Partial transpose of two disjoint blocks in XY spin chains
PPartial transpose of two disjoint blocks in XY spinchains
Andrea Coser, Erik Tonni and Pasquale Calabrese
SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy.
Abstract.
We consider the partial transpose of the spin reduced density matrix of twodisjoint blocks in spin chains admitting a representation in terms of free fermions, such asXY chains. We exploit the solution of the model in terms of Majorana fermions and showthat such partial transpose in the spin variables is a linear combination of four Gaussianfermionic operators. This representation allows to explicitly construct and evaluate theinteger moments of the partial transpose. We numerically study critical XX and Isingchains and we show that the asymptotic results for large blocks agree with conformalfield theory predictions if corrections to the scaling are properly taken into account. a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r artial transpose of two disjoint blocks in XY spin chains
1. Introduction
In the last decade, the understanding of the entanglement content of extended quantumsystems boosted an intense research activity at the boundary between condensed matter,quantum information and quantum field theory (see e.g. Refs. [1] as reviews). Thebipartite entanglement for an extended system in a pure state is measured by the well-known entanglement entropy which is the von Neumann entropy corresponding to thereduced density matrix of one of the two parts. One of the most remarkable results inthis field is the logarithmic divergence of the entanglement with the subsystem size inthe case when the low energy properties of the extended critical quantum systems aredescribed by a 1+1 dimensional conformal invariant theory [2, 3, 4, 5].Conversely, when an extended quantum system is in a mixed state (or one considersa tripartition of a pure state and is interested in the relative entanglement between twoof the three parts) the quantification of the entanglement is much more complicated. Avery useful concept is that of partial transposition. Indeed it has been shown that thepresence of entanglement in a bipartite mixed state is related to occurrence of negativeeigenvalues in the spectrum of the partial transpose of the density matrix [6]. Thisled to the proposal of the negativity [7] (or the logarithmic negativity) which was latershown to be an entanglement monotone [8], i.e. a good entanglement measure from aquantum information perspective. Compared to other entanglement measurements formixed states, the negativity has the important property of being easily calculable for anarbitrary quantum state once its density matrix is known (and indeed for this reason ithas been named a “computable measure of entanglement” [7]).Recently, a systematic path integral approach to construct the partial transposeof the reduced density matrix has been developed and from this the negativity in 1+1dimensional relativistic quantum field theories is obtained via a replica trick [9]. Thisapproach has been successfully applied to the study of one-dimensional conformal fieldtheories (CFT) in the ground state [9, 10], in thermal state [11, 12], and in non-equilibriumprotocols [12, 13, 14, 15], as well as to topological systems [16, 17]. The CFT predictionshave been tested for several models [10, 11, 13, 18, 19, 20], especially against exact results[10, 11, 13, 21] for free bosonic systems (such as the harmonic chain). Indeed for freebosonic models, the partial transposition corresponds to a time-reversal operation leadingto a partially transposed reduced density matrix which is Gaussian [22] and that can bestraightforwardly diagonalised by correlation matrix techniques [23, 24, 25]. It shouldbe also mentioned that there exist some earlier results for the negativity in many bodysystems [21, 22, 24, 26, 27, 28, 29, 30, 31].In the case of free fermionic systems (such as the tight-binding model and XYspin chains) the calculation of the negativity is instead much more involved. Indeedthe partial transpose of the reduced density matrix is not a Gaussian operator andstandard techniques based on the correlation matrix [25] cannot be applied. In view of theimportance that exact calculations for free fermionic systems played in the understandingof the entanglement entropy [3, 32, 33, 34, 35, 36], it is highly desirable to have an exact artial transpose of two disjoint blocks in XY spin chains n = 5 for thecritical Ising model and XX chain and carefully compare them with CFT predictions bytaking into account corrections to the scaling. In Sec. 6 we numerically evaluate andstudy the moments of the partial transpose for two disjoint blocks of fermions and againwe compare with new CFT predictions. Finally in Sec. 7 we draw our conclusions. Inappendix A we report all the CFT results which we needed in this manuscript.
