Continuum limits of sparse coupling patterns
PPUPT-2554
Continuum limits of sparse coupling patterns
Steven S. Gubser, Christian Jepsen, Ziming Ji, and Brian Trundy
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA
Abstract
We exhibit simple lattice systems, motivated by recently proposed cold atom experiments,whose continuum limits interpolate between real and p -adic smoothness as a spectral ex-ponent is varied. A real spatial dimension emerges in the continuum limit if the spectralexponent is negative, while a p -adic extra dimension emerges if the spectral exponent ispositive. We demonstrate H¨older continuity conditions, both in momentum space and inposition space, which quantify how smooth or ragged the two-point Green’s function is as afunction of the spectral exponent. The underlying discrete dynamics of our model is definedin terms of a Gaussian partition function as a classical statistical mechanical lattice model.The couplings between lattice sites are sparse in the sense that as the number of sites be-comes large, a vanishing fraction of them couple to one another. This sparseness propertyis useful for possible experimental realizations of related systems.May 2018 a r X i v : . [ h e p - t h ] M a y ontents p -dic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 The main model of interest: Sparse coupling . . . . . . . . . . . . . . . . . . 11 Introduction
The p -adic numbers (for any fixed choice of a prime number p ) are an alternative way of fillingin the “gaps” between rational numbers in order to form a complete set, or continuum. Theyhave been studied for over a hundred years, and one of many mathematical introductions tothe subject is [1]. The key ingredient is the p -adic norm. Briefly, one defines | a | p ≡ p − v ( a ) fornon-zero a ∈ Z , where v ( a ) (the so-called valuation of a ) is the number of times p divides a .Then | a/b | p ≡ p − v ( a )+ v ( b ) for non-zero integers a and b . By fiat, | | p = 0. This norm is verydifferent from the usual absolute value, which is denoted | x | ∞ to avoid any ambiguity. Justas the real numbers R are the completion of the rationals Q with respect to | · | ∞ , so the p -adic numbers Q p are the completion of Q with respect to |·| p for any fixed p . We will oftendescribe the real numbers as Archimedean because the norm | · | ∞ satisfies the Archimedeanproperty, namely that if 0 < | a | ∞ < | b | ∞ , then for some n ∈ Z we have | na | ∞ > | b | ∞ .This property fails for the p -adic norm because it enjoys instead the so-called ultra-metricinequality, | a + b | p ≤ max {| a | p , | b | p } , which implies in particular that | na | p ≤ | a | p for all n ∈ Z .Field theories (in the physics sense of “field”) over the p -adic numbers have been studiedextensively, starting with Dyson’s hierarchical model [2] and continuing with the rigorousresults of [3], with the field theory perspective emerging clearly in [4]. Recent reviews include[5]. The essential features that we will use in this work can already be understood, for p = 2,in terms of a slight rephrasing of Dyson’s original work, as follows. Consider the “furthestneighbor” Ising model. By this we mean that starting with 2 N Ising spins, numbered 0through 2 N −
1, we strongly couple the spins which are as far apart as possible as measuredthrough sequential counting. Thus spin 0 couples to spin 2 N − , spin 1 to spin 2 N − +1, and soforth. Arranging the spins on a circle, we are coupling pairs of spins which are diametricallyopposite. If we stopped there, we would have 2 N − strongly coupled pairs of spins, with eachpair entirely decoupled from every other pair. We want a more interesting thermodynamiclimit, so we keep going by coupling each pair of spins with the pair furthest from it (againin the sense of sequential counting). Then we couple pairs of pairs, and so on. At each stage Let’s briefly review what completion means. A Cauchy sequence { x n } ∞ n =1 with respect to a norm | · | is one where for any real number (cid:15) >
0, we have | x n − x m | < (cid:15) provided n and m are larger than someminimum value N (which usually depends on (cid:15) ). The reals R can be understood as the set of Cauchysequences of rational numbers, modulo an equivalence relation defined by considering two Cauchy sequencesequivalent iff combining them by alternating terms gives again a Cauchy sequence. The p -adic numbers Q p are the completion of Q with respect to the p -adic norm | · | p . The four field operations, namely addition,subtraction, multiplication, and division by non-zero elements, are defined on Q p by continuity from theirstandard definition on Q . Complications arise if one attempts to proceed similarly with non-prime p : Inparticular, the obvious Cauchy construction results in a ring, not a field—in fact, a ring in which one canhave xy = 0 with both x and y non-zero.
000 100 010 110 001 101 011 111000 001 010 011 100 101 110 111
Monna map M h M ( h ) Figure 1:
Left: A furthest neighbor coupling pattern among eight spins. The thicknessof lines indicates the strength of the coupling between spin 0 and the otherspins. The coupling pattern is invariant under shifting by a lattice spacing, sofor example spins 1 and 5 are as strongly coupled as spins 0 and 4. The bluecircle is to guide the eye and does not indicate additional couplings.Right: A hierarchical representation of the couplings between spins. Aboveeach spin’s label we have given the base 2 presentation of the spin number, andwe have shown how the Monna map acts on these numbers by reversing digitsin the base 2 presentation. we reduce the coupling strength by a fixed factor 2 s +1 , where s ∈ R is what we will call thespectral parameter. This coupling pattern can be expressed concisely in terms of the 2-adicnorm of the separation of the spins, as we will specify in more detail in section 2.4. Theoverall picture is illustrated in figure 1.A natural way to understand the furthest neighbor model is in terms of a hierarchy ofclusters of spins, as also illustrated in figure 1. The hierarchical tree of these clusters givesa particularly clear understanding of the 2-adic distance, because if we define d ( i, j ) as thenumber of steps required to go from point i to point j on the tree, then for boundary points i and j taking integer values between 0 and 2 N −
1, we have | i − j | = 2 − N + d ( i,j ) / .A further step is to send the position h of each spin through the discrete Monna map M , which takes as its argument an integer between 0 and 2 N − M need not have been hierarchically closebefore applying M .Not surprisingly, Green’s functions of spins in the furthest neighbor Ising model dependon the locations i and j of the spins only through the 2-adic norm | i − j | . Formally, this is a3onsequence of invariance of the partition function under relabeling all lattice sites accordingto i → ui + b where u and b are elements of Z / N Z and | u | = 1. Note that the statement | u | = 1 is well defined in Z / N Z : it amounts to requiring u to be an odd number. Intuitively, i → ui + b is like a rotation followed by a translation. Translational invariance means thatthe Green’s function can depend on i and j only through their difference i − j . “Rotational”invariance implies that the dependence on i − j must be only through the norm | i − j | .The sparse coupling pattern that we want to study eliminates couplings between spins i and j unless i − j is a power of 2, or minus a power of 2, modulo 2 N . So for the case N = 3shown in figure 1, we drop the coupling between spins 0 and 3, and between 0 and 5, andbetween all translated copies of these pairs, for example the pairs (1 ,
