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Commun Math Phys
Commun Math Phys
Communications in Mathematical Physics
0010-3616
1432-0916
Springer Berlin Heidelberg Berlin/Heidelberg

5092
10.1007/s00220-024-05092-6
Article
Universality in the 2d Quasi-periodic Ising Model and Harris–Luck Irrelevance
http://orcid.org/0000-0003-3999-8982
Gallone Matteo matteo.gallone@sissa.it

1
http://orcid.org/0000-0002-6885-1448
Mastropietro Vieri 2
1 https://ror.org/004fze387 grid.5970.b 0000 0004 1762 9868 International School for Advanced Studies - SISSA, Via Bonomea 265, 34136 Trieste, Italy
2 grid.4708.b 0000 0004 1757 2822 Dipartimento di Matematica F. Enriques, Università di Milano, Via C. Saldini 50, 20129 Milan, Italy
Communicated by A. Giuliani.

16 9 2024
16 9 2024
2024
405 10 2355 4 2023
6 7 2024
© The Author(s) 2024
2024
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We prove that in the 2D Ising model with a weak bidimensional quasi-periodic disorder in the interaction, the critical behavior is the same as in the non-disordered case; that is, the critical exponents for the specific heat and energy-energy correlations are identical, and no logarithmic corrections are present. The disorder produces a quasi-periodic modulation of the amplitude of the correlations and a renormalization of the velocities, that is, the coefficients of the rescaling of positions, and of the critical temperature. The result establishes the validity of the prediction based on the Harris–Luck criterion, and it provides the first rigorous proof of universality in the Ising model in the presence of quasi-periodic disorder in both directions and for any angle. Small divisors are controlled assuming a Diophantine condition on the frequencies, and the convergence of the series is proved by Renormalization Group analysis.

http://dx.doi.org/10.13039/100019180 HORIZON EUROPE European Research Council European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program ERC StG MaMBoQ, n.802901 Gallone Matteo http://dx.doi.org/10.13039/501100003407 Ministero dell’Istruzione, dell’Università e della Ricerca MIUR-PRIN 2017 project MaQuMA cod. 2017ASFLJR Mastropietro Vieri Scuola Internazionale Superiore di Studi Avanzati - SISSAOpen access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement.

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pmcIntroduction

Universality and Harris–Luck criterion

A certain number of macroscopic properties close to phase transitions show a remarkable independence from microscopic details. In particular, it is both predicted theoretically and observed experimentally that the critical exponents, describing the singularities of thermodynamic functions, are the same in systems sharing only a few general features but having different inter-molecular forces, atomic weights, or lattice structures. This phenomenon is known as universality, and the Renormalization Group, introduced by Kadanoff [41] and Wilson [65], provides an explanation by introducing the concepts of scaling dimension, dimensionally relevant, marginal, or irrelevant interactions, and universality classes. The fact that interactions are dimensionally relevant or marginal does not by itself imply that they can change the critical behavior; the precise effect on critical exponents is governed by an effective dimension, which can be different from the scaling dimension due to cancellations or other mechanisms.

A paradigmatic model where universality can be investigated is the Ising model, which describes a system of spins with nearest-neighbor interactions and shows a phase transition in dimensions d≥2 characterized by certain values of the critical exponents. One can perturb this model with finite-ranged or higher spin interactions, or consider it on different lattices, and ask what happens to the critical behavior. In d≥4, universality is proven in the context of the closely related ϕ4 models (see, e.g., [8] and references therein), where it has been rigorously shown that the values of the exponents are equal to the mean-field ones, e.g., the correlation length exponent is ν=1/2 and the specific heat exponent α=(4-d)/2. We remark, however, that while in d≥5 the behavior is exactly the same as in the mean-field theory, in d=4 logarithmic corrections are present; the difference is that in the first case the interaction is irrelevant in the Renormalization Group sense, while in the second it is marginal (or, more precisely, marginally irrelevant).

In d=2, the Ising model with nearest-neighbor interaction on a square lattice was solved by Onsager [60]. His solution proves that the value of the critical exponents (ν=1, α=0) is different from the ones obtained by approximate methods, such as the mean-field. With universality in mind, it is natural to ask whether these values are robust under perturbations. One can ask, for example, if the addition of a next-to-nearest neighbor interaction or a non-quadratic one leaves the system in the Onsager universality class or not. In this case, it is not convenient to use ϕ4 models, but one can use the representation in terms of Grassmann integrals, at the basis of the exact solution, and analyze it using Renormalization Group methods. This strategy was proposed in [64] and applied to the computation of the specific heat and energy correlations in [63] and in Appendix N of [44]. The Grassmann integral representation was then used in [43, 44] for the case of two Ising models coupled to each other by a quartic interaction, which can be mapped into models like the Eight-vertex, Six-vertex, or the Ashkin-Teller model.

Even if single or coupled Ising models have the same exponents in the absence of quartic interaction, when the interaction is present they belong to different universality classes. In the first case, the interaction is dimensionally irrelevant, implying that, when the strength of the interaction is small enough, the exponents are the same as in the pure Ising model (e.g. ν=1, α=0) and no logarithmic corrections are present. In the second case, the interaction is marginal, and its flow is controlled thanks to the complicated cancellations related to emergent symmetries. The exponents are continuous functions of the strength of the coupling [44], verifying suitable Kadanoff extended scaling relations [11, 12]. Continuous exponents also appear in the transition between the two universality classes in the Ashkin-Teller model [32, 46].

Subsequently, the Renormalization Group approach to interacting Ising models was used in the proof of the universality of the central charge [33], the scaling limit of all the energy correlations [31], and to analyze the role of non-periodic boundary conditions [6]. Interacting dimer models, which are in the same universality class as coupled Ising models in some parameter regions, were studied in [35, 37]. This approach typically requires a small value of the coupling.

Other approaches, different from the Renormalization Group, lead to universality results for the Ising model, like those in [16, 17], with nearest-neighbor interactions on different planar graphs. In [4], the Ising model with non-planar, or alternatively some non-nearest-neighbor pair interactions, was considered, proving the Gaussianity of correlations without a smallness condition but without providing information on exponents.

Another situation where the issue of universality can be posed in the Ising model is when disorder is considered. Disorder can be introduced either in the magnetic field [1–3, 39] or in the interaction, and we focus here on this second case, for which much less is known at a rigorous level. Typically, one can consider two kinds of disorder in the interaction: random or quasi-periodic. The first describes the effect of impurities, while the second is realized in quasi-crystals or cold atoms experiments. Early investigations were done in the 2D random Ising model; in particular, the Ising model with a layered disorder (that is, constant in one direction) was considered in [57] (see also [38] and [24]), and the specific heat was found continuous (instead of logarithmically divergent), while with a bidimensional random disorder, a double logarithmic behavior in the specific heat [21] was found.

In more general cases, Harris [38] proposed a criterion to predict when random disorder is irrelevant or not; if ξ is the correlation length and Δ2 is the covariance of the disorder, the condition for irrelevance is Δ2/ξd≪|β-βc|, where the left-hand side is (roughly) the ratio between typical fluctuation of the sum of disorder terms within a distance given by the correlation length ξ and the mean (βc is the critical inverse temperature). As close to criticality ξ∼|β-βc|-ν, with ν being the critical exponent, irrelevance is predicted for νd/2>1, see [38], while relevance is expected for νd/2<1. According to this criterion, irrelevance is predicted for d≥5 (ν=1/2>2/d) and relevance for d=3 (conformal bootstrap predicts ν=0.627⋯<2/3, see [62]). In the marginal cases d=4 (ν=1/2) and d=2 (ν=1), Harris’s criterion gives no predictions in general.

On the rigorous side, a generalization of Harris’s result was proved in [15], where it was shown that in all systems with continuous transitions ν~≥2/d, with ν~ being the index of the disordered system. In the case of layered disorder in d=2, the system is effectively one-dimensional as far as the ratio between mean and fluctuations is concerned, so the relevance of disorder is predicted in agreement with [57]. A rigorous proof is still lacking, despite progress being made in this direction in [18, 30]. In addition, the Harris criterion has been verified in simplified models of a probabilistic nature [29].

While the Harris criterion regards the case of random hopping, the case of quasi-periodic disorder was considered by Luck [42] (Harris–Luck criterion). In the case of the 2D Ising model with layered quasi-periodic disorder, the condition for irrelevance was generalized to 1/ξ∑x=0ξδx≪|β-βc|, where δx is a suitable function measuring the fluctuation of the quasi-periodic hopping, see [42]. Since ν=1, the condition for irrelevance requires that ∑x=0ξδx is bounded and small uniformly in ξ, a condition verified in the case of weak quasi-periodic modulation, while it is violated for strong quasi-periodic disorder.

Such conjectures were checked in [42] by a perturbative method, but the issue of convergence of the series was not addressed; they have also been confirmed by numerical investigations, see e.g. [19, 36]. In particular, in [19] it was numerically found that the Ising model with weak quasi-periodic disorder remains in the Onsager class, while evidence of a new universality class is found at stronger disorder. Finite difference equations for the spin correlations have been derived in [14, 61] from which low and high temperatures expansions are obtained.

In this paper, we finally prove that the critical exponents for the specific heat and energy-energy correlations in the weak quasi-periodic Ising model are identical to the Onsager ones, both for layered and non-layered disorder, in agreement with the Harris–Luck criterion. The result is based on convergent series expansions in the disorder, and the small-divisor problem is addressed via Renormalization Group analysis. This provides one of the very few cases in which a rigorous understanding of the critical behavior of the 2D Ising model with disorder is achieved and universality is proven.

Main result

The Hamiltonian of the 2D quasi-periodic Ising model is1.1 H=-∑x∈Λi[Jx(1)σxσx+e1+Jx(0)σxσx+e0]

where e0=(1,0), e1=(0,1), x=(x0,x1), σx=± and: For i∈N, x∈Λi, Λi=(-L0,i/2,L0,i/2]×(-L1,i/2,L1,i/2]∩Z2, σx=± and periodic boundary conditions are imposed.

The interaction is given by 1.2 Jx(j)=(1+λϕ(j)(2πω0,ix0+θj,0,2πω1,ix1+θj,1))J(j),j=0,1

where ϕ(j)(y) is such that 1.3 ϕ(j)(y)=∑n0=-⌊L0,i/2⌋⌊(L0,i-1)/2⌋∑n1=-⌊L1,i/2⌋⌊(L1,i-1)/2⌋ϕ^n(j)ei(n0y0+n1y1),

with ϕ^n(j)=(ϕ^-n(j))∗, n=(n0,n1) and y=(y0,y1); moreover, for suitable real constants A,η>01.4 |ϕ^n(j)|≤Ae-η|n|.

{ω0,i}i∈N,{ω1,i}i∈N are the best approximants ω0,i=p0,i/q0,i and ω1,i=p1,i/q1,i of two irrational numbers ω0,ω1<1. For j=0,1, the latter are obtained starting from the continuous fraction representation ωj=aj,0+1aj,1+1aj,2+1aj,3+⋯ from which, one has pj,1qj,1=aj,0+1aj,1, pj,2qj,2=aj,0+1aj,1+1aj,2 with ωj-pj,iqj,i≤Cqj,i2 (see e.g. Section IV.7 in [20]).

ω1,ω0 are irrational numbers verifying a Diophantine condition, that is, for j=0,1, 1.5 |2πωjn|T≥cj|n|-ρj∀n∈Z\{0},

where |·|T:=infm∈Z|·+2mπ| and ρj≥1, cj>0.

The side lengths of the boxes are chosen so that 1.6 L1,i=q1,i,L0,i=q0,i,

and limi→∞L1,i/L0,i=c with 0<c<∞.

Remark 1.1

The energy correlations of the quasi-periodic Ising model are obtained as the limit of the energy correlations of a sequence of Ising models in boxes with interactions periodic in space with a period equal to the side of the boxes. In the limit i→∞ the modulation becomes ∑n0,n1=-∞∞ϕ^n(j)ei(n0(2πω0x0+θj,0)+n1(2πω1x1+θj,1)), that is quasi-periodic in both directions. While in principle other ways to define a quasi-periodic Ising model can be imagined, this is the one chosen in numerical simulations in the physical literature, see e.g. [19].

The quasi-periodic Ising model has been considered up to now only with layered disorder, corresponding e.g. to ϕ(0)=0; for instance Jx(0)=J and Jx(1)=J(1+λcos(2πω1x1+θ)). In contrast, we consider a rather more general situation including interactions of the form, for instance, Jx(0)=(1+λcos(2πω0x0+θ)cos(2πω1x1+ϕ))J(0), Jx(1)=(1+λ(cos(3πω0x0+ψ)cos(6πω0x0+2ψ)cos(2πω1x1+ξ)))J(1), with θ,φ,ψ,ξ phases: that is the interaction is different in any bond, and quasi periodically modulated in both directions.

The form of disorder we are considering breaks essentially all the symmetries present in the non-disordered case other than spin-flip symmetry; in particular translation invariance and inversion symmetry xj→-xj in both directions. Less general forms of disorder preserve some symmetry; in particular, in the case of layered disorder, translation invariance and inversion in one space direction is preserved.

The truncated energy correlations are defined for x1,x2∈Λi and j1,j2∈{±} as1.7 Si(x1,j1;x2,j2)=⟨σx1σx1+ej1σx2σx2+ej2⟩i-⟨σx1σx1+ej1⟩i⟨σx2σx2+ej2⟩i,

with1.8 Oi=1Z∑{σx}∈{±}Λie-βHO,Z=∑{σx}∈{±}Λie-βH,

where Z is the partition function at inverse temperature β>0.

If λ=0, for β≠βc, with βc given by1.9 sinh(2βcJ(0))sinh(2βcJ(1))=1,

the thermodynamic limit i→+∞ of the truncated energy correlations exists and is denoted by S(x1,j1;x2,j2). Such limit decays exponentially for large distances with correlation length ξ diverging at βc as ξ=O(|β-βc|-1); βc is therefore the critical temperature. Moreover, in the limit β→βc one has1.10 S(x1,j1;x2,j2)=Zj1Zj2g+0(x1-x2)g-0(x2-x1)+Rj1,j2(x1,x2)

with g±0(x-x2)=(v1(x1,1-x2,1)±i(v0(x1,0-x2,0))-1, Zj,v1,v0 real constants, |Rj1,j2(x1,x2)|≤C|x1-x2|2+θ for |x1-x2|→∞, θ=14 and a real constant C. βc is therefore the critical temperature, defined as the temperature at which the correlation length diverges. Note that one is taking the |Λi|→∞ limit at β≠βc, so that terms O(e-Lic|β-βc|) vanishes in the limit, see Sect. 5 below, if c is a constant and Li=min{L0,i,L1,i} is the shorter side of Λi. Note that v1,v0 are the coefficients of the anisotropic rescaling of positions g+(x)=g¯(v1x1,v0x0) with g¯(x1,x0)=1x1+ix0 (and similar for g-); they will be also called velocities. Our main result describes the long-distance decay of correlations in the interacting case.

