Microscopic origin of abrupt mixed-order phase transitions

Abstract

We suggest a possible origin for abrupt mixed-order transitions in physical systems and demonstrate it on three different Ising models with additional different types of interactions. We identify a plausible microscopic origin of this abrupt transition. It is driven by long-term microscopic cascades of changes in the underlying interaction network due to the additional interaction. These spontaneous cascades of microscopic changes accumulate over macroscopic time, resulting in a long-term metastable cascading plateau that ultimately causes an abrupt transition of the system. We also calculate the critical exponents for the cascading, magnetization, convergence slowing down, and the typical fluctuations of single-trajectory critical temperature and magnetization. The developed approach and our findings can shed light on the microscopic mechanism at the origin behind many abrupt transitions in nature and technology.

Introduction

Many physical, biological, and socio-economic systems exhibit phase transitions between their macroscopic states^1,2. One distinguishes between two main types of phase transitions³, first-order (also called abrupt) transitions characterized by abrupt changes of an order parameter and second-order transitions that exhibit continuous changes of an order parameter and thus perform smoother transitions. Typically, a second-order transition exhibits diverging correlation length at the critical point. Since, in many systems, the correlation length is typically related to susceptibility, a second-order transition also exhibits divergent susceptibility at the critical point.

Mixed-order transitions are phase transitions that show the properties of both first-order and second-order transitions—they exhibit an abrupt change of the order parameter while the correlation length also diverges. This phenomenon is also called the Thouless effect and was first discovered in long-range spin systems⁴. These transitions recently received much attention in various applications⁵, long-range spin models⁶, phase transitions of colloidal crystals⁷, as well as percolation in complex networks⁸ or in the dynamics of social networks⁹.

While both types of phase transitions have been extensively studied in the context of many physical¹⁰ and other complex systems^11,12, the microscopic mechanism of these transitions and, in particular, the mechanism origin of the abrupt transition has not yet been fully understood. Here, we demonstrate that a possible origin of abrupt transitions is due to an additional long-range interaction, and the mechanism is long-term microscopic changes.

While in physics, many phenomena are dominated by a single type of interaction, still many natural complex systems are often characterized by multiple interaction types between their components². One example is social systems, where individuals interact through transferring information, exchange of goods and services, financial transactions, etc. Other examples with two types of interaction include adaptive epidemic spreading dynamics¹³, higher-order interactions of social contagion model¹⁴, or emergent complex interactions in colloidal systems⁷, evolutionary and catalytic sets (that are nowadays sometimes referred to as higher-order networks or hypergraphs)¹⁵, or infrastructures and interdependent superconducting networks with dependency and connectivity links^16,17.

Recently, the conditions for the transition between first-order and second-order transition have been studied in the context of dynamical systems¹⁸, where it has been argued that such behavior can be realized by adding a second parameter to a generic set of dynamic equations. The presence of two parameters is necessary to observe the shift in the order of the phase transition: In systems with a single parameter, the change of the parameter can alter the critical point but not the order of the transition.

While the presence of the first-order or second-order transition can be well predicted, their microscopic origin remains unclear. In the case of the so-called mixed-order transitions, it has been argued¹⁹ that often, such a transition might have a microscopic origin. This is connected to the fact that the system undergoes a series of microscopic changes that cannot be revealed at first sight from macroscopic quantities but finally cause the system to collapse into another phase.

In this paper, we provide a study of three examples of systems with two types of interaction, all showing the clear microscopic origin of the mechanism that yields the abrupt transition as well as a transition from the second-order to the first-order transition by strengthening one of the interactions of the system. To this end, we employ three simple models based on the regular Ising model and a second type of interaction, which is different for each model. In the first example, we assume a regular Ising model on a random network, in which the second type of interaction is acting on the underlying connectivity network, where any two particles can form a spin-neutral molecule that does not interact with the rest of the system anymore. We call this model the Ising model with molecule formation.

In the second example, we investigate again the Ising model on a random network, but particles of the system have the possibility to change their state to yet another set of hidden states, where they do not interact with the rest of the particles anymore. This model is a special type of a Potts model, so we call it the Potts model with hidden (or invisible) states.

In the final example, we consider the nearest-neighbor Ising model on a one-dimensional chain. The second type of interaction is obtained by allowing long-range interactions, but only on subchains of the same spins. Therefore, by changing a spin on a chain, we also change the effective interaction network of the system. We call this system the Truncated long-range Ising model.

In the first two examples, we derive the phase transition by using the mean-field approach, which is exact on random networks and the configuration model. In the third case, we use the theory presented by Bar and Mukamel in²⁰ to derive the value of the critical temperature in the thermodynamic limit. In all three examples, we find that when the coupling of the second interaction is strong enough, we observe the mixed-order phase transition, i.e., a transition with both a sudden change of the order parameter as well as diverging susceptibility.

