PHYSICAL REVIEW B 71, 155115 共2005兲
Sufficient condition for absence of the sign problem in the fermionic quantum Monte Carlo
algorithm
Congjun Wu and Shou-Cheng Zhang
Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA
共Received 11 July 2004; revised manuscript received 24 January 2005; published 19 April 2005兲
Quantum Monte Carlo 共QMC兲 simulations involving fermions have a notorious sign problem. Some wellknown exceptions to the auxiliary field QMC algorithm rely on the factorizibility of the fermion determinant.
Recently, a fermionic QMC algorithm 关C. Wu, J. Hu, and S. Zhang, Phys. Rev. Lett. 91, 186402 共2003兲兴 has
been found in which the fermion determinant may not necessarily be factorizable, but can instead be expressed
as a product of complex conjugate pairs of eigenvalues, thus eliminating the sign problem for a much wider
class of models. In this paper, we present the general conditions for the applicability of this algorithm and point
out that it is deeply related to the time-reversal symmetry of the fermion matrix. We apply this method to
various models of strongly correlated systems at all doping levels and lattice geometries, and show that many
phases can be simulated without the sign problem.
DOI: 10.1103/PhysRevB.71.155115
PACS number共s兲: 71.10.Fd, 02.70.Ss
I. INTRODUCTION
Understanding the physics of strongly correlated manybody systems is a main focus of condensed matter physics
today. However, most models with strong interactions cannot
be solved exactly except in one dimension. Presently, there
are no systematic, nonperturbative, analytic methods which
work in higher dimensions. Largely because of this reason,
numerical simulations such as exact diagonalization 共ED兲,
density-matrix renormalization group 共DMRG兲, and quantum Monte Carlo 共QMC兲 are extensively performed to study
strongly correlated systems. However, each of the numerical
methods has its own limitations. The ED can only be performed on a very small sample size and the DMRG method
is largely restricted to one-dimensional systems. In contrast,
QMC simulation is the only systematic and scalable method
with sufficient numerical accuracy for higher-dimensional
problems. However, QMC also has the notorious fermion
sign problem which makes low-temperature properties inaccessible.
In lattice systems, a particular version of QMC uses the
auxiliary-field method of Blankenbecler, Scalapino, and
Sugar,1 with fruitful results. Because one cannot directly
sample the fermionic Grassmann fields, the standard process
is to perform a Hubbard-Stratonovich 共HS兲 transformation to
decouple the four fermion interaction terms and then to integrate out the fermions.1 The resulting fermion functional determinant works as the statistical weight for sampling the
auxiliary fields. However, generally speaking, the fermion
determinant may not be positive and can even be complex in
some cases. The sign or the phase of the fermion determinants can lead to dramatic cancellations which makes statistical errors scale exponentially as the inverse of the temperature and size of the system. This notorious sign problem is
the major obstacle in applying QMC to fermionic systems. A
successful solution to the sign problem would obviously lead
to great advances in quantum many-body physics.
There are a few exceptions where the sign problem is
absent, such as the negative U Hubbard model and the posi1098-0121/2005/71共15兲/155115共14兲/$23.00
tive U Hubbard model in a bipartite lattice at the half filling.2
In both cases, the fermion determinant after the HS decomposition can be factorized into two real parts with the same
sign. It is therefore positive definite. Unfortunately, general
fermion determinants may not be factorizable for more complicated models and the majority of models do have the sign
problem. In recent years, several other algorithms have been
proposed which partially solve the minus sign problem.3–7
Recently, it has been shown that the minus sign problem
can be eliminated without relying on the factorizibility of the
fermion determinant; therefore, a broader class of models
can be simulated by the QMC algorithm.8 The fermion determinant can always be expressed as a product of its eigenvalues; under certain conditions, the eigenvalues of the fermion determinant always appear in complex conjugate pairs,
thus making the fermion determinant positive definite. In this
paper, we shall show that the property of conjugate eigenvalue pairs follows from the time-reversal symmetry of the
HS-decoupled Hamiltonian and can be viewed as a generalization of the Kramer’s theorem in quantum mechanics. We
shall call this method the T-invariant decomposition 共timereversal invariant decomposition兲. This method does not lead
to any improvement for the single-band Hubbard model, but
significantly extends the applicability of the QMC to multiband, multilayer, or higher-spin models. This algorithm is
particularly useful for Hubbard models with higher spins,
which can be accurately realized in systems of cold atoms.
Recently, Assaad et al.9 applied the QMC to generalized
Hubbard models with more bands. Imposing the factorizibility condition of the fermion determinant, they found that
they could extend the parameter regime for QMC free of the
sign problem only by scarifying the spin-rotational invariance. However, applying our method of T-invariant decomposition without requiring factorizibility, we shall show that
multi-band or higher spin Hubbard models can be simulated
for an extended parameter regime without scarifying the spin
rotational invariance. This QMC algorithm based on
T-invariant decomposition has been recently applied to conclusively demonstrate the staggered current-carrying ground
state in a bilayer model.10
155115-1
©2005 The American Physical Society
C. WU AND S.-C. ZHANG
The rest of this paper is outlined as follows. In Sec. II, the
sign problem for the spin- 21 Hubbard model is reviewed. In
Sec. III, we prove the fundamental theorem of T-invariant
decomposition and show the absence of the sign problem. In
Sec. IV, we employ the algorithm to the spin- 23 Hubbard
model and the generalized arbitrary spin-n − 21 fermionic
Hubbard model. In Sec. V, we apply it to a bilayer model
introduced by Scalapino, Zhang, and Hanke,11 which can be
mapped into the spin- 23 Hubbard model. In Sec. VI, we discuss the algorithm in the model Hamiltonians with bond interactions and various exotic phases. Our final conclusions
are presented in Sec. VII.
=
冕
再
Dn exp −
冉
冊
冉
n↑共i兲 −
1
2
冊
共1兲
with t the hopping integral, ␮ the chemical potential, ␴
= ↑ , ↓, n␴共i兲 = ci†␴ci␴, and n共i兲 = n↑共i兲 + n↓共i兲. At half filling and
on a bipartite lattice, the particle-hole symmetry ensures that
␮ = 0.
To perform the QMC simulation, we first need to decouple the four-fermion interaction terms using the HS transformations by the Gaussian integral,
冉 冊 冕 冉
冉 冊冑 冕 冉
1
exp A2 = 冑2␲
2
HK =
冊
共2兲
冊
1
dx exp − x2 − ixA .
2
=
再
冕 冉
冕
Dc†Dc exp −
冕
DnDc†Dc exp −
再冕
⫻exp −
␤
0
␤
0
再
d␶ c␴†
兩U兩
2
冕
␤
0
⳵
c␴ + H
⳵␶
d␶
冎
冊冎
兺ij ci†␴hij,K ␴␴⬘c j␴⬘,
det兵I + B其,
HI =
兺i ci†␴hij,I ␴␴⬘c j␴⬘ ,
共4兲
K
I
Here hij,
␴␴⬘ and hij,␴␴⬘ are defined for both spin components
on each site. After integrating out the fermions, we obtain
再
I + B = I + T exp −
冕
␤
冎
d␶关hK + hI共␶兲兴 .
0
I + B = I + e−⌬␶HKe−⌬␶Hi共␶l兲e−⌬␶HKe−⌬␶Hi共␶l−1兲 ¯
⫻ ¯ e−⌬␶HKe−⌬␶Hi共␶1兲 ,
where ⌬␶ = ␤ / l is the discretized time slice.
Similarly, at U ⬎ 0, Eq. 共1兲 can be decomposed in the
spin-density channel as
冕
再
DSz exp − 2U
冕
␤
0
d␶
兺i Sz2共i, ␶兲
冎
det兵I + B其,
共3兲
共7兲
with the same expression for B as in Eq. 共5兲, but with HI
replaced by
z
ci␤共␶兲其Sz共i, ␶兲.
兺i 兵ci†␣共␶兲␴␣␤
共8兲
It is well known that the spin- 21 Hubbard model is free of
the sign problem either for U ⬍ 0 or for U ⬎ 0 at half filling
and in a bipartite lattice.2,12 The usual proof is based on the
factorization of the fermion determinant as
det兵I + B其 = det兵I + B↑其det兵I + B↓其.
冎
共5兲
K
I
Note that the matrix kernels hij,
␴␴⬘ and hij,␴␴⬘ entering in Eq.
共5兲, as well as the I + B matrix itself, are 2N ⫻ 2N matrices, if
the lattice system under simulation has N = Lx ⫻ Ly sites. In
the subsequent discussions, we shall simply use the second
quantized operators HK and HI interchangeably with the firstquantized matrix kernels hK and hI to save some writing,
whenever their meanings are obvious from the context.
In practice, I + B needs to discretized as
HI共␶兲 = − 2U
,
兺i 关n共i, ␶兲 − 1兴2
d␶关HK + HI共␶兲兴 ,
冎
K
Z=
Various HS decoupling schemes are discussed in Ref. 12. For
U ⬍ 0, it is convenient to decouple Eq. 共1兲 in the density
channel and then integrate out the fermions. The resulting
partition function is given by
Z=
0
兺i 关n共i, ␶兲 − 1兴2
hij,␴␴⬘ = 兵− t共␦i,j+x̂ + ␦i,j−x̂ + ␦i,j+ŷ + ␦i,j−ŷ兲 − ␮␦ij其␦␴␴⬘ ,
1
dx exp − x2 − xA ,
2
1
exp − A2 = 2␲
2
d␶
I
1
,
2
⫻ n↓共i兲 −
冕
␤
hij,␴␴⬘ = Un共i, ␶兲␦ij␦␴␴⬘ .
