Introduction

Lecture 3 established the mathematical framework of statistical mechanics. The partition function \(Z\) as a generating functional encodes all statistical information of the system, the free energy \(F = -k_B T \ln Z\) constitutes the thermodynamic potential energy landscape, and the fluctuation-dissipation theorem reveals the intrinsic connection between response functions and fluctuation variance. The core conclusion of Lecture 3 is that when the system approaches the critical point of a continuous phase transition, the correlation length \(\xi \to \infty\), and traditional mean-field approximation and perturbation expansion both fail simultaneously, making direct calculation of the partition function infeasible.

However, the questions not yet answered in Lecture 3 are: what quantitative structure do the power-law divergences of physical quantities near the critical point possess, whether there are intrinsic constraints among these power-law exponents, and why systems with vastly different microscopic mechanisms exhibit the same scaling behavior near the critical point.

The answers to these questions constitute the theme of this lecture. In the logical chain of the renormalization group tutorial series, this lecture plays the key role of transitioning from phenomenological description to a quantifiable operational language, transforming the qualitative picture revealed in Lecture 3—that \(\xi \to \infty\) leads to computational collapse—into a system of critical exponents that is measurable, classifiable, and predictable, while laying the conceptual foundation for the subsequent formal introduction of Landau theory, the Ginzburg criterion, and the Wilson renormalization group.

Critical behavior is not exclusive to condensed matter physics. During the North American blackout of August 2003, a local grid failure expanded through cascade effects to the entire northeastern grid within seconds, causing 50 million people to lose power. Post-event analysis showed that the power network was operating near the critical edge under high load conditions, with load correlations between nodes spanning the entire network scale, and the influence of single-point disturbances was no longer confined to neighboring areas but could propagate to arbitrary distances. This phenomenon shares profound structural similarity with the collective motion of bird flocks discussed in Lecture 2—both have correlation lengths approaching the system size, with strong coupling between local fluctuations and global response.

Similar critical behavior also appears in long-range correlations of volatility before financial market crashes, percolation phase transitions in infection networks near disease transmission thresholds, and power-law distributions of neural avalanches in self-organized critical states of the nervous system. The reason these superficially unrelated systems exhibit similar statistical characteristics is that they share the same mathematical structure near the critical point—this is precisely the core meaning of the concept of universality.

This lecture will unfold at three levels. First is the definition and measurement of critical exponents. Physical quantities near the critical point, such as specific heat \(C\), susceptibility \(\chi\), order parameter \(M\), and correlation length \(\xi\), all exhibit power-law behavior, with exponents \(\alpha, \beta, \gamma, \nu\) etc. constituting the basic parameter set describing critical phenomena. This lecture will give rigorous definitions of these exponents and discuss how to extract them from experimental or simulation data.

Second is the mathematical structure of scaling laws. These critical exponents are not mutually independent fitting parameters. Based on the scaling hypothesis—that the singular free energy density satisfies the form of a generalized homogeneous function—one can derive scaling identities such as Rushbrooke, Widom, and Fisher, compressing six exponents into only two independent parameters. The existence of this mathematical structure indicates that there are deeper symmetry constraints behind critical phenomena.

Finally is the physical origin of universality classes. Why do different physical systems share the same set of critical exponents? This lecture will provide an answer at the phenomenological level: the factors determining universality class include only the spatial dimension \(d\), the symmetry of the order parameter, and the range of interactions, independent of the specific parameters of the microscopic Hamiltonian. The deep mechanism behind this conclusion will be rigorously proven through the complete framework of the renormalization group in subsequent lectures.

Through these three levels of exposition, this lecture will transform the physical question of what happens at the critical point into the operational quantitative framework of what values the critical exponents take, what relationships they satisfy, and what factors determine them, providing the necessary conceptual preparation for the formal introduction of the renormalization group.

1. Classification of Phase Transitions: First-Order and Continuous

There are two fundamentally different types of phase transitions in physics. The first type is abrupt phase transitions. When water is heated to \(100°\text{C}\) at standard atmospheric pressure, it boils, and liquid water suddenly becomes steam. During this process, even with continuous heating, the temperature does not rise—all the heat is used to break the bonds between liquid molecules. This heat is called latent heat, because it "hides" in the phase transition process without manifesting as a temperature increase. At the transition point, liquid and gas can coexist, with boiling water and rising steam existing simultaneously in the pot. Ice melting into water, metal solidification, and crystal melting all belong to this type of phase transition.