4) and (1 , N , obviously the “sparse” coupling pattern is still nearly all-to-all. But forlarge N , the number of spins coupling to spin 0 increases linearly with N while the totalnumber of spins is 2 N . This type of coupling pattern was first brought to our attention indiscussions about proposed cold atom experiments [6].If the spectral exponent s is large and positive, we expect to recover nearly the sameresults as if we had used a truly 2-adic all-to-all coupling as in the furthest neighbor modeldescribed previously. Here’s why. When s is large and positive, the coupling between spins0 and 2 N − produce very tightly coupled pairs, and the pairs of pairs are also pretty tightlycoupled. This tight coupling means that when we proceed to the next level down the treeand couple 0 relatively weakly to spins ± N − but not ± × N − , all that matters to agood approximation is that we are coupling the quartet { , N − , ± N − } to the quartet {± N − , ± × N − } . Likewise, as we go further down the hierarchy of couplings, while it’strue that we couple spins in previously established 2-adic clusters unequally, the clusters ateach step are so tightly bound within themselves relative to their coupling with each otherthat they act almost like single spins.Meanwhile, as we will see, when the spectral parameter s is made large and negative,we recover nearest neighbor interactions. The two-point Green’s function of the nearestneighbor model with 2 N spins is then well-approximated at large N by a continuum Green’sfunction that we can extract from field theory over R . This Green’s function is smooth in anArchimedean sense, except at zero separation: In fact, if we are considering the model withpure nearest neighbor interactions, the Green’s function away from zero separation is C ∞ .The smoothness of the continuum limit of the Green’s function is a good way to understandhow continuous quantities emerge from a discrete lattice description. Poetically, a continuousspatial dimension emerges from nearest neighbor interactions on a large discrete lattice.A natural question to follow up the discussion of the previous paragraph is, what counts asa smooth continuum Green’s function from a 2-adic point of view? Let’s revert to discussing4 -adic smoothness for any prime p , since it is no more difficult than for p = 2. Continuity iseasy to understand over the p -adic numbers: If a function G maps Q p to R , then we can define G as continuous at x if for any (cid:15) > δ > y with | x − y | p < δ has | G ( x ) − G ( y ) | ∞ < (cid:15) . It is harder to find the proper analog of a C ∞ condition on G ,because derivatives of G with respect to x are tricky to define. (Heuristically, that’s because dG/dx is neither real nor p -adic, but apparently some ratio of the two, which doesn’t quitemake sense.) In fact, the accepted analog of a C ∞ condition is to require that a map G from Q p to R is locally constant. For a function to be locally constant at a point x , we must beable to find some δ > y with | x − y | p < δ has G ( x ) = G ( y ). Surprisingly, afunction from Q p to R which is everywhere locally constant need not be globally constant (asit would for a function from R to R ). For example, the function which is 1 on Z p and 0 overthe rest of Q p is locally constant everywhere, but obviously not globally constant. Green’sfunctions in models with perfectly p -adic coupling are also locally constant except at zeroseparation, as we will see in examples soon. When we turn to sparse coupling patterns, wewill recognize that we are recovering 2-adic continuity precisely when the two-point Green’sfunction is well approximated by a locally constant function. This is exactly what happensin the limit of large positive s for the 2-adic statistical mechanical models that we will studyexplicitly.In short, as the spectral exponent s ranges from large negative to large positive values,the Green’s functions we study transition from showing emergent Archimedean continuityto showing emergent p -adic continuity. How this transition occurs is slightly subtle, butwe will combine some numerical results with analytical reasoning to characterize it both inmomentum space and position space.The sparse coupling pattern we consider was suggested to us in connection with prospec-tive cold atom experiments in which coupling patterns of at least approximately the form weconsider may be realized [6], using techniques along the lines of [8]. It is outside our presentscope to provide a detailed account of these experiments, but let us mention three salientpoints:1. Translational invariance of the coupling (except for endpoint effects) is a natural featureof the experimental setup.2. While it is possible in principle to arrange a wide variety of couplings, it is useful tofocus on sparse couplings, because every time a coupling is introduced between spinsat fixed separation, it increases dissipative tendencies in the system. A more complete introduction to smooth test functions over the p -adic numbers than we will providecan be found, for example, in [7].
5. The most straightforward models to realize in the cold atom system are XXZ Heisen-berg models with no on-site terms, i.e. with hopping terms only. We will be dealingwith a substantially simpler statistical mechanical model in this paper but hope toreturn to the more complicated dynamics of the XXZ model in future work.The organization of the rest of this paper is as follows. In section 2 we describe theclass of statistical mechanical, one-dimensional spins chains that we will study, and we givea general account of how to compute Green’s functions before treating in turn four modelswithin this class: Nearest neighbor interactions, power-law interactions, p -adic interactions(in principle for any prime p though we eventually focus on p = 2), and finally sparsecouplings, which interpolate between nearest neighbor and p -adic behavior. In section 3,starting from field theory, we obtain H¨older continuity bounds on the continuum limit ofGreen’s functions computed in section 2. In section 4, we show through numerical studiesthat the smoothness of Green’s functions in momentum space is well captured by the H¨oldercontinuity bounds derived in section 3. Position space continuity is more complex, withdifferent H¨older exponents depending on whether one is looking at global or local smoothnessproperties. Our aim is to work out the statistical mechanics of models with a variety of non-localcouplings. We want our results to be as explicit as possible, and to have as few parametersas we can arrange. Consider therefore the following Hamiltonian for a lattice with L sites: H ≡ − (cid:88) i,j J ij φ i φ j − (cid:88) j h j φ j , (1)where the φ i are still commuting real numbers. Clearly J ij = J ji because φ i φ j is symmetric.Let us also assume translational invariance: That is, J ij = J i − j , where arithmetic operationslike i − j are carried out modulo L . Define L -dimensional vectors (cid:126)v κ by v κ,j ≡ √ L e πiκj/L for κ = 0 , , , . . . , L − . (2)Any quantity X j depending on j ∈ { , , , . . . , L − } can be Fourier transformed accordingto X j = L − (cid:88) κ =0 ˜ X κ v κ,j . (3)6n easy calculation shows that J (cid:126)v κ = √ L ˜ J κ (cid:126)v κ , (4)where J without indices means the symmetric matrix J ij , and ˜ J κ is the Fourier transform ofthe coupling strengths J h . Using (3)-(4), we have immediately H = − √ L L − (cid:88) κ =0 ˜ J κ ˜ φ − κ ˜ φ κ − L − (cid:88) κ =0 ˜ h − κ φ κ . (5)We now make two assumptions: • ˜ J = 0. We understand this as a consequence of assuming the existence of a symmetrywhere all the φ i are shifted by a common value. • ˜ J κ < κ (cid:54) = 0. This