Understanding the dynamical universality classes of systems with long-range interactions remains a key challenge in statistical physics. In this Letter, we analytically and numerically investigate the non-equilibrium critical dynamics of the one-dimensional spin- $1/2$ Nagle-Kardar model, which is characterized by the competition between short- and long-range interactions and the presence of a tricritical point. We focus on the slowing-down of the magnetization $m$ at criticality under Glauber dynamics. Starting from the corresponding master equation, we perform a coarse-graining procedure to obtain a Fokker-Planck equation for the macroscopic variables. Then, the asymptotic decay of the magnetization is derived using central manifold theory. We find that $m$ decays as $t^{-1/2}$ along the critical line and as $t^{-1/4}$ precisely at the tricritical point. This finding confirms that the dynamical critical exponent is $z=2$ as for mean-field models, proving that the macroscopic critical dynamics of the Nagle-Kardar model falls within the dynamical universality class of purely relaxational, non-conserved order parameters (model A). While Kardar proved that the equilibrium Curie-Weiss theory extends to Ising models where nearest-neighbor interactions are included, we here show that such result is valid also for critical dynamics. Our work provides the semi-analytical solution for the critical dynamics of a model with mixed-range interactions, assigning its universality class.Understanding the dynamical universality classes of systems with long-range interactions remains a key challenge in statistical physics. In this Letter, we analytically and numerically investigate the non-equilibrium critical dynamics of the one-dimensional spin-$1/2$ Nagle-Kardar model, which is characterized by the competition between short- and long-range interactions and the presence of a tricritical point. We focus on the slowing-down of the magnetization $m$ at criticality under Glauber dynamics. Starting from the corresponding master equation, we perform a coarse-graining procedure to obtain a Fokker-Planck equation for the macroscopic variables. Then, the asymptotic decay of the magnetization is derived using central manifold theory. We find that $m$ decays as $t^{-1/2}$ along the critical line and as $t^{-1/4}$ precisely at the tricritical point. This finding confirms that the dynamical critical exponent is $z=2$ as for mean-field models, proving that the macroscopic critical dynamics of the Nagle-Kardar model falls within the dynamical universality class of purely relaxational, non-conserved order parameters (model A). While Kardar proved that the equilibrium Curie-Weiss theory extends to Ising models where nearest-neighbor interactions are included, we here show that such result is valid also for critical dynamics. Our work provides the semi-analytical solution for the critical dynamics of a model with mixed-range interactions, assigning its universality class.

arXiv:2511.07207Nov 10, 20250 citations

Spin dynamics near the critical point

雄一奥田

Sep 28, 19780 citations

Teaching spontaneous magnetization by using a probability distribution

J. Güémez, Santiago Velasco, Juan A. White

The equilibrium magnetization distribution for an Ising-spin model in the mean-field approximation is used to present a simple study of the ferromagnetic phase transition in the absence of external magnetic field (spontaneous magnetization). For temperatures below a critical one (the Curie Temperature) this distribution exhibits two peaks, showing that two states of magnetization are possible. At the Curie temperature the distribution presents a plateau, showing large fluctuations in the magnetization. A scaling-like analysis of the distribution shows that the Curie point can be characterized as an unstable fixed point of a recursion relation.The equilibrium magnetization distribution for an Ising-spin model in the mean-field approximation is used to present a simple study of the ferromagnetic phase transition in the absence of external magnetic field (spontaneous magnetization). For temperatures below a critical one (the Curie Temperature) this distribution exhibits two peaks, showing that two states of magnetization are possible. At the Curie temperature the distribution presents a plateau, showing large fluctuations in the magnetization. A scaling-like analysis of the distribution shows that the Curie point can be characterized as an unstable fixed point of a recursion relation.

European Journal of PhysicsJul 1, 19913 citations

Critical behavior of the driven Curie-Weiss model

Ruohan Xu, Faezeh Khodabandehlou, Christian Maes

We complete the phase diagram of the macroscopic Curie-Weiss magnet in a time-periodic external field, as a function of temperature and driving parameters. There is a regime (large enough driving amplitude and frequency, at low temperatures) where stable paramagnetic and ferromagnetic phases coexist. In particular, we present a new detailed analysis of the (nonequilibrium) specific heat, diverging at the same critical inverse temperature $β_c$ as the magnetic susceptibility. The new Curie temperature decreases with the driving, and we find critical exponent $α=1$ for $β\downarrow β_c$ , and $α\simeq 0.86$ for $β\uparrow β_c$ , even for small driving. A Floquet analysis shows the nature of the criticality, which is dynamical, with implications that remain unseen and are mostly impossible when the system is in thermal equilibrium.We complete the phase diagram of the macroscopic Curie-Weiss magnet in a time-periodic external field, as a function of temperature and driving parameters. There is a regime (large enough driving amplitude and frequency, at low temperatures) where stable paramagnetic and ferromagnetic phases coexist. In particular, we present a new detailed analysis of the (nonequilibrium) specific heat, diverging at the same critical inverse temperature $β_c$ as the magnetic susceptibility. The new Curie temperature decreases with the driving, and we find critical exponent $α=1$ for $β\downarrow β_c$, and $α\simeq 0.86$ for $β\uparrow β_c$, even for small driving. A Floquet analysis shows the nature of the criticality, which is dynamical, with implications that remain unseen and are mostly impossible when the system is in thermal equilibrium.

arXiv:2607.05130Jul 6, 20260 citations

Quantum Critical Points in Magnetic Systems

ScienceDec 24, 19990 citations

1/2

Nagle-Kardar model, which is characterized by the competition between short- and long-range interactions and the presence of a tricritical point. We focus on the slowing-down of the magnetization

m

at criticality under Glauber dynamics. Starting from the corresponding master equation, we perform a coarse-graining procedure to obtain a Fokker-Planck equation for the macroscopic variables. Then, the asymptotic decay of the magnetization is derived using central manifold theory. We find that

m

decays as

t^{-1/2}

along the critical line and as

t^{-1/4}

precisely at the tricritical point. This finding confirms that the dynamical critical exponent is

z=2

as for mean-field models, proving that the macroscopic critical dynamics of the Nagle-Kardar model falls within the dynamical universality class of purely relaxational, non-conserved order parameters (model A). While Kardar proved that the equilibrium Curie-Weiss theory extends to Ising models where nearest-neighbor interactions are included, we here show that such result is valid also for critical dynamics. Our work provides the semi-analytical solution for the critical dynamics of a model with mixed-range interactions, assigning its universality class.

β_c

as the magnetic susceptibility. The new Curie temperature decreases with the driving, and we find critical exponent

α=1

for

β\downarrow β_c

, and

α\simeq 0.86

for

β\uparrow β_c

, even for small driving. A Floquet analysis shows the nature of the criticality, which is dynamical, with implications that remain unseen and are mostly impossible when the system is in thermal equilibrium.