Long-range charge order in the extended Holstein--Hubbard model

0.27 MB·

by  Tadahiro Miyao

Long-range charge order in the extended Holstein–Hubbard
model
Tadahiro Miyao
Department of Mathematics, Hokkaido University,
6 Sapporo 060-0810, Japan
1
E-mail: [email protected]
0
2

p
e Abstract
S
This study investigated the extended Holstein–Hubbard model at half-filling
2
1 as a model for describing the interplay of electron-electron and electron-phonon
couplings. When the electron-phonon and nearest-neighbor electron-electron in-

] teractions are strong, we prove the existence of long-range charge order in three
h
p or more dimensions at a sufficiently low temperature, As a result, we rigorously
- justify the phase competition between the antiferromagnetism and charge orders.
h
t
a
m 1 Introduction
[

Electron-phonon coupling plays an essential role in the electron-pairing mechanism in
3
v the Bardeen–Cooper–Schrieffer theory [1]. Recently, strong electron-phonon coupling
5 was observed in high-T cuprates [18] and strong electron-phonon interactions were
c
6
reported in alkali-doped fullerides and aromatic superconductors [3, 14, 16, 30, 33].
7
0 These examples suggest that electron-phonon coupling has received much attention in
0 the field of superconductivity.
.
1 In the presence of strong electron-electron Coulomb and electron-phonon inter-
0
actions, correlated electron systems provide an attractive field of study exhibiting a
6
1 competition among various phases. Despit the extensive research regarding the compe-
: tition between these phases, only few exact results are currently known. The Holstein–
v
i Hubbard model is a simple model that enables us the exploration of the interplay of
X
electron-electron and electron-phonon interactions. Our aim is to rigorously study the
r
a competition between the phases in the system described by the this model.
Rigorous study of the Holstein model was initiated by Lo¨wen [23]. Later, Freericks
and Lieb proved that the ground state of the Holstein model is unique and has a
total spin S = 0 [6]. However, their studies focused on electron-phonon interaction
only and did not consider the interplay between electron-electron and electron-phonon
interactions. Taking this interplay into account, Miyao proved the following [27]:
If the electron-phonon coupling is weak (U νV > 0), there is no long-range
eff
• −
charge order in the Holstein–Hubbard system at half-filling.
Iftheelectron-phononcouplingisweak,thegroundstateoftheHolstein–Hubbard
•
model is unique and exhibits antiferromagnetism.
1
More precise statements of these two principles are provided in Section 2. The achieve-
mentofthisstudyistheproofthatthereexistsalong-rangechargeorderatasufficiently
lowtemperatureprovidedthattheelectron-phononinteractionisstrong(U νV < 0).
eff
−
The obtained phase diagram is compatible with the previous results conjectured by
heuristic arguments [2, 29]. To prove the main result, we apply the method of reflec-
tion positivity.
Reflection positivity originates from axiomatic quantum field theory [31]. Glimm,
Jaffe and Spencer first applied reflection positivity to the study of phase transition
[10, 11]. This idea was further developed by Dyson, Fro¨hlich, Israel, Lieb, Simon and
Spencer in [4, 7, 8, 9] and applications of reflection positivity to the Hubbard model
are given in [13, 15, 19]. In the present study, we further develop the method used in
[8] to apply reflection positivity to the Holstein–Hubbard model which is more difficult
to analyze than the Hubbard model.
Usually, the hopping matrix elements of the Hubbard model are real numbers.
BecauseofpastsuccessesintheresearchofthephasetransitionsoftheHubbardmodel,
it appears that reflection positivity was inapplicable to the case where the hopping
matrixelementsarecomplexnumbers. InthestudyoftheHolstein–Hubbardmodel,the
Lang–Firsov transformation is known to be very useful. However, this transformation
changes the hopping matrix elements from real into complex numbers. Therefore, at
first glance, it appears that reflection positivity is unsuitable for the study of phase
transitions of the Holstein–Hubbard model. On the other hand, in a series of papers
[27, 28], Miyao has shown that reflection positivity is still applicable to several models
withcomplex hoppingmatrixelements (also see[26]).1 Inthepresentpaper,wefurther
extend this idea and adapt reflection positivity to a rigorous analysis of the phase
transitions of the Holstein–Hubbard model.