In this section, we review the sign problem in the spin- 21
Hubbard model and interpret its absence in the negative U
case as due to its time-reversal properties of the HS decomposition. The Hubbard model on the lattice is commonly
defined as
共ci†␴c j␴ + H.c.兲 − ␮ 兺 n共i兲 + U 兺
兺
ij,␴
i
i
兩U兩
2
where n共i , ␶兲 is a real HS Bose density field. The imaginarytime-independent kinetic energy term HK and the imaginarytime-dependent decoupled interaction term HI共␶兲 can be expressed as
1
II. THE SIGN PROBLEM IN THE SPIN - 2 HUBBARD
MODEL
H=−t
PHYSICAL REVIEW B 71, 155115 共2005兲
共9兲
In the negative U case, the HS decomposition in Eq. 共4兲
enables such a factorization, and B↑ is identical to B↓ for any
HS field configurations. Therefore det兵I + B其 is the square of
a real number and thus positive definite. Generally speaking,
in the positive U case, the HS decomposition in Eq. 共8兲 still
enables factorization, but det兵I + B↑其 is different from det兵I
+ B↓其, and thus the sign problem appears. However, at half
filling and on a bipartite lattice, it is possible to change the
155115-2
PHYSICAL REVIEW B 71, 155115 共2005兲
SUFFICIENT CONDITION FOR ABSENCE OF THE…
sign of U while keeping the kinetic-energy part invariant by
a partial particle-hole transformation only on spin down particles,
ci↑ → ci↑,
†
ci↓ → 共− 兲ici↓
,
共10兲
then the above algorithm is also applicable. Nevertheless,
this transformation cannot be applied to lattices which are
not bipartite or away from the half filling 共␮ ⫽ 0兲, thus the
sign problem remains in general.
Recently, an anisotropic two-band model explicitly breaking the spin rotational symmetry was also shown to be free
of the sign problem.9 The Hamiltonian is defined by
H=−t
共ci†␴c j␴ + H.c.兲 − ␮ 兺 n␴共i兲 − 兩U兩 兺 关n1共i兲 − n2共i兲
兺
ij,␴
i,␴
i
+ n3共i兲 − n4共i兲兴 ,
共11兲
2
where n␴共i兲 = c␴† 共i兲c␴共i兲 are the particle densities for each spin
component ␴ = 1 , 2 , 3 , 4. The interaction part can be decoupled as
冕
再 冕
再冕
DS exp − 兩U兩
0
⫻exp −
␤
0
H I共 ␶ 兲 =
␤
d␶
冎
冎
兺i S 共i, ␶兲
2
THIT−1 = HI,
T2 = − 1,
兺i 共ci,1†ci,1 − ci,2†ci,2 + ci,3†ci,3 − ci,4†ci,4兲S共i, ␶兲.
This HS decomposition enables the factorization of the fermion determinant as
共12兲
where det兵I + B其12 and det兵I + B其34 for spin components 1,2
and 3,4, respectively, are identical and real. Therefore, the
fermion determinant is positive in this case as well. However, a disadvantage of this model is the explicit breaking of
the spin-rotational symmetry.
det共I + B兲 =
兿i 兩␭i兩2 艌 0.
共14兲
Proof. From the condition of the theorem stated in Eq.
共13兲, it obviously follows that T共I + B兲T−1 = 共I + B兲. For simplicity, we first consider the case where I + B is an n ⫻ n,
dimensional, diagonalizable matrix, i.e., there exists a nonsingular matrix P satisfying
P−1共I + B兲P = diag兵␭1,␭2, . . . ,␭n其.
共15兲
The n columns of P can be viewed as a set of linearly independent state vectors,
P = 兵兩⌿1典,兩⌿2典, . . . ,兩⌿n典其.
共16兲
Suppose that 兩⌿i典 is an eigenvector with eigenvalue ␭i, i.e.,
共I + B兲兩⌿i典 = ␭i兩⌿i典. Using the antiunitary property of T, we
see that
共17兲
Therefore, T兩⌿i典 is also an eigenvector, with eigenvalue ␭*i .
Since T2 = −1, T兩⌿i典 and 兩⌿i典 are orthogonal to each other.
This shows that ␭i and ␭*i are two different eigenvalues, thus
the eigenvalues of I + B appear in complex conjugate pairs as
stated in the theorem. If I + B is Hermitian, our theorem reduces to Kramer’s theorem on the time-reversal symmetry in
quantum mechanics, stating that the eigenvalues of I + B are
real and twofold degenerate.
In the general case, I + B may not be diagonalizable; instead it can always be transformed into the Jordan normal
form as diagonal blocks,
P−1共I + B兲P = diag兵J1,J2, . . . ,Jk其,
共18兲
where P is an n ⫻ n nonsingular matrix as before, and Ji is an
li ⫻ li bidiagonal matrix as
III. FUNDAMENTAL THEOREM OF T-INVARIANT
DECOMPOSITION
We now show that the condition of factorizibility of the
fermion determinant is unnecessarily restrictive, and a more
general condition can be precisely stated. The fermion determinant is a product of all the eigenvalues. Since I + B involves a time-ordered product, it may not be Hermitian, and
the eigenvalues may be complex in general. Because the ensemble of HS field configurations is arbitrary, one would
naively not expect any special relations among the eigenvalues. Surprisingly, the time-reversal symmetry provides an
important relationship among the eigenvalues. To formulate
the fundamental theorem, we consider HK and HI in the I
+ B matrix of Eq. 共6兲 to be the HS-decomposed singleparticle Hamiltonian matrix derived from a general Hamiltonian, not necessarily the s = 21 Hubbard model.
Theorem: If there exists an antiunitary operator T, such
that
共13兲
then the eigenvalues of the I + B matrix always appear in
complex conjugate pairs, i.e., if ␭i is an eigenvalue, then ␭*i
is also an eigenvalue. If ␭i is real, it is twofold degenerate. In
this case, the fermion determinant is positive definite,
共I + B兲T兩⌿i典 = T共I + B兲T−1T兩⌿i典 = ␭*i T兩⌿i典.
d␶关H0 + HI共␶兲兴 ,
det兵I + B其 = det兵I + B其12 det兵I + B其34 ,
THKT−1 = HK,
Ji =
冢
␭i
冣
1
¯
¯
¯
␭i
1
␭i
.
共19兲
The determinant of I + B is still the product of all the eigenvalues,
k
det共I + B兲 =
共␭i兲l .
兿
i=1
i
共20兲
As in Eq. 共16兲, P can be viewed as n linearly independent
column state vectors as
P = 兵P1, P2, . . . . , Pk其,
共21兲
where each Pi is an n ⫻ li matrix containing li column state
vectors,
155115-3
PHYSICAL REVIEW B 71, 155115 共2005兲
C. WU AND S.-C. ZHANG
i−1
Pi = 兵兩⌿m+1典, . . . ,兩⌿m+li典其,
m=
lj .
兺
j=1
共22兲
For each Jordan block Ji, it satisfies
共I + B兲Pi = PiJi ,
共23兲
thus among the li state vectors in Pi, 兩⌿m+1典 is the only
eigenvector with eigenvalule ␭i. It is straightforward to show
that
共I + B兲共TPi兲 = 共TPi兲J*i ,
共24兲
where 共TPi兲 is defined as
共TPi兲 = 兵T兩⌿m+1典, . . . ,T兩⌿m+li典其,
共25兲
and T兩⌿m+1典 is the only eigenvector with eigenvalue ␭*i in
共TPi兲. Again since 兩⌿m+1典 and T兩⌿m+1典 are orthogonal to
each other, 共TPi兲 contains different state vectors from what
Pi does. As a result, Ji and J*i are different Jordan blocks. As
before, the Jordan blocks appear in complex conjugate pairs,
and so do the eigenvalues. This completes the proof for the
general case of I + B.
Since the antiunitary operator T used in our theorem
shares similar properties with the time-reversal transformation in quantum mechanics, we call our method T-invariant
decomposition. However, it is important to emphasize that
any antiunitary operator with the stated mathematical properties could work here. In some examples we shall discuss, T
does not have the explicit physical meaning of the timereversal transformation.
It is also important to point out that the T2 = −1 condition
is essential for our theorem. In the case in which the fermion
matrix is real, one can define a trivial antiunitary operator
T = C, where C denotes the complex conjugation. In this case,
if the eigenvalue ␭i is complex, i.e., ␭i ⫽ ␭*i , then ␭*i must
also be an eigenvalue. However, when ␭i is real, it is in
general not twofold degenerate, since 兩⌿典 and T兩⌿典 may not
be orthogonal for the case of T2 = 1. In this case, an odd
number of negative eigenvalues would lead to a negative
determinant. The distribution of eigenvalues in the complex
plane for a fermion matrix satisfying the condition of our
theorem and the eigenvalues of a generic real-fermion matrix
is illustrated in Fig. 1. When the conditions of our theorem
are violated, either the complex conjugate eigenvalue pairs
collide on the real axis and move off from each other along
the real axis, or the twofold degenerate eigenvalues move off
directly from each other along the real axis.
