Density matrix elements and entanglement entropy for
the spin-1/2 XXZ chain at ∆=1/2
Jun Sato 1 ∗, Masahiro Shiroishi 1 †
1 Institute for Solid State Physics, University of Tokyo,
Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan
November 9, 2018
Abstract
We have analytically obtained all the density matrix elements up to six lattice
sites for the spin-1/2 Heisenberg XXZ chain at ∆ = 1/2. We use the multiple inte-
gral formula of the correlation function for the massless XXZ chain derived by Jimbo
and Miwa. As for the spin-spin correlation functions, we have newly obtained the
fourth- and fifth-neighbour transverse correlation functions. We have calculated all
the eigenvalues of the density matrix and analyze the eigenvalue-distribution. Using
these results the exact values of the entanglement entropy for the reduced density ma-
trix up six lattice sites have been obtained. We observe that our exact results agree
quite well with the asymptotic formula predicted by the conformal field theory.
∗junji@issp.u-tokyo.ac.jp
†siroisi@issp.u-tokyo.ac.jp
http://arxiv.org/abs/0704.0850v1
1 Introduction
The spin-1/2 antiferromagnetic Heisenberg XXZ chain is one of the most fundamental models
for one-dimensional quantum magnetism, which is given by the Hamiltonian
Sxj S
j+1 + S
j+1 +∆S
, (1.1)
where Sαj = σ
j /2 with σ
j being the Pauli matrices acting on the j-th site and ∆ is the
anisotropy parameter. For ∆ > 1, it is called the massive XXZ model where the system is
gapful. Meanwhile for −1 < ∆ ≤ 1 case, the system is gapless and called the massless XXZ
model. Especially we call it XXX model for the isotropic case ∆ = 1.
The exact eigenvalues and eigenvectors of this model can be obtained by the Bethe Ansatz
method [1, 2]. Many physical quantities in the thermodynamic limit such as specific heat,
magnetic susceptibility, elementary excitations, etc..., can be exactly evaluated even at finite
temperature by the Bethe ansatz method [2].
The exact calculation of the correlation functions, however, is still a difficult problem. The
exceptional case is ∆ = 0, where the system reduces to a lattice free-fermion model by the
Jordan-Wigner transformation. In this case, we can calculate arbitrary correlation functions
by means of Wick’s theorem [3, 4]. Recently, however, there have been rapid developments in
the exact evaluations of correlation functions for ∆ 6= 0 case also, since Kyoto Group (Jimbo,
Miki, Miwa, Nakayashiki) derived a multiple integral representation for arbitrary correlation
functions. Using the representation theory of the quantum affine algebra Uq(ŝl2), they first
derived a multiple integral representation for massive XXZ antiferromagnetic chain in 1992
[5, 6], which is before long extended to the XXX case [7, 8] and the massless XXZ case
[9]. Later the same integral representations were reproduced by Kitanine, Maillet, Terras
[10] in the framework of Quantum Inverse Scattering Method. They have also succeeded in
generalizing the integral representations to the XXZ model with an external magnetic field
[10]. More recently the multiple integral formulas were extended to dynamical correlation
functions as well as finite temperature correlation functions [11, 12, 13, 14]. In this way
it has been established now the correlation functions for XXZ model are represented by
multiple integrals in general. However, these multiple integrals are difficult to evaluate both
numerically and analytically.
For general anisotropy ∆, it has been shown that the multiple inetegrals up to four-
dimension can be reduced to one-dimensional integrals [15, 16, 17, 18, 19, 20, 21]. As a
result all the density matrix elements within four lattice sites have been obtained for general
anisotropy [21]. To reduce the multiple integrals into one-dimension, however, involves hard
calculation, which makes difficult to obtain correlation functions on more than four lattice
sites. On the other hand, at the isotropic point ∆ = 1, an algebraic method based on
qKZ equation has been devised [22] and all the density matrix elements up to six lattice
sites have been obtained [23, 24]. Moreover, as for the spin-spin correlation functions, up to
seventh-neighbour correlation 〈Sz1Sz8〉 for XXX chain have been obtained from the generating
functional approach [25, 26]. It is desirable that this algebraic method will be generalized to
the case with ∆ 6= 1. Actually, Boos, Jimbo, Miwa, Smirnov and Takeyama have derived an
exponential formula for the density matrix elements of XXZ model, which does not contain
multiple integrals [27, 28, 29, 30, 31]. It, however, seems still hard to evaluate the formula
for general density matrix elements.
Among the general ∆ 6= 0, there is a special point ∆ = 1/2, where some intriguing prop-
erties have been observed. Let us define a correlation function called Emptiness Formation
Probability (EFP) [8] which signifies the probability to find a ferromagnetic string of length
P (n) ≡
+ Szj
. (1.2)
The explicit general formula for P (n) at ∆ = 1/2 was conjectured in [33]
P (n) = 2−n
(3k + 1)!
(n + k)!
, (1.3)
which is proportional to the number of alternating sign matrix of size n × n. Later this
conjecture was proved by the explicit evaluation of the multiple integral representing the
EFP [34]. Remarkably, one can also obtain the exact asymptotic behavior as n → ∞
from this formula, which is the unique valuable example except for the free fermion point
∆ = 0. Note also that as for the longitudinal two-point correlation functions at ∆ = 1/2,
up to eighth-neighbour correlation function 〈Sz1Sz9〉 have been obtained in [32] by use of the
multiple integral representation for the generating function. Most outstanding is that all the
results are represented by single rational numbers. These results motivated us to calculate
other correlation functions at ∆ = 1/2. Actually we have obtained all the density matrix
elements up to six lattice sites by the direct evaluation of the multiple integrals. All the
results can be written by single rational numbers as expected. A direct evaluation of the
multiple integrals is possible due to the particularity of the case for ∆ = 1/2 as is explained
below.
