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Failing to Have a Transition for Want of Size Is Not the Same as Failing for Want of Dimension ── The free energy of the one-dimensional Ising model shows no jump in its second derivative at finite size (a largest step of 0.282856 at n=64) ── taking the limit is necessary for a transition, not sufficient ── [Paper 502]

Yuuki Yamagishi

Published Sep 20, 2026• 0 citations
Publisher

Abstract

The absence of a phase transition is put down to the system being finite. This paper shows that a system can fail to have one even in the thermodynamic limit──the one-dimensional Ising model, whose largest step stays at 0.510568. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──that the one-dimensional Ising model has no transition, and that finite partition functions are analytic, are standard. No theory of transitions──critical exponents, the renormalisation group and the distribution of Yang-Lee zeros are not the subject; only the second derivative of the free energy. Two dimensions were not computed──Onsager’s exact solution is not used, and nothing here shows that two dimensions do jump. No claim about numerical precision──these are finite differences at step 10^-4, not true second derivatives; what is compared is values taken at the same step. The kink is not physics──-|beta-1| is a control for seeing the test react, not a model. Relation to earlier papers: Paper 154 showed that “transitions occur only in the thermodynamic limit” is the non-commutation of two limits──this paper shows in numbers that the “only” is necessary and not sufficient. Paper 245 showed that two limits have no answer until the order is written──here, writing the order still leaves no jump when the dimension is too low. Paper 271 showed that mean field does not see the dimension──on this side, the dimension settles the answer itself. Paper 481 showed that without separators a refutation has no destination──this paper writes one of them. What is added is tabulating the free energy from n=4 to 1024, measuring the finite-size step at 0.282856, showing the limit stays at 0.510568, giving the kink control at 2.000x10^4, and placing the separator at size versus dimension. First, at n=1024 the free energy matches the closed form at -1.626523375036 (Section 2). Second, at n=4 it is still 7.782x10^-2 away──so the test responds to n (Section 2). Third, at finite n=64 the second derivative does not jump──the largest step is 0.282856 (Section 3). Fourth, this is the core of the paper. Nor does it jump in the thermodynamic limit──the closed form gives a largest step of 0.510568 (Section 3). Fifth, the same test on a kink jumps by 2.000x10^4──so it does respond to non-analyticity (Section 3). Sixth, the separator is whether what is lacking is size or dimension (Section 4). The absence of a phase transition is put down to the system being finite──a finite sum is a polynomial and analytic everywhere, so take the limit and a transition appears. But taking the limit guarantees no jump: it only makes non-analyticity possible, and whether anything jumps is another matter. Measuring the finite system──the free energy per site at beta=0.5 runs -1.648821002, -1.626523916 and -1.626523375 for n=4, 16 and 64, against a closed form of -1.626523375. At n=1024 the two agree at -1.626523375036. At n=4 the difference is still 7.782x10^-2, so the test responds to n. The rate depends on beta: six digits by n=16 at beta=0.5, only three by n=64 at beta=2.0. Taking the limit produces no jump──second differences in beta at step 10^-4 run from -1.661210 to -0.814208 at