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In the range correlation, and the correlation entropy. In Sec. information theory, the entropy is used to quantify the IV, we study the properties of the correlation entropy physical resource (in unit of classical bit due to log2 in its in thermal phase transition, as illustrated by classical expression) needed to store information. For example, in two-dimensional Ising model. In Sec. V, we address the the exact diagonalization approach, if we want to diago- properties of the correlation entropy in a simple quantum nalize a Hamiltonian of 10-site spin-1 chain and no sym- phase transition of one-dimensional transverse field Ising metry can be used to reduce the dimension of the Hilbert model. In Sec. VI, some discussions and prospects are 10 space, we need at least S = log2 3 = 10 log2 3 bits to presented. Finally, a brief summary is given in Sec. VII store a basis. Therefore, the correlation entropy actu- ally measures the additional physical resource required if we store two subsystems respectively rather than store II. TOY MODEL: HEISENBERG DIMER them together. As a simple example, let us consider a two-qubit system in a singlet state ( )/√2, | ↑↓i−| ↓↑i For the pedagogical purpose and making our motiva- we have S(A) = 1,S(B) = 1,S(AB) = 0, which leads tion more clear, we first have a look on a very simple to S(A: B) = 2. Obviously, there is no information in model: a Heisenberg dimer. Its Hamiltonian reads a given singlet state. However, each spin in this state is completely uncertain. So we need two bits to store H = σ1 σ2 (2) them respectively. On the other hand, the correlation · x y z entropy is simply the twice of the entanglement, as mea- where σi(σ , σ , σ ) are Pauli matrices at site i, sured by partial entropy, between two systems because of S(A) = S(B) for pure state. The reason that we inter- 0 1 0 i 1 0 σx = , σy = , σz = (3) pret the correlation entropy in this way is that besides 1 0 i −0 0 1 the quantum correlation the state also has classical cor- − relation. From this point of view, the correlation entropy and the coupling between two sites is set to unit for sim- is just a measure of total correlation, including quantum plicity. The Hamiltonian can be diagonalized easily. Its and classical correlation, between two subsystems. They ground state is a spin singlet state go halves with each other in correlation entropy for pure 1 state. For a mixed state, the correlation entropy also Ψ0 = [ ] , (4) measures the amount of the uncertainty of one system √2 | ↑↓i−| ↓↑i before we learn one from another. From the above inter- with eigenvalue E = 3, while three degenerate excited pretation for pure state, it is then not surprising that the 0 − correlation entropy fails to measure the entanglement [6]. states are In the statistical physics, the critical phenomena is the 1 Ψ1 = [ + ] , central topic. To have a complete understanding on the √2 | ↑↓i | ↓↑i critical behavior, various methods, such as renormaliza- Ψ2 = , Ψ3 = . (5) tion group [7], Monte-Carlo simulation [8], and mean- | ↑↑i | ↓↓i
field approach etc., have been developed and applied to with higher eigenvalue E1,2,3 = 1. Therefore, accord- many kinds of systems. In recent years, the study on the ing to the statistical physics, the thermal entropy of the role of entanglement in the quantum critical behavior [9] system vanishes at zero temperature. When the system have established a bridge between quantum information is contacted with a thermal bath with temperature T , theory and condensed matter physics, and shed new light the thermal state of the system is described by a density