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Fast algorithm for finding the eigenvalue distribution of very large matrices
Hams, A.H; de Raedt, Hans
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10.1103/PhysRevE.62.4365
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Hams, A. H., & de Raedt, H. A. (2000). Fast algorithm for finding the eigenvalue distribution of very large
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PHYSICAL REVIEW E
VOLUME 62, NUMBER 3
SEPTEMBER 2000
Fast algorithm for finding the eigenvalue distribution of very large matrices
Anthony Hams and Hans De Raedt
Institute for Theoretical Physics and Materials Science Centre, University of Groningen, Nijenborgh 4,
NL-9747 AG Groningen, The Netherlands
共Received 11 April 2000兲
A theoretical analysis is given of the equation of motion method, due to Alben et al. 关Phys. Rev. B 12, 4090
共1975兲兴, to compute the eigenvalue distribution 共density of states兲 of very large matrices. The salient feature of
this method is that for matrices of the kind encountered in quantum physics the memory and CPU requirements
of this method scale linearly with the dimension of the matrix. We derive a rigorous estimate of the statistical
error, supporting earlier observations that the computational efficiency of this approach increases with the
matrix size. We use this method and an imaginary-time version of it to compute the energy and specific heat
of three different, exactly solvable, spin-1/2 models, and compare with the exact results to study the dependence of the statistical errors on sample and matrix size.
PACS number共s兲: 05.10.⫺a, 05.30.⫺d, 03.67.Lx
I. INTRODUCTION
The calculation of the distribution of eigenvalues of very
large matrices is a central problem in quantum physics. This
distribution determines the thermodynamic properties of the
system 共see below兲. It is directly related to the single-particle
density of states 共DOS兲 or Green’s function. In a one-particle
共e.g., one-electron兲 description, knowledge of the DOS suffices to compute the transport properties 关1兴.
The most direct method to compute the DOS, i.e., all the
eigenvalues, is to diagonalize the matrix H representing the
Hamiltonian of the system. This approach has two obvious
limitations: The number of operations increases as the third
power of the dimension D of H, and, perhaps most importantly, the amount of memory required by state-of-the-art
algorithms grows as D 2 关2,3兴. This scaling behavior limits
the application of this approach to matrices of dimension D
⫽O(10000), which is too small for many problems of interest. What is needed are methods that scale linearly with D.
There has been considerable interest in developing ‘‘fast’’
关i.e., O(D)] algorithms to compute the DOS and other similar quantities. One such algorithm and an application of it to
electron motion in disordered alloy models was given by
Alben et al. 关4兴. In this approach the DOS is obtained by
solving the time-dependent Schrödinger equation 共TDSE兲 of
a particle moving on a lattice, followed by a Fourier transform of the retarded Green’s function 关4兴. Using the unconditionally stable split-step fast Fourier transform method to
solve the TDSE, it was shown that the eigenvalue spectrum
of a particle moving in continuum space can be computed in
the same manner 关5兴. Fast algorithms of this kind proved
useful to study various aspects of localization of waves 关6–8兴
and other one-particle problems 关9–14兴.
Application of these ideas to quantum many-body systems triggered further development of flexible and efficient
methods to solve the TDSE. Based on Suzuki’s product formula approach, an unconditionally stable algorithm was developed and used to compute the time-evolution of twodimensional S⫽1/2 Heisenberg-like models 关15兴. Results for
the DOS of matrices of dimension D⬇1000000 were reported 关15兴. A potentially interesting feature of these fast
1063-651X/2000/62共3兲/4365共13兲/$15.00
PRE 62
algorithms is that they may run very efficiently on a quantum
computer 关16,17兴.
A common feature of these fast algorithms is that they
solve the TDSE for a sample of randomly chosen initial
states. The efficiency of this approach as a whole relies on
the hypothesis 共suggested by the central limit theorem兲 that
satisfactory accuracy can be achieved by using a small
sample of initial states. Experience not only shows that this
hypothesis is correct; it strongly suggests that for a fixed
sample size the statistical error on physical quantities such as
the energy and specific heat decreases with the dimension D
of the Hilbert space 关16兴.
In view of the general applicability of these fast algorithms to a wide variety of quantum problems, it seems warranted to analyze in detail their properties and the peculiar D
dependence in particular. In Secs. II and III we recapitulate
the essence of the approach. We present a rigorous estimate
for the mean square error 共variance兲 on the trace of a matrix.
In Sec. IV we describe the imaginary-time version of the
method. The statistical analysis of the numerical data is discussed in Sec. V. Section VI describes the model systems
that are used in our numerical experiments. The algorithm
used to solve the TDSE is reviewed in Sec. VII. In Sec. VIII
we derive rigorous bounds on the accuracy with which all
eigenvalues can be determined, and demonstrate that this accuracy decreases linearly with the time over which the TDSE
is solved. The results of our numerical calculations are presented in Sec. IX, and our conclusions are given in Sec. X.
II. THEORY
The trace of a matrix A acting on a D-dimensional Hilbert
space spanned by an orthonormal set of states 兵 兩 ␾ n 典 其 is
given by
D
Tr A⫽
兺
n⫽1
具 ␾ n兩 A ␾ n典 .
共1兲
Note that according to Eq. 共1兲 we have Tr 1⫽D. If D is very
large, one might think of approximating Eq. 共1兲 by sampling
over a subset of K (KⰆD)‘‘important’’ basis vectors. The
problem with this approach is that the notion ‘‘important’’
4365
©2000 The American Physical Society
4366
ANTHONY HAMS AND HANS DE RAEDT
may be very model dependent. Therefore, it is better to
sample in a different manner. We construct a random vector
兩 ␺ 典 by choosing D complex random numbers, c n ⬅ f n
⫹ig n , with mean 0, for n⫽1 . . . D, so
p, the c n,p are now correlated, but that does not cause problems 共see Appendix A兲. Second it follows that E( 兩 c 兩 2 )
⫽1/D.
Obviously the error can be written as
D
兩␺典⫽
兺
n⫽1
c n兩 ␾ n典 ,
共2兲
and we calculate
PRE 62
Tr A⫺
D
S
S
兺
p⫽1
具 ␺ p 兩 A ␺ p 典 ⫽Tr RA,
where
D
具␺兩A␺典⫽
兺
n,m⫽1
*c n具 ␾ m兩 A ␾ n典 .
cm
If we now sample over S realizations of the random vectors
S
兺
p⫽1
具 ␺ p兩 A ␺ p典 ⫽
1
S
S
D
兺 兺
p⫽1 n,m⫽1
* c n,p 具 ␾ m 兩 A ␾ n 典 . 共4兲
c m,p
Assuming that there is no correlation between the random
numbers in different realizations, and that the random numbers f n,p and g n,p are drawn from an even and symmetric
共both with respect to each variable兲 probability distribution
共see Appendix A for more details兲, we have
1
lim
S
S→⬁
S
兺
p⫽1
* c n,p ⫽E 共 兩 c 兩 2 兲 ␦ m,n ,
c m,p
共5兲
where E(•) denotes the expectation value with respect to the
probability distribution used to generate the c n,p ’s. On the
right hand side of Eq. 共5兲 the subscripts of c n,p have been
dropped to indicate that the expectation value does not depend on n or p. It follows immediately that
1
S→⬁ S
lim
S
兺
R m,n ⬅ ␦ m,n ⫺
共3兲
兵 ␺ 其 and calculate the average, we obtain
1
S
具 ␺ p 兩 A ␺ p 典 ⫽E 共 兩 c 兩 2 兲 Tr A⫽E 共 兩 c 兩 2 兲 兺 具 ␾ n 兩 A ␾ n 典 ,
n⫽1
共6兲
showing that we can compute the trace of A by sampling
over random states 兵 ␺ p 其 , provided there is an efficient algorithm to calculate 具 ␺ p 兩 A ␺ p 典 共see Sec. VII兲.
According to the central limit theorem, for a large but
finite S, we have
1
S
S
兺
p⫽1
* c n,p ⫽E 共 兩 c 兩 兲 ␦ m,n ⫹O
c m,p
2
冉 冊
1
冑S
E 共 兩 Tr RA 兩 2 兲 ⫽
共7兲
meaning that the statistical error on the trace vanishes like
1/冑S, which is not surprising. What is surprising is that one
can prove a much stronger result as follows. Let us first
normalize the c n,p ’s so that, for all p,
D
兺
n⫽1
兩 c n,p 兩 2 ⫽1.
共8兲
This innocent looking step has far reaching consequences.
