How models become gapless in the thermodynamic limit? Given an Hamiltonian on some finite lattice, it lives in a finite-dimensional Hilbert space with a finite number of eigenvalues, so obviously there is a gap between the lowest value of an eigenvalue and the second lowest value. Therefore gaplessness (or a gap) of a model is an asymptotic property. A model is gapped if in the thermodynamic limit a gap is formed, and gapless if it does not form.
Formally - Denote by $E_0(N)$ the ground state energy of a model, and by $E_1(N)$ the energy of the first excited states, both for a model on $N$ sites. The model is gapless iff when $N\rightarrow \infty, N/V=const$ it holds that $E_0-E_1\rightarrow0$.
How can this limit occur if $E_0,E_1$ are independent of $N$? For example consider the Ising model with transverse field at the critical point.
 A: 
How can this limit occur if $E_0,E_1$ are independent of $N$? For example consider the Ising model with transverse field at the critical point.

As pointed out by Norbert Shuch in the comments, the $E_n$ generally are not independent of $N$. Let's consider the model you mention in 1+1 dimensions with periodic boundary conditions. Then conformal field theory predicts
$$
E_0 = \epsilon_0 N - \frac{\pi c}{6N} + \mathcal{O}\left( \frac{1}{N^2}\right),
$$
and
$$
\Delta E_n \equiv E_n - E_0 = \frac{2\pi x_n}{N},
$$
where $c=1/2$ is the central charge, $\epsilon_0$ is the energy density in the thermodynamic limit, and $x_n$ is a quantity called scaling dimension. You can easily verify that this gap exists and depends on $N$ by doing small-scale numerical calculations using e.g. exact diagonalization. If it didn't exist, we would not have to worry nearly as much about finite-size scaling in such computations.
In the vicinity of the critical point, the system may instead be described by a massive field theory, the mass of which determines the gap. This mass also has a finite size correction, which decays exponentially with $N$, according to Lüscher's formula. Oshikawa wrote more on the topic in this preprint.
