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- Holstein model and Peierls instability in 1D boson-fermion lattice gases
- Phase Separation close to the density-driven Mott transition in the Hubbard-Holstein model
- Superconducting instability in the Holstein-Hubbard model A numerical renormalization group
- Strong-coupling expansions for the anharmonic Holstein model and for the Holstein-Hubbard m
- Anderson Localization for the Holstein model G. Gentile
- Many-body CPA for the Holstein-DE model
- Exact diagonalization study of the two-dimensional t-J-Holstein model
- Peierls instability and optical response in the one--dimensional half--filled Holstein mode
- Extended Gutzwiller wavefunction for the Hubbard-Holstein model

arXiv:cond-mat/9803036v1 [cond-mat.stat-mech] 3 Mar 1998

Peierls instability for the Holstein model with rational density

G. Benfatto?

Dipartimento di Matematica, Universit` a di Roma “Tor Vergata” Via della Ricerca Scienti?ca, I-00133, Roma

G. Gentile?

Dipartimento di Matematica, Universit` a di Roma Tre Largo San Leonardo Murialdo 1, I-00146 Roma

V. Mastropietro?

Dipartimento di Matematica, Universit` a di Roma “Tor Vergata” Via della Ricerca Scienti?ca, I-00133, Roma

Abstract. We consider the static Holstein model, describing a chain of Fermions interacting with a classical phonon ?eld, when the interaction is weak and the density is a rational number. We show that the energy of the system, as a function of the phonon ?eld, has two stationary points, de?ned up to a lattice translation, which are local minima in the space of ?elds periodic with period equal to the inverse of the density.

?

Supported by MURST, Italy.

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1. Introduction

1.1. The Holstein model [P,H] was introduced to represent the interaction of electrons with optical phonons in a crystal. In the original model the phonons are represented in terms of quantum oscillators but the di?culty of understanding such a fully quantum model has led to a modi?cation of it, called static Holstein model (or adiabatic Holstein model), in which the phonons are classical oscillators. This corresponds to neglect the vibrational kinetic energy of the phonons, an approximation which can be justi?ed in physical models as the atom mass is much larger than the electron mass. The Hamiltonian of the model, if we neglect all internal degrees of freedom (the spin, for example, which play no role at zero external magnetic ?eld) is given by

el H ≡ HL +

1 2

?2 x

x ∈Λ + ? ψx ψx ? λ + ? ?x ψx ψx + x ∈Λ

=

x,y ∈Λ

+ ? txy ψx ψy ? ?

x ∈Λ

1 2

(1.1) ?2 x ,

x ∈Λ

where x, y are points on the one-dimensional lattice Λ with unit spacing, length L and periodic boundary conditions; we shall identify Λ with {x ∈ Z : ?[L/2] ≤ x ≤ [(L ? 1)/2]}. Moreover the matrix txy is de?ned as txy = δx,y ? (1/2)[δx,y+1 + δx,y?1 ], where δx,y is the Kronecker delta, ? is the chemical potential and λ is the interaction strength. The ?elds ± ψx are creation (+) and annihilation (?) fermionic ?elds, satisfying periodic boundary ± ± ± tH ± ?Ht conditions: ψx = ψx ψx e , with x = (x, t), ?β/2 ≤ t ≤ β/2 +L . We de?ne also ψx = e for some β > 0; on t antiperiodic boundary conditions are imposed. The potential ?x is a real function representing the classical phonon ?eld. At ?nite L, the fermionic Fock space is ?nite dimensional, hence there is a minimum el el eigenvalue EL (?, ?) of the operator HL , for each given phonon ?eld ? and each value of ?; let ρL (?, ?) be the corresponding fermionic density. The aim is to minimize the functional

el FL (?, ?) = EL (?, ?) +

1 2

?2 x ,

x ∈Λ

(1.2)

subject to the condition ρL (?, ?) = ρL , (1.3) where ρL is a ?xed value of the density, converging for L → ∞, say to ρ. The model (1.1) can be considered as an approximation of a more realistic continuous model containing also the interaction with a ?xed external periodic potential of period one. Then the discreteness is not a pure mathematical arti?ce, but it has a precise physical interpretation: the properties of the two models are expected to be the same, and we think that this could be easily proven along the lines of the present paper. 1.2. It is generally believed that, as a consequence of Peierls instability argument, [P,F], there is a ?eld ?(0) , uniquely de?ned up to a spatial translation, which minimizes (1.2), (1.3), in the limit L → ∞, and is a function of the form ? ?(2πρx), where ? ?(u) is a 2π periodic function. This is physically interpreted by saying that one-dimensional metals are unstable at low temperature, in the sense that they can lower their energy through a periodic distortion of the “physical lattice” with period 1/ρ (in the continuous version of the model, since 1/ρ is not an integer in general): such a distortion is called a charge density wave, as the physical lattice and the electronic charge density form a new periodic structure with period bigger than the original lattice period. The argument in [P,F] is quite simple: the periodic potential ? ?(2πρx) opens a gap in the electronic dispersion relation in correspondence of the Fermi momentum, and a trivial computation using degenerate perturbation theory shows that the elastic energy increase is less than the fermionic energy decrease. However, see [LRA], in this argument one does

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not take into account the e?ects due to the discreteness of the lattice, in particular the fact that the momenta are conserved modulo 2π (Umklapp). Neglecting the discreteness of the lattice one loses the di?erence between commensurate or incommensurate charge density wave (i.e. rational or irrational ρ) in the in?nite volume limit, whose properties are supposed to be di?erent, especially concerning the conductivity [F,LRA]. Note also that, even if the argument in [P,F] is perturbative, Peierls instability is expected to arise also for large interaction strength, [AL]. An exact result, [KL,LM], makes rigorous the theory of Peierls instability for the model (1.1) in the case ρ = ρL = 1/2 (half ?lled band case), for any value of λ. In fact, in this case it has been proved that there is a global minimum of FL (?) of the form ε(λ)(?1)x , where ε(λ) is a suitable function of λ. This means that the periodicity of the ground state phonon ?eld is 2 (recall that in our units 1 is just the lattice spacing): this phenomenon is called dimerization. The proof heavily relies on symmetry properties which hold only in the half ?lled case. In [AAR,BM] Peierls instability for the Holstein model is proven assuming λ large enough: in that case the Fermions are almost classical particles and the quantum e?ects are treated as perturbations. The results hold for the commensurate or incommensurate case; in particular in the incommensurate case the function ? ?(u), related to the minimizing ?eld through the relation ?x = ? ?(2πρx), has in?nite many discontinuities. On the contrary, in the small λ case, according to numerical results, ? ?(u) has been conjectured to be an analytic function of its argument, both for the commensurate and incommensurate cases, [AAR]. In this paper we study the case of small λ and any rational density, for which there are, to our knowledge, no results in the literature besides the simulations in [AAR]. Analytical results in the small λ case can be found for a related model, the Falikov-Kimball model, described by a Hamiltonian of the form (1.1), in which the continuous ?x is replaced by a discrete function taking only the values 0 or 1; see [FGM]. Let ρ = P/Q, with P, Q relative prime integers, and let L = Li ≡ iQ, i ∈ N; we shall prove that, if λ is small enough, there are two stationary points ?(±,i) of FLi (?, ?), de?ned up to a lattice translation, satisfying (1.3) with ρL = ρ. These stationary points are periodic functions on Li of period Q (the smallest multiple of 1/ρ which is an integer, hence a multiple of the unit lattice spacing), converging for i → ∞. Moreover, if we restrict FLi (?, ?) to functions such that ?x = ?x+Q , ?(±,i) are local minima in the norm ||?|| = supx∈L |?x |, uniformly in i. The presence of the lattice has the e?ect that we need the smaller λ the bigger Q is, see (1.16). In particular we are not able to draw conclusions about the incommensurate case neither we know if this is a technical limitation or there is some physical reason behind it, so that we can not draw any conclusions about the analyticity conjecture in [AAR]. 1.3. Let hxy = txy ? λ?x δxy be the one-particle Hamiltonian and e1 (?) ≤ e2 (?) ≤ . . . ≤ eL (?) its eigenvalues. We have

el EL (?, ?) =

n:en (?)≤?

[en (?) ? ?] = Tr([h ? ?]P? ) ,

(1.4)

where P? is the projector on the subspace spanned by the eigenvectors of h with eigenvalues el (?, ?) is a di?erentiable function of ? ≤ ?. As it is well known (see, for example [BM]), EL and, since Tr(h? P? /??x ) = 0, ? E el (?, ?) = Tr ??x L ?h P? = ?λρx (?, ?) , ??x (1.5)

where ρx (?, ?) = (P? )xx is the density of the electrons in the point x. Let us now suppose that ? is not equal to any eigenvalue of h. In this case, given ? ?, also ρL (?, ?) is di?erentiable in a neighborhood small enough of ? ? (so small that en (?) ? ?

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stays di?erent from zero, for any n, see again [BM]) and ?ρL (?, ?)/??x = 0. Hence, a local minimum of (1.2) satisfying (1.3) must satisfy the conditions ?x = λρx (?, ?) , 1 ρL = ρx (?, ?) , L x Mxy ≡ δxy ? λ ? ρy (?, ?) ??x is positive de?nite .

(1.6)

(1.7)

Note that, given ? ?, the previous condition on ? can be in general satis?ed only if ||? ? ? ?|| is of order 1/L, so that a solution of (1.6) de?nes in general a local minimum only in a neighborhood of size 1/L. It follows that the only solutions which are interesting in the limit L → ∞ are those associated with a gap of h around ?, whose size is independent of L. 1.4. If ? is a solution of (1.6), it must satisfy the condition ? ?0 = L?1 x ?x = λρL . On the other hand, if we de?ne χx = ?x ?? ?0 , we can see immediately that ρL (?, ?) = ρL (χ, ?+λ? ?0 ). It follows that we can restrict our search of local minima of (1.2) to ?elds ? with zero mean, satisfying the conditions ?x = λ(ρx (?, ?) ? ρL ) , 1 (1.8) ρL = ρx (?, ?) , L x and condition (1.7). Of course, if the ?eld ?x satis?es (1.8), the same is true for the translated ?eld ?x+n , for any integer n. On the other hand, one expects that the solutions of (1.8) are even with respect to some point of Λ; hence we can eliminate the trivial source of non-uniqueness described above by imposing the further condition ?x = ??x . We shall then consider only ?elds of the form

[(L?1)/2]

?x =

n=?[L/2]

? ?′ ne

i2nπx L

,

? ?′ ?′ ?n = ? n ∈ R,

? ?0 = 0 .

(1.9)

As we said in §1.2, we want to consider the case of rational density, ρ = P/Q, P and Q relatively prime, and we want to look for solutions such that ?x = ?x+Q . Hence, we shall look for solutions of (1.8) with L = Li = iQ, ρL = ρ and

[(Q?1)/2]

?x =

n=?[Q/2]

? ?n ei2πρnx ,

? ?n = ? ??n ∈ R ,

? ?0 = 0 .

