This suggests a possible dependence of the where scaling variable on these parameters. To obtain this dependence, one needs, at least, to compute the correction Φsing(h 1,γ) (h−1)ξΦh(1−γ)ξΦγ (ζ), (57) oforder1/N tothe entropy. AsexplainedinRef. [36],it N ≃ ∼ NnΦ GΦ isobtainedbyTaylorexpandingexpression(7a)atorder and ζ = N(h 1)3/2(1 γ)−1/2 is the scaling variable. 1/N where µ is also computed at this order (see Ap- The exponents−ξh, ξγ a−nd n are characteristics of the pendix C). This latter step requires to diagonalize H at Φ Φ Φ observables Φ. Thus, at the critical point, one expects order 1/N. Using the canonical tranformation method described inRef. [29], one then obtains the correctionof GΦ(ζ)∼ζ−2ξΦh/3, so that order 1/N to µ Φsing(h=1,γ) N−(nΦ+2ξΦh/3)(1 γ)ξΦγ+ξΦh/3. (58) N ∼ − µ(1) = τ(1−τ)(1−γ)2 4γ−h(1+2h+γ) + h . We emphasize that the scaling form (A3), can be 2 2∆4 ∆3 checked up to high order from the expansion of the (cid:20) − −(cid:21) (53) groundstateenergy,thegap,themagnetization,andthe AsalreadystatedinSec. IV,thecorrectionoforder1/N spin-spin correlation functions given in Ref. [50]. 8 1111 Therefore, one first has to perform a rotation around the y-axis to bring the z-axis along the classical ground state magnetization Scl. = N/2( sinθ ,0,cosθ ) with 0 0 ± θ =arccosh[50]. TousetheHPmapping,wethusneed 0 to rotate the spin operators according to 0000....8888 Sx cosθ0 0 sinθ0 Sx 2χτ  Sy = 0 1 0  Sy  , (60) 6χ Sz −sinθ0 0 cosθ0 Sez γ     e  0000....66660000 2222111111112222 2222111111110000 1111////NNNN 222211118888.... Ntuoatteiotnhsatartohuisncdhooniceeocfotrhreestpwoondcslatsosisctauldmyeiqnuimanatu(smeefl[u5c0-] for discussion). Next, one uses the HP representation of S which is exactly the same as in Eqs.(44)-(47). The FIG. 6: (Color online) Exponents χγ and χτ as a function Hamiltonian at order N0 then reads: of 1/N obtained from numerical diagonalization of H. For clarity,weplotted2χτ and6χγ whichareexpectedtobeequal e N(1+h2) h2+γ 2 h2 γ to 1 in the thermodynamic limit (dotted lines are guides for H = + − − a†a+b†b + theeyes). − 4 − 4 2 γ−h2 τ a†2+a2 +(1 τ) b†2+(cid:16)b2 + (cid:17) 11..22 4 − h (cid:16) (cid:17) (cid:16) (cid:17) 2 τ(1 τ) a†b†+ab + (1/N). (61) 11 − O γγ==00 p (cid:16) (cid:17)i Ascanbeeasilychecked,thisHamiltoniancoincideswith 00..88 ττ == 11 22 (48)forh=1asitshouldforasecondorderQPT.Using the general results of Sec. II, one straightforwardly gets EE 00..66 the two excitation energies 00..44 2 h2 γ ∆ = − − , (62) + 2 00..22 22 44 66 88 1100 ∆− = (1 h2)(1 γ), (63) − − llnnNN and p FIG.7: (Coloronline)EntropyasafunctionoflnN forγ =0 and τ =1/2. The blueline has a slope χN =1/6. 1 h2 1/4 1 γ 1/4 2 µ(0) = 1+τ(1 τ) − − . vuu − "(cid:18) 1−γ (cid:19) −(cid:18)1−h2(cid:19) # Thescalinglaw(58)appliedtotheobservableeE leads t (64) to the following critical behavior of the entropy It is worth noting that for h = √γ, one has µ(0) = 1 so that the entropy vanishes (see Fig. 4). At this point, (h=1) χ lnN+χ ln(1 γ)+χ ln[τ(1 τ)], (59) the ground state (which is two-fold degenerate in this N γ τ E ∼ − − phase) can indeed be chosen as separable [50]. Such a with χN = χγ = 1/6 and χτ = 1/2. As can be seen point is often present in anisotropic systems [53]. in Figs. 6 and 7, an excellent agreement between these predictionsandnumericsisobservedwhenincreasingthe system sizes. Such a behavior has striking similarities with the 1D The isotropic case XY model[4,5,7,8,11,52]. Thisisduetothefactthat the groundstateofthe LMGmodelliesinthe maximum For γ = 1, the above expression of the gaps are still spin sector S = N/2. As a consequence, the relevant valid in the thermodynamical limit and one recovers subsystem Hilbert space is spanned by the L+1 eigen- the Goldstone mode predicted by the classicalapproach. statesof (SA)2,SA withSA =L/2. Theentanglement This gapless excitation prevents from using the scaling { z } entropy as a function of the subsystem size L can thus hypothesistodeterminethebehavioroftheentropy. For- notexceedthe maximumvalue ln(L+1). However,note tunately,theisotropiccaseistrivialsincetheeigenstates thatthe physicsofthe LMGmodelisverydifferentfrom