The hyperscaling relation, deeply rooted in the theory of Critical Phenomena, provides a pivotal connection between the Singular Behavior of the free energy near a critical point and the spatial dimensions of the system. Specifically, it relates the critical exponents (\nu), (d), and (\alpha) through the equation:
Here, (\nu) is the critical exponent that describes the divergence of the correlation length (\xi) as the critical point is approached, according to (\xi \sim |T - T_c|^{-\nu}), where (T_c) is the critical temperature. The exponent (\alpha) characterizes the divergence of the specific heat (C) near the critical point as (C \sim |T - T_c|^{-\alpha}), and (d) represents the dimensionality of the system.
Physical Interpretation
The hyperscaling relation embodies the notion that the singular part of the free energy density (f_s), which diverges as the critical point is approached, scales with the correlation volume (\xi^d). This suggests that the critical behavior of the system can be understood in terms of fluctuations that occur over regions of size (\xi), with these fluctuations becoming correlated over the entire system as (\xi) diverges at (T_c).
Derivation and Implications
While a rigorous derivation of the hyperscaling relation involves advanced statistical mechanics and field theory, a heuristic argument can be made by considering the scaling behavior of the free energy near the critical point. The free energy density (f) in the vicinity of (T_c) can be expressed as a function of the correlation length, implying that the singular part scales as (f_s \sim \xi^{-d}). From thermodynamics, we know that the specific heat is related to the second derivative of the free energy with respect to temperature, leading to the conclusion that the specific heat scales as (C \sim \xi^{\alpha/\nu}). Equating these expressions gives the hyperscaling relation.
Hyperscaling is crucial for understanding the universality of critical phenomena. It implies that the critical behavior of physical systems is not merely a local phenomenon but is deeply connected to the global, system-wide properties.
Breakdown in Higher Dimensions
The hyperscaling relation holds in systems below their upper critical dimension (d_c), where fluctuations play a significant role in determining the critical behavior. Above (d_c), mean-field theory becomes applicable, and fluctuations are no longer critical in shaping the system’s behavior near (T_c). For example, for the Ising model, (d_c = 4), and above this dimension, the mean-field predictions for critical exponents ((\alpha = 0), (\beta = 1/2), (\gamma = 1), and (\nu = 1/2)) become exact, and hyperscaling does not hold.
Hyperscaling and Renormalization Group (RG) Theory
RG theory provides a formal framework for understanding why hyperscaling holds in lower dimensions and breaks down above (d_c). Through the process of coarse-graining and rescaling, RG transformations reveal how physical quantities like the correlation length and free energy density scale with system size. Fixed points of the RG flow correspond to phase transitions, and the scaling dimensions of relevant operators at these fixed points determine the critical exponents. Hyperscaling then follows from the assumption that the free energy is a homogeneous function of the relevant scaling fields near the critical point.
In summary, the hyperscaling relation is a fundamental aspect of the theory of critical phenomena, encapsulating how macroscopic properties near phase transitions are influenced by the spatial dimensions of the system and the scaling of fluctuations. Its validity in lower dimensions underscores the critical role of fluctuations, while its breakdown in higher dimensions highlights the transition to mean-field-like behavior.