Complex Bosonic Many-body Models: Overview of the Small Field Parabolic Flow
Tadeusz Balaban, Joel Feldman, Horst Knörrer, Eugene Trubowitz
aa r X i v : . [ m a t h - ph ] S e p Complex Bosonic Many–body Models:
Overview of the Small Field Parabolic Flow
Tadeusz Balaban , Joel Feldman ∗ , Horst Kn¨orrer , and EugeneTrubowitz Department of MathematicsRutgers, The State University of New [email protected] ∼ feldman/ ∼ knoerrer/ October 3, 2018
Abstract
This paper is a contribution to a program to see symmetry breaking in aweakly interacting many Boson system on a three dimensional lattice at lowtemperature. It provides an overview of the analysis, given in [13, 14], ofthe “small field” approximation to the “parabolic flow” which exhibits theformation of a “Mexican hat” potential well. ∗ Research supported in part by the Natural Sciences and Engineering Research Council ofCanada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich.
1t is our long term goal to rigorously demonstrate symmetry breaking in a gasof bosons hopping on a three dimensional lattice. Technically, to show that thecorrelation functions decay at a non–integrable rate when the chemical potentialis sufficiently positive, the non–integrability reflecting the presence of a long rangeGoldstone boson mediating the interaction between quasiparticles in the superfluidcondensate. It is already known [19, 20] that the correlation functions are expo-nentially decreasing when the chemical potential is sufficiently negative. See, forexample, [22] and [30, §
19] for an introduction to symmetry breaking in general, and[1, 18, 23, 28] as general references to Bose-Einstein condensation. See [17, 21, 26, 29]for other mathematically rigorous work on the subject.We start with a brief, formula free, summary of the program and its current state.Then we’ll provide a more precise, but still simplified, discussion of the portion ofthe program that controls the small field parabolic flow.The program was initiated in [3, 4], where we expressed the positive temperaturepartition function and thermodynamic correlation functions in a periodic box (adiscrete three–dimensional torus) as ‘temporal’ ultraviolet limits of four–dimensional(coherent state) lattice functional integrals (see also [27]). By a lattice functionalintegral we mean an integral with one (in this case complex) integration variable foreach point of the lattice. By a ‘temporal’ ultraviolet limit, we mean a limit in whichthe lattice spacing in the inverse temperature direction (imaginary time direction) issent to zero while the lattice spacing in the three spatial directions is held fixed.In [7] , by a complete large field/small field renormalization group analysis, weexpressed the temporal ultraviolet limit for the partition function , still in a periodicbox, as a four–dimensional lattice functional integral with the lattice spacing in allfour directions being of the order one, preparing the way for an infrared renormal-ization group analysis of the thermodynamic limit.This overview concerns the next stage of the program, which is contained in[13, 14] and the supporting papers [15, 9, 10, 12, 16, 11]. There we initiate theinfrared analysis by tracking, in the small field region, the evolution of the effec-tive interaction generated by the iteration of a renormalization group map that istaylored to a parabolic covariance : in each renormalization group step the spatiallattice directions expand by a factor L >
1, the inverse temperature direction ex-pands by a factor L and the running chemical potential grows by a factor of L ,while the running coupling constant decreases by a factor of L − . Consequently, the See also [8] for a more pedagogical introduction. A similar analysis will yield the corresponding representations for the correlation functions. Morally, the 1 + 3 dimensional heat operator. L is a fixed, sufficiently large, odd natural number. ǫ ’ power of the coupling constant.Then we can no longer base our analysis on expansions about zero field, becausethe renormalization group iterations have moved the effective model away from thetrivial noninteracting fixed point.In the next stage of the construction, we plan to continue the parabolic evolutionin the small field regime, but expanding around fields concentrated at the bottom ofthe (Mexican hat shaped) potential well rather around zero (much as is done in theBogoliubov Ansatz) and track it through an additional finite number of steps untilthe running chemical potential is sufficiently larger than one. At that point we willturn to a renormalization group map with a scaling taylored to an elliptic covariance,that expands both the temporal (inverse temperature) and spatial lattice directionsby the same factor L . It is expected that the elliptic evolution can be controlledthrough infinitely many steps, all the way to the symmetry broken fixed point. Thesystem is superrenormalizable in the entire parabolic regime because the runningcoupling constant is geometrically decreasing. However in the elliptic regime, thesystem is only strictly renormalizable.The final stage(s) of the program concern the control of the large field contribu-tions in both the parabolic and elliptic regimes.The technical implementation of the parabolic renormalization group in [13, 14]proceeds much as in [6, 7], except that we are restricting our attention to the smallfield regime and ◦ we use 1 + 3 dimensional block spin averages, as in [25, 2, 24]. In [7], we hadused decimation, which was suited to the effectively one dimensional problemof evaluating the temporal ultraviolet limit. ◦ Otherwise, the stationary phase calculation that controls oscillations is simi-lar, but technically more elaborate. ◦ The essential complication is that the critical fields and background fields arenow solutions to (weakly) nonlinear systems of parabolic equations. ◦ The Stokes’ argument that allows us to shift the multi dimensional integrationcontour to the ‘reals’ and ◦ the evaluation of the fluctuation integrals is similar. ◦ However, there is an important new feature: the chemical potential has to berenormalized. 3o analyze the output of the block spin convolution (a single renormalizationgroup step), it is de rigueur for the small field/large field style of renormalizationgroup implementations to introduce local small field conditions on the integrand andthen decompose the integral into the sum over all partitions of the discrete torusinto small and large field regions on which the conditions are satisfied and violated,respectively. Small field contributions are to be controlled by powers of the couplingconstant v (a suitable norm of the two body interaction) uniformly in the volumeof the small field region. Large field contributions are to be controlled by a factor e − / v ε , ε > do have positivity properties and consequentlythere is at least one factor e − / v ε whenever there is a large field region. A strongerbound of a factor per point of a large field region is reasonable and would be themain ingredient for controlling the full parabolic renormalization group flow in thisregime.We now formally introduce the main objects of discussion and enough machineryto allow technical (but simplified) statements of the main results of [13, 14] and themethods used to establish them.One conclusion of our previous work in [7] is that the purely small field contri-bution to the partition function for a gas of bosons hopping on a three dimensionaldiscrete torus X = Z /L sp Z (where L sp , a power of L , is the spatial infrared regu-lator which will ultimately be sent to infinity) takes the form Z S Y x ∈ X dψ ( x ) ∗ ∧ dψ ( x )2 πı e A ( ψ ∗ , ψ ) (1)where • X = Z /L tp Z × X is a 1 + 3 dimensional discrete torus with points x = ( x , x ) .Here, L tp ≈ kT , also a power of L , is the inverse temperature infrared regulator,which can ultimately be sent to infinity to get the temperature zero limit.4 ψ ∈ C X is a complex valued field on X , ψ ∗ is the complex conjugate field and,for each x ∈ X , dψ ( x ) ∗ ∧ dψ ( x )2 ı is the standard Lebesgue measure on C . • S = (cid:8) ψ ∈ C X (cid:12)(cid:12) | ψ ( x ) | ≤ v − / + ǫ , | ∂ ν ψ ( x ) | ≤ v − / + ǫ , ν = 0 , , , , x ∈ X (cid:9) ,where the small ‘coupling constant’ v is an exponentially, tree length weighted L – L ∞ –norm (see the discussion of norms at the end of this overview or [13,Definition 1.9]) of an effective interaction V (see [13, Proposition D.1]). Here, ∂ ν , ν = 0 , , , x ν direction. • Let ψ ∗ be another arbitrary element of C X . ( ψ ∗ is not to be confused with thecomplex conjugate ψ ∗ of ψ .) • A ( ψ ∗ , ψ ) = − A ( ψ ∗ , ψ ) + p ( ψ ∗ , ψ, ∇ ψ ∗ , ∇ ψ ) . The action A ( ψ ∗ , ψ ) deter-mining the partition function is the restriction A ( ψ ∗ , ψ ) = A ( ψ ∗ , ψ ) (cid:12)(cid:12) ψ ∗ = ψ ∗ of A ( ψ ∗ , ψ ) to the ‘real’ subspace ψ ∗ = ψ ∗ of C X × C X . Here, ∇ is the (fourdimensional) discrete gradient operator. • ‘Morally’, A ( ψ ∗ , ψ ) = h ψ ∗ , ( − ∂ + h ) ψ i + V ( ψ ∗ , ψ ) − µ h ψ ∗ , ψ i , where ◦ h f , g i = P x ∈ X f ( x ) g ( x ) is the natural real inner product on C X ◦ h is a nonnegative, second order, elliptic (lattice) pseudodifferential operatoracting on X — for example, a constant times minus the spatial discrete lapla-cian ◦ V ( ψ ∗ , ψ ) = P X V ( x , x , x , x ) ψ ∗ ( x ) ψ ( x ) ψ ∗ ( x ) ψ ( x ) is a quartic mono-mial whose kernel V is translation invariant