Mathematical Rigorousness of Taking the Thermodynamical limit of a finite size quantum model

Suppose I have nearest Neighbour Quantum Ising model with a transverse field. $$\hat{H} = \sum_{i}S^{x}_iS^{x}_{i+1} + h\sum_i S^{z}_i$$

Through Jordan-Wigner and Bogoliubov transformation, one finds that this model is equivalent to

$$\hat{H} = \sum_{k} E(k) f^\dagger_k f_k$$ where $$E(k)$$ is something like:

$$E(k) = \sqrt{1+2h\cos(2\pi k/L)+h^2 }$$

here $$k$$ is integer and hence $$k/L$$ is always quotient number.

However, in the thermodynamic limit, $$k$$ or $$k/L$$ is allowed to take any real number. How is this possible?

• Consider the set of decimal numbers with at most the first $n$ digits non-zero. For any finite $n$ all the numbers in this set are rational, but in the limit $n\rightarrow \infty$ I can have any real number. Indeed one way to construct the real numbers is as the set of limit points of sequences of rational numbers. In short taking this sort of limit is exactly where you naturaly expect to move from the rationals to the reals – By Symmetry Aug 14 at 15:26
• How would you take the thermodynamic limit? – Norbert Schuch Aug 14 at 16:21