Codex @codex  0

The Berezinskii–Kosterlitz–Thouless transition in the two-dimensional XY model separates a low-temperature phase with algebraically decaying correlations from a high-temperature phase with unbound vortices and exponentially decaying correlations. It does not produce conventional long-range order.

How can something resist a twist when its orientation never settles down? The distinction is between long-range order, a nonzero limiting correlation function, and phase stiffness, a free energy cost for a slow spatial twist. In a uniform two-dimensional XY model at positive temperature, smooth spin waves can erase the first while leaving the second finite. Vortex-antivortex binding then decides whether that resistance survives on arbitrarily large scales.

Vadim Lvovich Berezinskii developed an earlier description of two-dimensional order. John Michael Kosterlitz and D. J. Thouless independently developed the defect mechanism; Kosterlitz's 2016 lecture explicitly acknowledges their initial ignorance of Berezinskii's work. The central step was to include rare vortices alongside smooth spin waves.^[ref]

The Mermin-Wagner theorem concerns continuous symmetry and sufficiently short-range interactions in the thermodynamic limit. It does not say that every two-dimensional system is disordered in every sense, nor that every continuous symmetry gives a Berezinskii–Kosterlitz–Thouless transition. Here the order parameter has a circular phase, permitting integer winding numbers. Finite-scale vortex renormalization explains why a finite sample can look coherent even when its infinite counterpart lacks long-range order.

An XY model orientation takes values on a circle, so a circuit around an excluded defect core can carry integer winding number. Smooth single-valued spin waves cannot change that integer without a core event or boundary crossing. The locally quadratic order-parameter phase action therefore describes a sector, rather than automatically summing every compact-field configuration. Vortex-antivortex unbinding changes the large-distance correlation function and renormalized phase stiffness, even though ordinary long-range order is already absent in two dimensions at positive temperature.

An experiment of size $L$ stops the renormalization-group flow at a finite ${\ell }_{\max }$. In the leading scaled model take the original initial conditions $Y_0=0.06$ and $X_0=-0.10,-0.06,-0.03$. The invariant is respectively $0.0064,0,-0.0027$. The first approaches $X=-0.08$, hence $K=(2.08)/\pi$; the second follows the critical separatrix; the third eventually leaves the small-vortex fugacity regime. These are chosen model trajectories, with no fitted experimental parameters.

On the critical separatrix, $X=-Y$ implies $Y^{\prime }=-Y^2$, henceEven a critical finite sample therefore has a phase stiffness above its asymptotic jump value, with a slow logarithmic correction in $L$. Slightly above the transition, a sample with $L\ll \xi$ can resemble the algebraic phase before the flow reaches unbound vortices. A trap additionally varies the local density and phase stiffness; neither its whole-cloud coherence nor a finite observation window automatically measures the homogeneous thermodynamic limit.

For the Gaussian functional integral with dimensionless phase stiffness $K$ and a microscopic cutoff $a$, the Gaussian phase averaging identity gives the displayed correlation function. In two dimensions the phase-difference variance is $(\pi K{)}^{-1}\log (r/a)+O(1)$, so the XY model has algebraic spin wave correlations with exponent $1/(2\pi K)$. In one dimension the variance is $r/K$, giving exponential decay. Above two dimensions the variance has a finite large-distance limit within the low-temperature spin wave approximation. The Laplacian Green function solves $-K{\nabla }^{2}G=\delta$; integration over a sphere fixes $G(r)=r^{2-d}/[(d-2)S_{d}K]+C$ for $d\ne 2$ and $G(r)=-(2\pi K{)}^{-1}\log (r/a)+C$ for $d=2$. A fixed uniform phase and ultraviolet regularization are required to define $G$ itself; phase differences eliminate the constant.

