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Thermo quantum dynamics in the transverse Ising chain
ARTICLE IN PRESS
Physica A 368 (2006) 449–458 www.elsevier.com/locate/physa
Thermo quantum dynamics in the transverse Ising chain Asuka Sugiyama, Hidenori Suzuki, Masuo Suzuki Department of Applied Physics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan Received 3 October 2005; received in revised form 13 December 2005 Available online 25 January 2006
Abstract Thermo quantum dynamics, which is formulated by using the eigenstate jCðmÞ max ðbÞi of the quantum transfer-matrix with the maximum eigenvalue (where m denotes the Trotter number and b ¼ 1=kB T), is applied to a transverse Ising chain. In order to exemplify the formulation of the thermo quantum dynamics for ‘‘local’’ operators, we show how to evaluate both the thermal average hsxj i and the correlation function hsxj sxjþr i. Furthermore, it is demonstrated for the first time that the limit m ! 1 of the thermal state vector jCðmÞ max ðbÞi exists in the diagonalized representation and that this thermal state vector can be regarded as the ground state vector of the corresponding virtual system. r 2006 Elsevier B.V. All rights reserved. Keywords: Quantum transfer-matrix; Thermo quantum dynamics; Transverse Ising chain; Correlation function
1. Introduction Thermodynamical properties of quantum systems have been studied for long years. There are many methods to solve this problem. The Suzuki–Trotter transformation (ST-transformation) [1–3] of a quantum system to the corresponding classical system has been effectively applied for both analytical and numerical studies. This general scheme is based on the equivalence theorem [3] that a d-dimensional quantum system with finite-range interactions is equivalent to the corresponding ðd þ 1Þ-dimensional classical system with finite interactions. The quantum Monte–Carlo method [3–5] is one of the applications of the above equivalence theorem. Another application yields the quantum transfer-matrix method [6–8], which has been also used effectively for both analytical study [9–22] and numerical study [23–30]. Recently, Suzuki proposed the thermo quantum dynamics [31,32] on the basis of the quantum transfermatrix. By this formulation, the thermal average of a ‘‘local’’ operator in the thermodynamic limit is expressed in terms of the expectation value on a vector in single Hilbert space, in contrast to the usage of the double Hilbert space in thermo field dynamics [33–35]. In the present paper, we formulate it explicitly in the transverse Ising chain. In Section 2, we give a brief summary of the quantum transfer-matrix method, and summarize the formulation of thermo quantum dynamics. In Section 3, an explicit application of the thermo quantum dynamics to a transverse Ising chain is presented. Discussion about the thermal state vector which plays an Corresponding author. Tel.: +81 3 3260 4271; fax: +81 3 3260 9020.
E-mail addresses: [email protected] (A. Sugiyama), [email protected] (H. Suzuki), [email protected] (M. Suzuki). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.12.055
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important role in the thermo quantum dynamics is given in Section 4. Finally, we summarize the present work in Section 5. 2. Brief review of thermo quantum dynamics As is well known, the transfer-matrix method was introduced to solve analytically Ising models and other classical systems in one and two dimensions [36–38]. According to the equivalence theorem [3], a ddimensional quantum system is mapped to the corresponding ðd þ 1Þ-dimensional classical system. Then, two kinds of transfer-matrices are defined in this ðd þ 1Þ-dimensional classical system [6–8], namely a ‘‘real-space’’ transfer-matrix defined in the real space and a ‘‘virtual-space’’ transfer-matrix defined in the virtual space. The partition function of the original quantum system described by Hamiltonian H can be written by using the virtual space transfer-matrix Tm as follows [6–8]: ZðbÞ ¼ Tr ebH ¼ lim ZðmÞ N ðbÞ; m!1
N Z ðmÞ N ðbÞ ¼ Tr ðTm Þ ,
(2.1)
where N denotes the system size, m is the Trotter number and b ¼ 1=kB T. Only the maximum eigenvalue lðmÞ max of the transfer-matrix Tm is sufficient to describe the free energy in the thermodynamic limit [6,7]: f ¼
kB T log Z ¼ lim log lðmÞ max . m!1 N
(2.2)
Here, we have used the interchangeability theorem [6,7] of the limits N ! 1 and m ! 1. The correlation length of the system is given by [8] ! lðmÞ 1 max x ¼ lim log ðmÞ , (2.3) m!1 l2 where lðmÞ 2 denotes the second largest eigenvalue of Tm . The thermo quantum dynamics [31,32] is formulated by using the eigenvector jCðmÞ max ðbÞi of the quantum transfer-matrix Tm with the maximum eigenvalue lðmÞ : max ðmÞ ðmÞ Tm jCðmÞ max ðbÞi ¼ lmax jCmax ðbÞi,
(2.4)
with the normalization ðmÞ hCðmÞ max ðbÞjCmax ðbÞi ¼ 1.
(2.5)
The thermal average hQi of a ‘‘local’’ operator Q in the thermodynamic limit N ! 1 is expressed as [31,32] ^ ðmÞ hQi ¼ lim hCðmÞ max ðbÞjQjCmax ðbÞi. m!1
(2.6)
Note that the local operator Q is also transformed into an operator Q^ which operates to the thermal state jCðmÞ max ðbÞi, as will be shown explicitly in the case of the transverse Ising chain in the succeeding section. 3. Explicit formulation in the transverse Ising chain In the present section, we consider the transverse Ising chain [39–42] described by the following Hamiltonian: bH ¼ K
N X i¼1
szi sziþ1 þ g
N X i¼1
sxi ¼ bH1 bH2 ,
(3.1)
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where sai denotes the a-component of the Pauli matrices at site i. Using the ST-transformation [1–3], the partition function can be represented by the two-dimensional Ising model as follows [6–8]: bH1 =m bH2 =m m Z ðmÞ e Þ N ðbÞ ¼ Tr ðe " # N X m X X K Nm 0 si;j siþ1;j þ gm si;j si;jþ1 , ¼ ðAm Þ exp m fs ¼1g i¼1 j¼1
ð3:2Þ
i;j
where fsij g denotes the Ising variables, and the parameters Am and g0m are defined by 1=2 n g o 1 2g 1 sinh and g0m ¼ log coth Am ¼ . 2 m 2 m
(3.3)
Then the quantum transfer-matrix is given by [7] 1=2
1=2
Tm ¼ V 2 V 1 V 2 ,
(3.4)
where V1 ¼
" # m=2 m X 2K 0 z and sinh exp K m sj m j¼1
" V 2 ¼ exp
g0m
m X
# sxj sxjþ1
.
(3.5)
j¼1
Here, saj denotes the a-component of the Pauli matrices at site j in the virtual space and the parameter K 0m is defined by 1 K 0 K m ¼ log coth . (3.6) 2 m Using the above quantum transfer-matrix, the partition function Z ðmÞ N ðbÞ can be expressed as 1=2
1=2
Nm Z ðmÞ Tr ðV 2 V 1 V 2 ÞN N ðbÞ ¼ ðAm Þ
¼ ðAm ÞNm Tr ðTm ÞN .
