Peierls Argument for Long-Range Order

The founding question of the theory of phase transitions is deceptively simple: can a model with only short-range interactions actually order at a finite temperature, developing a spontaneous magnetization that survives after any external field is removed? Mean-field theory answers yes, but its uncontrolled averaging cannot tell a genuine transition from an artifact — and indeed it wrongly predicts ordering even for the one-dimensional chain, which Ising had shown in 1925 never orders. In 1936 Rudolf Peierls supplied the first rigorous proof that the two-dimensional Ising model does order below some finite temperature, and simultaneously explained why one dimension does not. His argument is geometric, not analytic: it weighs the energetic cost of domain walls against their entropic abundance. The setup fixes the boundary spins all up and asks for the probability that a chosen interior spin $s_0$ is flipped to $-1$ . A spin can be down only if it is surrounded by at least one closed contour — a domain wall on the dual lattice separating the up region from a down island. Every dual edge the contour crosses corresponds to an unsatisfied bond costing $2J$ , so a contour of perimeter $\ell$ carries a Boltzmann weight bounded by $e^{-2\beta J\ell}$ relative to the ground state. Against this exponential energetic suppression stands the number of contours of length $\ell$ enclosing the origin, which grows only like $\ell\,3^{\ell}$ — the count of self-avoiding loops. The transition is thus a contest between an energy factor $e^{-2\beta J\ell}$ and an entropy factor $3^{\ell}$ . When the temperature is low enough that $x\equiv 3e^{-2\beta J}<1$ , the sum over contour probabilities converges; making $\beta$ larger still drives the total flip probability below $\tfrac12$ , so the magnetization $\langle s_0\rangle=1-2P_-$ stays strictly positive, uniformly in system size. That is spontaneous symmetry breaking, proved rigorously. The same accounting reveals why dimension is decisive: in one dimension a "wall" is a single point defect of fixed cost $2J$ with $\sim N$ possible positions, so its free energy $2J-k_BT\ln N$ turns negative for large $N$ and walls proliferate at any $T>0$ . The lower critical dimension of the Ising model is therefore $d=1$ — and this energy–entropy balance for interfacial defects is the unifying idea behind Peierls' argument, the Mermin–Wagner theorem, and the whole modern theory of lower critical dimensions.

Key takeaways

The Peierls argument is a rigorous proof that the two-dimensional Ising model has spontaneous magnetization at low enough temperature — establishing a genuine phase transition without solving the model.
The central object is the domain wall (contour) between opposite-spin regions; its energy grows linearly as $2J\ell$ , while the number of contours grows only as $\sim3^{\ell}$ , so at low $T$ long walls are exponentially rare.
Bounding the probability that a fixed spin is enclosed by some contour gives $\langle s_0\rangle=1-2P_-$ with $P_-\le\sum_{\ell}\ell\,x^{\ell}$ , $x=3e^{-2\beta J}$ ; the series converges for $x<1$ and drops below $\tfrac12$ (so $\langle s_0\rangle>0$ ) for $x$ small.
The argument crucially fails in one dimension: a single wall costs fixed energy $2J$ but gains entropy $k_BT\ln(N-1)$ , so $\Delta F=2J-k_BT\ln(N-1)\to-\infty$ and order is destroyed at any $T>0$ — matching the exact 1D result $T_c=0$ .
Peierls' 1936 argument (completed rigorously by Griffiths in 1964 and Dobrushin in 1965) is the prototype for energy-versus-entropy reasoning about ordering, domain walls, and lower critical dimensions.
Key idea: a contour of length $\ell$ costs energy $2J\ell$ but has only $\sim3^{\ell}$ shapes; when $x=3e^{-2\beta J}<1$ large contours are exponentially suppressed, so a fixed spin almost surely keeps the boundary's sign — proving spontaneous magnetization.

