Multicritical Point Principle (MPP)

• Multicritical Point Principle (MPP) is defined as the occurrence of points in a system's phase diagram where distinct critical lines intersect, leading to novel scaling laws from simultaneously diverging fluctuations.
• Analytical methods and large-scale Monte Carlo simulations demonstrate activated scaling laws, characterized by logarithmic time dependencies and precise observable exponents.
• The universality class of MPP bridges classical stochastic models and quantum systems, enabling cross-disciplinary applications such as mapping epidemic spreading phenomena to quantum disorder frameworks.

The multicritical point principle (MPP) describes the emergence and universal properties of special points in the parameter space of models (statistical or quantum) where several distinct critical lines or transition surfaces meet. These multicritical points (MCPs) are characterized by the simultaneous divergence of distinct types of fluctuations, resulting in novel critical behavior and scaling laws not reducible to those at generic (single) critical points. The MPP manifests broadly across equilibrium and non-equilibrium settings, as well as in quantum and classical contexts, including disordered and clean systems. In models ranging from quantum spin systems to random infection processes, MCPs encode the interplay between disparate ordering mechanisms and remain a focal theme for theorists seeking to uncover universal organizing principles of phase diagrams.

1. Definition and Context of the Multicritical Point Principle

The multicritical point principle is exemplified wherever the phase diagram of a system exhibits loci where two or more critical (or transition) lines intersect, producing behavior that cannot be described by any one of the critical lines alone. At such multicritical points, different types of fluctuations—be they geometric, quantum, or disorder-driven—become simultaneously critical, and the underlying theory exhibits more than a single diverging length or time scale.

In classical settings, this includes systems with competing types of order (e.g., magnetism and nematicity), whereas in out-of-equilibrium or disordered models such as the contact process (CP), the MPP arises due to simultaneous onset of generic (dynamical or quantum) and percolation transitions. In the specific context of infection spreading models, the CP under disorder can exhibit two generic transition lines: one driven by dilution (percolation) and another by stochastic local variation in infection rates. The multicritical point arises where both drive the transition on an equal footing (Luzzatto et al., 28 Aug 2025).

2. Methodologies for Identifying and Characterizing Multicriticality

Quantitative investigation of multicriticality employs a combination of analytical and computational tools tailored to the system in question:

• Large-scale Monte Carlo simulations provide direct access to observables (density of infected sites, cluster sizes, survival probabilities) for large disordered CP lattices at or near criticality. For the multicritical CP, lattices are generated at the percolation threshold, and dynamics are run with variable local infection rates (Luzzatto et al., 28 Aug 2025).
• Scaling analysis with activated dynamics: Unlike ordinary transitions characterized by algebraic power-law scaling, the multicritical regime can display activated (ultra-slow) scaling. Observable quantities, such as the density ρ(t), number of active sites N(t), survival probability P(t), and rms radius R(t), exhibit scaling with logarithmic time dependence:

$$\rho(t) \sim \left[\ln(t/t_0)\right]^{-\bar{\delta}}$$

$$N(t) \sim \left[\ln(t/t_0)\right]^{\bar{\Theta}}, \quad R(t) \sim \left[\ln(t/t_0)\right]^{1/\psi}$$

The values of the exponents are determined from fits of these forms over long simulation times.

• Scaling relations between observables: To remove the influence of unknown nonuniversal time scales ($t_0$), scaling plots of one observable versus another (e.g., $N$ vs $R$, or $R$ vs $P$) are used. According to the strong disorder renormalization group (SDRG) theory, these plots collapse for correct exponent combinations:

$$R \sim P^{-1/(\psi \bar{\delta})}, \quad N \sim P^{-\bar{\Theta}/\bar{\delta}}, \quad N \sim R^{\psi\bar{\Theta}}$$

• Comparisons with SDRG predictions: The values of exponents and scaling relations are compared to those predicted by SDRG, which was originally developed for quantum disordered models (e.g., the random transverse-field Ising model).

3. Universal Scaling Behavior and Infinite Disorder Fixed Points

The defining feature of the multicritical CP is that its scaling behavior is governed by an infinite disorder fixed point (IDFP). At an IDFP, disorder strength increases without bound under renormalization, and the system’s dynamics become ultra-slow—no longer characterized by conventional power-law time scaling but by scaling in terms of logarithm of time. This structure leads to universal exponent combinations ($\bar{\delta}$, $\bar{\Theta}$, $\psi$) with values independent of the microscopic details of the disorder, such as whether it is in the infection rate or the geometry of the underlying lattice (Luzzatto et al., 28 Aug 2025).

The activated dynamic scaling at the MCP strongly contrasts with generic (single-disorder) critical behavior, making the multicritical point both theoretically significant and practically distinguishable by simulation.

4. Universality Class and Mapping to Quantum Models

Monte Carlo and SDRG results indicate that the universal properties of the multicritical CP coincide with those of the multicritical random transverse-field Ising model (RTIM) (Kovács, 2021). This includes the precise form of activated scaling and the associated critical exponents. Despite fundamental differences (the RTIM being an equilibrium quantum model, the CP an out-of-equilibrium classical stochastic process), their multicritical points are described by the same universality class.

This cross-connection enables transfer of techniques and interpretation. For example, concepts developed for quantum entanglement scaling in the RTIM may be implemented in classical CP simulations by exploiting the universality at their respective MCPs.

5. Implications and Prospects

The universal multicritical scaling laws identified in the CP are robust in both two and three dimensions, and simulations confirm that their exponents satisfy hyperscaling relations demanded by the theory. Practical consequences include:

• Experimental Relevance: The scaling predictions provide targets for experimental realizations of epidemic spreading in disordered media or analogous systems (e.g., percolating Josephson arrays, ultracold atom lattices with engineered disorder).
• Theoretical Trajectory: The equivalence of universality classes between classical and quantum models suggests that quantum phase transition properties (including entanglement) may be probed through classical simulations. Future work may focus on mapping quantum entanglement measures to observables accessible in large-scale classical Monte Carlo simulations, leveraging the established parallelism.

6. Representative Scaling Relations and Observable Exponents

The table below summarizes key activated scaling laws and their associated exponents at the multicritical point:

Observable	Scaling Law	Physical Meaning
Infection Density ρ(t)	$[\ln(t/t_0)]^{-\bar{\delta}}$	Decay of activity
Number N(t)	$[\ln(t/t_0)]^{\bar{\Theta}}$	Growth from initial seed
Radius R(t)	$[\ln(t/t_0)]^{1/\psi}$	Spreading of cluster
Survival Prob. P(t)	$[\ln(t/t_0)]^{-\bar{\delta}}$	Probability still active

Correct combinations of these exponents are extracted from scaling collapses in simulation, and agree with SDRG predictions for the IDFP universality class.

7. Summary

The multicritical point principle, as seen in the disordered contact process, highlights the emergence of universal behavior where different forms of disorder jointly control the critical properties. The resultant critical points are not simply generically critical, but display distinct scaling (ultraslow, activated dynamics) and belong to the universality class of the multicritical quantum Ising model. This profound connection between different classes of models enables both cross-disciplinary transfer of analytical and computational tools and opens new prospects for probing quantum properties using large-scale classical simulations (Luzzatto et al., 28 Aug 2025).