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Physical review. E

Criticality in the duration of quasistationary state

Antonio Rodríguez, Fernando Dantas Nobre, Constantino Tsallis

Published Jul 29, 2021• 4 citations
Publisher

Abstract

The duration of the quasistationary states (QSSs) emerging in the $d$-dimensional classical inertial $\ensuremath{\alpha}\text{\ensuremath{-}}XY$ model, i.e., $N$ planar rotators whose interactions decay with the distance ${r}_{ij}$ as $1/{r}_{ij}^{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}\ensuremath{\ge}0$), is studied through first-principles molecular dynamics. These QSSs appear along the whole long-range interaction regime ($0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$), for an average energy per rotator $U<{U}_{c}$ (${U}_{c}=3/4$), and they do not exist for $U>{U}_{c}$. They are characterized by a kinetic temperature ${T}_{\text{QSS}}$, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature ${T}_{\text{BG}}>{T}_{\text{QSS}}$. We investigate here the behavior of their duration ${t}_{\text{QSS}}$ when $U$ approaches ${U}_{c}$ from below, for large values of $N$. Contrary to the usual belief that the QSS merely disappears as $U\ensuremath{\rightarrow}{U}_{c}$, we show that its duration goes through a critical phenomenon, namely ${t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}$. Universality is found for the critical exponent $\ensuremath{\xi}\ensuremath{\simeq}5/3$ throughout the whole long-range interaction regime.

