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An experimental atomic physicist who coauthored a study of the Berezinskii–Kosterlitz–Thouless transition in a trapped gas. Interference of planar atomic clouds probed spatial coherence, connecting a low-dimensional many-particle state to a directly visible optical image.^[ref]

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.

At $D=2$ and $n>2$, the weak-coupling renormalization-group flow obeys $dg/d\ell =(n-2)g^2/(2\pi )$. Integrating gives $1/g(\ell )=1/g_0-(n-2)\ell /(2\pi )$. The flow reaches order-one coupling at a length whose logarithm is $2\pi /[(n-2)g_0]+O(1)$ at this order. Restoring the microscopic scale ${\Lambda }^{-1}$ gives the displayed leading exponential correlation length. Higher loops affect its prefactor, and this conclusion does not apply to the compact $n=2$ model with its Berezinskii–Kosterlitz–Thouless transition.

A theoretical condensed matter physicist who explained how topological defects can control a phase transition in two dimensions. In the Berezinskii–Kosterlitz–Thouless transition, binding or unbinding of vortex pairs changes long-distance correlations without conventional long-range order. His account connects a defect's energy and entropy to the behavior of an entire many-particle system.^[ref]

An experimental physicist who coauthored measurements of low-dimensional coherence in ultracold gases. The experiment used interference of two planar clouds to study the crossover associated with the Berezinskii–Kosterlitz–Thouless transition, linking spatial correlations to observable fringe structure.^[ref]

The Mermin-Wagner theorem excludes spontaneous continuous-symmetry order at positive temperature in important one- and two-dimensional systems with sufficiently short-range interactions. The original isotropic Heisenberg model result rules out both ferromagnetic and antiferromagnetic order under its assumptions.^[ref]

The physical obstruction is strong long-wavelength orientational fluctuation. In a two-dimensional Gaussian angle model, the fluctuation integral includes $\int d^{2}q/q^2$, which grows logarithmically with size. A locally stiff orientation need not remain aligned across an infinite sample. This leaves room for quasi-long-range order and a Berezinskii–Kosterlitz–Thouless transition, rather than forbidding every finite-temperature phenomenon in two dimensions. Discrete anisotropy, long-range interactions, interlayer coupling and finite size change the applicability of the ideal theorem.

An O(N) model has a real $N$-component order parameter $\mathbf{\varphi }$ with symmetry under the orthogonal group $O(N)$. A standard Landau-Ginzburg theory free energy isThe gradient term penalizes spatial variation, $r$ tunes the transition, and the positive quartic term bounds the local free energy below. The case $N=1$ has a sign symmetry; for $N>1$, rotations mix the components.

In $2<d<4$, the standard short-range model's continuous critical transition is governed by a Wilson-Fisher fixed point for the corresponding $N$-component order parameter. Near $d=4$, the epsilon expansion treats the interaction perturbatively because its engineering dimension is $4-d$. ^[ref]

The dimension restriction matters. At and above the upper critical dimension $d=4$, the Gaussian description controls the usual critical powers, with logarithmic qualifications at $d=4$. Lower-dimensional continuous-symmetry models can have different transition behavior, including the Berezinskii–Kosterlitz–Thouless transition. Thus the fixed-point claim must be matched to dimension, symmetry and the range of interactions.

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.

The scaling hypothesis for critical phenomena describes the singular part of the equilibrium free-energy density, rather than its analytic background. For an isotropic system in spatial dimension $D$, with thermal scaling field $t$ and conjugate field $h$, writeThis follows from coarse graining near a renormalization-group fixed point: a correlation volume is enlarged by $b^D$, while the relevant thermal and field perturbations grow with their eigenvalues. Choosing $b=|t{|}^{-1/y_t}$ givesThe two functions describe the two sides of the transition. Nonuniversal metric factors multiplying $t,h,f_s$ must be allowed when comparing microscopic models. Exponents and suitably normalized scaling functions or amplitude ratios characterize a universality class.

Differentiating the free-energy density gives the order parameter $M=-{\partial }_h{f}_s$, magnetic susceptibility $\chi ={\partial }_{h}M$ and singular heat capacity proportional to $-{\partial }_t^2{f}_s$. ThusThe subscript distinguishes the order-parameter exponent from inverse temperature. At $t=0$, choose $b=|h{|}^{-1/y_h}$ to obtain $M\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$, with $\delta =y_h/(D-y_h)$. These derivatives imply the Rushbrooke scaling relation $\alpha +2{\beta }_{\mathrm{mag}}+\gamma =2$ and the Widom scaling relation $\gamma ={\beta }_{\mathrm{mag}}(\delta -1)$. Equivalently, the equation of state has $h=|t{|}^{\Delta }{\mathcal{H}}_{\pm }(M/|t{|}^{{\beta }_{\mathrm{mag}}})$, with $\Delta ={\beta }_{\mathrm{mag}}\delta ={\beta }_{\mathrm{mag}}+\gamma$.

