- Email: [email protected]
The fermionic correlation functions of the Falicov–Kimball model
Physica A 279 (2000) 408–415
www.elsevier.com/locate/physa
The fermionic correlation functions of the Falicov–Kimball model Alain Messager ∗ Centre Physique Theorique au CNRS, 13288 Marseille, France
Abstract We estimate the fermionic correlation functions of the Falicov-Kimball model for large repulsion. In particular we show that the two point functions decay exponentially in the distance. c 2000 Elsevier Science B.V. All rights reserved.
Dedicated to Joel Lebowitz on the occasion of his 70th birthday
1. Introduction The Falicov–Kimball model [1] is a lattice model of quantum particules (fermions) in interaction with a classical eld. The FK model has been extensively studied these last 15 years. We refer to the nice review of C Gruber [8] for the main results, and to the references therein. We rst mention the seminal paper of Kennedy and Lieb [2], in which a Neel phase transition was proven. This result was extended by Lebowitz and Macris [3], and then by Messager and Miracle Sole [4]. One of the main feature of this model is that, for some value of the chemical potentials, the classical ground states coming from the diagonal part of the hamiltonian w.r.t the number of electrons are in nitely degenerated, then the quantum uctuations will remove this degeneration leading to the coexistence of a nite number of phases. This is a quantum phase transition. More precisely, the method used in Refs. [4,5] inherited from Ref. [2] consists in tracing over the electrons’ operators, then the quantum uctuations are represented in term of closed loops in the space time. Next we deduce the existence, at every temperature, of an e ective hamiltonian Hÿ expressed in term of the ions’ variables [5], which comes out from the quantum uctuations, which governs the behaviour of the ∗
Fax: +33-4-91-26-95-53. E-mail address: [email protected] (A. Messager)
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 5 8 3 - X
A. Messager / Physica A 279 (2000) 408–415
409
ions. Then the main advantage of this approach is that we can use the well-developed results of the classical statistical mechanics [6] to study the phase transitions [4,5], the interfaces [7], etc. Nevertheless, we did not able to get the fermionic correlation functions. The purpose of this note is to compute the fermionic correlation functions for large repulsion by enlarging the loop’s space composed of closed loops supplemented by a family of open loops. We de ne the hamiltonian of the FK model: HV = t
X
{Ca? Cb + Cb? Ca } +
ha; bi∈V
U X i X e X (x) (x) − (x) − (x) ; 2 x∈V 2 x∈V 2 x∈V
h:; :i refers to the nearest-neighbor lattice sites. t is called the hopping intensity, e and i are the chemical potentials of the electrons and respectively of the ions. Ca∗ (resp. Ca ) is the fermionic creation (resp. annihilation) operator. x is the random variable which is +1 if there is an ion at x and −1 otherwise. a = 2Ca∗ Ca − 1. On a bipartite lattice we have two additional conditions: the electron to the non-electron symmetry: Ca∗ → Ca , the transformation Ca → −Ca on one part of the bipartite lattice. We will use the parameters ÿ for ÿt, U for U=t, i for i =t, and e for e =t.
2. Loop’s representation and Feynman–Kac formula We build a twofold Feynman–Kac integral representation for the matrix elements of the operator exp(−ÿHV ). The rst one is expressed in term of the electrons’ trajectories. We de ne the extended space time =V ×[0; ÿ), in which the hyperplanes t =0 and ÿ V labelled are identi ed. V ⊂ Z is called the basis. We x the orthonormal basis in K by the pairs (FV ; V ) of classical {0; 1}-con gurations in V , and by the boundary conditions V ⊗ FV in V . By S(FV0 ; FV00 ), we denote the set of all one-to-one mappings between the electronic con gurations FV0 ; FV00 with |FV0 | = |FV00 | and by par the parity of the mapping : FV0 → FV00 . D((FV0 ⊗ V ); (FV00 ⊗ V ) | V ⊗ FVf ) is the matrix element of exp(−ÿHV ) in the reference basis: D((FV0 ⊗ V ); (FV00 ⊗ V ) | V ⊗ FVf ) = 0 D((FV0 ⊗ V ); (FV00 ⊗ V ) | V ⊗ FV ) ! X e − i X x = exp − 2 0 x⊂V
if |FV0 | = 6 |FV00 | ;
par
(−1)
×
∈S(FV ; FV00 )
×exp−(U + e )
X j∈V
Y j∈FV0
(1.1)
!