The second type is gradual phase transitions. Iron is a typical ferromagnet, possessing spontaneous magnetization at room temperature—even without an external magnetic field, the atomic magnetic moments inside an iron block tend to align parallel. But when the temperature rises to near the Curie temperature \(T_c \approx 770°\text{C}\), thermal fluctuations gradually destroy this ordered arrangement, and the spontaneous magnetization continuously approaches zero rather than suddenly disappearing. There is no latent heat in this process, no phase coexistence, and the temperature can smoothly pass through \(T_c\). However, the system's response to external disturbances becomes extraordinarily violent: near \(T_c\), a weak external magnetic field can cause a huge magnetization response. The \(\lambda\) transition of superfluid helium, the normal-superconducting transition of superconductors, and the isotropic-nematic transition of liquid crystals all belong to this type.

These two types of phase transitions differ not only in experimental phenomena but also in their underlying mathematical structures. What the renormalization group theory deals with is precisely the latter type of phase transition—continuous phase transitions. This section strictly distinguishes these two types of phase transitions from a thermodynamic perspective and reveals why only continuous phase transitions require the renormalization group as a completely new theoretical tool.

1.1 Thermodynamic Definition of Phase Transitions

The rigorous definition of phase transitions is built on the mathematical properties of free energy. In the thermodynamic limit (system degrees of freedom \(N \to \infty\)), a phase transition corresponds to the free energy \(F(T, h, \ldots)\) losing analyticity at a certain point, meaning it cannot be expanded as a convergent Taylor series at that point.

This definition seems abstract, but its physical origin is quite clear. For finite systems (i.e., systems with finite particle number \(N\) or finite linear size \(L\), such as a \(100 \times 100\) lattice in numerical simulations), the partition function \(Z = \sum_{\{\sigma\}} e^{-\beta H[\sigma]}\) is a sum of a finite number of analytic terms, so both \(Z\) and \(F = -k_B T \ln Z\) are analytic. Only when \(N \to \infty\) does the number of summation terms tend to infinity, making non-analyticity possible. This means that true phase transitions only exist in infinitely large systems; finite systems can only exhibit "precursors" of phase transitions—response function peaks that grow taller and sharper as the system size increases, but never truly diverge. This fact directly leads to the necessity of finite-size scaling theory later on.

According to the order at which non-analyticity appears, phase transitions are divided into two major categories. If the first derivative of the free energy is discontinuous, it is called a first-order phase transition; if the first derivative is continuous but the second derivative diverges, it is called a continuous phase transition (or second-order phase transition). This classification was proposed by Ehrenfest in 1933, and although actual situations are more complex, the distinction between first-order and continuous phase transitions remains the basic starting point for understanding critical phenomena.

1.2 First-Order Phase Transitions: Barrier Crossing and Finite Correlation

The core characteristic of first-order phase transitions is discontinuous jump of the order parameter. Taking a temperature-driven phase transition as an example, the entropy \(S = -\partial F / \partial T\) is discontinuous at the transition temperature \(T_{\text{tr}}\), and the entropy difference between the two phases \(\Delta S\) corresponds to latent heat \(L = T_{\text{tr}} \Delta S\). The order parameter \(M\) jumps from one non-zero value to another without smooth transition in between.

From the perspective of the free energy landscape, a first-order phase transition corresponds to a double-well structure. A potential well refers to a local minimum point on the free energy curve, where the system tends to stay at the bottom of the well, just like a ball rolling into a bowl and staying stable; a potential barrier refers to the raised portion between two wells, and the system must acquire enough energy to cross the barrier to transition from one stable state to another. In a first-order phase transition, the two wells represent two thermodynamically stable phases, separated by a finite barrier. When the temperature passes through the transition point, the depths of the two wells are exchanged, and the system "jumps" from one well to another. It is precisely the existence of this barrier that leads to dynamical phenomena such as metastable states, nucleation, and phase coexistence.

The key point is: the correlation length \(\xi\) at a first-order transition point remains finite. Although the two phases can coexist to form a macroscopic interface, fluctuations within each phase remain short-range correlated. This means that although first-order phase transitions appear dramatic macroscopically, their microscopic physics can still be handled by standard statistical mechanics methods, without needing the renormalization group.

1.3 Continuous Phase Transitions: Flat Landscape and Divergent Correlation

The physical picture of continuous phase transitions is completely different from first-order phase transitions. The order parameter does not jump, but continuously approaches zero; there is no latent heat, no phase coexistence; but response functions—susceptibility \(\chi\), specific heat \(C\)—diverge to infinity at the critical point.

\[ \chi = -\frac{\partial^2 F}{\partial h^2} \to \infty, \quad C = -T\frac{\partial^2 F}{\partial T^2} \to \infty \quad (T \to T_c) \]

The origin of this divergence lies in the shape of the free energy landscape. At the critical point, the bottom of the potential well becomes extremely flat, with curvature approaching zero. The order parameter can fluctuate almost without cost, leading to fluctuation amplitude approaching infinity. More crucially, this fluctuation is not local, but a collective behavior spanning the entire system. The correlation length \(\xi\) diverges at the critical point:

\[ \xi \sim |T - T_c|^{-\nu} \]

where \(\nu > 0\) is the correlation length critical exponent. When \(\xi \to \infty\), the system loses its intrinsic length scale, and the influence of local disturbances can propagate to arbitrary distances. This is precisely the core characteristic that distinguishes critical phenomena from general thermodynamic behavior, and is the physical root of the "computational collapse" revealed in Lecture 3—when fluctuations at all scales are strongly coupled, traditional mean-field approximation and perturbation expansion both fail simultaneously.

