Vanishing of Drude weight in interacting fermions on Zd with quasi-periodic disorder
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Vanishing of Drude weight in interacting fermions on Z d withquasi-periodic disorder Vieri Mastropietro University of Milano, Department of Mathematics “F. Enriquez”, Via C. Saldini 50,20133 Milano, ItalyFebruary 17, 2021
Abstract
We consider a fermionic many body system in Z d with a short range interaction andquasi-periodic disorder. In the strong disorder regime and assuming a Diophantine conditionon the frequencies and on the chemical potential, we prove at T “ The conductivity properties in fermionic systems, describing electrons in metals, are stronglyaffected by the presence of disorder, which breaks the perfect periodicity of an ideal lattice andis unavoidable in real systems. Disorder can be represented either by a random variable or bya quasi-periodic potential; the first description is more suitable for impurities in solids whilethe second appears naturally in quasi-crystals or cold atoms experiments. In absence of manybody interaction disorder produces the phenomenon of Anderson localization [1], consisting inan exponential decay of all eigenstates and in an insulating behavior with vanishing conductivity.Such a phenomenon relies on the properties of the single particle Schroedinger equation and ithas been the subject of a deep mathematical investigation. With random disorder Andersonlocalization was established for strong disorder in any dimension [2], [3] and in one dimensionwith any disorder. In the case of quasi-periodic disorder localization in one dimension is presentonly for large disorder [4], [5], while for weak disorder is absent; in higher dimensions localizationwas proved for strong disorder in d “ d in [8].The interplay between disorder and interaction has been deeply analyzed in the physicalliterature soon after [1]. The presence of many body interaction induces new processes whichcan indeed destroy localization. At zero temperature T “ d “ d “ T “
0. In morerecent times the properties at T ą T in any dimension were given (many bodylocalized phase). Subsequently numerical simulations found localization in certain systems inall the spectrum and vanishing of conductivity for any T , a phenomenon called many bodylocalization , see [15] for random and [16] for quasi-periodic disorder. If all states are localized1ne expects, in a non-equilibrium setting, that interaction is unable to produce thermalization inan isolated quantum system, a phenomenon that in classical mechanics is due to closeness to anintegrable system. Interacting quantum systems with quasi-periodic disorder have been realizedin cold atoms experiments [17], [18],[19] ; quasi-periodic disorder with many body interactionhas been extensively numerically analyzed [20]-[28].While the above works suggest that localization persists in presence of interaction, resultsbased on numerical or perturbative analysis cannot be conclusive. In particular the presence of small divisors has the effect that physical informations are difficult to be extracted by lower orderanalysis but are typically encoded in convergence or divergence of the whole series. This is a wellknown phenomenon in classical mechanics; the Birkoff series for prime integrals in Hamiltoniansystems are generically diverging while Lindsdtet series for Kolomogorov-Arnold-Moser (KAM)tori converge, even if both series are order by order finite and present similar small divisors.Therefore, even if perturbative analysis in [14] or [29] get localization at finite temperature andin any dimension, one cannot exclude that the series are divergent and localization eventuallydisappear (this would say that thermalization in experiments is eventually reached, even if atlong times). A non-perturbative proof of many body localization for all eigenstates has beenindeed finally obtained in d “ d ą T “ d ą f p ~ω~x q with f periodic, as the one considered in [6] for the single particle Schroedingerequation ; more general forms of disorder are however possible, as f p ~ω ~x, ~ω ~x q considered in[6]. The second aim is to compute the T “ § § § § § § § Interacting fermions with quasi-periodic disorder
We introduce the Fock space F L “ À N ě h ^ NL where the N particle Hilbert space h ^ NL is theset of the totally antisymmetric square integrable functions in Λ L : “ t ~x P Z d | ~x “ n ~e ` n ~e ` ... , ´ L { ď n i ď L { , i “ , , .., d u where ~e i are unit vectors. The a ˘ ~x are fermioniccreation or annihilation operators sending an element of h ^ NL in h ^ N ` L (creation) or h ^ N ´ L (annihilation) and t a ` ~x , a ´ ~y u “ δ ~x,~y , t a ` ~x , a ` ~y u “ t a ´ ~x , a ´ ~y u “