2 Analytical evaluation of multiple integral
Here we shall describe how we analytically obtain the density matrix elements at ∆ = 1/2
from the multiple integral formula. Any correlation function can be expressed as a sum of
density matrix elements P
,··· ,ǫ′n
ǫ1,··· ,ǫn , which are defined by the ground state expectation value of
the product of elementary matrices:
,··· ,ǫ′n
ǫ1,··· ,ǫn ≡ 〈E
1 · · ·Eǫ
n 〉, (2.1)
where E
j are 2× 2 elementary matrices acting on the j-th site as
E++j =
+ Szj , E
− Szj ,
E+−j =
= S+j = S
j + iS
j , E
= S−j = S
j − iS
The multiple integral formula of the density matrix element for the massless XXZ chain
reads [9]
,··· ,ǫ′n
ǫ1,··· ,ǫn =(−ν)−n(n−1)/2
· · ·
sinh(xa − xb)
sinh[(xa − xb − ifabπ)ν]
sinhyk−1 [(xk + iπ/2)ν] sinh
n−yk [(xk − iπ/2)ν]
coshn xk
, (2.2)
where the parameter ν is related to the anisotropy as ∆ = cosπν and fab and yk are
determined as
fab = (1 + sign[(s
′ − a + 1/2)(s′ − b+ 1/2)])/2,
y1 > y2 > · · · > ys′, ǫ′yi = +
ys′+1 > · · · > yn, ǫn+1−yi = −. (2.3)
In the case of ∆ = 1/2, namely ν = 1/3, the significant simplification occurs in the multiple
integrals due to the trigonometric identity
sinh(xa−xb) = 4 sinh[(xa−xb)/3] sinh[(xa−xb+iπ)/3] sinh[(xa−xb−iπ)/3]. (2.4)
Actually if we note that the parameter fab takes the value 0 or 1, the first factor in the
multiple integral at ν = 1/3 can be decomposed as
sinh(xa − xb)
sinh[(xa − xb − iπ)/3]
= 4 sinh
xa − xb
xa − xb + iπ
= −1 + ωe
(xa−xb) + ω−1e−
(xa−xb), (2.5)
sinh(xa − xb)
sinh[(xa − xb)/3]
= 4 sinh
xa − xb + iπ
xa − xb − iπ
= 1 + e
(xa−xb) + e−
(xa−xb), (2.6)
where ω = eiπ/3. Expanding the trigonometoric functions in the second factor into exponen-
tials
sinhy−1 [(x+ iπ/2)/3] sinhn−y [(x− iπ/2)/3]
= 21−n
ω1/2ex/3 − ω−1/2e−x/3
)y−1 (
ω−1/2ex/3 − ω1/2e−x/3
= 21−n
(−1)l+m
y − 1
ωy−l+m−(n+1)/2e
(n−2l−2m−1)x, (2.7)
we can explicitly evaluate the multiple integral by use of the formula
eαxdx
coshn x
= 2n−1B
, Re(n± α) > 0, (2.8)
where B(p, q) is the beta function defined by
B(p, q) =
tp−1(1− t)q−1dt, Re(p),Re(q) > 0. (2.9)
Table 1: Comparison with the asymptotic formula of the transverse correlation function
〈Sx1Sx2 〉 〈Sx1Sx3 〉 〈Sx1Sx4 〉 〈Sx1Sx5 〉 〈Sx1Sx6 〉
Exact −0.156250 0.0800781 −0.0671234 0.0521997 −0.0467664
Asymptotics −0.159522 0.0787307 −0.0667821 0.0519121 −0.0466083
In this way we have succeeded in calculating all the density matrix elements up to six lattice
sites. All the results are represented by single rational numbers, which are presented in
Appendix A. As for the spin-spin correlation functions, we have newly obtained the fourth-
and fifth-neighbour transverse two-point correlation function
〈Sx1Sx2 〉 = −
= −0.15625,
〈Sx1Sx3 〉 =
= 0.080078125,
〈Sx1Sx4 〉 = −
65536
= −0.0671234130859375,
〈Sx1Sx5 〉 =
1751531
33554432
= 0.0521996915340423583984375,
〈Sx1Sx6 〉 = −
3213760345
68719476736
= −0.046766368104727007448673248291015625.
The asymptotic formula of the transverse two-point correlation function for the massless
XXZ chain is established in [35, 36]
〈Sx1Sx1+n〉 ∼ Ax(η)
(−1)n
− Ãx(η)
+ · · · , η = 1− ν,
Ax(η) =
8(1− η)2
sinh(ηt)
sinh(t) cosh[(1− η)t]
− ηe−2t
Ãx(η) =
2η(1− η)
cosh(2ηt)e−2t − 1
2 sinh(ηt) sinh(t) cosh[(1− η)t]
sinh(ηt)
η2 + 1
, (2.10)
which produces a good numerical value even for small n as is shown in Table 1. Note that the
longitudinal correlation function was obtained up to eighth-neighbour correlaion 〈Sz1Sz9〉 from
the multiple integral representation for the generating function [32]. Note also that up to
third-neighbour both longitudinal and transverse correlation functions for general anisotropy
∆ were obtained in [21].
3 Reduced density matrix and entanglement entropy
Below let us discuss the reduced density matrix for a sub-chain and the entanglement entropy.
The density matrix for the infinite system at zero temperature has the form
ρT ≡ |GS〉〈GS|, (3.1)
0 10 20 30 40 50 60
0 10 20 30
Figure 1: Eigenvalue-distribution of density matrices
Table 2: Entanglement entropy S(n) of a finite sub-chain of length n
S(1) S(2) S(3) S(4)
1 1.3716407621868583 1.5766810784924767 1.7179079372711414
S(5) S(6)
1.8262818282012363 1.9144714710902746
where |GS〉 denotes the ground state of the total system. We consider a finite sub-chain of
length n, the rest of which is regarded as an environment. We define the reduced density
matrix for this sub-chain by tracing out the environment from the infinite chain
ρn ≡ trEρT =
,··· ,ǫ′n
ǫ1,··· ,ǫn
ǫj ,ǫ
. (3.2)
We have numerically evaluate all the eigenvalues ωα (α = 1, 2, · · · , 2n) of the reduced density
matrix ρn up to n = 6. We show the distribution of the eigenvalues in Figure 1. The
distribution is less degenerate comapared with the isotropic case ∆ = 1 shown in [24]. In the
odd n case, all the eigenvalues are two-fold degenerate due to the spin-reverse symmetry.
