Constraint correlation functions of the one-dimensional Ising model in the scaling limit

TL;DR

This work analyzes the constraint correlation functions of the one-dimensional Ising model at fixed magnetization in the scaling limit near the zero-temperature fixed point. Using transfer-matrix methods, it derives finite-size constraint quantities $Z_N(S)$ and $G_N(r;S)$ and then takes the scaling limit with $N\to\infty$, $K\to\infty$, keeping $\zeta=Ne^{-2K}$ fixed to obtain $z_\zeta(s)$ and $g_\zeta(x;s)$. The key finding is that, for small $\zeta$, the constraint correlation function in momentum space exhibits oscillations as a function of the magnetization, with a period that scales as $2/n$ in the limit $\zeta\to0$, driven by a domain-wall–generated length scale; explicit expressions involve Bessel functions and integrals. This oscillatory behavior is in sharp contrast with the correlation function at fixed magnetic field and informs the interpretation of related Monte Carlo results in higher dimensions, with important implications for functional methods like the FRG that aim to incorporate droplet or domain-wall physics under magnetization constraints.

Abstract

We study the correlation function of the one-dimensional Ising model at fixed magnetization. Focusing on the scaling limit close to the zero-temperature fixed point, we show that this correlation function, in momentum space, exhibits surprising oscillations as a function of the magnetization. We show that these oscillations have a period inversely proportional to the momentum and give an interpretation in terms of domain walls. This is in sharp contrast with the behavior of the correlation function in constant magnetic fields, and sheds light on recent results obtained by Monte Carlo simulations for the correlation functions of the critical two-dimensional Ising model at fixed magnetization.

Figures (5)

• Figure 1: Constraint correlation function in the scaling limit as a function of $s$ at fixed $n=2$ for increasing values of $\zeta\in[0,8]$ (from dark to light colors, same legend as Fig. \ref{['fig:g_x']}).
• Figure 2: Constraint correlation function in the scaling limit as a function of $s$ at fixed $\zeta=1$ for increasing values of $n\in\{1,2,3,4,5\}$ (from dark to light colors).
• Figure 3: Constraint correlation function in real space in the scaling limit as a function of $x$ at fixed magnetization $s=0.4$ for increasing values of $\zeta\in[0,8]$ (from dark to light colors).
• Figure 4: Comparison between the constraint (at $s=0$, full lines) and unconstrained (at $m=0$, dashed lines) correlation functions in real space in the scaling limit as a function of $x$ at small and large $\zeta$.
• Figure 5: Constraint correlation function of the critical 2D Ising model as a function of $s$ for increasing momenta $(p_x,p_y)=(\frac{2\pi n}{L},0)$ with $n\in \{1,2,3,4,5\}$ (from dark to light colors). The simulations were performed on a square lattice of linear size $L=256$ with nearest-neighbor interactions at $T=T_c$Rose2025. The correlation function at $n=1$ has been divided by a factor of $6$ for better visibility.