Z Mj; ( j)
X T (•j ; −j ) − (U − e ) T (∅k ; ∅k )
Pj; ( j) (d!j )
if |FV0 | = |FV00 | :
k∈V
(1.2)
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Mj; j0 (=Mj;ÿ j0 ) is the set of right-continuous paths ! of electron trajectories of time
length ÿ in Z with !(0) = lims→0 !(s) = j, !(ÿ) = lims→ÿ !(s) = j 0 , and Pj; j0 (=Pj;ÿ; j0 ) denotes the path distribution restricted to Mj; j0 of the Markov jump process on Z with jump’s intensity x → y given by t, where y is one of the nearest neighbor site of x. T (•j; j ) is the time which the path !j and the vertical ion line spend together above the site j. T (∅k ; ∅k ) is the time during which, a hole is living above the site k. We notice that the electron’s trajectories considered on the torus are closed loops. Next we de ne in the same way a twofold Feynman–Kac integral representation for the matrix elements of the operator exp(−ÿHV )[Ca∗1 : : : Ca∗m × Cbm : : : Cb1 ], where the sets A = {a1 : : : am } ⊂ V , B = {b1 : : : bm } ⊂ V are disjoint and V˜ = V=(A ∪ B). Then ˆ is the tensor product of an electron’s con guration in F 0˜ (resp. FV0˜ ⊗ Aˆ (resp. FV00˜ ⊗ B) V 00 in FV˜ ) with the electrons’ con guration sitting at the vertices of A (resp B). The rst Feynman–Kac formula is written in term of ions’ con gurations and of electrons’ trajectories: D((FV0˜ ⊗ Aˆ ⊗ V ); (FV00˜ ⊗ Bˆ ⊗ V ) | V ⊗ FV ) = 0 D(FV0˜ ⊗ Aˆ ⊗ V ; FV00˜ ⊗ Bˆ ⊗ V | V ⊗ FV ) ! X e − i X x = exp − 2 0 00 x⊂V
if |FV0˜ | = 6 |FV00˜ | ;
(1.3)
(−1)par
ˆ ∈S(F ˜ ⊗A;ˆ F ˜ ⊗B) V
×
Y j∈F 0˜ ⊗Aˆ
!
Z M˜ j; ( j)
V
Pj; ( j) (d!j )
V
× exp−(U + e )
X j∈V
T (•j ; −j ) − (U − e )
X
T (∅k ; ∅k )
k∈V
if |FV0˜ | = |FV00˜ | : (1.4)
The main di erence between (1.2) and (1.4) is the existence of families of open electrons’ trajectories connecting the points of A and of B. 2.1. The geometric construction of the loop’s space L() We represent our Feynman–Kac formulas in term of ions’ con gurations and in term of loops. First, we enlarge the loop’s representation given in Ref. [5]. The loops are geometrical objects describing a quantum con guration in ≡ V × (0; ÿ). One of the following four situations occurs. Case (i): There is an electron: we draw on the vertical lines a dashed line with an up arrow. Case (ii): There is an ion: we draw a dashed line with an down arrow.