By carefully analyzing the origin of the mixed-order phase transition, we find out that the mechanism leading to the abrupt transition is caused by a long-term microscopic cascade of changes in the underlying interaction network that occurs spontaneously during the abrupt transition, which is represented by a long-term plateau of the order parameter. These spontaneous cascading changes during the plateau effectively sparsify the underlying interaction network, which finally leads to a sudden drop in the order parameter. In all three examples, the phase transition is from the ordered phase to the disordered phase. We also find that while the macroscopic critical exponents are the same for all three models, the critical exponents related to the time necessary to converge (plateau time) from the ordered phase to the disordered phase above the critical temperature depends on the specific microscopic interactions of each system.

The structure of the paper is as follows. We start by introducing the three different models based on the Ising model. We derive the equilibrium distribution and the corresponding phase diagrams for all three models. We also confirm the theoretical results by the Monte Carlo simulations based on the Metropolis–Hastings algorithm. First, we introduce the Ising model with molecule formation. For this model, we derive the entropy describing the molecule formation on the random network. Second, we introduce the Potts model with hidden states, where the presence of hidden (i.e., non-interacting) states effectively changes the underlying interaction network. Finally, we present the one-dimensional Ising model with long-range interactions restricted to the subchains of the same spin.

To identify and understand the underlying microscopic mechanisms behind the phase transitions, we follow and identify the dynamics of the cascading failures (i.e., microscopic gradual changes of the underlying network along a plateau until the system suddenly changes the value of the order parameter) and compute the scaling exponents of their convergence times (plateau length) above the critical temperature, which we find to be of macroscopic time scales. We also study the scaling of the standard deviation of the critical temperature and critical magnetization at the transition point obtained from single-trajectory realizations. Finally, we discuss the generality of the model.

The importance of this work arises from the fact that many complex systems have multiple interaction types. If these systems generally have an abrupt transition, our message is that these transitions are controlled by a cascade of microscopic changes that last for macroscopic time scales that scale with the system size. The scaling exponents related to the transition time (duration of the plateau) are determined by the microscopic interactions of the system and its macroscopic size.

Results

Ising model with molecule formation

As a first transparent example system that manifests the more general behavior of systems with two types of interactions, we employ an Ising-like interaction model. Let us consider a system of n particles on a random network. They can be in one of two free states: either a positive spin, denoted by s_↑ = +1, or a negative spin, denoted by s_↓ = −1. Alternatively, they can become a part of a molecule. For simplicity, we assume that the particles can only form two-particle molecules of one type, denoted as s_∥. However, generalization to larger molecules and more molecule types is straightforward. Let us assume that at a given snapshot, there are n_↑ particles with a positive spin, n_↓ particles with a negative spin, and n_∥ molecules. By summing up all particles, we have n_↑ + n_↓ + 2n_∥ = n.

Any two particles joined by a link on the underlying network can then form a molecule that is effectively spinless, i.e., the molecule does not interact with other particles in a spin–spin interaction and effectively reduces the strength of the spin–spin coupling by reducing the average degree of the network. For simplicity, they can form these molecules regardless of their actual spin value, and once particles form a molecule, they no longer interact with the rest. Formally, we set the spin of the particles in the molecules as s_i = 0.

Further, the system can evolve by one of the following state transitions: any two particles can join and form a new molecule, an existing molecule can fall apart to form two free particles, and a free particle can flip a spin. An example of the system, as well as its evolution, is depicted in Fig. 1.

Fig. 1: Ising model with molecule formation.

Demonstration of the possible states of the system’s initial configuration and possible evolution steps of the Ising model with molecule formation. Each particle can be in three possible states: spin up (yellow), spin down (blue), or be part of a molecule (red) with effectively zero spin. Molecules are formed on the underlying network; any two particles joined by a link can form a molecule. The underlying network is effectively evolving due to the formation of molecules. While the Ising-interaction links between nearest-neighbor-free particles (green) are active, links to particles that are part of a molecule are inactive (gray). The system can evolve by randomly choosing two neighboring particles and creating a new molecule, destroying a molecule and making the particles free, or changing the spin of one of the particles. The first two types of evolution effectively change the interaction network of the system.

To obtain the equilibrium distribution of the system, we calculate the multiplicity of structure-forming systems on a network. We then use it to calculate the entropy using Boltzmann’s formula to obtain the equilibrium distribution by maximizing the entropy of the system.

The multiplicity W of a state characterized by {n_↑, n_↓, n_∥} can be calculated as a product of two terms: the first term, denoted by Ω, corresponds to how many ways we can divide n particles to n_i particles of state s_i and n_∥ molecules, the second term, denoted by M, corresponds to how many ways we can make n_∥ molecules from 2n_∥ particles so that the particles forming the molecules are connected by a link in the underlying network. The first term is a multinomial factor $\Omega ({n}_{\uparrow },{n}_{\downarrow },{n}_{\parallel })=\left(\begin{array}{c}n\\ {n}_{\uparrow },{n}_{\downarrow },2{n}_{\parallel }\end{array}\right)$.