ð3:7Þ
This transfer-matrix Tm can be diagonalized by the following well-known procedures [43]: (i) Jordan–Wigner transformation on the fermi operators al and ayl : l 1 Y
½szi s l
ayl ¼ sþ l
l 1 Y
½szi
(3.8)
eip=4 X iql e ðcos fq xq sin fq xyq Þ al ¼ pffiffiffiffi m q
(3.9)
eip=4 X iql ayl ¼ pffiffiffiffi e ðcos fq xyq sin fq xq Þ, m q
(3.10)
al ¼
and
i¼1
i¼1
and (ii) the canonical transformation
and
where xq and xyq are fermi operators, and fq is given [43] in 1=2 ð1 eiq tanh K 0m tanh g0m Þ ð1 eiq tanh K 0m coth g0m Þ iðqþ2fq Þ e ¼ ð1 eiq tanh K 0m tanh g0m Þ ð1 eiq tanh K 0m coth g0m Þ
(3.11)
and q¼
p 3p 5p ðm 1Þp ; ; ;...; m m m m
(3.12)
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assuming that m takes an even value. Then the quantum transfer-matrix Tm can be diagonalized as [43] " m=2 # X 2K 1 1=2 1=2 y exp eq;m xq xq , (3.13) Tm V 2 V 1 V 2 ¼ sinh m 2 q where eq;m is given by the larger root of the following equation: 2K 2g 2K 2g cosh eq;m ¼ coth coth cosech cosech . m m m m The maximum eigenvalue of the quantum transfer-matrix Tm can be written as [43,44] " # m=2 2K 1X ðmÞ exp eq;m . lmax ¼ 2 sinh m 2 q
(3.14)
(3.15)
ðmÞ The eigenvector jCðmÞ max ðbÞi of Tm with the maximum eigenvalue lmax is given by the x-vacuum j0i:
for arbitrary q.
xq j0i ¼ 0
(3.16)
Now we can calculate the thermal average of any ‘‘local’’ operators Q in the thermodynamic limit by using the thermo quantum dynamics [31,32]. Next we show how to evaluate both the expectation value hsxj i and the correlation function hsxj sxjþr i in order to exemplify the formulation of the thermo quantum dynamics for local operators. 3.1. Thermal average hsxj i Noting the following matrix elements: 0
0
hsjsx exp½gsx =mjs0 i ¼ e2gm ss hsj exp½gsx =mjs0 i,
(3.17) 2g0m sj;l sj;lþ1
we find that the local operator sxj can be expressed as e in the mapped two-dimensional Ising model. Here, the index l of the spin variable sj;l can be chosen an arbitrary number, because of the translation invariance in the Trotter direction. Then we can show the following relation: X 0 e2gm sj;l sj;lþ1 Tr sxj ðebH1 =m ebH2 =m Þm ¼ ðAm ÞNm "
fsk;i g
# N X m X K sk;i skþ1;i þ g0m sk;i sk;iþ1 exp m k¼1 i¼1 0
x x
¼ ðAm ÞNm Tr e2gm sl slþ1 ðTm ÞN .
ð3:18Þ
Thus, the operator sxj is transformed into the following form: 0
x x
sxj ! e2gm sl slþ1 ¼ cosh 2g0m sxl sxlþ1 sinh 2g0m .
ð3:19Þ
By the formulation of the thermo quantum dynamics, the thermal average hsxj i is given by hsxj i ¼ lim hsxj im m!1
0 x x 0 ðmÞ ¼ lim hCðmÞ max ðbÞjðcosh 2gm sl slþ1 sinh 2gm ÞjCmax ðbÞi m!1
¼ lim ½cosh 2g0m h0jsxl sxlþ1 j0i sinh 2g0m . m!1
ð3:20Þ
ðmÞ Here, h im ¼ hCðmÞ max ðbÞj jCmax ðbÞi denotes the average in the mth approximated system. The nearest neighbour correlation function, h0jsxl sxlþ1 j0i, in the mapped two-dimensional Ising model can be written in the
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following form [43]: h0jsxl sxlþ1 j0i ¼ h0jðayl al Þðaylþ1 þ alþ1 Þj0i 1 X iðqþ2fq Þ e . ¼ m q Thus, we obtain the following expression of hsxj im , sinh 2g0m X iðqþ2fq Þ hsxj im ¼ cosh 2g0m þ e m q " # X 2 2g c ¼ cosh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , m sinhð2g=mÞ q40 m a2 b2 using Eq. (3.11), where a, b and c are given by 2K 2g a ¼ cosh cosh cos q, m m b ¼ sinh
2K 2g sinh , m m
2K c ¼ cosh m
ð3:21Þ
ð3:22Þ
(3.23)
(3.24)
2g cosh cos q. m
Using the following formula: Z 1 p A þ B cos y B bA aB dy ¼ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , p 0 a þ b cos y b b ða þ bÞða bÞ
(3.25)
(3.26)
with A ¼ cosh
2K 2g sinh2 m m
(3.27)
and
2K 2g 2g B ¼ sinh cosh sinh , m m m the average hsxj im can be rewritten as Z pX 2 a1 x dy, hsj im ¼ mp 0 q40 a0 cos q
(3.28)
(3.29)
where 2K 2g 2K 2g a0 ¼ cosh cosh þ sinh sinh cos y, m m m m
(3.30)
2K 2g 2K 2g sinh þ sinh cosh cos y. m m m m
(3.31)
a1 ¼ cosh
Here, q is given by Eq. (3.12), so that we can sum over q in the following form: m=2 X j¼1
m 1 m cosh1 a0 . ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi tanh a0 cosðð2j 1=mÞpÞ 2 a2 1 2 0
(3.32)
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Then we arrive at the expression Z m 1 p a1 qffiffiffiffiffiffiffiffiffiffiffiffiffi tanh cosh1 a0 dy. hsxj im ¼ p 0 2 a20 1
(3.33)
Now the limit m ! 1 can be taken as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m lim cosh1 a0 ¼ K 2 þ g2 þ 2Kg cos y m!1 2 and
(3.34)
a1 g þ K cos y lim qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 K þ g2 þ 2Kg cos y a20 1
(3.35)
m!1
Finally, we arrive at the integral representation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z p 1 ðg þ K cos yÞ tanh K 2 þ g2 þ 2Kg cos y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hsxj i ¼ lim hsxj im ¼ dy. m!1 p 0 K 2 þ g2 þ 2Kg cos y We can confirm that this result agrees with the formula directly derived from the free energy [40,41]. 3.2. Correlation function hsxj sxjþr i The operator sxj sxjþr can be transformed into the following form: 0
x x
0
x x
sxj sxjþr ! e2gm sl slþ1 ðTm Þr e2gm sl slþ1 ðTm Þr
(3.36)
by using Eq. (3.19). The correlation function hsxj sxjþr i can be expressed as hsxj sxjþr i ¼ lim hsxj sxjþr im m!1
0
x x
0
x x
2gm sl slþ1 ¼ lim hCðmÞ ðTm Þr e2gm sl slþ1 ðTm Þr jCðmÞ max ðbÞj e max ðbÞi m!1
¼ lim h0jðcosh2 2g0m cosh 2g0m sinh 2g0m sxl sxlþ1 m!1
cosh 2g0m sinh 2g0m ðTm Þr sxl sxlþ1 ðTm Þr þ sinh2 2g0m sxl sxlþ1 ðTm Þr sxl sxlþ1 ðTm Þr Þj0i
ð3:37Þ
by the formulation of the thermo quantum dynamics. Let fjnigPbe a complete set composed of eigen vectors of Tm with eigenvalues flðmÞ jnihnj into the last term of Eq. (3.37), n g. Inserting the unit operator we have hsxj sxjþr im ¼ cosh2 2g0m 2 cosh 2g0m sinh 2g0m h0jsxl sxlþ1 j0i !r X lðmÞ
2 0 n
h0jsx sx jni 2 . þ sinh 2gm l lþ1 ðmÞ lmax n
ð3:38Þ
Noting that the operator sxl sxlþ1 can be written as a quadratic form of x-fermion representation, we find that the matrix element h0jsxl sxlþ1 jni has a non-zero value only when the state jni is equal to the vacuum state j0i or y y the two-particle excited state xyq xyq0 j0i ðq4q0 Þ. The eigenvalue lðmÞ qq0 of the excited state xq xq0 j0i has the following form: " # m=2 2K 1X ðmÞ lqq0 ¼ sinh exp eq eq0 þ eq00 . (3.39) m 2 q00 Thus, we can rewrite Eq. (3.38) as hsxj sxjþr im ¼ fcosh 2g0m sinh 2g0m h0jsxl sxlþ1 j0ig2 þ sinh2 2g0m
X q4q0
erðeq þeq0 Þ jh0jsxl sxlþ1 xyq xyq0 j0ij2 .
