January 01, 1970
Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field \(\phi(\mathbf{x})\) that depends on the spatial position \(\mathbf{x}\), with itself (autocorrelation function) at another position \(\phi(\mathbf{x}_0)\). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding the second-order phase transition in equilibrium. However, his analysis is restricted to a Euclidean space of dimension \(d\), and an exponent \(\eta\) is introduced to correct the spatial behavior of the correlation function in \(T=T_c\). In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at \(T_c\). In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models — the Ising, Potts, XY, and Heisenberg systems.
Phase transitions are typically described by symmetry breaking, which is reflected in the order parameter and the correlation length. This becomes particularly clear for a second-order phase transition, where the order parameter vanishes at the critical temperature \(T_c\), recovering the symmetry, and the correlation length diverges, thereby exhibiting scale invariance. Thus, it is a very broad phenomenon that appears in many branches of physics. To explore critical phenomena, we use lattice models that enrich statistical physics and quantum field theory. Lattice models are usually defined in Euclidean integer dimension \(d\), while critical exponents are usually given by fractal numbers. An important quantity to evaluate is the time-independent correlation function \[\label{Cdef} C(r)=\frac{1}{L^d} \sum_j \phi(\mathbf{j}+\mathbf{r}) \phi(\mathbf{j}),\tag{1}\] where the sum is over a hypercube lattice of dimension \(d\) and side \(L\). Here \(\phi(i)\) is the fluctuation of the order parameter at the lattice site \(i\). This function can be evaluated numerically and hopefully analytically. For a second-order phase transition, a mean-field solution is given by \[\label{Cd1} C(r)=C_1r^{2-d} \exp(-r/\rho),\tag{2}\] where \(C_1\) is a constant, and \(\rho\) the correlation length [1]. For \(T\) close to the critical temperature \(T_c\), the correlation length diverges as \[\label{rhodivergence} \rho \propto |T-T_c|^{-\nu}.\tag{3}\] In this limit, Eq. (2 ) clearly fails for \(d=2\), showing a correlation function independent of \(r\), which is physically inconsistent. This mean field solution breaks down at the transition, necessitating an empirical correction through the introduction of Fisher exponent \(\eta\) [2] \[\label{G2} C(r) \propto \begin{cases} r^{2-d} \exp(-r/\rho) , &\text{ if~~ } r>\rho,\\ r^{2-d-\eta}, &\text{ if~~ } r \ll\rho.\\ \end{cases}\tag{4}\] The homogeneous continuous medium function Eq. (4 ) is a formal solution of the equation \[\label{G} (-\nabla^2 +\rho^{-2})C(r)=\delta^{(d)}(r) \;.\tag{5}\] Correlation functions represent a simple form of the fluctuation-dissipation relation (see reviews [3], [4]) and are easily associated with susceptibility [5], [6]. The violation of the Fluctuation-Dissipation Theorem (FDT) is well known in the literature for structural glasses [7]–[12], random exchange Heisenberg chain [13], proteins [14], KPZ dynamics [15], [16], mesoscopic radioactive heat transfer [17], [18], and ballistic diffusion as well [19]–[22]. The FDT breaks down when ergodicity is violated [19], [23], [24].
The FDT fails for critical points in the sense that it cannot explain Eq.(4 ) for \(r \ll\rho\), where the \(\eta\) exponent appears as a need for consistency with the exponents equalities, e.g., the Fisher scaling relation [2] \[\label{gamma} \gamma =(2-\eta)\nu.\tag{6}\] In fact, the response function in the Euclidean formulation yields \(\eta=0\) in disagreement with reality, thus showing a failure of the FDT. Previous results for growth dynamics showed that when moving to a fractal formulation, the FDT was recovered [23], [25]–[27]. Moreover, fractal dynamics at phase transitions shares important features with growth phenomena, the scale invariance [28], [29] and the attempt to determine whether an upper critical dimension exists [30]–[34]. As well, the critical exponent of the correlation length \(\nu\) is related to the critical exponent of specific heat \(\alpha\) through the hyperscale relation [1] \[\label{alpha} \alpha=2-d\nu,\tag{7}\] thus associating a thermodynamic variable with the divergence of the correlation length. In other words, a relationship between thermodynamics and geometry.