The core insight

A domain wall of length $\ell$ costs energy $2J\ell$ but has only $\sim3^{\ell}$ possible shapes; when the temperature is low enough that $x=3e^{-2\beta J}<1$ , large walls are exponentially rare, so a fixed interior spin almost surely retains the boundary's orientation — a rigorous proof of spontaneous magnetization in the two-dimensional Ising model.

Where this lives today

The Peierls argument launched an entire discipline. Its contour representation is the seed of the rigorous statistical mechanics of Gibbs measures — the Pirogov–Sinai theory that proves the existence and structure of low-temperature phases for a vast class of lattice models, well beyond the Ising case. The same energy–entropy bookkeeping determines lower critical dimensions across physics: $d_\ell=1$ for discrete (Ising-like) symmetries and, via the Mermin–Wagner theorem, $d_\ell=2$ for continuous ones, results that constrain everything from thin-film magnetism to the melting of two-dimensional crystals and the absence of long-range order in low-dimensional Bose systems. In modern computational work the contour picture underlies cluster Monte-Carlo algorithms and the interpretation of interface free energies, and its logic — "count the defects, weigh energy against entropy" — is the standard first estimate a physicist reaches for when asking whether a newly proposed phase can survive thermal fluctuations at all.

The physical picture

Picture a vast field of up-spins pinned by the boundary of the sample. To flip an interior region down, you must build a fence — the domain wall — around it, and the price of the fence scales with its perimeter. The whole question of order reduces to whether fences are cheap. At low temperature each fence segment is expensive (a broken bond costs $2J$ ), so fences stay small and local, and the bulk of the sample remembers the boundary's orientation. That persistent memory is the spontaneous magnetization.

Figure

A down-island inside an up-sea is bounded by a domain-wall contour; each crossed bond costs 2J

The deep tension is geometry against thermodynamics. A wall of perimeter $\ell$ has energy strictly linear in $\ell$ , but the number of distinct walls of that length grows exponentially in $\ell$ . Entropy wants to multiply walls; energy wants to forbid them. In two dimensions the exponential entropy base ( $\sim3$ ) can be beaten by the exponential energy suppression ( $e^{2\beta J}$ ) once $T$ is low enough. In one dimension the "wall" collapses to a single point — its energy is a constant $2J$ while its entropy grows with the system size — and entropy always wins, killing order at any $T>0$ . The argument is a probabilistic bound, not an exact calculation. It never computes $m$ ; it certifies $m>0$ by showing the probability of a flipped central spin is strictly below one-half. That is exactly what "rigorous existence of a phase transition" demands: a one-sided inequality, valid uniformly in system size, that survives the thermodynamic limit. And because the contour sum is dominated by small loops near the origin rather than by the distant boundary, the conclusion is a genuine bulk property, independent of the sample's shape.

Figure

Why dimension decides — the energy and entropy of a wall versus its length in 1D and 2D

The core equations

Everything starts from the ferromagnetic Ising Hamiltonian on the square lattice; all-up boundary conditions select one ordered phase so that a nonzero magnetization can appear.

H=-J\sum_{\langle ij\rangle}s_i s_j,\qquad s_i=\pm1,\ J>0 \qquad (1)

where: $J>0$ is the ferromagnetic coupling, $\langle ij\rangle$ sums over nearest-neighbour bonds, and $s_i$ is the spin at site $i$ . Physically, the ground state has all spins aligned; any departure creates unsatisfied bonds. A domain wall is a set of dual-lattice edges crossing unsatisfied bonds. Each such bond raises the energy from $-J$ (satisfied) to $+J$ (unsatisfied), so the excess energy of a configuration is simply $2J$ times the total contour length.

E[\{s\}]-E_0=2J\,(\text{number of unsatisfied bonds})=2J\,L \qquad (2)

where: $E_0$ is the all-up ground-state energy, and $L$ is the total length of all contours. Reading it physically: energy is a perimeter cost, linear in the length of the interface — the crucial fact that lets entropy compete only in $d\ge2$ . The probability that a specific contour $\Gamma$ is present is bounded, independently of system size, by mapping every configuration that contains $\Gamma$ to a lower-energy one by flipping its interior.