n=64, with a largest step of 0.282856: smooth. The closed form in the thermodynamic limit runs from -1.661210 to -0.814202, a largest step of 0.510568──with n sent to infinity, the same order as at finite n. The same test applied to the kink -|beta-1| jumps by 2.000x10^4, four orders above, so it does pick up non-analyticity. So “finite, therefore no jump” is not the reason here: the limit was taken and still nothing jumped. That elsewhere is the dimension: in one dimension the cost of a single domain wall is finite, so arbitrarily many enter and no order forms at finite temperature. One thing separates them──is what is lacking the size of the system, or the dimension of space. If size, the limit should produce a jump──but it stayed at 0.510568. If dimension, the limit changes nothing──and nothing jumped. The two are distinct roots by the criterion of Paper 300: both are stated as “there is no phase transition”, but one is a reason the limit removes and the other a reason it does not. Paper 154 spoke to the first, that transitions occur only in the thermodynamic limit──a necessary condition and not a sufficient one, and this paper gives the difference in numbers. It neighbours Paper 245 as well: two limits have no answer until the order is written, yet here writing the order still gives “no jump” when the dimension is too low. This is one of the separators of Paper 481: when a model shows no transition, whether the system is small, the dimension too low, or the interaction too short-ranged sends the repair elsewhere. To be honest──two dimensions were not computed. That one dimension does not jump is shown; that two do is not. Onsager’s exact solution would show it, but it does not enter the numbers here, so “raise the dimension and it jumps” is not claimed. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: phase transition, one-dimensional Ising model, finite system, partition function, Lee-Yang theorem. ----- 相転移が無いのは系が有限だからだと言われる。本稿が示すのは、熱力学極限を取っても転移しない場合があることである──一次元イジングがそれで、跳びは 0.510568 にとどまる。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──一次元イジングに相転移が無いことも、有限系の分配関数が解析的であることも標準的である。相転移の理論を書かない──臨界指数、くりこみ群、ヤン=リーの零点の分布は主題ではない。自由エネルギーの二階微分だけを見る。二次元を計算していない──オンサーガーの厳密解は使っておらず、二次元で跳ぶことは本稿の数では示していない。数値微分の精度を主張しない──刻み 10^-4 の差分であり、真の二階微分ではない。比べているのは同じ刻みで測った値どうしである。折れ線は物理ではない──-|beta-1| は検査が反応することを見るための対照であって、模型ではない。既刊との関係:論文154 は「相転移は熱力学極限でだけ起きる」が二つの極限の非交換だと示した──本稿はその「だけ」が必要条件であって十分条件でないことを、数で示す。論文245 は二つの極限が順序を書くまで答を持たないと示した──ここでは順序を書いても、次元が足りなければ跳ばない。論文271 は平均場が次元を見ないと示した──本稿の側では、次元が答そのものを決めている。論文481 は分離子が無いと反証の行き先が決まらないと示した──本稿はその分離子を一本書く。加えたのは、n=4 から 1024 まで自由エネルギーを出したこと、有限の跳びを 0.282856 と測ったこと、極限でも 0.510568 にとどまることを示したこと、対照の折れ線で 2.000x10^4 を出したこと、分離子を大きさと次元に置いたことである。 第一に、n=1024 の自由エネルギーが閉じた式に -1.626523375036 で一致する(第2節)。 第二に、n=4 ではまだ 7.782x10^-2 離れている──検査が n に反応する(第2節)。 第三に、有限の n=64 では二階微分が跳ばない──最大の跳びは 0.282856 である(第3節)。 第四に、これが本稿の芯である。熱力学極限でも跳ばない──n->infinity の閉じた式でも最大の跳びは 0.510568 である(第3節)。 第五に、同じ検査を折れ線に当てると 2.000x10^4 跳ぶ──検査は非解析性に反応している(第3節)。 第六に、分離子は「足りないのが大きさか、次元か」である(第4節)。 相転移が無いのは系が有限だからだと言われる──有限個の項の和は多項式でどこでも解析的だから、極限を取れば転移する、という像である。だが極限を取ることは跳ぶことを保証しない。解析的でなくなりうるというだけで、実際に跳ぶかは別の問題である。有限系を測る──一次元イジングの一サイトあたり自由エネルギーは、beta=0.5 で n=4 が -1.648821002、n=16 が -1.626523916、n=64 が -1.626523375、閉じた式が -1.626523375 である。 n=1024 では -1.626523375036 対 -1.626523375036 で一致する。 n=4 ではまだ 7.782x10^-2 離れており、検査が n に反応していることの確認になる。寄り方は beta に依り、beta=0.5 では n=16 で 6 桁合うのに、beta=2.0 では n=64 でも 3 桁である。極限を取っても、跳ばない──beta について刻み 10^-4 の二階差分を取ると、n=64 では beta=0.90 から 1.10 にかけて -1.661210 から -0.814208 へ、最大の跳びは 0.282856 で滑らかである。熱力学極限の閉じた式でも -1.661210 から -0.814202 へ、最大の跳びは 0.510568──n を無限にしたのに、有限のときと同じ桁のままである。