Physical limitations of the Hohenberg–Mermin–Wagner theorem

Let us consider a system of localized spins described by the XXZ model:

$$\begin{equation}\mathcal{H}=-\sum\limits _{ij}{J}_{ij}\left({S}_{i}^{x}{S}_{j}^{x}+{S}_{i}^{y}{S}_{j}^{y}\right)-\sum\limits _{ij}{J}_{ij}^{z}{S}_{i}^{z}{S}_{j}^{z}-B\enspace V{M}_{x},\end{equation} \tag{ 1 }$$

where B is the external symmetry-breaking field and ${M}_{x}={V}^{-1}{\sum }_{i}{\mathrm{e}}^{-\mathrm{i}\boldsymbol{K}\cdot {\boldsymbol{R}}_{i}}{S}_{i}^{x}$ the staggered magnetization. The Bogoliubov inequality, in the form first used by Mermin and Wagner [2], is

$$\begin{equation}{\left\vert \left\langle [A,Q]\right\rangle \right\vert }^{2}{\leqslant}\frac{1}{2}\beta \enspace \left\langle \left\{A,{A}^{{\dagger}}\right\}\right\rangle \enspace \left\langle \left[[Q,\mathcal{H}],{Q}^{{\dagger}}\right]\right\rangle ,\end{equation} \tag{ 2 }$$

where braces are anticommutators, β is the inverse temperature, and Q and A are arbitrary operators. To probe IR fluctuations of some broken symmetry, Q is chosen to be a modulation of its generator and A is tuned so that $\left\langle [A,Q]\right\rangle$ gives the corresponding order parameter. For the XXZ model, the appropriate choices are ${Q}_{\boldsymbol{k}}={\sum }_{i}{\mathrm{e}}^{-\mathrm{i}\boldsymbol{k}\cdot {\boldsymbol{R}}_{i}}{S}_{i}^{z}$ and ${A}_{\boldsymbol{k},\boldsymbol{K}}={V}^{-1}{\sum }_{i}{\mathrm{e}}^{\mathrm{i}(\boldsymbol{k}-\boldsymbol{K})\cdot {\boldsymbol{R}}_{i}}{S}_{i}^{y}$ which yield [A _{k , K} , Q _k ] = iM_x .

To obtain a useful bound on ${\left\langle {M}_{x}\right\rangle }_{B}$ from (2), one has to divide it by $\left\langle \left[[Q,\mathcal{H}],{Q}^{{\dagger}}\right]\right\rangle$ and sumover momenta k in the first Brillouin zone. In that way, the orthogonality of the ${\mathrm{e}}^{\mathrm{i}(\boldsymbol{k}-\boldsymbol{K})\cdot {\boldsymbol{R}}_{i}}$ factors from A _{k , K} ensures the intensive bound ${\sum }_{\boldsymbol{k}\in \mathrm{B}\mathrm{Z}}{\langle \left\{{A}_{\boldsymbol{k},\boldsymbol{K}},{A}_{\boldsymbol{k},\boldsymbol{K}}^{{\dagger}}\right\}\rangle }_{B}{\leqslant}2{n}^{2}{S}^{2}$, where $n=\mathcal{N}/V$ is the concentration and $\mathcal{N}$ the number of unit cells. That Q _k is a generator of symmetry implies that $\left[[{Q}_{\boldsymbol{k}},\mathcal{H}],{Q}_{\boldsymbol{k}}^{{\dagger}}\right]\propto {\boldsymbol{k}}^{2}$, and indeed a rigorous bound ${V}^{-1}{\langle [[{Q}_{\boldsymbol{k}},\mathcal{H}],{Q}_{\boldsymbol{k}}^{{\dagger}}]\rangle }_{B}{\leqslant}nJ({\boldsymbol{k}}^{2}/{k}_{\text{BZ}}^{2})+\left\vert B{\left\langle {M}_{x}\right\rangle }_{B}\right\vert$ can be derived in which k_BZ is the effective radius of the first Brillouin zone and $J=\frac{S(S+1)}{\mathcal{N}}{\sum }_{ij}{k}_{\text{BZ}}^{2}{({\boldsymbol{R}}_{i}-{\boldsymbol{R}}_{j})}^{2}\left\vert {J}_{ij}\right\vert$ the effective spin–spin coupling constant. Thus we obtain the HMW inequality [2]

$$\begin{equation}{\left\vert {\left\langle {M}_{x}\right\rangle }_{B}\right\vert }^{2}{\leqslant}\frac{\beta n{S}^{2}}{\frac{1}{\mathcal{N}}\sum\limits _{\boldsymbol{k}\in \mathrm{B}\mathrm{Z}}\frac{1}{nJ({\boldsymbol{k}}^{2}/{k}_{\text{BZ}}^{2})+\left\vert B{\left\langle {M}_{x}\right\rangle }_{B}\right\vert }}.\end{equation} \tag{ 3 }$$

By taking the thermodynamic limit V → ∞, the sum in the denominator diverges in the limit B → 0 in 1D and 2D, forbidding a finite value of the magnetization ${\left\langle {M}_{x}\right\rangle }_{0}$ that is within the XY plane of symmetry.

