Study of correction terms for higher-order decompositions of exponential operators

ELSEVIER Physica A 211 (1994) 234-254

PHYSICA

Study of correction terms for higher-order decompositions of exponential operators

Hiroto Kobayashi, Naomichi Hatano, Masuo Suzuki Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan

Received 9 June 1994

Abstract

Many kinds of higher-order decompositions are applied to an exactly solvable model, and the Trotter-number dependence of the correction terms is evaluated systematically. Practically useful decompositions are explored.

1. Introduct ion

As computer facilities have been improved, numerical calculations have very much contributed to the development of various areas of physics, i.e., statistical mechanics, quantum mechanics, etc. Precise calculations have been carried out and essentially new facts in physics have been discovered thereby. The Monte Carlo method [ I ] is one of the useful methods to deal with considerably large systems.

Many methods of studying quantum spin systems are based on the Suzuki-Trotter formula [2-6] , that is,

= r X-A ~-B] n e x ( t + B ) lim [e, e, j (1.1) n - - - - ~ o o

According to this formula, we can use decomposed exponentials instead of the original exponential operator e - t ~ . Furthermore, the fractal decomposition has recently been introduced by one of the present authors [7-12]. By this decomposition method, we can obtain more accurate approximations to the original operator. That is the reason why various studies based on such fractal decompositions have been performed not only in statistical mechanics [ 13] but also in other areas, i.e., Hamiltonian dynamics [ 14,15], time-dependent quantum mechanics [ 16-19], etc.

Suzuki and Umeno [14] applied higher-order decompositions to the Hamiltonian dynamics. If we use the ordinary fourth-order Runge-Kutta formula, the energy deviates

0378-4371/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved SSDI0378-4371 ( 9 4 ) 0 0 1 8 1 - 2

H. Kobayashi et al. / Physica A 211 (1994) 234-254 235

monotonically from the initial value. In contrast, the approximate energy calculated in the fractal-decomposition scheme barely deviates from the initial value.

In order to improve the degree of approximation, it is necessary to increase the number of multiplications of the decomposed exponential operators [ 7]. The convergence of such fractal decompositions have been proved [20-22]. However, it is not yet known how to select the most useful approximation to the original exponential operator in a practical sense. It is not clear which approximation is a good compromise between accuracy and the computational time required to calculate it. This is because the actual magnitude of the correction term has not been discussed yet.

In this paper, as an example of studying this problem, we apply many kinds of higher-order decompositions to an exactly solvable model, namely, the one-dimensional quantum XY-model, and we evaluate the Trotter-number dependence of the correction terms systematically [23-25]. The relative error becomes smaller as the Trotter number increases, as shown in Fig. 2. It is advantageous to use higher-order decompositions, if we can carry out a large number of multiplications and if we require high precision.

The one-dimensional XY-model has an infinite number of degrees of freedom; in other words, its Hilbert space is infinite-dimensional. Confirmation of the effectiveness of higher-order decompositions in an infinite-dimensional space is qualitatively different from that in a finite-dimensional space, for example, computational confirmation by means of matrix multiplication.

In Section 2, we review the way to construct higher-order decompositions. In Sec- tions 3 and 4, the fractal decomposition and the improved Trotter-like formula [26] are applied to the one-dimensional quantum XY-model. We evaluate correction terms of the free energy in the thermodynamic limit for finite Trotter numbers. We compare the magnitudes of the correction terms obtained by means of various kinds of approximations.

2. Higher-order decompositions of exponential operators

In this section, we review the construction scheme of the higher-order decompositions. Approximants obtained from the series expansion of an exponential operator do not conserve symmetries of the original operator. The method based on the Baker- Campbell-Hausdorff formula is not practical since calculation of many commutators is required in this scheme. The fractal decomposition [7], which is constructed by means of multiplication of the operators e p'A and e pjB, conserves symmetries of the original operator e x(a+B) and is also very convenient for practical use.

First we consider the exponential operator exp[x(A + B) ], where A and B are non- commutable. In many cases, it is difficult to diagonalize the sum of the operators, A ÷ B. Nevertheless, we can evaluate exp[x(A + B)] approximately for finite n from the following formula [4]:

e x (a+B)= lim [e~ae~B] n , (2.1)

236 H. Kobayashi et al. / Physica A 211 (1994) 234-254

if it is practically possible to diagonalize each of A and B. Since the correction term in

the above formula is of the first order of 1 In, i.e.,

r X-A xB~n ( ~ ) e x(a+B) = [e, e, J + O , (2.2)

we can estimate the left-hand side by means of extrapolation from data obtained for

finite Trotter number n. If we have an approximation to exp[x(A + B) ] correct up to a certain higher order

of x for small x, then we have the following generalized Trotter formula [5]:

eX(a+B)= [ O m ( X ) ] n - F O \ nm , ] (2.3)

for finite x, where

a m ( x ) = e x(A+B) q- O( xm+l ). (2.4)

We refer to Qm(x), which is correct up to the mth order of x, as an "mth-order

decomposition" of exp[x(A + B) ]. Now we consider how to construct higher-order decompositions of the exponential

operator. We may obtain approximants to e x(a+B) from the following series expansion:

e x ( a + B ) = 1 + x ( A + B ) + ~ ( A + B ) 2 + . . . . (2.5)

Approximants of this type do not conserve the unitarity when the variable x is imaginary. We can obtain a higher-order decomposition, using the Baker-Campbell-Hausdorff

(BCH) formula, i.e.,

exp(A) exp(B) = exp(C) , (2.6)

where

C = A + B + ½[A,B] + - ~ [ A , [ A , B ] ] + ~ [ B , [ B , A ] ] + . . . . (2.7)

For instance, a fourth-order symmetric decomposition is written [27] in the form

eX(a+ B) = e~ae~B eX3ICe~B e~A + O(x 5) (2.8)

with R = ~ [ [B, A], A + 2B]. We can construct higher-order decompositions taking

into account higher-order terms of Eq. (2.7). However, it is not practical to use them because calculations of many commutators are required in this approximation.