(1.10)

Note that the condition on L allows to rewrite in a trivial way the ?eld ?x of (1.10) in the general form (1.9), by putting ? ?′ n = 0 for all n such that (2nπ )/L = 2πρm, ?m, and by relabeling the other Fourier coe?cients. The conditions (1.8) can be easily expressed in terms of the variables ? ?n ; if we de?ne ρ ?n so that

[(Q?1)/2]

ρx (?, ?) =

n=?[Q/2]

ρ ?n (?, ?)ei2nπρx ,

(1.11)

we get ? ?n = λρ ?n (?, ?) , n=0, n = ?[Q/2], . . . , [(Q ? 1)/2] , (1.12) (1.13) ρ ?0 (?, ?) = ρL .

Also the minimum condition (1.7) can be expressed in terms of the Fourier coe?cients; we get that the L × L matrix ? nm ≡ δnm ? λ ? ρ M ?′ m (?, ?) ?? ?′ n

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(1.14)

4

has to be positive de?nite, if the ?eld ? satis?es (1.12) and (1.13) and ρ ?′ m (?, ?) is de?ned ′ analogously to ? ?m in (1.9). Hence, if we restrict the space of phonon ?elds to those of the form (1.10), we have to show that the Q × Q matrix ? nm ≡ δnm ? λ ? ρ M ?m (?, ?) ?? ?n has to be positive de?nite, if the ?eld ? satis?es (1.12) and (1.13). 1.5. Remark. It is easy to show (by using the expansion described in §3, for example) ? nm can be di?erent from zero only if 2π (n ? m)/L is of the form 2πρk for some k . that M ? nm not belonging However, we are not able to get good bounds on all matrix elements M (up to a relabeling of indices) also to the matrix (1.15); therefore, in studying the minimum condition, we restrict ourselves to the ?elds of the form (1.10). 1.6. Theorem. Let ρ = P/Q, with P, Q relative prime integers, L = Li ≡ iQ. Then, for any positive integer N , there exist positive constants ε, ε ?, c and K , independent of i, ρ and N , such that, if ?1 ?1 4πv0 v 2 (1 + log v0 ) 0≤ ≤ λ2 ≤ ε 0N (1.16) 4) , log(? εv0 L) K N ! log(c Q/v0 where v0 = sin(πρ) , (1.17) there exist two solutions ?(±) of (1.8), with L = Li , 1 ? ? = cos(πρ) and ρL = ρ, of the form ? corresponding to these solutions, de?ned as in (1.15), are positive (1.10). The matrices M de?nite. (±) Moreover, the Fourier coe?cients ? ?n verify, for |n| > 1, the bound

(±) |? ?n |≤

(1.15)

λ2 v0 |n|

N

|? ?1 | .

(±)

(1.18)

Finally, λ? ?1

(±)

is of the form λ? ?1

(±) 2 = ±v0 exp

?

2πv0 + β (±) (λ, L) λ2 1 v0 ,

,

(1.19)

with |β (±) (λ, L)| ≤ Cλ2 1 + log

(1.20)

where C is a suitable constant. The one-particle Hamiltonian h corresponding to this solution has a gap of order |λ? ?1 | around ?, uniformly on i. 1.7. The above theorem proves that there are two stationary points of the ground state energy in correspondence of a periodic function with period equal to the inverse of the density, if the coupling is small enough and the density is rational, and that these stationary points are local minima at least in the space of periodic functions with that period. The energies associated to such minima are di?erent so that the ground state energy is not degenerate. The theorem is proved by writing ρx (?, ?) as an expansion convergent for small λ and solving the set of equations (1.12) by a contraction method. As a byproduct we prove that the ? ?n are fast decaying, (see (1.18)), so that ?x is really well approximated by its ?rst harmonics (this remark is important as the number of harmonics could be very large). The results are uniform in the volume, so they are interesting from a physical point of view (a solution de?ned only for |λ| ≤ O(1/L) should be outside any reasonable physical value for λ). The case in [KL] for the half ?lled case is contained in Theorem 1.6, but in

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[KL] it is also proved that the solution is a global minimum. On the other hand this case is quite special (see Remarks 2.5 in §2). Finally the lower bound in (1.16) is a large volume condition: this is not a technical condition as, if the number of Fermions is odd, there is Peierls instability only for L large enough. The upper bound for λ in (1.16) requires λ to decrease as Q increases: in particular irrational density are forbidden. This requirement is due to the discreteness of the lattice and to Umklapp phenomena. Note that the dependence of the maximum λ allowed on Q is not very strong as it is a logarithmic one. The case of irrational densities (possible only in the in?nite volume limit), excluded by our theorem, is physically interesting, but the existence of Peierls instability in this case is proven only for large λ, [AL,BM]. In [BGM] ρx (?, ?) is shown to be well de?ned for small λ not only in the rational density case, (in which the proof is almost trivial), but also in the irrational case: in fact the small divisor problem due to the irrationality of the density can be controlled thanks to a Diophantine condition. However to solve the set of equations (1.12) we use a contraction method which is not trivially adaptable in the latter case (see Remarks 2.5 in §2). The same kind of problem arises in proving the positive de?niteness of ? nm in the rational case (and this is the reason why we are able to prove that the stationary M points are local minima only in the space of periodic functions with pre?xed period). As we said above, we do not know if such problems are only technical or there is some physical reason for this to happen.

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2. Solution of the self-consistence equation

2.1. Let ρ = P/Q, with P, Q relatively prime integers such that 0 < P < Q, and L = Li ≡ iQ; we have to look for a solution of (1.12) and (1.13), which is well de?ned for |λ| ≤ ε0 , with ε0 independent of i (otherwise our solution is meaningless from a physical point of view). As discussed in §1.3, this means that our solution has to be looked for in a class of functions for which the one-particle Hamiltonian h has a gap around ? of width independent of L. This class of functions is described by the following lemma, to be proved in §5.4. 2.2. Lemma. Let ?x be a ?eld of the form (1.10), L = Li , 1 ? ? = cos(πρ), |λ? ?1 | > 0 and |λ? ?n | ≤ a|λ? ?1 |/|n|N for some positive constants a and N . Then there exists ε0 > 0, 4 independent of i and ρ, such that, if |λ? ?1 | ≤ ε0 v0 /Q, with v0 = sin(πρ), the one-particle Hamiltonian h has a gap of width ≥ |λ? ?1 |/2 around ?. Moreover, ρ ?n (?, ?) is a continuous function of λ, which converges to a continuous function of λ as i → ∞, and ρ ?0 (?, ?) = ρ. 2.3. We can write the self-consistence equation (1.12) as ? ?n = ?λ2 cn (σ )? ?n + λρ ?n (σ, Φ) , σ ≡ λ? ?1 , Φ ≡ {λ? ?n }|n|>1 , (2.1)

where cn (σ ) depends on ? only through σ . We write ρ ?n as a perturbative expansion in λ (di?erent from the power expansion in λ); this expansion is described in §3. If |n| > 1, we are here de?ning ?λcn (σ )? ?n the contribution to ρ ?n proportional to ? ?n of order 1 in the expansion, while ?σc1 (σ ) is the contribution to ρ ?1 proportional to σ of order ≤ 1 in the expansion (explicit expressions for cn (σ ) and c1 (σ ) will be given in (4.16) and (4.39) respectively); ρ ?n takes into account all the remaining terms of ?rst order plus all terms of order higher than 1. The equation (2.1) has of course the trivial solution ? ?n = 0, ?n, but it is easy to see that this is not a local minimum, by using the expansion for ρ ?n of §3. Therefore we shall look for solutions such that σ = 0, so that we can rewrite (2.1) as λ2 ρ ?1 (σ, Φ) , σ 2 λ ρ ?n (σ, Φ) Φn ≡ λ? ?n = , (1 + λ2 cn (σ )) (1 + λ2 c1 (σ )) = (2.2) |n| > 1 . (2.3)

Note that the equation for n = ?1 does not appear simply because ρ?1 = ρ1 , as a consequence of the condition ? ?n = ? ??n ∈ R, see (1.10). 2.4. We prove Theorem 1.6 in three steps as follows. ? We ?rst study the self-consistence equation (2.3), considering σ as a variable belonging to the interval J = ( ? exp(?π v0 /λ2 ) , exp(?π v0 /λ2 ) ) . (2.4) We ?nd a solution, that we denote Φ(σ ), if λ is small enough. ? We then prove that, if L is large enough, the equation (in λ) 1 + λ2 c1 (σ ) = λ2 ρ ?1 (σ, Φ(σ )) σ (2.5)

has two solutions σ (±) ∈ J , of the form (1.19). Therefore (σ (±) (λ)/λ, Φ(σ (±) (λ))/ λ) turn out to be solutions of (1.12), which verify, thanks to Lemma 2.2, (1.13) with L = Li . ? We ?nally prove that the Hessian matrices (1.15) corresponding to these two solutions are positive de?nite. 2.5. Remarks. The coe?cient ? ?1 has a privileged role with respect to the other coe?cients. In fact, as we shall see in §5, the properties of the system when only ? ?1 is di?erent from 0 are very close to the properties of the case in which all the coe?cients are non vanishing.

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This suggests that the “important” equation is (2.2), so explaining the strategy outlined above. The previous remark also implies that 1 + λ2 c1 (σ ) ? 0. It follows that 1 + λ2 cn (σ ) ? 0, for all n such that 2πρn ? 2πρ (mod 2π ). Since min|n|>1 |2πρn ? 2πρ| = 2π/Q, we can expect that our bounds will not be uniform in Q. This is the reason why Theorem 1.6 can not be extended to irrational density; at most one can hope that a Diophantine condition on ρ is needed, but we have only been able to prove that the Q dependence can be substituted with a dependence on the Diophantine constants in some of the bounds described below. Note also that, if Q = 2, the only equation to discuss is just the equation (2.2) with Φ = 0 and the r.h.s. equal to zero; its solution is well known in this case, see [KL,LM] for example. If Q = 3, again (2.2) is the only equation to discuss, but the r.h.s. is di?erent from zero; however it is easy to prove that the solution has essentially the same properties as in the case Q = 2. Hence, in the following we shall consider only the case Q ≥ 4. The following lemma, furnishing a bound on the constants cn (σ ) and their derivatives, is proven in §4.9. 2.6. Lemma. There exists a constant C , independent of i and ρ, such that, if |n| ≥ 2, |cn (σ )| ≤ C v0 1 + log 1 v0 log Q , (2.6)

C ?cn (σ ) , ≤ ?σ v0 |σ |

(2.7)

2.7. Fixed L = Li , Φ is a ?nite sequence of Q ? 3 elements, which can be thought as a Q?3 vector in R , which is a function of σ . In order to study the equation (2.2) for σ , we shall need a bound on Φ and on the derivative of Φ with respect to σ . Hence we consider Q?3 Q?3 the space F = C 1 ( J , R ) of C 1 -functions of σ ∈ J with values in R ; the solutions of (2.3) can be seen as ?xed points of the operator Tλ : F → F , de?ned by the equation: [Tλ (Φ)]n (σ ) = λ2 ρ ?n (σ, Φ(σ )) , (1 + λ2 cn (σ )) (2.8)

We shall de?ne, for each positive integer N , a norm in F in the following way: Φ We shall also de?ne B = {Φ ∈ F : Φ

F F

≡

sup

|n|>1,σ∈J

|n|N |σ |?1 |Φn (σ )| +

? Φn (σ ) ?σ

.

