Even-odd entanglement in boson and spin systems
aa r X i v : . [ qu a n t - ph ] A p r Even-odd entanglement in boson and spin systems
R. Rossignoli, N. Canosa, J.M. Matera
Departamento de F´ısica-IFLP, Universidad Nacional de La Plata, C.C. 67, La Plata (1900), Argentina (Dated: June 7, 2018)We examine the entanglement entropy of the even half of a translationally invariant finite chainor lattice in its ground state. This entropy measures the entanglement between the even andodd halves (each forming a “comb” of n/ XY couplings indicate that while at strong magnetic fields this entropy isstrictly extensive, at weak fields important deviations arise, stemming from parity-breaking effectsand the presence of a factorizing field (in which vicinity it becomes size-independent and identicalto the entropy of a contiguous half). Exact numerical results for small spin s chains are shown tobe in agreement with the bosonic RPA prediction. PACS numbers: 03.67.Mn, 03.65.Ud, 75.10.Jm
I. INTRODUCTION
The entanglement properties of many-body systemsare of great interest for both quantum information theory[1] and condensed matter physics [2–4]. Their knowledgeenables, on the one hand, to assess the potential of agiven many-body system for quantum information pro-cessing tasks such as quantum teleportation [5] and quan-tum computation [1, 6, 7]. On the other hand, it providesa deep understanding of quantum correlations and theirrelation with criticality [2–4, 8, 9]. In non-critical sys-tems with short range couplings, i.e., local couplings inboson or spin lattices, ground state entanglement is be-lieved to satisfy a general area law by which the entropyof the reduced state of a given region, which measures itsentanglement with the rest of the system, scales as thearea of its boundary as the system size increases [4, 10].This behavior is quite different from that of standardthermodynamic entropy which scales as the volume. Inone dimensional systems this statement has been quitegenerally and rigorously proved [4, 11] and simply meansthat the entropy of a contiguous section saturates, i.e.,approaches a size independent constant, as the size in-creases. Violation of this scaling is therefore an indica-tion of criticality [8, 9, 12]. The exact expression of theentropy of a contiguous block in a one-dimensional XYspin 1/2 chain in the thermodynamic limit has been ob-tained [13–15] and confirms the previous behavior.The conventional area law holds for contiguous sub-systems. For non-contiguous regions it actually impliesthat the entropy is proportional to the number of cou-plings broken by the partition. For instance, for comb -like regions like the subset of all even sites in a chain, theentropy should scale as the total number n of sites forfirst neighbor or short range couplings. This was in factverified in [11] for the harmonic cyclic chain, where thecorresponding logarithmic negativity was calculated, andalso verified numerically in [16] for some spin arrays anda 1- d half-filled Hubbard model, where the even entan- glement entropy was computed. An exact treatment ofgeneral comb entropies for a large one-dimensional criti-cal XX spin 1 / L plus a logarithmic correction.The aim of this work is to analyze in detail the en-tanglement entropy of all even sites in finite boson andspin arrays, both in one dimension as well as in general d -dimensions. Such bipartition can be normally expectedto be the maximally entangled bipartition at least for uni-form nearest neighbor