2. Order Parameter, Conjugate Field, and Response Functions

The previous section distinguished first-order and continuous phase transitions from the analytic properties of free energy. However, this classification is at the mathematical level. Experimental physicists in the laboratory do not directly measure whether the derivatives of free energy are continuous, but measure a series of observable quantities—magnetization, density, specific heat, susceptibility—and infer the existence and type of phase transition from the behavior of these quantities. This section will establish the observable quantity system for describing phase transitions and reveal the intrinsic connections among these quantities.

2.1 Order Parameter: Macroscopic Indicator Distinguishing Order from Disorder

The essence of a phase transition is the transformation of a system from one macroscopic state to another. To quantitatively describe this transformation, we need to introduce a physical quantity that can distinguish different phases—this is the order parameter.

Taking a ferromagnet as an example. At low temperatures, the atomic magnetic moments inside an iron block tend to align parallel, producing a macroscopic spontaneous magnetization \(M\). Even without an external magnetic field, \(M\) is not zero. When the temperature rises above the Curie temperature \(T_c\), thermal fluctuations destroy this ordered arrangement, and \(M\) drops to zero. Therefore, spontaneous magnetization \(M\) is the order parameter for the ferromagnetic-paramagnetic phase transition: in the ordered phase \(M \neq 0\), in the disordered phase \(M = 0\), and the value of \(M\) marks which phase the system is in.

The choice of order parameter is not arbitrary. In liquid-gas phase transitions, the order parameter is the density difference between liquid and gas phases \(\Delta\rho = \rho_l - \rho_g\); in superfluid helium, the order parameter is the wavefunction amplitude of Bose-Einstein condensation; in superconductors, the order parameter is the complex order parameter of Cooper pair condensation \(\psi = |\psi|e^{i\phi}\); in percolation systems, the order parameter is the ratio of the largest connected cluster to the total number of lattice sites \(P_\infty = s_{\max}/N\). Order parameters of different systems have different mathematical structures—scalar, vector, or complex number—and this structural difference will play a key role in the subsequent discussion of universality classes.

From the perspective of statistical mechanics, the order parameter is not defined by intuition, but is naturally derived from the partition function. Lecture 3 pointed out that the partition function \(Z\) is a generating functional, and taking its logarithm and differentiating with respect to external parameters yields macroscopic observables. For magnetic systems, if the Hamiltonian contains an external magnetic field term \(-hM\), then:

\[ \langle M \rangle = \frac{1}{\beta}\frac{\partial \ln Z}{\partial h} \]

This relation closely links the order parameter with the structure of the partition function, and also provides the mathematical foundation for subsequently defining critical exponents.

2.2 Conjugate Field: External Means of Driving Order Parameter Changes

The order parameter describes the intrinsic state of the system, but experimental physicists need an externally controllable means to probe the system's response to perturbations. The external parameter corresponding to the order parameter is called the conjugate field, which plays a role conjugate to the order parameter in thermodynamics.

For magnetization \(M\), its conjugate field is the external magnetic field \(h\); for density difference \(\Delta\rho\), its conjugate field is the chemical potential difference \(\Delta\mu\); for strain, its conjugate field is external stress. The product of conjugate field and order parameter has the dimension of energy and appears in the interaction terms of the Hamiltonian. For example, the Zeeman energy of a magnetic system is \(-hM\), and the pressure work of a fluid system is \(-pV\).

The physical significance of the conjugate field is: it provides an external mechanism for breaking symmetry. In zero external field, the free energy of a ferromagnet is symmetric under \(M \to -M\), and the system has no preference between the \(+M\) and \(-M\) states. Once a weak external magnetic field \(h > 0\) is applied, this symmetry is broken, and the system tends to choose the \(M > 0\) state.

Why do we need to break symmetry? This is not just a theoretical trick, but a practical requirement for experimental measurement. Imagine a perfectly symmetric system: in zero external field, a ferromagnet might randomly choose upward or downward magnetization direction, with uncertain results each time. This uncertainty prevents us from quantitatively studying the system's response behavior. By applying a small external field, we "guide" the system to choose a specific direction, thereby enabling stable measurement of the system's sensitivity to external perturbations. It is like using a flashlight to illuminate one direction in the dark—although the flashlight's light is weak, it breaks the spatial symmetry, allowing us to see the details in that direction.