If we can construct higher-order decompositions of the exponential operator exp[x(A + B)] by means of multiplication of the operators e p'A and e p:8, it is very convenient for practical use. Moreover, decomposed exponentials preserve symmetries

of the original operator such as the unitarity in quantum mechanics. The following decomposition of exp[x(A + B)] is of the second order [28]:

S2(x) = e~aexBe ~a. (2.9)


A third-order decomposition of e x p [ x ( A + B ) ] has been found by Ruth [29] as follows:

7 A 2 B 3 , 2 n 1 . Q3(x) = e ~ x e ~x e4X~e-3X"e -~X%X" . (2.10)

A fourth-order decomposition [ 7,12,30,31 ] is given by

s 1 --s 1 --s s ~xA sxB ---f--xA ( l - - 2 s ) x B xA sxB - x A a 4 ( x ) = e e e e e-r- e e2 , (2.11)

where

1 s - 2 - 21/3. (2.12)

It is easily shown by computer-analytical calculation with the REDUCE language that

a 4 ( x ) is correct up to the fourth order of x. However, a new systematic scheme is necessary in order to construct general higher-order decompositions.

Recently Suzuki [7] proposed the fractal decomposition of exponential operators,

which is a recursive scheme to construct Qm(x) for any integer m. For instance, a

decomposition S2m(X) correct up to the 2mth order of x is obtained [7,12] from

S2m-2(x) by

S 2 m ( X ) = S 2 m - 2 ( p m x ) S 2 m - 2 ( ( 1 - 2 p m ) x ) S 2 m - 2 ( p m x ) , ( 2 . 1 3 )

where

2pro 2m-1 + ( 1 - - 2 p r o ) 2m-1 = 0 , (2.14)

and Sz(x) = e~aexBe~ A. The approximant Q4(x) given by Eqs. (2.11) and (2.12)

is actually included in the series {Szm(X)} obtained from the above scheme, that is

Q4(x) = S4(x) •

In general, we obtain an mth-order decomposition [7] of exp[x ~q=l Ak] in the form

r

Qm(X) = 1-I Qm-1 (Pm,jX) , (2.15) j = l

where

and

r r

ZPmj--I j = l j = l

Q2(x) = e~Ale ~A2 . . . e~Aq-lexAqe ~Aq-1 . . . e ~A' .

(2.16)

In order to obtain real solutions of Eqs. (2.16), it is necessary that the inequality r > 3 holds and m is an odd integer [7] . It seems impossible to construct a real decomposition correct up to any order of x in this scheme. However, Q2m-1 (x ) is correct up to the 2mth order of x [7] , if Q2m-1 (x ) is symmetric, i.e.,

P2m- | , r+ l - j = P2m-l,j for 1 < j _< r. (2.18)

(2.17)


Therefore we can obtain a real mth-order decomposition Qm(x) for any integer m.

For practical convenience, we concentrate on symmetric decompositions in the present paper.

3. Fractal decomposition

In this section, we calculate the free energy of the one-dimensional XY-model using the fractal decomposition [7] with finite Trotter numbers. Then we compare the magnitudes of correction terms obtained by various kinds of approximations for a given Trotter number. This model has an infinite number of degrees of freedom and is a non-trivial system. It is essential that we confirm the effectiveness of higher-order decompositions for such an infinite system.

There are many construction schemes to obtain higher-order decompositions. For example, the fourth-order decomposition is obtained by

S4(x) = [S2(px)]tP[S2(qx)]lq[S2(px)] tp, (3.1)

p ( lp, lq) = [ 2lp - (21p) t/312/3 ] - t a n d q( lp , lq) = [ lq - (2 lp ) 2/311/3 ] - 1 f o r a n y i n t e g e r s

Ip and lq (lq 4= 21p), using the second-order decomposition S2(x). In Section 3.2 we obtain the free energy using the fourth-, sixth- and eighth-order fractal decompositions for Ip = 2 and lq = 1. Then we compare, in Section 3.3, the magnitudes of correction terms due to various degrees of approximations fixing lp = 2 and lq = 1. In Section 3.4 we evaluate correction terms using the fourth-order decomposition for various values of lp and lq. ThUS we determine the optimal approximation. In Section 3.5 we discuss the behavior of the relative error of the free energy in the low-temperature limit.

3.1. Calculation of the free energy of the one-dimensional XY-model for the second-order decomposition

The Hamiltonian of the one-dimensional quantum XY-model without an external field

is given by

2L

= - s + ( 3 . 2 )

i=l

We take the periodic boundary condition, i.e., o-~zL+ 1 = o-~ and O'~L+l = O'~1. Following Inoue and Suzuki [23,24], we calculate the free energy of the model for a finite Trotter

number. First we divide the Hamiltonian 7-I into two parts as

-/37-/= K ( ~ I + 7-(2), (3.3)

where


L Y

7"/1 = E ( 0 ~ 2 i 0 ~ 2 i + I "~ 0"2i0"~i+1), (3.4) i=1

L ,~...~0. x o.X Y 2 = ,,7 [ 2i--1 2i "~ 0~2i--10"2i), ( 3 . 5 ) i=1

fl = 1/kBT and K = flJ. After the calculation using the second-order decomposition shown in Appendix A, we

obtain the approximate free energy in the following form:

0

1 4 K 3 1 ( k ) k k ( ~ ) +~-~ ~ dktanh 2Kcos cos ~ sin 2 ~ + O . (3.6)

0

The first term of the right-hand side of Eq. (3.6) corresponds to the exact value of the free energy of the present model [32]. The second term gives the second-order correction to it.

