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When liquid helium is cooled below a certain temperature, it undergoes a second phase transition from He I to He II. The specific heat spikes at the famous lambda point. Is this spike in the specific heat understood? I've read sources that say it is essentially not theoretically understood from first principles, but my sources are a little old. Are there any references that explain it?

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Depends on what you mean by understood. I would argue that the answer is basically "yes", although helium (like water) is a complicated substance, and many details can only be understood by numerical simulation.

1) The lambda transition is in the universality class of the three dimensional $xy$ model, and the critical exponents are well known (from the epsilon expansion, and numerical simulation). this means, in particular, that the "lambda" shape of $c_v$ is understood.

2) Why the transition temperature is 2.2K, and why helium is an ordinary liquid above $T_c$ (and not a solid, for example) can only be understood using Quantum Monte Carlo simulations. The pioneering work in the area goes back to Ceperley.

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