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481

Progress of Theoretical Physics, Vol. 85, No.3, March 1991

Transfer-Matrix Calculations of the Spin 1/2 Antiferromagnetic XXZ Model on the 4 x 2 Triangular Lattice Using the Fractal Decomposition

N aomichi HAT ANO and Masuo SUZUKI

Department of Physics, University of Tokyo, Tokyo 113

(Received November 7, 1990)

The fractal decomposition of exponential operators proposed by Suzuki, which is a new category of the generalized Trotter decomposition valid up to higher orders, is examined with transfer-matrix calculations of the spin 1/2 antiferromagnetic XXZ model on the 4 x 2 triangular lattice. The dependence of correction terms on the Trotter number and temperature are studied. This dependence confirms rapid convergence of the fractal decomposition. The negative-sign problem arising in quantum Monte Carlo simulations is also discussed ·from the present new point of view.

§ 1. Introduction

Owing to a remarkable progress of computers, Monte Carlo calculations have become one of the most powerful methods to investigate many-body system, especially quantum ones. I)

One of the difficulties encountered in simulating quantum many-body systems at finite temperatures is that the Boltzmann weight <¢Iexp( - ,eJC)I¢> cannot be calculated locally_ The standard method to resolve this difficulty is to make use of the following generalized Trotter decomposition2) of the density matrix:

(1·1)

where ,e denotes inverse temperature (kB T)-I and n is called the "Trotter number". It enables one to map2) approximately ad-dimensional quantum system into a (d + I)-dimensional classical system, for which the Boltzmann weight can be calculated locally_ The extrapolation n~= of physical quantities {Q(n)} measured on the mapped classical system reproduces the required physical properties of the original quantum system, where correction terms vanish as n-m

_ On the other hand, the size of the mapped system increases linearly in the Trotter number n, and so does the CPU-time of computers for simulating it. Thus approximants of the type (1·1) with large m may improve the procedure of the extrapolation n~=_

Recently, one of the present authors3) discovered a new way to construct step by step appr0ximants correct up to arbitrarily higher orders_ An appropriate combination of lower-order approximants yields a higher-order approximant in such a way that the lowest-order corrections of the former cancel out with each other. Since coefficients of the decomposition,eJC ~{,eAI, ,eA2, ---, ,eAq} become fractal3

) for higherorder approximants, it is called the "fractal decomposition".

In the present paper, prior to quantum Monte Carlo simulations, transfer-matrix calculations were made to confirm the rapid convergence n ~ = of approximants of the fractal decomposition. The system discussed here is the spin 1/2 antiferromagnetic XXZ model on the triangular lattice of size 4 x 2. Approximants and

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482 N. Hatano and M. Suzuki

algorithm employed here are presented in § 2. The Trotter-number and temperature dependence of correction terms to approximants in the region of /3/n~I are studied in §§ 3 and 4 respectively: Some discussion is made in § 5, concerning the negative-sign problem, which arises unavoidably in simulations at low temperatures using the fractal decomposition.

§2. Method of calculations

The system and algorithm employed here are explained in the present section. As mentioned above, all the calculations in the present paper were performed for

the spin 1/2 anti ferromagnetic XXZ model on the triangular lattice of size 4 x 2:

117 -...., [ J (- x - x + - y - Y) J - z - Z] Jttotal= "'-' - xy (J i (Jj (J i (Jj - z(J i (Jj <i,j>

- ...., [ 2J (- + - - + - - - +) J - z - Z] - "'-' - xy (Ji (Jj (Ji (Jj - z(Ji (Jj , <i,j>

(2·1)

where (J denotes the Pauli matrices

0) and -1

-x+ '-Y (J - Z6 , (2.2)

2

and ~ (t,J) denotes the summation over all the nearest neighbours on the lattice with periodic boundary conditions, as illustrated in Fig. I(a). Hereafter the following three cases of the parameters ]xy and ]zare examined; namely the Ising-like case (Jxy=-1/4, ]z=-I), the isotropic Heisenberg case (JXy=]z=-I), and the XY-like case (JXY= -1, ]z= -1/4).

