The Average Field Approximation for Almost Bosonic Extended Anyons

Abstract

Anyons are 2D or 1D quantum particles with intermediate statistics, interpolating between bosons and fermions. We study the ground state of a large number N of 2D anyons, in a scaling limit where the statistics parameter $\alpha$ is proportional to $N ^{-1}$ when $N\rightarrow \infty$. This means that the statistics is seen as a “perturbation from the bosonic end”. We model this situation in the magnetic gauge picture by bosons interacting through long-range magnetic potentials. We assume that these effective statistical gauge potentials are generated by magnetic charges carried by each particle, smeared over discs of radius R (extended anyons). Our method allows to take $R\rightarrow 0$ not too fast at the same time as $N\rightarrow \infty$. In this limit we rigorously justify the so-called “average field approximation”: the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.

Appendix: Properties of the Average-Field Functional

In this appendix we establish some of the fundamental properties of the functional (1.10) and its limit $R \rightarrow 0$.

For $\beta \in \mathbb {R}$ and $V: \mathbb {R}^2 \rightarrow \mathbb {R}^+$ we define the average-field energy functional

$$\begin{aligned} \mathcal {E}^{\mathrm {af}}[u] := \int _{\mathbb {R}^2} \left( \left| \left( \nabla + i \beta \mathbf {A}[|u|^2] \right) u \right| ^2 + V|u|^2 \right) , \end{aligned}$$

(3.28)

with the self-generated magnetic potential

$$\begin{aligned} \mathbf {A}[\rho ] := \nabla ^{\perp }w_0 * \rho = \int _{\mathbb {R}^2} \frac{(x-y)^\perp }{|x-y|^2} \rho (y) \,dy, \qquad \mathrm {curl}\,\mathbf {A}[\rho ] = 2\pi \rho . \end{aligned}$$

The functional is certainly well-defined for $u \in C_c^\infty (\mathbb {R}^2)$, but we should ask what its natural domain is. We then have to make a meaning of $\mathcal {E}^{\mathrm {af}}[u]$ for general $u \in L^2(\mathbb {R}^2)$ and the problem is that it is not certain that $\mathbf {A}[|u|^2] \in L^2_{{{\mathrm{loc}}}}$ even though $u \in L^2$ (see [12] for an example³), so the product $\mathbf {A}[|u|^2]u$ may not be well-defined as a distribution (while $\nabla u$ certainly is). One way around this is to reconsider the form of the functional when acting on regular enough functions such that we can write $u = |u|e^{i\varphi }$ where $\varphi$ is real. Then

$$\begin{aligned} \left| (\nabla + i\beta \mathbf {A}[|u|^2])u \right| ^2= & {} \left| \nabla |u| + i|u|(\nabla \varphi + \beta \mathbf {A}[|u|^2]) \right| ^2 \\= & {} \left| \nabla |u| \right| ^2 + \left| |u|\nabla \varphi + \beta \mathbf {A}[|u|^2]|u| \right| ^2, \end{aligned}$$

where also $\nabla \varphi = |u|^{-2}\mathfrak {I}\bar{u}\nabla u$ and $\nabla |u| = |u|^{-1}\mathfrak {R}\bar{u}\nabla u$. Hence, an alternative definition is given by

$$\begin{aligned} \mathcal {E}^{\mathrm {af}}[u] := \int _{\mathbb {R}^2} \left( \left| \nabla |u| \right| ^2 + \left| \mathfrak {I}\frac{\bar{u}}{|u|}\nabla u + \beta \mathbf {A}[|u|^2]|u| \right| ^2 + V|u|^2 \right) , \end{aligned}$$

(3.29)

and the advantage of this formulation is that it makes clear that we actually demand $|u| \in H^1(\mathbb {R}^2)$ in order for $\mathcal {E}^{\mathrm {af}}[u] < \infty$. We can then use the following lemma to see that in fact $\mathbf {A}[|u|^2]u \in L^2(\mathbb {R}^2)$, and hence also $\nabla u \in L^2(\mathbb {R}^2)$. (And conversely this also shows that if $\mathbf {A}[|u|^2]u \notin L^2(\mathbb {R}^2)$ then we have no chance of making sense out of $\mathcal {E}^{\mathrm {af}}[u]$.)

