Entanglement entropy of excited states
EEntanglement entropy of excited states
Vincenzo Alba , Maurizio Fagotti , and PasqualeCalabrese Scuola Normale Superiore and INFN, Pisa, Italy. Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Pisa, Italy.
Abstract.
We study the entanglement entropy of a block of contiguous spins in excitedstates of spin chains. We consider the XY model in a transverse field and theXXZ Heisenberg spin-chain. For the latter, we developed a numerical applicationof algebraic Bethe Ansatz. We find two main classes of states with logarithmic andextensive behavior in the dimension of the block, characterized by the propertiesof excitations of the state. This behavior can be related to the locality propertiesof the Hamiltonian having a given state as ground state. We also provide severaldetails of the finite size scaling. a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t ntanglement entropy of excited states
1. Introduction
The study of the entanglement in the ground-states of extended quantum systemsbecame a major enterprise in recent times, mainly because of its ability in detectingthe scaling behavior in proximity of quantum critical points (see e.g. Refs. [1, 2, 3]as reviews). The most studied measure of entanglement is surely the entanglemententropy S A , defined as follows. Let ρ be the density matrix of a system, which we taketo be in the pure quantum state | Ψ (cid:105) , ρ = | Ψ (cid:105)(cid:104) Ψ | . Let the Hilbert space be writtenas a direct product H = H A ⊗ H B . A ’s reduced density matrix is ρ A = Tr B ρ . Theentanglement entropy is the corresponding von Neumann entropy S A = − Tr ρ A log ρ A , (1)and analogously for S B . When ρ corresponds to a pure quantum state S A = S B .The entanglement entropy is one of the best indicators of the critical propertiesof an extended quantum system when A and B are a spatial bipartition of the system.Well-known and fundamental examples are critical one-dimensional systems in thecase when A is an interval of length (cid:96) in a system of length N with periodic boundaryconditions. In this case, the entanglement entropy follows the scaling [4, 5] S A = c (cid:18) Nπ sin π(cid:96)N (cid:19) + c (cid:48) N →∞ −→ c (cid:96) + c (cid:48) , (2)where c is the central charge of the underlying conformal field theory and c (cid:48) a non-universal constant (the behavior for N → ∞ is known from Refs. [6, 7]). Away fromthe critical point, S A saturates to a constant value [7] proportional to the logarithm ofthe correlation length [4]. This scaling allows to locate the position (where S A divergesby increasing (cid:96) ) and a main feature (the value of the central charge c ) of quantumcritical points displaying conformal invariance. The entanglement entropy of disjointintervals gives also information about other universal features of the conformal fixedpoint related to the full operator content of the theory [8].Conversely, only little attention has been devoted to the entanglement propertiesof excited states (with the exception of few manuscripts [9, 10, 11, 12]), although itis a very natural problem. Here we consider two topical spin-chains [13] to addressthis issue. We first consider the XY model in a transverse magnetic field. We employthe well-known mapping of the model to free fermions to reduce the calculation ofthe entanglement entropy to that of the eigenvalues of a Toeplitz matrix on the linesof the ground-state case [7, 14, 15, 16, 17, 18, 19]. In the present computation, theproperties of the excitations above the ground-state will strongly affect the form ofthe reduced density matrix and of the entanglement entropy. Then, to consider atruly strongly interacting quantum model, we address the same problem for the XXZchain, always remaining in the realm of integrable systems. In fact, this model isexactly solvable by means of Bethe Ansatz [20, 21]. This provides a classification ofall eigenstates and their energies, but no information about dynamical properties. Toovercome this limit, we take advantage of recent progresses in the algebraic BetheAnsatz [22, 23] that provides all elements of the reduced density matrix as a (huge)sum of determinants whose entries are functions of the Bethe rapidities. However, inthis approach an inhomogeneous coupling must be considered and the homogeneouslimit (in which we are interested) is recovered in a cumbersome manner.In the study of the entanglement properties of excited states, a first subtlepoint is the choice of the basis of the Hilbert space. In fact, while the ground-state of a local Hamiltonian is usually unique (or with a finite small degeneracy, ntanglement entropy of excited states quantum quenches . In fact, it has been argued thatthe post-quench state is a time-dependent superposition of eigenstates that in thethermodynamic limit have the same energy [24]. It is known that for a global quench,the entanglement entropy first increases linearly with the time and then saturates toa values proportional to the length of the block (cid:96) [25, 26, 27]. We will indeed find afull class of excited states having an extensive entanglement entropy and those couldbe the relevant ones for quench problems. Oppositely in local quantum quenches theasymptotic state displays a logarithmic entanglement entropy [28] and a different classof states should be relevant. We also mention that some of the features we find havesimilarities with what obtained in some non-equilibrium steady states [29].The manuscript is organized as follows. In the next section 2 we study the XYmodel and we find two main classes of excited states, corresponding the extensive andlogarithmic behavior of the entanglement entropy. In Sec. 3 we consider the XXZmodel and the algebraic Bethe ansatz approach. We find that the states that have alogarithmic behavior in the XX limit conserve this property with the same prefactorof the logarithm and with a constant term slightly depending on ∆. Finally in Sec. 4we summarize our main results and discuss problems deserving further investigation.
2. The XY model in a transverse magnetic field
We start our analysis by considering the XY spin chain of length N with periodicboundary conditions, whose Hamiltonian is given by H XY = − N (cid:88) l =1 (cid:20) J (cid:18) γ σ xl σ xl +1 + 1 − γ σ yl σ yl +1 (cid:19) + h σ zl (cid:21) , (3)where σ αl are the Pauli matrices at the site l . h is the transverse magnetic field and γ the anisotropy parameter. For γ = 1 the Hamiltonian reduces to the Ising model, whilefor γ = 0 to the XX model. The diagonalization of this Hamiltonian is a standardtextbook exercise. First a Jordan-Wigner transformation c l = (cid:32) (cid:89) m 2, with iν j the ‡ If we would be pedantic in defining this limit, we can think to (1 + m ( ϕ )) / E x with a Gaussian of zero mean and standard deviation that must beput to zero at the end of any computation. Since in the sum in Eq. (12) there is almost everywhere(everywhere in non-critical regions) a regular function of ϕ , the regularization in the definition of m ( ϕ ) is perfectly well-defined. ntanglement entropy of excited states Γ · · · Γ (cid:96) − ... . . . ...Γ − (cid:96) · · · Γ , Γ l = (cid:68)(cid:18) A xs A ys (cid:19) (cid:0) A xs + l A ys + l (cid:1)(cid:69) − δ l . (11)The two-by-two matrices Γ l are easily computed observing that the generic eigenstatein the Slater-determinant basis (9) is the vacuum of the fermionic operators˜ b † k , = (cid:40) b k , k ∈ E x , b † k otherwise.After simple algebra one obtainsΓ ( E x ) l = Γ ( GS ) l + 2 iN (cid:88) k ∈ E x (cid:18) sin( lϕ k ) − cos( lϕ k − θ k )cos( lϕ k + θ k ) sin( lϕ k ) (cid:19) , (12)where θ k is the Bogolioubov angle of the transformation that diagonalizes theHamiltonian in Eq. (7) and Γ ( GS ) l the corresponding matrix in the ground-state [7, 14].As explained in the previous subsection, when | E x | ∼ N , we can substitute inequation (12) the sum with an integral1 N (cid:88) k ∈ E x → π (cid:90) π − π d ϕ m ( ϕ )2 ϕ k → ϕ , (13)where (1 + m ( ϕ )) / E x introducedabove. Substituting in Eq. (12) this regularization we haveΓ ( E x ) l = 12 π (cid:90) π − π d ϕe − ilϕ Γ ( E x ) ϕ , with (14)Γ ( E x ) ϕ = 12 (cid:18) m ( − ϕ ) − m ( ϕ ) − i [ m ( ϕ ) + m ( − ϕ )] e iθ i [ m ( ϕ ) + m ( − ϕ )] e − iθ m ( − ϕ ) − m ( ϕ ) (cid:19) . (15)The entanglement entropy can be expressed as a complex integration over acontour C that encircles the segment [ − , 1] at the infinitesimal distance η as in Ref.[17] S (cid:96) = lim η → + πi (cid:73) C d λe (1 + 2 η, λ ) dd λ log det | λ − Π | , (16)where e ( x, y ) = − x + y x + y − x − y x − y . A similar expression is easily written for all R´enyi entropies for general n . Applyingthe Sz¨ego lemma (see e.g. Ref. [31]) to the determinant of the block Toeplitz matrix λ − Π, we obtain the leading order in (cid:96) of the entanglement entropy S (cid:96) = (cid:96) π (cid:90) π − π d ϕ H ( m ( ϕ )) + O (log (cid:96) ) , (17)with H ( x ) = e (1 , x ).This first result is very suggestive: the entanglement entropy of a class of excitedstates in the XY model is extensive, in contrast with the logarithmic behavior of theground state. However, every time that m ( ϕ ) (cid:54) = 1 only in a region of vanishingmeasure of the domain (as in the ground state) this leading term vanishes, and oneshould go beyond the Sz¨ego lemma to derive the first non-vanishing order of the ntanglement entropy of excited states m ( ϕ ) = 1almost everywhere, m ( ϕ ) can be re-written in the following form, that is particularlyuseful to apply Fisher-Hartwig ( ϕ ∈ ] − π, π [) m ( ϕ ) = e i arg m ( π ) n (cid:89) j =1 e i arg( ϕ − ϕ j ) , (18)where 2 (cid:100) n/ (cid:101)§ is the number of the discontinuities of m ( ϕ ) and ϕ j are the discontinuitypoints (the term 2 (cid:100) n/ (cid:101) takes into account an eventual discontinuity in π that is notcounted by considering the open interval ϕ ∈ ] − π, π [). We prove analytically in thenext subsection that S (cid:96) ∝ log (cid:96) in the XX chain ( γ = 0) and then we show that thisis not a peculiar feature of the isotropic model. In the XX spin chain the Bogolioubov angle reduces to e iθ k = sign( J cos ϕ k − h )and the Fisher-Hartwig conjecture is sufficient to prove the following result: theentanglement entropy of the excited states described by the multi-step function (18)grows logarithmically with the width of the block. The coefficient in front of thelogarithm is 1 / | h | > | h | < 1) to take into account the modeswith zero energy. For | h | < ± ϕ F ( ϕ F = arccos | h/J | )define the function˜ m ( ϕ ) = (cid:40) m ( ϕ ) , ϕ ∈ [ − ϕ F , ϕ F ] , − m ( − ϕ ) , otherwise, (19)that substitutes m ( ϕ ) when counting discontinuities. The importance of the numberof discontinuities was firstly stressed in [32] in a different context. In Fig. 1 a directcomputation shows the importance of the position of the modes with zero energy. The proof of therelation between the entanglement entropy and the discontinuities of m ( ϕ ) when Eq.