(3.40)
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The first term of Eq. (3.40) is obviously equal to hsxj i2m . By using the transformation of sxl into the x-fermion, the matrix element h0jsxl sxlþ1 xyq xyq0 j0i can be written as 0
i eiðqþq Þl iðqþfq fq0 Þ 0 ðe eiðq fq þfq0 Þ Þ. (3.41) m Substituting this matrix element into Eq. (3.40) and noting that eq is an even function and fq is an odd function with respect to q, we can rewrite the second term of Eq. (3.40) as X erðeq þeq0 Þ jh0jsxl sxlþ1 xyq xyq0 j0ij2 sinh2 2g0m h0jsxl sxlþ1 xyq xyq0 j0i ¼
q4q0
sinh 2g0m X rðeq þeq0 Þ iðqþfq fq0 Þ 0 0 e ðe eiðq fq þfq0 Þ Þ ðeiðqþfq fq0 Þ eiðq fq þfq0 Þ Þ 2 m q4q0 2
¼
sinh2 2g0m X rðeq þeq0 Þ 0 e ð1 eiðqþ2fq Þiðq þ2fq0 Þ Þ m2 0 q;q 8 !2 !2 9 = 2 0 < X X sinh 2gm req req þiðqþ2fq Þ e e ¼ : q ; m2 q ( ) sinh2 2g0m Y X req ¼ e 1 eiðqþ2fq Þ . m2 q ¼
Thus, we obtain the following expression of hsxj sxjþr im : ( ) sinh2 2g0m Y X req x x x 2 iðqþ2fq Þ hsj sjþr im ¼ hsj im þ e ð1 e Þ m2 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!r " !# Y X a a2 b2 1 c x 2 ¼ hsj im þ 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , b m2 sinh2 ð2g=mÞ a2 b2 q40
ð3:42Þ
ð3:43Þ
using Eqs. (3.11) and (3.14), where a, b and c are given by Eqs. (3.23)–(3.25). Using the following formula pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!r Z 1 p cosðryÞ 1 a þ a2 b2 dy ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (3.44) p 0 ða þ bÞ cos y b a2 b2 the correlation function hsxj sxjþr im is represented as 1 hsxj sxjþr im ¼ hsxj i2m þ 2 m sinh2 ð2g=mÞ " # Y X ð1Þr Z p ða þ cÞ cosðryÞ þ b cosfðr 1Þyg dy . 2 ða þ bÞ cos y p 0 q40 Here, Eq. (3.32) can be used to sum over q, and then we have X b b cos q 2 ðb a0 b1 Þ tanhððm=2Þ cosh1 a0 Þ 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi b1 þ 0 ¼ , a0 cos q m a0 1 q40 where a0 is given by Eq. (3.30) and 2K 2g 2K 2g b0 ¼ cosh 1 þ cosh cosðryÞ þ sinh sinh cosfðr 1Þyg, m m m m b1 ¼
1 þ cosh
2g cosðryÞ. m
ð3:45Þ
(3.46)
(3.47)
(3.48)
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Thus, we obtain hsxj sxjþr im
¼
hsxj i2m
Z m 1Y p b0 a0 b1 qffiffiffiffiffiffiffiffiffiffiffiffiffi tanh cosh1 a0 dy. p 0 sinhð2g=mÞ a2 1 2
(3.49)
0
Now we can take the limit m ! 1. This limit yields the expression lim
m!1
b0 a0 b1 K cosfðr 1Þyg þ g cosðryÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 K 2 þ g2 þ 2Kg cos y sinhð2g=mÞ a0 1
(3.50)
Finally, the correlation function hsxj sxjþr i can be expressed as hsxj sxjþr i ¼ lim hsxj sxjþr im m!1 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p K cosfðr 1Þyg þ g cosðryÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh K 2 þ g2 þ 2Kg cos y dy ¼ hsxj i2 p 0 K 2 þ g2 þ 2Kg cos y Z p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 K cosfðr þ 1Þyg þ g cosðryÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh K 2 þ g2 þ 2Kg cos y dy. p 0 K 2 þ g2 þ 2Kg cos y
ð3:51Þ
This explicit result can be derived, as a special limit, from the general formula of correlation functions for the XY chain [45] in the presence of the transverse field.
4. Thermal state vector of the transverse Ising chain In the present section, we give discussions about the thermal state jCðmÞ max ðbÞi. As is illustrated in Fig. 1, we can construct the virtual transverse Ising chain described by the following Hamiltonian: HðmÞ v ¼ J v
m X
szj szjþ1 Gv
j¼1
m X
sxj þ const.
(4.1)
j¼1
Here, j denotes the site defined in the virtual space. The partition function of this virtual system can be written as follows: bHv ZðmÞ v ¼ Tr e
¼ C Tr eAv þBv ¼ lim C Tr ðeAv =N eBv =N ÞN , N!1
Fig. 1. Transformation from an original system to a virtual system.