Theorem 1.2

Consider the Hamiltonian (1.1) and assume (1)–(5). There exist λ0,C,κ>0, functions b:(-λ0,λ0)→R, ξj:(-λ0,λ0)×T2→R and αj:(-λ0,λ0)→C for j=0,1, with supλ|b(λ)|,supλ|αj(λ)|,supλ,ϑ|ξj(λ,ϑ)|<C such that the following holds. For any |λ|<λ0 there exists βc(λ)=βc+b(λ) such that for β≠βc(λ) the limit limi→∞Si(x1,j1;x2,j2)=S(x1,j1;x2,j2) exists and is finite.

For β≠βc(λ)1.11 |S(x1,j1;x2,j2)|≤Ce-κ(|β-βc(λ)||x1-x2|)12.

For β→βc(λ)1.12 limβ→βc(λ)S(x1,j1;x2,j2)=Zj1,x1(λ)Zj2,x2(λ)g+(x1-x2)g-(x2-x1)+Rj1,j2(x1,x2)

with 1.13 g+(x)=1v1(λ)x1+iv0(λ)x0,g-(x)=1(v1(λ))∗x1-i(v0(λ))∗x0,

and |Rj1,j2(x1,x2)|≤C|x1-x2|2+θ for |x1-x2|→∞, θ=1/4 and 1.14 Zj,x(λ)=Zj+λξj(λ,2πω0x0,2πω1x1)vj(λ)=vj+λαj(λ)

with Zj,vj defined in (1.10).

Remark 1.3

The asymptotic behavior of the 2-point correlation (1.12) at criticality is similar to the one of the unperturbed case, with the main difference that the amplitude is the product of two quasi-periodic functions Zj1,x1(λ) and Zj2,x2(λ). The velocities and the critical temperature are also modified. In contrast, the exponents are universal and no logarithmic corrections are present; this provides a rigorous confirmation of the Harris–Luck criterion. Outside the critical temperature a stretched exponential decay is found, but this is just for technical reasons and exponential decay is expected. The analysis could be easily extended to the n-point energy correlations.

The proof is based on the convergence of the series for the correlations, showing a small-divisor problem similar to the one appearing in perturbation of integrable Hamiltonian systems, see e.g. [25]. Convergence is shown assuming only a Diophantine condition on the frequencies, the smallness of the coupling and a fast decay property of the harmonics; without such assumptions a different behavior is expected.

The result holds for any angle θj, including cases where inversion or translation invariance is broken in both directions. This is a peculiar fact since in many similar models with small-divisor problems, extra conditions are usually required.

Sketch of the proof

The starting point of the analysis is the exact representation of the quasi-periodic Ising model as a Grassmann integral, which is an immediate consequence of the dimer representation, see e.g. [57], and the fact that Pfaffians can be expressed as Gaussian Grassmann integrals, see e.g. [56]. The energy correlations can be written as the sum of terms of the form (the exact expressions are in Sect. 2)1.15 ∫Pψ(dψ)Pξ(dξ)eVO∫Pψ(dψ)Pξ(dξ)eV

where Pψ(dψ),Pξ(dξ) are Grassmann Gaussian integrations, O is a quartic monomial in the Grassmann variables, and V is a sum of monomials in ψ,ξ and vanishes for λ=0. The propagator (or covariance) of Pξ(dξ) is g^ξ(k), given by1.16 g^ξ(k):=-it(1)sink1+t(0)sink0imξ(k)-imξ(k)-it(1)sink1-t(0)sink0-1,

with t(j)=1|Λ|∑x∈Λtanh(βJx(j)) and mξ=mχ=O(1). From the explicit expression given below in (2.22), mχ(k)=mχ(0)+F(k) with mχ(0)=O(1) and F(k)=0 at k=0, and bounded away from zero uniformly in β in the other three poles of the diagonal elements of g^ξ(k). One recognizes in (1.16) the propagator of a lattice Dirac fermion with a mass mχ(0) and Wilson term F(k).

The propagator g^ψ(k) of Pψ(dψ) has a similar expression with a mass that can vanish as a function of temperature. The variables ξ, being associated with a bounded propagator (called non-critical variables for this reason), can be integrated out (see Sect. 3), expressing the energy correlations as Grassmann integrals of the form1.17 ∫Pψ(dψ)eV~O~∫Pψ(dψ)eV~

with V~=1|Λi|∑n∑kψ-kW^n(k)ψk-2πΩn, where W^n(k) is a matrix with elements exponentially decaying in n and analytic in λ. Here, ψ=(ψ+,ψ-), Ω=ω000ω1, and O~ is still quartic in ψ. This representation is an immediate consequence of Wick’s theorem, allowing us to represent Wn(k) as a sum of chain graphs, that is, products of propagators of the form g^ξ(k)g^ξ(k-2πΩn1)g^ξ(k-2πΩn2)⋯. Convergence follows from the exponential decay of ϕ^n and the boundedness of g^ξ.

One could perform the integration in ψ (critical variables) in a similar way, obtaining an expansion for the correlations still expressed in terms of graphs. In this case, however, the propagator of the ψ-variables is unbounded, and at criticality there are graphs that are naively bounded by O(n!α) if n is the order and α a constant, due to the presence of small divisors. To achieve convergence, one needs to improve the bounds, showing that such factorials are indeed not present.

To show this, a multiscale analysis is required, as described in Sect. 4. One decomposes the propagator as a sum of propagators supported at different momentum shells with scale h, that is |k|∼γh, γ>1, with h=1,0,-1,-2,⋯. In other words, g^ψ(k)=∑h=-∞1g^(h)(k) with g^(h)(k)=O(γ-h). Integrating the higher momentum scales, we obtain1.18 ∫Pψ(≤h)(dψ(≤h))eV(h)(ψ(≤h))O~(h)∫Pψ(≤h)(dψ(≤h))eV(h)(ψ(≤h))

with Pψ(≤h)(dψ(≤h)) a Gaussian Grassmann integration corresponding to scales ≤h and againV(h)=1|Λi|∑n∑kψ-k(≤h)W^n(h)(k)ψk-2πΩn(≤h)

with W^n(h) depending on the scale h. Using that, for Gaussian Grassmann integrals, Pψ(≤h)(dψ(≤h))=Pψ(≤h-1)(dψ(≤h-1))Pψ(h)(dψ(h)), we can integrate the ψ(h) variable iteratively; this again produces chain graphs as a product of propagators of arbitrarily large size O(γ-h) times products of the W^n(h). In Renormalization Group terminology, the terms in V(h) are relevant perturbations that could alter the critical behavior.

To show that this is not the case, one needs to distinguish between the case n=0, which are called resonant terms or resonances, and the non-resonant case n≠0. In the first case, one gets an accumulation of identical small divisors in the perturbative expansion, ending with a non-summable behavior. Such a phenomenon is avoided by modifying the expansion, introducing a counterterm to account for the modification of the critical temperature, and by modifying the velocities at each iteration step, see Sect. 4.1. That is, the propagator of the ψ(≤h) close to k=0 acquires the form ∼χh(k)-iv1,hk1+v0,hk0-imim-iv1,h∗k1-v0,h∗k0-1 where χh(k)≠0 for |k|≤γh. Note that reabsorbing certain terms in the propagator is possible only if the W^0(h) have a suitable form that does not change the qualitative structure of the propagator; this is indeed what happens. When the angles θj are generic, the breaking of symmetries does not allow us to conclude the reality of velocities (which turn out to be real in the layered case).

One has then to deal with the terms in V(h) with n≠0; in that case, the repeated small divisors are not identical and they cannot be reabsorbed into the propagator. If the disorder was periodic, that is, Ω is rational so that 2πΩn mod 2π is bounded, this would mean that there is a scale h¯ so that such terms are not present for h≤h¯; hence, they could be easily bounded. In contrast, if Ω is irrational, that is in the quasi-periodic case, such terms appear at any scale h, and the propagators associated with fields multiplying W^n(h) are as large as O(γ-h). One needs therefore, to achieve convergence, to prove that W^n(h)(k) has a fast decay in h compensating for the small divisor γ-h. This follows from the Diophantine condition, as it implies that if k and k-2πΩn are O(γh), then n is large, that is |n|≥γ-hτ for a suitable constant τ. The decay in n of W^n(h)(k) can therefore be converted into a decay in γ-h compensating for the γ-h of the propagator.

However, the gain must be obtained at every iteration step and one has to check that no non-summable combinatorial factors are present; this is done using the cluster structure of graphs (see Sect. 4 and in particular Lemma 4.7 where the convergence of the series expansion is proved). The series obtained is in λ and in the running coupling constants (corresponding to the renormalizations of the temperature and of the velocities); one has to show that it is possible to fine-tune a parameter, corresponding to the shift of the critical temperature, to prove that they remain small at any iteration, as proved in Sect. 4.

Finally, in Sect. 5, the full expansion for the energy correlations is considered. In this case, after the integration of the fields of scales 1,0,-1,-2,⋯,h, one gets source terms of the form 1|Λi|2∑n,j,k,pZh,n(j)ψ-k(≤h)σ2ψk+p-2πΩn(≤h)A^p(j) where Zh,n(j) are running coupling constants associated with the source terms in the generating function for correlations and A^p(j) is the Fourier coefficient of an external field (see (2.1) below). In this case, there are running coupling constants corresponding to n≠0 as there is no gain due to the Diophantine condition. They have a finite limit as h→-∞, and this implies that the critical exponents are the same as in the unperturbed case, and they produce the quasi-periodic amplitude of the energy correlations.

Comparison with previous results

The paper uses a fermionic Renormalization Group approach to the Grassmann representation of the Ising model, previously used in the case of non nearest neighbor perturbations, see [31, 33, 44], or for coupled Ising and related models like Six-vertex, Ashkin-Teller or dimer models [11, 12, 35, 44]. In such cases, the starting point is a Grassmann integral similar to (1.15) but with V a quartic or higher order translation invariant interaction.

In the case of the quasi-periodic Ising model, the situation is different: the interaction in the Grassmann integral is quadratic but the modulation of the potential breaks translation invariance and it requires the use of KAM methods to solve the small-divisor problem.

The relation with KAM appears from (1.15); as the exponent of the integrand is quadratic in the Grassmann variables, the energy correlations could, in principle, be deduced by a suitable lattice Dirac equation in a quasi-periodic potential, essentially given byσ2(ψx+e0-ψx+λϕx(0)ψx)+σ1(ψx+e1-ψx+λϕx(1)ψx)+imσ3ψx=Eσ1ψx,

with σ1,σ2,σ3 being the Pauli matrices. Indeed, such an equation has not been studied, but an extensive literature has been instead devoted to the related problem of the lattice Schrödinger equation with a quasi-periodic potential (which is strictly related to a KAM problem), likeψx+1+ψx-1+λϕxψx=Eψx

where x∈Z and ϕx=ϕ¯(2πωx+θ) with ϕ¯ 2π-periodic. For small λ, the eigenvalues and eigenfunctions of the above equation were studied in [22] where two Diophantine conditions are assumed, one over the frequency and the other over the energy, using KAM methods. In particular, it was required that |2πωn|T≥C|n|-τ and |2πωn±2ρ|T≥C|n|-τ, with E=cosρ (first and second Melnikov condition). In [59] instead, the case ρ=nπω was studied, corresponding to the gaps in the spectrum. Several attempts were made to improve such conditions, culminating in [23], where the second Melnikov condition was removed, and in [7] where ω was assumed to be any irrational. In higher dimensions, for the strong coupling regime, results on localization are in [10, 40].

An important related issue is the computation of the correlations of a system of several particles (fermions in particular) in a quasi-periodic potential, with a single-body interaction described by (1.4). In the absence of a many-body interaction, the knowledge of the single particle properties of (1.4) could be sufficient to determine the properties of the ground state correlations. If ϕx in (1.4) is random, this was indeed done in [5], and with a periodic potential (in the continuum) it was done in [9], where indeed the asymptotic properties of correlations were determined only by a very precise knowledge of the singularities of the eigenvalues (branch points) in the complex plane.

In the quasi-periodic case, a derivation of the asymptotic behavior of fermionic correlations directly from the Schrödinger equation (1.4) has never been attempted. However, such asymptotic decay has been derived by writing the fermionic correlations as Grassmann integrals similar to (1.15), with interacting measure P(dψ)eV, propagator (ik0+cos(k1+nω)-E)-1 and V sum of monomials ψk0,k+ψk0,k+2πnω-. The long-distance behavior of the non-interacting ground state correlations in d=1 has been determined using a multiscale analysis in [13] via fermionic Renormalization Group methods, inspired by the ones used in KAM Lindstedt series [26, 27]. The result was valid for E=cosm¯πω, m¯∈N, that is assuming a gap condition like the one in [59]; the ground state correlations decay exponentially both in space and Euclidean time. Note that there are infinitely many gaps with size O(λϕ^m¯), the spectrum being a Cantor set.

Later on, the RG methods were extended to include the presence of a weak many-body interaction (and weak quasi-periodic potential): it was shown in [45] that the gaps are not closed by the interaction (if the corresponding harmonic is present in the potential), but are strongly modified via the presence of a critical interaction-dependent exponent; the gaps become O((λϕ^m¯)1+η), η=aU+O(U2), where U is the coupling of the many-body interaction and η is a critical exponent. A similar phenomenon was also shown to happen in the interacting Aubry-André model where only one harmonic is present in the initial potential [49] and in the interacting Hofstadter model [50] for the Hall effect. In higher dimensions, a class of fermionic systems in d=2,3 known as Weyl semimetals have been considered [55] in presence of a quasi-periodic disorder and interaction in the weak coupling regime; by assuming a first and second Melnikov condition restricting densities, it was shown the stability of the Weyl phase, that is the absence of localization.

While the above-mentioned results regard the case of fermions on a lattice with a weak quasi-periodic potential and a many-body interaction, the case of strong potential has a different behavior, manifesting the phenomenon of Anderson localization. In this case, one considers the kinetic energy as a perturbation of the quasi-periodic potential, and not the opposite as in the previous case. In [28], localization without many-body interaction was shown, and later the proof of T=0 many-body localization of interacting fermions [47, 48, 51–54] was established. It should be remarked that at the moment, such RG methods are the only ones allowing us to take into account rigorously the interaction in the thermodynamic limit.

At the mathematical level, the Renormalization Group methods used to analyze the above fermion systems in the weakly disordered regime are related to the ones used here for the quasi-periodic Ising model, but there are important differences. First of all, in fermionic systems one has to restrict the values of the chemical potential either to ensure the validity of a gap condition, as in [45, 49, 50], or a second Melnikov condition [55]. There is no analogue of chemical potential in the Ising model, but we can solve the small-divisor problem without imposing any condition. In addition, in fermionic models considered so far, the 2-point fermionic correlation was studied, while here the energy correlations are considered, quartic in the fermions, a fact producing new (infinitely many) marginal operators and the quasi-periodic modulation of the amplitude. Moreover, the quasi-periodic disorder is bidimensional in space and Euclidean time and all possible choices of angles are considered, while previously the only layered or bidimensional cases with angles chosen equal to zero were treated [54]. The general form of the disorder considered here breaks the inversion symmetries, an important property to prove the reality of the velocities.