amounts to saying that the interactions among the φ i areferromagnetic.It is useful to note that the second assumption follows from the first together with therequirement that all J h ≥ h (cid:54) = 0, with not all of them equal to zero.In order to make the statistical mechanics of H well-defined, we insert a factor of δ ( ˜ φ )into the partition function: Z [ h ] ≡ (cid:32) L − (cid:89) j =0 (cid:90) ∞−∞ dφ j (cid:33) δ ( ˜ φ ) e − βH = Z [0] exp (cid:40) − β √ L L − (cid:88) κ =1 J κ ˜ h − κ ˜ h κ (cid:41) . (6)We are interested in the two-point function G ij = (cid:104) φ i φ j (cid:105) = 1 β Z [0] ∂ Z [ h ] ∂h i ∂h j (cid:12)(cid:12)(cid:12)(cid:12) h =0 . (7)From J ij = J i − j it follows that G ij = G i − j . A short calculation starting with (7) leads to G h = − βL / L − (cid:88) κ =1 J κ e πiκh/L . (8)The factor of δ ( ˜ φ ) in the partition function may seem undesirable, especially from the pointof view of constructing Hamiltonians with only sparse couplings among the spins, because δ ( ˜ φ ) can be viewed as the K → ∞ limit of e − K ˜ φ , and this amounts to a strong all-to-allcoupling among spins (though of a very particular form). In fact, we could achieve theessentially the same results by omitting the factor of δ ( ˜ φ ) while sending J → J − µ where µ is small and positive. Then ˜ J ∝ − µ , while the other ˜ J κ would scarcely be affected since7hey are finite and negative already at O ( µ ). Use of (7) would then lead to the same G h asin (8), up to an overall constant proportional to 1 /µ . Discarding this uninteresting constantand then taking the limit µ → p -adic statistical models that the sparse coupling modelinterpolates between as the spectral parameter ranges from negative to positive values. As an extremal case of an Archimedean statistical model, we consider the model with nearestneighbor coupling specified by J NN h = J ∗ ( δ h +1 + δ h − − δ h ) , (9)which leads to G NN h = 14 βJ ∗ L L − (cid:88) κ =1 e πiκh/L sin πκL . (10)If L is large, then we can approximate sin πκL ≈ πκL and extend the sum to infinity: G NN h ≈ LβJ ∗ ∞ (cid:88) κ = −∞ , κ (cid:54) =0 e πiκh/L π κ = LβJ ∗ G ( h/L ) , (11)where the continuum two-point function G ( x ) takes the form G ( x ) = 12 (cid:18) x − (cid:19) −
124 for x ∈ [0 , . (12)Properly speaking, G ( x ) is defined on a circle with x ∼ x + 1, with periodic boundaryconditions, and it satisfies d Gdx = − δ ( x ) + 1 and (cid:90) dx G ( x ) = 0 . (13)8f instead of nearest neighbor coupling we have some generic finite-range J h satisfying J h = J − h > h (cid:54) = 0 and ˜ J = 0, then we get essentially the same result:˜ J κ ≈ − π κ L / J ∗ for (cid:12)(cid:12)(cid:12) κL (cid:12)(cid:12)(cid:12) ∞ (cid:28) J ∗ , and so for large L , G h ≈ LβJ ∗ G ( h/L ) (15)with the same continuum function G ( x ) given in (12).It is worth noting that if we focus in on small | h/L | ∞ , then we are mostly insensitive tothe fact that the system is at finite volume, and we find G ( x ) ≈ G (0) − | x | ∞ / For comparison with the sparse coupling model to be defined in section 2.4, we will eventuallyneed to adjust the nearest neighbor model so as to have it include an adjustable exponentthat tunes the strength of the coupling, analogously to the spectral parameter of the sparsecoupling model. To this end we define˜ J power κ ≡ − J ∗ s √ L (cid:104) sin (cid:16) πκL (cid:17)(cid:105) − s (16)so that ˜ G power κ = 2 s βJ ∗ √ L (cid:104) sin (cid:16) πκL (cid:17)(cid:105) s . (17)For s = −
2, this model reduces to the nearest neighbor coupling model. In general for s < h/L → J power h ∼ − J ∗ π Γ(1 − s ) sin( πs/ h + s/ h − s/ . (18)By additionally invoking Sterling’s formula, it becomes apparent that in the regime 1 (cid:28) h (cid:28) L , the model we are considering does indeed couple the spins according to a power law: J power h ∼ − J ∗ π Γ(1 − s ) sin ( πs/ h s − . (19)For s < −
1, the large L limit of the position space Green’s function asymptotes to G power h = 2 s π s βJ ∗ L s (cid:2) Li − s ( e πih/L ) + Li − s ( e − πih/L ) (cid:3) , (20)9here Li n ( x ) denotes the polylogarithm function. p -dic coupling Choose a prime number p and a positive integer N , and assume L = p N . (21)Then an all-to-all coupling of spins can be defined based on the p -adic norm: J p − adic h = J ∗ | h | − s − p if h (cid:54) = 0 − J ∗ L ζ p ( − s ) ζ p (1) ζ p ( − N s ) if h = 0 . (22)Here we have used the local zeta function ζ p ( s ) ≡ − p − s , (23)so named because the usual Riemann zeta function is ζ ( s ) = (cid:81) p ζ p ( s ) where the product isover all prime numbers.To analyze (22), it is useful first to work out the Fourier transform of the followingfunction: f h = A | h | − s − p (1 − δ h ) + B + Cδ h . (24)A tedious but straightforward calculation suffices to show that˜ f κ = ˜ A | κ | s (1 − δ κ ) + ˜ B + ˜ Cδ κ (25)where˜ A = L s + ζ p ( − s ) ζ p (1 + s ) A ˜ B = C √ L − L s + ζ p ( − s ) ζ p (1) A ˜ C = √ L (cid:18) B + ζ p ( − s ) ζ p (1) A (cid:19) . (26)With the help of (26) one can see immediately that J p − adic0 was chosen in (22) precisely soas to have ˜ J p − adic0 = 0. Indeed,˜ J p − adic κ = J ∗ √ L (cid:20) ζ p ( − s ) ζ p (1 + s ) (cid:12)(cid:12)(cid:12) κL (cid:12)(cid:12)(cid:12) sp − ζ p ( − s ) ζ p (1) (cid:21) (1 − δ κ ) . (27)While ˜ J p − adic κ < κ (cid:54) = 0 for any s ∈ R , we are mostly interested in the regime s >
0, in10hich case the absolute value of the first term in square brackets in (27) is larger than theabsolute value of the second. Thus we may expand˜ G p − adic κ = − βL ˜ J κ (1 − δ κ ) = − ζ p (1) /ζ p ( − s ) βL / J ∗ ∞ (cid:88) n =1 (cid:18) ζ p (1 + s ) ζ p (1) L − s (cid:19) n | κ | − nsp (1 − δ κ ) . (28)The expansion is useful because it allows us to apply the Fourier transform (24)-(26) andobtain G p − adic h = − ζ p (1) /ζ p ( − s ) βL J ∗ ∞ (cid:88) n =1 ζ p (1 + s ) n ζ p (1) n (cid:34) (cid:18) ζ p ( − ns + 1) ζ p ( ns ) | h | ns − p − ζ p ( − ns + 1) ζ p (1) (cid:19) (1 − δ h ) − ζ p ( − ns + 1) ζ p (1) ζ p ( N ( ns − δ h (cid:35) . (29)In a sense, the result (29) is more complicated than necessary, because by adding a constantto J p − adic h for h (cid:54) = 0 and adjusting J p − adic0 to keep ˜ J p − adic0 = 0, we could have arranged to have˜ J p − adic κ = J ∗ √ L ζ p ( − s ) ζ p (1+ s ) (cid:12)(cid:12) κL (cid:12)(cid:12) sp , which would result in the same result (29) for G p − adic h , exceptwith the infinite sum replaced by its first term: That is, G p − adic h = A | h | s − p + B + Cδ h for someconstants A , B , and C depending on s and proportional to βL J ∗ . However, for purposes ofanalyzing the next example, the alterations in J p − adic h just described are undesirable.Note that if we hold L J ∗ fixed, then except at h = 0 there is no L dependence at all in G p − adic h ; the only thing that changes is the range of allowed h . Taking L large means thatthe range of h becomes p -adically dense in the p -adic integers Z p , defined as the subset of Q p consisting of elements whose norm is no greater than 1. Z p can be understood as the p -adicanalog of the interval [ − , ⊂ R . Because G p − adic h is a function of h only through its p -adicnorm | h | p , we see that its continuum limit is locally constant everywhere on Z p , except at h = 0.The results of this section are perhaps not too surprising when compared with power-lawinteractions in real field theories. Indeed, a power law 1 / | x | α ∞ in the action leads to a powerlaw 1 / | x | ˜ α ∞ in the Green’s functions, where ˜ α + α = 2 d and d is the dimension of the fieldtheory (c.f. results in section 2.2.) The current setup is essentially the same, except that theordinary absolute value has been