Two stroke Pumping Technique for Many-Body Systems

Abstract

I introduce a new analytical framework for estimating critical temperatures in interacting many-body systems, focusing on the Ising model. Combining the Bethe cluster setting, the Metropolis update, and the Galam Majority Model developed in sociophysics, I build a two stroke pumping technique (TSP). Applied to the Ising model in dimensions d=2, 3, 4, TSP yields values of T_c which are all at an excess of +0.03 from exact estimates. At d=1 the exact value T_c=0 is obtained. In addition, TSP indicates analytically the practical impossibility to reach full symmetry breaking at T=0. The results are thus found in good agreement with numerical findings while requiring significantly fewer computational resources than Monte Carlo sampling. Calculations are computationally efficient and transparent. The framework is general and can be extended to a broad class of discrete spin models. This positions TSP as an intermediate yet scalable tool for studying cooperative behavior in many body interacting systems.

Figures (11)

• Figure 1: Up left: a cluster of one spin and its 4 nearest-neighbors with probability $p_0$ to have $+1$ for each of them. Up center: The central spin has been updated with a probability $p_1$ to be $+1$. That defines the first stroke of the pumping technique. Up right: the probability to $+1$ is now $p_1$ for all 5 spins defining the second stroke. Down left: the central spin is updated with probability $p_2$ to be $+1$. Down center: The probability to be $+1$ is $p_2$ for all 5 spins. Down right: the central spin is updated with probability $p_3$ to $+1$. and so forth till an attractor is reached.
• Figure 2: Five fixed points $p_{1,2,3,4,5}$ from Eq. (\ref{['pp4']}) as a function of $K$. For $0 < K < K_{c1}$ the fixed point $1/2$ is attractor while it turns to a tipping for $K> K_{c1}$. For $K<K_{c2}$ the phase is always disordered. Upper blue and lower red parts are attractors while magenta and green parts are tipping points.
• Figure 3: Update probabilities $p_{n+1}$ and $p^{GM}_{n+1}$ given respectively by Eqs. (\ref{['pp4']}) in red and Eq. (\ref{['m5']}) in blue, as a function of $p_n$ for $K=1.5$ (upper left), $K=0.65$ (upper right), $K=0.5$ (lower right), $K=0.4$ (lower right). TSP and GM are the associated fixed points, which are found to be identical.
• Figure 4: Iterated updates from a series of initial values $p_0$ covering the full spectrum of values from 0 to 1, using Eqs. (\ref{['pp4']}) (left side) and (\ref{['m5']}) (right side). Three values of $K$ are shown with $K=0.65$ illustrating the case of full ordering (low temperatures), $K=0.5$ illustrating the case of the first oder region with either ordering or disorder as a function of the initial value $p_0$, $K=0.4$ illustrating the case of full disorder (high tempratures).
• Figure 5: Up left: a cluster of one spin and its 2 nearest-neighbors with probability $p_0$ to have $+1$ for each of them. Up center: The central spin has been updated with a probability $p_1$ to be $+1$. That defines the first stroke of the pumping technique. Up right: the probability to $+1$ is now $p_1$ for all 3 spins defining the second stroke. Down left: the central spin is updated with probability $p_2$ to be $+1$. Down center: The probability to be $+1$ is $p_2$ for all 3 spins. Down right: the central spin is updated with probability $p_3$ to $+1$. and so forth till an attractor is reached.