In addition to such technical improvements, it should be also remarked that the application of direct methods, previously developed for apparently unrelated problems like KAM series or non-relativistic fermions, to the quasi-periodic Ising model is a major novelty of this paper and it produces the first rigorous proof of the Harris–Luck criterion, and a natural starting point for the inclusion of next to nearest neighbor interactions.

Grassmann Representation

From the dimer representation of the Ising model, see e.g. [58], one can express the energy correlations, which are expressed in terms of four Pfaffians, using Grassmann integrals; see e.g. [56]. The energy correlations can therefore be written as2.1 S(x1,j1;x2,j2)=∂2∂Ax1,j1∂Ax2,j2logZ(A)A=0,

with2.2 Z(A)=12∑α∈{±}2ταZα(A),

where τ+,-=τ-,+=τ-,-=-τ+,+=1 and2.3 Zα(A)=∏x∈Λi∏j=01cosh(βJx(j)+Ax,j)∫DΛiΦeSΛi(Φ,A),

with2.4 SΛi(Φ,A):=∑x∈Λitanh(βJx(1)+Ax,1)H¯xHx+e1+tanh(βJx(0)+Ax,0)V¯xVx+e0+∑x∈ΛiH¯xHx+V¯xVx+V¯xH¯x+VxH¯x+HxV¯x+VxHx.

Here, H¯x,Hx,V¯x,Vx are independent Grassmann variables, four for each lattice site, and Ex,1:=H¯xHx+e1, while Ex,0:=V¯xVx+e0. Moreover, Φ:={H¯x,Hx,V¯x,Vx}x∈Λi denotes the collection of all these Grassmann variables, and DΛiΦ is a shorthand for ∏x∈ΛidH¯xdHxdV¯xdVx. The Grassmann integration is defined so that, for all x∈Λi,2.5 ∫dH¯xdHxdV¯xdVx=0,∫dH¯xdHxdV¯xdVx(VxV¯xHxH¯x)=1.

The label α=(α1,α2), with α1,α2∈{±}, refers to the boundary conditions, which are periodic or antiperiodic in the horizontal (resp. vertical) direction. Letting Z=∑α∈{±}2ταZα with Zα=Zα(0), the truncated energy correlation (2.1) can be written as2.6 S(x1,j1;x2,j2)=∑α∈{±}2ταZα2ZEx1,j1;Ex2,j2α,iT,

where ·α,i is the average with respect to the Grassmann “measure” DΛiΦeSΛi(Φ,0)/Zα with α boundary conditions.

Let us consider first the case A=0.

We perform the (well-known) change of variables2.7 H¯x+iHx=eiπ/4ψ+,x-eiπ/4χ+,x,H¯x-iHx=e-iπ/4ψ-,x-e-iπ/4χ-,x,V¯x+iVx=ψ+,x+χ+,x,V¯x-iVx=ψ-,x-χ-,x.

We set Ξα=∫DΛiΦeSΛi(Φ,0), and, for j=0,1, t(j)=1|Λi|∑x∈Λitanh(βJx(j)), we define Vx(j) as2.8 tx(j)=tanh(βJx(j))=tanhβJ(j)(1+λϕ(j)(2πω0,ix0+θj,0,2πω1,ix1+θj,1))≡t(j)+Vx(j)

so that ∑x∈ΛiVx(j)=0. We can write2.9 Ξα=∫∏x∈Λidψ+,xdψ-,xdχ+,xdχ¯-,xeS(χ)(χ)+S(ψ)(ψ)+Q(ψ,χ)

where, denoting with · the Euclidean scalar product,2.10 S(χ)(χ):=-14∑x∈Λitx(1)χ+,xχ-,x·-1+i-i-1χ+,x+e1χ-,x+e1+-14∑x∈Λitx(0)χ+,xχ-,x·-i+i-i+iχ+,x+e0χ-,x+e0+-14∑x∈Λi2i(2+1)(χ+,xχ-,x-χ-,xχ+,x).

2.11 S(ψ)(ψ):=-14∑x∈Λitx(1)ψ+,xψ-,x·-1+i-i-1ψ+,x+e1ψ-,x+e1+-14∑x∈Λitx(0)ψ+,xψ-,x·-i+i-i+iψ+,x+e0ψ-,x+e0+-14∑x∈Λi[-2i(2-1)](ψ+,xψ-,x-ψ-,xψ+,x).

2.12 Q(ψ,χ):=14∑x∈Λitx(1)ψ+,xψ-,x·-1i-i-1χ+,x+e1χ-,x+e1+14∑x∈Λitx(0)ψ+,xψ-,x·i-ii-iχ+,x+e0χ-,x+e0+(ψ↔χ).

Note that, by (2.8), Vx(j) is a 2π-periodic function in 2πΩx+ϑj with Ω=ω0,i00ω1,i, and with zero mean so that we can write2.13 Vx(j)=∑nV^n(j)ein·ϑjei2πΩn·x,withV^n(j):=1|Λi|∑x∈ΛiVx(j)e-in·(2πΩx+ϑj),

where Vx(j) is defined in (2.8), n takes values as in (1.3),2.14 |V^n(j)|≤C|λ|e-η|n|

with C and η independent of i, and V^n(j)=(V^-n(j))∗, V^0(j)=0; these properties follow from (1.4), (2.8) and by analyticity of Vx(j) as a function of the coordinates.

Denoting by ζ±=ψ±,χ±,2.15 ζ±,x:=1|Λi|∑k∈Dαζ^±,keik·x,

with2.16 Dα=k=(k0,k1)∈R2|kj=πLj(2κj+1-αj1)κj∈{-⌊Lj+12⌋,⋯,0,1,⋯,⌊Lj2⌋}.

Note that2.17 ∑x∈ΛiVx(1)χ^+,xχ^-,x·-1i-i-1χ^+,x+e1χ^-,x+e1=12|Λi|∑k∈Dαn∈Z2V^n(1)ein·ϑ1χ^+,-kχ^-,-k·ei(k1-2πω1n1)-1i-i-1χ^+,k-2πΩnχ^-,k-2πΩn+12|Λi|∑k∈Dαn∈Z2V^n(1)ein·ϑ1χ^+,k-2πΩnχ^-,k-2πΩn·e-ik1-1i-i-1χ^+,-kχ^-,-k=12|Λi|∑k∈Dαn∈Z2V^n(1)ein·ϑ1χ^+,-kχ^-,-k·ei(k1-2πω1n1)-1i-i-1-e-ik1-1-ii-1χ^+,k-2πΩnχ^-,k-2πΩn=1|Λi|∑k∈Dαn∈Z2V^n(1)e-πiω1n1ein·ϑ1χ^+,-kχ^-,-k·-isin(k1-πω1n1)icos(k1-πω1n1)-icos(k1-πω1n1)-isin(k1-πω1n1)χ^+,k-2πΩnχ^-,k-2πΩn.

and similar expressions hold for the other quadratic expressions. By setting2.18 A^n(j)=V^n(j)e-iπωjnjein·ϑj,

we finally obtain2.19 Ξα=∫∏k∈Dαdψ^+,kdψ^-,kdχ^+,kdχ^-,keSfree(χ)(χ)+Sfree(ψ)(ψ)+Qfree(ψ,χ)+Sint(χ)(χ)+Sint(ψ)(ψ)+Qint(ψ,χ)

where, if ψ^k=(ψ^k,+,ψ^k,-) and χ^k=(χ^k,+,χ^k,-).2.20 Sfree(ζ)(ζ)=-14|Λi|∑k∈Dαζ^-k·Cζ(k)ζ^k,

2.21 Cζ(k):=-it(1)sink1-t(0)sink0-imζ(k)imζ(k)-it(1)sink1+t(0)sink0,

2.22 mχ(k):=t(1)cosk1+t(0)cosk0+2(2+1),

2.23 mψ0(k):=t(1)cosk1+t(0)cosk0-2(2-1).

and2.24 Qfree(ψ,χ)=14|Λi|∑k∈Dα[ψ^-k·Q(k)χ^k+χ^-k·Q(k)ψ^k],

with2.25 Q(k):=it(1)sink1-t(0)sink0i(t(1)cosk1-t(0)cosk0)-i(t(1)cosk1-t(0)cosk0)it(1)sink1+t(0)sink0.

Moreover,2.26 Sint(ζ)=-14|Λi|∑k∈Dαn∈Z2∑j=0,1A^n(j)ζ^-k·P(j)(k,n)ζ^k-2πΩn,

2.27 Qint(ψ,χ)=14|Λi|∑k∈Dαn∈Z2∑j=0,1A^n(j)ψ^-k·Q(j)(k,n)χ^k-2πΩn+(ψ↔χ),

with2.28 P(1)(k,n)=-isin(k1-πω1n1)icos(k1-πω1n1)-icos(k1-πω1n1)-isin(k1-πω1n1)(1-δn1,0),

2.29 P(0)(k,n)=sin(k0-πω0n0)icos(k0-πω0n0)-icos(k0-πω0n0)-sin(k0-πω0n0)(1-δn0,0),

and2.30 Q(1)(k,n)=P(1)(k,n),Q(0)(k,n)=-P(0)(k,n).

Finally, we introduce new Grassmann variables ξ^k2.31 χ^k=ξ^k+Cχ-1(k)Q(k)ψ^k

and with a straightforward computation yields2.32 Sfree=Sfree(ξ)+Sfree(ψ),Sint=Sint(ξ)+Sint(ψ)+Qint(ψ,ξ).

Explicitly, we obtain Sfree(ξ)(ξ)=Sfree(χ)(ξ) and2.33 Sfree(ψ)(ψ)=-14|Λi|∑k∈Dαψ^-k·(gψ(k))-1ψ^k

with2.34 (g^ψ(k))-1=Cψ(k)-Q(k)Cχ-1(k)Q(k),

2.35 Q(k)Cχ-1(k)Q(k)=Mψ+R(k)

where, if we denote with |M|:=∑a,b|Ma,b| the chosen norm on the space of matrices, we have |R(k)|≤C|k|, Mψ=-((t(0)-t(1))2/mχ)σ2 and, if mχ(0)=:mχ and mψ0(0)=:mψ0,2.36 mψ=mψ0-(t(0)-t(1))2/mχ=1mχ0[(t(0)+t(1))-2(2-1))(t(0)+t(0))+2(2+1))-(t(0)-t(0))2)=1mχ((t(0)+t(1))2-4+4(t(0)+t(0))-(t(0)-t(0))2)=4mχ(t(0)t(1)+t(0)+t(1)-1).

In conclusion,2.37 Ξα=N∫Pξ(dξ)∫Pψ(dψ)eV(ψ,ξ)

where N is a normalization constant and Pξ(dξ) is the Gaussian Grassmann integration, see e.g. Section 4.1 of [27], with propagator g^ξ(k)≡Cξ-1(k)2.38 gξ(x-y)=2|Λi|∑k∈Dαeik·(x-y)g^ξ(k),

Pψ(dψ) is the Grassmann integration with propagator gψ(x-y)=2|Λi|∑keik·(x-y)g^ψ(k) and V(ψ,ξ)=Sint(ξ)(ξ)+Sint(ψ)(ψ)+Qint(ψ,ξ) where2.39 Sint(ξ)(ξ)=-14|Λi|∑k∈Dαn∈Z2∑j=0,1A^n(j)ξ^-k·P(j)(k,n)ξ^k-2πΩn,

2.40 Sint(ψ)(ψ)=-14|Λi|∑k∈Dαn∈Z2ψ^-k·∑j=0,1A^n(j)Pψ(j)(k,n)ψ^k-2πΩn,

and2.41 Qint(ψ,ξ)=14|Λi|∑k∈Dαn∈Z2∑j=0,1A^n(j)ψ^-k·Qψ(j)(k,n)ξ^k-2πΩn+(ψ↔χ),

with2.42 Qψ(j)(k,n)=Q(j)(k,n)-Q(k)Cξ-1(k)P(j)(k,n),

and2.43 Pψ(j)(k,n)=P(j)(k,n)-Q(j)(k,n)Cξ-1(k-2πΩn)Q(k-2πΩn)-Q(k)Cξ-1(k)Q(j)(k,n)+Q(k)Cξ-1(k)P(j)(k,n)Cξ-1(k-2πΩn)Q(k-2πΩn).

Remark 2.1

The partition function is written in terms of Grassmann integrals (see (2.37)), and a similar representation holds for the energy correlations (see Sect. 5 below). The propagator gζ(x-y) decays exponentially with a rate proportional to mζ(0), where ζ=ψ,χ and mχ(0)=O(1). We call ψ and χ (or ξ) respectively critical and non-critical, or massless and massive variables.

If there is no disorder (i.e., λ=0 and t(j)=tanh(βJ(j))), the critical temperature βc, which is the temperature at which the correlation length diverges, is given by the condition mψ=0. Indeed, one finds that this happens when sinh2βcJ(1)sinh2βcJ(0)=1, noting that 4t(1)1-(t(1))2t(0)1-(t(0))2=1 is true for t(0)=1-t(1)1+t(1). As we will see below, the critical temperature when λ≠0 is different.

Integration of Non-critical Variables

Series expansion

We define3.1 eEξ+V(ψ)=∫Pξ(dξ)eV(ψ,ξ)=e∑q=1∞1q!EξT(V(ψ,·);q)=exp(Eξ+14|Λi|∑k∈Dαn∈Z2ψ-k·V^n(k)ψk-2πΩn)

where EξT(V(ψ,·);q) are the truncated expectations with respect to Pξ(dξ) defined as3.2 EξT(V;q)=∂q∂αqlog∫Pξ(dξ)eαV(ψ,ξ)|α=0

where α∈R and Eξ is constant. V^n(k) is a 2×2 matrix which can be expressed as sum of connected graphs defined as follows.

Definition 3.1

A graph with q vertices and index n is defined, see Fig. 1, as a chain of q lines ℓ1,⋯,ℓq+1 connecting points (vertices) v1,⋯,vq, so that ℓi enters vi and ℓi+1 exits from vi; ℓ1 and ℓq+1 are external lines of the graph and both have a free extreme, while the others are the internal lines. A labeled graph Γ is defined from the graph defined above by associating the following labels: To each point v is associated a label jv∈{0,1} and a momentum label nv∈Z2 with the constraint that ∑v=1qnvi=n.

To each line ℓ is associated a momentum kℓ with the constraint that kℓi+1-kℓi=-2πΩnvi; moreover, kℓ1=k and kℓq+1=k-2πΩn.