A restricted version of our theorem was originally discussed in the context of nuclear physics.4 However, these
authors overlooked the case that I + B may not be diagonalizable, thus their proof was not complete. In addition, our T
transformation is not restricted to the physical time-reversal
transformation as in Ref. 4, thus the theorem applies to a
much wider class of models.
We now illustrate this general theorem for the case of the
s = 21 Hubbard model. For the spin- 21 system on each site, the
time-reversal transformation T is defined as T共i兲 = R共i兲C, satisfying T2共i兲 = −1, where
FIG. 1. Distribution of eigenvalues in the complex plane. 共a兲
Eigenvalues of a fermion matrix satisfying the conditions of our
theorem are always paired. 共b兲 Complex eigenvalues of a generic
real matrix are paired, but real eigenvalues are not twofold degenerate in general, leading to negative determinants.
R = − i␴ y =
冉 冊
0 −1
1
0
共26兲
.
For the entire system, the time-reversal operator is defined as
the direct product T = 关⌸i 丢 R共i兲兴C. The four independent fermion bilinears in the particle-hole channel can be classified
as the particle number n共i兲 = ␺i,† ␣␺i,␣ and spin Sជ 共i兲
= ␺i,† ␣共␴ / 2兲␣␤␺i,␤, which are even and odd under the T transformation, respectively,
Tn共i兲T−1 = n共i兲,
TSជ 共i兲T−1 = − Sជ 共i兲.
共27兲
Now we can understand the absence of the sign problem in
the negative U case as follows. The density channel decomposition is T invariant, namely, T关HK + HI共␶兲兴T−1 = HK
+ HI共␶兲. The conditions of our theorem are satisfied and the
fermion determinant is thus positive. For U ⬎ 0, the Hamiltonian can be decoupled in the density channel at the cost of
involving the imaginary number i or decoupled in the spin
channel with only real numbers. In either case, while HK is
still even under T, HI is odd. The conditions of our theorem
do not apply, and the sign problem appears in general.
For a general interacting fermion model, we can always
express T = RC, where RR* = −1 and R* is the complex conjugate of R. In many cases, R is purely real, and it reduces to
R2 = −1. The general condition for our theorem then reads
R共HK + HI兲R−1 = 共HK + HI兲* ,
*
共28兲
with the unitary matrix R satisfying RR = −1 for any configurations of the HS field. Again we emphasis that the precise form for R in Eq. 共26兲 is not necessary.
While our method does not lead to any improvement of
the sign problem for the s = 21 Hubbard model, we shall show
now that it significantly improves the QMC algorithm for
multiband, multilayer, and higher-spin models, since the conditions for our theorem are far less restrictive than the condition for the factorizibility of the fermion determinant. Let
155115-4
PHYSICAL REVIEW B 71, 155115 共2005兲
SUFFICIENT CONDITION FOR ABSENCE OF THE…
us illustrate the general idea here by looking at the example
of a two-band spin- 21 model or a spin- 23 model. In this case,
we have the fermion operators ␺i,␤ within one unit cell,
where ␤ = 1 , 2 , 3 , 4. Therefore, there are 16 fermion bilinears,
I
␺i,␤, where I = 1 , . . . , 16. The 16
of the form M I = ␺i,† ␣M ␣␤
I
M ␣␤ matrices can in general be expressed in a complete basis
in terms of the product of s = 23 matrices Si,
generated by the transformation ci,␴ → 共−兲ici,† ␴. Because of
the Pauli’s exclusion principle, only on-site interactions in
the total spin singlet 共ST = 0兲 and the quintet 共ST = 2兲 channels
†
are the singlet and quintet pairing operaare allowed. P†0,P2m
tors defined by
†
共i兲兴 =
P†0共i兲关P20
I,
i
S,
a = 1, . . ,5,
␰aij = ␰aji,
␰LijkSiS jSk,
L = 1, . . ,7,
␰Lijk = ␰Ljik,
␰aii = 0,
共29兲
L
␰iik
= 0,
where the ␰’s are fully symmetric, traceless tensors. If one
insists on the factorizibility of the fermion determinant, one
could only perform the HS decomposition in the density
channel using the identity matrix I. However, since the
␰aijSiS j matrix contains an even power of spin matrices, it is
also even under time reversal. HS decomposition in this
channel does not lead to factorization of the fermion determinant, but according to our general theorem, it does lead to
paired eigenvalues, and therefore, a positive fermion determinant. As we see from this nontrivial example, our method
of T-invariant decomposition is indeed more general and
more powerful compared with the traditional method of factorization. We shall show the enlarged parameter space for
the QMC algorithm explicitly in the Sec. IV.
3
In this section, we apply the method of T-invariant decomposition to the s = 23 model as an explicit example, and
discuss the sign problem accordingly. After that, we generalize it to arbitrary fermionic Hubbard models with s = n − 21 .
These models are not of only academic interest. In fact, the
rapid progress in ultracold atomic systems provides an opportunity to study higher-spin fermions. The simplest cases
are the spin- 23 atoms, such as the 9Be, 132Cs, 135Ba, and 137Ba
atoms. Another important research direction is the trapped
atoms in an optical lattice, formed by the standing-wave laser beams, where the Hubbard model is a good approximation for these neutral atoms.
⌫1 =
冉
0
iI
− iI 0
共31兲
†
†
†
P2,−2
共i兲 = ci,−1/2
ci,−3/2
.
,
⌫2,3,4 =
冉
共32兲
冊
␴ជ 0
,
ជ
0 −␴
⌫5 =
冉
0
−I
−I
0
冊
,
R† = R−1 = tR = − R
R⌫aR−1 = t⌫a,
R⌫abR−1 = − t⌫ab .
In our explicit representation,
R = ⌫ 1⌫ 3 =
冉
0
− i␴2
− i␴2
0
冊
.
共33兲
共34兲
Using the R matrix, the s = 23 Hubbard interaction can be
written in an explicitly SO共5兲 symmetric fashion as
+
with ␮0 = 共U0 + 5U2兲 / 4. ␮ is fixed to be zero at half filling on
a bipartite lattice, to ensure the particle-hole 共p-h兲 symmetry
冊
R2 = − 1,
Hubbard model is defined as8
共30兲
†
†
†
P2,1
共i兲 = ci,3/2
ci,−1/2
,
ជ are the 2 ⫻ 2 unit and Pauli matrices. The ten
where I and ␴
SO共5兲 generators are defined as ⌫ab = −i / 2关⌫a , ⌫b兴共1 艋 a , b
艋 5兲. Since the SO共5兲 group is equivalent to the Sp共4兲 group,
there exists a symplectic matrix R, with the properties,11
H=−t
†
共i兲P2m共i兲,
兺i P†0共i兲P0共i兲 + U2i,m=±2,±1,0
兺 P2m
†
In this representation, we define five 4 ⫻ 4 Dirac ⌫a 共1 艋 a
艋 5兲 matrices to construct the Sp共4兲 or SO共5兲 algebra as
ci†␴ci␴
兺 兵ci†␴c j␴ + H.c.其 − 共␮ + ␮0兲 兺
具ij典,␴
i␴
+ U0
†
␺共i兲 = „c3/2共i兲,c1/2共i兲,c−1/2共i兲,c−3/2共i兲…T .
3
H=−t
†
The s = 23 Hubbard model has an exact SO共5兲 or equivalently,
Sp共4兲 symmetry, without any fine tuning of the parameters.8
This follows from the fact that singlet and quintet channel
interactions can also be interpreted as the SO共5兲 group’s singlet and five-vector representations. When U0 = U2, the
model has a larger symmetry, namely the SU共4兲 symmetry.
The SU共4兲 symmetric Hubbard model has been extensively
studied in the transition metal oxides with double orbital
degeneracy.13
To illustrate the T-invariant decomposition for this model,
we first define the four-component spinor,
A. The s = 2 Hubbard model
The
†
†
†
†
共i兲 = ci,1/2
ci,−3/2
,
P2,−1
1
IV. APPLICATION IN SPIN 2 AND n − 2 HUBBARD
MODEL
spin- 23
冑2 兵ci,3/2ci,−3/2 ⫿ ci,1/2ci,−1/2其,
†
†
†
共i兲 = ci,3/2
ci,1/2
,
P2,2
i = 1,2,3,
␰aijSiS j,
1
兺ij 关␺†共i兲␺共j兲 + H.c.兴 − 共␮ + ␮0兲 兺i ␺†共i兲␺共i兲
U0
2
U
␹†,a共i兲␹a共i兲,
兺i ␩†共i兲␩共i兲 + 22 兺
i,a
共35兲
where ␩†共i兲 = ␺†共i兲共R / 2兲␺†共i兲 is the singlet pairing operator,
and ␹a,†共i兲 = ␺†共i兲共⌫aR / 2兲␺†共i兲 are the polar forms of the
quintet pairing operators in Eq. 共31兲.