The second term, M, can be calculated as follows: we randomly choose two adjacent particles to form a molecule. This is equivalent to randomly selecting a link in the underlying network and removing these two nodes from the network. Similarly, we perform the same procedure with the second molecule until we form all the n_∥ molecules. We assume that the underlying interaction network is a random network of n nodes (also called Erdös–Rényi graph) where each link has the same probability of occurring between any pair of nodes. As a result, a random network can be entirely determined by defining a single parameter, i.e., the average degree k of the network. In the Supplementary Information (SI), we show that under the assumption of a random network, M can be expressed as

$$M({n}_{\parallel })=\frac{(2{n}_{\parallel })!}{{n}_{\parallel }!}{\left(\frac{k}{2(n-1)}\right)}^{{n}_{\parallel }}\,.$$

(1)

The mechanism of link deletion is illustrated in  Supplementary material (Supplementary Fig. 1). The corresponding entropy is $S\equiv \log W=\log (\Omega \cdot M)$ (setting k_B = 1). By using the Stirling approximation, we end with

$$S({\wp }_{\uparrow },{\wp }_{\downarrow },{\wp }_{\parallel })= -{\wp }_{\uparrow }(\log {\wp }_{\uparrow }-1)-{\wp }_{\downarrow }(\log {\wp }_{\downarrow }-1)\\ -{\wp }_{\parallel }(\log {\wp }_{\parallel }-1)-{\wp }_{\parallel }\log \left(\frac{2(n-1)}{kn}\right)$$

(2)

where ℘_↑ = n_↑/n, ℘_↓ = n_↓/n and ℘_∥ = n_∥/n. We define the effective degree as $\kappa=k\frac{n}{n-1}$ that describes the finite-size correction to the degree, and ${\lim }_{n\to \infty }\kappa (n)=k$. For the limiting case of a fully connected network, i.e., k = n − 1, we get κ = n, and the multiplicity boils down to the multiplicity derived in ref. ²¹.

We assume that the spin–spin interaction is driven by the Ising-like Hamiltonian, i.e., $H(s)=-\frac{J}{k}{\sum }_{i\ne j}{A}_{ij}{s}_{i}{s}_{j}$, where k is the average degree of the underlying network, and A_ij is the adjacency matrix. The Hamiltonian is normalized by k so that the Currie–Weiss temperature without molecule formation is independent of k, i.e., is T_C = J. In addition, any two neighboring particles can form two-particle molecules, as described above.

To calculate the equilibrium distribution, we use the Hamiltonian mean-field approach and approximate the Hamiltonian by the mean-field version H(s) = − Jm∑_is_i, where m = 〈s_i〉 is the average magnetization. We also define the total magnetization of the system, which is simply defined as M = m ⋅ n. Furthermore, to decouple the Hamiltonian into the single-spin contributions, we use the so-called configuration model in which the actual adjacency matrix is approximated by a probability that nodes i and j are linked based on the respective degrees of i and j. Therefore, the adjacency matrix is approximated in the configuration model as ${A}_{ij}\approx \frac{{k}_{i}{k}_{j}}{2k}$. These approximations are particularly valid in the thermodynamic limit, i.e., when the number of particles and the number of links goes to infinity, such as their ratio is a constant. Since the average degree is directly two times the number of links over the number of nodes, by keeping the average degree constant, we do not change the density of the interaction network.

The equilibrium distribution is obtained by maximizing the entropy (2) with respect to the average energy of the system. By considering α as the Lagrange multiplier corresponding to the normalization constraint ℘_↑ + ℘_↓ + 2℘_∥ = 1 and β is the inverse temperature, we obtain

$${\wp }_{\uparrow } =\exp (-\alpha+\beta Jm)\\ {\wp }_{\downarrow } =\exp (-\alpha -\beta Jm)\\ {\wp }_{\parallel } = \frac{\kappa }{2}\exp (-2\alpha )$$

(3)

where α ensures the normalization of the distribution, and β is the inverse temperature, i.e., β = 1/T. ℘_↑ is the probability of a particle of having spin s = +1, ℘_↓ is the probability of a particle of having spin s = −1, and ℘_∥ is the “probability” of observing a molecule (s = 0). α can be calculated from the normalization condition, which leads to a quadratic equation in e^−α that can be solved as

$${e}^{-\alpha }=\left(-\cosh (\beta Jm)+\sqrt{{\cosh }^{2}(\beta Jm)+\kappa }\right)/\kappa \,.$$

(4)

The self-consistent equation for the magnetization m of the system can be obtained either from the standard procedure of adding an external field to the Hamiltonian—h∑_is_i and then expressing $m=-\frac{\partial F}{\partial h}{| }_{h=0}$. Nevertheless, here, the magnetization can also be calculated in a straightforward way using m = ∑_k℘_ks_k = ℘_↑ − ℘_↓, where we plug in the probabilities from Eq. (3) which leads to the self-consistent equation

$$m=2{e}^{-\alpha }\sinh (\beta Jm)\,$$

(5)

where the term e^−α is from Eq. (4).