The role of breaking symmetry is also reflected in probing critical behavior. Near the critical point, the system is in a state of "hesitation," and tiny perturbations can cause huge responses. The role of the conjugate field is like gently placing a feather on a balance—although the feather itself is very light, at the extremely sensitive moment of the critical point, it is enough to tip the balance, thereby revealing the system's intrinsic instability.

For example, in power networks, a slight load change at one node (equivalent to a conjugate field) in a critical state can trigger a cascade response across the entire network; in financial markets, a tiny price signal near the critical point can trigger fluctuations across the entire market. It is precisely through this mechanism of "breaking symmetry" that we can observe the system's abnormal sensitive behavior near the critical point.

It is precisely through this approach that experimental physicists can control the state of the system and measure the system's response to external fields. This controllability enables us to quantitatively study phase transition behavior, rather than just passively observing the system's spontaneous evolution.

2.3 Response Functions: The Most Sensitive Experimental Signal at the Critical Point

In experiments, directly measuring the absolute value of the order parameter is often difficult, while measuring the response intensity of the system to external perturbations is relatively easy. Such quantities are called response functions, and they are the most commonly used probes in phase transition experiments.

Response functions have a direct definitional relationship with conjugate fields: a response function is the rate of change of the order parameter with respect to the conjugate field. When we apply a small change in the conjugate field, the order parameter changes accordingly, and the response function quantifies the sensitivity of this change. This relationship makes response functions the bridge connecting external control (conjugate field) and system state (order parameter).

Susceptibility \(\chi\) is defined as the response of magnetization to external magnetic field

\[ \chi_T = \left(\frac{\partial M}{\partial h}\right)_T \]

It reflects the sensitivity of the system to changes in external magnetic field at a given temperature. Heat capacity \(C\) is defined as the response of internal energy to temperature

\[ C_V = \left(\frac{\partial \langle E \rangle}{\partial T}\right)_V \]

It reflects the ease with which the system's temperature changes after absorbing heat. Isothermal compressibility \(\kappa_T\) is defined as the response of volume to pressure

\[ \kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T \]

It reflects the degree to which a fluid's volume changes under compression.

These response functions exhibit a common characteristic near the critical point of continuous phase transitions: divergence to infinity. This means that the system's response to weak perturbations becomes abnormally violent—a tiny external magnetic field can cause huge magnetization changes, a tiny temperature fluctuation can cause violent energy redistribution. This "hypersensitive" state is precisely the signature characteristic of critical phenomena.

Returning to the examples mentioned in the introduction, we can understand these phenomena using the concept of response functions. In the 2003 North American blackout, the "response function" of the power network—the range of impact of a local failure on the entire network—became extremely large in the critical state, and a tiny perturbation at one node could trigger a cascade response across the entire system. In financial markets, the response function of price to information (similar to susceptibility) increased sharply on the eve of a crash, and tiny negative news could trigger violent fluctuations across the entire market. In disease transmission, the response of infection rate to contact frequency becomes abnormally sensitive near the transmission threshold, and a slight increase in social activity can lead to exponential growth in infections. These seemingly different systems all exhibit the same characteristic near the critical point: extraordinary sensitivity to tiny perturbations, i.e., divergence of response functions.

So where does this extraordinary sensitivity come from? Why does the system become so "fragile" near the critical point that tiny external perturbations can cause huge responses? To answer this question, we need to deeply understand the microscopic origin of response functions.

2.4 Fluctuation-Dissipation Theorem: The Microscopic Origin of Response Functions

The divergence of response functions does not appear out of nowhere; it has deep microscopic roots. Lecture 3 derived the fluctuation-dissipation theorem (FDT) in detail, which establishes a precise connection between macroscopic response functions and microscopic fluctuations.

For susceptibility, the FDT gives

\[ \chi_T = \frac{1}{k_B T}\text{Var}(M) = \frac{1}{k_B T}\left(\langle M^2 \rangle - \langle M \rangle^2\right) \]

For heat capacity, the FDT gives

\[ C_V = \frac{1}{k_B T^2}\text{Var}(E) = \frac{1}{k_B T^2}\left(\langle E^2 \rangle - \langle E \rangle^2\right) \]

These two formulas reveal a key fact: the divergence of response functions is equivalent to the divergence of fluctuation variance. When \(\chi \to \infty\), the fluctuation of magnetization \(\text{Var}(M) \to \infty\); when \(C \to \infty\), the fluctuation of energy \(\text{Var}(E) \to \infty\).