To apply' the generalized Trotter decomposition to the density matrix exp( - /3Jltotal), the total Hamiltonian (2·1) is decomposed into the following four parts:4

) Jltotal=Jll +Jl2+Jl3 +Jl4 , where '

with

Jl1=Jl(a, b, /, e)+Jl(c, d, h, g), Jl2=Jl(b, c, g, /)+Jl(d, a, e, h) ,

Jl3=Jl(/, g, c, b)+Jl(h, e, a, d) and Jl4 =Jl(e, /, b, a) + Jl(g, h, d, c) (2·3)

117( ) - 1 ...., [ J (- x - x + - y -- Y) J - z - Z] Jt p, q, r, S =-2 .. "'-' - xy (Ji (Jj (Ji (Jj - z(Ji (Jj <Z,J>=<p,q>,<q,r>.<r,s>,<s,p>

+ [ J ( - x - x + - y - Y) J - z - Z] - xy (Jp (Jr (Jp (Jr - z(Jp (Jr , (2·4)

as illustrated in Fig. I(b). The following two kinds of the generalized Trotter decomposition are used and

compared in the present paper; (a) the second-order decomposition2)-3)

(2·5)

where

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The AFXXZ Using the Fractal Decomposition 483

s r

q a

(a) (b)

Fig. 1. (a) All the calculations were performed on the triangular lattice of size 4 x 2 with periodic boundary conditions. (b) A piece of the Hamiltonian into which the total Hamiltonian is decomposed.

and (b) the fourth-order decomposition3)

where

with

exp( -/1JCtotal)=[Si -/1/n)]n+ 0(/15/n4) , (2-7)

54(X) = 52(P2X) 52(P2X) 52((1-4P2)x) 5zCP2X) 5zCP2X) =exp(xJCtotal) + 0(x5)

(2-8)

4P23 +(1-4P2)3=0 or P2=(4-\I4)-1~0.41449···. (2-9)

The parameter P2 is chosen to cancel out the third-order corrections of the product of the five operators 52 in Eq. (2-8). The fourth-order correction proves to vani:;h automatically.3) Note that the propagator 54(-/1) yields a to-and-fro path3) in the direction of the imaginary time r=O--+ /1, owing to the part (1-4P2)< 0 in Eq. (2-8).

When these decompositions are applied to quantum Monte Carlo simulations/) the present two-dimensional quantum system is mapped to a three-dimensional classical Ising-spin system as follows: In the case of the second-order decomposition (2-5) with the Trotter number n=n2, the partition functions of the corresponding mapped classical system can be written2

) as follows:

=~W2({a}; /1/n2) , {O"}

(2-10)

where every I{ah> denotes an eigenstate of the operators {a'Z} of the spins on the eight lattice-points, and ~{O"} denotes the summation over all the complete sets Wah>}, k=l, 2, "', 6n2. The factors <{ahlexp( -xJCj)l{ah+l> in Eq. (2-10) can be interpreted as elements of a transfer matrix from Ising-spin configurations on the k-th layer to ones on the (k+ l)-th layer. Then W2({a}; /1/n2) is the Boltzmann weight of a configuration

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{a} = {{O-h, {O-}Z, ... , {a}6nz} of a three-dimensional Ising-spin system defined by these transfer matrices. The additional third dimension is called the "Trotter direction". The same technique can be used for the fourth-order decomposition

(2·11)

With the bases {I{ah>}, the transfer matrix <{ahlexp( -xJ(JI{O-h+1> is blockdiagonalized into the subspaces of the total magnetization Mz=4, 3, 2, ... , -3, -4. The maximal size of the blocks of the matrix is 70 x 70, and the products of the blocks can be easily taken on computers. Once the matrices S2 and S4 are calculated as in Eqs. (2·6) and (2·8), thenthe Trotter number n=2 q can be achieved only by q times operations of matrix-product. On the other hand, the matrix of the original Hamiltonian <{a}lJ(totaMa'}> has the same size as the transfer matrix has. Exact diagonalization of J(total can be made on computers. Corrections to the appro ximants Z2 and Z4 in Eqs. (2·10) and (2·11) are thus calculated.