Although the use of periodic boundary conditions apparently limits the above argument to very large systems, finite, and even fractal [32], lattices can be treated with minimal technical modification. One simply introduces a basis φ _k ( R _i ) in A _{k , K} and C _k which is not a plane wave, but tuned to the lattice [32]. The key formal properties of φ _k ( R _i ) needed in the above argument are that φ _k ( R _i ) → 1 as k → 0, and that the density of the discrete wave-vectors k near 0 grows sufficiently fast in the thermodynamic limit. Thus equation (3) still holds, with an appropriate understanding of ∑ _{k ∈BZ} and k_BZ, and the only physical effect of finiteness on the HMW arguments is the appearance of an IR cutoff, namely ${k}_{\mathrm{min}}={k}_{\text{BZ}}/\sqrt{\mathcal{N}}$ in 2D. The sum in the denominator can then be replaced with an integral for finite 2D samples as well,

$$\begin{equation}\frac{1}{\mathcal{N}}\sum\limits _{\boldsymbol{k}}\to \frac{1}{\mathcal{N}}{\left\vert \right.}_{\boldsymbol{k}=\boldsymbol{0}}+\frac{2}{{k}_{\text{BZ}}^{2}}{\int }_{{k}_{\mathrm{min}}}^{{k}_{\text{max}}}dkk,\end{equation} \tag{ 4 }$$

turning equation (3) into an inequality with $\left\vert {\left\langle {M}_{x}\right\rangle }_{B}\right\vert$ on both sides. This inequality is equivalent to $\left\vert {\left\langle {M}_{x}\right\rangle }_{B}\right\vert {\leqslant}{M}_{\mathrm{max}}$, where M_max is determined by a transcendental equation which can be solved in the limit B → 0, giving:

$$\begin{equation}{M}_{\mathrm{max}}^{2}\approx \frac{2\beta J}{\mathrm{ln}(\mathcal{N}/2)+\mathrm{ln}(\beta J)}\cdot {M}_{\text{sat}}^{2}.\end{equation} \tag{ 5 }$$

In 1D, the corresponding B → 0 solution is given by

$$\begin{equation}{M}_{\mathrm{max}}^{3}\approx \frac{4\beta J}{{\pi }^{2}\sqrt{\mathcal{N}}}\cdot {M}_{\text{sat}}^{3},\end{equation} \tag{ 6 }$$

where M_sat = nS is the saturation magnetization. In the derivation of the above, let us only note that one needs to use a finite, but physically infinitesimal, value of B to probe symmetry breaking. Technically, we have used $B=\sqrt{\mathcal{N}}\cdot ({k}_{\mathrm{B}}T/V{M}_{\text{sat}})$ so that $\mathcal{N}J\gg VB{M}_{\text{sat}}\gg {k}_{\mathrm{B}}T$. Clearly, finite-size effects do not affect the HMW outcome in 1D, because $\sqrt{\mathcal{N}}$ ∼ 10⁴ is still a very large number for macroscopic $\mathcal{N}$ ∼ ${N}_{A}^{1/3}$. In 2D, however, finite size affects the HMW outcome qualitatively, because $\mathrm{ln}\enspace {N}_{A}^{2/3}$ ≈ 37, effectively replacing the IR divergence with an order-of-magnitude reduction. Given that magnetic critical temperatures are typically a few 100 Kelvins in transition-metal compounds, in 2D the HMW argument only predicts a marked suppression of magnetic order, far from vanishing.

When any perturbation δH is added to the XXZ Hamiltonian (1), the original denominator in equation (3) becomes

$$\begin{equation}\frac{1}{\mathcal{N}}\sum\limits _{\boldsymbol{k}\in \mathrm{B}\mathrm{Z}}\frac{1}{(nJ{\boldsymbol{k}}^{2}/{k}_{\text{BZ}}^{2})+{{\Delta}}_{B}(\boldsymbol{k})+\left\vert B{{< }{M}_{x}{ >}}_{B,\delta \;H}\right\vert },\end{equation} \tag{ 7 }$$

where ${{\Delta}}_{B}(\boldsymbol{k})={V}^{-1}\left\vert \right.{\langle [[{Q}_{\boldsymbol{k}}, \;\delta H],{Q}_{\boldsymbol{k}}^{{\dagger}}]\rangle }_{B, \;\delta H}\left\vert \right.$ and the index δH indicates thermal averaging in the presence of δH. Thus, formally, any perturbation that breaks the continuous symmetry of the magnetization, generated by Q₀ , gives a finite Δ_B (0) ≠ 0, which cuts off the k ² term, invalidating the HMW theorem.