with P X V (0 , x , x , x ) > ◦ µ is essentially the chemical potential. • Let ψ ∗ ν , ψ ν , ν = 0 , , , C X .The perturbative correction p (cid:0) ψ ∗ , ψ, { ψ ∗ ν } ν =0 , { ψ ν } ν =0 (cid:1) , to the principal contri-bution − A , in A , is a power series in the ten variables ψ ∗ , ψ, { ψ ∗ ν , ψ ν } ν =0 ,with no ψ ∗ ( x ) ψ ( y ) terms, such that each nonzero term has as many factors withasterisks as factors without asterisks. That is, p conserves particle number. Itconverges on n(cid:0) ψ ∗ , ψ, { ψ ∗ ν , ψ ν } ν =0 (cid:1) ∈ C X (cid:12)(cid:12)(cid:12) | ψ ( ∗ ) ( x ) | , | ψ ( ∗ ) ν ( x ) | ≤ v − / + ε , ≤ ν ≤ , x ∈ X (cid:9) where “( ∗ )” means “either with ∗ or without ∗ ”.See [13, Proposition D.1] for more details.For convenience, set F ( ψ ∗ , ψ ) = e A ( ψ ∗ , ψ ) χ S ( ψ )5ith this notation the partition function is Z Y x ∈ X dψ ( x ) ∗ ∧ dψ ( x )2 πı F ( ψ ∗ , ψ ) + O (cid:0) e − / v ε (cid:1) (2)It is natural to study the partition function using a steepest descent/stationaryphase analysis. The exponential e h ψ ∗ , ∂ ψ i is purely oscillatory because the quadraticform h ψ ∗ , ∂ ψ i is pure imaginary. Fortunately, our partition function, Z , has the es-sential feature that there is an analytic function A ( ψ ∗ , ψ ) on a neighborhood of theorigin in C X × C X whose restriction to the real subspace is the ‘small field’ action.Our renormalization group analysis of the oscillating integral defining Z is basedon the critical points of A ( ψ ∗ , ψ ) = h ψ ∗ , ( − ∂ + h ) ψ i + V ( ψ ∗ , ψ ) − µ h ψ ∗ , ψ i in C X × C X that typically do not lie in the real subspace, and a multi dimen-sional Stokes’ contour shifting construction that is only possible because p ( ψ ∗ , ψ )is analytic.We now formally introduce the ‘block spin’ renormalization group transforma-tions that are used in this paper. Let X − be the subgroup L Z /L tp Z × L Z /L sp Z of X . Observe that the distance between points of X − on the inverse temperatureaxis is L and on the spatial axes is L , and that |X − | = L − |X | . Also, let Q (0) : C X → C X − be a linear operator that commutes with complex conjugation.We will make a specific choice of Q (0) later. It will be a ‘block spin averaging’ op-erator with, for each y ∈ X − , (cid:0) Q (0) ψ (cid:1) ( y ) being ‘morally’ the average value of ψ inthe L × L × L × L block centered on y . Insert into the integral of (2)1 = N (0) Z C X− Y y ∈ X − dθ ( y ) ∗ ∧ dθ ( y )2 πı e − L h θ ∗ − Q (0) ψ ∗ , θ − Q (0) ψ i − where h f , g i − = L P y ∈ X − f ( y ) g ( y ) is the natural real inner product on C X − and N (0) is a normalization constant. Then exchange the order of the ψ and θ integrals. This gives Z Y x ∈ X dψ ( x ) ∗ ∧ dψ ( x )2 πı F ( ψ ∗ , ψ ) = Z Y y ∈ X − dθ ( y ) ∗ ∧ dθ ( y )2 πı B ( θ ∗ , θ )where, by definition, the block spin transform of F ( ψ ∗ , ψ ) associated to Q (0) withexternal fields θ and θ ∗ is B ( θ ∗ , θ ) = N (0) Z C X Y x ∈ X dψ ( x ) ∗ ∧ dψ ( x )2 πı e − L h θ ∗ − Q (0) ψ ∗ , θ − Q (0) ψ i − F ( ψ ∗ , ψ )6ere θ , θ ∗ are two arbitary elements of C X − .It can be awkward to compare functions defined on discrete tori with differentlattice spacings. So, we scale X − down to the unit discrete torus X (1)0 = Z / L tp L Z × Z / L sp L Z using the ‘parabolic’ scaling map x ∈ X (1)0 → ( L x , L x ) ∈ X − , which is anisomorphism of Abelian groups. Abusing notation, we consciously use the symbol ψ ( x ) as the name of a field on the unit torus X (1)0 even though it was used before asthe name of a field on the unit torus X . By definition, the block spin renormalizationgroup transform of F ( ψ ∗ , ψ ) associated to Q (0) with external fields ψ and ψ ∗ in C X (1)0 is F ( ψ ∗ , ψ ) = B (cid:0) S − ψ ∗ , S − ψ (cid:1) where (cid:0) S − ψ (cid:1) ( y , y ) = L − / ψ (cid:0) y L , y L (cid:1) (3)for any ψ ∈ C X (1)0 . The ‘parabolic’ exponent − / has been chosen so that h S θ ∗ , ( ∂ + ∆) S θ i = h θ ∗ , ( ∂ + ∆) θ i − . We now have L − |X (1)0 | Z C X (1)0 Y x ∈ X (1)0 dψ ( x ) ∗ ∧ dψ ( x )2 πı F ( ψ ∗ , ψ ) = Z C X Y x ∈ X dψ ( x ) ∗ ∧ dψ ( x )2 πı F ( ψ ∗ , ψ )the original small field part of the partition function.Repeat the construction. ◦ Let X (2) − be the subgroup L Z / L tp L Z × L Z / L sp L Z of X (1)0 and ◦ let Q (1) : C X (1)0 → C X (2) − be a linear ‘block averaging’ operator that com-mutes with complex conjugation. ◦ Introduce the unit discrete torus X (2)0 = Z / L tp L Z × Z / L sp L Z and ◦ the isomorphism x = ( x , x ) ∈ X (2)0 → ( L x , L x ) ∈ X (2) − .As before, integrate against the normalized Gaussian to obtain the block spin trans-form of F associated to Q (1) B ( θ ∗ , θ ) = N (1) Z C X (1)0 Y x ∈ X (1)0 dψ ( x ) ∗ ∧ dψ ( x )2 πı e − L h θ ∗ −Q (1) ψ ∗ , θ −Q (1) ψ i − F ( ψ ∗ , ψ )and then rescale to obtain the block spin renormalization group transform F ( ψ ∗ , ψ ) = B (cid:0) S − ψ ∗ , S − ψ (cid:1) In h θ ∗ , ( ∂ + ∆) θ i − , ∂ is the forward difference operator on X − . That is, ( ∂ f )( y ) = f ( y + L , y ) − f ( y , y ) L . Similarly, for spatial difference operators. (cid:0) S − ψ (cid:1) ( y , y ) = L − / ψ ( y L , y L ) for any ψ ∈ C X (2)0 . Interchanging the orderof integration, L − |X (2)0 | L − |X (1)0 | Z C X (2)0 Y x ∈ X (2)0 dψ ( x ) ∗ ∧ dψ ( x )2 πı F ( ψ ∗ , ψ ) = Z C X Y x ∈ X dψ ( x ) ∗ ∧ dψ ( x )2 πı F ( ψ ∗ , ψ )We keep repeating the construction to generate a sequence F n ( ψ ∗ , ψ ) , n ≥ C X ( n )0 × C X ( n )0 . [13, 14] concerns a sequence F ( SF ) n ( ψ ∗ , ψ )of ‘small field’ approximations to the F n ’s. We expect, and provide some supportingmotivation for, but do not prove, that F n = F ( SF ) n + O (cid:0) e − / v ε (cid:1) . For the precise defini-tion, see [13, § Q (0) , · · · , Q ( n ) , · · · of block averaging operators, let q ( x ) be a nonnegative, compactly supported,even function on Z × Z and Q the associated convolution operator ( Qψ )( y ) = X x ∈ Z × Z q ( x ) ψ (cid:0) y + [ x ] (cid:1) , ψ ∈ C X ( n )0 , y ∈ X ( n +1) − ⊂ X ( n )0 where [ x ] is the point in the quotient X ( n )0 = Z / L tp L n Z × Z / L sp L n Z represented by x ∈ Z × Z . By construction, Qψ ∈ C X ( n +1) − . We fix q ( x ) to be the convolution ofthe indicator function of the (discrete) rectangle [ − L − , L − ] × [ − L − , L − ] in Z × Z convolved with itself four times and normalized so that its sum over Z × Z is one. In [13, 14] the basic objects are the ‘small field’ block spin renormalizationiterates F ( SF ) n ( ψ ∗ , ψ ) , where at each step Q is chosen to be convolution with thefixed kernel q .If we had defined Q by convolving just with the indicator function of the rect-angle itself, properly normalized, then ( Qψ )( y ) would be the usual average of ψ ( x )over the rectangular box in X ( n )0 centered at y with sides L and L . We work withthe smoothed averaging kernel rather than the sharp one for technical reasons: com-mutators [ ∂ ν , Q ] are routinely generated and are small enough when Q is smoothenough. For the rest of this overview we will pretend that q is just the indicatorfunction of the rectangle and formulate our results as if this were the case. We willalso pretend that the operator h on X appearing in the action A ( ψ ∗ , ψ ) is (minus)the lattice Laplacian. Full, technically complete, statements are in [13, § By abuse of notation, we use the same symbol Q for the convolution operator acting on all ofthe spaces C X ( n )0 . ǫ > v are small enough and L is large enough,there exists a µ ∗ = O ( v ) , such that for all µ ∗ + v / < µ < v / and all n <
25 log / v log L , the ‘small field approximations’ F ( SF ) n to the F n ’s are F ( SF ) n ( ψ ∗ , ψ ) = Z n exp (cid:8) − A n (cid:0) ψ ∗ , ψ, φ ∗ n ( ψ ∗ , ψ ) , φ n ( ψ ∗ , ψ ) (cid:1) + p n ( ψ ∗ , ψ, ∇ ψ ∗ , ∇ ψ ) (cid:9) A n = a n h ( ψ ∗ − Q n φ ∗ n ) , ( ψ − Q n φ n ) i + h φ ∗ n , ( − ∂ − ∆) φ n i n − µ n h φ ∗ n , φ n i n + V n ( φ ∗ n , φ n )on the domain S n = n ( ψ ∗ , ψ ) ∈ C X ( n )0 (cid:12)(cid:12)(cid:12) | ψ ( ∗ ) ( x ) | ≤ κ n , | ∂ ν ψ ( ∗ ) ( x ) | ≤ κ ′ n , ≤ ν ≤ , x ∈ X ( n )0 (cid:9) and zero on its complement. Here, • you can think of the radii κ n and κ ′ n as being roughly L n v − + ǫ and L n v − + ǫ ,respectively. Explicit expressions for κ n and κ ′ n are given in [13, Definition 1.11.a]. • φ ∗ n ( ψ ∗ , ψ ) and φ n ( ψ ∗ , ψ ) are (nonlinear) maps from an open neighborhood ofthe origin in C X ( n )0 × C X ( n )0 to C X n , where X n is the discrete torus, isomorphic to X , but scaled down to have lattice spacing L − n in the time direction and L − n in the spatial directions . We say more about them in the last of this sequence ofbullets. Given ‘external fields’ ψ ∗ , ψ , the functions φ ∗ n ( ψ ∗ , ψ )( u ) , φ n ( ψ ∗ , ψ )( u )on X n are referred to as the “background fields” at scale n . • h f , g i = P x ∈ X ( n )0 f ( x ) g ( x ) and h f , g i n = L − n P u ∈ X n f ( u ) g ( u ) are the natural realinner products on C X ( n )0 and C X n . • Q n : C X n → C X ( n )0 is the linear map for which ( Q n f )( x ) is the average of f ∈ C X n over the square box in X n centered at x ∈ X ( n )0 with sides 1. (Thisbox contains L n × ( L n ) points of X n .) • a n = − L − − L − n • − ∂ − ∆ is the natural heat operator on the ‘fine’ discrete torus X n . • For each f ∗ , f ∈ C X n V n ( f ∗ , f ) = (cid:0) L n (cid:1) X uj ∈ X nj =1 , , , V n ( u , u , u , u ) f ∗ ( u ) f ( u ) f ∗ ( u ) f ( u ) An explicit formula for µ ∗ is given in [13, (1.19)]. We are weakening some of the statements, for