For the second-order decomposition, the lattice size in the Trotter directions of the mapped Ising~spin system is L2=6n2, as is shown in Eq. (2·10). In the case of the fourth-order decomposition (2·7) with the Trotter number n=n4, the lattice size is L =30n4, because five operators of. S2 are combined into one operator of S4 in Eq. (2·8). The CPU-time of computers for Monte Carlo simulations depends on the lattice size. From this point of view, comparisons between these two decompositions should be made under the condition L2=L4 or n2=5n4. Hereafter the Trotter number n is chosen so as to satisfy this condition for the fourth-order decomposition, while for the second-order decomposition we set n2=5n except the cases mentioned explicitly.

.... ()

'" >< Q)

N I '" N

§ 3. Dependence on the Trotter number

The Trotter-number dependence of correction terms in highctemperature regions

10-2

.... ()

'" 10-4 >< Q)

I:J::l I '" 10-6 I:J::l

10-3 10-2 10- 1

1/n4 (5/n2)

(a)

10-4

10-6

10-8

/

'"

10-3 10-2 10-1

1/n4 (5/n2)

(b)

Fig. 2. The Trotter·number dependence of corrections to (a) the partition functions and (b) the energies at the temperature T=lO.O in the Ising·like case, calculated with the second-order decomposition (dashed lines) and the fourth-order decomposition (solid lines) respectively. These

plots confirm the behaviour shown in Eq. (3 ·1).

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106

---:;:; ---:;:; () ()

C1l C1l :<: :<: 1018

'" '" N 103 N

I I '" " '" '" N " '"

N '"

'" '" 1015 .... '" - .. '" -

'" '" +' 100 '" +'

¥ () .. ()

'" C1l '" C1l ¥ :<: ¥ :<: ./

'" '" '" '" ¥

N ¥ N '" 1012 "'...-I ",¥ I ",¥ '<I' 10-3 "'...- '<I'

",¥ N "'...- N ;/

10-4 10-2 10-1 100 10-4 10-2 10-1 100

1/n4 (5/n2) 1/n4 (5/n2) (a) (b)

Fig. 3. The Trotter-number dependence of corrections to the partition functions of the system at the temperatures (a) T=1.0, (b) T=O.25 in the Ising-like case, calculated with the second-order decomposition (dashed lines) and the fourth-order decomposition (solid lines) respectively. For a small Trotter-number, it occurs that IZ. - Zexactl > IZ2- Zexactl.

is discussed in the present section. The partition functions and the energies of the system calculated with Eqs. (2·5)

and (2·7) at the temperature T=10.0 are plotted in Fig. 2 in the Ising-like case, for example. The behaviour in the other two cases is quite similar to it. Since the inverse temperature ,8=0.1 (with kB=l) is rather small, the data of the partition functions (Fig. 2(a)) can be fitted well to the following functions even for rather small Trotter numbers n ~ 1:

with some parameters A2 and A. The data of the energies (Fig. 2(b)) show the same behaviour. In any case, the correction to Z4 in Eq. (3·1) is less than to Z2 for n;:?::1. From this point of view, the fourth-order decomposition 54 is more advantageous than the second-order decomposition 52 in order to calculate physical quantities in this temperature region.

At lower temperatures, however, it occurs for small Trotter number that

!Z2(,8;n)- Zexact(,8)! < !Zi,8;n) -Zexact(,8)!; (3·2)

see Fig. 3 in the Ising-like case. It is observed in most cases that the relation !Z2-Zexact!>!Z4-Zexact! is satisfied only for ,8/n~1/4 (in some cases, for ,8/n~1/8, or ,8/n~1/2). It is the situation as expected, because the correction term to the approximant 54 is of higher order with respect to ,8 than that to 52; see Eqs. (2·5) and (2·7). It is supposed that the approximant 54 is more advantageous than 52 is, when the criterion3

)

(3·3)

is satisfied. The above observation confirms this criterion.