Physically, let us consider the site-disorder model

$$\begin{equation} \;\delta H=-\sum\limits _{i}{\boldsymbol{h}}_{i}\cdot {\boldsymbol{S}}_{i},\end{equation} \tag{ 8 }$$

where h _i is a random local field. For definiteness, its planar component is considered to lie on a circle, | h _i∥| = const., placing the model in the RFIO limit [22]. The crossover to the Imry–Ma limit [23] by widening the amplitude distribution is beyond the scope of this article. In the present case, ${{\Delta}}_{B}={V}^{-1}\left\vert \right.{{< }\delta {H}_{{\Vert}}{ >}}_{B,\delta }\left\vert \right.$, where δH_∥ includes planar components of the h _i only, and δ indicates disorder-averaging. The salient observation is that the disorder effectively competes with the interaction scale J in cutting off the IR divergence:

$$\begin{equation}{\left\vert \left\langle \hspace{-3.0pt}\langle {M}_{x}\rangle \hspace{-3.0pt}\right\rangle \right\vert }^{2}{\leqslant}\frac{\beta J}{\enspace \mathrm{ln}(1+nJ/{{\Delta}}_{0})}\cdot {M}_{\text{sat}}^{2}\end{equation} \tag{ 9 }$$

in 2D, and

$$\begin{equation}{\left\vert \left\langle \hspace{-3.0pt}\langle {M}_{x}\rangle \hspace{-3.0pt}\right\rangle \right\vert }^{2}{\leqslant}\frac{2}{\pi }\sqrt{\frac{{{\Delta}}_{0}}{nJ}}\cdot \beta J\cdot {M}_{\text{sat}}^{2}\end{equation} \tag{ 10 }$$

in 1D. The double angular brackets denote the Bogoliubov quasi-average $\left\langle \hspace{-3.0pt}\langle M\rangle \hspace{-3.0pt}\right\rangle =\underset{B\to 0}{\mathrm{lim}}\underset{V\to \infty }{\mathrm{lim}}{\left\langle M\right\rangle }_{B}$. A similar pattern is observed as with finite-size effects, with the role of the size of the system taken over by the ratio between the magnetic coupling and the disorder scale. This ratio is evidently much smaller than Avogadro's number, so it is easy to imagine a 2D system with a small amount of disorder, say nJ/Δ₀ ∼ 100, for which IR fluctuations suppress T_c by less than an order of magnitude below J.

The competition between disorder and finite-size effects appears in equation (7), before the integration which introduces dimensionality. Disorder cuts off the quadratic term, meaning that dirty samples can be larger and still avoid the asymptotic HMW regime, irrespective of dimension. In the context of simulations, this result is a model-independent validation of the observation that weak disorder, or even numerical error, efficiently stabilizes the order parameter in 2D.

In a layered system, the order parameter can also be restored by a weak interlayer coupling J_⊥ ≪ J, for which the denominator in equation (9) similarly reads ln(1 + J/J_⊥). Suppression of the isotropic 3D T_c by weak interlayer coupling competes with the 2D disorder effect on an equal footing, because the ratio J/J_⊥ can easily be both greater and smaller than nJ/Δ₀.

Because the competition between mechanisms occurs in the denominator of the sum (7), which is itself in the denominator of equation (3), the largest scale ends in the numerator. Suppose, for illustration, that two mechanisms are at work in the same sample, one of which would limit kT_c < Δ₁, and the other kT_c < Δ₂, if acting alone. The two acting together have the net effect

$$\begin{equation}k{T}_{\mathrm{c}}{< }\mathrm{max}({{\Delta}}_{1},{{\Delta}}_{2}),\end{equation} \tag{ 11 }$$

because the larger scale gaps the other. This result runs contrary to the naive impression that the 'stricter' criterion should be applied.

We consider a layered two-dimensional system with localized impurities, electron–electron, and electron–phonon interactions, whose Hamiltonian is

$$\begin{equation}\mathcal{H}={H}_{0}+{V}_{\text{int}}.\end{equation} \tag{ 12 }$$

The non-interacting part is

$$\begin{equation}{H}_{0}=\sum\limits _{\boldsymbol{k}\sigma }{{\epsilon}}_{\boldsymbol{k}\sigma }{c}_{\boldsymbol{k}\sigma }^{{\dagger}}{c}_{\boldsymbol{k}\sigma }+\sum\limits _{\boldsymbol{\varrho }{\sigma }_{1}{\sigma }_{2}}{\varepsilon }_{\boldsymbol{\varrho }{\sigma }_{1}{\sigma }_{2}}{f}_{\boldsymbol{\varrho }{\sigma }_{1}}^{{\dagger}}{f}_{\boldsymbol{\varrho }{\sigma }_{2}}+\sum\limits _{\boldsymbol{k}\lambda }\hslash {\omega }_{\boldsymbol{k}\lambda }{b}_{\boldsymbol{k}\lambda }^{{\dagger}}{b}_{\boldsymbol{k}\lambda },\end{equation} \tag{ 13 }$$

where the three terms refer respectively to mobile carriers, impurities at fixed arbitrary positions ϱ , and phonons. V_int contains all possible hybridizations and interactions among them, subject to the limitation that they are local in real space and time, i.e. admit the usual fermion continuity equation. From the microscopic point of view, this limitation is quite mild, because all bare interactions considered in standard many-body theory are local in this sense.