on the quantum phase transitions due to its interesting matrix: behavior around the critical point [10, 11, 12, 13, 14, 15]. However, the entanglement is fragile under the thermal e−2/T 0 0 0 fluctuation and can be suppressed to zero at finite tem- 0 cosh(2/T ) sinh(2/T ) 0 ρ = ζ (6) peratures. Then it is difficult to witness a generalized 0 sinh(2/T )− cosh(2/T ) 0 thermal phase transition in terms of quantum entangle- 0 0 0 e−2/T ment. In this paper, we are going to study the role of correla- where tion entropy in both thermal and quantum phase transi- 1 ζ = . (7) tions. Like the fundamental role of two point correlation 3e−2/T + e2/T function in the statistical physics, we are interested in the universal role of two-point correlation entropy in the Because of SU(2) symmetry of the Heisenberg dimer, the present work. The paper is organized as follows. In Sec. single-site entropy S(i) of the system is always unity, and II, for the pedagogical purpose, we study a toy model and the entropy of the whole system is show that the thermal entropy is not an extensive pa- rameter in such a simple system. In Sec. III, we discuss log (3e−2/T + 1) 3 log (3 + e4/T ) S(12) = 2 2 2 . (8) the relation between the reduced density matrix, long- − 3e−2/T +1 − 3+ e4/T 3
Then we can see that S(12) = S(1) + S(2) both at zero then Eq. (12) can be simplified as temperature and finite temperatures.6 In high temper- ature limit, the asymptotic behavior of the correlation u+ 0 0 0 entropy is S(1 : 2) 1/T . So only when T , 0 w1 z 0 ∝ → ∞ ρij = ∗ (13) S(1 : 2) 0, the extensive property of the entropy holds. 0 z w2 0 → − The physics behind this fact is quite clear. It is the inter- 0 0 0 u action between two sites that establishes a kind of cor- in the basis of σzσz: , , , . Here the relation and then breaks the extensive property of the i j {| ↑↑i | ↑↓i | ↓↑i | ↓↓i} entropy. matrix elements can be calculated from the correlation For a large system, however, this correlation usually function, decays with the increasing of distance between two parts, + − 1 z z and the entropy becomes an extensive quantity beyond u = u = (1 + σi σj ), a definite scale. Only around the critical point where 4 h i 1 z z the phase transition happens, the system behaves like a w1 = w2 = (1 σi σj ), whole and can not be divided into two part, and then the 4 − h i 1 correlation entropy has long range behaviors. z = ( σxσx + σyσy ). (14) 4 h i j i h i j i With the help of the Jordan-Schwinger mapping [16], III. REDUCED DENSITY MATRIX, LONG-RANGE ORDER, AND CORRELATION + † σ = a aj↓, ENTROPY j j↑ − † σj = aj↓aj↑, In many-body physics, the reduced density matrix of z † † σj = (aj↑aj↑ aj↓aj↓) (15) a one-body and two-body subsystem can be, in general, − written as † where ajα stands for the pseudo fermionic creation op- ′ † erator for single particle state j, α at the position j, α ρi α = tr(aiα′ ρa ), | i h | | i iα the element in the reduced density matrix (12) can be ′ ′ † † α β ρ αβ = tr(a ′ a ′ ρa a ), (9) reexpressed in the form of Eq. (9). For example, h | ij | i iα jβ jβ iα + − † † respectively. Here aiα,ajβ are annihilation operators for σi σj = ai↑aj↓ai↓aj↑ . (16) states α , β localized at site i, j respectively, and satisfy h i −h i commutation| i | i (anti-commutation) relation for bosonic Therefore, we can explore the property of long-range cor- (fermionic) states. The reduced density matrix is usu- relation in spin system through pseudo fermion systems. ally normalized as We first consider the long-range correlation in classical systems, e.g. Ising model, in which the reduced density tr(ρi)=1, tr(ρij )=1, (10) matrix takes the diagonal form, i.e.