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---:;:; ---:;:; () ()

<IS 10-,6

<IS X X OJ OJ

N I'i! I I (\) (\)

N 10-9 I'i!

- ",'" -- ---:;:; ..., A/ () .a/"

()

<IS <IS X "

X OJ

10- 12 OJ

N I'i! I I "<I' "<I'

N I'i!

10- 15 0.01 0.05 0.1

Fig. 4. The inverse-temperature dependence in high-temperature regions of the partition functions (solid lines) and the energies (dashed lines) are calculated in the Ising-like case with the second-order decomposition (crosses) and the fourth-orde'r decomposition (circles) respectively. The Trotter number is fixed at n

= 1024. The plots confirm the behaviour shown in Eqs. (4·2) and (4-3) respectively.

§ 4. Dependence on temperature in . high-temperature regions

The temperature dependence of corrections in high-temperature regions is discussed in the present section.

At high temperatures (3/n<l, correction terms to the density matrix exp ( - (3 j() under a fixed Trotter number can be written as follows:

[Sz( - (3/n)]n= e-P.§( + ((33Q3+ 0((34»/nZ

+ 0((34/n3) ,

[Si - (3/n)]n= e-P.§( + ((35 R5+ 0((36»/n4

+ 0(/36/n5) . (4 ·1)

It can be generally proved,5) however, that the traces of the lowest-order corrections in Eq.(4 ·1) always vanish, namely,

Tr Q3=Tr R5=0. Then, as far as the partition functions are concerned, corrections come to be of higher order by one with respect to (3 as follows:

Zz((3;n)=Tr [Sz( - (3/n)]n~Zexact((3)+ Bz(34/n2 ,

(4·2)

for (3/n<l, with some constants B2 and B4. This behaviour is confirmed by the results, for example, in the Ising-like case plotted in Fig_ 4(a)_ In high-temperature regions, correction to the partition function is suppressed by this effect

Sinc,e Tr (j(Q3) and Tr (j(R5) are generally non-vanishing, corrections to the energies are fitted to the following functions:

E2((3;n)- Eexact~ Cz(33/n2 and Ei(3;n)- Eexact~ C(35/n4

with some constants Cz- and C; see Fig_ 4(b), for example.

§ 5. Low-temperature regions and the negative-sign problem

(4·3)

The temperature dependence of the partition functions in low-temperature regions is discussed in connection with the negative-sign problem_

The partition functions Zz and Z4 of mapped classical systems are always positive, while each Boltzmann weight w({o};(3/n) in Eqs_ (2·10) and (2·11) is not necessarily positive. Indeed, negative weights appear for frustrated systems such as the present one. All the diagonal elements of the transfer matrices <{Ohlexp( -xj(j)1 {O"h+l> in Eq. (2·10) are positive, while some of the off-diagonal elements have negative signs owing to jXY<O. There do exist4

) some configurations {O"} whose

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weights w({o}; /3ln) are negative. Furthermore for the fractal decomposition, it is essentiaP) that negative temperatures appear at some interactions, e.g., we have the negative coefficient (1-4P4)/3-::::=. -0.658/3<0 in the decomposition (2·8). Then negative Boltzmann weights appear even in non-frustrated systems, as well as in frustrated ones.