(2.9)

≤ 1} ; |n| ≥ 2 .

(2.10) (2.11)

R(Φ)n (σ ) = ρ ?n (σ, Φ(σ )) ,

The following two lemmata, to be proved in §5.5 and §5.6, respectively, resume the main properties of R(Φ). 2.8. Lemma. There are two constants C1 > 1 and C2 , independent of i, ρ and N , such that, if Φ, Φ′ ∈ B and ?4 2 C1 Qv0 |σ |[1 + log(v0 /|σ |)] ≤ 1 , (2.12) then R(Φ) ? R(Φ′ )

F

≤

C2 3N N ! v0

1 + log

1 v0

Φ ? Φ′

F

.

(2.13)

2.9. Lemma. There is C > 1, such that, if

?3 2 CQv0 |σ |1/2 [1 + log(v0 /|σ |)] ≤ 1 ,

1/f ebbraio/2008; 6:36

(2.14)

8

then R(0)

F

C ≤ v0

1 1 + log v0

sup

|n|>1

|n|

N

|σ | 2 v0

|n| 10

.

(2.15)

2.10. Lemma. There are ε, c, K , independent of i, ρ and N , such that, if σ ∈ J and λ2 ≤ ε

?1 ?1 2 v0 (1 + log v0 ) 4) , N K N ! log(c Q/v0

(2.16)

there exists a unique solution Φ ∈ B of (2.3); moreover the solution satis?es the bound Φ

F

≤

λ2 v0

N

.

(2.17)

2.11. Proof of Lemma 2.10. It is easy to see that, if σ ∈ J , the conditions on σ of Lemma 2.8 and Lemma 2.9 are satis?ed, if

4 λ2 ≤ ε0 / log(cQ/v0 ),

(2.18)

?1 ?1 ) and ε1 is chosen with suitable values of ε0 and c. Moreover, if ε0 ≤ ε1 v0 (1 + log v0 2 small enough, (2.6) and (2.18) imply that λ |cn (σ )| ≤ 1/2, so that, by using (2.7), (2.8) and Lemma 2.8, we have that, if Φ ∈ B ,

Tλ (Φ)

F

≤ 4λ2 1 + λ2

C v0

R(0)

F

+

C2 v0

1 + log

1 v0

3N N ! Φ

F

.

(2.19)

Therefore, by (2.15) and (2.4), there exist constants C3 and C4 , such that, if ε1 ≤ N εv0 (C4 N !)?1 and ε is small enough, Tλ (Φ)

F

≤

C3 λ2 2 v0

1 + log

1 v0

3N N ! + sup |n|N exp ?

|n|>1

πv0 |n| 10λ2

≤1.

(2.20)

Moreover, by (2.13), if Φ, Φ′ ∈ B and similar conditions on λ are satis?ed, we have Tλ (Φ) ? Tλ (Φ′ )

F

≤

N C5 N !λ2 2 v0

1 + log

1 v0

Φ ? Φ′

F

≤

1 Φ ? Φ′ 2

F

.

(2.21)

The bounds (2.20) and (2.21) imply that B is invariant under the action of Tλ and that Tλ is a contraction on B . Hence, by the contraction mapping principle, there is a unique ? of Tλ in B , which can be obtained as the limit of the sequence Φ(k) de?ned ?xed point Φ through the recurrence equation Φ(k+1) = Tλ (Φ(k) ), with any initial condition Φ(0) ∈ B . If we choose Φ(0) = 0, we get, by (2.21),

∞ ∞

? Φ

F

≤

i=1

Φ(i) ? Φ(i?1)

F

≤

1 2i?1

Φ(1)

F

i=1

≤ Φ(1)

F

.

(2.22)

On the other hand, by (2.15), Φ(1)

F

= Tλ (0)

F

≤

N C6 N !λ2 2 v0

1 + log

1 v0

λ2 v0

N

,

(2.23)

N which immediately implies the bound (2.17), if ε1 ≤ εv0 (C6 N !)?1 , with ε small enough.

2.12. Let us now consider the equation (2.5). We want to prove that it has two solutions of the form (1.19), if σ ∈ J and Li is large enough. In order to achieve this result, we need

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9

some detailed properties of the function c1 (σ ), which are described in the following Lemma 2.13, to be proved in §4.10. We need also the bounds on ρ ?1 (σ, Φ(σ )) and its derivative with respect to σ , contained in Lemma 2.14, to be proved in §5.7. 2.13. Lemma. There is a constant C , such that, if v0 1 ≤ε ?≤ , Li |σ | 8π then ?c1 (σ ) = with |σ | 2 ≤1 , v0 (2.24)

1 v2 log 0 + r1 (σ ) , 2πv0 |σ | 1 , v0 ε ? 1 + . 2 v0 |σ |

(2.25)

|r1 (σ )| ≤ C 1 + log ?r1 (σ ) ≤C ?σ

(2.26)

2.14. Lemma. If σ ∈ J , λ satis?es the inequality (2.16), with ε small enough, Φ(σ ) is the solution of the equation (2.3) described in Lemma 2.10 and r2 (σ ) ≡ then there is a constant C , such that |r2 (σ )| ≤ C C ?r2 (σ ) ≤ ?σ |σ | |σ | 2 v0 |σ | 2 v0

1/4

2πv0 ρ ?1 (σ, Φ(σ )) , σ

(2.27)

+

1/4

λ2 v0 λ2 v0

N

,

N

(2.28) .

+

2.15. Lemma. There exist positive constants ε, ε ?, c and K , independent of i, ρ and N , such that, if λ satis?es the inequalities (1.16), there are two solutions σ (±) (λ) ∈ J of equation (2.5) of the form (1.19). 2.16. Proof of Lemma 2.15. By using the de?nitions of r1 (σ ) and r2 (σ ) given in (2.25) and (2.27), we can write the equation (2.5) in the form F (σ ) ≡ log

2 v0 2πv0 ? 2 + r(σ ) = 0 , |σ | λ

(2.29)

where r(σ ) = r1 (σ ) + r2 (σ ). Let us now suppose that λ satis?es the inequalities (2.16) and that σ belongs to the interval

2 ?4πv0 /λ2 2 ?πv0 /λ2 ? = v0 ?J . J e , v0 e

(2.30)

If Li is large enough and the constant ε in (2.16) is chosen small enough, the conditions ?, and (2.24) of Lemma 2.13 are satis?ed, for σ ∈ J 4πv0 ≤ log(? εv0 Li ) . λ2

?2 Moreover, if ε ? and ε (hence |σ |v0 ) are small enough,

(2.31)

1 ? v2 ?r(σ ) ≤ log 0 ; ?σ 2 ?σ σ

1/f ebbraio/2008; 6:36

(2.32)

10

?. If we de?ne hence F (σ ) is a monotone decreasing function of σ in J

2 ?2πv0 /λ σ ? = v0 e ,

2

M = sup |r(σ )| ,

? σ ∈J

(2.33)

we have that F (σ ? exp(?2M )) > 0 and F (σ ? exp(2M )) < 0. Moreover, the interval ?, if ε is small enough, since the bounds (2.26) (σ ? exp(?2M )), σ ? exp(2M ))) is contained in J ?1 and (2.28) imply that M ≤ C (1+log v0 ). Hence there is a unique solution σ (+) (λ) of (2.29) ?, which can be written as in J

2 ? e σ (+) (λ) = v0

2πv0 +β (+) (λ) λ2

,

(2.34)

?1 with |β (+) (λ)| ≤ Cλ2 (1 + log v0 ). In the same manner, we can show that there is solution σ (?) (λ) in the interval 2 ? (?v0 e

πv0 λ2

2 ? , ?v0 e

4πv0 λ2

)?J ,

(2.35)

with the same properties. 2.17. Lemma. The constants ε, ε ?, c and K , appearing in (1.16), can be chosen so that the Hessian matrix (1.15) is positive de?nite. 2.18. The proof of Lemma 2.17 is in §5.8. This completes the proof of Theorem 1.6.

1/f ebbraio/2008; 6:36

11

3. Graph formalism

3.1. In this section we shall describe the expansion of ρx (?, ?), used to get the results of this paper. ± ± ?Ht Let us consider the operators ψx = etH ψx e , with x = (x, t), ?β/2 ≤ t ≤ β/2 for some β > 0; on t antiperiodic boundary conditions are imposed. As explained, for example, in [BGM], there is a simple (well known) relation between ρx (?, ?) and the two-point Schwinger function, de?ned by S L,β (x; y) = given by ρx = ? lim lim

β →∞ τ →0? ? + ? + Tr exp(?βH ) θ(x0 > y0 )ψx ψy ? θ(x0 < y0 )ψx ψy Tr [exp(?βH )]

,

(3.1)

1 L,β S (x, τ ; x, 0) . L

(3.2)

In [BGM], which we shall refer to for more details, it is also explained that the two-point Schwinger function can be written as S L,β (x; y) = lim

? + P (dψ ) eV (ψ) ψx ψy , P (dψ ) eV (ψ)

M →∞

(3.3)

± where ψx are now anticommuting Grassmanian variables and P (dψ ) is a Grassmanian Gaussian measure, formally de?ned by ? ? ? ? + ? P (dψ ) = (Lβ g ?(k))?1 ψk ψk dψ ? dψ + , (3.4) (Lβ g ?(k)) exp ? ? ? k∈DL,β k∈DL,β

k = (k, k0 ), DL,β ≡ DL × Dβ , DL ≡ {k = 2πn/L, n ∈ Z, ?[L/2] ≤ n ≤ [(L ? 1)/2]}, Dβ ≡ {k0 = 2(n + 1/2)π/β, n ∈ Z, ?M ≤ n ≤ M ? 1}, in the limit M → ∞, g ?(k) = 1 ?ik0 + cos pF ? cos k (3.5)

is the propagator or the covariance of the measure, pF = πρ is the Fermi momentum, de?ned so that cos pF = 1 ? ?, and

β/2

V (ψ ) =

x ∈Λ

?β/2

+ ? ψx . dx0 λ?x ψx

(3.6)

If we insert (1.10) in the r.h.s. of (3.6), we get

[(Q?1)/2]

V (ψ ) =

n=?[Q/2]

1 Lβ

+ ? λ? ?n ψk ψk+2npF , k∈DL,β

(3.7)

where pF = (pF , 0) and k + 2npF is of course de?ned modulo 2π . 3.2. Note that g ?(k)?1 is small for k ? ±pF . Hence there is no hope to treat perturbatively the terms with n = ±1 and k near ?pF , but we can at most expect that the interacting measure is a perturbation of the measure (whose covariance is not singular at k = ±pF ) ?λ (dψ ) ≡ 1 P (dψ ) P N ? ? 1 exp λ? ?1 ? β

k0 ∈Dβ

1 L

? + + ? ψk ψk+2pF + ψk +2pF ψk k∈I?