A. Messager / Physica A 279 (2000) 408–415
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Case (iii): There is a pair: we draw a continuous segment with an up arrow. Case (iv): There is a hole: we draw a continuous segment with an down arrow. We complete the Feynman–Kac representation in terms of dashed and of continuous arrows with additional segments. When an electron hops from the site x to one of its nearest-neighbor sites y, the points x to y are connected by an horizontal segment of the same type and with the same arrow as the vertical lines coming at the sites x and y. All the vertical lines are covered with parts of oriented loops, two loops of di erent kinds meet along the additional lines of the loops, and secondly that some loops may wind around the torus. Notice that two loops of the same kind cannot intersect along the vertical parts exept at the boundary points of their vertical segments, they are said to be compatible. A con guration is now described by: (see the picture, in which one open loop connects the point a to the point b) • A family of continuous closed loops and a family of closed dashed loops. A generic closed continuous loop will be denoted by . They are characterized by their jumps’ times: s0 is the Feynman–Kac time of occurence of an arbitrary jump of the loop, which is taken as the origin for the other jumps’ times {s1 ; : : : ; sn }, which are counted along the orientation of the loop. • A family of pairs of open loops composed of one open dashed loop and one continuous loop, which boundaries a ∈ A; b ∈ B belongs to the basis. An open continuous loop will be denoted by {a; b} . They are also characterized by their jumps’ times together with their boundaries. The loops are also classi ed by their homotopic properties into four classes (see Plate 1). (1) The non-winding closed loops. (2) The non-winding open loops. (3) The winding closed loops with winding number ±w according to the orientation of the loops. (4) The winding open loops. The “signed densities” (s.d.) of a non-winding loop and of a winding loop with winding number ±w are de ned in Refs. [4,5]. We give the example of the s.d. of a closed non-winding loop: ’() = ()t n+1 exp − [Us] × (s1 ; : : : ; sn ) :
(1.2a)
(s1 ¡ · · · ¡ sn ) is the characteristic function of the ordered jumps’ times counted along the orientation of the loop with as origin the time s0 , s is the sum of the vertical lengths of the loop. () = ±1 comes from the Fermi statistic. ˆ B; ˆ V |V ) is the subset of the Deÿnition. (1) The conditional ensemble COND(A; compatible loops of L{} (intersecting V ), which are built from the ion con gurations V in V , and V in V , and from the sets A and B. This set of compatible loops contain a set of closed loops together with the set of the pairs of open loops which connect ˆ B; ˆ V |V ) is the family of the points of the set A to the set B (|A| = |B|). (2) (A; ˆ B; ˆ V |V ). (3) The the open loops belonging to the conditionnal ensemble COND(A; conditional partition function ZV [V |V ] is the restriction of the partition function to the conditional ensemble COND(V |V ).
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A. Messager / Physica A 279 (2000) 408–415
Plate 1. This picture represents a quantum con guration containing an open loop (a; b) , two non winding loops 3 ; 4 , and the three winding loops 1 ; 2 ; 5 .
Proposition (The loop’s representation). The conditionnal partition functions; the partition function; and the correlation functions are written as: " # X e − i × x (V ) (i) ZV [V |V ] = exp − ÿ 2 x⊂V r Y X ’(i ) ; × {1 ::: r } ⊂ COND(V |V ) i=1
(ii) ZV (V ) =
X
ZV [V |V ] ;
V
P (iii) hX i(V ) =
V
ZV [V |V ]X (V ) ; Z(V )
A. Messager / Physica A 279 (2000) 408–415
413
(iv) hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V |V ] " # e i X − × x (V ) = ZV−1 (V |V )exp − ÿ 2 x⊂V # " r Y X ’(i ) × ˆ V |V ]) i=1 (1 :::r ;{ai(1) ; bi(1) } ;:::;{ai(m) ; bi(m) } ⊂ COND(A;ˆ B;[
×
j=m
Y
’({ai( j) ; bi( j) } );
j=1
(v) hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V ] =
X
hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V |V ] :
V
(1.5) Proof (sketch). The representation of the conditional partition function is the transcription of the Feynman–Kac formula, previously written in terms of electrons’ trajectories, into the Feynman–Kac formula written in terms of closed loops. The representation of the correlation functions of fermionic operators is deduced in the same way in the extended loop’s space, supplemented by the weights of the open loops connecting the points of the set A to the points of B. We consider the di erent possible sums of the distances between the pairs of points belonging to A and B. Let dm [A; B] be the minimal sum.
