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The dot product of the two magnetizations is ${\bar{m}}^2\cos [\theta (x)-\theta (0)]$. For the Gaussian spin wave field, the Gaussian phase averaging identity $⟨e^{iX}⟩=e^{-⟨X^2⟩/2}$ for a centered Gaussian variable givesKeeping this exponential is essential: expanding the cosine to quadratic order would fail when phase differences grow with separation. Equivalently, the phase-difference variance isFor $d>2$, the integral defining the absolute phase variance is finite at small momentum after fixing the uniform mode; its microscopic ultraviolet cutoff still matters. Its nonconstant Green function decays as $r^{2-d}$, so $D(r)$ tends to a finite constant $D_{\infty }$ andIn two dimensions, the logarithmic Green function gives $D(r)=(\pi \bar{K}{)}^{-1}\log (r/a)+O(1)$. ConsequentlyFor $1\le d<2$, $D(r)$ grows as $2r^{2-d}/[(2-d)S_d\bar{K}]$, up to bounded microscopic terms, and the correlation function again vanishes. In particular, in one dimension $S_1=2$ gives $D(r)=r/\bar{K}$ and exponential decay ${\bar{m}}^2{e}^{-r/(2\bar{K})}$. Therefore there is no conventional long-range order for $d\le 2$ at positive temperature in this short-range continuous-symmetry model. In two dimensions, algebraic decay can still support quasi-long-range order.

These infrared spin-wave correlations of the XY model explain the mechanism behind the Mermin-Wagner theorem. They are a test of the low-temperature spin wave approximation, not a claim that every $d>2$ XY model is ordered at every temperature. Large local fluctuations and defects destroy the ordered phase at sufficiently high temperature. Also, the assumed small angle is a local smooth-field approximation; phase differences at arbitrarily large separation need not remain small.

The Berezinskii–Kosterlitz–Thouless transition is a defect-unbinding transition of a two-dimensional system with a compact continuous phase, such as the short-range XY model. At long wavelengths, smooth spin waves have energywhere ${\rho }_s$ is the phase stiffness. Gaussian averaging givesThus smooth fluctuations already destroy true nonzero magnetization in an infinite two-dimensional system, in accordance with the Mermin-Wagner theorem. The low-temperature phase instead has quasi-long-range order, with algebraically decaying correlations. The stiffness in its asymptotic exponent is the renormalized long-distance stiffness, not merely the microscopic coupling.

The phase is periodic modulo $2\pi$, so it also permits topological defects with integer winding $n$. A phase vortex has $|\nabla \theta |=|n|/r$, and integration givesA singly charged phase vortex can be placed in roughly $(L/a{)}^2$ locations, giving entropy $2k_B\log (L/a)$. This energy-entropy balance suggests that isolated vortices become favorable when the effective stiffness falls to $2k_{B}T/\pi$. Below the transition, vortex-antivortex binding predominates; above it their unbinding screens the logarithmic interaction and produces a finite correlation length. A single-vortex estimate identifies the mechanism but does not include screening by smaller pairs.

The dilute-vortex vortex-pair renormalization flow incorporates this screening. With $K={\rho }_s/(k_{B}T)$ and dimensionless core fugacity $y$, one conventional leading-order normalization givesThe first equation says polarizable phase vortex pairs reduce stiffness. The second compares the phase vortex's position scaling dimension two with its logarithmic energy cost $\pi K$. Numerical coefficients in $y$ depend on its definition, but the zero-fugacity threshold $K=2/\pi$ is invariant. For $K>2/\pi$, sufficiently small fugacity flows to zero, leaving a line of Gaussian spin-wave renormalization-group fixed points with varying algebraic exponent. On the other side of the separatrix, fugacity grows and the dilute approximation eventually ceases to apply.

