Renormalization Group Solution of the One-Dimensional Ising Model with a magnetic Field

Michael Nauenberg

Raghav G. Jha

We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field $h$, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to $2.6\%$, has no spontaneous magnetization, and gives an effective critical-isotherm exponent $δ\to 1$. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at $βh = iπ/2$) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to $0.15\%$ for all $β$ and $h$, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm $M\propto h^{1/δ}$ with $δ= 15.2$ and amplitude $1.055$, compared with the exact $δ=15$ and $1.059$. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, $δ=15$) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to $4\times 10^{-5}$.

A. Rabinovitch, Rathindra Nath Sen

The recursion relations obtained by applying the renormalization group technique to a finite one-dimensional Ising chain with nearest-neighbor interaction and constant external magnetic field are decoupled and solved in closed form.

S. Moss de Oliveira, M. A. Contínentino, P.M.C. de Oliveira

I. P. Fittipaldi, Douglas F. de Albuquerque

Robert Konik, Yury Adamov

We present a renormalization group (RG) procedure which works naturally on a wide class of interacting one-dimension models based on perturbed (possibly strongly) continuum conformal and integrable models. This procedure integrates Wilson's numerical renormalization group with Zamolodchikov's truncated conformal spectrum approach. The key to the method is that such theories provide a set of completely understood eigenstates for which matrix elements can be exactly computed. In this procedure the RG flow of physical observables can be studied both numerically and analytically. To demonstrate the approach, we study the spectrum of a pair of coupled quantum Ising chains and correlation functions in a single quantum Ising chain in the presence of a magnetic field.