However, "fluctuation divergence" here needs to be carefully understood. It does not mean that the system becomes more chaotic or noisier, but means that fluctuations become coordinated across larger spatial scales. Far from the critical point, fluctuations in different regions are independent and local, and they cancel each other out, resulting in small variance of macroscopic quantities. Near the critical point, fluctuations develop long-range correlations, fluctuations in distant regions are no longer independent, but fluctuate in coordination, causing the variance of macroscopic quantities to increase sharply. This "collective fluctuation" is the direct manifestation of correlation length \(\xi \to \infty\), and is the core content that will be analyzed in depth when discussing correlation functions in the next section.

3. Correlation Functions, Correlation Length, and "Scale-Free"

The previous section revealed the microscopic origin of response function divergence through the fluctuation-dissipation theorem: near the critical point, fluctuations become coordinated across larger spatial scales, causing the variance of macroscopic quantities to increase sharply. However, how exactly is this "coordination" transmitted in space? Why are fluctuations local and independent far from the critical point, while forming long-range correlations spanning the entire system near the critical point? To answer these questions, we need to introduce a physical quantity that describes the spatial structure of fluctuations—the correlation function.

The correlation function not only reveals how fluctuations are transmitted in space, but more importantly, it closely links macroscopic response with microscopic correlation. Through the correlation function, we can quantitatively understand why the divergence of correlation length \(\xi\) leads to the divergence of response functions, and why power-law behavior rather than exponential decay appears near the critical point. This bridge from spatial correlation to macroscopic response is the key to understanding critical phenomena.

3.1 Correlation Function: How Fluctuations Propagate in Space

Imagine a ferromagnet system, where atomic magnetic moments (spins) constantly flip under thermal fluctuations. Far from the critical point, if you observe a spin at a certain position, its flip only affects a few neighboring spins, and distant spins are almost unaffected. But near the critical point, the situation is completely different: a spin flip "infects" distant spins, forming collective fluctuations spanning the entire system. How does the strength of this "infection" decay with distance? This is precisely the question that correlation functions answer.

Taking the Ising spin model as an example, each lattice site has a spin \(s_i = \pm 1\). Define the two-point correlation function (connected correlation function):

\[ G(r) = \langle s(0)s(r) \rangle - \langle s \rangle^2 \]

The physical meaning of this definition needs to be carefully understood. \(\langle s(0)s(r) \rangle\) is the statistical average of the product of two spins separated by distance \(r\). If the two spins are always aligned, this average is close to \(+1\); if always anti-aligned, close to \(-1\); if completely uncorrelated, then close to \(\langle s \rangle^2\) (because \(\langle s(0)s(r) \rangle = \langle s(0) \rangle \langle s(r) \rangle = \langle s \rangle^2\)). Therefore, \(G(r) = \langle s(0)s(r) \rangle - \langle s \rangle^2\) measures the degree of correlation between two spins, removing the contribution of the mean and keeping only the correlation of the fluctuation part.

The physical meaning of the correlation function can be understood as follows:

• \(G(r) \approx 0\): Two spins separated by \(r\) are nearly independent; a fluctuation in one spin does not affect the other. It is like two cities very far apart—weather changes in one city do not affect the other.

• \(G(r) > 0\): Two spins tend to be aligned; when one spin is up, the other also tends to be up. It is like in a power network, an increase in load at one node "infects" neighboring nodes, causing them to also increase load.

• \(G(r) < 0\): Two spins tend to be anti-aligned; when one spin is up, the other tends to be down. This occurs in certain antiferromagnetic systems.

More intuitively, if \(\langle s(0)s(r) \rangle\) quickly approaches \(\langle s \rangle^2\), it means distant locations almost do not "know" what fluctuations are happening nearby, and the influence range of fluctuations is limited; if it decays slowly, distant locations will "oscillate together with nearby locations," and the influence range of fluctuations is large. This is exactly what we saw at the end of Section 2: far from the critical point, fluctuations are local; near the critical point, fluctuations are long-range correlated.

3.2 Non-Critical: Exponential Decay and Finite Correlation Length

Far from the critical point, the correlation function exhibits typical exponential decay behavior:

\[ G(r) \sim e^{-r/\xi}, \qquad (T \neq T_c) \]

The physical meaning of this formula is very clear. The exponential factor \(e^{-r/\xi}\) tells us that correlation strength decays exponentially with distance \(r\), and the characteristic decay length is \(\xi\). When \(r = \xi\), correlation strength decays to \(1/e \approx 0.37\) of the original; when \(r = 2\xi\), it decays to \(1/e^2 \approx 0.14\); when \(r = 3\xi\), it decays to \(1/e^3 \approx 0.05\). Therefore, \(\xi\) gives a natural length scale: beyond this distance, correlations become very weak.

