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The term ‘free energy’ confused me for a long time. One one hand, when we learn thermodynamics, people talk about the Helmholtz free energy, $F = U - TS$, which is somehow ‘minimised’ at constant temperature. On the other hand, in the literature, people seem to describe the logarithm of a partition function to be the ‘free energy’, too. They must be related, but how?

The conventions and relationships between discrete Fourier transforms (DFT), Fourier series, and continuous Fourier transforms (FT), are confusing enough that I decided to write a reference on it.

The transverse-field Ising (TFI) model is a prototypical model of a quantum magnet, where quantum fluctuations are used to generate paramagnetism instead of temperature.

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