Inspired by this, Lima et al. [6] showed that at \(T=T_c\), where \(\rho \rightarrow \infty\), we have the self-organization of the system into fractal clusters and the correlation function \(C(r)\) does not satisfy Eq. (5 ), since it is in a Euclidian space, rather it obeys the equation \[\label{G3} (-\nabla^2)^\zeta C(r)=\delta^{d_R}(r),\tag{8}\] to reflect the dynamics being restricted to a fractal structure characterized by the Riesz fractal dimension1 \(d_R\), \(d-1 \leq d_R \leq d\) in the Riesz fractional derivative [37] of order \(\zeta\), \(1/2 \leq \zeta \leq 1\). The solution of this fractional differential equation, associated with the proposed solution (4 ), leads us to [6] \[\label{eta} \eta=d-d_R=1-\zeta.\tag{9}\] Now the Fisher exponent is not an ad hoc imposition; it arises naturally from the solution of (8 ). In the renormalization-group framework, the anomalous dimension \(\eta\) — which is non-zero for interacting critical theories in dimensions less than \(4\) — is obtained as the fixed-point value of the field anomalous dimension [38]. Equation (3 ) highlights the scaling invariance at the transition, where the dynamics occur not in the Euclidean dimension of the lattice but in the fractal structure of clusters of all sizes. Furthermore, it shows the exponent \(\eta\) as the deviation from the integer dimension to the fractal dimension, or, equivalently, from an integer derivative to a fractional Riesz derivative.
The above result is not the only one to interpret the scaling behavior seen at critical points as manifestations of fractal geometries. The Fortuin-Kasteleyn clusters [39] or Coniglio-Klein droplets [40] (FKCK) allowed to unify geometry and thermodynamics. The fractal geometry of FKCK droplets is the fractal, percolating clusters that form exactly at the critical point. When such a mapping is available, it provides the deepest link between critical phenomena and fractal geometry, and it has paved the way for the most efficient computational algorithms for studying spin models [41]. The FKCK mapping has been investigated in particular in the context of percolation [42]–[44]. For example, the relation between the exponent of the order parameter \(\beta\), and the fractal dimension of the ordered phase \(d_f\) was proposed by Suzuki, who also made a connection to the renormalization-group framework [42] \[\label{df} d_f=d-\frac{\beta}{\nu}\;.\tag{10}\] If the critical point is the percolation one, the fractal structure is the infinite percolating cluster at the transition [45], [46]. In general, it is associated with the largest ordered cluster at the critical point [43]. Combining Eqs. (6 ), (7 ), (9 ) and (10 ), and using the Rushbrooke equality \(\alpha+2\beta+\gamma=2\), we obtain an important connection between \(d_R\) and \(d_f\) as [6] \[\label{dR2} d_R=2(d_f-1).\tag{11}\] The fractal dimension predicted by Eq. (10 ), \(d_f=15/8\) for the 2D Ising model, coincides with that of Coniglio for the two-state Potts model based on a mapping to the two-dimensional Coulomb gas [47]. This yields \(d_R=7/4\) and \(\eta=1/4\), from Eqs. (11 ) and (9 ), respectively, in agreement with the known values of the Ising model in two dimensions. Subsequently [35], [36], these relationships were verified for non-integer dimensions in the range \(1 \leq d \leq 4\). Furthermore, it was verified for disordered Ising systems [48].
Note that according to Eq. (9 ) the geometric limit \(1/2 \leq \zeta \leq 1\), imposes a maximum value \(\eta=1/2\). However, it is possible to overcome this limitation using a scaling method [36]. In this work, our goal is to verify the relationships (9 ) and (11 ) in more complex systems, such as the Potts, XY, and Heisenberg models, and to explore how far our interpretation can take us. This work is organized as follows: Section 2 presents our discussion of the two-dimensional Potts model, for which we can obtain exact results. In Section 3, we present results and discussion for the two-dimensional XY model, for which our relations do not apply, and the three-dimensional Ising, XY, and Heisenberg models, with results that are not exact, but highly precise. We conclude in Section 4.