2. The model and the quantities of interest
In this manuscript we consider the XY spin chains with Hamiltonian H XY = − L (cid:88) j =1 (cid:18) γ σ xj σ xj +1 + 1 − γ σ yj σ yj +1 + h σ zj (cid:19) , (1)where σ αj are the Pauli matrices at the j -th site and we assume periodic boundaryconditions σ αL +1 = σ α . For γ = 1 Eq. (1) reduces to the Hamiltonian of the Ising modelin a transverse field while for γ = 0 to the one of the XX spin chain. The Hamiltonian (1)is a paradigmatic model for quantum phase transitions [38]. In fact, it depends on twoparameters: the transverse magnetic field h and the anisotropy parameter γ . The systemis critical for h = 1 and any γ with a transition that belongs to the Ising universalityclass. It is also critical for γ = 0 and | h | < artial transpose of two disjoint blocks in XY spin chains Figure 1.
We consider the entanglement between two disjoint spin blocks A and A embedded in a spin chain of arbitrary length. The reminder of the system is denoted by B which is also composed of two disconnected pieces B and B . The Jordan-Wigner transformation c j = (cid:16) (cid:89) m 3. R´enyi entropies In this section we review the results of Ref. [41], where the R´enyi entropies of two disjointblocks A = A ∪ A for the XY spin chains have been computed. These results are themain ingredients to construct the integer powers of the partial transpose which will bederived in the following section. We will denote the number of spins in A and A with (cid:96) and (cid:96) respectively and the remainder of the system B contains a region separating A and A denoted as B , as pictorially depicted in Fig. 1.In a general spin 1 / ρ A = Tr B | Ψ (cid:105)(cid:104) Ψ | of A = A ∪ A can be computed by summing all the operators in A as follows [3] ρ A = 12 (cid:96) + (cid:96) (cid:88) ν j (cid:68) (cid:89) j ∈ A σ ν j j (cid:69) (cid:89) j ∈ A σ ν j j , (13)where j is the index labelling the lattice sites and ν j ∈ { , , , } , with σ = the identitymatrix and σ = σ x , σ = σ y and σ = σ z the Pauli matrices. The multipoints correlatorsin (13) are very difficult to compute, unless there is a representation of the state in termsof free fermions.For the single interval case, the Jordan-Wigner string (cid:81) m 5, where we have computed also T , in addition to the other ones already reported in Ref. [41]: • n = 2: T = { Γ } + { Γ , Γ } + δ B (cid:16) { Γ } − { Γ , Γ } (cid:17) ; (33) artial transpose of two disjoint blocks in XY spin chains • n = 3: T = { Γ } + 3 { Γ , Γ } + 3 δ B (cid:16) { Γ , Γ } + { Γ , Γ } − { Γ , Γ , Γ } (cid:17) ; (34) • n = 4: T = { Γ } + { Γ , Γ , Γ , Γ } + 4 { Γ , Γ } + 2 { Γ , Γ } (35)+ 2 δ B (cid:16) { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + 2 { Γ , Γ } + 2 { Γ , Γ } + 2 { Γ , Γ , Γ , Γ } + 4 { Γ , Γ , Γ } − 2[ 2 { Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } ] (cid:17) + δ B (cid:16) { Γ } + 2 { Γ , Γ } + { Γ , Γ , Γ , Γ } − { Γ , Γ } (cid:17) ; • n = 5 T = { Γ } + 5 (cid:16) { Γ , Γ } + { Γ , Γ } + { Γ , Γ , Γ , Γ } (cid:17) (36)+ 5 δ B (cid:16) { Γ , Γ } + { Γ , Γ } + 2 { Γ , Γ , Γ } + 2 { Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + 2 { Γ , Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + 2 { Γ , Γ , Γ , Γ , Γ }− { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ } + { Γ , Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ , Γ } ] (cid:17) + 5 δ B (cid:16) { Γ , Γ } + { Γ , Γ } + 2 { Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + 2 { Γ , Γ , Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } − { Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } + { Γ , Γ , Γ } + { Γ , Γ , Γ , Γ } ] (cid:17) . We notice that the algebraic sum of the integer coefficients occurring in any termmultiplying a power δ pB with p > δ B in T n , the sum of their coefficients is 2 n − . 