Subsequently we exactly evaluate the von Neumann entropy (Entanglement entropy)
defined as
S(n) ≡ −trρn log2 ρn = −
ωα log2 ωα. (3.3)
The exact numerical values of S(n) up to n = 6 are shown in Table 2. By analyzing the
behaviour of the entanglement S(n) for large n, we can see how long quantum correlations
reach [37]. In the massive region ∆ > 1, the entanglement entropy will be saturated as n
grows due to the finite correlation length. This means the ground state is well approximated
by a subsystem of a finite length corresponding to the large eigenvalues of reduced density
matrix. On the other hand, in the massless case −1 < ∆ ≤ 1, the conformal field theory
predict that the entanglement entropy shows a logarithmic divergence [38]
S(n) ∼ 1
log2 n + k∆. (3.4)
1 2 3 4 5 6
Exact
Asymptotics
Figure 2: Entanglement entropy S(n) of a finite sub-chain of length n
Our exact results up to n = 6 agree quite well with the asymptotic formula as shown in Figure
2. We estimate the numerical value of the constant term k∆=1/2 as k∆=1/2 ∼ S(6)− 13 log2 6 =
1.0528. This numerical value is slightly smaller than the isotropic case ∆ = 1, where the
constant k∆=1 is estimated as k∆=1 ∼ 1.0607 from the exact data for S(n) up to n = 6 [24].
At free fermion point ∆ = 0, the exact asymptotic formula has been obtained in [39]
S(n) ∼ 1
log2 n+ k∆=0,
k∆=0 = 1/3−
t sinh2(t/2)
− cosh(t/2)
2 sinh3(t/2)
/ ln 2. (3.5)
In this case the numerical value for the constant term is given by k∆=0 = 1.0474932144 · · · .
4 Summary and discussion
We have succeeded in obtaining all the density matrix elements on six lattice sites for XXZ
chain at ∆ = 1/2. Especially we have newly obtained the fourth- and fifth-neighbour
transverse spin-spin correlation functions. Our exact results for the transverse correlations
show good agreement with the asymptotic formula established in [35, 36]. Subsequently we
have calculated all the eigenvalues of the reduced density matrix ρn up to n = 6. From these
results we have exactly evaluated the entanglement entropy, which shows a good agreement
with the asymptotic formula derived via the conformal field theory. Finally, we remark
that similar procedures to evaluate the multiple integrals are also possible at ν = 1/n for
n = 4, 5, 6, · · · , since there are similar trigonometric identities as (2.4). We will report the
calculation of correlation functions for these cases in subsequent papers.
Acknowledgement
The authors are grateful to K. Sakai for valuable discussions. This work is in part sup-
ported by Grant-in-Aid for the Scientific Research (B) No. 18340112. from the Ministry of
Education, Culture, Sports, Science and Technology, Japan.
Appendix A Density matrix elements up to n = 6
In this appendix we present all the independent density matrix elements defined in eq. (2.1)
up to n = 6. Other elements can be computed from the relations
,··· ,ǫ′n
ǫ1,··· ,ǫn = 0 if
ǫj 6=
ǫ′j , (A.1)
,··· ,ǫ′n
ǫ1,··· ,ǫn = P
ǫ1,··· ,ǫn
,··· ,ǫ′n
,··· ,−ǫ′n
−ǫ1,··· ,−ǫn
ǫ′n,··· ,ǫ
ǫn,··· ,ǫ1 , (A.2)
,··· ,ǫ′n
+,ǫ1,··· ,ǫn
,··· ,ǫ′n
−,ǫ1,··· ,ǫn
,··· ,ǫ′n,+
ǫ1,··· ,ǫn,+
,··· ,ǫ′n,−
ǫ1,··· ,ǫn,−
,··· ,ǫ′n
ǫ1,··· ,ǫn , (A.3)
and the formula for the EFP [33, 34]
P (n) = P
+,··· ,+
+,··· ,+ = 2
(3k + 1)!
(n+ k)!
. (A.4)
Appendix A.1 n ≤ 4
P−++− = −
= −0.3125, P−++++− =
= 0.0800781,
P−++++−++ = −
= −0.0269775, P−+++++−+ =
65536
= 0.0240936,
P−++++++− = −
32768
= −0.00881958, P+−+++−++ =
16384
= 0.0632935,
P+−++++−+ = −
32768
= −0.0611877, P−−+++−+− = −
65536
= −0.0583038,
P−−++++−− =
65536
= 0.0212555, P−+−++−+− =
32768
= 0.149017,
P−++−+−−+ =
32768
= 0.0943298.
Appendix A.2 n = 5
P−+++++−+++ = −
14721
8388608
= −0.00175488, P−++++++−++ =
37335
16777216
= 0.00222534,
P−+++++++−+ = −
48987
33554432
= −0.00145993, P−++++++++− =
13911
33554432
= 0.00041458,
P+−++++−+++ =
179699
33554432
= 0.00535545, P+−+++++−++ = −
120337
16777216
= −0.00717264,
P+−++++++−+ =
165155
33554432
= 0.004922, P++−++++−++ =
168313
16777216
= 0.0100322,
P−−++++−−++ =
31069
2097152
= 0.0148149, P−−++++−+−+ = −
411583
16777216
= −0.0245323,
P−−++++−++− =
196569
16777216
= 0.0117164, P−−+++++−+− = −
281271
33554432
= −0.00838253,
P−−++++++−− =
79673
33554432
= 0.00237444, P−+−+++−−++ = −
1441787
33554432
= −0.0429686,
P−+−+++−++− = −
1261655
33554432
= −0.0376002, P−+−++++−+− =
59459
2097152
= 0.0283523,
P−++−++−++− =
1575515
33554432
= 0.046954, P−+++−+−−++ = −
696151
33554432
= −0.0207469,
P−+++−+−+−+ =
1366619
33554432
= 0.0407284.