By solving the self-consistent equation (5) for each combination of T, and κ, we obtain the phase diagram of the system. Let us discuss the dependence of the phases on κ. To explore the phase diagram of the system, we approximate the right-hand side of Eq. (5) by its Taylor expansion around m =  0 to the second non-zero term (that corresponds to m³ since the right-hand side is an odd function) and obtain

$$m\approx \, \, 2J\beta \frac{\sqrt{\kappa+1}-1}{\kappa }m \\ +\frac{{J}^{3}{\beta }^{3}}{3\kappa \sqrt{\kappa+1}}(\sqrt{\kappa+1}-1)(\sqrt{\kappa+1}-3){m}^{3}\,.$$

(6)

By solving this equation for m, we obtain three solutions, one trivial m₀ = 0 and two non-trivial solutions  ± m(β, κ). The solutions are presented in the SM.

By comparing where the trivial and the non-trivial solution coincide, i.e., taking m(β, κ) = 0, we get ${T}_{S}(\kappa )=\frac{2J}{\sqrt{\kappa+1}+1}$, where T_S denotes the spinodal temperature. Further, one can (in analogy to ref. ¹⁸) determine the order of the phase transition from the sign at the cubic term in Eq. (6). The system exhibits a first-order transition if the coefficient is positive and a second-order transition if the coefficient is negative. Here, we obtain that the coefficient is zero if $\sqrt{\kappa+1}=3$, i.e., the transition between the second-order transition and the first-order transition happens for κ = 8. Below this degree, no abrupt first-order transition can occur.

This is shown in Fig. 2a, where we depict the phase diagram of the Ising model with molecule formation. In Fig. 2b–e, we compare the theoretical results with the Monte Carlo simulations based on the Metropolis–Hastings algorithm described in  Supplementary Information. We present two specific values of k, i.e., k = 6 (Fig. 2b, d) and k = 50 (Fig. 2c, e) for n = 100 in order to demonstrate that the first-order transition happens only for the latter case. In the case of the first-order transition, we observe a hysteresis, where the convergent state depends on the initial condition. We see that the transitions are different for cooling the system down (lower transition) or heating it up (higher transition). This is in agreement with theoretical results where we observe the presence of two stable states in the hysteresis region.

Fig. 2: Phase diagram of the Ising model with molecule formation.

a We depict the phase diagram for the Ising model with molecule formation as a function of two main parameters of the system, i.e., temperature T and network degree k. b–e We compare the theoretical solution (red line for a stable solution, gray line for an unstable solution) with the MC simulation with n = 100 particles for specific values of average degree k (for b, d k = 6, for c, e k = 50). MC simulations are initialized in two states: ordered state ℘_↑(0) = 1, ℘_↓ = 0 (b, c) and disordered state ℘_↑ = ℘_↓ = 0.5 (d, e). b, c We observe that for low k, we observe a second-order continuous phase transition with no hysteresis appearing. d, e We see that for large k, we observe an abrupt first-order phase transition with the presence of hysteresis, where the system converges abruptly to a different state, depending on whether the system is cooling or heating.

In the Supplementary Information, we also discuss the situation where the number of particles that can form molecules is restricted. This can be interpreted as the situation when the molecule formation is mediated by a catalyst whose amount is restricted. In this case, we observe yet a richer phase diagram, which can be derived from combining the unrestricted regime with the regime when the number of molecules is fixed at the maximum amount. The phase diagram and various MC simulations for various configurations are depicted in Supplementary Figs. 2 and  3.

Potts model with hidden states

As a second example, we investigate the Potts model with hidden states, as demonstrated in Fig. 3, see also ref. ²². The system consists of n particles with possible spins s_i ∈ { − 1, 0, 1}. Again, we denote the positive spin as s_↑ and the negative spin as s_↓. Here, the hidden state (i.e., spin 0) has a non-trivial degeneracy, i.e., there are g different states (which can be represented as {0₁, 0₂, …, 0_g}) which do not interact with other particles. The illustration of the system’s possible states and possible evolution is shown in Fig. 3. Similar to the previous example, the Hamiltonian can be written in the same form, i.e., $H(s)=-\frac{J}{k}{A}_{ij}{s}_{i}{s}_{j}$, where A_ij is the adjacency matrix. The Hamiltonian is again normalized by the average degree k so that the Currie–Weiss temperature without the hidden states is T_C = J. Again, we assume that the underlying interaction network is an Erdös–Rényi random network so that the adjacency matrix can be approximated by the configuration model. In the mean-field approximation, the Hamiltonian can be rewritten as H(s) = − Jm∑s_i and the equilibrium distribution is

$${p}_{\uparrow } =\exp (-\alpha+\beta Jm)\\ {p}_{\downarrow } =\exp (-\alpha -\beta Jm)\\ {p}_{0} =g\exp (-\alpha )$$

(7)

The normalization is then ensured by $\exp (\alpha )=2 \cosh (\beta Jm)+g$, which is a partition function.

Fig. 3: Potts model with hidden states.