′ † so that one has a probability explanation for their diag- α ρi α = δα′αtr(aiα′ ρa ), h | | i iα † † onal elements in the corresponding eigenstate space. ′ ′ ′ ′ ′ ′ α β ρij αβ = δα αδβ βtr(aiα ajβ ρajβ aiα). (17) In spin system, the reduced density matrix of a single h | | i spin at position i takes the form Then, if there is no long-range correlation
† † 1 y y ρ = 1+ σx σx + σ σ + σz σz . (11) αβ ρij αβ = tr(ajβ aiαaiαajβ ρ), i 2 h i i i h i i i h i i i h | | i = a† a† a a , h jβ iα iα jβ i For two arbitrary spins at position i and j, the two-site a† a a† a , (18) reduced density matrix generally takes the form, → h iα iαih jβ jβ i 1 1 for i j , the two-site entropy becomes ρ = + σα σα + σα σα | − | → ∞ ij 4 4 h i i i h j i j α S(ij) S(i)+ S(j), (19) X 1 → + σασβ σασβ. (12) 4 h i j i i j where Xαβ S(l)= a† a log a† a , l = i, j (20) − h lβ lβi 2h lβ lβi Obviously, under some symmetry, the above reduced den- β sity matrix can be simplified. For example, if the state of X N spins is also an eigenstate of z component of total spins where the normalization conditions of the ρi(j) and ρij z z S = Si = 0 and possesses the exchange symmetry, have been used. Obviously, we have S(i : j) = 0, which P 4 means that if there is no long-range correlation, the cor- Moreover, Pα′β′,αβ will become unit matrix if and only if relation entropy vanishes at long distance. φαβ ϕαϕβ = 1. Then On the other hand, if there exists long-range correla- h | i tion, for example S(i : j) (30)
Giα, jβ niαnjβ niα njβ = C, (21) ′ ′ ′ ′ ≡ h i − h ih i = qαβ log2 qαβ Pα β ,αβ log2(pα pβ ) . − ′ ′ for i j and a constant C, then Xαβ αXβ | − | → ∞ S(i : j) > 0. (22) Taking into account the concavity property of log func- tion, i.e. Now we study the correlation entropy in the quantum systems, in which the reduced density matrix usually is ′ ′ ′ ′ ′ ′ not diagonal. Then, if the system does not have long- Pα β ,αβ log2(pα pβ ) log2 Pα β ,αβpαpβ , ′ ′ ≤ ′ ′ range correlation, αXβ αXβ ′ ′ † † α β ρ αβ = a a ′ a a ′ , (23) we can have h | ij | i h iα iα ih jβ jβ i S(i : j) q log Q for i j , the reduced density matrix can be writ- ≥ αβ 2 αβ | − | → ∞ αβ ten into a direct product form, i.e. X 1 qαβ (1 Qαβ)=0, (31) ρij = ρi ρj . (24) ≥ ln 2 − ⊗ αβ X Then the reduced density matrices ρ ,ρ can be diago- i j where nalized in their own subspace. So in principle, we can have ρ = p ϕ ϕ , and ρ = p ϕ ϕ , qαβ i α α α ii α j β β β jj β Qαβ = , (32) | i h | | i h | ′ ′ where pα,pβ are the probability distribution for ρi and α′β′ Pα β ,αβpαpβ P P ρj respectively. As we have done for the classical system, we then have The above inequalityP becomes an equality if and only if Pα′β′,αβ is an unit matrix. Therefore, if S(i : j)= S(ρ )+ S(ρ ) S(ρ )=0. (25) i j ij † † − ′ ′ ′ ′ α β ρij αβ = aiαaiα ajβ ajβ + C, (33) In order to study its relation to the long-range corre- h | | i h ih i lation, now we express the correlation entropy in term of for i j , P is not diagonal, then we have the de- | − | → ∞ the relative entropy [17] sired result S(i : j) = tr(ρ log ρ ) tr(ρ log ρ ρ ) (26) S(i : j) > 0. (34) ij 2 ij − ij 2 i ⊗ j between the whole system and direct product form of two In the information theory, the inequality similar to the † † subsystems where (ρ ρ ) ′ ′ = a a ′ a a ′ . above result is called Klein inequality [18]. Therefore the i ⊗ j αβ,α β h iα iα ih jβ jβ i For Hermition operators ρij , the eigen-equation existence of the long-range correlation will lead to a posi- ρij φαβ = qαβ φαβ gives a representation of the cor- tive correlation entropy (34). This observation is very im- relation| i entropy| in thei basis φ . Therefore, in the portant in understanding critical phenomena. According {| αβi} eigenstate space of φαβ , we have to the theory of either thermal or quantum phase transi- | i tion, the presence of the long-range correlation is crucial.
