Landau free energy and the absence of spontaneous magnetization of the one-dimensional Ising model

J. M. Zhang; R. K. Lin; Z. F. Zheng

arxiv: 2605.06691 · v1 · pith:O5HOG5Q2new · submitted 2026-05-02 · ❄️ cond-mat.stat-mech

Landau free energy and the absence of spontaneous magnetization of the one-dimensional Ising model

Z. F. Zheng , R. K. Lin , J. M. Zhang This is my paper

Pith reviewed 2026-05-11 00:56 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords Ising modelspontaneous magnetizationLandau free energyone-dimensional systemsthermodynamic limitdensity of statesparamagnetismmaximum-term approximation

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The pith

The one-dimensional Ising model has no spontaneous magnetization at any finite temperature because its Landau free energy is minimized at zero magnetization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the one-dimensional Ising model never develops spontaneous magnetization at finite temperature by deriving its Landau free energy from an exact count of spin configurations. The authors first compute the density of states for fixed magnetization following Ising's original technique, then apply the maximum-term approximation in the thermodynamic limit to obtain a closed-form free energy. This free energy rises steadily with absolute magnetization and its second derivative at zero magnetization stays positive though non-analytic in temperature. A reader would care because the result supplies an independent, rigorous confirmation that the model stays paramagnetic for all finite temperatures, with the order parameter remaining strictly zero in the absence of an external field.

Core claim

We define and calculate the density of states of the one-dimensional Ising model. In the thermodynamic limit the maximum-term approximation yields the exact Landau free energy, which is an increasing function of |m| whose second derivative at m = 0 is positive and non-analytic in temperature. These properties prove rigorously that spontaneous magnetization is absent at any finite temperature.

What carries the argument

The density of states for configurations with fixed magnetization, used to construct the Landau free energy via the maximum-term approximation in the thermodynamic limit.

If this is right

Magnetization remains zero for all finite temperatures when the external field is zero.
The model exhibits no phase transition at any nonzero temperature.
The magnetic susceptibility diverges as temperature approaches zero from above.
The free-energy curvature at m = 0 supplies a temperature-dependent quantity that is non-analytic yet always positive.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same density-of-states route could be tested on other exactly solvable one-dimensional models to see whether their Landau free energies likewise forbid spontaneous order.
The non-analytic temperature dependence of the curvature at m = 0 may connect to the essential singularity that appears in the exact partition function of the model.
If the maximum-term method works here because the density of states is analytically simple, the approach might still give useful bounds in higher-dimensional or disordered Ising systems even when the approximation is no longer exact.

Load-bearing premise

The maximum-term approximation becomes exact in the thermodynamic limit because the density of states has a simple analytical form that lets the single largest term dominate without appreciable corrections from other terms.

What would settle it

A direct evaluation or simulation that finds the Landau free energy decreasing with increasing |m| at some finite temperature T > 0 would contradict the central claim.

Figures

Figures reproduced from arXiv: 2605.06691 by J. M. Zhang, R. K. Lin, Z. F. Zheng.

**Figure 1.** Figure 1: (Color online) Density of states D(E, M) of a 1D Ising model with the open boundary condition. Only nonzero values are shown. The total number of spins is L = 21. Each curve is for a different value of the magnetization M. We see that the curve corresponding to M + 2 is always below that to M, in accord with (9). E \ M -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 -13 1 1 -11 2 2 2 2 2 2 2 2 2 2 2 2 2 -9 12 1… view at source ↗

**Figure 2.** Figure 2: (Color online) Sketch of the Landau free energy of the 1D Ising model based on the monotonicity property of the density of states revealed in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: (Color online) Heatmap of the function w(d, m) as defined in (23). The parameters are (β = 0.8, α = βh = 0). The red solid line indicates the maxima of w for fixed m, which are points satisfying (24a). The red dot indicates the global maximum (d ∗ , m∗ ), which corresponds to the stable thermodynamic equilibrium state of the model. Taking logarithm gives 1 L ln Wmax ≤ 1 L ln Z ≤ 1 L ln Wmax + 1 L ln[L(L + … view at source ↗

read the original abstract

We revisit the problem of spontaneous magnetization of the one-dimensional Ising model from the Landau free energy perspective. To this end, we define and calculate the density of states of the one-dimensional Ising model following a technique introduced by Ising. The observed monotonicity property of the density of states suggests heuristically that the model does not exhibit spontaneous magnetization at any finite temperature. Subsequently, we solve the model exactly in the thermodynamic limit by employing the maximum-term approximation, which is feasible due to the simple analytical expression of the density of states. We also show that the Landau free energy is an increasing function of $|m|$ and its second derivative at $m=0$ is positive and non-analytic in temperature, proving rigorously the absence of spontaneous magnetization of the model at any finite temperature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a clean re-derivation of the textbook fact that the 1D Ising model has no spontaneous magnetization at finite T, done via explicit density of states and Landau free energy.