3.2. Correction terms for the fractal decomposition

Now we calculate the free energy of the model using the fractal decomposition. Here we construct higher-order decompositions according to the following construction scheme:

S2m(X) = [ S2m--2(pmX) ]2S2m-2( qmx) [ S2m-2(pmx) ]2 ,

where

and

(3.7)

_x B ~_ S2(x) = e2 eXAe2 B, (3.9)

that is, we fix lp = 2 and lq = 1 in Eq. ( 3 . 1 ) . The fourth-order decomposition $4 (K) of this series is given by

S4(K) = [ S2(p2K) ]2S2(q2K) [ S2(p2K) ] 2, (3.10)

with

4p2+q2 =1 , 4p23+q3=0,

o r

1 1 P2 = 4 - 41/3 = 0.414 . . . . q2 = 1 - 42/3

(3.11)

= -0.657 . . . . (3.12)

4pro + qm = 1, 4pro 2m-I + qm 2m-1 = 0 (3.8)


As shown in Appendix A, we obtain the approximate free energy in the following form:

7 r

/ [ ( _ f i f O ) = 17r d k l o g 2cosh 2Kcos

o 7","

4n--4 45-----~ dk tanh 2Kcos cos

0

[ ' ' ] × ( c e 4 - - 1 ) c o s 4 ~ - J - c ~ 2 c o s 2 ~ - l - a 0 - t -O , (3.13)

with

~4 = ~ ( 7 + 1 5 6 p 2 - 3 6 9 p ~ ) = 0.330 . . . .

1 123p~) 0.697 . . . . a2 = g (6 - 52p2 + = 1 123p~) - 0 . 0 2 7 . . . . Go = - N - 6 ( 6 - 52p2 + =

(3.14)

In a similar way, we can explicit ly calculate the sixth- and eighth-order approximants

of the free energy using the decomposit ions

S6(K) ~ [ S4(p3K) ]2S4( q3K) [ S4(p3K) ]2,

with

1 p3 - 4 - 41/-------~

and

(3.15)

1 = 0.373 . . . . q3 = 1 - 44/5 = - 0 . 4 9 2 . . . . (3.16)

Ss( K) =-- [ S6(p4K) ]2S6( q4K) [ S6(paK) ] 2, (3.17)

with

1 1 - - _ . . . P4 = 4 - 41/7 = 0.359 . . . . q4 = 1 - 46/7 - 0 . 4 3 8 . (3.18)

3.3. Comparison of the correction terms

We define the relative error of the approximate free energy _flf(2m) compared to the exact one _flf(exact) as r2m, that is,

_flf(2rn) = _flf(exact) ( 1 + r2m). (3.19)

The exact value of the free energy [32] is given by

7 r

- f l f ( e x a c t ) = l / d k l ° g [ 2 c ° s h ( 2 K c ° s k ) } (3.20)

0

which is the first term of the right-hand side of (3.6) and (3.13). In the infinite-Trotter-

number l imit n --~ oc, the free energy _ f l f ( 2 m ) converges to the exact one, namely,

r2m ~ 0 as n --~ oc, (3.21)


0.0100

f .... ' / , , " " 7 7 / ooo5o t / , ' / , ' /

E /, ' / , ' / , ' L .Y.. l,' / , ' / ..... /K : ;o ............... : /L2 .... / /

L °°°'°k,,'/ ,,'1 '- l , ' - oo0oo K / , ' / ;I ~:o.~

v/ ,' ~ , ' / ~.

o.ooo1 // / ' 0.005 0.01 0.05 0.1 0.5

-1 Nmul

Fig. 1. The relative errors r4 (solid lines) and r2 (dashed lines) at the temperatures K = 0.5, 2 and 10.

as is expected from the definition of S2m(K).

Here we introduce the number of multiplications Nmul to compare various kinds of approximations. The 2ruth-order decomposition defined by Eq. (3.7) has 5 m-m operators

of $2. Therefore, the total number of multiplications of $2 is given by

Nmul = 5 m - i n (3.22)

for the Trotter number n. Since the computational time is proportional to Nmub compar- isons between two kinds of approximations should be made for a fixed value of Nmul.

For the second-order decompos i t ion , Nmu I is equal to n.

Now we compare with r2 the relative error r4 obtained from the fourth-order decomposition. The error r4 is of the fourth order of 1/Nmul and the error r2 is of the second order as shown in Eqs. (3.6) and (3.13), namely,

r 4 = O ~ , r 2 = O ~ , (3.23) ul ul

for a large value of Nmul- It may happen, however, that the absolute value of r4 is larger than that of r2 is, for a small value of Nmul- As the value of Nmul increases, the value of Ir4I decreases faster than that of Jr21 does, and they coincide for a certain value of Nmul as shown in Fig. 1.