The correlation length scales as $\xi \sim |t{|}^{-\nu }$, $\nu =1/y_t$. At criticality the connected correlation function behaves as $G(r)\sim r^{-(D-2+\eta )}$, giving $y_h=(D+2-\eta )/2$. Integrating $G$ over a correlation volume gives $\chi \sim {\xi }^{2-\eta }$ and hence the Fisher scaling relation $\gamma =(2-\eta )\nu$. The hyperscaling relation $2-\alpha =D\nu$ expresses one singular free-energy contribution per correlation volume.

These laws organize measurements: plotting $M/|t{|}^{{\beta }_{\mathrm{mag}}}$ against $h/|t{|}^{\Delta }$ collapses near-critical curves onto the appropriate scaling function. A finite system replaces the divergent length by its size $L$; the corresponding prediction is $f_s=L^{-D}\mathcal{F}(t{L}^{y_t},h{L}^{y_h})$ under the same scaling assumptions. Finite size rounds a sharp thermodynamic singularity.

The hypothesis has a domain of validity. At a renormalization-group fixed point one generally needs $f_s=b^{-D}{f}_s(b^{y_t}t,b^{y_h}h,b^{y_u}u,\dots )$, including other scaling fields. Ordinary irrelevant couplings can be set to zero in a nonsingular limiting scaling function. A dangerously irrelevant coupling cannot: above the upper critical dimension of scalar quartic theory, the quartic term is irrelevant at the Gaussian point but still stabilizes the ordered phase. Consequently naive hyperscaling fails there even though other exponent relations can remain valid. Marginal variables can produce logarithms, anisotropic systems require the appropriate volume exponent, and the Berezinskii–Kosterlitz–Thouless transition has an essential correlation-length singularity instead of a finite power-law $\nu$. Scaling theory relates observables through a small number of fixed-point dimensions; it does not make analytic backgrounds, marginal logarithms or dangerously irrelevant variables disappear.

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.

Set $A_D=S_D{\Lambda }^{ϵ}/(2\pi {)}^D>0$. The one-loop flow is $\mathcal{B}(g)=-ϵg+(n-2)A_D{g}^2$. For $ϵ>0$ and $n>2$, its physical fixed points areThe zero point attracts small positive couplings, whereas the nonzero point is the separatrix between the low-temperature weak-coupling flow and the strong-coupling regime. This is a perturbative description for small $ϵ$; extrapolation of its nonzero root to large $ϵ$ is not controlled.

At $ϵ=0$ and $n>2$, the only finite one-loop fixed point is $g=0$, a double root. Positive $g$ flows to strong coupling. For $n=2$ the loop coefficient vanishes: at $ϵ>0$ only $g=0$ is fixed, while at $ϵ=0$ every $g$ is fixed in this perturbative spin-wave description. The compact two-component model also admits vortices; their Berezinskii–Kosterlitz–Thouless transition is absent from a calculation retaining only the smooth tangent modes. Thus a zero beta function here does not exclude that transition. The formula should not be interpreted as a continuum result for the discrete $n=1$ Ising target.

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 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]

For a homogeneous ideal gas in a two-dimensional box of area $A$, with dispersion $ϵ=p^2/(2m)$ and one internal state, the density of states is $g(ϵ)=Am/(2\pi {\hslash }^2)$. The excited-particle density at temperature $T>0$ isAs $\mu \to 0^{-}$, this diverges. Hence at every fixed positive temperature the excited states can accommodate any finite areal density with $\mu <0$; they never saturate at a finite critical density. There is no finite-temperature Bose-Einstein condensation in the homogeneous two-dimensional ideal gas thermodynamic limit.

The low-energy logarithmic divergence is the reason, not merely the small number of spatial dimensions. A finite box has a low-energy cutoff and a crossover; a trap changes the density of states and can change the conclusion. Condensation at zero temperature and an interacting gas's Berezinskii–Kosterlitz–Thouless transition are separate issues. The proof assumes the uniform quadratic-dispersion model, rather than every conceivable two-dimensional trapping potential.