pedagogical reasons. In particular, the sets ofallowed µ ’s and n ’s are a bit larger than the sets specified here. X n = L n Z / L tp L n Z × L n Z / L sp L n Z and the map u ∈ X n x = ( L n u , L n u ) ∈ X is anisomorphism of Abelian groups. V n ( u , u , u , u ) is close to V ( u ) n ( u , u , u , u ) = L n ( L n ) V ( U , U , U , U ) , U j = ( L n u j , L n u j ) (4) • The perturbative correction p n (cid:0) ψ ∗ , ψ, { ψ ∗ ν } ν =0 , { ψ ν } ν =0 (cid:1) is a power series in theten variables ψ ∗ , ψ, { ψ ∗ ν , ψ ν } ν =0 ∈ C X ( n )0 , with no ψ ∗ ( x ) ψ ( y ) or constant terms,such that each nonzero term has as many factors with asterisks as factors withoutasterisks. It converges when | ψ ( ∗ ) ( x ) | ≤ κ n and | ψ ( ∗ ) ν ( x ) | ≤ κ ′ n for all 0 ≤ ν ≤ x ∈ X ( n )0 . • Z n is a normalization constant . • µ n is the ‘renormalized’ chemical potential . It is close to L n µ . • For each pair in the polydisc n ( ψ ∗ , ψ ) ∈ C X ( n )0 × C X ( n )0 (cid:12)(cid:12)(cid:12) | ψ ( ∗ ) ( x ) | ≤ κ n for all x ∈ X ( n )0 o the fields φ ∗ n ( ψ ∗ , ψ )( u ) , φ n ( ψ ∗ , ψ )( u ) on X n are critical points of the functional( φ ∗ , φ ) A n ( ψ ∗ , ψ, φ ∗ , φ )= a n h ( ψ ∗ − Q n φ ∗ ) , ( ψ − Q n φ ) i − h φ ∗ , ( ∂ + ∆ + µ n ) φ ) i n + V n ( φ ∗ , φ )The maps φ ∗ n ( ψ ∗ , ψ ) , φ n ( ψ ∗ , ψ ) are holomorphic on that polydisc.In practical terms, what have we achieved? If ψ = z is a constant field on X ,then the dominant part of the initial effective potential is A ( ψ ∗ , ψ ) = V ( ψ ∗ , ψ ) − µ h ψ ∗ , ψ i = |X | (cid:16) v | z | − µ | z | (cid:17) = |X | v h(cid:16) | z | − µ v (cid:17) − µ v i where, v = P X V (0 , x , x , x ) . The graph of the real valued functionv h(cid:16) | z | − µ v (cid:17) − µ v i It is necessary to measure the size of p n by introducing an appropriate norm. See the lastparagraphs of this overview. When we take logarithms and ultimately differentiate with respect to an external field to obtaincorrelation functions, it will disappear. We will describe the inductive construction of µ n later on in this overview. The dependence of p n on the derivatives of the fields arises because of the renormalization of the chemical potential. z = x + ıx is a surface of revolution around the x –axiswith the circular well of absolute minima | z | = q µ v . Our hypothesis on µ impliesthat the radius and depth of the well are of order one and order v respectively. After n renormalization group steps, the effective potential becomes A n (cid:0) ψ ∗ , ψ, φ ∗ n ( ψ ∗ , ψ ) , φ n ( ψ ∗ , ψ ) (cid:1) (cid:12)(cid:12) ψ = z ≈ |X ( n )0 | v L n h(cid:16) | z | − µ n v / L n (cid:17) − µ n ( v / L n ) i (5)since, by [16, Remark 1.1], φ n ( ψ ∗ , ψ ) | ψ = z ≈ z and φ ∗ n ( ψ ∗ , ψ ) | ψ = z ≈ z ∗ . Thegraph is again a surface of revolution with the circular well of absolute minima | z | = q µ n v0 / Ln , but now the radius and depth are of order L n and order L n v respectively; the well is developing. We stop the flow when the well becomes so wideand so deep that we can no longer construct background fields by expanding around ψ ∗ , ψ = 0 . This happens as µ n approaches order one.If the power series expansion of the perturbative correction p n had a quadraticpart P x,y ∈X ( n )0 K ( x, y ) ψ ∗ ( x ) ψ ( y ) the discussion of the evolving well in the last para-graph would be misleading, because the minimum of the total action A n − p n wouldnot be close enough to the minimum of the dominant part A n . The requirementthat p n must not contain quadratic terms is the renormalization condition for thechemical potential. (See, Step 9 below.) Under the scaling map (3), the local mono-mials h ψ ∗ , ψ i h ψ ∗ , ∂ ν ψ i ≤ ν ≤ h ψ ∗ , ∂ ψ i h ∂ ν ψ ∗ , ∂ ν ′ ψ i ≤ ν, ν ′ ≤ h ψ ∗ , ∂ ν ψ i , 1 ≤ ν ≤
3, do not appear, becauseof reflection invariance. See [13, Definition B.1 and Lemma B.4]. So p n does notcontain any relevant monomials.The parabolic renormalization group flow drives the system away from the trivial(noninteracting) fixed point. To continue, we will have to construct background fieldsby expanding about configurations supported near the bottom of the developingwell, analogously to the ‘Bogoliubov Ansatz’. At present, we expect to continue theparabolic flow, but expanding about configurations supported near the bottom of thewell, through a transition regime (which overlaps with the regime of [13, 14]) until µ n becomes large enough (but still of order one), and then switch to a new ‘elliptic’renormalization group flow for the push to the symmetry broken, superfluid fixed11oint. In Appendix A, below, we perform several model computations that contrastthe parabolic nature of the early renormalization group steps with the ellipticalnature of the late renormalization group steps.The next part of this overview is an outline, in nine steps, of the inductive con-struction that uses a steepest descent/stationary phase calculation to build the de-sired form for F n +1 ( ψ ∗ , ψ ) = B n +1 (cid:0) S − ψ ∗ , S − ψ (cid:1) , from that of F n ( ψ ∗ , ψ ) , n ≥ B n +1 ( θ ∗ , θ ) = N ( n ) Z C X ( n )0 Y x ∈ X ( n )0 dψ ( x ) ∗ ∧ dψ ( x )2 πı e − L h θ ∗ − Qψ ∗ , θ − Qψ i − F n ( ψ ∗ , ψ )We are expecting that, by induction, B n +1 ( θ ∗ , θ ) = N ( n ) Z S n Y x ∈ X ( n )0 dψ ( x ) ∗ ∧ dψ ( x )2 πı e − L h θ ∗ − Qψ ∗ , θ − Qψ i − F ( SF ) n ( ψ ∗ , ψ ) + O (cid:0) e − / v ε (cid:1) = N ( n ) Z n Z S n Y x ∈ X ( n )0 dψ ( x ) ∗ ∧ dψ ( x )2 πı e − L h θ ∗ − Qψ ∗ , θ − Qψ i − − A n ( ψ ∗ ,ψ, φ ∗ n ,φ n ) + p n + O (cid:0) e − / v ε (cid:1) = Dominant Part + Non Perturbative Correction (6)We emphasise that Steps 1 and 6, which control the difference between F n +1 ( ψ ∗ , ψ )and its, dominant, ‘small field’, part F ( SF ) n +1 ( ψ ∗ , ψ ), have not been proven, though wedo supply some motivation in [15]. Step 1 (Large field generates small factors). If Ψ ∈ C X ( n +1)0 is ‘large field’, that isΨ / ∈ S n +1 , then we expect B n +1 ( S − Ψ ∗ , S − Ψ) = O (cid:0) e − / v ε (cid:1) , since the real part of theexponent appearing in the integrand of (6) is of order − v ε . See [15, Proposition 1,“Corollary” 2 and the subsequent Steps ∈ C X ( n +1)0 is ‘small field’, that isΨ ∈ S n +1 , and therefore construct holomorphic functions of (Ψ ∗ , Ψ) on the product S n +1 × S n +1 . Let θ ∗ = S − Ψ ∗ and θ = S − Ψ. Step 2 (Holomorphic form representation). We wish to analyze the integral in (6)by a steepest descent/stationary phase argument. Recall that a critical point of afunction f ( z ) of one complex variable z = x + iy , that is not analytic in z , is a pointwhere both partial derivatives ∂f∂x and ∂f∂y , or equivalently, both partial derivatives ∂f∂z = (cid:0) ∂∂x − i ∂∂y (cid:1) f and ∂f∂ ¯ z = (cid:0) ∂∂x + i ∂∂y (cid:1) f vanish. We prefer the latter formulation.12o we rewrite the integral in (6) in a form that allows us to treat ψ and its complexconjugate as independent fields. For each fixed ( θ ∗ , θ ) , the ‘action’ A n ( θ ∗ , θ, ψ ∗ , ψ ) = n − L h θ ∗ − Qψ ∗ , θ − Qψ i − − A n (cid:0) ψ ∗ , ψ, φ ∗ n ( ψ ∗ , ψ ) , φ n ( ψ ∗ , ψ ) (cid:1)o + p n ( ψ ∗ , ψ, ∇ ψ ∗ , ∇ ψ )= − A n, eff ( θ ∗ , θ, ψ ∗ , ψ ) + p n ( ψ ∗ , ψ, ∇ ψ ∗ , ∇ ψ ) (7)is a holomorphic function of ( ψ ∗ , ψ ) on S n × S n . By design, the Dominant Part of B n +1 ( θ ∗ , θ ) in (6) is expressed as (a constant times) the integral of the holomorphicform e A n ( θ ∗ ,θ, ψ ∗ ,ψ ) ^ x ∈ X ( n )0 dψ ∗ ( x ) ∧ dψ ( x )2 πı (8)of degree 2 |X ( n )0 | over the real subspace in S n × S n given by ψ ∗ = ψ ∗ . We shallsee below that, typically, the critical point does not lie in the real subspace and sois not in the domain of integration. This representation permits us to use Stokes’theorem , to shift the contour of integration to a non real contour that does containthe critical point of (the principal terms of) the action. The shift will be implementedin Step 6. Step 3 (Critical Points). Our next task is to find critical points. In (7), above,we wrote the exponent, A n ( θ ∗ , θ, ψ ∗ , ψ ), as the sum of a very explicit, main, part − A n, eff and a not very explicit, smaller, part p n . We just find the critical points of A n, eff rather than the full A n . Indeed, there is a unique pair of holomorphic maps ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) from ( S − S n +1 ) × ( S − S n +1 ) to S n such that the gradient (cid:0) ∇ ψ ∗ ∇ ψ (cid:1) of A n, eff ( θ ∗ , θ, ψ ∗ , ψ ) vanishes when ψ ∗ = ψ ∗ cr ( θ ∗ , θ ) , ψ = ψ cr ( θ ∗ , θ ) . Thispair of ‘critical field maps’ can be constructed by solving the critical point equations,a nonlinear parabolic system of (discrete) partial difference equations, using thenatural contraction mapping argument to perturb off of the linearized equations .The analysis of the linearized equations is based on a careful examination of somelinear operators given in [10]. Beware that, in general, ψ ∗ cr ( θ ∗ , θ ) = ψ cr ( θ ∗ , θ ) ∗ . The argument is similar to the use of Cauchy’s theorem in stationary phase arguments forfunctions of one variable. In [13, 14] these maps are called ψ n ∗ , ψ n . In [13, 14, 16] we take another route to the critical field maps. The background fields φ ( ∗ ) n ( ψ ∗ , ψ ) are constructed first, using the natural contraction mapping argument to perturb offof the linearized background field equations. See [16, Proposition 2.1]. The critical fields can thenbe expressed as functions of the background fields. See [13, Proposition 3.4].