\(\xi\) is called the correlation length, and it tells you "the typical size of a fluctuation cluster." Imagine in a ferromagnet, due to thermal fluctuations, spins in some regions temporarily flip, forming a fluctuation cluster. Far from the critical point, the typical size of these clusters is \(\xi\). If the distance between two spins is less than \(\xi\), they likely belong to the same fluctuation cluster, so correlation is strong; if the distance is greater than \(\xi\), they belong to different clusters, and correlation is weak.

The physical origin of this exponential decay lies in locality. Far from the critical point, the system's interactions are short-range, and a spin flip can only be transmitted to distant locations through "relay" by neighboring spins. With each transmission step, the signal decays, so correlation strength decays exponentially with distance. It is like sound propagating in air: the farther the distance, the weaker the sound, and the decay law is exponential.

In this case, the system is "scale-dependent": you have a natural length \(\xi\) that controls all spatial structure. All physical quantities related to space, such as correlation functions, fluctuation cluster sizes, and the response range of the system, are determined by \(\xi\). This "scale-dependent" characteristic allows us to use standard statistical mechanics methods, because the system's behavior is "cut off" beyond the characteristic length \(\xi\).

However, distinguishing "scale-dependent" from "scale-free" is not always straightforward. An example comes from the study of human mobility. For a long time, numerous studies based on mobile phone location data found that the displacement distribution of human mobility follows a power-law form, seemingly indicating that human mobility is "scale-free"—no characteristic distance, with short-distance and long-distance movements equally common. However, a 2020 study published in Nature [Alessandretti et al., Nature 587, 402-407 (2020)] revealed a profound contradiction: although the displacement distribution appears scale-free, human mobility actually has meaningful spatial scales—from neighborhoods, cities, to regions and countries, these "containers" constrain mobility behavior. The scale-free result comes from aggregating displacements across containers: when you mix all movements within containers and movements between containers together, you get a seemingly scale-free distribution. But if these containers are correctly identified, you find that each container has typical scales internally, and the distribution of container sizes also follows a log-normal distribution rather than a power-law distribution.

This example reveals the complexity behind "scale-free" phenomena. In phase transition theory, the system is "scale-dependent" far from the critical point, with correlation length \(\xi\) giving a clear characteristic scale; but the "scale-free" caused by \(\xi \to \infty\) near the critical point is an intrinsic property of the system, not an artifact of the analysis method. The human mobility example shows another possibility: seemingly scale-free power-law distributions may actually originate from the aggregation of multiple scale-dependent distributions. How to distinguish these two cases? The key is to understand the intrinsic structure of the system: if there exists some "container" or "hierarchical structure" that can explain the observed power-law behavior, then the system may still be "scale-dependent"; if the power-law behavior comes from the system's own critical characteristics (such as \(\xi \to \infty\)), then the system is truly "scale-free." This distinction is crucial for understanding scaling behavior in complex systems.

3.3 Critical: Power-Law Decay and ξ → ∞

When the system approaches the critical point, a fundamental change occurs. The correlation length \(\xi\) begins to increase sharply, diverging according to a power law:

\[ \xi \sim |T - T_c|^{-\nu} \]

where \(\nu > 0\) is the correlation length critical exponent. When \(T \to T_c\), \(\xi \to \infty\), meaning that the system loses its intrinsic length scale. Without a characteristic length, the exponential factor \(r/\xi\) in the exponential decay \(e^{-r/\xi}\) approaches zero as \(\xi \to \infty\), and the mechanism of exponential decay fails.

At the critical point \(T = T_c\), the correlation function transforms from exponential decay to power-law decay:

\[ G(r) \sim \frac{1}{r^{d-2+\eta}}, \qquad (T = T_c) \]

Here \(d\) is the spatial dimension, and \(\eta\) is a key critical exponent, often called the anomalous dimension. The origin of this name will be explained in detail in subsequent lectures, reflecting the correction to the scaling dimension of field operators from renormalization in field theory.

The essential difference between power-law decay and exponential decay is: exponential decay has a characteristic length \(\xi\), beyond which correlation becomes very weak; while power-law decay has no characteristic length, correlation strength decays slowly with distance, with non-zero correlation at any distance. It is like in a power network, when the system operates in a critical state, a failure at one node is no longer confined to a local area, but can propagate to arbitrary distances, affecting the entire network.

3.3.1 Why Must Scale-Free Systems Exhibit Power-Law Distributions?

When reading papers on critical phenomena, we almost always see figures like this: the horizontal axis is some physical quantity \(x\) (such as distance \(r\), cluster size \(s\), energy \(E\)), the vertical axis is probability density \(P(x)\) or cumulative distribution, both axes are on logarithmic scales, and then the data points magically line up in a straight line. The slope of this line is the power-law exponent. Why do scale-free systems exhibit this power-law behavior? Why must log-log plots be used? There are deep mathematical and physical reasons behind this.