couplings, as it will there break allcoupling links. We first analyze the bosonic case withgeneral quadratic couplings, where a fully analytic treat-ment of this entropy is shown to be feasible and allowsto derive simple general expressions in the weak couplinglimit. Comparison with single site and block entropiesis also made. The bosonic treatment is then applied tofinite spin s arrays with anisotropic ferromagnetic-type XY couplings in a uniform transverse field through theRPA approach [18]. This allows to predict in a simpleway the main properties of the total even entropy in thesesystems. Comparison with exact numerical results indi-cate that the RPA prediction, while qualitatively correct,is also quite accurate outside the critical region alreadyfor low spin s &
2, representing the high spin limit. Re-sults corroborate that for strong fields, the total evenentropy in these systems is extensive, i.e., directly pro-portional to the total number n of sites. However, forlow fields B < B c , this entropy has an additive con-stant, which arises in the RPA from parity restoration[18]. Moreover, in the immediate vicinity of the factoriz-ing field B s < B c [19–22], extensivity is fully lost and thetotal even entropy reduces to this constant, which is thesame as that for the block entropy and is exactly eval-uated. The exact bosonic treatment is described in sec.II, whereas its application to spin systems is discussed insec. III. Conclusions are finally drawn in IV. II. ENTANGLEMENT ENTROPY IN BOSONICSYSTEMS
We start by considering a system of n bosonic modesdefined by boson creation operators b † i ([ b i , b † j ] = δ ij ),interacting through a general quadratic coupling. TheHamiltonian can be written as H = X i,j ( λ i δ ij − ∆ + ij )( b † i b j + δ ij ) − (∆ − ij b † i b † j + ¯∆ − ij b j b i )(1)= Z † HZ , Z = (cid:18) bb † (cid:19) , H = (cid:18) Λ − ∆ + − ∆ − − ¯∆ − Λ − ¯∆ + (cid:19) , where Z † = ( b † , b ), Λ ij = λ i δ ij and the 2 n × n matrix H is hermitian. The system is assumed stable, such thatthe matrix H is positive definite . We may then also write(1) in the standard diagonal form H = X k ω k ( b ′† k b ′ k + ) , (2)where ω k are the symplectic eigenvalues of H , i.e., thepositive eigenvalues of the matrix MH , with M = ( − ),which come in pairs of opposite sign and are all real non-zero when H is positive definite [23], and b ′† k are thenormal boson operators determined by the diagonaliz-ing Bogoliubov transformation [23] Z = WZ ′ satisfying W † MW = M and ( W † HW ) kk ′ = ω k δ kk ′ . The groundstate is the vacuum | ′ i of the operators b ′ k and is non-degenerate.Ground state entanglement properties can be eval-uated through the general Gaussian state formalism[11, 24, 25], which we here recast in terms of the con-traction matrix [18, 23] D = hZZ † i ′ − M = W (cid:18) (cid:19) W † (3)= (cid:18) F + F − ¯ F − I + ¯ F + (cid:19) , F + ij = h b † j b i i ′ F − ij = h b j b i i ′ = h b † i b † j i ∗ ′ . (4)This hermitian matrix determines, through application ofWick’s theorem [23], the average of any many-body oper-ator. In particular, the reduced state ρ A = Tr ¯ A | ′ ih ′ | ofa subsystem A of n A modes ( ¯ A denoting the complemen-tary subsystem and Tr ¯ A the partial trace) is fully deter-mined by the corresponding sub-matrix D A = h Z A Z † A i −M A (Eq. (4) with i, j ∈ A ) and can be written as [18] ρ A = exp[ − Z † A ˜ H A Z A ] / Tr exp[ − Z † A H A Z A ] , (5)where ˜ H A = M A ln[ I + M A D − A ]. Eq. (5) represents athermal-like