Lemma 3.4

(Bound on the magnetic term) We have for any $u \in L^2(\mathbb {R}^2)$ that

$$\begin{aligned} \int _{\mathbb {R}^2} \left| \mathbf {A}[|u|^2] \right| ^2 |u|^2 \le \frac{3}{2} \Vert u\Vert _{L^2(\mathbb {R}^2)}^4 \int _{\mathbb {R}^2} \left| \nabla |u| \right| ^2. \end{aligned}$$

Proof

This follows from symmetry and from the three-body Hardy inequality of Lemma 2.5:

$$\begin{aligned}&\int _{\mathbb {R}^2} \left| \mathbf {A}[|u|^2](x) \right| ^2 |u(x)|^2 \,dx \\&\quad =\int \!\!\int \!\!\int _{\mathbb {R}^6} \frac{x-y}{|x-y|^2} \cdot \frac{x-z}{|x-z|^2} |u(x)|^2 |u(y)|^2 |u(z)|^2 \,dxdydz \\&\quad = \frac{1}{6} \int _{\mathbb {R}^6} \frac{1}{\mathcal {R}(X)^2} \left| |u|^{\otimes 3} \right| ^2 dX \le \frac{1}{2} \int _{\mathbb {R}^6} \left| \nabla _X |u|^{\otimes 3} \right| ^2 dX = \frac{3}{2} \int _{\mathbb {R}^2} \left| \nabla |u| \right| ^2 dx \left( \int _{\mathbb {R}^2} |u|^2 dx \right) ^2. \end{aligned}$$

$\square$

We can therefore define the domain of $\mathcal {E}^{\mathrm {af}}$ to be (and otherwise let $\mathcal {E}^{\mathrm {af}}[u] := +\infty$)

$$\begin{aligned} \mathscr {D}^{\mathrm {af}}:= \left\{ u \in H^1(\mathbb {R}^2) : \int _{\mathbb {R}^2} V|u|^2 < \infty \right\} , \end{aligned}$$

and we find using Cauchy-Schwarz, Lemma 3.4, and $|\nabla |u|| \le |\nabla u|$ that for $u \in \mathscr {D}^{\mathrm {af}}$

$$\begin{aligned} 0\le & {} \mathcal {E}^{\mathrm {af}}[u] \le 2 \Vert \nabla u\Vert ^2 + 2\beta ^2 \Vert \mathbf {A}[|u|^2] u \Vert ^2 + \int V|u|^2 \le (2 + 3\beta ^2 \Vert u\Vert ^4) \Vert \nabla u\Vert ^2 \\&+ \int V|u|^2 < \infty . \end{aligned}$$

The ground-state energy of the average-field functional is then given by

$$\begin{aligned} E ^{\mathrm {af}}:= \inf \left\{ \mathcal {E}^{\mathrm {af}}[u] : u \in \mathscr {D}^{\mathrm {af}}, \int _{\mathbb {R}^2} |u|^2 = 1 \right\} . \end{aligned}$$

For convenience we also make the assumption on V that $V(x) \rightarrow +\infty$ as $|x| \rightarrow \infty$ and that $C_c^\infty (\mathbb {R}^2) \subseteq \mathscr {D}^{\mathrm {af}}$ is a form core for $\left||u \right||_{L^2_V}^2 := \int _{\mathbb {R}^2} V|u|^2$, with $-\Delta + V$ essentially self-adjoint and with purely discrete spectrum (see, e.g., [46, Theorem XIII.67]). This is then also a core for $\mathcal {E}^{\mathrm {af}}$:

Proposition 3.5

(Density of regular functions in the form domain) $C_c^\infty (\mathbb {R}^2)$ is dense in $\mathscr {D}^{\mathrm {af}}$ w.r.t. $\mathcal {E}^{\mathrm {af}}$, namely for any $u \in \mathscr {D}^{\mathrm {af}}$ there exists a sequence $(u_n)_{n \rightarrow \infty } \subset C_c^\infty (\mathbb {R}^2)$ such that

$$\begin{aligned} \Vert u-u_n\Vert _{H^1} \rightarrow 0 \text{ and } \mathcal {E}^{\mathrm {af}}[u_n] \rightarrow \mathcal {E}^{\mathrm {af}}[u] \text{ as } n \rightarrow \infty . \end{aligned}$$