(18) holds (i.e. when m ( ϕ ) = ± 1) in an XX chain is a slight modification of the proofgiven by Jin and Korepin in Ref. [17] for a critical XX ground-state. For γ = 0, thematrix (15) can be written in terms of the Pauli matrix σ y asΓ( ϕ ) = ± σ y m ( ∓ σ y ϕ ) , (20)with the upper (lower) sign if the momentum ϕ is below (above) the Fermi level ofthe Jordan-Wigner fermions. As a consequence the block Toeplitz matrix (11) can bereduced to a standard Toeplitz matrix with symbol γ ( ϕ ) = (cid:40) , (cid:0) e ϕ > ∧ m ( − ϕ ) = 1 (cid:1) ∨ (cid:0) e ϕ < ∧ m ( ϕ ) = − (cid:1) , − , otherwise , (21) § Here and below, (cid:100) x (cid:101) stands for the closest integer larger than x and (cid:98) x (cid:99) for the closest integersmaller than x . ntanglement entropy of excited states Figure 1. The entanglement entropy as a function of the block length for theexcited state with characteristic function m ( ϕ ) = sign(( ϕ − π )( π − ϕ )) of twocritical XX chains. The different behavior is caused by the position of the zeromodes ( ϕ F = π/ π/ a = 1 or 2. The straight lines are the analytic prediction for large (cid:96) given by Eqs. (25) and (26). with e ϕ = J cos ϕ − h . The reduced correlations matrix λ − Π is generated by thesymbol t ( ϕ ) = λ − n (cid:89) j =1 e i arg[ ϕ − ϕ j ] , where the ϕ j ’s are the momenta corresponding to the n discontinuities of γ ( ϕ ). Theground state has two symmetric discontinuities at ± ϕ F . The symbol admits thecanonical Fisher-Hartwig factorization [33] t ( ϕ ) = ( λ + 1) a ( λ − b n (cid:89) j =1 t j ( ϕ ) , with t j ( ϕ ) = e − iβ j ( π − ϕ + ϕ j ) , ϕ j < ϕ < ϕ j + 2 π, (22) β j ( λ ) = ( − j − πi log λ + 1 λ − , − π ≤ arg (cid:104) λ + 1 λ − (cid:105) < π, (23)and the two exponents are b = 1 − a = 12 π n (cid:88) j =1 ( − j − ϕ j . Defining k F ≡ (cid:80) nj =1 ( − j − ϕ j / 2, the Fisher-Hartwig conjecture (that for thiscase with | λ | > 1, i.e. | Re( β j ) | < / 2, has been proved by Basor [33]) readsdet | λ − Π | ∼ n (cid:89) i The entanglement entropy as a function of the block length for twoexcited states of the infinite critical Ising chain with 4 discontinuities (Left) atmomenta { , . , . , . } and {− . , . , . , . } . The different slopes are causedby the zero mode. Right: Two excited states of a non-critical XY chain in finitesize. subsection we show that a commuting set of local operators of the XY chain can beused to prove that all these logarithmic excited states are ground-states of properlydefined local conformal Hamiltonians. Eq. (25) can be exploited to deduce the centralcharge of this local Hamiltonian c = a = n/ It is straightforward from Eq. (12) to calculate the spectrum of the reduced densitymatrix and the entanglement entropy for any eigenstate at any value of γ and h .We calculated the entanglement entropy numerically for several different cases and wealways find a logarithmic behavior with (cid:96) every time m ( ϕ ) = 1 almost everywhere (seee.g. Fig. 2). To get a proof similar to the one of the previous section for the generalXY model, one should generalize the methods in Ref. [18] mapping the computationto a Riemann-Hilbert problem. This way of proceeding is very complicated andwe take here a different route based on the considerations we reported at the endof last subsection. In fact, this general logarithmic behavior of the entanglemententropy suggests that this type of excited states can be the ground states of criticalHamiltonians. We explicitly build these critical, translational invariant, and localHamiltonians, proving the logarithmic behavior, with the correct prefactor.The excited state | E x (cid:105) in Eq. (9) is the ground state of all free-fermionicHamiltonians of the form˜ H = (cid:88) k ˜ ε ( ϕ k ) b † k b k , with ˜ ε ( ϕ k ) < ⇔ k ∈ E x , (28)for any choice of the function ˜ ε ( ϕ k ). In particular we could choose ˜ ε ( ϕ k ) = − f ( ϕ k ) m ( ϕ k ), with f ( x ) an arbitrary positive function. The choice of ˜ ε ( ϕ k )determines the locality properties of ˜ H : most of the choices of ˜ ε ( ϕ k ) would produce anon local ˜ H (while by construction ˜ H is always hermitian and translational invariantbecause it is built by Fourier transform).To understand the locality of this effective Hamiltonian it is useful to introduce ntanglement entropy of excited states A x,y are the Majorana operators introduced above from Ref. [7]) G ( r ) = i (cid:88) l A xl A yl + r , and F x ( y ) ( r ) = i (cid:88) l A x ( y ) l A x ( y ) l + r . In fact, by separating ˜ ε ( ϕ k ) in its even and odd part (˜ ε ( ϕ k ) = ˜ ε e ( ϕ k ) + ˜ ε o ( ϕ k )), wecan rewrite the effective Hamiltonian as the sum ˜ H = H e + H o where H e = (cid:88) r (cid:104) N N − (cid:88) k = − N ˜ ε e ( ϕ k ) e iθ k e − iϕ k r (cid:105) G r ≡ (cid:88) r g e ( r ) G r ,H o = i (cid:88) r (cid:104) N N − (cid:88) k = − N ˜ ε o ( ϕ k ) e − iϕ k r (cid:105)(cid:0) F xr + F yr (cid:1) ≡ (cid:88) r g o ( r )( F xr + F yr ) , (29)where we defined the complex couplings g e ( r ) and g o ( r ).The locality of ˜ H is related to the long distance behavior of these complexcouplings g e/o ( r ). From a standard theorem in complex analysis, we know that g e/o ( r ) decay faster than any power (and so results in local couplings) if their Fouriertransforms in the above equations are C ∞ (i.e. with all derivatives being continuousfunctions; often we will refer to these functions simply as regular). When Eq. (18)holds, that is m ( ϕ ) = ± non-critical system (i.e. when e − iθ k is regular), the arbitrariness in the choice of ˜ ε allows us totake it among the C ∞ functions. This conclude the proof for non-critical systems.For the critical case, a slight modification is enough to give the correctHamiltonian. In the XX spin chain e − iθ = sign( J cos ϕ − h ) so that we can makethe two above functions regular simply defining the characteristic function ˜ m ( ϕ )˜ m ( ϕ ) = (cid:40) m ( ϕ ) ϕ ∈ [ − ϕ F , ϕ F ] , − m ( − ϕ ) otherwise ,as we have already done in Eq. (19). The critical XY ( | h | = 1) is more involvedbecause e − iθ can be made regular only after imposing anti-periodic conditions to themode of zero energy. It is then convenient to extend the definition of ˜ ε to the interval[0 , π ] ˜ ε (4 π ) ( ϕ ) = (cid:40) ˜ ε ( ϕ ) ϕ ∈ [0 , π ] − ˜ ε (4 π − ϕ ) ϕ ∈ [2 π, π ] . (30)˜ ε (4 π ) can be chosen C ∞ because it has at most 2 n + 2 zeros, where n is the numberof the discontinuities corresponding to the excited state. The constructed functionrestricted to [0 , π ] has the correct regularity properties. Regardless of the presenceof a discontinuity in ϕ = 0 the dispersion law must vanish in ϕ = 0 ( see Eq. (30)),thus the number of chiral modes is the number of discontinuities, plus 1 if there is nota discontinuity in ϕ = 0. This ends the construction of the local Hamiltonian for allthe XY models.And this is not yet the end of the story. We can in fact use the arbitrariness wehave in the choice of ˜ ε k to fix it in such a way that it crosses the zero-energy line witha non-vanishing slope. The low-energy properties of the resulting Hamiltonian canbe then studied by linearizing the dispersion relation close to the zeros in a canonicalmanner. Each zero gives a chiral mode with central charge 1 / ntanglement entropy of excited states Figure 3. Two 5-folded wrapped chains of 60 spins. The thick green linerepresents the subsystem (6 spins on the left and 18 spins on the right) whilethe red links give weight to the interaction between the subsystem and the restof the chain. If the “area law” holds the entanglement entropy is proportional tothe number of the links. charge will be n/ 2, with n the number of zeros, i.e. the number of discontinuities of m ( ϕ ) for non-critical systems, or the proper variation for critical ones (when thezero mode gives one additional contribution). This agrees with all the specific casesin the previous section. In particular if m ( ϕ ) is discontinuous in ϕ = 0, the zeromode contributes only once. In Fig. 2 we report some specific examples stressing theimportance of the critical modes and of the location of discontinuities. When the width of the block (cid:96) is comparable with the length of the chain N , thecharacterization of the entanglement becomes tricky. When an excited state | E x (cid:105) can be associated to the ground-state of a local Hamiltonian ˜ H with central charge a = n/ 2, i.e. when the entropy grows logarithmically with (cid:96) with a prefactor givenby a , the constructive proof of previous subsection in the thermodynamic limit is stillvalid. Thus, in this case, the entanglement entropy has the finite size scaling given byEq. (2) with c replaced by a . This is shown in the right panel of Fig. 2.A more intriguing problem is to understand the finite size scaling of excited statesthat have an extensive entanglement entropy in the thermodynamic limit. The resultfor N → ∞ only predicts the derivative of the entropy for small subsystems. Increasing (cid:96) peculiar finite size behaviors must emerge, because the chain is finite and the entropymust be symmetric around (cid:96) = N/ | E x (cid:105) = | (cid:81) dj =1 ↑ n j ↓ m j (cid:105) , where n j and m j are all O ( N )and d is a finite number. States with m ( ϕ ) (cid:54) = 1 (that have extensive entanglemententropy) do not fall in this category as evident in the definition (10). They can berealized by joining in a regular fashion small blocks κ made by a given sequence ofpopulated or empty energy levels (e.g. κ = {↑↓} or κ = {↑ ↓} etc.). Thus