13o start the stationary phase calculation, we factor the integral of the holomorphicform (8) over the real subspace (cid:8) ( ψ ∗ , ψ ) ∈ S n × S n (cid:12)(cid:12) ψ ∗ = ψ ∗ (cid:9) as the product of e A n ( θ ∗ ,θ, ψ ∗ cr ( θ ∗ ,θ ) ,ψ cr ( θ ∗ ,θ ) ) and the ‘fluctuation integral’ Z real subspace of Sn × Sn e A n ( θ ∗ ,θ, ψ ∗ ,ψ ) − A n ( θ ∗ ,θ, ψ ∗ cr ( θ ∗ ,θ ) ,ψ cr ( θ ∗ ,θ ) ) ^ x ∈ X ( n )0 dψ ∗ ( x ) ∧ dψ ( x )2 πı (9) Step 4 (The Value of the Action at the Critical Point). We would expect that thebiggest contribution to the integral would come from simply evaluating the exponentat the critical point, and that the biggest contribution to the value of the exponent A n at the critical point would come from evaluating − A n, eff at the critical point. By[13, Proposition 3.4.c] A n, eff ( θ ∗ , θ, ψ ∗ , ψ ) (cid:12)(cid:12) ψ ∗ = ψ ∗ cr ( θ ∗ , θ ) , ψ = ψ cr ( θ ∗ , θ ) = ˇ A n +1 (cid:0) θ ∗ , θ, ˇ φ ∗ n +1 ( θ ∗ , θ ) , ˇ φ n +1 ( θ ∗ , θ ) (cid:1) whereˇ A n +1 ( θ ∗ , θ, f ∗ , f ) = a n +1 L h θ ∗ − QQ n f ∗ , θ − QQ n f i − − h f ∗ , ( ∂ + ∆ + µ n ) f i n + V n ( f ∗ , f )and the ‘checked’ fieldˇ φ ( ∗ ) n +1 ( θ ∗ , θ ) = φ ( ∗ ) n (cid:0) ψ ∗ cr ( θ ∗ , θ, ψ cr ( θ ∗ , θ ) (cid:1) is the background field evaluated at the critical point. Consequently, e A n ( θ ∗ ,θ, ψ ∗ cr ( θ ∗ , θ ) ,ψ cr ( θ ∗ , θ )) = e − ˇ A n +1 ( θ ∗ , θ, ˇ φ ∗ n +1 ( θ ∗ , θ ) , ˇ φ n +1 ( θ ∗ , θ )) + p n ( ψ ∗ cr ,ψ cr , ∇ ψ ∗ cr , ∇ ψ cr ) Remark
Bear in mind that the checked fields depend implicitly on µ n . In the nextsteps, we will build a new ‘renormalized’ chemical potential µ n +1 that will appearin A n +1 . If, for the purposes of discussion, we ignored the effects of renormalization, A n +1 would just be a rescaled ˇ A n +1 (see [13, Definition 2.3 and Lemma 2.4.c]) andthe new background field φ ( ∗ ) n +1 would just be a rescaled ˇ φ ( ∗ ) n +1 (see [13, Definition3.2 and Proposition 3.4.b]). So, we are not far off.14 tep 5 (Diagonalization of the Quadratic Form in the Fluctuation Integral). Nextconsider the fluctuation integral (9). Make the change variables( ψ ∗ , ψ ) → (cid:0) δψ ∗ = ψ ∗ − ψ ∗ cr ( θ ∗ , θ ) , δψ = ψ − ψ cr ( θ ∗ , θ ) (cid:1) to shift the critical point to δψ ∗ = δψ = 0. Substitute ψ ( ∗ ) = ψ ( ∗ )cr ( θ ∗ , θ ) + δψ ( ∗ ) intothe main part A n, eff ( θ ∗ , θ, ψ ∗ , ψ ) − A n, eff (cid:0) θ ∗ , θ, ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) (cid:1) of the exponent and expand in powers of δψ ( ∗ ) . The constant and, by criticality,linear parts vanish. The quadratic term has a dominant part (see [13, Lemma 4.1,(4.13)] and [14, Lemma 5.5]), that is independent of θ ∗ , θ . All of the eigenvaluesof the kernel of that dominant part are bounded away from the negative real axis,uniformly in n . So, it is invertible and its inverse, C ( n ) , has a square root D ( n ) , allof whose eigenvalues have strictly positive real parts. See [10, Corollary 4.5]. Now,the Taylor expansion of the above difference of effective actions in the new variables δψ ∗ = D ( n ) T ζ ∗ , δψ = D ( n ) ζ becomes h ζ ∗ , ζ i + smaller terms of degree 2 in ζ ∗ , ζ + terms of degree at least 3 in ζ ∗ , ζ and the fluctuation integral (9) becomes Z Ω n ( θ ∗ , θ ) e −h ζ ∗ ,ζ i + q n ( θ ∗ , θ, ζ ∗ , ζ ) det( D ( n ) ) ^ x ∈ X ( n )0 dζ ∗ ( x ) ∧ dζ ( x )2 πı (10)where the domain of integration Ω n ( θ ∗ , θ ) consists of the set of all pairs ( ζ ∗ , ζ ) ∈ C X ( n )0 × C X ( n )0 such that (cid:0) ψ ∗ cr ( θ ∗ , θ ) + D ( n ) T ζ ∗ , ψ cr ( θ ∗ , θ ) + D ( n ) ζ (cid:1) is in the realsubspace of S n × S n . The term q n is holomorphic on the complex domain of allquadruples ( θ ∗ , θ, ζ ∗ , ζ ) with ( θ ∗ , θ ) ∈ ( S − S n +1 ) × ( S − S n +1 ) and (cid:0) ψ ∗ cr ( θ ∗ , θ ) + D ( n ) T ζ ∗ , ψ cr ( θ ∗ , θ ) + D ( n ) ζ (cid:1) ∈ S n × S n See [13, (4.4) and Corollary 4.3].