ð4:2Þ
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where Av ¼ bJ v
m X
szj szjþ1 ;
j¼1
Bv ¼ bGv
m X
sxj
(4.3)
j¼1
and C is constant. The operators exp½Av =N and exp½Bv =N can be identified as V 2 and V 1 , respectively, with setting the parameters bJ v 1 g ¼ log coth 2 m N
(4.4)
and bGv 1 K (4.5) ¼ log coth . 2 m N Because the maximum eigenvalue of the above transfer-matrix gives the ground energy of the virtual system Hv , the thermal state jCðmÞ max i can be identified with the ground state of the above virtual Hamiltonian in the thermodynamic limit ðN ! 1Þ. With diagonalizing the virtual Hamiltonian, we can find that the ground state can be written as the x-vacuum state. Then we find that the single Hilbert space which was introduced in the formulation of thermo quantum dynamics is described by the above virtual space Hamiltonian. Although the parameters J v and Gv diverge in the limit m ! 1, there is no problem in the formulation of thermo quantum dynamics with the correct procedure. For example, the quantum transfer-matrix in the spin fsx;z j g representation has singular elements in the limit m ! 1 (see Eqs. (3.3)–(3.6)). However, the diagonalized form has no singular factor in the limit m ! 1. This fact can be easily shown from the following form of the diagonalized quantum transfer-matrix: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y y ðAm Þm TðdÞ ¼ fa þ ð1 2x x Þ a2 b2 g1=2 , (4.6) q q m q
where a and b are given by Eqs. (3.23) and (3.24), respectively. Although the singularity of Tm in the spin representation may leads to the nonexistence of the limit m ! 1 of the thermal state, the state associated to the diagonalized quantum transfer-matrix do exist as a x-vacuum, as was mentioned by Suzuki [31]. The local operator sxj also has singular elements in the spin fsx;z j g representation (Eq. (3.19)). However, it is easy to find that the following x-fermion representation: sinh 2g0m X iq2ifq y y ½e ðixq xq þ xyq xq xq xyq þ ixq xq Þ (4.7) sxj ! cosh 2g0m m q has no singular elements in the limit m ! 1. Here, we have to mention that the limit m ! 1 has to be taken after the summation over all q, because q also depends on the Trotter number m. 5. Summary In the present paper, we have applied the formulation of the thermo quantum dynamics to the transverse Ising model, and have shown how to evaluate the average values of local operators explicitly. It has been shown that the thermal state jCðmÞ max ðbÞi is described as the ground state of the corresponding virtual Hamiltonian. Although the quantum transfer-matrix has singular properties in some representations in the limit m ! 1, these singularities disappear after diagonalizing the quantum transfer-matrix. This is the first explicit demonstration of the existence of the limit m ! 1 of the thermal state vector jCðmÞ max ðbÞi. All these properties of the thermal state will be expected to be relevant in other models. References [1] M. Suzuki, J. Phys. Soc. Jpn. 21 (1966) 2274; See also H.F. Trotter, Proc. Am. Math. Soc. 10 (1959) 545.
ARTICLE IN PRESS 458 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
A. Sugiyama et al. / Physica A 368 (2006) 449–458 M. Suzuki, Commun. Math. Phys. 51 (1976) 183. M. Suzuki, Prog. Theor. Phys. 56 (1976) 1454. M. Suzuki, S. Miyashita, A. Kuroda, Prog. Theor. Phys. 58 (1977) 1377. M. Suzuki, Quantum Monte Carlo Method in Condensed Matter Physics, World Scientific, Singapore, 1993. M. Suzuki, Phys. Rev. B 31 (1985) 2957. M. Suzuki, M. Inoue, Prog. Theor. Phys. 78 (1987) 787. M. Inoue, M. Suzuki, Prog. Theor. Phys. 79 (1988) 645. J. Suzuki, Y. Akutsu, M. Wadati, J. Phys. Soc. Jpn. 59 (1990) 2667. M. Takahashi, Phys. Rev. B 43 (1991) 5788. J. Suzuki, T. Nagao, M. Wadati, Int. J. Mod. Phys. B 6 (1992) 1119. C. Destri, H.J. de Vega, Phys. Rev. Lett. 69 (1992) 2313. C. Destri, H.J. de Vega, Nucl. Phys. B 504 (PS) (1997) 621. A. Klu¨mper, Z. Phys. B 91 (1993) 507. H. Mizuta, T. Nagao, M. Wadati, J. Phys. Soc. Jpn. 63 (1994) 3951. A. Klu¨mper, Eur. Phys. J. B 5 (1998) 677. G. Ju¨ttner, A. Klu¨mper, Euro. Phys. Lett. 37 (1997) 335. G. Ju¨ttner, A. Klu¨mer, J. Suzuki, Nucl. Phys. B 487 (1997) 650. G. Ju¨ttner, A. Klu¨mper, J. Suzuki, J. Phys. A 30 (1997) 1881. A. Kuniba, T. Nakanishi, J. Suzuki, Int. J. Mod. Phys. A 9 (1994) 5215. A. Kuniba, K. Sakai, J. Suzuki, Nucl. Phys. B 525 (FS) (1998) 597. Y. Umeno, J. Phys. Soc. Jpn. 70 (2001) 2531. H. Betsuyaku, Prog. Theor. Phys. 73 (1985) 319; H. Betsuyaku, Prog. Theor. Phys. 75 (1986) 774, 808. T. Tsuzuki, Prog. Theor. Phys. 73 (1985) 1352. H. Betsyaku, T. Yokota, Prog. Theor. Phys. 75 (1986) 808. K. Kubo, Takada, J. Phys. Soc. Jpn. 55 (1986) 438. T. Koma, Prog. Theor. Phys. 71 (1993) 269. K. Kubo, Phys. Rev. B 46 (1992) 866. T. Delica, K. Kopinga, H. Leschke, K.K. Mon, Europhys. Lett. 15 (1991) 55. N. Hatano, M. Suzuki, Prog. Theor. Phys. 85 (1991) 481. M. Suzuki, J. Stat. Phys. 110 (2003) 945. M. Suzuki, Physica A 321 (2003) 334. H. Umezawa, H. Matsumoto, M. Tachiki, Thermo Field Dynamics and Condensed States, North-Holland, Amsterdam, 1982 and references cited therein. M. Suzuki, J. Phys. Soc. Jpn. 54 (1985) 4483. M. Suzuki, Phys. Lett. A 111 (1985) 440. H.A. Kramers, G.H. Wannier, Phys. Rev. 60 (1941) 252. E.W. Montroll, J. Chem. Phys. 9 (1941) 706; E.W. Montroll, J. Chem. Phys. 10 (1942) 61. R. Kubo, Busseiron Kenkyu 1 (1943) 1. E.H. Lieb, T.D. Shultz, D.C. Mattis, Ann. Phys. 16 (1961) 941. S. Katsura, Phys. Rev. 127 (1962) 1508. P. Pfeuty, Ann. Phys. 57 (1970) 79. B.K. Chakrabarti, A. Dutta, P. Sen, Quantum Ising Phases and Transitions in Transverse Ising Models, Springer, Berlin, 1996. T.D. Shultz, D.C. Mattis, E.H. Lieb, Rev. Mod. Phys. 36 (1964) 856. L. Onsager, Phys. Rev. 65 (1944) 117. E. Barouch, B.M. McCoy, Phys. Rev. A 3 (1971) 786.