In order to implement the method of T-invariant decomposition, we first need to express the interaction terms in the
particle-hole channel, rather than the particle-particle chan-
155115-5
PHYSICAL REVIEW B 71, 155115 共2005兲
C. WU AND S.-C. ZHANG
nel. In the particle-hole channel, there are 16 bilinear fermionic operators, which can be classified into the scalar, vector,
and antisymmetric tensors 共generators兲 of the SO共5兲 group as
n共i兲 = ␺␣† 共i兲␺␣共i兲,
1
a
na共i兲 = ␺␣† 共i兲⌫␣␤
␺␤共i兲,
2
1
ab
␺␤共i兲,
Lab共i兲 = − ␺␣† 共i兲⌫␣␤
2
1艋a艋5
共36兲
1 艋 a ⬍ b 艋 5,
second and the fourth rows of Eq. 共29兲. Thus, the n共i兲 operator describes a particle-hole pair with total spin zero, the five
na共i兲 operators describe five particle-hole pair states with total spin two, and the ten Lab operators include the degenerate
three spin-1 and seven spin-3 particle-hole pair states. From
this point of view, the physical meaning of Eqs. 共39兲 and 共29兲
becomes transparent; operators with even total spins are even
under T, while operators with odd total spins are odd under
T.
Using the identities,
5
3 a a
1 ab ab
共⌫aR兲␣␤共R⌫a兲␥␦ = ␦␣␥␦␤␦ − ⌫␣␥
⌫␤␦ − ⌫␣␥
⌫ ␤␦ ,
4
4
4
where n共i兲 is the particle number operator and na共i兲 and
Lab共i兲 represent the spin degrees of freedom. The ten SO共5兲
generators are often conveniently denoted as
Lab共i兲 =
冢
0 Re ␲x Re ␲y Re ␲z
0
Q
− Sz
Sy
Im ␲x
0
− Sx
Im ␲y
0
Im ␲z
0
冣
.
共37兲
兺
兺
and the Firez identity 关Eq. 共38兲兴, we can now express the s
= 23 Hubbard model in the following form:
HK = − t
Although similar symbols are used in the SO共5兲 algebra in
the high Tc cuprates,14 the operators here have different
physical meanings.
These 16 fermion bilinears are related through the Fierz
identity,
5
2
Lab
共i兲 +
n2a共i兲 + 关n共i兲 − 2兴2 = 5.
4
1艋a⬍b艋5
1艋a艋5
共42兲
1
1 a a
1 ab ab
⌫␤␦ − ⌫␣␥
⌫ ␤␦ ,
R␣␤R␥␦ = ␦␣␥␦␤␦ + ⌫␣␥
4
4
4
TnaT−1 = na,
TLabT−1 = − Lab .
兺
HI = −
i,1艋a艋5
共38兲
共39兲
On the other hand, we can relate the above ⌫-matrices
with the usual spin-SU共2兲 operators Ji, which form a subgroup of the SO共5兲 group,
J± = Jx ± iJy
共44兲
g␯ = 共U2 − U0兲/2.
B. Absence of the sign problem
After a series of transformations, we arrived at a form of
the s = 23 Hubbard which is suitable for the T-invariant decomposition method. The interactions in Eq. 共43兲 are fully
expressed in the T-invariant fermion operators in the
particle-hole channel. When gc, g␯ 艌 0, i.e.,
− 3/5U0 艌 U2 艌 U0 ,
共40兲
Z=
=
共41兲
where ␰aij is a rank-two, symmetric, traceless, tensor given in
Eq. 共29兲 and discussed more explicitly in Ref. 15. The ten
antisymmetric tensor ⌫ab matrices contain both the three linear 共rank 1兲 Si and seven cubic, symmetric, traceless 共rank 3兲
combinations of SiS jSk operators, and they correspond to the
共45兲
the partition function can be expressed using the T-invariant
decomposition as
It is easy to check that the Si operators form the s = 23 representation of the SU共2兲 algebra. While the above equation
expresses the spin operators in terms of the ⌫ matrices of the
SO共5兲 algebra, the reverse can also be accomplished. The
five ⌫a matrices can actually be expressed in terms of the
quadratic forms of the spin matrices,
⌫a = ␰aij共SiS j + S jSi兲,
冎
gc = − 共3U0 + 5U2兲/8,
= 冑3共− L34 ± iL24兲 + 共L12 ± iL25兲 ⫿ i共L13 ± iL35兲,
Jz = − L23 + 2L15 .
再
共43兲
gc
g␯
关n共i兲 − 2兴2 + n2a共i兲 ,
2
2
where
Defining the time-reversal operator as T = RC, and using the
properties of the R matrix given in Eq. 共33兲, it can be shown
that n共i兲,na共i兲 are even while Lab共i兲 is odd under T,
TnT−1 = n,
兺i 共␺i,†␴␺ j,␴ + H.c.兲 − 兺i ␮␺i†␴␺i␴ ,
冕
0
冕 冕
冕
Dn
−
g␯
2
再冕
再 冕
冎
␤
D␺†D␺ exp −
␤
0
Dna exp −
d␶
n2a共i, ␶兲
兺
i,a
␤
gc
2
d␶␺␴†
␤
0
d␶
冉
冊
⳵
+ H ␺␴
⳵␶
冎
兺i 关n共i, ␶兲 − 2兴2
det兵I + B其.
共46兲
Again I + B = I + Te−兰0 d␶关HK+Hi共␶兲兴 is obtained from the integration of the fermion fields, n and na are real HS Bose fields.
155115-6
PHYSICAL REVIEW B 71, 155115 共2005兲
SUFFICIENT CONDITION FOR ABSENCE OF THE…
†
ci共−1/2兲 → 共− 兲ici共−1/2兲
,
†
ci共−3/2兲 → 共− 兲ici共−3/2兲
,
共49兲
while keeping the ci共3/2兲, ci共1/2兲 operators unchanged. The
kinetic-energy part is invariant under the above transformation, while the interaction part is changed into Hint
2
= 2U兺iL15
共i兲. It can be decomposed using the imaginary
number as
Z=
冕
再
DQ exp − 2U
冕
␤
冎
d␶Q2共i兲 det兵I + B其,
0
␤
FIG. 2. Within the method of T-invariant decomposition, the
shaded area marks the parameter region without the sign problem in
3
the s = 2 Hubbard model at any doping level and lattice geometry.
The fermion determinant can only be factorized along the SU共4兲
line with U0 = U2 ⬍ 0, where the traditional algorithms can be applied without the sign problem. The sign problem along the line
with U0 = U2 ⬎ 0 only disappears at the half filling and on a bipartite
lattice.
with
H i共 ␶ 兲
where
B = Te−兰0 d␶HK+Hi共␶兲
†
15
= iU兺i␺i 共␶兲⌫ ␺i共␶兲Q共i , ␶兲. Because T共iL15兲T−1 = iL15, the
det共I + B兲 is positive definite. However, we did not succeed in
generalizing this at negative values of gc or g␯ away from the
SU共4兲 line at half filling.
In practice, it is more efficient to sample with discrete HS
transformation using two Ising-like fields ␩, s for each quartic fermion term as in Ref. 16 instead of using the continuous
HS boson field. For any bilinear fermionic operator O共i兲, the
decomposition below has the numerical precision at the order of O共⌬␶兲4 as
H I共 ␶ 兲 = − g c
− g␯
兺
i,a
兺i ␺i,␴共␶兲␺i,␴共␶兲n共i, ␶兲
␺i,† ␴共␶兲⌫␴a ,␴⬘␺i,␴⬘共␶兲na共i, ␶兲.
We see that HI共␶兲 mixes the four spin components together, therefore the fermion determinant is factorizable if
and only if g␯ = 0, which is the SU共4兲 line with U0 = U2 ⬍ 0.
We define the time-reversal transformation T for the entire
lattice as T = 关兿i 丢 R共i兲兴C. From Eq. 共39兲 we see that both
terms are T-invariant,
T共HK + HI兲T−1 = HK + HI ,
2
兺 l es␩ 冑⌬␶gO共i,␶兲 + O共⌬␶4兲,
l,s=±1 4
ˆ
2
兺 l eis␩ 冑⌬␶gO共i,␶兲 + O共⌬␶4兲,
l,s=±1 4
e−g⌬␶O共i, ␶兲 =
共47兲
ˆ
l
␥
l
ˆ
where g ⬎ 0 and ␥l = 1 + 共冑6 / 3兲l, ␩l = 冑2共3 − 冑6l兲. The above
proof for the positive definite of det共I + B兲 applies equally
well in this scheme.
We only used the time-reversal properties of the SO共5兲
algebra in the above proof; the exact SO共5兲 symmetry is
useful for transforming the model expressed in the particleparticle channel to the particle-hole channel, but it is not
essential. A general anisotropic spin- 23 lattice model is defined by
H=−
共48兲
and all other conditions of our theorem are met. Therefore,
the minus sign is absent as long as Eq. 共45兲 is satisfied. This
is a much broader parameter range 共shown in Fig. 2兲, compared to the conventional factorizibility condition U0 = U2
⬍ 0. Our algorithm therefore enables us to study the s = 23
Hubbard model away from the SU共4兲 line. Our proof is valid
for any filling level and lattice topology. In this parameter
range, it is shown in Ref. 8 that a number of interesting
competing orders such as the staggered order of na, singlet
superconductivity, and the charge-density wave can exist
there.