S(i : j)= qαβ log2 qαβ However, different phase transition depends on the differ- αβ ent long-range correlation. For example, in the superfluid X phase of 4He, the off-diagonal-long-range order, as sug- φ ρ log (ρ ρ ) φ . (27) − h αβ| ij 2 i ⊗ j | αβ i gested by Yang [19], is necessary; while in anther kind Xαβ of condensate of exciton, it may require diagonal-long- range order [20]. Then the above results show that the Insert the identity ϕαϕβ ϕβ ϕα = 1, it becomes αβ | ih | non-vanishing positive defined correlation is a universal
P ′ ′ and necessary condition for all critical phenomena [21]. S(i : j)= qαβ log2 qαβ Pα β ,αβqαβ log2(pαpβ) − ′ ′ Xαβ αβαXβ where IV. THERMAL PHASE TRANSITION: TWO-DIMENSIONAL ISING MODEL P ′ ′ = φ ϕ ′ ϕ ′ ϕ ′ ϕ ′ φ (28) α β ,αβ h αβ | α β ih α β | αβ i and satisfies The physics in the above toy model is quite limited. In order to verify our analytical result and see the signif- Pα′β′,αβ > 0, Pα′β′,αβ =1, Pα′β′,αβ =1. (29) icance of the correlation entropy in the critical phenom- ′ ′ ena, let us first study its properties in a thermodynamical αXβ Xαβ 5 system. One of typical examples is the two-dimensional Ising model, which is certainly the most thoroughly re- searched model in statistical physics [22, 23]. In the absence of external field, the model Hamiltonian 0.35 defined on a square lattice field reads 0.3
z z 0.25 H = σi σj , (35) 0.2 − S hiji X 0.15 where the sum is over all pairs of nearest-neighbor sites i 0.1 and j, and the coupling is set to unit for simplicity. Since 0.05 the Ising model is a classical model, the reduced density 0 0 matrix of two arbitrary sites then takes the form 10 20 2.28 2.26 + r 30 u 0 0 0 2.24T 40 2.22 0 w1 0 0 50 2.2 ρij = (36) 0 0 w2 0 − 0 0 0 u FIG. 1: (color online) The correlation entropy as a function of temperature T (in unit of Ising coupling) and the distance in which the elements can be calculated from Eq. (14), r (in unit of √2 lattice constant). and single-site reduced density matrix
z 0.07 1 1+ σi 0 ρi = h i z , (37) 2 0 1 σ − h i i 0.06 For simplicity, we only consider the correlation entropy numerical results 0.05 2 1/2 along (1, 1) direction, because the long distance behavior S=A / 2N ln2 of the correlation entropy should be independent of the 0.04 direction. According to the exact solution of the two-dimensional S 0.03 Ising model [23], the magnetization per site of the system is 0.02 −4 1/8 z 1 sinh (2/T ) T
2000 0 Therefore, in the critical region below Tc, the dominant S(0, 0; 1, 1) z z z S(0, 0; 2, 2) term in ∂S(i, j)/∂T is 2 σi σj σi ∂ σi /∂T, which leads S(0, 0; 5, 5) h ih i h i S(0, 0; 10, 10) to that ∂S(i, j)/∂T diverges as T Tc, and scales like 1500 S(0, 0; 20, 20) -10 → S(0, 0; 50, 50)
T i j T ∂S( : ) −3/4 d 8 d T Tc , (48) S/ S/ )
d ∂T ∝ | − | d T 7 d
1000 S/ 6 -20 0 as we can see from the left picture in Fig. 3. Then the 5 ln(d critical exponents of ∂S(i, j)/∂T below T is 3/4, which is T -10 c
4 d
-12 -11 -10 S/ consistent with the critical exponent 1/8 of σi . While in ln(T -T) d -20 z h i 500 c -30 the critical region above Tc, σi vanishes and the domi- -30 h i -12 -11 -10 nating term in S(i : j) becomes the correlation function. ln(T-Tc) Then ∂S(i, j)/∂T scales like 0 -40 -0.0001 -5e-05 0 0 5e-05 0.0001 ∂S(i : j) T-T T-T ln T T , (49) c c ∂T ∝ | − c| FIG. 3: (color online) The critical behavior of the correlation which is the same as the specific heat Cv. Therefore, the entropy. LEFT: dS(0, 0 : N, N)/dT below Tc, and the inset is critical exponent now becomes 0 (See the right picture in to explore its critical exponent. RIGHT: dS(0, 0 : N, N)/dT Fig. 2 ). Moreover, we also note that the slope of lines above Tc. in the right inset of Fig. 2 is not the same. This is due to