state of suitable n A independent modes de-termined by the effective Hamiltonian ˜ H A . The entan-glement entropy of the ( A, ¯ A ) partition, S ( ρ A ) = S ( ρ ¯ A ),is then determined by the symplectic eigenvalues f Ak of D A (i.e., the positive eigenvalues of the matrix D A M A , which has eigenvalues f Ak and − − f Ak ), and given by S ( ρ A ) = − Tr ρ A ln ρ A = n A X k =1 h ( f Ak ) , (6) h ( f ) = − f ln f + (1 + f ) ln(1 + f ) . (7)For instance, the entanglement of a single mode i withthe rest of the system is just S ( ρ i ) = h ( f i ) , f i = q ( F + ii + ) − | F − ii | − , (8)where f i , the symplectic eigenvalue of the single modecontraction matrix D i , represents the deviation fromminimum uncertainty of the mode: ( F + ii + ) − | F − ii | = h q i i ′ h p i i ′ − [Re( h q i p i i ′ )] ≥ q i = b i + b † i √ , p i = b i − b † i √ i . A. Finite translationally invariant systems
Let us now associate each bosonic mode with a givensite in a cyclic chain and consider a translationally in-variant system of n sites, such that λ i = λ and ∆ ± ij =∆ ± ( i − j ), with ∆ ± ( − l ) = ∆ ± ( n − l ). We first con-sider for simplicity the one-dimensional case. Througha discrete Fourier transform b † i = √ n P n − k =0 e i πki/n b † k ,we can diagonalize H analytically and obtain an explicitexpression for the contractions F ± ij . We will assume∆ ± ( l ) = ∆ ± ( − l ) ∀ l , in which case the energies ω k in(2) adopt the simple form [18] ω k = q ( λ − ∆ + k ) − (∆ − k ) , (9)where ∆ ± k are the Fourier transforms of the couplings:∆ ± k = n − X l =0 e i πkl/n ∆ ± ( l ) . (10)The contractions F ± ij depend just on the separation l ≡| i − j | and are given by F ± l ≡ F ± j + l,j = 1 n n − X k =0 e − i πkl/n f ± k , (11) f + k = h b † k b k i ′ = λ − ∆ + k ω k − , f − k = h b k b − k i ′ = ∆ − k ω k . (12)The symplectic eigenvalues of the full contraction matrix(4) are of course f k = q ( + f + k ) − ( f − k ) − = 0 ∀ k .In the weak coupling limit | ∆ ± k | ≪ λ ∀ k , f ± k becomesmall and up to lowest non-zero order we obtain f − k ≈ ∆ − k λ , f + k ≈ (∆ − k ) λ ≈ ( f − k ) , (13)which leads to F − l ≈ ∆ − ( l )2 λ , F + l ≈ P l ′ ∆ − ( l ′ )∆ − ( l − l ′ )4 λ . (14) FIG. 1. (Color online) Even-odd partitions of one- and two-dimensional arrays
At this order just sites linked by ∆ − ( l ) or its convolutionare correlated. The eigenvalues f Ak of subsystem contrac-tion matrices will depend up to lowest non-zero order on F + l and ( F − l ) , being then O (∆ − /λ ) for ∆ − ( l ) ∝ ∆ − .We can then use in (6) the approximation h ( f ) ≈ − f (ln f −
1) + O ( f ) , (15)such that S ( ρ A ) = O ( ∆ − λ ln ∆ − λ ).On the other hand, it is seen from Eq. (9) that thepresent system is stable provided λ ≥ ∆ + k + | ∆ − k | ∀ k .For attractive couplings ∆ + ( l ) ≥ ∀ l , with all ∆ − ( l )of the same sign, the strongest condition is obtained for k = 0, so that stability occurs for λ > λ c = ∆ +0 + | ∆ − | = X l ∆ + ( l ) + | ∆ − ( l ) | . (16)For λ → λ c , ω → ω k remain finite ina finite system), implying a divergence of f ± (Eq. (12)): | f − | ≈ s | ∆ − | λ − λ c ) , f +0 ≈ | f − | − / , (17)plus terms O ( λ/λ c − / . This entails in turn a di-vergence f A ∝ ( λ/λ c − − / of the largest eigenvalueof a subsystem contraction matrix D A , with S ( ρ A ) ≈ ln f A + 1 ≈ − ln( λ/λ c −