0. The Hamiltonian is H “ ´ ε ÿ ~x d ÿ i “ p a ` ~x ` ~e i a ´ ~x ` a ` ~x a ´ ~x ` ~e i q ` u ÿ ~x φ ~x a ` ~x a ´ ~x ` λ ÿ ~x d ÿ i “ a ` ~x a ´ ~x a ` ~x ` ~e i a ´ ~x ` ~e i (1)where a ` ~x must be interpreted as zero for ~x R Λ L and φ ~x “ ¯ φ p ~ω~x q with ¯ φ p t q : T Ñ R periodicof period 1. In order to describe a quasi-periodic disorder we impose that ~ω is rationallyindependent and ”badly” approximated by rationals (Diophantine condition). The first term in(1) represents the kinetic energy of the fermions hopping on a lattice, the second represents theinteraction with a quasi-periodic potential and the last term represents a 2 body interaction.There are several interesting limits; λ “ λ “ u “ λ “ ε “ λ, ε are small with respect to u , and we set u “ N “ ř ~x a ` ~x a ´ ~x we define x¨y β,L “ Tr F L ¨ e ´ β p H ´ µN q Z β,L , Z β,L “ Tr F L e ´ β p H ´ µN q (2)where µ is the chemical potential, which is fixed by the density in the Grand-Canonical ensamble,and Z β,L is the partition function. In the limit β Ñ 8 they provide information on the groundstates. We define x¨y “ lim β Ñ8 lim L Ñ8 x¨y β,L (3)The imaginary-time (or Euclidean) evolution of the fermionic operators is a ˘ x “ e x p H ´ µN q a ˘ ~x e ´ x p H ´ µN q (4)with x “ p x , ~x q with x P r , β q , The 2-point function is given by S β,L p x , y q “ @ T a ´ x a ` y D β,L (5)and T is the time order product. We also consider the truncated expectations x T A ; B y β,L “x T AB y β,L ´ x T A y β,L x T B y β,L . The density and the current are given by ρ ~x “ a ` ~x a ´ ~x j i~x “ ε i p a ` ~x ` ~e i a ´ ~x ´ a ` ~x a ´ ~x ` ~e i q (6)The (Euclidean) conductivity density in the zero temperature limit is defined by Kubo formula σ i~y “ lim p Ñ p lim β Ñ8 lim L Ñ8 r ÿ ~x P Λ L ż β dx e ip x @ T j i~x,x ; j i~y, D β,L ` ă τ i~y ą β,L s (7)where τ i~y “ ´ ε p a ` ~y ` ~e i a ´ ~y ` a ` ~y a ´ ~y ` ~e i q (8)3he conductivity can be equivalently expressed in terms of the Fourier transform which is,in the β Ñ 8 , L
Ñ 8 limit , i “ , , d p H ii p p , ~y q “ ÿ ~x P Λ ż R dx e i px ă T j i~x,x ; j i~y, ą (9)and similarly we define p H µν p p , ~y q , with µ “ , , ...d ( µ “ µ “ , ..., d thecurrent component). We can rewrite (7) as σ i~y “ lim p Ñ lim ~p Ñ p r p H ii p p , ~y q` ă τ i~y ąs (10)Finally the (zero temperature) Drude weight, see eg [39], [40] , is defined as D i~y “ lim p Ñ lim ~p Ñ r p H ii p p , ~y q` ă τ i~y ąs (11)In a perfect metal at equilibrium the Drude weight is non-vanishing implying that the conduc-tivity is infinite; a vanishing Drude weight signals a non-metallic behavior.In the above definitions of conductivity the order in which the limits are taken is essential;already in the integrable limit u “ λ “ In the anti-integrable limit λ “ ε “ ~x P Λ L H “ ÿ ~x P Λ L ¯ φ p ~ω~x q n ~x n ~x “ , δ ~x,~y . The 2-point function is given by g p x , y q “ δ ~x,~y e p φ ~x ´ µ qp x ´ y q r θ p x ´ y q ` e β p φ ~x ´ µ q ´ θ p y ´ x q e β p φ ~x ´ µ q ` e β p φ ~x ´ µ q s (13)which can be equivalently written as g p x , y q “ δ ~x,~y β ÿ k “ πβ p n ` q e ´ ik p x ´ y q p g p ~x, k q “ δ ~x,~y ¯ g p ~x ; x ´ y q (14)with p g p ~x, k q “ ´ ik ` φ ~x ´ µ (15)We define µ “ ¯ φ p α q (16)and the occupation number on the ground state is θ p ¯ φ p ~ω~x q ´ ¯ φ p α qq ; the choice of µ fixes theaveraged density. The conductivity is exactly vanishing as the is proportional to ε . The densitycorrelation is ă ρ x ; ρ y ą“ δ ~x,~y ¯ g p ~x ; x ´ y q ¯ g p ~x ; y ´ x q (17)We want to investigate what happens when we consider a non-vanishing hopping ε “ λ “
0. As usual in small divisor problems, we need to impose a Diophantinecondition on the frequencies ~ω of the quasi-periodic disorder that is ||p ~ω~x q|| T ě C | ~x | ´ τ ~x P Z d { ~ | . || being the norm on the one dimensional torus with period 1; we require also a Diophantinecondition on the chemical potential, that is ||p ~ω~x q ˘ α || T ě C | ~x | ´ τ ~x P Z d { ~ ω, α verifying the diophantine conditions for some C has measure O p C q , see eg [41].In general the value of the chemical potential is modified by the interaction; in order to fixthe interacting chemical potential to the value ¯ φ p α q we choose the bare one to µ “ ¯ φ p α q ` ν with ν chosen properly.Our main result is the following Theorem 3.1.