Physica A 368 (2006) 449–458 www.elsevier.com/locate/physa
Thermo quantum dynamics in the transverse Ising chain Asuka Sugiyama, Hidenori Suzuki, Masuo Suzuki Department of Applied Physics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan Received 3 October 2005; received in revised form 13 December 2005 Available online 25 January 2006
Abstract Thermo quantum dynamics, which is formulated by using the eigenstate jCðmÞ max ðbÞi of the quantum transfer-matrix with the maximum eigenvalue (where m denotes the Trotter number and b ¼ 1=kB T), is applied to a transverse Ising chain. In order to exemplify the formulation of the thermo quantum dynamics for ‘‘local’’ operators, we show how to evaluate both the thermal average hsxj i and the correlation function hsxj sxjþr i. Furthermore, it is demonstrated for the first time that the limit m ! 1 of the thermal state vector jCðmÞ max ðbÞi exists in the diagonalized representation and that this thermal state vector can be regarded as the ground state vector of the corresponding virtual system. r 2006 Elsevier B.V. All rights reserved. Keywords: Quantum transfer-matrix; Thermo quantum dynamics; Transverse Ising chain; Correlation function
1. Introduction Thermodynamical properties of quantum systems have been studied for long years. There are many methods to solve this problem. The Suzuki–Trotter transformation (ST-transformation) [1–3] of a quantum system to the corresponding classical system has been effectively applied for both analytical and numerical studies. This general scheme is based on the equivalence theorem [3] that a d-dimensional quantum system with finite-range interactions is equivalent to the corresponding ðd þ 1Þ-dimensional classical system with finite interactions. The quantum Monte–Carlo method [3–5] is one of the applications of the above equivalence theorem. Another application yields the quantum transfer-matrix method [6–8], which has been also used effectively for both analytical study [9–22] and numerical study [23–30]. Recently, Suzuki proposed the thermo quantum dynamics [31,32] on the basis of the quantum transfermatrix. By this formulation, the thermal average of a ‘‘local’’ operator in the thermodynamic limit is expressed in terms of the expectation value on a vector in single Hilbert space, in contrast to the usage of the double Hilbert space in thermo field dynamics [33–35]. In the present paper, we formulate it explicitly in the transverse Ising chain. In Section 2, we give a brief summary of the quantum transfer-matrix method, and summarize the formulation of thermo quantum dynamics. In Section 3, an explicit application of the thermo quantum dynamics to a transverse Ising chain is presented. Discussion about the thermal state vector which plays an Corresponding author. Tel.: +81 3 3260 4271; fax: +81 3 3260 9020.
E-mail addresses: [email protected] (A. Sugiyama), [email protected] (H. Suzuki), [email protected] (M. Suzuki). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.12.055
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important role in the thermo quantum dynamics is given in Section 4. Finally, we summarize the present work in Section 5. 2. Brief review of thermo quantum dynamics As is well known, the transfer-matrix method was introduced to solve analytically Ising models and other classical systems in one and two dimensions [36–38]. According to the equivalence theorem [3], a ddimensional quantum system is mapped to the corresponding ðd þ 1Þ-dimensional classical system. Then, two kinds of transfer-matrices are defined in this ðd þ 1Þ-dimensional classical system [6–8], namely a ‘‘real-space’’ transfer-matrix defined in the real space and a ‘‘virtual-space’’ transfer-matrix defined in the virtual space. The partition function of the original quantum system described by Hamiltonian H can be written by using the virtual space transfer-matrix Tm as follows [6–8]: ZðbÞ ¼ Tr ebH ¼ lim ZðmÞ N ðbÞ; m!1
N Z ðmÞ N ðbÞ ¼ Tr ðTm Þ ,
(2.1)
where N denotes the system size, m is the Trotter number and b ¼ 1=kB T. Only the maximum eigenvalue lðmÞ max of the transfer-matrix Tm is sufficient to describe the free energy in the thermodynamic limit [6,7]: f ¼
kB T log Z ¼ lim log lðmÞ max . m!1 N
(2.2)
Here, we have used the interchangeability theorem [6,7] of the limits N ! 1 and m ! 1. The correlation length of the system is given by [8] ! lðmÞ 1 max x ¼ lim log ðmÞ , (2.3) m!1 l2 where lðmÞ 2 denotes the second largest eigenvalue of Tm . The thermo quantum dynamics [31,32] is formulated by using the eigenvector jCðmÞ max ðbÞi of the quantum transfer-matrix Tm with the maximum eigenvalue lðmÞ : max ðmÞ ðmÞ Tm jCðmÞ max ðbÞi ¼ lmax jCmax ðbÞi,
(2.4)
with the normalization ðmÞ hCðmÞ max ðbÞjCmax ðbÞi ¼ 1.
(2.5)
The thermal average hQi of a ‘‘local’’ operator Q in the thermodynamic limit N ! 1 is expressed as [31,32] ^ ðmÞ hQi ¼ lim hCðmÞ max ðbÞjQjCmax ðbÞi. m!1
(2.6)
Note that the local operator Q is also transformed into an operator Q^ which operates to the thermal state jCðmÞ max ðbÞi, as will be shown explicitly in the case of the transverse Ising chain in the succeeding section. 3. Explicit formulation in the transverse Ising chain In the present section, we consider the transverse Ising chain [39–42] described by the following Hamiltonian: bH ¼ K
N X i¼1
szi sziþ1 þ g
N X i¼1
sxi ¼ bH1 bH2 ,
(3.1)
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where sai denotes the a-component of the Pauli matrices at site i. Using the ST-transformation [1–3], the partition function can be represented by the two-dimensional Ising model as follows [6–8]: bH1 =m bH2 =m m Z ðmÞ e Þ N ðbÞ ¼ Tr ðe " # N X m X X K Nm 0 si;j siþ1;j þ gm si;j si;jþ1 , ¼ ðAm Þ exp m fs ¼1g i¼1 j¼1
ð3:2Þ
i;j
where fsij g denotes the Ising variables, and the parameters Am and g0m are defined by 1=2 n g o 1 2g 1 sinh and g0m ¼ log coth Am ¼ . 2 m 2 m
(3.3)
Then the quantum transfer-matrix is given by [7] 1=2
1=2
Tm ¼ V 2 V 1 V 2 ,
(3.4)
where V1 ¼
" # m=2 m X 2K 0 z and sinh exp K m sj m j¼1
" V 2 ¼ exp
g0m
m X
# sxj sxjþ1
.
(3.5)
j¼1
Here, saj denotes the a-component of the Pauli matrices at site j in the virtual space and the parameter K 0m is defined by 1 K 0 K m ¼ log coth . (3.6) 2 m Using the above quantum transfer-matrix, the partition function Z ðmÞ N ðbÞ can be expressed as 1=2
1=2
Nm Z ðmÞ Tr ðV 2 V 1 V 2 ÞN N ðbÞ ¼ ðAm Þ
¼ ðAm ÞNm Tr ðTm ÞN .
