🔬 condensed matter

Exact expressions of correlation functions between two spins in the boundary row of the two-dimensional rectangular Ising model with periodic-free boundary conditions and finite size

This paper derives and analyzes exact expressions for spin-spin correlation functions in the boundary row of a finite two-dimensional rectangular Ising model with periodic-free boundary conditions, demonstrating how these results converge to known thermodynamic limit forms while highlighting the dependence of long-range order on the order of limits and finite-size effects.

Original authors: Mei Tao

Published 2026-08-21✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Mei Tao

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, flat grid made of tiny magnets, each one capable of pointing either up or down. This is the Ising model, a simplified but powerful way physicists describe how materials like iron behave when they are heated or cooled. In this grid, the magnets do not act alone; they whisper to their neighbors, trying to align their directions. When the material is hot, this whispering is drowned out by thermal chaos, and the magnets point in random directions. But as the temperature drops, a remarkable transformation occurs: the whispers grow louder, and the magnets begin to organize, eventually locking into a unified state where most point the same way. This shift from disorder to order is known as a phase transition, and understanding exactly how the magnets influence each other across the grid is central to predicting how real materials behave.

For decades, scientists have studied these magnetic grids, but they have often relied on approximations that assume the grid is infinitely large. While these approximations are useful for quick calculations, they can hide subtle details about how the system behaves when it is finite, or when the edges of the material are treated differently from the middle. A specific type of grid, shaped like a cylinder where the top and bottom edges are free but the sides wrap around, presents a unique challenge. In this setup, the behavior of magnets along the very edge of the cylinder is difficult to pin down with standard methods, especially when looking at how two magnets far apart on that edge might still be connected.

In this work, researchers have derived exact mathematical expressions for the connection between any two magnets located on the boundary row of such a finite, cylindrical grid. They did not rely on approximations or simulations; instead, they used a rigorous method involving complex matrix calculations to find the precise formula for how these spins correlate. The study reveals that the way we imagine the grid growing to an infinite size matters profoundly. If we imagine the grid growing infinitely wide first and then infinitely tall, the long-range connections between distant magnets on the edge disappear completely when the material is cold. However, if we reverse the order—growing the grid infinitely tall first and then infinitely wide—the long-range connections persist, revealing a hidden order that the other method misses.

This finding clarifies a long-standing ambiguity in how we define the "thermodynamic limit," the point where a material is considered large enough to be treated as a continuous substance. The researchers proved that for short distances between magnets, the order in which we take these limits does not change the result. But for magnets far apart, the order is critical. When the material is cooled below a specific critical temperature, the system develops a long-range order. If the limits are taken in the wrong order, this long-range order is not revealed by the equations, leading to an incomplete picture of the material's true state, even though the order itself physically exists.

The study also examined what happens when the temperature is very close to this critical point. The researchers found that the distance over which magnets influence each other grows dramatically as the temperature approaches the critical value. This growth follows a specific pattern, indicating that the system is on the verge of a major transformation. By providing these exact expressions, the work confirms that the long-range order observed in previous studies of infinite systems is real, but only if the mathematical limits are handled with care. It serves as a reminder that in the microscopic world of magnets, the path we take to understand the infinite can change what we see, and that the edges of a material hold secrets that are only revealed when we look at them with exact precision.

Technical Summary: Exact Expressions of Correlation Functions in the 2D Rectangular Ising Model with Periodic-Free Boundary Conditions

Problem Statement
This paper addresses the calculation of exact spin-spin correlation functions, specifically $\langle \sigma_{1,1} \sigma_{1,1+P} \rangle$ , for two spins located in the boundary row of a two-dimensional rectangular Ising model. The model is defined with finite size dimensions $L$ (horizontal) and $N$ (vertical) and features periodic boundary conditions in the horizontal direction and free boundary conditions in the vertical direction. While asymptotic expressions for correlation functions in the thermodynamic limit are well-documented in existing literature, exact expressions for finite systems with these specific boundary conditions, particularly covering both short-range and long-range orders, remain a subject of detailed investigation. The study focuses on the absence of an external magnetic field.