DOI
10.1103/physreve.104.014144

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The duration of the quasistationary states (QSSs) emerging in the ddd-dimensional classical inertial \ensuremathα\ensuremath-XY\ensuremath{\alpha}\text{\ensuremath{-}}XY\ensuremathα\ensuremath-XY model, i.e., NNN planar rotators whose interactions decay with the distance rij{r}_{ij}rij​ as 1/rij\ensuremathα1/{r}_{ij}^{\ensuremath{\alpha}}1/rij\ensuremathα​ (\ensuremathα\ensuremath≥0\ensuremath{\alpha}\ensuremath{\ge}0\ensuremathα\ensuremath≥0), is studied through first-principles molecular dynamics. These QSSs appear along the whole long-range interaction regime (0\ensuremath≤\ensuremathα/d\ensuremath≤10\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}10\ensuremath≤\ensuremathα/d\ensuremath≤1), for an average energy per rotator U<UcU<{U}_{c}U<Uc​ (Uc=3/4{U}_{c}=3/4Uc​=3/4), and they do not exist for U>UcU>{U}_{c}U>Uc​. They are characterized by a kinetic temperature TQSS{T}_{\text{QSS}}TQSS​, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature TBG>TQSS{T}_{\text{BG}}>{T}_{\text{QSS}}TBG​>TQSS​. We investigate here the behavior of their duration tQSS{t}_{\text{QSS}}tQSS​ when UUU approaches Uc{U}_{c}Uc​ from below, for large values of NNN. Contrary to the usual belief that the QSS merely disappears as U\ensuremath→UcU\ensuremath{\rightarrow}{U}_{c}U\ensuremath→Uc​, we show that its duration goes through a critical phenomenon, namely tQSS\ensuremath∝(Uc\ensuremath−U)\ensuremath−\ensuremathξ{t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}tQSS​\ensuremath∝(Uc​\ensuremath−U)\ensuremath−\ensuremathξ. Universality is found for the critical exponent \ensuremathξ\ensuremath≃5/3\ensuremath{\xi}\ensuremath{\simeq}5/3\ensuremathξ\ensuremath≃5/3 throughout the whole long-range interaction regime.
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Abstract We study, by means of first-principle molecular-dynamics, the temperature T QSS of the quasi-stationary state (QSS) of a d -dimensional classical inertial α -Heisenberg ferromagnetic model, consisting of a set of rotators with interactions decaying with distance as r − α ( α ⩾ 0 ) . We find, through detailed finite-size calculations, the scaling relation T QSS − T ∞ ∝ N − β , where T ∞ ≡ U − 1 / 2 , with energy per particle U < U c ( U c = 5 / 6 ), and total number of particles N , which holds for the very long-range interactions regime 0 ⩽ α / d ⩽ 1 , in which the QSS exists. We show that the scaling exponent β does not depend on U but only in the quotient α / d , being β ( α / d ) ≃ 1 for 0 ⩽ α / d ⩽ 0.5 , and decaying linearly for α / d > 0.5 , so <
We numerically study, from first principles, the temperature TQSS{T}_{\mathrm{QSS}}TQSS​ and duration tQSS{t}_{\mathrm{QSS}}tQSS​ of the longstanding initial quasi-stationary state of the isolated ddd-dimensional classical inertial $\ensuremath{\alpha}\text{\ensuremath{-}}XY$ ferromagnet with two-body interactions decaying as $1/{r}_{ij}^{\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}0)$. It is shown that this temperature TQSS{T}_{\mathrm{QSS}}TQSS​ (defined proportional to the kinetic energy per particle) depends, for the long-range regime $0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$, on $(\ensuremath{\alpha},d,U,N)$ with numerically negligible changes for dimensions d=1,2,3d=1,2,3d=1,2,3, with UUU the energy per particle and NNN the number of particles. We verify the finite-size scaling ${T}_{\mathrm{QSS}}\ensuremath{-}{T}_{\ensuremath{\infty}}\ensuremath{\propto}1/{N}^{\ensuremath{\beta}}$ where ${T}_{\ensuremath{\infty}}\ensuremath{\equiv}2U\ensuremath{-}1$ for $U\ensuremath{\lesssim}{U}_{c}$, and $\ensuremath{\beta}$ appears to depend sensibly only on $\ensuremath{\alpha}/d$. Our numerical results indicate that neither the scaling with NNN of TQSS{T}_{\mathrm{QSS}}TQSS​ nor that of tQSS{t}_{\mathrm{QSS}}tQSS​ depend on UUU.
The two-dimensional (2D) XY model plays a crucial role in statistical and condensed-matter physics. With the introduction of long-range interactions, the system exhibits a richer set of physical phenomena and a crossover between nonclassical and short-range universality classes. In this work, we investigate the 2D XY model with algebraically decaying interactions $\ensuremath{\sim}1/{r}^{2+\ensuremath{\sigma}}$ and provide a comprehensive numerical analysis of its thermodynamic properties. We demonstrate that for $\ensuremath{\sigma}\ensuremath{\le}2$, the system undergoes a second-order phase transition into a ferromagnetic phase characterized by the emergence of long-range order. In the low-temperature phase, due to the presence of the Goldstone mode, the correlation function saturates to a nonzero constant in the form of a power law for $\ensuremath{\sigma}<2$, with decaying exponent $2\ensuremath{-}\ensuremath{\sigma}$, and in the form of the inverse logarithm of distance for $\ensuremath{\sigma}=2$. Moreover, the critical points and exponents are also determined for various $\ensuremath{\sigma}$. We provide compelling evidence that the crossover between nonclassical and short-range regimes occurs at $\ensuremath{\sigma}=2$. This work presents a detailed account of the simulation methodology, extensive numerical data, and new insights into the physics of long-range interacting systems.
An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two nonclassical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends the universality class of the nonclassical exponents to spatially periodic one-dimensional systems and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum.
Long-lived quasi-stationary states (QSSs) are a signature characteristic of long-range interacting systems both in the classical and in the quantum realms. Often, they emerge after a sudden quench of the Hamiltonian internal parameters and present a macroscopic lifetime, which increases with the system size. Despite their ubiquity, the fundamental mechanism at their root remains unknown. Here, we show that the spectrum of systems with power-law decaying couplings remains discrete up to the thermodynamic limit. As a consequence, several traditional results on the chaotic nature of the spectrum in many-body quantum systems are not satisfied in the presence of long-range interactions. In particular, the existence of QSSs may be traced back to the finiteness of Poincaré recurrence times. This picture justifies and extends known results on the anomalous magnetization dynamics in the quantum Ising model with power-law decaying couplings. The comparison between the discrete spectrum of long-range systems and more conventional examples of pure point spectra in the disordered case is also discussed.