Rényi entropy and subsystem distances in finite size and thermal states in critical XY chains
RR´enyi entropy and subsystem distances in finite size andthermal states in critical XY chains
Ra´ul Arias , ∗ , Jiaju Zhang † SISSA and INFN, Via Bonomea 265, 34136 Trieste, Italy Instituto de F´ısica La Plata - CONICET and Departamento de F´ısica, Universidad Nacional de LaPlata C.C. 67, 1900, La Plata, Argentina
Abstract
We study R´enyi entropy and subsystem distances of one interval in finite size and thermal statesin critical XY chains, focusing on critical Ising chain and XX chain with zero transverse field. Weconstruct numerically the reduced density matrices and calculate the von Neumann entropy, R´enyientropy, subsystem trace distance, Schatten two-distance and relative entropy. As the continuumlimit of the critical Ising chain and XX chain with zero field are, respectively, two-dimensional freemassless Majorana and Dirac fermion theories, which are conformal field theories, we compare thespin chain numerical results with the analytical results in conformal field theories and find perfectmatches in the continuum limit. ∗ [email protected] † [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r ontents (cid:96)
18B Thermal RDM in XY chains 20
B.1 Gapped XY chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22B.2 Critical Ising chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23B.3 XX chain with zero field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
C Relative entropy among RDMs in low-lying energy eigenstates 25
C.1 Free massless Majorana fermion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25C.2 Free massless Dirac fermion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Quantum entanglement has become one of the key tools to the understanding of quantum many-bodysystems and quantum field theories [1–5]. For a quantum system in a state with density matrix ρ ,one could choose a subsystem A and trace out the degrees of freedom of its complement ¯ A to get thereduced density matrix (RDM) of the subsystem ρ A = tr ¯ A ρ . With the RDM ρ A , one could computethe von Neumann entropy S A = − tr A ( ρ A log ρ A ) , (1.1)and R´enyi entropy S ( n ) A = − n − A ρ nA . (1.2)2he n → S A = lim n → S ( n ) A . (1.3)When the whole system is in a pure state ρ = | Ψ (cid:105)(cid:104) Ψ | , the von Neumann entropy is a rigorous measureof entanglement that is usually called the entanglement entropy but in cases where the whole systemis in a mixed state neither the von Neumann entropy or R´enyi entropy are good entanglement mea-sures. Nevertheless they are still interesting quantities that characterize to some extent the amount ofentanglement.The continuum limit of one-dimensional quantum spin chains can be described by two-dimensional(2D) conformal field theories (CFTs) [6–10]. For example, the continuum limit of the critical Isingchain gives 2D free massless Majorana fermion theory, which is a 2D CFT with central charge c = ,and the continuum limit of the XX chain with zero transverse filed gives 2D free massless Dirac fermiontheory, or equivalently 2D free massless compact boson theory with the unit radius target space, whichis a 2D CFT with central charge c = 1. It is interesting to compare von Neumann and R´enyi entropiesin critical spin chains with those in the corresponding CFTs. Some examples are the case of oneinterval in the ground state [11–14] and excited states [15,16], and the cases of multiple intervals in theground state [17–30]. In this paper, we consider the case of one interval in a state with both finite sizeand finite temperature in critical XY chains. We focus on two special critical points of the spin- XYchain, i.e. the critical Ising chain and the XX chain with zero field. In 2D CFT, the state with both afinite size and a finite temperature is described by the theory on a torus. To calculate R´enyi entropyon torus in 2D free massless boson and fermion theories, one needs to take into account properly thevarious spin structures on the replicated multi-genus Riemann surface. The final complete results weregiven in [31, 32], and previous results could be found in [33–45].Sometimes knowing the entanglement is not enough to characterise the system. It is also interestingto know quantitatively the distance between two density matrices. There are many objects that dothis job [46–48] but in the present work we will just analyse some of them, the trace distance, theSchatten n -distance and the relative entropy. For two density matrices ρ, σ , the trace distance isdefined as [46–48] D ( ρ, σ ) = tr | ρ − σ | . (1.4)Subsystem trace distances in low-lying energy eigenstates and states after local operator quench in 2DCFTs and one-dimensional quantum spin chains have been investigated [49–51]. In these works thereplica trick was used tr | ρ − σ | = lim n e → tr( ρ − σ ) n e , (1.5)and one firstly evaluates the right hand side for a general even integer n e and then makes the analyticcontinuation to one n e →
1. For n ≥
1, one could also define the Schatten n -distance D n ( ρ, σ ) = (tr | ρ − σ | n ) /n /n . (1.6)3n 2D CFT, the Schatten n -distance defined above for two reduced density matrices (RDMs) ρ A , σ A depends on the UV cutoff, and we will add the normalization as D n ( ρ A , σ A ) = (cid:16) tr A | ρ A − σ A | n A ( ρ A ( ∅ ) n ) (cid:17) /n . (1.7)Here ρ A ( ∅ ) is the RDM of the subsystem A on an infinite system in the ground state. Another quantitythat characterizes the difference between two states ρ, σ is the relative entropy S ( ρ (cid:107) σ ) = tr( ρ log ρ ) − tr( ρ log σ ) . (1.8)In this paper we will consider a subsystem A that is an interval of length (cid:96) , and it has differentRDMs ρ A in different states ρ of the total system. The most general case we will consider is an intervalon a torus with spatial circumference L and imaginary temporal period β , which is a finite system in athermal state. We denote the RDM of the interval in such a state as ρ A ( L, β ). Taking β → ∞ limit weget an interval on a vertical cylinder with spatial period L , which is a finite system in the ground state.We denote the RDM in such a state as ρ A ( L ). On the other hand, taking L → ∞ for the torus, weget an interval on a horizontal cylinder with imaginary temporal period β , which is an infinite systemin a thermal state. We denote the RDM in such a state as ρ A ( β ). Taking both L → ∞ and β → ∞ limit, we get an interval on a complex plane, which is an infinite system in the ground state. We willdenote the RDM in such a state as ρ A ( ∅ ). In this paper we are going to compute the numerical spinchain von Neumann entropy and R´enyi entropy and compare them with those in CFT. Moreover wewill calculate the subsystem trace distance, the Schatten two-distance and the relative entropy amongthese RDMs ρ A ( L, β ), ρ A ( L ), ρ A ( β ), ρ A ( ∅ ) in both CFTs and spin chains and compare the results.We find perfect matches in the continuum limit.The remaining part of the paper is arranged as follows. In section 2, we consider the critical Isingchain and 2D free massless Majorana fermion theory. In section 3, we consider the XX chain withzero field and 2D free massless Dirac fermion theory. In these two sections, we compare the CFTand spin chain results of von Neumann entropy, R´enyi entropy, subsystem trance distance, Schattentwo-distance, and relative entropy, and find perfect matches in the continuum limit. We conclude withdiscussions in section 4. In appendix A, we show that the method of twist operators cannot give thecorrect short interval R´enyi entropy on torus at order (cid:96) in some specific 2D CFTs, including 2D freemassless Majorana and Dirac fermion theories. In appendix B, we elaborate how to construct thenumerical RDMs in finite size and thermal states in XY chains, especially in critical Ising chain andXX chain with zero field. In appendix C we compare the CFT and spin chain results of subsystemrelative entropy among low-lying energy eigenstates. We consider critical Ising chain, whose continuum limit gives the 2D free massless Majorana fermiontheory, a 2D CFT with central charge c = . 