of the LMG Hamiltonian (43) are then the eigenstates 1D models. S,M of S2 and S . The ground state is obtained for z | i Thissurprisinganalogywith1DXY modelcarriesover S = N/2 and M < N/2 for h < 1, whereas for h > 1 to the broken phase. Indeed, at order N0, one can com- one has M = N/2. Thus, in the symmetric phase, the putetheentropyinthesamewayasinthebrokenphase. entropy vanishes since the state N/2,N/2 is separable, | i 9 but in the broken phase, the entropy of the Dicke state operatorsand the scaling of the entropy would be differ- N/2,M < N/2 is known to diverge as 1lnN [37, 38]. ent [55]. Note alsothat in the isotropic LMG model, the | i 2 This meansthatthe entropyis discontinuousatthe crit- entropy vanishes at the critical point even though there ical point h = 1 (as the concurrence [50]) although this are two vanishing gaps at the critical point. transition is often considered as second-order. Theanalysisperformedhereforsimplemodelscorrob- orates the results obtained in many 1D systems where it has been also found that diverging entropies are always VII. CONCLUSION AND PERSPECTIVES associatedto gapless excitations [4, 5, 7, 8, 13, 17]. This should constitute a strong motivation to further investigate collective models which are simpler to treat than The results presented in this paper show that the en- low-dimensional ones and already exhibit nontrivial be- tanglement entropy in collective models displays a large haviors of the entanglement entropy. variety of behaviors. First of all one has to distinguish with respect to the number of Goldstone modes in the system (0, 1 or 2). Acknowledgments 1. The classical ground state has a finite degeneracy, i.e.noGoldstonemode: thissituationismetinthe T. B. thanks the DFG and the Studienstiftung des broken phases of the Dicke and anisotropic LMG Deutschen Volkes for financial support. model, as well as in the symmetric phases of the models studied here. In this case, and away from critical points, both excitations around any classi- APPENDIX A: THE SCALING HYPOTHESIS cal states are gapful and the entropy is finite (pos- sibly vanishing). This appendix is dedicated to a basic description of the scaling hypothesis. This hypothesis is useful in col- 2. TheclassicalgroundstatehasoneGoldstonemode: lectivemodelswherea1/N expansionismeaningfulsuch this situation is met in the broken phases of the as those considered in this paper. It relies on the fact two-levelBCSmodelandtheisotropicLMGmodel. that, in these models, the 1/N expansion of the expec- In this case, one excitation energy is finite and the tationvalue ofany physicalobservablecanbe written as other vanishes and the entropy diverges as 1lnN. 2 the sum of a regular and a singular contribution, 3. The classical ground state has two Goldstone Φ (h,γ)=Φreg(g)+Φsing(g). (A1) N N N modes: this situation is met in the antiferromagnetic Lieb-Mattis model. In this case, the entropy Here,singularmeansthatthefunctionand/oritsderiva- diverges as lnN. tives with respectto the controlparameterg driving the transition diverge at the critical point g , following a c In case 1, where we have no Goldstone mode in the sys- power law. In addition, one has tem, the collective spins, SA and SB, of the two sub- systems may be expanded in 1/N around their classical Φsing(g g ) (g−gc)ξΦ [N(g g )α]. (A2) orientation via Holstein-Primakoff transformation. The N ∼ c ∼ NnΦ FΦ − c systemis thusmappedtoatwo-modebosonmodel. The The exponents ξ and n are characteristics of the ob- general analysis of the order N0 (at which the effective Φ Φ servables Φ but the exponent α depends on the model bosonic models are quadratic) in Sec. II shows how the consideredasearlyunderstoodbyBotetandJullien[45]. entanglemententropydivergeswhenexcitationgapsvan- For the models considered in this paper, we have explic- ish. Its scaling can thus be classified according to the itly checked that α = 3/2 which is reminiscent of the number vanishing of gaps, when a critical point is ap- behavior of the gap in the vicinity of the critical point proached. Inthe anisotropicLMGmodelasinthe Dicke which vanishes as (g g )1/2. c