\mathrm{Prob}(\Gamma)\le e^{-2\beta J\,|\Gamma|} \qquad (3)

where: $|\Gamma|$ is the contour length and $\beta=1/k_BT$ . The bound comes from an injection: flipping all spins inside $\Gamma$ removes it and lowers the energy by $2J|\Gamma|$ , so $\sum_{\{s\}\ni\Gamma}e^{-\beta E}\le e^{-2\beta J|\Gamma|}Z$ ; dividing by $Z$ gives (3), with no reference to the lattice size. Against this stands the number of contours of length $\ell$ that enclose the origin — a self-avoiding-loop count that grows only geometrically.

N(\ell)\le \ell\,3^{\ell} \qquad (4)

where: $\ell$ is the (even) contour length, $\ell\ge4$ on the square lattice; the base $3$ counts the at-most-three forward directions at each step (no immediate backtrack), and the prefactor $\ell$ counts where the enclosing loop may sit relative to the origin. This is a deliberate overcount; the true growth base is the connective constant $\mu<3$ . Combining the energy bound (3) with the entropy bound (4) through a union bound gives the probability that the central spin is flipped, hence a lower bound on the magnetization.

\langle s_0\rangle=1-2P_-\ge 1-2\sum_{\ell\ge4,\,\text{even}}\ell\,x^{\ell},\qquad x\equiv 3e^{-2\beta J} \qquad (5)

where: $P_-=\mathrm{Prob}(s_0=-1)$ , and $x$ is the effective contour fugacity. The series $\sum_\ell\ell\,x^\ell$ converges for $x<1$ and tends to $0$ as $\beta\to\infty$ , so $\langle s_0\rangle\to1$ at low temperature. The transition to order is guaranteed once the series drops below $\tfrac12$ . In one dimension the same accounting gives the opposite verdict: a single wall costs a fixed energy but has a system-size entropy.

\Delta F_{\text{wall}}^{(1\mathrm D)}=2J-k_BT\ln(N-1)\xrightarrow{N\to\infty}-\infty \qquad (6)

where: $2J$ is the energy of one broken bond and $k_BT\ln(N-1)$ is the entropy of placing that wall at any of the $N-1$ bonds. Because this is negative for large $N$ at any $T>0$ , walls proliferate, the correlation length stays finite ( $\xi\sim e^{2\beta J}$ ), and $m=0$ — exactly the exact transfer-matrix result $T_c=0$ .

Derivation APeierls' proof of spontaneous magnetization in two dimensions

The goal is to prove that the two-dimensional Ising model has $\langle s_0\rangle>0$ at low temperature, uniformly in system size, by combining three ingredients: an exact energy accounting of contours, a size-independent probability bound for a single contour, and a geometric bound on how many contours can enclose a point. We work on a finite square lattice with all-up boundary spins, which explicitly break the up–down symmetry and make $\langle s_0\rangle$ nonzero; the final inequality holds independently of the lattice size, so it survives the thermodynamic limit and certifies genuine long-range order. The strategy converts a hard statement about a many-body Gibbs measure into a convergent sum over loop lengths.

Step 1. Contour representation: every configuration corresponds uniquely to a set of closed, non-crossing contours on the dual lattice that separate up from down regions. With all-up boundaries, $s_0=-1$ requires the origin to be enclosed by at least one contour. Why: a flipped interior spin must be cut off from the up-boundary by a closed interface.

Step 2. Energy accounting: $E-E_0=2J\sum_\Gamma|\Gamma|$ , since each contour edge is one unsatisfied bond costing $2J$ . Why: the excess energy is a pure perimeter cost, additive over disjoint contours.

Step 3. Single-contour bound: flipping all spins inside $\Gamma$ is an injection onto configurations of energy lower by $2J|\Gamma|$ , so $\mathrm{Prob}(\Gamma)=Z^{-1}\!\!\sum_{\{s\}\ni\Gamma}e^{-\beta E}\le e^{-2\beta J|\Gamma|}$ . Why: the flipped configurations form a subset of all configurations, so their partition sum is $\le Z$ ; the suppression is purely local in $|\Gamma|$ , hence size-independent.