Proof

Take $u \in \mathscr {D}^{\mathrm {af}}$, then $\Vert \nabla u\Vert _{L^2} < \infty$ and hence also $\Vert u\Vert _{L^p} < \infty$ for any $p \in [2,\infty )$ by Sobolev embedding. We use that $C_c^\infty (\mathbb {R}^2)$ is dense in $H^1(\mathbb {R}^2)$, so there exists a sequence $(u_n)_{n \rightarrow \infty } \subset C_c^\infty$ s.t. $\Vert u-u_n\Vert _{H^1} \rightarrow 0$. Also,

$$\begin{aligned}&\left| \left|| (\nabla + i\beta \mathbf {A}[|u|^2])u \right||_2 - \left|| (\nabla + i\beta \mathbf {A}[|u_n|^2])u_n \right||_2 \right| \\&\quad \le \left|| (\nabla + i\beta \mathbf {A}[|u|^2])u - (\nabla + i\beta \mathbf {A}[|u_n|^2])u_n \right||_2 \\&\quad \le \left|| \nabla (u-u_n) \right||_2 + |\beta | \Vert (\mathbf {A}[|u|^2] - \mathbf {A}[|u_n|^2])u + \mathbf {A}[|u_n|^2](u-u_n) \Vert _2 \\&\quad \le \left||u-u_n \right||_{H^1} + |\beta | \left|| \mathbf {A}[|u|^2-|u_n|^2] u \right||_2 + |\beta |\left|| \mathbf {A}[|u_n|^2](u-u_n) \right||_2, \end{aligned}$$

where by Hölder’s and generalized Young’s inequalities

$$\begin{aligned}&\left||\mathbf {A}[|u|^2-|u_n|^2] u \right||_2 \le \left||\mathbf {A}[|u|^2-|u_n|^2] \right||_{4} \left||u \right||_{4} \le C \left|||u|^2-|u_n|^2 \right||_{4/3} \left||\nabla w_0 \right||_{2,w} \left||u \right||_4 \\&\quad \le C' \left||u-u_n \right||_{8/3} \le C'' \left||u-u_n \right||_{H^1} \rightarrow 0, \end{aligned}$$

and similarly

$$\begin{aligned} \left||\mathbf {A}[|u_n|^2](u-u_n) \right||_2 \le C\left||u-u_n \right||_{H^1} \rightarrow 0, \end{aligned}$$

as $n \rightarrow \infty$.

We also have continuity for $\left||u \right||_{L^2_V}$ here since we assumed that $C_c^\infty (\mathbb {R}^2)$ is a form core. $\square$

Lemma 3.6

(Basic magnetic inequalities) We have for $u \in \mathscr {D}^{\mathrm {af}}$ that (diamagnetic inequality)

$$\begin{aligned} \int _{\mathbb {R}^2} \left| (\nabla + i\beta \mathbf {A}[|u|^2])u \right| ^2 \ge \int _{\mathbb {R}^2} \left| \nabla |u| \right| ^2, \end{aligned}$$

(3.30)

and

$$\begin{aligned} \int _{\mathbb {R}^2} \left| (\nabla + i\beta \mathbf {A}[|u|^2])u \right| ^2 \ge 2\pi |\beta | \int _{\mathbb {R}^2} |u|^4. \end{aligned}$$

(3.31)

Proof

By density we can w.l.o.g. assume $u \in C_c^\infty (\mathbb {R}^2)$. We then have $\mathbf {A}[|u|^2] \in C^\infty (\mathbb {R}^2) \subseteq L^2_{{{\mathrm{loc}}}}(\mathbb {R}^2)$ and hence the first inequality follows by the usual diamagnetic inequality (see e.g. Theorem 2.1.1 in [14]). Furthermore, by e.g. Lemma 1.4.1 in [14],

$$\begin{aligned} \int _{\mathbb {R}^2} \left| (\nabla + i\beta \mathbf {A}[|u|^2])u \right| ^2 \ge \pm \int _{\mathbb {R}^2} \mathrm {curl}\left( \beta \mathbf {A}[|u|^2] \right) |u|^2, \end{aligned}$$

which proves the second inequality since $\mathrm {curl}\mathbf {A}[|u|^2] = 2\pi |u|^2$. Instead of using density we could also have used the formulation (3.29) or the fact that $u \in H^1 \Rightarrow \mathbf {A}[|u|^2] \in L^p$, $p \in (2,\infty )$ by generalized Young. $\square$

Proposition 3.7

(Existence of minimizers) For any value of $\beta \in \mathbb {R}$ there exists $u ^{\mathrm {af}}\in \mathscr {D}^{\mathrm {af}}$ with $\int _{\mathbb {R}^2} |u ^{\mathrm {af}}|^2 = 1$ and $\mathcal {E}^{\mathrm {af}}[u ^{\mathrm {af}}] = E ^{\mathrm {af}}$.