First we note that the normalization renders the method exact
in the 共rather trivial兲 case when the matrix A is proportional
to the unit matrix. The price we pay for this is that for fixed
S
兺
p⫽1
* c n,p
c m,p
共10兲
D Tr A † A⫺ 兩 Tr A 兩 2
,
S 共 D⫹1 兲
共11兲
an exact expression for the variance in terms of the sample
size S, the dimension D of the matrix A, and the 共unknown兲
constants Tr A † A, and 兩 Tr A 兩 .
Invoking a generalization of Markov’s inequality 关18兴
P共 兩 X 兩 2 ⭓a 兲 ⭐
E共兩X兩2兲
, ᭙ a⬎0,
a
共12兲
where P(Q) denotes the probability for the statement Q to be
true. We find that the probability that 兩 Tr RA 兩 2 exceeds a
fraction a of 兩 Tr A 兩 2 is bounded by
P
冉
兩 Tr RA 兩 2
兩 Tr A 兩 2
冊
⭓a ⭐
D Tr A † A⫺ 兩 Tr A 兩 2
1
;
a S 共 D⫹1 兲
兩 Tr A 兩 2
᭙ a⬎0,
共13兲
or, in other words, the relative statistical error e A on the
estimator of the trace of A is given by
e A⬅
,
D
S
is a traceless 关due to Eq. 共8兲兴 Hermitian matrix of random
numbers. We put X⫽Tr RA, and compute E( 兩 X 兩 2 ). The result for the general case can be found in Appendix A. For a
uniform distribution of the c n,p ’s on the hypersphere defined
D
兩 c n,p 兩 2 ⫽1, the expression simplifies considerably,
by 兺 n⫽1
and we find
D
p⫽1
共9兲
冑
D Tr A † A⫺ 兩 Tr A 兩 2
S 共 D⫹1 兲 兩 TrA 兩 2
共14兲
if 兩 Tr A 兩 ⬎0. We see that e A ⫽0 if A is proportional to a unit
matrix. From Eq. 共14兲 it follows that, in general, we may
expect e A to vanish with the square root of SD. The prefactor is a measure for the relative spread of the eigenvalues of
A, and is obviously model dependent. The dependence of e A
on S, D, and the spectrum of A is corroborated by the numerical results presented below.
It is also of interest to examine the effect of not normalizing the c n,p ’s. A calculation similar to the one that led to
the above results yields
e A⫽
冑
Tr A † A
S 兩 Tr A 兩 2
.
共15兲
PRE 62
4367
FAST ALGORITHM FOR FINDING THE EIGENVALUE . . .
Clearly, this bound is less sharp and does not vanish if A is
proportional to a unit matrix.
mined by examination of the bottom of spectrum. To compute Z, E, or C, we simply replace the interval 关 ⫺⬁,⫹⬁ 兴 by
关 ⑀ 0 ,⫹⬁ 兴 .
III. REAL-TIME METHOD
The distribution of eigenvalues or DOS of a quantum system is defined as
兺 ␦ 共 ⑀ ⫺E n 兲 ⫽ 2 ␲ 冕⫺⬁ e it ⑀ Tr e ⫺itH dt,
n⫽1
D
D共 ⑀ 兲 ⫽
⬁
1
共16兲
where H is the Hamiltonian of the system and n runs over all
the eigenvalues of H. The DOS contains all the physical
information about the equilibrium properties of the system.
For instance the partition function, the energy, and the heat
capacity are given by
Z⫽
E⫽
C⫽ ␤ 2
1
Z
冕
⫺⬁
冕
d ⑀ D共 ⑀ 兲 e ⫺ ␤ ⑀ ,
共17兲
⬁
⬁
⫺⬁
d ⑀ ⑀ D共 ⑀ 兲 e ⫺ ␤ ⑀ ,
共18兲
冊
d ⑀ ⑀ 2 D共 ⑀ 兲 e ⫺ ␤ ⑀ ⫺E 2 ,
共19兲
respectively. Here ␤ ⫽1/k B T and k B is Boltzmann’s constant
共we put k B ⫽1 and ប⫽1 from now on兲.
As explained above, the trace in integral 共16兲 can be estimated by sampling over random vectors. For the statistical
error analysis discussed below it is convenient to define a
DOS per sample by
d p共 ⑀ 兲 ⬅
1
2␲
The real-time approach has the advantage that it yields
information on all eigenvalues and can be used to compute
both dynamic and static properties without suffering from
numerical instabilities. However for the computation of the
thermodynamic properties, the imaginary-time version is
more efficient. We will use the imaginary-time method as an
independent check on the results obtained by the real-time
algorithm.
Repeating the steps that lead to Eq. 共17兲, we find
Z⫽Tr exp共 ⫺ ␤ H 兲
⫽ lim
S→⬁
⫺⬁
冉冕
1
Z
⬁
IV. IMAGINARY-TIME METHOD
冕
⬁
⫺⬁
e it ⑀ 具 ␺ p 兩 e ⫺itH ␺ p 典 dt,
共20兲
where the subscript p labels the particular realization of the
random state 兩 ␺ p 典 . The DOS is then given by
D共 ⑀ 兲 ⫽ lim
S→⬁
1
S
S
1
S
兺
p⫽1
兺
d p共 ⑀ 兲 .
共21兲
Schematically the algorithm to compute d p ( ⑀ ) consists of
the following steps: 共1兲 Generate a random state 兩 ␺ p (0) 典 ,
and set t⫽0. 共2兲 Copy this state to 兩 ␺ p (t) 典 . 共3兲 Calculate
具 ␺ p (0) 兩 ␺ p (t) 典 and store the result. 共4兲 Solve the TDSE for a
small time step ␶ , replacing 兩 ␺ p (t) 典 by 兩 ␺ p (t⫹ ␶ ) 典 共see Sec.
VII for model specific details兲. 共5兲 Repeat N times from step
共3兲. 共6兲 Perform a Fourier transform on the tabulated result,
and store d p ( ⑀ ).
In practice the Fourier transform in Eq. 共16兲 is performed
by the fast Fourier transform 共FFT兲. We use a Gaussian window to account for the finite time ␶ N used in the numerical
time integration of the TDSE. The number of time step N
determines the accuracy with which the eigenvalues can be
computed. In Sec. VIII we prove that this systematic error in
the eigenvalues vanishes as 1/␶ N.
Since for any reasonable physical system 共or finite matrix兲
the smallest eigenvalue E 0 is finite, for all practical purposes
d p ( ⑀ )⫽0 for ⑀ ⬍ ⑀ 0 ⬍E 0 . The value of ⑀ 0 is easily deter-
共22兲
with similar expressions for E and C.
Furthermore we have
具 ␺ p 兩 H n e ⫺ ␤ H ␺ p 典 ⫽ 具 e ⫺ ␤ H/2␺ p 兩 H n e ⫺ ␤ H/2␺ p 典 ,
共23兲
assuming H is Hermitian. Therefore we only need to propagate the random state for an imaginary time ␤ /2 instead of ␤ .
Furthermore we do not need to perform a FFT. Disregarding
these minor differences, the algorithm is the same as in the
real-time case with ␶ replaced by ⫺i ␶ .
V. ERROR ANALYSIS
Estimating the statistical error on the partition function Z
is easy because it depends linearly on the trace of the 共imaginary兲 time evolution operator. However, the error on E and C
depends on this trace in a more complicated manner, and this
fact has to be taken into account.
