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Entanglement of two blocks of spins in the critical Ising model P. Facchi,1,2 G. Florio,3,2 C. Invernizzi,4 and S. Pascazio3,2 1Dipartimento di Matematica, Università di Bari, I-70125 Bari, Italy 2INFN, Sezione di Bari, I-70126 Bari, Italy 3Dipartimento di Fisica, Università di Bari, I-70126 Bari, Italy 4Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy We compute the entropy of entanglement of two blocks of L spins at a distance d in the ground state of an Ising chain in an external transverse magnetic field. We numerically study the von Neumannentropyfordifferentvaluesofthetransversefield. Atthecriticalpointweobtainanalytical results for blocks of size L=1 and L=2. In the general case, the critical entropy is shown to be additivewhend→∞. Finally,based onsimplearguments,wederiveanexpression fortheentropy 9 at the critical point as a function of both L and d. This formula is in excellent agreement with 0 numerical results. 0 2 PACSnumbers: 03.67.Mn;73.43.Nq;75.10.Pq;03.67.-a n a J I. INTRODUCTION spins and the rest of the chain are studied as a function of the distance d between the blocks and their common 1 2 A comprehension of the features of entanglement in size L. This entropy of entanglement will be denoted systems with manydegreesof freedom,suchasquantum S(L,d)andwillbe analyzedbyanalyticalandnumerical ] methods. spinchains,iscurrentlyoneofthemostchallengingprob- h p lems, at the borderline of quantum information science The physical system we shall consider is the Ising - [1] and statistical physics. In the last few years several model in a transverse magnetic field, since it fulfills a nt measures of entanglement have been proposed [2, 3, 4] convenient combination of requirements. It is solvable, a and calculated (analytically in the simplest cases, oth- its ground state can be computed by using well-known u erwise numerically) for the ground states of many-body analyticaland numericaltechniques [7] and, at the same q systems [5]. time, it successfully describes a rich spectrum of physi- [ Despite accurate investigations and different propos- cal phenomena, that include the ordered and disordered 2 als, there is still no consensus on the correct character- magnetic phases, connected by a quantum phase transi- v ization of the multipartite entanglement of the ground tion [13]. 0 state of a many-body system. We will consider here the We will analytically compute the entropy of entangle- 0 entanglement entropy, a measure that can sometimes be ment at the quantum critical point (QCP) for blocks of 6 tackledbyanalyticinvestigationsandforwhichquantum L = 1 and L = 2 spins, and will tackle the problem nu- 0 . fieldtheoreticalmethodscanbeemployed. Theentangle- merically for larger values of L. We will first study the 8 mententropyisjustthevonNeumannentropyassociated behavior of the entropy as a function of the magnetic 0 with the reduceddensity matrix,that is the entropyofa field λ, then at the QCP, λ = 1, as a function of the 8 0 subsystem of the chain, and was explicitly evaluated for distance d between the blocks and their size L. We will : quantum spin chains [4, 7, 8, 9, 10, 11]. investigate the limits d 0 and d . Our results v → → ∞ One of the most striking features of the entanglement will include as a particular case (d = 0) the logarith- i X entropy is its universal behavior at and close to a quan- mic behaviour of the entropy of a single block of L spins ar tnuomn-cprhitaiscealtrsaynstseitmiosng.eInnedreaeldly, ittenisdfsotuondsatthuartateenttroowpayrdins [a4t,c7r,it8ic,a9li,ty10S,L11=, 1614l,og15L,+16K]., wWheerweilKl ailssoaschoonwstathnet a finite value as the size of the subsystem increases, but additivity of S(L,d) at the critical point as d . Fi- → ∞ this value (logarithmically) diverges with the size of the nally, we will plot S(L,d) as a function of both L and d, subsystem as the system approaches a quantum critical getting anaccurateidea ofthe features ofthe entropyat point. Close to a quantum critical point, where the cor- the QCP. relation length ξ is much larger than the lattice spacing, This paper is divided in six sections. In Section correlationsaredescribedbya1+1dimensionalquantum