Appendix A.3 n = 6
P−++++++−++++ = −
1546981
34359738368
= −0.0000450231, P−+++++++−+++ =
5095899
68719476736
= 0.0000741551,
P−++++++++−++ = −
2366275
34359738368
= −0.0000688677, P−+++++++++−+ =
2455833
68719476736
= 0.0000357371,
P−++++++++++− = −
284577
34359738368
= −8.28228× 10−6, P+−+++++−++++ =
2927709
17179869184
= 0.000170415,
P+−++++++−+++ = −
20086627
68719476736
= −0.000292299, P+−+++++++−++ =
19268565
68719476736
= 0.000280395,
P+−++++++++−+ = −
10295153
68719476736
= −0.000149814, P++−+++++−+++ =
17781349
34359738368
= 0.000517505,
P++−++++++−++ = −
35087523
68719476736
= −0.000510591, P−−+++++−−+++ =
48421023
34359738368
= 0.00140924,
P−−+++++−+−++ = −
214080091
68719476736
= −0.00311528, P−−+++++−++−+ =
88171589
34359738368
= 0.00256613,
P−−+++++−+++− = −
57522267
68719476736
= −0.000837059, P−−++++++−−++ =
56776545
34359738368
= 0.00165241,
P−−++++++−+−+ = −
154538459
68719476736
= −0.00224883, P−−++++++−++− =
60809571
68719476736
= 0.000884896,
P−−+++++++−−+ =
6708473
8589934592
= 0.000780969, P−−+++++++−+− = −
33366621
68719476736
= −0.000485548,
P−−++++++++−− =
3860673
34359738368
= 0.00011236, P−+−++++−−+++ = −
85706851
17179869184
= −0.0049888,
P−+−++++−+−++ =
12211375
1073741824
= 0.0113727, P−+−++++−++−+ = −
332557469
34359738368
= −0.0096787,
P−+−++++−+++− =
56183761
17179869184
= 0.00327033, P−+−+++++−−++ = −
430452959
68719476736
= −0.00626391,
P−+−+++++−+−+ =
606065059
68719476736
= 0.00881941, P−+−+++++−++− = −
123612511
34359738368
= −0.0035976,
P−+−++++++−−+ = −
108202041
34359738368
= −0.00314909, P−+−++++++−+− =
70061315
34359738368
= 0.00203905,
P−++−+++−−+++ =
7860495
1073741824
= 0.00732066, P−++−+++−+−++ = −
591759525
34359738368
= −0.0172225,
P−++−+++−++−+ =
1044016671
68719476736
= 0.0151924, P−++−+++−+++− = −
367905053
68719476736
= −0.00535372,
P−++−++++−−++ =
676957849
68719476736
= 0.00985103, P−++−++++−+−+ = −
988973861
68719476736
= −0.0143915,
P−++−++++−++− =
6581795
1073741824
= 0.00612977, P−++−+++++−−+ =
363618785
68719476736
= 0.00529135,
P−+++−++−−+++ = −
185522333
34359738368
= −0.00539941, P−+++−++−+−++ =
901633567
68719476736
= 0.0131205,
P−+++−++−++−+ = −
103539423
8589934592
= −0.0120536, P−+++−++−+++− =
38524625
8589934592
= 0.00448486,
P−+++−+++−−++ = −
267901987
34359738368
= −0.00779697, P−+++−+++−+−+ =
12750645
1073741824
= 0.011875,
P−+++−++++−−+ = −
309855965
68719476736
= −0.004509, P−++++−+−−+++ =
29410257
17179869184
= 0.0017119,
P−++++−+−+−++ = −
296882461
68719476736
= −0.00432021, P−++++−+−++−+ =
35985105
8589934592
= 0.00418922,
P−++++−++−−++ =
92176287
34359738368
= 0.00268268, P+−−++++−−+++ =
202646807
34359738368
= 0.0058978,
P+−−++++−+−++ = −
972245985
68719476736
= −0.014148, P+−−++++−++−+ =
217687057
17179869184
= 0.0126711,
P+−−+++++−+−+ = −
211696415
17179869184
= −0.0123224, P+−−++++++−−+ =
78922695
17179869184
= 0.00459391,
P+−+−+++−+−++ =
1196499417
34359738368
= 0.0348227, P+−+−+++−++−+ = −
2209522727
68719476736
= −0.0321528,
P+−+−++++−+−+ =
1108384987
34359738368
= 0.0322582, P+−++−++−++−+ =
530683585
17179869184
= 0.0308899,
P+−++−+++−−++ =
347202525
17179869184
= 0.0202098, P−−−++++−−++− = −
268623007
68719476736
= −0.00390898,
P−−−++++−+−+− =
46285135
8589934592
= 0.0053883, P−−−++++−++−− = −
136974885
68719476736
= −0.00199325,
P−−−+++++−+−− =
19939391
17179869184
= 0.00116063, P−−−++++++−−− = −
18442085
68719476736
= −0.000268368,
P−−+−+++−−++− =
1018463205
68719476736
= 0.0148206, P−−+−+++−+−+− = −
1454513249
68719476736
= −0.021166,
P−−+−+++−++−− =
277721503
34359738368
= 0.00808276, P−−+−++++−+−− = −
335265249
68719476736
= −0.00487875,
P−−++−++−−++− = −
369408975
17179869184
= −0.0215024, P−−++−++−+−+− =
1104236607
34359738368
= 0.0321375,
P−−++−++−++−− = −
880560357
68719476736
= −0.0128138, P−−++−+++−−+− = −
876924641
68719476736
= −0.0127609,
P−−+++−+−−−++ =
113631201
17179869184
= 0.00661421, P−−+++−+−−+−+ = −
292857807
17179869184
= −0.0170466,
P−−+++−+−+−−+ =
548645951
34359738368
= 0.0159677, P−−+++−++−−−+ = −
377925345
68719476736
= −0.00549954,
P−+−+−++−−++− =
1719255909
34359738368
= 0.0500369, P−+−+−++−+−+− = −
5350158879
68719476736
= −0.0778551,
P−+−++−+−−+−+ =
1565770597
34359738368
= 0.0455699, P−+−++−+−+−−+ = −
3059753503
68719476736
= −0.0445253,
P−++−−++−−++− = −
2117554719
68719476736
= −0.0308145.