Step 4. Union bound + entropy: $P_-\le\sum_{\Gamma\ni0}\mathrm{Prob}(\Gamma)\le\sum_{\ell\ge4}N(\ell)e^{-2\beta J\ell}\le\sum_{\ell\ge4}\ell\,3^{\ell}e^{-2\beta J\ell}=\sum_{\ell}\ell\,x^{\ell}$ with $x=3e^{-2\beta J}$ . Why: the probability of at least one enclosing contour is at most the sum of individual probabilities, and there are at most $\ell\,3^\ell$ enclosing contours of length $\ell$ .

Step 5. Conclusion: $\langle s_0\rangle=(+1)(1-P_-)+(-1)P_-=1-2P_-$ . The geometric-type series $\sum_{\ell\ge4,\text{even}}\ell\,x^\ell=4x^4+6x^6+\cdots$ converges for $x<1$ and can be made $<\tfrac12$ by taking $\beta$ large (numerically for $x\lesssim0.4$ , since then the sum is $\approx0.13$ ), giving $\langle s_0\rangle>0$ uniformly in $L$ . Letting $L\to\infty$ proves spontaneous magnetization persists in the bulk. Why: a uniform lower bound on a bulk observable, independent of boundary distance, is exactly the definition of long-range order. *Verified: verify_aq-sm-peierls-argument.py confirms the $2J$ bond cost, the series $\sum\ell x^\ell=2[y/(1-y)^2-y]$ ( $y=x^2$ ) with leading terms $4x^4+6x^6+8x^8$ , its convergence for $x<1$ , the value $\approx0.1335<\tfrac12$ at $x=0.4$ giving $\langle s_0\rangle\gtrsim0.73$ , and the sharper order threshold $x^\star\approx0.525$ where the sum equals $\tfrac12$ .*

Go deeper · Derivation B · 1 minFailure of order in one dimension

Derivation BFailure of order in one dimension

The goal is to show that the very same energy–entropy accounting predicts no order in one dimension, so that the Ising lower critical dimension is $d=1$ , in exact agreement with the transfer-matrix solution. We take an open chain of $N$ up-spins, insert a single domain wall, and compare the energy it costs with the entropy it gains from the many places it can sit. The free-energy change turns out to be negative for large $N$ at any positive temperature, which means a single wall always lowers the free energy, walls proliferate, and long-range order cannot exist — the mirror image of the 2D result, and the clean explanation of why dimension is decisive.

Step 1. Energy of one wall: $\Delta E=2J$ , a single broken bond, independent of where the wall sits. Why: in 1D the interface between an up-block and a down-block is a single bond, a point defect of fixed cost.

Step 2. Entropy of one wall: $\Delta S=k_B\ln(N-1)$ , because the wall may occupy any of the $N-1$ bonds of the open chain. Why: all $N-1$ positions are energetically equivalent, so the multiplicity is $N-1$ .

Step 3. Free-energy balance: $\Delta F=\Delta E-T\Delta S=2J-k_BT\ln(N-1)$ . For any fixed $T>0$ this is negative once $\ln(N-1)>2J/k_BT$ , i.e. for $N-1>e^{2J/k_BT}$ . Why: the logarithmic entropy, though slowly growing, eventually beats the constant energy cost for a large enough chain.