? ? ?

(3.8) ,

1/f ebbraio/2008; 6:36

12

where N is a normalization constant and I? is a small interval centered in ?pF , so small ? + 2 pF , k ? ∈ I? }. that I? ∩ I+ = ?, if I+ ≡ {k = k This remark suggests to apply a multiscale expansion to the integral (3.3), in order to treat in a di?erent way the momenta near ±pF and the others. This procedure was applied in [BGM] to study systems of electrons in presence of a potential of the form ? ?(2px), with ? ? 2π -periodic, p/π an irrational Diophantine number and pF = mp, m arbitrary integer. In [BGM] the aim was mainly to get the best possible results about the dependence of the two-point Schwinger function on λ and we found it useful to realize the multiscale expansion by dividing the momenta near pF into a number of “slices” of order | log λ|. This expansion could be applied also the case pF = p with ρ = pF /π rational, without no important di?erence, and we could get immediately Lemma 2.2. However, in this paper we prefer to use a simpler expansion into only two scales; this expansion gives weaker results about the dependence on λ, but it is su?cient in order to prove Lemma 2.2 and it is more suitable for studying the equation (1.12). 3.3. Let us introduce a smooth positive function f0 (k ′ ) on the one dimensional torus T1 , such that 1, if k ′ T1 ≤ t0 /2 , f0 (k ′ ) = (3.9) 0, if k ′ T1 ≥ t0 , where t0 = min{pF /2, (π ? pF )/2} .

′ ′

(3.10)

′

and the norm k T1 on T is de?ned so that k T1 = |k |, if k ∈ [?π, π ). Then we write ? ? ? f f (3.11) 1 (k ) = 1 ? f0 (k + pF ) ? f0 (k ? pF ) , 0 (k ) = 1 ? f1 (k ) , so that (3.5) becomes g ?(k) = g ?(1) (k) + g ?(0) (k) =

h=0,1

′

1

? f h (k ) , ?ik0 + cos pF ? cos k

(3.12)

and, for h = 0, we de?ne g ?(0) (k) =

ω =± 1 (0) g ?ω (k) ,

(3.13)

where, if k′ = k ? ω pF ,

(0) ′ (0) ′ g ?ω (k + ω pF ) ≡ g ?ω (k ) =

f0 (k ′ ) , ?ik0 + cos pF ? cos(k ′ + ωpF )

(3.14)

(1) ? with k′ = (k ′ , k0 ); we set also g ?1,1 (k′ ) ≡ g ?(1) (k), with k = k ′ + pF , and f1 (k ′ ) = f 1 (k ), in order to simplify the notations in the following sections.

3.4. We can associate with the decomposition (3.12) of g ?(k) a decomposition of the Grassmanian Gaussian measure P (dψ ) into a product of two independent Grassmanian Gaussian measures: P (dψ ) = P (dψ (1) )P (dψ (0) ) , (3.15) if P (dψ (i) ) is de?ned as in (3.4), with g ?(i) (k) in place of g ?(k). If we insert (3.15) in (3.3) and we perform the integration over the ?eld ψ (1) , it is easy to show (see [BGM], §4) that S L,β (x; y) = g (1) (x; y) + Kφ,φ (x; y) + S (0) (x; y) , where g (1) (x; y) = (Lβ )?1 S (0) (x; y) = e

1/f ebbraio/2008; 6:36

(0)

(3.16)

k∈DL,β

g ?(1) (k) exp[?ik · (x ? y)] and P (dψ (0) )

(0)+

?2

? ?φ+ x ?φy d x φ+ x ψx

(0)?

1 N0

(3.17)

φ? x

+ψ x

eV 13

(0)

(ψ (0) )+W (0) (ψ (0) ,φ) φ+ =φ? =0

;

in (3.17)

dx is a shortcut for

± N0 = P (dψ (0) ) exp[V (0) (ψ (0) )], with {ψx } (0) (0) and V (ψ ), the e?ective potential on the small momenta scale, can be easily represented (0) as a series in λ, as well as the function W (0) (ψ (0) , φ) and the function Kφ,φ (x; y) appearing in (3.16). A precise description of these series in terms of Feynman graphs can be found in [BGM]; see in particular the equations (4.3), (4.4) and (4.6) in [BGM]. Here we want only to stress that the involved graphs are chains formed by vertices, associated with the Fourier components of the potential ? ?n , connected through lines, associated to the propagators g (1) ; hence by using the bound (3.27) below, it is easy to prove that these series are convergent, uniformly in L and β . On the contrary, the series obtained by integrating the ?eld ψ (0) do not have this property, for the reason explained before, and we have to look for a di?erent expansion, based on the idea outlined in §3.2. (0) (0) We can associate with the decomposition (3.13) of g ?(0) (k) a decomposition ψk = ψk,+ +

β/2 x∈Λ ?β/2 dx0 , {φ± x } are Grassmanian variables anticommuting

ψk,? of the ?eld ψk . The support properties of f0 (k ′ ), see (3.9), and the de?nition (3.14) supports of ψk,+ and ψk,? are disjoint. The idea is to modify the free measure P (dψ (0) ) by multiplying it by the terms present in (0) (0) (0) V (ψ (0) ), (see [BGM], §3), which couple the variables ψk,? and ψk+2pF ,+ ; then we expand the integral by using the new measure as the free measure. The new graphs di?er from the previous ones for two respects; ?rst of all they are not singular anymore at k = ±pF , but they are bounded by C |λ? ?1 |?1 , see below; moreover the two propagators exiting and entering in the same vertex can not have both the momentum equal to ±pF . As we shall see, these two properties are su?cient to control the expansions. The two properties of the new free measure described above are realized also if we only (0) (0) extract from V (0) (ψ (0) ) the ?rst order terms coupling ψk,? and ψk+2pF ,+ . It is easy to see that these terms are equal to λ? ?1 Fσ (ψ (0) ), with

(0) Fσ (ψ (0) ) = (0)

(0)

(0)

imply that the ?eld ψk,ω has support on the set {k = k′ + ω pF : f0 (k ′ ) = 0} and that the

(0) (0)

(0)

1 Lβ ω =± 1

ψk′ +ωpF ,ω ψk′ ?ωpF ,?ω .

k′ ∈DL,β

(0)+

(0)?

(3.18)

Hence we de?ne

(0) ?1 Fσ (ψ (0) ) ? (dψ (0) ) = 1 P (dψ (0) )eλ? , P N where N is a suitable constant, and

(3.19)

(0) ? (0) (ψ (0) ) = V (0) (ψ (0) ) ? λ? V ?1 Fσ (ψ (0) ) .

(3.20)

? (dψ (0) ) By proceeding as in [BGM], §3, one can show that the Grassmanian integration P has propagator ′ (0) g (0) (x; y) = e?i(ωx?ω y)pF gω,ω′ (x; y) , (3.21)

ω,ω ′ =±1

with gω,ω′ (x; y) =

(0) g ?ω,ω (k′ ) = (0)

1 Lβ

e?ik ·(x?y) g ?ω,ω′ (k′ ) ,

k′ ∈D

L,β

′

(0)

(3.22)

σ0 (k ′ ) = σf0 (k ′ ), if σ = λ? ?1 = 0, and (see (1.17) for the de?nition of v0 ) F1 (k ′ ) = (cos k ′ ? 1) cos pF ,

1/f ebbraio/2008; 6:36

[?ik0 ? F1 (k ′ ) ? ωF2 (k ′ )] f0 (k ′ ) 2 (k ′ ) + σ 2 (k ′ )] , [?ik0 ? F1 (k ′ )]2 ? [F2 0 ′ ′ [ σ ( k )] f ( k ) 0 0 (0) g ?ω,?ω (k′ ) = 2 (k ′ ) + σ 2 (k ′ )] , [?ik0 ? F1 (k ′ )]2 ? [F2 0

(3.23)

F2 (k ′ ) = v0 sin k ′ .

(3.24)

14

3.5. Remark. Note that, if |k ′ | ≤ t0 , 2(1 ? cos k ′ )/| sin k ′ | = 2| tan(k ′ /2)| ≤ 2 tan(t0 /2) ≤ | tan pF |; hence 1 for |k ′ | ≤ t0 . (3.25) |F1 (k ′ )| ≤ |F2 (k ′ )| , 2 It immediately follows that

2 + (v k ′ )2 k0 0 f0 (k ′ ) , + (v0 k ′ )2 + σ 2 |σ | (0) f0 (k ′ ) . |g ?ω,?ω (k′ )| ≤ C 2 k0 + (v0 k ′ )2 + σ 2 (0) |g ?ω,ω (k′ )| ≤ C 2 k0

(3.26)

where C denotes a suitable constant. From now on, for simplifying the notation, the symbol C will be used everywhere to denote a generic constant, that we do not need to better specify. It is also easy to prove that |g ?1,1 (k′ )| ≤ C

(1)

|k0 |2 + (v0 k ′ )2

1 ? f0 (k ′ )

,

(3.27)

3.6. We now insert (3.19) and (3.20) in (3.17) and represent the result of the integration ? (dψ (0) ) as the free measure and V ? (0) (ψ (0) ) as the in terms of Feynman graphs, by using P e?ective potential; then we apply (3.2). By proceeding as in [BGM], it is easy to show that we get an expansion for ρx , which can be described in the following way. 3.7. A graph ? of order q ≥ 1 is a chain of q + 1 lines ?1 , . . . , ?q+1 connecting a set of q ordered points (vertices) v1 , . . . , vq , so that ?i enters vi and ?i+1 exits from vi , i ≤ q ; the lines ?1 and ?q+1 are the external lines of the graph and both have a free extreme, while the others are the internal lines; we shall denote int(?) the set of all internal lines. We say that ′ vi < vj if vi precedes vj and we denote vj the vertex immediately following vj , if j < q . We denote also by ?v the line entering the vertex v , so that ?i ≡ ?vi , 1 ≤ i ≤ q . We say that a line ? emerges from a vertex v if ? either enters v (? = ?v ) or exits from v (? = ?v′ ). By a ′ the line ?q +1 exiting from vq even slight abuse of notation, if v = vq , we still denote by ?vq if there is no vertex vq+1 . We shall say that ? is a labeled graph of order q ≥ 1, if ? is a graph of order q , to which the following labels are associated: ? a label nv for each vertex, ? a frequency (or scale) label h? for each (internal or external) line, with the constraint that, if nv = ±1 for some v , then h?v = h?v′ = 0 is not possible, 1 2 2 1 = 1 if h? = 1, = ω? ? for each line ?, two labels ω? , ω? , such that ω? ′ ? a momentum k?1 = k = k + ω1 pF for the ?rst line, ? a momentum 1 2 (3.28) k?v = k ′ + ? ω? )pF 2 nv ? pF + (ω?v ? v ?′

v ?<v

for each internal line, ? a momentum

k?q+1 = k ′ +

v ∈?