4. Traces of integer powers of the partial transpose of the spin reduceddensity matrix In this section we move to the main objective of this paper which is to give a representationof the integer powers of the partial transpose of the spin reduced density matrix of twodisjoint blocks with respect to A . Eisler and Zimboras in Ref. [37] showed how to obtainthe partial transpose of a fermionic Gaussian density matrix, a procedure which can beapplied to the spin reduced density matrix in Eq. (19) using the linearity of the partialtranspose as we are going to show. We mention that in Ref. [37] the moments of thepartial transpose for two adjacent intervals were studied in details using the property thatfermionic and spin reduced density matrices are equal for this special case.Given a Gaussian density matrix ρ W written in terms of Majorana fermions in A = A ∪ A , the partial transposition with respect to A leaves invariant the modes in A and acts only on the ones in A . Furthermore, the partial transposition with respect artial transpose of two disjoint blocks in XY spin chains A of ρ A in (17) leaves the operator P B unchanged (because it does not contain modesin A ), therefore we have ρ T A = ρ T even + P B ρ T odd = + P B ρ T + + − P B ρ T − , (37)where ρ T ± = ρ T even ± ρ T odd , (38)as clear from Eq. (19) because of the linearity of the partial transpose. The partialtransposition of an arbitrary product of Majorana fermions A (denoted shortly as O like in the previous section) is given by the following map [37] R ( O ) ≡ ( − τ ( µ ) O , τ ( µ ) ≡ (cid:110) µ mod 4) ∈ { , } , µ mod 4) ∈ { , } , (39)where we recall that µ is the number of Majorana operators in O . Then, applying Eq.(39) to (18), we find ρ T even = 12 (cid:96) + (cid:96) (cid:88) even ( − µ / (cid:104) O O (cid:105) O † O † ,ρ T odd = 12 (cid:96) + (cid:96) (cid:88) odd ( − ( µ − / (cid:104) O P B O (cid:105) O † O † , (40)which gives the desired fermionic representation of the partial transpose of the spinreduced density matrix.At this point the moments of ρ T A can be obtained following the same reasoning as forthe moments of ρ A . Indeed, since ρ T ± are unitarily equivalent ( P A ρ T ± P A = ρ T ∓ because P A ρ T even P A = ρ T even and P A ρ T odd P A = − ρ T odd ) and P B commutes with them, startingfrom (37) and repeating the same observations that lead to (21), one getsTr( ρ T A ) n = Tr( ρ T ± ) n . (41)Similarly to the case of the R´enyi entropies considered in Sec. 3 (see Eq. (21)), thematrices ρ T ± are fermionic but not Gaussian. In the following we write them as sums offour Gaussian matrices, as done in (17) and (27) for ρ A . In particular, by introducing˜ ρ A ≡ (cid:96) + (cid:96) (cid:88) evenodd i µ (cid:104) O O (cid:105) O † O † , ˜ ρ B A ≡ (cid:96) + (cid:96) (cid:88) evenodd i µ (cid:104) O P B O (cid:105)(cid:104) P B (cid:105) O † O † , (42)one has that the matrices in (40) become ρ T even = ˜ ρ A + P A ˜ ρ A P A , ρ T odd = (cid:104) P B (cid:105) ˜ ρ B A − P A ˜ ρ B A P A , (43)telling us that ρ T ± in (38) are linear combinations of four Gaussian fermionic matricesoccurring in the r.h.s.’s of (43). Notice that ρ T even and ρ T odd are Hermitian but the matricesdefining them are not since( ˜ ρ A ) † = P A ˜ ρ A P A , ( ˜ ρ B A ) † = P A ˜ ρ B A P A . (44) artial transpose of two disjoint blocks in XY spin chains (cid:102) M ≡ (cid:32) (cid:96) i (cid:96) (cid:33) . (45)Then, the correlation matrices associated to ˜ ρ A , P A ˜ ρ A P A , ˜ ρ B A and P A ˜ ρ B A P A are givenby (cid:101) Γ k ≡ (cid:102) M Γ k (cid:102) M , k ∈ { , , , } . (46)In analogy to Eq. (32), we write the moments of ρ T A asTr( ρ T A ) n = (cid:101) T n n − . (47)From Eqs. (38), (41) and (43), we have that (cid:101) T n is a linear combination of 4 n terms. Thenet effect is that (cid:101) T n can be written by taking T n and replacing Γ i with (cid:101) Γ i and δ B with − δ B . The latter rule comes from the imaginary unit in the denominator of ρ T odd in Eq.