3. The fermionic correlation functions Theorem. There exists a positive constant 0 ¡ A ¡ 1. Then if the following condition holds: |˜e | ¡ U˜ − 2) where ˜e = A−1 e ; U˜ = A−1 U , we have the bounds: hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V ]6C
1 : m [U˜ − ˜e − 2]d (A; B)
Proof (sketch). We start from one conditional correlation function de ned in (1.5). We decompose the conditional correlation function into a sum over the s.p. of the open loops. We factorize the s.p. of each family of open loops {{ai( j) ; bi( j) } ; : : : ; {ai(m) ; bi(m) } } with the conditional partition function restricted to the s.p. of the closed loops compatible with the family of open loops {{ai( j) ; bi( j) } ; : : : ; {ai(m) ; bi(m) } }. Then we are
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A. Messager / Physica A 279 (2000) 408–415
left with the quotient of two partition functions:
j=m
X
hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V |V ] =
{ai( j) ; bi( j) } ∈(A; B;V |V )
×
Y
’({ai( j) ; bi( j) } )
j=1
Z{V={{ai(1) ; bi(1) } ;:::;{ai(m) ; bi(m) } } [V |V ] ZV [V |V ]
:
Next we use the cluster expansion contained in Appendix C of Ref. [5] to evaluate the quotient of the partition functions in a standard way. We partition the sum of the s.d. of the open loops in two sums. (1) We will say that two loops are equivalent if they have the same set of jumps i.e., if they are projected on the basis along the same directed graph g[ai( j) ; bi( j) ]. (2) The class of equivalent loop to {ai(1) ; bi(1) } ˆ A; ˆ B; ˆ V |V ) is the set of classes of open loop of is denoted by ˆ{ai(1) ; bi(1) } . (3) ( ˆ B; ˆ V |V ). This sum is done in two steps. First, we integrate over the s.p. COND(A; of the open loops belonging to the same class. This is done by using the computations contained in the Appendix A of [5]. hCa∗1 : : : Ca∗m Cbm : : : Cb1 i[V |V ] j=m
X
X
6
Y
’[{ai( j) ; bi( j) } ]
ˆ B;V | )} {{a ; b } ∈ˆ{a ; b } } j=1 {ˆ{ai( j) ; bi( j) } ∈(A; V i( j) i( j) i( j) i( j)
×exp
j=m X
|{ai( j) ; bi( j) } | ×
j=1
X
6
1 U˜ 0 − ˜e0 − 2 j=m
Y
ˆ B;V | )} j=1 {ˆ{ai( j) ; bi( j) } ∈(A; V
1 [U˜ 0 − ˜e0 ]|g[ai( j) ; bi( j) ]|
:
ˆ B; V |V ). This is Secondly, we sum over the di erent classes of open loops of (A; equivalent to sum over the set of directed graphs g[ai( j) ; bi( j) ] connecting the pairs of vertices of A and B. We get the uniform bound w.r.t. the ion’s con guration, again we use Appendix A of Ref. [5]: hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V |V ]6
1 : m [U˜ − ˜e − 2]d (A; B)
4. Conclusion We are able to describe all the correlation functions of the FK model for large repulsion. In particular, we have shown that the two-point fermionic correlation function
A. Messager / Physica A 279 (2000) 408–415
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decays exponentialy in the distance. These results can be obtained in the same way for more general models including arbitrary hoppings decaying exponentially and fermionic interactions [5]. I believe that this approach can be extended to the static Holstein model studied by Lebowitz and Macris [9]. Acknowledgements I am pleased to thank Joel for its constant support, and for the pleasure that I have had in reading his very nice papers. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Falicov, J. Kimball, Phys. Rev. Lett. 22 (1969). T. Kennedy, E. Lieb, Physica A 138 (1986). J. Lebowitz, N. Macris, Rev. Math. Phys. 6 (5a) (1994). A. Messager, S. Miracle Sole, Rev. Math. Phys. 8 (2) (1996). A. Messager, On quantum phase transition I and II, J. Stat. Phys. (1999), submitted for publication in J. Stat. Phys. (2000). Ja. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon Press, Gordon, 1982. N. Datta, A. Messager, B. Nachtergaele, Rigidity of interfaces in the Falicov–Kimball model. J. Stat. Phys., in press. C. Gruber, Falicov–Kimball model: A partial review of the ground states problem, Proceedings of the Marseille’s Conference, 1999. J. Lebowitz, N. Macris, J. Stat. Phys. 76 (1–2) (1994).
www.elsevier.com/locate/physa
The fermionic correlation functions of the Falicov–Kimball model Alain Messager ∗ Centre Physique Theorique au CNRS, 13288 Marseille, France
Abstract We estimate the fermionic correlation functions of the Falicov-Kimball model for large repulsion. In particular we show that the two point functions decay exponentially in the distance. c 2000 Elsevier Science B.V. All rights reserved.