1) plus constant terms.For example, the single site entropy (8) becomes S ( ρ i ) = h ( f ) , f = q ( + F +0 ) − ( F − ) − , (18)with F ± = n P k f ± k (Eq. (11)). For weak coupling, f ≈ F +0 − ( F − ) ≈ P l =0 (∆ − ( l )) λ , (19)which involves just the couplings ∆ − ( l ) connecting thesite with the rest of the system. On the other hand, for λ → λ c , f ∝ q f − n ∝ ( λ/λ c − − / . B. Even-odd entanglement entropy
We now evaluate the entropy of the reduced state ofall even sites, S ( ρ E ) = S ( ρ O ), which measures their en-tanglement with the complementary set of odd sites (Fig. 1 left). We will assume n even, such that the even sub-system, defined by ( − i = +1, is again translationallyinvariant. The ensuing contraction matrix D E can be ob-tained by removing contractions between even and oddsites in the full matrix (4) and extracting then the evenpart. This leads to elements˜ F ± ij = F ± ij (1 + e iπ ( i − j ) ) , (20)whose Fourier transforms are, using Eq. (11),˜ f ± k = ( f ± k + f ± k + n/ ) . (21)The final symplectic eigenvalues of D E then become˜ f k = q ( + ˜ f + k ) − ( ˜ f − k ) ] − , (22)for k = 0 , . . . , n/ −
1. We then obtain S ( ρ E ) = n/ − X k =0 h ( ˜ f k ) = n − X k =0 h ( ˜ f k ) . (23)Whenever ˜ f k can be approximated by a smooth function˜ f (˜ k ) of ˜ k ≡ k/n , we may replace (23) by the integral S ( ρ E ) ≈ n Z h [ ˜ f (˜ k )] d ˜ k . (24)In these cases, we may then expect S ( ρ E ) extensive , i.e.,proportional to the number n/ sin-gle non-zero symplectic eigenvalue f n A for any subsystem[18], including the whole even set, and S ( ρ E ) = h ( f n/ ) isno longer proportional to n . A similar lack of extensivityholds in a finite system in the vicinity of the instability( λ → λ c , see below).For weak coupling , Eqs. (13), (14) and (21) lead to˜ f k ≈ (∆ − k − ∆ − k + n/ ) λ = ( P l odd e i πkl/n ∆ − ( l )) λ , (25)which involves again just the couplings ∆ − ( l ) connectingthe even and odd subsystems. On the other hand, for λ → λ c (Eq. (16)), ˜ f ≈ q (1 + 2 f + n/ ) | f − | − f − n/ f − diverges as ( λ/λ c − − / whereas all other ˜ f k remainfinite, and extensivity is lost. C. First neighbor coupling
Let us now examine in detail the first neighbor case∆ ± ( l ) = ∆ ± ( δ l + δ l, − ), where Eq. (10) becomes∆ ± k = ∆ ± cos(2 πk/n ) . (26)The exact S ( ρ E ) can be obtained from Eqs. (21)–(23).In the weak coupling limit, Eqs. (19) and (25) lead to f ≈ (∆ − ) λ , (27)˜ f k ≈ (∆ − k ) λ ≈ f cos (2 πk/n ) . (28)Using Eqs. (15)–(24), the single site and the total evenentropies can then be expressed just in terms of f : S ( ρ i ) ≈ − f (ln f − , (29) S ( ρ E ) ≈ − nf Z cos (2 π ˜ k ) { ln[2 f cos (2 π ˜ k )] − } d ˜ k = − n f (ln f − ln 2) . (30)Hence, in this limit S ( ρ E ) is extensive , becoming n/ O ( nf ) correc-tion accounting for the interaction between even sites: S ( ρ E ) ≈ n S ( ρ i ) − n f (1 − ln 2) . (31)The last term represents the even mutual entropy n S ( ρ i ) − S ( ρ E ), which is always a positive quantity andbecomes here also extensive in this limit.In contrast, the block entropy S ( ρ L ), where ρ L denotesa contiguous block of L < n spins, rapidly saturates as L increases [11]. In the weak coupling limit, it is ver-ified that the ensuing contraction matrix D L possesses,up to lowest non-zero order, just two positive non-zerosymplectic eigenvalues f ± L ≈ f for any L ≥
2, such that S ( ρ L ) ≈ − f (ln f / − ≈ S ( ρ i ) + f ln 2 , (32)for 2 ≤ L ≤ n −
2, i.e., it saturates already for L = 2.Hence, in this limit, S ( ρ E ) ≈ n S ( ρ L ) − n f . (33)Assuming ∆ + > + < b i → − b i at odd sites) the present systemis stable for λ > λ c = ∆ + + | ∆ − | (Eq. (16)). For λ → λ c , ω → f ≈ [ s | ∆ − | λ c ∆ + ( λ − λ c ) − , being then verified that S ( ρ E ) ≈ − ln( λ/λ c −