read the letter

The paper starts from Ising's original counting of states and builds the density of states as a function of magnetization. They observe that it is monotonic in |m|, then apply the maximum-term approximation in the thermodynamic limit to obtain the Landau free energy. Because the DOS has a simple closed form, the largest term dominates exactly with no sub-extensive corrections that could shift the minimum, so the free energy is strictly increasing in |m| and its second derivative at zero is positive and non-analytic in temperature. That part is solid and makes the usual result rigorous in this language without extra assumptions. What is new is the explicit route through the DOS rather than the transfer matrix; it is not a new physical claim. The result itself is already in the literature they cite, so the contribution is mainly an alternative proof and a way to connect to Landau theory on a solvable model. The only soft spot is that the initial monotonicity is described as observed before the rigorous step, and a referee would want to see the monotonicity proven independently of the final free-energy argument rather than inferred from the counting. Overall the derivations look internally consistent and the approximation is justified by the analytic form. This is useful for readers who teach or study stat mech and want to see the same fact from the free-energy side, or for anyone who likes explicit DOS calculations. It is not aimed at people hunting for new phenomena. I would send it to peer review; the math is straightforward enough that referees can check the steps quickly and the alternative derivation has some pedagogical value even if the conclusion is old.

Referee Report

1 major / 1 minor

Summary. The manuscript revisits the one-dimensional Ising model by constructing its density of states following Ising's technique and deriving the Landau free energy. Using the maximum-term approximation, which is justified by the analytical form of the density of states, it solves the model exactly in the thermodynamic limit. The authors show that the resulting Landau free energy increases with |m| and has a positive second derivative at m=0 that is non-analytic in temperature, providing a rigorous proof of the absence of spontaneous magnetization at any finite temperature.

Significance. This work offers an insightful re-derivation of the absence of spontaneous magnetization in the 1D Ising model through the lens of Landau free energy and explicit density of states. The approach leverages the model's solvability to demonstrate key properties of the free energy, including its monotonicity and curvature, which aligns with the exact transfer-matrix solution. Such explicit constructions can enhance understanding of phase transitions and Landau theory in low-dimensional systems. The non-analyticity in the second derivative is a notable feature that underscores the T=0 transition.

major comments (1)

[Abstract] The abstract describes the monotonicity of the density of states as 'observed' leading to a 'heuristic' suggestion, while claiming a 'rigorous' proof via the maximum-term approximation. The manuscript should specify whether the rigorous demonstration of the free energy being increasing in |m| and the positivity of its second derivative at m=0 depends on the observed monotonicity or is established independently through the approximation.

minor comments (1)

A reference to the original Ising paper would clarify the technique used for the density of states.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the constructive comment on the abstract. We address the point below.

read point-by-point responses

Referee: [Abstract] The abstract describes the monotonicity of the density of states as 'observed' leading to a 'heuristic' suggestion, while claiming a 'rigorous' proof via the maximum-term approximation. The manuscript should specify whether the rigorous demonstration of the free energy being increasing in |m| and the positivity of its second derivative at m=0 depends on the observed monotonicity or is established independently through the approximation.

Authors: We appreciate the referee highlighting this potential ambiguity. The monotonicity of the density of states is observed from the explicit calculation following Ising's technique and serves only as the basis for the heuristic suggestion in the first part of the abstract. The rigorous demonstration that the Landau free energy increases with |m| and has a positive second derivative at m=0 is obtained independently via the maximum-term approximation in the thermodynamic limit. This approximation is justified by the closed analytical form of the density of states and yields an explicit expression for the free energy as a function of m; the monotonicity and curvature properties are then verified directly from that expression without further reference to the observed monotonicity. We will revise the abstract to make this separation explicit and thereby address the referee's request for clarification. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from exact DOS enumeration

full rationale

The paper starts from the exact combinatorial density of states obtained by Ising's counting technique applied to the 1D Ising Hamiltonian, which is an input definition independent of the target magnetization result. The maximum-term approximation is then invoked in the N→∞ limit; because the DOS admits a closed analytic form, the largest term dominates with sub-extensive corrections that cannot alter the location of the global minimum or the sign of the second derivative at m=0. The claimed monotonicity of the resulting Landau free energy and its positive curvature at m=0 therefore follow directly from this enumeration without any fitted parameters, self-citation chains, or redefinition of the conclusion as an input. The derivation is thus non-circular and reproduces the known absence of spontaneous magnetization from first principles.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the density of states being monotonic (observed heuristically) and on the maximum-term approximation becoming exact in the thermodynamic limit. No free parameters are introduced; the only background assumptions are standard properties of the thermodynamic limit and the definition of the density of states following Ising.

axioms (2)

domain assumption The maximum-term approximation is exact for the partition function in the thermodynamic limit when the density of states has a simple analytical expression.
Invoked to obtain the exact free energy from the density of states.
ad hoc to paper The observed monotonicity of the density of states implies the system does not favor nonzero magnetization.
Used heuristically before the rigorous second-derivative argument.