1 2 ? ω? )pF 2 nv ? pF + (ω?v ? v ?′

(3.29)

for the last line. (h ) (k′ If g ?ω1? ? ) denotes the propagator associated with the line ?, we will use the shorthand ,ω 2 g ?? = g ?ω1? (k′ ? ). ,ω 2

? ? 1/f ebbraio/2008; 6:36

(h )

?

?

15

Let us call Tn,q the set of the labeled graphs of order q and such that 2 nv pF +

v ∈? ?∈? 2 1 ω? ? ω? pF = 2npF

mod 2π .

(3.30)

Then, if ρ ?n (?, ?) is de?ned as in (1.11), we have 1 ρ ?n (?, ?) = lim ?? β →∞ Lβ =

?∈Tn,q

?

∞

δn,ω g ??ω,ω (k′ ) +

k′ ∈DL,β ω =±1 q=1

ρq n (σ, Φ) where

Val(?) ,

? , ρq n (σ, Φ)

?

(3.31)

Val(?) = ?

1 Lβ

q+1

g ??i

k′ ∈DL,β i=1 v ∈?

λ? ?nv

.

(3.32)

Hence, the function ρ ?n (σ, Φ) de?ned in (2.1) can be written as

∞

ρ ?n (σ, Φ) = lim

β →∞

ρ ?q n (σ, Φ)

q=1

q ρ ?q n (σ, Φ) = ρn (σ, Φ) ,

ρ ?1 n (σ, Φ) =

?∈Tn,1

if q ≥ 2

(3.33)

(1 ? δn,nv )Val(?) .

after substituting in the r.h.s. of (3.32) λ? ?nv either with Φnv , if |nv | > 1, or with σ , if |nv | = 1.

1/f ebbraio/2008; 6:36

16

4. First order graphs

4.1. In this section we study the ?rst order contributions to the density, i.e. the terms corresponding to graphs with only one vertex in the perturbative expansion (3.31), calculated in the limit β → ∞. For these graphs we have, if L = Li = iQ,

β →∞

lim Val(?) = ? λ? ?m 1 L

∞

′ k′ ∈DL

?∞

dk0 (h) (h ′ ) ′ ? ω2 )pF ) , ?ω2 ,ω′ (k′ + (2m + ω1 g ?ω1 ,ω′ (k′ ) g 2 1 2π

(4.1)

′ where DL is the set of possible values of the variable k ′ introduced before (3.14) as the di?erence between the “space momentum” k and ±pF . Since pF = πρ = πP/Q = (2π/L)(iP/2), we have 2π ′ (n + δ/2), n ∈ Z, ?[L/2] ≤ n ≤ [(L ? 1)/2]} , (4.2) DL = {k ′ = L where δ = 1, if iP (the number of particles) is odd, while δ = 0, if iP is even. Note that the value of δ will be in general not relevant, except in the proof of Lemma 2.13 in §4.10, the only place were there is a non trivial dependence on the volume. Note also that, if the graph value (4.1) contributes to ρ ?n (?, ?), then ′ ′ 2mpF = 2npF + (ω1 ? ω1 + ω2 ? ω2 )pF mod 2π .

(4.3)

The r.h.s of (4.1) can be easily bounded, by using (3.27) and (3.26) and the remark that limpF →0 t0 /v0 = 1/2. If h = h′ = 1, one gets, for any integer r, 1 L C

?π 2π ∞

′ k′ ∈DL

?∞

dk0 (1) ′ (1) ′ g ? (k ) g ?1,1 (k + 2rpF ) ≤ 2π 1,1 dk0 [1 ? f0 (k ′ )]

2 + (v k ′ )2 k0 0

π

dk ′ dk ′

∞ ?∞ ∞ 0

[1 ? f0 (k ′ + 2rpF )] 1 + log 1 v0 .

2 + v 2 (k ′ + 2rp )2 k0 F 0

≤

(4.4)

C

t0

C dk0 2 + (v k ′ )2 ≤ v k0 0 0

∞

If h = 0, h′ = 1, for any ω, ω ′ , one gets 1 L C

0

′ k′ ∈DL

?∞

dk0 (0) (1) g ? ′ (k′ ) g ?1,1 (k′ + 2rpF ) ≤ 2π ω,ω dk0

2 + (v k ′ )2 k0 0

t0

dk ′ 2π

∞ 0

C . ≤ 2 + v4 v 0 k0 0

(4.5)

The bound (4.5) can be improved for σ → 0, if ω ′ = ?ω ; we have 1 L

∞

′ k′ ∈DL

?∞

dk0 (0) (1) g ? (k′ ) g ?1,1 (k′ + 2rpF ) ≤ 2π ω,?ω

∞ 0

t0

C |σ |

0

dk ′ 2π

C |σ | ≤ 3 2 2 4 ′ 2 2 v0 [k0 + (v0 k ) + σ ] k0 + v0 dk0

v2 1 + log 0 |σ |

(4.6) .

′ Let us now consider the case h = h′ = 0; for any ωi , ωi , we get

1 L C

∞

′ k′ ∈DL

?∞

dk0 (0) (0) ′ ?ω2 ,ω′ (k′ + 2rpF ) ≤ g ? ′ (k ) g 2 2π ω1 ,ω1 dk0 f0 (k ′ )

2 + (v k ′ )2 + σ 2 k0 0

π ?π t0

dk ′ dk ′

∞ ?∞ ∞ 0 2 k0

f0 (k ′ + 2rpF )

2 + v 2 (k ′ + 2rp )2 + σ 2 k0 F 0 2 v0 |σ |

≤

(4.7)

C

0

1/f ebbraio/2008; 6:36

C dk0 ≤ + (v0 k ′ )2 + σ 2 v0 17

1 + log

.

′ ′ If h = h′ = 0 and ω1 = ω1 or ω2 = ω2 , the last bound can be improved; in fact we get

1 L

∞

′ k′ ∈DL

?∞

dk0 (0) (0) ′ g ? ?ω2 ,ω′ (k′ + 2rpF ) ≤ ′ (k ) g 2 2π ω1 ,ω1

∞ 0

t0

(4.8) C dk0 . ≤ 2 ′ 2 2 3 / 2 v0 [k0 + (v0 k ) + σ ]

C |σ |

dk ′

0

The previous bound can be further improved, if we suppose also that r = 0, by taking into account that, in this case, max{|k ′ |, |k ′ + 2rpF |} ≥ π/Q. Let us suppose, for example, ′ that ω2 = ?ω2 ; we have 1 L

∞

′ k′ ∈DL

?∞

dk0 (0) (0) ′ ?ω2 ,?ω2 (k′ + 2rpF ) ≤ g ? ′ (k ) g 2π ω1 ,ω1

∞

π

C |σ |

dk ′

0 t0 0

dk0

∞ 0 2 k0

?π

f0 (k ′ + 2rpF ) ≤ 2 + (v k ′ )2 + σ 2 [k 2 + v 2 (k ′ + 2rpF )2 + σ 2 ] k0 0 0 0 C |σ |Q dk0 ≤ 2 + (v0 k ′ )2 + σ 2 v0 1 + log

2 v0 |σ |

f0 (k ′ )

(4.9)

C |σ |Q v0

dk ′

.

In the following we shall need also the bounds of the expression obtained substituting in the r.h.s. of (4.1) one of the two propagators with its derivative with respect to σ . By proceeding as before, one can easily prove that, for any ω and any integer r, 1 L 1 L

∞

′ k′ ∈DL

∞

′ k′ ∈DL

?∞ (0)

?ω,ω (k′ ) (1) ′ C dk0 ? g g ?1,1 (k + 2rpF ) ≤ 3 , 2π ?σ v0

2 v0 |σ |

(0)

(4.10)

?∞

?ω,?ω (k′ ) (1) ′ C dk0 ? g g ?1,1 (k + 2rpF ) ≤ 3 2π ?σ v0

1 + log

;

(4.11)

′ that, for any ωi , ωi and any integer r,

1 L

∞

′ k′ ∈DL

?∞

?ω1 ,ω′ (k′ ) (0) C dk0 ? g 1 g ?ω2 ,ω′ (k′ + 2rpF ) ≤ , 2 2π ?σ v0 |σ |

(0)

(4.12)

′ and ?nally that, for any ω1 , ω1 , ω and any integer r = 0,

1 L 1 L

∞

′ k′ ∈DL

?∞

?ω1 ,ω′ (k′ ) (0) CQ dk0 ? g 1 g ?ω,?ω (k′ + 2rpF ) ≤ 2 , 2π ?σ v0 ?g ?ω,?ω ′ CQ dk0 (0) g ?ω1 ,ω′ (k′ ) (k + 2rpF ) ≤ 2 . 1 2π ?σ v0

(0)

(0)

(4.13)

∞

′ k′ ∈DL

(4.14)

?∞

4.2. Remark. All the previous bounds are valid also if we exchange in the l.h.s. k′ with k′ + 2rpF ; this immediately follows from the observation that the variable k ′ is de?ned modulo 2π . 4.3. We shall now consider the graphs contributing to the constants cn (σ ) introduced in (2.1), in order to prove Lemma 2.6. We can write ?λ? ?n cn (σ ) =

1/f ebbraio/2008; 6:36

δn,nv Val(?) .

?∈Tn,1

(4.15)

18

The equations (4.15), (4.1) and (4.3) imply, if |n| > 2, cn (σ ) = +

ω =± 1

1 L

∞

′ k′ ∈DL

?∞

dk0 (1) ′ (1) ′ g ? (k ) g ?1,1 (k + 2npF ) 2π 1,1 (4.16)

(0) (0)

(0) g ?1,1 (k′ ) g ?ω,ω (k′ + 2npF + (1 ? ω )pF ) (1)

(1)

(0) +? gω,ω (k′ ) g ?1,1 (k′ + 2npF ? (1 ? ω )pF ) + g ?ω,?ω (k′ ) g ??ω,ω (k′ + 2npF ) (0) (0) (0) +? gω,ω (k′ ) g ?ω,ω (k′ + 2npF ) + g ?ω,ω (k′ ) g ??ω,?ω (k′ + (2n + 2ω )pF ) (0)

.