Step 6 (Stokes’ Theorem). For each pair ( θ ∗ , θ ) ∈ ( S − S n +1 ) × ( S − S n +1 ) , weconstruct, in [15, following (22)], a 2 |X ( n )0 | + 1 (real) dimensional “cylinder”, insidethe ( ζ ∗ , ζ ) domain of analyticity of q n , whose boundary consists of A major part of [10] is devoted to proving this vital technical statement. the original domain of integration Ω n ( θ ∗ , θ ) (which typically does not containthe critical point ζ = ζ ∗ = 0), ◦ the desired new domain of integration D n = (cid:8) ( ζ ∗ , ζ ) (cid:12)(cid:12) ζ ∗ = ζ ∗ , | ζ ( x ) | < (cid:0) L n +1 v (cid:1) ε / for all x ∈ X ( n )0 (cid:9) (which does contain the critical point ζ = ζ ∗ = 0) ◦ and components on which e − h ζ ∗ ,ζ i + q n ( θ ∗ , θ, ζ ∗ , ζ ) is O ( e − / v ε ) .See [15, (23)]. The holomorphic differential form in Step 5 has maximal rank and istherefore closed. It follows from Stokes’ theorem that the fluctuation integral (10) isequal to the small field contributionˇ F n ( θ ∗ , θ ) = det( D ( n ) ) Z D n Y x ∈ X ( n )0 dζ ( x ) ∗ ∧ dζ ( x )2 πı e − h ζ ∗ ,ζ i + q n ( θ ∗ , θ, ζ ∗ , ζ ) (11)plus corrections that are expected to be nonperturbatively small. Step 7 (The Logarithm of the Fluctuation Integral). In [5] we developed a simplevariant of the polymer expansion that can be directly applied to the integral in (11) toobtain the logarithm Log h ˇ F n ( θ ∗ , θ )ˇ F n (0 , i as an analytic function on ( S − S n +1 ) × ( S − S n +1 ).See [14, Proposition 5.6]. Step 8 (Rescaling). To this point we have determined that the small field part of B n +1 ( θ ∗ , θ ) is a constant times the exponential of the sum of ◦ the contribution which comes from simply evaluating A n at the critical point —in Step 4 we saw that this was − ˇ A n +1 ( θ ∗ , θ, ˇ φ ∗ n +1 ( θ ∗ , θ ) , ˇ φ n +1 ( θ ∗ , θ )) + p n ( ψ ∗ cr , ψ cr , ∇ ψ ∗ cr , ∇ ψ cr ) ◦ and an analytic function that came, in Step 7, from the fluctuation integral.We are now ready to scale to get the small field part of F n +1 (Ψ ∗ , Ψ) = B n +1 (cid:0) S − Ψ ∗ , S − Ψ (cid:1) Using that L (cid:10) S − Ψ ∗ , S − Ψ (cid:11) − = h Ψ ∗ , Ψ i S QQ n S − = Q n +1 (cid:10) S − f ∗ , S − f (cid:11) n = L h f ∗ , f i n +1 (cid:10) S − f ∗ , ( ∂ +∆) S − f (cid:11) n = h f , , ( ∂ +∆) f i n +1 (12)16see [13, Remark 2.2.c and Lemma 2.4.a,b]) we have thatˇ A n +1 ( θ ∗ , θ, ˇ φ ∗ n +1 ( θ ∗ , θ ) , ˇ φ n +1 ( θ ∗ , θ )) (cid:12)(cid:12)(cid:12) θ ( ∗ ) = S − Ψ ( ∗ ) = A ′ n +1 (Ψ ∗ , Ψ , φ ′∗ n +1 (Ψ ∗ , Ψ) , φ ′ n +1 (Ψ ∗ , Ψ))where A ′ n +1 (cid:0) Ψ ∗ , Ψ , f ∗ , f (cid:1) = ˇ A n +1 (cid:0) S − Ψ ∗ , S − Ψ , S − f ∗ , S − f (cid:1) = a n +1 h Ψ ∗ − Q n +1 f ∗ , Ψ − Q n +1 f i − (cid:10) f ∗ , ( ∂ +∆+ L µ n ) f (cid:11) n +1 + V ′ n +1 ( f ∗ , f ) φ ′ ( ∗ ) n +1 (Ψ ∗ , Ψ) = S ˇ φ ( ∗ ) n +1 ( S − Ψ ∗ , S − Ψ)and if the kernel V n of V n were exactly the V ( u ) n of (4), then the kernel of V ′ n +1 wouldbe exactly V ( u ) n +1 . See [13, Remark 2.2.h]. Renormalization is going to tweak, forexample, the value of the chemical potential. As a result A ′ n +1 is not quite A n +1 and φ ′ ( ∗ ) n +1 is not quite φ ( ∗ ) n +1 . That’s the reason for putting the primes on.Similarly, the contributions from p n ( ψ ∗ cr , ψ cr , ∇ ψ ∗ cr , ∇ ψ cr ) and from the fluctua-tion integral get scaled to p ′ n +1 (cid:0) Ψ ∗ , Ψ , { Ψ ∗ ν } ν =0 , { Ψ ν } ν =0 (cid:1) = (cid:20) p n (cid:0) ψ ∗ cr ( θ ∗ , θ ) , ψ cr ( θ ∗ , θ ) , ∇ ψ ∗ cr ( θ ∗ , θ ) , ∇ ψ cr( θ ∗ ,θ ) (cid:1) + Log (cid:16) ˇ F n ( S − Ψ ∗ , S − Ψ)ˇ F n (0 , (cid:17)(cid:21) θ ( ∗ ) = S − Ψ ( ∗ ) and we have that, renaming Ψ ( ∗ ) to ψ ( ∗ ) , the small field part of F n +1 ( ψ ∗ , ψ ) is F ( SF ) n +1 ( ψ ∗ , ψ ) = e − A ′ n +1 ( ψ ∗ ,ψ , φ ′∗ n +1 ( ψ ∗ ,ψ ) , φ ′ n +1 ( ψ ∗ ,ψ )) + p ′ n +1 ( ψ ∗ ,ψ , ∇ ψ ∗ , ∇ ψ ) on S n +1 × S n +1 . Step 9 (Renormalization of the Chemical Potential). At this point, we are closeto the end of the induction step, but not there yet because the power series p ′ n +1 contains (renormalization group) relevant contributions, in particular a quadraticterm h ψ ∗ , Kψ i , where K is a translation and (spatial) reflection invariant linearoperator mapping C X ( n +1)0 to itself. If such a term were to be left in p n +1 it would, bythe third line of (12), grow by roughly a factor of L in each future renormalizationgroup step. So we need to move (at least the local part of) this term out of p n +1 A n +1 . By the discrete fundamental theorem of calculus, for any translationinvariant K , h ψ ∗ , Kψ i = K h ψ ∗ , ψ i + X ν =0 h ψ ∗ , K ν ( ∂ ν ψ ) i where K ∈ C and K ν , ν = 0 , , , C X ( n +1)0 . See [14,Corollary B.2]. By reflection invariance, K is real and P ν =1 h ψ ∗ , K ν ( ∂ ν ψ ) i can berewritten as a sum of marginal and irrelevant monomials. See [14, Lemma B.3.c].So we would like to move K h ψ ∗ , ψ i out of p n +1 into A n +1 . There are two factorsthat complicate (but not seriously) this move. ◦ The chemical potential term in A ′ n +1 ( ψ ∗ , ψ , φ ′∗ n +1 ( ψ ∗ , ψ ) , φ ′ n +1 ( ψ ∗ , ψ )) is L µ n (cid:10) φ ′∗ n +1 ( ψ ∗ , ψ ) , φ ′ n +1 ( ψ ∗ , ψ ) (cid:11) n +1 It is expressed in terms of φ ′ ( ∗ ) n +1 ( ψ ∗ , ψ ) rather than directly in terms of ψ ( ∗ ) . ◦ The prime fields φ ′∗ n +1 ( ψ ∗ , ψ ), φ ′ n +1 ( ψ ∗ , ψ ) are background fields with chemicalpotential L µ n , not with the chemical potential µ n +1 that we are going to end upwith (and which we do not yet know).To deal with the first complication, we use that φ ′ ( ∗ ) n +1 ( ψ ∗ , ψ ) = B ( ∗ ) ψ ( ∗ ) plus terms ofdegree at least three in ( ψ ∗ , ψ ) (see [16, Proposition 2.1.a]). Because the linear opera-tors B ( ∗ ) have left inverses (see [10, Lemma 5.7] and the beginning of the proof of [14,Lemma 6.3]), one can show that K h ψ ∗ , ψ i = K ′ (cid:10) φ ′∗ n +1 ( ψ ∗ , ψ ) , φ ′ n +1 ( ψ ∗ , ψ ) (cid:11) n +1 plus a power series in ψ ∗ , ψ, ∇ ψ ∗ , ∇ ψ that converges on the desired domain of an-alyticity and that does not contain any relevant contributions. See [14, Lemma 6.3].Thus p ′ n +1 = K ′ (cid:10) φ ′∗ n +1 , φ ′ n +1 (cid:11) n +1 + p ′′ n +1 where p ′′ n +1 has no h ψ ∗ , ψ i term. Moving K ′ (cid:10) φ ′∗ n +1 , φ ′ n +1 (cid:11) n +1 from p ′ n +1 into A ′ n +1 ,we obtain − A ′ n +1 (cid:0) ψ ∗ , ψ, φ ′∗ n +1 , φ ′ n +1 (cid:1) + p ′ n +1 = − A ′′ n +1 (cid:0) ψ ∗ , ψ, φ ′∗ n +1 , φ ′ n +1 (cid:1) + p ′′ n +1 with A ′′ n +1 (cid:0) ψ ∗ , ψ, f ∗ , f (cid:1) = a n +1 h ψ ∗ − Q n +1 f ∗ , ψ − Q n +1 f i − (cid:10) f ∗ , ( ∂ + ∆ + ( L µ n + K ′ ) f (cid:11) n +1 + V ′ n +1 ( f ∗ , f ) For reasons that will be explained shortly, we do not actually use this fact expressed in thisway. φ ′∗ n +1 ( ψ ∗ , ψ ), φ ′ n +1 ( ψ ∗ , ψ ) are background fields for chemical potential L µ n , and not for chemical potential L µ n + K ′ . That is, the prime fields are criticalfor f ∗ , f A ′ n +1 (cid:0) ψ ∗ , ψ, f ∗ , f (cid:1) and not for f ∗ , f A ′′ n +1 (cid:0) ψ ∗ , ψ, f ∗ , f (cid:1) , as theymust be to have A n +1 = A ′′ n +1 . The way out of this is of course a (straightforward)fixed point argument that yields a self consistent µ n +1 ≈ L µ n . See [14, Lemmas6.2 and 6.6].So far we have skirted the issue of bounding the perturbative correction p n inour main result. To measure the size of p n , we introduce a norm whose finitenessimplies that all the kernels in its power series representation are small with v anddecay exponentially as their arguments separate in X ( n )0 . For pedagogical simplicitypretend that p n is a function of only two fields — ψ and one derivative field ψ ν . Ithas a power series expansion p n ( ψ, ψ ν ) = X r,s ∈ N r + s> X x ∈ ( X ( n )0 ) r y ∈ ( X ( n )0 ) s p n r s ( x , y ) ψ ( x ) ψ ν ( y )with the notations, N = N ∪ { } , and ψ ( x ) = ψ ( x ) · · · ψ ( x r ). Each p n r s ( x , y ) isseparately invariant under permutations of the components of x and under permu-tations of the components of