(43).In the following we write explicitly (cid:101) T n for 2 (cid:54) n (cid:54) • n = 2 (cid:101) T = { (cid:101) Γ } + { (cid:101) Γ , (cid:101) Γ } + δ B (cid:16) { (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ } (cid:17) ; (48) • n = 3 (cid:101) T = { (cid:101) Γ } + 3 { (cid:101) Γ , (cid:101) Γ } + 3 δ B (cid:16) { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ } (cid:17) ; (49) • n = 4 (cid:101) T = { (cid:101) Γ } + { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 4 { (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ } (50)+ 2 δ B (cid:16) { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 4 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ }− { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ } (cid:17) + δ B (cid:16) { (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ } + { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ } (cid:17) ; • n = 5 (cid:101) T = { (cid:101) Γ } + 5 (cid:16) { (cid:101) Γ , (cid:101) Γ } + { (cid:101) Γ , (cid:101) Γ } + { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } (cid:17) (51)+ 5 δ B (cid:16) { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ }− { (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ }− { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ }− { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } (cid:17) + 5 δ B (cid:16) { (cid:101) Γ , (cid:101) Γ } + { (cid:101) Γ , (cid:101) Γ } + { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 2 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } (cid:17) . artial transpose of two disjoint blocks in XY spin chains T n , also in (cid:101) T n the algebraic sum of the integer coefficients occurring in any termmultiplying a power δ pB with p > 5. Numerical results for the ground state of the critical Ising and XX model The results of the previous section for the moments of the partial transpose of the reduceddensity matrix of two disjoint blocks are valid for arbitrary configurations of the XY spinchain: equilibrium, non-equilibrium, finite and infinite systems, critical and non-criticalvalues of the parameters γ and h . In this section we evaluate numerically these momentsfor the configurations that so far attracted most of the theoretical interest, namely thecritical points of the XY Hamiltonian, whose scaling properties are described by conformalfield theories. A great advantage of the present approach compared to purely numericalmethods such as exact diagonalization or tensor networks techniques is that it allows todeal directly with infinite chains without any approximations, reducing the systematicerrors in the estimates of asymptotic results. Indeed, all the numerical results presentedin the following are obtained for infinite chains.We will consider two particular points of the XY Hamiltonian, namely the criticalIsing model for γ = h = 1 and the zero field XX spin chain (corresponding to fermionsat half-filling) obtained for γ = h = 0. The scaling limit of the former is the Ising CFTwith central charge c = 1 / 2, while the scaling limit of the latter is a compactified bosonat the Dirac point with c = 1.The CFT predictions for the moments of both reduced density matrix and its partialtranspose have been derived in a series of manuscripts and they are reviewed in AppendixA. For both models we consider the case of two disjoint blocks of equal length (cid:96) embeddedin an infinite chain and placed at distance r . We numerically evaluate the moments of ρ A and ρ T A using the trace formulas of the previous sections for n = 2 , , , R n (defined in Eq. (11)): R n ≡ Tr( ρ T A ) n Tr ρ nA , (52)whose (unknown) analytic continuation for n e → R n = (cid:101) T n /T n . In the scaling limit (i.e. (cid:96), r → ∞ with ratio fixed) the ratio R n converges to the CFT prediction (cf. Eq. (81) in Appendix)written in terms of the four-point ratio x , which is x = (cid:18) (cid:96)(cid:96) + r (cid:19) , (53)when specialised to the case of two intervals of equal length (cid:96) at distance r . The negativity and the moments of ρ T A for the critical Ising chain in a transverse fieldhave been already numerically considered in Ref. [18] by using a tree tensor networkalgorithm and in Ref. [19] by Monte Carlo simulations of the two-dimensional classical artial transpose of two disjoint blocks in XY spin chains Figure 2. The ratio R n between the integer moments of ρ A and ρ T A for two disjointblocks of length (cid:96) at distance r embedded in an infinite critical Ising chain. We reportthe results for n = 3 , , x for various values of (cid:96) (andcorrespondingly of r ). For large (cid:96) , the data approach the CFT predictions (solid lines).The extrapolations to (cid:96) → ∞ –done using the scaling form (54)– are shown as crossesand they perfectly agree with the CFT curves for n = 3 and 4, while for n = 5 the fitsare unstable and the extrapolations are not shown. The last panel shows explicitly theextrapolating functions for two values of x and n = 3 , problem in the same universality class. However, the finiteness of the chain length did notallow to obtain very precise extrapolations to the scaling theory for all values of n and ofthe four-point ratio x . We found, as generally proved [9], that R is identically equal to 1.In Fig. 2 we report the obtained values of R n for n = 3 , , x for differentvalues of (cid:96) . It is evident that increasing (cid:96) the data approach the CFT predictions (thesolid curves). We can also perform an accurate scaling analysis to show that indeed thedata converge to the CFT results when the corrections to the scaling are properly takeninto account.It has been argued on the basis of the general CFT arguments [44], and shownexplicitly in few examples [35, 45, 46] both analytically and numerically, that Tr ρ nA displays‘unusual’ corrections to the scaling which, at the leading order, are governed by theunusual exponent δ n = 2 h/n where h is the smallest scaling dimension of a relevantoperator which is inserted locally at the branch point [44]. For the Ising model it hasbeen found that, in the case of two intervals, h = 1 / ρ T A ) n becausethey are only due to the conical singularities. Unfortunately, the corrections to the scaling artial transpose of two disjoint blocks in XY spin chains n increases, as already pointed out in Ref. [18]. Indeed,corrections of the form (cid:96) − m/n for any integer m are know to be present [35, 41, 57]. Thusthe most general finite- (cid:96) ansatz is of the form R n = R CFT n ( x ) + r (1) n ( x ) (cid:96) /n + r (2) n ( x ) (cid:96) /n + r (3) n ( x ) (cid:96) /n + · · · . (54)The variables r n ( x ) are used as fitting parameters in the extrapolation procedure. Thenumber of terms that we should keep in order to have a stable fit depends both on n and on x . For each case we keep a number of terms such that the extrapolated value at (cid:96) → ∞ isstable. In any case we never keep corrections beyond the order O ( (cid:96) − ). The results of thisextrapolation procedure for n = 3 and 4 are explicitly reported in Fig. 2. The agreementof the extrapolations with the CFT predictions is really excellent, at an unprecedentedprecision compared with fully numerical computations [18, 19]. Conversely, we find thatfor n = 5 the extrapolations are still unstable because of the large number of terms weshould keep in order to have a precise enough extrapolation. We now move to the study of the powers of ρ T A for the XX model in zero field. There areno previous numerical studies of this paradigmatic model. We again consider the ratios R n for n = 2 , , , R is identically equal to 1, as it should be.In Fig. 3 we report the obtained values of R n for n = 3 , , x for differentvalues of (cid:96) . It is evident that increasing (cid:96) the data approach the CFT predictions (the solidcurves). We should however mention a very remarkable property. It has been observedthat Tr ρ nA shows oscillating corrections to the scaling [35, 45, 41], which for zero magneticfield, are of the form ( − (cid:96) . These oscillations however cancel in the ratio R n and thecorrections to the scaling are monotonous, a property which makes the extrapolation toinfinite (cid:96) slightly simpler.Also in this case we can perform an accurate scaling analysis to show how the dataconverge to the CFT results when the corrections to the scaling are properly taken intoaccount. For the XX model, the leading correction to the scaling is governed by anexponent δ n = 2 /n , which means that they are less severe than in the case of the Isingmodel as it is also qualitatively clear from the figure. We then use the general finite- (cid:96) ansatz R n = R CFT n ( x ) + r (1) n ( x ) (cid:96) /n + r (2) n ( x ) (cid:96) /n + r (3) n ( x ) (cid:96) /n + · · · , (55)and, as in the case of the Ising model, we keep a number of fitting parameters whichmake stable the