Assume that µ “ ¯ φ p α q ` ν and φ x “ ¯ φ p ~ω~x q with ¯ φ : T Ñ R , even, differentiableand such that v “ B ¯ φ p α q “ : in addition ~ω verifies (18) and α verifies (19). There exists ε and a suitable choice of ν “ O p ε q such that, for | λ | ď | ε | ď ε in the zero temperature andinfinite volume limit1. The 2-point correlation verifies, for any N | S p x , y q| ď | log ∆ ~x,~y | C N e ´ | log | ε ||| ~x ´ ~y | ` p ∆ ~x,~y | x ´ y |q N (20) with ∆ ~x,~y “ p ` min p| ~x | , | ~y |qq ´ τ (21)
2. The density and current correlations verify | H µ,ν p x , y q| ď ∆ ´ ~x,~y C N e ´ | log | ε ||| ~x ´ ~y | ` p ∆ ~x,~y | x ´ y |q N (22)
3. The Drude weight is vanishing D i~x “ φ p ~ω~x q in any dimension. Moreover the Drude weight at T “ ε is proportional to C to some power, with fixed ε, λ weget a large measure set of densities for which localization is present (but not on an interval).Information on the conductivity are obtained by combining the Ward Identities followingfrom the conservation of the current with regularity properties of the Fourier transform ofthe correlations, which are related to the decay in the coordinate space. In the case of non-interacting fermions, or for 1 d interacting fermions without disorder, the slow power law decayof correlations implies a non vanishing Drude weight, see [42]. In the present case, the decayin space is exponentially fast but the decay in the imaginary time has rate not uniform in ~x, ~y ,due to the lack of translation invariance. As a consequence, we can deduce the vanishing of theDrude weight but not of the conductivity.The analysis is based on an extension of the Lindstedt series approach to KAM tori withexact Renormalization Group methods for fermions. The correlations are expressed by a seriesexpansion showing a small divisor problem, as in the Lindstedt series for KAM, in graphswith loops, which are a peculiarity of quantum physics. Small divisors are controlled by the5iophantine conditions and the huge number of loop graphs is compensated by cancellationsdue to anticommutativity.While we have proved here the vanishing of the Drude weight, it would be interesting tounderstand if also the conductivity is vanishing or if a zero result is found only by a suitableaveraging over the phase, as is done in numerical simulations [27].The effective interaction is irrelevant in the Renormalization Group sense, as consequenceof Diophantine conditions and by cancellations due to anticommutativity. The presence of spin[43] and an anisotropic hopping [44] produce extra marginal couplings. They can in principledestroy the convergence result of the present paper, and it is interesting to observe that nu-merical [45] or cold atoms experiments [19] have found evidence of delocalization is such cases.Another important point would be to extend the analysis to a more general kind of disorder like f p ~ω ~x, ~ω ~x q . The condition of strong disorder is non technical; in the case of weak quasiperiodicdisorder there is no localization; in particular, this is the case of the interacting Aubry-Andre’model [46], of the bidimensional Hofstadter model [47] or of three dimensional Weyl semimetals[48]. Finally, we stress that a rigorous understanding of T “ T “
0, this isexpected to be a major difficulty for T ą
0. Another difficulty is due to the fact that we donot get ground state localization in an interval of densities, but only in a large measure set.The absence of thermalization in the classical case is considered related to KAM theorem; it isinteresting to note that the persistence of localization in a quantum system, which is consideredan obstruction to thermalization, is also obtained via the generalization of KAM methods in aquantum context.
We show that the vanishing of Drude weight (23) is consequence of the bound (22) combined withWard Identities. Note first that the Fourier transform in the infinite volume limit is continuousas | p H µ,ν p p , ~y q| ď ÿ ~x ż dx | H µ,ν p x , y q| ď ÿ ~x ż dx ∆ ´ ~x,~y C N e ´ | log | ε || ~x ´ ~y | ` p ∆ ~x,~y | x |q N ď (24) C ÿ ~x p| ~x ` ~y | τ ` | ~y | τ q e ´ | log | ε || ~x || ď C ÿ ~x e ´ | log | ε || ~x || p| ~x | τ ` | ~y | τ q ď C | ~y | τ {p| log | ε ||q d ` τ Ward identities can be deduced from the continuity equation, B ρ x “ r H, ρ x s “ ´ i ÿ i p j i x ´ j i x ´ e i q (25)we get, setting B i j x ” j x ´ j x ´ e i , i “ , ..., d , e i “ p , ~e i qB ă T ρ x ; ρ y ą“ ´ i ÿ i B i ă T j i x ; ρ y ą ` δ p x ´ y q ă r ρ x , ρ y s ąB ă T ρ x ; j j y ą“ ´ i ÿ i B i ă T j i x ; j j y ą ` δ p x ´ y q ă r ρ x , j j y s ą (26)Note that r ρ ~x,x , ρ ~y,x s “ r ρ ~x,x , j j~y,x s “ ´ iδ ~x,~y τ j~x ` iδ ~x ´ ~e j ,~y τ j~y (27)6o that, in the L, β
Ñ 8 limit B ă T ρ x ; ρ y ą“ ´ i ÿ i B i ă T j i x ; ρ y ą (28) B ă T ρ x ; j j y ą“ ´ i ÿ i B i ă T j i x ; j j y ą ´ iδ p x ´ y qp´ δ ~x,~y ă τ j~y ą ` δ ~x ´ ~e j ,~y ă τ j~y ąq Taking the Fourier transform in x we get, using translation invariance in time and setting y “ ÿ ~x ż dx e i px pB ă T ρ x ; j j~y ą ` i ÿ i B i ă T j i x ; j j~y ą ` iδ p x qp´ δ ~x,~y ă τ j~y ą ` δ ~x ´ ~e j ,~y ă τ j~y ąq “ p P R and ~p P r´ π, π q d so that ´ ip p H ,j p p , ~y q ` i ÿ i p ´ e ´ ip i qp p H i,j p p , ~y q ` e ´ i~p~y ă τ jy, ąq “ j “ ~p “ p p , , q so that ´ ip p H , p ¯ p , ~y q ` i p ´ e ´ ip qp p H , p ¯ p , ~y q ` e ´ ip y ă τ y,y ąq “ p Ñ p p H , p , p , ~y q ` e ´ ip y ă τ y,y ąq “ p Ñ p e ´ ip y ´ q “