1) plus aconstant term up to leading order. Hence, in this limit S ( ρ E ) /S ( ρ i ) → n = 36 sites with ∆ − = ∆ + /
3, where λ c = 4∆ + / D. Even-Odd entropy in d-dimensions
The whole previous treatment can be directly extendedto a translationally invariant cyclic array in d dimen-sions. We should just replace l, k, n by vectors l =( l , . . . , l d ), k = ( k , . . . , k d ) and n = ( n , . . . , n d ), with l i , k i = 0 , . . . , n i −
1. We will assume couplings satisfying∆ ± i , j = ∆ ± ( i − j ), with ∆ ± ( − l ) = ∆ ± ( n − l ) = ∆ ± ( l ).The same previous expressions (9)–(12) then hold, with∆ ± k = X l e i π ˜ k · l ∆ ± ( l ) , (34) F ± l = 1 n X k e − i π ˜ k · l f ± k , (35)where ˜ k = ( k /n , . . . , k d /n d ) and n = Q di =1 n i is thetotal number of sites. Eqs. (13)–(14) remain unchangedwith i, k, l → i , k , l .The subsystem of all even sites, like that formed bythe blue sites in Fig. 1 right, is defined by( − i + ... + i d = +1 . Its contraction matrix will then be the even block of˜ F ± ij = F ± ij (1 + e iπ ( i − j ) · ) (36)where = (1 , . . . , n i even ∀ i , its Fouriertransform is then given again by˜ f ± k = [ f ± k + f ± k + n / ] , (37)where k i + n i / → k i − n i / k i ≥ n i /
2. The symplecticeigenvalues of D E are then given again by Eq. (22) with k → k , and the even-odd entanglement entropy reads S ( ρ E ) = X k h ( ˜ f k ) ≈ n Z h [ ˜ f ( ˜ k )] d d ˜ k , (38)where k i = 0 , . . . , n i − ≤ ˜ k i ≤ f k is asmooth function ˜ f (˜ k ) of ˜ k .In the case of first neighbor couplings∆ ± ( l ) = 12 d X i =1 ∆ ± i ( δ l , e i + δ l , − e i ) , where e i = (0 , . . . i , . . . ± k = d X i =1 ∆ ± i cos(2 πk i /n i ) . (39)with ∆ ± k + n / = − ∆ ± k . In the weak coupling limit wethen obtain f ≈ | ∆ − | λ , | ∆ − | = d X i =1 (∆ − i ) , (40)˜ f k ≈ u ( ˜ k ) f , u ( ˜ k ) = 2( X i ∆ − i | ∆ − | cos 2 πk i n i ) . (41) Λ (cid:144) Λ c S H Ρ A L S H Ρ i L S H Ρ L L S H Ρ E L n=36 Λ (cid:144) Λ c n/4n/2 S H Ρ A L (cid:144) S H Ρ i L S H Ρ L L(cid:144) S H Ρ i L S H Ρ E L(cid:144) S H Ρ i L Λ (cid:144) Λ c S H Ρ A L S H Ρ i L S H Ρ L L S H Ρ E L n=6x6 Λ (cid:144) Λ c n/4n/2 S H Ρ A L (cid:144) S H Ρ i L S H Ρ L L(cid:144) S H Ρ i L S H Ρ E L(cid:144) S H Ρ i L FIG. 2. (Color online) Top left: Entanglement entropies in the ground state of a one-dimensional bosonic chain of n = 36sites described by the Hamiltonian (1) with first neighbor cyclic couplings and ∆ − = ∆ + / S ( ρ i ), S ( ρ L ) and S ( ρ E ) denote,respectively, the entropy of the reduced state of a single site, a block of L = n/ S ( ρ E ) /S ( ρ i ) and S ( ρ L ) /S ( ρ i ). Dotted lines depict the ratios determined by the asymptotic expressions(29)–(32), For large λ these ratios are then close to n/ λ → λ c they all approach 1. The rightpanels depict the same quantities for a two-dimensional square array of n = 6 × − / ∆ + . S ( ρ L ) denotes the entropy of a contiguous half of 6 × λ/λ c , S ( ρ E ) and S ( ρ i ) are now roughly half the value of the left panel (Eq. (45)), while S ( ρ L ) /S ( ρ i ) isproportional to √ n (Eq. (46)). Hence, the single site entropy is again S ( ρ i ) ≈ − f (ln f −
1) while Eq. (38) yields S ( ρ E ) ≈ − n f (ln f − α ) (42) ≈ n S ( ρ i ) − n f α , (43)where α is a geometric entropy factor: α = Z u ( ˜ k ) ln u ( ˜ k ) d d ˜ k , (44)( u ( ˜ k ) ≥ R u ( ˜ k ) d d ˜ k = 1). In the isotropic case ∆ − i =∆ − ∀ i , we have α = α d , with α = 1 − ln 2 ≈ . α = 2 α ≈ .