At half filling and on a bipartite lattice where ␮ = 0, the
sign problem also disappears along the SU共4兲 line at U
= U0 = U2 ⬎ 0. Similar to the spin- 21 case, after performing a
partial particle-hole transformation,
␥
ˆ
eg⌬␶O共i, ␶兲 =
The time-dependent interaction HI共␶兲 after the HS transformation is
a
␺ j␤ + H.c.其
兵t␺i†␣␺ j␣ + ta␺i†␣⌫␣␤
兺
具ij,a典
+
兵hana共i兲 − ␮n共i兲其 + 兺
兺
i,a
i,a⬍b
−
ga 2
gab 2
na共i兲 +
L 共i兲 ,
2
2 ab
冎
再
−
gc
关n共i兲 − 2兴2
2
共50兲
where ta is the spin-dependent hopping amplitude, ha is the
analogy of the Zeeman field coupling to the na共i兲 field, and
ga and gab are coupling constants in corresponding channels.
When gc,ga,gab are arbitrary positive-interaction parameters,
we can perform the same decomposition process as before.
By using the fact that T共iLab兲T−1 = iLab, we again reach the
positive definite fermion determinant. This conclusion also
holds for any valid representation of ⌫ matrices, with the
redefined n共i兲, na共i兲, Lab共i兲 and time-reversal operations accordingly.
155115-7
PHYSICAL REVIEW B 71, 155115 共2005兲
C. WU AND S.-C. ZHANG
C. General higher-spin Hubbard models
spin- 23
We can generalize the results in the
case to any
fermionic system with spin s = n − 21 . The spin s = n − 21 Hubbard model can be written as
H=−t
+
兵ci†␴c j␴ + H.c.其 − 共␮ + ␮0兲 兺 ci†␴ci␴
兺
具ij典,␴
i␴
†
UJ PJJ
共i兲PJJ 共i兲,
兺
i,J,J
z
z
共51兲
z
where J = 0 , 2 , . . . , 2n − 2 are the total spin of the particle†
particle pairs, Jz = 0 , ± 1 , . . . , ± J. The pairing operators PJ,J
z
are defined through the Clebsch-Gordan coefficient for two
indistinguishable particles as
†
PJ,J
共i兲 =
z
具J,Jz兩s,s; ␣␤典c␣† 共i兲c␤† 共i兲.
兺
␣␤
共52兲
The total spin of the particle-particle pair takes only even
integer values so that the Pauli principle is satisfied on every
site. At half filling and on a bipartite lattice, ␮ = 0 ensures the
particle-hole symmetry, and ␮0 = 1 / 共2n兲兺J共2J + 1兲UJ.
The general strategy to implement the method of
T-invariant decomposition is to first transform the interaction
terms originally expressed in the particle-particle channel to
the particle-hole channel. In this case, we have the fermion
operators ␺i,␤ within one unit cell, where ␤ = 1 , . . . , 共2s + 1兲.
Therefore, there are 共2s + 1兲2 fermion bilinears of the form
I
I
␺i,␤, where I = 1 , . . . , 共2s + 1兲2. The 共2s + 1兲2 M ␣␤
M I = ␺i,† ␣M ␣␤
matrices can in general be expressed in a complete basis in
terms of the product of spin s matrices Si,
In the following, we shall illustrate this general procedure
more explicitly for a special case of the higher-spin Hubbard
model where
U2 = U4 = ¯ = U2n−2 ⬅ U⬘ .
The generic higher-spin Hubbard model only has the spin
SU共2兲 symmetry for s ⫽ 23 . However, under the above condition, the higher-spin Hubbard has the Sp共2n兲 symmetry.
When an additional condition, namely, U0 = U⬘, is imposed,
the model has a larger, SU共2n兲 symmetry. In the Appendix,
an introduction to the Sp共2n兲 algebra is given. As shown
there, the singlet-pairing operator is also the singlet of the
Sp共2n兲 group, while all other 2n2 − n − 1 pairing operators
with J = 2 , 4 , . . . , 2n − 2 together form a representation for the
Sp共2n兲 group. Thus we conclude that Eq. 共51兲 is
Sp共2n兲-symmetric if and only if the coupling constants satisfy Eq. 共51兲. For n = 1 and 2, the Sp共2n兲 symmetry is generic
and does not need any fine tuning. Actually, Sp共2兲 is isomorphic to SU共2兲, while Sp共4兲 is isomorphic to SO共5兲. This is
consistent with our earlier finding that the s = 23 Hubbard
model has the SO共5兲 or the Sp共4兲 symmetry without any
conditions on the parameters.8
To show the Sp共2n兲 symmetry explicitly, we can rewrite
the Hamiltonian in Eq. 共51兲 as
H=−t
␰aijSiS j,
+ c0
a = 1, . . . ,5,
...
=
␰iL2,i1,. . .iJ ,
共54兲
or any other permutation of indices, and
␰iL1,i1,. . .iJ = 0.
共2s + 1兲2 = 1 + 3 + 5 + ¯ + 共4s + 1兲.
U0 艋 U⬘ 艋 −
共56兲
According to the method of T-invariant decomposition, any
negative interaction terms in the even-spin channel like 1,
␰aijSiS j , . . . or any positive interaction terms in the odd-spin
channel like Si, ␰LijkSiS jSk , . . . can be simulated by our algorithm without the sign problem.
共58兲
n+1
U0 .
2n − n − 1
2
共59兲
Because the Y a 共1 艋 a 艋 2n2 − n − 1兲 are even under timereversal transformation while the spin operators Si共i
= x , y , z兲 are odd, Y a can be expanded in the basis of Eq. 共53兲,
including all terms with even powers of spin matrix.
共55兲
Spherical harmonics can be used to explicitly construct these
tensors.17 This decomposition is obviously complete, since
U0 − U⬘
.
c2 =
2n
The expression for Y a 共1 艋 a 艋 2n2 − n − 1兲 is given in the Appendix, where it is also shown that they are even under the
time-reversal transformation. By the same reasoning before,
we perform the HS decoupling in the above two channels.
Then the sign problem is absent when both c1 and c2 are
negative, i.e.,
L = 1, . . . ,共4s + 1兲,
where the ␰’s are fully symmetric, traceless tensors, satisfying
␰iL1,i2,. . .iJ
a
␺i,␤兲2 ,
兺i 共␺i,†␣␺i,␣ − n兲2 + c2兺i 共␺i,†␣Y ␣␤
n+1
2n2 − n − 1
c0 =
U ⬘,
2 U0 +
4n
4n2
共53兲
␰iL1,i2,. . .iJSi1Si2 ¯ SiJ,
兺 兵␺i,†␣␺ j,␣ + H.c.其 − ␮ 兺i ␺i,†␣␺i,␣
具ij典,␴
1,
Si, i = 1,2,3,
共57兲
1
V. BILAYER s = 2 MODELS
The spin- 23 Hubbard model has a close relationship with a
bilayer model introduced by Scalapino, Zhang, and Hanke
共SZH兲.11 This model was constructed and extensively investigated because of the exact particle-particle channel SO共5兲
symmetry between the antiferromagnetism and superconductivity when the coupling constants satisfy a simple
relation.18–21 The original SZH model was introduced on a
two-leg ladder 共Fig. 3兲, and it is straightforward to generalize
it to a bilayer system as
155115-8
PHYSICAL REVIEW B 71, 155115 共2005兲
SUFFICIENT CONDITION FOR ABSENCE OF THE…
FIG. 3. The SZH model defined on a two-leg ladder segment of
1
the double-layer spin- 2 system.
H = − t储
兵ci†␴c j␴ + di†␴d j,␴ + H.c.其 − t⬜ 兺 兵ci,† ␴di,␴ + H.c.其
兺
具ij典
i
−␮
冋
1
+ n↑,d共i兲 −
2
+V
再冋
册冎
兺i 兵nc共i兲 + nd共i兲其 + U 兺i
册冋
1
n↓,d共i兲 −
2
n↑,c共i兲 −
1
2
册冋
兺i „nc共i兲 − 1…„nd共i兲 − 1… + J 兺i Sជ i,c · Sជ i,d ,
n↓,c共i兲 −
1
2
册
共60兲
employ the method of T-invariant decomposition, we adopt
the view from the SO共5兲 symmetry in the particle-hole channel in this section.