Dedicated to Joel Lebowitz on the occasion of his 70th birthday
1. Introduction The Falicov–Kimball model [1] is a lattice model of quantum particules (fermions) in interaction with a classical eld. The FK model has been extensively studied these last 15 years. We refer to the nice review of C Gruber [8] for the main results, and to the references therein. We rst mention the seminal paper of Kennedy and Lieb [2], in which a Neel phase transition was proven. This result was extended by Lebowitz and Macris [3], and then by Messager and Miracle Sole [4]. One of the main feature of this model is that, for some value of the chemical potentials, the classical ground states coming from the diagonal part of the hamiltonian w.r.t the number of electrons are in nitely degenerated, then the quantum uctuations will remove this degeneration leading to the coexistence of a nite number of phases. This is a quantum phase transition. More precisely, the method used in Refs. [4,5] inherited from Ref. [2] consists in tracing over the electrons’ operators, then the quantum uctuations are represented in term of closed loops in the space time. Next we deduce the existence, at every temperature, of an e ective hamiltonian Hÿ expressed in term of the ions’ variables [5], which comes out from the quantum uctuations, which governs the behaviour of the ∗
Fax: +33-4-91-26-95-53. E-mail address: [email protected] (A. Messager)
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 5 8 3 - X
A. Messager / Physica A 279 (2000) 408–415
409
ions. Then the main advantage of this approach is that we can use the well-developed results of the classical statistical mechanics [6] to study the phase transitions [4,5], the interfaces [7], etc. Nevertheless, we did not able to get the fermionic correlation functions. The purpose of this note is to compute the fermionic correlation functions for large repulsion by enlarging the loop’s space composed of closed loops supplemented by a family of open loops. We de ne the hamiltonian of the FK model: HV = t
X
{Ca? Cb + Cb? Ca } +
ha; bi∈V
U X i X e X (x) (x) − (x) − (x) ; 2 x∈V 2 x∈V 2 x∈V
h:; :i refers to the nearest-neighbor lattice sites. t is called the hopping intensity, e and i are the chemical potentials of the electrons and respectively of the ions. Ca∗ (resp. Ca ) is the fermionic creation (resp. annihilation) operator. x is the random variable which is +1 if there is an ion at x and −1 otherwise. a = 2Ca∗ Ca − 1. On a bipartite lattice we have two additional conditions: the electron to the non-electron symmetry: Ca∗ → Ca , the transformation Ca → −Ca on one part of the bipartite lattice. We will use the parameters ÿ for ÿt, U for U=t, i for i =t, and e for e =t.
2. Loop’s representation and Feynman–Kac formula We build a twofold Feynman–Kac integral representation for the matrix elements of the operator exp(−ÿHV ). The rst one is expressed in term of the electrons’ trajectories. We de ne the extended space time =V ×[0; ÿ), in which the hyperplanes t =0 and ÿ V labelled are identi ed. V ⊂ Z is called the basis. We x the orthonormal basis in K by the pairs (FV ; V ) of classical {0; 1}-con gurations in V , and by the boundary conditions V ⊗ FV in V . By S(FV0 ; FV00 ), we denote the set of all one-to-one mappings between the electronic con gurations FV0 ; FV00 with |FV0 | = |FV00 | and by par the parity of the mapping : FV0 → FV00 . D((FV0 ⊗ V ); (FV00 ⊗ V ) | V ⊗ FVf ) is the matrix element of exp(−ÿHV ) in the reference basis: D((FV0 ⊗ V ); (FV00 ⊗ V ) | V ⊗ FVf ) = 0 D((FV0 ⊗ V ); (FV00 ⊗ V ) | V ⊗ FV ) ! X e − i X x = exp − 2 0 x⊂V
if |FV0 | = 6 |FV00 | ;
par
(−1)
×
∈S(FV ; FV00 )
×exp−(U + e )
X j∈V
Y j∈FV0
(1.1)
!