II we review previous works on spin chains, following field theory and at the critical point, where ξ diverges, [4, 7, 17, 18, 19, 20], where the ground state of the Ising the field theory is also a conformal field theory [12]. In modeliscomputed: theexplicitexpressionsobtainedwill thelattercase,thebehaviorofentropycalculatedbyana- be used in the following sections in order to obtain the lyticaland numericaltechniques for severalspin systems reduced density matrix ̺ and the Von Neumann en- L isconfirmedbythepredictionsofthecorrespondingfield tropyS ofL contiguousspins. In SectionIII we extend L theory. the definition of the correlation matrix given in [4, 7] In this work we extend the characterization of the to a bipartition of two blocks of L spins separated by a entanglement entropy to a more general subsystem, in generic distance d. We define here the entropy S(L,d), which the correlations between two disjoint blocks of describing the entanglement of the two blocks with the 2 restofthe chain. Bymakinguseofthe newly definedre- L d L duceddensitymatrix̺ ,inSectionIV, weanalytically L,d compute the entropy for blocks of one and two spins at the critical point. In Section V we carry out numerical computations of S(L,d) for several sizes of the blocks. Figure 1: (Color online) Two blocks of L adjacent spins at a Finally, we plot the entropy as a function of the size L of the blocks and their reciprocal distance d, in order distance d. The state ̺L,d is obtained from the ground state |ψ0i of the spin chain by tracing out the spins that do not to get a general idea of the features of the entropy of belong to theblocks. entanglement at the critical point. Our results are sum- marized and discussed in Section VI. In the Appendices we included, for self consistency,additionalmaterialand suchaninvariance: aˇ aˇ = aˇ aˇ =0, m,n 2m 2n 2m−1 2n−1 explicit calculations. h i h i ∀ with m = n, while aˇ aˇ = ig depend only on 2m−1 2n m−n 6 h i the difference m n. Therefore, the block correlation matrix ΓA becom−es independent of k and reads II. GROUND STATE OF THE ISING MODEL L Π Π ... Π 0 −1 −L+1 The Ising chain in a tranverse field consists of 2N +1 . . spins with nearest neighbor interactions and an external ΓA = Π1 Π0 . , (7) magnetic field, described by the Hamiltonian L .. .. ..  . . .    HI =−J (λσiz +σixσix+1). (1)  ΠL−1 ... ... Π0  −NX≤i≤N with Here i labels the spins (we take an odd number of spins 0 i aˇ aˇ 0 g for simplicity), J > 0 and we consider open boundary Πl = i aˇ aˇ − h 20l−1 0i = g 0l , conditions, σx = 0. σµ (µ = x,y,z) are the Pauli (cid:18)− h 2l −1i (cid:19) (cid:18)− −l (cid:19) N+1 i (8) matrices acting on spin i. The determination of the where the real coefficients g are given, for an infinite ground state proceeds with the Jordan-Wigner transfor- l chain, by mation in terms of Dirac or Majorana fermionic operators[17,18,19,20]. Hereitwillbeconvenienttoconsider 1 2π (cosφ λ)+isinφ Majorana fermions, whose operators are defined by g = dφeiφl − . (9) l 2π Z0 (cosφ λ)2+sin2φ − aˇ σz σx, aˇ σz σy, (2) q 2l−1 ≡ m! l 2l ≡ m! l Thus, ΓA is a real, skew-symmetic 2L 2L matrix, i.e. mY<l mY<l (ΓA)T =L ΓA, since all the blocks Π h×ave the property with N l N. They are hermitian and obey anti- (ΠL)T = −ΠL. l comm−utat≤ion r≤elations, l − −l aˇ† =aˇ , aˇ ,aˇ =2δ , (3) m m { m n} mn III. ENTROPY OF TWO BLOCKS OF SPINS and their expectation values in the ground state ψ 0 | i The entropy of a single block of L contiguous spins in hψ0|aˇmaˇn|ψ0i=haˇmaˇni=δmn+iΓAmn, (4) the critical regime can be obtained by very accurate nu- mericalresults[4,7]andanalyticalconformalfieldtheory with 2N 1 m,n 2N, completely characterize − − ≤ ≤ calculations[8,9]. Giventhegroundstate ψ0 ,onefinds ψ . Consider now a block of L contiguos spins labeled | i | 0i the reduced density matrix by i with ̺ =Tr ψ ψ , (10) k i k+L 1, (5) L ¬L{| 0ih 0|} ≤ ≤ − wherethetraceisoverallspinsthatdonotbelongtothe withk > N andk+L 1<N. Theexpectationvalues − − block, and its entropy oftheMajoranaoperatorsoftheblockareencodedinthe 2L 2L submatrix × S = Tr ̺ log̺ . (11) L L L − { } ΓA = i( aˇ aˇ δ ), (6) L mn − h m ni− mn For definiteness, in this paper we will fix the base of log- with 2k 1 (cid:0)m,(cid:1)n 2k+2L 2. arithms to 2. The calculationsof Refs.