There are four single fermion states per unit cell in both
the s = 23 Hubbard model and the s = 21 bilayer model. A mapping between them can be established through ␺i
= 共ci,3/2 , ci,1/2 , ci,−1/2 , ci,−3/2兲T ↔ 共ci,↑ , ci,↓ , di,↑ , di,↓兲T. We denote
the time-reversal operator defined in Eq. 共34兲 for the s = 23
system as T1, and the usual definition for the s = 21 system as
T2. T1 actually is the combined operation of T2 and the interchange between the upper and lower layers,
T1 =
␮ = 0,
t⬜ = 0,
I 0
共63兲
T2 .
n共i兲 = ci†␴ci␴ + di†␴di␴ ,
n1共i兲 = − i共di†␴ci␴ − H.c.兲/2,
n5共i兲 = 共di†␴ci␴ + H.c.兲/2,
冉冊 冉冊
冉冊
冋 冉冊
册
冉冊 冉冊
n2,3,4共i兲 = ci†␣
共61兲
which unifies the antiferromagnetism with the
superconductivity.11 Remarkably, there exists another exact
SO共5兲 symmetry in the particle-hole channel when
J = 4共U − V兲,
0 I
The 16 p - h channel fermionic bilinear forms are mapped
onto
where c† and d† are creation operators in the upper and lower
layers, respectively, t储 and t⬜ are the hopping amplitudes in
the layer and across the rung, respectively, U is the on-site
interaction, and V and J are the charge and Heisenberg exchange interactions across the rung, respectively. The SZH
model is known to have an exact SO共5兲 symmetry when
J = 4共U + V兲,
冉 冊
ជ 共i兲 =
Re ␲
共62兲
␴ជ
2
␣␤
ci†␣
ជ 共i兲 = − i ci†␣
Im ␲
and the symmetry is valid for all filling factors. We denote
the former particle-particle SO共5兲 symmetry as SO共5兲 pp and
the later particle-hole 共p-h兲 SO共5兲 symmetry as SO共5兲 ph. The
two SO共5兲 symmetric lines are shown in Fig. 4. In order to
␴ជ
Sជ i = ci†␣
2
␣␤
ci␤ − di†␣
␴ជ
2
␣␤
␴ជ
2
␴ជ
2
␣␤
d i␤ ,
共64兲
di␤ + H.c.,
␣␤
ci␤ − di†␣
di␤ − H.c. ,
␴ជ
2
␣␤
d i␤ ,
Q = 关nc共i兲 − nd共i兲兴/2,
where n1,5 are the singlet rung-current and rung-bond order
ជ and Re ␲
ជ are their triplet
parameters, respectively, and Im ␲
counterparts; n2,3,4 are the rung-Néel order parameter, Sជ is
the total rung spin, and Q is the charge-density wave order
parameter. n,n1⬃5 are even under the T1 transformation, and
ជ , and Q are even
the others are odd. In contrast, n, n5, Im ␲
under the usual definition, and T2 and the others are odd.
The general SZH model can be mapped into an anisotropic SO共5兲 model in the following form:
H = − t储
+
FIG. 4. Two SO共5兲 lines are shown in the SZH model as well as
in the QMC region without the sign problem for any filling 共hashed
area兲: g␯1 ⬎ 0, g␯2 ⬎ 0, and gc ⬎ 0. There is another region with V
⬍ 0 共not shown兲.
5
␺i†␣␺ j␣ + t⬜ 兺 ␺i†␣⌫␣␤
␺ i␤ − ␮ 兺 n i
兺
i
具ij典
i
兺i
再
−
gc
g ␯1 2
g ␯2
关n1共i兲 + n25共i兲兴 −
关n共i兲 − 2兴2 −
2
2
2
冎
⫻关n22共i兲 + n23共i兲 + n24共i兲兴 ,
with
155115-9
共65兲
PHYSICAL REVIEW B 71, 155115 共2005兲
C. WU AND S.-C. ZHANG
other in two dimensions. To the best of our knowledge, this
is the first time a current-carrying ground state has been conclusively demonstrated in a two-dimensional 共2D兲 system.
The mapping between the SZH model and the s = 23 model
is not unique. More generally the SZH model can be written
as
H = − t储
+
FIG. 5. 共a兲 Sketch of a staggered interlayer current phase from
Ref. 10. For clarity, we do not show the bottom layer current. 共b兲
Top view of the bilayer current. 共c兲 Sketch of the D-density wavecurrent pattern for comparison.
3
4gc = J − U − 3V,
4
兺i
再
5
␺i†␣␺ j␣ + t⬜ 兺 ␺i†␣⌫␣␤
␺ i␤ − ␮ 兺 n i ,
兺
i
具ij典
i
共68兲
−
gc
g ␯1 2
g ␯2
关n共i兲 − 2兴2 −
关n1共i兲 + n25共i兲兴 −
2
2
2
⫻关n22共i兲 + n23共i兲 + n24共i兲兴
+
3
4g␯1 = J − U + V,
4
gt2
gt3
gt1 2
ជ 共i兲 · Re ␲
ជ 共i兲
Q 共i兲 + 关Sជ 共i兲 · Sជ 共i兲兴 + 关Re ␲
2
2
2
冎
ជ 共i兲 · Im ␲
ជ 共i兲兴 .
+ Im ␲
共66兲
J
4g␯2 = + U − V.
4
The particle-hole channel SO共5兲 ph symmetry is restored at
g␯1 = g␯2 and t⬜ = 0, i.e., when the conditions of Eq. 共62兲 are
satisfied. At this point, the SZH model expressed in Eq. 共65兲
takes exactly the same form as the s = 23 Hubbard model expressed in Eq. 共43兲. The equivalence between the two models
is therefore rigorously established.
For the general SZH model, the interactions can be expressed purely in terms of the fermion bilinears which are
invariant under the T1 transformation from Eq. 共65兲, by virtue of Eq. 共39兲. We perform the T1-invariant decomposition
of the interactions in the region of gc , g␯1 , g␯2 艌 0, i.e.,
1
3
J + V 艌 U 艌 − J + V,
4
4
共67兲
3
J 艌 U + 3V
4
as shown in Fig. 4, and then the sign problem is absent. In
this region with positive g␯1,2 and gc, we expect the competing orders of the five-vector channel and the superconductivity, i.e., the antiferromagnetism, the staggered current, and
the rung-singlet superconductivity, which can be investigated
systematically with high numerical accuracy, at and away
from the half filling.
The above algorithm has been applied to demonstrate the
existence of the two-dimensional staggered-current phase
conclusively at half filling with t储 = 1 , t⬜ = 0.1, U = 0 , V
= 0.5, J = 2.10 The current pattern is illustrated in Fig. 5 with
staggered interlayer currents 共SIC兲 between the bilayers and
the alternating source to drain currents within the bilayers.
Viewed from the top, this current pattern has a s-wave symmetry. While the D-density wave22 currents are divergencefree within the layer, the SIC current is curl-free within the
layer. These two patterns can be considered as dual to each
Only three out of the six coupling constants are independent,
as shown here in the correspondence to the U , V , J parameters,
U = − 4gc + 3g␯2 + gt1 − 3gt2 ,
V = − 4gc + g␯1 − gt1 − 3gt3 ,
共69兲
J = 4共g␯1 + g␯2 + gt2 + gt3兲.
If the gt1 , gt2 , gt3 are set to zero, it returns to Eq. 共65兲. For
any given values for U , V , J, if we can find a set of values of
gc , g␯1 , g␯2 , gt1 , gt2 , gt3 艌 0, then we can perform the HS
transformation, keeping the invariance under the T1 operation, and arrive at the absence of the sign problem regardless
of the doping and lattice topology. This general decoupling
scheme extends the valid parameter region in Eq. 共67兲. On
the other hand, we can also consider performing the HS decoupling with the invariance under the usual definition of the
time-reversal operation T2. After setting g␯1 , gt3 = 0, we have
the condition that gc 艌 0 , g␯2 艋 0 , gt1 艋 0 , gt2 艌 0. This
decoupling-scheme-based T1 also enlarges the region of Eq.
共67兲. For example, the usual bilayer negative U Hubbard
model with U ⬍ 0 , V = J = 0 is out of that region. Nevertheless, we can still show the absence of the sign problem by
setting gc = −gt1 / 4 = −U / 2 ⬎ 0 and all other parameters to
zero.
In contrast, the conventional algorithm based on the factorization of the fermion determinant works only at either
g␯1 = g␯2 = 0, gc ⬍ 0 or the usual negative U Hubbard model
U ⬍ 0 , V = J = 0. This parameter set is included in the above
HS decomposition schemes, respecting either the T2 or the T1
time-reversal symmetry. We therefore see the significant improvement provided by the method of the T-invariant decomposition.
VI. MODELS WITH BOND INTERACTIONS
So far, the models we considered only have on-site interactions. In this section, we will generalize them to include
155115-10
PHYSICAL REVIEW B 71, 155115 共2005兲
SUFFICIENT CONDITION FOR ABSENCE OF THE…
interactions defined on the bond. Such models can have
many exotic phases.
We first consider the following general single-layer spin- 21
Hamiltonian with bond interactions:
H=−t
+
共ci†␴c j␴ + H.c.兲 − ␮ 兺 n共i兲 + 兺
兺
具ij典,␴
i
具ij典
再
gsbd
M ij M ij
2
−
冎
gscur
gtbd ជ
ជ − gtcur N
ជ ·N
ជ ,
NijNij +
M ij · M
ij
ij
ij
2
2
2
冉冊
ជ = c† ␴ជ
M
ij
i␣
2
␣␤
再 冉冊
ជ = i c† ␴ជ
N
ij
i␣
2
c j␤ + H.c.,
M ij = ci†␴c j␴ + H.c.,
␣␤
共70兲
冎
c j␤ − H.c. ,
Nij = i兵ci†␴c j␴ − H.c.其,
where M ij and Nij are the singlet bond and current operators
ជ and N
ជ are their triplet counterparts,
on the bond 具ij典, M
ij
ij
and gsbd, gscur, gtbd, and gtcur are the coupling constants in the
corresponding channels. The sites i and j forming the bond
具ij典 are not necessarily nearest neighbors, but can be at an
arbitrary distance apart. Under the time-reversal transformaជ are even while M
ជ and N are odd.
tion T, M ij, and N
ij
ij
ij
These four interactions are not independent, and can be
reorganized into
Hint =
兵−
兺
具ij典
† †
Jc共ci↑
ci↓c c
j↓ j↑
FIG. 6. Four possible density-wave phases can be simulated
without a sign problem. 共a兲 singlet spin-Peierls 共p-density-wave兲,
共b兲 Triplet dx2−y2 density-wave, 共c兲 singlet dxy density-wave, 共d兲
triplet diagonal-current.