replaced by the p -adic norm. Now let L = 2 N (30)11or some positive integer N . Then we can consider a sparse coupling of the form J sparse h = J ∗ N − (cid:88) n =0 ns ( δ h − n + δ h +2 n − δ h ) . (31)By sparse we mean that out of L independent values of J h , only O (log L ) are non-vanishing.We could generalize from p = 2 to other values of p , but some unobvious complications arisein doing so which we prefer to postpone.The main qualitative features of G sparse h are: • For sufficiently negative s , G sparse h closely approximates G NN h . This makes sense becausewhen s is large and negative, only the first few terms in the sum matter. • For sufficiently positive s , G sparse h closely approximates G − adic h . This is less obvious andwill be investigated further in the next section. • As s crosses from negative to positive values, G sparse h undergoes a transition from beingcloser to a smooth function in an Archimedean sense to being closer to a smoothfunction in a 2-adic sense.To visualize the behavior of G sparse h , we have found it helpful to employ a discrete versionof the Monna map, introduced for p = 2 already in figure 1. For completeness, we recordhere its definition for any p . Let any h ∈ { , , , . . . , L − } be expressed as h = N − (cid:88) n =0 h n p n where each h n ∈ { , , , . . . , p − } . (32)Then the image of h under the Monna map is M ( h ) ≡ N − (cid:88) n =0 h N − − n p n . (33)That is, we reverse the digits in the base p expansion of h . Clearly (this version of) theMonna map is an involution. By inspection, we see that if i and j are p -adically close, then M ( i ) and M ( j ) are sequentially close.In figure 2 we show G sparse h and G − adic h , the former as a function of both h and log M ( h ),for various values of s , to confirm the qualitative features listed above. The standard Monna map from Q p to the non-negative reals is defined similarly, by expanding x ∈ Q p as x = (cid:80) ∞ n = v ( x ) x n p n and then defining M ( x ) ≡ (cid:80) ∞ n = v ( x ) x n p − − n . This map is continuous, volume-preserving,and surjective, but not quite injective: For example, M maps both − /p to 1. We will not have needof this continuous version of the Monna map. = - G h sparse G h power
10 20 30 40 50 60 h - G h / G s = - G h sparse G h - adic log ( h ) - G h / G s = - G h sparse G h power
10 20 30 40 50 60 h - G h / G s = - G h sparse G h - adic log ( h ) - G h / G s = G h sparse G h power
10 20 30 40 50 60 h - G h / G s = G h sparse G h - adic log ( h ) - G h / G s = G h sparse G h power
10 20 30 40 50 60 h - G h / G s = G h sparse G h - adic log ( h ) - G h / G Figure 2:
Left: G sparse h and G power h versus h . This column shows how close G sparse h is to asmooth function in the usual Archimedean sense, and confirms that G sparse h ≈ G power h when s is sufficiently negative.Right: G sparse h and G − adic h versus log M ( h ). This column shows how close G sparse h is to a smooth function in the 2-adic sense, and confirms that G sparse h ≈ G − adic h when s is sufficiently positive. Continuity bounds
Having observed an apparent change from Archimedean to 2-adic continuity in the exampleof section 2.4, we are naturally led to investigate continuum theories with similar couplingpatterns. We start in section 3.1 with p -adic field theories, since they are actually easierto deal with once one understands the rules than Archimedean field theories. We deriveH¨older continuity bounds for the two-point Green’s function both in momentum space andreal space. Then in section 3.2 we derive analogous bounds for bilocal Archimedean fieldtheories.Before getting into the main field theory calculations, let’s review what H¨older continuitybounds are in general. Let F be either Q p or R , and denote the norm on F as | · | . Let f be a map from some subset D ⊂ F to R . Usually, if F = R , then for us D will be an openinterval, while if F = Q p , then D will be an affine copy of Z p . Let O be any subset of D (and again we usually have in mind simple choices of O like open intervals or affine copiesof Z p ). Then f satisfies a H¨older continuity condition over O with positive real exponent α iff there is some positive real number K such that | f ( x ) − f ( x ) | ∞ < K | x − x | α (34)for all x and x in O . If O = D , then we would say that f is globally α -H¨older continuous.We say that f is locally α -H¨older continuous at x iff there exists some open set I containing x such that f is α -H¨older continuous on I . And we describe f as a whole as locally α -H¨oldercontinuous if it is locally α -H¨older continuous at every point in its domain (assumed to bean open set).A H¨older continuous function with any positive exponent α is continuous in the usualsense. How big we can make α is an indication of how much “better” our function is thanmerely continuous. If F = R , then we don’t usually expect to find α bigger than 1, becauseif we do, then f must be constant over its connected components. But if F = Q p , then it ispossible to have non-constant functions with arbitrarily positive H¨older continuity exponent.A useful example of an α -H¨older continuous function f ( x ) is a linear combination of functions | x − x i | α where the x i are constants.The distinction between global and local α -H¨older continuity is important to us becausewe are going to argue, through a combination of analytic and numerical means, that thecontinuum limit of the two-point function G sparse h is, in some cases, globally H¨older continuouswith one exponent and locally H¨older continuous away from the origin with a larger exponent.14 .1 -adic field theories The standard integration measure on Q p satisfies two key properties: • The measure of Z p is 1. • If S ⊂ Q p has measure (cid:96) (a real number), then the set aS + b has measure | a | p (cid:96) for any a, b ∈ Q p .The Fourier transform on Q p is defined by f ( x ) = (cid:90) Q p dk χ ( kx ) ˜ f ( k ) (35)where χ ( kx ) = e πi { kx } . The notation { ξ } means the fractional part of ξ ∈ Q p : that is, { ξ } = ξ + n for the unique element n ∈ Z p that leads to { ξ } ∈ [0 , R , we have χ ( ξ + ξ ) = χ ( ξ ) χ ( ξ ); in technical terms, χ is an additivecharacter. Note that χ ( ξ ) = 1 precisely if ξ ∈ Z p .Specializing now to p = 2, consider the bilocal field theory S = − (cid:90) Q dxdy φ ( x ) J ( x − y ) φ ( y ) (36)where J ( x ) = J ∗ (cid:88) n ∈ Z ns [ δ ( x − n ) + δ ( x + 2 n ) − δ ( x )] , (37)and δ ( x ) is defined as usual by the relation (cid:82) Q dx f ( x ) δ ( x ) = f (0) for any continuous function f . The action (36) becomes diagonal in Fourier space: S = − (cid:90) Q dk
12 ˜ φ ( − k ) ˜ J ( k ) ˜ φ ( k ) , (38)The two-point function is defined as G ( x ) = (cid:104) φ ( x ) φ (0) (cid:105) ≡ (cid:82) D φ e − S φ ( x ) φ (0) (cid:82) D φ e − S , (39)and one straightforwardly finds ˜ G ( k ) = − J ( k ) . (40)For explicit calculations, it is convenient to set J ∗ = 1 /
4. Then˜ J ( k ) = 14 (cid:88) n ∈ Z ns [ χ (2 n k ) + χ ( − n k ) −
2] = − (cid:88) n ∈ Z ns sin ( π { n k } ) . (41)15he infinite sums in (41) may be restricted to n < − v ( k ), because only then is { n k } non-zero. We immediately see that it is necessary to choose s > s >