614 and α ≈ . ≈ ln 2 for large d .At fixed λ , and for ∆ ± i = ∆ ± , f = (∆ − ) d/ (8 λ )and hence both S ( ρ i ) and S ( ρ E ) increase as d increases,reflecting the larger number of links. However, and as-suming again ∆ + ≥ λ c = d (∆ + + | ∆ − | ) also increases,entailing that at fixed λ/λ c , f (and so S ( ρ i ) and S ( ρ E )) decreases: f ≈ [∆ − / (∆ + + | ∆ − | )] d ( λ/λ c ) . (45)For example, the right panels in Fig. 2 depict S ( ρ E ) and S ( ρ i ) in an isotropic square lattice of 6 × − / ∆ + = 1 /
3. At fixed λ/λ c ,their values are verified to be roughly half that of thesimilar one-dimensional case (Eq. (45)). Their ratio isalso slightly smaller due to the increase in the parameter α in (43). On the other hand, for λ → λ c there is againa single vanishing energy ω , so that all entropies behaveas − ln( λ/λ c −
1) up to leading order, with all ratiosapproaching 1.We also depict there the entropy S ( ρ L ) of a contiguoushalf-size block ( n x × n y / × n x . For λ ≫ λ c , it is ver-ified that the number of non-zero positive eigenvalues ofthe corresponding contraction matrix D L is just the num-ber of couplings “broken” by the partition (2 n x ), beingall approximately equal to f / S ( ρ L ) ≈ − n x f (ln f / − ≈ n x S ( ρ i ) + 2 f ln 2] , (46)whence S ( ρ L ) /S ( ρ i ) ∝ n x / III. APPLICATION TO SPIN SYSTEMS
The previous bosonic formalism can be directly appliedto interacting spin s systems in an external magnetic fieldthrough the RPA approximation [18]. Denoting with s iµ the dimensionless spins S iµ / ~ at site i , we will consider acyclic translationally invariant finite array which can bedescribed by an XY Hamiltonian of the form H = B X i s iz − s X i = j ( J xij s ix s jx + J yij s iy s jy ) (47a)= B X i s iz − s [ X i = j ∆ + ij s i + s j − + ∆ − ij ( s i + s j + + h.c. )](47b)where s j ± = s jx ± is jy , J ijµ = J µ ( i − j ) and∆ ± ij = ( J xij ± J yij ) = ∆ ± ( i − j ) . (48)We note that x, y, z may in principle also denote local in-trinsic axes at each site, in which case the field is assumedto be directed along the local z axis. The s − scaling ofthe couplings ensures a spin-independent mean field andeffective RPA boson Hamiltonian (see below). Normal RPA . For sufficiently strong field B , the low-est mean field state (i.e., the separable state with lowestenergy) is the aligned state | i = | i ⊗ . . . ⊗ | n i , where | i i denotes the local state with maximum spin along the − z axis ( s iz | i i = − s | i i ). In such a case, RPA impliesthe approximate bosonization [18] s i + → √ sb † i , s i − → √ sb i , s iz → b † i b i − , (49)which is similar to the Holstein-Primakoff bosoniztion[23, 26] and leads to the quadratic boson Hamiltonian(1) with the parameters (48) and λ = B . We may thendirectly apply all previous expressions.The bosonic RPA scheme becomes exact for strongfields | B | ≫ B c for any size n , spin s , geometry or in-teraction range, since for weak coupling it correspondsto the exact first order perturbative expansion of theground state wave function [18]. As a check, in the caseof the spin 1 / XY coupling, an analytic expression of the block entropyin the limit n → ∞ has been obtained in [13–15]. For λ = B > ∆ + , it is given in present notation by [13] S ( ρ L ) = [ln αα ′ + ( α − α ′ ) I ( α ) I ( α ′ ) π ] , (50)where α = ∆ − / p λ + ∆ − − ∆ +2 , α ′ = √ − α and I ( α ) = R dx/ p (1 − x )(1 − α x ) is the elliptic integral of the first kind. An expansion of (50) for λ ≫ ∆ ± leadsexactly to present Eq. (32), with f given by (27). We canthen expect the asymptotic expressions (30) and (43) for S ( ρ E ) to be exact in this limit also in spin systems. Parity breaking RPA . Considering now the anisotropicferromagnetic-type case | J y ( l ) | ≤ J x ( l ) ∀ l in (47a), theprevious normal RPA scheme will hold, according to Eq.