Proof

First note that for $u \in \mathscr {D}^{\mathrm {af}}$, by Lemma 3.4 and Lemma 3.6,

$$\begin{aligned} \left||\nabla u \right||_2&= \left||\nabla u + i\beta \mathbf {A}[|u|^2] u - i\beta \mathbf {A}[|u|^2] u \right|| _2 \le \mathcal {E}^{\mathrm {af}}[u]^{1/2} + |\beta | \left||\mathbf {A}[|u|^2]u \right||_2 \\&\le \mathcal {E}^{\mathrm {af}}[u]^{1/2} + |\beta | \sqrt{\frac{3}{2}} \left||u \right||_2^2 \left||\nabla |u| \right||_2 \le \left( 1 + |\beta | \sqrt{\frac{3}{2}} \left||u \right||_2^2 \right) \mathcal {E}^{\mathrm {af}}[u]^{1/2}. \end{aligned}$$

Now take a minimizing sequence

$$\begin{aligned} (u_n)_{n \rightarrow \infty } \subset \mathscr {D}^{\mathrm {af}},\, \left||u_n \right||_2 = 1,\, \lim _{n \rightarrow \infty } \mathcal {E}^{\mathrm {af}}[u_n] = E ^{\mathrm {af}}. \end{aligned}$$

Then clearly $(u_n)$ is uniformly bounded in both $L^2(\mathbb {R}^2)$, $L^2_V$, and $H^1(\mathbb {R}^2)$ (and hence in $L^p(\mathbb {R}^2)$, $p \in [2,\infty )$), and therefore by the Banach–Alaoglu theorem there exists $u ^{\mathrm {af}}\in \mathscr {D}^{\mathrm {af}}$ and a weakly convergent subsequence (still denoted $u_n$) such that

$$\begin{aligned} u_n \rightharpoonup u ^{\mathrm {af}}\ \text {in} \ L^2(\mathbb {R}^2) \cap L^2_V \cap L^p(\mathbb {R}^2), \quad \nabla u_n \rightharpoonup \nabla u ^{\mathrm {af}}\ \text {in} \ L^2(\mathbb {R}^2). \end{aligned}$$

Moreover, since $(-\Delta + V + 1)^{-1/2}$ is compact we have that

$$\begin{aligned} u_n = (-\Delta +V + 1)^{-1/2}(-\Delta +V + 1)^{1/2}u_n \end{aligned}$$

is actually strongly convergent (again extracting a subsequence), hence

$$\begin{aligned} u_n \rightarrow u ^{\mathrm {af}} \text{ in } L^2(\mathbb {R}^2). \end{aligned}$$

Also, $\mathbf {A}[|u_n|]$ converges pointwise a.e. to $\mathbf {A}[|u|^2]$ by weak convergence of $u_n$ in $L^p$ and, by the trick of Lemma 3.4,

$$\begin{aligned} \left||\mathbf {A}[|u_n|^2]u_n \right||_2^2= & {} \frac{1}{6} \int _{\mathbb {R}^6} \mathcal {R}(X)^{-2} \left| |u_n|^{\otimes 3} \right| ^2 dX \rightarrow \frac{1}{6} \int _{\mathbb {R}^6} \mathcal {R}(X)^{-2} \left| |u|^{\otimes 3} \right| ^2 dX \\= & {} \left||\mathbf {A}[|u|^2]u \right||_2^2 \end{aligned}$$

by dominated convergence. The functions $\mathbf {A}[|u_n|^2]u_n$ are therefore even strongly converging to $\mathbf {A}[|u|^2]u$ in $L^2(\mathbb {R}^2)$ by dominated convergence. It then follows that