[4, 7], yielding an − ≤ ≤ − We are interested in the thermodynamic limit of an expression of the reduced density matrix ̺ and the en- L infinite chain, N . In such a limit, the ground state tanglemententropyS ofLadjacentspinsintheground L →∞ becomestranslationinvariant,andallcorrelationsinherit state ψ are reviewed in Appendix A. The key point 0 | i 3 is the following: the block entropy (11) is the sum of L with L x < L+d in the (2L+d) (2L+d) matrix terms ΓA , ≤ × 2L+d S = L H 1+νl , (12) A(0L) A(−LL,d) A(−LL)−d L 2 Xl=1 (cid:18) (cid:19) ΓA = A(d,L) A(d) A(d,L) . (19) 2L+d L 0 −d   where    A(L) A(L,d) A(L)  H(x)= xlogx (1 x)log(1 x) (13)  L+d d 0  − − − −   For example, let us consider the case of two blocks of is the Shannon entropy of a bit, and iνl, with 1 l L = 2 spins at a distance d = 3. In this case the 7 7 L, are the pairs of (purely imaginary)±eigenvalues o≤f th≤e matrix ΓA =ΓA reads × 2L+d 7 block correlation matrix ΓA of Eq. (7). L In the continuous limit Π0 Π−1 Π−2 Π−3 Π−4 Π−5 Π−6 Π Π Π Π ... Π 1 0 −1 −2 −5 S = 1logL+ (L), (14)  .. ..  L 6 K Π2 Π1 . .   where 1/6 derives from the central charge c = 1/2 of a ΓA7 =Π3 Π2 ... ...  (20) free massless fermionic field and  .. ..  Π . .   4  1 Π ... ...  (L)= +O , L , (15)  5  K K L →∞ Π ... ... ... Π  (cid:18) (cid:19)  6 0    being a constant. In this section we want to extend andwe haveto cancelthe columns whosefirstelementis K this approachandconstruct the density matrix ̺ of a labelled by 2, 3, 4 and the rowswhose firstelement L,d − − − subsystem of two blocks of L adjacent spins situated at is labelled by 2, 3, 4, obtaining a distance d, studying their entanglement with the rest Π Π Π Π of the chain. See Fig. 1. To this aim, one starts by com- 0 −1 −5 −6 puting the matrix ΓAL of a single block of adjacent spins ΓA2,3 =ΠΠ1 ΠΠ0 ΠΠ−4 ΠΠ−5 , (21) (7), and then traces out the central d spins as follows. 5 4 0 −1 We define the 4L 4L correlation matrix ΓA of two Π6 Π5 Π1 Π0  × L,d   blocks,eachofLspins,situatedatadistanced(d,likeL, that is again a real skew-symmetrix matrix. are expressed in units of the distance between adjacent The entanglement of the two blocks of spins reads spins, and are therefore dimensionless) S(L,d)= Tr(̺ log̺ ), (22) L,d L,d A(L) A(L) ΓA A(L) − ΓA = 0 −L−d = L −L−d ,(16) where ̺L,d is the density matrix of two blocks of L adja- L,d     cent spins at a distance d. Exactly as for a single block, A(L) A(L) A(L) ΓA  L+d 0   L+d L  the above entropy can be given an explicit expression in     terms of the eigenvalues iν, with 1 l 2L, of ΓA , (L) (L,L) ± l ≤ ≤ L,d where Ax =Ax with analogous to (12), Πx Πx−1 ... Πx−M+1 2L 1+ν l Π Π ... Π S(L,d)= H , (23) A(xL,M) =  x...+1 x ... x−...M+2 .(17) Xl=1 (cid:18) 2 (cid:19)   Πx+L−1 Πx+L−2 ... Πx−M+L  with H given by (13).     Before investigating the behavior of Eq. (22), it is in- T structive to look first at some simple examples. The matrix A(L) = A(L) is a Toeplitz matrix, and x − −x ΓA has the propert(cid:16)y (cid:17) L,d IV. ANALYTICAL RESULTS ΓA =ΓA , (18) L,0 2L A. Entanglement of Two Single Spins i.e., when the distance between the two blocks is zero, ΓAL,d becomes equal to the matrix (7) of a single block We consider the Ising chain in a critical transverse of size 2L. The matrix ΓA in Eq. (16) is obtained by magnetic field λ = 1. At λ = λ , the coefficients g of L,d c c l tracing out the d rows and d columns that are labeled the reduced correlation matrix ΓA , defined in Section L,d 4 III, and given by Eq. (9), can be computed analytically, O(1/d2) coincide in the limit d + , we have → ∞ yielding π+2 1 S(1,d) = 2H +O gl = 21π 2πdφeiφlieiφ2 =−π(l1+ 1). (24) 2 (cid:18) 2ππ (cid:19)2 (cid:18)d2(cid:19)4π2 1 Z0 2 = log − +log +O . π π+2 π2 4 d2 (cid:18) (cid:19) (cid:18) − (cid:19) (cid:18) (cid:19) In orderto compute the entanglemententropyS(L,d) (30) of two single spins at a distance d, we have to calculate theeigenvaluesofthecorrelationmatrixΓA ,whereL= More on this phenomenon later. L,d 1 and d 0. Equation (16) reads ≥ Π Π B. Entanglement of two blocks of L=2 spins ΓA = 0 −(1+d) 1,d Π Π 1+d 0 (cid:18) (cid:19) If L=2, we have to compute the eigenvalues of 0 g 0 g 