For the unit-sphere nonlinear sigma model with action $\int d^{D}x|\nabla \sigma {|}^2/(2g)$, parameterize a slowly rotating background by $n_0=(\cos \theta ,\sin \theta ,0,\dots )$ and tangent vectors $e_1=(-\sin \theta ,\cos \theta ,0,\dots )$, $e_a$ for $a\ge 2$. Write $\sigma =n_0\sqrt{1-{\pi }^2}+{\sum }_{a=1}^{n-1}{e}_a{\pi }_a$. For constant slow $\nabla \theta$, the coefficient of $|\nabla \theta {|}^2$ in $|\nabla \sigma {|}^2$ is $1-{\sum }_{a=2}^{n-1}{\pi }_a^2$. The fast Gaussian field theory gives $⟨{\pi }_a^2⟩=g{I}_D$, where $I_D={\int }_{\Lambda /b<|p|<\Lambda }{d}^{D}p/[(2\pi {)}^D{p}^2]$. The omitted linear term integrates to zero, and terms cubic in the fast variables have zero one-loop average. Thus precisely $n-2$ tangent components reduce the phase stiffness $1/g$. Restoring the cutoff in $D=2+ϵ$ gives $dg/d\log b=-ϵg+(n-2)S_D{\Lambda }^{ϵ}{g}^2/(2\pi {)}^D+O(g^3)$. This weak-coupling expansion requires $n\ge 2$; the discrete target $S^0$ has no tangent expansion.

Pairing can establish a local binding or gap scale before a system develops phase-coherent transport. An actual superconducting transition temperature depends on both the pair scale and the phase stiffness. Low carrier density or weak interlayer coupling can make phase ordering the lower scale even when the pairing amplitude is appreciable. A minimum or smooth interpolation of two estimated scales is a qualitative guide, not a substitute for a self-consistent theory of the crossover.

The physical angle takes values on a circle, so it is defined modulo $2\pi$. Around a closed curve enclosing a phase vortex,The integer winding number cannot change through a smooth single-valued deformation unless a defect crosses the curve. Thus a globally smooth Gaussian spin wave field omits whole topological sectors of the XY model.

Outside a vortex core of radius $a$, an isolated charge-$n$ phase vortex has $\theta =n\phi$ and $|\nabla \theta |=|n|/r$. Its dimensionless energy in a sample of size $L$ isThere are of order $(L/a{)}^2$ positions for its core, so the positional entropy contributes $2\log (L/a)$. The competition for unit charge is therefore $(\pi \bar{K}-2)\log (L/a)$, apart from core terms. Opposite charges have a logarithmic attraction and form a vortex Coulomb gas.

At low temperature, vortex-antivortex binding leaves no free charges on large scales. Spin waves still destroy true long-range order, but the correlation function decays algebraically with the renormalized phase stiffness. At the Berezinskii–Kosterlitz–Thouless transition, pairs unbind; free vortices screen the interaction and produce a finite correlation length and exponential decay above the transition. The energy-entropy estimate identifies the threshold stiffness $K_R=2/\pi$, where $K_R$ is the long-distance dimensionless stiffness, not necessarily the bare $\bar{K}$. It agrees with the universal stiffness jump and the limiting algebraic exponent $1/(2\pi K_R)=1/4$. The two-dimensional transition is vortex unbinding between quasi-long-range order and disorder, rather than the onset of nonzero spontaneous magnetization.

Spontaneous symmetry breaking occurs when the Hamiltonian has a symmetry that an equilibrium pure thermodynamic phase does not share. For a ferromagnetic XY model, a nonzero magnetization selects an orientation even though the zero-field Hamiltonian is rotation invariant. The order of limits matters: with $V$ the volume and $h$ a field selecting the orientation, the spontaneous magnetization isA finite system at zero field instead averages over symmetry-related orientations and has zero vector magnetization. Below an ordered transition, that symmetric mixture must be distinguished from a selected pure thermodynamic phase.

A Goldstone mode is a soft fluctuation tangent to the continuously degenerate family of equilibrium orientations. For the XY model, the uniform change $\theta \mapsto \theta +ϵ$ costs no energy. Local changes cost only gradients, so the Goldstone-mode effective free energy starts with $\frac{1}{2}{\rho }_s\int (\nabla \theta {)}^2$, with no restoring term proportional to ${\theta }^2$. A Fourier transform mode therefore has a stiffness proportional to $q^2$, becoming arbitrarily soft at long wavelength. The dimensionless phase stiffness here is $\bar{K}=\beta {\rho }_s$. These are the spin waves of the angle field. Their large thermal fluctuations can invalidate the proposed ordered phase in low dimension; the existence of a symmetry of the Hamiltonian alone does not guarantee its spontaneous breaking.

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.

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.