H I共 ␶ 兲 = −
+ H.c.兲 + V关n共i兲 − 1兴关n共j兲 − 1兴
+i
+ JsSជ 共i兲 · Sជ 共j兲其,
⫻ ci†␣
Jc = 2共gsbd + gscur兲 + 3共gtbd + gtcur兲,
+
gsbd − gscur 3
− 共gtbd − gtcur兲,
4
2
Js = 2共gsbd − gscur兲 + 共gtbd − gtcur兲.
The Jc term is the pair hopping, V is the charge interaction
between sites i and j, and Js is the Heisenberg exchange.
When all of gsbd , gscur , gtbd , gtcur are positive, we perform the
HS decomposition in each channel, respectively, as
Z=
冕
再冕
ជ DNDN
ជ exp −
DMDM
␤
0
d␶
gsbdM 2ij共␶兲
兺
具ij典
ជ 2 共␶兲 + g N
ជ2 2
+ gscurN2ij共␶兲 + gtbdM
tcur ij共␶兲
ij
⫻det兵I + B其,
␤
冎
共72兲
where I + B = I + Te−兰0 d␶HK+HI共␶兲. HI共␶兲 after the HS decoupling
is given by
ជ 共␶兲
gscurNij共␶兲i共ci,† ␴c j,␴ − H.c.兲 − i 兺 gtbdM
兺
ij
具ij典
具ij典
冋 冉冊
共71兲
V=
gsbdM ij共␶兲共ci,† ␴c j,␴ + H.c.兲
兺
具ij典
␴ជ
2
␣␤
册
c j␤ + H.c.
ជ 共␶兲i
gtcurN
兺
ij
具ij典
冋 冉冊
ci,† ␣
␴ជ
2
␣␤
册
c j,␤ − H.c. .
共73兲
Therefore, HI and I + B are even under the time-reversal
transformation and the sign problem is absent.
The valid parameter region for the above algorithm is
very general as long as all gsbd , gscur , gtbd , gtcur 艌 0. As a result, V and J can be either positive or negative while Jc has to
be positive. Many interesting competing orders are supported
in this parameter region. For example, various density-wave
states exist on a square lattice near half filling,23 as shown in
Fig. 6. With gsbd , gtcur ⬎ 0, the above algorithm provides a
good opportunity to study the singlet bond and the triplet
ជ , while it is
current order parameters formed by M ij and N
ij
not good for studying the singlet current and the triplet bond
ជ , because g , g
order parameters formed by Nij and M
ij
tbd scur
⬎ 0. After setting gtbd = gscur = 0 for the bond 具ij典 connecting
the nearest sites, the gsbd term favors the p-density wave
共spin-Peierls兲 phase, and the gtcur term favors the triplet
channel dx2−y2-density wave. The latter order is recently proposed as the origin of the pseudogap in the high Tc
cuprates.24 For the bond interaction between the next nearest
155115-11
PHYSICAL REVIEW B 71, 155115 共2005兲
C. WU AND S.-C. ZHANG
†
M ab
ij = ␺i
FIG. 7. The Fermi surface instability in the Fa1 channel, with
dashed lines marking the Fermi surface before symmetry breaking.
In the ␣ phase, the anisotropic Fermi surface distortion appears for
two spin components. In the ␤ phase, the spin-orbital coupling is
generated dynamically and two Fermi surfaces are characterized by
helicity.
bond, i.e., the diagonal bond, the gsbd term leads to the singlet dxy order, and the gtcur term leads to the triplet diagonalcurrent order. The triplet diagonal-current phase was studied
in the two-leg ladder system using the bosonization method
in Ref. 25 and also under the name of the triplet F-density
wave in Ref. 26.
When the Fermi surface nesting effect is not important
either at large doping or in the nonbipartite lattice, the gtcur
term can lead to the Fa1 channel of the Landau-Pomeranchuk
instability on the Fermi surface, which was studied recently
in the continuum model in Ref. 27. After the symmetry
breaking, two possible phases are named as ␣ and ␤ phases
in analogy to the A and B phases in the triplet p-wave channel superfluid phase in 3He, as shown in Fig. 7. The ␣ phase
was studied by Hirsch28,29 under the name of spin-split phase
on the lattice system with an opposite anisotropic Fermi surface distortion for two-spin components. In contrast, the
Fermi surface distortion is isotropic and a spin-orbit coupling
is dynamically generated in the ␤ phase. The two singleparticle bands are characterized by the helicities. It would be
interesting to study these exotic phases in our version of the
QMC algorithm free of the sign problems.
Bond interactions can also be added into the spin- 23 Hubbard model of Eq. 共35兲 as
Hbond =
兺
具ij典
+
再
−
gsbd
gscur
M ij M ij +
NijNij +
2
2
兺a −
gvbd a a
M ij M ij
2
冎
gtbd ab ab gtcur ab ab
gvsur
M ij M ij −
NijNij +
N N
,
2
2 ij ij
a⬍b 2
兺
M ij = ␺†i ␺ j + H.c.,
M aij = ␺†i
⌫a
␺ j + H.c.,
2
Nij = i兵␺†i ␺ j − H.c.其,
再
Naij = i ␺†i
⌫ab
␺ j + H.c.,
2
再
†
Nab
ij = i ␺i
冎
⌫ab
␺ j − H.c. ,
2
where M ij and Nij are the singlet bond and current operators
ab
on the bond 具ij典,M aij,Naij,M ab
ij ,Nij are their five-vector and tentensor
channel
counterparts,
respectively,
and
gsbd , gvbd , gtbd , gscur , gvcur , gtcur are the coupling constants in
the corresponding channels. Again the sites i and j forming
the bond 具ij典 can be at an arbitrary distance apart. The bond
interactions can be decoupled by introducing the HS field in
each channel respectively. Following the same reasoning as
in the case of the spin- 21 , the bond-interactions keep the fermion determinant positive definite, provided all of these coupling constants are nonnegative. Similarly, with
gsbd , gvbd , gtcur ⬎ 0, the algorithm can be applied to study the
singlet, quintet bond orders and the tenfold current order,
while with gscur , gvcur , gtbd ⬎ 0, it is not useful for study applied to the singlet, quintet current orders and the tenfold
bond order.
VII. CONCLUSION
The sign problem of the fermionic QMC algorithm is one
of the most important problems in theoretical physics. Its
solution would practically give a universal computational
method to solve models with strong correlations. The rigorous theorem established in this work shows that the minus
sign problem can be eliminated for a much wider class of
models than before, in which the fermion matrix is invariant
under an antiunitary symmetry similar to the time-reversal
symmetry in quantum mechanics. The method of T-invariant
decomposition does not only provide a deep connection between the sign problem and the time-reversal symmetry, it
also leads to practical algorithms which can be applied to
many interesting models with strong correlations. Using this
algorithm, a class of models with strong correlations can be
simulated, and some exotic ground states have been firmly
established.
We conclude this paper with an optimistic outlook. Even
though our method can only be applied presently to models
with definite constraints among the interaction parameters,
we believe that the deep symmetry connections revealed in
this paper could guide us in future works and might eventually lead to the complete elimination of the sign problem.
Note added in proof. Recently, we learned that a similar
version of the theorem of T-invariant decomposition had
been discussed in the context of lattice gauge theory.32 However, they did not consider the case that I + B is not diagonalizable. Our proof is valid regardless of whether I + B is diagonalizable or not, thus is more complete.
共74兲
ACKNOWLEDGMENTS
冎
⌫a
␺ j − H.c. ,
2
共75兲
We thank Dr. B. A. Bernevig, Dr. S. Capponi, Dr. D.
Ceperley, Dr. S. Chandrasekharan, Dr. J. P. Hu, Dr. D. Scalapino, and Dr. T. Xiang for helpful discussions. This work is
supported by the NSF under Grant Nos. DMR-0342832 and
the U.S. Department of Energy, Office of Basic Energy Sciences under Contract No. DE-AC03-76SF00515. C.W. is
155115-12
PHYSICAL REVIEW B 71, 155115 共2005兲
SUFFICIENT CONDITION FOR ABSENCE OF THE…
also supported by the Stanford Graduate Fellowship program.
1
APPENDIX: Sp„2n… ALGEBRA IN THE SPIN s = n − 2
FERMION SYSTEM
We give a brief introduction to the Sp共2n兲 algebra here.