Demonstration of the possible states, system’s initial configuration, and possible evolution steps of the Potts model with hidden states. Each particle can be in different possible states: spin up (yellow), spin down (blue), or in one of the g hidden states with effectively zero spin (shades of gray). The underlying network is effectively evolving due to the presence of hidden states. While the Ising-interaction links between the particles with spin up and down (green) are active, links to particles that are in a hidden state are inactive (gray). The system can evolve by changing one hidden spin to an active state (spin up or down), changing one active state to a hidden state, or flipping one active (or hidden) state to another active (or hidden) state. Again, the first two types of evolution effectively change the effective interaction network while the other two types leave the interaction network intact.

From Eq. (7), it is straightforward to obtain the self-consistent equation

$$m=\frac{2\sinh (\beta Jm)}{g+2\cosh (\beta Jm)}\,.$$

(8)

We can again expand the self-consistent equation around m = 0

$$m\approx \frac{2J\beta }{g+2}m+\frac{{J}^{3}{\beta }^{3}(g-4)}{3{(g+2)}^{2}}{m}^{3}\,.$$

(9)

In the SM, we derive that the spinodal temperature is equal to ${T}_{S}(g)=\frac{2J}{2+g}$. From the cubic term, we can finally deduce that the system changes the order of the phase transition when g = 4. For g < 4, we observe a second-order transition, while for g > 4, we observe a first-order, abrupt transition.

In Fig. 4a, we depict the phase diagram of the Potts model with hidden states. In Fig. 4b–e, we compare the theoretical results with the Monte Carlo simulations based on the Metropolis–Hastings algorithm. We again show two specific choices of hidden states g, i.e., g = 3 (Fig. 4b, d) and g = 10 (Fig. 4c, e) for n = 100 to demonstrate that the first-order transition happens only for the latter case. In the case of the first-order transition, we again observe a hysteresis, where the convergent state depends on the initial condition.

Fig. 4: Phase diagram of the Potts model with hidden states.

a We depict the phase diagram for the Potts model with hidden states as a function of two main parameters of the system, i.e., temperature T and number of hidden states g. b–e We compare the theoretical solution (red line for a stable solution, gray line for an unstable solution) with the MC simulation with n = 100 particles for specific values of g (for b, d, g = 3, for c, e, g = 10). MC simulations are initialized in two states: ordered state ℘_↑(0) = 1, ℘_↓ = 0 (b, c), and disordered state ℘_↑ =  ℘_↓ = 0.5 (d, e). b, c We observe that for low k, we observe a second-order continuous phase transition with no hysteresis. d, e We observe that for large k, we observe an abrupt first-order phase transition with the presence of hysteresis, where the system converges to a different state, depending on whether the system is cooling or heating.

Finite-size truncated long-range Ising model

As our final example, we consider yet another version of the Ising model: the Truncated long-range Ising model. Ising models with long-range interactions have been extensively studied for a long time²³. While solving the Ising model with long-range interactions in multiple dimensions is a very challenging task, it can be solved in one dimension, as shown by Thouless and others⁴. One particular version of it has been studied and solved by Bar et al.⁶, where the long-range interactions are confined to the domains of constant spin; therefore, the model is called the Truncated inverse-squared distance Ising model. In this model, the long-range coupling has a strength proportional to the inverse-squared distance between the spins on the chain, i.e.,  − J(∣i − j∣)s_is_jI(s_i, s_j) where J(r) = C/r² and I(x, y) = 1 if x = y and 0 otherwise. This model can be related to the popular model of DNA denaturation, also called the Poland–Scheraga model²⁴. Interestingly, the Truncated inverse-squared Ising model exhibits an abrupt, first-order transition in magnetization while simultaneously exhibiting a second-order transition in another order parameter, namely the number of domains (i.e., the chains of constant spin.)

Here, we investigate a slightly more general version of the Truncated long-range Ising model, which can be defined by the Hamiltonian

$$H(s)=-J{\sum }_{i=1}^{N}{s}_{i}{s}_{i+1}-{\sum}_{i < j}\frac{C}{| i-j{| }^{\alpha }}{s}_{i}{s}_{j}I({s}_{i},{s}_{j})\,.$$

(10)

This system is called Truncated long-range Ising model. We depict its possible evolution in Fig. 5. The whole system can be represented in terms of domains, i.e., chains of constant spin, which are tight together by the long-range interactions. Therefore, in the truncated long-range Ising model, the structure of the interaction network changes with the spin flips.

Fig. 5: Truncated long-range Ising model.

Demonstration of initialization and possible evolution of the Truncated long-range Ising model. Each particle can be in two possible states: spin up (yellow) or spin down (blue). Here, the particles are on a one-dimensional chain, where the nearest-neighbor interactions are present (blue). Furthermore, for the subchains of the same spin, the long-range interactions (red) of the strength are proportional to $\frac{1}{{r}^{\alpha }}$, where r is the distance between the particles. Thus, the interaction network is effectively evolving due to the spin flips. In the right-hand figure, we see simulations of the phase transition of the finite system with n = 400 spins for C = 1, J = 1, and α = 1.5. Again, we observe an abrupt phase transition, but since the interactions are long-range, the critical temperature is size-dependent. We show the estimated critical temperature for the finite size of the system, and for comparison, we also depict the theoretical critical temperature that can be obtained from the theory in the thermodynamic limit.

