Thermodynamic Game and the Kac Limit in Quantum Lattices
A mathematically rigorous computation of the pressure and equilibrium states of important short-range quantum models on lattices (like the Hubbard model) to show possible phase transitions is generally elusive, beyond perturbative arguments, even after decades of mathematical studies. By contrast, s...
| Autores: | , , |
|---|---|
| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| País: | España |
| Recursos: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/1895 |
| Acesso em linha: | http://hdl.handle.net/20.500.11824/1895 |
| Access Level: | acceso embargado |
| Palavra-chave: | Kac limit Lattice fermion systems Thermodynamic game |
| Resumo: | A mathematically rigorous computation of the pressure and equilibrium states of important short-range quantum models on lattices (like the Hubbard model) to show possible phase transitions is generally elusive, beyond perturbative arguments, even after decades of mathematical studies. By contrast, such a question can be solved for mean-field models. This is done by using some form of the Bogoliubov approximation, leading to the thermodynamic game introduced in Bru and de Siqueira Pedra (Non-cooperative Equilibria of Fermi Systems with Long Range Interactions. Memoirs AMS, vol. 224, no. 1052. American Mathematical Society, Providence, 2013). Here we illustrate this abstract result on a specific, albeit still general, example. We then state recent results contributing a precise mathematical relation between mean-field and short-range models via the long-range limit that is known in the literature as the the Kac or van der Waals limit. This paves the way for studying phase transitions, or at least important fingerprints of them like strong correlations at long distances, for models having interactions whose ranges are finite, but very large as compared to the lattice constant. It also sheds a new light on mean-field models. If both attractive and repulsive long-range forces are present then it turns out that the limit mean-field model is not necessarily what one traditionally guesses. |
|---|