ð3:7Þ
This transfer-matrix Tm can be diagonalized by the following well-known procedures [43]: (i) Jordan–Wigner transformation on the fermi operators al and ayl : l 1 Y
½szi s l
ayl ¼ sþ l
l 1 Y
½szi
(3.8)
eip=4 X iql e ðcos fq xq sin fq xyq Þ al ¼ pffiffiffiffi m q
(3.9)
eip=4 X iql ayl ¼ pffiffiffiffi e ðcos fq xyq sin fq xq Þ, m q
(3.10)
al ¼
and
i¼1
i¼1
and (ii) the canonical transformation
and
where xq and xyq are fermi operators, and fq is given [43] in 1=2 ð1 eiq tanh K 0m tanh g0m Þ ð1 eiq tanh K 0m coth g0m Þ iðqþ2fq Þ e ¼ ð1 eiq tanh K 0m tanh g0m Þ ð1 eiq tanh K 0m coth g0m Þ
(3.11)
and q¼
p 3p 5p ðm 1Þp ; ; ;...; m m m m
(3.12)
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assuming that m takes an even value. Then the quantum transfer-matrix Tm can be diagonalized as [43] " m=2 # X 2K 1 1=2 1=2 y exp eq;m xq xq , (3.13) Tm V 2 V 1 V 2 ¼ sinh m 2 q where eq;m is given by the larger root of the following equation: 2K 2g 2K 2g cosh eq;m ¼ coth coth cosech cosech . m m m m The maximum eigenvalue of the quantum transfer-matrix Tm can be written as [43,44] " # m=2 2K 1X ðmÞ exp eq;m . lmax ¼ 2 sinh m 2 q
(3.14)
(3.15)
ðmÞ The eigenvector jCðmÞ max ðbÞi of Tm with the maximum eigenvalue lmax is given by the x-vacuum j0i:
for arbitrary q.
xq j0i ¼ 0
(3.16)
Now we can calculate the thermal average of any ‘‘local’’ operators Q in the thermodynamic limit by using the thermo quantum dynamics [31,32]. Next we show how to evaluate both the expectation value hsxj i and the correlation function hsxj sxjþr i in order to exemplify the formulation of the thermo quantum dynamics for local operators. 3.1. Thermal average hsxj i Noting the following matrix elements: 0
0
hsjsx exp½gsx =mjs0 i ¼ e2gm ss hsj exp½gsx =mjs0 i,
(3.17) 2g0m sj;l sj;lþ1
we find that the local operator sxj can be expressed as e in the mapped two-dimensional Ising model. Here, the index l of the spin variable sj;l can be chosen an arbitrary number, because of the translation invariance in the Trotter direction. Then we can show the following relation: X 0 e2gm sj;l sj;lþ1 Tr sxj ðebH1 =m ebH2 =m Þm ¼ ðAm ÞNm "
fsk;i g
# N X m X K sk;i skþ1;i þ g0m sk;i sk;iþ1 exp m k¼1 i¼1 0
x x
¼ ðAm ÞNm Tr e2gm sl slþ1 ðTm ÞN .
ð3:18Þ
Thus, the operator sxj is transformed into the following form: 0
x x
sxj ! e2gm sl slþ1 ¼ cosh 2g0m sxl sxlþ1 sinh 2g0m .
ð3:19Þ
By the formulation of the thermo quantum dynamics, the thermal average hsxj i is given by hsxj i ¼ lim hsxj im m!1
0 x x 0 ðmÞ ¼ lim hCðmÞ max ðbÞjðcosh 2gm sl slþ1 sinh 2gm ÞjCmax ðbÞi m!1
¼ lim ½cosh 2g0m h0jsxl sxlþ1 j0i sinh 2g0m . m!1
ð3:20Þ
ðmÞ Here, h im ¼ hCðmÞ max ðbÞj jCmax ðbÞi denotes the average in the mth approximated system. The nearest neighbour correlation function, h0jsxl sxlþ1 j0i, in the mapped two-dimensional Ising model can be written in the
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following form [43]: h0jsxl sxlþ1 j0i ¼ h0jðayl al Þðaylþ1 þ alþ1 Þj0i 1 X iðqþ2fq Þ e . ¼ m q Thus, we obtain the following expression of hsxj im , sinh 2g0m X iðqþ2fq Þ hsxj im ¼ cosh 2g0m þ e m q " # X 2 2g c ¼ cosh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , m sinhð2g=mÞ q40 m a2 b2 using Eq. (3.11), where a, b and c are given by 2K 2g a ¼ cosh cosh cos q, m m b ¼ sinh
2K 2g sinh , m m
2K c ¼ cosh m
ð3:21Þ
ð3:22Þ
(3.23)
(3.24)
2g cosh cos q. m
Using the following formula: Z 1 p A þ B cos y B bA aB dy ¼ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , p 0 a þ b cos y b b ða þ bÞða bÞ
(3.25)
(3.26)
with A ¼ cosh
2K 2g sinh2 m m
(3.27)
and
2K 2g 2g B ¼ sinh cosh sinh , m m m the average hsxj im can be rewritten as Z pX 2 a1 x dy, hsj im ¼ mp 0 q40 a0 cos q
(3.28)
(3.29)
where 2K 2g 2K 2g a0 ¼ cosh cosh þ sinh sinh cos y, m m m m
(3.30)
2K 2g 2K 2g sinh þ sinh cosh cos y. m m m m
(3.31)
a1 ¼ cosh
Here, q is given by Eq. (3.12), so that we can sum over q in the following form: m=2 X j¼1
m 1 m cosh1 a0 . ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi tanh a0 cosðð2j 1=mÞpÞ 2 a2 1 2 0
(3.32)
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Then we arrive at the expression Z m 1 p a1 qffiffiffiffiffiffiffiffiffiffiffiffiffi tanh cosh1 a0 dy. hsxj im ¼ p 0 2 a20 1
(3.33)
Now the limit m ! 1 can be taken as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m lim cosh1 a0 ¼ K 2 þ g2 þ 2Kg cos y m!1 2 and
(3.34)
a1 g þ K cos y lim qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 K þ g2 þ 2Kg cos y a20 1
(3.35)
m!1
Finally, we arrive at the integral representation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z p 1 ðg þ K cos yÞ tanh K 2 þ g2 þ 2Kg cos y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hsxj i ¼ lim hsxj im ¼ dy. m!1 p 0 K 2 þ g2 þ 2Kg cos y We can confirm that this result agrees with the formula directly derived from the free energy [40,41]. 3.2. Correlation function hsxj sxjþr i The operator sxj sxjþr can be transformed into the following form: 0
x x
0
x x
sxj sxjþr ! e2gm sl slþ1 ðTm Þr e2gm sl slþ1 ðTm Þr
(3.36)
by using Eq. (3.19). The correlation function hsxj sxjþr i can be expressed as hsxj sxjþr i ¼ lim hsxj sxjþr im m!1
0
x x
0
x x
2gm sl slþ1 ¼ lim hCðmÞ ðTm Þr e2gm sl slþ1 ðTm Þr jCðmÞ max ðbÞj e max ðbÞi m!1
¼ lim h0jðcosh2 2g0m cosh 2g0m sinh 2g0m sxl sxlþ1 m!1
cosh 2g0m sinh 2g0m ðTm Þr sxl sxlþ1 ðTm Þr þ sinh2 2g0m sxl sxlþ1 ðTm Þr sxl sxlþ1 ðTm Þr Þj0i
ð3:37Þ
by the formulation of the thermo quantum dynamics. Let fjnigPbe a complete set composed of eigen vectors of Tm with eigenvalues flðmÞ jnihnj into the last term of Eq. (3.37), n g. Inserting the unit operator we have hsxj sxjþr im ¼ cosh2 2g0m 2 cosh 2g0m sinh 2g0m h0jsxl sxlþ1 j0i !r X lðmÞ
2 0 n
h0jsx sx jni 2 . þ sinh 2gm l lþ1 ðmÞ lmax n
ð3:38Þ
Noting that the operator sxl sxlþ1 can be written as a quadratic form of x-fermion representation, we find that the matrix element h0jsxl sxlþ1 jni has a non-zero value only when the state jni is equal to the vacuum state j0i or y y the two-particle excited state xyq xyq0 j0i ðq4q0 Þ. The eigenvalue lðmÞ qq0 of the excited state xq xq0 j0i has the following form: " # m=2 2K 1X ðmÞ lqq0 ¼ sinh exp eq eq0 þ eq00 . (3.39) m 2 q00 Thus, we can rewrite Eq. (3.38) as hsxj sxjþr im ¼ fcosh 2g0m sinh 2g0m h0jsxl sxlþ1 j0ig2 þ sinh2 2g0m
X q4q0
erðeq þeq0 Þ jh0jsxl sxlþ1 xyq xyq0 j0ij2 .