$$\begin{aligned} \left|| (\nabla + i\beta \mathbf {A}[|u|^2])u \right||_2&= \sup _{\Vert v\Vert =1} |\langle \nabla u + i\beta \mathbf {A}[|u|^2]u, v \rangle | \\&= \sup _{\Vert v\Vert =1} \lim _{n \rightarrow \infty } |\langle \nabla u_n + i\beta \mathbf {A}[|u_n|^2]u_n, v \rangle | \\&\le \liminf _{n \rightarrow \infty } \sup _{\Vert v\Vert =1} |\langle \nabla u_n + i\beta \mathbf {A}[|u_n|^2]u_n, v \rangle | \\&= \liminf _{n \rightarrow \infty } \left|| (\nabla + i\beta \mathbf {A}[|u_n|^2])u_n \right||_2, \end{aligned}$$

and since $\left||\cdot \right||_{L^2_V}$ is also weakly lower semicontinuous (see, e.g., [44, Supplementto IV.5]), we have $\liminf _{n \rightarrow \infty } \mathcal {E}^{\mathrm {af}}[u_n] \ge \mathcal {E}^{\mathrm {af}}[u ^{\mathrm {af}}]$. Thus, with $\Vert u ^{\mathrm {af}}\Vert = \lim _{n \rightarrow \infty } \Vert u_n\Vert = 1$, we also have $\mathcal {E}^{\mathrm {af}}[u ^{\mathrm {af}}] = E ^{\mathrm {af}}$. $\square$

Proposition 3.8

(Convergence to bosons) Let $E_0$ resp. $u_0$ denote the ground-state eigenvalue resp. normalized eigenfunction of the non-magnetic Schrödinger operator $H_1 = -\Delta + V$, with $V \in L^{\infty }_{\mathrm{loc}}$. We have

$$\begin{aligned} E ^{\mathrm {af}}_{(\beta )} \underset{\beta \rightarrow 0}{\rightarrow } E_0, \end{aligned}$$

and that given an arbitrary sequence $(u_\beta )$ of minimizers for $\mathcal {E}^{\mathrm {af}}_{(\beta )}$

$$\begin{aligned} u_\beta \underset{\beta \rightarrow 0}{\rightarrow } u_0 \text{ in } L^2(\mathbb {R}^2) \end{aligned}$$

up to a subsequence and a constant phase.

Proof

Note that under our conditions for V, $u_0 \in \mathscr {D}^{\mathrm {af}}$ is the unique minimizer of $\mathcal {E}_0 = \mathcal {E}^{\mathrm {af}}_{(\beta =0)}$ and can be taken positive (see, e.g., [29, Theorem 11.8]). By the diamagnetic inequality (3.30), and by taking the trial state $u_0 = |u_0|$ in $\mathcal {E}^{\mathrm {af}}_{(\beta \ne 0)}$, we find

$$\begin{aligned} E_0 \le E ^{\mathrm {af}}_{(\beta )} \le \mathcal {E}^{\mathrm {af}}_{(\beta )}[u_0] = \mathcal {E}_0[u_0] + \beta ^2\left||\mathbf {A}[|u_0|^2]u_0 \right||_2^2 \le (1 + C\beta ^2)E_0 \end{aligned}$$

(where we also used Lemma 3.4), and hence $E ^{\mathrm {af}}_{(\beta )} \rightarrow E_0$ as $\beta \rightarrow 0$. Now consider a sequence $(u_\beta ) \subset \mathscr {D}^{\mathrm {af}}$ of minimizers as $\beta \rightarrow 0$ with $\mathcal {E}^{\mathrm {af}}[u_\beta ] \rightarrow E_0$, $\Vert u_\beta \Vert =1$. Then, because of uniform boundedness and as in the proof of Proposition 3.7, we have after taking a subsequence that $u_\beta \rightarrow u$ for some $u \in \mathscr {D}^{\mathrm {af}}$, $\Vert u\Vert =1$, and also

$$\begin{aligned} \Vert \nabla u\Vert&= \sup _{\Vert v\Vert =1} \left| \langle \nabla u,v \rangle \right| \\&= \sup _{\Vert v\Vert =1} \lim _{\beta \rightarrow 0} \left| \langle \nabla u_\beta + i\beta \mathbf {A}[|u_\beta |^2]u_\beta ,v \rangle \right| \\&\le \liminf _{\beta \rightarrow 0} \left|| \nabla u_\beta + i\beta \mathbf {A}[|u_\beta |^2]u_\beta \right||, \end{aligned}$$

so

$$\begin{aligned} E_0 \le \mathcal {E}_0[u] \le \liminf _{\beta \rightarrow 0} \mathcal {E}^{\mathrm {af}}_{(\beta )}[u_\beta ]. \end{aligned}$$

It follows that $\mathcal {E}_0[u] = E_0$ and hence $u=u_0$ up to a constant phase.