First we define
S
p⫽1
具 ␺ p 兩 exp共 ⫺ ␤ H 兲 ␺ p 典 ,
z p⬅
冕
⬁
d ⑀ d p 共 ⑀ 兲 e ⫺ ␤⑀ ,
共24兲
d ⑀ d p 共 ⑀ 兲 ⑀ e ⫺ ␤⑀ ,
共25兲
d ⑀ d p 共 ⑀ 兲 ⑀ 2 e ⫺ ␤⑀
共26兲
⑀0
h p⬅
冕
w p⬅
冕
⬁
⑀0
⬁
⑀0
for the real-time method, and
z p⬅ 具 ␺ p兩 e ⫺␤H␺ p典 ,
共27兲
h p ⬅ 具 ␺ p 兩 He ⫺ ␤ H ␺ p 典 ,
共28兲
w p⬅ 具 ␺ p兩 H 2e ⫺␤H␺ p典
共29兲
for the imaginary-time method. For each value of ␤ we generate the data 兵 z p 其 , 兵 h p 其 , and 兵 w p 其 , for p⫽1, . . . ,S. For
both cases we have
4368
ANTHONY HAMS AND HANS DE RAEDT
PRE 62
A. Spin chains
共30兲
Z⫽ lim z̄,
S→⬁
Open spin chains of L sites described by the Hamiltonian
L⫺1
h̄
E⫽ lim
S→⬁
C⫽ lim ␤ 2
S→⬁
z̄
共31兲
,
冉 冊
w̄
z̄
⫺
h̄ 2
z̄ 2
共32兲
,
i⫽1
␦ z⫽
冑
␦ h⫽
冑
var共 h 兲
,
S⫺1
␦ w⫽
冑
var共 w 兲
,
S⫺1
共33兲
L
H⫽
兺
i, j⫽1
冋
␦ Z 2⫽
␦ E 2⫽
1
S⫺1
1
␦C ⫽
S⫺1
2
1
␦ z 2,
S⫺1
3
共36兲
册
dĒ dĒ
k
A i, j ⫽⫺J 共 1⫹⌬ 兲共 ␦ i, j⫺1 ⫹ ␦ i⫺1,j 兲 ⫺2h ␦ i, j ,
共41兲
B i, j ⫽⫺J 共 1⫺⌬ 兲共 ␦ i, j⫺1 ⫺ ␦ i⫺1,j 兲
共42兲
are L⫻L matrices. By further canonical transformation, this
Hamiltonian can be written as
L
兺 ⌳k
k⫽1
冉 冊
n k⫺
1
1
⫹ Tr A⫹hL,
2
2
共43兲
where n k is the number operator of state k, and the ⌳ k ’s are
given by the solution of the eigenvalue equation
共 A⫺B 兲共 A⫹B 兲 ␾ k ⫽⌳ 2k ␾ k .
共44兲
In the general case this eigenvalue problem of the L⫻L Hermitian matrix (A⫺B)(A⫹B) is most easily solved numerically. In the present paper we confine ourselves to two limiting cases: the XY model (⌬⫽1), and the Ising model in a
transverse field (⌬⫽0).
The Hamiltonian of the mean-field model reads
共37兲
l
dC̄ dC̄
兺 M k,l d¯x d ¯x ,
k,l⫽1
k
H⫽⫺
共38兲
2
J
L
L
L
兺
i⬎ j⫽1
␴ជ i • ␴ជ j ⫺h
兺 ␴ zi ,
i⫽1
共45兲
and can be rewritten as
l
3
J
H⫽⫺2 Sជ •Sជ ⫺2hS z ⫹ J,
L
2
where Ē⫽h̄/z̄ and and C̄⫽ ␤ (w̄/z̄⫺h̄ /z̄ ).
2
共40兲
B. Mean-field model
兺 M k,l d¯x d ¯x ,
k,l⫽1
3
共39兲
1
c †i A i, j c j ⫹ 共 c †i B i, j c †j ⫹c j B *j,i c i 兲 ⫹hL,
2
H⫽
where var(x)⬅x̄ 2 ⫺x̄ 2 denotes the variance on the data
兵 x p 其 . However, the sets of data 兵 z p 其 , 兵 h p 其 , and 兵 w p 其 are
correlated, since they are calculated from the same set
兵 兩 ␺ p 典 其 . These correlations in the data are accounted for by
calculating the covariance matrix M k,l (k,l⫽1, . . . ,3), the
¯k¯
x l , where 兵 x 1 其 ,
elements of which are given by ¯
x k x l ⫺x
兵 x 2 其 , and 兵 x 3 其 are a shorthand for 兵 z p 其 , 兵 h p 其 , and 兵 w p 其 ,
respectively. The estimates for the errors in Z, E, and C are
given by
兺 ␴ zi
i⫽1
where c †i and c i are spinless fermion operators and
共34兲
共35兲
L
x
y
⫹⌬ ␴ iy ␴ i⫹1
兲 ⫺h
共 ␴ xi ␴ i⫹1
—where ␴ xi , ␴ iy , and ␴ zi denote the Pauli matrices, and J, ⌬,
and h are model parameters—can be solved exactly. They
can be reduced to diagonal form by means of the JordanWigner transformation 关19兴. We have
where x̄⬅S ⫺1 兺 Sp⫽1 x p . The standard deviations on z̄, h̄, and
w̄ are given by
var共 z 兲
,
S⫺1
兺
H⫽⫺J
2
VI. EXACTLY SOLVABLE SPIN 1Õ2 MODELS
The most direct way to assess the validity of the approach
described above is to carry out numerical experiments on
exactly solvable models. In this paper we consider three different exactly solvable models, two spin-1/2 chains and a
mean-field spin-1/2 model. The former have a complicated
spectrum, the latter has a highly degenerate eigenvalue distribution. These spin models differ from those studied elsewhere 关15,16兴, in that they belong to the class of integrable
systems.
共46兲
with
Sជ ⫽
1
2
L
兺 ␴ជ i .
i⫽1
共47兲
The single spin-L/2 Hamiltonian has eigenvalues
3
E l,m ⫽⫺2Jl 共 l⫹1 兲 /L⫺2hm⫹ J,
2
with degeneracy
共48兲
PRE 62
FAST ALGORITHM FOR FINDING THE EIGENVALUE . . .
n l,m ⫽
冉 冊
L
2l⫹1
.
L/2⫹l⫹1 L/2⫺l
共49兲
This rather trivial model serves as a test for the case of
highly degenerate eigenvalues.
VII. TIME EVOLUTION
For the approach outlined in Secs. III and IV to be of
practical use, it is necessary that the matrix elements of the
exponential of H can be calculated efficiently. The purpose
of this section is to describe how this can be done.
The general form of the Hamiltonians of the models we
study is
L
H⫽⫺
兺 兺
i, j⫽1 ␣ ⫽x,y,z
L
J ␣i, j ␴ ␣i ␴ ␣j ⫺
兺 兺
i⫽1 ␣ ⫽x,y,z
h ␣i ␴ ␣i , 共50兲
where the first sum runs over all pairs P of spins, ␴ ␣i ( ␣
⫽x,y,z) denotes the ␣ th component of the spin-1/2 operator
representing the ith spin. For both methods, we have to
calculate the evolution of a random state, i.e., U( ␶ ) 兩 ␺ 典
⬅exp(⫺i␶H)兩␺典 or U( ␶ ) 兩 ␺ 典 ⬅exp(⫺␶H)兩␺典 for the real and
imaginary time methods, respectively. We will discuss the
real-time case only; the imaginary-time problem can be
solved in the same manner.
Using the semigroup property U(t 1 )U(t 2 )⫽U(t 1 ⫹t 2 ),
we can write U(t)⫽U( ␶ ) m where t⫽m ␶ . Then the main
step is to replace U( ␶ ) by a symmetrized product-formula
approximation 关20兴. For the case at hand it is expedient to
take
U 共 ␶ 兲 ⬇Ũ 共 ␶ 兲 ⬅e ⫺i ␶ H z /2e ⫺i ␶ H y /2e ⫺i ␶ H x e ⫺i ␶ H y /2e ⫺i ␶ H z /2,
共51兲
where
L
H ␣ ⫽⫺
兺
i, j⫽1
L
J ␣i, j ␴ ␣i ␴ ␣j ⫺
兺 h ␣i ␴ ␣i ,
i⫽1
␣ ⫽x,y,z.
共52兲
Other decompositions 关15,21兴 work equally well, but are
somewhat less efficient for the cases at hand. In the real-time
approach Ũ( ␶ ) is unitary, and hence the method is unconditionally stable 关20兴 共also the imaginary-time method can be
made unconditionally stable兲. It can be shown that
円兩 U( ␶ )⫺Ũ( ␶ ) 兩 円⭐s ␶ 3 (s⬎0 a constant兲 关22兴, implying that
the algorithm is correct to second order in the time step ␶
关20兴. Usually it is not difficult to choose ␶ so small that for
all practical purposes the results obtained can be considered
as being ‘‘exact.’’ Moreover, if necessary, Ũ( ␶ ) can be used
as a building block to construct higher-order algorithms 关23–
26兴. In Appendix B we will derive bounds on the error in the
eigenvalues when they are calculated using a symmetric
product formula.
As basis states 兵 兩 ␾ n 典 其 we take the direct product of the
eigenvectors of the S zi 共i.e., spin-up 兩 ↑ i 典 and spin-down 兩 ↓ i 典 ).
In this basis, e ⫺i ␶ H z /2 changes the input state by altering the
phase of each of the basis vectors. As H z is a sum of pair
interactions, it is trivial to rewrite this operation as a direct
product of 4⫻4 diagonal matrices 共containing the
4369
interaction-controlled phase shifts兲 and 4⫻4 unit matrices.