同じ検査を折れ線 -|beta-1| に当てると 2.000x10^4 跳ぶ。一次元イジングの 0.510568 より 4 桁大きく、検査は非解析性を拾っている。だから「有限だから跳ばない」は、この場合の理由ではない。その別のところが次元で、一次元では境界一つを作る代価が有限なので、いくらでも入り、有限の温度で秩序が立たない。分けているものは一つ──足りないのが系の大きさか、空間の次元か。大きさが足りないなら極限で跳ぶはずだが、0.510568 にとどまった。次元が足りないなら極限でも跳ばず、実際に跳ばなかった。二つは論文300 の基準で別根であり、どちらも「相転移が無い」と言えるが、一方は極限を取れば消える理由、一方は取っても消えない理由である。論文154 が言ったのは前者の側で、相転移は熱力学極限でだけ起きるという言明である。これは必要条件であって十分条件ではなく、本稿はその差を数で出した。論文245 とも隣り合い、二つの極限は順序を書くまで答を持たないが、ここでは順序を書いても、次元が足りなければ答は「跳ばない」である。これは論文481 でいう分離子の一本で、「模型に相転移が出ない」という報告が来たとき、系が小さいのか、次元が足りないのか、相互作用が短距離すぎるのかで直す先が違う。正直に言えば──二次元を計算していない。一次元で跳ばないことは示したが、二次元で跳ぶことは示しておらず、オンサーガーの厳密解を使えば示せるはずだが本稿の数には入っていない。だから「次元が上がれば跳ぶ」までは、ここでは主張していない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:相転移、一次元イジング模型、有限系、分配関数、リー・ヤンの定理。

DOI
10.5281/zenodo.22861634

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The absence of a phase transition is put down to the system being finite. This paper shows that a system can fail to have one even in the thermodynamic limit──the one-dimensional Ising model, whose largest step stays at 0.510568. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──that the one-dimensional Ising model has no transition, and that finite partition functions are analytic, are standard. No theory of transitions──critical exponents, the renormalisation group and the distribution of Yang-Lee zeros are not the subject; only the second derivative of the free energy. Two dimensions were not computed──Onsager’s exact solution is not used, and nothing here shows that two dimensions do jump. No claim about numerical precision──these are finite differences at step 10^-4, not true second derivatives; what is compared is values taken at the same step. The kink is not physics──-|beta-1| is a control for seeing the test react, not a model. Relation to earlier papers: Paper 154 showed that “transitions occur only in the thermodynamic limit” is the non-commutation of two limits──this paper shows in numbers that the “only” is necessary and not sufficient. Paper 245 showed that two limits have no answer until the order is written──here, writing the order still leaves no jump when the dimension is too low. Paper 271 showed that mean field does not see the dimension──on this side, the dimension settles the answer itself. Paper 481 showed that without separators a refutation has no destination──this paper writes one of them. What is added is tabulating the free energy from n=4 to 1024, measuring the finite-size step at 0.282856, showing the limit stays at 0.510568, giving the kink control at 2.000x10^4, and placing the separator at size versus dimension. First, at n=1024 the free energy matches the closed form at -1.626523375036 (Section 2). Second, at n=4 it is still 7.782x10^-2 away──so the test responds to n (Section 2). Third, at finite n=64 the second derivative does not jump──the largest step is 0.282856 (Section 3). Fourth, this is the core of the paper. Nor does it jump in the thermodynamic limit──the closed form gives a largest step of 0.510568 (Section 3). Fifth, the same test on a kink jumps by 2.000x10^4──so it does respond to non-analyticity (Section 3). Sixth, the separator is whether what is lacking is size or dimension (Section 4). The absence of a phase transition is put down to the system being finite──a finite sum is a polynomial and analytic everywhere, so take the limit and a transition appears. But taking the limit guarantees no jump: it only makes non-analyticity possible, and whether anything jumps is another matter. Measuring the finite system──the free energy per site at