0, we may rewrite (41) for non-zero k as˜ J ( k ) = −| k | s Ψ(ˆ k ) (42)where ˆ k = | k | k and Ψ(ˆ k ) ≡ ∞ (cid:88) n =1 − ns sin ( π { − n ˆ k } ) . (43)The following features of Ψ(ˆ k ) are at the center of our analysis:1. Ψ(ˆ k ) is bounded above and below by positive constants which depend on s but not ˆ k .2. Ψ(ˆ k ) is globally s -H¨older continuous over U .The first of these properties is easily demonstrated:2 − s + 2 − s − ≤ Ψ(ˆ k ) ≤ − ζ ( − s ) , (44)where the first inequality comes from dropping all but the first two terms in the sum (43),and the second inequality comes from replacing sin ( π { − n ˆ k } ) by 1 in all terms of the sum.The second property requires more care, and it turns on observing that if n ≤ v (ˆ k − ˆ k ),then sin ( π { − n ˆ k } ) = sin ( π { − n ˆ k } ). (This follows because if n ≤ v (ˆ k − ˆ k ), then 2 − n ˆ k and 2 − n ˆ k differ by a 2-adic integer, so χ (2 − n ˆ k ) = χ (2 − n ˆ k ).) Therefore, when computingΨ(ˆ k ) − Ψ(ˆ k ), only the terms with n > v (ˆ k − ˆ k ) contribute, and if we replace sin ( π { − n ˆ k } )by 1 in these terms we arrive at the desired H¨older inequality with K = − ζ ( − s ) . (45) An amusing connection to population dynamics can be observed at this point. Recall the logistical map, x → rx (1 − x ). If { x n } n ∈ Z is a solution to this iterated map, then we can think of x n as (proportionalto) the population of a species at generation number n . For r = 4, a solution is x n = sin ( π n k ) where k is a real number. However, this is not the most general solution, because it has the property x n → n → −∞ . Consider instead x n = sin ( π { n k } ) where k is a 2-adic number. Then x n = 0 for all n ≥ − v ( k ),but we need not have x n → n → −∞ . Thus we see that the 2-adic number k parametrizes the routesto extinction under the r = 4 logistical map, and the 2-adic norm of k predicts the moment of extinction: n ∗ = log | k | . To make the discussion simple, suppose now that k is a 2-adic integer, so that extinctionhas occurred by the time n = 0. Further suppose that each generation leaves an imprint on its environmentproportional to x n , and that this imprint dissipates over time with a half life of 1 /s generations. So theenvironmental imprint at time 0 of generation n (with n < I n = α ns x n , where α is the constant of proportionality. Then I = − α ˜ J ( k ) as computed in (41) is the totalenvironmental imprint of the species, summed across all generations and measured at time 0. k ) implies that 1 / Ψ(ˆ k ) is also globally s -H¨older contin-uous. Since the Green’s function ˜ G ( k ) = 1 | k | s Ψ(ˆ k ) , (46)is the product of a locally constant factor and a H¨older-continuous factor, we conclude thataway from k = 0, ˜ G ( k ) is locally s -H¨older continuous.Turning to position space, our intuitive understanding is that G ( x ) will be continuouseverywhere iff ˜ G ( k ) is integrable at large k , which is the case iff s >
1. Let us focus thereforeon the regime s >
1. There is a complication in defining G ( x ) when s >
1: The integral G ( x ) = (cid:90) Q dk χ ( kx ) | k | s Ψ(ˆ k ) (47)is infrared divergent. An efficient way to handle this divergence is to alter (47) to G ( x ) ≡ (cid:90) Q dk χ ( kx ) − | k | s Ψ(ˆ k ) = | x | s − g (ˆ x ) , (48)where, by calculation, g (ˆ x ) = ζ (1 − s ) (cid:90) U d ˆ k Ψ(ˆ k ) + ∞ (cid:88) n =1 (1 − s ) n (cid:90) U d ˆ k χ (2 − n ˆ k ˆ x )Ψ(ˆ k ) . (49)Other approaches to regulating the infrared divergence give substantially the same result. We can conclude from (48) that G ( x ) is globally H¨older continuous with exponent s − g (ˆ x ) is globally H¨older continuous with the same exponent. (Notethat the H¨older bound for G ( x ) can be made global rather than local because | x | s − is itselfglobally H¨older continuous with exponent s − g (ˆ x ) − g (ˆ x ), we can restrict the sum in (49) to n > v (ˆ x − ˆ x ). The remainingterms can be bounded using | χ (2 − n ˆ k ˆ x ) − χ (2 − n ˆ k ˆ x ) | ∞ ≤
2, and the desired H¨older conditionfollows. Note that our final position space continuity condition is significantly weaker thanthe one in momentum space, because the H¨older exponent, which was s in momentum space,is now s −
1. In section 4, we will in fact find numerical evidence that a stronger H¨older For example, instead of (48) we could stick with (47) but exclude from the domain of integration all k with | k | < | k IR | , where k IR = 2 v IR is an infrared regulator (with v IR large and positive). Then we wouldfind G ( x ) = ζ ( s − | k IR | − s (cid:90) U d ˆ k Ψ(ˆ k ) + | x | s − g (ˆ x ) , (50)and upon dropping the first term we are back to (48). locally in position space, away from x = 0. No improvement to the global H¨older continuity exponent is possible, though, because if it were we could demonstrate afaster fall-off of ˜ G ( k ) at large | k | than the one that follows from (46). A similar analysis can be carried out on the Archimedean side, starting with the field theory S = − (cid:90) R dxdy φ ( x ) J ( x − y ) φ ( y ) = − (cid:90) R dk
12 ˜ φ ( − k ) ˜ J ( k ) ˜ φ ( k ) , (51)where the Fourier transform is f ( x ) = (cid:90) R dk e πikx ˜ f ( k ) , (52)and we use precisely the same form of J ( x ) as in (37). The general analysis (39)-(40) oftwo-point functions holds unaltered, now leading to˜ J ( k ) = − ψ ( k ) (53)where we have set J ∗ = 1 / ψ ( k ) ≡ (cid:88) n ∈ Z ns sin ( π n k ) . (54)If s ≤ −
2, the infinite sum in (54) diverges at large negative n . But this only means that thecoupling function J ( x ) is overwhelming concentrated near x = 0. If a cutoff is imposed onthe sum, and then J ∗ is rescaled as this cutoff is gradually removed, one can show that J ( x )converges precisely to − δ (cid:48)(cid:48) ( x ), resulting in a perfectly local theory. If instead s ≥
0, then thesum in (53) diverges at large positive n , signaling that arbitrarily long-ranged interactionsdominate.The interesting regime, then, is − < s <
0. Here the sum (54) converges, and we canask what properties the function ψ ( k ) satisfies analogous to the ones enumerated below (43)for Ψ(ˆ k ) in the 2-adic case. In fact, we claim1. ψ ( k ) ≈ | k | − s ∞ , meaning that there exist positive constants K and K , independent of Again a population dynamical narrative can be attached to (a slight variant of) (54): Regarding x n =sin ( π n k ) as a solution to the r = 4 logistical map, and supposing that each generation “eats” an amount αx n of a resource which, when undisturbed, grows exponentially with doubling time − /s , we see that a cutoffversion of the sum, ψ > ( k ) ≡ (cid:80) ∞ n =0 ns sin ( π n k ), computes for us the total quantity of resources I = αψ > ( k )required at time 0 to feed the species for all future time. Here α is some constant of proportionality. , such that K | k | − s ∞ < ψ ( k ) < K | k | − s ∞ for all k ∈ R \{ } .2. For − < s < ψ ( k ) is globally H¨older continuous with exponent − s .3. For − < s < −
1, the derivative ψ (cid:48) ( k ) = dψ ( k ) /dk is globally H¨older continuouswith exponent − s −