Let us define N2m,2m+2 as the value of Nmul at the point Ir2mI = [r2m+2[, and r2m,2m+2 as the absolute value of r2m at that point. It is advantageous to use the fourth-order decomposition in the region where the inequality [r4[ < ]r21, or Nmul > N2,4 holds. We numerically calculated N2,4 at various temperatures; all the data obtained are listed


Table 1 The number N2,4 defined by the value of Nmu I at the point Ir21 = Ir4l, and the number N4,6 defined by the value of Nmu I at the point Ir4] = Ir61, at various temperatures

K 0.5 1.0 2.0 5.0 10.0 20.0 50.0 100.0

N2,4 5.7 11.1 21.6 52.9 105.4 210.7 526.6 1053.2 N4,6 21.2 42.3 84.5 211.4 423.0 846.2 2115.6 4231.3

lo_ 3 r2.,: .... , . . , .,..,..,'1 . . . . . . . . . . ' . . . . . . ' ..... ' . . " ' " ~ . . . . F2

F 4

- ~ _ r4,e

1 0 - 6

1 0 - 9 - - L~

~lr~l ~ 1

10-12 , / . . . . . , ] . . . . . . . I 0.005 0.01 0.05 0.1

-1 Nmul

Fig. 2. The Nmu! dependence of the relative errors of the free energy at the temperature K = 1.

in Table 1. The number N2,4 is approximately proportional to K at low temperatures,

namely,

N2,4 "-' 10K. (3.24)

This temperature dependence is explained below in Section 3.5. We also calculate the approximate free energy of the model using the sixth-order

decomposition S6(K) given by Eqs. (3.15) and (3.16) and the eighth-order decomposition Ss(K) given by Eqs. (3.17) and (3.18). We show in Fig. 2 the Nmul dependence of the relative errors r2, r4, r6 and r8 for K = 1. (Here we note that the relative errors r6 and r8 are negative for a large value of Nm,l at some temperatures.) We numerically calculated N4,6; all the data obtained are listed in Table 1. At low temperatures, the

number N4,6 is approximately proportional to K, namely,

N4,6 "-~ 40K. (3.25)

From Eqs. (3.24) and (3.25), we find that N4,6 is about four times as large as N2,4,

and consequently that r4,6 is 44 times as small as r2,4.

H. Kobayashi et al. / Physica A 211 (1994) 234-254 2 4 3

Here we put a question: Which one is the optimal approximation of all degrees of the

fractal decompositions given by Eq. (3.7)? The answer depends on available computer facilities and on the required precision of the calculation. In preliminary calculations, the analysis as shown in Fig. 2 is quite useful to select the optimal decomposition [33]. From Fig. 2, we can decide the optimal approximation which gives the smallest relative error for a fixed value of Nmul, and the one which is the most efficient for a fixed value of the allowed error. For instance, if we cannot spare the computer CPU time enough to carry out the matrix multiplications N4,6 times, the fourth-order fractal decomposition is the optimal approximation. For another example, if we allow the error r which satisfies

r4,6 > r > r6,8, the optimal approximation is the sixth-order one. It is advantageous to use higher-order decompositions if we require high precision.

Incidentally, the convergence of the fractal decompositions has been proved generally by Suzuki [21]. The relative error r2m for a given Trotter number n converges to zero as we improve the approximation, namely,

r2m -'-+ 0 as m ~ c~, (3.26)

since the approximant S2m(K) converges to the original operator e - ~ . The present data numerically confirm the convergence of the fractal decompositions to the original one. We obtained the relative errors for a Trotter number n = 1 and K = 1 as follows:

r2 = 1.02 x 10 -1, r4 = 6.11 x 10 -2, Ir6] = 8.40 x 10 -5,

IrsI -- 2.55 × 10 -9. (3.27)

These data suggest that the relative error r2m decreases faster than e -2m does as the degree of approximation goes to infinity.

3.4. Optimization of the fractal decomposition for various types of construction schemes

In the previous subsection we discussed the optimal degree of approximation, con- centrating on the construction scheme (3.7). Now we calculate the approximate free energy for various values of lp and lq using the fourth-order decomposition (3.1). Then we discuss which construction scheme is the optimal one.

The approximate free energy of the one-dimensional XY-model for the Trotter number n is obtained as

- -~ f (4 )" L / d k l ° g [ 2 c ° s h ( ~ - ) ] o

with

(3.28)

cosh y2 = cosh(2lpyp) cosh(lqyq) + cos(0p - - Oq) sinh(21pyp) sinh(lqyq). (3.29)


0.0100

0.0050 / /

0.0010 /" / / / / /

0.0005 / . / / / / / / / /

0.0001 0.03 0.04 0.05 0.06 0 .070 .080 .090 .1 0.2

- 1 N m u l

Fig. 3. The Nmul dependence of the relative errors r4 (solid lines) and r2 (dashed line) for K = 1, lq = 1, lp = 1, 2, 3 and 6.

Here we have defined yp and Op in Eqs. (A.20) and (A.21) with P2 replaced by

p(lp, lq) . We have also defined yq and Oq in Eqs. (A.20) and (A.21) with p replaced

by q and p2 replaced by q(Ip,lq). Note that the number of multiplications of the operators $2 (px) and Sz (qx) is given

by

Nmul = (21p + lq)n. (3.30)

We numerically calculated the relative error r4 of the free energy and obtained the

number N2,4(lp, lq) defined by the value of Nmul at the point [r2[ = Ir41. All the data thus obtained are listed in Table 2. In Fig. 3 we show the Nmul dependence of the

relative errors for K = 1, lq = 1, Ip = 1, 2, 3 and 6. We can intuitively find that the approximation for lp = 2 and lq = 1, i.e.,

[Sz(px)]ZS2(qx)[Sz(px)] 2, is more advantageous than that for lp = 1 and lq = 1, since the decomposition parameters p and q in the case (Ip = 2, lq = 1) are all less than unity. In Fig. 3 we observe that, for a fixed value of Nmul, the relative error due to the approximation in the c a s e (lp = 1, lq = 1) is about ten times as large as that due to the approximation in the case (lp = 2, Iq = 1 ) . This is one of the reasons why we

use the construction scheme (3.7) in Section 3.2. At any temperature, we obtain the smallest value of N2,4(lp, lq) for lp = 6 and lq = 1. It is not clear at present why the decomposition [ $2 (px) ] 6 Sz (qx) [ $2 (px) ] 6 gives the smallest relative error.

