Across dimensions: Two- and three-dimensional phase transitions from the iterative renormalization-group theory of chains

Sharp two- and three-dimensional phase transitional magnetization curves are obtained by an iterative renormalization-group coupling of Ising chains, which are solved exactly. The chains by themselves do not have a phase transition or nonzero magnetization, but the method reflects crossover from temperaturelike to fieldlike renormalization-group flows as the mechanism for the higher-dimensional phase transitions. The magnetization of each chain acts, via the interaction constant, as a magnetic field on its neighboring chains, thus entering its renormalization-group calculation. The method is highly flexible for wide application.

Figure 1

From left to right, magnetizations for $d=2$ and 3 Ising models obtained by coupling exact solutions of Ising chains. The magnetization of each chain acts, via the coupling constant $J$, as a magnetic field entering the renormalization-group calculation of its neighboring chains. This procedure is iterated until the magnetization curve converges, as seen in Fig. 2, in fact creating order out of the renormalization-group flows within a disordered phase, reflecting a crossover from fieldlike (at lower temperatures) to temperaturelike flows. The magnetization curves in this figure are actually smooth on reaching zero, but this cannot be seen on the scale of the figure.

Figure 2

Iterations in the magnetization calculations. Starting with a small seed of $M=0.00001$ and proceeding with the renormalization-group calculated magnetizations, after each iteration, each site applies a magnetic field of $JM$ to its neighbors on the neighboring chains. The chains are arrayed to give a $d>1$-dimensional system. The leftmost curve is the result of the first iteration. Distinctly, for both $d=2$ and 3, the magnetizations quickly converge (to the rightmost curve) and give the higher-dimensional phase transitions.

Figure 3

Lower panel: Renormalization-group flows [Eq. (2)] of the $d=1$ Ising model [Eq. (1)]. Trajectories originating at the small $H=0.001$ and the entire breadth of temperature are given, all terminating at different locations on the fixed line of the renormalization-group flows at $J_z=0$. Upper panel: The derivative of the renormalized magnetic field $H^{\prime }$ with respect to the unrenormalized $H$, at $H=0$. Its values stay around its maximal value of 2 (at strong coupling the magnetic field of a decimated spin adds to the magnetic field of a remaining spin) until temperatures around $1/J_z=3$ and cross to its minimal value of 1 (at weak coupling the magnetic field of a decimated spin does not affect the magnetic field of a remaining spin). In the lower panel, we compare the magnetic fields acquired by the renormalization-group trajectories originating at $tanh\left(J_z\right)=0.9$ and 0.1. The renormalization-group trajectories originating at higher temperatures end on the fixed line at a small value of $H$. This mechanism thwarts the lateral couplings of the chains and ushers the high-temperature disordered phase. The calculated transition temperatures for $d=2$ (on left) and 3 are consistently shown on the middle axis.

Figure 4

Phase boundaries for the isotropic and anisotropic Ising models: From left to right, the $d=2$ exact boundary $exp(-2J)=tanh\left(J_z\right)$ [8] and the $d=2$ and 3 boundaries calculated by our method. $J_z$ is the nearest-neighbor interaction along the chains and $J$ is the nearest-neighbor interaction lateral to the chains. The exact phase transition points for the isotropic systems, $J_z=J$, are given by the circle $(d=2)$ and square $(d=3)$ [9, 10] data points.

Figure 5

Power-law $M\sim {(T_C-T)}^{\beta }$ fit to our $d=2$ result. A fit over six decades, with a quality of fit $R=99.6$, gives the critical exponent $\beta =0.43$, lower than the mean-field value of $1/2$. However, this value is quite far from the exact value of $\beta =0.125$.

Figure 6

Power-law $M\sim {(T_C-T)}^{\beta }$ fit to our $d=3$ result. A fit over six decades, with a quality of fit $R=99.5$, gives the critical exponent $\beta =0.40$, lower than the mean-field value of $1/2$. However, this value is quite far from the exact value of $\beta =0.326$.