⚛️ lattice

On Ising model in magnetic field on the lattice

This paper proposes and refines a conjecture for the free energy of the square-lattice Ising model in a magnetic field by expressing it as a superposition of one-dimensional chains weighted by a distribution of Lee--Yang gaps, achieving high-precision agreement with numerical data and reproducing exact critical exponents and spontaneous magnetization.

Original authors: Raghav G. Jha

Published 2026-10-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Raghav G. Jha

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast grid of tiny magnets, each pointing either up or down, interacting with their neighbors. This simple setup, known as the Ising model, is one of the most important tools in physics for understanding how matter changes state, such as when iron becomes magnetic or water turns to steam. For decades, physicists have been able to solve the mathematics of this grid perfectly when no external force is applied, revealing exactly how the magnets align and how the system behaves at the tipping point between order and chaos. However, the moment a real magnetic field is turned on, pushing all the magnets in one direction, the mathematical equations become so tangled that no exact solution has ever been found. This missing piece has remained one of the most famous unsolved problems in statistical physics, leaving scientists to rely on powerful computer simulations to guess the system's behavior.

A researcher at the Thomas Jefferson National Accelerator Facility has now taken a fresh look at this decades-old puzzle, proposing a new way to describe the energy of this magnetic grid under a real magnetic field. The work does not claim to have found the final, perfect mathematical answer, but it offers a remarkably accurate approximation that bridges the gap between what we know and what we can calculate. By combining known results from simpler, one-dimensional versions of the problem with deep theoretical insights about how the system's "zeros" behave, the author constructed a formula that predicts the system's energy with unprecedented precision. The study suggests that the complex two-dimensional grid can be understood as a weighted mixture of many simpler, one-dimensional chains, each responding to the magnetic field in a slightly different way.

The journey began with an initial guess, a formula built by stitching together the solution for a single line of magnets and a famous solution for the grid with no magnetic field. When the author compared this first guess against high-precision computer simulations, the results were encouraging but flawed. The formula was off by as much as 2.6 percent in certain conditions. More critically, it failed to capture a fundamental physical reality: it predicted that the magnets would not spontaneously align when the field was removed, a phenomenon known as spontaneous magnetization that is known to happen in the real world. Furthermore, the formula suggested that the system's response to the field at the critical temperature followed a pattern that was mathematically too simple, missing the complex, sharp behavior that nature actually exhibits.

To fix these errors, the author turned to a powerful mathematical principle called the Lee-Yang theorem, which describes how the mathematical roots of the system's energy are arranged. This theorem allowed the researcher to reframe the problem entirely. Instead of looking for a single, rigid formula, the new approach treats the two-dimensional grid as a superposition, or a blend, of many one-dimensional chains. Each chain in this blend has a specific "gap" in its energy levels, and the author proposed a specific distribution of these gaps to represent the whole system. This distribution was not guessed at random; it was carefully constructed to satisfy several known facts, such as the exact energy of the system when there is no magnetic field and the exact behavior when the magnetic field is extremely strong. Crucially, this new construction included a special component that accounts for the spontaneous magnetization, ensuring the formula respects the laws of physics even when the field is tiny.

The results of this refined approach were striking. The new formula agreed with the computer simulations to within 0.15 percent across a wide range of temperatures and field strengths, a massive improvement over the initial guess. It successfully reproduced the exact amount of spontaneous magnetization that occurs when the field is turned off, a feat the previous version could not achieve. When the author examined how the system behaves right at the critical temperature, the new formula predicted a specific exponent, a number that describes how sharply the magnetization rises, to be 15.2. This is incredibly close to the exact theoretical value of 15, differing by only 1.5 percent, and the predicted strength of the magnetization was accurate to within 0.4 percent.

The author went a step further by creating an even more detailed version of the formula, incorporating additional theoretical inputs and fine-tuning a few constants against the simulation data. This final, refined form matched the computer simulations to an astonishing degree of accuracy, differing by only four parts in one hundred thousand. While the paper does not claim to have solved the problem with a rigorous mathematical proof, it provides a highly reliable map of the terrain. The work demonstrates that by viewing the complex two-dimensional grid as a collection of simpler one-dimensional chains, weighted by a specific distribution of energy gaps, physicists can capture the essential physics of the system with remarkable fidelity. This approach not only corrects the errors of earlier guesses but also offers a clear, concrete path forward for understanding one of the most stubborn problems in the study of magnetism.

Technical Summary: Free Energy of the 2D Ising Model in a Magnetic Field

Problem Statement
The two-dimensional Ising model on a square lattice is a cornerstone of statistical mechanics, famously solved by Onsager in zero magnetic field. However, no exact closed-form solution exists for the model in the presence of a non-zero real magnetic field ( $h \neq 0$ ). While the scaling limit is described by integrable Ising field theory (providing exact results for the critical isotherm and vacuum energy), the lattice model in a real field remains unsolved. The challenge is to construct an expression for the free energy that respects the known exact constraints (Onsager's zero-field solution, Lee-Yang theorems, strong-field expansions) while accurately approximating the lattice behavior across all temperatures and field strengths.