4 .1 von Neumann and R´enyi entropies We will first review the result for the R´enyi entropy of one interval A = [0 , (cid:96) ] on a torus in 2D freemassless Majorana fermion theory [32], and then recalculate it using twist operators [14, 52, 53] andtheir operator product expansion (OPE) [23, 25, 54–59]. We get the same R´enyi entropy to order (cid:96) from OPE of twist operators as the result from the expansion of the exact result in [32]. The shortinterval expansion of the R´enyi entropy allows us to do the analytic continuation n → (cid:96) .In the critical Ising chain, we construct numerically the RDMs in finite size and thermal statesand compute the von Neumann entropy for a short interval and the R´enyi entropy for a relativelylong interval. We compare analytical CFT results with numerical spin chain results and find perfectmatches in the continuum limit. Details of 2D free massless Majorana fermion theory can be found in the books [60, 61]. Except theidentity operator 1 in the Neveu-Schwarz (NS) sector, there is primary operator σ with conformalweights ( , ) in the Ramond (R) sector and a primary operator ε with conformal weights ( , ) inthe NS sector.The state with both finite size and finite temperature in 2D CFT corresponds to a torus whichin our case has spatial period L and temporal period β , and the interval A has length (cid:96) . The R´enyientropy of one interval on torus was calculated in [32] from higher genus partition function, and it wasargued in [32, 62] that the method of twist operators cannot give the correct result in a fermion theory.The result can be written in terms of the ratio x = (cid:96)/L and the torus modulus τ = i β/L . The R´enyientropy of the interval A = [0 , (cid:96) ] on torus is [32] S ( n ) A = n + 112 n log (cid:12)(cid:12)(cid:12) L(cid:15) θ ( x | τ ) θ (cid:48) (0 | τ ) (cid:12)(cid:12)(cid:12) − n − (cid:20) (cid:80) (cid:126)α,(cid:126)β (cid:12)(cid:12)(cid:12) Θ (cid:104) (cid:126)α(cid:126)β (cid:105) (0 | Ω) (cid:12)(cid:12)(cid:12)(cid:0) (cid:81) n − k =1 | A k | (cid:1) / (cid:0) (cid:80) ν =2 | θ ν (0 | τ ) | (cid:1) n (cid:21) , (2.1)with the period matrix of the higher genus Riemann surfaceΩ ab ( x, τ ) = 1 n n − (cid:88) k =0 cos (cid:104) π ( a − b ) kn (cid:105) C k ( x, τ ) , C k ( x, τ ) = B k ( x, τ ) A k ( x, τ ) , (2.2)and A k ( x, τ ) = (cid:90) τ τ ω ( z, x, τ )d z, B k ( x, τ ) = (cid:90) + τ ω ( z, x, τ )d z,ω ( z, x, τ ) = θ ( z | τ ) θ (cid:0) z + kn x | τ (cid:1) − kn θ (cid:0) z − (1 − kn ) x | τ (cid:1) kn . (2.3)In A k , B k , we have shifted the integral ranges to make them convenient for numerical evaluation.The genus- n Siegel theta function is defined asΘ (cid:104) (cid:126)α(cid:126)β (cid:105) ( (cid:126)z | Ω) = (cid:88) (cid:126)m ∈ Z n exp (cid:2) π i( (cid:126)m + (cid:126)α ) · Ω · ( (cid:126)m + (cid:126)α ) + 2 π i( (cid:126)m + (cid:126)α ) · ( (cid:126)z + (cid:126)β ) (cid:3) , (2.4)5ith · being multiplications between vectors and matrices. The entries of the n -component vectors (cid:126)α, (cid:126)β are chosen independently from 0 and and the sum of (cid:126)α, (cid:126)β in (2.1) is over all the possible spinstructures. The Jacobi theta function is θ (cid:2) αβ (cid:3) ( z | τ ) = (cid:88) m ∈ Z exp (cid:2) π i τ ( m + α ) + 2 π i( m + α )( z + β ) (cid:3) , (2.5)and, as usual, we have the relations θ ( z | τ ) = − θ (cid:2) / / (cid:3) ( z | τ ) , θ ( z | τ ) = θ (cid:2) / (cid:3) ( z | τ ) , θ ( z | τ ) = θ (cid:2) (cid:3) ( z | τ ) , θ ( z | τ ) = θ (cid:2) / (cid:3) ( z | τ ) . (2.6)Following [57], we use OPE of twist operators and get the short interval expansion of the R´enyientropy S ( n ) A = n + 112 n log (cid:96)(cid:15) − ( n + 1) (cid:96) n (cid:16) (cid:104) T (cid:105) + 14 (cid:104) ε (cid:105) (cid:17) + O ( (cid:96) ) . (2.7)The expectation values on torus [60] read (cid:104) T (cid:105) = − π qL ∂ q Z ( q ) Z ( q ) , (cid:104) ε (cid:105) = πL η ( τ ) Z ( q ) , (2.8)where we set q = e π i τ and the partition function can be written as Z ( q ) = 12 η ( τ ) (cid:2) θ (0 | τ ) + θ (0 | τ ) + θ (0 | τ ) (cid:3) . (2.9)The short interval expansion of R´enyi entropy (2.7) is consistent with the small (cid:96) expansion of theexact result (2.1), which is S ( n ) A = n + 112 n log (cid:96)(cid:15) + ( n + 1) (cid:96) nL (cid:104) θ (cid:48)(cid:48)(cid:48) (0 | τ ) θ (cid:48) (0 | τ ) − (cid:80) ν =2 θ (cid:48)(cid:48) ν (0 | τ ) (cid:80) ν =2 θ ν (0 | τ ) − (cid:16) θ (cid:48) (0 | τ ) (cid:80) ν =2 θ ν (0 | τ ) (cid:17) (cid:105) + O ( (cid:96) ) . (2.10)Note that θ (0 | τ ) = θ (cid:48) ν (0 | τ ) = θ (cid:48)(cid:48) (0 | τ ) = θ (cid:48)(cid:48)(cid:48) ν (0 | τ ) = 0 with ν = 2 , , θ (cid:48) (0 | τ ) = 2 πη ( τ ) , q∂ q θ ν ( z | τ ) = − π θ (cid:48)(cid:48) ν ( z | τ ) , ν = 1 , , , , (2.11)we can show that the expressions (2.7) and (2.10) are in fact the same. This means that the methodof short interval expansion from OPE of twist operators is valid at order (cid:96) . However, it breaks downat order (cid:96) , as we show in appendix A. For a short interval, we compare the exact R´enyi entropy andthe short interval expansion one in Fig. 1. In the figure we have subtracted the R´enyi entropy on aninfinite straight line in the ground state to make it independent of the UV cutoff, i.e. that we use∆ S ( n ) A = S ( n ) A − n + 112 n log (cid:96)(cid:15) . (2.12)We see good matches for the exact and leading order short interval results. This indicates that thesmall (cid:96) expansion for the R´enyi entropy is a good approximation in the regime of parameters weconsider.The short interval result (2.7) remarks the validity of the method of twist operators at the order (cid:96) in a small (cid:96) expansion. Furthermore, it is convenient to do the analytic continuation n → S A = 16 log (cid:96)(cid:15) − (cid:96) (cid:16) (cid:104) T (cid:105) + 14 (cid:104) ε (cid:105) (cid:17) + O ( (cid:96) ) . (2.13) In this paper we only consider the case without chemical potential, i.e. that τ is purely imaginary, and so ¯ q = q . Wehave the partition function Z ( q ) = Z ( q, ¯ q = q ), and (cid:104) T (cid:105) = (cid:104) ¯ T (cid:105) . xact resultshort interval0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.000.010.020.030.04 β / L with L = ℓ = Δ S A ( ) exact resultshort interval0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.000.010.020.030.04 β / L with L = ℓ = Δ S A ( ) exact resultshort interval0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 - - - - - L / β with β = ℓ = Δ S A ( ) exact resultshort interval0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 - - - - L / β with β = ℓ = Δ S A ( ) Figure 1: The comparison of the exact R´enyi entropy with the short interval expansion one in freemassless Majorana fermion theory. We use ∆ S ( n ) A = S ( n ) A − n +112 n log (cid:96)(cid:15) to make it independence of theUV cutoff. We compare the R´enyi entropy on a torus in a Majorana fermion theory with R´enyi entropy in athermal state in periodic critical Ising chain. We construct the numerical RDM of one interval in finitesize and thermal states in critical Ising chain following [12, 13, 22, 63, 64], as detailed in appendix B.To handle the zero modes in the R sector in critical XY chains, we need a special trick as in [22]. Tocalculate the von Neumann entropy, we need the explicit numerical RDMs, and we can only calculateit for a short interval. For the R´enyi entropy, only the correlation matrices are enough, and we cancalculate it for a relatively long interval.On the CFT side, we use the short interval expansion of the von Neumann entropy (2.13) and theexact R´enyi entropy (2.1). We denote the CFT von Neumann and R´enyi entropies as S CFT ( L, β ) and S ( n )CFT ( L, β ). We denote the spin chain von Neumann and R´enyi entropies as S SC ( L, β ) and S ( n )SC ( L, β ).The CFT and spin chain results are compared in Fig. 2. Note that in CFT we have the subtractedCFT results of the von Neumann and R´enyi entropies on an infinite line in the ground state and get∆ S CFT ( L, β ) and ∆ S ( n )CFT ( L, β ), and in spin chain we have the subtracted spin chain results of thevon Neumann and R´enyi entropies on an infinite chain in the ground state and get ∆ S SC ( L, β ) and∆ S ( n )SC ( L, β ). In other words, ∆ S CFT ( L, β ) and ∆ S ( n )CFT ( L, β ) are pure CFT results, and ∆ S SC ( L, β )and ∆ S ( n )SC ( L, β ) are pure spin chain results, and we have compared results independently obtainedin CFT and spin chain. Unfortunately, in Fig. 2 there are generally no good matches between theanalytical CFT results and numerical spin chain results. As L (cid:29) β and L (cid:28) β , the matches are good,but for general L, β , especially for