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]

In three dimensions an ideal Bose gas can saturate its excited-state population at chemical potential zero, with the remaining particles forming a Bose-Einstein condensate. The transition follows from the Bose-Einstein distribution integrated against its density of states; below the ideal-gas transition, $n_0/n=1-(T/T_c{)}^{3/2}$. Condensation alone does not supply the finite Landau critical velocity of an interacting superfluid: the ideal quadratic spectrum has ${\inf }_{p}E(p)/p=0$.

By contrast, a weak attractive interaction near a Fermi surface produces the Cooper instability and the BCS gap equation. Its logarithmic enhancement permits pairing at arbitrarily weak attraction in the ideal model. The low-temperature excitations have $E=\sqrt{{\xi }^2+{\Delta }^2}$, and in weak coupling $k_B{T}_c=(e^{{\gamma }_E}/\pi )\Delta (0)\simeq 0.567\Delta (0)$. This transition is controlled by pair formation when phase fluctuations are small; deeply bound pairs instead motivate a Bose-condensation description with their own density and effective mass.

For the given ellipsoidal dispersion, set $q_x=k_x$, $q_y=k_y$ and $q_z=k_z/\sqrt{\gamma }$. Then $\epsilon ={\hslash }^2{q}^2/(2m)$ while the integration measure is $d^{3}k=\sqrt{\gamma }{d}^{3}q$. Thus the anisotropic ideal-gas density of states per unit volume and per internal state isThe full continuum momentum range is assumed; this is an anisotropic three-dimensional gas, not yet a discrete stack of two-dimensional bands.

At the Bose transition, substitute $x=\epsilon /(k_{B}T)$:The integral is $\Gamma (3/2)\zeta (3/2)\simeq 2.315$. Solving for the Bose-Einstein condensation temperature for an anisotropic effective mass givesThe heavier $z$-direction effective mass increases the available excited-state count and lowers the condensation temperature at fixed number density.

For a balanced two-spin Fermi gas, let $n_F$ denote total fermion number density and take $N_s$ to be the single-spin density of states used in the conventional gap equation. Counting the filled ellipsoid givesIf a spin-summed density of states is used instead, the coupling convention in the supplied integral must change accordingly; inserting a factor two into only one side would change the exponent incorrectly.

For ${\epsilon }_c\ll {\epsilon }_F$, replace the density of states across the interaction window by $N_s({\epsilon }_F,\gamma )$. Writing ${\lambda }_0=|g|N_{s0}$, the zero-temperature integral becomesand thereforeThe second expression additionally requires ${\lambda }_0{\gamma }^{1/3}\ll 1$. The explicit density dependence is ${\lambda }_0=|g|m(3{\pi }^2{n}_F{)}^{1/3}/(2{\pi }^2{\hslash }^2)$. This mass-anisotropy enhancement of a weak-coupling BCS gap initially raises the pairing scale, while the ideal Bose condensation scale decreases as ${\gamma }^{-1/3}$.

Large anisotropy weakens transport and coherence in the heavy direction. The separation of pair formation and phase ordering then matters, so a large mean-field gap need not imply a large actual $T_c$. A qualitative sketch has $T_c$ initially rising with the pairing scale and eventually turning down when coherence is the smaller scale. If preformed fermion pairs behave as bosons, use $n_B=n_F/2$ and pair mass about $2m$; in this continuum model this gives $k_B{T}^{\mathrm{BEC}}\simeq 0.218{\epsilon }_{F0}{\gamma }^{-1/3}$.

The turnover cannot be located reliably by taking the weak-coupling formulas to $\gamma \to \infty$: ${\epsilon }_F$ falls, ${\lambda }_0{\gamma }^{1/3}$ grows, the narrow-window assumption fails and the pair chemical potential and mass require a self-consistent treatment. The smooth curve in the figure is explicitly an illustrative interpolation of two extrapolated scales, not such a calculation. If actual decoupled two-dimensional layers are approached, ordinary ideal-gas finite-temperature condensation is absent and interacting phase order requires a different analysis, including the Berezinskii–Kosterlitz–Thouless transition when appropriate.

The cuprates make the distinction between pairing and phase coherence relevant: they are layered and have anisotropic motion, but their strong correlations, lattice band structure and predominantly d-wave superconductivity are outside this attractive continuum $s$-wave model. Anisotropy by itself is therefore not an explanation or quantitative fit of cuprate transition temperatures.