Gn,q is the set of all possible graphs with q vertices and index n.

The value of the labeled graph Γ is defined as3.3 WΓ(k):=Fv1(k)∏i=2qg^ξ(kℓi)Fvi(kℓi)

where3.4 Fv(kℓ)=A^nv(jv)×Qψ(jv)(kℓ,nv),ifv=1,qP(jv)(kℓ,nv),ifv=2,3,⋯,q-1

with the definitions in (2.8), (2.18), (2.42) and (2.43).

Lemma 3.2

The effective potential V(ψ) admits the representation 14|Λi|∑k∈Dα∑n∈Z2ψ-k·V^n(k)ψk-2πΩn with3.5 V^n(k)=∑q=1∞∑Γ∈Gn,qWΓ(k).

For the proof, see Appendix A.Fig. 1 A graph Γ with q=4

We denote by |A|:=∑i,j|Ai,j|, if A is a square matrix. Note that WΓ(k) depends on n.

Lemma 3.3

There exist C,λ0>0 independent of i such that for |λ|≤λ0, V^n(k) and its derivatives satisfy, for s≤2,3.6 |∂ksV^n(k)|≤C|λ|e-η2|n|.

Moreover,3.7 V^0(k)=a(k)ib(k)-ib(k)-a∗(k)

with a(k)=-a(-k)∈C and b(k)=b(-k)∈R.

Proof

Using that |∂jsgξ(k)|≤Gξ and recalling that by (2.8) and (2.14) one has |Fv(kin)|≤|λ|C1e-η|nv|, and by (2.18), (3.3) and (3.4) we get, for suitable constants Gξ,C1>0 independent of i,3.8 |∂ksWΓ(k)|≤9q|λ|qGξq-1C1q∏ve-η|nv|≤9q|λ|qGξq-1C1qe-η2|n|∏ve-η2|nv|

where 9 is an upper bound for the number of derivatives on the propagators and on the Fv’s. The sum over graphs consists simply in the sums over all possible jv and nv so that, using that ∑nve-η2|nv|≤4(1-e-η2)2 and the sum over jv is bounded by 2, one gets3.9 |∂ksVn(k)|≤∑q=1∞|λ|q1Gξ(72C1Gξ(1-e-η2)2)qe-η2|n|

and the sum over q≥1 is convergent for |λ|<(1-e-η2)2144C1Gξ. The proof of (3.7) is in Appendix B. □

Integration of Critical Modes

Multiscale decomposition

We write4.1 Ξα=N∫Pψ(dψ)exp{14|Λi|∑k∈Dαn∈Z2ψ^-k·V^n(k)ψ^k-2πΩn}=N1∫P(≤1)(dψ)exp{14|Λi|∑k∈Dαψ^-k·γ2νσ2ψ^k+14|Λi|∑k∈Dαn∈Z2ψ^-k·V^n(k)ψ^k-2πΩn}

where P(≤1)(dψ):=NN1P(dψ)exp{-14|Λi|∑k∈Dαψ^-k·γ2νσ2ψ^k}. Note that, in writing the above expression we have added and subtracted a counterterm proportional to ν, which will be suitably chosen below.

As we noticed, in the integration over the ψ we cannot repeat the analysis done for the ξ because the propagator is unbounded. The integration of ψ in (4.1) is done via a multiscale analysis. We introduce a Gevrey class 2 function χ (see e.g. [34, Appendix A]) such that χ′(|k|T)≤0 and4.2 χ(k)=χ(|k|T)=1,if|k|T<γ-1π20,if|k|T≥π2

with T denoting the two dimensional torus of length 2π, |k|T:=|k0|T2+|k1|T2 with |k|T:=infm∈Z|k+2mπ|. We also define, if γ>1, h≤04.3 χh(k):=χ(γ-hk),

and χ1(k)=1. The functions fh(k):=χh(k)-χh-1(k) and f~h(k):=χh(k)(1-χh-1(k)) are Gevrey class 2 compact support functions with support π2γh-2≤|k|T≤π2γh, see Fig. 2.Fig. 2 Plot of the function χ and some of the fh

The integration is defined recursively in the following way. Suppose we have just integrated the field on scale h, h=1,0,-1,-2,⋯ obtaining4.4 Ξα=Nh∫P(≤h)(dψ(≤h))eV(h)(ψ(≤h)),

with Nh constant in ψ and P(≤h)(dψ(≤h)) a Grassmann Gaussian integration with propagator4.5 g(≤h)(k)=χh(k)Ah+1(k)

with4.6 Ah(k)=-ia1(h)k1-a0(h)k0-b1(k)-iμ-ib2(k)iμ+ib2(k)-i(a1(h))∗k1+(a0(h))∗k0+b1∗(k)-1

and |b1(k)|,|b2(k)|≤C|k|2; moreover4.7 V(h)(ψ(≤h))=14|Λi|∑k∈Dαn∈Z2ψ-k(≤h)·V^n(h)(k)ψk-2πΩn(≤h).

If h=1, (4.5) holds with χ1(k)=1; moreover μ=mψ+γ2ν and V(1) given by the exponent of the second line of (4.1).

Remark 4.1

We will show in the following that ν has to be chosen as a suitable non trivial function of λ,μ,β; the condition for criticality, that is so that the correlation length diverges, is given by μ=0 and not by mψ=0 as in the non disordered case.

We define a localization operation as4.8 LV(h)(ψ(≤h)):=14|Λi|∑k∈Dαψ^-k(≤h)·(V^0(h)(0)+∑j=01kj∂jV^0(h)(0))ψ^k(≤h),

and4.9 RV(h)(ψ(≤h))=V(h)(ψ(≤h))-LV(h)(ψ(≤h)).

We move the second term of LV(h)(ψ(≤h)) in the Gaussian integration and by the change of integration property of Gaussian Grassmann Integrals [27, Eq. 2.24], we have for suitable N¯h∈R,4.10 Nh∫P(≤h)(dψ(≤h))eLV(h)(ψ(≤h))+RV(h)(ψ(≤h))=N¯h∫P¯(≤h)(dψ(≤h))e14|Λi|∑k∈Dαψ^-k(≤h)·γhνhσ2ψ^k(≤h)+RV(h)(ψ(≤h)),

where P¯(≤h)(dψ(≤h)) has propagator4.11 g¯(≤h)(k)=χh(k)A¯h(k)

with4.12 A¯h(k):=-ia1(h)(k)k1-a0(h)(k)k0-b1(k)-iμ-ib2(k)iμ+ib2(k)-i(a1(h)(k))∗k1+(a0(h)(k))∗k0+b1∗(k)-1

and4.13 a1(h)(k)=a1(h+1)+iχh(k)[∂1V^0(h)(0)]1,1,a0(h)(k)=a0(h+1)-χh(k)[∂0V^0(h)(0)]1,1

where aj(h+1):=aj(h+1)(0) for any j=0,1 and for any h, and with νhσ2=γ-hV^0(h)(0). To begin the iteration, one can define4.14 a0(2):=-[∂0(g^(≤1))-1(0)]1,1,a1(2):=i[∂1(g^(≤1))-1(0)]1,1.

We can write4.15 P¯(≤h)(dψ(≤h))=P(≤h-1)(dψ(≤h-1))P(h)(dψ(h))

where P(≤h-1)(dψ(≤h-1)) has propagator4.16 g(≤h-1)(k)=χh-1(k)Ah(k)

with Ah(k) being defined in (4.6). P(h)(dψ(h)) has propagator4.17 g(h)(k)=g¯(≤h)(k)-g(≤h-1)(k)

where the analogous of (3.7) has been used. We can integrate P(h)(dψ(h)) and the procedure can be iterated.

The single scale propagator

Inserting (4.16) and (4.11) in (4.17) one obtains4.18 g(h)(k)=fh(k)Ah(k)+f~h(k)(A¯h(k)-Ah(k)),

where fh and f~h are defined after (4.3). It is important to notice that suppχh(k)(A¯h(k)-Ah(k))⊆[π2γh-1,π2γh] (therefore we can multiply for free with (1-χh-1(k) to obtain f~h) and therefore g(h)(k) is a Gevrey compact support function, with suppg(h)⊆[π2γh-2,π2γh]. Note also that in the expression of g(1) the second term is not present because χ1(k)=1.

Assuming iteratively (what will be proved inductively below in Lemma 4.7 for |λ| small enough) that 78aj(2)≤aj(h)≤98aj(2), we can show that for s=0,1,2,4.19 |∂ksg(h)(k)|≤C1γ-h(1+s).

Indeed,4.20 |detAh-1(k)|=|ia1(h)k1+a0(h)k0+b1(k)|2+|μ+b2(k)|2

with b1(k),b2(k)=O(|k|2) as k→0. Then, by algebraic manipulations, one obtains4.21 |detAh-1(k)|≥|a1(h)|2k12+|a0(h)|2k02+2Im(a1(h)a0(h)∗)k0k1+F(k)

with F(k)=O(|k|3) as k→0. Using now that a1(2),a0(2)∈R and the iterative hypothesis on aj(h), one has4.22 |Im(a1(h)a0(h)∗)|=|Im(a1(h)a0(h)∗-a1(2)a0(2))|=|Im((a1(h)-a1(2))a0(h)∗-a1(2)(a0(h)∗-a0(2)))|≤|a1(h)-a1(2)||a0(h)|+|a1(2)||a0(h)-a0(2)|≤18·98+18|a0(2)||a1(2)|≤1764|a0(2)||a1(2)|.

Thus, (4.21) can be estimated as4.23 |detAh-1(k)|≥|a1(h)|2k12+|a0(h)|2k02-2|Im(a1(h)a0(h)∗)k0k1|-|F(k)|≥4964|a1(2)|2k12+4964|a0(2)|2k02-1732|a0(2)||a1(2)||k0||k1|-|F(k)|≥12((a1(2))2k12+(a0(2))2k02)-|F(k)|,

where in the last step we used |(a0(2)k0)(a1(2)k1)|≤12((a0(2))2k02+(a1(2))2k12).

Graphs and clusters

The outcome of the multiscale integration described above is again a representation of the effective potential in terms of graphs, which are called renormalized graphs.

Definition 4.2

Gn,qR,h is the set of renormalized graphs Γ, which are defined starting from the graphs defined in Definition 3.1 by associating the following labels To each point v is associated a label nv and a label iv∈{ν,V}, with the constraint that ∑i=1qnvi=n.

To each line ℓ is associated a momentum kℓ with the constraint that kℓi+1-kℓi=-2πΩnvi; moreover kℓ1=k and kℓq+1=k-2πΩn.

To each line ℓ is associated a scale index hℓ=1,0,⋯,-∞; if ℓ is an internal line hℓ≥h+1; the minimal scale of the internal lines is hΓ. To each external line is associated a scale and hext≤h is the greatest of such scales.

Given a renormalized graph, we associate a set of clusters defined in the following way.

Definition 4.3

Given a renormalized Graph ΓA non-trivial cluster T is defined as a nonempty connected subset of internal lines and points attached to them such that if hT is the minimum of the scales of the lines of T, then hT>hText, where hText is the maximal of the scales of the external lines of T (the lines ∉T attached to a single point of T). The points are trivial clusters and Γ is also a cluster.

The difference of the momenta of the external lines of T is given by 2πΩnT with nT=∑v∈Tnv. If nT=0 then T is a resonant cluster (or resonance), otherwise is a non-resonant cluster. An inclusion relation is established between clusters and we say that T~⊂T if all the elements of T~ belong also to T. T~ is a maximal cluster (trivial or not trivial) contained in T if T~⊊T and there is no other cluster T¯ such that T~⊊T¯⊊T.

QT is the number of maximal clusters in T, MT is the number of the maximal non-resonant clusters contained in T; RT is the number of the maximal resonant clusters contained T; QT=MT+RT; MTν (MTI ) is the set of resonant (non-resonant) maximal trivial clusters (i.e. points) in T.

Given a cluster, we can associate a value in the following way.

Definition 4.4

The value of a cluster T with maximal clusters T~w, w=1,⋯,QT is given by4.24 WT(k)=∏w=1QT-1W¯T~w(kw)g(hT)(kw+1)W¯T~QT(kQT),

where kw-kw-1=2πΩnT~w-1, k1=k and W¯T~w(kw) is defined as If T~w is a trivial cluster, then by Definition 4.2 (item (1)) it has two labels iw and nw. If iw=ν, then nw=0 and WT~w(kw)=γhTνhTσ2; if iw=V then either nw=0 and then WT~w(kw)=RV^0(kw) or nw≠0 and then WT~w(kw)=V^nw(kw) defined in (3.1).

If T~w is a non-trivial cluster then W¯T~w=RWT~w with R=1-L defined in (4.8).

Remark 4.5

Let v be a maximal trivial cluster v∈T. If v is a resonant V-point (i.e. nv=0), then by (4.9) and Lemma 3.3, we have4.25 |χh(k)χh(k-2πΩn)RV^0(k)|≤γ2hTC|λ|.

With the above definitions, the following lemma holds.

Lemma 4.6

V^n(h)(k) in (4.7) can be written as4.26 V^n(h)(k)=∑q=1∞∑Γ∈Gn,qR,hWΓ(k).

Similarly, the running coupling constants verify4.27 νh-1=γνh+βν,haj(h-1)=aj(h)+βaj,h

with4.28 βν,h=iγ-h+1∑q=2∞∑Γ∈G0,qR,h-1,hΓ=h[WΓ(0)]1,2,βa1,h=-i∑q=2∞∑Γ∈G0,qR,h-1,hΓ=h[∂k1WΓ(0)]1,1,βa0,h=∑q=2∞∑Γ∈G0,qR,h-1,hΓ=h[∂k0WΓ(0)]1,1.

The proof is an immediate consequence of Appendix A and Sect. 3.

An example of a renormalized graph with its clusters is given in Fig. 3: in Fig. 4 is represented the same graph with only its maximal clusters.Fig. 3 Graphical representation of a renormalized graph Γ: q=11, h5<h4<h3<h2<h1, hΓ=h5. QΓ=4 (with 2 non-trivial clusters, i.e. T1 and T2, and two trivial ones, i.e. the points 1 and 9). QT1=3 (with 2 non-trivial clusters T3 and T4 and a trivial one, v=6). T3 has two maximal clusters, a trivial one v=2 and a non-trivial one T5. T5 has two maximal clusters, a non-trivial one T6 and a trivial one, n5. T6, T4 and T2 have two maximal trivial clusters each

Note that a set of clusters can be equivalently represented as a Gallavotti-Nicolò tree, see e.g. [56].Fig. 4 The same graph as in Fig. 3 with only its maximal clusters represented. Trivial clusters are represented by dots, non-trivial clusters by ellipses

If we consider as first non-trivial cluster T=Γ and we use the above definition we get an expression similar to the graphs defined in Sect. 3 with the difference that (a) the propagators associated to the lines ℓ are g(hℓ); (b) to each resonant cluster is associated the R operation; (c) the vertices are of type ν or V; (d) the vertices do not have a jv index. In contrast with the expansion in λ seen in Sect. 3, the renormalized expansion is in λ and in the running coupling constants νh.