1. (Note that ψ ( k ) itself cannot have H¨older continuity exponentgreater than 1 without being constant. So the derivative condition we claim here isthe best that can be expected.) It follows that ψ ( k ) is globally 1-H¨older continuous onany bounded domain.To arrive at the estimate ψ ( k ) ≈ | k | − s ∞ , we define n k ≡ − log ( π | k | ∞ ) . (55)Then we have ψ ( k ) = (cid:88) n 2) + ζ ( − s )] . (56)To derive the H¨older condition on ψ for − < s < 0, set δ = | k − k | ∞ and note that | sin ( π n k ) − sin ( π n k ) | ∞ ≤ min { , π n δ } . (57)Defining n δ = − log ( πδ ) , (58)we see that | ψ ( k ) − ψ ( k ) | ∞ ≤ (cid:88) n ∈ Z ns min { , π n δ } = (cid:88) n 1) + ζ ( − s )] , (59)where again ≈ means equality to within fixed multiplicative factors, independent in this caseof δ . The last expression in (59) is the desired H¨older bound, valid when − < s < 0. Ifinstead − < s < − 1, then we may calculate ψ (cid:48) ( k ) = π (cid:88) n ∈ Z n ( s +1) sin( π n +1 k ) . (60)By the same method as in (59) we arrive at the H¨older continuity condition for ψ (cid:48) ( k ) withexponent − s − 1. 19y combining the property ψ ( k ) ≈ | k | − s ∞ with the H¨older bounds, we see that ˜ G ( k ) islocally H¨older with exponent − s for − < s < 0. Also, ˜ G (cid:48) ( k ) is locally H¨older away from k = 0 with exponent − s − − < s < − 1, implying that ˜ G ( k ) is locally 1-H¨older awayfrom k = 0.Now let’s investigate smoothness of the Green’s function in position space. We naivelydefine G ( x ) = − (cid:90) R dk e πikx ˜ J ( k ) = (cid:90) R dk e πikx ψ ( k ) . (61)As in the 2-adic case, our intuitive understanding is that G ( x ) will be continuous everywhereiff ˜ G ( k ) is integrable at large k , which is the case iff s < − 1. Because ψ ( k ) ≈ | k | s ∞ , the UV-integrable regime is − < s < − G . Itdoes not matter much how this constant is removed; one option is to alter (61) to G ( x ) ≡ (cid:90) R dk e πikx − ψ ( k ) . (62)For the purposes of a H¨older continuity condition we must estimate G ( x ) − G ( x ) = (cid:90) R dkψ ( k ) ( e πikx − e πikx ) . (63)Setting δ = | x − x | ∞ , we have | G ( x ) − G ( x ) | ∞ ≤ (cid:90) R dkψ ( k ) min { , π | k | ∞ δ } = 2 π (cid:90) | k | < /δ dkψ ( k ) | k | ∞ δ + 2 (cid:90) | k | > /δ dkψ ( k ) ≈ (cid:90) /δ dk k s +1 δ + (cid:90) ∞ /δ k s ≈ δ − s − (cid:20) s + 2 − s + 1 (cid:21) . (64)In short, for − < s < − 1, we have obtained a global H¨older bound with exponent − s − The first question we wish to ask of numerics is how well the two-point Green’s functionderived from sparse coupling approximates the one derived from 2-adic coupling with thesame value of s . Based on the rigorous field theory results of section 3, we expect that, for20 og max G ˜ κ sparse / G ˜ κ - adic log min G ˜ κ sparse / G ˜ κ - adic s - N log max G ˜ κ sparse / G ˜ κ - adic log min G ˜ κ sparse / G ˜ κ - adic N - s Figure 3: Left: Optimal values of the constants K and K appearing in (65) as functionsof s for fixed N .Right: Optimal values of the constants K and K as function of N for fixed s . The expectation is that provided s > K and K asymptote to constantsat sufficiently large N . s > 0, the answer in momentum space is that K < ˜ G sparse κ / ˜ G − adic κ < K (65)for some positive constants K and K which may depend on s . Numerical support for thisconclusion is shown in figure 3, where we show optimal values of K and K as functions of s for various N . As s → 0, the evidence that K remains bounded as N increases becomestenuous. We are limited ultimately by our ability to go to sufficiently high values of N .Away from small positive s , ˜ G sparse κ ≈ ˜ G − adic κ is evidently an excellent approximation. Basedon empirically examining the curves on the left side of figure 3, we find K i ≈ − s κ i ( s )where the functions κ i ( s ) vary relatively slowly with s , possibly as a negative power of s , orpossibly as a small positive power of 2 − s . In order to obtain K and K as functions of N and s , the actual procedure was as follows:1. For fixed N and s , compute ˜ G sparse κ using the methods of section 2, and adjust theoverall coupling strength J ∗ so that G sparse h = 1 when h = 0. (In other words, thenormalization condition is implemented in position space .)2. Likewise compute ˜ G − adic κ with G − adic0 = 1.21. Compute K and K as K = min (cid:32) ˜ G sparse κ ˜ G − adic κ (cid:33) ≡ min κ (cid:54) =0 ˜ G sparse κ ˜ G − adic κ K = max (cid:32) ˜ G sparse κ ˜ G − adic κ (cid:33) ≡ max κ (cid:54) =0 ˜ G sparse κ ˜ G − adic κ . (66) Next we would like to understand how well the local H¨older continuity bounds in momentumspace are reflected in the numerics. We also want to quantify how ragged the Green’sfunctions become in momentum space in regimes where we couldn’t derive any continuitybound (by methods developed in the current work). The H¨older bounds, as derived in fieldtheory in section 3, are approximately as follows: • | ˜ G sparse ( k ) − ˜ G sparse ( k ) | ∞ < K | k − k | s when s > 0. More precisely, ˜ G sparse ( k ) as amap from Q to R is locally s -H¨older continuous away from k = 0. • | ˜ G sparse ( k ) − ˜ G sparse ( k ) | ∞ < K | k − k | − s ∞ when − < s < 0. More precisely, ˜ G sparse ( k )as a map from R to R is locally − s -H¨older continuous away from k = 0 when − 0, and locally 1-H¨older continuous away from k = 0 when − < s < − p -adic side, we first adjust the overall coupling strength J ∗ so that G sparse0 = 1, and likewise G − adic0 = 1. Then we define˜ A − adic ( N, s ) ≡ log max κ odd (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G sparse κ ˜ G − adic κ − ˜ G sparse κ + L/ ˜ G − adic κ + L/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ , (67)where on the right hand side we understand that ˜ G sparse κ and ˜ G − adic κ are computed using thesame values of N and s . We find numerically that A − adic ( N, s ) exhibits linear trajectories:˜ A − adic ( N, s ) ≈ − s ( N − 1) + log ˜ K − adic ( s ) , (68)where ˜ K − adic ( s ) is N -independent. These linear trajectories persist even at negative s , after2-adic continuity is lost. See figure 4.In formulating the definition of ˜ A − adic ( N, s ), we chose to focus on differences betweensite κ and κ + L/ N - - A ˜ - adic s - - - N - - A ˜ power s - - - Figure 4: Left: 2-adic smoothness in momentum space. The dots are evaluations of˜ A − adic ( N, s ) in (67), and the lines are plots of the linear trajectories indicatedin (68), with K ( s ) chosen so that the line goes through the last data point.Right: Archimedean smoothness in momentum space. The dots are evaluationsof ˜ A power ( N, s ) in (71), and the lines are plots of the linear trajectories indicatedin (72), with K ( s ) chosen so that the line goes through the last data point. are equivalent to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G sparse κ ˜ G − adic κ − ˜ G sparse κ ˜ G − adic κ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ ≤ ˜ A − adic ( N,s ) ≈ ˜ K − adic ( s ) | κ − κ | s (69)for all odd κ and κ with κ − κ = L/ 2. The inequality (69) is clearly a close relative of thelocal s -H¨older continuity condition on ˜ G ( k ). We could make an even closer connection tothis continuity condition if we generalized ˜ A − adic ( N, s ) to a quantity that would track alsothe separation between κ and κ . Doing so would allow us to check the H¨older condition on˜ G sparse κ / ˜ G − adic κ more thoroughly; however, our explorations in this direction seem to indicatethat the final results are unaffected by such a generalization.In light of the approximately linear trajectories (68), it is convenient to define˜ α − adic ( N, s ) ≡ − ˜ A − adic ( N, s ) + ˜ A − adic ( N − , s ) . (70)Then, recalling that ˜ G − adic κ is a 2-adically smooth function, we arrive at our main numericalresult on 2-adic smoothness of momentum space Green’s functions: ˜ G sparse κ satisfies a localH¨older condition whose best (i.e. most positive) exponent is approximately ˜ α − adic ( N, s ) ≈ s ,in agreement with our field theory expectations.On the Archimedean side, in order to pursue a similar strategy, we need some standard23f comparison analogous to ˜ G − adic κ . We define˜ A power ( N, s ) ≡ log max L ≤ κ< L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G sparse κ ˜ G power κ − ˜ G sparse κ +1 ˜ G power κ +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ , (71)where ˜ G power κ is given in (17) as usual we can adjust J ∗ so that G power h = 1 when h = 0 inposition space.Because ˜ G power is C ∞ away from κ = 0, forming the ratio ˜ G sparse κ / ˜ G power κ doesn’t affectthe local smoothness properties of ˜ G power κ . However, this ratio does cancel out part of theoverall trend whereby ˜ G sparse κ gets bigger near κ = 0 and κ = L . As a result, studying˜ G sparse κ / ˜ G power κ rather than ˜ G