From the bound (3.31) we observe that the self-generated magnetic interaction is stronger than a contact interaction of strength $2\pi |\beta |$ (despite the fact that we already removed a singular repulsive interaction in the initial regularization step for extended anyons). Hence we have not only $E ^{\mathrm {af}}\ge E_0$ by the diamagnetic inequality, but also

$$\begin{aligned} E ^{\mathrm {af}}\ge \min _{\rho \ge 0,\ \int _{\mathbb {R}^2} \rho = 1} \int _{\mathbb {R}^2} \left( 2\pi |\beta |\rho ^2 + V\rho \right) , \end{aligned}$$

(3.32)

which can be computed for given V by straightforward optimization.

Let us now consider the corresponding situation for the regularized functional (extended anyons)

$$\begin{aligned} \mathcal {E}^{\mathrm {af}}_R[u] := \int _{\mathbb {R}^2} \left( \left| \left( \nabla + i \beta \mathbf {A}^R[|u|^2] \right) u \right| ^2 + V|u|^2 \right) , \quad \mathbf {A}^R[\rho ] := \nabla ^{\perp }w_R * \rho , \quad R > 0. \end{aligned}$$

Since $\nabla w_R \in L^\infty (\mathbb {R}^2)$ we have $\mathbf {A}^R[|u|^2] \in L^\infty (\mathbb {R}^2)$ with

$$\begin{aligned} \left||\mathbf {A}^R[|u|^2] \right||_\infty \le \frac{C}{R}\Vert u\Vert _2^2 \end{aligned}$$

and instead of Lemma 3.4 we have

$$\begin{aligned} \left||\mathbf {A}^R[|u|^2]u \right||_2 \le C\Vert u\Vert _2^2 \Vert |u|\Vert _{H^1} \end{aligned}$$

using Lemma 2.4. Hence the natural domain is again $\mathscr {D}^{\mathrm {af}}$ and all properties established above for $\mathcal {E}^{\mathrm {af}}$ are also found to be valid for $\mathcal {E}^{\mathrm {af}}_R$ (except (3.31) and (3.32) which now have regularized versions). Denoting

$$\begin{aligned} E ^{\mathrm {af}}_R := \min \left\{ \mathcal {E}^{\mathrm {af}}_R[u] : u \in \mathscr {D}^{\mathrm {af}}, \Vert u\Vert _2 = 1\right\} , \end{aligned}$$

we furthermore have the following relationship:

Proposition 3.9

(Convergence to point-like anyons) The functional $\mathcal {E}^{\mathrm {af}}_R$ converges pointwise to $\mathcal {E}^{\mathrm {af}}$ as $R \rightarrow 0$. More precisely, for any $u\in \mathscr {D}^{\mathrm {af}}$

$$\begin{aligned} \left| \mathcal {E}^{\mathrm {af}}_R [u] - \mathcal {E}^{\mathrm {af}}[u] \right| \le C_u |\beta |(1+\beta ^4) (1+\mathcal {E}^{\mathrm {af}}[u])^{3/2} R, \end{aligned}$$

(3.33)

where $C_u$ depends only on $\left||u \right||_2$. Hence,

$$\begin{aligned} E ^{\mathrm {af}}_R \underset{R\rightarrow 0}{\rightarrow } E ^{\mathrm {af}}, \end{aligned}$$

and if $(u_R)_{R \rightarrow 0} \subset \mathscr {D}^{\mathrm {af}}$ denotes a sequence of minimizers of $\mathcal {E}^{\mathrm {af}}_R$, then there exists a subsequence $(u_{R'})_{R' \rightarrow 0}$ s.t. $u_{R'} \rightarrow u ^{\mathrm {af}}$ as $R' \rightarrow 0$, where $u ^{\mathrm {af}}$ is some minimizer of $\mathcal {E}^{\mathrm {af}}$.

