Low-temperature Glauber dynamics under weak competing interactions

We consider the low but nonzero-temperature regimes of the Glauber dynamics in a chain of Ising spins with first- and second-neighbor interactions $J_1,J_2$. For $0<-J_2/\left|J_1\right|<1$ it is known that at $T=0$ the dynamics is both metastable and noncoarsening, while being always ergodic and coarsening in the limit of $T\to 0^{+}$. Based on finite-size scaling analyses of relaxation times, here we argue that in that latter situation the asymptotic kinetics of small or weakly frustrated $-J_2/\left|J_1\right|$ ratios is characterized by an almost ballistic dynamic exponent $z\simeq 1.03\left(2\right)$ and arbitrarily slow velocities of growth. By contrast, for noncompeting interactions the coarsening length scales are estimated to be almost diffusive.

Figure 1

Normalized spectral gaps ${\Lambda }_1=e^{4r{K}_1}{\lambda }_1$ with frustrated interactions for $L=22$ on approaching low-temperature regimes and taking $r=$ 0.3, 0.2, 0.1375, 0.1, 0.075, 0.05, 0.025 (from top to bottom, alternating between dashed and solid lines). All $\Lambda$'s saturate towards a common value $\propto L^{-z}$ (see Fig. 2). For comparison, the lowermost dotted curve denotes the soluble case $r=0$. The straight solid lines in the inset, closely following the asymptotic behavior of the corresponding data, are fitted with slopes $-4r$.

Figure 2

Finite-size scaling of normalized spectral gaps ${\Lambda }_1=e^{4r{K}_1}{\lambda }_1$ for (a) $r=0.1$ and (b) $r=0.3$. In the main panels, sizes increase downwards as $L=2k$ with $5\le k\le 12$ alternating between dashed and solid lines. At low temperatures the data collapse of larger sizes was attained upon setting a common dynamic exponent $z\sim 1.1$ (however, see extrapolations of Sec. 3c). The inset of (a) shows the plain spectral gap behavior for the temperatures indicated by vertical lines below, always consistent with an $\sim L^{-1.1}$ decrease. Similarly, the inset of (b) exhibits the typical finite-size decay of ${\Lambda }_1$ at saturation [Eq. (19)] for all $r$'s considered in Fig. 1. For comparison, the diffusive case of $J_2=0$ is also shown here.

Figure 3

Gaps of noncompeting chains with $L=22$. From top to bottom the main panel depicts the branches corresponding, respectively, to (i) $-r=J_2/\left|J_1\right|>1$  (2, 1.35, 1.2, 1.1, 1.05), (ii) $r=-1,$ and (iii) $0<-r<1$  (0.95, 0.9, 0.8, 0.6, 0.3, 0.2, 0.1), alternating between solid and dashed lines. Alike the frustrated regime (Fig. 1), at low temperatures each branch saturates towards common values $\propto L^{-z}$ (see Fig. 4). The inset exhibits the discontinuous tendency of these latter as temperature is lowered ($T/J_1=0.15,0.07,0.01$). Filled circles stand at the intersection of cases $r=0$ and $r=-1$. Gaps at the leftmost regime (weakly competing, $r>0$) decay as $e^{-4r{K}_1}$ (see inset of Fig. 1).

Figure 4

Typical finite-size scaling of gaps for nonfrustrated coupling regimes (Fig. 3), taking (a) $J_2/\left|J_1\right|=0.9$, (b) 1, and (c) 1.1. Lengths, alternating between solid and dashed lines, increase downwards as $L=2k$ with $6\le k\le 12$. At low temperatures the data collapse of larger sizes in each branch was obtained upon choosing $z\sim$ 2.3, 2.4, and 2.5, respectively. In turn, these values are used as slopes to fit the typical finite-size decay of these gaps at saturation (denoted by ${\lambda }_1^{*}$ in the insets). See, however, extrapolations of Sec. 3c.

Figure 5

Extrapolations of effective dynamic exponents [Eq. (20)] within the regions $0<r=-J_2/\left|J_1\right|\le 0.3$ (rhomboids), $r=0$ (dots), $0<-r<1$ (circles), $r=-1$ (triangles), and $r<-1$ (squares). Nonlinear fittings in the weak frustrated regime $(a\sim 1.45)$ yield a dynamic exponent $z\simeq 1.03\pm 0.02$, while those for circles $(a\sim 1.40)$, triangles $(a\sim 1.47)$, and squares $(a\sim 1.64)$ yield slightly subdiffusive exponents within the confidence interval $[2.02,2.11]$. The latter and that of the weak competing situation are depicted by rightmost horizontal lines. The usual $r=0$ case converges faster $(a\sim 1.95)$ towards the exact exponent within the same range of sizes.