extrapolation at (cid:96) → ∞ . The results of this procedure for n = 3 and4 are explicitly reported in Fig. 3. The agreement of the extrapolations with the CFTpredictions is excellent. Also for the XX chain we find that for n = 5 the fits are unstable. artial transpose of two disjoint blocks in XY spin chains Figure 3. The ratio R n between the integer moments of ρ A and ρ T A for two disjointblocks of length (cid:96) at distance r embedded in an infinite XX chain at zero field. We reportthe results for n = 3 , , x for various values of (cid:96) (andcorrespondingly of r ). For large (cid:96) , the data approach the CFT predictions (solid lines).The extrapolations to (cid:96) → ∞ –done using the scaling form (55)– are shown as crosseswhich perfectly agree with the CFT curves for n = 3 and 4, while for n = 5 the fits areunstable. The last panel shows explicitly the extrapolating functions for two values of x and n = 3 , 6. Two disjoint intervals for free fermions In this section we consider the partial transposition for two disjoint blocks in the fermionicvariables. This problem was already addressed by Eisler and Zimboras [37], but a detailednumerical analysis was not presented. For fermionic variables there is no string in B connecting the two blocks (cf. Eq. (16)). Thus the partial transpose of fermions canbe obtained from the formulas derived in the previous sections by discarding the stringof Majorana operators (16), i.e. by replacing P B with . Performing this replacement,many simplifications occur in the formulas found in Sec. 3 and Sec. 4 as we will discussin the following. By definition the fermionic reduced density matrix is Gaussian with correlation matrixΓ defined in the Sec. 3, i.e.Tr ρ nA = { Γ n } . (56) artial transpose of two disjoint blocks in XY spin chains P B → , the reduced density matrix of the two disjointblocks given in Eqs. (17) and (18) becomes ρ A = ρ even + ρ F odd , (57)where § ρ even = 12 (cid:96) + (cid:96) (cid:88) even (cid:104) O O (cid:105) O † O † , ρ F odd = 12 (cid:96) + (cid:96) (cid:88) odd (cid:104) O O (cid:105) O † O † . (58)Moreover, ρ B A defined in Eq. (23) is replaced as ρ B A → ρ A and therefore, from Eqs. (27)and (57) we conclude that ρ F + = ρ A . As for the correlation matrices Γ i , since ρ B A → ρ A ,it is obvious that Γ → Γ and Γ → Γ .Summarising, we conclude that the fermionic Tr ρ nA is found by making in Eq. (32)the following replacements δ B → , Γ → Γ , Γ → Γ . (59)Performing these substitutions in the explicit examples given in Sec. 3 for 2 (cid:54) n (cid:54) .As a further check of our numerical codes, we numerically calculated Tr ρ nA using Eq.(56) (as was already done in Ref. [43]), obtaining that on the critical lines in the scalinglimit it converges toTr ρ nA → c n [ (cid:96) (cid:96) (1 − x )] n , (60)that corresponds to F n ( x ) = 1 identically in the general CFT formula (73). Indeed thisresult was already proven in the continuum free fermion theory [47]. We are now ready to set up the formulas for the moments of the partial transpose, asalready derived by Eisler and Zimboras [37], but numerically studied only for the case ofadjacent intervals.Once again, Tr( ρ T A ) n in the fermionic variables is obtained by replacing P B with in the formulas reported in Sec. 4. Performing this replacement in Eq. (37) we get ρ T A = ρ T even + ( ρ F odd ) T = ρ T + and in Eq. (42) it gives ˜ ρ B A → ˜ ρ A . These observationstogether with Eq. (43) lead to ρ T A = 1 − i2 ˜ ρ A + 1 + i2 P A ˜ ρ A P A , (61)which is exactly the same result obtained in [37] (for direct comparison, set ( O + ) there =( ˜ ρ A ) here and ( O − ) there = ( P A ˜ ρ A P A ) here ). § To compare our notation with the one used in [37], set ( ρ + ) there = ( ρ even ) here and ( ρ − ) there = ( ρ F odd ) here . artial transpose of two disjoint blocks in XY spin chains Figure 4. The ratio R n between the integer moments of ρ A and ρ T A for two disjointintervals of length (cid:96) at distance r for the tight-binding model at half-filling. We reportthe results for n = 3 , , x for various values of (cid:96) (and correspondingly of r ). For large (cid:96) , the data approach the