y . The norm of p n is k p n k ( n ) = X r,s ∈ N r + s> k p n r s k m κ rn κ ′ ns For a translation invariant kernel with four arguments, like the interaction kernel V ( x , x , x , x ) , k V k m is the (mass m ) exponentially weighted L – L ∞ norm of V : k V k m = max j =1 , , , sup x j ∈X X xk ∈X k = j | V ( x , x , x , x ) | e m τ ( x ,x ,x ,x ) where τ ( x , x , x , x ) is the minimal length of a tree graph in X that has x , x , x , x among its vertices and m ≥ v = 2 k V k m .) The norm k w k m of a kernel w with an arbitrary numberof arguments is defined in much the same way. For details see [13, § k p n k ( n ) would be bounded (and in fact small) uniformly in n . Unfortu-nately, such a bound is too naive to achieve the upper limit on n stated in our mainresult. The reason is that, while the coefficient of an irrelevant monomial decreases19s the scale n increases, the maximum allowed size of fields in the domain S n alsoincreases, so the monomial as a whole can be relatively large. So we have chosen • to move all quartic (cid:0) ψ ∗ ψ ) monomials out of p n into A n , i.e. to also renormalizethe interaction V n , and • to split p n into two parts, ◦ one, called E n ( ψ ∗ , ψ ), is an analytic function whose size is measured in terms ofa norm like k · k ( n ) and is small (and decreasing with n ) and ◦ the other, called R n , is a polynomial of fixed degree, the size of whose coefficientkernels are measured in terms of a norm like k · k m .The details are stated in our main result, [13, Theorem 1.17].20 Seeing the Parabolic and Elliptic Regimes
In this appendix we perform several model computations that contrast the parabolicnature of the early renormalization group steps with the elliptical nature of the laterenormalization group steps. We imagine that after n (block spin) renormalizationgroup steps we have an action whose dominant part (that we are simplifying a bit )is A n (cid:0) ψ ∗ , ψ, φ ∗ n ( ψ ∗ , ψ ) , φ n ( ψ ∗ , ψ ) (cid:1) where A n ( ψ ∗ , ψ, φ ∗ , φ ) = h ( ψ ∗ − Q n φ ∗ ) , ( ψ − Q n φ ) i + h φ ∗ , ( − d n ∂ − ∆) φ i n − µ n h φ ∗ , φ i n + v n h φ ∗ φ, φ ∗ φ i n (A.1)Here ◦ h f , g i = P x ∈ Y f ( x ) g ( x ) and h f , g i n = ˜ ε n ε n P u ∈ Y n f ( u ) g ( u ) are the natural realinner products on C Y and C Y n , where the fine lattice Y n is a finite periodicbox in ˜ ε n Z × ε n Z (the lattice spacings ˜ ε n and ε n are small) and the unit lattice Y is a finite periodic box in Z × Z and is a sublattice of Y n . ◦ Q n : C Y n → C Y is the linear map for which ( Q n f )( x ) is the average of f ∈ C Y n over the square box in Y n centered at x ∈ Y with sides 1. This box contains ε n ε n points of Y n . ◦ ∂ and ∆ are the discrete forward time derivative and Laplacian on Y n , respectively. ◦ µ n > n > d n > ◦ For each ψ ∗ , ψ ∈ C Y the fields φ ∗ n ( ψ ∗ , ψ ) , φ n ( ψ ∗ , ψ ) on Y n are critical pointsof the functional ( φ ∗ , φ ) A n ( ψ ∗ , ψ, φ ∗ , φ )They obey the background field equations δδφ ∗ A n ( ψ ∗ , ψ, φ ∗ , φ ) = Q ∗ n ( Q n φ − ψ ) + D n φ + (v n φ ∗ φ − µ n ) φ = 0 δδφ A n ( ψ ∗ , ψ, φ ∗ , φ ) = Q ∗ n ( Q n φ ∗ − ψ ∗ ) + D ∗ n φ ∗ + (v n φ ∗ φ − µ n ) φ ∗ = 0 (A.2)with D n = − d n ∂ − ∆. In particular, for pedagogical purposes, we have replaced a n by 1 and replaced V n by a localinteraction. The fine lattice Y n is a rescaled version of the original lattice X of (1). .1 Constant Field Background Fields To start getting a feel for the background field equations (A.2) we consider the casethat ψ ∗ and ψ are constant fields with ψ ∗ = ψ ∗ . We’ll look for solutions φ ( ∗ ) whichare also constant fields with φ ∗ = φ ∗ . Since both Q n and Q ∗ n map the constantfunction 1 to the constant function 1, the constant field background fields obey φ + (cid:0) v n | φ | − µ n (cid:1) φ = ψ This is of the form “real number times φ equals real number times ψ ” so the phaseof φ and ψ will be the same (modulo π ). So it suffices to consider the case that ψ and φ are both real and obey φ + (cid:0) v n φ − µ n (cid:1) φ = ψ Since ddφ (cid:2) φ + (cid:0) v n φ − µ n (cid:1) φ (cid:3) = 1 − µ n + 3v n φ ≥ µ n ≤ > µ n > | φ | > q µ n − n < µ n > | φ | < q µ n − n there is always exactly one solution when µ n ≤
1, but the solution can be nonuniquewhen µ n >
1. For example, when µ n > ψ = 0 the solutions are φ = 0 and φ = ± q µ n − n . A.2 The Background Field in the Parabolic Regime
Imagine that we wish to solve the background field equations (A.2) for φ ( ∗ ) as analyticfunctions of ψ ( ∗ ) , in the parabolic regime, when µ n is small, so that the minimum ofthe effective potential is still near the origin — see (5). Then (cid:0) Q ∗ n Q n + D n − µ n (cid:1) φ = Q ∗ n ψ − v n φ ∗ φ (cid:0) Q ∗ n Q n + D ∗ n − µ n (cid:1) φ ∗ = Q ∗ n ψ ∗ − v n φ ∗ φ and, to first order in ψ ( ∗ ) , φ = (cid:0) Q ∗ n Q n − µ n − d n ∂ − ∆ (cid:1) − Q ∗ n ψ + O (cid:0) ψ ∗ ) (cid:1) φ ∗ = (cid:0) Q ∗ n Q n − µ n − d n ∂ ∗ − ∆ (cid:1) − Q ∗ n ψ ∗ + O (cid:0) ψ ∗ ) (cid:1) (A.3)We are interested in small ψ ( ∗ ) , so the O (cid:0) ψ ∗ ) (cid:1) corrections are unimportant. We heresee the parabolic (discrete) differential operators d n ∂ ( ∗ )0 + ∆.22 .3 The Background Field in the Elliptic Regime Imagine that we again wish to solve the background field equations (A.2), but thistime in the elliptic regime when µ n is large, v n is small and the effective potential hasa deep well, whose minima form a circle in the complex plane of radius r n = q µ n v n .We are interested in ψ ( ∗ ) and φ ( ∗ ) near the minimum of the effective potential. Thatis, with (cid:12)(cid:12) ψ ( ∗ ) (cid:12)(cid:12) , (cid:12)(cid:12) φ ( ∗ ) (cid:12)(cid:12) ≈ r n . We write ψ = r n e R + i Θ ψ ∗ = r n e R − i Θ φ = r n e X + iH φ ∗ = r n e X − iH (A.4)and look for solutions when R, Θ are small. Substitute into (A.2) and divide by r n .This gives D n (cid:2) e X + iH (cid:3) + Q ∗ n ( Q n e X + iH − e R + i Θ ) + µ n (cid:0) e X − (cid:1) e X + iH = 0 D ∗ n (cid:2) e X − iH (cid:3) + Q ∗ n ( Q n e X − iH − e R − i Θ ) + µ n (cid:0) e X − (cid:1) e X − iH = 0Expand the exponentials, keeping only terms to first order in (cid:8) R, Θ , X, H (cid:9) , to get D n ( X + iH ) + Q ∗ n Q n ( X + iH ) + 2 µ n X = Q ∗ n ( R + i Θ) D ∗ n ( X − iH ) + Q ∗ n Q n ( X − iH ) + 2 µ n X = Q ∗ n ( R − i Θ) (A.5)Now simplify, by adding together the two equations of (A.5) and dividing by 2,and then subtracting the second equation of (A.5) from the first and dividing by2 i . Pretend that ∂ is a continuum partial derivative rather than a discrete forwardderivative. Then ( D n + D ∗ n ) = − d n ( ∂ + ∂ ∗ ) − ∆ = − ∆ i ( D n − D ∗ n ) = i d n ( ∂ − ∂ ∗ ) = i d n ∂ and (A.5) gives (cid:2) µ n − ∆ + Q ∗ n Q n (cid:3) X − i d n ∂ H = Q ∗ n Ri d n ∂ X + (cid:2) − ∆ + Q ∗ n Q n (cid:3) H = Q ∗ n Θor, in matrix form, (cid:3) (cid:20) XH (cid:21) = Q ∗ n (cid:20) R Θ (cid:21) or (cid:20) XH (cid:21) = (cid:3) − Q ∗ n (cid:20) R Θ (cid:21) (A.6)where (cid:3) = " µ n − ∆ i d n ∂ ∗ i d n ∂ − ∆ + Q ∗ n Q n (A.7)23he Q ∗ n Q n provides a mass which makes (cid:3) boundedly invertible. But, the presenceof this mass is a consequence of our having rescaled the original unit lattice downto the very fine lattice Y n . To invert (cid:3) , ignoring the Q ∗ n Q n , we have to divide,essentially, bydet " µ n − ∆ i d n ∂ ∗ i d n ∂ − ∆ = d n (cid:8) ∂ ∗ ∂ + 2 µ n d n ( − ∆) + d n ( − ∆) (cid:9) • In the parabolic regime, µ n is small and d n is essentially one so that the operatorin the curly brackets is approximately ∂ ∗ ∂ + ( − ∆) , which is parabolic. • In the elliptic regime, µ n and d n are both very large with µ n d n > n . So the operator in the curly brackets is approximately ∂ ∗ ∂ + +2 µ n d n ( − ∆), which is elliptic. A.4 The Quadratic Approximation to the Action
For the remaining model computations, we study the quadratic approximation tothe action (A.1).