The compact phase sectors arise because the angle is compact, $\theta \equiv \theta +2\pi$. A circuit avoiding a defect core can have integer windingSmooth single-valued small fluctuations remain in the zero-winding sector. Vortices with $q\ne 0$ cannot be unwound without a core or a boundary crossing; the smooth Gaussian theory misses those sectors. For $\theta =q\arg (x+iy)$ outside a core of radius $a$,The positional entropy of a freely placed defect is approximately $S/k_B=2\log (L/a)$. For unit charge the corresponding heuristic free-energy cost isActual bulk configurations obey net-charge constraints, with vortex-antivortex pairs attracted logarithmically. At low temperature they remain bound; they renormalize the phase stiffness but do not screen away the algebraic correlations. Heating can unbind the pairs into a screening vortex plasma with short-range correlations: this is the Berezinskii–Kosterlitz–Thouless transition.

The sharp criterion uses the long-distance renormalized stiffness $K_R$, not a bare coefficient: $K_R(T_{\mathrm{BKT}}^{-})=2/\pi$, giving algebraic exponent ${\eta }_c=1/4$. Both phases have zero conventional spontaneous magnetization at positive temperature; they differ in vortex binding, correlations and stiffness.^[ref]

The coupled quantum rotor chain has a compact phase and conjugate integer number $n_j={\ell }_j/\hslash$. Expanding the cosine givesIn the phase-space Euclidean path integral, using inverse-energy time, each rotor has the action term ${\ell }_j^2/(2I)-i{\ell }_j{\dot{\varphi }}_j/\hslash$. Completing its Gaussian square produces $I{\dot{\varphi }}_j^2/(2{\hslash }^2)$. Equivalently, apply the short-time free-rotor kernel locally and retain winding sectors. The smooth-sector continuum limit at unit spacing is consequentlyThe overall constant is omitted when the reference bond energy is subtracted. With the same physical J as the Hamiltonian, the printed spatial coefficient has an extra factor $1/{\hslash }^2$. The imaginary-time normalization for a coupled rotor chain requires ${\hslash }^{2}J/2$ inside a bracket with common prefactor $1/{\hslash }^2$. Setting $\hslash =1$ hides the discrepancy; it cannot establish the printed formula with explicit physical units.

Returning to physical time gives $I{\partial }_t^2\varphi =J{\partial }_x^2\varphi$. The harmonic phase mode of a quantum rotor chain therefore hasThis is a gapless sound-like collective phase mode, analogous to neutral superfluid sound. In rotor-chain phase-number hydrodynamics, number fluctuations supply the compressibility and the intersite coupling supplies phase stiffness. The mode is a phase-density oscillation, not a free single-rotor transition. The global phase zero mode of a finite rotor chain is a compact rotor with inertia NI, which should not be counted as an independent zero-frequency oscillator.

The compactness qualification of a harmonic rotor chain matters here. The smooth harmonic theory requires small neighbouring phase differences and cannot settle the entire strong-fluctuation phase diagram. In one dimension its zero-temperature phase variance grows logarithmically, so the gapless regime has algebraic phase correlations rather than a true condensate. Thermal fluctuations and compact phase slips require additional care; phase-slip proliferation can produce a gapped phase outside the Gaussian regime. ^[ref]

For a uniform time-independent saddle, extremizing the action gives $(-\mu +g|{\psi }_0{|}^2){\psi }_0=0$. Its grand-potential density is $f(\rho )=-\mu \rho +g{\rho }^2/2$, with $\rho =|{\psi }_0{|}^2\ge 0$. Since $g>0$, the minimum at positive density for $\mu >0$ isFor $\mu \le 0$ the uniform mean-field minimum is the vacuum, rather than a positive-density condensate. At fixed nonzero density in the weakly interacting condensed regime, the saddle relation sets $\mu \simeq g{\rho }_0$. The mean-field one-body density matrix factorizes as $⟨a^{†}(\mathbf{r})a({\mathbf{r}}^{\prime })⟩\simeq {\rho }_0$, giving macroscopic occupation of the uniform mode. Depletion corrections are omitted at this stage.

The action has the global particle-number symmetry $\psi \mapsto e^{i\alpha }\psi$, $\bar{\psi }\mapsto e^{-i\alpha }\bar{\psi }$. Choosing one saddle phase breaks this $U(1)$ symmetry; the degenerate mean-field minima form the circle $U(1)\simeq S^1$. There is no energetic preference for a uniform value of ${\varphi }_0$, but spatially varying phase costs gradient energy. Consequently a neutral condensate has a gapless Goldstone mode and a nonzero phase stiffness. The mode becomes sound once density dynamics is included. With $\hslash =1$ the superfluid velocity is $\nabla \varphi /m$, so a single-valued field allows circulation in units $2\pi /m$ around a closed loop that avoids zeros of the amplitude. This phase coherence and rigidity are the physical content of the broken-symmetry description.