L/β ∼
1, there are large deviations. We believe the derivations aredue to finite values of
L, β, (cid:96) .To better see the continuum limit of the critical Ising chain, we fix the ratios L : β : (cid:96) , which makethe scale invariant CFT result ∆ S ( n )CFT a constant, and look into the difference of the von Neumann andR´enyi entropies in spin chain and CFT with the increase of (cid:96) . We plot the results in Fig. 3. We seethat the differences of spin chain and CFT results decrease monotonically. Furthermore, by numericalfit, we get approximately | ∆ S ( n )SC − ∆ S ( n )CFT | ∝ (cid:96) − /n . (2.14)7hus we obtain perfect matches between the CFT and spin chain results of the von Neumann andR´enyi entropies in the continuum limit of the spin chain. Δ S CFT Δ S SC β / L with L = ℓ = Δ S CFT ( ) Δ S SC ( ) β / L with L = ℓ = Δ S CFT ( ) Δ S SC ( ) β / L with L = ℓ = Δ S CFT Δ S SC - - - - - - L / β with β = ℓ = Δ S CFT ( ) Δ S SC ( ) - - - - L / β with β = ℓ = Δ S CFT ( ) Δ S SC ( ) - - - - L / β with β = ℓ = Figure 2: We compare the von Neumann and R´enyi entropies in free massless Majorana fermion theorywith the numerical results in critical Ising chain. We see deviations of the results that we attribute tofinite values of
L, β, (cid:96) . △ △ △ △ △ △△△△△△△△△△△△△△△△△△△△△△△△△△△○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○▽ ▽ ▽ ▽ ▽ ▽▽ △ n = ○ n = ▽ n =
11 2 5 10 200.0050.0100.020 ℓ | Δ S S C ( n ) - Δ S C FT ( n ) | L : β : ℓ = △ △ △ △ △ △△△△△△△△△△△△△△△△△△△△△△△△△△△○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○▽ ▽ ▽ ▽ ▽ ▽▽ △ n = ○ n = ▽ n =
11 2 5 10 200.0050.0100.020 ℓ | Δ S S C ( n ) - Δ S C FT ( n ) | L : β : ℓ = △ △ △ △ △ △△△△△△△△△△△△△△△△△△△△△△△△△△△○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○▽ ▽ ▽ ▽ ▽ ▽▽ △ n = ○ n = ▽ n =
11 2 5 10 200.010.020.05 ℓ | Δ S S C ( n ) - Δ S C FT ( n ) | L : β : ℓ = △ △ △ △ △ △△△△△△△△△△△△△△△△△△△△△△△△△△△○ ○ ○ ○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○▽ ▽ ▽ ▽ ▽ ▽▽ △ n = ○ n = ▽ n =
11 2 5 10 200.0050.0100.020 ℓ | Δ S S C ( n ) - Δ S C FT ( n ) | L : β : ℓ = Figure 3: The difference of the von Neumann and R´enyi entropy in critical Ising chain and free masslessMajorana fermion theory with increase of (cid:96) . We see perfect matches in the continuum limit of the spinchain. The thin solid red lines are proportional to (cid:96) − /n . We consider short interval expansion of the subsystem trace distance. The leading trace distance oftwo RDMs ρ A , σ A depends on the quasiprimary operators with the lowest scaling dimension that havedifferent expectations in the two states ρ, σ . Among the states on plane and cylinder ρ ( ∅ ), ρ ( L ), and ρ ( β ), the quasiprimary operators with the lowest scaling dimension that have different expectationsare the stress tensor T , ¯ T . Furthermore, they always have the same expectation values (cid:104) T (cid:105) ρ = (cid:104) ¯ T (cid:105) ρ inone of such states ρ ( ∅ ), ρ ( L ), and ρ ( β ) that we denote generally by ρ and the same difference of theexpectation values in two such states ρ, σ (cid:104) T (cid:105) ρ − (cid:104) T (cid:105) σ = (cid:104) ¯ T (cid:105) ρ − (cid:104) ¯ T (cid:105) σ . (2.15)8ollowing [49, 50], we use OPE of twist operators and get the leading order of the short intervalexpansion of the trace distance D ( ρ A , σ A ) = y T (cid:96) √ c |(cid:104) T (cid:105) ρ − (cid:104) T (cid:105) σ | + o ( (cid:96) ) . (2.16)We have the coefficient y T = lim p → (cid:16) c (cid:17) p (cid:88) S⊆S (cid:104)(cid:68) (cid:89) j ∈S [ f j T ( f j )] (cid:69) C (cid:68) (cid:89) j ∈ ¯ S [ ¯ f j ¯ T ( ¯ f j )] (cid:69) C (cid:105) , f j = e π i jp , ¯ f j = e − π i jp , (2.17)where the sum of S is over all the subsets of S = { , , · · · , p − } , including the empty set ∅ and S itself, and ¯ S is the complement set ¯ S = S / S . One needs to first evaluate the right hand side of (2.17)for a general positive integer p and then take the analytic continuation p → . Unfortunately, we donot know how to evaluate y T . In the following we will fit numerically the coefficient y T from the specialcase D ( ρ A ( ∅ ) , ρ A ( L )) in the spin chain results and check the coefficient in the other cases. Since OPEof twist operators has been used, for this equation (2.16) being valid we need that the interval length (cid:96) is much smaller than any characteristic length of the two states L , i.e. (cid:96) (cid:28) L , which includes boththe size of the total system L and the inverse temperature β .In the ground state on a circle ρ ( L ) we have the expectation value of the stress tensor (cid:104) T (cid:105) ρ ( L ) = π c L . (2.18)Combining both the CFT and spin chain results, we get D ( ρ A ( ∅ ) , ρ A ( L )) ≈ . (cid:96) L + o (cid:16) (cid:96) L (cid:17) . (2.19)In CFT we know the leading order trace distance is proportional to (cid:96) L , and we obtain the approximateoverall coefficient 0 .
126 from numerical fit of the spin chain results. This gives the approximate valueof (2.17) y T ≈ . In the thermal state on an infinite line ρ ( β ), we have the expectation values ofthe stress tensor (cid:104) T (cid:105) ρ ( β ) = − π c β . (2.20)Based on (2.16) and (2.19), we further get D ( ρ A ( L ) , ρ A ( L )) ≈ . (cid:96) (cid:12)(cid:12)(cid:12) L − L (cid:12)(cid:12)(cid:12) + o ( (cid:96) ) . (2.21) D ( ρ A ( β ) , ρ A ( β )) ≈ . (cid:96) (cid:12)(cid:12)(cid:12) β − β (cid:12)(cid:12)(cid:12) + o ( (cid:96) ) . (2.22) D ( ρ A ( L ) , ρ A ( β )) ≈ . (cid:96) (cid:16) L + 1 β (cid:17) + o ( (cid:96) ) . (2.23)Some of the results are plotted in Fig. 4. We see perfect matches of the CFT and spin chain resultsfor (cid:96)/ L (cid:28) L being all values of L and β . The formula (2.16) also applies to the trace distance D ( ρ A ( L ) , ρ A,ε ( L )), with ρ A,ε ( L ) being the RDM of the energyeigenstate ρ ε ( L ). The state ρ ε ( L ) represents a vertical cylinder with spatial circumference L and ε inserted at its two ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ CFT ◦ SC ℓ =
45 10 50 10010 - L D ( ρ A ( ∅ ) , ρ A ( L )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ L = L = L = L = L =
128 CFT ◦ SC ℓ =
45 10 50 10010 - - L D ( ρ A ( L ) , ρ A ( L )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ β = β = β = β = β =
128 CFT ◦ SC ℓ =
45 10 50 10010 - - - β D ( ρ A ( β ) , ρ A ( β )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ L = L = L = L = L =
128 CFT ◦ SC ℓ =
45 10 50 1001. × - × - β D ( ρ A ( L ) , ρ A ( β )) Figure 4: Trace distance of the RDMs in states on cylinder in free massless Majorana fermion theory(solid line) and critical Ising chain (empty circle).When one of the two states ρ, σ are on torus with (cid:104) ε (cid:105) ρ (cid:54) = (cid:104) ε (cid:105) σ , the leading order short intervalexpansion of the trace distance is [49, 50] D ( ρ A , σ A ) = (cid:96) π |(cid:104) ε (cid:105) ρ − (cid:104) ε (cid:105) σ | + o ( (cid:96) ) . (2.24)However, when |(cid:104) ε (cid:105) ρ − (cid:104) ε (cid:105) σ | is exponentially small while |(cid:104) T (cid:105) ρ − (cid:104) T (cid:105) σ | is not, the dominate contributionto the trace distance would be (2.15). When the terms (2.24) and (2.15) are at the same order, wedo not have a reliable CFT result. In the critical Ising chain, we could calculate numerically the tracedistance for such states. As we do no have reliable CFT results to be compared with, we will not showthese spin chain results here. We define the subsystem Schatten two-distance of two RDMs ρ A , σ A as D ( ρ A , σ A ) = (cid:115) tr A ( ρ A − σ A ) A ( ρ A ( ∅ ) ) . (2.25)Note that in the ground state of the 2D CFT on the plane [11, 14]tr A ( ρ A ( ∅ ) ) = c (cid:16) (cid:96)(cid:15) (cid:17) − , (2.26) ends in the infinity. In [50] it was obtained numerically D ( ρ A ( L ) , ρ A,ε ( L )) ≈ .
153 2 π (cid:96) L + o (cid:16) (cid:96) L (cid:17) , which gives y T ≈ . y T ≈ .