model, there is only one vanishing gap and the entropy − The behavior of Φsing at the critical point g is then diverges as 1lnN in both models. N c 6 obtained by noting that since no divergence can occur In the two-level BCS model, there are two vanishing gapsandweobtainedanentropythatdivergesas 1lnN. at finite N, one must have FΦ(x),∼ x−ξΦ/α and conse- 3 quently However,these behaviorsarefound usingthe scalinghy- (psoeteheAspispwenhdicixhAre)l.ieIsnomnothdeelsscsatluindgievdairniatbhleisNpa(gpe−r,gocn)αe ΦsNing(g =gc)∼N−(nΦ+ξΦ/α). (A3) alwayshasα=3/2duetotheirrelationshipwithaquar- Finally, since we are mainly interested here in the en- ticoscillator[54]. Nevertheless,thisvaluedependsonthe tropy, let us simply note that to obtain a power-law Hamiltonianandwouldbedifferentif,forinstance,atthe behavior, the observable to be considered is eE which critical point, the terms of order N0 and 1/N vanish si- indeed bevaves as suggested in (A2). As a result, if multaneously. In this case, the Hamiltonian would be ξEln(g gc) near the critical point, the scaling E ∼ − − sextic instead of quartic in the creation and annihilation hypothesis predicts that it diverges as ξE lnN. α 10 This scaling hypothesis has been checked up to high APPENDIX C: FIRST-ORDER CORRECTION OF order in the several models and allowedus to obtain the THE ENTANGLEMENT ENTROPY IN A finite-size scalingexponents in many systems[29, 34, 49, TWO-MODE BOSON SYSTEM 50, 51, 56, 57]. We give the exact expression of the first-order correction of the entanglement entropy in a two-mode boson APPENDIX B: LINEAR ENTROPY FOR system. This correction is derived in Ref. [36] for the QUADRATIC TWO-MODE BOSON SYSTEMS general case. Let us consider a Hamiltonian of the following form In this Appendix, we derive the expressionfor the lin- earentropyforthegroundstate ψ ofageneralquadratic | i two-modebosonHamiltonianoftheform(2). Thislinear H =H +gH , (C1) 0 1 entropy is defined as =1 Tr(ρ2)=1 Tr(ρ2), (B1) where H is a quadratic two-mode boson Hamiltonian Elin. − A − B 0 written in terms of bosons a and b, and where g 1. where ρA,B = TrB,A(ψ ψ ). Here, we suppose that UsingthesamenotationsasinAppendixB,weintro≪duce | ih | bosons a and b define subsets and . the parameter A B AsshowninRef.[1],thereduceddensitymatricesρ A,B can always be written as the exponential of a quadratic formintheoperatorsrelatedtothecorrespondingsubset. µ = a†+a 2 a† a 2 (C2) Concretely, tracing out over ,one has ρ =e−K where −h ih − i B A = qµ(0)+(cid:0)gµ(1)(cid:1)+ (cid:0)(g2), (cid:1) (C3) O K =κ +κ a†a+κ a†2+a2 . (B2) 0 1 2 (cid:16) (cid:17) Then, as detailed in Ref. [36], the entanglement entropy The coefficients κi’s are determined from the conditions at order g0 and g is obtained by expanding the function Tr ρ =1, a†a = a†a and a†2 = a†2 , (B3) A A hh ii h i hh ii h i µ+1 µ+1 µ 1 µ 1 F(µ)= ln − ln − , (C4) where Ω = ψ Ωψ and Ω =Tr (e−KΩ). 2 2 − 2 2 A h i h | | i hh ii The next step is to diagonalize K which becomes at the correspondingorder. One then obtains the entan- K =E0+∆c†c, (B4) glement entropy between both modes where, as shown in Ref. [18], = Tr (ρ lnρ ) , (C5) A A A E − µ+1 ∆=ln , (B5) = (0)+g (1)+ (g2), (C6) µ 1 E E O − and µ = a†+a 2 a† a 2 . The normalization with −h ih − i ceo−nEd0it=io2n/q(TµrA+ρ(cid:0)1A).=Itei(cid:1)s−tEh0(cid:0)e/n(1st−rae(cid:1)ig−h∆t)for=wa1rdthtoencoimmppluietes (0) = µ(0)+1lnµ(0)+1 µ(0)−1lnµ(0)−1,(C7) E 2 2 − 2 2 = 1 Tre−2(E0+∆c†c), (B6) µ(1) µ(0)+1 Elin. − (1) = ln . (C8) 1 E 2 µ(0) 1 = 1 . (B7) (cid:18) − (cid:19) − µ Note that it is trivial to generalize this derivation for Theonlydifficultyisthustocomputeµatorderg and anarbitrarynumberofbosonicorfermionicmodesusing one readily has the correction of order g to the entropy. the expression of the reduced density matrices given in Note finally that the second order correction is not ob- Ref. [18]. tained by expanding F at order g2. [1] L. Bombelli, R. Koul, J. Lee, and R. Sorkin, Phys. Rev. (London) 416, 608 (2002). D 34, 373 (1986). [3] T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, [2] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 032110 (2002).