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Classification of critical phenomena in hierarchical small-world networks

A classification of critical behavior is provided in systems for which the renormalization group equations are control-parameter dependent. It describes phase transitions in networks with a recursive, hierarchical structure but appears to apply also to a wider class of systems, such as conformal field theories. Although these transitions generally do not exhibit universality, three distinct regimes of characteristic critical behavior can be discerned that combine an unusual mixture of finite- and infinite-order transitions. In the spirit of Landau's description of a phase transition, the problem can be reduced to the local analysis of a cubic recursion equation, here, for the renormalization group flow of some generalized coupling. Among other insights, this theory explains the often-noted prevalence of the so-called inverted Berezinskii-Kosterlitz-Thouless transitions in complex networks. As a demonstration, a one-parameter family of Ising models on hierarchical networks is considered.

The renormalization group (RG) [1,2] is by now a method found in any "classical" statistical physics text book [3][4][5]. It has allowed to categorize broad classes of equilibrium systems into an enumerable set of universality classes, each characterized by discrete features, such as their dimension and the symmetries adhered to by their Hamiltonians. Such universality is made possible through the property of scaling that is an inherent feature near phase transitions [6] exhibited by these systems in regions of the space spanned by their physical couplings. Scaling entails that system-specific details on the microscopic level become irrelevant, as the behavior over many orders in the range of the interactions become self-similar. In this framework, analogous behavior in a surprisingly wide set of phenomena, such as the condensation of fluids, spontaneous magnetization of materials, or the generation of particle masses in the early universe, can be described with a few effective theories -a major intellectual accomplishment of modern physics [3].
Unlike the Euclidean arrangement of atoms in a lattice, agents in biological or social systems may exhibit complex networks of mutual interactions [7][8][9]. The study of critical phenomena is inseparable from the understanding of the geometry of the network [10]. It has been realized that many of the networks that are engineered by some natural or human activity themselves exhibit emergent complex properties, for instance, as found in the scale-free degree distribution of the internet. While these networks, and dynamical systems on them, may behave critically, those phenomena were soon found to be non-universal [11], i.e., they are intimately tied to intricate details of the specific system. In this sense, it would seem unlikely that a sweeping classification could be devised. In this Letter, we will categorize equilibrium phenomena observed on a large set of networks having hierarchical structure. Our discussion explains, for example, the robustness of infinite-order transitions in a range of quite distinct network models summarized in Ref. [9], or in conformal field theory, where it signals the loss of conformality [12]. Our study shows that criticality in these models is generally non-universal but falls into three generic regimes, here characterized by their divergence in the correlation length [13]. One of these regimes is an infinite-order transition reminiscent of that described by Berezinskii and Kosterlitz and Thouless (BKT) [4]. It is flanked on one side by a transition with a weaker, algebraic divergence, similar to a second order transition (albeit non-universal), and on the other by a regime with an even stronger essential singularity.
Our classification immediately applies to hierarchical networks on which RG is exact and transitions can be studied in great detail. There, these regimes are characterized by the relative strength of long-range, smallworld bonds [7]. Metric version of such networks, like the Migdal-Kadanoff RG, provide classical text-book examples of RG and universality [4,14,15]. But in the advent of complex networks, many hierarchical designs with non-metric (i.e., small-world or scale-free) properties have been devised and studied [15][16][17][18][19][20][21].
The central tenant of real-space RG consists of a procedure whereby the partition function of the original system is mapped recursively onto itself after tracing out a fraction 1 − 1/b of the dynamic variables, in some form of "blocking". Prior couplings κ n between those variables combine non-trivially to produce new, effective couplings κ n+1 between the remaining variables after the n-th RGstep while leaving the Hamiltonian form-invariant. This mapping constitutes the RG-flow where R indicates a (typically non-linear) set of recursions. In the thermodynamic limit, n ∼ log b N → ∞, phase transitions are characterized purely by a local analysis for κ n ∼ κ n+1 ∼ κ ∞ near the fixed points (FP) of independent of κ 0 . Here, κ 0 represents the "raw" couplings of the original system that carry the dependence on the system's control parameter µ ∈ [0, 1], referring to a temperature, κ 0 (µ) = µ = e −βJ in units of J = 1, or a bond-percolation probability, p 0 = p = 1 − µ, say.
Since κ 0 expresses microscopic details of a potentially large family of conceivable systems adhering to Eq. (1), the insensitivity of κ ∞ on κ 0 is an expression of universality: only certain symmetry properties of the original systems remain preserved by R. Therefore, the linearized expansion, κ n ∼ κ ∞ + ǫ n with small ǫ n for large n, near the FP provides a full accounting of the macroscopically observable properties of any such system via the eigenvalue problem obtained from The eigenvalues λ of the Jacobian ∂R ∂ κ ( κ ∞ ) and their eigenvectors u λ provide the scaling exponents and scaling fields observed in the phase transition.