Z Mj; ( j)
X T (•j ; −j ) − (U − e ) T (∅k ; ∅k )
Pj; ( j) (d!j )
if |FV0 | = |FV00 | :
k∈V
(1.2)
410
A. Messager / Physica A 279 (2000) 408–415
Mj; j0 (=Mj;ÿ j0 ) is the set of right-continuous paths ! of electron trajectories of time
length ÿ in Z with !(0) = lims→0 !(s) = j, !(ÿ) = lims→ÿ !(s) = j 0 , and Pj; j0 (=Pj;ÿ; j0 ) denotes the path distribution restricted to Mj; j0 of the Markov jump process on Z with jump’s intensity x → y given by t, where y is one of the nearest neighbor site of x. T (•j; j ) is the time which the path !j and the vertical ion line spend together above the site j. T (∅k ; ∅k ) is the time during which, a hole is living above the site k. We notice that the electron’s trajectories considered on the torus are closed loops. Next we de ne in the same way a twofold Feynman–Kac integral representation for the matrix elements of the operator exp(−ÿHV )[Ca∗1 : : : Ca∗m × Cbm : : : Cb1 ], where the sets A = {a1 : : : am } ⊂ V , B = {b1 : : : bm } ⊂ V are disjoint and V˜ = V=(A ∪ B). Then ˆ is the tensor product of an electron’s con guration in F 0˜ (resp. FV0˜ ⊗ Aˆ (resp. FV00˜ ⊗ B) V 00 in FV˜ ) with the electrons’ con guration sitting at the vertices of A (resp B). The rst Feynman–Kac formula is written in term of ions’ con gurations and of electrons’ trajectories: D((FV0˜ ⊗ Aˆ ⊗ V ); (FV00˜ ⊗ Bˆ ⊗ V ) | V ⊗ FV ) = 0 D(FV0˜ ⊗ Aˆ ⊗ V ; FV00˜ ⊗ Bˆ ⊗ V | V ⊗ FV ) ! X e − i X x = exp − 2 0 00 x⊂V
if |FV0˜ | = 6 |FV00˜ | ;
(1.3)
(−1)par
ˆ ∈S(F ˜ ⊗A;ˆ F ˜ ⊗B) V
×
Y j∈F 0˜ ⊗Aˆ
!
Z M˜ j; ( j)
V
Pj; ( j) (d!j )
V
× exp−(U + e )
X j∈V
T (•j ; −j ) − (U − e )
X
T (∅k ; ∅k )
k∈V
if |FV0˜ | = |FV00˜ | : (1.4)
The main di erence between (1.2) and (1.4) is the existence of families of open electrons’ trajectories connecting the points of A and of B. 2.1. The geometric construction of the loop’s space L() We represent our Feynman–Kac formulas in term of ions’ con gurations and in term of loops. First, we enlarge the loop’s representation given in Ref. [5]. The loops are geometrical objects describing a quantum con guration in ≡ V × (0; ÿ). One of the following four situations occurs. Case (i): There is an electron: we draw on the vertical lines a dashed line with an up arrow. Case (ii): There is an ion: we draw a dashed line with an down arrow.