the fact that the exponent of the correlation function ν introduce the distance dependence in the ∂S(i : j)/∂T perature and higher temperature agree with the analytic above Tc. result in thermodynamic limit excellently. However, in the critical region, as we can see from Fig. 1, the corre- lation entropy decays in a power-law way. This fact not V. QUANTUM PHASE TRANSITION: only tells us a strong dependence between arbitrary two ONE-DIMENSIONAL TRANSVERSE FIELD sites in the system, but also manifests the integrality of ISING MODEL the whole system. It is also the reason of the difficulty of the Monte-Carlo simulation around the critical region. We now study the correlation entropy in one- Moreover, since the correlation entropy comprises all dimensional transverse field Ising model whose Hamil- kinds of two-point correlation function in its expression, tonian reads it can tell us more details about the critical phenomena. z N At the critical point, S(i) = 1 because of σi = 0, and x x z h i HIsing = λσ σ + σ , the correlation function behaviors like − j j+1 j j=1 z z 1/4 X SN σ σ A/N , (43) σ = σ (50) ≡ h 0,0 N,N i≃ 1 N+1 where A 0.645. Then the two-site entropy can be where λ is an Ising coupling in unit of the transverse field. simplified≃ as The Hamiltonian changes the number of down spins by 1 two, the total space of system then can be divided by S(ij) = 2 [(1 + SN ) log2(1 + SN ) the parity of the number of down spins. That is the − 2 Hamiltonian and the parity operator P = σz can be +(1 S ) log (1 S )] , (44) j j − N 2 − N simultaneously diagonalized and the eigenvalues of P is 1. We confine our interesting to the correlationQ entropy in the large N limit, and the correlation entropy behav- ± iors like between two spins at position i and j in the chain. There- fore we need to consider both single-site reduced density A2 S(0, 0 : N,N)= , (45) matrix ρi obtained from the ground-state wave function 2N 1/2 ln 2 by tracing out all spins except the one at site i, and the as has been shown in Fig. 2. On the other hand, around two-site reduced density matrix ρij obtained by tracing the critical point, the correlation entropy can be written out all spins except those at site i and j. Then if there as is no symmetry broken, such as in a finite-size system, according to the parity conservation, ρi has a diagonal 1 z z 2 z z z 2 S(i : j) σi σj σi σj σi (46) form Eq. (37), and the reduced density matrix of two ≃ 2h i − h ih i spins on a pair of lattice sites i and j can be put into the Then following block-diagonal form z z + − ∂S(i : j) z z z 2 ∂ σi σj u 0 0 z = σi σj σi h i + ∂T h i − h i ∂T 0 w1 z 0 z ρij = + (51) z z z ∂ σi 0 z w2 0 2 σi σj σi h i (47) − − − h ih i ∂T z 0 0 u 7
3 3 N=12 2.5 1 2.5 N=40 0.9 N=100 C 2 0.8 N=500 λ=λ
0.7 N=5000 λ | 1.5 0.6 2 /d S λ
d 1 0.5 /d S 0.4 S
d 0.5 0.3 1.5 0 0.2 0 1 2 3 4 5 6 7 0.1 lnN 0 0 1
10
20
r 30 1.8 2 0.5 1.4 1.6 0.9 0.95 1 1.05 1.1 1.15 40 1 1.2 λ 0.6 0.8 λ 50 0.4 0 0.2 FIG. 5: The scaling behavior the correlation entropy between FIG. 4: (color online) The correlation entropy as a function two neighboring sites. of λ and the distance r (in unit of lattice constant) at T = 0 for a system with N = 5000. 15 15 in the basis , , , . The elements in the density matrix| ↑↑iρ |can ↑↓i be| ↓↑i calculated| ↓↓i from the correlation ij N=12 10 function. 10 N=40 λ N=100 d S/
1 N=200 d ± z z z λ u = (1 2 σi + σi σj ), d N=400 5