to studythe finite size scaling of “extensive” states, we concentrate on those of the form | E x (cid:105) = | d (cid:89) j =1 κ n j ¯ κ m j (cid:105) , (31) ntanglement entropy of excited states κ is the set obtained interchanging ↑ with ↓ . The entanglement entropy ofthis type of states in the thermodynamic limit has an extensive behavior because κ averages to give m ( ϕ ) = ( u − d ) / ( u + d ), where u ( d ) is the number of up (down) arrowsin κ , while ¯ κ gives m ( ϕ ) = ( d − u ) / ( u + d ): the regularized characteristic function is amulti-step function but with modulus different from 1 and Eq. (17) gives the leadingterm of the entanglement entropy.In order to have a quantitative prediction for the finite size scaling, we follow theideas in the previous subsection by looking at the effective Hamiltonian obtained bythe construction in Eq. (28). The resulting couplings in Eq. (29) could never givea finite-range Hamiltonian because the entanglement entropy is not logarithmic. Wecan make a local choice of the sign that makes ˜ ε a regular function (that we call ¯ ε )giving the coupling (cid:107) g ( r ) ≡ N N − (cid:88) k = − N e irϕ k ˜ ε ( ϕ k ) (32)= − N (cid:88) ϕ q ∈ ] − π | κ | , π | κ | [ e − i | κ | rϕ q (cid:20) ¯ ε ( | κ | ϕ q ) + O (cid:0) N (cid:1)(cid:21) | κ | (cid:88) n =1 κ n e − i ( n − n ) ϕ r , and the interaction is not local anymore. The O (1 /N ) term comes from the seriesexpansion of ¯ ε . The first factor in equation (32) is periodic of period N/ | κ | while thesecond one is a modulation. The coupling decays faster than any power for r < N/ | κ | ,but it explodes (i.e. it grows faster than a power) up to N/ | κ | when g ( r ) becomesagain of order 1. The behavior for large distances is determined only by the first region g (cid:16) r + j N | κ | (cid:17) ≈ (cid:80) | κ | n =1 κ n e − i πnj | κ | e − inϕ r (cid:80) | κ | n =1 κ n e − inϕ r g ( r ) , < r < N | κ | . The interaction is localized within a distance jN/ | κ | , so the Hamiltonian can beinterpreted as a local one in a | κ | -folded wrapped 1D chain. If we assume the “area law”to be valid for the wrapped chain (i.e. that only a shell of mutually interacting spinscontributes to the entanglement [34]), we can predict the behavior of the entanglemententropy: each spin strongly interacts with the neighborhood spins and with the | κ | spins of the other wrappings (see Figure 3). Thus the entanglement entropy is apiece-wise function of (cid:96) that changes slope at jN/ | κ | . We can find excited states withanalogous properties considering any finite partition of unity of the circle ] − π, π ],with the property that all functions of the set are regular and approach step functionsin the limit of large N . We associate a small block κ ( i ) to each function of the set andwe write the coupling as a sum of terms of the form (32) g ( r ) = n (cid:88) i =1 g iκ ( i ) ( r )that we obtain identifying the regularized ¯ ε with the given function of the partition.In the scaling limit the characteristic function is m ∼ (cid:81) di =1 κ n i ( i ) . Each g iκ i has thebehavior previously described, so the entanglement entropy is a piece-wise function (cid:107) This coupling is a slight modification of the ones in Eq. (29). It has the advantage to make allthe formulas simpler, but it applies only to non-critical systems. The generalization to critical onesis straightforward, but long and we do not report it here for clarity. However all results (except forthe ground state) are independent of this choice, as in the previous section. ntanglement entropy of excited states Figure 4. Examples of 3- and 4-folded states. Left: The 3-folded excited state | ↓ {↑↓ } ↓ (cid:105) for the non critical chain ( h = 0 . , γ = 0 . N/ m ( ϕ ). Right: The 4-folded excited state | ↓ {↑↓ } ↓ (cid:105) . For (cid:96) < N/ 2, the entropy always grows linearly, but with achange of slope close to (cid:96) ∼ N/ of (cid:96) changing slope in jN/ | κ | , where | κ | is the least common multiple of the {| κ | ( i ) } .Two examples of 3-folded and 4-folded states are reported in Fig. 4.To give the details of a specific example, we report the 3-folded case κ (1) = {↑ ↓} and κ (0) = {↓} with ϕ ∈ I ⇔ cos ϕ ≥ / ϕ ∈ I ⇔ cos ϕ < / 2, in otherwords the set E x is made of the quasiparticles with momenta (2 π (3 k + q )) /N with | k | ≤ N/ 12 and q ∈ { , } | E x (cid:105) = N (cid:89) k ≈− N b † k b † k +1 | (cid:105) . The excited state | E x (cid:105) is the ground state of the Hamiltonian˜ H = N − (cid:88) k = − N (cid:104)(cid:16) − cos ϕ k (cid:17) ( − (cid:100) k (cid:101) + ( − (cid:98) k (cid:99) + ( − (cid:98) k +1)3 (cid:99) (cid:105) b † k b k , and if N is divisible by 3 the coupling is different from 0 only in 9 points g ( r ) = r = 0 , − r = ± − ± √ i r = ± N + q q ∈ {− , , } .The effective Hamiltonian ˜ H is local on the 3-folded wrapped chain. The entropy growslinearly with the width of the block up to (cid:96) = N/ 3, after that the interaction surfacedoes not further increase and the entanglement entropy does not depend anymore onthe width of the block, see Figure 4 (left). Notice on the same figure (right), thechange of slope in the 4-folded case. ntanglement entropy of excited states Figure 5. Rescaled half-chain entanglement entropy (cid:96) = ( N − / N = 15. Each pointcorresponds to an excited state with energy (in unit of J ) on the real axis. Thered curves are the “2-folded” estimations of the envelope. To have a general picture of the scaling of the entanglement for all excited states andnot only in the particular classes considered so far, we study here the entanglemententropy in a small enough chain to be able to calculate it for all the 2 N states. Wemainly concentrate on blocks with maximal entropy, i.e. with length equal to half-chain (actually ( N − / N odd). Drawing general conclusionsin an analytic manner for finite systems is not easy, so we mainly analyze numericalresults. The plots in Fig. 5 suggest that some regularities are general features ofexcited states and not only of the classes we can compute analytically. In these plots(and in all those relative to this section) we always consider the rescaled entropyrescaled entropy = S (cid:96) S GS(cid:96) , with S GS(cid:96) = 13 log (cid:0) Nπ sin π(cid:96)N (cid:1) , (33)so that, for states with a critical-like behavior (for large enough (cid:96) and N ) we have adirect estimation of the effective central charge. We found particular instructive toplot the (rescaled) entanglement entropy as function of the energy of the eigenstates.In Fig. 5, we considered chains of 15 spins and we plot the rescaled S for all the 2 eigenstates. Similar plots can be done as function of total momentum instead of theenergy. ntanglement entropy of excited states Figure 6. Histograms for the number of the states with a given entanglemententropy for a non-critical XY chain of 23 spins, after cutting the Hilbert space inan energy shell. Main plot: rescaled S . Inset : rescaled S . The band-structureis evident only for (cid:96) = 11. A first feature that is particularly evident from the plots is the band-like structureof the entanglement entropy (notice that this is independent of the use of the energy onthe horizontal axis, any other conserved quantity would result in qualitative similarplots). This means that the entanglement entropy of excited states distributes atroughly integer (or half-integer for critical XY at h = 1) multiples of S GS(cid:96) . For stateswith a small number of discontinuities (compared to N ), this phenomenon is clearlydue to the quantization of the prefactor of the logarithm. However, in general this bandstructure cannot be so easily explained: the excited states with a logarithmic behaviorare expected to be negligible in number compared to all the others. Increasing thenumber of discontinuities at fixed N , the crossover to extensive behavior takes placeand eventually it deteriorates the bands. This last phenomenon is not evident inFig. 5 because the band structure persists up to the maximum allowed number ofdiscontinuities. The simplest explanation is that also extensive states should roughlybe quantized but within a scale different from S GS(cid:96) , that in particular does not growwith N . To check this, we should increase N , but in doing so, the dimension ofthe Hilbert space grows exponentially and it becomes soon prohibitive to plot (andunderstand) so many points in an readable graph. For this reason we considered a non-critical chain of 23 spins, and to reduce the number of states, we limited to states withenergy in the interval 4 . < E − E < . (cid:96) = 11, the band structure is evident and the points distribute in analmost Gaussian fashion around some discrete values of the entanglement entropy, butthe distance between them becomes smaller S GS(cid:96) , confirming that the origin of thisphenomenon in the upper part of the band has nothing to do with logarithmic states.For (cid:96) = 6 (inset of Fig. 6) the band structure disappears completely, confirming thatmost of the states are extensive. We checked that still increasing N , this scenario isconsistent.Another very interesting feature is that in all the plots, the entanglement entropyhas a maximum value that seems to be a regular function of the energy (that is the ntanglement entropy of excited states Figure 7. Rescaled entanglement entropy for small blocks. Left: (cid:96) = 4 in anon-critical XY-chain of 15 spins; The continuous curve is Eq. (34) giving a goodestimation of the envelope. Right: (cid:96) = 5 in a non-critical Ising chain of 15 spins;The “3-folded” envelop (in red) of the envelope is in good agreement with thedata. For high energies, when the “3-folded” approximation is not defined, Eq.(34) (in green) works well. final reason why we made this kind of plots). We argue here that these envelopes havea characteristic dependence on the energy that in the scaling limit is determinedby excited states with extensive behavior. We already derived the entanglemententropy for the excited states that are equivalent to the ground state of n -foldedwrapped Hamiltonians. Eq. (17) characterizes the scaling regime, e.g. for the 2-folded case the entanglement entropy increases linearly up to N/ 2, while in the 3-folded it increases up to N/ (cid:96)/N ≥ H (1 / / (2 H (0)) = 0 . . . . that the 2-folded case is more entangled than the3-folded one. This suggests that the 2-folded states can explain the envelopes in Fig.