A.4.a Expanding Around Zero Field
We first consider the parabolic regime as studied in [13, 14]. Substitute the linearapproximation to the background fields φ ( ∗ ) (as functions on ψ ( ∗ ) ) of (A.3) into theaction (A.1), keeping only terms that are of degree at most two in ψ ( ∗ ) . Writing S n ( µ n ) = (cid:0) Q ∗ n Q n − µ n + D n (cid:1) − (A.3) becomes φ = S n ( µ n ) Q ∗ n ψ + O ( ψ ∗ ) ) φ ∗ = S n ( µ n ) ∗ Q ∗ n ψ ∗ + O ( ψ ∗ ) )so that A n = h ( ψ ∗ − Q n φ ∗ ) , ( ψ − Q n φ ) i + h φ ∗ , ( D n − µ n ) φ i n + O ( ψ ∗ ) )= h ψ ∗ , ψ i − h ψ ∗ , Q n φ i − h Q n φ ∗ , ψ i + h φ ∗ , ( Q ∗ n Q n + D n − µ n ) φ i n + O ( ψ ∗ ) )= h ψ ∗ , ψ i − h ψ ∗ , Q n S n ( µ n ) Q ∗ n ψ i − h Q n S n ( µ n ) ∗ Q ∗ n ψ ∗ , ψ i + h S n ( µ n ) ∗ Q ∗ n ψ ∗ , Q ∗ n ψ i n + O ( ψ ∗ ) )= h ψ ∗ , (1l − Q n S n ( µ n ) Q ∗ n ) ψ i + O ( ψ ∗ ) ) (A.8)24e now analyse the operator 1l − Q n S n ( µ n ) Q ∗ n in momentum space, in the special casethat µ n = 0, and see that it is basically a (discrete) parabolic differential operator.Set ∆ ( n ) = (cid:0)
1l + Q n D − n Q ∗ n (cid:1) − (A.9)Substituting in the definitions and simplifying, we see that S n (0) − D − n Q ∗ n ∆ ( n ) = Q ∗ n ,so that Q n S n (0) Q ∗ n = Q n D − n Q ∗ n ∆ ( n ) (A.10)By [10, Remark 2.1.e], with q = 1, \ ( Q n φ )( k ) = X ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ φ ( k + ℓ ) \ ( Q ∗ n ψ )( k + ℓ ) = u n ( k + ℓ ) ˆ ψ ( k )where u n ( p ) = sin (cid:0) p (cid:1) ε n sin (cid:0) ˜ ε n p (cid:1) Y ν =1 sin (cid:0) p ν (cid:1) ε n sin (cid:0) ε n p ν (cid:1) Here k runs over the dual lattice of Y and k + ℓ runs over the dual lattice of Y n . Wedo not need to know much about these dual lattices, except that the dual lattice of Y is a discretization of (cid:0) R / π Z (cid:1) × (cid:0) R / π Z (cid:1) , the dual lattice of Y n is a discretizationof (cid:0) R / π ˜ ε n Z (cid:1) × (cid:0) R / πε n Z (cid:1) , and ℓ runs overˆ B n = (cid:0) π Z / π ˜ ε n Z (cid:1) × (cid:0) π Z / πε n Z (cid:1) So, by [10, Lemmas 2.2.b,c, 3.2.d and 4.2.b, and Remark 4.1.a], with q = 1 and Q n replaced by 1l, the operator Q n S n (0) Q ∗ n has Fourier transform X ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ D − n ( k + ℓ ) u n ( k + ℓ ) ˆ∆ ( n ) ( k )= u n ( k ) ˆ D − n ( k ) ˆ∆ ( n ) ( k ) + X = ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ D − n ( k + ℓ ) ˆ∆ ( n ) ( k )= u n ( k ) u n ( k ) + ˆ D n ( k ) + O (cid:0) | k | (cid:1) + X = ℓ ∈ ˆ B n O (cid:0) | k | (cid:1) Y ν =0 (cid:2) | ℓ ν | + π (cid:3) O (1) O ( | k | )= 11 + ˆ D n ( k ) u n ( k ) − + O (cid:0) | k | (cid:1) + X = ℓ ∈ ˆ B n O (cid:0) | k | (cid:1) Y ν =0 (cid:2) | ℓ ν | + π (cid:3) = 11 + ˆ D n ( k ) + O (cid:0) | k | (cid:1) + O (cid:0) | k | (cid:1) − Q n S n (0) Q ∗ n has Fourier transform1 −
11 + ˆ D n ( k ) + O (cid:0) | k | (cid:1) + O (cid:0) | k | (cid:1) = ˆ D n ( k ) + O (cid:0) ˆ D n ( k ) (cid:1) + O (cid:0) | k | (cid:1) = − id n k + k + O (cid:0) k (cid:1) + O (cid:0) | k | (cid:1) (A.11)and so is a parabolic operator. A.4.b Expanding Around the Bottom of the Effective Potential
For all µ n = 0 it is appropriate to expand the action about the bottom of the effectivepotential, rather than about the origin. That is, rather than in powers of ψ ( ∗ ) . Sowe rewrite the action (A.1) A n = h ( ψ ∗ − Q n φ ∗ ) , ( ψ − Q n φ ) i + h φ ∗ , ( − d n ∂ − ∆) φ i n + v n (cid:10) [ φ ∗ φ − r n ] , (cid:11) n − r n v n h , i n and then substitute the representations (A.4) of ψ ( ∗ ) and φ ( ∗ ) in terms of radial andtangential fields. Note that when R = Θ = X = H = 0, the field magnitudes | ψ ( ∗ ) | = | φ ( ∗ ) | = r n and ψ ( ∗ ) and φ ( ∗ ) are at the bottom of the effective potential.Still pretending that ∂ is a continuous derivative, and using the notation O [3] = O ( X + R + H + Θ ), we get the following representation of the action, which isreminiscent of (A.8). Lemma A.1. r n A n = (cid:28)(cid:20) R Θ (cid:21) , (cid:8) − Q n (cid:3) − Q ∗ n (cid:9) (cid:20) R Θ (cid:21)(cid:29) − r n v n h , i n + O [3] Proof.
The three main terms in A n are h ( ψ ∗ − Q n φ ∗ ) , ( ψ − Q n φ ) i = r n (cid:10) ( e R − i Θ − Q n e X − iH ) , ( e R + i Θ − Q n e X + iH ) (cid:11) = r n h R − i Θ − Q n ( X − iH ) , R + i Θ − Q n ( X + iH ) i + O [3]= r n (cid:28)(cid:20) R Θ (cid:21) , (cid:20) R Θ (cid:21)(cid:29) − r n (cid:28)(cid:20) R Θ (cid:21) , Q n (cid:20) XH (cid:21)(cid:29) + r n (cid:28)(cid:20) XH (cid:21) , Q ∗ n Q n (cid:20) XH (cid:21)(cid:29) n + O [3]and h φ ∗ , ( − d n ∂ − ∆) φ i n = r n (cid:10) e X − iH , ( − d n ∂ − ∆) e X + iH (cid:11) n = r n h X − iH, ( − d n ∂ − ∆)( X + iH ) i n + O [3]= r n (cid:28)(cid:20) XH (cid:21) , (cid:20) − ∆ i d n ∂ ∗ i d n ∂ − ∆ (cid:21) (cid:20) XH (cid:21)(cid:29) + O [3]26nd v n D [ φ ∗ φ − r n ] , E n = v n r n D [ e X − , E n = 2 r n µ n h X, X i n + O [3]= r n (cid:28)(cid:20) XH (cid:21) , (cid:20) µ n
00 0 (cid:21) (cid:20) XH (cid:21)(cid:29) + O [3]So all together r n A n = (cid:28)(cid:20) R Θ (cid:21) , (cid:20) R Θ (cid:21)(cid:29) − (cid:28)(cid:20) R Θ (cid:21) , Q n (cid:20) XH (cid:21)(cid:29) + (cid:28)(cid:20) XH (cid:21) , (cid:26)(cid:20) µ n − ∆ i d n ∂ ∗ i d n ∂ − ∆ (cid:21) + Q ∗ n Q n (cid:27) (cid:20) XH (cid:21)(cid:29) n − r n v n h , i n + O [3]= (cid:28)(cid:20) R Θ (cid:21) , (cid:20) R Θ (cid:21)(cid:29) − (cid:28)(cid:20) R Θ (cid:21) , Q n (cid:20) XH (cid:21)(cid:29) + (cid:28)(cid:20) XH (cid:21) , (cid:3) (cid:20) XH (cid:21)(cid:29) n − r n v n h , i n + O [3]Substituting in (A.6), we have r n A n = (cid:28)(cid:20) R Θ (cid:21) , (cid:20) R Θ (cid:21)(cid:29) − (cid:28)(cid:20) R Θ (cid:21) , Q n (cid:3) − Q ∗ n (cid:20) R Θ (cid:21)(cid:29) + (cid:28) (cid:3) − Q ∗ n (cid:20) R Θ (cid:21) , (cid:3)(cid:3) − Q ∗ n (cid:20) R Θ (cid:21)(cid:29) n − r n v n h , i n + O [3]= (cid:28)(cid:20) R Θ (cid:21) , (cid:8) − Q n (cid:3) − Q ∗ n (cid:9) (cid:20) R Θ (cid:21)(cid:29) − r n v n h , i n + O [3] (A.12)as desired.So now we should analyse the operator 1l − Q n (cid:3) − Q ∗ n in momentum space. Wefollow the pattern of the computation from (A.9) through (A.11). Define D n = " µ n − ∆ id n ∂ ∗ id n ∂ − ∆ and, analogously to (A.9), ˜ D n = (cid:0)
1l + Q n D − n Q ∗ n (cid:1) − (cid:3) D − n Q ∗ n ˜ D n and simplifying yields Q ∗ n so that Q n (cid:3) − Q ∗ n = Q n D − n Q ∗ n ˜ D n The Fourier transform of Q n (cid:3) − Q ∗ n is X ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ D − n ( k + ℓ ) u n ( k + ℓ ) b ˜ D n ( k )= u n ( k ) ˆ D − n ( k ) b ˜ D n ( k ) + X = ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ D − n ( k + ℓ ) b ˜ D n ( k ) (A.13)where, pretending that we have continuum, rather than discrete, differential opera-tors, ˆ D n ( p ) = (cid:20) µ n + p d n p − d n p p (cid:21) and b ˜ D n ( k ) = (cid:16)
1l + X ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ D − n ( k + ℓ ) (cid:17) − During the course of the upcoming computation we shall use the following facts. • The parameter d n ≥
1. For small n it takes the value 1 and for large n it decaysquickly approaching 0 as n → ∞ . • The parameter µ n >