153 in [50] nor the value y T ≈ .
154 is this paper is of high precision,mainly due to the small value of
L, (cid:96) . In the following we will use y T ≈ .
154 in the free massless Majorana fermiontheory, which is precise enough for us in the paper. n = c ( n − n . (2.27)We have normalized the Schatten two-distance so that it is scale invariant and does not depend on theUV cutoff. Short interval expansion of Schatten two-distance could be calculated from OPE of twistoperators [59, 65]. For the finite size and thermal states, including states on plane, cylinder and torus,we get D ( ρ A , σ A ) = 116 (cid:113) (cid:96) ( (cid:104) ε (cid:105) ρ − (cid:104) ε (cid:105) σ ) + 7 (cid:96) ( (cid:104) T (cid:105) ρ − (cid:104) T (cid:105) σ ) + O ( (cid:96) ) . (2.28)Note that (cid:104) T (cid:105) ρ = (cid:104) ¯ T (cid:105) σ and the contributions from the anti-holomorphic sectors have been included.As in the case of the R´enyi entropy, we do not need the explicit RDMs to calculate the Schattendistance in spin chain, and correlation matrices are enough. This allows us to calculate the Schattentwo-distance for a relatively large (cid:96) and compare the CFT and spin chain results in Fig. 5. ◦ ◦ ◦ ◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ L = L = L = L = L =
512 CFT ◦ SC ℓ = - - - L D ( ρ A ( L ) , ρ A ( L )) 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L = L = L = L = L =
512 CFT ◦ SC ℓ = × - × - β D ( ρ A ( L ) , ρ A ( β )) ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ β = β = β = β = β =
512 CFT ◦ SC ℓ = - - - - L D ( ρ A ( β ) , ρ A ( L , β )) 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β = β = β = β = β =
512 CFT ◦ SC L = ℓ = - - β D ( ρ A ( L , β ) , ρ A ( L , β )) Figure 5: Schatten two-distance of the RDMs in states on cylinder and torus in the free masslessMajorana fermion theory (solid line) and critical Ising chain (empty circle).
For two density matrices ρ, σ the relative entropy is defined as S ( ρ (cid:107) σ ) = tr( ρ log ρ ) − tr( ρ log σ ) . (2.29)The replica trick to calculate the subsystem relative entropy in 2D CFT was developed in [66, 67]. ForRDMs on cylinder, there are analytical CFT results [68] which are valid for an interval A = [0 , (cid:96) ] with11n arbitrary length S ( ρ A ( L ) (cid:107) ρ A ( L )) = c L sin π(cid:96)L L sin π(cid:96)L + c (cid:16) − L L (cid:17)(cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) ,S ( ρ A ( β ) (cid:107) ρ A ( β )) = c β sinh π(cid:96)β β sinh π(cid:96)β + c (cid:16) − β β (cid:17)(cid:16) − π(cid:96)β coth π(cid:96)β (cid:17) ,S ( ρ A ( L ) (cid:107) ρ A ( β )) = c β sinh π(cid:96)β L sin π(cid:96)L + c (cid:16) β L (cid:17)(cid:16) − π(cid:96)β coth π(cid:96)β (cid:17) ,S ( ρ A ( β ) (cid:107) ρ A ( L )) = c L sin π(cid:96)L β sinh π(cid:96)β + c (cid:16) L β (cid:17)(cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) . (2.30)For two Gaussian sates in spin chain, the subsystem relative entropy [69] can be written in termsof the correlation matrix Γ defined in (B.13) S ( ρ Γ (cid:107) ρ Γ ) = tr (cid:16) (cid:17) − tr (cid:16) (cid:17) , (2.31)This means the we just need to compute the correlation matrix Γ, rather than the explicit RDM ρ Γ , toobtain the relative entropy which allows us to check the CFT analytical results (2.30) for long intervals.We show some of them in the top two panels of Fig. 6. As the CFT results are exact, we see matchesof the CFT and spin chain results not only for short intervals with (cid:96) (cid:28) L , (cid:96) (cid:28) β but also for longintervals with (cid:96) ∼ L , (cid:96) ∼ β .For RDMs on torus we have to make short interval expansion of the relative entropy. The shortinterval expansion of subsystem relative entropy from OPE of twist operators was developed in [59]and we get the result for the RDMs on torus S ( ρ A (cid:107) σ A ) = (cid:96)
12 ( (cid:104) ε (cid:105) ρ − (cid:104) ε (cid:105) σ ) + 2 (cid:96)
15 ( (cid:104) T (cid:105) ρ − (cid:104) T (cid:105) σ ) + (cid:96)
15 ( (cid:104) ε (cid:105) ρ − (cid:104) ε (cid:105) σ ) (cid:2) (cid:104) T (cid:105) ρ ( (cid:104) ε (cid:105) ρ + (cid:104) ε (cid:105) σ ) − (cid:104) T (cid:105) σ (cid:104) ε (cid:105) σ (cid:3) + (cid:96)
120 ( (cid:104) ε (cid:105) ρ − (cid:104) ε (cid:105) σ ) (cid:0) (cid:104) ε (cid:105) ρ + 2 (cid:104) ε (cid:105) ρ (cid:104) ε (cid:105) σ + 3 (cid:104) ε (cid:105) σ (cid:1) + O ( (cid:96) ) . (2.32)In critical Ising chain the states with both finite size and finite temperature are not Gaussian, andwe cannot use the formula (2.31) to calculate the relative entropy in the spin chain. In that case weneed to construct explicitly the numerical RDMs and calculate the relative entropy from the definition(2.29). We compare the CFT and spin chain results in bottom two panels of Fig. 6. In this section we consider the XX chain with zero transverse field, and its continuum limit gives the2D free massless Dirac fermion theory, or equivalently the 2D free massless compact boson theory withunit target space radius, which is a 2D CFT with central charge c = 1. The calculations and resultsare parallel to those in critical Ising chain and 2D free massless Majorana fermion theory, and we willkeep it brief in this section. Subsystem relative entropy on torus could also be calculated from modular Hamiltonian [45], which we will considerin this paper. ◦◦ ◦◦ ◦◦ ◦◦ 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L = L = L = L = L =
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512 CFT ◦ SC ℓ = - - - - β S ( ρ A ( L ) || ρ A ( β )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ β = β = β = β = β =
128 CFT ◦ SC ℓ =
45 10 50 10010 - - - - - L S ( ρ A ( β ) || ρ A ( L , β )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ β = β = β = β = β =
128 CFT ◦ SC L = ℓ =
45 10 50 10010 - - - - - β S ( ρ A ( L , β ) || ρ A ( L , β )) Figure 6: Relative entropy of the RDMs in states on cylinder and torus in the free massless fermiontheory (solid line) and critical Ising chain (empty circle).