In Monte Carlo simulations, however, all the Boltzmann weights must be positive. A new system defined by Boltzmann weights of Iw({o}; /3ln) I should be simulated. These weights I wi can be generated as follows: In Eq. (2 ·10), negative signs of the weights W2 originate from the off-diagonal elements of the transfer matrices <{oh lexp( -x3UI{Oh+1> as mentioned above. Define new transfer matrices I<{ohlexp(-x 3lj )I{Oh+1>1 whose off-diagonal elements are absolute values of the original transfer matrices. The following operator can be defined by these new transfer matrices:

< {Ohl Sz( - /3)I{oh>

= 1< {ohl e-P.9l'I{ oh>ll< {Ohl e-P.9l,/2I{oh>I·· ·1< {o}6Ie-P.9l,/zl{ oh>1 . (5·1)

(Hereafter quantities and operators of the system with positive weights Iwl are denoted by primes'.) The operator (5·1) gives the partition function of the new system to be simulated in the following form:

(5·2)

As for the fractal decomposition (2·8), the following operator should be used instead of Eq. (5·1):

S4( - /3)= Sz( - P2f3)SZ( - P2/3)SZ( -(1-4P2)f3) Sz( - P2/3) Sz( - P2/3) . (5·3)

Note that the part Sz( -(1-4P2)/3) must be a non-prime-operator, because the coupling (1-4P2)Jxy turns positive here owing to the inequality (1-4P2)<0. The partition function to be simulated is given by

(5·4)

The ratio of the original partition function Z and the partition function with a prime Z' can be measured through simulations as follows:

Z(/3;n) Z'(/3;n)

where s denotes

~{(1}s({O})lw({o}; /3ln) I ~{(r]lw({o}; /3ln) I

s({o})=sgn(w({o}; /3ln)) ,

<s({o}»' , (5·5)

(5·6)

and < ... >' denotes the thermal average with respect to the prime-system Z'. On the other hand, one can also measure the relevant physical quantity Q multiplied by the sign s as follows:

<sQ>' ~{(1}Q({o}; /3ln)s({o})lw({o}; /3ln) I ~{(1}lw({o}; /3ln) I (5·7)

Combination of them yields the thermal average with respect to the non-prime-system

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Z as follows:

<Q> L:{(l"}Q({O"}; p/n)w({O"}; p/n) L:{O"}w({O"}; p/n)

<sQ>' <s>' . (5·8)

What is the Hamiltonian §t, of the prime-system Z' in Eqs. (5·2) and (5·4)? Suppose that the relevant operator fl(x) is expanded with respect to x in the form

(5·9)

where I denotes the identity-operator. The following limit gives an exponential operator:

(5·10)

This is nothing but the definition of the exponential function: limn-oo (I +x/n)n=ex.

When one performs quantum Monte Carlo simulations using the operator fl(x) of Eq. (5·9), the simulated system is, in the limit n-HXJ , described by the Hamiltonian §t of the r.h.s. of Eq. (5·9).

Of course, the expansion (5·9) applied to the operators 52 (Eq. (2·6» and 54 (Eq. (2·8» gives the original Hamiltonian §ttotal. The expansions of the operators 52 (Eq. (5·1» and S4 (Eq. (5·3» are made as follows: The original transfer matrix <kh lexp( - p§tj)I{O"h+l> is expanded in the following form:

<{O"hle-p.9l;I{O"h+l>= I +<khl( - p§tJI{O"h+l>+ O(P2) . (5 ·11)

The diagonal elements of the matrix on the r.h.s., <{O"hl( - p§tj)I{O"h+l>, should take the form PJz multiplied by some factors, while the off-diagonal elements should be PJxy multiplied by some positive factors. Changing the off-diagonal elements of the transfer matrix iI). the l.h.s. into the absolute values is equivalent, in the order of p, to

. switching the coupling Jxy<O to its absolute value IJxyl. Therefore, the operator 52( - P) defined by Eq. (5·1) is expanded in the following form>

with

lir, - '" [ IJ I( - x - x+ - Y - Y) J - z - Z] J1, (2) = £..... - xy 0" i (J j 0" i 0" j - z 0" i 0" j . <i.i>

As for the operator 54( - P) defined by Eq. (5·3), the expansion gives

54( - P)=( 1-P2P§t(2)Y( 1-(1-4P2)fJ§ttotal)( 1-P2P§t(2»2+ O(P2)