(5) for (cid:96) = ( N − / E < S MAX N ∼ log 24 π (cid:90) π − π d ϕ θ ( µ − ε ) ,EN ∼ − π (cid:90) π − π d ϕ ε θ ( ε − µ ) . In Fig. 5 this analytical result is compared with the numerical data for a critical XX,a critical Ising and two non critical XY spin chains: the approximated envelope is ingood agreement with the numerical data also for small chains. We also notice that d S ( MAX ) d E = log 2 µ and sod S MAX d E ≤ log 2∆ = (cid:16) d S ( MAX ) d E (cid:17) G.S. , where ∆ is the gap in the dispersion law: if the system is critical then the “2-folded”approximation of the envelope has infinite derivative in E = E G.S. (cf. Fig. 5).In the opposite limit of small (cid:96) , the band structure is practically lost (see leftpanel of Fig. 7) and for most of the states Eq. (17) gives a good estimate of S (cid:96) so ntanglement entropy of excited states S MAX (cid:96) ∼ π (cid:90) π − π d ϕ H (tanh( βε )) ,EN ∼ − π (cid:90) π − π d ϕ ε tanh( βε ) , (34)and the loss of the band structure can be seen as a consequence of a “pure” extensivebehavior of the entropy. Eq. (34), in the scaling limit, is always an upper bound forthe entanglement entropy because entropy is a concave function of (cid:96) . In Fig. 7 (left)we compare this analytical curve with the data for N = 15 and (cid:96) = 4 in a non-criticalXY-chain.Considering blocks of intermediate lengths the parametric equations (34) definea too high bound (see right of Fig. 7). At the same time the band structure startsemerging. We can improve our estimation considering a generalization of the “2-foldedapproximation” of the envelope: the “ n -folded approximation” (that makes sense onlyfor (cid:96) ≤ N/n ). The maximal entanglement entropy in the n -folded family of excitedstates is S MAX N ∼ H (cid:0) − n (cid:1) nπ (cid:90) π − π d ϕ θ ( µ − ε ) ,EN ∼ nπ (cid:90) π − π d ϕ ε (cid:0) θ ( µ − ε ) − n (cid:1) . (35)In Fig. 7 (right) we report S for a non-critical Ising chain of 15 spins (so the maximumallowed n is 3). It is evident that up to the point where it exists the 3-folded curve isa good approximation of the actual envelope, while for larger values Eq. (34) workswell.All the plots in this subsection are relative to the Slater-determinant basis.We have checked that considering linear combinations of eigenstates with the sameenergies, these envelopes remain unchanged, while the band-structure disappears (asmaybe could have been expected).Lack of space prevents us to show many other similar plots about the distributionin the energy of excited states for the entanglement entropy. The main featuresabout appearance and disappearance of the band-structure and the envelopes (thatwe showed here with few examples) are always true. 3. The XXZ model and the algebraic Bethe Ansatz approach to reduceddensity matrices We consider the anisotropic spin-1 / z direction, with Hamiltonian H XXZ = N (cid:88) m =1 (cid:110) σ xm σ xm +1 + σ ym σ ym +1 + ∆( σ zm σ zm +1 − − h σ zm (cid:111) , (36)and periodic boundary conditions. The model is solvable by means of the BetheAnsatz for any real value of the anisotropy parameter ∆ [21, 35], but we will considerhere only the antiferromagnetic critical regime 0 < ∆ ≤ ntanglement entropy of excited states / N odd, thanks to very peculiarcombinatorial properties [48], some exact results are known also for finite chains[49, 50].The content of next subsections is highly technical. We first review the mainresults of Ref. [22, 23] (to make this paper self-consistent and to fix the notations)and then we explain the technical tricks to adapt these fundamental results to thecomputation of the reduced density matrix. We remand the reader interested only inthe results to the final subsection 3.5. In the algebraic Bethe Ansatz approach (see the book [35] for an introduction to thesubject), the dynamics of the model is encoded in the so called R matrix R ( λ, µ ) = b ( λ, µ ) c ( λ, µ ) 00 c ( λ, µ ) b ( λ, µ ) 00 0 0 1 , (37)where b ( λ, µ ) = sinh( λ − µ )sinh( λ − µ + η ) , c ( λ, µ ) = sinh η sinh( λ − µ + η ) . Here the parameter η is related to ∆ by the relation∆ = 12 ( e η + e − η ) . (38)Now we introduce the monodromy matrix T ( λ ) = R N ( λ − ξ N ) . . . R ( λ − ξ ) R ( λ − ξ ) = (cid:18) A ( λ ) B ( λ ) C ( λ ) D ( λ ) (cid:19) , where ξ i are arbitrary parameters sitting on each site of the spin chain. The role ofthe inhomogeneities ξ i will become clear in the following. We introduce the transfermatrix as trace of the monodromy matrix T ( λ ) = Tr T ( λ ) that satisfies[ lim (cid:126)ξ → (cid:126)α T ( λ, (cid:126)ξ ) , lim (cid:126)ξ → (cid:126)β T ( λ, (cid:126)ξ )] = 0 , with (cid:126)α = ( α, . . . , α ) , (39)where we denoted with (cid:126)ξ the vector with components ξ i . In the approach of Ref. [22]keeping the ξ i different helps in deriving general results. Only at the end, to recover ntanglement entropy of excited states (cid:126)ξ → (cid:126)α .Every eigenstate of the Hamiltonian (36) can be written as | { λ i } (cid:105) = M (cid:89) k =1 B ( λ k ) | (cid:105) , (cid:104) { λ i } | = (cid:104) | M (cid:89) k =1 C ( λ k ) , (40)where we denoted with | (cid:105) the reference state with all spins up | (cid:105) = N (cid:79) k =1 | + (cid:105) k . (41)The parameter M is such that M ≤ N/ M = N/ { λ , . . . , λ M } are called rapidities. We alsointroduce d ( λ ) d ( λ ) = N (cid:89) i =1 b ( λ, ξ i ) , for which d ( ξ i ) = 0 ∀ i . (42)Not all states of the form Eq. (40) are eigenstates of the Heisenberg Hamiltonian: therapidities λ i must satisfy a set of non-linear equation known as Bethe equations thatfor the Heisenberg chain can be written as1 d ( λ j ) M (cid:89) k =1 k (cid:54) = j b ( λ j , λ k ) b ( λ k , λ j ) = 1 , ≤ j ≤ M. (43)We need the commutation relations (cid:2) B ( λ ) , B ( µ ) (cid:3) = (cid:2) C ( λ ) , C ( µ ) (cid:3) = 0 , for all λ, µ , (44)to derive the action of the operators A , B , C , D on an arbitrary state | { λ i } (cid:105) [23] (cid:104) | M (cid:89) k =1 C ( λ k ) A ( λ M +1 ) = M +1 (cid:88) a (cid:48) =1 a ( λ a (cid:48) ) M (cid:81) k =1 sinh( λ k − λ a (cid:48) + η ) M +1 (cid:81) k =1 k (cid:54) = a (cid:48) sinh( λ k − λ a (cid:48) ) (cid:104) | M +1 (cid:89) k =1 k (cid:54) = a (cid:48) C ( λ k ); (45) (cid:104) | M (cid:89) k =1 C ( λ k ) D ( λ M +1 ) = M +1 (cid:88) a =1 d ( λ a ) M (cid:81) k =1 sinh( λ a − λ k + η ) M +1 (cid:81) k =1 k (cid:54) = a sinh( λ a − λ k ) (cid:104) | M +1 (cid:89) k =1 k (cid:54) = a C ( λ k ) , (46) (cid:104) | M (cid:89) k =1 C ( λ k ) B ( λ M +1 ) = M +1 (cid:88) a =1 d ( λ a ) M (cid:81) k =1 sinh( λ a − λ k + η ) M +1 (cid:81) k =1 k (cid:54) = a sinh( λ a − λ k ) × M +1 (cid:88) a (cid:48) =1 a (cid:48)(cid:54) = a a ( λ a (cid:48) )sinh( λ M +1 − λ a (cid:48) + η ) M +1 (cid:81) j =1 j (cid:54) = a sinh( λ j − λ a (cid:48) + η ) M +1 (cid:81) j =1 j (cid:54) = a,a (cid:48) sinh( λ j − λ a (cid:48) ) (cid:104) | M +1 (cid:89) k =1 k (cid:54) = a,a (cid:48) ] C ( λ k ) . (47)We can fix a ( λ ) = 1 for all λ . ntanglement entropy of excited states 21A fundamental ingredient is the formula for the scalar product between twoarbitrary states. Given a set { λ , . . . , λ M } that is solution to the Bethe equations(43) and another set of arbitrary numbers { µ , . . . , µ M } , the scalar product of statesof the form (40) is given by the so called Slavnov formula [51] (cid:104) | M (cid:89) j =1 C ( µ j ) M (cid:89) k =1 B ( λ α ) | (cid:105) = det H ( { λ α } , { µ j } ) (cid:81) j>k sinh( µ k − µ j ) (cid:81) α<β sinh( λ β − λ α ) , (48)where we defined H ab = sinh( η )sinh( λ a − µ b ) (cid:32) d ( µ b ) (cid:89) m (cid:54) = a sinh( λ m − µ b + η ) − (cid:89) m (cid:54) = a sinh( λ m − µ b − η ) (cid:33) . (49)When { λ i } = { µ i } , Eq. (48) gives the Gaudin formula for the norm of a Bethe state[21, 52] (cid:104) | M (cid:89) j =1 C ( λ j ) M (cid:89) j =1 B ( λ j ) | (cid:105) = sinh M η M (cid:89) a,b =1 a (cid:54) = b sinh( λ a − λ b + η )sinh( λ a − λ b ) det M H (cid:48) ( { λ } ) . (50)where H (cid:48) is H (cid:48) jk ( { λ } ) = − δ jk (cid:34) d (cid:48) ( λ j ) d ( λ j ) − M (cid:88) a =1 K ( λ j − λ a ) (cid:35) − K ( λ j − λ k ) , (51)and K ( λ ) = sinh(2 η )sinh( λ + η ) sinh( λ − η ) . (52)Notice that for the following manipulations, it is fundamental that the set of numbers { µ i } could not be solution of some Bethe equations. Let us consider a given Bethe state | { λ i } (cid:105) , and let us select a block of length (cid:96) as asubsystem of the spin chain. Every element of the reduced density matrix of these (cid:96) contiguous spins can be written as P (cid:15) (cid:48) ,...,(cid:15) (cid:48) (cid:96) (cid:15) ,...,(cid:15) (cid:96) ≡ (cid:104) Ψ | E (cid:15) (cid:48) (cid:15) · · · E (cid:15) (cid:48) (cid:96) (cid:15) (cid:96) l | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) , (53)where the indices (cid:15) can have the values { + , −} and the matrices E (cid:15),(cid:15) (cid:48) are E ++ j = (cid:18) (cid:19) [ j ] = 12 + S zj , E −− j = (cid:18) (cid:19) [ j ] = 12 − S zj ,E + − j = (cid:18) (cid:19) [ j ] = S xj + iS yj , E − + j = (cid:18) (cid:19) [ j ] = S xj − iS yj . Once we know the reduced density matrix, any multi-point correlation function builtwithin the (cid:96) spins can be found by considering the appropriate linear combinations.The most general object we need is F (cid:96) ( { (cid:15) j , (cid:15) (cid:48) j } ) = (cid:104) Ψ | (cid:96) (cid:81) j =1 E (cid:15) (cid:48) j ,(cid:15) j j | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) , (54) ntanglement entropy of excited states F (cid:96) ( { (cid:15) j , (cid:15) (cid:48) j } ) = φ (cid:96) ( { λ } ) (cid:104) Ψ | T (cid:15) ,(cid:15) (cid:48) ( ξ ) . . . T (cid:15) (cid:96) ,(cid:15) (cid:48) (cid:96) ( ξ (cid:96) ) | Ψ (cid:105)(cid:104) Ψ | Ψ (cid:105) , (55)where φ (cid:96) ( { λ } ) = (cid:96) (cid:89) j =1 M (cid:89) a =1 sinh( λ a − ξ j )sinh( λ a − ξ j + η ) . Before reporting the main result of [23], we have to define the following two sets ofindices α + = { j : 1 ≤ j ≤ (cid:96), (cid:15) j = + } , (56) α − = { j : 1 ≤ j ≤ (cid:96), (cid:15) (cid:48) j = −} . (57)We denote with d + ( d − ) the dimension of the set α + ( α − ). For each j ∈ α ± it isnecessary to define a set a j (if j ∈ α − ) and a set α (cid:48) j (if j ∈ α + ) such that1 ≤ a j ≤ M + j, a j ∈ A j , ≤ a (cid:48) j ≤ M + j, a (cid:48) j ∈ A (cid:48) j . where we introduced A j = { b : 1 ≤ b ≤ M + (cid:96), b (cid:54) = a k , a (cid:48) k , k < j } , (58) A (cid:48) j = { b : 1 ≤ b ≤ M + (cid:96), b (cid:54) = a (cid:48) k , k < j, b (cid:54) = a k , k ≤ j } . (59)Now we need only the redefinition { λ k } → { λ k , ξ . . . ξ (cid:96) } , (60)to write [23] (cid:104) | M (cid:89) k =1 C ( λ k ) T (cid:15) ,(cid:15) (cid:48) ( λ M +1 ) . . . T (cid:15) (cid:96) ,(cid:15) (cid:48) (cid:96) ( λ M + (cid:96) ) = (61) (cid:88) { a j ,a (cid:48) j } G { a j ,a (cid:48) j } ( λ , . . . , λ M + (cid:96) ) (cid:104) | (cid:89) b ∈ A l +1 C ( λ b ) , where G { a j ,a (cid:48) j } ( λ , . . . , λ M + (cid:96) ) = (cid:89) j ∈ α − d ( λ a j ) M + j − (cid:81) b =1 b ∈ A j sinh( λ a j − λ b + η ) M + j (cid:81) b =1 b ∈ A (cid:48) j sinh( λ a j − λ b ) ×× (cid:89) j ∈ α + M + j − (cid:81) b =1 b ∈ A (cid:48) j sinh( λ b − λ a (cid:48) j + η ) M + j (cid:81) b =1 b ∈ A j +1 sinh( λ b − λ a (cid:48) j ) . (62)An important simplification comes from the relation (42) which allows a j ≤ M ∀ j . ntanglement entropy of excited states how many terms are involved in the summation inEq. (61). By simple counting we get M !