Still working in the same representation, the action of
e ⫺i ␶ H y /2 can be written in a similar manner, but the matrices
that contain the interaction-controlled phase-shift have to be
replaced by nondiagonal matrices. Although this does not
present a real problem, it is more efficient and systematic to
proceed as follows. Let us denote by X(Y ) the rotation by
␲ /2 of each spin about the x(y) axis. As
e ⫺i ␶ H y /2⫽XX † e ⫺i ␶ H y /2XX † ⫽Xe ⫺i ␶ H z⬘ /2X † ,
共53兲
it is clear that the action of e ⫺i ␶ H y /2 can be computed by
applying to each spin, the inverse of X followed by an
interaction-controlled phase-shift and X itself. The prime in
Eq. 共53兲 indicates that J zi, j and h zi in H z have to be replaced
by J i,y j and h iy respectively. A similar procedure is used to
compute the action of e ⫺i ␶ H x . We only have to replace X by
Y.
VIII. ACCURACY OF THE COMPUTED EIGENVALUES
First we consider the problem of how to choose the number of time steps N to obtain the DOS with acceptable accuracy. According to the Nyquist sampling theorem employing
a sampling interval ⌬t⫽ ␲ /maxi 兩Ei兩 is sufficient to cover the
full range of eigenvalues. On the other hand, the time step
also determines the accuracy of the approximation Ũ( ␶ ). Let
us call the maximum value of ␶ which gives satisfactory
accuracy ␶ 0 共for the imaginary-time method, this is the only
parameter兲. For the examples treated here ␶ 0 ⬍⌬t), implying
that we have to use more steps to solve the TDSE than we
actually use to compute the FFT. Eigenvalues that differ less
than ⌬ ⑀ ⫽ ␲ /N⌬t cannot be identified properly. However,
since ⌬ ⑀ ⬀N ⫺1 we only have to extend the length of the
calculation by a factor of 2 to increase the resolution by the
same factor.
At first glance the above reasoning may seem to be a little
optimistic. It apparently overlooks the fact that if we integrate the TDSE over longer and longer times the error on the
wave function also increases 共although it remains bounded
because of the unconditional stability of the product formula
algorithm兲. In fact it has been shown that, in general 关20兴,
円兩 e ⫺itH 兩 ␺ 共 0 兲 典 ⫺Ũ m 共 ␶ 兲 兩 ␺ 共 0 兲 典 兩 円⭐c ␶ 2 t,
共54兲
where t⫽m ␶ , suggesting that the loss in accuracy on the
wave function may well compensate for the gain in resolution that we obtain by using more data in the Fourier transform. Fortunately this argument does not apply when we
want to determine the eigenvalues as we now show. As before, we will discuss the real-time algorithm only because
the same reasoning 共but different mathematical proofs兲 holds
for the imaginary-time case.
Consider the time-step operator 共52兲. Using the fact that
any unitary matrix can be written as the matrix exponential
of a Hermitian matrix, we can write
Ũ 共 ␶ 兲 ⫽e ⫺i ␶ H z /2e ⫺i ␶ H y /2e ⫺i ␶ H x e ⫺i ␶ H y /2e ⫺i ␶ H z /2⬅e ⫺i ␶ H̃( ␶ ) .
共55兲
ANTHONY HAMS AND HANS DE RAEDT
4370
PRE 62
FIG. 1. The density of states 共DOS兲 as obtained from the real-time algorithm for spin chains of length L⫽15 and for S⫽20 random initial
states. Left: XY model; middle: Ising model in a transverse field; right: mean-field model. For the mean-field model a logarithmic scale was
used to show the highly degenerate spectrum more clearly.
It is clear that in practice the real-time method yields the
spectrum of H̃( ␶ ), not the one of H. Therefore the relevant
question is: How much do the spectra of H̃( ␶ ) and H differ?
In Appendix B we give a rigorous proof that the difference
between the eigenvalues of H̃( ␶ ) and H vanishes as ␶ 2 . In
other words the value of m 共or t⫽m ␶ ) has no effect whatsoever on the accuracy with which the spectrum can be determined. Therefore, the final conclusion is that the error in
the eigenvalues vanishes as ␶ 2 /N where N is the number of
data points used in the Fourier transform of Tr e ⫺itH̃( ␶ ) .
IX. RESULTS
We write our results in units of J and take h⫽0, except
for the Ising model in a transverse field, where we take h
⫽0.75J. The random numbers c n,p are generated such that
the Eqs. 共A3兲 and 共A4兲 are satisfied. We use two different
techniques to generate these random numbers.
共1兲 A uniform random number generator produces 兵 f n,p 其
and 兵 g n,p 其 with ⫺1⭐ f n,p ,g n,p ⭐1. We then normalize the
vector 关see Eq. 共8兲兴.
共2兲 c n,p ’s are obtained from a two-variable 共real and
imaginary part兲 Gaussian random number generator and the
resulting vector is normalized.
Both methods satisfy the basic requirements Eqs. 共A3兲
and 共A4兲 but because the first samples points out of a
2D-dimensional hypercube and subsequently projects the
vector onto a sphere, the points are not distributed uniformly
over the surface of the unit hypersphere. The second method
is known to generate numbers which are distributed uni-
formly over the hypersurface. Although the first method does
not satisfy all the mathematical conditions that lead to error
共14兲, our numerical experiments with both generators give
identical results, within statistical errors of course. Also,
within the statistical errors, the results from the imaginary
and real-time algorithm are the same. Therefore, we only
show some representative results as obtained from the realtime algorithm.
In Fig. 1 we show a typical result for the DOS D( ⑀ ) of
the XY model, the Ising model in a transverse field, and the
mean-field model, all with L⫽15 spins and using S⫽20
samples. Because of the very high degeneracy we plotted the
DOS for the mean-field model on a logarithmic scale.
In Fig. 2 we show the relative error ␦ Z/Z based on Eq.
共36兲 for the three models of various size, as obtained from
the simulation 共symbols兲. For these figures we used the
imaginary-time algorithm, because then the statistical error
can be related to e A directly 关see Eq. 共14兲, with A
⫽exp(⫺␤H)]. The theoretical results 共lines兲 for the error estimate, obtained by a direct exact numerical evaluation of
Eq. 共14兲 are shown as well. We conclude that for all systems,
lattice sizes, and temperatures there is very good agreement
between numerical experiment and theory.
Results for the energy E and specific heat C are presented
in Fig. 3 the (XY model兲, 4 共the Ising model in a transverse
field兲, and 5 共the mean-field model兲. The solid lines represent
the exact result for the case shown. Simulation data as obtained from S⫽5 and 20 samples are represented by symbols, and the estimates of the statistical error by error bars.
We see that the data are in excellent agreement with the
FIG. 2. The relative error ␦ Z/Z 关see Eq. 共36兲兴 on a logarithmic scale as a function of temperature T⬅1/␤ and for various system sizes.
Left: XY model; middle: Ising model in a transverse field; right: mean-field model. Solid lines: e A 关with A⫽e ⫺ ␤ H ; see Eq. 共14兲兴 for L
⫽6; dashed lines: e A for L⫽10; dash-dotted line: e A for L⫽15. Crosses: simulation data for S⫽20 and L⫽6; squares: simulation data for
S⫽20 and L⫽10; circles: simulation data for S⫽20 and L⫽15.
PRE 62
FAST ALGORITHM FOR FINDING THE EIGENVALUE . . .
4371
FIG. 3. Energy 共top兲 and specific heat 共bottom兲 of the XY model 关see Eq. 共39兲兴, with ⌬⫽1 and h⫽0. Left: L⫽6; middle: L⫽10; right:
L⫽15. Solid lines: exact result; crosses: simulation data using S⫽5 samples; squares: simulation data using S⫽20 samples. Error bars: One
standard deviation.
exact results and, equally important, the estimate for the error captures the deviation from the exact result very well. We
also see that in general the error decreases with the system
size. Both the imaginary- and real-time methods seem to
work very well, yielding accurate results for the energy and
specific heat of quantum spin systems with modest amounts
of computational effort.
X. CONCLUSIONS
The theoretical analysis presented in this paper gives a
solid justification of the remarkable efficiency of the real-
time equation-of-motion method for computing the distribution of all eigenvalues of very large matrices. The real-time
method can be used whenever the more conventional,
Lanczos-like, sparse-matrix techniques can be applied:
Memory and CPU requirements for each iteration 共time-step兲
are roughly the same 共depending on the actual implementation兲 for both approaches.