References
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http://arxiv.org/abs/hep-th/0507290
	Introduction
	Analytical evaluation of multiple integral
	Reduced density matrix and entanglement entropy
	Summary and discussion
	Density matrix elements up to n=6

  We have analytically obtained all the density matrix elements up to six
lattice sites for the spin-1/2 Heisenberg XXZ chain at $\Delta=1/2$. We use the
multiple integral formula of the correlation function for the massless XXZ
chain derived by Jimbo and Miwa. As for the spin-spin correlation functions, we
have newly obtained the fourth- and fifth-neighbour transverse correlation
functions. We have calculated all the eigenvalues of the density matrix and
analyze the eigenvalue-distribution. Using these results the exact values of
the entanglement entropy for the reduced density matrix up six lattice sites
have been obtained. We observe that our exact results agree quite well with the
asymptotic formula predicted by the conformal field theory.

Introduction
The spin-1/2 antiferromagnetic Heisenberg XXZ chain is one of the most fundamental models
for one-dimensional quantum magnetism, which is given by the Hamiltonian
Sxj S
j+1 + S
j+1 +∆S
, (1.1)
where Sαj = σ
j /2 with σ
j being the Pauli matrices acting on the j-th site and ∆ is the
anisotropy parameter. For ∆ > 1, it is called the massive XXZ model where the system is
gapful. Meanwhile for −1 < ∆ ≤ 1 case, the system is gapless and called the massless XXZ
model. Especially we call it XXX model for the isotropic case ∆ = 1.
The exact eigenvalues and eigenvectors of this model can be obtained by the Bethe Ansatz
method [1, 2]. Many physical quantities in the thermodynamic limit such as specific heat,
magnetic susceptibility, elementary excitations, etc..., can be exactly evaluated even at finite
temperature by the Bethe ansatz method [2].
The exact calculation of the correlation functions, however, is still a difficult problem. The
exceptional case is ∆ = 0, where the system reduces to a lattice free-fermion model by the
Jordan-Wigner transformation. In this case, we can calculate arbitrary correlation functions
by means of Wick’s theorem [3, 4]. Recently, however, there have been rapid developments in
the exact evaluations of correlation functions for ∆ 6= 0 case also, since Kyoto Group (Jimbo,
Miki, Miwa, Nakayashiki) derived a multiple integral representation for arbitrary correlation
functions. Using the representation theory of the quantum affine algebra Uq(ŝl2), they first
derived a multiple integral representation for massive XXZ antiferromagnetic chain in 1992
[5, 6], which is before long extended to the XXX case [7, 8] and the massless XXZ case
[9]. Later the same integral representations were reproduced by Kitanine, Maillet, Terras
[10] in the framework of Quantum Inverse Scattering Method. They have also succeeded in
generalizing the integral representations to the XXZ model with an external magnetic field
[10]. More recently the multiple integral formulas were extended to dynamical correlation
functions as well as finite temperature correlation functions [11, 12, 13, 14]. In this way
it has been established now the correlation functions for XXZ model are represented by
multiple integrals in general. However, these multiple integrals are difficult to evaluate both
numerically and analytically.
For general anisotropy ∆, it has been shown that the multiple inetegrals up to four-
dimension can be reduced to one-dimensional integrals [15, 16, 17, 18, 19, 20, 21]. As a
result all the density matrix elements within four lattice sites have been obtained for general
anisotropy [21]. To reduce the multiple integrals into one-dimension, however, involves hard
calculation, which makes difficult to obtain correlation functions on more than four lattice
sites. On the other hand, at the isotropic point ∆ = 1, an algebraic method based on
qKZ equation has been devised [22] and all the density matrix elements up to six lattice
sites have been obtained [23, 24]. Moreover, as for the spin-spin correlation functions, up to
seventh-neighbour correlation 〈Sz1Sz8〉 for XXX chain have been obtained from the generating
functional approach [25, 26]. It is desirable that this algebraic method will be generalized to
the case with ∆ 6= 1. Actually, Boos, Jimbo, Miwa, Smirnov and Takeyama have derived an
exponential formula for the density matrix elements of XXZ model, which does not contain
multiple integrals [27, 28, 29, 30, 31]. It, however, seems still hard to evaluate the formula
for general density matrix elements.
Among the general ∆ 6= 0, there is a special point ∆ = 1/2, where some intriguing prop-
erties have been observed. Let us define a correlation function called Emptiness Formation
Probability (EFP) [8] which signifies the probability to find a ferromagnetic string of length
P (n) ≡
+ Szj
. (1.2)
The explicit general formula for P (n) at ∆ = 1/2 was conjectured in [33]
P (n) = 2−n
(3k + 1)!
(n + k)!
, (1.3)
which is proportional to the number of alternating sign matrix of size n × n. Later this
conjecture was proved by the explicit evaluation of the multiple integral representing the
EFP [34]. Remarkably, one can also obtain the exact asymptotic behavior as n → ∞
from this formula, which is the unique valuable example except for the free fermion point
∆ = 0. Note also that as for the longitudinal two-point correlation functions at ∆ = 1/2,
up to eighth-neighbour correlation function 〈Sz1Sz9〉 have been obtained in [32] by use of the
multiple integral representation for the generating function. Most outstanding is that all the
results are represented by single rational numbers. These results motivated us to calculate
other correlation functions at ∆ = 1/2. Actually we have obtained all the density matrix
elements up to six lattice sites by the direct evaluation of the multiple integrals. All the
results can be written by single rational numbers as expected. A direct evaluation of the
multiple integrals is possible due to the particularity of the case for ∆ = 1/2 as is explained
below.