0 −l g 0 g 0 =  −00 g −0l g  Π0 Π−1 Π−(2+d) Π−(3+d) l 0 Π Π Π Π g 0 g 0 ΓA = 1 0 −(1+d) −(2+d)  − −l − 0  2,d Π(2+d) Π(1+d) Π0 Π−1  0 −2 0 −2 Π Π Π Π π π(−2l+1)  (3+d) (2+d) 1 0  2 0 2 0  Π Π Π Π  =  π π(2l+1) ,(25) 0 −1 −(1+l) −(2+l) 0 −2 0 −2 Π Π Π Π π(2l+1) π = 1 0 −l −(1+l) , (31)  2 0 2 0   Π(1+l) Πl Π0 Π−1   π(−2l+1) π  Π Π Π Π    (2+l) (1+l) 1 0    where l=d+1. The eigenvaluesare the solutions to the with l = d+1. The characteristic equation (26) is of characteristic equation 8th degree, but can be reduced to a quartic equation in t=µ2 det(ΓA µ)=0. (26) L,d− t4+pt3+qt2+rt+s=0, (32) For L = 1, there are four eigenvalues iν and iν , 1 2 ± ± with whichhasanexactsolution. Thecoefficientsp,q,rands are functions of the distance d and are explicitly written 2 (4l2 1)2+4l2 1 in Appendix B. ν = − ± . (27) 1,2 π 4l2 1 ThematrixΓA willhavetheeighteigenvalues µ = p − L,d ± k iν , with k = 1,2,3,4, two for each (negative) root of k The eigenvalues of the reduced density matrix ̺ are ± L,d t. The eigenvaluesofthe reduceddensity matrix̺ are 2,d (1 ν )/2 and the von Neumann entropy reads k (1 ν )/2 and the entropy reads ± k ± S(1,d)=H 1+ν1 +H 1+ν2 , (28) 4 1+νk 2 2 S(L,d)= H , (33) (cid:18) (cid:19) (cid:18) (cid:19) 2 k=1 (cid:18) (cid:19) X where H is given by (13). with H given by (13). It is interesting to consider the cases d = 0 and d 1. In the former case we have ν = 2(√13 1)/(3π≫), In Fig. 2 we plot the eigenvalues νk versus the dis- 1,2 ± tance d. Note that the eigenvalues quickly saturate at a whence distance d 5 between the blocks. This means that the ≃ entanglement between the two L = 2 blocks reaches its √13+2 √13 S(1,0) = H +H asymptotic value for d&5. 3π ! 3π ! The asymptotic values of the eigenvalues are solutions 1 1 16 7π2 to the equation obtained by taking the limit d of → ∞ = log + − (32). One gets −2 16 9π4 (cid:18) (cid:19) √13 8√13π 2√13+1 log 1+ ν1( ) = ν3( )= , − 3π 16 4√13π+3π2! ∞ ∞ π 3 − 2√13 1 1 8π ν ( ) = ν ( )= − , (34) log 1+ . (29) 2 ∞ 4 ∞ π 3 −3π 3π2 4π 16 (cid:18) − − (cid:19) in agreement with Fig. 2. Note that they coincide with In the latter case, since the eigenvalues ν = 2/π + (27) evaluated at l = 1, i.e. with the eigenvalues of 1,2 5 ν k Π = σ∓ = (σ iσ )/2, Eqs. (16) and (17) greatly ±1 1 2 ∓ simplify. In particular, for any L > 1, the off-diagonal 1 blocks in (16) read, for d=0, 0.5 0 0 ... Π+1 A(LL) =− A(−LL) T = ... ... ...  (35) 0 (cid:16) (cid:17) 0 0 ... 0     -0.5 and yield a tridiagonal block matrix -1 0 Π−1 0 ... 0 Π 0 Π ... 0 +1 −1 0 5 10 15 20 d ΓAL,0 =ΓA2L = ... ... ... ... , (36)    0 0 0 ... 0 Figure 2: (Color online) Eigenvalues of the reduced density   matrix of two blocks of L = 2 spins versus their distance   T d at the critical point λc = 1. Note that the νk’s reach a while, for any d> 0, A(L) = A(L) = 0 and one saturation value at d ≃5. Here and in the following figures, L+d − −L−d disexpressedinunitsofthedistancebetweenadjacentspins, gets (cid:16) (cid:17) and is therefore dimensionless. ΓA =ΓA ΓA. (37) L,d L ⊕ L Thus, at d=0, the characteristic polynomial is the reduced density matrix of two spins at a distance d = l 1 = 0. Therefore, for d the eigenvalues − → ∞ det(ΓA µ)=det(ΓA µ)=µ2(µ2+1)2L−1, (38) coalesce into pairs and the spectrum of two blocks (of L,0− 2L− twospins)coincideswiththe spectrum(withdegeneracy hence ν =0 and ν =1 for 2 l 2L, so that 2) of a single block. This phenomenon, which implies 1 l ≤ ≤ the asymptotic additivity of block entropy, is indepen- 2L 1 1 dent of the blocks dimension L and will be discussed in S(L,0)=H + H(1)=H =log2=1. full generality in Sec. VB2. 