XY Model with Persistent Noise

Abstract

We consider a 2D XY model subjected to time-correlated noise, a model of direct relevance to active crystals, which were shown recently to be able to support very large deformations without melting in the presence of persistent fluctuations. We find that our persistent XY model can remain quasi-ordered in spite of correlations decaying much faster than allowed in equilibrium. We then investigate theoretically and numerically the order-disorder transition and conclude that it remains of the Berezinskii-Kosterlitz-Thouless type, but with scaling exponents that vary with the persistence time of the noise.

Figures (5)

• Figure 1: Quasi-ordered phase of our persistent XY model. (a) phase diagram in the ${(\tau_0,T\tau_0)}$ plane. Black points and dashed line linking them: order-disorder transition defined to be at inflection points of polar order parameter curves $p(T)$ in a system of size $L=448$; orange points: asymptotic BKT-like transition points determined in the second part of the paper, as summarized in Table \ref{['Exponents']}. (b) Angular correlation function $g_\theta(r)$ for $\tau_0=20$, $T=0.15$ and various system sizes. (c) Spatial spectra of angular field at various $\tau_0$ values ($T=0.175$). From bottom to top, $\tau_0 =0.06$, $0.12$, $0.25$, $0.5$, $1$, $2$, $4$, $8$, $12$, $18$. (d) Same spectra as in (c), but rescaled as explained in the main text. (e) Variation of exponent $\eta$ with $\tau_0$ from direct measurement as in (b), and from expression $c^* {k^*}^2/2\pi$. System size in (c-e) is $L=448$. Typical simulation time for each run in (b-e) is of the order of $10^7$.
• Figure 2: Order-disorder transition for $\tau_0=6$. (a) $\chi(T)$ curves at different system sizes $L=56, 112, 224, 448, 896$ from blue to violet, varying $T$ around the susceptibility peak. Typical simulation time for each run is of the order of $5\times10^8$. (inset: corresponding order parameter curves $p(T)$). (b) Same data as in (a), but rescaled by exponent $\eta=0.35$ (inset: rescaling by equilibrium value $\eta=\tfrac{1}{4}$). (c) Optimal scaling of the susceptibility peak location $T^*(L)$ yielding $T_c=0.295(5)$. (d) Decay of $p(t)$ from ordered initial conditions at different $T>T_c$ values (superimposed dashed lines: portions fitted by $p(t)\sim t^{-\lambda}\exp(-t/\tau)$). (e) Decay time $\tau$ and exponent $\lambda$ (inset) vs $T$ from data and fits shown in (d). (f) Best fit of divergence of $\tau$ near $T_c$ ($T-T_c \sim (\ln\tau+b)^{-2}$ yielding $T_c=0.297(3)$. (g) Correlation function $g_\theta(r)$ at $T=0.295$ for various system sizes, yielding $\eta=0.341(1)$ (inset: $\eta$ for various $T$ values for $L=448$). (h) Decay of $p(t)$ for $T=0.295$ and $L=3360$ giving $\lambda\simeq 0.089(1)$ (inset: $\lambda$ for various $T$ values).
• Figure 3: Quenches into the quasi-ordered phase. (a) Time-decay of $p$ for various $\tau_0$ values ($T=0.175$, $L=3360$); superimposed dashed lines are powerlaw fits giving the estimates of $\lambda$ indicated in the legends. (b) Variation with $\tau_0$ of $\lambda$ [data from (a)] and $\eta/4$ [from decay of $g_\theta(r)$].
• Figure 4: BKT transition in the equilibrium limit $\tau_0=0$, varying ${\cal T}$. (a) $\chi({\cal T})$ curves at different system sizes $L=56, 112, 224, 448$ from blue to red, varying ${\cal T}$ around the susceptibility peak (inset: corresponding order parameter curves $p({\cal T})$). Typical simulation time for each run is $5\times10^8$ with $dt=0.02$. (b) Same data as in (a), but rescaled by exponent $\eta=0.25$ (inset: rescaling by the value $\eta=0.35$ found for $\tau_0=6$). (c) Optimal scaling of the susceptibility peak location ${\cal T}^*(L)$ yielding ${\cal T}_c=0.726(5)$. (d) Decay of $p(t)$ from ordered initial conditions at different ${\cal T}>{\cal T}_c$ values (superimposed dashed lines: portions fitted by $p(t)\sim t^{-\lambda}\exp(-t/\tau)$). (e) Decay time $\tau$ and exponent $\lambda$ (inset) vs ${\cal T}$ from data and fits shown in (d). (f) Best fit of divergence of $\tau$ near ${\cal T}_c$ (${\cal T}-{\cal T}_c \sim (\ln\tau+b)^{-2}$ yielding ${\cal T}_c=0.727(2)$. (g) Correlation function $g_\theta(r)$ at ${\cal T}=0.73$ for various system sizes, yielding $\eta=0.236(1)$ (inset: $\eta$ for various ${\cal T}$ values for $L=448$). (h) Decay of $p(t)$ for ${\cal T}=0.73$ and $L=3360$ giving $\lambda\simeq 0.060(1)$ (inset: $\lambda$ for various ${\cal T}$ values).
• Figure 5: Varying the value of the $\nu$ exponent used in fits to estimate $T_c$ for our $\tau_0=6$ data. ($T^*(L)-T_c \sim (\ln L + a)^{-1/\nu}$ for susceptibility peak data, and $T-T_c \sim (\ln \tau + b)^{-1/\nu}$) for quench data). (a) Variation of estimated $T_c$ with $\nu$. (b) Variation of the prefactors $a$ and $b$ with $\nu$. Insets: same but for our equilibrium ($\tau_0=0$) data.