3.1.1. The HMW argument

The operators to be used in Bogoliubov's inequality (2) are:

$$\begin{equation}{Q}_{\boldsymbol{k}}=\sum\limits _{\boldsymbol{R}\sigma }{\mathrm{e}}^{-\mathrm{i}\boldsymbol{k}\cdot \boldsymbol{R}}{c}_{\boldsymbol{R}\sigma }^{{\dagger}}{c}_{\boldsymbol{R}\sigma }+\sum\limits _{\boldsymbol{\varrho }\sigma }{\mathrm{e}}^{-\mathrm{i}\boldsymbol{k}\cdot \boldsymbol{\varrho }}{f}_{\boldsymbol{\varrho }\sigma }^{{\dagger}}{f}_{\boldsymbol{\varrho }\sigma },\end{equation} \tag{ 14 }$$

$$\begin{equation}{A}_{\boldsymbol{k},\boldsymbol{r}}={\mathcal{N}}^{-1}\sum\limits _{\boldsymbol{R}}{\mathrm{e}}^{\mathrm{i}\boldsymbol{k}\cdot \boldsymbol{R}}{c}_{\boldsymbol{R}{\sigma }_{1}}{c}_{\boldsymbol{R}+\boldsymbol{r}{\sigma }_{2}},\end{equation} \tag{ 15 }$$

where $\mathcal{N}$ is the number of unit cells within the 2D layers, while σ₁, σ₂, and r are fixed and arbitrary. Clearly, Q _k is a modulation of the number operator, while the commutator of A _k with Q _k produces the microscopic gap operator

$$\begin{equation}[{A}_{\boldsymbol{k}-\boldsymbol{K},\boldsymbol{r}},{Q}_{\boldsymbol{k}}]=(1+{\mathrm{e}}^{-\mathrm{i}\boldsymbol{k}\cdot \boldsymbol{r}}){{\Delta}}_{{\sigma }_{1}{\sigma }_{2}}^{\boldsymbol{K}}(\boldsymbol{r}).\end{equation} \tag{ 16 }$$

The operator ${{\Delta}}_{{\sigma }_{1}{\sigma }_{2}}^{\boldsymbol{K}}(\boldsymbol{r})$ can be summed over K , r , σ₁, and σ₂ to obtain every possible (singlet, triplet, s, p, ${d}_{{x}^{2}-{y}^{2}}$, pair momentum K , etc) superconducting gap operator Δ, thus generalizing previous results [8, 26–30]. Summing Bogoliubov's inequality (2) over k , one obtains the SC analogue of equation (3):

$$\begin{equation}{\left\vert {\left\langle {{\Delta}}_{{\sigma }_{1}{\sigma }_{2}}^{\boldsymbol{K}}(\boldsymbol{r})\right\rangle }_{\kappa }\right\vert }^{2}{\leqslant}\frac{\ell \enspace \beta }{\frac{2}{\mathcal{N}}\sum\limits _{\boldsymbol{k}}\frac{{\left\vert 1+{\mathrm{e}}^{\mathrm{i}\boldsymbol{k}\cdot \boldsymbol{r}}\right\vert }^{2}}{({{\epsilon}}_{\text{IR}}{k}^{2}/{k}_{\text{BZ}}^{2})+\left\vert \kappa \right\vert {\mathcal{D}}_{\kappa }}},\end{equation} \tag{ 17 }$$