Details of 2D free massless Dirac fermion theory and compact boson theory could be found in [60, 61].In the NS sector of the 2D free massless Dirac fermion theory there are nonidentity primary operators J = i ψ ψ , ¯ J = i ¯ ψ ¯ ψ , K = J ¯ J , (3.1)with conformal weights (1 , , ,
1) respectively. In the R sector there are primary operators σ , σ with the same conformal weights ( , ). In the NS and R sectors, there are also other primaryoperators with larger conformal weights, which are irrelevant to our low order computations in thispaper.The exact R´enyi entropy for one interval A = [0 , (cid:96) ] on torus with spatial circumference L andtemporal period β is [32] S ( n ) A = n + 16 n log (cid:12)(cid:12)(cid:12) L(cid:15) θ ( x | τ ) θ (cid:48) (0 | τ ) (cid:12)(cid:12)(cid:12) − n − (cid:20) (cid:80) (cid:126)α,(cid:126)β (cid:12)(cid:12)(cid:12) Θ (cid:104) (cid:126)α(cid:126)β (cid:105) (0 | Ω) (cid:12)(cid:12)(cid:12) (cid:0) (cid:81) n − k =1 | A k | (cid:1)(cid:0) (cid:80) ν =2 | θ ν (0 | τ ) | (cid:1) n (cid:21) . (3.2)Again we have defined x = (cid:96)L , τ = i βL and the rest of the functions involved are in (2.2), (2.3). Onecould also see the R´enyi entropy of one interval on torus in 2D free massless compact boson theoryin [31].From OPE of twist operators we get the short interval expansion of the R´enyi entropy on torus S ( n ) A = n + 16 n log (cid:96)(cid:15) − ( n + 1) (cid:96) n (cid:104) T (cid:105) + O ( (cid:96) ) , (3.3)with the expectation value (cid:104) T (cid:105) = − π qL ∂ q log Z ( q ) , Z ( q ) = 12 η ( τ ) (cid:2) θ (0 | τ ) + θ (0 | τ ) + θ (0 | τ ) (cid:3) . (3.4)13 xact resultshort interval0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.000.020.040.060.080.10 β / L with L = ℓ = Δ S A ( ) exact resultshort interval0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.000.020.040.060.08 β / L with L = ℓ = Δ S A ( ) exact resultshort interval0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 - - - - - L / β with β = ℓ = Δ S A ( ) exact resultshort interval0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 - - - - - L / β with β = ℓ = Δ S A ( ) Figure 7: The comparison of the exact R´enyi entropy with the short interval expansion one in freemassless Dirac fermion theory. We use ∆ S ( n ) A = S ( n ) A − n +16 n log (cid:96)(cid:15) to eliminate the dependence on theUV cutoff.Note q = ¯ q = e − πβ/L . The contributions from ¯ T have also been included. Short interval expansion ofthe exact result (3.2) gives S ( n ) A = n + 16 n log (cid:96)(cid:15) + ( n + 1) (cid:96) nL (cid:16) θ (cid:48)(cid:48)(cid:48) (0 | τ ) θ (cid:48) (0 | τ ) − (cid:80) ν =2 θ ν (0 | τ ) θ (cid:48)(cid:48) ν (0 | τ ) (cid:80) ν =2 θ ν (0 | τ ) (cid:17) + O ( (cid:96) ) , (3.5)which is the same as the short interval expansion result from twist operators (3.3). This indicates thatthe method of short interval expansion from OPE of twist operators is valid to order (cid:96) , but as weshow in appendix A the method fails to give the correct R´enyi entropy at order (cid:96) . We compare theexact R´enyi entropy and short interval one in Fig. 7. We see that the short interval expansion R´enyientropy is a good approximation in the parameter regimes we consider. Taking n → S A = 13 log (cid:96)(cid:15) − (cid:96) (cid:104) T (cid:105) + O ( (cid:96) ) . (3.6)In XX chain with zero field, we construct numerically the RDMs of one interval in finite size andthermal states as detailed in appendix B. In XX chain with total number of sites L that is four times ofan integer there are two zero modes in the R sectors, and we need to use the trick in [22]. We computethe von Neumann entropy for a short interval from the explicit numerical RDM, and calculate theR´enyi entropy for a relatively long interval from the correlation matrices. We compare the CFT andspin chain results in Fig. 8. On the CFT side, we use the short interval expansion of the von Neumannentropy (3.6) and the exact R´enyi entropy (3.2). We see perfect matching between the CFT and spinchain results. We calculate the trace distance among the RDMs in states on plane and cylinder in 2D free masslessDirac fermion theory. The trace distance D ( ρ A ( ∅ ) , ρ A ( L )) can be written as (2.16) with the coefficient(2.17) that we cannot evaluate analytically in the CFT. By fitting of the numerical results in XX chainwith (cid:96) = 4, we obtain the trace distance D ( ρ A ( ∅ ) , ρ A ( L )) ≈ . (cid:96) L + o (cid:16) (cid:96) L (cid:17) , (3.7)14 Δ S CFT Δ S SC β / L with L = ℓ = Δ S CFT ( ) Δ S SC ( ) β / L with L = ℓ = Δ S CFT ( ) Δ S SC ( ) β / L with L = ℓ = Δ S CFT Δ S SC - - L / β with β = ℓ = Δ S CFT ( ) Δ S SC ( ) - - - - - - L / β with β = ℓ = Δ S CFT ( ) Δ S SC ( ) - - - - - L / β with β = ℓ = Figure 8: We compare the von Neumann and R´enyi entropies in free massless Dirac fermion theoryand those in XX chain with zero field.which gives the approximate coefficient y T ≈ . We will use this approximate value in the freemassless Dirac fermion theory. For the RDMs of one interval in states on cylinder we further get D ( ρ A ( L ) , ρ A ( L )) ≈ . (cid:96) (cid:12)(cid:12)(cid:12) L − L (cid:12)(cid:12)(cid:12) + o ( (cid:96) ) ,D ( ρ A ( β ) , ρ A ( β )) ≈ . (cid:96) (cid:12)(cid:12)(cid:12) β − β (cid:12)(cid:12)(cid:12) + o ( (cid:96) ) ,D ( ρ A ( L ) , ρ A ( β )) ≈ . (cid:96) (cid:16) L + 1 β (cid:17) + o ( (cid:96) ) . (3.8)These analytical CFT results and numerical spin chain results are compared in Fig. 9.For two states ρ, σ on torus, there are generally three quasiprimary operators at level two K , T ,¯ T that have different expectation values. Using the method in [49, 50], we cannot calculate the tracedistance among the RDMs on torus in free massless Dirac fermion theory. As there are no CFT resutlsto be compared with, we will not show the trace distance involving the RDMs in states with both finitesize and finite temperature in XX chain in this paper. In free massless Dirac fermion theory we get the short interval expansion of the Schatten two-distancefrom OPE of twist operators D ( ρ A , σ A ) = (cid:96) √ (cid:113) ( (cid:104) K (cid:105) ρ − (cid:104) K (cid:105) σ ) + 10( (cid:104) T (cid:105) ρ − (cid:104) T (cid:105) σ ) + O ( (cid:96) ) . (3.9) In 2D free massless Dirac fermion theory, the formula (2.16) also applies to the trace distance D ( ρ A ( L ) , ρ A,K ( L )),with ρ A,K ( L ) being the RDM of the energy eigenstate ρ K ( L ). In [50] it was obtained numerically D ( ρ A ( L ) , ρ A,K ( L )) ≈ .
166 2 √ π (cid:96) L + o (cid:16) (cid:96) L (cid:17) , which gives y T ≈ . y T ≈ .
166 in [50] nor the value y T ≈ .
164 is this paper is of high precision,due to the small value of (cid:96) . ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ CFT ◦ SC ℓ =
45 10 50 10010 - L D ( ρ A ( ∅ ) , ρ A ( L )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ L = L = L = L = L =
128 CFT ◦ SC ℓ =
45 10 50 10010 - - L D ( ρ A ( L ) , ρ A ( L )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ β = β = β = β = β =
128 CFT ◦ SC ℓ =
45 10 50 10010 - - β D ( ρ A ( β ) , ρ A ( β )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ L = L = L = L = L =
128 CFT ◦ SC ℓ =
45 10 50 1005. × - β D ( ρ A ( L ) , ρ A ( β )) Figure 9: Trace distance of the RDMs on cylinder in the free massless Dirac fermion theory (solid line)and XX chain with zero field (empty circle).Note that on torus with q = ¯ q = e π i τ = e − πβ/L we have the expectation of stress tensor (3.4) and theexpectation value (cid:104) K (cid:105) = 4 π L q∂ q log θ (0 | τ ) θ (0 | τ / . (3.10)The contributions from ¯ T have also been included. We compare the analytical results of Schattentwo-distance in free massless Dirac fermion theory and the numerical results in XX chain with zerofield in Fig. 10. The results of relative entropy of RDMs on cylinder (2.30) are universal and apply to any 2D CFT.For RDMs on torus, we get the short interval expansion of the relative entropy from OPE of twistoperators S ( ρ A (cid:107) σ A ) = (cid:96)
60 ( (cid:104) K (cid:105) ρ − (cid:104) K (cid:105) σ ) + (cid:96)
15 ( (cid:104) T (cid:105) ρ − (cid:104) T (cid:105) σ ) + O ( (cid:96) ) , (3.11)with the expectation values (3.4), (3.10). The contributions from the anti-holomorphic sector havebeen included. We compare the CFT and spin chain results in Fig. 11. In this paper, we have constructed the numerical RDM of an interval in finite and thermal states incritical XY chains, especially for the states with both a finite size and a finite temperature, focusing oncritical Ising chain and XX chain with zero transverse field. With the numerical RDM, we calculatedthe subsystem von Neumann entropy, R´enyi entropy, trace distance, Schatten two-distance, and relativeentropy, and compared the results with those in 2D free massless Majorana and Dirac fermion theories,16 ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ L = L = L = L = L =
512 CFT ◦ SC ℓ = - - L D ( ρ A ( L ) , ρ A ( L )) 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L = L = L = L = L =
512 CFT ◦ SC ℓ = × - β D ( ρ A ( L ) , ρ A ( β )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ β = β = β = β = β =
512 CFT ◦ SC ℓ = - - - - - L D ( ρ A ( β ) , ρ A ( L , β )) 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β = β = β = β = β =
512 CFT ◦ SC L = ℓ = - - - β D ( ρ A ( L , β ) , ρ A ( L , β )) Figure 10: Schatten two-distance of the RDMs on cylinder and torus in the free massless Dirac fermiontheory (solid line) and XX chain with zero field (empty circle). 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◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ L = L = L = L = L =
512 CFT ◦ SC ℓ = - - - L S ( ρ A ( L ) || ρ A ( L )) 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◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ L = L = L = L = L =
512 CFT ◦ SC ℓ = - - - - β S ( ρ A ( L ) || ρ A ( β )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ β = β = β = β = β =
128 CFT ◦ SC ℓ =
45 10 50 10010 - - - - - L S ( ρ A ( β ) || ρ A ( L , β )) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ β = β = β = β = β =
128 CFT ◦ SC L = ℓ =
45 10 50 10010 - - - β S ( ρ A ( L , β ) || ρ A ( L , β )) Figure 11: Relative entropy of the RDMs on cylinder and torus in the free massless Dirac fermiontheory (solid line) and XX chain with zero field (empty circle).17hich are respectively the continuum limits of the critical Ising chain and XX chain with zero field.We found perfect matches of the numerical spin chain and analytical CFT results in the continuumlimit.There are several interesting generalizations of the present results. In CFT, we only got shortinterval expansion of von Neumann entropy of a length (cid:96) interval to order (cid:96) , and it is interesting tocalculate higher order results. We cannot calculate subsystem trace distance for RDMs in states withboth finite size and finite temperature in CFT, and other methods to calculate the subsystem tracedistance are needed. The states with both finite length and finite temperature in XY spin chains arenot Gaussian, and we can only calculate the von Neumann entropy, trace distance and relative entropyfor a short interval. It would be interesting to calculate those quantities for a long interval in spinchains. Acknowledgements
We thank P. Calabrese and E. Tonni for helpful discussions, comments, and suggestions. We acknowl-edge support from ERC under Consolidator grant number 771536 (NEMO).