2 2 (cid:18) (cid:19) l=2 (cid:18) (cid:19) X (39) On the other hand, for d 1, one gets ≥ V. ENTROPY OF TWO BLOCKS OF SPINS det(ΓA µ)=det(ΓA µ)2 =µ4(µ2+1)2L−2, (40) L,d− L − In this section we look at the entanglement entropy hence ν =ν =0 and ν =1 for 3 l 2L and S(L,d) of two blocks of spins when the magnetic field 1 2 l ≤ ≤ varies. At the critical point, λ = 1, we will find an ex- c 1 pressionforS(L,d)intermsoftheentropyS ofasingle S(L,d)=2S(L,0)=2H =2. (41) L 2 block, given in Eqs. (11)-(14), and investigate its limits (cid:18) (cid:19) d 0, d . We will combine numerical estimates → → ∞ This is intuitively clear: at zero transverse field S(L,d) with analytical methods. follows exactly an area law [27] and in 1 dimension the presenceofa gapdoublesthe areaofthe boundary,dou- bling the entropy. See Fig. 3. For nonzero values of the A. Entropy versus λ magnetic field, there are correction to the area law, due to correlations between the two blocks. An entropy in- We start by evaluating the entanglement entropy creaseis stillnatural,butit turnsoutto be smallerthan S(L,d) versus the magnetic field λ. The entanglement the factor 2 that one would naively expect for a doubled betweentwo single spins ata distance d andthe remain- boundary. We will show that the factor 2 can be recov- ingpartofthechainwasinvestigatedinRefs.[23,24,25] ered in the limit of large gap d: this is the phenomenon asafunctionofthe magneticfieldλ. Ageneralizationto ofasymptoticadditivityofentropymentionedattheend a comb of m spins, spaced d sites apart can be found in of Sec. IVB. See Sec. VB. [26]. We now generalize these results to the case of two We show the results of some numerical investigations arbitrary blocks of L spins at a distance d. inFig. 3. LetusfirstlookatthecaseofsmallL( 2 3). ≃ ÷ In general, the presence of a gap between the blocks For d > 0 entropy is not maximum at the critical point yields a larger entropy for all values of the magnetic λ = 1, but rather for some λ < 1. On the other hand, field λ. Let us start examining the situation at zero for bigger L, the maximum entropy is always reached at magnetic field. At λ = 0, from (9) one gets that the critical point λ=1 and its value growswith the size g = δ . Therefore, since Π = 0 for l = 1 and L of the blocks. l l,−1 l 6 ± 6 One also notices that, far from criticality, the entropy 2.0 ìàìàìàìàìàìà has a very weak dependence on the value of the gap d ìà ≥ ìà 1, and thus an area law is a very good approximation. 1.5 HaL The largestdeviationsareatthe criticalpoint, whenthe ìà correlation length diverges. At fixed L, the ordinate of ìà the cusp at λ = 1 is always an increasing function of S 1.0 ææææææææ ìàìà æ ìà d. We now turn to the study of the critical case and æ ìàìà endeavor to find some interesting analytical expressions. 0.5 æææææìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæà 0.0 B. Critical chain 0.5 1.0 1.5 2.0 2.5 3.0 Λ 1. Blocks of contiguous spins (d→0) 2.0 ìàìàìàìàìàìàìà In the limit d 0, the entropy of entanglement (22) ìà → must reproduce the single-block result (11) as a particu- 1.5 ìà HbL lar case: ìà S(L,0)=S2L. (42) S 1.0 æææææææææ ìàìàìà æ ìàìà Tvaehbriiysfipeirsdoadwushciemtn,pttlhehiescomennasaigbsntleeentsiccuysficethlodecokibstacanriindtictwhaale,svλnaculum=eero1if.catAlhlyes 0.5 æææææìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæà 0.0 constant via the logarithmic fit 0.5 1.0 1.5 2.0 2.5 3.0 K Λ 1 1 S(L,0)= log2L+ +O . (43) 6 K L (cid:18) (cid:19) 2.5 We obtain ìàìàìà 2.0 ìàìàìàìàìà ìà =0.690413, (44) K 1.5 ìà HcL æ with an error 9 10−6, corroboratingthe results in [4]. S æ ìà An accurate fi≃t en·ables us to give a precise estimate of 1.0 ææææææ æ ìàìà the corrections in 1/L, but more on this later. 0.5 ææææìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæà 0.0 2. Asymptotic additivity of entropy (d→∞) 0.5 1.0 1.5 2.0 2.5 3.0 Λ The plot in Fig. 4 shows that in the limit d the → ∞ two-block entropy is accurately fitted by ì 2.5 à 1 ìà S(L,∞)= 3logL+2K(L)=2SL. (45) 2.0 ìàìàìàìàìàìà ìà Entropy becomes therefore additive at large distances d. S 1.5 æ ìà HdL The physical meaning of this result is that the entangle- æ ìà mententropyoftwoseparatedblocksofLspinsbecomes 1.0 ææææææ æ ìàìà btweticweeetnhethenetbrolopcyksofapapsrionagclehebslo∞ck, iL.e.wihtebnetchoemdesismtauncche 0.5 ææææìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæà larger than the size of a block. Therefore, at the crit- 0.0 ical point, the quantum correlations between two finite 0.5 1.0 1.5 2.0 2.5 3.0 blocks of spins saturate at a certain distance. We now Λ explain this result, which turns out to be valid for every value of the magnetic field λ. Figure 3: (Color online) Entropy of the reduced density ma- Note that from (9) one gets trix for blocks of (a) L = 2, (b) L = 3, (c) L = 8 and (d) L=15 spins versus the external magnetic field λ for several lim gl =0, (46) distancesdbetweentheblocks. Circlesd=0,squaresd=10, l→∞ diamonds d = 50. Squares and diamonds are indistinguish- able in (a) and (b), and are barely distinguishable in (c) and (d) only around thecritical value λ=1. 