It is natural to expect that the universal physics is insensitive to the precise form of the regularization. One specific and possibly physical choice is to use the lowest-Landau-Level projected density operator in Eq. (1) and the short-distance length scale is the magnetic length. We will comment on its effect at the end of the manuscript.

Equation (14) implicitly depends on the short-distance cutoff $a$. First, it holds only when the distances between these pairs are larger than $a$. Second, we omit a prefactor $(\kappa a^2{)}^N$ that accounts for the residual relative motion of each pair. Namely, the two particles in each pair can still move relative to each other within a disk of radius $a$. And $\kappa$ is a nonuniversal geometric factor.

The fugacity comes from the unbinding energy as well as the entropic effect analyzed above, i.e., $e^{-\overline{\mu }}\sim e^{-\mu }\times \sqrt{N/A{a}^2}$. We also omitted an $O(1)$ prefactor that depends on the details of the screening due to the charge-$2\sqrt{m}$ plasma and the short-distance cutoff.

Coarse grained because the noise inevitably has a short-range cutoff $a\ll {\ell }_B$.

To understand this equality, we first note ${\sum }_{(b,c)}|{\psi }_b{|}^2+|{\psi }_c{|}^2=2{\sum }_{(b,c)}|{\psi }_b{|}^2$. We then compute ${\sum }_{(b,c)}|{\psi }_b{|}^2$ by counting electron configurations $c$ that differ from a given configuration $b$ by exactly one electron position. There are $N$ choices for which electron is moved, and the allowed area available to that electron is $c_0{L}^2$ with a nonuniversal factor $c_0<1$.

Any separable state across $R$ and $Q$ can be written as a convex sum ${\rho }_{QR}={\sum }_k{p}_k|Q_k⟩⟨Q_k|\otimes |R_k⟩⟨R_k|,p_k>0$, where $|Q_k⟩$ and $|R_k⟩$ are sets of generically nonorthogonal but normalized pure states on $Q$ and $R$. It is then easy to see that $\mathrm{tr}{\rho }_{QR}^2\le \mathrm{tr}{\rho }_Q^2$ and $I_c^{(2)}$ cannot be positive.

Specifically, the symmetry is generated by the following line operators $$U_i(C)=\exp (i\frac{k_i}{m}{\oint }_C(\nabla {\theta }_i+e\mathbf{A})·d\ell ),$$(78)where $C$ is a loop and $\mathbf{A}$ is the vector potential for the background magnetic field. They are related to the worldline of charge-$e/m$ Laughlin quasiholes. The partition function $Z_{{\mathrm{MR}}_{\infty }}$ with $U_i(C)$ inserted is invariant under continuous deformation of $C$, i.e., the line operators (78) are topological, ensuring the symmetry.

The ${\mathrm{U}(1)}_m$ and ${\mathrm{U}(1)}_w$ symmetries have a mixed anomaly. Therefore, the coexistence of $\mathrm{\Phi }$ and $\mathrm{\Theta }$ condensates is also a coexistence of the ordered and disordered phase for each of the U(1) symmetries, which typically happen at first order transitions. The current situation is even more special as the Mermin-Wagner theorem forbids continuous symmetry breaking in usual 2D theories. Our theory, not described by a local Lagrangian, can evade this no-go constraint.