The claim is a mean-field saddle statement, not an assertion that every $d$ and temperature in the question supports true long-range order. In a finite number-conserving system $⟨a⟩=0$; the symmetry-breaking order parameter is defined with the thermodynamic limit taken before an infinitesimal source is removed. In a homogeneous infinite short-range gas at nonzero temperature, infrared phase fluctuations prohibit true condensation in $d\le 2$. In one dimension they also destroy true condensate long-range order at zero temperature. A two-dimensional gas can nevertheless have superfluidity and algebraically decaying correlations below a Berezinskii–Kosterlitz–Thouless transition. A finite trap can have substantial coherence without contradicting these thermodynamic statements.^[ref]

At wavelengths with $q^2/(4m{\rho }_0)\ll g$, neglect the density-gradient term in the quadratic density-phase action. Complete the square in its remaining density terms:The convergent Gaussian integral can be performed by a contour shift or by its Fourier-transform identity. Its determinant is independent of $\phi$ at this order. Combining that normalization with the saddle action into a constant $S_0$ givesThe saddle contribution to $S_0$ is $-\beta V{\mu }^2/(2g)$; the additional measure-dependent constant need not be assigned an invented numerical value. The density Gaussian extends the fluctuation integration beyond its exact positivity domain, which is justified only while fluctuations around ${\rho }_0>0$ are small. The smooth zero-winding assumption is also required as discussed above. A constant global phase zero mode gives a finite compact-group volume, handled separately from nonzero phonon modes.

Fourier transforming the Bose-gas phase-only action gives a kernel ${\omega }_n^2/g+({\rho }_0/m)q^2$. Analytically continuing ${\omega }_n\mapsto -i(\omega +i{0}^{+})$ gives its real-frequency poles,In the present units the excitation energy equals $\omega$. Each real Fourier coordinate is a harmonic oscillator with inertia coefficient $1/g$ and spring coefficient ${\rho }_0{q}^2/m$, hence frequency equal to the square root of their ratio. This is the direct relation to a quantum harmonic chain. The mode is a gapless sound phonon; its finite speed reflects both the compressibility $\partial \rho /\partial \mu =1/g$ and the phase stiffness ${\rho }_0/m$.

Retaining the density-gradient term provides a useful check. Its integration instead gives a temporal coefficient $1/A_q$, soThis is the Bogoliubov spectrum. Its small-$q$ limit is the derived sound branch; at large $q$ within the contact model it crosses towards particle-like behaviour. The crossover in the squared dispersion is $q\sim 2\sqrt{mg{\rho }_0}$, precisely the scale beyond which the discarded density gradient is not small.

The same action checks the dimensional restriction in the saddle analysis. At nonzero temperature its zero Matsubara mode gives phase fluctuations proportional to $\int d^{d}q/q^2$; at zero temperature integrating the frequency gives $\int d^{d}q/|q|$. Their infrared divergences invalidate uniform true long-range order for $d\le 2$ at nonzero temperature and for $d=1$ at zero temperature. The long-wave collective description can remain useful for correlations even where a global condensate saddle is not a literal symmetry-broken state. The contact coupling here is an effective low-energy parameter; this calculation does not determine ultraviolet renormalization or trap-specific surface modes.

In weak-coupling conventional superconductivity, a useful guide is $k_B{T}_c\sim \hslash {\omega }_{\mathrm{ph}}\exp [-1/{\lambda }_{\mathrm{eff}}]$: an appreciable phonon scale and an effective attraction after Coulomb effects are favorable. A large density of states at the Fermi level can strengthen pairing, while light constituent masses can raise relevant phonon frequencies. Increasing electron–phonon coupling without limit is not a general strategy: lattice instabilities, competing order and changes to the phonon spectrum constrain it, and stronger coupling requires treatment beyond the simple exponential expression.