Incidentally, we can show in the following way that the approximation for lp = 2 and Iq = 2 gives the upper bound of that for Ip = 1 and lq = 1 for a fixed value of Nmul. The

approximation for Ip = 1 and lq = 1 is given by

Table 2 The number

H. Kobayashi et al. / Physica A 211 (1994) 234-254

N2,4 (lp, lq) defined by the value of Nmul at the point Ir21 = Ir41, at the temperature K = 1

245

lp \ lq 1 2 3 4 5

1 29.85 - - - 2 11.15 33.56 - - 3 9.10 18.40 38.16 - 4 8.50 14.83 24.99 43.02 - 5 8.29 13.37 20.51 31.10 47.93 6 8.24 12.63 18.35 26.07 36.88 7 8.26 12.23 17.13 23.36 31.48 8 8.32 12.00 16.37 21.70 28.33 9 8.40 11.87 15.87 20.61 26.30 10 8.50 11.81 15.54 19.85 24.90

• r n

with p = 1 / (2 - 21/3) and q = 1/(1 - 22/3). The approximation for lp = 2 and lq = 2 is given by

Tr [ [ s 2 ( P ' X ~ ] 2 [ s 2 ( q ' x ~ ] Z [ s z ( P ' X ~ ] 2 ] n' \ n' ] J \ n ' J J \ n' ] J J ' (3.32)

with pl = p/2 and ql = q/2. When the approximant (3.32) has the same number of the operators {$2} as the approximant (3.31), we can rewrite (3.32) in the following form:

Tr [[S2 ( ~ ) ] 2 [$2 ( ~ ) ] 2 [S2 ( ~ ) ] 2 ] n/2. (3.33)

Using Thompson's lemma [34],

Tr (AB) 2n < Tr (A2B2) n, (3.34)

we obtain

Tr [$2 ( ~ ) $ 2 (q~)$2 ( ~ ) ] n

_<Tr [[$2 ( ~ ) ] 2 [$2 ( ~ ) ] 2 [$2 ( ~ ) ] 2 ] n/2 (3.35)

This inequality has been confirmed from the obtained data; namely, the inequality

N2,4(1, 1) < N2,4(2, 2) holds.

3.5. Behavior of the relative error in the low-temperature limit

The error of the free energy due to the 2mth-order decomposition is approximately proportional to K2m+2/n 2m at high temperatures [35]. In this region we have to increase n faster than to increase K in order to keep the error constant.


Here we argue in general that at low temperatures the error can be written as a

function only of the variable Kin. Hence we have only to take the Trotter number n

proportional to K in order to suppress the error. We show that the relative error r2m

of the free energy calculated from the 2mth-order fractal decomposition is given in the

low-temperature limit as follows:

r2m ~ ~-~ c(m) K Z.., 2j as K ~ c~ with - - << 1. (3.36) n

j=m

It is obvious that the largest eigenvalue e y" of S2m(K/n) is a function of Kin for any temperatures. We obtain the free energy in the low-temperature limit from the largest

eigenvalue as follows:

[ - / 3 f (2m) = logTr Szm ~ log( e ny" ) = nym. (3.37)

Since Ym is a function of K/n, and since the approximant _~f (2m) is correct up to the

2ruth order of 1/n, we can represent Ym in the following form:

Ym = y(exaet) ~_ ~ ,,(m) -- ~ 62j+1 ' (3.38) j=m

where the first term of the right-hand side corresponds to the exact value of the free energy, and {g~)~+)l} are coefficients independent of K and n. In the low-temperature limit, the exact free energy f(exact) converges to the ground state energy. If we define a

factor t'(m) independent of K and n as ~2j

c(m) (m) _/~f(exact) t_(m) / ~e(exact) 2j = Kg2j+l/( ) = - a g 2 j + l / J ' (3.39)

the approximate free energy _~/f(2m) is given by

~x~ K2J+ 1 _/3f(2m) ~ _/~f(exact) + Z _ _ g ( m )

n2 j 2j+ 1 j=m

= - B f (exact) 1 + Z c ( m ) . ~2j • (3.40)

./=m

Thus we obtain the relative error r:m in the form (3.36) in the low-temperature limit.

Therefore, we can keep the relative error rzm small at low temperatures, if we take the

Trotter number n proportional to K. From Eqs. (3.36) and (3.22), we find that the solution of the equation Ir=ml = Ir:m+=l

should be expressed in the form K/Nmu~ =const. Hence the number N:m,:m+: defined by the value of Nmul at the point Ir=ml = Ir=m+=[ is proportional to K in the low-temperature

limit. This gives an explanation of Eqs. (3.24) and (3.25).


4. Baker-Camphell-Hausdorff formula and improved Trotter-like formula

247

In this section, we construct a fourth-order approximation to Tr e x(A+B), namely, the improved Trotter-like formula [26], which is introduced by Suzuki on the basis of the Baker-Campbell-Hausdorff (BCH) Formula. We calculate the approximate free energy of the one-dimensional quantum XY-model, and compare the relative error due to the improved Trotter-like formula with that due to the ordinary Trotter decomposition.

