Topological Dimensions from Disorder and Quantum Mechanics?

We have recently shown that the critical Anderson electron in D=3 dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension dIR≈8/3. Here, we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the points comprising a region are of similar relevance, and calculate the IR scaling dimension d of each. This allows us to infer the probability density p(d) for dimension d to be accessed by the electron. We find that p(d) has a strong peak at d very close to two. In fact, our data suggest that p(d) is non-zero on the interval [dmin,dmax]≈[4/3,8/3] and may develop a discrete part (δ-function) at d=2 in the infinite-volume limit. The latter invokes the possibility that a combination of quantum mechanics and pure disorder can lead to the emergence of integer (topological) dimensions. Although dIR is based on effective counting, of which p(d) has no a priori knowledge, dIR≥dmax is an exact feature of the ensuing formalism. A possible connection of our results to the recent findings of dIR≈2 in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.


Introduction.
Understanding spatial geometry of Anderson transitions [1] is an intriguing problem.Indeed, although studied quite extensively, the complicated structure of critical electronic states (see e.g.[2]) leaves room for new insights.Novel characterization may reveal unknown details of disorder-driven metal-insulator transitions and, for example, lead to deeper understanding of their renormalization group description [3].
Another reason to study the geometry of Anderson transitions arises by seeing them as quantum dimension transitions, a viewpoint taken in Ref. [4].Using effective number theory (ENT) [5,6], which entails a unique measure-based dimension d IR [7,8] for spaces with probabilities, it showed that the transition is a two-step dimension reduction Here the flow is from extended to critical to localized state, and exponential localization was assumed.Remarkable property of the above is that these reductions are complete [9].Indeed, probability doesn't leak away from subdimensional effective supports, and electron is fully confined to them in infinite volume.It is thus meaningful to say that the space available to quantum particle collapses into lower dimensional one under the influence of strong enough disorder.As such, it represents a mechanism for generating lower-dimensional spaces by simple combination of quantum mechanics and disorder.While dimension is the most basic characteristic of space available to critical electron, this space may contain subsets with dimensions d < d IR .Such substructure may be physically significant if electron mostly resides there.The aim of this work is to characterize the critical spatial geometry in such manner: we will compute the probability distribution p(d) that electron is present in space of dimension d.We refer to p(d) as dimension content of Anderson criticality or that of probability distribution in general.
Critical states at Anderson transitions were recognized to have fractal-like features long ago, first interpreting them in analogy to scale-invariant fractals [10,11] and later to more complex multifractals [12][13][14][15].Formalism used in the latter mimics one that describes ultraviolet (UV) measure singularities occurring in turbulence and strange attractors (see e.g.[16,17]).More recent works in the Anderson context are [18][19][20][21][22].However, the focus of multifractal analysis doesn't make it convenient for computing p(d).We thus proceed by proposing a method that organizes the calculation in terms of probabilities from the outset and zooms in on dimensions by degree of their actual presence.Moreover, d involved is simply the IR Minkowski dimension of a subset, and thus manifestly a measure-based dimension of space.In the ensuing multidimensionality formalism, wave function is where d max = sup {d | p(d) > 0}, D = 3 is the IR dimension of the underlying space, and d IR ≥ d max holds in general.
Before proceeding to define p(d), we illustrate the idea on a "shovel" in R D=3 space (Fig. 1).The shovel consists of 2d square blade and 1d handle with uniformly distributed masses M b > 0 and M h > 0 respectively.If the relevance of space points is set by mass they carry, the probability of encountering the handle, the blade and the rest of space is P = M h /(M b +M h ), 1−P and 0 respectively.Note that UV cutoff a and IR cutoff L are also indicated.
Above we implicitly assumed that d is the usual UV dimension (a → 0 at fixed L) in which case we have by inspection p(d) = P δ(d − 1) + (1 − P) δ(d − 2).But how would this p(d) be concluded by a computer that cannot "see" and only processes regularized probability vectors P (a) = (p 1 , p 2 , . . ., p N (a) )?Here N (a) = (L/a) 3 , p i is the probability within elementary cube at point x i of latticized space, and a ∈ {L/k | k = 2, 3, . ..}.
Anticipating that any number J of discrete dimensions 0 ≤ d 1 < d 2 < . . .< d J ≤ 3 with probabilities P j > 0 could be present, computer first orders p i in each P (a) so that p 1 ≥ p 2 ≥ . . .≥ p N (a) .The rationale is that, with decreasing a, this increasingly better separates out populations related to different d j .Indeed, the typical size of p associated with d j is ∝ a dj and so P (a) gradually organizes into J sequential blocks starting with d 1 .The above ordering in P will always be assumed from now on.