The Potts model is a generalization of the Ising model to more than two components [49], [50]. Although intensively studied, it is a very active field of modern research [51]. A simple Hamiltonian for it can be expressed as \[H=- J \sum_{(i,j)}\delta_{s_i,s_j}-h_0 \sum_i \delta_{s_i,0}\] where the double sum \((i,j)\) runs over nearest neighbor spins, \(\delta\) is the delta of Kronecker, which is \(1\) for \(s_i=s_j\) and zero otherwise. \(J\) is the spin-spin coupling constant and \(h_0\) is an external magnetic field. Spins \(s_i\) take on values \(s_i=(0,1,2,\ldots, q-1)\). It is thus defined as the \(q\) state Potts model. It is now known that the Potts model is related to a number of outstanding problems in lattice statistics; the critical behavior has also been shown to be richer and more general than that of the Ising model. For example, \(q=1\) is used for the study of percolation and complex networks, while for \(q=2\) we have the Ising model, and for \(q=4\) we have the Baxter-Wu model. In subsequent efforts to explore its properties, the Potts model has become an important test ground for different methods and approaches in the study of critical point theory [50].
An useful representation for the \(q\)-state Potts model local order parameter \(\mathbf{m}_i\) at site \(i\) is a \((q-1)\)-component vector. In this simplicial representation, each one of the \(q\) states, \(\sigma_i\), is mapped onto a unit vector \(\mathbf{e}_{\sigma_i}\) pointing to one of the \(q-1\) vertices of a simplex. Thus, the order parameter at site \(i\) is given by \(\mathbf{m}_i = \mathbf{e}_{\sigma_i}\). The global order parameter reads \[\mathbf{M} = \frac{1}{N} \sum_i \mathbf{m}_i,\] and the corresponding fluctuation field, such as in Eq. (1 ), is \(\mathbf{\phi}_i = \mathbf{m}_i - \langle \mathbf{M} \rangle\).
| \(q\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|
| \(\beta\) | \(1/6\) | \(5/36\) | \(1/8\) | \(1/9\) | \(1/12\) |
| \(\nu\) | \(\infty\) | \(4/3\) | \(1\) | \(5/6\) | \(2/3\) |
| \(d_f\) | \(2\) | \(91/48\) | \(15/8\) | \(28/15\) | \(15/8\) |
| \(d_R\) | \(2\) | \(43/24\) | \(7/4\) | \(26/15\) | \(7/4\) |
| \(\eta\) | \(0\) | \(5/24\) | \(1/4\) | \(4/15\) | \(1/4\) |
| \(\zeta\) | \(1\) | \(19/24\) | \(3/4\) | \(11/15\) | \(3/4\) |
| Model | Ising (3D) | XY (2D) | XY (3D) |
|---|---|---|---|
| \(\beta\) | \(0.326418(2)\) | 0 | \(0.3485(2),\) |
| \(\nu\) | \(0.629971(4)\) | \(\infty\) | \(0.67155(27)\) |
| \(d_{f}\) | \(2.481856(2)\) | 2 | \(2.481(2)\) |
| \(d_R\) | \(2.963713(5)\) | 2 | \(2.962(5)\) |
| \(\zeta\) | \(0.963713(5)\) | \(1\) | \(0.9619(4)\) |
| \(\eta*\) | \(0.0362978(20)\) | \(1/4\) | \(0.0380(4)\) |
| \(\eta\) | \(0.036288(5)\) | 0 | \(0.0380(5)\) |
In Table 1, we present the values of \(d_f\) and \(d_R\) for the \(2D\) Potts model universality class for \(q=0,1,2,3\), and \(4\), alongside the corresponding critical exponents. From [50] we obtain \(\beta\) and \(\nu\), and \(d_f\) from Eq. (10 ). The values agree with those of Coniglio [47]. We note that the equation is fulfilled for all \(q\) shown. From this, \(d_R\), \(\zeta\) and \(\eta\) can be independently determined. Furthermore, note that the values of \(\eta\) obtained this way agree with the values obtained exactly previously [50]. Notice that for \(q=4\), \(\beta\) and \(\nu\) are different from the values for \(q=2\). However, the ratio \(\beta/\nu\) is the same. Thus, from Eq. (10 ) we obtain the same \(d_f\), from Eq. (10 ) and (11 ) the same \(d_R\) and the same \(\eta\). The identical values of \(\eta\) for \(q=2\) and \(q=4\) cannot be interpreted as a coincidence, but as an imposition of fractal geometry, which has the same \(d_f\) and \(d_R\) for both cases. Finally, note that our values of \(\eta\) agree with the exact values reported in the literature. This reinforces the value of Eq. (9 ) for both the determination and the interpretation of the exponent \(\eta\).