sparse κ by itself makes it easier to accurately pick out the localsmoothness properties from a finite sampling of points. As on the 2-adic side, the numericaldata approximately follow exponential trajectories:˜ A power ( N, s ) ≈ s ( N − 1) + log K power ( s ) , (72)where K ( s ) is N -independent. These trajectories persist even at positive s , after Archimedeancontinuity is lost. So we can usefully define˜ α power ( N, s ) ≡ − ˜ A power ( N, s ) + ˜ A power ( N − , s ) , (73)and then ˜ α power ( N, s ) ≈ − s for large N is our numerical estimate of the best (i.e. mostpositive) exponent appearing in a local Archimedean H¨older condition for ˜ G sparse κ . Position space smoothness can be studied using quantities analogous to the ones used insection 4.2 for momentum space. Specifically, we define A − adic ( N, s ) ≡ log max h odd (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G sparse h G − adic h − G sparse h + L/ G − adic h + L/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ α − adic ( N, s ) ≡ − A − adic ( N, s ) + A − adic ( N − , s ) , (74)and then, assuming α − adic ( N, s ) is nearly constant for large N , its large N limit is ournumerical estimate of the best possible local H¨older exponent for G sparse h in a 2-adic setting.24 - a d i c f i e l d t h e o r y r ea l f i e l d t heo r y • α - adic • α power - - s - α - a d i c f i e l d t h e o r y real field t heo r y • α ˜ - adic • α ˜ power - - s - α ˜ Figure 5: α versus s and ˜ α versus s in the 2-adic and Archimedean settings. Field theorybounds derived in section 3 are shown in dashed black and dashed blue. Dottedblack and dotted blue show the naive extrapolations of these bounds to negative α and ˜ α . Red and green dots are numerical evaluations of α and ˜ α as defined insections 4.3 and 4.2, respectively, with N = 20. Solid red and green lines showthe obvious piecewise linear trends which approximately match the numericalevaluations. Open circles denote evaluations in which we restricted L ≤ h < L ; otherwise we use half the available points as explained in the main text. For s ≤ − 2, convergence of the sparse model to the nearest neighbor model impliesthat α = ˜ α = 1, but our numerical scheme for picking out α and ˜ α becomesless reliable in this region due to difficulty normalizing G sparse and G power in amutually consistent way. Likewise, we define A power ( N, s ) ≡ log max L ≤ h< L (cid:12)(cid:12)(cid:12)(cid:12) G sparse h G power h − G sparse h +1 G power h +1 (cid:12)(cid:12)(cid:12)(cid:12) ∞ α power ( N, s ) ≡ − A power ( N, s ) + A power ( N − , s ) . (75)The large N limit of α power ( N, s ) (assuming it exists) is our numerical estimate of the bestpossible local H¨older exponent for G sparse h in an Archimedean setting.We find good evidence that α − adic ( N, s ) and α power ( N, s ) have finite large N limits.Our numerical results are well described by piecewise linear dependence of α on s , and inparticular by α power = − s + 1 / 2) for − < s < α − adic = 2( s − / 2) for 0 < s < . (76)See figure 5. Two caveats on our numerical results can be summarized as follows: • When | s | > 1, it becomes harder to get good numerical results, particularly on theArchimedean side, because the functions under consideration are quite smooth, and we25ave to compute very small differences accurately. Even apart from issues of numericalaccuracy, it becomes challenging on the Archimedean side to distinguish between rapidbut smooth variation and the slightly non-smooth behavior that determines the H¨olderexponent. • Numerical evaluations of H¨older exponents diverge a bit from expectations at s = 0,and also at s = − 1. This is not too surprising, given that our estimates of the prefactors K in the H¨older inequalities show divergences at these values of s : See for example (45),(59), and (64). Possibly at these special values we need logarithmic corrections to therelevant H¨older condition. It is also possible that simple piecewise linear functions onlyapproximately fit the dependence of α on s . More extensive and accurate numericalinvestigations are needed in order to establish fully reliable results. The most interesting regime in position space is − < s < 1, where we are losing Archimedeancontinuity and gaining 2-adic continuity. We focus in this section entirely on this regime,and we present the simplest account of the transition from Archimedean to ultrametric con-tinuity which is consistent with our numerics. Due to finite numerical resolution, we cannotrigorously determine the measure-theoretic behavior of the position space Green’s functionsin regions where the Green’s functions are very ragged. We attempt to qualify our claimsbelow with the appropriate level of confidence.In momentum space, our numerics are consistent with there being a single exponent onthe 2-adic side, ˜ α − adic = s , which describes both the global H¨older continuity conditionover all k and the local continuity at each possible value of k . In other words, as far as wecan tell, the function ˜ G ( k ) is equally ragged everywhere. A similarly uniform story applieson the Archimedean side, with ˜ α power = − s . Numerical results are fully in accord withexpectations from field theory, where we were able to compute ˜ α − adic and ˜ α power analytically.The upshot is that the transition from Archimedean to ultrametric continuity happens rathersimply, with ordinary continuity failing just as 2-adic continuity emerges: i.e. ˜ α power becomesnegative just as ˜ α − adic becomes positive, at s = 0.The field theory estimates of the H¨older exponents for the position-space Green’s functionwere s − − s − − < s < 1) were based entirely on the average scaling of ˜ G ( k ) as a powerof | k | far from k = 0. As such, they tell us the global H¨older exponent, which we believecharacterizes the behavior of G ( x ) close to x = 0: That is, G ( x ) ≈ | x | s − on the 2-adicside, while G ( x ) ≈ | x | − s − ∞ on the Archimedean side. The surprise we get from numerics is26 = ( h )/ L - - G h sparse / G h - adic s = ( κ )/ L G ˜ κ sparse / G ˜ κ - adic s = - ( h )/ L - - - G h sparse / G h - adic s = - ( κ )/ L G ˜ κ sparse / G ˜ κ - adic Figure 6: Plots of G sparse h /G − adic h and ˜ G sparse κ / ˜ G − adic κ over the Monna map of the oddintegers. As s becomes more positive, the numerical data is closer to a 2-adically continuous curve when N is large. Blue points are for N = 6, whilethe red curves are for N = 10. that away from x = 0, a more complicated dependence of H¨older smoothness on s emerges,with local H¨older exponents α somewhat more positive than the field theory bounds: Thatis, G ( x ) seems to be somewhat smoother away from the origin than its behavior right near x = 0. Our numerical results are consistent with there being a piecewise linear dependenceof α on s , as summarized in particular by (76). These results (76) indicate that Archimedeancontinuity of G sparse h is lost at s = − / 2, but 2-adic continuity doesn’t emerge until s = 1 / − / < s < / 2, when both α power and α − adic arenegative?To better understand the region of transition between the Archimedean and 2-adicsmoothness, it is instructive to inspect overlaid plots of the Green’s function for differentsystem sizes, see figures 6 and 7.Based on these figures and related studies, the scenario we regard as most likely is thatfor − / < s < 0, the continuum limit of G power h defines an absolutely continuous measure, G ( x ) dx , with respect to ordinary Lebesgue measure dx , but for s > = h / L - - - G h sparse / G h power s = κ / L G ˜ κ sparse / G ˜ κ power s = - h / L - - - - G h sparse / G h power s = - κ / L G ˜ κ sparse / G ˜ κ power Figure 7: Plots of G sparse h /G power h and ˜ G sparse κ / ˜ G power κ over the middle half of points. As s becomes more negative, the numerical data is closer to a continuous curve when N is large. Blue points are for N = 6, while the red curves are for N = 10. 