4.1. Improved Trotter-l ike f o r m u l a

We obtain higher-order decompositions, using the BCH formula given in Eqs. (2.6) and (2.7). It is not practical to use them as they are, because calculations of many commutators are required. However, we can construct a more practical approximation, which is correct up to the fourth order after taking the trace of it.

First we consider the fourth-order decomposition based on the BCH formula, which

is given [27] by

e x(A+B) = e~Ae~BeX3Re~Be~A + O(xS), (4.1)

with R = ~4 [ [B, A], A + 2B]. We rewrite e x(a+B) for the Trotter number n as follows:

e x(A+B)= e2.Aes, Be~Re~,Be 2.A -q-O

[ ~ ~ ( x 3 ) xB xA]n (x5) = e2,,ae z.n 1 + ~-~R e~ e~ ] + O ~-g . (4.2)

We expand the first term of the right-hand side of Eq. (4.2). Taking the trace of e x(a+B),

we obtain the following formula:

r ~A ~B ~A nn Tr eX(a+B)=Tr [e ~' e" e2, ]

X 3 [: XA XB xa\n--1 XB xa\'] (X5~ (e ~ e ~ R e ~ e r, )] + 0 - ~ +~-~ Tr [re r', e r' e~ ) " x a x B

\ /

t.[: xm XB xa'n--1 ( ) )1 (x5) -~ =Tr e; e~ e2,-Sae-~neSRe~Ber,,a + 0

[ , : (:) =Tr ( e ~ B e ~ A e ~ B ) " e ~ ] + O ~-~ . (4.3)

This is called the improved Trotter-like formula [26]. It is practically convenient to calculate the commutator R only once. In the trace of the operator, we put the commutators R together into one exponential operator.

4.2. Correct ion terms f o r the improved Trotter-like f o rmu la

Now we calculate the free energy of the model using the improved Trotter-like formula above introduced.

248 H. Kobayashi et al.

Table 3 The number NT,IT at various temperatures

/ Physica A 211 (1994) 234-254

K 0.5 1.0 2.0 5.0 10.0 20.0 Nv,rr 1.1 2.6 6.4 22.9 62.7 174.4

First we calculate the commutator R defined by

R = ~ [ [7-/2,7-/1],7-/1 + 27-(2], (4.4)

where 7-/i and 7"(2 are given by Eqs. (3.4) and (3.5). From the definition of the commutator, we obtain

L R = - + - + z z - 20"2i0r2i+ 1 -- O'2i--2Or2i - 1 O'2iO'2i+ 1

i=1

+ z z - "+20"2i_10"2i0"2i+10"2i+2 -4- h.c.), (4.5)

with o-~ = ½ ( ~ + itrY).

After the calculation shown in Appendix A, we obtain the approximate free energy

in the following form:

K5 K6 ( ~ 6 ) _ f l f ( I T ) = _/~f(exact) + --n4 a + ~ - b + O , (4.6)

where 7T

/ [ _/~f(exact) = 1 dklog 2cosh 2Kcos ~ , 77"

0

4 : ( ~ ) k k ( k) a = ~ dktanh 2Kcos cos ~ sin 2 ~ 1 + 21cos 2 , (4.7)

0 7/"

b= 8 dk l + t a n h 2 2Kcos cos 4~s in e - 7r 2

0

Now we compare the relative error rrr due to the improved Trotter-like formula with

r2 due to the ordinary Trotter decomposition. Let us define NT,rr by the value of n at the point Irel = ]rlTI. We numerically calculated NT,IT at various temperatures. All the data thus obtained are listed in Table 3. From Tables 1 and 3, we find that the improved Trotter-like formula is more advantageous at some temperatures than the fourth-order

fractal decomposition is. At low temperatures, the number NT,IT is approximately proportional to K 3/2 as shown

in Fig. 4. This is because the leading correction term in (4.6) is proportional to K6/n 4, while that in (3.6) is proportional to K3/n 2.

It is one of the merits that the number of multiplications Nmul is equal to the Trotter number n, and yet the improved Trotter-like formula is the fourth-order approximation to Tr e -#7~. However, its correction becomes relatively large at low temperatures.

H. Kobayashi et al. I Physica A 211 (1994) 234-254 249

0 .050

F / / / / ................. • " K=2

0 . 0 1 0 / /

/ / . . . " / / /

0 . 0 0 5 .. • I i / /

/ / / / /

/ / / /

/ F/ / / / ~ I / / / , / , '

I I I I I L i o.oo, I/ , ' , / , , ' , , , / , , i L 0.01 0.05 0.1 0.5

- 1 N m u l

Fig. 4. The relative errors r r r (sol id l ines) and r2 (dashed l ines) at the temperatures K = 1, 2, 5 and 10.

5. Summary and discussions

In the present paper, we have applied many kinds of higher-order decompositions to an exactly solvable model. We have evaluated the Trotter-number dependence of the correction terms systematically. We have discussed which kind of higher-order decomposition is the most useful in practical use.

We have confirmed the effectiveness of higher-order decompositions by calculating the free energy of the one-dimensional quantum XY-model. They are quite efficient, in particular, if we require high precision. The analysis shown in Fig. 2 is quite useful, when we select the optimal decomposition for calculations in other systems. We expect that the fractal decomposition can be successfully applied to various kinds of systems.

Acknowledgement

We would like to thank Dr. Akira Terai for his useful comments.