| Ref. | [55] | [56] | [57] | [54] |
|---|---|---|---|---|
| \(\beta\) | \(0.3645(25)\) | \(0.368(4)\) | 0.362(4) | \(0.3689(3)\) |
| \(\nu\) | \(0.705(3)\) | \(0.710(7)\) | \(0.706(9)\) | \(0.71120(5)\) |
| \(d_{f}\) | \(2.483(5)\) | \(2.482(11)\) | \(2.49(1)\) | \(2.4813(5)\) |
| \(d_R\) | \(2.97(1)\) | \(2.963(22)\) | \(2.97(2)\) | \(2.9626(9)\) |
| \(\zeta\) | \(0.965(10)\) | \(0.963(22)\) | \(0.975(2)\) | \(0.9626(9)\) |
| \(\eta*\) | \(0.033(4)\) | \(0.040(3)\) | \(0.027(2)\) | \(0.0375(3)\) |
| \(\eta\) | \(0.035(10)\) | \(0.037(40)\) | \(0.025(2)\) | \(0.0374(9)\) |
Following the order of complexity, we first consider the Potts model as a natural extension of the Ising model, then investigate the XY and Heisenberg models. The Ising 3D model is mentioned because its critical exponents are known with high precision.
Unlike the Potts model, the classical XY and Heisenberg models have continuous symmetry. Now the dynamical degrees of freedom consist of unity 2- and 3-component vectors \(\mathbf{S}_i\). The Hamiltonian for these model systems may be written \[H=- J \sum_{(i,j)} \left( \Delta S_i^z S_j^z + S_i^x S_j^x + S_i^y S_j^y\right),\] where \(J\) sets the overall energy scale, and \(\Delta\) is an anisotropy parameter. The XY model [53] is recovered setting \(\Delta = 0\), and the isotropic Heisenberg model [58] corresponds to \(\Delta = 1\). For \(\Delta \ne 1\), the anisotropic Heisenberg system generally lies in a different universality class from the isotropic one [59] (see subsection B); consequently, we will not discuss the anisotropic case here.
In Table 2, we present the values of \(d_f\) and \(d_R\) for the Ising 3D model and the XY model in 2 and 3 dimensions, along with the corresponding critical exponents. Here we use the notation \(\eta*\) obtained from the literature to distinguish it from our \(\eta\), Eq.(9 ). The values of \(\beta\), \(\nu\) and \(\eta*\), are obtained from: Ising 3D ref [52], XY 2D ref [60] and XY 3D [52]. We obtain \(d_f\) from Eq. (10 ), and \(d_R\) from Eq. (11 ), while \(\eta\) from Eq.(9 ). From this, \(d_R\), \(\zeta\) and \(\eta\) can be independently determined. Furthermore, note that the values of \(\eta\) obtained this way agree with the values of \(\eta*\), within the error for the three-dimensional Ising and XY model.
An intriguing result emerges for the two-dimensional XY model, which does not exhibit Mermin-Wagner symmetry breaking, but possesses a BKT (Berezinskii-Kosterlitz-Thouless) topological transition [44], [53], [61]–[63]. The \(\eta\) parameter behaves as \[\label{etaR} \eta(T) = \begin{cases} 0 , &\text{ if~~ } T < T_{BKT},\\ 1/4 , &\text{ if~~ } T = T_{BKT},\\ \infty , &\text{ if~~ } T > T_{BKT},\\ \end{cases}\tag{12}\] Where \(T_{BKT}\) is the topological transition temperature. The main characteristic is the universal value of \(\eta=1/4\) which, since it is not an order-to-disorder transition, does not agree with our result. Curiously, under this extreme condition, our approach predicts flat surfaces with \(d_r=d_f=d=2\) and a return to the Euclidean mean-field results, \(\eta=0\) and fractional derivative order \(\zeta=1\). Consequently, our formulation does not apply to topological phase transitions.