CFT predictions (solidlines). The extrapolations to (cid:96) → ∞ –done using the scaling form (55)– are shown ascrosses and they perfectly agree with the CFT curves for n = 3 , , 5. The last panelshows explicitly the extrapolating functions for one value of x and n = 3 , , The last remaining step is just to write these formulas in terms of correlation matrices.Given that ˜ ρ B A → ˜ ρ A , it follows that we should perform the replacements (cid:101) Γ → (cid:101) Γ and (cid:101) Γ → (cid:101) Γ in order to get the moments of the partial transpose in terms of the correlationmatrices. Summarising, the fermionic Tr( ρ T A ) n are given by the formulas in Sec. 4performing the replacements δ B → , (cid:101) Γ → (cid:101) Γ , (cid:101) Γ → (cid:101) Γ . (62)Writing Tr( ρ T A ) n = (cid:101) T Fn / n − , and performing the replacements in the formulas for2 (cid:54) n (cid:54) (cid:101) T F = 2 { (cid:101) Γ , (cid:101) Γ } , (63) (cid:101) T F = − { (cid:101) Γ } + 6 { (cid:101) Γ , (cid:101) Γ } , (64) (cid:101) T F = − { (cid:101) Γ } + 4 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } + 8 { (cid:101) Γ , (cid:101) Γ } , (65) (cid:101) T F = − { (cid:101) Γ } − { (cid:101) Γ , (cid:101) Γ } + 20 { (cid:101) Γ , (cid:101) Γ } + 20 { (cid:101) Γ , (cid:101) Γ , (cid:101) Γ , (cid:101) Γ } . (66)Notice that the final expressions are very compact compared to the much morecumbersome spin counterparts. artial transpose of two disjoint blocks in XY spin chains We are now going to evaluate numerically the moments of the reduced density matrix andits partial transpose. We can study the problem for arbitrary values of h and γ enteringin Eq. (3), but in the following we focus on the most physically relevant fermionic systemwith h = γ = 0, i.e. the tight binding model H = 12 L (cid:88) i =1 (cid:104) c † i c i +1 + c † i +1 c i (cid:105) , (67)at half filling ( k F = π/ c = 1.The numerical results for the ratios R n = Tr( ρ T A ) n / (Tr ρ nA ) are reported in Fig. 4 asfunction of the four-point ratio x for different (cid:96) and for n = 3 , , R = 1identically, as it should). We also derived asymptotic CFT predictions for the fermionicmoments of the partial transpose, but their derivation is too cumbersome and beyond thegoals of this manuscript. We will report the derivation in a forthcoming publication [48]and we limit here to give the final results for n = 2 , , , 5. In order to have manageableformulas we introduce the shorts (cid:104) ε δ (cid:105) τ = | Θ[ e ]( | τ ( x )) | | Θ( | τ ( x )) | , (68)where Θ is the Riemann Theta function defined in Appendix A. In terms of the Θ functionthe fermionic ratios R n are given by [48]2 R (1 − x ) −→ (cid:104) (cid:105) ˜ τ , (69)4 R (1 − x ) −→ − (cid:104) (cid:105) ˜ τ , (70)8 R (1 − x ) −→ − (cid:104) (cid:105) ˜ τ + 4 (cid:104) (cid:105) ˜ τ , (71)16 R (1 − x ) −→ − (cid:104) (cid:105) ˜ τ + 20 (cid:104) (cid:105) ˜ τ − (cid:104) (cid:105) ˜ τ , (72)where the matrix ˜ τ has been defined in Eq. (80) and the exponent ∆ n in App. A.It is evident from Fig. 4 that the lattice numerical results approach the CFTpredictions depicted as solid lines for all n . As in the spin case, we can perform a carefulfinite (cid:96) analysis to take into account corrections to the scaling. The leading correction isexpected to be of the form (cid:96) − /n and subleading ones to be integer powers of the leadingone. The finite (cid:96) ansatz is then given by Eq. (55) and again to have an accurate descriptionof the data we keep a number of fitting parameters which make stable the extrapolation at (cid:96) → ∞ . The results of this extrapolation procedure for n = 3 , , n = 5 as a difference compared to the spin counterpart. artial transpose of two disjoint blocks in XY spin chains 7. Conclusions We have shown that the partial transpose of the reduced density matrix of two disjointspin blocks in the XY spin chain can be written as a linear combination of fourGaussian fermionic operators, fully specified by their correlation matrices (denoted as (cid:101) Γ i , i = 1 , , , Acknowledgments We are grateful to Maurizio Fagotti for many discussions on the subject of this paper. ETis grateful to