We do not recommend using the real-time method if one
is interested in the smallest 共or largest兲 eigenvalue only.
Then the Lanczos method is computationally more efficient
because it needs less iterations 共time steps兲 than the real-time
approach. However, if one needs information about all ei-
FIG. 4. Energy 共top兲 and specific heat 共bottom兲 of the Ising model in a transverse field 关see Eq. 共39兲兴 with ⌬⫽0 and h⫽0.75J. Left:
L⫽6; middle: L⫽10; right: L⫽15. Solid lines; exact result; crosses: simulation data using S⫽5 samples; squares: simulation data using
S⫽20 samples. Error bars: one standard deviation.
ANTHONY HAMS AND HANS DE RAEDT
4372
PRE 62
FIG. 5. Energy 共top兲 and specific heat 共bottom兲 of the mean-field model 关see Eq. 共45兲兴 with h⫽0. Left: L⫽6; middle: L⫽10; right:
L⫽15. Solid lines: exact result; crosses: simulation data using S⫽5 samples; squares: simulation data using S⫽20 samples. Error bars: one
standard deviation.
genvalues and direct diagonalization is not possible 共because
of memory/CPU time兲 there is as yet no alternative to the
real-time method. The matrices used in this example 共up to
32768⫻32768) are not representative in this respect: The
real-time method has been used to compute the distribution
of eigenvalues for matrices of dimension 16777216
⫻16777216 关15兴.
Once the eigenvalue distribution is known the thermodynamic quantities directly follow. However, if one is interested in the accurate determination of the temperature dependence of thermodynamic 共and static correlation functions兲
properties but not in the eigenvalue distribution itself, the
imaginary-time method is by far the most efficient method to
compute these quantities. For instance the calculation of the
thermodynamic properties for ␤ J⫽0, . . . ,10 of a 15-site
spin-1/2 system 共i.e. implicitly solving the full 32768
⫻32768 eigenvalue problem兲 takes 1410 sec per sample on a
Mobile Pentium III 500 MHz system.
E 共 兩 Tr RA 兩 2 兲 ⫽E
⫽
冉冏
1
S2
1
S
S
兺 兺
兺
ACKNOWLEDGMENTS
Support from the Dutch ‘‘Stichting Nationale Computer
Faciliteiten 共NCF兲’’ and the Dutch ‘‘Stichting voor Fundamenteel Onderzoek der Materie 共FOM兲’’ is gratefully acknowledged.
APPENDIX A: EXPECTATION VALUE CALCULATION
In this Appendix we calculate the expectation value of the
error squared, as defined in Sec. II. By definition we have
冏冊
2
D
p⫽1 m,n⫽1
S
Finally we remark that although we used quantum-spin
models to illustrate various aspects, there is nothing in the
real or imaginary-time method that is specific to the models
used. The only requirement for these methods to be useful in
practice is that the matrix is sparse and 共very兲 large.
* c n,p 兲 A m,n
共 ␦ m,n ⫺D c m,p
D
兺
p,p ⬘ ⫽1 k,l,m,n⫽1
* c n,p 兲 ⫺D ␦ m,n E 共 c k,p ⬘ c *l,p 兲
„␦ k,l ␦ m,n ⫺D ␦ k,l E 共 c m,p
⬘
* 兲 … A k,l
* c n,p c k,p ⬘ c l,p
* A m,n ,
⫹D 2 E 共 c m,p
⬘
共A1兲
where p and p ⬘ label the realization of the random numbers c n,p ⬅ f n,p ⫹ig n,p .
First we assume that different realizations p⫽p ⬘ are independent implying that
* 兲 p⫽p ⬘ ⫽E 共 c m,p
* c n,p c k,p ⬘ c l,p
* c n,p 兲 E 共 c k,p ⬘ c *l,p 兲 .
E 共 c m,p
⬘
⬘
共A2兲
Second we assume that the random numbers are drawn from a probability distribution that is an even function of each variable,
PRE 62
4373
FAST ALGORITHM FOR FINDING THE EIGENVALUE . . .
P 共 f 1,p ,g 1,p , f 2,p ,g 2,p , . . . , f k,p ,g k,p , . . . , f D,p ,g D,p 兲 ⫽ P 共 f 1,p ,g 1,p , f 2,p ,g 2,p , . . . ,⫺ f k,p ,g k,p , . . . , f D,p ,g D,p 兲
⫽ P 共 f 1,p ,g 1,p , f 2,p ,g 2,p , . . . , f k,p ,⫺g k,p , . . . , f D,p ,g D,p 兲 ,
共A3兲
and symmetric under interchange of any two variables,
P 共 f 1,p ,g 1,p , . . . , f i,p ,g i,p , . . . , f j,p ,g j,p , . . . , f D,p ,g D,p 兲 ⫽ P 共 f 1,p ,g 1,p , . . . , f j,p ,g i,p , . . . , f i,p ,g j,p , . . . , f D,p ,g D,p 兲
⫽ P 共 f 1,p ,g 1,p , . . . ,g i,p , f i,p , . . . , f j,p ,g j,p , . . . , f D,p ,g D,p 兲 ,
共A4兲
for all i, j,k⫽1, . . . ,D. This is most easily done by drawing individual numbers from the same even probability distribution,
i.e.,
D
P 共 f 1,p ,g 1,p , . . . , f j,p ,g i,p , . . . , f i,p ,g j,p , . . . , f D,p ,g D,p 兲 ⫽
兿
n,m⫽1
共A5兲
P 共 f n,p 兲 P 共 g n,p 兲 ,
D
where P(x)⫽ P(⫺x). Normalizing the vector ( f 1,p ,g 1,p , . . . , f D,p ,g D,p ) such that 兺 i⫽1
兩 c n,p 兩 2 ⫽1 共for p⫽1, . . . ,S), does not
affect the basic requirements 共A3兲 and 共A4兲.
Making use of the above properties of P( f 1 ,g 1 , . . . , f D ,g D ), we find that
* c n,p 兲 ⫽ ␦ m,n E 共 兩 c m,p 兩 2 兲 ⫽ ␦ m,n E 共 兩 c 兩 2 兲 ,
E 共 c m,p
共A6兲
where in the last equality we omitted the subscripts of c m,p to indicate that the expectation value does not depend on m or p.
An expectation value of a product of two c * ’s and two c’s can be written as
* 兲 ⫽ 共 1⫺ ␦ p,p ⬘ 兲 ␦ m,n ␦ k,l E 共 兩 c m,p 兩 2 兲 E 共 兩 c m,p ⬘ 兩 2 兲 ⫹ ␦ p,p ⬘ ␦ m,n ␦ k,l 共 1⫺ ␦ mk 兲 E 共 c m* c m c k c k* 兲
* c n,p c k,p ⬘ c l,p
E 共 c m,p
⬘
* c n c m c *n 兲 ⫹ ␦ p,p ⬘ ␦ m,l ␦ n,k 共 1⫺ ␦ m,n 兲 E 共 c m* c n c n c m* 兲
⫹ ␦ p,p ⬘ ␦ m,k ␦ n,l 共 1⫺ ␦ m,n 兲 E 共 c m
* c m c m c m* 兲
⫹ ␦ p,p ⬘ ␦ m,l ␦ n,k ␦ m,n E 共 c m
⫽ 共 1⫺ ␦ p,p ⬘ 兲 ␦ m,n ␦ k,l E 共 兩 c 兩 2 兲 2 ⫹ ␦ p,p ⬘ ␦ m,n ␦ k,l 共 1⫺ ␦ m,k 兲 E 共 兩 c m,p 兩 2 兩 c k,p 兩 2 兲
* c n,p c n,p c m,p
* 兲
⫹ ␦ p,p ⬘ ␦ m,k ␦ n,l 共 1⫺ ␦ m,n 兲 E 共 兩 c m,p 兩 2 兩 c n,p 兩 2 兲 ⫹ ␦ p,p ⬘ ␦ m,l ␦ n,k 共 1⫺ ␦ m,n 兲 E 共 c m,p
⫹ ␦ p,p ⬘ ␦ m,l ␦ n,k ␦ m,n E 共 兩 c m,p 兩 4 兲 .