By using the bounds (4.4), (4.5) and (4.8), we see that the ?rst four terms in the r.h.s. of ?1 (4.16) are bounded by (C/v0 )(1 + log v0 ). However, the remaining terms, i.e. those with ′ ′ ′ h = h = 0 and ω1 ? ω1 = ω2 ? ω2 = 0, need a more careful analysis; these terms will be denoted as ∞ 1 dk0 (0) ′ (0) ′ an ≡ g ? (k ) g ?ω,ω (k + 2npF ) , (4.17) L ′ ′ ?∞ 2π ω,ω

k ∈DL

when ω1 = ω2 , and bn,ω = 1 L

∞

′ k′ ∈DL

?∞

dk0 (0) ′ (0) g ? (k ) g ??ω,?ω (k′ + (2n + 2ω )pF ) . 2π ω,ω

(4.18)

when ω1 = ?ω2 . The following two Lemmata 4.5 and 4.7 show that the dimensional bounds which would follow from (4.7) in fact can be improved. 4.4. Remark. Note that an is a ω -independent quantity, so that we can set ω = 1 in (0) (0) (4.17); this property easily follows from the observation that gω,ω (k ′ , k0 ) = g?ω,?ω (?k ′ , k0 ), see (3.23). It is also easy to prove that bn,1 = b?n,?1 . 4.5. Lemma. Let |n| ≥ 2 and let an be de?ned as in (4.17); then |an | < C/v0 . 4.6. Proof of Lemma 4.5. By Remark 4.4, it is enough to study the case ω = 1 in (4.17). De?ne f0 (k ′ ) (0) g ?ω,ω (k′ ) = , ?ik0 + F (ωk ′ ) (4.19)

2 (k ′ ) + σ 2 (k ′ ) ? F (k ′ ) , F (k ′ ) ≡ sign (k ′ ) F2 1 0

and a ?n ≡ =

1 L 1 L

∞

′ k′ ∈DL

?∞

dk0 (0) ′ (0) ′ g ? (k )? g1,1 (k + 2npF ) 2π 1,1 (4.20)

′ k′ ∈DL

f0 (k ′ ) f0 (k ′ + 2npF ) An (k ′ ) .

where An (k ′ ) is obtained by explicitly performing the integral on k0 . It is easy to see that, de?ning s(k ′ ) = sign F (k ′ ) , (4.21) if s(k ′ ) = s(k ′ + 2npF ), one has An (k ′ ) = 0, while, if s(k ′ ) = ?s(k ′ + 2npF ), one has An (k ′ ) = s(k ′ ) F (k ′ + 2npF ) ? F (k ′ )

?1

.

(4.22)

Note that, by (3.25), s(k ′ ) = sign (k ′ ), if |k ′ | ≤ t0 , i.e. on the support of f0 (k ′ ); hence we have f0 (k ′ ) f0 (k ′ + 2npF ) 1 , (4.23) a ?n = ? L ′ ′ ′ |F (k ′ )| + |F (k ′ + 2npF )|

k ∈DL ∩D?

1/f ebbraio/2008; 6:36

19

where

′ D? = {k ′ ∈ [?t0 , t0 ] : sign (k ′ ) = ?sign (k ′ + 2npF )} .

(4.24)

We want to show that max{|F (k ′ )|, |F (k ′ + 2npF )|} ≥ c2 2npF T1 ≡ ?1 , 2 (4.25)

√ ′ if k ′ ∈ D? , k ′ + 2npF ∈ [?t0 , t0 ] and c2 = ( 2/π )v0 . If |F (k ′ )| ≥ ?1 , (4.25) is immediately veri?ed. Let us suppose now that |F (k ′ )| < ?1 ; then, by using (3.25), we get |F (k ′ )| ≥ |F2 (k ′ )| ? |F1 (k ′ )| ≥ so that |k ′ | < implying k ′ + 2npF T1 ≥ Moreover, since k ′ + 2npF 1 |F2 (k ′ )| > c2 |k ′ | , 2 (4.26)

?1 1 = 2npF T1 , c2 2

(4.27)

1 2npF T1 . 2npF T1 ? |k ′ | ≥ 2 T1 ≤ t0 , then

(4.28)

|F (k ′ + 2npF )| > c2 k ′ + 2npF T1 ; hence, by using (4.28) and (4.29), we get |F (k ′ + 2npF )| ≥ which implies (4.25) also when |F (k ′ )| < ?1 . Inserting (4.25) into (4.23) leads to |a ?n | ≤ √ dk ′ 2 , ≤ πc2 2npF T1 v0 c2 2npF T1 = ?1 , 2

(4.29)

(4.30)

(4.31)

′ D?

′ as the size of the set D? is bounded by 2 2npF T1 .

In order to complete the proof of Lemma 4.5, we note that an ? a ?n = + 1 L

∞

′ k′ ∈DL

?∞

dk0 2π

g ?1,1 (k′ ) ? g ?1,1 (k′ ) g ?1,1 (k′ + 2npF )

(0) g ?1,1 (k′

(0)

(0)

(0)

(4.32)

(0) g ?1,1 (k′ )

(0) g ?1,1 (k′

+ 2npF ) ?

+ 2npF )

.

Moreover, by (3.23) and (4.19), ?1,1 (k′ )| = |g ?1,1 (k′ ) ? g

(0) (0) 2 (k ′ ) ? |F (k ′ )| F2 (k ′ )2 + σ0 2 |f0 (k ′ )| , 2 (k ′ ) + σ 2 (k ′ )] [?ik0 ? F1 (k ′ )]2 ? [F2 0

(4.33)

so that, by using also (3.25), we get

t0

|an ? a ?n | ≤ C

dk ′

0

∞

dk0

0

σ2 C . ≤ 2 ′ 2 2 ′ 2 2 3 / 2 v (v0 k ) + σ [k0 + (v0 k ) + σ ] 0

(4.34)

The bounds (4.31) and (4.34) imply Lemma 4.5. 4.7. Lemma. Let |n| ≥ 2, and let bn,ω be de?ned as in (4.18); then |bn,ω | ≤ (C log Q)/v0 . 4.8. Proof of Lemma 4.7. Let us de?ne ? bn,ω as 1 ? bn,ω = L

1/f ebbraio/2008; 6:36

∞

′ k′ ∈DL

?∞

dk0 (0) ′ (0) g ? (k ) g ??ω,?ω (κ′ ω , k0 ) , 2π ω,ω 20

(4.35)

(0) ′ bn,ω + I ′ . where κ′ ?ω,ω (k′ ) is de?ned in (4.19), and set bn,ω = ? ω = k + (2n + 2ω )pF and g By dimensional bounds analogous to those which led to (4.34), it is easy to prove that |I ′ | ≤ C/v0 . Moreover, by proceeding as in §4.6, we see that

1 ? bn,ω = ? L

′ ∩D ′ k′ ∈DL ω

f0 (k ′ ) f0 (κ′ ω) , ′ |F (ωk )| + |F (?ωκ′ ω )|

(4.36)

′ where Dω = {k ′ ∈ [?t0 , t0 ] : sign (ωk ′ ) = sign (ωκ′ ω )}. By using the bound (4.26), we have ′ ′ |F (ωk ′ )| + |F (?ωκ′ ω )| ≥ c2 (|k | + κω T1 )

(4.37)

Moreover, since |n| ≥ 2, 2npF ± 2pF T1 ≥ |? bn,ω | ≤ 2πc2

2π Q;

hence (4.38)

′ Dω

(|k ′ |

dk ′ C log Q . ≤ ) + κ′ v0 ω T1

This completes the proof of Lemma 4.7. 4.9. Proof of Lemma 2.6. The bound (2.6) immediately follows from the remark after (4.16), Lemma 4.5 and Lemma 4.7. The bound (2.8) is easily proven from (4.16) by using the bounds (4.10)÷(4.12). 4.10. Proof of Lemma 2.13. The de?nition of c1 (σ ) in §2.3, (3.31), (4.1) and (4.3) imply that c1 (σ ) = +

ω =± 1

1 L

∞

′ k′ ∈DL

?∞

??1,1 (k′ ) dk0 g (1) (1) +g ?1,1 (k′ ) g ?1,1 (k′ + 2pF ) 2π σ

(1)

(0)

(0) (0) g ?1,1 (k′ ) g ?ω,ω (k′ + (3 ? ω )pF ) + g ?ω,ω (k′ ) g ?1,1 (k′ + (1 + ω )pF )

(1)

≡ ?F (σ, L) + c ?1 (σ ) ,

(4.39)

where ?F (σ, L) denotes the ?rst term in the r.h.s. of (4.39), while c ?1 (σ ) is the sum of the other ones. It turns out that ?F (σ, L) is the leading term for σ → 0; moreover it is the only term whose dependence on L is not trivial, hence we decide to indicate it explicitly. By using the de?nition (3.23) and by performing the integration over k0 , we get F (σ, L) = 1 2L f0 (k ′ )2

′ k′ ∈DL

2 sin2 k ′ + σ 2 f (k ′ )2 v0 0

.

(4.40)

′ The de?nition (4.2) of DL implies that, for any ?nite volume L ≡ Li , the r.h.s. is singular for σ → 0, only if δ = 0, that is only if the number of Fermions is even; in that case, in fact, ′ k ′ = 0 belongs to the set DL . It follows that, if δ = 1, the equation (2.5) has no solution for 2 λ very small, how small depending on L; this is the main source of the lower bound on λ of Theorem 1.6. [Equivalently, for ?xed λ verifying the inequality to the right in (1.16), which is uniform in L, L has to be large enough so that also the inequality to the left in (1.16) can be ful?lled.] We separate the term with k ′ = 0, if it is present, by writing

F (σ, L) = It is easy to see that

1?δ + F0 (σ, L) . 2Lσ

(4.41)

F0 (σ, L) = F1 (σ, L) + d1 (σ, L) + d2 (σ, L) ,

1/f ebbraio/2008; 6:36

(4.42)

21

where F1 (σ, L) =

n=0:|2πL?1 (n+δ/2)|≤t0 /2

(2πv0

)2 (n

1 , + δ/2)2 + (σL)2 ,

(4.43)

d1 (σ, L) =

1 2L

f0 (k ′ )2

k′ ∈D′ L |k′ |≥t0 /2

2 sin2 k ′ + σ 2 f (k ′ )2 v0 0

(4.44)

1 d2 (σ, L) = 2L

k′ ∈D′ L 0=|k′ |≤t0 /2

? ?

1

2 v0

sin

2

k′

+

σ2

?

1 (v0 k ′ )2 + σ 2

?

? .

(4.45)

Note that the sum in the r.h.s. of (4.43) is empty, if [t0 L/(4π ) + δ/2] < 1; in that case the equation (2.5) may have a solution, for λ small enough, only if δ = 0. Hence we shall suppose that: t0 L ≥ 4 π , (4.46) a condition which is certainly veri?ed, if the conditions (2.24) are satis?ed, since 4 ≤ v0 /t0 ≤ 2 . π . By using (4.47) and supposing that |σ | 2 ≤1 , v0 it is easy to show that

2 i=1

(4.47)

(4.48)

|di (σ )| ≤

C , v0

2 i=1

?di (σ ) C ≤ 3 . ?σ v0

(4.49)

By substituting the sum in the r.h.s. of (4.43) with an integral, we can write F1 (σ, L) = F2 (σ, L) + d3 (σ, L) , where F2 (σ, L) =

1?δ/2 t0 L/(4π )

(4.50)

dx (2πv0 x)2 + (σL)2

.