where ℓ is the number of layers, κ is the symmetry-breaking field conjugate to Δ, same as B in equation (3), and ${\mathcal{D}}_{\kappa }$ is a corresponding expectation value of Δ. The infrared energy scale _IR is determined by the bound ${(\mathcal{N}\ell )}^{-1}\left\vert \right.\langle [[{Q}_{\boldsymbol{k}},\mathcal{H}],{Q}_{\boldsymbol{k}}^{{\dagger}}]\rangle \left\vert \right.{\leqslant}{{\epsilon}}_{\text{IR}}{k}^{2}/{k}_{\text{BZ}}^{2}$. It is a sum of two positive terms: one due to particle dispersions and another from hybridization with impurities. The particle-dispersion term is always finite, while the impurity hybridization ${V}_{{\sigma }_{1}{\sigma }_{2}}^{\boldsymbol{\varrho }\boldsymbol{R}}({f}_{\boldsymbol{\varrho }{\sigma }_{1}}^{{\dagger}}{c}_{\boldsymbol{R}{\sigma }_{2}}+\mathrm{c}.\mathrm{c}.)$ must fall off sufficiently rapidly, ${\mathcal{N}}^{-1}{\sum }_{\boldsymbol{\varrho },\boldsymbol{R}}{\left\vert \boldsymbol{\varrho }-\boldsymbol{R}\right\vert }^{2}\left\vert {V}_{{\sigma }_{1}{\sigma }_{2}}^{\boldsymbol{\varrho }\boldsymbol{R}}\right\vert {< }\infty$, for the impurity term to be finite. The electron–electron and electron–phonon interactions drop out in $[Q,\mathcal{H}]$ because they are local in space. Because ${\left\vert 1+{\mathrm{e}}^{\mathrm{i}\boldsymbol{k}\cdot \boldsymbol{r}}\right\vert }^{2}\ne 0$ at k = 0, the IR divergence is unchecked in 1D and 2D when κ → 0, so the HMW theorem follows.

3.1.2. Physical considerations

Technically, hopping terms and couplings to them are non-local, but even such interactions do not invalidate the above conclusion. They are merely subject to the same caution as impurity hybridization, namely that their range falls off fast enough. Hence the HMW theorem in its original formulation is quite robust with respect to model variations. To ascertain how physically relevant it is, we rewrite the bound (17) in a more intuitive form:

$$\begin{equation}\left\vert {\left\langle {{\Delta}}_{{\sigma }_{1}{\sigma }_{2}}^{\boldsymbol{K}}(\boldsymbol{r})\right\rangle }_{\kappa }\right\vert {\leqslant}\sqrt{\frac{{T}_{\text{HMW}}}{T}}\cdot 1,\end{equation} \tag{ 18 }$$

where T_HMW is the HMW temperature and 1 is the saturation value of $\left\langle {{\Delta}}_{{\sigma }_{1}{\sigma }_{2}}^{\boldsymbol{K}}(\boldsymbol{r})\right\rangle$. Specializing to 2D, assuming a parabolic dispersion, and taking some care with numerical factors, one finds ${k}_{\mathrm{B}}{T}_{\text{HMW}}\approx (\ell /4)({\hslash }^{2}{k}_{\text{BZ}}^{2}/2{m}_{{\ast}})\left\langle n\right\rangle /\mathrm{ln}({k}_{\text{BZ}}L)$, where $\left\langle n\right\rangle =\left\langle N\right\rangle /(\mathcal{N}\ell )$ is the total number of carriers per unit cell and L is the linear size of the sample. Therefore, the HMW bound is physically relevant only for temperatures T > T_HMW, when enough IR modes are excited to suppress the SC order. The conclusion, which applies to all previous HMW arguments concerning SC [1, 7, 8, 26–30] as well, is that in 2D

$$\begin{equation}{T}_{\mathrm{c}}{< }4{T}_{\text{HMW}}=\ell \cdot {T}_{\mathrm{F}}/\mathrm{ln}({k}_{\text{BZ}}L),\end{equation} \tag{ 19 }$$

where T_F is the Fermi temperature. Inserting T_F ∼ 10⁴ K, k_BZ ∼ Å⁻¹, and L ∼ 1 cm, one finds the bound to be in the ∼500 K range, which is comparable to the bound k_B T_c < E_F/8, obtained by considering phase stiffness in an infinite system [31].

The true significance of the finite-size effect becomes evident in the converse exercise: k_BZ L < exp(T_F/T_c). One would need L ∼ 10³³ m, much larger than the observable Universe (∼10²⁷ m), for T_c in a single layer to be forced below ∼100 K, the value observed in optimally doped cuprates.

The critical junction in the above derivation is that the commutator [Q _k , V_int] vanishes, because interactions are local and therefore commute with the local number operators appearing in Q _k (14). The f-sum rule ${(\mathcal{N}\ell )}^{-1}\left\langle [[{Q}_{\boldsymbol{k}},\mathcal{H}],{Q}_{\boldsymbol{k}}^{{\dagger}}]\right\rangle \lesssim {{\epsilon}}_{\text{IR}}{k}^{2}/{k}_{\text{BZ}}^{2}$ follows, which is physically the statement that boosting the electrons by ℏ k results in a second-order-in- k energy increase of the form $\ell \mathcal{N}{{\epsilon}}_{\text{IR}}{k}^{2}/{k}_{\text{BZ}}^{2}$. As a quick way to derive this quadratic dependence, write ${\hat{Q}}_{\boldsymbol{k}}=\hat{N}-\mathrm{i}\boldsymbol{k}\cdot \hat{\boldsymbol{X}}+\mathcal{O}({k}^{2})$, where $\hat{N}={\hat{Q}}_{\boldsymbol{0}}={\sum }_{\boldsymbol{R}}{\hat{n}}_{\boldsymbol{R}}$ and $\hat{\boldsymbol{X}}={\sum }_{\boldsymbol{R}}\boldsymbol{R}\enspace {\hat{n}}_{\boldsymbol{R}}$. The double commutator is immediately