0. In conclusionlim p Ñ p p H , p , p , ~y q` ă τ y,y ąq “ p H , p p , ~y q is continuous in p so that we can exchange the limitslim p Ñ lim ~p Ñ p p H , p p , ~y q` ă τ y,y ąq “ D ~x “ The starting point of the analysis consists in expanding around the anti-integrable limit (12);defining H ´ µN “ H ` V (35) H “ ÿ ~x p φ ~x ´ ¯ φ p α qq a ` ~x a ´ ~x V “ ε ÿ ~x,i p a ` ~x ` ~e i a ´ ~x ` a ` ~x a ´ ~x ` ~e i q ` λ ÿ ~x,i a ` ~x a ´ ~x a ` ~x ` ~e i a ´ ~x ` ~e i ` ν ÿ ~x a ` ~x a ´ ~x (36)and using the Trotter formula one can write the partition function and the correlations as a powerseries expansion in λ, ε . The correlations can be equivalently written in terms of Grassmannintegrals. We can write e W p η,J q “ ż P p dψ q e ´ V p ψ q´ B p ψ,J,η q (37)7 ˘ e i x ˘ e i x x x x ˘ e i x x νελ Figure 1: Graphical representation of the three terms in V p ψ q eq.(38)with e i “ p , ~e i q V p ψ q “ ε ÿ i ż d x p ψ ` x ` e i ψ ´ x ` ψ ` x ´ e i ψ ´ x q ` λ ż d x ÿ i ψ ` x ψ ´ x ψ ` x ` e i ψ ´ x ` e i ` ν ż d x ψ ` x ψ ´ x (38)where ş d x “ ř x P Λ L ş β ´ β dx and ψ ˘ x is vanishing outside Λ L ; moreover B p ψ, J, η q “ ż d x r η ` x ψ ´ x ` ψ ` x η ´ x ` d ÿ µ “ J µ p x q j µ p x qs (39)with j p x q “ ψ ` x ψ ´ x j i p x q “ ε p ψ ` x ` e i ψ ´ x ´ ψ ` x ψ ´ x ` e i q (40)The 2-point and the current correlations are given by S L,β p x , y q “ B B η ` x B η ´ y W p η, J q| , H µ,ν p x , y q “ B B J µ, x B J ν, y W p η, J q| , (41)By expanding in λ, ε, ν one can write the correlations as a series expansion, which can beexpressed in terms of Feynman graphs obtained contracting the half lines of vertices, see Fig.1, and associating to each line the propagator g p x , y q . There is a basic difference between theperturbative expansion in the non interacting case λ “ λ “
0. Inthe first case there are only chain graphs, while in the second there are also loops, producingfurther combinatorial problems. One can verify that the perturbative expansions obtained byTrotter formula for (2) and by the Grassmann generating functions are the same (this is trueup to the so called ”tadpoles” which can be easily taken into account, see § L, β , as proven in the following sections. Indeedat finite
L, β the partition function in (2) is entire and it coincides order by order with theGrassmann representation, which is analytic in a disk independent on the volume, so theycoincide. As the denominator of the correlations is non vanishing in this finite disk and thenumerator is entire at finite β, L , also the correlations (2) is analytic and coincide with theGrassmann representation, and the identity holds also in the limit.
The difficulty in controlling the perturbative expansion is due to a ”small divisor problem”related to the size of the propagator; the denominator of p g p ~x, k q can be arbitrarily small if ~ω~x is8lose to ˘ α , a fact which can produce in principle O p n ! q -terms which could destroy convergence.The starting point of the analysis is to separate the propagator in two terms, one containing thequasi-singularity and a regular part; we write g p x , y q “ g p q p x , y q ` ÿ ρ “˘ g pď q ρ p x , y q (42)where g p q p x , y q “ δ ~x,~y β ÿ k χ p q p ~ω~x, k q e ´ ik p x ´ y q ´ ik ` ¯ φ p ~ω~x q ´ ¯ φ p α q “ δ ~x,~y g p q p ~x, x ´ y q g pď q ρ p x , y q “ δ ~x,~y β ÿ k χ p q ρ p ~ω~x, k q e ´ ik p x ´ y q ´ ik ` ¯ φ p ~ω~x q ´ ¯ φ p α q “ δ ~x,~y g pď q ρ p ~x, x ´ y q (43)with χ p q ρ p ~ω~x, k q “ r θ ρ p ~ω~x q ¯ χ p b k ` p ¯ φ p ~ω~x q ´ ¯ φ p α qq q with r θ ρ is the periodic theta function( r θ ˘ “ ~ω~x mod. 1 is positive/negative and zero otherwise) and ¯ χ such that C p R ` q Ñ R such that ¯ χ p t q “ t ď χ p t q “ t ě γ ą
1; moreover χ p q ` ř ρ “˘ χ ρ “
1. The”infrared” propagator g pď q p x , y q has denominator arbitrarily small. We can further decomposethe infrared propagator as sum of propagators with smaller and smaller denominators g pď q ρ p ~x, x ´ y q “ ÿ h “´8 g p h q ρ p ~x, x ´ y q (44)with g p h q ρ similar g pď q ρ witrh f h replacing ¯ χ with f h “ ¯ χ p γ h b k ` p ¯ φ p ~ω~x q ´ ¯ φ p α qq q ´ ¯ χ p γ h ´ b k ` p ¯ φ p ~ω~x q ´ ¯ φ p α qq q (45)For any integer N one has | g p h q ρ p ~x, x ´ y q| ď C N ` p γ h | x ´ y |q N (46)if C N is a suitable constant.The integration of (37) is done iteratively by using two crucial properties of Grassmannintegrations. If P p dψ p q q and P p dψ pď q q are gaussian Grassmann integrations with propagators g p q and g pď q , we can write P p dψ q “ P p dψ p q q P p dψ pď q q so that e W p η,J q “ ż P p dψ p q q P p dψ pď q q e ´ V p ψ p q ` ř ρ “˘ ψ pď q ρ q´ B p ψ p q ` ř ρ “˘ ψ pď q ρ ,η,J q “ ż P p dψ pď q q e ´ V p q p ψ pď q ρ ,η,J q (47)with V p q p ψ pď q ρ , η, J q “ ÿ n “ n ! E T p V ` B ; n q (48)and E T are fermionic truncated expectations with propagator g p q . By integrating ψ p q , ψ p´ q , .., ψ p h ` q one obtains a sequence of effective potentials V p h q , h “ , ´ , ´ , .. . The way in which we definethe integration is dictated by the scaling dimension which is, as we will see below, D “
1; thatis all terms are relevant in the Renormalization Group sense.