(4.51)

It is easy to see that, if the condition (4.48) is veri?ed, together with v0 ≤ε ?≤ 1 , L|σ | then |d3 (σ )| ≤ C , v0 ?d3 (σ ) Cε ? . ≤ ?σ v0 |σ | (4.52)

(4.53)

The integral de?ning F2 (σ, L) can be explicitly calculated; we get 1 log F2 (σ, L) = 2πv0 If we write F2 (σ, L) =

1/f ebbraio/2008; 6:36

v0 t0 2|σ| 2πv0 L |σ |

+ +

v0 t0 2 2σ 2πv0 L |σ | 2

+1 1?

δ 2 2

. +1

(4.54)

1?

δ 2

1 v2 log 0 + d4 (σ, L) , 2πv0 |σ | 22

(4.55)

it is easy to prove, using (4.47), (4.48) and (4.52), that |d4 (σ )| ≤ C , v0 ?d4 (σ ) C ≤ ?σ v0 1+ ε ? |σ | . (4.56)

Finally, the function c ?1 (σ ) introduced in (4.39) and its derivative can be bounded, by using (4.4), (4.5) and (4.10), as |c ?1 (σ )| ≤ It is now su?cient to de?ne

4

C v0

1 + log

1 v0

,

C ?c ?1 (σ ) ≤ 3 . ?σ v0

(4.57)

r1 (σ ) = 2πv0

i=1

di (σ, L) +

1?δ ?c ?1 (σ ) 2Lσ

,

(4.58)

to complete the proof of Lemma 2.13.

1/f ebbraio/2008; 6:36

23

5. Bounds on the density perturbative expansion

5.1. In this section we give some bounds about the perturbative expansion (3.33) of the function ρ ?n (σ, Φ), introduced in (2.1), and we prove Lemmata 2.2, 2.8, 2.9 and 2.16. Given Φ ∈ F , let us de?ne

(q ) R(Φ)n (σ ) = ρ ?q n (σ, Φ(σ )) ,

|n| > 0,

q>0.

(5.1)

Moreover, if J is the space of the C 1 -functions of σ ∈ J with values in we shall de?ne, in agreement with (2.9), r

J

R, and r(σ) ∈ J ,

(5.2)

≡ sup |σ |?1 |r(σ )| +

σ ∈J

?r (σ ) ?σ

.

2 5.2. Lemma. If Φ ∈ B and |σ | ≤ v0 , then, for any n = 0 and q > 0, (q ) R(Φ)n J

≤ C 2 v0

Dv0 1 + log

1 v0

q

q2

3q |n|

N

1 + (1 ? δq,1 ) log

2 v0 (|σ |Q)[q/2] , |σ |

(5.3)

where C and D are suitable constants. 5.3. Proof of Lemma 5.2. In order to bound ρ ?q n (σ, Φ), we shall use the expansion in (3.33). Let ? ∈ Tn,q be one of the graphs contributing to ρ ?q n (σ, Φ) and v one of its vertices. If |nv | = 1, one has (see §3 for notations) 2π 1 2 ′ ′ 1 = ? ω? )p T1 ≥ 2nv p + (ω? ? k? k? , v v v′ v′ T Q so that π ′ ′ 1, k . max{ k? ?v′ T1 } ≥ v T Q (5.4)

(5.5)

Then there is a constant C2 such that, ?v ∈ ?, |nv | = 1, if |σ | ≤ 1, C2 Q 4 |σ | , v0 C2 Q ??v′ | ≤ 2 , min |g ??v |, |g v0 ??v′ ≤ g ??v g (5.6) (5.7)

by (5.5), (3.27) and (3.26), since v0 ≤ 1. Note that (5.6) and (5.7) still hold for |nv | = 1, as, in such a case, h?v = h?v′ = 0 is not allowed (see §3.7) and the support properties imply that both propagators are bounded by 2 C/v0 . Note also that, thanks to (3.30), |n| ≤ q + 1 + |nv | ≤ 3 |nv | ? ?v ? : |nv? | ≥ |n| . 3q (5.8)

v

v

Let us now suppose that q = 2? q, with q ? ≥ 1. It is easy to see that, in this case, it is possible to couple 2? q among the 2? q + 1 propagators appearing in the expression of Val(?), ??(1) , ?(1) = ?v? , the propagator left alone ??v′ with v = v ? ; let g see (3.32), in q ? pairs g ??v , g after this coupling operation. We select in an arbitrary way one of the q ? couples and we use the bound (5.7) for one of the propagators belonging to it; let g ??(2) the other propagator of the selected couple. The propagators of all the other couples will be bounded by (5.6). We get 2? q ?q ?+1 1 ? |σ | |Φnvi | |Val(?)| ≤ (C2 Q)q |g ??(1) g ??(2) | . (5.9) 4? q ?2 Lβ ′ v0 i=1

k ∈DL,β

1/f ebbraio/2008; 6:36

24

2 Let us now suppose that |σ | ≤ v0 ; then we can use the bounds (4.4)–(4.8), valid also for ?nite β , to prove that, for any choice of g ??(1) and g ??(2) ,

1 Lβ

k′ ∈DL,β

|g ??(1) g ??(2) | ≤

D2 v0

1 + log

1 v0

1 + log

2 v0 |σ |

.

(5.10)

Hence, if Φ ∈ B , by using (2.10), (5.8) and (5.10), we get |Val(?)| ≤ D2 2q?1 v0 1 + log 1 v0 v2 1 + log 0 |σ | 3q |n|

N

?∈Tn,2q ?

(5.11) ,

q q ? ? D1 qC2 |σ |(|σ |Q)q

q 1 2 q takes into account the fact that there are 5 possible choices for the ω? , ω? , h? where D1 ? labels for each line, and q possible choice for the vertex v ; then the bound (5.3) is proved for even q . The case q = 2? q + 1, with q ? ≥ 1, can be treated in a similar way. We note that it is always possible to couple 2? q among the 2? q + 2 propagators appearing in the expression of Val(?) ??(1) and g ??(2) be the propagators left alone after this ??v′ with v = v ? ; let g in q ? pairs g ??v , g coupling operation. Then we use (5.6) for all the couples and the bound (5.10) for the two ? ?+1 remaining propagators. We get a bound similar to (5.9), with |σ |?q in place of |σ |?q , but the ?nal bound is the same as before. We still have to consider the case q = 1. We could of course get again the previous bound with q ? = 0, but there is now an improvement, which will play an important role. The improvement follows from the observation that, if q = 1, the graphs contributing to ρ ?1 n have only one vertex with Fourier index nv1 = n, so that at least one of the two propagators must have di?erent ω -indices. By using the bound (4.8), this implies that the bound (5.10) can 2 be improved by erasing the factor [1 + log(v0 /|σ |)]. In order to complete the proof of (5.3), we have to bound also ?ρ ?q n (σ, Φ(σ ))/ ?σ . We can proceed as before, by noticing that ? Val(θ )/?σ can be written as the sum of 2q + 1 terms, each term di?ering from Val(θ) only because there is the derivative acting on a single propagator or a single vertex function. If the derivative acts on one of the coupled propagators, one can use the bounds (5.6) and (5.7) modi?ed so that the r.h.s. is multiplied by |σ |?1 ; if the derivative acts on a vertex function, since Φ ∈ B , one can use the bound |? Φn (σ )/?σ | ≤ |n|?N ; if the derivative acts on one of the propagators left alone after 2 the coupling operation, one can use the bound, following from (4.10)÷(4.12), if |σ | ≤ v0 ,

1 Lβ We get, for any q > 0,

k′ ∈DL,β

D3 ?g ??(1) g ??(2) ≤ . ?σ v0 |σ |

(5.12)

?R(Φ)n (σ ) ≤ ?σ

(q )

?∈Tn,2q ?

? Val(?) ≤ ?σ D3

2q?1 v0 q q ? ? D1 qC2 (|σ |Q)q

(2q + 1)

3q |n|

N

(5.13) ,

with q ? = [q/2]. This complete the proof of (5.3). 5.4. Proof of Lemma 2.2. The bound (5.3) immediately implies that ρ ?q n (σ, Φ) is summable 4 over q , for σQ/v0 small enough, uniformly in i, ρ and β . On the other end, it is easy to see that the bound is valid also if we substitute in the expression (3.32) of Val(?) the sum ?q over k0 with the integral on the real axis and that limβ →∞ ∞ n (σ, Φ) is obtained from q=1 ρ

1/f ebbraio/2008; 6:36

25

ρ ?q n (σ, Φ) by doing this substitution. The claim of the Lemma about the continuous dependence on λ of ρ ?n (?, ?) is an easy consequence of this remark and (3.31). In a similar way, one can see that limi→∞ ρ ?n (?, ?) is obtained by substituting in the ′ expression of Val(?) the sum over k with an integral over the interval [?π, π ] and that this limit is also continuous in λ near 0. The other claims of the Lemma about the density and the gap of h can be proved as in [BGM], §4.5 and §4.6. In [BGM] a more complicated expansion was used (involving a further decomposition of the ?eld ψ (0) ), but the proof of these two points can be even more simply reached by using the expansion of this paper and bounds of the graphs similar to (5.11). We shall not give here the details, but we only remark that the main point in the proof is the remark that the propagators (3.23) are analytic in k0 in the strip |Im k0 | ≤ |σ |/2. 5.5. Proof of Lemma 2.8. By the remark in §5.4, limβ →∞ can be exchanged with the sum over q in (3.31) and (3.33). In the following, for simplicity, we shall use the notation ρ ?q n (σ, Φ) q to identify limβ →∞ ρ ?n (σ, Φ). Let us suppose that Φ, Φ′ ∈ B and q ≥ 2; then, by (3.31)

′ ρ ?q ?q n (σ, Φ ) ? ρ n (σ, Φ) =

∞ q=1

Val(?) ,

?∈Tn,q (Φ,Φ′ )

(5.14)

where Tn,q (Φ, Φ′ ) is a set of labeled graphs whose de?nition di?ers from the de?nition of Tn,q , see §3.7, only because there is a new label αv ∈ {0, 1, 2} for each vertex; moreover Val(?) = ? 1 L

∞

′ k′ ∈DL

?∞

dk0 2π

q+1

g ??i

i=1 v ∈?

αv Fn v

,

(5.15)

αv Fn v

and there is the constraint that at least one vertex has label αv = 2. By proceeding as in the proof of Lemma 5.2 and using the de?nitions (2.9) and (5.1), we get, if q ≥ 2, R(Φ′ )(q) ? R(Φ)(q) D v0 1 1 + log v0 C 2 v0

F q

? ?σ = Φnv ? Φ′ ? Φ nv nv

if αv = 0 , if αv = 1 , if αv = 2 ,

(5.16)

≤ Φ′ ? Φ

F

q 2 (3q )N

1 + log

2 v0 |σ |

(|σ |Q)[q/2] .

(5.17)

?4 ?4 2 Hence, if Qv0 /|σ |)] ≤ 1 and |σ |Qv0 |σ |[1 + log(v0 ≤ 1/(2C 2 ), we have ∞ q=2

R(Φ′ )(q) ? R(Φ)(q)

F

≤

C1 v0

1 + log

1 v0

3 N N ! Φ′ ? Φ

F

.