Notethatanapplicationofreflectionpositivity totheHubbardmodelwithcomplex
hopping matrix elements was first discussed by Lieb [20] in his solution of the flux-
phase conjecture (also see [12, 21, 24, 28]). Our present paper aims to apply reflection
positivity to the study of phase transitions of a model of interacting electrons with
complex hopping matrix elements.
The rest of the paper is organized as follows: In Section 2, we define the Holstein–
Hubbard model and state the main results. We also compare the obtained results with
those of previous studies as well. Section 3is devoted to theproof of the main theorem.
InAppendixA,weshowthatoursystemishalf-filledwithelectrons; inAppendixB,we
give an extension of the Dyson–Lieb–Simon inequality; and in Appendix C, we prove
a useful inequality. Appendix D is devoted to construct an antiunitary transformation
which plays an important role in Section 3.
Acknowledgments. This work was partially supported by KAKENHI (20554421)
and KAKENHI(16H03942). I would be grateful to the anonymous referees for useful
comments.
1Namely,heappliedthespinreflectionpositivitytotheHolstein–HubbardandSu–Schrieffer–Heeger
models and investigated their ground state properties.
2
2 Main results
Let Λ= [ L,L)ν Zν. The extended Holstein–Hubbard model on Λ is given by
− ∩
H = ( t)(c∗ c +c∗ c )
Λ − xσ yσ yσ xσ
hx;yiσ=↑,↓
X X
+U (n 1l)2+V (n 1l)(n 1l)
x x y
− − −
x∈Λ hx;yi
X X
+g (n 1l)(b +b∗)+ω b∗b , (2.1)
x− x x x x
x∈Λ x∈Λ
X X
where n = n +n with n = c∗ c . Here, x;y refers to a sum over nearest-
x x↑ x↓ xσ xσ xσ h i
neighbor pairs. We impose periodic boundary conditions, so L L. H acts in the
Λ
≡ −
Hilbert space according to
H = F P. (2.2)
⊗
The electrons exist in the fermionic Fock space F given by F := F (ℓ2(Λ) ℓ2(Λ)) :=
as ↑ ⊕ ↓
n(ℓ2(Λ) ℓ2(Λ)), where ℓ2(Λ) = ℓ2(Λ) = ℓ2(Λ), and n is the n-fold antisym-
n≥0∧ ↑ ⊕ ↓ ↑ ↓ ∧
metric tensor product. The phonons exist in the bosonic Fock space P defined by
L
P = nℓ2(Λ), where n is the n-fold symmetric tensor product, c is the elec-
n≥0⊗s ⊗s xσ
tronannihilationoperator,andb isthephononannihilationoperator. Theseoperators
x
L
satisfy the following relations:
c ,c∗ = δ δ , [b ,b∗ ] = δ . (2.3)
{ xσ x′σ′} σσ′ xx′ x x′ xx′
tisthehoppingmatrixelement,andgisthestrengthoftheelectron-phononinteraction.
Theon-site and nearest-neighbor repulsionsare denoted by U and V, respectively. The
phonons are assumed to be dispersionless with energy ω. Henceforth, we assume the
following:
g R, t > 0, U > 0, V >0, ω > 0.
• ∈
L is an odd number.
•
The thermal expectation value is defined by
A = Tr Ae−βHΛ Z , Z =Tr e−βHΛ . (2.4)
β,Λ β,Λ β,Λ
h i
.
We restrict ourselves to the ca(cid:2)se of half(cid:3)-filling. In fact, we s(cid:2)how th(cid:3)at
n = 1 (2.5)
x β,Λ
h i
in Appendix A. We let q = n 1l and define the two-point correlation function as2
x x
−
q q = lim q q . (2.6)
x o β x o β,Λ
h i L→∞h i
The effective interaction strength is defined as
2g2
U = U . (2.7)
eff
− ω
In [27], the following theorem is proven provided that νV U < 0.
eff
−
2In this paper, we simply assume that the right hand side of (2.6) exists. Alternatively, we choose
a subsequencesuch that theright hand side of (2.6) exists.
3
Theorem 2.1 [27] Suppose that νV U < 0. Then the following is obtained:
eff
−
(i) For all β 0, we have
≥
lim q q = 0. (2.8)
x o β
kxk→∞h i
Hence, there is no long-range charge order.
(ii) Let H be the M-subspace3 and let H = H ↾ H , the restriction of H to
M Λ,M Λ M Λ
H . The ground state of H is unique for all possible values of M.