Step 4. Proliferation: once one wall is free to appear, many do, breaking the chain into domains of typical size $\xi\sim e^{2\beta J}$ — a finite correlation length. Long-range order is destroyed, $m=0$ for all $T>0$ , and $T_c=0$ . Why: with a finite density of walls the boundary orientation is forgotten beyond $\xi$ , so no bulk magnetization survives — precisely the exact 1D transfer-matrix result. *Verified: verify_aq-sm-peierls-argument.py confirms $\Delta F=2J-k_BT\ln(N-1)\to-\infty$ as $N\to\infty$ for any $T>0$ , evaluates the crossover $N=e^{2J/k_BT}+1$ (e.g. $\approx4.85\times10^{8}$ at $k_BT=0.1J$ ), and the 1D correlation length $\xi\sim e^{2\beta J}$ finite for $T>0$ .*

Limiting cases

The single dimensionless number $x=3e^{-2\beta J}$ (2D) or the ratio $2J/k_BT\ln N$ (1D) controls the outcome:

Case	Result	Meaning
Low- $T$ , 2D (convergence)	$x<1\Leftrightarrow k_BT<2J/\ln3\approx1.82\,J$	The contour series converges; a well-defined bound exists. This alone does not yet prove $\langle s_0\rangle>0$ .
Low- $T$ , 2D (order)	$\sum_\ell\ell\,x^\ell<\tfrac12$ (e.g. $x\lesssim0.4$ , $k_BT<2J/\ln7.5\approx0.99\,J$ ; sharper $x^\star\approx0.525$ )	Flip probability $<\tfrac12$ , so $\langle s_0\rangle>0$ — a rigorous lower bound on $T_c$ , below the exact $2.269\,J/k_B$ .
One dimension	$\Delta F=2J-k_BT\ln(N-1)\to-\infty$	A point wall in $N$ places always wins for large $N$ : $T_c=0$ , no order at any $T>0$ (matches transfer matrix).
General $d\ge2$	$E_{\text{wall}}\sim\sigma L^{d-1},\ S_{\text{wall}}\sim L^{d-1}$	Surface energy and entropy scale identically, so a finite $T_c$ exists for all $d\ge2$ ; the Ising lower critical dimension is $d_\ell=1$ .

Worked examples

Example 1 — the magnetization floor at low temperature. Take the 2D square-lattice Ising model at a temperature where $x=3e^{-2\beta J}=0.4$ . Bound the probability that the central spin is flipped and hence the spontaneous magnetization.

Step 1. Sum the contour series exactly. With $y=x^2$ , $\sum_{\ell\ge4,\text{even}}\ell\,x^{\ell}=2\big[y/(1-y)^2-y\big]$ . At $x=0.4$ , $y=0.16$ , giving $2\big[0.16/0.7056-0.16\big]=2\big[0.22676-0.16\big]=\mathbf{0.1335}$ .

Step 2. Therefore $P_-\le0.1335$ , so the magnetization is bounded below: $\langle s_0\rangle=1-2P_-\ge1-2(0.1335)=\mathbf{0.733}$ . The central spin keeps the boundary's sign more than $73\%$ of the time — a strictly positive magnetization, uniformly in system size.

Step 3. Convert to a temperature. $x=0.4$ means $3e^{-2\beta J}=0.4$ , i.e. $e^{-2\beta J}=0.1333$ , so $2\beta J=\ln(7.5)=2.015$ and $k_BT=2J/2.015=\mathbf{0.99\,J}$ . So the argument rigorously proves order at least up to $k_BT\approx0.99\,J$ — comfortably below, but the same side of zero as, the exact $2.269\,J$ .

Example 2 — how large a 1D chain destroys its own order. For the one-dimensional Ising chain at $k_BT=0.1\,J$ , find the chain length at which a single domain wall becomes free-energetically favourable.

Step 1. The single-wall free energy is $\Delta F=2J-k_BT\ln(N-1)$ . It vanishes when $k_BT\ln(N-1)=2J$ , i.e. $\ln(N-1)=2J/k_BT=2/0.1=20$ .

Step 2. Solve: $N-1=e^{20}\approx4.85\times10^{8}$ , so $N\approx\mathbf{4.85\times10^{8}}$ . For chains longer than this the wall lowers the free energy and order is lost.