(5.18)

with a suitable constant C1 . In order to complete the proof of the lemma, we have to estimate R(Φ′ )(1) ? R(Φ)(1) F . The bound (5.17), with q = 1, is still valid, but it is not su?cient; however there is the improvement with respect to (5.17) due to the fact that, if ? is a graph contributing to ′ ρ ?1 ?1 n (σ, Φ ) ? ρ n (σ, Φ), the only vertex belonging to ? has a Fourier index nv1 = n. As in the 2 proof of Lemma 5.2, this remark allows to eliminate the factor [1 + log(v0 /|σ |)] in the bound (5.17) for any value of n. The previous remark implies that R(Φ′ )(1) ? R(Φ)(1)

1/f ebbraio/2008; 6:36

F

≤

C v0 26

1 + log

1 v0

Φ′ ? Φ

F

.

(5.19)

This bound and (5.18) immediately imply Lemma 2.8. 5.6. Proof of Lemma 2.9. The graph expansion of ρ ?q n (σ, 0) has the property that, given a graph ? ∈ Tn,q with Val(?) = 0, each vertex of ? has Fourier index nv = ±1. This implies that, for any v ∈ ? (see §3.7), h?v = h?v′ = 0 is not allowed, so that the number of non diagonal propagators is less or equal of q ? + 1, if q ? = [q/2]. Hence (3.30) implies that |n| ≤ q + q ?+ 1 . (5.20)

We can bound Val(?) = 0 as in §5.3, by choosing in an arbitrary way the vertex v ? (since we do not need now to extract the factor |n|?N ), and we get |Val(?)| ≤ D

2q?1 v0

1 + log

θ ∈Tn,q

1 v0

? ? 1 + log 5q+1 C q |σ |(|σ |Q)q

2 v0 |σ |

.

(5.21)

?3 2 It is easy to see that, if q ≥ 2, q ? ≥ |n|/5; hence, if Qv0 |σ |1/2 [1+log(v0 /|σ |)] is small enough, ∞ q=2 (q ) ||R(0)n ||J

C ≤ v0

1 1 + log v0

|σ | 2 v0

|n| 10

.

(5.22)

In order to complete the proof of Lemma 2.9, we have to improve the bound (5.21) in the case q = 1. Note that ρ ?1 n (σ, 0) is di?erent from 0 only if |n| = 2 (only one propagator may have frequency label h = 0) and it is given, if n = 2 (the case n = ?2 is similar), by σ 1 L

∞

′ k′ ∈DL

?∞

dk0 (1) ′ (0) (1) (0) ?1,1 (k′ + 2pF ) . ??1,1 (k′ ) g ??1,1 (k′ + 4pF ) + g g ? (k ) g 2π 1,1

(5.23)

Hence, by using (4.6) and (4.11), we get ||R(0)2 ||J ≤

(1)

v2 C |σ | 1 + log 0 3 v0 |σ |

≤

C v0

|σ | 2 v0

2/10

.

(5.24)

This bound and the bound (5.22) imply (2.15).

?3 2 5.7. Proof of Lemma 2.14. By Lemma 5.2, if Qv0 |σ |1/2 [1 + log(v0 /|σ |)] is small enough, which is certainly true if condition (2.16) is satis?ed, with ε small enough, we have

q ≥2

|ρ ?q 1 (σ, Φ)| ≤ |σ |

CN N ! v0

1 + log

1 v0

|σ | 2 v0

1/2

.

(5.25)

Moreover, since the graphs contributing to ρ ?1 1 (σ, Φ) have only one vertex with index nv = ±2, ±3, at least one of its two propagators has frequency index h = 0 and di?erent ω -indices. It follows, by (4.6), (4.8) and Lemma 2.10, that |ρ ?1 1 (σ, Φ)| ≤ C C (|Φ2 (σ )| + |Φ3 (σ )|) ≤ |σ | v0 v0

1/2

λ2 v0

N

.

(5.26)

?1 Hence, if |σ |1/4 C N N !(1 + log v0 ) ≤ v0 , which is certainly true if condition (2.16) is satis?ed, with ε small enough, and r2 (σ ) is de?ned as in (2.27), we get

ρ ?1 1 (σ ) ≤ C

|σ | v0

|σ | 2 v0

1/4

+

λ2 v0

N

,

(5.27)

which implies the bound in the ?rst line of (2.28).

1/f ebbraio/2008; 6:36

27

?3 2 Let us now consider the derivative of r2 (σ ). By Lemma 5.2, if |σ |1/2 [1 + log(v0 /|σ |)] Qv0 is small enough, we have

q ≥2

?ρ ?q CN N ! 1 (σ, Φ(σ )) ≤ ?σ v0

1 + log

1 v0

|σ | 2 v0

1/2

.

(5.28)

Moreover, by Lemma 2.10 and the remark preceding (5.26), ?ρ ?1 C 1 (σ, Φ(σ )) Φ ≤ ?σ v0

1/2

F

≤

C v0

λ2 v0

N

.

(5.29)

?1 It follows that, if |σ |1/4 C N N !(1 + log v0 ) ≤ v0 , which is certainly true if condition (2.16) is satis?ed, with ε small enough,

C 2πv0 ?r2 (σ ) ≤ ||R(Φ)1 ||J ≤ ?σ |σ | |σ |

|σ | 2 v0

1/4

+

λ2 v0

N

,

(5.30)

which immediately implies the bound in the second line of (2.28). ? , de?ned in (1.15), is a real matrix; hence, 5.8. Proof of Lemma 2.17. The Hessian matrix M we have to show that

[(Q?1)/2]

? nm xm > 0 , xn M

n,m=?[Q/2]

(5.31)

for any {xn }n=?[Q/2] ∈ R

[(Q?1)/2]

Q?3

. This will be done by writing

[(Q?1)/2]

n,m=?[Q/2]

? nm xm ≥ xn M 1 ?? x2 n ?Mnn ? 2 ?

[(Q?1)/2]

[(Q?1)/2]

n=?[Q/2]

m=?[Q/2] m= n

? nm | + |M ? mn | ? |M ? ,

?

(5.32)

and showing that the right hand side of the above equation is strictly positive. ? nn . If |n| = 1, by (1.15), (1.12) and (2.1) we have Let us ?nd ?rst a lower bound for M ?n ? nn = 1 + λ2 cn (σ ) ? λ2 ? ρ M , ? Φn (5.33)

where 1 + λ2 cn (σ ) ≥ 1/2, see §2.11, and ? ρ ?n /? Φn obeys to the same bound of ? ρ ?n /?σ , see §5.3, up to the factor |n|?N : simply note that the derivatives can act only on the vertex functions (and not on the propagators), and |? Φn /?σ | ≤ |n|?N has to be replaced with |? Φn /? Φn | ≤ 1. Then, analogously to (5.13), we obtain, for any q > 0, λ2 ?ρ ?q n ≤ λ2 Dv0 ? Φn C 2 v0

q

q 2 (3q )N (|σ |Q)[q/2] ,

(5.34)

?4 so that, if |σ |Qv0 is small enough, we have

λ2 It follows that

?ρ ?n ?1 N ≤ Cλ2 v0 3 N! . ? Φn

(5.35)

? nn ≥ 1 , M 3 28

|n| = 1 ,

(5.36)

1/f ebbraio/2008; 6:36

for λ satisfying (1.16), with ε small enough and K ≥ 3. In the case n = 1 (the case n = ?1 is discussed in the same way) we have ?1 ? 11 = 1 + λ2 c1 (σ ) + λσ ?c1 (σ ) ? λ ? ρ M . ?? ?1 ?? ?1 (5.37)

Note that our de?nitions of c1 (σ ) and ρ ?n (σ, Φ) do not distinguish the dependence on ? ?1 and ? ??1 , which are equal in the ?xed points we are studying (see discussion in §1.4). However, ?, ? in the de?nition of M ?1 and ? ??1 have to be treated as independent variables. By taking ?2 into account this remark and by using Lemmata 2.13 and 2.14, with ε ? and |σ |v0 small enough, we get λσ ?c1 (σ ) 1 ?c1 (σ ) 1 λ2 = λ2 σ , ≥ ?? ?1 2 ?σ 6π v0 Cλ2 λ2 ρ ?1 ≤ σ v0 |σ | 2 v0 |σ | 2 v0

1/4

1 + λ2 c1 (σ ) = λ so that

+

1/4

λ2 v0 λ2 v0

N

,

N

(5.38) ,

?ρ ?1 ?ρ ?1 Cλ2 ≤ λ2 ≤ ?? ?1 ?σ v0

+

2 ? nn | ≥ 1 λ , n = ±1 , |M 8π v0 under the hypotheses of Theorem 1.6. The non diagonal terms (n = m) are of the form

(5.39)

?m ? nm = λ2 Φm ?cm (σ ) δn,1 + δn,?1 ? λ ? ρ M . ?σ 2 ?? ?n

(5.40)

By using (2.7) and (2.17), the ?rst term in the r.h.s. of (5.40), where m = n implies |m| > 1, can be bounded as N +1 ?cm (σ ) δn,1 + δn,?1 λ2 λ2 Φm ≤C . (5.41) ?σ 2 v0 Moreover, by proceeding again as in §5.3, we obtain λ so that ?ρ ?q m ≤ λ2 Dv0 ?? ?n

∞

C 2 v0

q

q 2 (3q )N (|σ |Q)[q/2] ,

(5.42)

[(Q?1)/2]

λ

n=?[Q/2];n=m q=2

?ρ ?q |σ |Q2 m ≤ Cλ2 3N N ! 5 . ?? ?n v0

(5.43)

The contributions with q = 1 need an improved bound. Let us ?rst suppose that |n| > 1; in this case the derivative can act only on the vertex function of the graphs contributing to ρ ?1 m . Then, if the derivative is di?erent from 0, the vertex function is equal to Φn and, since m ? n = 0, at least one of the two propagators must have di?erent ω indices; this follows from (4.3), which also implies that the integer which multiplies pF in the value (4.1) of the graph is di?erent from 0. Hence, by using (4.6), (4.9) and the fact that |n ? m| ≤ 2, we get

[(Q?1)/2]

λ2

n=?[Q/2];n=m

|σ |Q ?ρ ?1 m ≤ Cλ2 3 ? Φn v0

1 + log

2 v0 |σ |

.

(5.44)

The case q = 1 and |n| = 1 can be treated in a similar way; the main di?erence is that the derivative can act also on the propagators of the graphs contributing to ρ ?1 m , but it is still true that the integer which multiplies pF in the value (4.1) of each graph is di?erent from 0, an essential point in the previous bound, since it allowed to use the improved bound (4.9) in place of (4.8). By using again (4.6) and (4.9), as well as the improved bounds (with respect to (4.12)) (4.13) and (4.14), we get again the bound (5.44). The r.h.s. of (5.41), (5.43) and (5.44) can be made arbitrarily small with respect to λ2 /v0 , by suitably choosing the constants in (1.16); hence Lemma 2.17 is proved.

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29

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