Metastable and scaling regimes of one-dimensional Kawasaki dynamics

We investigate the large-time scaling regimes arising from a variety of metastable structures in a chain of Ising spins with both first- and second-neighbor couplings while subject to Kawasaki dynamics. Depending on the ratio and sign of these former, different dynamic exponents are suggested by finite-size scaling analyses of relaxation times. At low but nonzero temperatures these are calculated via exact diagonalizations of the evolution operator in finite chains under several activation barriers. In the absence of metastability the dynamics is always diffusive.

Figure 1

Normalized spectral gaps ${\mathrm{\Lambda }}_1\equiv e^{E_b/T}{\lambda }_1$ of the evolution operator for $L=20$ on approaching low-temperature regimes in sectors (a) and (b) of Table 2. From top to bottom, alternating solid and dashed lines stand for coupling ratios $r\equiv J_2/J_1=0.95,0.9,0.8,0.7,0.5,0.4,0.3,0.2,0.1$. Upper- and lowermost dotted curves denoting respectively the cases $r=1$ [sector (c)] and $r=0$, are shown for comparison. Details of that latter standard case are displayed in Fig. 2.

Figure 2

Finite-size scaling of normalized gaps $e^{4(K_1+K_2)}{\lambda }_1$ for (a) $r=0$ (usual ferromagnetic dynamics), (b) $r=0.3$, and (c) $r=0.7$. Solid and dashed lines show alternately the cases of $L=12,14,16,18,20,22,24$. In (a) sizes increase in upward order, while in (b) and (c) they do so from top to bottom. The data collapse of larger sizes was attained using dynamic exponents read off from the slope of the insets. These latter estimate the finite-size decay of spectral gaps close to $T\to 0^{+}$ [Eq. (18)]. In that limit, the decay of (b) (not shown) is indistinguishable from that of (c) due to the common saturation trends of Fig. 1. For displaying convenience vertical scales of main panels were normalized by a factor ${24}^z$.

Figure 3

Normalized gaps $e^{E_b/T}{\lambda }_1$ for $L=20$ within regime (d) of Table 2. From top to bottom solid and dashed lines refer in turn to coupling ratios $r=1.5,1.3,1.2,1.1,1.05$. For comparison, also the case $r=1$ [regime (c)] is shown by the dotted line. The inset estimates the typical finite-size decay of these gaps in the low-temperature limit [(Eq. (18)]. Filled triangles are representative of $r>1$ in sector (d), while open circles stand for $r=1$. However, see extrapolations of Sec. 4d.

Figure 4

(a) Normalized gaps $e^{E_b/T}{\lambda }_1$ in regime (e) of Table 2 using $L=20$. From left to right solid and dashed lines denote in turn the cases of $-r=0.1,0.2,0.3,0.5,0.8,1.5$. Cusps stem from level crossings in the spectrum of the evolution operator. The dotted line shows for comparison the standard case of $r=0$ with $J_1<0$.  (b) Finite-size scaling of normalized gaps for $r=-0.5$ upon identifying $z$ with the slope of the inset. Alternating solid and dashed lines in downward direction indicate sizes $L=12,14,16,18,20,22,24$. For displaying purposes the vertical scale was normalized by a factor ${24}^z$. The inset estimates the typical finite-size decay of these gaps within sector (e) in the limit of $T\to 0^{+}$. See however extrapolations of Sec. 4d.

Figure 5

(a) First excitation level of the evolution operator for $J_1,J_2<0$ and $L=20$ in the nonmetastable region $0\le r<1/2$. From left to right solid and dashed lines refer alternately to coupling ratios $r=0.45,0.4,0.3,0.2,0.1$. The case $r=0$ is also shown for comparison (dotted line).  (b) Scaling plot of these levels for $r=0.3$ and $L=24,22,20,18,16,14,12$ (solid and dashed lines from top to bottom) using $z=2$. For convenience the vertical scale was normalized by a factor ${24}^2$. The slope of the inset corroborates a common diffusive decay in the low-temperature limit of all ratios considered in panel (a).

Figure 6

Normalized gaps $e^{E_b/T}{\lambda }_1$ for $L=20$ in sectors (f) and (g) of Table 2 [panels (a) and (b) respectively, exhibiting results of several coupling ratios]. From top to bottom solid and dashed lines stand in turn for (a), $r=0.7,0.712,0.725,0.75,0.8,0.9,0.95,1$, and (b), $r=1.5,1.4,1.3,1.2$. The inset shows the case of $r=1$ for which these gaps become size-independent in nearing the Arrhenius regime. Here, lengths $L=24,20,16,12,8$ (in upward direction) match the fourfold periodicity of the ground state alluded to in the text.

Figure 7

Normalized gaps $e^{E_b/T}{\lambda }_1$ in regime (h) of Table 2, using $L=16$ for $-r=0.1,0.2,0.3,0.4,0.55,1.5$ (solid and dashed lines from left to right).

Figure 8

Extrapolations of effective dynamic exponents (a), and amplitudes (b) defined in Eqs. (19) and (20). Data of regimes (a) and (b) (listed in Table 2) are represented by circles, regimes (c) and (d) by triangles, and regime (e) by rhomboids. Rightmost horizontal lines depict the confidence intervals arising from nonlinear fittings in those regimes (see text for details). Leftmost dashed lines indicate the limiting values of the rapidly converging approximants for the usual case of $J_2=0,J_1>0$. Small dots stand for results in the nonmetastable sector $0\le r<1/2$ with $J_1<0$.