Methodology
The author employs a combination of exact theoretical constraints and high-precision numerical data to construct and validate conjectures for the free energy:

Numerical Benchmarking: The author utilizes the Higher-Order Tensor Renormalization Group (HOTRG) method to compute the free energy and magnetization on a dense grid of temperatures ( $\beta \in [0.05, 2]$ ) and magnetic fields ( $h \in [0, 5]$ ). This data serves as the ground truth for testing conjectures, with relative errors estimated to be $\sim 10^{-7}$ .
Theoretical Framework (Lee-Yang Representation): Leveraging the Lee-Yang circle theorem, the author demonstrates that the exact free energy of the 2D model can be represented as a superposition of free energies of one-dimensional (1D) Ising chains in the same field. This superposition is weighted by a temperature-dependent distribution, $W_\beta(v)$ $W_{β} (v)$ , of squared Lee-Yang gaps ( $v$ $v$ ).
- The 1D chain free energy is known exactly.
- The distribution $W_\beta(v)$ must satisfy specific sum rules derived from known exact results: Onsager's zero-field free energy, the strong-field expansion coefficients, and the Lee-Yang solution at the imaginary field $\beta h = i\pi/2$ .
Conjecture Construction: The author proposes specific functional forms for the distribution $W_\beta(v)$ that satisfy these sum rules and the physical requirements of the model (e.g., spontaneous magnetization, critical exponents).

Key Contributions and Conjectures

1. Conjecture I (Original Proposal)
The author first revisits a previous conjecture where the free energy is modeled as a single 1D Ising chain with an effective coupling derived from the field and temperature.

Performance: This expression deviates from HOTRG data by up to 2.6%.
Limitations: It fails to capture essential physics: it is an even function of $h$ (analytic at $h=0$ ), resulting in zero spontaneous magnetization. Consequently, it predicts a critical isotherm exponent $\delta \to 1$ (behavior $M \sim h \ln(1/h)$ ) instead of the exact $\delta = 15$ .

2. Conjecture II (Improved Proposal)
To correct the deficiencies of Conjecture I, the author proposes a distribution $W_\beta(v)$ that is a superposition of:

An "infinitely coupled" chain (an atom at $v=0$ ) carrying the weight of Yang's spontaneous magnetization ( $M_0$ ).
A continuous distribution (Beta-like) for the remaining weight, shaped to satisfy Onsager's zero-field free energy and the strong-field expansion.
Specific terms to handle the Yang-Lee edge singularity above the critical temperature ( $T > T_c$ ).
Parameters: The shape parameters of the distribution are fixed without fitting to numerical data, relying solely on Onsager's free energy and the single-flipped-spin energy coefficient ( $L_1 = e^{-8\beta}$ ).

3. Conjecture II' (Refined Form)
A further refinement incorporates additional inputs from Ising field theory (specifically the exact critical isotherm $\delta=15$ ) and adjusts a few constants to match transfer-matrix and TRG data.

Results

Accuracy:
- Conjecture I: Maximum deviation from TRG is 2.6%; RMS deviation is 0.9%.
- Conjecture II: Maximum deviation reduces to 0.15%; RMS deviation is 0.03%. It reproduces Yang's spontaneous magnetization exactly.
- Conjecture II': Matches TRG free energy to $4 \times 10^{-5}$ .
Critical Isotherm:
- Conjecture II predicts the critical exponent $\delta = 15.2$ (within 1.5% of the exact $\delta=15$ ) and an amplitude of 1.0547 (0.4% error compared to the exact 1.0590).
- Conjecture II' enforces the exact $\delta=15$ .
Physical Consistency:
- The improved expressions correctly exhibit spontaneous magnetization below $T_c$ .
- The specific heat remains positive.
- The Lee-Yang zero density derived from Conjecture II matches TRG calculations at complex fields.
- The strong-field expansion is satisfied to the first order by construction and agrees well at higher orders.

Significance and Claims
The paper claims that while the results are not exact proofs, they represent a significant step toward solving this long-standing problem. The primary achievement is demonstrating that the complex free energy of the 2D Ising model in a magnetic field can be effectively approximated as a superposition of 1D chains, provided the distribution of "gaps" is chosen to satisfy exact sum rules.

The author emphasizes that Conjecture II achieves high precision (0.15% deviation) using no fitted parameters to the numerical data, relying only on exact theoretical inputs (Onsager's solution and strong-field limits). This suggests that the structure of the solution is deeply rooted in the interplay between 1D chain physics and the Lee-Yang theorem. The refined form (Conjecture II') demonstrates that incorporating field-theoretic constraints can push the agreement to the level of numerical precision ( $4 \times 10^{-5}$ ), offering a highly accurate practical formula for the free energy where an exact closed form is currently unknown.

Technical Summary: Free Energy of the 2D Ising Model in a Magnetic Field

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