pith-pipeline@v0.9.0 · 5436 in / 1437 out tokens · 76136 ms · 2026-05-11T00:56:41.311138+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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[2]

An English translation of Ising’s original paper can be found here: https://www.hs-augsburg.de/ ~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html

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Peliti L 2011 Statistical Mechanics in a Nutshell (Princeton: Princ eton University Press)

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[5]

Landau L D 1936 The theory of phase transitions Nature 138 840

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[6]

Semenov S and Rubtsov A N 2024 Landau free energy of small clus ters beyond mean-ﬁeld approach Phys. Rev. E 110 L012101

work page 2024
[7]

stars and bars

This standard combinatorial problem now has a standard solution known to all: the “stars and bars” method. It is interesting to note, however, that this techn ique appears to have been 6 CONCLUSIONS AND DISCUSSIONS 13 unknown to David Hilbert. He considered the problem in his book Theory of Algebraic Invariants but solved it using mathematical induction, ...

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[8]

Hill T L 1987 An Introduction to Statistical Thermodynamics (New York: Dover Publications)

work page 1987
[9]

Dominguez T and Mourrat J C 2023 Statistical mechanics of mean- ﬁeld disordered systems: A Hamilton-Jacobi approach arXiv:2311.08976

work page arXiv 2023
[10]

Seth S 2017 Combinatorial approach to exactly solve the 1D Isin g model Eur. J. Phys. 38 015104

work page 2017
[11]

Magare S, Roy A K and Srivastava V 2022 1D Ising model using the Kronecker sum and Kronecker product Eur. J. Phys. 43 035102

work page 2022
[12]

Kastner M 2008 Phase transitions and conﬁguration space top ology Rev. Mod. Phys. 80 167

work page 2008
[13]

Mattis D C and Swendsen R H 2008 Statistical Mechanics Made Simp le (Singapore: World Scientiﬁc)

work page 2008
[14]

Kardar M 2007 Statistical Physics of Fields (Cambridge: Cambrid ge University Press)

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[15]

Part I Phys

Kramers H A and Wannier G H 1941 Statistics of the Two-Dimension al Ferromagnet. Part I Phys. Rev. 60 252

work page 1941
[16]

Tanaka T 2002 Methods of Statistical Physics (Cambridge: Cam bridge University Press)

work page 2002

[1] [1]

Ising E 1925 Beitrag zur Theorie des Ferromagnetismus Z. Phys. 31 253

work page 1925

[2] [2]

An English translation of Ising’s original paper can be found here: https://www.hs-augsburg.de/ ~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html

work page

[3] [3]

Peliti L 2011 Statistical Mechanics in a Nutshell (Princeton: Princ eton University Press)

work page 2011

[4] [4]

Landau L D and Lifshitz E M 1986 Statistical Physics Part 1 (Singap ore: World Scientiﬁc)

work page 1986

[5] [5]

Landau L D 1936 The theory of phase transitions Nature 138 840

work page 1936

[6] [6]

Semenov S and Rubtsov A N 2024 Landau free energy of small clus ters beyond mean-ﬁeld approach Phys. Rev. E 110 L012101

work page 2024

[7] [7]

stars and bars

This standard combinatorial problem now has a standard solution known to all: the “stars and bars” method. It is interesting to note, however, that this techn ique appears to have been 6 CONCLUSIONS AND DISCUSSIONS 13 unknown to David Hilbert. He considered the problem in his book Theory of Algebraic Invariants but solved it using mathematical induction, ...

work page

[8] [8]

Hill T L 1987 An Introduction to Statistical Thermodynamics (New York: Dover Publications)

work page 1987

[9] [9]

Dominguez T and Mourrat J C 2023 Statistical mechanics of mean- ﬁeld disordered systems: A Hamilton-Jacobi approach arXiv:2311.08976

work page arXiv 2023

[10] [10]

Seth S 2017 Combinatorial approach to exactly solve the 1D Isin g model Eur. J. Phys. 38 015104

work page 2017

[11] [11]

Magare S, Roy A K and Srivastava V 2022 1D Ising model using the Kronecker sum and Kronecker product Eur. J. Phys. 43 035102

work page 2022

[12] [12]

Kastner M 2008 Phase transitions and conﬁguration space top ology Rev. Mod. Phys. 80 167

work page 2008

[13] [13]

Mattis D C and Swendsen R H 2008 Statistical Mechanics Made Simp le (Singapore: World Scientiﬁc)

work page 2008

[14] [14]

Kardar M 2007 Statistical Physics of Fields (Cambridge: Cambrid ge University Press)

work page 2007

[15] [15]

Part I Phys

Kramers H A and Wannier G H 1941 Statistics of the Two-Dimension al Ferromagnet. Part I Phys. Rev. 60 252

work page 1941

[16] [16]

Tanaka T 2002 Methods of Statistical Physics (Cambridge: Cam bridge University Press)

work page 2002