28e suggest that for 0 < s < / 2, the continuum limit of G − adic h defines an absolutelycontinuous measure with respect to the standard Haar measure on Q while for s < Q ) inits Radon-Nikodym decomposition. We find support for the claim of absolutely continuousmeasures in the above-mentioned regimes when we study the scaling of the height of thespikes in figures 6 and 7 as a function of N : the weight of each spike (meaning the integralover a small region including the spike) distinctly appears to tend to zero with increasing N .When singular terms in Radon-Nikodym decompositions do exist, we conjecture that theyhave as their support sets which are dense in position space.One way in which singular terms in Radon-Nikodym decompositions could arise is forthe continuum limit G ( x ) to include delta functions. Inspection of figure 6 is consistentwith there being a dense set of delta function spikes in G ( x ) as a function of 2-adic x when s = − . 3, but none when s = 0 . 3. Similarly, figure 7 is consistent with there being a denseset of delta function spikes in G ( x ) as a function of real x when s = 0 . 3, but none with s = − . 3. The discerning reader may note, however, that the spikes on the Archimedeanside are stronger at s = 0 . s = − . 3. This asymmetrymanifests itself in the scaling of the height of these spikes with N , for the weight of eachspike grows with N on the Archimedean side for s = 0 . 3, but may be trending very slowlytoward 0 on the 2-adic side at s = − . 3. A related effect appears in figure 5: α − adic ≈ − s < 0, while α power ≈ − − s for s > For decades, p -adic numbers have been considered as an alternative to real numbers as anotion of continuum which could underlie fundamental physics at a microscopic scale; seefor example [9]. The current study shows how the large system size limit of an underlyingdiscrete system naturally interpolates between a one-dimensional Archimedean continuumand a 2-adic continuum as we vary a spectral exponent. By focusing a free field example,we are able to solve the model through essentially trivial Fourier space manipulations. Thecorrelators of the theories we study are all determined in terms of the two-point functionthrough application of Wick’s theorem. The two-point function is smooth in an Archimedeansense when s is sufficiently negative, and in a 2-adic sense when s is sufficiently positive. The29ransition from these two incompatible notions of continuity can be precisely characterizedin terms of H¨older exponents characterizing the smoothness of the two-point function. Wehave found the dependence of these exponents on s through a combination of analytical fieldtheory arguments and numerics on finite but large systems.Quite a wide range of generalizations of our basic construction can be contemplated:1. We can generalize to primes p > 2. One significant subtlety arises when doing so,namely the structure within Z /p Z of sparse couplings. The simplest alternative is forspin 0 to couple to spins ± θp n with a strength p ns , where θ runs over all elementsof { , , , . . . , p − } . This coupling pattern is featureless within Z /p Z because ittreats all values of θ the same. We could however contemplate other possibilities. Forexample, if p = 5, an interesting alternative is to introduce couplings only for θ = 1and θ = 4 (the quadratic residues). More generally, one could expand the dependenceof couplings on θ in a sum of multiplicative characters over Z /p Z .2. We focused entirely on bosonic spins φ i , but there is no reason not to consider fermions c i instead. Then the coupling matrix J ij would have to be anti-symmetric, and like-wise the two-point Green’s function would be odd. Within this framework one couldconsider a variety of sparse coupling patterns.3. Higher-dimensional examples are not hard to come by. Consider real bosonic spins φ (cid:126)ı labeled by a two-dimensional vector (cid:126)ı = i ˆ1 + i ˆ2, where i and i are in Z / N Z .Suppose we establish a coupling matrix J (cid:126)ı(cid:126) = J (cid:126)ı − (cid:126) where J (cid:126)h = min { n ,n } s if h = ± n and h = ± n n s if h = 0 and h = ± n n s if h = 0 and h = ± n , (77)with all other entries vanishing except J , whose value we choose in order to havethe Fourier coefficient ˜ J (cid:126)κ vanish when (cid:126)κ = 0. Then for sufficiently negative s wehave effectively a nearest neighbor model which approximates the massless field theory S = (cid:82) d x ( ∇ φ ) . For s sufficiently positive, one obtains instead a continuum theoryover Z × Z , which can be understood as the ring of integers in the unramified quadraticextension of Q .All the examples above remain within the paradigm of free field theory. Still easy to for-mulate, but obviously much harder to solve, are interacting theories with sparse couplings.For example, we could start with any of the models introduced in section 2 and add a30 = 0 1 12 23 − − − φ r e l e v a n t φ r e l e v a n t φ r e l e v a n t φ r e l e v a n t No relevant deformations ULTRAMETRICARCHIMEDEANGaussian Gaussian s Figure 8: Conjectured pattern of fixed points of the renormalization group for interactingfield theories of a single bosonic scalar field with φ → − φ symmetry. term (cid:80) i V ( φ i ) to the Hamiltonian describing arbitrary on-site interactions. To get somefirst hints of what to expect these interactions to do, recall in 2-adic field theory that G ( x ) ≈ | x | s − at small x . Comparing this to the standard expectation G ( x ) ≈ | x | φ ,we arrive at ∆ φ = (1 − s ) / φ . When describing perturba-tions of the Gaussian theory, we can use normal UV power counting: [ φ n ] = n ∆ φ . Thus φ n isrelevant when s > − /n . If we impose Z symmetry, φ → − φ , then in the region s < / s increases from 1 / φ and then higher powers of φ become relevant. It is reasonable to expect some analogof Wilson-Fisher fixed points to appear. Possibly as s → s in the range ( − , G ( x ) ≈ | x | − s − ∞ andtherefore ∆ φ = (1 + s ) / 2. See figure 8.The sparse coupling theories are sufficiently similar to 2-adic field theories for s > s < − / < s < / φ higher than φ are relevant—according at least tonaive power counting as presented here. Acknowledgments We thank S. Hartnoll for getting us started on this project by putting us in touch withM. Schleier-Smith’s group, and we particularly thank G. Bentsen and M. Schleier-Smith for31xtensive discussions. This work was supported in part by the Department of Energy underGrant No. DE-FG02-91ER40671, and by the Simons Foundation, Grant 511167 (SSG). Thework of C. Jepsen was supported in part by the National Science Foundation under GrantNo. PHY-1620059. 32 eferences [1] F. Q. Gouvˆea, p-adic Numbers . Springer, 1997.[2] F. J. Dyson, “Existence of a phase transition in a one-dimensional Ising ferromagnet,” Commun. Math. Phys. (1969) 91–107.[3] P. M. Bleher and J. G. Sinai, “Investigation of the critical point in models of the typeof Dyson’s hierarchical models,” Comm. Math. Phys. (1973), no. 1 23–42.[4] E. Yu. Lerner and M. D. Missarov, “Scalar Models of p -adic Quantum Field Theoryand Hierarchical Models,” Theor. Math. Phys. (1989) 177–184.[5] M. Missarov, “ p -Adic Renormalization Group Solutions and the EuclideanRenormalization Group Conjectures,” P -Adic Numbers, Ultrametric Analysis, andApplications (2012), no. 2 109–114.[6] G. Bentsen, E. Davis, and M. 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