Across Dimensions: Two- and Three-Dimensional Phase Transitions
from the Iterative Renormalization-Group Theory of Chains

It is well-known and quickly shown that one-dimensional models $(d=1)$ with finite-range interactions are exactly solvable and do not have a phase transition at non-zero temperature Frobenius . Nevertheless, the phase transitions of the $d>1$ models can be distinctively recovered from the correlations in the exactly solved $d=1$ chains, as we show in the present study. Specifically, using the exact renormalization-group solution of the $d=1$ Ising chain (which at non-zero temperatures has no phase transition and zero magnetization), the finite-temperature phase transitions and entire magnetization curves of the $d=2$ and $d=3$ Ising models are recovered distinctively (Fig. 1). Our method is an approximation and is not obviously systematically improvable towards the exact results. The method is general and flexible, and thus can be applied to a wide range of systems, such as with random fields and/or random bonds, more complicated local degrees of freedom such as spin-s Ising or q-state Potts.

The systems that we study are defined by the Hamiltonian

where at each site $i$, the spin is $s_{i}=\pm 1$ and the first sum is over all pairs of nearest-neighbor sites $<ij>$. We obtain the phase transitions and magnetizations of these Ising systems in spatial dimensions $d=2,3$ at magnetic field $H=0$, based on the renormalization-group solution of the $d=1$ system with $H\neq 0$, as given in Eq.(3).

The Ising model of Eq. (1) with non-zero magnetic field can be subjected, in a $d=1$ chain, to exact renormalization-group transformation NelsonFisher ; Grinstein by performing the sum over every other spin (aka, decimating, actually a misnomer).

The couplings of the remaining spins (of the thus renormalized system) are given by the recursion relations:

where the primes refer to the quantities of the renormalized system, $b=2$ is the length rescaling factor, $d=1$ is the dimensionality, $\sigma_{i}$ is the sign of $s_{i}$ and, for calculational convenience, the Hamiltonian of Eq.(1) has been rewritten in the equivalent form of

where $J_{z}$ is the interaction strength between neighboring spins within the $d=1$ chain.

In the present calculation, the magnetic field $H$ in Eqs.(2,3) represents the lateral interactions to the $d=1$ chain: Its starting value, before renormalization-group is applied, is

where $q=2,4$ respectively for $d=2,3$ is the number of lateral (off-chain) neighbors of each spin in the chain, $J$ is the lateral interaction, $M$ is the magnetization, and we divide by 2 because each spin gets counted twice in the sum in Eq.(3). The lateral mean-field approximation of Eq.(4) is applied only at the initial point of the renormalization-group trajectory. This magnetic field $H$ representing the lateral interactions gets renormalized under the renormalization-group transformations, together with the in-chain interaction $J_{z}$, as given in the recursion relations in Eq.(2).

In Eq.(3), $G$ is the additive constant per bond, unavoidably generated by the renormalization-group transformation, not entering the recursion relations as an argument (therefore a captive variable), but crucial to the calculation of all the thermodynamic densities, as seen in Sec. III below.BerkerOstlund ; Ilker2 ; Atalay ; Artun

Typical calculated renormalization-group flows of $(J_{z},H)$ are given in the lower panel of Fig. 3. All flows are to infinite temperature $1/J_{z}=\infty$ (with the exception of the unstable critical fixed point at zero temperature, zero field $(1/J_{z}=0,H=0))$. At infinite temperature (zero coupling, $J_{z}=0$) a fixed line occurs in the $H$ direction and is the sink of the disordered phase, which attracts everything in $(J_{z},H)$ except for the single critical point. However, we shall see in Sec. IV below that this disordered phase engenders the ordered phases of $d=2$ and 3.