The 2n-dimensional Hilbert space on each site can be arranged as a direct product between an n-dimensional and a
two-dimensional space. The complete basis of the eigenstates
of Sz are labeled in the sequence of 兩1典 = 兩n − 21 典, 兩2典 = 兩−n
+ 23 典 , . . . , 兩n典 = 兩共−兲n−1 / 2典,
and
兩1̄典 = 兩−n + 21 典,
兩2̄典 = 兩n
− 23 典 , . . . , 兩n̄典 = 兩共−兲n / 2典. The Sp共2n兲 spinor is defined as
␺ = 共cn−1/2,c−n+1/2,c−n+3/2,cn−3/2, . . . 兲T .
共A1兲
Group elements of Sp共2n兲 include any 2n ⫻ 2n unitary matrix U satisfying UTRU = R or, equivalently, R−1UR = U* 共Ref.
30兲 with the R matrix,
R = In 丢 共− i␴2兲.
共A2兲
The R matrix is a straightforward generalization of the
R = −i␴2 in the spin- 21 case, which also satisfies RT = R−1
= R† = −R. Clearly, the Sp共2n兲 group is a subgroup of the
SU共2n兲 group defined in the 2n-dimensional space.
In the particle-hole channel, there are 4n2 independent
bilinear operators as ␺␣† ␺␤ 共␣ = 1 , . . . , 2n , ␤ = 1 , . . . , 2n兲.
Among them, the particle-density operator n = ␺␣† ␺␣ is a singlet under both the SU共2n兲 and the Sp共2n兲 group. The timereversal transformation T = CR is defined as usual, and it satisfies T2 = −1. The other 4n2 − 1 bilinear operators form the
generators 共adjoint representation兲 for the SU共2n兲 group.
They can be decomposed into two classes according to their
transformation properties under the T operation. The first
class contains n共2n + 1兲 elements which forms the generators
b
␺␤共b = 1 ⬃ 2n2 + n兲.
of the Sp共2n兲 group as denoted by ␺␣† X␣␤
b
X can be expressed in terms of a direct product between the
SU共n兲 and SU共2兲 generators. We define the SU共n兲 generators
as
1
关M 共1兲
ij 兴lk = 共␦il␦ jk + ␦ik␦ jl兲 共1 艋 i ⬍ j 艋 n兲,
2
关M 共2兲
ij 兴lk =
1 R.
−i
共␦il␦ jk − ␦ik␦ jl兲 共1 艋 i ⬍ j 艋 n兲,
2
共A3兲
Blankenbecler, D. J. Scalapino, and R. L. Sugar, Phys. Rev. D
24, 2278 共1981兲.
2 J. E. Hirsch, Phys. Rev. B 31, 4403 共1985兲.
3 S. Chandrasekharan and U. J. Wiese, Phys. Rev. Lett. 83, 3116
共1999兲.
4 S. E. Koonin, D. J. Dean, and K. Langanke, Phys. Rep. 278, 1
共1997兲.
5 S. Rombouts, K. Heyde, and N. Jachowicz, Phys. Rev. C 58,
M 共3兲
j =
diag共1, . . . ,1,− 共j − 1兲,0, . . . ,0兲
冑2j共j − 1兲
共2 艋 j 艋 n兲,
共2兲
共3兲
are the n ⫻ n-dimensional generalizawhere M 共1兲
ij ,M ij ,M j
tion of the SU共2兲 Pauli matrices ␴x,y,z, respectively. Counting
the numbers of SU共2n兲 generators, there are n共n − 1兲 / 2 real,
symmetric M 1ij’s, n共n − 1兲 / 2 imaginary, antisymmetric M 2ij’s,
and n − 1 real, diagonal M 3ij’s. Then the Sp共2n兲 generators Xb
can be expressed as
M 共2兲
ij 丢 I2,
M 共1兲
ij ’s
ជ,
M 共1兲
ij 丢 ␴
ជ,
M 共3兲
j 丢 ␴
M 共3兲
l ’s
ជ,
In 丢 ␴
共A4兲
M 共2兲
ij ’s
and
are real and
are purely
Because
imaginary, the Sp共2n兲 generators are odd under the time rea
versal: T−1XbT = −Xb. The second class bilinears ␺␣† Y ␣␤
␺␤
2
a
2
have 2n − n − 1 elements. Y 共a = 1 , . . . , 2n − n − 1兲 are given
by
M 共2兲
ij 丢 ␴i,
M 共1兲
j 丢 I 2,
M 共3兲
j 丢 I2 ,
共A5兲
which are even under the time reversal: T−1Y aT = Y a.
These 4n2 bilinear operators are not independent of each
other, but are related by the Fierz identity. The total Hilbert
space for one site has the dimension of 22n, which can be
decomposed into subspaces with different particle numbers
r共0 艋 r 艋 2n兲. Each of them form the totally antisymmetric
representation of the SU共2n兲 group 1r. The Casimir value of
the SU共2n兲 group in such representations is r共2n + 1兲共2n
− r兲 / 共2n兲. Thus we arrive at the Fierz identity for the spinn − 21 system as
b
a
␺i␤兲2 + 兺 共␺i†␣Y ␣␤
␺ i␤兲 2 +
兺b 共␺i†␣X␣␤
a
=
2n2 + n
.
2
2n + 1 †
共␺␣␺␣ − n兲2
2n
共A6兲
The on-site pairing operators can be easily formed by using the R matrix. Due to the Pauli’s exclusion principle, the
total spin for a s-wave pair can only be 0 , 2 , . . . , 2n − 2. The
singlet-pair operator is also the Sp共2n兲 singlet operator. It can
be written as ␺␣† R␣␤␺␤† , which was studied extensively in a
Sp共2n兲 generalization of the Heisenberg antiferromagnet.31
The other 2n2 − n − 1 pairing operators with a total spin of
2 , 4 , . . . , 2n − 2 together form a representation of Sp共2n兲 as
␺␣† 共i兲共RY a兲␣␤␺␤† 共i兲. When all the interaction parameters are
equal, these n共2n − 1兲 pairing operators together form an antisymmetric representation of SU共2n兲 of 1r 共r = 2兲.
3295 共1998兲.
Imada and T. Kashima, J. Phys. Soc. Jpn. 69, 2723 共2000兲.
7 T. Kashima and M. Imada, J. Phys. Soc. Jpn. 70, 2287 共2001兲.
8 C. J. Wu, J. P. Hu, and S. C. Zhang, Phys. Rev. Lett. 91, 186402
共2003兲.
9 F. F. Assaad, V. Rousseau, F. Hebert, M. Feldbacher, and G. G.
Batrouni, Europhys. Lett. 63, 569 共2003兲.
10 S. Capponi, C. J. Wu, and S. C. Zhang, Phys. Rev. B 70,
6 M.
155115-13
PHYSICAL REVIEW B 71, 155115 共2005兲
C. WU AND S.-C. ZHANG
220505共R兲 共2004兲.
Scalapino, S. C. Zhang, and W. Hanke, Phys. Rev. B 58, 443
共1998兲.
12
J. E. Hirsch, Phys. Rev. B 28, 4059 共1983兲.
13
Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev. Lett. 81,
3527 共1998兲.
14 E. Demler, W. Hanke, and S. C. Zhang, Rev. Mod. Phys. 76, 909
共2004兲.
15
S. Murakami, N. Nagaosa, and S. C. Zhang, Phys. Rev. B 69,
235206 共2004兲.
16 Y. Motome and M. Imada, J. Phys. Soc. Jpn. 66, 1872 共1997兲.
17
T. L. Ho and S. Yip, Phys. Rev. Lett. 82, 247 共1999兲.
18 P. Bouwknegt and K. Schoutens, Phys. Rev. Lett. 82, 2757
共1999兲.
19 D. Duffy, S. Haas, and E. Kim, Phys. Rev. B 58, R5932 共1998兲.
20 H. H. Lin, L. Balents, and M. P. A. Fisher, Phys. Rev. B 58, 1794
共1998兲.
11 D.
Frahm and M. Stahlsmeier, Phys. Rev. B 63, 125109 共2001兲.
Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys.
Rev. B 63, 094503 共2001兲.
23
C. Nayak, Phys. Rev. B 62, 4880 共2000兲.
24 W. V. Liu and F. Wilczek, cond-mat/0312685 共unpublished兲.
25
C. J. Wu, W. V. Liu, and E. Fradkin, Phys. Rev. B 68, 115104
共2003兲.
26
M. Tsuchiizu and A. Furusaki, Phys. Rev. B 66, 245106 共2002兲.
27
C. J. Wu and S. C. Zhang, Phys. Rev. Lett. 93, 036403 共2004兲.
28
J. E. Hirsch, Phys. Rev. B 42, 4774 共1990兲.
29 J. E. Hirsch, Phys. Rev. B 41, 6820 共1990兲.
30 M. Hamermesh, Group Theory and Its Application to Physical
Problems 共Dover, New York, 1989兲.
31 S. Sachdev and Z. Wang, Phys. Rev. B 43, 10 229 共1991兲.
32 S. Hands, I. Montvay, S. E. Morrison, M. Oevers, L. Scorzato,
and J. Skellurd, Eur. Phys. J. C 17, 285 共2000兲.
21 H.
22 S.
155115-14