0. For small n it is very small and for large n it is very large,with d − n µ n bounded uniformly in n . When d n > d − n µ n is bounded away fromzero. • By [10, Lemma 2.2.b,c]. ◦ u n ( k ) = 1 + O ( | k | ) and ◦ if ℓ = 0, (cid:12)(cid:12) u n ( k + ℓ ) (cid:12)(cid:12) ≤ h Q ≤ ν ≤ ℓν =0 | k ν | i Q ν =0 24 | ℓ ν | + π .The dominant term in (A.13) is u n ( k ) ˆ D − n ( k ) b ˜ D n ( k ) = u n ( k ) ˆ D − n ( k ) n
1l + X ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ D − n ( k + ℓ ) o − = n
1l + ˆ D n ( k ) u n ( k ) − + X = ℓ ∈ ˆ B n u n ( k + ℓ ) u n ( k ) ˆ D − n ( k + ℓ ) ˆ D n ( k ) o − (cid:26)
1l + ˆ D n ( k ) u n ( k ) − + O (cid:18) | k | (cid:20) d − n µ n + | k | d − n | k | d − n µ n + d n | k | | k | (cid:21)(cid:19) (cid:27) − (by Lemma A.2.b)= (cid:20) µ n u n ( k ) + k + O ( d − n µ n | k | ) + O ( | k | ) d n k u n ( k ) + O ( d − n | k | ) − d n k + O ( d − n µ n | k | ) + O ( d n | k | ) 1 + k + O ( | k | ) (cid:21) − = (cid:20) µ n (1 + q ( k )) + k + O ( | k | ) d n k + O ( d n | k | ) − d n k + O ( d − n µ n | k | ) + O ( d n | k | ) 1 + k + O ( | k | ) (cid:21) − (A.14.a)with q ( k ) = O ( | k | ). The determinant of the matrix to be inverted in the last lineof (A.14.a) isdet (cid:20) µ n (1 + q ( k )) + k + O ( | k | ) d n k + O ( d n | k | ) − d n k + O ( d − n µ n | k | ) + O ( d n | k | ) 1 + k + O ( | k | ) (cid:21) = 1 + 2 µ n (cid:0) q ( k )) (cid:1) + 2(1 + µ n ) k + d n k + O (cid:0) | k | (cid:1) + O (cid:0) µ n | k | (cid:1) + O (cid:0) d n | k | (cid:1) = d n n d − n (cid:2) µ n (cid:0) q ( k ) (cid:1)(cid:3) + k + 2 d − n (1 + µ n ) k + O (cid:0) | k | (cid:1)o = d n n d − n + 2 d − n µ n + q ( k ) + O (cid:0) | k | (cid:1)o (A.14.det)where q ( k ) = k + 2 d − n k + 2 d − n µ n (cid:0) k + q ( k ) (cid:1) The tail of (A.13) is, by Lemma A.2.d, X = ℓ ∈ ˆ B n u n ( k + ℓ ) ˆ D − n ( k + ℓ ) b ˜ D n ( k ) = O (cid:18) | k | (cid:20) d − n µ n + d − n | k | d − n | k | d − n µ n + d − n | k | | k | (cid:21)(cid:19) (A.14.b)Combining (A.13) and the three (A.14)’s we have that the Fourier transform of Q n D − n Q ∗ n ˜ D n = Q n (cid:3) − Q ∗ n is ◦ d − n n d − n + 2 d − n µ n + q ( k ) + O (cid:0) | k | (cid:1)o − = d − n d − n +2 d − n µ n n − q ( k ) d − n +2 d − n µ n + O (cid:0) | k | (cid:1)o ◦ times (cid:20) k + O ( | k | ) − d n k + O ( d n | k | ) d n k + O ( d − n µ n | k | ) + O ( d n | k | ) 1 + 2 µ n (cid:0) q ( k ) (cid:1) + k + O ( | k | ) (cid:21) plus O (cid:18) | k | (cid:20) d − n µ n + d − n | k | d − n | k | d − n µ n + d − n | k | | k | (cid:21)(cid:19) which is " − d − n µ n d − n +2 d − n µ n − q ( k ) + O (cid:0) µ n | k | d n (cid:1) + O (cid:0) | k | d n (cid:1) − d − n k d − n +2 d − n µ n + O (cid:0) | k | d n (cid:1) d − n k d − n +2 d − n µ n + O (cid:0) µ n | k | d n (cid:1) + O (cid:0) | k | d n (cid:1) − q ( k ) + O (cid:0) | k | (cid:1) with q ( k ) = d − n d − n +2 d − n µ n n q ( k ) d − n +2 d − n µ n − k o q ( k ) = q ( k ) d − n +2 d − n µ n − d − n µ n q ( k )+ d − n k d − n +2 d − n µ n So the Fourier transform of 1l − Q n (cid:3) − Q ∗ n is " d − n µ n d − n +2 d − n µ n + q ( k ) + O (cid:0) µ n | k | d n (cid:1) + O (cid:0) | k | d n (cid:1) d − n k d − n +2 d − n µ n + O (cid:0) | k | d n (cid:1) − d − n k d − n +2 d − n µ n + O (cid:0) µ n | k | d n (cid:1) + O (cid:0) | k | d n (cid:1) q ( k ) + O (cid:0) | k | (cid:1) (A.15)Unraveling the definitions and simplifying gives q ( k ) = d − n ( d − n +2 d − n µ n ) (cid:8) k + d − n k + 2 µ n d n q ( k ) (cid:9) q ( k ) = d − n +2 d − n µ n (cid:8) k + d − n k + 2 µ n d n k ) (cid:9) When n is large, that is, deep in the “elliptic regime”, the parameter d n ≫ d − n µ n is essentially constant and the Fourier transform of 1l − Q n (cid:3) − Q ∗ n is roughly,for small k " k d − n µ n + k We see an elliptic operator in the tangential direction and a mass in the radialdirection.On the other hand, when n is small, that is, early in the “parabolic regime”, theparameter d n = 1 and µ n ≪ − Q n (cid:3) − Q ∗ n is roughly,for small k " k + k k − k k + k ± ik + k + k ≈ ± ik + k which are parabolic operators. A.4.c Some Operators in Momentum Space
We here gather together some momentum space properties of the operators D n and˜ D n that are used in the computations leading up to (A.15). Lemma A.2. (a) If p is bounded away from zero, then ˆ D − n ( p ) = O (cid:18)(cid:20) d − n d − n d − n (cid:21)(cid:19) (b) If ℓ = 0 , then ˆ D − n ( k + ℓ ) ˆ D n ( k ) = O (cid:18)(cid:20) d − n µ n + | k | d − n | k | d − n µ n + d n | k | | k | (cid:21)(cid:19) (c) b ˜ D n ( k ) = O (cid:18)(cid:20) d − n µ n + | k | d − n | k | d − n | k | | k | (cid:21)(cid:19) (d) If ℓ = 0 , then ˆ D − n ( k + ℓ ) b ˜ D n ( k ) = O (cid:18)(cid:20) d − n µ n + d − n | k | d − n | k | + d − n | k | d − n µ n + d − n | k | d − n | k | + | k | (cid:21)(cid:19) Proof. (a) If p is bounded away from zero, thenˆ D − n ( p ) = (cid:20) µ n + p d n p − d n p p (cid:21) − = d − n p +2 d − n µ n p + d − n p (cid:20) p − d n p d n p µ n + p (cid:21) = O (cid:18)(cid:20) d − n d − n d − n (cid:21)(cid:19) ℓ = 0, then k + ℓ is bounded uniformly away from zero andˆ D − n ( k + ℓ ) ˆ D n ( k ) = O (cid:18)(cid:20) d − n d − n d − n (cid:21)(cid:19) (cid:20) µ n + k d n k − d n k k (cid:21) = O (cid:18)(cid:20) d − n µ n + | k | d − n | k | d − n µ n + d n | k | | k | (cid:21)(cid:19) (c) Using line 4 of (A.14.a), b ˜ D n ( k ) = u n ( k ) − ˆ D n ( k ) (cid:8) u n ( k ) ˆ D − n ( k ) b ˜ D n ( k ) (cid:9) = (cid:0) O ( | k | (cid:1) (cid:20) µ n + k d n k − d n k k (cid:21)(cid:20) µ n u n ( k ) + k + O ( d − n µ n | k | ) + O ( | k | ) d n k u n ( k ) + O ( d − n | k | ) − d n k + O ( d − n µ n | k | ) + O ( d n | k | ) 1 + k + O ( | k | ) (cid:21) − Next using (A.14.det) b ˜ D n ( k ) = (cid:0) O ( | k | (cid:1) (cid:20) µ n + k d n k − d n k k (cid:21) d − n d − n +2 d − n µ n n − q ( k ) d − n +2 d − n µ n + O (cid:0) | k | (cid:1)o" k + O ( | k | ) − d n k u n ( k ) + O ( d − n | k | ) d n k + O ( µ n | k | d n ) + O ( d n | k | ) 1 + µ n u n ( k ) + k + O ( µ n | k | d n ) + O ( | k | ) = O (1) d − n (cid:20) µ n + k d n k − d n k k (cid:21)" k + O ( | k | ) − d n k u n ( k ) + O ( d − n | k | ) d n k + O ( µ n | k | d n ) + O ( d n | k | ) 1 + µ n u n ( k ) + k + O ( µ n | k | d n ) + O ( | k | ) = O (1) d − n (cid:20) µ n + O ( d n | k | + µ n | k | ) d n k + O ( d n | k | + µ n d n | k | ) − d n k + O ( d − n µ n | k | + d n | k | ) O ( d n | k | + µ n | k | ) (cid:21) So b ˜ D n ( k ) = O (cid:18)(cid:20) d − n µ n + | k | d − n | k | d − n | k | | k | (cid:21)(cid:19) ℓ = 0, thenˆ D − n ( k + ℓ ) b ˜ D n ( k ) = O (cid:18)(cid:20) d − n d − n d − n (cid:21) (cid:20) d − n µ n + | k | d − n | k | d − n | k | | k | (cid:21)(cid:19) = O (cid:18)(cid:20) d − n µ n + d − n | k | d − n | k | + d − n | k | d − n µ n + d − n | k | d − n | k | + | k | (cid:21)(cid:19) eferences [1] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski. Methods of QuantumField Theory in Statistical Physics . Dover Publications, 1963.[2] T. Balaban. A low temperature expansion for classical N -vector models. I. Arenormalization group flow. Comm. Math. Phys. , 167:103–154, 1995.[3] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. A Functional IntegralRepresentation for Many Boson Systems. I: The Partition Function.
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