Horacio Casini for discussions. PC and ET thank GGI and the organisersof the workshop Holographic Methods for Strongly Coupled Systems for hospitality duringpart of this work. PC and ET have been supported by the ERC under Starting Grant279391 EDEQS. artial transpose of two disjoint blocks in XY spin chains A. CFT results for entanglement entropy and negativity of two disjointintervals The moments of the reduced density matrix of two disjoint intervals for CFTs have beenstudied in a series of manuscripts [47, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63].These results have been derived using earlier findings for the partition functions of CFTson Riemann surfaces with non vanishing genus [64]. In this appendix we review themain results (especially from Refs. [10, 53, 56]) which are useful for the comparisonwith numerical results. We mention that some universal results are also known in higherdimensions both from field theory [65] and holography [63, 66, 67].From global conformal invariance, we know that Tr ρ nA for two disjoint intervals admitsthe general scaling form (choosing, without loss of generality, the endpoints of the intervalsin the order u < v < u < v ):Tr ρ nA = c n (cid:18) ( u − u )( v − v )( v − u )( v − u )( v − u )( u − v ) (cid:19) n F n ( x ) , (73)where ∆ n = c ( n − /n ) / 12, being c the central charge. The variable x is the four-pointratio x = ( u − v )( u − v )( u − u )( v − v ) . (74)Given the order of the points we have 0 ≤ x ≤ 1. The prefactor c n is non-universal,but can be exactly fixed from the exact calculation of the entanglement entropy of oneinterval.The difficult task of CFT is to have an exact representation for the universal function F n ( x ) normalised so that F n (0) = 1. This universal function has been analytically derivedfor the compactified boson (with central change c = 1) [52, 53] and for the Ising CFT (with c = 1 / 2) [56, 57], as well as for other conformal theories which however are not of interestfor this paper. Concerning the compactified boson, we are only interested in the value ofthe compactfication ratio corresponding to the scaling limit of the XX spin chain whichis the so called Dirac point. For the Ising CFT and at the Dirac point (which describerespectively the scaling limit of the critical Ising chain and the critical XX model), thefunction F n ( x ) reads [56] F Ising n ( x ) = (cid:80) e | Θ[ e ]( τ ( x )) | n − | Θ( τ ( x )) | , F Dirac n ( x ) = (cid:80) e | Θ[ e ]( τ ( x )) | n − | Θ( τ ( x )) | , (75)where Θ[ e ](Ω) is the Riemann theta function, which is defined as follows [68]Θ[ e ](Ω) ≡ (cid:88) m ∈ Z n − e i π ( m + ε ) t · Ω · ( m + ε )+2 π i ( m + ε ) t · δ , [ e ] ≡ (cid:104) εδ (cid:105) ≡ (cid:104) ε , . . . , ε n − δ , . . . , δ n − (cid:105) , (76)being Ω a ( n − × ( n − 1) symmetric complex matrix with positive imaginary part and e is the characteristic of the Riemann theta function, which is defined by a pair of n − artial transpose of two disjoint blocks in XY spin chains ε i , δ i ∈ { , / } . In Eq. (76) we have to sum over all thecharacteristics e . The elements of the matrix τ ( x ) in Eq. (75) read [53] τ ( x ) rs = i 2 n n − (cid:88) k =1 sin( πk/n ) F ( k/n, − k/n ; 1; 1 − x ) F ( k/n, − k/n ; 1; x ) cos (cid:20) π kn ( r − s ) (cid:21) , (77)where x ∈ (0 , 1) and F is the hypergeometric function.Also the moments of the partial transpose correspond to a four-point function oftwist fields in which two of them have been interchanged [9]. Consequently also thesemoments admit the universal scaling formTr( ρ T A ) n = c n (cid:18) ( u − u )( v − v )( v − u )( v − u )( v − u )( u − v ) (cid:19) n G n ( x ) , (78)with c n the same non-universal constant appearing in Eq. (73) and G n ( x ) a new universalscaling function. Exploiting the fact that the above moments correspond to the exchangeof two twist fields, it has been shown that G n ( x ) and F n ( x ) are related as [9, 10] G n ( x ) = (1 − x ) n F n (cid:18) xx − (cid:19) , (79)but some care is needed to take the analytic continuation of the function F n ( y ) to negativeargument y (see for details [10]). 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