2 Analytical evaluation of multiple integral
Here we shall describe how we analytically obtain the density matrix elements at ∆ = 1/2
from the multiple integral formula. Any correlation function can be expressed as a sum of
density matrix elements P
,··· ,ǫ′n
ǫ1,··· ,ǫn , which are defined by the ground state expectation value of
the product of elementary matrices:
,··· ,ǫ′n
ǫ1,··· ,ǫn ≡ 〈E
1 · · ·Eǫ
n 〉, (2.1)
where E
j are 2× 2 elementary matrices acting on the j-th site as
E++j =
+ Szj , E
− Szj ,
E+−j =
= S+j = S
j + iS
j , E
= S−j = S
j − iS
The multiple integral formula of the density matrix element for the massless XXZ chain
reads [9]
,··· ,ǫ′n
ǫ1,··· ,ǫn =(−ν)−n(n−1)/2
· · ·
sinh(xa − xb)
sinh[(xa − xb − ifabπ)ν]
sinhyk−1 [(xk + iπ/2)ν] sinh
n−yk [(xk − iπ/2)ν]
coshn xk
, (2.2)
where the parameter ν is related to the anisotropy as ∆ = cosπν and fab and yk are
determined as
fab = (1 + sign[(s
′ − a + 1/2)(s′ − b+ 1/2)])/2,
y1 > y2 > · · · > ys′, ǫ′yi = +
ys′+1 > · · · > yn, ǫn+1−yi = −. (2.3)
In the case of ∆ = 1/2, namely ν = 1/3, the significant simplification occurs in the multiple
integrals due to the trigonometric identity
sinh(xa−xb) = 4 sinh[(xa−xb)/3] sinh[(xa−xb+iπ)/3] sinh[(xa−xb−iπ)/3]. (2.4)
Actually if we note that the parameter fab takes the value 0 or 1, the first factor in the
multiple integral at ν = 1/3 can be decomposed as
sinh(xa − xb)
sinh[(xa − xb − iπ)/3]
= 4 sinh
xa − xb
xa − xb + iπ
= −1 + ωe
(xa−xb) + ω−1e−
(xa−xb), (2.5)
sinh(xa − xb)
sinh[(xa − xb)/3]
= 4 sinh
xa − xb + iπ
xa − xb − iπ
= 1 + e
(xa−xb) + e−
(xa−xb), (2.6)
where ω = eiπ/3. Expanding the trigonometoric functions in the second factor into exponen-
tials
sinhy−1 [(x+ iπ/2)/3] sinhn−y [(x− iπ/2)/3]
= 21−n
ω1/2ex/3 − ω−1/2e−x/3
)y−1 (
ω−1/2ex/3 − ω1/2e−x/3
= 21−n
(−1)l+m
y − 1
ωy−l+m−(n+1)/2e
(n−2l−2m−1)x, (2.7)
we can explicitly evaluate the multiple integral by use of the formula
eαxdx
coshn x
= 2n−1B
, Re(n± α) > 0, (2.8)
where B(p, q) is the beta function defined by
B(p, q) =
tp−1(1− t)q−1dt, Re(p),Re(q) > 0. (2.9)
Table 1: Comparison with the asymptotic formula of the transverse correlation function
〈Sx1Sx2 〉 〈Sx1Sx3 〉 〈Sx1Sx4 〉 〈Sx1Sx5 〉 〈Sx1Sx6 〉
Exact −0.156250 0.0800781 −0.0671234 0.0521997 −0.0467664
Asymptotics −0.159522 0.0787307 −0.0667821 0.0519121 −0.0466083
In this way we have succeeded in calculating all the density matrix elements up to six lattice
sites. All the results are represented by single rational numbers, which are presented in
Appendix A. As for the spin-spin correlation functions, we have newly obtained the fourth-
and fifth-neighbour transverse two-point correlation function
〈Sx1Sx2 〉 = −
= −0.15625,
〈Sx1Sx3 〉 =
= 0.080078125,
〈Sx1Sx4 〉 = −
65536
= −0.0671234130859375,
〈Sx1Sx5 〉 =
1751531
33554432
= 0.0521996915340423583984375,
〈Sx1Sx6 〉 = −
3213760345
68719476736
= −0.046766368104727007448673248291015625.
The asymptotic formula of the transverse two-point correlation function for the massless
XXZ chain is established in [35, 36]
〈Sx1Sx1+n〉 ∼ Ax(η)
(−1)n
− Ãx(η)
+ · · · , η = 1− ν,
Ax(η) =
8(1− η)2
sinh(ηt)
sinh(t) cosh[(1− η)t]
− ηe−2t
Ãx(η) =
2η(1− η)
cosh(2ηt)e−2t − 1
2 sinh(ηt) sinh(t) cosh[(1− η)t]
sinh(ηt)
η2 + 1
, (2.10)
which produces a good numerical value even for small n as is shown in Table 1. Note that the
longitudinal correlation function was obtained up to eighth-neighbour correlaion 〈Sz1Sz9〉 from
the multiple integral representation for the generating function [32]. Note also that up to
third-neighbour both longitudinal and transverse correlation functions for general anisotropy
∆ were obtained in [21].
3 Reduced density matrix and entanglement entropy
Below let us discuss the reduced density matrix for a sub-chain and the entanglement entropy.
The density matrix for the infinite system at zero temperature has the form
ρT ≡ |GS〉〈GS|, (3.1)
0 10 20 30 40 50 60
0 10 20 30
Figure 1: Eigenvalue-distribution of density matrices
Table 2: Entanglement entropy S(n) of a finite sub-chain of length n
S(1) S(2) S(3) S(4)
1 1.3716407621868583 1.5766810784924767 1.7179079372711414
S(5) S(6)
1.8262818282012363 1.9144714710902746
where |GS〉 denotes the ground state of the total system. We consider a finite sub-chain of
length n, the rest of which is regarded as an environment. We define the reduced density
matrix for this sub-chain by tracing out the environment from the infinite chain
ρn ≡ trEρT =
,··· ,ǫ′n
ǫ1,··· ,ǫn
ǫj ,ǫ
. (3.2)
We have numerically evaluate all the eigenvalues ωα (α = 1, 2, · · · , 2n) of the reduced density
matrix ρn up to n = 6. We show the distribution of the eigenvalues in Figure 1. The
distribution is less degenerate comapared with the isotropic case ∆ = 1 shown in [24]. In the
odd n case, all the eigenvalues are two-fold degenerate due to the spin-reverse symmetry.