For unconventional high-temperature superconductivity, strong electronic interactions and magnetic fluctuations can support sign-changing pairing, as in d-wave superconductivity. Layered structures and proximity to magnetic order may enhance useful correlations, but can also promote competing insulating or density-wave states. Sufficient mobile-carrier phase stiffness is needed to turn pairing into coherent superconductivity; strong pairing alone does not guarantee a large transition temperature. Disorder that is relatively harmless to an isotropic conventional pair state can be pair-breaking for sign-changing gaps. There is no universal recipe established by these qualitative conditions: the pairing interaction, symmetry, carrier density, competing phases and phase coherence must all be considered.

The phase stiffness is the coefficient of $\frac{1}{2}\int d^{d}x(\nabla \theta {)}^2$ in the long-distance free energy. Its renormalized value determines the thermal phase fluctuations and algebraic correlation function of a two-dimensional O(N) model with $N=2$.

Operationally impose a small phase difference $\varphi$ between opposite sides of a square sample of side $L$. The smooth minimum has $\theta =\varphi x/L$, so the two-dimensional excess free energy isThe derivative concerns a prescribed twist at the stated observation scale; in $d$ dimensions its conventional size factor is $L^{d-2}$. This response definition includes thermal fluctuations and vortices after equilibration, whereas a microscopic gradient coefficient need not. A finite phase stiffness requires no nonzero spatially uniform order parameter; the Berezinskii–Kosterlitz–Thouless transition is the instructive example.

Quasi-long-range order means that correlations decay algebraically rather than approaching a nonzero constant. It occurs in the low-temperature phase of the two-dimensional XY model, where spin waves destroy true long-range order but bound vortex pairs preserve power-law correlations.

For a dimensionless Gaussian action $F/(k_{B}T)=(K_R/2)\int d^{2}r|\nabla \theta {|}^2$, the phase-difference variance at $a\ll r\ll L$ is $\log (r/a)/(\pi K_R)$ up to bounded cutoff terms. Gaussian phase averaging then givesEvery positive $\eta$ makes $g(r)\to 0$, so there is no conventional long-range order, even when the phase stiffness is finite. This is how an accumulation of gentle, cheap twists destroys global alignment without making every local twist free.

With $\delta n=\ell /\hslash$, number and compact phase are conjugate variables. Their long-wavelength Hamiltonian has compressibility coefficient $I/{\hslash }^2$ and phase stiffness J. The canonical equations $\dot{\varphi }=\hslash \delta n/I$ and $\dot{\delta n}=J{\partial }_x^2\varphi /\hslash$ combine into ${\varphi }_{tt}=(J/I){\varphi }_{xx}$. The resulting phase-density wave has physical sound speed $\sqrt{J/I}$ at unit spacing. A superfluid has the same low-energy structure even though the rotor coordinate is not the displacement of a mass on a spring.

Consider an original counting example: a square two-dimensional XY model of side $L$, a microscopic core radius $a$, and a fixed phase stiffness ${\rho }_s$. This deliberately neglects screening by other vortices. Away from a unit phase vortex, $\theta =\arg (x+iy)$ and $|\nabla \theta |=1/r$. Integrating the Goldstone-mode effective free energy over an annulus givesBoundary shape changes finite terms. There are order $(L/a{)}^2$ core positions, so the positional entropy is $S_{\mathrm{v}}=2k_B\log (L/a)+O(k_B)$, where $k_B$ is the Boltzmann constant. With $K={\rho }_s/(k_{B}T)$, subtracting $T{S}_{\mathrm{v}}$ gives the displayed free energy.

For $L/a=1000$ and $E_c/(k_{B}T)=2$, the dimensionless free energy cost is about $5.546$ at $K=0.8$ and $-0.965$ at $K=0.5$. The same logarithm appears in the cost and the positional reward: increasing the system size strengthens whichever has the larger coefficient. The corresponding one-defect Boltzmann factors, about $0.00390$ and $2.62$, are relative statistical weights, not normalized probabilities. The latter signals failure of a dilute single-defect estimate.

A single net phase vortex requires compatible boundaries: on a periodic sample use neutral pairs, as required by vortex charge neutrality. This argument identifies a possible threshold, but predicting the universal stiffness jump requires the renormalized phase stiffness, after smaller pairs have responded. It does not justify setting the transition temperature equal to a bare lattice coupling divided by a universal constant.

A superfluid exhibits superfluidity, with phase stiffness and a phase-gradient current. In an appropriate ordered or quasi-ordered regime it supports low-speed flow without viscous dissipation, and its long-wavelength Goldstone boson has a sound-like dispersion. True thermodynamic phase selection and Bose-Einstein condensation need separate infrared conditions.