( M − d − )! d + (cid:89) i =1 (cid:0) M − d − + α + i − i + 1 (cid:1) , (63)that in the limit of large M behaves as M (cid:96) .We stress now one of the main features of this approach for the calculation of thereduced density matrix. The computational resources we need for the algorithm growexponentially with (cid:96) , so it would be comparable to exact diagonalization, but at fixed (cid:96) they only grows algebraically with M (but with a power equal to (cid:96) ). Thus we canexpect that for relative small (cid:96) we can calculate the reduced density matrix for verylarge systems, while exact diagonalization can work with at most about 30 spins.Thanks to the invariance under permutations of the set { λ , . . . , λ k } in theSlavnov formula, the number of determinants we need to calculate can be reducedto d + (cid:88) i =1 (cid:26) d + (cid:88) j 0) + f (cid:48) ( a, x + 12! f (cid:48)(cid:48) ( a, x + . . . + 1 n ! f ( n ) ( a, x n + . . . . (77)Gauss manipulations on the matrix above give the same result on each column exceptthe index a which distinguishes the different columns. Therefore we can restrict toone column and construct the following matrix f i (0) f (cid:48) i (0) (cid:15) j . . . (cid:96) ! f ( (cid:96) ) i (0) (cid:15) (cid:96)j . . .f i (0) f (cid:48) i (0) (cid:15) j . . . (cid:96) ! f ( (cid:96) ) i (0) (cid:15) (cid:96)j . . . ... . . . ... f i (0) f (cid:48) i (0) (cid:15) j n . . . (cid:96) ! f ( (cid:96) ) i (0) (cid:15) (cid:96)j n . . . . ntanglement entropy of excited states f ( k ) i (0) we can neglect it (we willrestore it at the end of the manipulations) and consider the matrix (cid:15) j (cid:15) j . . . (cid:96) ! (cid:15) (cid:96)j . . . (cid:15) j (cid:15) j . . . (cid:96) ! (cid:15) (cid:96)j . . . ... . . . ...1 (cid:15) j n (cid:15) j n . . . (cid:96) ! (cid:15) (cid:96)j n . . . . (78)By mean of rows manipulations it is possible to put the last matrix in a triangularform g ( (cid:15) j ) g ( (cid:15) j ) . . . (cid:96) ! g ( (cid:15) (cid:96)j )0 g ( (cid:15) j , (cid:15) j ) g ( (cid:15) j , (cid:15) j ) . . . (cid:96) ! g ( (cid:15) (cid:96)j , (cid:15) (cid:96)j )0 0 g ( (cid:15) j , (cid:15) j , (cid:15) j ) . . . (cid:96) ! g ( (cid:15) (cid:96)j , (cid:15) (cid:96)j , (cid:15) (cid:96)j )0 0 . . . , (79)where we have for instance g ( x ) = x , g ( x, y ) = x − y and more complicatedexpressions for the other functions. In order to obtain the homogeneous limit inthe Slavnov formula, we need one more step: since we know that the limit exists, wehave to choose in a convenient way the variables (cid:15) j i . One possible choice is ξ j = (cid:26) η j = 1 η + (cid:15) exp( πin − j ) j > (cid:15) j k = (cid:15) exp( πin − j k ). It is useful to consider the simpler case inwhich (cid:15) j i = (cid:15) i . Substituting in (78) we obtain that the matrix (79) has a simple form.Indeed it is easy to see that (79) becomes proportional to the identity matrix (up tothe n -th order) K , with K = det (cid:20) exp( πin − jk ) j ! (cid:21) j,k . (80)In conclusion this means that to have the lowest order in (cid:15) for the Slavnov determinantwe can write the matrix (75) as T ij = H (cid:48) ij j ≤ M − n ,f i (0) j = M − n + 1 ,Kf ( j − M + n − i (0) j > M − n + 1 . (81)Moreover we have to consider the contribution given by n (cid:89) j,k =1 j>k sinh( ξ k − ξ j ) = ( − (cid:15) ) n ( n − / n − (cid:89) k =1 j>k (cid:18) e i πjn − − e i πkn − (cid:19) . (82)This concludes the calculation of the homogeneous limit in the Slavnov formula.We can now ask how many terms is it possible to obtain with the algorithmdeveloped so far . The answer can be given examining the function G in Eq. (61). If thefunction G has no poles, then the procedure just outlined works with no modification.Unfortunately this happens only in very few cases, for example for the first elementof the reduced density matrix, that is the so called emptiness formation probability τ ( (cid:96) ) = (cid:104) ψ g | (cid:96) (cid:81) j =1 12 (1 − σ zj ) | ψ g (cid:105)(cid:104) ψ g | ψ g (cid:105) . (83) ntanglement entropy of excited states (cid:96) Here DMRG1 0.49999999999999 0.52 0.17659666969479 0.176596669694683 0.04110985506014 0.041109855060124 0.00595577151455 0.005955771514555 0.00050690054232 0.000506900542326 0.00002367077112 0.000023670771127 0.00000055351689 0.00000055351689 Table 1. Emptiness formation probability of a chain of 20 spins in the groundstate for ∆ = 0 . (On passing it is worth mentioning that this element can be computed in thethermodynamic limit [53, 54, 35], basically because of this simplification.) In thiscase Eq. (61) simplifies to τ ( (cid:96) ) = φ (cid:96) ( { λ } ) (cid:104) | M (cid:81) a =1 C ( λ a ) (cid:96) (cid:81) j =1 D ( ξ j ) M (cid:81) a =1 B ( λ a ) | (cid:105)(cid:104) | M (cid:81) a =1 C ( λ a ) M (cid:81) a =1 B ( λ a ) | (cid:105) , (84)that can be written as (cid:104) | M (cid:89) k =1 C ( λ k ) (cid:96) (cid:89) j =1 D ( λ M + j ) = M +1 (cid:88) a =1 M +2 (cid:88) a a (cid:54) = a . . . (85) . . . M + (cid:96) (cid:88) al =1 al (cid:54) = a ,...,al − G a ...a l ( λ . . . λ M + (cid:96) ) (cid:104) | M + (cid:96) (cid:89) k =1 k (cid:54) = a ,...,al C ( λ k ) , where G is G a ...a (cid:96) ( λ , . . . la M + (cid:96) ) = (cid:96) (cid:89) j =1 d ( λ a j ) M + j − (cid:81) b =1 b (cid:54) = a ,...,aj − sinh( λ a j − λ b + η ) M + j (cid:81) b =1 b (cid:54) = a ,...,aj sinh( λ a j − λ b ) . (86)By definition G cannot diverge, then all the machinery developed so far is enoughto compute τ ( (cid:96) ). In Table 3.3 we report some results for the emptiness formationprobability for a chain of length L = 20 at ∆ = 0 . ρ (cid:96) accessible without further manipulations.In Eq. (61) the term that is easily manipulated is M + j (cid:89) b =1 b ∈ A j +1 sinh( λ b − λ a (cid:48) j ) , (87)thus the only class with no poles is when in the sets A j α j . . . we have α + = { } , (88) ntanglement entropy of excited states P (cid:15) (cid:48) ,...,(cid:15) (cid:48) m , ,..., , which is one particular column of the reduceddensity matrix. For the other 2 (cid:96) − The problems in the general case arise from the divergencies of the term (87). Let usstart with some preliminary observations. First, it is important to know the maximumdegree of the poles in (87). Given a term of the summation in (61), the order of thepole is the order of the zero in (cid:89) j ∈ α + M + j (cid:89) b = M +1 b ∈ A j +1 sinh( λ b − λ a (cid:48) j ) , (89)that is given by a (cid:48) j >M (cid:88) i =1 ( α + i − i ) , (90)where a (cid:48) j is the number of elements j ∈ α + such that a (cid:48) j > M . It is easy to maximizethe last expression to find d + (cid:88) i =1 (cid:48) ( α + d + − i +1 − i ) , (91)where the prime means that the summation is restricted to the i such that α + d + − i +1 − i > ρ (cid:96) . If we would have been able to find a “shortest”representation of the same elements, we could have been able to describe much larger (cid:96) . Further developments in this direction would allow this method to be competitiveeven with DMRG [56] for the ground-state.Since the determinant in front of the pole is in general finite and the final resultmust be finite, all the coefficients multiplying each pole must sum to zero in (61).Furthermore this implies that we can ignore these terms (because we know in advancethat they give zero) and concentrate on the important ones. To proceed, it is necessaryto reshuffle the various terms in (61). Let us defineˆ G = M (cid:81) j =1 M (cid:81) i =1 sinh( λ i − µ j + η ) (cid:81) j>k sinh( µ k − µ j ) (cid:81) α<β sinh( λ β − λ α ) G , (92)and ˆ T = det T . (93)We know that ˆ T ∼ (cid:15) n ( n − / in the homogeneous limit. However here we have ingeneral a pole of order n ( n − / G , then we have to expand both ˆ T andthe nonsingular part of ˆ G up to the order n ( n − / T , we developed thefollowing procedure. Instead of doing the substitution (81), we put the higher orders ntanglement entropy of excited states ρ Tr ρ N Here Exact [49] Here Exact [49]27 0.4130835714633 0.4130835714633 0.1879727171090 0.187972717109051 0.4108297243638 0.4108297243637 0.1851632322689 0.1851632322688101 0.4101798729742 0.4101798729745 0.1843536264631 0.1843536264633151 0.4100571750358 0.4100571750361 0.1842007880727 0.1842007880729201 0.4100139161598 0.4100139161591 0.1841469044722 0.1841469044717 Table 2. Tr ρ (left) and Tr ρ (right) for the ground-state of ∆ = 0 . in (cid:15) up to n + q. Doing so we know that the determinant gives a polynomial in (cid:15) withlowest degree is m d = n ( n − / 2, and we indicate with M d the highest degree. Thusthe determinant is a polynomial of the formˆ T = a m d (cid:15) m d + . . . + a M d (cid:15) M d (94)We can calculate all the coefficients a i