Subsequently we exactly evaluate the von Neumann entropy (Entanglement entropy)
defined as
S(n) ≡ −trρn log2 ρn = −
ωα log2 ωα. (3.3)
The exact numerical values of S(n) up to n = 6 are shown in Table 2. By analyzing the
behaviour of the entanglement S(n) for large n, we can see how long quantum correlations
reach [37]. In the massive region ∆ > 1, the entanglement entropy will be saturated as n
grows due to the finite correlation length. This means the ground state is well approximated
by a subsystem of a finite length corresponding to the large eigenvalues of reduced density
matrix. On the other hand, in the massless case −1 < ∆ ≤ 1, the conformal field theory
predict that the entanglement entropy shows a logarithmic divergence [38]
S(n) ∼ 1
log2 n + k∆. (3.4)
1 2 3 4 5 6
Exact
Asymptotics
Figure 2: Entanglement entropy S(n) of a finite sub-chain of length n
Our exact results up to n = 6 agree quite well with the asymptotic formula as shown in Figure
2. We estimate the numerical value of the constant term k∆=1/2 as k∆=1/2 ∼ S(6)− 13 log2 6 =
1.0528. This numerical value is slightly smaller than the isotropic case ∆ = 1, where the
constant k∆=1 is estimated as k∆=1 ∼ 1.0607 from the exact data for S(n) up to n = 6 [24].
At free fermion point ∆ = 0, the exact asymptotic formula has been obtained in [39]
S(n) ∼ 1
log2 n+ k∆=0,
k∆=0 = 1/3−
t sinh2(t/2)
− cosh(t/2)
2 sinh3(t/2)
/ ln 2. (3.5)
In this case the numerical value for the constant term is given by k∆=0 = 1.0474932144 · · · .
4 Summary and discussion
We have succeeded in obtaining all the density matrix elements on six lattice sites for XXZ
chain at ∆ = 1/2. Especially we have newly obtained the fourth- and fifth-neighbour
transverse spin-spin correlation functions. Our exact results for the transverse correlations
show good agreement with the asymptotic formula established in [35, 36]. Subsequently we
have calculated all the eigenvalues of the reduced density matrix ρn up to n = 6. From these
results we have exactly evaluated the entanglement entropy, which shows a good agreement
with the asymptotic formula derived via the conformal field theory. Finally, we remark
that similar procedures to evaluate the multiple integrals are also possible at ν = 1/n for
n = 4, 5, 6, · · · , since there are similar trigonometric identities as (2.4). We will report the
calculation of correlation functions for these cases in subsequent papers.
Acknowledgement
The authors are grateful to K. Sakai for valuable discussions. This work is in part sup-
ported by Grant-in-Aid for the Scientific Research (B) No. 18340112. from the Ministry of
Education, Culture, Sports, Science and Technology, Japan.
Appendix A Density matrix elements up to n = 6
In this appendix we present all the independent density matrix elements defined in eq. (2.1)
up to n = 6. Other elements can be computed from the relations
,··· ,ǫ′n
ǫ1,··· ,ǫn = 0 if
ǫj 6=
ǫ′j , (A.1)
,··· ,ǫ′n
ǫ1,··· ,ǫn = P
ǫ1,··· ,ǫn
,··· ,ǫ′n
,··· ,−ǫ′n
−ǫ1,··· ,−ǫn
ǫ′n,··· ,ǫ
ǫn,··· ,ǫ1 , (A.2)
,··· ,ǫ′n
+,ǫ1,··· ,ǫn
,··· ,ǫ′n
−,ǫ1,··· ,ǫn
,··· ,ǫ′n,+
ǫ1,··· ,ǫn,+
,··· ,ǫ′n,−
ǫ1,··· ,ǫn,−
,··· ,ǫ′n
ǫ1,··· ,ǫn , (A.3)
and the formula for the EFP [33, 34]
P (n) = P
+,··· ,+
+,··· ,+ = 2
(3k + 1)!
(n+ k)!
. (A.4)
Appendix A.1 n ≤ 4
P−++− = −
= −0.3125, P−++++− =
= 0.0800781,
P−++++−++ = −
= −0.0269775, P−+++++−+ =
65536
= 0.0240936,
P−++++++− = −
32768
= −0.00881958, P+−+++−++ =
16384
= 0.0632935,
P+−++++−+ = −
32768
= −0.0611877, P−−+++−+− = −
65536
= −0.0583038,
P−−++++−− =
65536
= 0.0212555, P−+−++−+− =
32768
= 0.149017,
P−++−+−−+ =
32768
= 0.0943298.
Appendix A.2 n = 5
P−+++++−+++ = −
14721
8388608
= −0.00175488, P−++++++−++ =
37335
16777216
= 0.00222534,
P−+++++++−+ = −
48987
33554432
= −0.00145993, P−++++++++− =
13911
33554432
= 0.00041458,
P+−++++−+++ =
179699
33554432
= 0.00535545, P+−+++++−++ = −
120337
16777216
= −0.00717264,
P+−++++++−+ =
165155
33554432
= 0.004922, P++−++++−++ =
168313
16777216
= 0.0100322,
P−−++++−−++ =
31069
2097152
= 0.0148149, P−−++++−+−+ = −
411583
16777216
= −0.0245323,
P−−++++−++− =
196569
16777216
= 0.0117164, P−−+++++−+− = −
281271
33554432
= −0.00838253,
P−−++++++−− =
79673
33554432
= 0.00237444, P−+−+++−−++ = −
1441787
33554432
= −0.0429686,
P−+−+++−++− = −
1261655
33554432
= −0.0376002, P−+−++++−+− =
59459
2097152
= 0.0283523,
P−++−++−++− =
1575515
33554432
= 0.046954, P−+++−+−−++ = −
696151
33554432
= −0.0207469,
P−+++−+−+−+ =
1366619
33554432
= 0.0407284.