4 ± h i h i S/ d 1 z z w1 = w2 = (1 σi σj ), 0 4 − h i 5 0 100 200 1 3 z± = ( σxσx σyσy ) (52) ln (N) 4 h i j i ± h i j i Otherwise, if the symmetry is broken at the ground state 0 of the ordered phase in the thermodynamic limit, i.e. 0.9 0.95 1 1.05 1.1 1.15 1.2 σx = 0, then the single-site reduced density matrix λ hbecomesi 6 z x FIG. 6: The scaling behavior the correlation entropy between 1 1+ σi σi two sites at the longest distance. ρi = h i h i , (53) 2 σx 1 σz h i i − h i i and the two-site reduced density matrix return to the original form (12) since no symmetry can be used to sim- where q is integer (half-odd integer) for parity P = ply it. Therefore, the correlation entropy between two 1(+1). The two-point correlation functions are calcu- − sites i and j becomes lated as [25]
S(i : j) = 2trρi log2 ρi trρij log2 ρij . (54) − a−1 a−2 a−N ··· Taking into account the translation invariance, the cor- a0 a−1 a−N+1 x x ··· relation entropy is simply a function of the distance be- σ0 σN = . . . . (57) h i . . . . tween two sites. aN−2 aN−3 a−1 The transverse field Ising model can be solved exactly ··· in terms of Jordan-Wigner transformation. The mean magnetization is given by [24] a a a 1 (1 λ cos φ) tanh[ω /T ] 1 0 −N+2 z φ a a ··· a σ = − , (55) y y 2 1 −N+3 h i N ωφ ··· φ σ0 σN = . . . . (58) X h i . . . . where ωφ is the dispersion relation, aN aN−1 a1 ···
ω = 1+ λ2 2λ cos(φ ), φ − q φ =2qπq/N, (56) σzσz =4 σz 2 a a (59) q h 0 N i h i − N −N 8 where
1 cos(φN)(λ cos φ 1) tanh[ωφ/T ] aN = − 1.0 N ωφ Xφ 0.9 0.8 λ sin(φN) sin(φ) tanh[ωφ/T ] (60) 0.7 −N ωφ φ 0.6 X 0.5 S We show the correlation entropy S(i : j) in the ground 0.4 state as a function of coupling λ and distance between 0.3 two site r = i j in Fig. 4. The result is also impressive. 0.2 − 0.1 As is well known [9], the ground state of the transverse 0 field Ising model consists of two different phases, whose 0 10 corresponding physical picture can be understood from 20 2 r 30 1.6 1.8 both weak and strong coupling limit. If λ 0, all spins 1.2 1.4 40 0.8 1 → 0.4 0.6 λ are polarized along z direction, the ground state then is 50 0 0.2 a paramagnet and in the absence of long-range correla- tion, while in the limit λ 1, the strong Ising coupling ≫ FIG. 7: (color online) The correlation entropy as a function of introduce magnetic long-range correlation in the order λ and the distance r (in unit of lattice constant) at T = 0.2. parameter σx to the ground state. The competition be- Obviously, at low temperature, the correlation entropy at long tween these two different order leads to a quantum phase distance is destroyed in the quantum critical region around transition at the critical point λc = 1. From Fig. 4, λ = 1. we can see that the correlation entropy tends to zero quickly as the distance between two sites increases in the paramagnetic phase. This phenomena can be well under- However, for those sites are separated far away, the stood from the fact that the ground state in this phase correlation entropy shows quite different scaling behav- is non-degenerate and almost fully polarized, therefore ior. For examples, in Fig. 6, we show the scaling behavior the knowledge of the state at one site i does not effect the correlation entropy between two sites at the longest the state of another site j far away, which leads to zero distance in a ring. This first observation is that when N , dS(0,N/2)/dλ becomes divergent. Moreover, information in common between two sites. However, this → ∞ scene is not true in another phase. When λ > 1, the detailed analysis reveal that the maximum value of the ground state is twofold degenerate and possess long-range first derivative of the correlation entropy between two correlation. Before the measurement, the uncertainty of farthest sites in a ring scales like the state at an arbitrary site is very large. However, if dS(0,N/2) we learn it from one site, the state at another site, even const ln3 N. (63) far away, is almost determined. Which leads to a finite dλ ≃ × correlation entropy between two