numerically: it is enough to calculate thedeterminant in D = M d − m d different points and then to invert the linear system.Moreover if we choose the points in a smart way p k = exp (cid:18) πiD k (cid:19) (95)the solution of the linear system is numerically trivial since the matrix of the systemis unitary. Using this procedure we are able to calculate, in principle, all the elementsof the reduced density matrix. In practice, our possibilities are limited by the size ofthe density matrix. Actually for small sizes we can go quite far and obtain the ρ (cid:96) withthree spins for chains with 200 spins, a task impossible with exact diagonalization.In Table 3.4 we show the quantity Tr ρ n for n = 2 and n = 3 for odd chains at∆ = 0 . elements in double precision 10 − ). The main advantage of the method we have developed in the previous subsectionis that we can exactly evaluate all elements of the reduced density matrix for anyeigenstate of the XXZ chain. Compared to exact diagonalization we do not need tofully diagonalize the 2 N × N matrix to find the eigenstates, we can just pick up ourdesired state by choosing the correct quantum numbers. As we have already pointedout, once the eigenstate has been chosen, the numerical complexity of the algorithmis only a power-law in N (actually M , but for the most interesting states they areproportional), but the exponent grows linearly in (cid:96) limiting the range of applicabilityof the method. If we would have been interested only in the ground-state properties,this method is less effective than DMRG or any method based on matrix product states[56]. In fact, these numerical methods require very little numerical effort to get thespectrum of the ρ (cid:96) at machine precision for the system sizes that are accessible to us.However it is hard, if not impossible, to calculate the entanglement properties of highlyexcited states with DMRG. Thus our method, based on algebraic Bethe Ansatz, is byfar the most effective available. We checked that our algorithm reproduces the known ntanglement entropy of excited states Figure 8. Entanglement entropy of the excited states of the XXZ spin-chainfor ∆ = 10 − , . , . , . N = 24 plotted against the logarithm of theconformal distance. Left: States that in the fermionic description for ∆ = 0 havetwo discontinuities. The slope agrees with effective central charge equal 1. Right:States that in the fermionic description have four discontinuities. The results arecompatible with an effective central charge equal to 2. The bottom blue-line isthe entropy of the ground-state at ∆ = 0 . results for the ground-state for several different ∆, but we do not find instructive toreport these results here.For the study of excited states, we consider spin-chains of length N = 24in the critical antiferromagnetic region (0 < ∆ ≤ 1) for four different values of∆ = 10 − , . , . , . 5. Using our algorithm we generate the full reduced densitymatrices with (cid:96) ≤ (cid:96) capture the asymptoticbehavior [49, 55]). The Bethe equations (43)can be re-casted in a form that is useful for numerical solutions and for a completeclassification of the states. For practical reasons, we consider only the number of sites N to be even. We recall that the Hilbert space separates in sector with defined numberof reversed spins M (with respect to the reference state cf. Eq. (41)), that gives thetotal spin of the state in the z direction S T OTz = N/ − M . Taking the logarithm ofEq. (43) and posing ζ = arccos(∆), we haveatan (cid:20) tanh( λ j )tan( ζ/ (cid:21) − N M (cid:88) k =1 atan (cid:20) tanh( λ j − λ k )tan ζ (cid:21) = π I j N . (96)Each set of distinct half-odd integer (integer) for M even (odd) numbers { I i } (definedmod( N )) specifies a set of rapidities, and therefore an eigenstate. For example, in theground state these numbers take the values I (0) j = − M + 12 + j , j = 1 , ..., M . (97)This ground state can be interpreted as the spinon vacuum. Spinons are theelementary excitations of the model. They have spin 1 / n + and down-spinon n − . The total number of ntanglement entropy of excited states n + + n − ≤ N is even when N is even, while n + − n − = N − M = 2 S T OTz .(Actually for the interacting model different values of S zT OT are possible when the statehas some higher strings, which are non-dispersive. This discussion is too technicalfor the goals of this manuscript and we remand the interested reader to Ref. [43]).However, we will see that the spinon content is not the most important quantity forthe entanglement entropy of excited states.Employing the property that the quantum numbers I j are defined mod( N ), wecan choose the allowed ones in the sets I (odd) = {− N , . . . , N − } for M odd ,I (even) = {− N , . . . , N − } for M even . (98)Only a subset of these numbers, bounded by a calculable I max function of ∆ and M (see again Ref. [43] for the technical details) provides real solutions for the rapidities λ , and we limited our attention to these states. Fixed the parity of M (i.e. of S zT OT ,since N is even), any state is defined by taking M numbers among the allowed onesin I (odd) or I (even) . The spinon content of the state then follows (see again [57]).Instead of using this standard I j notation to indicate the states, following Ref. [58],we adopt a more complicated one that is useful to recover the fermionic descriptionof the XX model when ∆ = 0 (because we want to compare with the results in theprevious section). We denote with (cid:32) +( − ) the spinons with polarization up (down)and with (cid:35) +( − ) the empty positions that can be occupied by spinons with up (down)polarization. We indicate with an exponent the number of consecutive symbols, forinstance (cid:32) (cid:35) stands for (cid:32) + (cid:32) + (cid:35) + (cid:35) + (cid:35) + (cid:35) + (cid:35) + (cid:35) + (cid:35) + (cid:35) + (cid:35) + (cid:35) + (cid:35) + . (99)To each sequence of N of these symbols (cid:32) ± and (cid:35) ± we can associate a singleconfiguration of I i . For example, the sequence (cid:35) − (cid:32) (cid:35) has two up-spinonsand no down ones; it follows that we need M = N/ − ( n + − n − ) = 11 quantumnumbers; the state is fixed by taking from the set I odd the last 11 numbers (see againRef. [58] for more details). The advantage of this maybe not really intuitive notationis that the rule to recover the fermionic description is very easy: given the sequenceone has to associate a fermion for every (cid:32) − or (cid:35) + [58].Once we have the set I i for each state we are interested in, we use the Newtonmethod to solve the Bethe equations. We limit ourself to the two-spinon and four-spinon sector of the spectrum (but we could easily consider other states). We alsoselect states with real rapidities, to avoid problems with strings contributions, thathowever can be handled following Ref. [43]. The main feature we want to check here is if the conformal scaling(2) with an effective central charge a is still valid for given excited states when we addthe interaction ∆ to the XX chain considered in the previous section. The predictionfor the XX is based on the discontinuities of ˜ m ( φ ) (cf. Eq. (19)). In order to predictthe result at ∆ (cid:54) = 0 we exploit the mapping between the fermionic description and thespinonic one at ∆ = 0. Once we have the fermionic picture associated to the state,we have ˜ m ( φ ) for ∆ = 0. In order to check if the logarithmic scaling is obeyed, weplot S (cid:96) against S GS(cid:96) in Eq. (33), so that if the dependence is linear, the slope givesautomatically the central charge a of the effective Hamiltonian. In Fig. 8 (left) we ntanglement entropy of excited states Figure 9. Entanglement entropy for the two- and four-spinon states with∆ = 10 − , . , . , . 5. Left: Summary of all the states we considered (for spaceproblems, the legend shows only states at ∆ = 0 . S zTOT = 0 , , 2. The slopedoes not depend on the polarization. The bottom-red line is the ground-state at∆ = 0 . display some states in the two-spinon and four-spinon sectors. We choose these statesin such a way that in the limit ∆ → 0, the corresponding fermionic structure has twodiscontinuities. For example, the state (cid:35) − (cid:32) (cid:35) (cid:35) − corresponds to the fermionrepresentation | ↓ ↑ ↓ (cid:105) , having two discontinuities in ˜ m ( ϕ ). For ∆ = 0, we knowfrom the previous section, that all these states are described by Eq. (2) with effectivecentral charge a = 1, as in the ground-state. Fig. 8 (left) provides a clear evidencethat the asymptotic behavior of the entropy for (cid:96) (cid:29) ∈ [0 , . . (cid:96) ≤ a gives a ∼ . 3, which is in agreement with the XXprediction a = 2. Moreover, the state (cid:35) − (cid:35) (cid:32) (cid:35) (cid:35) − shows that the additiveconstant c (cid:48) in Eq. (2) depends dramatically on the details of the state (as we alreadyknow in the XX model). In Fig. 9 (right) we show the dependence on the spinoncontribution of the additive constant. In Fig. 9 (left) we report the von Neumannentropy for all states and values of ∆ we calculated. The changing in behavior fordifferent numbers of discontinuities is clearly visible. In this figure we also reporttwo (almost indistinguishable) states that have six discontinuities in the fermionicdescription and so are expected to have a = 3. There are strong crossover effectspreventing us to extract clearly the value of a for such small subsystems, but thedata are clearly in the right direction. This crossover is expected from the resultsfor the XX model: when having 6 discontinuities in a chain of 24 spins, we expectapproximately linear behavior in (cid:96) up to (cid:96) ∗ ∼ N/ (cid:96) ∼ 4. A quantitative understanding of this crossover(even for more excited states) requires larger values of (cid:96) and N that are not currentlyaccessible to us.To conclude this section, we also report in Fig. 10 the data for log(Tr ρ (cid:96) ) plotted ntanglement entropy of excited states Figure 10. log(Tr ρ (cid:96) ) against the logarithm of conformal distance. In the legendwe only give the states for ∆ = 0 . against the logarithm of the conformal distance to check the conformal prediction [4] − log Tr ρ n(cid:96) = 1 + n n c log (cid:18) Nπ sin π(cid:96)N (cid:19) + c (cid:48) n , (100)for n = 2. In fact, if the slope of all the previous curves can be interpreted as thecentral charge of some effective critical Hamiltonian having this state as a ground-state, not only the entanglement entropy should follow the conformal prediction (2),but also all R´enyi entropies should scale according to Eq. (100). And in fact, as for S (cid:96) the curves arrange in sectors with approximately similar slopes. Strong even-oddoscillations of the R´enyi entropies prevent us from any reliable quantitative analysis,as it is the case in the ground-state [49]. Again it is visible the same structure observedfor the von Neumann entropy. However, na¨ıve fits give reasonable estimations of theeffective central charges for the two lowest sets, but the oscillations (combined withthe crossover previously mentioned) spoil the result for the last set for which a = 3 isexpect.The knowledge of the full reduced density matrix can be also used to calculate theentanglement spectrum (i.e. the distribution of its eigenvalues). However, because ofthe relative small values of (cid:96) we can access, this is not enough to check recent conformalpredictions for the spectrum [59]. 