M Λ,M
(iii) Let
S+ = c∗ c , S− = c∗ c . (2.10)
x x↑ x↓ x x↓ x↑
Let ϕ be the ground state of H . We obtain
M Λ,M
( 1)kxk ϕ S+S−ϕ >0 (2.11)
− h M| x o Mi
for all x Λ, where x = ν x . This means that the ground state is
∈ k k j=1| j|
antiferromagnetic.
P
It is logical and important to study the case where νV U > 0. Our main result
eff
−
in this paper is the following:
Theorem 2.2 Assume that νV U > 0. For each ν 3, we have
eff
− ≥
liminf( 1)kxk q q (2.12)
x o β
kxk→∞ − h i
1 β−1(νV U )−1ln4(1 e−βω)−1 8νt(νV U )−1 γ dpE(p)−1 γ ,
eff eff 1 2
≥ − − − − − − Tν −
Z
(2.13)
where T = ( π,π), E(p) = ν (1 cosp ) and
− j=1 − j
P
1 t 1/2 1 t 1/2
γ = (2π)−ν (βV)−1+ , γ = . (2.14)
1 2
2 V 4 V
n (cid:16) (cid:17) o (cid:16) (cid:17)
Corollary 2.3 Let ν 3. Assume that νV U > 0. If β,V,g are sufficiently large
eff
≥ −
such that the right-hand side of (2.13) is strictly positive, then we obtain
liminf( 1)kxk q q > 0. (2.15)
x o β
kxk→∞ − h i
Thus, a staggered long-range charge order exists.
3 To be precise, H is defined by
M
H ={ψ∈H|Nψ=|Λ|ψ, S3ψ=Mψ}, (2.9)
M
where N =N +N andS3 = 1(N −N ) with N =P n . Thecondition Nψ=|Λ|ψ indicates
↑ ↓ 2 ↑ ↓ σ x∈Λ xσ
that we consider thecase of half-filling.
4
3 Proof of Theorem 2.2
3.1 Lang–Firsov transformation
We set H = T +P +I +K, where
Λ
T = ( t) c∗ c +c∗ c , (3.1)
− xσ yσ yσ xσ
hx;yiσ=↑,↓
X X (cid:0) (cid:1)
P = U q2+V q q , (3.2)
x x y
x∈Λ hx;yi
X X
I = g q (b +b∗), (3.3)
x x x
x∈Λ
X
K = ω b∗b . (3.4)
x x
x∈Λ
X
For each x Λ, let
∈
1 ω
φ = (b∗ +b ), π = i (b∗ b ). (3.5)
x 2ω x x x 2 x− x
r r
Both φ and π are essentially self-adjoint and we denote their closures by the same
x x
symbols. Next, let
L = iω−3/2g q π . (3.6)
x x
−
x∈Λ
X
L is essentially antiself-adjoint. We also denote its closure by the same symbol. The
Lang-Firosov transformation is a unitary operator defined by
U = e−iπNp/2eL, (3.7)
where N = b∗b [17]. We can check the following:
p x∈Λ x x
P Uc U−1 = eiαφxc , α =√2ω−3/2g, (3.8)
xσ xσ
g
Ub U−1 = b q . (3.9)
x x x
− ω
Using these formulas, we obtain the following:
Lemma 3.1 Let H′ = UH U−1. We have
Λ Λ
H′ = T′+P′+K, (3.10)
Λ
where
T′ = ( t) e−iα(φx−φy)c∗ c +e+iα(φx−φy)c∗ c , (3.11)
− xσ yσ yσ xσ
hXx;yiσX=↑,↓ (cid:16) (cid:17)
P′ = U q2+V q q , (3.12)
eff x x y
x∈Λ hx;yi
X X
1
K = (π2 +ω2φ2). (3.13)
2 x x
x∈Λ
X
5
3.2 The Schr¨odinger representation
The bosonic Fock space can be identified as
P = L2( ,dµ ), (3.14)
Λ Λ
Q
where = RΛ and dµ = dφ is the Λ-dimensional Lebesgue measure. More-
QΛ Λ x∈Λ x | |
over, each φ can be regarded as a multiplication operator by the real-valued function,
x
Q
and π can be regarded as a partial differential operator i ∂ . This representation
x − ∂φx
of the canonical commutation relations is called the Schro¨dinger representation. In the
following section, we will focus on this representation.