A Break down of twist operators at order (cid:96) In this appendix, we show that the method of OPE of twist operators cannot give the correct shortinterval R´enyi entropy on torus at order (cid:96) in some specific 2D CFTs, including the 2D free masslessMajorana and Dirac fermion theories.In a general 2D unitary CFT, we consider the nonidentity primary operators φ i , i = 1 , , · · · , g with the smallest scaling dimension ∆. There is a degeneracy g at scaling dimension ∆. Each primaryoperator φ i has conformal weights ( h i , ¯ h i ). Note that ∆ = h i + ¯ h i for all i . We require that 0 < ∆ < φ i is non-chiral, i.e. both h i (cid:54) = 0 and ¯ h i (cid:54) = 0. Apparently,2D free massless Majorana and Dirac fermion theories belong to such theories. For 2D free masslessMajorana fermion theory, the operator is σ with conformal weights ( , ), and there is no degeneracy∆ = , g = 1. For 2D free massless Dirac fermion theory, the operators are σ , σ with the sameconformal weights ( , ), and there is double degeneracy ∆ = , g = 2.We consider the R´enyi entropy of one interval A = [0 , (cid:96) ] in the 2D CFT on a torus with spatialcircumference L and temporal period β . In the low temperature limit L (cid:28) β , the density matrix ofthe whole system could be written as an expansion in the variable q = e − πβ/L ρ = | (cid:105)(cid:104) | + q ∆ (cid:80) gi =1 | φ i (cid:105)(cid:104) φ i | + o ( q ∆ )1 + gq ∆ + o ( q ∆ ) . (A.1)We have the ground state | (cid:105) and the orthonormal primary excited states | φ i (cid:105) that satisfy (cid:104) φ i | φ j (cid:105) = δ ij .There is an universal single interval R´enyi entropy [39] in this case that reads S ( n ) A = c ( n + 1)6 n log (cid:16) Lπ(cid:15) sin π(cid:96)L (cid:17) − ngq ∆ n − (cid:104) n (cid:16) sin π(cid:96)L sin π(cid:96)nL (cid:17) − (cid:105) + o ( q ∆ ) . (A.2)18o compare, we compute the same R´enyi entropy using OPE of twist operators. In general onehas [57] S ( n ) A = c ( n + 1)6 n log (cid:96)(cid:15) − n − (cid:110) (cid:96) b T ( (cid:104) T (cid:105) + (cid:104) ¯ T (cid:105) )+ (cid:96) [ b A ( (cid:104)A(cid:105) + (cid:104) ¯ A(cid:105) ) + b T T ( (cid:104) T (cid:105) + (cid:104) ¯ T (cid:105) ) + b T (cid:104) T (cid:105)(cid:104) ¯ T (cid:105) ] + O ( (cid:96) )+ (cid:88) ψ [ (cid:96) ψ b ψψ (cid:104) ψ (cid:105) + O ( (cid:96) ψ )] (cid:111) , (A.3)with the coefficients b T = n − n , b A = ( n − n , b T T = ( n − c ( n + 1)( n − + 2( n + 11)]1440 cn , (A.4)and the level four quasiprimary operator A = ( T T ) − ∂ T. (A.5)It is similar for the anti-holomorphic quasiprimary operators ¯ T , ¯ A . The sum ψ is over all the nonidentityprimary operators in the theory. The following argument show the coefficient b ψψ will be irrelevantat the order of the expansion we are interested. In state (A.1), we have the expectation value for anarbitrary operator O (cid:104)O(cid:105) = (cid:104)O(cid:105) + q ∆ g (cid:88) i =1 ( (cid:104)O(cid:105) φ i − (cid:104)O(cid:105) ) + o ( q ∆ ) . (A.6)with (cid:104)O(cid:105) being expectation value in the ground state and (cid:104)O(cid:105) φ i being expectation value in the primaryexcited state. On the torus in the low temperature limit, for a primary operator ψ there is a leadingorder expectation value (cid:104) ψ (cid:105) ∼ q ∆ . As we focus on the q ∆ part of the R´enyi entropy, we do not needto consider the contributions from nonidentity primary operators, i.e. the terms with ψ in (A.3).On the torus in low temperature limit q (cid:28)
1, using (A.6) and (cid:104) T (cid:105) φ i , (cid:104)A(cid:105) φ i in [56] we get theexpectation values (cid:104) T (cid:105) = π [ c − Hq ∆ + o ( q ∆ )]6 L , (cid:104)A(cid:105) = π [ c (5 c + 22) − q ∆ (( c + 2) H − H ) + o ( q ∆ )]180 L , (cid:104) ¯ T (cid:105) = π [ c −
24 ¯ Hq ∆ + o ( q ∆ )]6 L , (cid:104) ¯ A(cid:105) = π [ c (5 c + 22) − q ∆ (( c + 2) ¯ H −
12 ¯ H ) + o ( q ∆ )]180 L , (A.7)with definitions H = g (cid:88) i =1 h i , H = g (cid:88) i =1 h i , ¯ H = g (cid:88) i =1 ¯ h i , ¯ H = g (cid:88) i =1 ¯ h i . (A.8)We compare the low temperature expansion of the R´enyi entropy (A.2) with the short intervalexpansion result (A.3) and focus on the q ∆ part of the R´enyi entropy. At order (cid:96) , they are the samebut at order (cid:96) , there is a non-vanishing difference π ( n − n + 1) ( H + ¯ H − g ∆ ) (cid:96) q ∆ n L . (A.9)It is essential that 0 < ∆ < B Thermal RDM in XY chains
The spin- XY chain with transverse field has the Hamiltonian H XY = − L (cid:88) j =1 (cid:16) γ σ xj σ xj +1 + 1 − γ σ yj σ yj +1 + λ σ zj (cid:17) , (B.1)where L is the total number of sites in the spin chain. In this paper, we only consider the cases that L are four times of integers. We consider the periodic boundary conditions σ x,y,zL +1 = σ x,y,z for the Paulimatrices σ x,y,zj . When γ = λ = 1 it defines the critical Ising chain, and its continuum limit givesthe 2D free massless Majorana fermion theory. When γ = λ = 0 it defines the XX chain with zerotransverse field, and its continuum limit gives the 2D free massless Dirac theory, or equivalently 2Dfree massless compact boson theory with the target space being a unit radius circle. The Hamiltonianof the XY chain can be exactly diagonalized [70–72] and the numerical RDMs in the ground state andexcited energy eigenstates could be constructed following [12, 13, 15, 16, 63, 64, 73, 74]. The constructionof RDM in thermal state on an infinite line can be found in [75]. In this appendix, we elaborate how toconstruct the numerical RDMs of one interval in a state with both finite size and finite temperature.In the construction, the trick in [22] will be extremely useful to us.The XY chain Hamiltonian can be exactly diagonalized by successively applying the Jordan-Wignertransformation, Fourier transforming, and Bogoliubov transformation. The Jordan-Wigner transfor-mation is a j = (cid:16) j − (cid:89) i =1 σ zi (cid:17) σ + j , a † j = (cid:16) j − (cid:89) i =1 σ zi (cid:17) σ − j , (B.2)with σ ± j = ( σ xj ± i σ yj ). In the NS sector there are antiperiodic boundary conditions a L +1 = − a , a † L +1 = − a † , and in the R sector there are periodic boundary conditions a L +1 = a , a † L +1 = a † . TheFourier transformation is b k = 1 √ L L (cid:88) j =1 e i jϕ k a j , b † k = 1 √ L L (cid:88) j =1 e − i jϕ k a † j , (B.3)with ϕ k = πkL . In this paper, we only consider the cases that L are four times of integers. Themomenta k ’s are half integers in the NS sector k = 1 − L , · · · , − , , · · · , L − , (B.4)and integers in the R sector k = 1 − L , · · · , − , , , · · · , L . (B.5)20he Bogoliubov transformation is c k = b k cos θ k b †− k