In the following we denote by ∏Tn.t.=∏T∈ΓTnon-trivial.

Bounds

We define4.29 ‖V^n(h)‖:=supk∈Dαχh(k)χh(k-2πΩn)|Vn(h)(k)|.

The following lemma holds. We denote with subscript l the infinite volume limit of a quantity.

Lemma 4.7

Let τ:=min{ρ1,ρ0}, take γ>4τ and assume that for h′>h one has |νh′|≤|λ|. Then, there exist λ0,C>0 independent of i and h such that, for any |λ|<λ0 one has (i) the limit V^n,l(h)(k):=limi→+∞V^n(h)(k) exists;

(ii) for s=0,1,2, the following estimates hold 4.30 ‖∂ksRV^n,l(h)‖≤γh(1-s)C|λ|e-η4|n|,

4.31 |βν,h|≤(Cλ)2γh,|βaj,h|≤(Cλ)2γh.

The bounds (4.30) and (4.31) are obtained by estimating the value of the graphs in (4.26) and (4.28). For clarity, we write in Remark 4.8 below and example: we show how the general procedure works in the particular case of the graph in Fig. 3.

Proof

WΓ,l is obtained by WΓ replacing Ωi with Ω, considering nv∈Z2 and k∈[-π,π)2. First, we show that we can multiply by χΓ, i.e. we show that4.32 χh(k)χh(k-2πΩn)WΓ,l(k)=χh(k)χh(k-2πΩn)χΓWΓ,l(k)

where χΓ=1 if, for any non-resonant cluster T in Γ, it is true that4.33 |nT|≥C0γ-hTextτ

and χΓ=0 otherwise. Indeed if kin and kout are the momenta associated to the external lines of T, then by the compact support properties of g(h)’s or χh, |kin|T≤π2γhText and |kout|T≤π2γhText (note that h≤hText). Therefore4.34 |kin-kout|T≤|kin|T+|kout|T≤2π2γhText

and by the Diophantine condition (1.5) we get4.35 2π2γhText≥|kin-kout|T=2πminm0,m1∈Z(ω1n1-m1)2+(ω0n0-m0)2≥maxj=0,1cj|nj|-ρj≥max(c1,c0)|nT|-min(ρ1,ρ0)

hence the l.h.s. of (4.32) is vanishing if for at least one non-resonant T, (4.33) is not true.

The proof proceeds then by induction. First, notice that the first step is a straightforward consequence of Lemma 3.3. By the inductive step, let us assume that (4.30) and (4.31) hold for any scale 2,⋯,h+1 and we prove that they hold at scale h. First of all by (4.31) we get4.36 |aj(h)-aj(2)|≤C2λ2∑k=h+12γk≤C2λ2γ3γ-1

hence for C2λ2γ3γ-1<minj18aj(2) we get 78aj(2)≤aj(h)≤98aj(2); this implies (4.19) for s=0,1,2. To estimate the quantities appearing in (4.24), we recall that from Lemma 3.3, there exists a constant C2 independent of i, such that ‖∂ksV^n‖≤C2|λ|e-η/2|n|, and from (4.19) there exists a constant C1 independent from i and h′ such that ‖∂ksg(h′)‖≤C1γ-h′(1+s). Moreover, by Remark 4.5, we can estimate resonant V vertices as |RV^0|≤|λ|γhT (see also Remark 4.9 below). Thus,4.37 ‖∂ksRχΓWΓ,l(k)‖≤(cC1C2)qγ-sh|λ|q×∏ve-η2|nv|∏Tn.t.γ-hT(MT+RT-1)(∏Tn.t.nT=0γ2(hText-hT))∏Tn.t.γhTMTν.

where c=9 counts the number of derivatives produced by R or ∂s, the factor γ2(hText-hT) is the result of the application of the R operation described in Appendix C and γ-hT(MT+RT-1) comes from the product of propagators. We can write4.38 ∏ve-η2|nv|≤e-η4|n|∏ve-η8|nv|∏ve-η8|nv|,

and e-η8|nv|=∏h=-∞0e-2hη16|nv| so that4.39 ∏ve-η8|nv|≤∏Tn.t.e-η162hText|nT|.

The presence of χΓ guarantees that when nT≠0, the estimate (4.33) holds and the assumption γ>4τ ensures that γ~:=γ1/τ2>1. Therefore,4.40 ∏ve-η8|nv|≤e-ζγ~-h∏Tn.t.e-ζMTγ~-hTifnΓ≠0∏Tn.t.e-ζMTγ~-hTifnΓ=0.

with ζ=η16C0 a constant independent of i and h. We get therefore4.41 ‖∂ksRχΓWΓ,l‖≤(cC1C2)qγ-sh|λ|qfexte-η4|n|∏ve-η8|nv|∏Tn.t.γ-hT(MT+RT-1)×(∏Tn.t.nT=0γ2(hText-hT))∏Tn.t.e-ζMTγ~-hT∏Tn.t.γhTMTν,

where4.42 fext:=e-ζγ~-hTextifnT≠01ifnT=0.

Using that for any M∈N, one has e-ζγ~-hT≤γ-MMlnγζMlnγγMhT (this is a consequence of the bound e-αxxM≤(Mα)Me-M) and ∑Tn.t.MT≤4q, we can bound4.43 ∏Tn.t.e-ζMTγ~-hT≤C3q∏Tn.t.γ2hTMT∏Tn.t.γhTMTI

by setting M=3 and with C3=γ-123lnγζ12lnγ. We bound MT with MTI, that is the number of non resonant maximal trivial clusters. Therefore4.44 ∏Tn.t.γ-hTMT∏Tn.t.e-ζMTγ~-hT≤C3q∏Tn.t.γhTMT∏Tn.t.γhTMTI

and4.45 ‖∂ksRχΓWΓ,l‖≤(cC1C2C3)qγ-sh|λ|qfexte-η4|n|∏ve-η8|nv|×∏Tn.t.γ-hT(RT-1)(∏Tn.t.nT=0γ2(hext-hT))∏Tn.t.γhTMT∏Tn.t.γhT(MTI+MTν).

Finally, using that RT=MTν+RTn.t. where RTn.t. is the number of non-trivial resonant maximal clusters in T we get4.46 (∏Tn.t.γ-hT(RT-1))(∏Tn.t.nT=0γhText-hT)∏Tn.t.γhTMTν≤γεΓh

with εΓ=1 if nΓ=0 and εΓ=0 otherwise. Equation (4.46) follows from the fact that4.47 (∏Tn.t.γ-hTRT)(∏Tn.t.nT=0γhText)∏Tn.t.γhTMTν=γεΓh(∏Tn.t.nT=0,T≠ΓγhText)(∏Tn.t.γ-hTRT)∏Tn.t.γhTMTν=γεΓh

and, moreover4.48 ∏Tn.t.γhT∏Tn.t.nT=0γ-hT≤1.

We define4.49 f~ext:=e-ζγ~-hifnΓ≠0γhifnΓ=0.

Inserting (4.46) and (4.49) in (4.45), we get4.50 ‖∂ksRχΓWΓ,l‖≤γ-sh(cC1C2C3)q|λ|qf~exte-η4|n|∏ve-η8|nv|×(∏Tn.t.nT=0γhText-hT)∏Tn.t.γhTMT∏Tn.t.γhTMTI.

We use the inequality e-ζγ~-h≤γ-1lnγζlnγγh, and we call C4:=max1,γ-1lnγζlnγ. If T~⊂T is maximal and T is non-resonant we have γhT=γhT~ext≤γhT~ext-hT~ and therefore4.51 f~ext(∏Tn.t.nT=0γhText-hT)∏Tn.t.γhTMT≤C4γh∏Tn.t.γhText-hT.

Inserting (4.51) in (4.50), bounding the factor ∏γhTMTI in (4.50) by a constant, we get4.52 ‖∂ksRχΓWΓ,l‖≤γh(1-s)(cC1C2C3C4)q|λ|qe-η4|n|∏ve-η8|nv|∏Tn.t.γhText-hT.

The sum over Γ consists in the sum over the label nv associated to the vertices and the sum over the scales. We use that4.53 ∏v∑nve-η8|nv|≤∏v4(∑n≥0e-η8n)2≤4q(1-e-η8)-2q.

The sum over the scale labels of the lines, hℓ can be controlled by summing over the scales of non-trivial clusters and keeping only the constraint that, for each non-trivial cluster, hText<hT:4.54 ∑{hℓ}∏Tn.t.γhText-hT=∏Tn.t.∑hT>hTextγhText-hT≤(∑r>0γ-r)∑TMT≤(1γ-1)4q,

where we used again ∑TMT≤4q. Inserting (4.53) and (4.54) in (4.52), we get4.55 ‖∂ksRχΓWΓ,l‖≤γ(1-s)he-η4|n||λ|qC¯q,

with C¯:=4cC1C2C3C4(1-e-η8)-2(γ-1)4 a constant independent on i and h. The sum over q is convergent if |λ|<C¯, therefore, if |λ|≤C¯2 one gets4.56 ‖∂ksRV^n,l(h)‖≤∑q=1+∞∑Γ∈Gn,q‖∂ksRχΓWΓ,l‖≤γ(1-s)h2C¯|λ|e-η4|n|.

To estimate βν,h and βaj,h, we have to bound WΓ(0) for Γ∈G0,qR,h, with hΓ=h (see (4.28)). In this case we have to consider only the case q≥2, since the sums in (4.28) start from q=2. Moreover, there must be at least two maximal non-resonant clusters in Γ, therefore MΓ≥1. Indeed, if this was not the case, then there must be an internal line with kℓ=0, implying g(hℓ)(kℓ)=0 by the support properties of g(h)’s, which yields WΓ(0)=0. Thus, in particular, we must have MΓ≠0 and q≥2.

One can repeat the same argument used to estimate ∂ksRχΓWΓ with the following difference. By construction, Γ is a resonant cluster on which no R operator acts. Therefore, analogously to (4.41), one obtains4.57 |βν,h|≤∑q=2∞∑Γ(cC1C2)q|λ|q(∏ve-η8|nv|)(∏Tn.t.γ-hT(MT+RT-1))×(∏Tn.t.nT=0T≠Γγ2(hText-hT))(∏Tn.t.e-ζMTγ~-hT)∏Tn.t.γhTMTν.

Using that MΓ≠0 and hΓ=h, one can replace (4.43) with4.58 ∏Tn.t.e-ζMTγ~-hT≤γ2hC~3q∏Tn.t.γ2hTMT∏Tn.t.γhTMTI

where C~3=maxC3,γ-205lnγζ20lnγ.

Moreover,4.59 (∏Tn.t.γ-hT(RT-1))(∏Tn.t.nT=0γhText-hT)∏Tn.t.γhTMTν≤1.

The sum over the graphs is done in the same exact way, with regard to the effective potential. To sum over q, one first notices that we have no graphs with q=1 and therefore the sum starts from q=2, and then one proceeds obtaining, for |λ|≤C¯′2,4.60 |βν,h|≤γh∑q=2∞λq(C¯′)q≤2(C¯′)2λ2γh

for a constant C¯′ independent of i and h. With the exactly same argument one proves4.61 |βaj,h|≤2(C¯′)2λ2γh.

We can therefore choose C=max{2C¯,2C¯′} so that (4.30) and (4.31) hold. Moreover,4.62 λ0=min{1C,1Cγ-18γ3minjaj(2)}

so that the inductive step is proved.

It remains to prove the existence of the limit i→+∞ where i is the index of the box side Li introduced in point (iii) after (1.3). the expression obtained replacing Li with ∞ and ωi with ω is finite.

Let us denote with L¯i:=min{L0,i,L1,i}. Define for shortness of notation k(t)=(k-ki)t+ki where k∈[-π,π)2 and ki∈D-- with |k-ki|≤2πL¯i, and Ω(t):=(Ω-Ωi)t+Ωi and χh(k,Ω)=χh(k)χh(k-2πΩn)χΓ, then let us consider the term with n≠0 and s=0. One has4.63 ‖V^l,n(k)-V^n(ki)‖≤∑q=1+∞∑Γ∈Gn,q‖WΓ(k,Ω)-WΓ,l(ki,Ωi)‖=∑q=1+∞∑Γ∈Gn,q∫01ddtχh(k(t),Ω(t))WΓ(k(t),Ω(t))dt.

By the Leibnitz rule there are three terms: one in which there is a difference k-ki which can be estimated using the same argument of eq. (4.37)–(4.56) with an additional term 2πγ-hL¯i. Therefore, at the end, this is bounded by C|λ|e-η4|n|1L¯i.

In the second term, the derivative can act either on a vertex or on a propagator producing terms that can be estimated as |Ωi-Ω||nv|γ-h. Then, the procedure to estimate the sum is again similar to (4.37)–(4.56) but in the sum over n (4.53) one sums ∑n≥0(|n|+1)e-η8n to absorb the term |nv|≤∏v′|nv′|. One then uses that |Ωi-Ω|≤CL¯i2 because the sizes of the lattice are the best approximants of the Diophantine numbers ω0 and ω1 (see Section IV.7 in [20]). Therefore, the second term can be estimated as C|λ|e-η4|n|1L¯i2.

To estimate the third term we use the same procedure from (4.37) to (4.56) and the fact that either |ki-k|≤2π/L¯i or |Ωi-Ω|≤C/L¯i2.

The last term involves graphs with at least a vertex with nv≥L¯i and this is O(e-L¯i).

Therefore, there exists a λ0>0 and C>0 independent of h such that, for any λ<λ0 one has4.64 ‖V^l,n(k)-V^n(ki)‖≤CL¯i.

This implies the existence of the limit. □

Remark 4.8

Take the Graph in Fig. 3 and consider the case in which the only resonant cluster is T2. We repeat the argument of Lemma 4.7, applied to this graph only, in order to clarify the procedure. One has,4.65 ∏ℓ∈Γ|g(hℓ)(kℓ)|≤C110γ-h5γ-h3γ-h1γ-h2γ-2h4γ-h3γ-2h5γ-h2=C110γ-h4(ST1-1)γ-h2(ST2-1)γ-h3(ST3-1)γ-h3(ST4-1)γ-h2(ST5-1)γ-h5(SΓ-1)

with QT1=3, QT2=2, QT3=2, QT4=2, QT5=2, QTΓ=3; moreover the action of the R operator on T2 produces a factor γ2(h5-h2), in agreement with (4.37).

Remark 4.9

Note that in (4.52) we have bounded the factor ∏γhTMTI in (4.50) with a constant, and an extra γhT coming from the analysis of Remark 4.5 has been estimated by a constant before (4.37). Such terms will be used after (5.32) in the proof of Lemma 5.5 and in the proof Corollary 5.6.