A much richer phenomenology arises when the RG-flow R itself becomes dependent on the control-parameter. In that case, Eq. (1) generalizes to κ n+1 = R ( κ n ; µ) with i.e., the FP κ ∞ becomes a nontrivial function of µ. The consequences of such behavior (for a single coupling) are depicted in Figs. 1. First, consider the case of constant FP shown in Fig. 1(a). Drawing constant FP as function of the control-parameter may seem redundant, however, it allows to illustrate the connection to the raw couplings κ 0 (µ) (green-dashed line), which must rise smoothly with µ from κ 0 (0) = 0 to κ 0 (1) = 1 to be physical, but are otherwise arbitrary (here, chosen as a straight line). Below (above) the point where κ 0 (µ) intersects the unstable FP, the RG-flow evolves toward the stable FP at κ ∞ = 0 (κ ∞ = 1). The arbitrariness in κ 0 (µ) reflects the universality in the family of systems obeying the same FP: no matter at which value of µ c the specific system κ 0 (µ) intersects the unstable FP with 0 < κ ∞ < 1, its µindependence guarantees identical critical behavior. We claim that the remaining three panels in Fig. 1 capture all generic features that can arise for µ-dependent FP. Variable FP κ ∞ (µ) can collide, either linearly or in a square-root branch point (BP); any other behavior (such as a higher-order BP) is exceptional. We can devise a simple theory [26] that reproduces these generic features, thereby demonstrating the generality of this classification, not only accounting for hierarchical networks but for any physical system described by parameter-dependent renormalization group equations [12,13,[15][16][17][18][19][20][21].
It is sufficient to consider the RG recursion for a single coupling κ n with some control parameter µ. We argue that FP in a real, µ-dependent RG-flow R in Eq. (4) will exhibit BP at some point(s) (µ B , κ B ). Near µ B we express generically R (κ; µ) ∼ a (µ) κ + b (µ) κ 2 + c (µ) κ 3 because the need for a strong-coupling solution κ 0 ∞ = 0 prevents a constant term and requires at least a cubic form to achieve BP. With generically analytic coefficients at µ B , we expect a (µ) . In (a), µ-independence of FP ensures universal critical behavior wherever the raw couplings κ0 (µ) (green-dashed line) intersects the unstable FP (red line); a blue dot and dashed line mark the critical point µc, and blue arrows indicate the RG-flow from κ0 (µ) toward the nearest stable FP (black lines). In (b), κ0 (µ) still intersects at an unstable FP below BP, such that the RG-flow does not pass BP, leading to quasi-conventional behavior but with µc-dependent exponents. In (c), κ0 (µ) is located above BP so that the RG-flow must pass near BP at µc, leading to BKTlike behavior. In (d), BP drops below the physical domain and only stable FP are accessible, resulting in an exponential divergence at the intersection µc of two FP-branches. c (µ) ∼ c 0 for µ → µ B . Locating BP at (µ B , κ B ) fixes a 0 = 1 + c 0 κ 2 B and b 0 = −2c 0 κ B . To orient BP correctly requires a 1 /c 0 < 0, and we set a 1 = −c 0 A 2 with A > 0. Finally, stability of the strong-coupling FP at κ 0 ∞ demands c 0 < 0, and we may set c 0 = −1. This yields the minimal model After extracting κ 0 ∞ ≡ 0 [27], the remaining FP equation indeed produces a BP at (µ B , κ B ): Local expansion near each FP as in Eq.
For the case of panel (c) where κ 0 (µ) passes above BP (e.g., for somewhat stronger long-range couplings [13]), the RG-flow must pass BP which now dominates criticality, i.e., µ c = µ B , with an infinite-order divergence characterizing this regime. Well below (above) µ B , the RG-flow evolves unperturbed to κ 0 ∞ (to κ + ∞ ), the closest stable FP. However, just below µ B the RG-flow gets ever more impeded near BP before it can reach κ 0 ∞ . Asymp-totically for µ ր µ B near κ n ∼ κ B + ǫ n with small ǫ n at large but intermediate n, Eq. (5) provides This relation exhibits a boundary layer, i.e., in the limit µ ր µ B the solution drastically changes behavior. With the methods of Ref. [22], we rescale ǫ n → γǫ n and n → δn to obtain a balance for δ ∼ 1/γ ∼ 1/ √ µ B − µ. Accordingly, the characteristic width of the boundary layer scales with n * ∼ 1/ √ µ B − µ, which leads to the divergence in the correlation length characteristic of BKT, Clearly, the physical origin of this singularity is not related to an actual BKT transition, with its formation of delicate topological structures [4]. In fact, instead of being rare, it is one of three generic types of transition often found in hierarchical networks [9]. The most unconventional behavior is depicted in panel (d) of Fig. 1, where κ B < 0 (e.g., when long-range couplings dominate [13]). No unstable FP can be reached for any physical choice of κ 0 (µ). The RG-flow always advances to the closest stable FP, either at strong coupling, κ 0 ∞ for 0 ≤ µ < µ c , or at patchy order, κ + ∞ (µ) for µ c < µ ≤ 1. Both lines of FP cross at µ c (> µ B ), defined by κ + ∞ (µ c ) = κ 0 ∞ ≡ 0. This condition implies that both their eigenvalues are equal and marginal, λ 0 (µ c ) = λ + (µ c ) = 1, as κ 0 ∞ must invert its stability at the intersection, making marginal stability inherent to any such system. In our model, κ + ∞ (µ c ) = 0 in Eq. (6) provides −κ B = √ µ c − µ B , hence, Eqs. (7) give λ 0,+ ∼ 1−A 2 |µ − µ c | for µ → µ c , see Fig. 2(a). Near κ 0,+ ∞ (µ c ) = 0, the local analysis on Eq. (5) according to Eq. (3) gives ǫ n+1 ∼ λ 0,+ ǫ n or ǫ n ∼ ǫ 0 exp −nA 2 |µ − µ c | with a cross-over at n * ∼ 1/ |µ − µ c | that is generic when λ 0 and λ + intersect linearly. Thus, the correlation length diverges as Again, the RG flow does not pass BP, since it is located below the physical domain. We can extend the discussion to include the effect of further control parameters, such as an external field η 0 = η = e −βh . We find that its generic RG-flow can be expressed asymptotically as with some µ c -dependent constant C > 0, near the critical point µ c and for sufficiently small κ + ∞ (µ c ). (E.g., in Supplement B we find C = 2µ c for the Ising model on HN5.) This satisfies the physical requirements on its FP, η ∞ = 0, 1; only for h = 0 the RG-flow remains at the unstable FP, i.e., η n ≡ 1 f. a. n, and for any h > 0 the stable strong-coupling FP at η ∞ = 0 is reached eventually. The eigenvalue near the unstable FP satisfies λ h ≤ 2 such that y h = log 2 λ h ≤ 1, as shown in Ref. [23]. There, a scaling theory is developed concerning the BKT regime based on the exponentially divergent correlation length in Eq. (10), leading to an order-parameter (magnetization, fraction of sites on percolating cluster, etc.) when y t → 0. Of course, for y t > 0 it is [5] m for small-world systems where N takes the role of L d [23]. Our theory in Eq. (5) not only explains the robustness of the value of x t = 1 2 conjectured in Ref. [23], but also broadens the scope to a total of three generic regimes in the divergence of ξ, as we have explained. With the addition of Eq. (12), we can account for the behavior of the order-parameter m. For the first two regimes where κ B > 0, it is λ h < 2 in Eq. (12) so that 1 − y h > 0. In the regime with the weakest distortion of the FP, see Fig. 1(b), we have shown in Eq. (8) that y t = 1/ν (µ c ) > 0, i.e., it is 0 < β (µ c ) < ∞ similar to an ordinary 2nd -order transition, except for its non-universal µ c -dependence. In the BKT-regime, see Fig. 1(c), it is µ c = µ B and λ + (µ c ) = 1 in Eq. (7) such that y t = 0 and β → ∞, which leads to Eq. (13) described in Ref. [23]. Finally, when κ B < 0, see Fig. 1(d), it is µ c > µ B and κ + ∞ (µ c ) = 0. And although λ + (µ c ) = 1 leads to the divergent correlation length in Eq. (11), it is λ h = 2 in Eq. (12) and β = 0, hence, the order-parameter m in Eq. (14) becomes discontinuous. Such an unusual discontinuity, originating in intersecting stable FP, has indeed been shown for ordinary percolation on a hierarchical network [24]. We plot the magnetization of the Ising model on a hierarchical network, which follows 2ndorder behavior, Eq. (14), in the first regime in Fig. 3(a), it has an infinite-order transition, Eq. (13), in the BKT regime in Fig. 3(b), and it develops a discontinuity in the regime of intersecting stable FP in Fig. 3(c).
As a demonstration for our theory, we revisit the Ising model on the Hanoi network HN5 previously considered for h = 0 in Ref. [13]. There, a one-parameter family of Ising models was conceived via the ratio between short-range and small-world coupling strengths that interpolates between all three regimes; Fig. 11 in Ref. [13] corresponds to Fig. 1(b-d) here. That system is far more complex than our model here in that there are two couplings and three fields (when h > 0, for site-, bond-, and three-point magnetizations) to be renormalized. Yet, the same three regimes in the divergence of the correlation length ξ and the magnetization m ensues, as our theory predicts. In Supplement B, a Mathematica script reproduces plots identical to Figs. 1 and 2 for this case; more examples will be presented elsewhere.
Conclusions: Ref. [23] has raised the question about the universality of the BKT result in Eq. (10). Our discussion shows that the robustness of this behavior derives from the fact that BP is most generically a square-root singularity. Only if for some reason the cubic and higherorder terms in the RG-flow near BP in Eq. (5) were absent, a higher order root-singularity could arise. The full exponential singularity in Eq. (11) is even more solid, as it does not depend on the nature of BP but merely in the fact that two intersecting lines of FP must switch stability, which implies marginal eigenvalues at the point of collision. While those eigenvalues scale linearly with µ there, the exponential divergence in Eq. (11) is robust. This work was supported by DMR-grant #0812204 from the NSF. SB would like to thank N. Berker and P. Phillips for helpful discussions.