(16), for B ≥ B c = J x ≡ P l J x ( l ), i.e., when the corre-sponding boson system is stable.For | B | < B c , the normal RPA becomes unstable ( ω becomes imaginary). The lowest mean field state corre-sponds here to degenerate states |± Θ i fully aligned alongan axis z ′ forming an angle ± θ with the z axis in the x, z plane: | Θ i = | θ i ⊗ . . . ⊗ | θ n i , with | θ i i = exp[ − iθs iy ] | i i .We are assuming here an anisotropic XY coupling suchthat H commutes with the S z parity P z = e iπ ( P i s iz + ns ) ,but not with an arbitrary rotation around the z axis (asin the XX case). Such states break then parity sym-metry, satisfying P z | Θ i = | − Θ i . The angle θ is to bedetermined from [18]cos θ = B/B c , B c = X l J x ( l ) . (51)For | B | < B c , the bosonization (49) is then to be appliedin the RPA to the rotated spin operators s iz ′ = s iz cos θ + s ix sin θ , s i ± ′ = s ix ′ ± is iy ′ , with s ix ′ = s ix cos θ − s iz sin θ and s iy ′ = s iy . This leads again to a stable Hamiltonianof the form (1) with [18] λ = B c , ∆ ± ( l ) = [ J x ( l ) cos θ ± J y ( l )] . (52)For | B | < B c we should also take into account the im-portant effects from parity restoration for a proper RPAestimation of entanglement entropies [18]. The exactground state in a finite array will have a definite parity P z outside crossing points [21], implying that the actualRPA ground state should be taken as a definite paritysuperposition of the RPA spin states constructed around | ± Θ i [18]. This leads to reduced RPA spin densitiesof the form ρ A ≈ [ ρ A ( θ ) + ρ A ( − θ )] if the complemen-tary overlap O ¯ A = h− Θ ¯ A | Θ ¯ A i can be neglected. If thesubsystem overlap O A = h− Θ A | Θ A i = cos n A s θ is alsonegligible, such that ρ A ( θ ) ρ A ( − θ ) ≈
0, then [18] S ( ρ A ) ≈ S ( ρ A ( θ )) + δ , (53)where δ = ln 2. The final effect is then the additionof a constant shift to the bosonic subsystem entropy for | B | < B c . This is applicable to both S ( ρ E ) and S ( ρ L ) if θ , n and the block size L are not too small.For first neighbor couplings with anisotropy χ = J y /J x ∈ (0 ,
1) (if χ > x, y axes)as well as for arbitrary range couplings with a commonanisotropy χ = J y ( l ) /J x ( l ) ∈ (0 , | B | < B c is the existence of a transversefactorizing field B s = B c √ χ where the mean field states | ± Θ i become exact ground states [19–22]. As seen from(52), at this field ∆ − ( l ) = 0 ∀ l , so that the RPA vac-uum remains the same as the mean field vacuum [18] and (cid:144) B c S H Ρ E L s = (cid:144) = = (cid:144) B c S H Ρ E L (cid:144) S H Ρ L L s = (cid:144) = = RPAn=8 (cid:144) B c S H Ρ L L s = (cid:144) = = (cid:144) B c S H Ρ E L s = (cid:144) = = n=4x2 FIG. 3. (Color online) Top: Exact entanglement entropy of all even sites (left) and of a contiguous block of n/ n = 8 spins with anisotropic XY first neighbor couplings ( J y /J x = )and spin s = 1 /