共A7兲
Furthermore, for m⫽n we have
2
2
2
2
* c n,p c n,p c m,p
* 兲 ⫽E„共 f m,p
E 共 c m,p
⫺2i f m,p g m,p ⫺g m,p
⫹2i f n,p g n,p ⫺g n,p
兲共 f n,p
兲…
2
2
2
2
2
2
f n,p
f n,p g n,p 兲 ⫺E 共 f m,p
g n,p
⫽E 共 f m,p
兲 ⫹2iE 共 f m,p
兲 ⫺2iE 共 f m,p g m,p f n,p
兲 ⫹4E 共 f m,p f n,p g m,p g n,p 兲
2
2
2
2
2
2
⫹2iE 共 f m,p g m,p g n,p
f n,p
f n,p g n,p 兲 ⫹E 共 g m,p
g n,p
兲 ⫺E 共 g m,p
兲 ⫺2iE 共 g m,p
兲
2
2
2
2
2
2
2
2
⫽E 共 f m,p
f n,p
f n,p
g n,p
g n,p
兲 ⫺E 共 g m,p
兲 ⫺E 共 f m,p
兲 ⫹E 共 g m,p
兲
⫽0.
By symmetry E( 兩 c m,p 兩 2 兩 c n,p 兩 2 ) does not depend on m, n, or
p, and the same holds for E( 兩 c m,p 兩 4 ).
The fact that the vector (c 1,p , . . . ,c D,p ) is normalized
yields the identities
冉兺 冊
D
E
n⫽1
and
冉冉 兺 冊 冊
D
E
兺
n⫽1
E 共 兩 c n,p 兩 2 兲 ⫽DE 共 兩 c 兩 2 兲 ⫽E 共 1 兲 ⫽1 共A9兲
2
兩 c n,p 兩
n⫽1
2
D
⫽
兩 c n,p 兩 2
D
⫽
共A8兲
兺
m,n⫽1
E 共 兩 c n,p 兩 2 兩 c m,p 兩 2 兲
D
⫽
兺
n⫽1
D
E 共 兩 c n,p 兩 4 兲 ⫹
兺
m,n⫽1
共 1⫺ ␦ m,n 兲 E 共 兩 c n 兩 2 兩 c m 兩 2 兲
⫽DE 共 兩 c 兩 4 兲 ⫹D 共 D⫺1 兲 E 共 兩 c 兩 2 兩 c ⬘ 兩 2 兲 ⫽E 共 1 兲 ⫽1,
共A10兲
4374
ANTHONY HAMS AND HANS DE RAEDT
where c and c ⬘ refer to two different complex random variables. Therefore, we have
E 共 兩 c 兩 2 兲 ⫽1/D
E 共 兩 Tr RA 兩 2 兲 ⫽
共A11兲
and
E 共 兩 c 兩 2兩 c ⬘兩 2 兲 ⫽
1⫺DE 共 兩 c 兩 4 兲
.
D 共 D⫺1 兲
⫽ 共 1⫺ ␦ p,p ⬘ 兲 ␦ m,n ␦ k,l D ⫺2
共 D⫹1 兲 D 2 E 共 兩 c 兩 4 兲 ⫺2D
D⫺1
共A13兲
and the final result for the variance reads
兺
n⫽1
兩 A n,n 兩 2 .
共A15兲
where we omitted the subscript p because it is irrelevant for
what follows. The even moments of 兩 c n 兩 ⫽( f 2n ⫹g 2n ) 1/2 are
defined by
2
2
⫹g D
⫺1 兲 d f 1 dg 1 d f 2 dg 2 •••d f D dg D
共 f 21 ⫹g 21 兲 M ␦ 共 f 21 ⫹g 21 ⫹ f 22 ⫹g 22 ⫹•••⫹ f D
冕
冊
D
2
2
⬀ ␦ 共 f 21 ⫹g 21 ⫹ f 22 ⫹g 22 ⫹•••⫹ f D
⫹g D
⫺1 兲 ,
⫹ ␦ m,k ␦ n,l 共 1⫺ ␦ m,n 兲 …⫹ ␦ p,p ⬘ ␦ m,l ␦ n,k ␦ m,n E 共 兩 c 兩 兲 ,
E 共 兩 c 兩 2M 兲 ⫽
⫹
P 共 f 1 ,g 1 , f 2 ,g 2 , . . . , f D ,g D 兲
4
⫺⬁
1⫺D 2 E 共 兩 c 兩 4 兲
兩 Tr A 兩 2
D⫺1
共A14兲
1⫺DE 共 兩 c 兩 4 兲
„␦ m,n ␦ k,l 共 1⫺ ␦ m,k 兲
D 共 D⫺1 兲
⬁
⫹
An expression for the fourth moment E( 兩 c 兩 4 ) cannot be
derived from general properties of the probability distribution or normalization of random vector. We can only make
progress by specifying the former explicitly. As an example
we take a probability distribution such that for each realization p the random numbers f n,p and g n,p are distributed uniformly over the surface of a 2D-dimensional sphere of radius 1. This probability distribution can be written as
* 兲
* c n,p c k,p ⬘ c l,p
E 共 c m,p
⬘
冕
冉
1 D⫺D 2 E 共 兩 c 兩 4 兲
Tr A † A
S
D⫺1
共A12兲
Substitution into Eq. 共A7兲 yields
⫹ ␦ p,p ⬘
PRE 62
⬁
⫺⬁
共A16兲
.
␦ 共 f 21 ⫹g 21 ⫹•••⫹ f D2 ⫹g D2 ⫺1 兲 d f 1 dg 1 •••d f D dg D
It is expedient to introduce an auxiliary integration variable X by
E 共 兩 c 兩 2M 兲 ⫽
冕
⬁
⫺⬁
2
2
X M ␦ 共 f 21 ⫹g 21 ⫺X 兲 ␦ „f 22 ⫹g 22 ⫹•••⫹ f D
⫹g D
⫺ 共 1⫺X 兲 …dXd f 1 dg 1 d f 2 dg 2 •••d f D dg D
冕
⬁
⫺⬁
.
共A17兲
␦ 共 f 21 ⫹g 21 ⫹•••⫹ f D2 ⫹g D2 ⫺1 兲 d f 1 dg 1 •••d f D dg D
We can perform the integration over X last and regard Eq. 共A17兲 as the M th moment of the variable X with respect to the
probability distribution:
P共 X 兲⫽
冕
⬁
⫺⬁
␦ 共 f 21 ⫹g 21 ⫺X 兲 ␦ „f 22 ⫹g 22 ⫹•••⫹ f D2 ⫹g D2 ⫺ 共 1⫺X 兲 …d f 1 dg 1 d f 2 dg 2 •••d f D dg D
冕
⬁
⫺⬁
.
The calculation of P(X) amounts to computing integrals of
the form
I N共 X 兲 ⫽
冕
⬁
⫺⬁
冉兺 冊
I N共 X 兲 ⫽
N
␦
n⫽1
共A18兲
␦ 共 f 21 ⫹g 21 ⫹•••⫹ f D2 ⫹g D2 ⫺1 兲 d f 1 dg 1 •••d f D dg D
x 2n ⫺X dx 1 dx 2 •••dx N .
Changing to spherical coordinates, we have
共A19兲
⫽
yielding
2 ␲ N/2
⌫ 共 N/2兲
冕
⬁
0
r N⫺1 ␦ 共 r 2 ⫺X 兲 dr
␲ N/2 N/2⫺1
X
␪共 X 兲,
⌫ 共 N/2兲
共A20兲
PRE 62
4375
FAST ALGORITHM FOR FINDING THE EIGENVALUE . . .
P共 X 兲⫽
I 2 共 X 兲 I 2D⫺2 共 1⫺X 兲
I 2D 共 1 兲
E 共 兩 Tr RA 兩 2 兲 ⫽
⫽ 共 D⫺1 兲共 1⫺X 兲 D⫺2 ␪ 共 X 兲 ␪ 共 1⫺X 兲 .