To detect possible blocks/dimensions, computer uses variable q ∈ [0, 1] for cumulative probability, and associates with each P (a) function ν(q, a), namely the number of first elements in P (a) (space points) whose probabilities add to q. Keeping track of fractional boundary contributions at each q makes it a continuous, convex, increasing, piecewise linear function such that ν(0, a) = 0 and ν(1, a) = N (a).Number of points in interval (q − , q] is ν(q, a)−ν(q − , a) and scales as a −d(q, ) for a → 0. When processing P (a) for the shovel, computer finds perfect scaling ( h /a) × /P for ≤ q ≤ P, and ( b /a) 2 × /(1 − P) for P + < q < 1.It will thus conclude d(q) shown in Fig. 1 upon → 0. Value at q = 1 represents the spatial complement of the shovel (zero probability).Collecting the probability of d, namely p(d) = 1 0 dq δ(d − d(q)) produces the inspected result.Two points are relevant here.(1) The above approach doesn't change if continuous set of dimensions is present.In that case the obtained d(q) is not piecewise constant, but rather a piecewise continuous non-decreasing function, possibly with constant parts identifying discrete dimensions.(2) IR case is fully analogous, but it is useful to recall the meaning of IR dimension (L → ∞, a fixed) which is somewhat non-standard.Thus, if both h and b are fixed as L → ∞ (usual case) then p(d) = δ(d) since populations at each q remain constant.However, if e.g.b is fixed while handle responds by h ∝ L (shovel reaches anywhere in space) then p(d) = (1 − P)δ(d) + Pδ(d − 1).
For N and ν of 1-particle eigenstates ψ at energy E we get Keeping the dependence on E, W implicit, L → ∞ behavior defines dimensional characteristics d IR and d(q) via with d(q) = lim →0 d(q, ).Due to convexity of cumulative counts, d(q, ) and d(q) are non-decreasing.Probability density of finding IR dimension d in a state is then The range of d(q), equal to support of p(d), specifies IR dimensions occurring with non-zero probability in states of interest.It is a subset of [d min , d max ] where Important feature in the ensuing formalism is that Here the inequalities involving D are obvious and the last one can be most easily seen in discrete case.Indeed, let and assume that d IR < d J = d max .Consider q such that 1−P J < q < 1.Then ν(q, L)−ν(q − , L) = v(q, , L)L d(q, ) for sufficiently small , where lim →0 d(q, ) = d J and lim →0 lim L→∞ v(q, , L) = v(q) > 0. The size of individual p = /(ν(q, L) − ν(q − , L)) in this population is then L −d(q, ) /v(q, L, ).Hence, if d J < D then min{1, N p} in definition of N yields 1 for sufficient L and , while if d J = D it yields 1/v(q).In both cases, the contribution of this population to N is ∝ L d J .Hence, d IR ≥ d J which contradicts the assumption and leads to (10).
Given that, we show d(q, L) at L = 40 and L = 144 in Fig. 2. Important feature of the obtained behavior is the flatness in the middle part of q, indicating large probabilities for dimensions in the corresponding range.Increase of L results in flatter d(q, L) and yet sharper range of prominent dimensions.Visible linear parts at small q mark regions where finite-size effects yield ν(q) non-convex.Their extent shrinks toward zero with growing L. Linearity was imposed to keep the behavior regular.
The corresponding p(d, L) obtained via (8) are shown in Fig. 3.We observe sharp peaks of decreasing width, centered at d m ≈ 2. The error bars, too small to be visible, were obtained via Jackknife procedure with respect to disorder samples.Stability of d m and its proximity to 2 is quite remarkable as shown in the inset for the largest sizes studied.Quoted values were obtained from quadratic fits in the displayed vicinity of the maximum.The constant parts at small d correspond to linear segments in Fig. 2.
Among key chracteristics of dimension content p(d) is its support, i.e. dimensions that can contribute to physical processes with non-zero probability density.Given the strong dominance of d m , the second key question is whether d m could be a discrete dimension in Anderson critical states.This would mean that, in L → ∞ limit, d(q, L) (see Fig. 2) develops a strictly constant part in certain range of q.We will test this possibility for the observed d m = 2 via the following procedure.Given a d(q, L), we find q 2 (L) such that d(q 2 , L) = 2 and calculate which is only zero if d(q, L) = 2 on the interval.For given ρ, we perform L → ∞ extrapolation via fit to a constant I(ρ) with general power correction.Fitting data for systems with L > 28 leads to results shown in Fig. 5  I ≈ 0 at ρ ≈ 0.4 with errors becoming large below this point.While this could simply indicate a very steep analytic behavior of I(ρ), further analysis suggests otherwise.Indeed, restricting fits to larger systems, namely L > 32 (diamonds) and L > 40 (triangles), results in increasingly steeper decay toward zero at yet larger ρ.Natural interpretation of these tendencies is that I(ρ) ≡ 0 for ρ < ρ 0 ≈ 0.5, pointing to discrete nature of d m .