4. Summary and discussions In this paper we considered the entanglement entropy of excited states in spin chains.We provided a full analytical study of the XY model in a transverse magnetic field.We found that the entanglement properties of the excited states depend stronglyon the distribution of excitations above the ground state. To characterize them inthe thermodynamic limit, we introduced the regularized characteristic function ofexcitation m ( ϕ ) ∈ [ − , ntanglement entropy of excited states N (cid:29) (cid:96) (cid:29) 1. When m ( ϕ ) (cid:54) = ± (cid:96) ).The analytic expression for such states is given by Eq. (17) as we proved by usingthe Sz¨ego lemma for block Toeplitz matrices. Oppositely when m ( ϕ ) = ± a ) is the central chargeof a critical, local, translational invariant Hamiltonian, that we built explicitly. Inthe case of the XX model we proved this result rigorously via the Fisher-Hartwigconjecture. These logarithmic states have a finite-size scaling that is by constructionthe conformal one in Eq. (2). Oppositely the extensive states have very peculiar finitesize scaling with slopes that changes according to the analytic properties of m ( ϕ ). Wehave been able to connect these features to the (non-)locality properties of an effectiveHamiltonian that can be made local on a wrapped chain.We also considered the XXZ spin chain, that is solvable by Bethe Ansatz. We usedthe algebraic construction to calculate exactly the reduced density matrix for finitechains with (cid:96) ≤ 6. The method we developed is ideal to obtain the entanglemententropy of excited states. In fact, while numerical methods based on MPS like DMRG[56] are very effective for the ground-state, they usually work bad for highly excitedones. Our method instead treats on the same foot any eigenstate, that is specified bythe quantum numbers related to the spinonic content of the state. This method hasthe numerical advantage that its complexity increases only in a polynomial way with N (while exact diagonalization is exponential). The drawback is that the complexityincreases exponentially with (cid:96) and limited our study to (cid:96) ≤ 6. We do not not knowwhether this is an intrinsic limit of the method, or if our representation of the reduceddensity matrix can be still drastically optimized to make the procedure more effective.The trickiest point in our derivation was to obtain the homogenous limit from theresults in Ref. [22, 23]. If we would have been able to find a more effective way toperform this limit, the method we propose could have been as effective as DMRG.However, even if we could study only subsystem with (cid:96) ≤ 6, we have been able toconclude that the main results obtained analytically for the XX model (at ∆ = 0)remain valid when interaction is turned on. We showed in fact (making the propermapping between spinonic and fermionic excitation at ∆ = 0) that all the statesthat are logarithmic for ∆ = 0 maintain this property with the same prefactor andwith a non-universal additive constant that depends very smoothly on ∆ (as for theground-state [55]).After this study, the characterization of the asymptotic block entanglement ofexcited states in these two chains is at an advanced level. Few unsolved problems arestill present, especially for the XXZ chain, as e.g. the understanding of the string-states and the quantitative description of the crossover between linear and logarithmicbehavior. However, the main question that still remains open is how general are theseresults. The fact that the division among extensive and logarithmic states is conservedwhen the interaction ∆ is introduced, strongly suggests that this phenomenon shouldbe expected for any local spin-chain, with a prefactor that can be predicted afterthat the relevant excitations have been identified. In fact, in the interacting system(especially for ∆ not small) the excited states are complicated linear combinations ofthe free-particle ones, several degenerations are also removed by ∆, and it is unlikelythat such result is only a coincidence. However, we do not have a general proof forthis statement. It might be that for low-lying excited states the generalization of the ntanglement entropy of excited states Acknowledgments We are extremely grateful to Jean-Sebastian Caux for his interest in this project andfor continuous fruitful discussions. We thank G. Sierra and M. Ibanez for sharingwith us their unpublished results and for useful discussions. We thank F. Colomoand F. Franchini for discussions. PC benefited of a travel grant from ESF (INSTANSactivity). References [1] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in Many-Body Systems, Rev.Mod. Phys. , 517 (2008) [quant-ph/0703044].[2] J. Eisert, M. Cramer, and M. B. Plenio, Area laws for the entanglement entropy - a review, Rev.Mod. Phys. XX , XXX (2009) [0808.3773].[3] Entanglement entropy in extended systems, P. Calabrese, J. Cardy, and B. Doyon Eds., J. Phys.A, Special issue, in preparation.[4] P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.P06002 (2004) [hep-th/0405152];P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory: a non-technicalintroduction, Int. J. Quant. Inf. , 429 (2006) [quant-ph/0505193].[5] P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, 0905.4013.[6] C. Holzhey, F. Larsen, and F. Wilczek, Geometric and Renormalized Entropy in ConformalField Theory, Nucl. Phys. B , 443 (1994) [hep-th/9403108].[7] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in quantum critical phenomena,Phys. Rev. Lett. , 227902 (2003) [quant-ph/0211074]J. I. Latorre, E. Rico, and G. Vidal, Ground state entanglement in quantum spin chains,Quant. Inf. and Comp. , 048 (2004) [quant-ph/0304098].[8] M. Caraglio and F. Gliozzi, Entanglement Entropy and Twist Fields, JHEP 0811: 076 (2008)[0808.4094];S. Furukawa, V. Pasquier, and J. Shiraishi, Mutual Information and Compactification Radiusin a c=1 Critical Phase in One Dimension, Phys. Rev. Lett. , 170602 (2009) [0809.5113];P. Calabrese, J. Cardy, and E. Tonni, Entanglement entropy of two disjoint intervals inconformal field theory, [0905.2069].[9] S. Das and S. Shankaranarayanan, How robust is the entanglement entropy-area relation?, Phys.Rev. D (2006) 121701 [gr-qc/0511066];S. Das and S. Shankaranarayanan, Where are the black hole entropy degrees of freedom ?,Class. Quant. Grav. , 5299 (2007) [gr-qc/0703082];S. Das, S. Shankaranarayanan, and S. Sur, Power-law corrections to entanglement entropy ofhorizons, Phys. Rev. D , 064013 (2008) [0705.2070].[10] M. Requardt, Entanglement Entropy for Ground states, Low lying and Highly ExcitedEigenstates of General (Lattice) Hamiltonians, hep-th/0605142.[11] F. C. Alcaraz and M. S. Sarandy, Finite-size corrections to entanglement in quantum criticalsystems, Phys. Rev. A 78, 032319 (2008) [0808.0020].[12] L. Masanes, An area law for the entropy of low-energy states, 0907.4672.[13] J. I. Latorre and A. Riera, A short review on entanglement in quantum spin systems, 0906.1499. ntanglement entropy of excited states [14] I. Peschel, M. Kaulke, and O. Legeza, Density-matrix spectra for integrable models, Ann. Physik(Leipzig) (1999) 153 [cond-mat/9810174];I. Peschel and M.-C. Chung, Density Matrices for a Chain of Oscillators, J. Phys. A , 064412 (2001) [cond-mat/0103301];I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A ,L205 (2003) [cond-mat/0212631];I. Peschel, On the reduced density matrix for a chain of free electrons, J. Stat. Mech. (2004)P06004 [cond-mat/0403048].[15] I. Peschel, On the entanglement entropy for a XY spin chain, J. Stat. Mech. (2004) P12005[cond-mat/0410416][16] I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free latticemodels, 0906.1663.[17] B.-Q. Jin, V. E. Korepin, Quantum Spin Chain, Toeplitz Determinants and Fisher-HartwigConjecture, J. Stat. Phys. , 79 (2004) [quant-ph/0304108].[18] A. R. Its, B.-Q. Jin, and V. E. Korepin, Entanglement in XY Spin Chain, J. Phys. A , 2975(2005) [quant-ph/0409027];F. Franchini, A. R. Its, and V. E. Korepin, R´enyi Entropy of the XY Spin Chain, J. Phys. A (2008) 025302 [0707.2534].[19] F. Igloi and R. Juhasz, Exact relationship between the entanglement entropies of XY andquantum Ising chains, Europhys. Lett. , 57003 (2008) [0709.3927].[20] H. Bethe, Zur theorie der metalle, Z. Phys. , 205 (1931).[21] M. Gaudin, “La fonction d’onde de Bethe”, Masson (Paris) (1983).[22] N. Kitanine, J. M. Maillet, and V. Terras, Form factors of the XXZ Heisenberg spin-1/2 finitechain Nucl. Physics B 647 (1999) [math-ph/9807020].[23] N. Kitanine, J. M. Maillet, and V. Terras, Correlation functions of the XXZ Heisenberg spin-1/2chain in a magnetic field, Nucl. Phys. B , 554 (2000) [math-ph/9907019].