Remark
Note that after the integration of ψ one gets a theory defined in terms of twofields ψ ` , ψ ´ . This is due to the fact that ¯ φ p t q “ ¯ φ p α q in correspondence of two points ˘ α . If9e consider more general forms of quasi periodic disorder, like ¯ φ p t , t q as the one in [7] , then¯ φ p t , t q ´ µ “ ψ ρ , with ρ a parameter parametrizing this curve, a situation somewhat analogue towhat happens in interacting fermions with extended Fermi surface.The multiscale integration is described iteratively in the following way. Assume that we havealready integrated the fields ψ p q , ψ p´ q , .., ψ p h ` q obtaining (we set η “ e W p ,J q “ ż P p dψ pď h q q e ´ V p h q p ψ pď h q ,J q (49)where P p dψ pď h q has propagator g pď h q ρ p x , y q “ δ ~x,~y β ÿ k χ p h q ρ p k , ~ω~x q e ´ ik p x ´ y q ´ ik ` ¯ φ p ~ω~x q ´ ¯ φ p α q “ δ ~x,~y g pď q ρ p ~x, x ´ y q (50)and V p h q p ψ pď h q , J q “ ÿ l ě ,m ě ÿ ε,ρ ż d x ...d x l d y ...d y m H hl,m p x , y q l ź i “ ψ ε i pď h q ρ i , x i m ź i “ l J y i (51)If there is a subset of ψ ε i ρ i , x i with the same ε, ρ and ~x i , by the anticommuting properties ofGrassmann variables we can write, if l ą l ź i “ ψ ε~x,x ,i “ ψ ε~x,x , l ź i “ D ε~x,x ,i ,x , D ε~x,x ,i ,x , “ ψ ε~x,x ,i ´ ψ ε~x,x , (52)We can therefore rewrite that effective potential in the following way V p h q p ψ pď h q , J q “ ÿ l ě ,m ě ÿ ε,ρ ż d x ...d x l d y ...d y m H hl,m p x , y q l ź i “ d σ i ψ ε i ρ i , x i m ź i “ l J y i (53)with σ “ , d ψ “ ψ and d ψ “ D .We define resonant the terms with fields with the same coordinate ~x , that is x i “ p x ,i , ~x q .Note that all the resonant terms with l ě D fields; thefields have the same ρ index as have the same ~ω~x .We define a renormalization operation R in the following way1. If l “ m “ R ÿ ~x ż dx , dx , H p h q , ψ `pď h q ~x,x , ,ρ ψ ´pď h q ~x,x , ,ρ “ ÿ ~x ż dx , dx , H p h q , ψ `pď h q ~x,x , ,ρ T ´pď h q ~x,x , ,x , ρ (54)with T ´pď h q ~x,x , ,x , ρ “ ψ ´pď h q ~x,x , ,ρ ´ ψ ´pď h q ~x,x , ,ρ ´ p x , ´ x , qB ψ ´pď h q ~x,x , ,ρ (55)2. R “ R “ ´ L and by definition LV p h q is given by the following expression LV p h q “ γ h F p h q ν ` F p h q ζ ` F p h q α (56)where, if H p h q , p ~x, x ´ y q ” ¯ H p h q , p ~ω~x, x ´ y q one has ν h “ ż dx ¯ H p h q , p ρα, x q ξ h p ~x q “ ż dx ¯ H p h q , p ~ω~x, x q ´ ¯ H p h q , p ρα, x q ~ω~x ´ ρα (57)10nd α h p ~x q “ ş dx x ¯ H p h q , p ~ω~x, x q ; moreover F p h q ν “ ÿ ρ ÿ ~x ż dx ν h ψ `pď h q x ,ρ ψ ´pď h q x ,ρ F p h q ζ “ ÿ ρ ÿ ~x ż dx pp ~ω~x q ´ ρα q ζ h,ρ p ~x q ψ `pď h q x ,ρ ψ ´pď h q x ,ρ F p h q α “ ÿ ρ ÿ ~x ż dx α h,ρ p ~x q ψ `pď h q x ,ρ B ψ ´pď h q x ,ρ (58)The running coupling constants ~v h “ p ν h , α h , ξ h q are independent from ρ , as (37) is invariantunder parity ~x Ñ ´ ~x . Note also that p p g p k q q ˚ p ~x, k q “ p g p k q p ~x, ´ k q so that p p H p h q ,ρ p ~x, k qq ˚ “ p H p h q ,ρ p ~x, ´ k q , and this implies that ν h is real. Remark
The R operation is defined in order to act non trivially on the resonant terms withtwo fields and no J fields; they are the only resonant terms with no D fields. This fact wouldbe not true of there is the spin or an extra degree of freedom, as in the case of lattice Weylsemimetals [48]. In that case the local part of the effective potential would contain also effectiveinteractions.With the above definitions we can write (49) e W p ,J q “ ż P p dψ pď h ´ q q ż P p dψ p h q q e ´ LV p h q p ψ pď h q ,J q´ RV p h q p ψ pď h q ,J q “ ż P p dψ pď h ´ q q e ´ LV p h q p ψ pď h ´ q ,J q (59)and the procedure can be iterated. The effective potential can be written as a sum over Gallavotti trees τ , see Fig.2 V p h q p ψ pď h q , J q “ ÿ n “ ÿ τ P T h,n V p h q p τ, ψ pď h q q (60)where τ are trees constructed adding labels to the unlabeled trees, obtained by joining a point,the root , with an ordered set of n ě endpoints , so that the root is not a branchingpoint.The set of labeled trees T h,n is defined associating a label h ď r h, s intersecting all the non-trivial vertices, the endpoints and other points called trivial vertices.To a vertex v is associated h v and, if v and v are two vertices and v ă v , then h v ă h v . Moreover, there is only onevertex immediately following the root, which will be denoted v and can not be an endpoint;its scale is h `
1. To the end-points are associated V ` B , and in such a case the scale is 2;or LV h v ´ p ψ pď h v ´ q , J q and in this case the scale is h v ď h v “ h ¯ v `