beta=0.5 runs -1.648821002, -1.626523916 and -1.626523375 for n=4, 16 and 64, against a closed form of -1.626523375. At n=1024 the two agree at -1.626523375036. At n=4 the difference is still 7.782x10^-2, so the test responds to n. The rate depends on beta: six digits by n=16 at beta=0.5, only three by n=64 at beta=2.0. Taking the limit produces no jump──second differences in beta at step 10^-4 run from -1.661210 to -0.814208 at n=64, with a largest step of 0.282856: smooth. The closed form in the thermodynamic limit runs from -1.661210 to -0.814202, a largest step of 0.510568──with n sent to infinity, the same order as at finite n. The same test applied to the kink -|beta-1| jumps by 2.000x10^4, four orders above, so it does pick up non-analyticity. So “finite, therefore no jump” is not the reason here: the limit was taken and still nothing jumped. That elsewhere is the dimension: in one dimension the cost of a single domain wall is finite, so arbitrarily many enter and no order forms at finite temperature. One thing separates them──is what is lacking the size of the system, or the dimension of space. If size, the limit should produce a jump──but it stayed at 0.510568. If dimension, the limit changes nothing──and nothing jumped. The two are distinct roots by the criterion of Paper 300: both are stated as “there is no phase transition”, but one is a reason the limit removes and the other a reason it does not. Paper 154 spoke to the first, that transitions occur only in the thermodynamic limit──a necessary condition and not a sufficient one, and this paper gives the difference in numbers. It neighbours Paper 245 as well: two limits have no answer until the order is written, yet here writing the order still gives “no jump” when the dimension is too low. This is one of the separators of Paper 481: when a model shows no transition, whether the system is small, the dimension too low, or the interaction too short-ranged sends the repair elsewhere. To be honest──two dimensions were not computed. That one dimension does not jump is shown; that two do is not. Onsager’s exact solution would show it, but it does not enter the numbers here, so “raise the dimension and it jumps” is not claimed. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: phase transition, one-dimensional Ising model, finite system, partition function, Lee-Yang theorem. ----- 相転移が無いのは系が有限だからだと言われる。本稿が示すのは、熱力学極限を取っても転移しない場合があることである──一次元イジングがそれで、跳びは 0.510568 にとどまる。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──一次元イジングに相転移が無いことも、有限系の分配関数が解析的であることも標準的である。相転移の理論を書かない──臨界指数、くりこみ群、ヤン=リーの零点の分布は主題ではない。自由エネルギーの二階微分だけを見る。二次元を計算していない──オンサーガーの厳密解は使っておらず、二次元で跳ぶことは本稿の数では示していない。数値微分の精度を主張しない──刻み 10^-4 の差分であり、真の二階微分ではない。比べているのは同じ刻みで測った値どうしである。折れ線は物理ではない──-|beta-1| は検査が反応することを見るための対照であって、模型ではない。既刊との関係:論文154 は「相転移は熱力学極限でだけ起きる」が二つの極限の非交換だと示した──本稿はその「だけ」が必要条件であって十分条件でないことを、数で示す。論文245 は二つの極限が順序を書くまで答を持たないと示した──ここでは順序を書いても、次元が足りなければ跳ばない。論文271 は平均場が次元を見ないと示した──本稿の側では、次元が答そのものを決めている。論文481 は分離子が無いと反証の行き先が決まらないと示した──本稿はその分離子を一本書く。加えたのは、n=4 から 1024 まで自由エネルギーを出したこと、有限の跳びを 0.282856 と測ったこと、極限でも 0.510568 にとどまることを示したこと、対照の折れ線で 2.000x10^4 を出したこと、分離子を大きさと次元に置いたことである。 第一に、n=1024 の自由エネルギーが閉じた式に -1.626523375036 で一致する(第2節)。 第二に、n=4 ではまだ 7.782x10^-2 離れている──検査が n に反応する(第2節)。 第三に、有限の n=64 では二階微分が跳ばない──最大の跳びは 0.282856 である(第3節)。 第四に、これが本稿の芯である。熱力学極限でも跳ばない──n->infinity の閉じた式でも最大の跳びは 0.510568 である(第3節)。 第五に、同じ検査を折れ線に当てると 2.000x10^4 跳ぶ──検査は非解析性に反応している(第3節)。 第六に、分離子は「足りないのが大きさか、次元か」である(第4節)。 相転移が無いのは系が有限だからだと言われる──有限個の項の和は多項式でどこでも解析的だから、極限を取れば転移する、という像である。だが極限を取ることは跳ぶことを保証しない。解析的でなくなりうるというだけで、実際に跳ぶかは別の問題である。