$$\begin{equation}[[{Q}_{\boldsymbol{k}},\mathcal{H}],{Q}_{\boldsymbol{k}}^{{\dagger}}]=[[\boldsymbol{k}\cdot \hat{\boldsymbol{X}},\mathcal{H}],\boldsymbol{k}\cdot \hat{\boldsymbol{X}}]+\mathcal{O}({k}^{3}),\end{equation} \tag{ 20 }$$

because $[\hat{N},\mathcal{H}]=0$ of course follows from local number-conservation. Therefore, the HMW argument is model-independent, as long as the interactions, or disorder, conserve the particle number locally, and boosting the electrons of the system increases the energy by an extensive amount. All later proofs thus necessarily reproduce its physically most counter-intuitive aspect, that one probes high-energy IR fluctuations of the non-interacting Fermi sea, instead of the superconducting mechanism.

Having understood that, we develop an alternative argument which probes fluctuations associated with Cooper pairing, as expressed in the reduced BCS Hamiltonian:

$$\begin{equation}\mathcal{H}=\sum\limits _{\boldsymbol{k}\sigma }{{\epsilon}}_{\boldsymbol{k}}{c}_{\boldsymbol{k}\sigma }^{{\dagger}}{c}_{\boldsymbol{k}\sigma }+\frac{1}{\mathcal{N}}\sum\limits _{{\boldsymbol{k}}_{1}{\boldsymbol{k}}_{2}}{V}_{{\boldsymbol{k}}_{1}{\boldsymbol{k}}_{2}}{c}_{{\boldsymbol{k}}_{1}{\uparrow}}^{{\dagger}}{c}_{-{\boldsymbol{k}}_{1}{\downarrow}}^{{\dagger}}{c}_{-{\boldsymbol{k}}_{2}{\downarrow}}{c}_{{\boldsymbol{k}}_{2}{\uparrow}}.\end{equation} \tag{ 21 }$$

The reduced BCS Hamiltonian breaks the standard HMW argument because the interaction is non-local in real space, violating local continuity. It causes _IR in equation (17) to diverge with ${({k}_{\text{BZ}}L)}^{2}$ in the thermodynamic limit, as typical for long-range interactions. In the remainder of this section, we adapt the technical steps of the original argument to the BCS pair interaction, leaving the physical discussion to the end.

The operator Q is constructed to commute with the kinetic energy now, and A adjusted so that [A, Q] gives the gap operator as before:

$$\begin{equation}{Q}_{\boldsymbol{R}}=\sum\limits _{\boldsymbol{k}}{\mathrm{e}}^{\mathrm{i}\boldsymbol{k}\cdot \boldsymbol{R}}\left({c}_{\boldsymbol{k}{\uparrow}}^{{\dagger}}{c}_{\boldsymbol{k}{\uparrow}}^{}+{c}_{-\boldsymbol{k}{\downarrow}}^{{\dagger}}{c}_{-\boldsymbol{k}{\downarrow}}^{}\right)\enspace ,\end{equation} \tag{ 22 }$$

$$\begin{equation}{A}_{\boldsymbol{R},\boldsymbol{K}}={\mathcal{N}}^{-1}\sum\limits _{\boldsymbol{k}}{\mathrm{e}}^{-\mathrm{i}\boldsymbol{k}\cdot \boldsymbol{R}}{S}_{\boldsymbol{K}}(\boldsymbol{k}){c}_{\boldsymbol{k}+\boldsymbol{K}{\uparrow}}{c}_{-\boldsymbol{k}{\downarrow}}\enspace ,\end{equation} \tag{ 23 }$$

$$\begin{equation}[{A}_{\boldsymbol{R},\boldsymbol{K}},{Q}_{\boldsymbol{R}}]=(1+{\mathrm{e}}^{\mathrm{i}\boldsymbol{K}\cdot \boldsymbol{R}}){{\Delta}}_{\boldsymbol{K}},\end{equation} \tag{ 24 }$$

where S _K ( k ) is a Cooper-pair structure factor which allows a finite pair momentum K for the sake of generality.