E 共 兩 c 兩 2M 兲 ⫽
冕
⫺⬁
APPENDIX B: ERROR BOUNDS
X M P 共 X 兲 dX
⫽ 共 D⫺1 兲
冕
1
0
⌫ 共 D 兲 ⌫ 共 1⫹M 兲
,
⌫ 共 D⫹M 兲
X M 共 1⫺X 兲 D⫺2 dX⫽
共A22兲
1
2
,
E 共 兩 c 兩 兲 ⫽1, E 共 兩 c 兩 兲 ⫽ , E 共 兩 c 兩 4 兲 ⫽
D
D 共 D⫹1 兲
⫽
2
共A23兲
where the first two results provide some check on the procedure used. Substituting Eq. 共A23兲 into Eq. 共A14兲 yields
円兩 R 共 ␶ 兲 兩 円⭐
1
4
冐冕 冕 冕
冐冕 冕 冕
␶
⭐
1
4
0
␶
␭
d␭
0
␮
d
0
0
d ␯ e ␭A/2e ␭B 兵 e ⫺ ␯ B †2B, 关 A,B 兴 ‡e ␯ B
⫹e ␯ A/2†A, 关 A,B 兴 ‡e ⫺ ␯ A/2其 e ␭A/2e ( ␶ ⫺␭)(A⫹B) ,
␮
0
1
4
1
4
1
4
共B1兲
a well-known result 关24兴. We have 关26兴
d ␯ e ␭A/2e (␭⫺ ␯ )B †2B, 关 A,B 兴 ‡e ␯ B e ␭A/2e ( ␶ ⫺␭)(A⫹B)
0
d␮
0
␮
␭
冐
d ␯ e ␭A/2e ␭B e ␯ A/2†A, 关 A,B 兴 ‡e (␭⫺ ␯ )A/2e ( ␶ ⫺␭)(A⫹B)
冐
0
0
␶
1
4
d e ␭円兩 A 兩 円/2e (␭⫺ ␯ )円兩 B 兩 円円兩 †2B, 关 A,B 兴 ‡兩 円␯ 円兩 B 兩 円e ␭円兩 A 兩 円/2e ( ␶ ⫺␭)(円兩 A 兩 円⫹円兩 B 兩 円)
d
0
␮
␭
d␭
d
0
0
0
d e ␭円兩 A 兩 円/2e ␭円兩 B 兩 円e ␯ 円兩 A 兩 円/2円兩 †A, 关 A,B 兴 ‡兩 円e (␭⫺ ␯ )円兩 A 兩 円/2e ( ␶ ⫺␭)(円兩 A 兩 円⫹円兩 B 兩 円)
冐冕 冕 冕
冐冕 冕 冕
冐冕 冕 冕
冐冕 冕 冕
⫺␶
␭
d␭
0
0
␭
d␭
0
0
␭
d␭
0
0
␭
d␭
0
0
0
d␮
␮
d␮
␶
1
4
␮
d␮
⫺␶
1
4
␶
0
d ␯ e ␭A/2e (␭⫺ ␯ )B †2B, 关 A,B 兴 ‡e ␯ B e ␭A/2e (⫺ ␶ ⫺␭)(A⫹B)
␮
0
共B2兲
冐
d ␯ e ␭A/2e ␭B e ␯ A/2†A, 关 A,B 兴 ‡e (␭⫺ ␯ )A/2e (⫺ ␶ ⫺␭)(A⫹B)
冐
d ␯ e ⫺␭A/2e (⫺␭⫹ ␯ )B †2B, 关 A,B 兴 ‡e ⫺ ␯ B e ⫺␭A/2e (⫺ ␶ ⫹␭)(A⫹B)
␮
d␮
0
冐
d ␯ e ⫺␭A/2e ⫺␭B e ⫺ ␯ A/2†A, 关 A,B 兴 ‡e (⫺␭⫹ ␯ )A/2e (⫺ ␶ ⫹␭)(A⫹B)
冐
冕 冕 ␮冕 ␯
冕 冕 ␮冕 ␯
⫹
⫽
0
d␭
⫹
⭐
␭
d␭
␶
⫹
⫽
␶
冕 冕 ␮冕
1 3 ␶ (円兩 A 兩 円⫹円兩 B 兩 円)
␶ e
共 円兩 †A, 关 A,B 兴 ‡兩 円⫹円兩 †2B, 关 A,B 兴 ‡兩 円兲 ,
24
and
円兩 R 共 ⫺ ␶ 兲 円兩 ⭐
0
1
4
␮
d␮
1
4
冕 冕 ␮冕 ␯
冕 冕 ␮冕 ␯
⫹
⫽
␭
d␭
⫹
Here we prove that the difference between the eigenvalues of the Hermitian matrix A⫹B and those obtained from
the approximate time-evolution exp(zA/2)exp(zB)exp(zA/2)
(z⫽⫺i ␶ ,⫺ ␶ ) is bounded by ␶ 2 . In the following we assume
A and B are Hermitian matrices and take ␶ , a real, nonnegative number. We start with the imaginary-time case.
We define the difference R( ␶ ) by
R 共 ␶ 兲 ⬅e ␶ (A⫹B) ⫺e ␶ A/2e ␶ B e ␶ A/2
and the values of interest to us are
0
共A24兲
共A21兲
The moments E( 兩 c 兩 2M ) are given by
⬁
D Tr A † A⫺ 兩 Tr A 兩 2
.
S 共 D⫹1 兲
␶
0
1
4
␮
␭
d␭
d
0
␶
0
0
␮
␭
d␭
d e ␭円兩 A 兩 円/2e (␭⫺ ␯ )円兩 B 兩 円円兩 †2B, 关 A,B 兴 ‡兩 円e ␯ 円兩 B 兩 円e ␭円兩 A 兩 円/2e ( ␶ ⫺␭)(円兩 A 兩 円⫹円兩 B 兩 円)
d
0
0
d e ␭円兩 A 兩 円/2e ␭円兩 B 兩 円e ␯ 円兩 A 兩 円/2円兩 †A, 关 A,B 兴 ‡兩 円e (␭⫺ ␯ )円兩 A 兩 円/2e ( ␶ ⫺␭)(円兩 A 兩 円⫹円兩 B 兩 円)
1 3 ␶ (円兩 A 兩 円⫹円兩 B 兩 円)
␶ e
共 円兩 †A, 关 A,B 兴 ‡兩 円⫹円兩 †2B, 关 A,B 兴 ‡兩 円兲 .
24
共B3兲
4376
ANTHONY HAMS AND HANS DE RAEDT
Hence the bound in R( ␶ ) does not depend on the sign of ␶ so
that we can write
円兩 R 共 ␶ 兲 兩 円⭐s 兩 ␶ 兩 3 e 兩 ␶ 兩 (円兩 A 兩 円⫹円兩 B 兩 円) ,
共B4兲
where
s⬅
1
円兩 †A, 关 A,B 兴 ‡兩 円⫹円兩 †2B, 关 A,B 兴 ‡兩 円.
24
e ␶ A/2e ␶ B e ␶ A/2⬅e ␶ C( ␶ ) ,
共B6兲
where C( ␶ ) is Hermitian. Clearly we have
e ␶ (A⫹B) ⫺e ␶ C( ␶ ) ⫽R 共 ␶ 兲 .
共B7兲
We already have an upper bound on R( ␶ ), and now want to
use this knowledge to put an upper bound on the difference
in eigenvalues of C( ␶ ) and A⫹B. In general, for two Hermitian matrices U and V with eigenvalues 兵 u n 其 and 兵 v n 其 ,
respectively, both sets sorted in nondecreasing order, we
have 关2兴
兩 u n ⫺ v n 兩 ⭐円兩 U⫺V 兩 円, ᭙ n.
共B8兲
Denoting the eigenvalues of A⫹B and C( ␶ ) by x n (0) and
x n ( ␶ ), respectively, combining Eqs. 共B4兲 and 共B8兲 yields
兩 e ␶ x n (0) ⫺e ␶ x n ( ␶ ) 兩 ⭐s 兩 ␶ 兩 3 e 兩 ␶ 兩 (円兩 A 兩 円⫹円兩 B 兩 円) .
共B9兲
To find an upper bound on 兩 x n (0)⫺x n ( ␶ ) 兩 we first assume
that x n (0)⭐x n ( ␶ ) and take ␶ ⭓0. It follows from Eq. 共B9兲
that
e
␶ „x n ( ␶ )⫺x n (0)…
3 ␶ (円兩 A 兩 円⫹円兩 B 兩 円)⫺ ␶ x n (0)
⫺1⭐s ␶ e
.
共B10兲
For x⭓0, e x ⫺1⭓x and we have ⫺x n (0)⭐円兩 A⫹B 兩 円⭐円兩 A 兩 円
⫹円兩 B 兩 円. Hence we find
x n 共 ␶ 兲 ⫺x n 共 0 兲 ⭐s ␶ 2 e 2 ␶ (円兩 A円兩 ⫹円兩 B 兩 円) .