The choice of the counterterm

In Lemma 4.7 we have proved the convergence of the expansion considering νh as parameters and provided that νh are small enough. νh are determined recursively by (4.27) starting from the initial value ν which is a free parameter; we show that there exists a unique choice of ν so that νh is bounded uniformly in h. We impose the condition ν-∞=0 choosing ν verifying βν,h=βν,h(νh,νh+1,⋯,ν;λ)4.66 ν=-∑k=-∞2γkβν,k(νk,νk+1,⋯,ν;λ)

from which4.67 νh=-∑k=-∞hγk-hβν,k(νk,νk+1,⋯,ν;λ)

and we want to show that (4.67) has a solution.

We define the Banach space M of sequences ν_={νk}k≤2 with norm ‖ν‖M:=∑k≤2|νk|γ-k/4γ1/2 and we consider the ball B⊂M of sequences ν_ such that ‖ν‖M≤|λ|. We define the map T:M→M as4.68 T(ν_)h=-∑k=-∞hγk-hβν,k(νk,νk+1,⋯,ν;λ).

Therefore, (4.67) can be rewritten as4.69 (ν_)h=T(ν_)h.

Lemma 4.10

For |λ|≤λ0, T:B→B is a contraction.

Proof

To prove that T leaves B invariant, we prove a stronger statement: if ν_ is such that |νh|≤|λ|, then T(ν_)∈B. Under these hypothesis, Lemma 4.7 holds and, using (4.31), we get4.70 ‖T(ν_)‖M≤∑h≤2∑k=-∞hγk-hγ-h4γ1/2|βν,k|≤γ154(γ-1)(γ34-1)(Cλ)2,

where C is the constant of Lemma 4.7. Choosing now λ0≤(γ-1)(γ34-1)γ154C2, we get ‖T(ν_)‖M≤|λ|. If ν_,ν′_∈M, then4.71 T(ν_)h-T(ν_′)h=-∑k=-∞hγk-h(βν,k(νk,νk+1,⋯,ν;λ)-βν,k(νk′,νk+1′,⋯,ν′;λ)).

The r.h.s. can be expressed as a sum of graphs identical to Γ with the difference that in a vertex instead of νk there is νk-νk′. Indeed, repeating the argument of Lemma 4.7, one gets4.72 |βν,h(ν_)-βν,h(ν′_)|≤∑q≥2∑Γ(cC1C2C¯3)q|λ|q-∑TMTνγh(∏ve-η8|nv|)×(∏Tn.t.γhText-hT)|∏Tn.t.νhTMTν-∏Tn.t.νhT′MTν|

Using now that ν_∈B, one has |∏Tn.t.νhTMTν-∏Tn.t.(νhT′)MTν|≤(2|λ|)MTν-1‖ν_-ν_′‖M. Therefore, summing over Γ as in Lemma 4.7, and calling C4:=8cC1C2C¯3(1-e-η8)-2(γ-1)-4 one gets4.73 |βν,h(ν_)-βν,h(ν′_)|≤∑q≥2C4q|λ|q-1γh‖ν_-ν_′‖M.

If |λ|≤12C4, then4.74 |βν,h(ν_)-βν,h(ν_′)|≤2C42|λ|γh‖ν_-ν_′‖M.

Using (4.74), we now have4.75 ‖T(ν_)-T(ν_′)‖M≤∑h≤2γ-h4γ1/2|T(ν_)h-T(ν_′)h|≤∑h≤2γ-h4∑k=-∞hγk-hγ1/2|βν,k(ν)-βν,k(ν′)|≤2γ1/2C42|λ|‖ν_-ν_′‖M∑h≤2γ3h4∑k=-∞hγ2(k-h).

Thus, choosing λ0≤(γ2-1)(γ3/4-1)γ19/42C42, T is a contraction on B. □

Remark 4.11

Since, by construction, ν_∈B, we have the bound ∑h≥2|νh|γ-h/4γ1/2≤|λ|, that implies4.76 |νh|≤γ(h-2)/4|λ|≤|λ|

which improves, and hence also justifies, the assumption in Lemma 4.7.

Remark 4.12

Equation (4.69) and Lemma 4.10 determines uniquely ν=ν(λ,μ,β) and proves the assumption |νh|≤C|λ| used in Lemma 4.7. From (4.6), μ=mψ+λγ2ν(λ,μ,β) with mψ≡mψ(β) given by (2.36). The criticality condition is imposed setting μ=0; from 0=mψ(β)+λγ2ν(λ,0,β) we determine the value of βc(λ)=βc(0)+O(λ) by the implicit function theorem as the derivative is non vanishing. In addition μ=O(|β-βc(λ)|).

Energy–Energy Correlations

Integration of ξ variables

The energy correlation (2.1) can be written as5.1 S(x1,j1;x2,j2)=∑α∈{±}2ταZα2Z∂2∂Ax1,j1∂Ax2,j2Wα(A)|A=0

with Z, τα and Zα defined as in (2.6),5.2 Wα(A):=log∫DΛiΦeS(Φ,0)+B(Φ,A)

where5.3 B(Φ,A)=∑x∈Λi[t¯x(1)(A)H¯xHx+e1+t¯x(0)(A)V¯xVx+e0]

and t¯x(j)(A)=tanh(βJx(j)+Ax,j)-tanh(βJx(j)). Proceeding as in Sect. 2 we perform the change of variables Φ=Φ(χ,ψ) defined in (2.7) and then (2.31) to get5.4 eWα(A)=∫Pψ(dψ)∫Pξ(dξ)eV(ψ,ξ)+B^(ψ,ξ,A)

where5.5 B^(ψ,ξ,A):=B¯(ψ,ξ+Cχ-1Qψ,A),B¯(ψ,χ,A):=B(Φ(ψ,χ),A).

Using the following representation in Fourier series for A5.6 Ax,j:=1|Λi|∑p∈D++A^p,jeip·x,

expanding in Taylor series t¯x(j)(A) around A=0 and denoting by ζ=ψ,ξ, one has5.7 B^(ψ,ξ,A)=∑s=1+∞14|Λi|1+s∑ζ1,ζ2=ψ,ξ∑k∈Dα,p_∈(D++)s,n∈Z2j_∈{0,1}sζ^1,-k·K^ζ1,ζ2,n(k,p_,j_)ζ^2,k-∑r=1spr-2πΩn∏r=1sA^pr,jr.

We can integrate over the ξ field obtaining5.8 eWα(A)=eN1(A)∫Pψ(dψ)eV(ψ)+B(1)(ψ,A)

where5.9 B(1)(ψ,A)=∑s=1+∞14|Λi|1+s∑k∈Dα,p_∈(D++)s,n∈Z2,j_∈{0,1}sψ^-k·K^n2,s,1(k,p_,j_)ψ^k-∑r=1spr-2πΩn∏r=1sA^pr,jr

where K^n2,s,1(k,p_,j_) can be expressed as sum over graphs Γ similar to the ones in Definition 3.1 with the following differences. To each point v of the graph Γ is associated a label jv∈{0,1,2} and momentum label nv∈Z2, if jv∈{0,1}, or pv if jv=2, with the constraint that ∑vnv=n and pv is equal to one of the p1,⋯,ps or a linear combination of them; the number of points with jv=0,1 is q. To each line ℓ is associated a momentum kℓ; if ki and ko are two lines attached to the same point v, then ki-ko=2πΩnv if jv=0,1 and ki-ko=pv if jv=2. The proof of Lemma 3.3 can be repeated up to some trivial modifications and we get, under the same conditions, the exponential decay of the kernels in B(1)(ψ,A):5.10 |K^n2,s,1(k,p_,j_)|≤C¯se-η2|n|

for a suitable constant C¯.

Multiscale analysis

The integration of (5.8) is done inductively, by a generalization of the analysis in Sects. 3 and 4. Suppose we have just integrated the scales 1,0,-1,-2,⋯,h+1 obtaining5.11 eWα(A)=eNh(A)∫P(≤h)(dψ(≤h))eV(h)(ψ(≤h))+B(h)(ψ(≤h),A),

with5.12 B(h)(ψ(≤h),A)=∑s=1+∞14|Λi|1+s∑k∈Dαn∈Z2∑p_∈(D++)sj_∈{0,1}sψ^-k(≤h)·K^n2,s,h(k,p_,j_)ψ^k-∑r=1spr-2πΩn(≤h)∏r=1sA^pr,jr

and5.13 Nh(A)=∑s=0+∞14|Λi|s-1∑p_∈(D++)sj_∈{0,1}s∑n∈Z2K^n0,s,h(p_,j_)δ∑rpr+2πΩn,0∏r=1sA^pr,jr

where δ denotes the Kronecker delta. We define a localization operation as5.14 LB^(h)(ψ,A):=14|Λi|2∑k∈Dα,p∈D++,n∈Z2,j∈{0,1}ψ^-k(≤h)·K^n2,1,h(0,0,j)ψk-p-2πΩn(≤h)A^p,j.

Note that, in contrast with the analysis in Sect. 4, the localization acts also on the terms n≠0. We get therefore5.15 eWα(A)=eNh(A)∫P¯(≤h)(dψ(≤h))e14|Λi|∑k∈Dαψ^-k(≤h)·γhνhσ2ψ^k(≤h)e14|Λi|2∑n∈Z2∑k,p,jψ^-k(≤h)·Zh,n(j)σ2ψk-p-2πΩn(≤h)A^p,j+RV(h)(ψ(≤h))+RB(h)(ψ(≤h),A)

with Zh,n(j)=K^n2,1,h(0,0,j). Note that, in writing the above expression, we have used that K^n2,1,h(0,0,j) is proportional to σ2. This latter fact can be checked simply using the anticommutation property of Grassmann variables. We can write P¯(≤h)(dψ(≤h))=P(≤h-1)(dψ(≤h-1))P(h)(dψ(h)) and integrate ψ(h) so that the procedure can be iterated as in Sect. 4.

Let us introduce the following definitions.

Definition 5.1

The special renormalized graphs are labeled graphs defined starting from the renormalized graphs in Definition 4.2 with the following additional labels and modifications if z=2 the first and the last line are attached to a single point while if z=0 there are no external lines.

Each point v is associated with a label Sv; if Sv=0 (normal point) v is associated with a label iv∈{ν,V} and a momentum label nv∈Z2; if Sv=1 (special point) it is associated with a momentum pv, an index jv∈J, a momentum label nv∈Z2 and an index i~v∈{z,B}. The normal points are q and the special ones are s.

Gn,qR,z,s,h,J is the set of special renormalized graphs Γ (here R stands for renormalized, z∈{0,2}, s∈{0,1,2}, h is the scale and J is the collection of jv of the special points).

Similarly to what we did in Sect. 4.3, to a special renormalized graph we associate a set of clusters in the following way.

Definition 5.2

Given a special renormalized graph Γ, we define clusters as in Definition 4.3. Then, a non-trivial cluster T is associated with ST=1,2 if it contains ST special end-point and ST=0 otherwise; in the first case the cluster is called special, and is associated with a momentum 2πΩnT+pT (where pT:=∑v∈Tpv), and in the second case is called normal, and it is associated with a momentum 2πΩnT. We call QT the number of maximal clusters in T; STn=MTn+RTn the number of normal maximal clusters and STsp the number of maximal special clusters; MTsp is the set of maximal special trivial clusters (i.e. points) in T. The scales are such that, when z=2, hΓ=h; when z=0 to each external line is associated a scale and h is the greatest of such scales (see Fig. 5).

Fig. 5 (Left) A graph Γ∈Gn,3R,2,2,h,{j1,j2}. (Right) A graph Γ∈GnR,0,2,h,{j1,j2}

Definition 5.3

The value of graph Γ∈Gn,qR,2,1,h,J with maximal clusters T~w, w=1,⋯,QΓ is defined as5.16 WΓ(p)=∏w=1QΓ-1W¯T~w(kw)g(hΓ)(kw+1)W¯T~QΓ(kQΓ)

where kw=kw-1-2πΩnT~w-1 if T~w-1 is a normal cluster, kw=kw-1+pT~w-1-2πΩnT~w-1 if T~w-1 is a special cluster k1=k. W¯T~w(kw) is defined as5.17 W¯T~w=ZhT,ns(jw)ifT~wis a specialz-point,RK^w2,1,1ifT~wis a specialB-point,γhTνhTσ2ifT~wis aν-point(nw=0),RV^0ifT~wis aV-point(nw=0),V^nwifT~wis aV-point(nw≠0),RWT~wifT~wis a non-trivial cluster.

Similarly, if the special renormalized graph is Γ∈Gn,qR,0,2,h,J5.18 WΓ(p)=14|Λi|∑k∈Dα∏w=1QΓ-1W¯T~w(kw)g(hΓ)(kw+1)W¯T~QΓ(kQΓ)g(hΓ)(kQΓ)

with k1=k.

Lemma 5.4

The kernels K can be written as a sum of graphs5.19 K^n2,s,h(k,p_,j_)=∑q=0∞∑Γ∈Gn,qR,2,s,h,JWΓ(k,p_,j_),K^n0,s,h(p_,j_)=∑q=0∞∑Γ∈Gn,qR,0,s,h,JWΓ(p_,j_)

and the running coupling constants verify5.20 Zh-1,n(j)=Zh,n(j)+βz,n,h(j),βz,n,h(j)=∑q=1∞∑Γ∈Gn,qR,2,1,h-1,jhΓ=hWΓ(0,0,j).

Also in this case, the proof follows along the lines Appendix A and Lemma 4.6.

Bounds

Let us now define5.21 ‖|K^n2,1,h‖|:=supj∈{0,1}supp∈D++supk∈Dα|χh(k)χh(k+p-2πΩn)K^n2,1,h(k,p,j)|.

We will denote by ∏vn.s.=∏v∈Γ,Sv=0.

Lemma 5.5

If |λ|≤λ0 and ν is chosen as in Sect. 4.5, then there exists a constant C independent of i,β and h such that5.22 ‖|K^n2,1,h‖|≤Ce-η4|n|,supp1,p2∈D++|K^n0,2,h(p1,p2,j1,j2)|≤Ce-η8|n|

and5.23 |βz,n,h(j)|≤C|λ|γhe-η4|n|.

Proof

Assume inductively that the statement is valid for k≥h+1; then |Zn,2(j)|≤C1e-η|n| by (5.10) and by induction5.24 Zn,h(j)=Zn,2(j)+∑r=h2βz,n,r(j)≤2CZe-η4|n|

assuming |λ|4C(1-e-η4)2≤C1.