7 by Riemann-Lesbegue lemma. Thus, from (8) SHL, ¥L 3.5 lim Π =0, (47) l l→∞ 3.0 and from (17) xl→im∞A(xL) =0, ∀L. (48) 2.5 Ax(L) accounts for the residual correlation of two blocks 2.0 of spin at a distance x. Therefore, the limit of the reduced correlation matrix 1.5 of two blocks (16) reads L 20 40 60 80 100 (L) (L) A A 0 −L−d ΓA = lim ΓA = lim L,∞ L,d   Figure4: (Coloronline)Saturationvalueofthecriticalblock d→+∞ d→+∞ A(L) A(L) entropy (λ= 1). The numerical values of S(L,∞) are fitted  L+d 0  by 1logL+2K(L). A(L) 0   3 0 = =ΓA ΓA, (49)  0 A(L)  L ⊕ L with 0 < α < 1 and β R. See Fig. 1. The quan- 0 ∈   tity α fixes the position of the end of each block, α = 0 for all L. The two blocks become independent and the correspondingto the centralpoint between two adjacent limiting spectrum of ΓA is given by the spectrum of a spins, while α = 1 to the position of the last (or first) L,d single block ΓA, with degeneracy 2. As a consequence, spin. Clearly, α detects granularity in the chain and the L the entropy (23) becomes additive in the limit correctionsduetoαwillbeimportantforsmallvaluesof L and/or d. We obtain 2L 1+ν (ΓA ) l L,∞ 1 1 α S(L, ) = H S(L,0) = log(2L α)+ log 1 ∞ 2 ! 6 − 3 − L l=1 X 1 (cid:16) (cid:17) L 1+ν(ΓA) + (logα+β) = 2 H l L =2S , (50) 6 L 2 1 1 Xl=1 (cid:18) (cid:19) log(2L)+ (logα+β), L , ∼ 6 6 →∞ where ±iνl(ΓAL,d) with 1 ≤ l ≤ 2L denote the eigenval- (52) ues of ΓA and iν (ΓA) with 1 m L denote the L,d ± m L ≤ ≤ whence eigenvalues of ΓA. We stress againthat these results are L valid for allvalues of the magnetic field λ. See the intro- logα+β =6 . (53) K ductory comments in Sec. VA. The area law discussed there is restoredforsufficiently larged whenthe correla- On the other hand, tions between the two blocks are negligible and the two 1 boundaries become “independent”. S(L,+ ) = 2log(L α)+β ∞ 6 − 1(cid:16) 1 (cid:17) log(L)+ β, L , (54) ∼ 3 6 →∞ 3. General behavior of S(L,d) at the critical point whence We now turn to the problem of describing the entan- β =12 . (55) glement of two blocks of L spin at an arbitrary distance K d with the remaining part of the critical Ising chain. In In conclusion, ordertofindafunctionofLanddwemaketwoassump- tions: werequirethattheentropybeafunctionofall the 1 S(L,d) = 2log(L α) 2log(L+d) scalesoftheproblem;moreover,thedependencemustbe 6 − − logarithmic. (cid:16) +log(2L+d α)+log(d+α) 2logα , Let therefore − − ((cid:17)56) 1 S(L,d) = 2log(L α) 2log(L+d) 6 − − with (cid:16) +log(2L+d α)+log(d+α)+β , (51) α=2−6K =0.0566226. (57) − (cid:17) 8 Notice that there are no free parameters. Moreover, in the realm of validity of CFT, when d,L 1, one gets ≫ 1 S(L,d) 2logL 2log(L+d) ∼ 6 − (cid:16) +log(2L+d)+logd + , S K (cid:17) (58) 3 that agrees with the results of Calabrese and Cardy [9] 2 150 when one adds a missing addendum in their formula (3.32). 1 From (56) we get 100 d 1 S(L,0)= log(2L)+ (L), (59) 10 6 K 50 L where 20 1 1 α 1 α (L) = logα+ log 1 + log 1 K −6 6 − 2L 3 − L 30 5 α 3 α(cid:16)2 17 α(cid:17)3 (cid:16) (cid:17) ∼ K− 12L − 16L2 − 144L3, L→∞. (60) Figure5: (Coloronline)Criticalentropy(λ=1)betweentwo blocksofLspinsatadistancedandtheremaningpartofthe This formula is in excellent agreement with numerical (infinite) chain. results. It provides the explicit expression in Eq. (15) and was used in Eqs. (43)-(44) and Fig. 4. See also the S discussion at the end of Sec. VB1. The global behavior of 3.0 1 S(L,d) = 2log(L α) 2log(L+d) 6 − − (cid:16) S +log(2L+d α)+log(d+α) 2.5 − 3.0 1 (cid:17) = 2logL 2log(L+d) 2.5 6 − +(cid:16)log(2L+d)+logd + (L), (61) 2.0 2.0 K with α and (L) given by (57) and (60(cid:17)), respectively, is 1.50 20 40 60 80 100 120 140d K displayed in Fig. 5. The fit is accurate up to one part in 1.5 d 103 for small L (< 10) and one part in 106 for L > 10. 