The two-dimensional Heisenberg model represents an even more restrictive case than the two-dimensional XY model, that is, it does not even present a topological transition. Consequently, we will not consider it here.
In Table 3, we present the values of the fractal dimensions \(d_f\) and \(d_R\) for the three-dimensional Heisenberg model, together with their corresponding critical exponents. Here we use the notation \(\eta*\) obtained from the literature to distinguish it from our \(\eta\), Eq.(9 ). The values of \(\beta\), \(\nu\), and \(\eta*\) were obtained from the cited references. We obtain \(d_f\) from Eq. (10 ), and \(d_R\) from Eq. (11 ), while \(\eta\) from Eq.(9 ). From this, \(d_R\), \(\zeta\) and \(\eta\) can be independently determined. Furthermore, note that the values of \(\eta\) obtained in this way agree with the values of \(\eta*\), within the error margins.
We have introduced a new fractal structure associated with the dynamic correlations present at equilibrium critical points. This structure is characterized by a correlation fractal dimension. We relate this dimension to the Fisher exponent \(\eta\), which characterizes the decay of the correlation function near criticality, via Eq.(9 ). Our approach replaces the traditional correlation function Eq.(5 ) with a version based on the Riesz fractional derivative Eq.(8 ) at the critical point. The Riesz fractional derivative is a mathematical tool that generalizes differentiation to non-integer orders, capturing the fractal nature of correlations. This equation defines the fractal subspace determined by the correlations. Using this substitution restores the form of the correlation function at the critical point. Therefore, the Fisher exponent in the correlation function \(C(r)\) indicates how much the correlation fractal dimension deviates from the system’s integer spatial dimension. Both \(\zeta\), a parameter controlling the decay rate, and \(d_R\), the Riesz fractal dimension or correlation fractal dimension, fall within previously established bounds: \(0<\zeta<1\) and \(d-1 \leq d_R \leq d\).
Using precise estimates of previously published critical exponents, we obtained the fractal dimensions \(d_R\) and \(d_f\), as well as the Fisher exponent \(\eta\) for the two-dimensional Potts model (Table 1, exact results). Similarly, we analyzed the three-dimensional Ising model and the XY models (Table 2), and the three-dimensional Heisenberg model (Table 3). In the latter cases, our results agree, within their precision, with previous literature, though we also discussed discrepancies, such as the topological transition in the XY 2D model.
In summary, this and prior work [6], [35], [36] provide strong evidence that the mean-field fractal formulation for the correlation function (8 ) is effective for describing equilibrium symmetry-breaking phase transitions.
However, our framework does not capture topological phase transitions in the two-dimensional XY model, indicating a need for further investigation to address this limitation. Another relevant question concerns the underlying geometric object with fractal dimension \(d_R\).
Since Eq. (9 ) was derived for a general continuous order parameter \(\phi(\mathbf{r})\), it is reasonable to anticipate its extension to non-equilibrium dynamics. Exploring these directions [64]–[66], especially in dynamic phase transitions [65], [67]–[71], requires including temporal dependence in the autocorrelation function (1 ). This will open new avenues for understanding the relationship between critical phenomena, fractal geometry, and universality.
Recent research related to this work, such as Schrödinger invariance in the voter model [64], conformal scaling [72], and Fisher curvature scaling at critical points [73], illustrates the field’s promise and open questions.
The authors acknowledge the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES. FAO acknowledges financial support from Fundação de Amparo à Ciência e Tecnologia de Pernambuco - FACEPE and CNPq (Grants APV-0064-1.05/25 and 303119/2022-5). ALRB acknowledges financial support from CNPq (Grants 302502/2025-4 and 406836/2022-1 INCT of Spintronics and Advanced Magnetic Nanostructures - SpinNanoMag).