有限系を測る──一次元イジングの一サイトあたり自由エネルギーは、beta=0.5 で n=4 が -1.648821002、n=16 が -1.626523916、n=64 が -1.626523375、閉じた式が -1.626523375 である。 n=1024 では -1.626523375036 対 -1.626523375036 で一致する。 n=4 ではまだ 7.782x10^-2 離れており、検査が n に反応していることの確認になる。寄り方は beta に依り、beta=0.5 では n=16 で 6 桁合うのに、beta=2.0 では n=64 でも 3 桁である。極限を取っても、跳ばない──beta について刻み 10^-4 の二階差分を取ると、n=64 では beta=0.90 から 1.10 にかけて -1.661210 から -0.814208 へ、最大の跳びは 0.282856 で滑らかである。熱力学極限の閉じた式でも -1.661210 から -0.814202 へ、最大の跳びは 0.510568──n を無限にしたのに、有限のときと同じ桁のままである。同じ検査を折れ線 -|beta-1| に当てると 2.000x10^4 跳ぶ。一次元イジングの 0.510568 より 4 桁大きく、検査は非解析性を拾っている。だから「有限だから跳ばない」は、この場合の理由ではない。その別のところが次元で、一次元では境界一つを作る代価が有限なので、いくらでも入り、有限の温度で秩序が立たない。分けているものは一つ──足りないのが系の大きさか、空間の次元か。大きさが足りないなら極限で跳ぶはずだが、0.510568 にとどまった。次元が足りないなら極限でも跳ばず、実際に跳ばなかった。二つは論文300 の基準で別根であり、どちらも「相転移が無い」と言えるが、一方は極限を取れば消える理由、一方は取っても消えない理由である。論文154 が言ったのは前者の側で、相転移は熱力学極限でだけ起きるという言明である。これは必要条件であって十分条件ではなく、本稿はその差を数で出した。論文245 とも隣り合い、二つの極限は順序を書くまで答を持たないが、ここでは順序を書いても、次元が足りなければ答は「跳ばない」である。これは論文481 でいう分離子の一本で、「模型に相転移が出ない」という報告が来たとき、系が小さいのか、次元が足りないのか、相互作用が短距離すぎるのかで直す先が違う。正直に言えば──二次元を計算していない。一次元で跳ばないことは示したが、二次元で跳ぶことは示しておらず、オンサーガーの厳密解を使えば示せるはずだが本稿の数には入っていない。だから「次元が上がれば跳ぶ」までは、ここでは主張していない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:相転移、一次元イジング模型、有限系、分配関数、リー・ヤンの定理。
Scaling, hyperscaling and finite-size scaling were long considered problematic in theories of critical phenomena in high dimensions. The scaling relations themselves form a model-independent structure that any model-specific theory must adhere to, and they are accounted for by the simple principle of homogeneity. Finite-size scaling is similarly founded on the fundamental idea that only two length scales enter the game — namely system length and correlation length. While all scaling relations are quite satisfactory for multitudes of physical systems in low dimensions, one fails in high dimensions. The aberrant scaling relation is called hyperscaling and involves dimensionality itself. Finite-size scaling also appears to fail in high dimensions. Developed in the 1930s, Landau mean-field theory is valid in such high-dimensional systems. However, it too does not accord with hyperscaling and finite-size scaling there. The advent of renormalization-group theory in the 1970s brought deeper fundamental insights into critical phenomena, allowing systems to be viewed at different scales. Above a critical dimensionality, higher-order Renormalization Group (RG) eigenvalues become irrelevant and scaling is governed by the Gaussian fixed point. Although obeying all scaling relations including hyperscaling, and although it appears to successfully explain scaling in the correlation sector, the Gaussian fixed point fails to capture the free energy and derivatives, even in infinite volume. In the 1980s, to fix this for the magnetisation, specific heat and susceptibility, Fisher introduced the notion of dangerous irrelevant variables. Since the correlation sector did not appear to be broken, no attempt was made to repair it and Fisher’s modified RG formalism worked quite well in the thermodynamic limit of infinite volume. However, finite-size scaling fails. Also in the 1980’s, Binder, Nauenberg, Privman and Young extended Fisher’s concept to the free energy itself and to finite-size systems. While putting Fisher’s ideas on a more fundamental footing, the failure of finite-size scaling there still presented a problem. This appeared to be resolved by the introduction of “thermodynamic length” to replace correlation length as the length scale that controls Finite-Size Scaling (FSS). Thus hyperscaling and FSS were both sacrificed in favour of the RG patched together by ad hoc