0 5 10 15 20 25 30 Notice the logarithmic L dependence for d = 0 and the saturation effect for L/d 1. A section of Fig. 5 is Figure 6: (Color online) Critical entropy (λ = 1) between ≪ displayed in Fig. 6. In particular, the inset shows the two blocks of L = 30 spins and the remaning part of the asymptotic behaviour of the entropy and its saturation. chain versus d. Inset: L=30 with 0<d<140. 4. Behavior of the critical entropy for small d whereweusedEq.(57)inthelastequalities. Thisagrees very well with Fig. 6 and explains why ∆S is largely independent of L in Fig. 5. More to this, the final result Both in Figs. 5 and 6 one notices for all values of L inEq.(62)yieldsasuggestiveinterpretationofthefitting a sharp entropy increase at small values of d from d = parameterαinEq.(57)andoftheconstant : theyturn 0 to d = 1. This corresponds to the two ordinates of K out to be related to the entropy increase ∆S associated the cusps in Fig. 3. Let us endeavor to interpret this withtheopeningofad=1gap(onequbit)inaninterval phenomenon on the basis of the formulas derived in this of2Lcontiguousspins. Itisthereforenotsurprisingthat section. Equation (56) yields = (1/6)logα, being O(∆S), be also necessarily of ∆S = S(L,1) S(L,0) oKrder−1. − 1 L+1 2L+1 α 1+α = 2log +log − +log 6 − L 2L α α (cid:18) − (cid:19) VI. CONCLUSIONS 1 1+α L≫1 log ∼ 6 α 1 We provided an analytic and numerical treatment of logα= (62) the entanglement entropy of two disjoint blocks of spins ≃ −6 K 9 as a function of their length and distance in the quan- Then, V defines a new set of Majorana operators, cˇ† = m tum Ising model with a transverse magnetic field. We cˇ , m gave an analytic expression of the entropy at the critical point, for two blocks of 1 and 2 spins at a generic 2L distance. We showed that the presence of a gap always cˇ V aˇ , (A2) m m,n n ≡ yields an entropy increase. At criticality, due to a log- n=1 X arithmic correction to the area law, this increase is less than a factor 2 for all values of d, and becomes 2 for that satisfy the same anticommutation relations as the larged, whenasymptoticadditivity takesplace. We also aˇ ’s, and have correlation matrix ΓC. The structure of n L showed that, interestingly, the entropy of the two blocks ΓC implies that mode cˇ is only correlated to mode L 2l−1 canbewrittenintermsoftheentropyofasingleblockof cˇ . In the language of fermionic operators, one gets L 2l spins,that,atthequantumphasetransition,growsloga- spinless fermionic modes rithmicallywiththesizeoftheblock. Wehavealsogiven anaccurateidea ofthe generalfeatures ofthe entropyof cˇ +icˇ 2l−1 2l two blocks as a function of their size and distance. cˆl , ≡ 2 The behavior of the entanglement in a critical spin cˆ,cˆ =0, cˆ†,cˆ =δ , (A3) chain agrees with well known results in conformal field { l m} { l m} mn theory, where the geometric entropy (analogous to the spinblockentropy,butdefinedinthe continuum)canbe that, by construction, fulfill computed for 1+1 dimensional theories [14, 15]. The translation of field theoretical methods and ideas in the cˆcˆ =0, cˆ†cˆ =δ 1+νl. (A4) languageofquantuminformationwillhopefullyenableus h l mi h l mi lm 2 tomakeuseofadditionalresultsforanarbitrarynumber of disjoint intervals [9, 21]. This would of great inter- Thus,theL(nonlocal)fermionicmodesareuncorrelated, est from the point of view of multipartite entanglement. so that the reduced density matrix can be written as a The study of the statistical distribution of bipartite en- product tanglement for different bipartitions [28] is a useful tool for the analysis of multipartite entanglement. A deeper ̺ =ρ ... ρ . (A5) L 1 L comprehensionofthedependenceofentropyondistances ⊗ ⊗ andsizesofblocksofspinscouldyieldinformationabout Now, the density matrix ρ , 1 l L, has eigenvalues l the role of quantum phase transitions in the generation ≤ ≤ (1 ν )/2 and entanglement entropy l of multipartite entangled states [29, 30]. ± 1+ν l S(ρ )=H , (A6) l Acknowledgments 2 (cid:18) (cid:19) We thank Matteo Paris for interesting discussions. where H is the Shannonentropyof a bit (13). Therefore This work is partly supported by the European Com- thespectrumof̺Lresultsfromtheproductofthespectra munity through the Integrated Project EuroSQIP. of the density matrices ρ , and the entropy of ̺ is the l L sum of the entropies of the L uncorrelated modes, Appendix A: COMPUTATION OF S L L 1+ν l S = H . (A7) L 2 We