3.3 The zigzag transformation
Following [8], we introduce the zigzag transformation as follows: Let
v = ( 1)nzσ (c∗ +c ). (3.15)
xσ − xσ xσ
" #
z6=x
Y
Note that v−1 = v . It is not hard to check that
xσ xσ
c∗ if (x,σ) = (x′,σ′)
v c v−1 = xσ . (3.16)
xσ x′σ′ xσ (cx′σ′ if (x,σ) = (x′,σ′)
6
Let Λ = x Λ x is even and let Λ = x Λ x is odd . Now, we set
e o
{ ∈ |k k } { ∈ |k k }
V = v v . (3.17)
x↑ x↓
xY∈Λo
We observe that
c∗ if x Λ
Vc V −1 = xσ ∈ o , Vq V−1 = ( 1)kxkq . (3.18)
xσ x x
(cxσ if x Λe −
∈
Lemma 3.2 Let H′′ = VH′ V −1. We have H′′ = T′′+P′′+K, where
Λ Λ Λ
ν
T′′ = ( t) e−iα(φx−φx+εδj)c∗ c∗ +h.c. , (3.19)
− xσ x+εδjσ
xX∈ΛeσX=↑,↓Xj=1εX=± (cid:16) (cid:17)
P′′ = U q2 V q q . (3.20)
eff x− x y
x∈Λ hx;yi
X X
Here, δ (j = 1,...,ν) is the unit vector in Zν defined by δ = (0,...,0, 1 ,0,...,0).
j j
j−th
Proof. T′ can be expressed as |{z}
ν
T′ = (−t) e−iα(φx−φx+εδj)c∗xσcx+εδjσ +h.c. . (3.21)
xX∈ΛeσX=↑,↓Xj=1εX=±1 (cid:16) (cid:17)
Thus, by using (3.18), we obtain (3.19), and similarly, (3.20). ✷
To show the main theorem, we introduce the following modified Hamiltonian:
6
Definition 3.3 For each h = h RΛ, we set
x x∈Λ
{ } ∈
V
P′′(h) = (U νV) q2+ (q h q +h )2 (3.22)
eff − x 2 x− x− y y
x∈Λ hx;yi
X X
and
H′′(h) = T′′+P′′(h)+K. (3.23)
Λ
Trivially, we have H′′ = H′′(0).
Λ Λ ♦
3.4 Reflection positivity
3.4.1 Overview
The hopping matrix elements in (3.19) are complex. In general, it is impossible to
apply reflection positivity (RP) to a fermionic system with complex hopping matrix
elements. However, a suitably modified RP can be still applicable to H′′ because these
Λ
complex phase factors (i.e., e−iα(φx−φx+εδj) ) are not random, but rather exhibit a
regular structure.
Next, webrieflyexplainthemodifiedRP.LetX andX becomplexHilbertspaces,
L R
and let ϑ bean antiunitary transformation from X onto X . In AppendixB, we prove
L R
the following:
Tr A ϑAϑ−1 = Tr [A] 2 0. (3.24)
XL⊗XR ⊗ XL ≥
h i (cid:12) (cid:12)
This is the basic idea of the modified RP. Thus, o(cid:12)ur probl(cid:12)em is reduced to constructing
a suitable ϑ. This formalism allows us to apply RP to H′′.
Λ
In Proposition 3.10 and Appendix D, we actually construct a suitable ϑ. Moreover,
in Lemmas 3.11 and 3.12, we prove that the extended RP can be applicable to our
model. In these arguments, we carefully use the regular structure of the phase factors
and the assumption that L is odd.
In the original paper [7, Section 3], the authors give several examples of how we
construct RP. Our formalism is different from these examples and more convenient for
studying the Holstein–Hubbard model.
3.4.2 Preliminaries
We divide Λ as Λ= Λ Λ , where
L R
∪
Λ = x = (x ,...,x ) Λ x < 0 , Λ = x = (x ,...,x ) Λ x 0 . (3.25)
L 1 ν 1 R 1 ν 1
{ ∈ | } { ∈ | ≥ }
Corresponding to this, we also divide ℓ2(Λ) as
ℓ2(Λ) = ℓ2(Λ ) ℓ2(Λ ). (3.26)
L R
⊕
Hence, we have the following identifications:
F = F F , (3.27)
L R
⊗
7
where F = F (ℓ2(Λ ) ℓ2(Λ )) and F = F (ℓ2(Λ ) ℓ2(Λ )), and
L as ↑ L ⊕ ↓ L R as ↑ R ⊕ ↓ R
P = P P , (3.28)
L R
⊗
where P = F (ℓ2(Λ )) = L2( ,dµ ) and P = F (ℓ2(Λ )) = L2( ,dµ ).