We start from the first of (5.22). Considering that the operator R acting on a special cluster gives a factor γhText-hT, one proceeds as in the proof of Lemma 4.7 to get (instead of (4.37))5.25 ‖|χΓWΓ,l‖|≤C¯1(cC1C2)q|λ|q(∏Tn.t.ST=1T≠ΓγhText-hT)e-η4|ns|∏vn.s.e-η2|nv|×∏Tn.t.γ-hT(MTn+RTn+STsp-1)(∏Tn.t.nT=0ST=0γ2(hText-hT))∏Tn.t.γhTMTν

where c=18 (up to 2 derivatives to points and vertex, with j=0,1), ns is the momentum label of the special point, STsp is the number of special end-points contained in T. We can write5.26 1=γ-hΓ(∏Tn.t.ST=1T≠ΓγhText-hT)∏Tn.t.γhTMTsp.

We get therefore5.27 ‖|χΓWΓ,l‖|≤(cC1C2)q|λ|qγ-hΓ(∏Tn.t.ST=1T≠Γγ2(hText-hT))∏vn.s.e-η2|nv|e-η4|ns|×∏Tn.t.γ-hT(MTn+RTn+STsp-1)(∏Tn.t.nT=0ST=0γ2(hText-hT))∏Tn.t.γhTMTν∏Tn.t.γhTMTsp.

We now use that5.28 ∏Tn.t.γ-hT(RTn+STsp-1)(∏Tn.t.ST=1T≠ΓγhText-hT)×(∏Tn.t.nT=0ST=0γhText-hT)∏Tn.t.γhTMTν∏Tn.t.γhTMTsp≤γhΓ

and following the same argument of Lemma 4.7 from (4.39) to (4.45) we get rid of all γ-hTMT’s and, since Γ is a special cluster, we finally obtain5.29 ‖|χΓWΓ,l‖|≤C¯1(cC1C2C3)q|λ|q(∏Tn.t.ST=1T≠ΓγhText-hT)(∏Tn.t.ST=0γhText-hT)×e-η4|ns|∏vn.s.e-η8|nv|∏vn.s.e-η4|nv|.

To handle the sum over Γ, we perform the sum over the scales as in Lemma 4.7, while in the sum over nv’s one uses that n is fixed, and ∑v=1qnv+ns=n: the sum over nv’s and ns can be performed only on n1,⋯,nq. Thus, using triangular inequality ∑v=1q|nv|+|ns|≥|n| in the last product of (5.29), one has5.30 e-η4|ns|∏vn.s.e-η8|nv|∏vn.s.e-η4|nv|≤e-η4|n|∏vn.s.e-η8|nv|

and then one can sum over n1,⋯,nq as in Lemma 4.7. Therefore5.31 ∑Γ∈Gn,qR,2,s,h,J‖|χΓWΓ,l‖|≤e-η4|n||λ|qC¯1C¯q

with C¯=(c×3)C1C2C3(1-e-η8)-2(1-1/γ)); by summing over q we get, for |λ|≤C¯/2 we get5.32 ∑q≥0∑Γ∈Gn,qR,2,s,h,J‖|χΓWΓ,l‖|≤e-η4|n|C¯12C¯.

We choose |λ|≤min{C/2,C1/(4C(1-e-η4)2}, with C=C¯14C1C2.

In order to prove (5.23), we note that we have to bound K^n2,1,h(0,0,j). This is exactly the same argument used to prove (4.31).

Finally we have to prove the second of (5.22). Using that Γ∈Gn,qR,0,2,h, the analogue of (4.37) becomes5.33 |WΓ,l(p1,p2,j1,j2)|≤(4C1C2)qγ2hΓe-η4|ns1|e-η4|ns2||λ|q(∏Tn.t.ST=1T≠ΓγhText-hT)×∏vn.s.e-η2|nv|∏Tn.t.γ-hT(MTn+RTn+STsp-1+δT)×(∏Tn.t.nT=0ST=0γ2(hText-hT))∏Tn.t.γhTMTν

where the extra γ2hΓ comes from the integration over k and the compact support properties of the propagators at scale hΓ; moreover δΓ=1 and δT=0 if T≠Γ. Note that, since Γ has two special points, we have5.34 1=γ-2hΓ(∏Tn.t.,T≠ΓγST(hText-hT))∏Tn.t.γhTMTsp.

To prove the second of (5.22), one repeats the argument used to prove the first of (5.22) with, instead of (5.28), the following5.35 (∏Tn.t.γ-hT(RTn+STsp+δT-1))(∏Tn.t.ST=1,2T≠ΓγhText-hT)×(∏Tn.t.nT=0ST=0γhText-hT)∏Tn.t.γhTMTν∏Tn.t.γhTMTsp≤1.

Then, to perform the sum over nv’s one now can not repeat the previous argument to isolate the ns as one has to sum to at least one of them. Thus, one has5.36 e-η4∑vspecial|nv|e-η4∑v=1q|nv|≤e-η8|n|e-η8∑vspecial|nv|∏vn.s.e-η8|nv|.

The rest of the proof proceeds as in the cases before. □

We now denote by K~n0,2,h(p1,p2,j1,j2) the contribution to K^n0,2,h(p1,p2,j1,j2) given by the graphs with at least one ν or V point:5.37 K~n0,s,h(p1,p2,j1,j2)=∑q=1∞∑Γ∈Gn,qR,0,s,h,JWΓ(p1,p2,j1,j2).

Corollary 5.6

Let |λ|<λ0 and let ν be chosen as in Sect. 4.5. Then,5.38 supp1,p2∈D++|K~n0,2,h(p1,p2,j1,j2)|≤Cγh4e-η8|n|.

Proof

Repeating the argument of Lemma 5.5, one has to estimate WΓ,l for graphs that have q≥1. One gets a bound identical to (5.33) with ∏Tn.t.γ54hTMTν replacing ∏Tn.t.γhTMTν (we used Remark 4.11 to estimate the ν vertices, i.e. |νh|≤γh4|λ| and Remarks 4.5 and 4.9 to estimate resonant V vertices as |RV^0|≤γ54hT). We also decompose the exponential as in (4.38) and from (4.43) we keep the factor ∏Tn.t.γhTMTI≤∏Tn.t.γ14hTMTI. Using now (5.36), and (5.35), we get5.39 |χΓWΓ,l(p1,p2,j1,j2)|≤(cC1C2C~3)q|λ|qe-η8|n|(∏vn.s.e-η8|nv|)(∏vspeciale-η8|nv|)×(∏Tn.t.γhText-hT)∏Tn.t.γhT4(MTν+MTI).

We now split5.40 ∏Tn.t.γhText-hT=(∏Tn.t.γ34(hText-hT))∏Tn.t.γ14(hText-hT),

and since, by hypothesis, q≥1 for at least one cluster, one has MTν+MTI≥1. Therefore, using the telescopic sum, we can bound5.41 (∏Tn.t.γ14(hText-hT))∏Tn.t.γ14hT(MTν+MTI)≤γhΓ4=γh4.

At this point, one has5.42 ‖|χΓWΓ,l‖|≤(cC1C2C~3)q|λ|qe-η8|n|γh4(∏vn.s.e-η8|nv|)(∏vspeciale-η8|nv|)∏Tn.t.γ34(hText-hT)

and one can sum over all nv’s and one proceeds as in the proofs of Lemma 4.7 to sum over scales to get (5.38). □

The decay of the energy correlations

Before starting the analysis, let us recall that μ=O(β-βc) and, in particular, μ=0 identifies the critical temperature (see Remark 4.12).

We have now to consider the energy correlation S(x1,j1;x2,j2) given by (2.6). We consider first the infinite volume limit ∂2∂Ax1,j1∂Ax2,j2Wα,l(A):5.43 ∂2∂Ax1,j1∂Ax2,j2Wα,l(A)|A=0=∑h=h∗2∑n∈Z2e-2πiΩn·x2Kn0,2,h(x1-x2,j1,j2).

where h∗=logγμ and5.44 Kn0,2,h(x1-x2,j1,j2):=1|Λi|∑pe-ip·(x1-x2)K^n0,2,h(p,p+2πΩn,j1,j2).

Indeed, one can write Kn0,2,h(x1-x2,j1,j2) as the sum over graphs in coordinate space. On each graph, the constraint between the labels nv and the scales hℓ remains unchanged, due to the presence of the χΓ function.

Due to the Gevrey regularity of the cutoff function χ defined in (4.2), there exist constants C,κ>0 such that, for k>h∗, the propagator obeys to the bounds, see e.g. Appendix A of [34]5.45 |g(k)(x)|≤Cγke-κ(γk|x|)12

and5.46 |g(≤h∗)(x)|≤Cγ∗e-κ(γ∗|x|)12.

Note that γ∗=O(|μ|) for small μ. In the analysis of the graphs in coordinate space, we use (5.45) to bound each propagator. Fixing x1, the L1 norm is therefore bounded exactly as in the proof of Lemma 5.5.

Regarding the bound on the point-wise norm (i.e. when both x1 and x2 are fixed), we can write, if h¯ is the scale of the smallest cluster T⊂Γ such that ST=2, for k≥h¯, e-κ(γk|x|)12≤e-κ/2(γh¯|x|)12e-κ/2(γk|x|)12 so that we can extract a factor e-κ/2(γh¯|x|)12 from each propagator. Moreover there is an extra γ2h¯ in the bound due to the lack of sum over the coordinates so that5.47 ∂2∂Ax1,j1∂Ax2,j2Wα,l(A)|A=0≤∑h¯=h∗2Cγ2h¯e-κ/2(γh¯|x1-x2|)12≤C1e-κ1(|μ||x1-x2|)12,

for some constant κ1>0. In deriving the above expression we have used that the sum over all the scales can be done fixing h¯ instead of h.

To get a sharper estimate in the case μ=0, we can split ∂2∂Ax1,j1∂Ax2,j2Wα,l(A)|A=0 in the contribution with q≥1 and in the contribution with q=0. The term with q≥1, according to Corollary 5.6, has an extra γh¯4. The term with q=0 contains two special vertices, each one of which is associated with a Zk,n(j), with k≥h.

In the term with q=0 we replace the velocities aj(k) appearing in the propagators g(k) with aj(∞) since the difference aj(-∞)-aj(k) is bounded by γk/4 by (4.31).

In the same way we can replace Zk,n(j) with Z-∞,n(j) and the difference is bounded by γk/4. Moreover, Z-∞,0(j)=Z-∞,0,λ=0(j)+λF0(λ) and Z-∞,n(j)=λFn(λ) for n≠0, with F0,Fn bounded. Therefore, we can write5.48 ∂2∂Ax1,j1∂Ax2,j2Wα,l(A)|A=0=Sa(x1,j1;x2,j2)+Sb(x1,j1;x2,j2),

where we have included in Sb(x1,j1;x2,j2) the contributions with q≥1 and the terms with q=0 and containing aj(-∞)-aj(k) or Zk,n(j)-Z-∞,n(j) so that5.49 |Sb(x1,j1;x2,j2)|≤C1∑h¯=-∞2γ2h¯+h¯/4e-κ(γh¯|x1-x2|)12≤C2|x1-x2|2+1/4.

In Sa(x1,j1;x2,j2) are collected the terms with q=0 and aj(k), Zn,k(j) replaced by their limiting values, so that, calling g¯(x1,x2) the propagator with velocities aj(-∞), we have5.50 Sa(x1,j1;x2,j2)=∑n1,n1∈Z2Z-∞,n1(j1)Z-∞,n2(j2)e2πiΩn1·x1e2πiΩn2·x2∑ω∈±g¯ω,ω(x1,x2)g¯-ω,-ω(x2,x1).

Finally, we have to perform the sum over α in (5.1). First note that Z is non-vanishing; we write5.51 Z=Z^--Z0+Z^--∑α∈{±}2ταZα0Z^αZ^---1

where Z0=Z|λ=0 denotes the partition function of the Ising model for λ=0, Z^α=Zα/Zα0 and Zα0=Zα|λ=0. In the limit i→∞, 1|Λi|log|Zα0| is independent of boundary conditions if β≠βc, see e.g. chapter IV in [58], and the limit is reached as O(e-c|μ|L¯i) if L¯i:=min{Li,0,Li,1}. Moreover, Z0 is non vanishing for β≠βc: indeed, for μ<0, Zα0 is positive for all α; for μ>0, Zα0 is negative for α=++ and positive for all other α’s.

We consider now Z^αZ^--; note that ωi is the same in Z^α for any α. logZ^αZ^-- is sum of graphs containing at least a difference of propagators with different boundary conditions. We choose a point x¯∈Λi and we decompose the graphs in a term in which all the sums are in a rectangle around x¯ of side Li,0/4 and Li,1/4 and a remainder. In the remainder there is a product of propagators connecting x¯ to a point distant O(L¯i), L¯i:=min{Li,0,Li,1}, hence such term is O(|λ||Λi|e-c|μ|L¯i). In the first term we use Poisson summation allowing us to write the propagator as the infinite volume limit plus a term depending on boundary conditions and exponentially decaying in x1-x2 when both x1 and x2 are in the rectangle around x¯ of side Li,0/4 and Li,1/4, hence again we get for it a bound O(|λ||Λi|e-c|μ|L¯i). Therefore,5.52 Z^αZ^---1≤C|λ||Λi|e-c|μ|L¯i

by using the uniform convergence, see Lemma 4.7. This says that5.53 c1|Z^--Z0|≤|Z|≤c2|Z^--Z0|

where c1,c2=1+O(λ) constants.

Using that 2Z=∑αταZα, we can write (5.1) as5.54 S(x1,j1;x2,j2)=∂2∂Ax1,j1∂Ax2,j2W--(A)|A=0+∑αταZα2Z[∂2∂Ax1,j1∂Ax2,j2Wα(A)|A=0-∂2∂Ax1,j1∂Ax2,j2W--(A)|A=0].

where in the first term Z cancels out by (5.53).

The graphs contributing to ∂2∂Ax1,j1∂Ax2,j2Wα(A)|A=0 can be also decomposed as the limit i→∞, independent from α and a difference which is vanishing. Indeed the difference contains a difference of propagators, whose contribution is vanishing at |μ|>0, and a difference of oscillating factors ei2πΩn·x which is bounded by |x||n||ω-ωi|; note that |x| produces an extra maxj{Lj,i} and |ω-ωi|≤C/L¯i2 (see Section IV.7 in [20]) while for the sum over n one uses the exponential decay of the Fourier coefficients of the potential. Hence the difference vanishes in the limit because we take the limit on sequences of Lj,i such that limi→+∞L1,i/L0,i=c>0. Moreover, if μ≠0, as a consequence of (5.53) and (5.52) we have that Z′0/Zα0=1+O(|Λi|e-c|μ|L¯i) and Z^α′/Z^α=1+O(|λ||Λi|e-c|μ|L¯i). Therefore the second term in (5.54) vanishes in the limit i→∞.

The first term in (5.54) can be decomposed according to (5.48) with Sa given by (5.50) and Sb satisfying (5.49). Therefore, the first term in the r.h.s. of (1.12) is given by Sa and the decay in |x1-x2| of Rj1,j2(x1,x2) is given by (5.49). This concludes the proof of Theorem 1.2.