review here the computation of the entropy of the l=1 (cid:18) (cid:19) X reduced density matrix ̺ Tr ψ ψ for L adja- L ¬L 0 0 ≡ | ih | cent spins [4, 7]. In the limit of infinite chain (N ), This is Eq. (12) of the text. Summarizing, for arbitrary →∞ a given finite section of the chain is fully translational values ofthe magnetic fieldλ andinthe thermodynamic invariant and ̺L describes the state of any block of L limit, N , the block entropy SL of the ground state →∞ contiguous spins. The density matrix ̺L can be recon- of the Ising model is given by the sum (A7), where iνl ± structed from the restricted 2L 2L correlation matrix arethe pairsofimaginaryeigenvaluesofthe block corre- ΓA of Eq. (7). In particular, a×direct way to compute lation matrix ΓA of Eq. (7). L L the spectrum of ̺ and its entropy S from ΓA is the L L L following. The matrix ΓA can be put into a block-diagonal form L by an orthogonal transformation and its eigenvalues are Appendix B: EIGENVALUES OF ΓA2,d purely imaginary and come in pairs, iν, and ν 1, l l ± | | ≤ with 1 l L. Let V SO(2L) be the special orthogonal ma≤trix≤such that Γ∈C =VΓAVT is block-diagonal The coefficients in Eq. (32) are functions of the dis- L L tance d between the blocks and read L 0 1 ΓC = ν . (A1) L l 1 0 l=1 (cid:20)− (cid:21) M 10 26(3+2d)4(5+2d)4 24314B 41528B2 10144B3 896B4 p(d) = 5303+ + + + , π2A 3 9 9 9 (cid:20) (cid:21) 212(3+2d)2(5+2d)2 2841841B2 q(d) = 203297+391466B+ π4A 9 (cid:20) 3652160B3 2617216B4 329984B5 17152B6 + + + + , 27 81 81 81 (cid:21) 222(2+d)4 68116B 158795B2 198074B3 139000B4 17312B5 896B6 r(d) = 12271+ + + + + + , π6A 3 9 27 81 81 81 (cid:20) (cid:21) 232 s(d) = (1+d)4(2+d)8(3+d)4, (B1) 34π8A where A = (1 + 2d)2(3 + 2d)6(5 + 2d)6(7 + 2d)2 and eigenvaluesof ΓA will be µ = iν , with 0 ν 1, 2,d ± k ± k ≤ k ≤ B = d(4+d). We obtain µ2 by solving (32). The eight for k =1,2,3,4. [1] M.A. Nielsen and I.L. Chuang, Quantum Computation [16] H.CasiniandM.Huerta,Phys.Lett.B600,142(2004). andQuantum Information(CambridgeUniversityPress, [17] E. Lieb, T. Schultz and D. Mattis, Annals of Phys. 16, Cambridge, 2000). 407 (1961). [2] A. Osterloh, L. Amico, G. Falci and R. Fazio, Nature [18] S. Katsura, Phys. Rev. 127 1508 (1962). 416, 608 (2002). [19] E. Barouch, B. McCoy, and M. Dresden, Phys. Rew. A [3] T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 2 1075 (1970). 032110 (2002). [20] E. Barouch and B. McCoy, Phys.Rev.A 3786 (1971). [4] J. I. Latorre, E. Rico, and G. Vidal, Quant. Inf. and [21] P.CalabreseandJ.Cardy,J.Stat.Mech.P04010(2005). Comp. 4 048 (2004). [22] L.CamposVenuti,C.degliEspostiBoschi,M.Roncaglia, [5] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. andA.Scaramucci,Phys.Rev.A,73,010303(R)(2006). Mod. Phys. 80, 517 (2008). [23] G. Rigolin, T. R. de Oliveira, and M. C. de Oliveira, [6] T. Prosen, I. Pizorn, Phys. Rev. Lett. 101, 105701 Phys. Rev.A 74, 022314 (2006). (2008); T. Prosen, New J. Phys. 10, 043026 (2008). [24] T. R. de Oliveira, G. Rigolin, and M. C. de Oliveira, [7] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Phys. Rev.A 73, 010305(R) (2006). Rev.Lett. 90, 227902 (2003). [25] T. R. de Oliveira, G. Rigolin, M. C. de Oliveira, and E. [8] B.-Q. Jin and V.E. Korepin, J. Stat. Phys. 116, 79 Miranda, Phys. Rev.Lett. 97, 170401 (2006). (2004). [26] J.P.Keating,F.Mezzadri,andM.Novaes,Phys.Rev.A [9] P.CalabreseandJ.Cardy,J.Stat.Mech.P06002(2004). 74, 012311 (2006). [10] J. E. Eisert and M. Cramer, Phys. Rev. A 72, 042112 [27] J. Eisert, M. Cramer, and M.B. Plenio, Area laws (2005). for the entanglement entropy - a review, Preprint [11] A. R. Its, B.-Q. Jin and V. E. Korepin, J. Phys. A, 38, arXiv:0808.3773 (2008). 2975 (2005). [28] P. Facchi, G. Florio and S. Pascazio, Phys. Rev. A 74, [12] P.DiFrancesco,P.Mathieu,andD.Senechal,Conformal 042331 (2006). Field Theory (Springer,Heidelberg, 1999). [29] G. Costantini, P. Facchi, G. Florio, and S. Pascazio, J. [13] S. Sachdev, Quantum Phase Transitions (Cambridge Phys. A: Math. Theor. 40, 8009 (2007). Univ.Press, 1999). [30] P.Facchi,G.Florio,G.Parisi,S.Pascazio,Phys.Rev.A [14] C.G.Callan andF.Wilczek, Phys.Lett.B, 333(1994). 77, 060304(R) (2008). [15] C. Holzhey, F. Larsen, and F. Wilczek, Nucl. Phys. B 424 44 (1994).