Appendix A.3 n = 6
P−++++++−++++ = −
1546981
34359738368
= −0.0000450231, P−+++++++−+++ =
5095899
68719476736
= 0.0000741551,
P−++++++++−++ = −
2366275
34359738368
= −0.0000688677, P−+++++++++−+ =
2455833
68719476736
= 0.0000357371,
P−++++++++++− = −
284577
34359738368
= −8.28228× 10−6, P+−+++++−++++ =
2927709
17179869184
= 0.000170415,
P+−++++++−+++ = −
20086627
68719476736
= −0.000292299, P+−+++++++−++ =
19268565
68719476736
= 0.000280395,
P+−++++++++−+ = −
10295153
68719476736
= −0.000149814, P++−+++++−+++ =
17781349
34359738368
= 0.000517505,
P++−++++++−++ = −
35087523
68719476736
= −0.000510591, P−−+++++−−+++ =
48421023
34359738368
= 0.00140924,
P−−+++++−+−++ = −
214080091
68719476736
= −0.00311528, P−−+++++−++−+ =
88171589
34359738368
= 0.00256613,
P−−+++++−+++− = −
57522267
68719476736
= −0.000837059, P−−++++++−−++ =
56776545
34359738368
= 0.00165241,
P−−++++++−+−+ = −
154538459
68719476736
= −0.00224883, P−−++++++−++− =
60809571
68719476736
= 0.000884896,
P−−+++++++−−+ =
6708473
8589934592
= 0.000780969, P−−+++++++−+− = −
33366621
68719476736
= −0.000485548,
P−−++++++++−− =
3860673
34359738368
= 0.00011236, P−+−++++−−+++ = −
85706851
17179869184
= −0.0049888,
P−+−++++−+−++ =
12211375
1073741824
= 0.0113727, P−+−++++−++−+ = −
332557469
34359738368
= −0.0096787,
P−+−++++−+++− =
56183761
17179869184
= 0.00327033, P−+−+++++−−++ = −
430452959
68719476736
= −0.00626391,
P−+−+++++−+−+ =
606065059
68719476736
= 0.00881941, P−+−+++++−++− = −
123612511
34359738368
= −0.0035976,
P−+−++++++−−+ = −
108202041
34359738368
= −0.00314909, P−+−++++++−+− =
70061315
34359738368
= 0.00203905,
P−++−+++−−+++ =
7860495
1073741824
= 0.00732066, P−++−+++−+−++ = −
591759525
34359738368
= −0.0172225,
P−++−+++−++−+ =
1044016671
68719476736
= 0.0151924, P−++−+++−+++− = −
367905053
68719476736
= −0.00535372,
P−++−++++−−++ =
676957849
68719476736
= 0.00985103, P−++−++++−+−+ = −
988973861
68719476736
= −0.0143915,
P−++−++++−++− =
6581795
1073741824
= 0.00612977, P−++−+++++−−+ =
363618785
68719476736
= 0.00529135,
P−+++−++−−+++ = −
185522333
34359738368
= −0.00539941, P−+++−++−+−++ =
901633567
68719476736
= 0.0131205,
P−+++−++−++−+ = −
103539423
8589934592
= −0.0120536, P−+++−++−+++− =
38524625
8589934592
= 0.00448486,
P−+++−+++−−++ = −
267901987
34359738368
= −0.00779697, P−+++−+++−+−+ =
12750645
1073741824
= 0.011875,
P−+++−++++−−+ = −
309855965
68719476736
= −0.004509, P−++++−+−−+++ =
29410257
17179869184
= 0.0017119,
P−++++−+−+−++ = −
296882461
68719476736
= −0.00432021, P−++++−+−++−+ =
35985105
8589934592
= 0.00418922,
P−++++−++−−++ =
92176287
34359738368
= 0.00268268, P+−−++++−−+++ =
202646807
34359738368
= 0.0058978,
P+−−++++−+−++ = −
972245985
68719476736
= −0.014148, P+−−++++−++−+ =
217687057
17179869184
= 0.0126711,
P+−−+++++−+−+ = −
211696415
17179869184
= −0.0123224, P+−−++++++−−+ =
78922695
17179869184
= 0.00459391,
P+−+−+++−+−++ =
1196499417
34359738368
= 0.0348227, P+−+−+++−++−+ = −
2209522727
68719476736
= −0.0321528,
P+−+−++++−+−+ =
1108384987
34359738368
= 0.0322582, P+−++−++−++−+ =
530683585
17179869184
= 0.0308899,
P+−++−+++−−++ =
347202525
17179869184
= 0.0202098, P−−−++++−−++− = −
268623007
68719476736
= −0.00390898,
P−−−++++−+−+− =
46285135
8589934592
= 0.0053883, P−−−++++−++−− = −
136974885
68719476736
= −0.00199325,
P−−−+++++−+−− =
19939391
17179869184
= 0.00116063, P−−−++++++−−− = −
18442085
68719476736
= −0.000268368,
P−−+−+++−−++− =
1018463205
68719476736
= 0.0148206, P−−+−+++−+−+− = −
1454513249
68719476736
= −0.021166,
P−−+−+++−++−− =
277721503
34359738368
= 0.00808276, P−−+−++++−+−− = −
335265249
68719476736
= −0.00487875,
P−−++−++−−++− = −
369408975
17179869184
= −0.0215024, P−−++−++−+−+− =
1104236607
34359738368
= 0.0321375,
P−−++−++−++−− = −
880560357
68719476736
= −0.0128138, P−−++−+++−−+− = −
876924641
68719476736
= −0.0127609,
P−−+++−+−−−++ =
113631201
17179869184
= 0.00661421, P−−+++−+−−+−+ = −
292857807
17179869184
= −0.0170466,
P−−+++−+−+−−+ =
548645951
34359738368
= 0.0159677, P−−+++−++−−−+ = −
377925345
68719476736
= −0.00549954,
P−+−+−++−−++− =
1719255909
34359738368
= 0.0500369, P−+−+−++−+−+− = −
5350158879
68719476736
= −0.0778551,
P−+−++−+−−+−+ =
1565770597
34359738368
= 0.0455699, P−+−++−+−+−−+ = −
3059753503
68719476736
= −0.0445253,
P−++−−++−−++− = −
2117554719
68719476736
= −0.0308145.
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	Introduction
	Analytical evaluation of multiple integral
	Reduced density matrix and entanglement entropy
	Summary and discussion
	Density matrix elements up to n=6