The synthesis of our results suggests the following form of spatial dimension content at Anderson criticality where π(d) is a continuous probability distribution with support on interval [d min , d max ].The parameters are where we estimate the accuracy of d m at couple ‰ and that of d min , d max at couple %.Graphical representation of this result in terms of d(q) and p(d) is shown in Fig. 6. 4. Discussion.We proposed to characterize probability distributions on metric spaces by their measure-based effective dimension (d UV or d IR ) [5][6][7][8] and the associated dimension content p(d).The method was applied to the structure of critical states in D = 3 Anderson transition (O class).Here p(d) identifies dimensions of regions where electron can in fact be found, i.e. those relevant to its physics.Critical wave functions are subdimensional, multidimensional and our new results are summarized by Eqs. ( 13) and (14).Few comments should be made.
(i) The picture of Anderson transition as spatial dimension transformation (1) receives key refinements by virtue of p(d).Indeed, although critical electron is fully confined to spatial effective support S of Minkowski dimension d IR ≈ 8/3 [4,9], its key substructure has d m ≈ 2, and the continuum of lower and higher-dimensional features is also present.Geometrically, S may thus also be viewed as surface-like structure endowed with complex lowerdimensional "hair" and higher-dimensional "halo".(ii) Our results suggest that d m is a discrete dimension and that it may assume an exactly topological value d m = 2.
[Mathematical meaning of "topological" in the context of IR dimension would of course need some clarification.]This invokes a possibility that quantum mechanics combined with pure disorder can lead to emergence of integer dimensions.Apart from understanding of Anderson transitions, variations on such dynamics could find relevance in modeling emergent space in early universe.More detailed description of this geometry would be needed.(iii) Connection between d IR and p(d) results from builtin additivity which makes them measure-based: in case of d IR it is additivity of effective counting with respect to combining the systems [5,6], and for d(q) the familiar additivity of ordinary counting.This aspect is key to interpretation of these concepts as spatial dimensions.Indeed, it is because Hausdorff measure and Minkowski count properly quantify volume that dimensions based on them became useful and accepted characteristics of space.(iv) It is natural to ask whether some features of the described spatial structure have analogues in the multi- fractal approach [16,17] adopted to IR Anderson setting via moment method [25].Here the focus is on the so called dimensional spectrum f (α).Inner workings of the method give special status to information dimension [26] in a way somewhat similar to d m .It would be interesting to study the possible association between the two in detail.(See also debate regarding d IR in Refs.[27,28].)(v) Our data is consistent with critical wave functions being of proper dimension (d IR = d max ).However, albeit state of the art, their statistical power is not sufficient to reach sharper conclusion at this point.
(vi) Our findings acquire another angle in light of recent results [7,29] in quantum chromodynamics (QCD).The original proposal that Anderson-like mobility edge λ A > 0 appears in QCD Dirac spectrum upon thermal chiral transition [30,31], worked out by Refs.[32][33][34], became more structured.Indeed, existence of a new mobility edge λ IR ≡ 0 has been concluded and its simultaneous appearance with λ A at temperature T IR was conjectured [29].
Here T IR marks the transition to phase featuring IR scale invariance of glue fields [35].Approach to IR criticality (λ → λ

FIG. 1 .
FIG.1.The "shovel" (left) and d(q) (right) associated with its UV dimension content in R 3 .See discussion in the text.
The above properties of p(d, L) imply that the support in fact spans [d min , d max ], and its specification thus reduces to finding d min and d max .To that effect, we evaluate probabilities p(d < d 0 , L) of dimensions smaller than d 0 , and vary d 0 upward.For each d 0 , p(d < d 0 , L) is L → ∞ extrapolated by fitting to a constant with general power correction.The result, shown in Fig. 4 panel (a), features a probability threshold turning on near d 0 = 1.3.We take d 0 = 4/3 as a reference value: in panel (c) we show its extrapolation leading to a clean statistical zero.Analogous procedure based on p(d > d 0 ) yields results shown in panels (b),(d) with d 0 = 8/3 referencing the other threshold.
+ IR ) was found to proceed via d IR ≈ 2 Dirac modes[7], with topological origin of the dimension suspected.Clarifying a possible relation of this to d m ≈ 2 found here may shed new light on QCD-Anderson localization connection.(vii) The proposed IR/UV guises of multidimensionality formalism easily extend to more general situations without metric.Here the sequence {O k } involving collections O k = (o k,1 , o k,2 , . . ., o k,N k ) with increasing number N k of arbitrary objects comes with associated sequence {P k } of relevance (probability) vectors.The role of d IR /d UV is taken by the effective counting dimension 0 ≤ ∆ ≤ 1 defined via scaling N[P k ] ∝ N ∆ k for k → ∞ [8].Dimension function d(q) is replaced by analogous γ(q) and dimension content p(d) by p(γ).The target (k → ∞) effective collection defined by{O k }, {P k } is then subdimensional if ∆ < 1 multidimensional if p(γ) = δ(γ − γ max ) of proper dimension if ∆ = γ max(15)where γ max = sup {γ | p(γ) > 0} and γ max ≤ ∆.P.M. was supported by Slovak Grant Agency VEGA, Project n. 1/0101/20.