Methodology
The author employs the method of spinor analysis, originally introduced in Ref. [4] and previously utilized in Refs. [5, 6] for column correlations in periodic-periodic models. The core of the methodology involves:

Matrix Formulation: Expressing the correlation function as a trace of products of transfer matrices ( $V_1, V_2$ ) and a partition function matrix ( $Z_1$ ).
Algebraic Simplification: Utilizing properties of the trace and commutators to reduce the complex matrix product into a form involving diagonal matrices and specific operators ( $\Gamma$ ).
Eigenvalue Analysis: Leveraging the eigenvalues and eigenvectors of the transfer matrices, specifically utilizing the spinor representation properties and orthogonal matrices ( $\mu$ ) to diagonalize the system.
Limiting Processes: Systematically evaluating the limits as $N \to \infty$ and $L \to \infty$ in different orders to determine the behavior of the correlation functions in the thermodynamic limit. The author distinguishes between short-range order (where the distance $P$ is small relative to $L$ ) and long-range order (where $P$ approaches $L/2$ ).

Key Contributions and Results
The paper derives exact analytical expressions for the correlation functions $\langle \sigma_{1,1} \sigma_{1,1+P} \rangle$ for all values of $P$ in a finite system. The primary results and findings include:

Exact Finite-Size Expressions: The author provides a closed-form expression (Eq. 44) for the correlation function in a finite system, expressed as a sum involving hyperbolic functions of the eigenvalues $\gamma_n$ and coefficients $\Omega_n$ .
Thermodynamic Limit and Order of Limits: A critical finding is the dependence of the thermodynamic limit results on the order in which the system dimensions $L$ $L$ and $N$ $N$ approach infinity:
- Case $T \geq T_c$ (Above or at Critical Temperature): The correlation functions vanish for long-range orders in the thermodynamic limit, regardless of the order of limits ( $N \to \infty$ then $L \to \infty$ , or vice versa). Short-range correlations remain non-zero.
- Case $T < T_c$ (Below Critical Temperature): The order of limits significantly impacts the results.
  - Taking the limit $N \to \infty$ first, followed by $L \to \infty$ , correctly reveals the emergence of long-range order (non-zero correlation at large distances), consistent with the physical expectation of spontaneous magnetization.
  - Taking the limit $L \to \infty$ first, followed by $N \to \infty$ , results in the vanishing of long-range order terms, failing to capture the physical long-range correlations present in the system.
Consistency with Previous Work: The author demonstrates that their derived expressions in the thermodynamic limit (under the correct limit order) are identical in form to those provided by previous researchers (specifically Ref. [1]).
Critical Behavior: Near the critical temperature $T_c$ , the paper analyzes the behavior of the correlation length. It identifies a term $\Lambda_P$ that exhibits behavior characteristic of a correlation length with a critical exponent $\nu = 1$ . However, the author explicitly notes that this exponent applies to the specific term $\Lambda_P$ and not necessarily to the full correlation function $\langle \sigma_{1,1} \sigma_{1,1+P} \rangle$ without further analysis of the $Y_P$ term.

Significance
The paper claims significance in providing exact solutions for a specific boundary condition (periodic-free) that allows for the calculation of both short-range and long-range correlations in finite systems. The primary theoretical contribution is the rigorous demonstration that the order of taking thermodynamic limits is not merely a mathematical technicality but physically crucial for correctly describing long-range order in the low-temperature phase ( $T < T_c$ ). The author concludes that the limit sequence $N \to \infty$ followed by $L \to \infty$ is necessary to correctly describe the properties of the correlation functions, particularly the emergence of long-range order as the temperature decreases below $T_c$ . This work clarifies the mathematical conditions required to recover the fundamental property of the Ising model regarding the emergence of long-range order.

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