(3.40)
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The first term of Eq. (3.40) is obviously equal to hsxj i2m . By using the transformation of sxl into the x-fermion, the matrix element h0jsxl sxlþ1 xyq xyq0 j0i can be written as 0
i eiðqþq Þl iðqþfq fq0 Þ 0 ðe eiðq fq þfq0 Þ Þ. (3.41) m Substituting this matrix element into Eq. (3.40) and noting that eq is an even function and fq is an odd function with respect to q, we can rewrite the second term of Eq. (3.40) as X erðeq þeq0 Þ jh0jsxl sxlþ1 xyq xyq0 j0ij2 sinh2 2g0m h0jsxl sxlþ1 xyq xyq0 j0i ¼
q4q0
sinh 2g0m X rðeq þeq0 Þ iðqþfq fq0 Þ 0 0 e ðe eiðq fq þfq0 Þ Þ ðeiðqþfq fq0 Þ eiðq fq þfq0 Þ Þ 2 m q4q0 2
¼
sinh2 2g0m X rðeq þeq0 Þ 0 e ð1 eiðqþ2fq Þiðq þ2fq0 Þ Þ m2 0 q;q 8 !2 !2 9 = 2 0 < X X sinh 2gm req req þiðqþ2fq Þ e e ¼ : q ; m2 q ( ) sinh2 2g0m Y X req ¼ e 1 eiðqþ2fq Þ . m2 q ¼
Thus, we obtain the following expression of hsxj sxjþr im : ( ) sinh2 2g0m Y X req x x x 2 iðqþ2fq Þ hsj sjþr im ¼ hsj im þ e ð1 e Þ m2 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!r " !# Y X a a2 b2 1 c x 2 ¼ hsj im þ 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , b m2 sinh2 ð2g=mÞ a2 b2 q40
ð3:42Þ
ð3:43Þ
using Eqs. (3.11) and (3.14), where a, b and c are given by Eqs. (3.23)–(3.25). Using the following formula pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!r Z 1 p cosðryÞ 1 a þ a2 b2 dy ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (3.44) p 0 ða þ bÞ cos y b a2 b2 the correlation function hsxj sxjþr im is represented as 1 hsxj sxjþr im ¼ hsxj i2m þ 2 m sinh2 ð2g=mÞ " # Y X ð1Þr Z p ða þ cÞ cosðryÞ þ b cosfðr 1Þyg dy . 2 ða þ bÞ cos y p 0 q40 Here, Eq. (3.32) can be used to sum over q, and then we have X b b cos q 2 ðb a0 b1 Þ tanhððm=2Þ cosh1 a0 Þ 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi b1 þ 0 ¼ , a0 cos q m a0 1 q40 where a0 is given by Eq. (3.30) and 2K 2g 2K 2g b0 ¼ cosh 1 þ cosh cosðryÞ þ sinh sinh cosfðr 1Þyg, m m m m b1 ¼
1 þ cosh
2g cosðryÞ. m
ð3:45Þ
(3.46)
(3.47)
(3.48)
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Thus, we obtain hsxj sxjþr im
¼
hsxj i2m
Z m 1Y p b0 a0 b1 qffiffiffiffiffiffiffiffiffiffiffiffiffi tanh cosh1 a0 dy. p 0 sinhð2g=mÞ a2 1 2
(3.49)
0
Now we can take the limit m ! 1. This limit yields the expression lim
m!1
b0 a0 b1 K cosfðr 1Þyg þ g cosðryÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 K 2 þ g2 þ 2Kg cos y sinhð2g=mÞ a0 1
(3.50)
Finally, the correlation function hsxj sxjþr i can be expressed as hsxj sxjþr i ¼ lim hsxj sxjþr im m!1 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p K cosfðr 1Þyg þ g cosðryÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh K 2 þ g2 þ 2Kg cos y dy ¼ hsxj i2 p 0 K 2 þ g2 þ 2Kg cos y Z p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 K cosfðr þ 1Þyg þ g cosðryÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh K 2 þ g2 þ 2Kg cos y dy. p 0 K 2 þ g2 þ 2Kg cos y
ð3:51Þ
This explicit result can be derived, as a special limit, from the general formula of correlation functions for the XY chain [45] in the presence of the transverse field.
4. Thermal state vector of the transverse Ising chain In the present section, we give discussions about the thermal state jCðmÞ max ðbÞi. As is illustrated in Fig. 1, we can construct the virtual transverse Ising chain described by the following Hamiltonian: HðmÞ v ¼ J v
m X
szj szjþ1 Gv
j¼1
m X
sxj þ const.
(4.1)
j¼1
Here, j denotes the site defined in the virtual space. The partition function of this virtual system can be written as follows: bHv ZðmÞ v ¼ Tr e
¼ C Tr eAv þBv ¼ lim C Tr ðeAv =N eBv =N ÞN , N!1
Fig. 1. Transformation from an original system to a virtual system.