A. Messager / Physica A 279 (2000) 408–415
411
Case (iii): There is a pair: we draw a continuous segment with an up arrow. Case (iv): There is a hole: we draw a continuous segment with an down arrow. We complete the Feynman–Kac representation in terms of dashed and of continuous arrows with additional segments. When an electron hops from the site x to one of its nearest-neighbor sites y, the points x to y are connected by an horizontal segment of the same type and with the same arrow as the vertical lines coming at the sites x and y. All the vertical lines are covered with parts of oriented loops, two loops of di erent kinds meet along the additional lines of the loops, and secondly that some loops may wind around the torus. Notice that two loops of the same kind cannot intersect along the vertical parts exept at the boundary points of their vertical segments, they are said to be compatible. A con guration is now described by: (see the picture, in which one open loop connects the point a to the point b) • A family of continuous closed loops and a family of closed dashed loops. A generic closed continuous loop will be denoted by . They are characterized by their jumps’ times: s0 is the Feynman–Kac time of occurence of an arbitrary jump of the loop, which is taken as the origin for the other jumps’ times {s1 ; : : : ; sn }, which are counted along the orientation of the loop. • A family of pairs of open loops composed of one open dashed loop and one continuous loop, which boundaries a ∈ A; b ∈ B belongs to the basis. An open continuous loop will be denoted by {a; b} . They are also characterized by their jumps’ times together with their boundaries. The loops are also classi ed by their homotopic properties into four classes (see Plate 1). (1) The non-winding closed loops. (2) The non-winding open loops. (3) The winding closed loops with winding number ±w according to the orientation of the loops. (4) The winding open loops. The “signed densities” (s.d.) of a non-winding loop and of a winding loop with winding number ±w are de ned in Refs. [4,5]. We give the example of the s.d. of a closed non-winding loop: ’() = ()t n+1 exp − [Us] × (s1 ; : : : ; sn ) :
(1.2a)
(s1 ¡ · · · ¡ sn ) is the characteristic function of the ordered jumps’ times counted along the orientation of the loop with as origin the time s0 , s is the sum of the vertical lengths of the loop. () = ±1 comes from the Fermi statistic. ˆ B; ˆ V |V ) is the subset of the Deÿnition. (1) The conditional ensemble COND(A; compatible loops of L{} (intersecting V ), which are built from the ion con gurations V in V , and V in V , and from the sets A and B. This set of compatible loops contain a set of closed loops together with the set of the pairs of open loops which connect ˆ B; ˆ V |V ) is the family of the points of the set A to the set B (|A| = |B|). (2) (A; ˆ B; ˆ V |V ). (3) The the open loops belonging to the conditionnal ensemble COND(A; conditional partition function ZV [V |V ] is the restriction of the partition function to the conditional ensemble COND(V |V ).
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A. Messager / Physica A 279 (2000) 408–415
Plate 1. This picture represents a quantum con guration containing an open loop (a; b) , two non winding loops 3 ; 4 , and the three winding loops 1 ; 2 ; 5 .
Proposition (The loop’s representation). The conditionnal partition functions; the partition function; and the correlation functions are written as: " # X e − i × x (V ) (i) ZV [V |V ] = exp − ÿ 2 x⊂V r Y X ’(i ) ; × {1 ::: r } ⊂ COND(V |V ) i=1
(ii) ZV (V ) =
X
ZV [V |V ] ;
V
P (iii) hX i(V ) =
V
ZV [V |V ]X (V ) ; Z(V )
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(iv) hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V |V ] " # e i X − × x (V ) = ZV−1 (V |V )exp − ÿ 2 x⊂V # " r Y X ’(i ) × ˆ V |V ]) i=1 (1 :::r ;{ai(1) ; bi(1) } ;:::;{ai(m) ; bi(m) } ⊂ COND(A;ˆ B;[
×
j=m
Y
’({ai( j) ; bi( j) } );
j=1
(v) hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V ] =
X
hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V |V ] :
V
(1.5) Proof (sketch). The representation of the conditional partition function is the transcription of the Feynman–Kac formula, previously written in terms of electrons’ trajectories, into the Feynman–Kac formula written in terms of closed loops. The representation of the correlation functions of fermionic operators is deduced in the same way in the extended loop’s space, supplemented by the weights of the open loops connecting the points of the set A to the points of B. We consider the di erent possible sums of the distances between the pairs of points belonging to A and B. Let dm [A; B] be the minimal sum.