1, if ¯ v is the first non trivial vertex immediately preceding v . The tree structureinduces a jerarchy of end-points which can be represented by clusters, see Fig.3.If v is the first vertex of τ and τ , .., τ s ( s “ s v ) are the subtrees of τ with root v , V p h q p τ, ψ pď h q q is defined inductively by the relation V p h q p τ, ψ q “ p´ q s ` s ! E Th ` r ¯ V p h ` q p τ , ψ pď h ` q q ; .. ; ¯ V p h ` q p τ s , ψ pď h ` q qs (61)where ¯ V p h ` q p τ i , ψ pď h ` q q it is equal to RV p h ` q p τ i , ψ pď h ` q q if the subtree τ i is non trivial;if τ i istrivial, it is equal to LV p h ` q . By iterating (61) we get a jerarchy of truncated expectations, with11 vv h v ðñ V p h q p τ, ψ pď h q q as sum over sets defined in the following way. We call I v the set of ψ associated to the end-pointsfollowing v and P v is a subset of I v denoting the external ψ . We denote by Q v i the intersectionof P v and P v i ; they are such that P v “ Y i Q v i and the union I v of the subsets P v i z Q v i is, bydefinition, the set of the internal fields of v , and is non empty if S v ą
1. The effective potentialcan be therefore written as V p h q p τ, ψ pď h q q “ ÿ P P P τ V p h q p τ, P q ¯ V p h q p τ, P q “ ż d x v r ψ pď h q p P v q K p h ` q τ, P p x v q , (62)where r ψ pď h q p P q “ ś f P P ψ x p f q . If we expand the truncated expectations by the Wick rule weget a sum of Feynman graphs with an associated cluster structure; an example is in Fig.4.The truncated expectations can be written by the Brydges-Battle-Federbush formula E Th v p r ψ p h v q p P { Q q , ¨ ¨ ¨ , r ψ p h v q p P s { Q s qqq “ ÿ T v ź l P T v “ δ ~x l ,~y l ¯ g p h v q p ~x l , x ,l ´ y ,l q ‰ ż dP T p t q det G h v ,T p t q , (63)where T v is a set of lines forming an anchored tree graph between the clusters of points x p i q Y y p i q ,that is T v is a set of lines, which becomes a tree graph if one identifies all the points in the same12igure 4: An example of graph with λ and ε vertices and the associated cluster structure;the propagator in the cluster, represented as a circle, has scale h smaller than the scales of thepropagators external to the cluster.cluster. Moreover t “ t t ii P r , s , ď i, i ď s u , dP T v p t q is a probability measure with supporton a set of t such that t ii “ u i ¨ u i for some family of vectors u i P R s of unit norm. G h,Tij,i j “ t ii δ ~x ij ,~y i j ¯ g p h q p ~x ij , x ,ij ´ y ,i j q , (64)We define ¯ T v “ Ť w ě v T w starting from T v and attaching to it the trees T v , .., T v Sv associatedto the vertices v , .., v S v following v in τ , and repeating this operation until the end-points of τ are reached. w w a w b w c w Figure 5: A tree ¯ T v with attached wiggly lines representing the external lines P v ; the linesrepresent propagators with scale ě h v connecting w , w a , w b , w c , w , representing the end-pointsfollowing v in τ .The tree ¯ T v connects the end-points w of the tree τ . To each end-point w we associate afactor ~δ i w w , and a) ~δ iw “ w corresponds to a ν h , α h , ζ h end-point; b) ~δ iw one among ˘ ~e i , i “ , , ε end-point; c) δ iw one among 0 , ˘ ~e i , i “ , , λ end-point. If ~x w and ~x w are coordinates of the external fields r ψ p P v q we have, see Fig.5 ~x w ´ ~x w “ ÿ w P c w ,w ~δ i w w (65)13here c w ,w is the set of endpoints in the path in ¯ T connecting w and w . The above relationimplies, in particular, that the coordinates of the external fields r ψ p P v q are determined once thatthe choice of a single one of them and of τ, ¯ T v and P is done. We can therefore write the effectivepotential as sum over trees T , setting the Kronecker deltas in the propagators in l P T equal to1 V p h q p τ, ψ pď h q q “ ÿ P P P τ ÿ T V p h q p τ, P , T q ¯ V p h q p τ, P , T q “ ÿ ~x ż dx ,v r ψ pď h q p P v q K p h ` q τ, P ,T p x v q , (66)where in K p h ` q τ, P ,T the propagators in T are g p h q p ~x, x ´ y q and the determinants are product ofdeterminats involving propagators with the same ~x . We can bound the propagators in T by ż dx | g p h q p ~x, x ´ y q| ď Cγ ´ h (67)Moreover the determinants in the BFF formula can be bounded by the Gram-Hadamard in-equality . We introduce an Hilbert space H “ R s b L p R q so that r G h,Tij,i j “ ´ u i b A p x ,ij ´ , x ij q , u i b B p y ,i j ´ , x ij q ¯ , (68)where u P