L s L QΛL ΛL R s R QΛR ΛR
Thus, the Hilbert space H can be identified as follows:
H = H H , (3.29)
L R
⊗
where H = F P and H = F P . Under the identification (3.29), we have the
L L L R R R
⊗ ⊗
following identifications:
c 1l if x Λ
xσ L
c = ⊗ ∈ (3.30)
xσ (( 1)NL cxσ if x ΛR,
− ⊗ ∈
where N = n , and
L x∈ΛL x
P
π 1l if x Λ φ 1l if x Λ
x L x L
π = ⊗ ∈ , φ = ⊗ ∈ . (3.31)
x x
(1l πx if x ΛR (1l φx if x ΛR
⊗ ∈ ⊗ ∈
Using these, we state the following lemmas:
Lemma 3.4 Under the identification (3.29), we have T′′ = T′′ 1l+1l T′′ +T′′ ,
L ⊗ ⊗ R LR
where
ν ′
T′′ = ( t) e−iα(φx−φx+εδj)c∗ c∗ +h.c. , (3.32)
L − xσ x+εδjσ
x∈ΛeX, x1≤−2σX=↑,↓Xj=1εX=± (cid:16) (cid:17)
ν ′′
T′′ = ( t) e−iα(φx−φx+εδj)c∗ c∗ +h.c. , (3.33)
R − xσ x+εδjσ
x∈ΛXe, x1≥0σX=↑,↓Xj=1εX=± (cid:16) (cid:17)
TL′′R = (−t) eiαφx−δ1(−1)NLc∗x−δ1σ ⊗ e−iαφxc∗xσ +h.c.
( )
x∈ΛXe, x1=0σX=↑,↓ h i h i
+ (−t) eiαφx+δ1(−1)NLc∗x+δ1σ ⊗ e−iαφxc∗xσ +h.c. .
( )
x∈Λe,Xx1=L−1σX=↑,↓ h i h i
(3.34)
′
Here, refers to a sum over pairs x;x+εδ such that x,x+εδ Λ . Similarly,
j j L
h i ∈
ε=±
X
′′
refers to a sum over pairs x;x+εδ such that x,x+εδ Λ .
j j R
h i ∈
ε=±
X
Remark 3.5 To obtain (3.34), we assume that L is odd.
♦
8
Lemma 3.6 For each h = h RΛ, we set h = h and h = h .
{ x}x∈Λ ∈ L { x}x∈ΛL R { x}x∈ΛR
We have P′′(h) = P′′(h ) 1l+1l P′′(h )+P′′ (h), where
L L ⊗ ⊗ R R LR
V
P′′(h )=(U νV) q2 + (q h q +h )2, (3.35)
L L eff − x 2 x− x − y y
xX∈ΛL hx;yi,Xx,y∈ΛL
V
P′′(h )=(U νV) q2 + (q h q +h )2, (3.36)
R R eff − x 2 x− x − y y
xX∈ΛR hx;yi,Xx,y∈ΛR
P′′ (h) = V (q h ) (q h )
LR − x−δ1 − x−δ1 ⊗ x− x
x∈ΛXe, x1=0
V (q h ) (q h ). (3.37)
− x+δ1 − x+δ1 ⊗ x− x
x∈Λe,Xx1=L−1
Lemma 3.7 We have K = K 1l+1l K , where
L R
⊗ ⊗
1 1
K = (π2 +ω2φ2), K = (π2 +ω2φ2). (3.38)
L 2 x x R 2 x x
xX∈ΛL xX∈ΛR
For all x Λ , we define
L
∈
a =c ( 1)NL. (3.39)
xσ xσ
−
In terms of a , T′′ and T′′ can be expressed as follows.
xσ L LR
Proposition 3.8 We obtain the following:
ν ′
T′′ = (+t) e−iα(φx−φx+εδj)a∗ a∗ +h.c. , (3.40)
L xσ x+εδjσ
x∈ΛeX, x1≤−2σX=↑,↓Xj=1εX=± (cid:16) (cid:17)
TL′′R = (−t) eiαφx−δ1a∗x−δ1σ ⊗ e−iαφxc∗xσ +h.c.