[24] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolatedquantum systems, Nature , 854 (2008) [0708.1324];A. Silva, The statistics of the work done on a quantum critical system by quenching a controlparameter, Phys. Rev. Lett. , 120603 (2008) [0806.4301];L. Campos Venuti and P. Zanardi, Unitary equilibrations: probability distribution of theLoschmidt echo, [0907.0683];G. Biroli, C. Kollath, A. Laeuchli; Does thermalization occur in an isolated system after aglobal quantum quench? [0907.3731];F. N.C. Paraan and A. Silva, Quantum quenches in the Dicke model: statistics of the workdone and of other observables, 0905.4833;M. Rigol, Quantum quenches and thermalization in one-dimensional fermionic systems,[0908.3188];M. Fagotti and P. Calabrese, to appear.[25] P. Calabrese and J. Cardy, Evolution of Entanglement entropy in one dimensional systems, J.Stat. Mech. P04010 (2005) [cond-mat/0503393].[26] G. De Chiara, S. Montangero, P. Calabrese, and R. Fazio Entanglement Entropy dynamics inHeisenberg chains, J. Stat. Mech. P03001 (2006) [cond-mat/0512586];J. Eisert and T. J. Osborne, General Entanglement Scaling Laws from Time Evolution, Phys.Rev. Lett. , 150404 (2006) [quant-ph/0603114];S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb-Robinson Bounds and the Generation ofCorrelations and Topological Quantum Order, Phys. Rev. Lett. , 050401 (2006) [quant-ph/0603121];A. Laeuchli and C. Kollath, Spreading of correlations and entanglement after a quench in theBose-Hubbard model, J. Stat. Mech. (2008) P05018 [0803.2947];V. Eisler and I. Peschel, Entanglement in a periodic quench, Ann. Phys. (Berlin) , 410(2008) [0803.2655];P. Calabrese, C. Hagendorf, and P. Le Doussal, Time evolution of 1D gapless models from adomain-wall initial state: SLE continued?, J. Stat. Mech. (2008) P07013 [0804.2431];M. Znidaric, T. Prosen, and I. Pizorn, Complexity of thermal states in quantum spin chains,Phys. Rev. A , 022103 (2008) [0805.4149];S. R. Manmana, S. Wessel, R. M. Noack, and A. Muramatsu, Time evolution of correlationsin strongly interacting fermions after a quantum quench, Phys. Rev. B , 155104 (2009)[0812.0561]; ntanglement entropy of excited states V. Eisler, F. Igloi, and I. Peschel, Entanglement in spin chains with gradients, J. Stat. Mech.(2009) P02011 [0810.3788].[27] M. Fagotti and P. Calabrese, Evolution of entanglement entropy following a quantum quench:Analytic results for the XY chain in a transverse magnetic field, Phys. Rev. A , 010306(R)(2008) [0804.3559].[28] V. Eisler and I. Peschel, Evolution of entanglement after a local quench, J. Stat. Mech. P06005(2007) [cond-mat/0703379];P. Calabrese and J. Cardy, Entanglement and correlation functions following a local quench:a conformal field theory approach, J. Stat. Mech. (2007) P10004 [0708.3750]; V. Eisler, D.Karevski, T. Platini, and I. Peschel, Entanglement evolution after connecting finite to infinitequantum chains, J. Stat. Mech. (2008) P01023 [0711.0289];A. Perales and G. Vidal, Entanglement growth and simulation efficiency in one-dimensionalquantum lattice systems, Phys. Rev A , 042337 (2008) [0711.3676];I. Klich and L. Levitov, Quantum Noise as an Entanglement Meter, Phys. Rev. Lett. ,100502 (2009) [0804.1377];I. Pizorn and T. Prosen, Operator Space Entanglement Entropy in XY Spin Chains, Phys.Rev. B 79, 184416 (2009) [0903.2432];B. Hsu, E. Grosfeld, and E. Fradkin, Quantum noise and entanglement generated by a localquantum quench, Phys. Rev. B to appear [0908.2622].[29] F. C. Alcaraz, V. Rittenberg, and G. Sierra, Entanglement in Far From Equilibrium StationaryStates, Phys. Rev. E , 030102(R) (2009) [0905.0211];V. Eisler and Z. Zimboras, Entanglement in the XX spin chain with an energy current, Phys.Rev A , 042318 (2005) [quant-ph/0412118].[30] M. Ibanez and G. Sierra, private communication.[31] H. Au-Yang and B. McCoy, Theory of layered Ising models. II. Spin correlation functions parallelto the layering, 1974 Phys. Rev. B (2004) 543 [quant-ph/0407047];J. P. Keating and F. Mezzadri, Entanglement in Quantum Spin Chains, Symmetry Classes ofRandom Matrices, and Conformal Field Theory, Phys. Rev. Lett. (2005) 050501 [quant-ph/0504179].[33] M. E. Fisher and R. E. Hartwig, Toeplitz determinants: some applications, theorems, andconjectures, Adv. Chem. Phys. , 333 (1968);P. J. Forrester and N. E. Frankel, Applications and generalizations of Fisher-Hartwigasymptotics, J. Math. Phys. , 2003 (2004) [math-ph/0401011];E. L. Basor and K. E. Morrison, Linear Algebra and Its Applications , 129 (1994).[34] M. Srednicki, Entropy and Area, Phys. Rev. Lett. (1993) 666 [hep-th/9303048];M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Area laws in quantum systems:mutual information and correlations, Phys. Rev. Lett. , 070502 (2008) [0704.3906].[35] V. E. Korepin, N. M. Bogoliubov and A. G. Izergin, Quantum Inverse Scattering Method andCorrelation Functions , Cambridge University Press (1993).[36] N. Kitanine, J.M. Maillet, N.A. Slavnov, V. Terras, Spin-spin correlation functions of the XXZ-1/2 Heisenberg chain in a magnetic field, Nucl. Phys. B (2002) 487 [hep-th/0201045];N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, Large distance asymptotic behaviorof the emptiness formation probability of the XXZ spin-1/2 Heisenberg chain, J.Phys. A (2002) L753 [hep-th/0210019];N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, Master equation for spin-spin correlationfunctions of the XXZ chain, Nucl.Phys. B (2005) 600 [hep-th/0406190];N. Kitanine, J. M. Maillet, N. A. Slavnov, V. Terras, Dynamical correlation functions of theXXZ spin-1/2 chain, Nucl.Phys. B (2005) 558 [hep-th/0407108];N. Kitanine, J.M. Maillet, N.A. Slavnov, V. Terras, On the spin-spin correlation functions ofthe XXZ spin-1/2 infinite chain, J.Phys. A (2005) 7441 [hep-th/0407223].[37] N. Kitanine, K. Kozlowski, J. M. Maillet, G. Niccoli, N. A. Slavnov, V. Terras, Correlationfunctions of the open XXZ chain I, J. Stat. Mech P10009, 2007 [0707.1995];N. Kitanine, K. Kozlowski, J. M. Maillet, G. Niccoli, N. A. Slavnov, V. Terras, Correlationfunctions of the open XXZ chain II, J. Stat. Mech. (2008) P07010 [0803.3305].[38] J.-S. Caux and J.-M. Maillet, Computation of dynamical correlation functions of Heisenbergchains in a field, Phys. Rev. Lett. , 077201 (2005);J.-S. Caux, R. Hagemans and J.-M. Maillet, Computation of dynamical correlation functionsof Heisenberg chains: the gapless anisotropic regime, J. Stat. Mech. P09003 (2005).[39] R. G. Pereira, J. Sirker, J.-S. Caux, R. Hagemans, J. M. Maillet, S. R. White, I. Affleck, The ntanglement entropy of excited states dynamical spin structure factor for the anisotropic spin-1/2 Heisenberg chain, Phys. Rev.Lett. 96, 257202 (2006) [cond-mat/0603681];R. G. Pereira, J. Sirker, J.-S. Caux, R. Hagemans, J. M. Maillet, S. R. White, I. Affleck,Dynamical structure factor at small q for the XXZ spin-1/2 chain, J. Stat. Mech. (2007)P08022 [0706.4327].[40] J.-S. Caux and P. Calabrese, Dynamical density-density correlations in the one-dimensional Bosegas, Phys. Rev. A , 031605 (2006);J.-S. Caux, P. Calabrese and N. A. Slavnov, One-particle dynamical correlations in the one-dimensional Bose gas, J. Stat. Mech. P01008 (2007).[41] A. Faribault, P. Calabrese, and J.-S. Caux, Exact mesoscopic correlation functions of the pairingmodel, Phys. Rev. B , 064503 (2008).[42] A. Faribault, P. Calabrese, and J.-S. Caux, Quantum quenches from integrability: the fermionicpairing model, J. Stat. Mech. (2009) P03018 [0812.1928];A. Faribault, P. Calabrese, and J.-S. Caux, Bethe Ansatz approach to quench dynamics inthe Richardson model, J. Math. Phys. 50, 095212 (2009) [0908.1675].[43] J.-S. Caux Correlation functions of integrable models: a description of the ABACUS algorithm,J. Math. Phys. 50, 095214 (2009) [0908.1660].[44] J. Sato and M. Shiroishi, Density matrix elements and entanglement entropy for the spin-1/2XXZ chain at ∆=1/2, J. Phys. A , 8739 (2007).[45] J. L. Jacobsen and H. Saleur, Exact valence bond entanglement entropy and probabilitydistribution in the XXX spin chain and the Potts model, Phys. Rev. Lett. , 087205(2008).[46] J. Damerau, F. G¨ohmann, N. P. Hasenclever, and A. Kl¨umper, Density matrices for finitesegments of Heisenberg chains of arbitrary length J. Phys. A , 4439 (2007);H. E. Boos, J. Damerau, F. G¨ohmann, A. Kl¨umper, J. Suzuki, and A. Weisse, Short-distancethermal correlations in the XXZ chain, J. Stat. Mech. P08010 (2008);C. Trippe, F. G¨ohmann, A. Kl¨umper, Short-distance thermal correlations in the massive XXZchain, [0908.2232].[47] J. Sato, M. Shiroishi, M. Takahashi, Exact evaluation of density matrix elements for theHeisenberg chain , J. Stat. Mech. P12017 (2006).[48] A. V. Razumov and Y. G. Stroganov, Spin chains and combinatorics, J. Phys. A , 3185 (2001);Y. G. Stroganov, The Importance of being Odd, J. Phys. A , L179 (2001).[49] B. Nienhuis, M. Campostrini, and P. Calabrese, Entanglement, combinatorics and finite-sizeeffects in spin-chains, J. Stat. Mech. (2009) P02063 [0808.2741].[50] L. Banchi, F. Colomo, and P. Verrucchi, When finite-size corrections vanish: The S=1/2 XXZmodel and the Razumov-Stroganov state, Phys. Rev. A , 022341 (2009) [0906.3703].[51] N. A. Slavnov, On scalar products in the algebraic Bethe ansatz, Teor. Mat. Fiz. , 232 (1989).[52] V. E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. , 391(1982).[53] A. G. Abanov and V. E. Korepin, On the probability of ferromagnetic strings inantiferromagnetic spin chains, Nucl. Phys. B , 565 (2002).[54] F. Colomo and A. G. Pronko, Emptiness formation probability in the domain-wall six-vertexmodel, Nucl. Phys. B , 340 (2008) [0712.152].[55] P. Calabrese, M. Campostrini, B. Nienhuis et al., in preparation.[56] U. Schollwoeck, The density-matrix renormalization group, Rev. Mod. Phys. , 259 (2005)[cond-mat/0409292];F. Verstraete, J.I. Cirac, and V. Murg, Matrix Product States, Projected Entangled PairStates, and variational renormalization group methods for quantum spin systems, Adv. Phys. , 143 (2008) [0907.2796];J.I. Cirac and F. Verstraete, Renormalization and tensor networks in spin chains and lattices,J. Phys. A, to appear.[57] M. Karbach, K. Hu, and G. Muller, Introduction to the Bethe ansatz II, Computers in Physics , 565 (1998) [cond-mat/9809163][58] M. Arikawa, M. Karbach, G. Muller, and K. Wiele, Spinon excitations in the XX chain: spectra,transition rates, observability, J. Phys. A (2006 10623 [cond-mat/0605345].[59] P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A , 032329 (2008).[60] G. Refael and J. E. Moore, Entanglement entropy of random quantum critical points in onedimension, Phys. Rev. Lett.93