Appendix A: Proof of Lemma 3.2

We begin by recalling that if f(ξ^) is a polynomial in the Grassmann variables ξ^, Eξ(f)=∫P(dξ)f(ξ^). We then recall that, for m∈N, Wick’s theorem states thatA.1 Eξ(ξ^k1,σ1ξ^p1,ρ1⋯ξ^km,σmξ^pm,ρm)=∑p∈Sm(-1)sgn(p)∏j=1mEξ(ξ^kj,σjξ^pp(j),ρp(j))

where for all j, (pj)0<0, (kj)0>0 and σj,ρj∈{±} and Sm denotes the set of permutations of m elements. From the definition of propagator of a Grassmann Gaussian measure (see e.g. [27, eq. (4.11)]), one hasA.2 Eξ(ξ^k,σξ^p,ρ)=[g(ξ)(k)]σ,ρδk,-p|Λi|.

For q∈N, let us compute Eξ(Vq). Using linearity of the expectation and the explicit form of V(ψ,ξ) (given in (2.40), (2.41), (2.42) and (2.43)), we can writeA.3 Eξ(Vq)=∑∗Eξ(∏r=1qM#r(kr,nr,σr,ρr,jr))

where ∑∗ is the sum over kr, nr, σr, ρr, #r∈{ψ,ξ,(Q,L),(Q,R)}, jr∈{0,1}. The monomials M♯ are defined asA.4 Mψ(k,n,σ,ρ,j)=1|Λi|ψ^-k,σAn(j)[Pψ(j)(k,n)]σ,ρψ^k-2πΩn,ρ,

A.5 Mξ(k,n,σ,ρ,j)=1|Λi|ξ^-k,σAn(j)[P(j)(k,n)]σ,ρξ^k-2πΩn,ρ,

A.6 MQ,L(k,n,σ,ρ,j)=1|Λi|ψ^-k,σAn(j)[Qψ(j)(k,n)]σ,ρξ^k-2πΩn,ρ,MQ,R(k,n,σ,ρ,j)=1|Λi|ξ^-k,σAn(j)[Qψ(j)(k,n)]σ,ρψ^k-2πΩn,ρ,

and each one of them can be represented as in Fig. 6 by associating a dashed line to each ψ variable and a solid line to each ξ variable.Fig. 6 Vertices associated to monomials in (A.4), (A.5) and (A.6)

To compute each of the expectations on the r.h.s. of (A.3) we use Wick’s theorem (A.1) and therefore Eξ(M#1⋯M#q) reduces to the sum over permutations of products of expectations of pairs of ξ variables. Each of such summand has a graphical interpretation obtained as follows. First, one draws the vertices in Fig. 6 corresponding to the monomials M#1,⋯,M#q. Second, one connects the solid lines corresponding to a pair (ξ^kj,σj,ξ^pi,ρi) whenever the expectation Eξ(ξ^kj,σjξ^pi,ρi) appears in the product of expectations of the summand. In this way one obtains a graphical object that we define as unordered graph. As a consequence of this correspondence, one writes Eξ(M#1⋯M#q) as the sum over all unordered graphs obtained by contracting all solid lines of the graphical elements associated to M#1,⋯,M#q. Note that there are two types of unordered graphs: connected and disconnected.

We now write truncated expectations in terms of expectations. Let us consider S={1,…,s} and let us denote by Pp the set of all possible partitions of S into p pairwise disjoint subsets. Define for any I⊆SA.7 ET(MI)=ET(Mi1;⋯;Mir),I={i1,i2,⋯,ir},

where each of the Mi’s is one of the monomials in (A.4), (A.5) and (A.6).

One can prove (see eq. 2.100 in [56]) that the following formula connects expectations and truncated expectations:A.8 EξT(M#1;⋯;M#s)=Eξ(M#1⋯M#s)-∑p=2s∑{I1,⋯,Ip}∈Pp∏j=1pEξT(MIj).

Using this formula, one can prove inductively that the truncated expectations are obtained as the sum over connected unordered graphs only.

Using multilinearity of the truncated expectations, one hasA.9 EξT(V;q)=∑∗EξT(M#1;⋯;M#q),

where the ∑∗ is the same as in (A.3) and the dependence on all parameters is understood. From (A.9) and the observation after (A.8) we obtain a representation of ET(V;q) in terms of connected unordered graphs.

When q=1, EξT(V(ψ,ξ);q)=Sint(ψ)(ψ) which gives the first term in the sum (3.5) with WΓ given by the definition (3.3) in the case q=1.

For the case q≥2, one notices that with the vertices in Fig. 6 one can make only two types of connected graphs: either one picks q vertices of type Mξ, or one picks two vertices of type MQ,L, MQ,R and q-2 vertices of type Mξ.

In the first case, the value of the associated truncated expectation does not depend on ψ and it contributes to Eξ.

In the second case, one first notices that each of the entries of the truncated expectation on the r.h.s. of (A.9) is a quadratic monomial in the Grassmann variables, and then it commutes with the monomials on each other entry. Fixing an order of the entries in the truncated expectation produces a q! in front and restricts the sum over the graphs of Definition 3.1. Last, the relation between EξT(M#1;⋯;M#q) and (3.5) (with WΓ defined as in (3.3)) follows from (A.4), (A.5), (A.6), (A.1) and (A.2) after noting that each unordered connected graph has the same sign. Indeed, it is sufficient to take the sign appearing in Wick’s theorem (A.1) and to note that to keep the graph connected one must always exchange an even number of Grassmann variables.

Appendix B: Derivation of (3.7)

The value of a graph WΓ is product of complex valued scalar functions An and matrices of the form an(k)bn(k)bn∗(k)-an∗(k) such that an(k)=-a-n(-k)∈C and bn(k)=b-n(-k)∈iR. Equation (3.7) will be proved by induction. For the graph with two vertices we have to consider the product of three matrices AnA(k),gχ(k-2πΩnA),BnB(k-2πΩnA). Here A and B can be either P(j) or Q(j). The following explicit computation yieldsAnA(k)gχ(k-2πΩnA)=anA(A)(k)bnA(A)(k)(bnA(A)(k))∗-(anA(A))∗a(ξ)(k-2πΩnA)b(ξ)(k-2πΩnA)(b(ξ)(k-2πΩnA))∗-(a(ξ)(k-2πΩnA))∗=β(k,nA)α(k,nA)-α∗(k,nA)β∗(k,nA).

where, explicitly,α(k,nA):=aA(A)(k)b(ξ)(k-2πΩnA)-bA(A)(k)(a(ξ)(k-2πΩnA))∗,β(k,nB):=anA(A)(k)a(ξ)(k-2πΩnA)+bnA(A)(k)(b(ξ)(k-2πΩnA))∗.

It follows now from the symmetry properties of a and b that α(k,nA)=-α(-k,-nA) and β(k,nA)=β(-k,-nA).

Computing the value of the graph, one hasAnA(k)gχ(k-2πΩnA)BnB(k-2πΩnA)=β(k,nA)α(k,nA)-α∗(k,nA)β∗(k,nA)aA(B)(k-2πΩnA)bB(B)(k-2πΩnA)(bB(B)(k-2πΩnA))∗-(aB(B)(k-2πΩnA))∗=an(k)bn(k)bn∗(k)-an∗(k)

withan(k)=β(k,nA)anA(B)(k-2πΩnA)+α(k,nA)(bB(B)(k-2πΩnA))∗,bn(k)=β(k,nA)bnB(B)(k-2πΩnA)-α(k,nA)(anB(B)(k-2πΩnA))∗.

From the symmetry properties of a,b,α and β it follows that under the exchange {nA,nB,k}↦{-nA,-nB,-k} one has that an(k)=-a-n(-k) and bn(k)=b-n(-k). This completes the proof for the graph with two vertices. By the inductive hypothesis we assume that the property holds for the product of the matrices associated to the sub-graph of the first q-1 points, and repeat the argument.

We note now that, calling Val(Γ{n_v}(k)) the value WΓ with Γ∈Gn,qB.1 ∑k∈Dαψ-k·V0(k)ψk=14∑k∈Dαψ-k·∑Γ{n_v}∈G0,qξVal(Γ{n_v}(k))-Val(Γ{-n_v}(-k))T+Val(Γ{-n_v}(k))-Val(Γ{n_v}(-k))Tψk.

We start by computingB.2 ValΓ{n_v}(k)-ValΓ{-nn}(-k)T=∏v∈ΓA^nv(jv)G{n_v}(k)-∏v∈ΓA^-n_v(jv)G{-nv}(-k)T

It is convenient to call fn:=∏v∈ΓA^nv(jv). Then, using that f-n=fn∗, we haveB.3 ValΓ{n_v}(k)-ValΓ{-n_v}(-k)T=fnan(k)bn(k)bn∗(k)-an∗(k)-fn∗a-n(-k)b-n∗(-k)b-n(-k)-a-n∗(-k)=fnan(k)bn(k)bn∗(k)-an∗(k)-fn∗-an(k)bn∗(k)bn(k)an∗(k)=(fn+fn∗)an(k)fnbn(k)-fn∗bn∗(k)fnbn∗(k)-fn∗bn(k)-(fn(k)+fn∗)an∗(k)=αn(k)βn(k)βn∗(k)-αn∗(k)

withB.4 αn(k)=(fn+fn∗)an(k)βn(k)=fnbn(k)-fn∗bn∗(k).

With those explicit expressions at hand, it is clear that βn(k)∈iR and αn(k)∈C, in general. FinallyB.5 ValΓ{n_v}(k)-ValΓ{-n_v}(-k)T+ValΓ{-n_v}(k)-ValΓ{-n_v}(-k)T=αn(k)βn(k)βn∗(k)-αn∗(k)+α-n(k)β-n(k)β-n∗(k)-α-n∗(k)=αn(k)+α-n(k)βn(k)+β-n(k)βn∗(k)+β-n∗(k)-αn∗(k)-α-n∗(k)=αn(k)-αn(-k)βn(k)+βn(-k)βn∗(k)+βn∗(-k)-αn∗(k)+αn∗(-k).

Remark .7

In the case of layered disorder constant in one direction, say e0, the theory is translation invariant in one direction and one has to the additional property that an∈iR, implying that the velocities are real. Indeed only P(1) matrices are present and, as we are interested in the case k0=k1=0 with ω1≠0. This implies that also the entries of the propagator are purely imaginary, and the product of an odd number of imaginary numbers is imaginary.

Remark .8

With respect to the symmetries of Section II.D in [31] we break manifestly symmetries 1)-3) (which are, in order, parity, diagonal reflection and orthogonal reflection). Moreover, note that all kernels Kn(k) appearing in Sect. 2 are such that Kn(k)=[K-n(-k)]∗ which is nothing but the symmetry by complex conjugation (i.e., the symmetry 4) in Section II.D of [31].

Appendix C: Action of the R Operation

In this appendix, for j∈N, we denote by Rj the set of resonant clusters strictly contained in Rj-1 and not in any other resonant cluster. (We also denote by R:=⋃j=1+∞Rj.) Denoting with R2 the set of maximal resonances contained in R1, the value of renormalized resonant cluster can now be estimated as, if T~ is a resonanceC.1 RWT~(hT~)(kT~)≤supt∈[0,1]d2dt2WT~(hT~)(tkT)

One has now to analyze what happens when a derivative acts on a renormalized cluster.

If two derivatives corresponding to a resonance T~ acts on the value of some renormalized resonant cluster T~′⊂T~, recalling that kT~′=tk+q for suitable q, one hasC.2 d2dt2RW′(hT~′)(tk+q)=d2dt2W′(hT~′)(tk+q)-W′(hT~′)(0)-(tk+q)·∂kW′(hT~′)(0)=d2dt2W′(hT~′)(tk+q).

If one derivative acts on a renormalized cluster, we have insteadC.3 ddtRW′(hT~′)(tk+q)=∫01ddtddsW′(hT~′)(s(tk+q))ds.

Whence we get the two boundsC.4 d2dt2RW′(hT~′)(tk+q)=d2dt2W′(hT~′)(tk+q),

C.5 ddtRW′(hT~′)(tk+q)≤sups,t∈[0,1]ddtddsW′(hT~′)(s(tk+q)).

So, summarizing, for the estimate we have the following:if two derivatives corresponding to a resonance T~ act on the value of some resonance T~′⊂T~, one can replace with 1 the R operator;

if one derivative corresponding to a resonance T~ acts on the value of some resonance T~′⊂T~, one can replace with dds the R operator and take the supremum over s∈[0,1];

if no derivatives act on a resonance, one can replace R with d2ds2 and take the supremum over s∈[0,1].

These remarks permit us to iterate this procedure considering the action of derivatives on resonances inside resonances. Proceeding in this way, we see that the R.H.S. of (C.1) can be bounded in the following way. We denote by f either a line or a vertex and with TR∈R a resonant cluster.There is one term for each ordered pair (f1,f2), with f1,f2∈TR, not necessarily different (i.e. it may happen that f1=f2).

If f1∈T~0 and T~ is a cluster contained in TR, then T~=T(r)⊂T(r-1)⊂⋯⊂T(1)=TR is the chain of clusters associated to f1 containing T~ and contained in TR. Similarly, if f2∈T^0 and T^ is a cluster contained in TR, one constructs the chain of clusters associated to f2 containing T^ and contained in TR.

At this point we replaced the R operator acting on the cluster TR with two derivatives.

If a resonant cluster belongs to both the chain of clusters (the one associated with f1 and the one associated with f2), then its R operator is removed.

If instead there is a cluster (say, TV) belonging to only one of the chain of clusters, then there is one term for any f3∈TV. If f3∈(TV′)0⊂TV, then one considers the chain of cluster associated to f3, containing TV′ and contained in TV. One replaced the R operator acting on TV.

This construction is repeated until all R operators are replaced. At this point each cluster inside a resonance belongs to two chains of vertices.

From their explicit expression, it is also obvious that one can estimate the action of a derivative on a vertex with the action of a derivative on a propagator on the same scale.

Last, the number of terms that are generated in this procedure is estimated by 9q (that is the number of terms generated when each vertex or each line can be derived zero, one or two times without any constraint).

Note that, an adaptation of this argument permits to treat the terms ∂ks appearing in (3.6) and (4.30).

Acknowledgements

We thank Rafael L. Greenblatt and Marcello Porta for fruitful discussions and the anonymous referees for their fruitful criticism. We acknowledge financial support of the MIUR-PRIN 2017 project MaQuMA cod. 2017ASFLJR, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program ERC StG MaMBoQ, n.802901. We also thank GNFM, the Italian National Group of Mathematical Physics. M.G. acknowledges Dipartimento di Matematica “F. Enriques”, University of Milan, where part of this work was carried out. V.M. acknowledges Institute for Advanced Studies (Princeton) where part of this work was carried out.

Funding

Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement.

Data Availability

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Publisher's Note

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