Near the endpoint set $X=2-\pi K$ and $Y=4\pi y$. To leading order $X^{\prime }=Y^2$ and $Y^{\prime }=XY$, so $X^2-Y^2$ is invariant. The critical separatrix approaches $(0,0)$. Just on the disordered side this invariant is $-{\epsilon }^2$, proportional to the reduced temperature deviation for a generic path through the transition. Hence $X^{\prime }=X^2+{\epsilon }^2$, whose integration gives a time in logarithmic scale of order $1/\epsilon$. Stopping when the fugacity becomes order one yieldswith nonuniversal $B>0$. This derives the essential singularity and explains why assigning a finite ordinary correlation-length exponent is inappropriate.

At the transition the limiting stiffness from below has the universal stiffness jumpAbove $T_c$, the thermodynamic long-distance stiffness vanishes and correlations decay exponentially. The singular free-energy contribution is of order ${\xi }^{-2}$ and is exponentially small in $1/\sqrt{\tau }$, so this infinite-order transition is unlike a Landau transition with a discontinuous order parameter or an ordinary power-law heat-capacity singularity. The renormalization mechanism is developed in Kosterlitz's original vortex analysis. The transition separates bound vortices and algebraic order from unbound vortices and short-range order, while neither finite-temperature phase has conventional spontaneous XY magnetization.

The vacuum of the diagonal quasiparticles contains original spin bosons: $⟨a_k^{†}{a}_k⟩=v_k^2$. Thus the quantum depletion of Néel order givesIn one dimension $\sqrt{1-{\gamma }_k^2}=|\sin k|$. Removing the global zero modes with an infrared cutoff, the continuum integral contains $\int dk/|k|$ near both gapless points and diverges logarithmically as the cutoff is removed. This is a breakdown of the assumed Néel order reference, not a physical infinitely negative magnetization. It indicates that the one-dimensional ordered spin-wave expansion cannot maintain a finite staggered moment; it does not determine the exact spectral gap or all properties of the spin chain for arbitrary $S$.

At zero temperature in $d$ dimensions the singular contribution scales as ${\int }_0{k}^{d-2}dk$. It is infrared finite for $d\ge 2$, permitting a finite quantum reduction and a self-consistent ordered spin-wave description in an appropriate regime. This differs from the positive-temperature contribution, where $n_B(\omega )\sim T/\omega$ yields ${\int }_0{k}^{d-3}dk$. Its divergence for $d\le 2$ agrees with the Mermin-Wagner theorem for short-range continuous-symmetry models. Finite-temperature nonordering and the zero-temperature one-dimensional depletion argument are separate statements.

Assume $g>0$ for stability. With equal quadratic coefficients and $\lambda =2g$, the internal symmetry is the full orthogonal group $O(2)$. In two dimensions at positive temperature, with short-range interactions, the Mermin-Wagner theorem forbids spontaneous symmetry breaking of this continuous symmetry. Thus $⟨{\varphi }_1⟩=⟨{\varphi }_2⟩=0$ in both phases in the infinite system; a nonzero average order parameter does not distinguish them.

Instead measure the correlation functionThe low-temperature phase has quasi-long-range order and algebraic correlations; the high-temperature phase has a finite correlation length and exponential correlations. More precisely,A finite order parameter amplitude allows $\mathbf{\varphi }\simeq v(\cos \theta ,\sin \theta )$. Its long-distance Goldstone-mode effective free energy is $F_{\theta }=({\rho }_s/2)\int d^{2}x(\nabla \theta {)}^2$. Restoring the Boltzmann constant and using the renormalized phase stiffness, the thermal phase fluctuations giveThe Berezinskii–Kosterlitz–Thouless transition separates vortex-antivortex binding from unbound vortex defects. At the boundary, the universal stiffness jump gives ${\rho }_s(T_{\mathrm{BKT}}^{-})=2k_B{T}_{\mathrm{BKT}}/\pi$ and $\eta (T_{\mathrm{BKT}})=1/4$. The low-temperature phase stiffness is nonzero; it vanishes at long distances in the disordered phase. The exact transition location is fluctuation dependent and cannot be obtained just by setting the bare ${\mu }^2$ to zero.