sin θ k , c † k = b † k cos θ k − i b − k sin θ k . (B.6)For critical Ising chain, we choose the angle θ k = − π − πkL k < k = 0 π − πkL k > . (B.7)For XX chain, the Bogoliubov transformation is not needed, and, in other words, there is always θ k = 0.Finally, the Hamiltonian becomes H = 1 + P H NS + 1 − P H R , H NS = (cid:88) k ∈ NS ε k (cid:16) c † k c k − (cid:17) , H R = (cid:88) k ∈ R ε k (cid:16) c † k c k − (cid:17) . (B.8)In critical Ising chain we have ε k = 2 sin π | k | L , (B.9)and in XX chain with zero transverse field we have ε k = − cos 2 πkL . (B.10)The projection operator is P = L (cid:89) j =1 σ zj = e π i (cid:80) Lj =1 a † j a j . (B.11)One can define the Majorana modes as d j − = a j + a † j , d j = i( a j − a † j ) . (B.12)For an interval with (cid:96) sites on the spin chain in a Gaussian state ρ , one defines the 2 (cid:96) × (cid:96) correlationmatrix (cid:104) d m d m (cid:105) ρ = δ m m + Γ m m , m , m = 1 , , · · · , (cid:96). (B.13)The 2 (cid:96) × (cid:96) RDM in state ρ is [12, 13] ρ A = 12 (cid:96) (cid:88) s , ··· ,s (cid:96) ∈{ , } (cid:104) d s (cid:96) (cid:96) · · · d s (cid:105) ρ d s · · · d s (cid:96) (cid:96) , (B.14)and the multi-point correlation functions (cid:104) d s (cid:96) (cid:96) · · · d s (cid:105) ρ are calculated from the correlation matrix (B.13)by Wick contractions.For the ground state on an infinite chain ρ ( ∅ ), the ground state on a length L circular chain ρ ( L ),and a thermal state with inverse temperature β on an infinite chain ρ ( β ), the nonvanishing componentsof the correlation matrix Γ could be written in terms of the function g j that is defined asΓ j − , j = − Γ j , j − = g j − j . (B.15)21n the critical Ising chain, we have in different states g j ( ∅ ) = − i π j + ,g j ( L ) = − i L π ( j + ) L ,g j ( β ) = − i π j + + 2i π (cid:90) π d ϕ sin[( j + ) ϕ ]1 + exp(2 β sin ϕ ) . (B.16)In the XX chain with zero field we obtain g j ( ∅ ) = 2i πj sin πj , g ( ∅ ) = 0 ,g j ( L ) = 2i L sin πj sin πjL , g ( L ) = 0 ,g j ( β ) = 2i πj sin πj − − ( − ) j ] π (cid:90) π d ϕ cos( jϕ )1 + exp( β cos ϕ ) , g ( β ) = 0 . (B.17)For a state with both finite size and finite temperature ρ ( L, β ), it is more complicated to constructthe numerical RDM ρ A ( L, β ). Depending on the number of zero modes, i.e. modes with zero energy,we consider three different cases in the following subsections. In gapped XY chain, there is no zeromode, in critical Ising chain and XX chain with zero field there are respectively one and two zeromodes.
B.1 Gapped XY chain
There is no zero mode in gapped XY chain. The normalized density matrix of the whole system inthermal state is ρ = e − βH tre − βH = e − βH NS + P e − βH NS + e − βH R − P e − βH R Z +NS + Z − NS + Z +R − Z − R , (B.18)with Z +NS = (cid:89) k ∈ NS (cid:16) βε k (cid:17) , Z − NS = (cid:89) k ∈ NS (cid:16) βε k (cid:17) ,Z +R = (cid:89) k ∈ R (cid:16) βε k (cid:17) , Z − R = (cid:89) k ∈ R (cid:16) βε k (cid:17) . (B.19)We rewrite the thermal density matrix as ρ = 1 Z +NS + Z − NS + Z +R − Z − R (cid:16) Z +NS ρ +NS + Z − NS ρ − NS + Z +R ρ +R − Z − R ρ − R (cid:17) ,ρ +NS = e − βH NS Z +NS , ρ − NS = P e − βH NS Z − NS , ρ +R = e − βH R Z +R , ρ − R = P e − βH R Z − R . (B.20)Note that all the four density matrices ρ +NS , ρ − NS , ρ +R , ρ − R are Gaussian and properly normalized, andso we can construct their RDMs ρ + A, NS , ρ − A, NS , ρ + A, R , ρ − A, R from the corresponding correlation matrices.Then we get the RDM of the thermal density matrix ρ A = 1 Z +NS + Z − NS + Z +R − Z − R (cid:16) Z +NS ρ + A, NS + Z − NS ρ − A, NS + Z +R ρ + A, R − Z − R ρ − A, R (cid:17) . (B.21)22or ρ + A, NS , ρ − A, NS , ρ + A, R , ρ − A, R , we have the correlation matrix with nonvanishing components (B.15) and g j = − i L (cid:88) k ∈ NS e i( jϕ k − θ k ) tanh βε k ,g j = − i L (cid:88) k ∈ NS e i( jϕ k − θ k ) coth βε k ,g j = − i L (cid:88) k ∈ R e i( jϕ k − θ k ) tanh βε k ,g j = − i L (cid:88) k ∈ R e i( jϕ k − θ k ) coth βε k . (B.22) B.2 Critical Ising chain
There is one zero mode in the R sector, i.e. ε = 0, which needs a careful treatment. We write thethermal density matrix as ρ = 1 Z +NS + Z − NS + Z +R (cid:16) Z +NS ρ +NS + Z − NS ρ − NS + Z +R ρ +R − Z − R σ z L ˜ ρ − R (cid:17) ,ρ +NS = e − βH NS Z +NS , ρ − NS = P e − βH NS Z − NS , ρ +R = e − βH R Z +R , ˜ ρ − R = σ z P e − βH R Z − R /L . (B.23)We have defined ˜ Z − R = (cid:89) k ∈ R ,k (cid:54) =0 (cid:16) βε k (cid:17) . (B.24)Note that the zero mode makes Z − R = 0. We have also defined ˜ ρ − R following Appendix D of [22]. TheRDM of the thermal density matrix is ρ A = 1 Z +NS + Z − NS + Z +R (cid:16) Z +NS ρ + A, NS + Z − NS ρ − A, NS + Z +R ρ + A, R − Z − R σ z L ˜ ρ − A, R (cid:17) . (B.25)All the RDMs ρ + A, NS , ρ − A, NS , ρ + A, R , ˜ ρ − A, R are Gaussian. The RDMs ρ + A, NS , ρ − A, NS , ρ + A, R can be constructedin the same way as that in the previous subsection. For ˜ ρ − A, R , we have the correlation matrix withcomponents Γ j − , j − = − Γ j − , j − = Γ j , j = − Γ j , j = f j j , Γ j − , j = − Γ j , j − = g j j , (B.26)and definitions f j j = − δ j + δ j , g j j = ˜ g j − j + ˜ g − ˜ g j − − ˜ g − j , ˜ g j = − i L (cid:88) k ∈ R ,k (cid:54) =0 e i( jϕ k − θ k ) coth βε k . (B.27)To confirm that the above trick works we compare the RDM in the gapped XY chain with γ = 1and λ →
1, i.e. gapped Ising chain with λ →
1, which we denote by ρ A ( λ ), with the RDM in criticalIsing chain, which we denote by ρ A (1). We plot the trace distance of ρ A ( λ ) and ρ A (1) in Fig. 12. We23ee that as λ → D ( ρ A ( λ ) , ρ A (1)) ∝ | λ − | (B.28)This indicates that the thermal RDM in critical Ising chain we have constructed is correct. △△△△△△△△△△△△ ▽▽▽▽▽▽▽▽▽▽▽▽ △ L = β = ℓ = ▽ L = β = ℓ = - - - - - - - - - - - λ D ( ρ A ( λ ) , ρ A ( )) △△△△△△△△△△△△ ▽▽▽▽▽▽▽▽▽▽▽▽ △ L = β = ℓ = ▽ L = β = ℓ = - - - - - - - - - - λ - D ( ρ A ( λ ) , ρ A ( )) Figure 12: Trace distance of the thermal RDM in gapped Ising chain ρ A ( λ ) and the thermal RDM incritical Ising chain ρ A (1). B.3 XX chain with zero field
There are two zero modes in the R sector, i.e. ε ± L/ = 0. Remember that in this paper we onlyconsider the cases that L are four times of integers. We write the thermal density matrix as ρ = 1 Z +NS + Z − NS + Z +R (cid:16) Z +NS ρ +NS + Z − NS ρ − NS + Z +R ρ +R −
16 ˜ Z − R σ z σ z L ˜ ρ − R (cid:17) ,ρ +NS = e − βH NS Z +NS , ρ − NS = P e − βH NS Z − NS , ρ +R = e − βH R Z +R , ˜ ρ − R = σ z σ z P e − βH R
16 ˜ Z − R /L , (B.29)with the new definition ˜ Z − R = (cid:89) k ∈ R ,k (cid:54) = ± L/ (cid:16) βε k (cid:17) . (B.30)The RDM of the thermal density matrix is ρ A = 1 Z +NS + Z − NS + Z +R (cid:16) Z +NS ρ + A, NS + Z − NS ρ − A, NS + Z +R ρ + A, R −