Step 3. Check a specific case: at $N=10^{6}$ , $\Delta F=2J-0.1J\ln(10^{6}-1)=2J-0.1J(13.8155)=2J-1.382J=\mathbf{0.618\,J}>0$ — still positive, so a chain of a million spins is not yet long enough at this temperature. But the crossover $N=e^{2J/k_BT}$ is finite for every $T>0$ : the thermodynamic limit always destroys 1D order, exactly as the transfer matrix shows.

Origins · 1 minlong-range order and how this was discovered

long-range order

the ordering concept · Long-range order is the property that distinguishes an ordered phase from a disordered one: the correlation

\langle s_0 s_r\rangle

approaches a nonzero constant as the separation

r\to\infty

, equivalently the spontaneous magnetization

m=\lim_{r\to\infty}\sqrt{\langle s_0 s_r\rangle}

is nonzero. The Peierls argument is precisely a proof that long-range order exists in the two-dimensional Ising model at low temperature and is absent in one dimension at any positive temperature. The concept sharpens the intuitive idea of "order" into a precise, boundary-independent statement about the persistence of correlations across the whole system, and it is the quantity whose onset defines the critical temperature. Because it is defined through the infinite-distance limit, long-range order is a genuine thermodynamic property that survives fluctuations, unlike short-range order which merely reflects local energetic preferences; the interplay of the two, and the dimensional threshold at which long-range order can first appear (the lower critical dimension), is the organising theme of the modern theory of phase transitions.

History

1920

— Lenz introduces the two-state spin lattice model

1925

— Ising proves the one-dimensional chain has no finite-temperature order

1936

— Peierls gives the contour argument for long-range order in the 2D Ising model

1944

— Onsager's exact solution fixes the true critical temperature 2J/[k ln(1+√2)]

1964

— Griffiths closes a gap in Peierls' original argument with a rigorous proof

1965

— Dobrushin independently establishes the rigorous contour bound

1966

— Mermin and Wagner extend energy–entropy reasoning to continuous symmetries

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., §13.4 (existence of long-range order) · Kardar, Statistical Physics of Fields, Ch. 3 (lower critical dimension; the Peierls argument) · R. Peierls, Proc. Camb. Phil. Soc. 32, 477 (1936) · R. B. Griffiths, Phys. Rev. 136, A437 (1964) · R. L. Dobrushin, Theory Probab. Appl. 10, 193 (1965) · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 8 · Landau & Lifshitz, Statistical Physics Part 1, §148.

Provenance. All structural facts are standard textbook material. The $2J$ cost per unsatisfied bond follows from $H=-J\sum s_is_j$ ; the contour Boltzmann-weight bound $e^{-2\beta J|\Gamma|}$ is the interior-flip injection (Kardar SPoF Ch. 3; Pathria §13.4); the self-avoiding-loop count $N(\ell)\le\ell\,3^\ell$ and the convergence of $\sum_\ell\ell\,x^\ell$ for $x=3e^{-2\beta J}<1$ are elementary. The order threshold is stated carefully: convergence needs $x<1$ ( $k_BT<2J/\ln3\approx1.82\,J$ ), while the sufficient condition for order needs the sum below $\tfrac12$ (conservatively $x\lesssim0.4$ , $k_BT\lesssim0.99\,J$ ; the exact threshold is $x^\star\approx0.525$ ). The exact comparison value $k_BT_c/J=2/\ln(1+\sqrt2)=2.2692$ is Onsager (1944). Historical attributions: Peierls (1936), rigorously completed by Griffiths (1964) and Dobrushin (1965). The 1D balance $\Delta F=2J-k_BT\ln(N-1)$ and its crossover, plus all worked numbers, are recomputed in verify_aq-sm-peierls-argument.py.

Builds on: The Ising Model: Definition and Hamiltonian · Order Parameter and Spontaneous Magnetization · Domain Walls and Interfacial Excitations · One-Dimensional Ising Model and Transfer Matrix Method. Leads to: Onsager Solution of the Two-Dimensional Ising Model · Kramers-Wannier Duality · High- and Low-Temperature Series Expansions.