3. The fermionic correlation functions Theorem. There exists a positive constant 0 ¡ A ¡ 1. Then if the following condition holds: |˜e | ¡ U˜ − 2) where ˜e = A−1 e ; U˜ = A−1 U , we have the bounds: hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V ]6C
1 : m [U˜ − ˜e − 2]d (A; B)
Proof (sketch). We start from one conditional correlation function de ned in (1.5). We decompose the conditional correlation function into a sum over the s.p. of the open loops. We factorize the s.p. of each family of open loops {{ai( j) ; bi( j) } ; : : : ; {ai(m) ; bi(m) } } with the conditional partition function restricted to the s.p. of the closed loops compatible with the family of open loops {{ai( j) ; bi( j) } ; : : : ; {ai(m) ; bi(m) } }. Then we are
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left with the quotient of two partition functions:
j=m
X
hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V |V ] =
{ai( j) ; bi( j) } ∈(A; B;V |V )
×
Y
’({ai( j) ; bi( j) } )
j=1
Z{V={{ai(1) ; bi(1) } ;:::;{ai(m) ; bi(m) } } [V |V ] ZV [V |V ]
:
Next we use the cluster expansion contained in Appendix C of Ref. [5] to evaluate the quotient of the partition functions in a standard way. We partition the sum of the s.d. of the open loops in two sums. (1) We will say that two loops are equivalent if they have the same set of jumps i.e., if they are projected on the basis along the same directed graph g[ai( j) ; bi( j) ]. (2) The class of equivalent loop to {ai(1) ; bi(1) } ˆ A; ˆ B; ˆ V |V ) is the set of classes of open loop of is denoted by ˆ{ai(1) ; bi(1) } . (3) ( ˆ B; ˆ V |V ). This sum is done in two steps. First, we integrate over the s.p. COND(A; of the open loops belonging to the same class. This is done by using the computations contained in the Appendix A of [5]. hCa∗1 : : : Ca∗m Cbm : : : Cb1 i[V |V ] j=m
X
X
6
Y
’[{ai( j) ; bi( j) } ]
ˆ B;V | )} {{a ; b } ∈ˆ{a ; b } } j=1 {ˆ{ai( j) ; bi( j) } ∈(A; V i( j) i( j) i( j) i( j)
×exp
j=m X
|{ai( j) ; bi( j) } | ×
j=1
X
6
1 U˜ 0 − ˜e0 − 2 j=m
Y
ˆ B;V | )} j=1 {ˆ{ai( j) ; bi( j) } ∈(A; V
1 [U˜ 0 − ˜e0 ]|g[ai( j) ; bi( j) ]|
:
ˆ B; V |V ). This is Secondly, we sum over the di erent classes of open loops of (A; equivalent to sum over the set of directed graphs g[ai( j) ; bi( j) ] connecting the pairs of vertices of A and B. We get the uniform bound w.r.t. the ion’s con guration, again we use Appendix A of Ref. [5]: hCa∗1 : : : Ca∗m × Cbm : : : Cb1 i[V |V ]6
1 : m [U˜ − ˜e − 2]d (A; B)
4. Conclusion We are able to describe all the correlation functions of the FK model for large repulsion. In particular, we have shown that the two-point fermionic correlation function
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decays exponentialy in the distance. These results can be obtained in the same way for more general models including arbitrary hoppings decaying exponentially and fermionic interactions [5]. I believe that this approach can be extended to the static Holstein model studied by Lebowitz and Macris [9]. Acknowledgements I am pleased to thank Joel for its constant support, and for the pleasure that I have had in reading his very nice papers. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Falicov, J. Kimball, Phys. Rev. Lett. 22 (1969). T. Kennedy, E. Lieb, Physica A 138 (1986). J. Lebowitz, N. Macris, Rev. Math. Phys. 6 (5a) (1994). A. Messager, S. Miracle Sole, Rev. Math. Phys. 8 (2) (1996). A. Messager, On quantum phase transition I and II, J. Stat. Phys. (1999), submitted for publication in J. Stat. Phys. (2000). Ja. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon Press, Gordon, 1982. N. Datta, A. Messager, B. Nachtergaele, Rigidity of interfaces in the Falicov–Kimball model. J. Stat. Phys., in press. C. Gruber, Falicov–Kimball model: A partial review of the ground states problem, Proceedings of the Marseille’s Conference, 1999. J. Lebowitz, N. Macris, J. Stat. Phys. 76 (1–2) (1994).