The analogue of equations (3) and (17) reads, in any dimension,

$$\begin{equation}{\left\vert {\left\langle {{\Delta}}_{\boldsymbol{K}}\right\rangle }_{\kappa }\right\vert }^{2}{\leqslant}{I}_{\boldsymbol{K}}\cdot \beta {\left\{\frac{1}{\mathcal{N}}\sum\limits _{\boldsymbol{R}}\frac{{\left\vert 1+{\mathrm{e}}^{\mathrm{i}\boldsymbol{K}\cdot \boldsymbol{R}}\right\vert }^{2}}{{\mathcal{V}}_{\kappa }(\boldsymbol{R})+\left\vert \kappa \right\vert {\mathcal{D}}_{\kappa }}\right\}}^{-1},\end{equation} \tag{ 25 }$$

where ${I}_{\boldsymbol{K}}={\mathcal{N}}^{-1}{\sum }_{\boldsymbol{k}}{\left\vert {S}_{\boldsymbol{K}}(\boldsymbol{k})\right\vert }^{2}$ is a kinematic factor. In contrast to small values of k in the HMW argument, there is nothing special about small R here, so the total sum does not diverge in the thermodynamic limit in any dimension. Thus, equation (25) does not constrain the order parameter to vanish. Nevertheless, the upper bound for T_c it provides is sharper than in the original formulation for samples of reasonable size when the Cooper pairs are small, as we show below.

The term ${\mathcal{V}}_{\kappa }(\boldsymbol{R})$ is an average of the interaction operator,

$$\begin{equation}{\mathcal{V}}_{\kappa }(\boldsymbol{R})=\frac{8}{{\mathcal{N}}^{2}}\sum\limits _{{\boldsymbol{k}}_{1}{\boldsymbol{k}}_{2}}\enspace {\mathrm{sin}}^{2}\left[({\boldsymbol{k}}_{1}-{\boldsymbol{k}}_{2})\cdot \boldsymbol{R}/2\right]\enspace \left\vert {V}_{{\boldsymbol{k}}_{1}{\boldsymbol{k}}_{2}}\right\vert {\langle {c}_{{\boldsymbol{k}}_{1}{\uparrow}}^{{\dagger}}{c}_{-{\boldsymbol{k}}_{1}{\downarrow}}^{{\dagger}}{c}_{-{\boldsymbol{k}}_{2}{\downarrow}}{c}_{{\boldsymbol{k}}_{2}{\uparrow}}\rangle }_{\kappa },\end{equation} \tag{ 26 }$$

while ${\mathcal{D}}_{\kappa }$ averages the order parameter like in equation (17). A more transparent form is obtained after taking an angular average of ${\left\vert 1+{\mathrm{e}}^{\mathrm{i}\boldsymbol{K}\cdot \boldsymbol{R}}\right\vert }^{2}$ in 2D and 3D,

$$\begin{equation}\left\vert \left\langle \hspace{-3.0pt}\langle {{\Delta}}_{\boldsymbol{K}}\rangle \hspace{-3.0pt}\right\rangle \right\vert {\leqslant}\sqrt{\frac{{T}_{\text{pair}}}{T}}\cdot {{\Delta}}_{\text{sat}}^{\boldsymbol{K}},\end{equation} \tag{ 27 }$$

where ${{\Delta}}_{\text{sat}}^{\boldsymbol{K}}=\left\vert {\mathcal{N}}^{-1}{\sum }_{\boldsymbol{k}}{S}_{\boldsymbol{K}}(\boldsymbol{k}){\left\langle {c}_{\boldsymbol{k}+\boldsymbol{K}{\uparrow}}{c}_{-\boldsymbol{k}{\downarrow}}\right\rangle }_{T=0}\right\vert$ is the saturation value of the order parameter at T = 0, while T_pair is the upper bound temperature

$$\begin{equation}{k}_{\mathrm{B}}{T}_{\text{pair}}=\frac{{I}_{\boldsymbol{K}}}{{({{\Delta}}_{\text{sat}}^{\boldsymbol{K}})}^{2}}\cdot \frac{8}{{\mathcal{N}}^{2}}\sum\limits _{{\boldsymbol{k}}_{1}{\boldsymbol{k}}_{2}}\left\vert {V}_{{\boldsymbol{k}}_{1}{\boldsymbol{k}}_{2}}\right\vert \langle {c}_{{\boldsymbol{k}}_{1}{\uparrow}}^{{\dagger}}{c}_{-{\boldsymbol{k}}_{1}{\downarrow}}^{{\dagger}}{c}_{-{\boldsymbol{k}}_{2}{\downarrow}}{c}_{{\boldsymbol{k}}_{2}{\uparrow}}\rangle .\end{equation} \tag{ 28 }$$

In the thermodynamic limit, both ${\mathcal{V}}_{\kappa }(\boldsymbol{R})$ and k_B T_pair tend to intensive constants. Thus the upper bound (27) is independent of the size of the sample. For more general interactions, the above is easily generalized to

$$\begin{equation}{k}_{\mathrm{B}}{T}_{\text{pair}}=\frac{{I}_{\boldsymbol{K}}}{{({{\Delta}}_{\text{sat}}^{\boldsymbol{K}})}^{2}}\cdot \frac{8}{{\mathcal{N}}^{2}}\sum\limits _{\boldsymbol{q}{\boldsymbol{k}}_{1}{\boldsymbol{k}}_{2}}\sum\limits _{{\sigma }_{1}{\sigma }_{2}}\left\vert {V}_{\boldsymbol{q}}\right\vert \langle {c}_{{\boldsymbol{k}}_{1}+\boldsymbol{q}{\sigma }_{1}}^{{\dagger}}{c}_{{\boldsymbol{k}}_{2}-\boldsymbol{q}{\sigma }_{2}}^{{\dagger}}{c}_{{\boldsymbol{k}}_{2}{\sigma }_{2}}{c}_{{\boldsymbol{k}}_{1}{\sigma }_{1}}\rangle .\end{equation} \tag{ 29 }$$