By denoting the lengths of a chain as l_i, the total length (corresponding to the number of particles) as L =  ∑_il_i, and the number of domains as N, the Hamiltonian can be conveniently rewritten as

$$H(l)=J(L+1-2N)-C{\sum }_{i=1}^{N}{\sum}_{k}\frac{{l}_{i}-k}{{k}^{\alpha }}\,.$$

(11)

As shown in ref. ²⁰, and briefly recalled in  Supplementary Information, it is possible to obtain the equation for the critical temperature T_C = 1/β_C in the thermodynamic limit, which reads:

$$\exp (2J{\beta }_{C})={\sum}_{l}\exp \left(-{\beta }_{C}C\left(\frac{1}{\alpha -1}-\frac{1}{\alpha -2}\right){l}^{2-\alpha }\right)\,.$$

(12)

However, due to the long-range nature of the interactions, the critical temperature will be slightly different for the finite version of the system. In Fig. 5, we depict the phase diagram for C = 1, J = 1, and α = 1.5 as a function of temperature T. We observe that the phase transition is of the first order and the critical temperature is slightly smaller (T_C(n = 400) ≈3.2) than the theoretical value obtained in the thermodynamic limit (T_C(n → ∞) ≈ 3.98).

Critical exponents

For all three aforementioned models, we run Monte Carlo simulations based on the Metropolis algorithm described in  Supplementary information (Supplementary Algorithm). The link to the simulation codes is provided at the end of this paper. We use the simulations to compute the critical behavior near the critical temperature. This allows us, as discussed below, to understand the detailed microscopic mechanism behind this phase transition, obtain the critical scaling exponent of the order parameter, and compare it among the three example systems to see whether they fall into the same universality class. The illustration of how the scaling exponents are estimated is depicted in Fig. 6. Furthermore, the microscopic origin of the mixed-order transition (as discussed below) is depicted in Fig. 7.

Fig. 6: Estimation of critical scaling exponents.

Left: Illustration of estimation of average time to converge (τ, plateau length). For a temperature slightly above the critical temperature, the system is initialized in an ordered state m = 1, and the temperature is shifted just above T_c. Then, the system evolves as a plateau until it spontaneously reaches a disordered state m ≈ 0. For each realization, we measure the length of the trajectory (i.e., the number of MC steps at the plateau). Right: Illustration of the estimation of the variance of single-trajectory critical temperature and magnetization. a We see an example of several trajectories starting in the ordered phase (m = 1) below the critical temperature. With increasing temperature to just above T_C, the trajectory eventually reaches the disordered phase phase (m ≈ 0). b We measure the critical temperature and magnetization at the abrupt phase change for each realization and make a scatter plot. We construct the histograms for both temperature and magnetization and calculate their standard deviation in (c, d), respectively.

Fig. 7: Microscopic origin of the abrupt, mixed-order phase transition.

Left: a typical trajectory of all accepted changes starting from the state ∣m∣ = 1 just above critical temperature T_c. Here, M denotes the total magnetization (M = m ⋅ n) and τ^ac is the number of accepted MC steps (most of the MC suggested MC steps are rejected, as discussed in the text). After the initial period of small microscopic fluctuations, the trajectory finally converges to its equilibrium state with m ≈ 0. We denote the different types of transitions by different colors: adding a molecule in red, removing a molecule in green, flipping a spin up in yellow, and flipping a spin down in purple. In this case, due to the large average degree (k = 50), most changes include adding a molecule or removing a molecule, which affects the underlying interaction network. These microscopic steps of adding and removing molecules change the effective network topology (green links) from a very dense (a) over less dense networks (b, c) to the final state (d) where most particles form molecules (in red) and the effective network becomes very sparse showing zero macroscopic magnetization.

The first critical exponent that is investigated is the critical exponent β for $| m(T)-m({T}_{C})| \propto {({T}_{C}-T)}^{\beta }$. For the standard mean-field Ising model, we have m(T_C) = 0 and $\beta=\frac{1}{2}$. For all three models, the transition is abrupt, and the scaling near the critical exponent is depicted in Figs. 8a, 9a, and 10a, respectively, where we observe that for all three models, the exponent is β ≈ 0.5.

Fig. 8: Scaling exponents for the Ising model with molecule formation.

In all cases, k = 49. a We see that the scaling exponent β related to the scaling of magnetization is β ≈0.5. b The scaling exponent δ related to the plateau time is δ ≈1.5. c The scaling exponent related to the plateau time as a function of system size is γ ≈1.0. Finally, in (d), we observe the scaling exponents related to the variance of single-trajectory temperature η ≈ 0.5. Error bars are defined as standard deviations between runs.

Fig. 9: Scaling exponents of the Potts model with hidden states.