Proof

We have for any $u \in \mathscr {D}^{\mathrm {af}}$ that

$$\begin{aligned}&\left| \left|| (\nabla + i\beta \mathbf {A}[|u|^2])u \right||_2 - \left|| (\nabla + i\beta \mathbf {A}^R[|u|^2])u \right||_2 \right| \\&\quad \le \left|| (\nabla + i\beta \mathbf {A}[|u|^2])u - (\nabla + i\beta \mathbf {A}^R[|u|^2])u \right||_2= |\beta | \left|| (\mathbf {A}[|u|^2] - \mathbf {A}^R[|u|^2])u \right||_2 \\&\quad \le |\beta | \left|| \mathbf {A}[|u|^2] - \mathbf {A}^R[|u|^2] \right||_4 \left||u \right||_4= |\beta | \left|| (\nabla w_0 - \nabla w_R) * |u|^2 \right||_4 \left||u \right||_4, \end{aligned}$$

where by Young

$$\begin{aligned} \left|| (\nabla w_0 - \nabla w_R) * |u|^2 \right||_4 \le \left|| \nabla w_0 - \nabla w_R \right||_1 \left|||u|^2 \right||_4 \le \left||\nabla w_0 \right||_{L^1(B(0,R))} \left||u \right||_8^2 \rightarrow 0, \end{aligned}$$

as $R \rightarrow 0$, since $\nabla w_0 \in L^1_{\mathrm{loc}}(\mathbb {R}^2)$. We deduce (3.33) by combining this with previous estimates of this appendix and Sobolev embeddings. It follows that $\mathcal {E}^{\mathrm {af}}_R[u] \rightarrow \mathcal {E}^{\mathrm {af}}[u]$ as $R \rightarrow 0$.

Let $(u_R)_{R \rightarrow 0}$ denote a sequence of minimizers of $\mathcal {E}^{\mathrm {af}}_R$:

$$\begin{aligned} E ^{\mathrm {af}}_R = \mathcal {E}^{\mathrm {af}}_R[u_R],\, \Vert u_R\Vert = 1, \end{aligned}$$

and take $u \in \mathscr {D}^{\mathrm {af}}$ an arbitrary minimizer of $\mathcal {E}^{\mathrm {af}}$. Then, since

$$\begin{aligned} E ^{\mathrm {af}}_R \le \mathcal {E}^{\mathrm {af}}_R[u] \underset{R\rightarrow 0}{\rightarrow } \mathcal {E}^{\mathrm {af}}[u] = E ^{\mathrm {af}}, \end{aligned}$$

we have that $E ^{\mathrm {af}}_R$ is uniformly bounded as $R \rightarrow 0$ and that

$$\begin{aligned} \limsup _{R \rightarrow 0} E ^{\mathrm {af}}_R \le E ^{\mathrm {af}}. \end{aligned}$$

Then $\mathcal {E}^{\mathrm {af}}_R[u_R]$, and hence also

$$\begin{aligned} \mathcal {E}^{\mathrm {af}}[u_R] \le C\left( \Vert u_R\Vert _{H^1}^2 + \Vert u_R\Vert _{L^2_V}^2 \right) \le C'(\mathcal {E}^{\mathrm {af}}_R[u_R] + 1), \end{aligned}$$

are uniformly bounded as well. As in the proof of Proposition 3.7, there then exists a strongly convergent subsequence $(u_{R'})_{R' \rightarrow 0}$, with $u_{R'} \rightarrow u_0 \in \mathscr {D}^{\mathrm {af}}$. Also, by weak lower semicontinuity $\mathcal {E}^{\mathrm {af}}[u_0] \le \liminf _{R' \rightarrow 0} \mathcal {E}^{\mathrm {af}}[u_{R'}]$, so that for any $\varepsilon > 0$ and sufficiently small $R'>0$,

$$\begin{aligned} E ^{\mathrm {af}}\le \mathcal {E}^{\mathrm {af}}[u_0] \le \mathcal {E}^{\mathrm {af}}[u_{R'}] + \varepsilon \le \mathcal {E}^{\mathrm {af}}_{R'}[u_{R'}] + 2\varepsilon = E ^{\mathrm {af}}_{R'} + 2\varepsilon , \end{aligned}$$

where we also used that the convergence is uniform for our uniformly bounded sequence $u_R$ by the bound (3.33). It follows that $E ^{\mathrm {af}}\le \mathcal {E}^{\mathrm {af}}[u_0] \le E ^{\mathrm {af}}+ 3\varepsilon$, and hence $u_0$ is a minimizer with $\Vert u_0\Vert =1$ and $E ^{\mathrm {af}}= \mathcal {E}^{\mathrm {af}}[u_0] = \lim _{R \rightarrow 0} E ^{\mathrm {af}}_R$. $\square$