Mathematical Expression of Scale Invariance

The rigorous mathematical definition of "scale-free" is scale invariance. If a distribution \(P(x)\) is scale-free, then when changing the observation scale, the shape of the distribution should remain unchanged. Specifically, if the variable \(x\) is scaled by a factor \(\lambda\) (i.e., \(x \to \lambda x\)), then the distribution should satisfy:

\[ P(\lambda x) = \lambda^{-\alpha} P(x) \]

Here \(\alpha\) is some constant. The physical meaning of this equation is: after changing the observation scale, the shape of the distribution (except for a normalization factor) remains unchanged. This is exactly the meaning of "scale-free"—no characteristic scale, all scales are equivalent.

Now the question is: what kind of function \(P(x)\) satisfies this scale invariance condition? The answer is: only power-law functions. Let us verify. Assume \(P(x) = C x^{-\beta}\) (where \(C\) is a normalization constant and \(\beta\) is the power-law exponent), then:

\[ P(\lambda x) = C (\lambda x)^{-\beta} = C \lambda^{-\beta} x^{-\beta} = \lambda^{-\beta} P(x) \]

This is exactly the form of scale invariance, where \(\alpha = \beta\). Conversely, it can be proven (by solving the functional equation) that only power-law functions satisfy scale invariance. Therefore, scale-free systems must exhibit power-law distributions—this is not a coincidence, but a mathematical necessity.

Why Use Log-Log Plots?

Log-log plots are ubiquitous in critical phenomena research for two reasons:

1. Power laws become straight lines on log-log coordinates

If \(P(x) = C x^{-\beta}\), then taking logarithms:

\[ \log P(x) = \log C - \beta \log x \]

This is a linear relationship: the plot of \(\log P\) versus \(\log x\) is a straight line with slope \(-\beta\). Therefore, on a log-log plot, power-law distributions appear as straight lines. This not only makes power-law behavior immediately apparent, but also makes extracting power-law exponents from data extremely simple—just measure the slope of the line.

2. Log-log plots can simultaneously display data spanning multiple orders of magnitude

An important characteristic of power-law distributions is spanning multiple orders of magnitude. For example, in critical percolation, cluster sizes can range from a few lattice sites to the entire system size, spanning \(10^0\) to \(10^6\) orders of magnitude. On ordinary linear coordinates, small cluster data points would be crowded together while large cluster data points would be sparsely distributed, making it difficult to observe both simultaneously. But on log-log coordinates, each order of magnitude occupies the same visual space, making the entire distribution structure clearly visible.

Intuitive Understanding of Scale Invariance

Scale invariance can be understood through a simple thought experiment. Imagine a snapshot of a system near the critical point (such as the spin configuration of an Ising model). Now, use a "magic magnifying glass" to enlarge the system by a factor of 2, but miraculously, the enlarged image looks almost exactly the same as the original—the same fractal structure, the same cluster size distribution, the same correlation function shape. This is scale invariance: the system looks the same at all scales, with no "characteristic scale."

This scale invariance directly leads to power-law distributions. Because if the system looks the same at all scales, then small clusters and large clusters should follow the same statistical laws. Assuming the cluster size distribution is \(P(s)\), then "enlarging by a factor of 2" means \(s \to 2s\), and scale invariance requires \(P(2s) \propto P(s)\). The only function form satisfying this condition is the power law \(P(s) \sim s^{-\tau}\).

Identifying Power Laws from Experimental Data

In practical research, how do we determine whether a system truly exhibits a power-law distribution? This is not always straightforward, because:

1. Finite system effects: Real systems are always finite, so power-law distributions always truncate at some upper cutoff. For example, cluster size cannot exceed the total system size \(N\).

2. Finite data: Experimental or simulation data is always finite, and statistical fluctuations make data points for low-probability events (large \(x\) values) sparse and unreliable.

3. Pseudo-power-laws: Some non-power-law distributions (such as log-normal distributions, exponentially truncated power laws) may also appear as straight lines on log-log plots, especially over a limited data range.

Therefore, simply seeing a straight line on a log-log plot is not sufficient to prove power-law behavior. More reliable methods include: - Goodness-of-fit tests: Using Kolmogorov-Smirnov tests or maximum likelihood estimation to evaluate the reasonableness of the power-law hypothesis.

• Finite-size scaling: Checking whether the power-law exponent is independent of system size (true power laws should be).

• Data collapse: Using scaling transformations to collapse data from different parameters onto the same curve.

Relationship Between Power-Law Distributions and Critical Exponents

In critical phenomena, power-law distributions are ubiquitous:

• Correlation function: \(G(r) \sim r^{-(d-2+\eta)}\) at the critical point.

• Cluster size distribution: \(P(s) \sim s^{-\tau}\) at the percolation threshold.