3.2.1. Evaluation of the upper bound temperature

First, we estimate the ratio

$$\begin{equation}\frac{{I}_{\boldsymbol{K}}}{{({{\Delta}}_{\text{sat}}^{\boldsymbol{K}})}^{2}}=\frac{{\mathcal{N}}^{-1}{\sum }_{\boldsymbol{k}}{\left\vert {S}_{\boldsymbol{K}}(\boldsymbol{k})\right\vert }^{2}}{\left\vert \right.{\mathcal{N}}^{-1}{\sum }_{\boldsymbol{k}}{S}_{\boldsymbol{K}}(\boldsymbol{k}){\left\langle {c}_{\boldsymbol{k}+\boldsymbol{K}{\uparrow}}{c}_{-\boldsymbol{k}{\downarrow}}\right\rangle }_{T=0}{\left\vert \right.}^{2}}.\end{equation} \tag{ 30 }$$

It is reasonable to set the Cooper-scattering structure factor S _K to zero outside a shell of thickness δk_gap on the surface of the Fermi sphere, leading to

$$\begin{equation}\frac{{I}^{\boldsymbol{K}}}{{({{\Delta}}_{\text{sat}}^{\boldsymbol{K}})}^{2}}\sim \frac{{k}_{\text{BZ}}}{\delta {k}_{\text{gap}}}\sim {\xi }_{\mathrm{P}},\end{equation} \tag{ 31 }$$

where ξ_P is the size of a Cooper pair (Pippard scale) in units of the lattice constant. Here, we have taken into account that ${\left\langle {c}_{\boldsymbol{k}+\boldsymbol{K}{\uparrow}}{c}_{-\boldsymbol{k}{\downarrow}}\right\rangle }_{T=0}\propto {{\Delta}}_{\mathrm{S}\mathrm{C}}/\sqrt{{({{\epsilon}}_{\boldsymbol{k}}-{E}_{\mathrm{F}})}^{2}+{{\Delta}}_{\mathrm{S}\mathrm{C}}^{2}}$ ∼ 1 within δk_gap from the Fermi surface, where the microscopic gap Δ_SC ≠ 0.

Next, we take the BCS interaction to be the usual schematic −V₀ within a range of ±ℏω_D around E_F, and introduce the dimensionless coupling constant λ = Ag(E_F)V₀, where A is the surface of the unit cell and g(E_F) the (intensive) level density at the Fermi energy. Then

$$\begin{equation}{k}_{\mathrm{B}}{T}_{\text{pair}}\sim {\xi }_{\mathrm{P}}\cdot 8\cdot {V}_{0}\cdot {[Ag({E}_{\mathrm{F}})\hslash {\omega }_{\mathrm{D}}]}^{2}\sim 10{\xi }_{\mathrm{P}}\cdot \frac{{\lambda }^{2}{(\hslash {\omega }_{\mathrm{D}})}^{2}}{{V}_{0}}.\end{equation} \tag{ 32 }$$

When the actual numbers for some classical 3D superconductors [33] are inserted, one finds that the intensive bound (32) still allows 2D SC by a wide margin, primarily because Cooper pairs are so large, ξ_P ∼ 1000. For hafnium, Ag(E_F) ≈ 0.8 eV⁻¹, λ ≈ 0.14 and ℏω_D ≈ 22 meV, which gives T_pair ≈ 4000 K, while for aluminum, Ag(E_F) ≈ 0.5 eV⁻¹, λ ≈ 0.4 and ℏω_D ≈ 36 meV, giving T_pair ≈ 15 000 K. The measured values of the SC T_c are 0.13 K and 1.2 K, respectively.

A similar conclusion pertains to high-T_c superconducting cuprates, even though their Cooper pairs are small, ξ_P ∼ 1–3, and the critical temperatures are much larger, in the ∼100 K range. The SC mechanism for them is not known at present. There is broad consensus that it is not phononic, but the schematic BCS Hamiltonian is still applicable if the SC involves Cooper pairs. One can replace ℏω_D with an electronic scattering scale ∼100 meV, V₀ with an effective Mott or charge-transfer local scale U_eff ∼ 3–5 eV, and λ with ∼1 for a strong coupling. One obtains an upper bound T_pair ∼ 1000 K, which is surprisingly reasonable, given the crudity of the estimates.