Appendix A. Calculation of the free energy

Here we present calculations of the free energy of the one-dimensional quantum XY- model for a finite Trotter number, using the second-order decomposition (Section 3.1), the fourth-order decomposition (Section 3.2) and the improved Trotter-like formula (Section 4).


Using the well-known Jordan-Wigner transformation

1--1 l--1

o'~- =exp[i~r)--~CJCjlCtt, o" l =exp[i~')--~C;CjlCl, j=l j=l

for o-~ = ' x io-Y), we (o- l + obtain

(A.1)

Y Y -~ 2 (C7Ci+ 1 q- h . c . ) . ( A . 2 ) orxo'~/+l --~ o" i ori+ 1

Here Cfi (Ci) is a fermion creation (annihilation) operator. We define ~:l,k and se2,k by means of the Fourier transformation as follows:

L L 1

SOl, k = ~ 1 ~j=l ei~Jc2j' ¢2,k = ~ ~j=l eikJc2j- l" (A.3)

The operators (l,k and ~:2,k satisfy the anticommutation relations

{•i,k, ¢~,k' } = •iJ akk' ; (A.4)

thus they are fermion operators. After the above two transformations, we can rewrite Eqs. (3.4) and (3.5) in the forms

= = ( A . 5 )

k k

where

7-(1,k = 2 (e - i k~[k~z , k + h.c.), ~2,k = 2(~:~,t~:z,k + h.c.). (A.6)

Thus we have made the block diagonalizafion with respect to k. Taking the basis { 11,2), ]1), 12), 10) } for each block with

I 1, 2) t t ~:~,kl0), se2t,kl0), (A.7) = ¢:l,k~:2,kl0), I1)= 12)=

we can represent "~l,k and 7-/2,k in the following matrix forms: (o) o) 0 2e -ik 0 2 (A.8)

7-[l,k = 2e ik 0 , 7-(2,k = 2 0 "

0 0

Let us define the transfer matrix T2,k(ff) by

T2,k = e ~ , k e Z ~ ' , ~ e ~ 2 , ~ , (A.9)

which is the second-order approximation (2.9). We obtain the exponential operators K K

e~ ~,k and e~ ~2,k in the following forms:


1

e~7~1, k = cosh(K/n) e ik sinh(K/n)

e-ik sinh( K/n) ) cosh(K/n)

1

e~ ~2.k = cosh(K/n) sinh(K/n) ) sinh(K/n) cosh(K/n) "

1

251

(A.10)

(A.11)

Hence the transfer matrix (A.9) is expressed as follows:

cosh y e-i° sinh y e iO sinhy cosh y

1

(A.12)

where

cos cosh y = cosh ~ + sin 2 z '

and

(A.13)

ei o = c o s h ( - ~ ) c o s k + / s i n k . (A.14)

~/cosh2 (_~) cos 2 k + sin 2 k

Since the eigenvalues of T2,k( ff ) are e +y and 1, we obtain the approximant Z (2) to the

partition function of the model as follows [24]:

Z(2)=Tr [ s E ( K ) l n = g T r [T2,~(K)] n

= I I ( 2 + 2 cosh(ny)) = I I [4cosh2 ( 7 ) ] . (A.15) k k

Then the free energy per site in the thermodynamic limit L ~ oo is given by

-- '°gz 2' -- 12-z Z l o g [4cos.2 k

7 r

___1 f dklog[2cosh(7)] " (A.16) , r r j

o

1 | f~_~ dk in the limit Here we have replaced the summation Z }-'~k by the integral ~ L ~ oc. If we expand the approximate free energy _flf(2) in (A.16) up to the second order of 1/n, we obtain Eq. (3.6).


Now we calculate the approximate free energy using the fourth-order fractal decomposition (3.10). If we define 7"4,k(K/n) in the form

k

then T4,k(K/n) is calculated from 7"2,k(K/n) as follows:

T4,k ( K ) . [Tz,k ( ~ - ) 1 2 T 2 , k (q-~)[T2,k ( P - ~ ) ] 2. (A.18)

We represent 7-2,k(p2K/n) in the form

( ) Tz,k (~_~K) cosh yp e-i°'sinhyp = e i°p sinh yp cosh yp '

1

where

and

cosh yp = cosh COS 2 ~ + sin 2 ~,

(A.19)

(A.20)

eiOp = cosh(-~-)cos~ +is in~ (A.21)

~/cosh2 ( -~ - ) cos2 ~ + sin2 ~ '

while T2,k(q2K/n) is represented by Eqs. (A.19), (A.20) and (A.21) with p replaced by q. Since the eigenvalues of 7"4,k(K/n) are e ±y2 and 1 with

cosh Y2 = cosh (4yp) cosh yq 9- cos ( Op -- Oq) sinh(4yp ) sinh yq, (A.22)

we obtain the approximant Z ~a) as follows:

Z O)~Tr $4 = Tr T4,k k

= H ( 2 9- 2 cosh(ny2) ) = 1--[ [4 cosh2 (~-2-)]. (A.23) k k

Then the free energy per site in the thermodynamic limit L ~ oo is given by

__flf(4) = __1 1ogZ(4) =--1 ~ l o g [4cosh 2 ( ~ ) ] 2L 2L k

----+ I j dk l°g [2 c°sh (-~) ] " ~r (A.24) o

If we expand the approximate free energy _f i fO) in (A.24) up to the fourth order of 1 n' we obtain Eq. (3.13).


Hereafter we calculate the approximate free energy using the improved Trotter-like formula.