ð4:2Þ
ARTICLE IN PRESS A. Sugiyama et al. / Physica A 368 (2006) 449–458
457
where Av ¼ bJ v
m X
szj szjþ1 ;
j¼1
Bv ¼ bGv
m X
sxj
(4.3)
j¼1
and C is constant. The operators exp½Av =N and exp½Bv =N can be identified as V 2 and V 1 , respectively, with setting the parameters bJ v 1 g ¼ log coth 2 m N
(4.4)
and bGv 1 K (4.5) ¼ log coth . 2 m N Because the maximum eigenvalue of the above transfer-matrix gives the ground energy of the virtual system Hv , the thermal state jCðmÞ max i can be identified with the ground state of the above virtual Hamiltonian in the thermodynamic limit ðN ! 1Þ. With diagonalizing the virtual Hamiltonian, we can find that the ground state can be written as the x-vacuum state. Then we find that the single Hilbert space which was introduced in the formulation of thermo quantum dynamics is described by the above virtual space Hamiltonian. Although the parameters J v and Gv diverge in the limit m ! 1, there is no problem in the formulation of thermo quantum dynamics with the correct procedure. For example, the quantum transfer-matrix in the spin fsx;z j g representation has singular elements in the limit m ! 1 (see Eqs. (3.3)–(3.6)). However, the diagonalized form has no singular factor in the limit m ! 1. This fact can be easily shown from the following form of the diagonalized quantum transfer-matrix: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y y ðAm Þm TðdÞ ¼ fa þ ð1 2x x Þ a2 b2 g1=2 , (4.6) q q m q
where a and b are given by Eqs. (3.23) and (3.24), respectively. Although the singularity of Tm in the spin representation may leads to the nonexistence of the limit m ! 1 of the thermal state, the state associated to the diagonalized quantum transfer-matrix do exist as a x-vacuum, as was mentioned by Suzuki [31]. The local operator sxj also has singular elements in the spin fsx;z j g representation (Eq. (3.19)). However, it is easy to find that the following x-fermion representation: sinh 2g0m X iq2ifq y y ½e ðixq xq þ xyq xq xq xyq þ ixq xq Þ (4.7) sxj ! cosh 2g0m m q has no singular elements in the limit m ! 1. Here, we have to mention that the limit m ! 1 has to be taken after the summation over all q, because q also depends on the Trotter number m. 5. Summary In the present paper, we have applied the formulation of the thermo quantum dynamics to the transverse Ising model, and have shown how to evaluate the average values of local operators explicitly. It has been shown that the thermal state jCðmÞ max ðbÞi is described as the ground state of the corresponding virtual Hamiltonian. Although the quantum transfer-matrix has singular properties in some representations in the limit m ! 1, these singularities disappear after diagonalizing the quantum transfer-matrix. This is the first explicit demonstration of the existence of the limit m ! 1 of the thermal state vector jCðmÞ max ðbÞi. All these properties of the thermal state will be expected to be relevant in other models. References [1] M. Suzuki, J. Phys. Soc. Jpn. 21 (1966) 2274; See also H.F. Trotter, Proc. Am. Math. Soc. 10 (1959) 545.
ARTICLE IN PRESS 458 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
A. Sugiyama et al. / Physica A 368 (2006) 449–458 M. Suzuki, Commun. Math. Phys. 51 (1976) 183. M. Suzuki, Prog. Theor. Phys. 56 (1976) 1454. M. Suzuki, S. Miyashita, A. Kuroda, Prog. Theor. Phys. 58 (1977) 1377. M. Suzuki, Quantum Monte Carlo Method in Condensed Matter Physics, World Scientific, Singapore, 1993. M. Suzuki, Phys. Rev. B 31 (1985) 2957. M. Suzuki, M. Inoue, Prog. Theor. Phys. 78 (1987) 787. M. Inoue, M. Suzuki, Prog. Theor. Phys. 79 (1988) 645. J. Suzuki, Y. Akutsu, M. Wadati, J. Phys. Soc. Jpn. 59 (1990) 2667. M. Takahashi, Phys. Rev. B 43 (1991) 5788. J. Suzuki, T. Nagao, M. Wadati, Int. J. Mod. Phys. B 6 (1992) 1119. C. Destri, H.J. de Vega, Phys. Rev. Lett. 69 (1992) 2313. C. Destri, H.J. de Vega, Nucl. Phys. B 504 (PS) (1997) 621. A. Klu¨mper, Z. Phys. B 91 (1993) 507. H. Mizuta, T. Nagao, M. Wadati, J. Phys. Soc. Jpn. 63 (1994) 3951. A. Klu¨mper, Eur. Phys. J. B 5 (1998) 677. G. Ju¨ttner, A. Klu¨mper, Euro. Phys. Lett. 37 (1997) 335. G. Ju¨ttner, A. Klu¨mer, J. Suzuki, Nucl. Phys. B 487 (1997) 650. G. Ju¨ttner, A. Klu¨mper, J. Suzuki, J. Phys. A 30 (1997) 1881. A. Kuniba, T. Nakanishi, J. Suzuki, Int. J. Mod. Phys. A 9 (1994) 5215. A. Kuniba, K. Sakai, J. Suzuki, Nucl. Phys. B 525 (FS) (1998) 597. Y. Umeno, J. Phys. Soc. Jpn. 70 (2001) 2531. H. Betsuyaku, Prog. Theor. Phys. 73 (1985) 319; H. Betsuyaku, Prog. Theor. Phys. 75 (1986) 774, 808. T. Tsuzuki, Prog. Theor. Phys. 73 (1985) 1352. H. Betsyaku, T. Yokota, Prog. Theor. Phys. 75 (1986) 808. K. Kubo, Takada, J. Phys. Soc. Jpn. 55 (1986) 438. T. Koma, Prog. Theor. Phys. 71 (1993) 269. K. Kubo, Phys. Rev. B 46 (1992) 866. T. Delica, K. Kopinga, H. Leschke, K.K. Mon, Europhys. Lett. 15 (1991) 55. N. Hatano, M. Suzuki, Prog. Theor. Phys. 85 (1991) 481. M. Suzuki, J. Stat. Phys. 110 (2003) 945. M. Suzuki, Physica A 321 (2003) 334. H. Umezawa, H. Matsumoto, M. Tachiki, Thermo Field Dynamics and Condensed States, North-Holland, Amsterdam, 1982 and references cited therein. M. Suzuki, J. Phys. Soc. Jpn. 54 (1985) 4483. M. Suzuki, Phys. Lett. A 111 (1985) 440. H.A. Kramers, G.H. Wannier, Phys. Rev. 60 (1941) 252. E.W. Montroll, J. Chem. Phys. 9 (1941) 706; E.W. Montroll, J. Chem. Phys. 10 (1942) 61. R. Kubo, Busseiron Kenkyu 1 (1943) 1. E.H. Lieb, T.D. Shultz, D.C. Mattis, Ann. Phys. 16 (1961) 941. S. Katsura, Phys. Rev. 127 (1962) 1508. P. Pfeuty, Ann. Phys. 57 (1970) 79. B.K. Chakrabarti, A. Dutta, P. Sen, Quantum Ising Phases and Transitions in Transverse Ising Models, Springer, Berlin, 1996. T.D. Shultz, D.C. Mattis, E.H. Lieb, Rev. Mod. Phys. 36 (1964) 856. L. Onsager, Phys. Rev. 65 (1944) 117. E. Barouch, B.M. McCoy, Phys. Rev. A 3 (1971) 786.