16 ˜ Z − R σ z σ z L ˜ ρ − A, R (cid:17) . (B.31)All the RDMs ρ + A, NS , ρ − A, NS , ρ + A, R , ˜ ρ − A, R are Gaussian. The RDMs ρ + A, NS , ρ − A, NS , ρ + A, R can be constructedthe same as before. We get ˜ ρ − A, R from the correlation functions (cid:104) d l − d l − (cid:105) = (cid:104) d l − d l − (cid:105) = (cid:104) d l − d l − (cid:105) = (cid:104) d l d l (cid:105) = δ j j + ( − ) l δ l − ( − ) l δ l , (cid:104) d l − d l − (cid:105) = (cid:104) d l − d l (cid:105) = 0 , (cid:104) d l − d l − (cid:105) = (cid:104) d l − d l (cid:105) = ˜ g l − l ) + ( − ) l ˜ g − l ) + ( − ) l ˜ g l − + ( − ) l + l ˜ g , (cid:104) d l − d l (cid:105) = ˜ g l − l )+1 + ( − ) l ˜ g − l + ( − ) l ˜ g l − + ( − ) l + l ˜ g , (cid:104) d l − d l − (cid:105) = ˜ g l − l ) − + ( − ) l ˜ g − l + ( − ) l ˜ g l − + ( − ) l + l ˜ g − , (B.32)24ith the definition of the function˜ g j = − i L (cid:88) k ∈ R ,k (cid:54) = ± L/ e i jϕ k coth βε k . (B.33)Note that (cid:104) d m d m (cid:105) = δ m m − (cid:104) d m d m (cid:105) .To confirm that the numerical RDM in XX chain with zero field is correct we compare it with theRDM in the gagged XY chain with λ = 0 and γ →
0, which we denote by ρ A ( γ ). We denote the RDMof the XX chain with no field as ρ A (0). We plot the trace distance of ρ A ( γ ) and ρ A (0) in Fig. 13. Wesee that as γ → D ( ρ A ( γ ) , ρ A (0)) ∝ | γ | . (B.34) △△△△△△△△△△ ▽▽▽▽▽▽▽▽▽▽ △ L = β = ℓ = ▽ L = β = ℓ = - - - - - - - γ D ( ρ A ( γ ) , ρ A ( )) △△△△△△△△△△ ▽▽▽▽▽▽▽▽▽▽ △ L = β = ℓ = ▽ L = β = ℓ = - - - - - - - - γ D ( ρ A ( γ ) , ρ A ( )) Figure 13: Trace distance of the thermal RDM in gapped XY chain ρ A ( γ ) and the thermal RDM inXX chain with no field ρ A (0). C Relative entropy among RDMs in low-lying energy eigenstates
We revisit the relative entropy among the RDMs of one interval A = [0 , (cid:96) ] on cylinder in various low-lying energy eigenstates, generalizing [50, 58, 76]. With the formula in [69], i.e. (2.31), we can calculatethe relative entropy of an interval with a relatively large length. This checks various results of theexact relative entropy, not only the leading order results in a short interval expansion. C.1 Free massless Majorana fermion theory
In CFT, we denote ρ A, O = tr ¯ A |O(cid:105)(cid:104)O| as the RDM of the excited state |O(cid:105) on cylinder. In free masslessMajorana fermion theory we consider primary operators 1 , σ, µ, ψ, ¯ ψ, ε with conformal weights (0,0),(1/16,1/16), (1/16,1/16), (1/2,0), (0,1/2), (1/2,1/2), respectively. There are exact results [50, 58, 76] S ( ρ A, (cid:107) ρ A,σ ) = S ( ρ A,σ (cid:107) ρ A, ) = S ( ρ A, (cid:107) ρ A,µ ) = S ( ρ A,µ (cid:107) ρ A, ) = 14 (cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) ,S ( ρ A,σ (cid:107) ρ A,µ ) = S ( ρ A,µ (cid:107) ρ A,σ ) = 1 − π(cid:96)L cot π(cid:96)L ,S ( ρ A,ψ (cid:107) ρ A, ) = S ( ρ A, ¯ ψ (cid:107) ρ A, ) = S ( ρ A,ε (cid:107) ρ A,ψ ) = S ( ρ A,ε (cid:107) ρ A, ¯ ψ ) = 1 − π(cid:96)L cot π(cid:96)L + sin π(cid:96)L + log (cid:16) π(cid:96)L (cid:17) + ψ (cid:16)
12 csc π(cid:96)L (cid:17) , ( ρ A,ε (cid:107) ρ A, ) = 2 (cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) + 2 (cid:104) sin π(cid:96)L + log (cid:16) π(cid:96)L (cid:17) + ψ (cid:16)
12 csc π(cid:96)L (cid:17)(cid:105) , (C.1) S ( ρ A,ψ (cid:107) ρ A,σ ) = S ( ρ A,ψ (cid:107) ρ A,µ ) = S ( ρ A, ¯ ψ (cid:107) ρ A,σ ) = S ( ρ A, ¯ ψ (cid:107) ρ A,µ ) = 54 (cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) + sin π(cid:96)L + log (cid:16) π(cid:96)L (cid:17) + ψ (cid:16)
12 csc π(cid:96)L (cid:17) ,S ( ρ A,ε (cid:107) ρ A,σ ) = S ( ρ A,ε (cid:107) ρ A,µ ) = 94 (cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) + 2 (cid:104) sin π(cid:96)L + log (cid:16) π(cid:96)L (cid:17) + ψ (cid:16)
12 csc π(cid:96)L (cid:17)(cid:105) . We compare some of the analytical CFT results with the numerical spin chain results in Fig. 14.Generally, we see good matches not only for a short interval, but also for a long interval. Especially,the relative entropies S ( ρ A,ε (cid:107) ρ A,σ ), S ( ρ A,ψ (cid:107) ρ A,µ ), S ( ρ A, (cid:107) ρ A,σ ) have the same leading order shortinterval expansion results, but they are different for a long interval, as we can see in both the CFTand spin chain results in the figure. In some cases there are mismatches as (cid:96)/L →
1, and we attributethem to numerical errors in the spin chain calculations. Actually, in the limit (cid:96)/L → ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ σ - με - σψ - μ - σε - ψ - - ℓ / L S ( ρ A || σ A ) ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ Figure 14: Relative entropy of the RDMs in low-lying energy eigenstates in the 2D free masslessMajorana fermion theory (solid line) and critical Ising chain (small empty circle). We have set L = 128. C.2 Free massless Dirac fermion theory
For 2D free massless Dirac fermion theory, it is convenient to use the language of 2D free masslesscompact boson theory. We consider of the RDMs in the excited states by the primary operators 1, V α, ¯ α , J , ¯ J , K = J ¯ J with conformal weights (0,0), ( α / , ¯ α / S ( ρ A,V α, ¯ α (cid:107) ρ A,V α (cid:48) , ¯ α (cid:48) ) = [( α − α (cid:48) ) + ( ¯ α − ¯ α (cid:48) ) ] (cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) ,S ( ρ A,J (cid:107) ρ A,V α, ¯ α ) = S ( ρ A, ¯ J (cid:107) ρ A,V α, ¯ α ) = (2 + α + ¯ α ) (cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) + 2 (cid:104) sin π(cid:96)L + log (cid:16) π(cid:96)L (cid:17) + ψ (cid:16)
12 csc π(cid:96)L (cid:17)(cid:105) , (C.2) S ( ρ A,K (cid:107) ρ A,V α, ¯ α ) = (4 + α + ¯ α ) (cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) + 4 (cid:104) sin π(cid:96)L + log (cid:16) π(cid:96)L (cid:17) + ψ (cid:16)
12 csc π(cid:96)L (cid:17)(cid:105) ,S ( ρ A,K (cid:107) ρ A,J ) = S ( ρ A,K (cid:107) ρ A, ¯ J ) = 2 (cid:16) − π(cid:96)L cot π(cid:96)L (cid:17) + 2 (cid:104) sin π(cid:96)L + log (cid:16) π(cid:96)L (cid:17) + ψ (cid:16)
12 csc π(cid:96)L (cid:17)(cid:105) .
26e compare the some of the analytical CFT results with the numerical CFT results in Fig. 15. ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ V - - V - - K - V - J - V - - V - - V K - K - J - ℓ / L S ( ρ A || σ A ) ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ Figure 15: Relative entropy of the RDMs in low-lying energy eigenstates in the 2D free massless Diracfermion theory (solid line) and XX chain with zero field (small empty circle). We have set L = 128. References [1] L. Amico, R. Fazio, A. Osterloh and V. Vedral,
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