( )
x∈ΛXe, x1=0σX=↑,↓ (cid:16) (cid:17) (cid:16) (cid:17)
+ (−t) eiαφx+δ1a∗x+δ1σ ⊗ e−iαφxc∗xσ +h.c. . (3.41)
( )
x∈Λe,Xx1=L−1σX=↑,↓ (cid:16) (cid:17) (cid:16) (cid:17)
Remark 3.9 Since q = a∗ a 1l, expressions of P′′(h ) and P′′ (h) are
x σ=↑,↓ xσ xσ − L L LR
unchanged if we write these in terms of a .
xσ
P ♦
3.4.3 Gaussian domination
We define the reflection map r : Λ Λ by
R L
→
r(x)= ( x 1,x ,...,x ), x Λ . (3.42)
1 2 ν R
− − ∈
We begin with the following proposition:
9
Proposition 3.10 There exists an antiunitary transformation4 ϑ from H to H such
L R
that
c =ϑa ϑ−1, φ = ϑφ ϑ−1, π = ϑπ ϑ−1, x Λ , (3.43)
xσ r(x)σ x r(x) x r(x) R
− ∈
ϑΩ = Ω , (3.44)
L R
where Ω is the Fock vacuum ΩL ΩL in H 5, and Ω can be defined in a similar
L f ⊗ b L R
manner.
Proof. See Appendix D. ✷
Lemma 3.11 We have the following:
(i) T′′ = ϑT′′ϑ−1.
R L
(ii)
TL′′R = (−t) eiαφx−δ1a∗x−δ1σ ⊗ϑ eiαφx−δ1a∗x−δ1σ ϑ−1+h.c.
( )
x∈ΛXe, x1=0σX=↑,↓ (cid:16) (cid:17) (cid:16) (cid:17)
+ (−t) eiαφx+δ1a∗x+δ1σ ⊗ϑ eiαφx+δ1a∗x+δ1σ ϑ−1+h.c. .
( )
x∈Λe,Xx1=L−1σX=↑,↓ (cid:16) (cid:17) (cid:16) (cid:17)
(3.45)
Proof. While (ii) is trivial, (i) has be addressed carefully. First, T′′ can be expressed as
L
ν ′
T′′ = (+t) e−iα(φx+εδj−φx)a∗ a∗ +h.c. . (3.46)
L x+εδjσ xσ
x∈ΛoX, x1≤−1σX=↑,↓Xj=1εX=± (cid:16) (cid:17)
Hence, by (3.33), we see that
ν ′
T′′ = ϑ ( t) e+iα(φr(x)−φr(x+εδj))a∗ a∗ +h.c. ϑ−1
R − r(x)σ r(x+εδj)σ
x∈ΛXe, x1≥0σX=↑,↓Xj=1εX=± (cid:16) (cid:17)
ν ′
= ϑ ( t) e+iα(φX−φX+εδj)a∗ a∗ +h.c. ϑ−1
− Xσ X+εδjσ
X∈ΛoX, X1≤−1σX=↑,↓Xj=1εX=± (cid:16) (cid:17)
ν ′
= ϑ (+t) e−iα(φX+εδj−φX)a∗ a∗ +h.c. ϑ−1
X+εδjσ Xσ
X∈ΛoX, X1≤−1σX=↑,↓Xj=1εX=± (cid:16) (cid:17)
= ϑT′′ϑ−1. (3.47)
L
Here, we use the fact that r maps even sites to odd sites; namely, if x Λ , then
e
r(x) Λ . Additionally, recall that ϑ is antilinear. ✷ ∈
o
∈
The following lemmas then immediately follow from (3.43).
4 Namely, ϑ is a bijective antilinear map which satisfies hϑϕ|ϑψi=(hϕ|ψi)∗ for all ϕ,ψ∈X .
L
2
5 IntheSchr¨odingerrepresentation,ΩL =(1)|ΛL|/4e−Px∈ΛLφx/2. ΩL isthestandardFockvacuum
b π f
in F . Note that ΩL is a real-valued function on Q .
L b ΛL
10