共B11兲
An upper bound on the difference in the eigenvalues between
C( ␶ ) and A⫹B can equally well be derived by considering
the inverse of the exact and approximate time-evolution operator 共B6兲. This is useful for the case x n (0)⬎x n ( ␶ ): Instead
of using Eq. 共B7兲 we start from exp„⫺ ␶ (A⫹B)…
⫺exp„⫺ ␶ C(⫺ ␶ )…⫽R(⫺ ␶ ) ( ␶ ⭓0). Note that the set of eigenvalues of a matrix and its inverse are the same and that
the matrices we are considering here, i.e., matrix exponentials, are nonsingular. Making use of Eq. 共B4兲 for R(⫺ ␶ )
gives
兩 e ⫺ ␶ x n (0) ⫺e ⫺ ␶ x n ( ␶ ) 兩 ⭐s 兩 ␶ 兩 3 e 兩 ␶ 兩 (円兩 A 兩 円⫹円兩 B 兩 円) ,
共B12兲
and proceeding as before we find
␶ „x n 共 0 兲 ⫺x n 共 ␶ 兲 …⭐e
Clearly Eq. 共B14兲 proves that the differences in the eigenvalues of A⫹B and C( ␶ ) vanish as ␶ 2 .
We now consider the case of the real-time algorithm (z
⫽⫺i ␶ ). For Hermitian matrices A and B the matrix exponentials are unitary matrices, and hence their norm equals 1.
This simplifies the derivation of the upperbound on
R(⫺i ␶ ). One finds 关20兴
共B5兲
For real ␶ we have
PRE 62
円兩 R 共 ⫺i ␶ 兲 兩 円E ⭐s 兩 ␶ 兩 3 ,
where 円兩 A 兩 円E2 ⬅Tr A † A denotes the Euclidean norm of the matrix A 关2兴. In general the eigenvalues of a unitary matrix are
complex valued, and therefore the strategy adopted above to
use the bound on R( ␶ ) to set a bound on the difference of the
eigenvalues no longer works. Instead we invoke the
Wielandt-Hoffman theorem 关27兴:
If U and V are normal matrices with eigenvalues u i and v i
respectively, then there exists a suitable rearrangement (a
permutation % of the numbers 1, . . . ,n) of the eigenvalues so
that
N
兺 兩 u j ⫺ v %( j)兩 2 ⭐円兩 U⫺V 兩 円E2 .
j⫽1
N
兺 兩 e i ␶ x (0) ⫺e i ␶ y ( ␶ )兩 2 ⭐円兩 R 共 ⫺i ␶ 兲 兩 円E2 ⭐s 2 ␶ 6 .
j
j
j⫽1
.
共B13兲
Putting the two cases together, we finally have
兩 x n 共 ␶ 兲 ⫺x n 共 0 兲 兩 ⭐s ␶ 2 e 2 ␶ (円兩 A 兩 円⫹円兩 B 兩 円) .
0⭐ 兩 ␶ „x j 共 0 兲 ⫺y j 共 ␶ 兲 …兩 ⭐ ␲ .
共B18兲
Rewriting the sum in Eq. 共B17兲, we have
N
兺 兩e
N
i ␶ x j (0)
j⫽1
⫺e
i ␶ y j(␶) 2
兩 ⫽
兺
j⫽1
兵 2⫺2 cos关 ␶ „x j 共 0 兲 ⫺y j 共 ␶ 兲 …兴 其
N
⫽4
兺
j⫽1
sin2 关 ␶ /2 „x j 共 0 兲 ⫺y j 共 ␶ 兲 …兴 .
共B19兲
As we have
4 x2
␲2
for 0⭐ 兩 x 兩 ⭐ ␲ /2,
共B20兲
the restriction Eq. 共B18兲 allows us to write
N
共B14兲
共B17兲
where y j ( ␶ )⫽x %( j) ( ␶ ), % being the permutation such that
inequality 共B17兲 is satisfied. We see that Eq. 共B17兲 only
depends on „␶ x j (0)mod 2 ␲ … and „␶ y j ( ␶ )mod 2 ␲ …, but so
does the DOS 关see Eq. 共16兲兴. Since inequality 共B17兲 and the
DOS only depend on these ‘‘angles’’ modulo 2 ␲ , there is no
loss of generality if we make the choice
3 2 ␶ (円兩 A 兩 円⫹円兩 B 兩 円)
⫺1⭐s ␶ e
共B16兲
Let U and V denote the exact and approximate real-time
evolution operators respectively. The eigenvalues of A⫹B
and C( ␶ ) are x n (0) and x n ( ␶ ), respectively. All the x n ’s and
x n ( ␶ )’s are real numbers. According to the WielandtHoffman theorem
sin2 x⭐
␶ „x n (0)⫺x n ( ␶ )…
共B15兲
兺
j⫽1
„x j 共 0 兲 ⫺y j 共 ␶ 兲 …2 ⭐
␲ 2s 2 4
␶ ,
4
共B21兲
PRE 62
FAST ALGORITHM FOR FINDING THE EIGENVALUE . . .
implying
␲s 2
␶ .
共B22兲
2
In summary, we have shown that in the real-time case there
exists a permutation of the approximate eigenvalues such
兩 x j 共 0 兲 ⫺y j 共 ␶ 兲 兩 ⭐
关1兴 G. D. Mahan, Many-Particle Physics 共Plenum Press, New
York, 1981兲.
关2兴 J. H. Wilkinson, The Algebraic Eigenvalue Problem 共Clarendon Press, Oxford, 1965兲.
关3兴 G. H. Golub and C. F. Van Loan, Matrix Computations 共John
Hopkins University Press, Baltimore, MD, 1983兲.
关4兴 R. Alben, M. Blume, H. Krakauer, and L. Schwartz, Phys.
Rev. B 12, 4090 共1975兲.
关5兴 M. D. Feit, J. A. Fleck, and A. Steiger, J. Comput. Phys. 47,
412 共1982兲.
关6兴 H. De Raedt and P. de Vries, Z. Phys. B: Condens. Matter 77,
243 共1989兲.
关7兴 T. Kawarabayashi and T. Ohtsuki, Phys. Rev. B 53, 6975
共1996兲.
关8兴 T. Ohtsuki and T. Kawarabayashi, J. Phys. Soc. Jpn. 66, 314
共1997兲.
关9兴 T. Iitaka, S. Nomura, H. Hirayama, X. Zhao, Y. Aoyagi, and
T. Sugano, Phys. Rev. E 56, 1222 共1997兲.
关10兴 S. Nomura, T. Iitaka, X. Zhao, T. Sugano, and Y. Aoyagi,
Phys. Rev. B 56, 4348 共1997兲.
关11兴 S. Nomura, T. Iitaka, X. Zhao, T. Sugano, and Y. Aoyagi,
Phys. Rev. B 59, 10 309 共1999兲.
关12兴 T. Iitaka and T. Ebisuzaki, Phys. Rev. E 60, 1178 共1999兲.
4377
that the difference with the exact ones vanishes as ␶ 2 .
Finally we note that both upper bounds 共B22兲 and 共B14兲
hold for arbitrary Hermitian matrices A and B and are therefore rather weak. Except for the fact that they provide a
sound theoretical justification for the real- and imaginary
time method, they are of little practical value.
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
关20兴
关21兴
关22兴
关23兴
关24兴
关25兴
关26兴
关27兴
T. Iitaka and T. Ebisuzaki, Microelectron. Eng. 47, 321 共1999兲.
T. Iitaka and T. Ebisuzaki, Phys. Rev. E 61, 3314 共2000兲.
P. de Vries and H. De Raedt, Phys. Rev. B 47, 7929 共1993兲.
H. De Raedt, A. Hams, K. Michielsen, S. Miyashita, and K.
Saito, Prog. Theor. Phys. 138, 489 共2000兲.
D. S. Abrams and S. Lloyd, Phys. Rev. Lett. 83, 5162 共1999兲.
G. R. Grimmet and D. R. Stirzaker, Probability and Random
Processes 共Clarendon, Oxford, 1992兲.
E. Lieb, T. Schultz, and D. C. Mattis, Ann. Phys. 共N.Y.兲 16,
407 共1961兲.
H. De Raedt, Comput. Phys. Rep. 7, 1 共1987兲.
M. Suzuki, S. Miyashita, and A. Kuroda, Prog. Theor. Phys.
58, 1377 共1977兲.
円兩 X 兩 円 denotes the spectral norm of the matrix X; see The Algebraic Eigenvalue Problem 共Ref. 关2兴兲 and Matrix Computations
共Ref. 关3兴兲.
H. De Raedt and B. De Raedt, Phys. Rev. A 28, 3575 共1983兲.
M. Suzuki, J. Math. Phys. 26, 601 共1985兲.
H. De Raedt and K. Michielsen, Comput. Phys. 8, 600 共1994兲.
M. Suzuki, J. Math. Phys. 61, 3015 共1995兲.
A. J. Hoffman and H. W. Wielandt, Duke Math. J. 20, 37
共1953兲.