2, 1 and 2, as a function of the transverse magnetic field. The dotted line depicts the bosonic RPA result, with B c = J x the mean field critical field. We have used base 2 logarithm in the entropy, such that all entropies approach 1 at thefactorizing field B s ≈ . B c . Bottom: Left: The corresponding ratio S ( ρ E ) /S ( ρ L ). Right: The entanglement entropy of alleven sites in a rectangular lattice of 4 × all contractions F ± ij vanish, implying S ( ρ A ( θ )) = 0. AllRPA entropies at B s reduce then to the correction term δ arising from parity restoration [18].This is essentially also the exact result at B s : Thetransverse factorizing field corresponds to the last groundstate parity transition as B increases from 0 [21] and theground state side-limits for B → B s are actually the def-inite parity combinations of the mean field states | ± Θ i [21, 28]. These definite parity states have Schmidt num-ber 2 for any bipartition, implying that the side-limits ofthe exact entropy of the reduced state of any subsystemat B s do not approach 0 but rather the values [28] S ( ρ ± A ( B s )) = X ν = ± q ± ν ln q ± ν , q ± ν = (1 + νO A )(1 ± νO ¯ A )2(1 ± O A O ¯ A ) , (54)where + ( − ) corresponds to positive (negative) parity, i.e.the right (left) side limit at B s [28]. Eq. (54) is valid forany size or spin. For small complementary overlap ¯ O ¯ A , q ± ν ≈ (1 + νO A ) and both side limits coincide, while if O A is also small, q ν ≈ / s = 1 / XY chain at B s [13].Illustrative exact results for the even-odd entanglemententropy in a finite linear cyclic spin s chain with firstneighbor couplings are plotted in the top left panel of Fig.3 for spins s = 1 /
2, 1 and 2, together with the bosonicRPA estimation. The exact definite parity ground statewas employed in all cases. We also depict for comparisonthe entropy of a contiguous half (top right), and the ratio S ( ρ E ) /S ( ρ L ) (bottom left). The anisotropy of the cou-pling is the same as in Fig. 2 (∆ − / ∆ + = 1 / s for the scaling used in (47)), rep-resents the large spin limit but is already quite close tothe exact results for s = 2 except in the vicinity of B c ,where the exact entropies remain of course finite in a fi-nite chain. The ratio S ( ρ E ) /S ( ρ L ) is nonetheless quiteaccurately reproduced and shows the extensive characterof S ( ρ E ) for B > B c , in agreement with (33), where theentropies for all spin values rapidly approach the RPAresult and become spin independent. For | B | < B c theshift δ in (53) ( δ = +1 in Figs 3–4 since base 2 logarithmwas employed) is essential for the agreement and explainsthe lack of direct extensivity in this region. The collapseof all entropies to the value δ at B s is also verified, and (cid:144) B c S H Ρ E L s = (cid:144) = (cid:144) B c S H Ρ E L (cid:144) n E n = n = (cid:144) B c @ S H Ρ E L - δ D (cid:144) n E FIG. 4. (Color online) Top: Exact entanglement entropy ofall n/ / n . Couplings are the same as in Fig. 3.Bottom: The intensive entropy S ( ρ E ) /n E ( n E = n/ B & ∆ + . The inset depicts the intensiveshifted entropy ( S ( ρ E ) − δ ) /n E , where δ = 0 for B > B s and δ = 1 for B < B s , which makes curves for n ≥ B < B s . for s = 1 / B s predictedby Eq. (54) can be appreciated (together with the otherparity transitions for B < B s ). The bottom right paneldepicts S ( ρ E ) in a 4 × − / ∆ + , where a similar behavior is obtained. Exact results for s = 2 are now even closer to the RPA prediction, indi-cating that the accuracy of the latter tends to improve,for stable mean fields, as the connectivity increases [29].Exact results for a spin 1 / S ( ρ E ) ∝ n is verified for strong fields B & ∆ + (bottom panel), whereas for B < B s it holds for theshifted entanglement entropy S ( ρ E ) − δ , as seen in theinset. Complete lack of extensivity takes place at the fac-torizing field B s , where the discontinuity implied by (54)is appreciable for n = 8 and becomes quite noticeable for n = 4. IV. CONCLUSIONS