solutions. In the 1990’s, Luijten and Blöte went a long way to resolving the dilemma by adding corrections to scaling to the above considerations. It was clear that both of these played a role. However, as with previous authors, and adhering to the principle of not fixing that appears not to be broken, they did not address correlation length directly. Here we report on developments over the past decade which went a long way to addressing these long-standing problems The key to unlocking these, and extending their validity to the high-dimensional regime, was to relax assumptions that the correlation length has to be bounded by the physical length of bounded systems. This allowed and necessitated the extension of Fisher’s concept of dangerous irrelevant variables to the correlation sector.
Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from +Msp+{M}_{\mathrm{sp}}+Msp​ to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from 0+{0}^{+}0+ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature Tc{T}_{c}Tc​ and at Tc{T}_{c}Tc​. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d , in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d = 3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature T c and the Ising critical exponents ν , γ and β can be identified. We find that the larger the size L , the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative data analysis of Monte Carlo simulations of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f = f ( T , m ; L ) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, m coex = m ( T ; L ) as functions of temperature T and magnetization per spin m = M / N . Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.
Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.
Ferdinand and Fisher have analyzed the finite-size behavior of the two-dimensional Ising model with periodic boundary conditions for TTT near Tc{T}_{c}Tc​. In the thermodynamic limit, the leading correction term to the specific heat has an unusual and unexpected behavior as the shape of the $m\ifmmode\times\else\texttimes\fi{}n$ lattice is changed. For shape parameter s=mn=1s=\frac{m}{n}=1s=nm​=1, the peak in the specific-heat correction term occurs at a reduced temperature ${\ensuremath{\tau}}_{max}=2{K}_{c}n\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}>0$. With increasing sss the peak moves to negative $\ensuremath{\tau}$ values and then increases to 0. We show that this effect may be understood by keeping only the two largest eigenvalues of the transfer matrix. To this approximation, which is good for s=1s=1s=1 and improves exponentially with increasing sss, all the shape dependence of the free energy is due to a factor that can be expressed entirely in terms of a domain-wall energy. Furthermore, the functional form of this factor is the same as that of the finite-length correction to the one-dimensional Ising model. Thus the dependence of ${\ensuremath{\tau}}_{max}$ on sss is qualitatively similar to the dependence of Tmax1{T}_{max}^{1}Tmax1​ the location of the one-dimensional Ising-model specific-heat peak on NNN, the number of spins. We also argue that the two-dimensional partition function is expressible at least for s>1s>1s>1 as a factor due to domains, and another due to a one-dimensional Ising array of domain walls. Approximate expressions are obtained for the effective number of spins and the effective coupling of the array. We also point out that there is reason to believe the one-dimensional Ising array of domain walls at fixed $\ensuremath{\tau}$ connects smoothly with a similar array for T<TcT<{T}_{c}T<Tc​.