In all cases, g = 10. a We see that the scaling exponent β related to the scaling of magnetization is β ≈0.5. b The scaling exponent δ related to the plateau time is δ ≈1.0. c The scaling exponent related to the plateau time as a function of system size is γ ≈1.0. Finally, in (d), we observe the scaling exponent related to the variance of single-trajectory temperature η ≈ 0.5. Error bars are defined as standard deviations between runs.

Fig. 10: Scaling exponents of the truncated long-range Ising model.

Here, C = 1, J. a We see that the scaling exponent β related to the scaling of magnetization is β ≈0.5. b The scaling exponent δ related to the plateau time is δ ≈ 0.8. Error bars are defined as standard deviations between runs.

In all three cases, the transition is simultaneously first and second-order (i.e., mixed-order) phase transition since we observe that m(T) becomes discontinuous at T_C and, additionally, the system exhibits the so-called square-root singularity, i.e., that the magnetization scales with $\beta=\frac{1}{2}$, for which $\frac{\partial m(T)}{\partial T}$ becomes infinite for $T\to {T}_{C}^{-}$. This value of $\beta=\frac{1}{2}$ seems to be universal for mixed-order transitions. Indeed, also for interdependent networks, it is found that $\beta=\frac{1}{2}$^16,17.

To understand the mechanism behind the emergence of the abrupt, first-order transition in all three cases, we investigate how the system evolves from one phase to another when the critical temperature is crossed. This behavior cannot be captured by the theoretical description of a system in a thermodynamic limit. One of the important characteristics of the system is to follow the process of crossing the critical point and measuring the time it takes to converge to a new equilibrium state. This process and its characteristic time can tell us both the mechanism and how fast the system collapses after it surpasses its tipping point. It has been demonstrated that, for the percolation of abstract interdependent networks, the convergence time gets longer when approaching the critical point in the case of mixed-order transitions due to criticality^19,25.

We initialize the system to have the temperature just above the critical temperature, where the equilibrium magnetization corresponds to the disordered phase, but we set the initial distribution as it is still in the ordered phase (∣m∣ ≈ 1), and ask how long it takes the system to reach equilibrium. In Fig. 7, we depict one sample trajectory of the MC simulation. We observe a long-term plateau where, due to the high degree (k = 49), most accepted transitions either add or remove a molecule, i.e., a microscopic change. These transitions lead to effective changes in the network topology (i.e., the network structure) by effectively decreasing the network average degree. We observe that for a long time, when the system fluctuates around m ≈ (0.8, 1), it forms a plateau without any clear sign of further decrease. After some time, the system suddenly drops to its equilibrium state m ≈ 0. This behavior is found for this critical transition when the convergence time near the critical temperature tends to diverge. Note that the microscopic changes both in the size of the network and its topology are the main drivers for the sudden transition.

We further investigate the properties of the plateau and its scaling properties. In Fig. 6, we show several trajectories evolving just above the critical temperature and their time (i.e., the number of MC steps) to converge. First, we mention that in the simulation, most of the MC steps are rejected, causing the ratio between all MC steps and accepted MC steps to be τ/τ^ac ∝ 10². This is caused by the fact that the number of possible state transitions is much larger than in the corresponding Ising model without the additional interaction. For example, for the Ising model with molecule formation, the ratio between possible spin transition and adding/removing molecules is roughly given by the average degree k. We observe that the length of the plateau dramatically increases when the system gets closer to its critical temperature. Furthermore, the fluctuations of the length of the plateau (corresponding to the convergence time) also increase with the decreasing distance from the critical temperature, causing different realizations to converge at significantly different times. This loss of characteristic time scale is typical for this type of critical transition.

We now estimate the average convergence time τ by running the systems repeatedly for 100 realizations and then average the convergence time over all realizations for each of the three models. Let us denote the scaling exponent of the average convergence time as $\tau \propto {(T-{T}_{C})}^{-\delta }$ and τ ∝ n^γ. In Figs. 8b, 9b, and 10b, we observe an important result of this paper. Our results suggest that the critical exponent δ is different for each system. For the Ising model with molecule formation, we observe δ ≈ 1.5, for the Potts model with hidden states δ ≈ 1, and for the Truncated long-range Ising model, δ ≈ 0.8. The reason for observing different critical exponents is possibly the different microscopic interactions in the system. For each model, the microscopic changes are different, and therefore, the number of possible transitions in each step is given by the precise microscopic mechanism underlying each model that eventually reflects the value of the scaling exponent δ.

In addition, in Figs. 8c and 9c, we see that the scaling exponent γ related to the scaling of the plateau with respect to the system size is for both cases of the Ising model with molecule formation and Potts model with hidden states equal to γ ≈ 1.