R s are unit vectors p u i , u i q “ t ii , and A, B p A, B q “ ż dz A p ~x, x ´ z q B ˚ p ~x, z ´ y q (69)given by A p ~x, x ´ z q “ β ÿ k e ´ ik p x ´ z q a f h B p ~x, y ´ z q “ β ÿ k e ´ ik p y ´ z q ? f h ´ ik ` ¯ φ p ~ω~x q ´ ¯ φ p α q Moreover || A h || “ ş dz | A h p x , z q| ď Cγ h and || B h || ď Cγ ´ h so that By Gram-Hadamardinequality we get: | det r G h v ,T v p t v q| ď C ř Svi “ | P vi |´| P v |´ p S v ´ q . (70)One get therefore the bound, for | λ | , | ~v h | ď ε , | K p h ` q τ, P ,T p x v q| ď C n ε n ź v S v ! γ ´ h v p S v ´ q (71)which is not suitable for summing over τ and P . In order to improve the above bound we needto implement in the bounds some constraints which have been neglected in the derivation of(71), and to take into account the effect of the presence of the D fields.We define V χ the set of non trivial vertices or the trivial ones with non zero internal lines;we define v the first vertex in V χ following v . We say that v is a non-resonant vertex if in r ψ p P v q there are at least two different coordinates, and a resonant vertex when all coordinatesare equal. We define S v “ S Lv ` S Hv where S Lv is the number of non resonant subtrees (includingtrivial ones) and S Hv the number of resonant ones (inluding trivial ones). We also call H theset of v P V χ which are resonant and L the v P V χ which are non resonant. Consider a nonresonant vertex v so that there are at least two fields in P v with different spatial coordinates ~x ,say ~x w “ ~x w . The fields r ψ pď h v q p P v q have scale ď γ h v , v P V χ the first vertex belonging to V χ after v so that ||p ~ω~x w q ´ ρ α || T ď cv ´ γ h v ´ ||p ~ω~x w q ´ ρ α || T ď cv ´ γ h v ´ (72)14o that2 cv ´ γ h v ě ||p ~ω~x w q ´ ρ α || T ` ||p ~ω~x w q ´ ρ α || T ě || ~ω p ~x w ´ ~x w q ´ p ρ ´ ρ q α || T (73)and by (65) 2 cv ´ γ h v ě || ~ω p ÿ w P c w ,w ~δ i w w q ` p ρ ´ ρ q α || T ě C | ř w P c w ,w ~δ i w w | τ (74)where the Diophantine conditions have been used. Therefore ÿ w P c w ,w | ~δ i w w | ě | ÿ w P c w ,w ~δ i w w | ě Cγ ´ h v { τ (75)and, if N v is the number of end-points following v in τ ÿ w P c w ,w | ~δ i w w | ď N v (76)as | ~δ i w w | “ , N v ě Cγ ´ h v { τ (77)Note that to each endpoint is associated a small factor ε and the fact that N v is large by (77)produces a gain for the v with the fields with different ~x . Of course there can be several ¯ T v with different v passing through the same end-points. Therefore, given a constant c ă
1, we canmultiply the contribution to each tree τ with n -endpoints by c ´ n c n (the factor c ´ n is of coursearmless); we can then write c “ ź h “´8 c h ´ (78)and associate to each v a factor c N v h ´ . If there are two fields in P v (that is external to thecluster v ) with different ~x we get in the bounds, by assuming γ τ { ” γ η ą Nc Aγ ´ hτ h “ e ´| log c | Aγ ´ ηh ď γ Nηh N r| log | c || A s N e N (79)as e ´ αx x N ď r Nα s N e ´ N , and we can choose N “ { η ; therefore given a couple of fields externalto a vertex v with different ~x , we can associate a factor γ h v in the bounds.On the other hand if there is a D field we get in the bound an extra γ h v ´ h v from theexpression¯ g p h v q p ~ω~x, x , ´ z q ´ ¯ g p h v q p ~ω~x, x , ´ z q “ p x , ´ x , q ż dt B ¯ g p h v q p ~ω~x, p x , , p t q ´ z q (80)where p x , , p t q “ x , ` t p x , ´ x , q . In conclusion1. To each non-resonant v we associate a factor (79) so that we get in the bound an extrafactor ś v P V χ γ h v S Lv
2. There is a factor ś ˚ v γ h v where v are the endpoints ν, α, ξ (it comes from the definition of ν and the presence p x ´ y q or p ~ω~x ´ ρα ).3. In the resonant v with l ě ś v P H γ p h v ´ h v q . For l “ R definition, for l ě | P v | ě ψ εx whose number is maximal; we cangroup them in couples connected by path in ¯ T non overlapping, and or have different ~x ,hence there is a path in ¯ T connecting them giving an extra γ h v , or they have the same ~x so that there is an extra γ p h v ´ h v q . This produces an extra γ ´ α | P v | , see § F in [36].We bound first the effective potential ( J “ τ P T h,n , the set of trees with n end-points anddefining || K p h ` q τ, P ,T || “ βL d ÿ ~x ż dx ,v | K p h ` q τ, P ,T | (81)we get || K p h ` q τ, P ,T || ď C n ε n ź v S v ! γ ´ h v p S v ´ q ź v P V χ γ h v S Lv ˚ ź v γ h v ź v P H γ p h v ´ h v q ź v P V χ γ ´ α | P v | (82)If the first vertex v P V χ is non