• Energy distribution: \(P(E) \sim E^{-\alpha}\) in certain critical systems.

These power-law exponents (\(\eta, \tau, \alpha\)) are precisely critical exponents, which are interconnected through scaling laws and constitute the complete parameter set describing critical phenomena. In Section 4 we will systematically define these critical exponents.

Now, let us return to the physical picture of "\(\xi \to \infty\)." Combined with the above discussion, we should understand "\(\xi \to \infty\)" as:

• There is no "typical size of fluctuation clusters": Cluster sizes no longer concentrate around some characteristic value, but are distributed across all scales, with contributions from microscopic to macroscopic scales. This is the direct manifestation of the power-law distribution \(P(s) \sim s^{-\tau}\)—clusters of all sizes appear according to the same statistical law.

• Cluster size distribution has contributions at all scales: Small clusters, medium-sized clusters, large clusters, even clusters spanning the entire system, all exist simultaneously. This multi-scale structure is precisely the characteristic of fractal geometry, and is the inevitable result of scale invariance.

• Enlarging/shrinking snapshots of the system, statistical structure remains approximately unchanged: If you enlarge or shrink snapshots of the system, the statistical structure (such as cluster size distribution, correlation function shape) basically remains unchanged. This is the meaning of the "scale-free world," and is why you see straight lines on log-log plots—because the system follows the same power-law rules at all scales.

This "scale-free" characteristic makes systems near the critical point exhibit fractal structures. Just like the 2003 North American blackout mentioned in the introduction, the power network in the critical state formed a correlation structure spanning the entire network scale, and the influence of local failures was no longer confined to neighboring areas, but could propagate to arbitrary distances. This long-range correlation is the direct manifestation of power-law behavior.

3.4 Key Bridge: The Geometric Origin of Response Function Divergence

Now we can answer the question posed at the end of Section 2: why do response functions diverge at the critical point? The answer is hidden in the precise relationship between correlation functions and response functions.

Lecture 3 already gave the static form of the fluctuation-dissipation theorem, which links susceptibility with the integral of the correlation function:

\[ k_B T \chi = \int d^d r \, G(r) \]

The physical meaning of this formula is very profound. The left side is the macroscopic response function \(\chi\), which measures the sensitivity of the system to external magnetic field; the right side is the integral of the correlation function over all space, which measures the cumulative effect of fluctuations in space. This formula tells us: the divergence of response functions is equivalent to the divergence of the spatial integral of the correlation function.

Let us carefully analyze the behavior of this integral. In \(d\)-dimensional space, the volume element \(d^d r\) can be written in spherical coordinates as \(r^{d-1} dr\) times the angular part of the integral. Therefore, the integral \(\int d^d r \, G(r)\) is actually computing \(\int_0^\infty r^{d-1} G(r) dr\).

Far from the critical point: The correlation function decays exponentially \(G(r) \sim e^{-r/\xi}\), and the integral \(\int_0^\infty r^{d-1} e^{-r/\xi} dr\) converges (because the exponential decay is fast enough), so \(\chi\) is finite. This matches our physical intuition: far from the critical point, the influence range of fluctuations is limited, and the system's response is also limited.

Near the critical point: The correlation function decays as a power law \(G(r) \sim 1/r^{d-2+\eta}\), and the integral becomes \(\int_0^\infty r^{d-1} \cdot r^{-(d-2+\eta)} dr = \int_0^\infty r^{1-\eta} dr\). When \(\eta < 1\) (which holds in most systems), this integral diverges as \(r \to \infty\), so \(\chi \to \infty\). This is precisely the geometric origin of response function divergence: the "long tail" of power-law decay causes the spatial integral of the correlation function to diverge, which in turn causes the response function to diverge.

This reasoning is very important because it locks three things together:

\[ \boxed{\text{Correlation length divergence} \Leftrightarrow \text{Power-law correlation} \Leftrightarrow \text{Response divergence}} \]

Correlation length \(\xi \to \infty\) causes exponential decay to fail, and the correlation function transforms from exponential decay to power-law decay; the long tail of power-law decay causes the spatial integral of the correlation function to diverge, which in turn causes the response function \(\chi \to \infty\). These three phenomena are equivalent; they describe the same physical essence near the critical point from different perspectives: the system loses its intrinsic length scale, and the influence range of fluctuations extends to the entire system.

From this moment on, "critical exponents" are no longer abstract mathematical parameters, but directly correspond to experimentally measurable divergence laws. \(\nu\) describes the divergence rate of the correlation length, \(\eta\) describes the power-law decay exponent of the correlation function, and \(\gamma\) describes the divergence rate of the response function. These exponents are interconnected through scaling laws and constitute the complete parameter set describing critical phenomena. In the next section we will systematically define these critical exponents and reveal their intrinsic connections.