Finally, we focus on the critical behavior for finite-size realizations near the critical temperature and the variation between specific realizations. Gross et al.¹⁹ suggested that there is a relation between the scaling exponent of the correlation length and the scaling exponent of the variance of single-realization critical temperature. This is related to the specific structure of fluctuations in the case of the mixed-order transition. In the case of mixed-order transitions, we observe (contrary to second-order transitions) that fluctuations of the critical control parameter (T_C) follow a scaling law where the scaling coefficient can be related to the correlation length. Moreover, the fluctuations of the single-realization critical order parameter (m_C) can be related to the dimension of the interaction network of the system. We quantify these relations as

$$\sigma ({T}_{C})\propto {n}^{-\eta }\,,$$

(13)

$$\sigma ({m}_{C})\propto {n}^{\xi }\,.$$

(14)

For mixed-order transitions in percolation of interdependent networks which follow a mixed-order transition, it has been suggested that the expected scaling exponents are $\eta=\frac{1}{2}$ and $\xi=\frac{3}{4}$¹⁹. As shown below, these values have also been found for our spin models.

Since the critical temperature of the Truncated long-range Ising model explicitly depends on system size, we focus on the first two examples. To estimate the scaling exponents of the variance single-trajectory temperature and magnetization, we run 100 realizations for each of the systems of characteristic size (n = {50, 100, 200, 400, 800}) with the same parameters (for the Ising model with molecule formation, k =  49 for the Potts model with hidden states g = 10). We start below the critical temperature and slowly increase the temperature until the system exhibits the phase transition. Once the simulations are run with increasing temperature, the critical temperature at which the trajectory switches from one phase to another is slightly different for every run. By repeating the run for many realizations, we obtain a distribution of the critical temperature and the order parameter. The whole procedure is illustrated in Fig. 6. We calculate the sample standard deviation of both critical temperature T_C and magnetization at the critical temperature m_C and analyze the scaling of the standard deviation with system size.

In Figs. 8c and 9c, we observe that for both examples η ≈ 0.5. In  Supplementary Information, we also show that in both examples, ξ ≈ 0.75, see Supplementary Fig. 4. Thus, these results further support the mixed-order transition nature of our system while they remain the same for both systems, despite the fact that δ was different. This shows again that while ξ and η are related to the type of the phase transition, δ reflects the specific microscopic dynamics of the underlying system.

Discussion

In this paper, we studied the origin of mixed-order phase transitions in the case of the Ising model with three types of additional interactions that effectively change the interaction network of the system. We studied three types of models: (i) the Ising model with molecule formation, (ii) the Potts model with hidden states, and (iii) the Truncated long-range Ising model. In each of the three examples, the additional type of interaction changed the order of the phase transition (from a second-order continuous transition observed in the standard Ising model without any additional interaction to an abrupt first-order transition) while keeping the transition critical.

We showed that the origin of the mixed-order transition is caused by long-term microscopic changes along the plateau that are weakening the spin–spin interaction of the system until the system finally collapses, see Fig. 6. This is supported by the analysis of the critical behavior of the plateau times. While the critical exponent β related to the macroscopic magnetization near the critical point is the same for all three models, the scaling exponent related to the length of the plateau, that is, the convergence time from the ordered phase to the disordered phase, slightly above the critical temperature is found different for each model.

Interestingly, in all three cases, the additional type of interaction has been able to change the order of the phase transition to an abrupt mixed-order transition. The critical exponent related to the order parameter remained the same as the value β = 1/2, which is found in the percolation of interdependent networks²⁵ as well as in experiments of interdependent superconducting networks¹⁶. This highly indicates that the universality of the critical macroscopic quantities remains the same for all known mixed-order transitions.

Here, an important question arises: Do different additional microscopic interactions preserve the scaling exponents related to microscopic changes, such as the plateau length? Our results on the three examples suggest that there is a deep relation between the size of microscopic changes per unit of time during the plateau and the duration of the plateau just before the system transitions to another phase. The results suggest that the more microscopic the changes are, the longer the plateau is. Understanding the connection between the microscopic is currently a challenge that should be clarified in future studies. Answering this might shed new light on better understanding and predicting sudden catastrophic failures in many physical, biological, or socio-economic systems.

Such theory will not only be beneficial for spin systems but also for a broad class of systems undergoing transition from second to first-order transition. Examples of such systems have been observed in network models, for instance, in connection with evolutionary systems²⁶, epidemic spreading¹³, attacks on critical infrastructure networks²⁵ explosive percolation^8,27, abrupt jamming²⁸ or sudden synchronization²⁹. As demonstrated, one of the reasons for observing the transition in networks (and generalized network structures as, e.g., hypergraphs³⁰) is the fact that the interactions in the system depend both on the system nodes (subsystems) as well as its (hyper)-edges. Examples of this behavior can be found in connection with percolation theory in higher-order networks^15,31, network of networks models^12,32 or in interdependent networks¹⁷. Also, similar plateaus and transitions have been observed in an overload model³³ and experimentally in coupled superconducting layers¹⁶.

Another important extension of the present study would be to investigate the system beyond the mean-field approximation and random-network approach, that is, as a spatial network, and see how the phase diagram and the overall behavior change due to an additional dependency interaction, analogous to the case of interdependent network lattices³⁴. Further, it would be interesting to see if similar microscopic cascading and a transition between first and second-order transition can be observed for other types of systems with non-multinomial multiplicities (e.g., systems following Fermi–Dirac or Bose–Einstein statistics³⁵ or systems with internal entropy³⁶).

Reporting summary

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