= 1- p§t(4) + O(P2)

with

§t(4)=4Pz§t(2) + (1-4P2)§ttotal

= L: [-(8P2-1)IJxyl(o?O'l+ 0'/0'/)- JzO'/O'/J. <i,j>

(5·12)

(5·13)

(5·14)

(5·15)

The coefficient (8P2-1) originates from the total length 41p21 + 11-4P21( > 1) of the

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The AFXXZ Usz"ng the Fractal Decomposz"tion 489

to-and-fro path mentioned below Eq. (2·9). It can be concluded from the above discussions that the systems Zi.({3;n) and ZI({3;n) defined by Eqs. (5·2) and (5·4) converge in the limit n ~ co as follows:

limZi.({3;n)=Tr exp( - (3§tb) and limZI({3;n)=Tr exp( - (3§t(4» . (5·16) n-oo n-oo

In low-temperature regions, it may happen that Z ~Z'. In such cases, the value of <s>'=Z/Z' in Eq. (5·5) becomes so small that the statistical error of it arising in Monte Carlo simulations may become larger than itself, namely L1<s>' > <s>'. The estimate of < Q> through such simulations makes no sense, because the denominator of the r.h.s. of Eq. (5·8) has a large error. This is the negative-sign problem.

Recently, for the system transformed from the Hubbard model by the HubbardStratonovich transformation, the following discussion concerning the (3-dependence of the ratio <s >' has been made:6

)-9) At low temperatures, namely kB T ~Eex - Eg with Eg and Eex denoting the ground-state and the first-excited-state energies respectively, the partition function Z is expected to show such a behaviour as

(5·17)

If the partition function of the prime-system Z' at low temperatures also shows a behaviour as

then the ratio of them behaves as

1038

.... N

1030

'" N

.;:, 1022 ()

~ Q)

N 10 14

Fig. 5. The inverse· temperature dependence in low·temperature regions of the partition functions of (i) the original system Zexact (dashdotted line) and (ij) the prime-system Zi (dashed line) and (iii) Z~ (solid line) are calculated in the XY-like case. The Trotter number is fixed at'n=128. The data behave as shown in Eqs. (5·17) and (5·18).

(5·18)

(5·19)

a

D b J C

Fig. 6. The XXZ model on a triangular,

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Table 1. The ground·state energies of the systems Zexact, Z2 and Z~ in the limit n-->OO.

Ising·like Heisenberg XY·like

Eg~r E~ Eg-E~ Eg or E~

Zexact -12.0 - -24.0

Z2 -13.3'" 1.3··· -33.6···

Z~ -22.0'" 10.0'" -70.4···

1.1 1.2 1.4 1.5

(3 Fig. 7. The ratio <s>' are calculated in the

XY-like case with the second-order decomposition <s>~=Z';Z2 (dashed lines) and the fourthorder decomposition <S>4=Z4/Z~ (solid lines) respectively. The Trotter number is fixed at n =128. This plot guarantee the behaviour shown in Eq. (5'19).

other two systems_

Eg-E~ Eg or E~ Eg-E~

- -18.0 -

9.6'" -29.4'" 11.4 .. ·

46.4'" -66.5'" 48.5'"

with L1=Eg- E~. This discussion may also hold for the present system.

"fheinverse-temperature dependence of Z2(fJ) ~ Zi(3) ~ ZexactC(3) , Zz(/3), ZI(/3} and the ratios <S>;=Z2/ZZ and <s>~ =Z4/ZI for a sufficiently large Trotternumber n (n~/3~(Eex- Eg) -1) are shown in Fig. 5 in the XY-like case, for example. The linearity of these data may guarantee the behaviour shown in Eqs. (5 -17) and (5 -18). The data becomes linear at /3 ~ 1 also in the other two cases. The ground-state energies of Jltotal, Jl(2) and Jl(4) calculated with exact diagonalization are listed in Table 1. The energy of the system ZI(/3) becomes far lower than those of the