resonant we get ź v P V χ γ ´ h v S v ź v γ h v S Lv ˚ ź v γ h v ź v P H,v “ v γ h v “ ź v P V χ γ h v ź v P H,v “ v γ ´ h v ď γ h v (83)We use that S v “ S Lv ` S Hv , ś v γ h v S Lv “ ś v P L γ h v ś ˚˚ v γ h v , with ś ˚˚ v is over the first vertex v P V χ after the ε, λ endpoints, and that ś v P L γ h v ď ś v P L γ h v ´ h v || K p h ` q τ, P ,T || ď C n ε n γ h v ź v S v ! ź v P V χ γ p h v ´ h v q ˚˚ ź v γ h v ź v P V χ γ ´ α | P v | (84)where ś ˚˚ v is over the vertices v P V χ following from the end-points associated to ε, λ . Notethat ř P r ś v P V χ γ ´ | P v | s ď C n ; moreover ř T r ś v S v ! s ď C n . The sum over the trees τ is doneperforming the sum of unlabeled trees and the sum over scales. The unlabeled trees can bebounded by 4 n by Caley formula, and the sum over the scales reduces to the sum over h v , with v P V χ , as given a tree with such scales assigned, the others are of course determined.Let us consider now the case in which the first vertex v is resonant; we can distinguish twocases. If we are considering the contribution to the beta function then there is no R applied in v so that the same bound as above is found with h v “ h `
1. Instead if R is applied we getinstead of (83), as there is an extra γ h v ´ h v ź v P V χ γ ´ h v S v ź v γ h v S Lv ˚ ź v γ h v ź v P H γ h v “ γ h v ź v P V χ γ h v ź v P H γ ´ h v ď h v “ h `
1. In conclusion we get ÿ τ P T h,n ÿ P ,T || K p h ` q τ, P ,T || ď C n ε n γ h (86)The running coupling constant α h , ξ h verify α h ´ “ α h ` O p ε γ h q ξ h ´ “ ξ h ` O p ε γ h q (87)where the factor γ h is due to the fact that the trees have at least an ε, λ endpoint, from thefactor ś ˚˚ v γ h v in (84) (short memory property). The flow of z h , α h is therefore summable; inaddition one can choose ν so that ν h is bounded, by proceeding as in Lemma 2.7 of citeM3.16 Decay of correlations
We consider now the current correlations, which can be written as H µ,ν p x , y q “ ÿ h,n ÿ τ P T h,n ` ÿ P ,T G τ, P ,T p x , y q (88)where T h,n ` is the set of trees with n ` J end-points.In the trees τ we can identify a vertex v x for the end-point corresponding to J x , and v y for theend-point corresponding to J y with h v x “ h v y “ `
2; we call p v , with scale p h , the first vertex v P V χ such that v x , v y follows p v , and v the first vertex P V χ , with scale h . There are severalconstraints.1. By (65) and using that ~x ´ ~y “ ř w P C vx,vy ~δ i w w we get n ě ř w P C vx,vy | ~δ i w w | ě | ~x ´ ~y | h ě ¯ h p n q with, if | ~z | “ ` min p| ~x | , | ~y |q γ ´ ¯ h ď sup ~q “ ř ni “ ~e i || ~ω p ~x ` ~q q ´ ρα || ď C p| ~z | ` n q τ (89)With respect to the bound for the J “ T p v is thetree connecting the 2 J endpoints, we have an extra γ p h due to the fact that we do not integrateover the coordinates of the J fields, and we can extract from the the propagators in ś l P ¯ T p v g p h l q , h l ě p h a decay factor 11 ` p γ p h | x ´ y |q N (90)Moreover there is no R in the resonant terms with one or two external J lines. We canmultiply and divide by γ ´ h γ h : we can select two paths in τ v ă v ă ..v x and v ă v ă ..v y ,writing γ h “ γ p ¯ h ´ h v q ...γ h v x γ h “ γ p ¯ h ´ h v q ...γ h v y (91)where v x , v y are the first vertex P V χ after v x , v y . We get therefore the following bound | G τ, P ,T p x , y q| ď γ ´ h C n | ε | n γ p h p γ p h | x ´ y |q N ź v S v ! γ ´ h v p S v ´ q ź v P V χ γ h v S Lv ˚ ź v γ h v ź v P H γ p h v ´ h v q ź v P V χ γ ´ α | P v | (92)where H now includes also resonant terms with one or two J fields. Proceeding as in § | x ´ y | ą
1, if T n are the trees with n end-points ÿ τ P T h,n ÿ P ,T | G τ, P ,T p x , y q| ď γ ´ h C n | ε | n ` p γ ¯ h | x ´ y |q N ď C n | ε | n | ~z | τ p| ~z | ´ τ | x ´ y |q N p ` n | ~z | q p N ` q τ (93)The sum over h ě ¯ h can be bounded by an an extra γ ´ ¯ h . As | ~z | ě n {| ~z | ď n ; we can sumover n obtaining, remembering the constraint n ě | ~x ´ ~y || H µ,ν p x , y q| ď C | ~z | τ p| ~z | ´ τ | x ´ y |q N | ε | | ~x ´ ~y |{ (94)The analysis of the 2-point function is done in a similar way; there are 2 endpoints associatedwith the externl fields, so with respect to the bound for the effective potential there is an extrafactor γ ´ h and an extra γ ¯ h from the lack of integration; the sum over the scales produces anextra | ¯ h | . Acknowledgements.
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