The derivatives of the renormalized magnetic field $H^{\prime}$ with respect to the unrenormalized $H$, at $H=0$, are shown in the upper panel of Fig. 3. 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 crosses 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 of Fig. 3, we compare the magnetic fields acquired by the renormalization-group trajectories originating at $\tanh(J_{z})=0.9$ and $\tanh(J_{z})=0.1$. 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 thermodynamic densities $\textbf{M}\equiv[1,<s_{i}s_{j}>,<(s_{i}+s_{j})>]$, which are the densities conjugate to the interactions $\mathbf{J_{z}}\equiv[G,J_{z},H]$ of Eq.(3), obey the density recursion relation

where the recursion matrix is $\textbf{T}=\partial\mathbf{J_{z}^{\prime}}/\partial\mathbf{J_{z}}$. Although the renormalization-group recursion relations [Eq.(2)] are nonlinear, this linear equation relating the renormalized and unrenormalized densities is exact. It is obtained by using the derivative chain rule on $\textbf{M}=(1/N)\partial\ln Z/\partial\mathbf{J_{z}}$, where $Z$ is the partition function and $N$ is the number of nearest-neighbor pairs of spins, and is used to calculate densities from renormalization-group theory BerkerOstlund ; Ilker2 ; Atalay ; Artun , as explained below.

The densities at the starting interactions of the renormalization-group trajectory are calculated by repeating Eq.(5) until the fixed-line is quasi-reached and applying the fixed-line densities, variable with respect to the terminus $H$, on the right side of the repeated Eq.(5):

where M(n) are the densities at the $(J_{z},H)$ location of the trajectory after the $(n)$th renormalization-group transformation and T(n) is the recursion matrix of the $(n)$th renormalization-group transformation BerkerOstlund ; Artun . Thus, M(0) are the densities of the $(J_{z},H)$ location where the renormalization-group trajectory originates and the aim of the renormalization-group calculation. Note that M(0) is obtained by doing a calculation along the entire length of the trajectory. As seen in Fig. 3, the trajectory closely approaches, after a few renormalization-group transformations, a point $(J_{z}=0,H)$ on the fixed line and $\textbf{M(n)}\simeq\textbf{M*}(H)$, where the latter density is calculated on the fixed line.

The densities $\textbf{M*}(H)$ on the fixed line are, by Eq.(5), the left eigenvector of the recursion matrix $\textbf{T*}(H)$ at the fixed line with eigenvalue $b^{d}$. (Since the recursion matrix is always non-symmetric, the left and right eigenvectors are different with the same eigenvalue.) In the present case, on the fixed line,

and the left eigenvector with eigenvalue $b^{d}=2$ is $\textbf{M*}(H)=[1,<s_{i}s_{j}>=(\tanh(2H))^{2},<(s_{i}+s_{j})>=2\,\tanh(2H)].$

The phase diagrams (Fig. 4) for the anisotropic and isotropic Ising models in $d=2$ and 3 are obtained by repeating our calculation for different values of the interactions $J_{z}$ along the chains and $J$ lateral to the chains, and compare well with the exact results also given in the figure. Critical exponents are obtained by power-law $M\sim(T_{C}-T)^{\beta}$ fitting simultaneously the exponent and the critical temperature to the curves in Fig. 1. As seen in Figs. 5 and 6, in both cases fitting over 6 decades with a quality of fit of $R=99.6$ and 99.5, the critical exponents $\beta=0.43$ and 0.40 are obtained, perhaps meaningfully lower than the mean-field value of $1/2$. However, these values are quite far from the exact values of $\beta=0.125$ and 0.326 in $d=2$ and 3 respectively McCoyWu ; 3dIsing ; Pelissetto , which indicates that our approach, although qualitatively not unreasonable and widely applicable, is not that accurate with respect to critical exponents.

We believe that our method could be easily and widely implemented, since complex systems (as long as the interactions are non-infinite ranged) can be solved in $d=1$ NelsonFisher ; Grinstein and applied to the higher dimensions as demonstrated here. Furthermore, random local densities can be obtained for quenched random systems Yesilleten , in $d=1$ using renormalization-group theory, and applied with our method to a variety of quenched random systems in $d>1$. It would also be interesting to apply to systems which show chaos under direct renormalization-group theory, obtaining an alternate path to study such chaos Atalay ; Fernandez2 ; Eldan .