sites even if they are which differs from ln N for dS(0, 1)/dλ. Obviously, these separated far away from each other. interesting scaling behavior enable us to learn the physics Obviously, the behavior of correlation entropy in the of real infinite system from the scaling analysis. transverse field Ising model is quite different from the On the other hand, there is no thermal phase transi- quantum entanglement. In the previous works [10, 11] tion in one-dimensional quantum spin system according on the pairwise entanglement in the ground state of this to the Mermin-Wagner theorem [26]. Thus let us study model, it has been shown that the concurrence vanishes the properties of the correlation entropy away from zero unless the two sites are at most next-nearest neighbors. temperature. The results are shown in Fig. 7 and 8 for In the paramagnetic phase, the correlation entropy share T = 0.2 and 0.5 respectively. We can see that at lower similar properties in common with the concurrence. In temperature T = 0.2, the correlation entropy is broken the ordered phase, however, the correlation entropy does only around the critical region λ 1. As the temperature not vanish even the distance between two sites becomes increases, the correlation entropy∼ in larger λ region is also very large, such as 50 lattice constant. Moreover, the destroyed. These observations tell us that the correla- correlation entropy also shows interesting scaling behav- tion entropy is a decreasing function of the temperature, ior, just as that of the concurrence, around the critical as we can see from Fig. 9. Since the broken symme- point, as is shown in Fig. 5. Moreover, we find that at try only exists in the ground state of an infinite system the critical point the first derivative of the correlation and the thermal fluctuation tends to destroy the correla- entropy between two neighboring sites scales like tion entropy, it is not possible to reestablish such a long- dS(0, 1) range correlation at finite temperatures. Therefore, no const ln N. (61) dλ ≃ × thermal phase transition happens in the one-dimensional λ=λc transverse field Ising model. This observation, based on or the numerical calculation, is consistent with the Mermin- S(0, 1) S(0, 1) c + const (λ λ ) ln N. (62) Wagner theorem. ≃ |λ=λ × − c 9
On the other hand, it also implies that the entropy at the critical point is no longer a linear function of the volume of the system, i.e. S = sV . In one dimensional system, it has already been noted6 that entropy of the subsystem satisfies S(x) ln(x) [28] where x is the length of sub- system in the critical∝ region of some spin systems. Then, from this point of view, the spatial degree of freedom of the system is suppressed around the critical point. This observation reminds us a well-known holographic prin- ciple on the entropy of the black hole which says that the entropy of the black hole is proportional to the sur- face. Though a rigorous prove is still not available, we are sure there there are something in common between the entropy in the critical phenomena and that in the black hole, and this relation deserve for further investigation. On the other hand, though we restrict ourselves to FIG. 8: (color online) The correlation entropy as a function the two-point correlation entropy in the above studies, of λ and the distance r (in unit of lattice constant) at T = if the system processes block-block order, such as dimer 0.5T . At higher temperature, the correlation entropy at long order [27], it may be useful to investigate the proper- distance is completely destroyed. ties of block-block correlation entropy. A simple ex- ample is Majumdar-Ghosh model with the Hamiltonian H = i (J1σi σi+1 + J2σi σi+2), where J2 is the cou- pling between· two next-nearest· neighbor sites. In this P model, if J2 = 1/2, the ground state is a uniformly weighted superposition of the two nearest-neighbor va- lence bond state [27]:
ψ = [1, 2][3, 4] [L 1,L] | 1i ··· − ψ = [L, 1][2, 3] [L 2,L 1] (64) | 2i ··· − − where 1 [i, j]= ( i j i j ). (65) √2 | ↑i | ↓i −| ↓i | ↑i Then the block-block correlation entropy can help us to understand such a dimer order. FIG. 9: (color online) The correlation entropy as a function of Moreover, from the definition of our correlation en- temperature T and the distance r (in unit of lattice constant) tropy, it is also useful to introduce a characteristic length at λ = 2.0. We can see from the figure that the correlation for the statistical system, below which the extensive entropy is protected by an energy gap at low temperature. properties of the entropy is violated. Such a characteris- tic length has a non-trivial meaning, since the extensive property of the entropy in the statistical physics is only VI. DISCUSSIONS AND PROSPECTS valid above this scale. A simple example is the molecule which is composed of some atoms. When we study the With analytical studies and numerical calculations, we physics of molecule gas, we have to regard a molecule have discovered the rigorous relation between the corre- as a whole because of its internal order. Only when the lation entropy and the long-range correlation. It strongly temperature is high enough to break its order, the atom indicates us the non-trivial role of the correlation entropy then plays the important role to the statistical properties in the critical phenomena. This discovery motivates us of the system. to study both the thermal and quantum phase transi- Though we verify the non-trivial behavior of the corre- tions from the point view of information theory, i.e. mu- lation entropy in terms of spin systems, the rigorous rela- tual information, whose non-vanishing behavior at long tion between the non-vanishing correlation entropy and distance really witnesses the violation of the extensive long-rang correlation is valid for all many-body systems. properties of the entropy in the statistical physics, i.e. Therefore, it is expected to provide more physical intu- S(AB) = S(A) + S(B). In the two models we studied ition into the critical phenomena, such as Bose-Einstein in this model, since the correlation length at the critical condensation and superconductivity from the point view point diverges, the physics of the system has a strong of the correlation entropy. Take the former as a simple dependence on the system size, i.e. the scaling behavior. example, the reduced density matrix between two sites in 10 the space-time can also be expressed in terms of bosons vanishing behavior not only help us have deep under- † operators ax,p 0 = exp( ipx) in space representation. standing to the entropy in the statistical physics, but | i − ′ ′ At high temperatures, p ρxx′ p vanishes as x x in- also shed light on the long-range correlation in the criti- h | | i − creases. Only when the condensation happens, 0 ρxx′ 0 cal behavior. leads to a non-vanishing correlation entropy, justh as| what| i we have shown for the correlation entropy in quantum spin system. This work is supported by the Earmarked Grant for Research from the Research Grants Council of VII. SUMMARY AND ACKNOWLEDGEMENT HKSAR, China (Project CUHK N CUHK204/05 and HKU 3/05C) and UGC of CUHK; and NSFC with In summary, the correlation entropy plays a univer- Grants No. 90203018, No. 10474104 and No. 60433050. sal role in understanding critical phenomena. Its non- We thank Kerson Huang for the helpful discussion.
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