When $m_1=m_2={\mu }^2$, the free energy has continuous symmetry $O(2)$. Its ordered minima satisfyContinuous phase fluctuations make the lower critical dimension $d_{\mathrm{l}}=2$, in agreement with the Mermin-Wagner theorem. In two dimensions write the complex order parameter as $\psi ={\rho }_0{e}^{i\theta }$. Neglecting the massive amplitude mode gives the Goldstone-mode effective free energyThe phase-difference variance isand thereforewithThus spin waves replace true long-range order by quasi-long-range order.

A vortex-antivortex pair of separation $R$ has the logarithmic energy $E_{\mathrm{pair}}\simeq 2\pi {\rho }_0^2\log (R/a)$. The number of pair separations below $R$ grows as $R^2$, giving the coarse entropy $S_{\mathrm{pair}}\simeq 2\log (R/a)$. Thusand widely separated pairs become favorable at ${\rho }_0^2\simeq 1/\pi$. Substituting the mean-field stiffness givessuggesting a Berezinskii–Kosterlitz–Thouless transition. The numerical location is only a bare-stiffness estimate: vortex-core fluctuations renormalize the stiffness near the transition.

For the $O(2)$ model writeAt long distances the massive radial field $\rho$ can be neglected, leaving the Goldstone-mode effective free energyThereforeThe mode is massless, so its correlation length is infinite. For $d>2$ its large-distance Green function isFor $d=2$ it is $-(2\pi \gamma v^2{)}^{-1}\log (r/a)$ up to an infrared-dependent constant, and in $d=1$ it is $-|r|/(2\gamma v^2)$ up to such a constant.

The invariant diagnostic is the phase-difference varianceIt diverges linearly in $d=1$ and logarithmically in $d=2$, destroying true long-range order, but approaches a finite infrared limit for $d=3,4$. Thus the continuous-symmetry ordered phase exists for $d=3,4$ and not for $d=1,2$, in agreement with the Mermin-Wagner theorem. The lower critical dimension is $d_{\mathrm{lc}}=2$; the two-dimensional $O(2)$ model can instead show quasi-long-range order below a BKT transition.

For finite $p=4$, the clock model has a discrete symmetry. Domain walls have finite energy per unit boundary area, so thermal disorder destroys long-range order in one dimension but a finite-temperature ordered phase can exist in two dimensions. Its lower critical dimension is therefore $\boxed{d_{\mathrm{l}}=1}$.

As $p\to \infty$, the permitted angles become continuous and the model becomes the XY model with $O(2)$ symmetry. The Mermin-Wagner theorem forbids spontaneous long-range order at positive temperature in two dimensions, so the lower critical dimension for conventional symmetry breaking is $\boxed{d_{\mathrm{l}}=2}$. The two-dimensional model can nevertheless undergo a Berezinskii–Kosterlitz–Thouless transition between algebraic and exponential correlation decay.

For bosonic Matsubara frequencies ${\omega }_n=2\pi nT$, set $a=vk/(2\pi T)$. Applying the residue theorem to $\pi \coth (\pi z)/(z^2+a^2)$, whose integer poles reproduce the desired summands, givesIt follows thatUsing the area $S_{d-1}=2{\pi }^{d/2}/\Gamma (d/2)$ of the unit sphere, the thermal phase fluctuation becomesAt small $k$, $\coth (vk/(2T))\sim 2T/(vk)$, so the infrared divergence is governed by ${\int }_{0}dk{k}^{d-3}$. It diverges for $d=1,2$ and is finite for $d=3$. Therefore short-range systems cannot have true finite-temperature breaking of this continuous symmetry in one or two dimensions, in agreement with the Mermin-Wagner theorem, whereas it is allowed in three dimensions. In two dimensions a Berezinskii–Kosterlitz–Thouless transition may still produce quasi-long-range order.

The long-wavelength Goldstone fluctuation is proportional towhich diverges in the infrared for $d\le 2$. These fluctuations destroy finite-temperature long-range order with a broken continuous symmetry, as formalized by the Mermin-Wagner theorem. Therefore the lower critical dimension is $d_l=2$: conventional spontaneous $O(N)$ order is possible for $d>2$ but not for $d\le 2$.