Using the Jordan-Wigner transformation given in Eq. (A . I ) and the Fourier transformation given in Eq. (A.3) , we rewrite R in Eq. (4.5) as follows:

= ~ Rk, Rk = ~l,k~2,~ + h.c., (A.25) R k

with a defined by

a = ~( 1 - 2 e - i k - e - Z i k + 2 e l k ) . (A.26)

Taking the basis { I1,2), I1), ]2), 10) }, we represent Rk and e (K3/n~)Rk in the following matrix forms:

( °° ) Rk = Ot ot 0

0 (' Rk cosh z

e - = e- i~ sinh z

(A.27)

) e 'q~ sinh z (A.28) c o s h Z

1

1 1 - - e - i ° U = - ~ e i° 1 ' (A.30)

v5

we obtain the approximate partition function Z (IT) as follows:

S'= r

= 1-I Tr e ~, ~2.k e ,r'~.k e 2, ~2,k e.-,r k

= 1 1 1 2 + 2 cosh z cosh(ny) + 2 cos(0 + ~o) sinh z s inh(ny) ]. k

(A.31)

Then we obtain the free energy per site in the thermodynamic limit L ~ ~ as

xT- t xT- t x7-¢ Since the transfer matrix T 2 , k ( K / n ) =-- e~, 2.kex ~.ke~ 2.k, or Eq. (A.12), is diagonal- ized with the unitary matrix

with

g3la[ . a

z = n2 , e'~ = / I '°t---v" (A.29)

254

I f we e x p a n d the

1/n, we ar r ive at


_//3f( iT ) 1 2L log Z (rr)

= 1 ~ log [ 2 + 2 cosh z cosh ( n y ) + 2 cos ( 0 + ~ ) s inh z s inh ( n y ) ] 2 / .

k q r

1 f dk log [ 2 + 2 cosh z cosh ( n y ) + 2 cos ( 0 + ~o) s inh z s i n h ( n y ) ]. --~ 2~r

o ( A . 3 2 )

a p p r o x i m a t e f ree energy _ /~f ( IT) in ( A . 3 2 ) up to the four th o rder o f

Eq. ( 4 . 6 ) .

References

[1] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087.

[2] H.E Trotter, Proc. Am. Math. Phys. 10 (1959) 545. [3] M. Suzuki, Prog. Theor. Phys. 56 (1976) 1454. [4] M. Suzuki, Commun. Math. Phys. 51 (1976) 183. [5] M. Suzuki, J. Stat. Phys. 43 (1986) 883. [6] M. Suzuki, ed., Quantum Monte Carlo Methods (Springer, Berlin, 1987). [7] M. Suzuki, Phys. Lett. A 146 (1990) 319. [8] M. Suzuki, J. Math. Phys. 32 (1991) 400. [9] M. Suzuki, Phys. Lett. A 165 (1992) 387.

[10] M. Suzuki, J. Phys. Soc. Jpn. 61 (1992) 3015. [11] M. Suzuki, Physica A 191 (1992)501. [12] H. Yoshida, Phys. Lett. A 150 (1990) 262. [13] N. Hatano and M. Suzuki, Prog. Theor. Phys. 85 (1991) 481. [14] M. Suzuki and K. Umeno, in: Computer Simulation Studies in Condensed-Matter Physics VI, D.P.

Landau, K.K. Mon and H.B. Schtittler, eds. (Springer, Berlin, 1993) p. 74. [15] K. Umeno and M. Suzuki, Phys. Lett. A 181 (1993) 387. [161 M. Suzuki, Proc. Japan Acad. 69 Ser. B (1993) 161. [17] A. Terai and Y. Ono, Prog. Theor. Phys. Suppl. 113 (1993) 177. [181 H. Yamada, K. lkeda and M. Goda, Phys. Lett. A 182 (1993) 77. [ 19] H. De Raedt and K. Michielsen, Algorithm to solve the time-dependent Schrrdinger equation for a

charged particle in a inhomogenous magnetic field, application to the Aharonov-Bohm effect, preprint. [20] M. Suzuki and T. Yamauchi, J. Math. Phys. 34 (1993) 4892. [21] M. Suzuki, Commun. Math. Phys. (1994), in press. [22] K. Aomoto, On a unitary version of Suzuki's exponential product formula, preprint. [23] M. Suzuki and M. lnoue, Prog. Theor. Phys. 78 (1987) 787. [24] M. Inoue and M. Suzuki, Prog. Theor. Phys. 79 (1988) 645. [25] M. Inoue and M. Suzuki, Prog. Theor. Phys. 79 (1988) 30. [26] M. Suzuki, Phys. Lett. A 180 (1993) 232. [27] H. De Raedt and B. De Raedt, Phys. Rev. A 28 (1983) 3575. [28] M. Suzuki, J. Math. Phys. 26 (1985) 601. [29] R.D. Ruth, IEEE Trans. Nucl. Sci. NS-30 (1983) 2669. [30] J. Candy and W. Rozmus, J. Comput. Phys. 92 (1991) 230. [31] E. Forest and R.D. Ruth, Physica D 43 (1990) 105. [32] S. Katsura, Phys. Rev. 127 (1962) 1508. [33] N. Hatano and M. Suzuki, in: Quantum Monte Carlo Methods in Condensed Matter Physics, M. Suzuki,

ed. (World Scientific, Singapore, 1993) p. 13. [34] C.J. Thompson, J. Math. Phys. 6 (1965) 1812. [35] N. Hatano and M. Suzuki, Phys. Lett. A 153 (1991) 191.

Study of correction terms for higher-order decompositions of exponential operators

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