Does the Peierls argument compute the critical temperature?

No — and this is its most misunderstood feature. It proves only that spontaneous magnetization exists below some finite temperature; it never computes $T_c$ . The temperature it produces (roughly $k_BT\lesssim0.99\,J$ for the sufficient condition) is a rigorous lower bound on the true $T_c$ , deliberately loose because the entropy estimate $\ell\,3^\ell$ overcounts contours. The exact value, $k_BT_c/J=2/\ln(1+\sqrt2)\approx2.269$ , comes from the Onsager solution. What Peierls delivers is existence — the certainty that a transition happens at all — on which the sharper tools (duality, series, the exact solution) then build.

Why are all-up boundary conditions essential?

They break the up–down symmetry and select one ordered phase, so that $\langle s_0\rangle$ can be nonzero. Without symmetry-breaking boundaries (or an external field $h\to0^+$ ), a finite system's magnetization is exactly zero by symmetry: the $+$ and $-$ states are visited with equal weight. The all-up boundary acts as an infinitesimal ordering field applied at the edge; the power of the proof is that it shows the order survives even as that boundary recedes to infinity — the flip probability bound is dominated by small loops near the origin, not by the distant boundary, so the magnetization is a genuine bulk quantity.

What is the connective constant, and why does it matter here?

The connective constant $\mu$ is the exponential growth rate of the number of self-avoiding walks: the count of length- $\ell$ walks grows like $\mu^\ell$ . Peierls bounds the number of enclosing contours crudely by $3^\ell$ (three forward choices per step), but self-avoidance and closure make the true base smaller — on the square lattice $\mu\approx2.638$ . Using the sharper $\mu$ in place of $3$ improves the temperature bound (raising the guaranteed $T_c$ ), though still below the exact Onsager value because the union bound and the neglect of contour interactions remain. The gap between $3$ , $\mu$ , and the exact $T_c$ is exactly why the argument proves existence but not the precise transition temperature.

How does this connect to the Mermin–Wagner theorem?

Both are energy-versus-entropy arguments about defects, differing in the symmetry and hence the defect type. Peierls treats a discrete ( $Z_2$ ) symmetry, whose defects are sharp domain walls with energy linear in their size; the balance gives a lower critical dimension $d_\ell=1$ , so 2D orders. Mermin–Wagner treats a continuous symmetry, whose cheap excitations are smooth, gapless spin waves (Goldstone modes); their entropy overwhelms their energy up through two dimensions, giving $d_\ell=2$ , so a 2D Heisenberg magnet does not order. The unifying principle is the same — count the cheapest disordering excitations and weigh their entropy against their energy — but continuous symmetries have much cheaper excitations, pushing the lower critical dimension up by one.

Why does the argument need

d\ge2

Because the whole method rests on contours being loops whose energy grows with their perimeter. In $d\ge2$ a domain wall is an extended interface of size $L$ : its energy scales as $\sigma L^{d-1}$ and its entropy also as $L^{d-1}$ , so the two can balance and produce a finite $T_c$ . In $d=1$ the "wall" degenerates to a single point of fixed energy $2J$ , with entropy $\ln N$ that grows without bound as the chain lengthens — there is no perimeter to make energy competitive, so entropy always wins and $T_c=0$ . Applying the contour counting to 1D is a category error: there are no loops to count, only point defects.

Is the argument specific to the square lattice, or more general?

The specific numbers ( $3^\ell$ , the $\ell\ge4$ minimum contour, the $0.99\,J$ bound) are for the square lattice, but the logic is general. On any lattice in $d\ge2$ with short-range ferromagnetic interactions, contours have energy proportional to their surface and a number growing at most geometrically in their surface, so the same convergence argument establishes long-range order at low temperature. Only the lattice-dependent constants (coordination number, connective constant, minimum contour size) change. Peierls-type arguments have been extended to antiferromagnets, Potts models, lattice gases, and many other systems — anywhere a phase can be described by domains with a well-defined interface cost.