This behaviour of the ground-state energies may be explained by an illustrative example, namely the three-site system on a triangle (Fig_ 6). The energy eigenvalues of the system are -3]z, ]z+2]xy and ]z-4]xy. The ground-state energy in the case ]z, ]Xy<O is E g=]z+2]xy. When the coupling ]Xy is switched tol]xyl as in Eq. (5-13), the ground-state energy becomes E~(2)=]z-41]xYI( <Eg). For the Hamiltonian such as Eq_ (5-15), the ground-state energy becomes E~(4)=]z-4(8p2-1)IJxYI, which is even lower than E~(2).

The above situation may also stand in the case of the present 4 x 2 system. At sufficiently low temperatures, the system Zz may be in a ground state different from that of the original system Zexact, ~hich results in the behaviour shown in Eq. (5 -19)_ The system ZI may be in the same ground state as Zz, while the energy of the state becomes still lower than that of Zz owing to the factor (8P2-1).

The low ground-state energy of the system ZI causes a rapid decay of the ratio <s>'; see Fig. 7, for example. Then the Monte Carlo simulations will be critical in low-temperature regions, unless some counterplots of the negative-sign problem are taken.

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Q(n) Q(n)

o o (a) (b)

Fig. 8. The schematic situations of the extrapolation n->OO for the relevant quantity Q(n) measured in quantum Monte Carlo simulations with (a) the decomposition 52 and (b) the fractal decomposition 54.

§ 6. Summary

The rapid convergence n ~ = of the fractal decomposition is confirmed for the present system. Also the temperature dependence of correction terms verifies the discussion given in the papers by SuzukL3

) As for transfer-matrix calculations, the fractal decomposition proves to be advantageous if the criterion /3/n4:.1/5 is satisfied. Quantum Monte Carlo simulations using the fractal decomposition may work well at high temperatures where the ratio <s>' is not so small, while at low temperatures the rapid decay of <s>' makes estimates of physical quantities difficult.

The schematic situations of the extrapolation n ~ = are shown in Fig. 8. The error bar for the finally estimated value Q(n==) results both from th.e finiteness of the Trotter number n and from statistical errors appearing in Monte Carlo simulations. It can be stated that the fractal decomposition has a merit with respect to the former, a demerit with respect to the latter.

Acknowledgements

The present authors are grateful to Dr. M. Takasu, Mr. N. Ito and Mr. N. Furukawa for useful discussions. One of the present authors (N. H.) is also grateful to Nihon-Sekiyu-Kagaku Scholarship for its financial support. The present calculations were performed, partially on the HIT AC M680 of the Computer Centre, University of Tokyo, partially on the V AX6440 of the Meson Science Laboratory, Faculty of Science, University of Tokyo. The authors would express their gratitude to Professor R. Hayano for making available the V AX machine.

References

1) Quantum Monte Carlo Methods, ed. M. Suzuki (Springer-Verlag, Berlin, 1987). 2) M. Suzuki, J. Stat. Phys. 43 (1986), 883, and references cited therein. 3) M. Suzuki, Phys. Lett. A146 (1990), 319; J. Math. Phys. 32 (1991), 400. 4) M. Takasu, S. Miyashita and M. Suzuki, in Ref. 1). 5) N. Hatano and M. Suzuki, to be published in Phys. Lett. A.

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6) E. Y. Loh, J. E. Gubematis, R. T. Scaletter, S. R. White, D. J. Scalapino and R. L. Sugar, Phys. Rev. B41 (1990), 930l.

7) D. R. Hamann and S. B. Fahy, Phys. Rev. B41 (1990), 11352. 8) S. Sorell a, Preprint, S.I.S.S.A. 147 eM (Dec. 89). 9) N. Furukawa and M. Imada, J. Phys. Soc. Jpn. 60 (1991), 810.

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acoustical properties of a kentucky a-style mandolin matthew

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