Abstract
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.
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Notes
- 1.
In fact, one can see the lattice points in a (space) period as a single point in an equivalent lattice on which particles have an enlarged spin set.
- 2.
For E is a metrizable compact space, any finite Borel measure is regular and tight. Thus, here, probability measures are just the same as normalized Borel measures.
- 3.
I.e., \(\int _{\mathbb {R}^{d}}\gamma _{\pm }^{d}f_{\pm }\left ( \gamma _{\pm }x\right ) \mathrm {d}x=\int _{\mathbb {R}^{d}}f_{\pm }\left ( x\right ) \mathrm {d}x\doteq \hat {f}_{\pm }(0)\).
- 4.
Mean-field repulsions have generally a geometrical effect by possibly breaking the face structure of the set of (generalized) equilibrium states (see [1, Lemma 9.8]). When this appears, we have long-range order of correlations without necessarily a non-unique equilibrium state (i.e., first order phase transition). See [1, Section 2.9].
References
Bru, J.-B., de Siqueira Pedra, W.: Non-cooperative Equilibria of Fermi Systems with Long Range Interactions. Memoirs AMS, vol. 224, no. 1052. American Mathematical Society, Providence (2013)
Bru, J.-B., de Siqueira Pedra, W., Rodrigues Alves, K.: From short-range to mean-field models in quantum lattices . Adv. Theoret. Math. Phys. To be published (2023). See arXiv:2203.01021 [math-ph] (52 pages)
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. I, 2nd edn. Springer, New York (1987)
Rudin, W.: Functional Analysis. McGraw-Hill Science. McGraw-Hill, New York (1991)
Bru, J.-B., de Siqueira Pedra, W.: Quantum dynamics generated by long-range interactions for Lattice-Fermion and quantum spins. J. Math. Anal. Appl. 493(1), 124517 (2021)
Zagrebnov, V.A., Bru, J.-B.: The bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350, 291–434 (2001)
Araki, H., Moriya, H.: Equilibrium statistical mechanics of fermion lattice systems. Rev. Math. Phys. 15, 93–198 (2003)
Bogoliubov, N.N., Jr., Brankov, J.G., Zagrebnov, V.A., Kurbatov, A.M., Tonchev, N.S.: Metod approksimiruyushchego gamil’toniana v statisticheskoi fizike. The Approximating Hamiltonian Method in Statistical Physics. Sofia: Izdat. Bulgar. Akad. Nauk. Publ. House Bulg. Acad. Sci. (1981)
Bogoliubov, N.N., Jr., Brankov, J.G., Zagrebnov, V.A., Kurbatov, A.M., Tonchev, N.S.: Some classes of exactly soluble models of problems in Quantum Statistical Mechanics: the method of the approximating Hamiltonian. Russ. Math. Surv. 39, 1–50 (1984)
Brankov, J.G., Danchev, D.M., Tonchev, N.S.: Theory of Critical Phenomena in Finite–size Systems: Scaling and Quantum Effects. Word Scientific, Singapore (2000)
Brankov, J.G., Tonchev, N.S., Zagrebnov, V.A.: A nonpolynomial generalization of exactly soluble models in statistical mechanics, Ann. Phys. (N. Y.) 107(1–2), 82–94 (1977)
Brankov, J.G., Tonchev, N.S., Zagrebnov, V.A.: On a class of exactly soluble statistical mechanical models with nonpolynomial interactions, J. Stat. Phys. 20(3), 317–330 (1979)
Komiya, H.: Elementary proof for Sion’s minimax theorem. Kodai Math. J. 11(1), 5–7 (1988)
Presutti, E.: Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Springer, Berlin (2009)
Lieb, E.: Quantum-mechanical extension of the Lebowitz-Penrose theorem on the Van Der Waals theory. J. Math. Phys. 7(6), 1016–1024 (1966)
de Smedt, P., Zagrebnov V. A.: van der Waals limit of an interacting Bose gas in a weak external field. Phys. Rev. A 35(11), 4763–4769 (1987)
Cooper, L.N.: Bound electron Pairs in a degenerate Fermi gas. Phys. Rev. 104, 1189–1190 (1956)
Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Microscopic theory of superconductivity. Phys. Rev. 106, 162–164 (1957)
Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957)
Bogoliubov, N.N.: On the theory of superfluidity. J. Phys. (USSR) 11, 23–32 (1947)
Bogoliubov, N.N.: On some problems of the theory of superconductivity. Physica 26, S1–S16 (1960)
Bogoliubov, N.N., Jr.: On model dynamical systems in statistical mechanics. Physica 32, 933 (1966)
Bogoliubov, N.N., Jr.: A Method for Studying Model Hamiltonians. Pergamon, Oxford (1977)
Lebowitz, J., Penrose, O.: A rigorous treatment of the Van der Waals-Maxwell theory of the vapor-liquid transition. J. Math. Phys. 7, 98 (1966)
Penrose, O., Lebowitz, J.L.: Rigorous treatment of metastable states in the van der Waals-Maxwell theory. J. Stat. Phys. 3(2), 211–236 (1971)
Hemmer, P. C., Lebowitz, J.L.: Systems with weak long-range potentials. In: Domb, C., Green, M.S. (eds.) Phase Transitions and Critical Phenomena, vol. 5b, pp. 107–203. Academic Press, Cambridge (1976)
Franz, S., Toninelli, F.L.: Kac limit for finite-range spin glasses. Phys. Rev. Lett. 92, 030602 (2004)
Franz, S., Toninelli, F.L.: Finite-range spin glasses in the Kac limit: free energy and local observables. J. Phys. A: Math. Gen. 37, 7433–7446 (2004)
Franz, S.: Spin glass models with Kac interactions. Eur. Phys. J. B 64, 557–561 (2008)
Bru, J.-B., de Siqueira Pedra, W.: Classical dynamics generated by long-range interactions for lattice fermions and quantum spins. J. Math. Anal. Appl. 493(1), 124434 (2021)
Bru, J.-B., de Siqueira Pedra, W.: Entanglement of classical and quantum short-range dynamics in mean-field systems. Ann. Phys. 434, 168643 (2021)
Acknowledgements
This work is supported by CNPq (309723/2020-5) as well as by the Basque Government through the grant IT1615-22 and the BERC 2022-2025 program, by the COST Action CA18232 financed by the European Cooperation in Science and Technology (COST), and by the Ministry of Science and Innovation via the grant PID2020-112948GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. We thank Domingos Marchetti for valuable discussions and hints.
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Bru, JB., Pedra, W.d.S., Alves, K.R. (2023). Thermodynamic Game and the Kac Limit in Quantum Lattices. In: Correggi, M., Falconi, M. (eds) Quantum Mathematics II. INdAM 2022. Springer INdAM Series, vol 58. Springer, Singapore. https://doi.org/10.1007/978-981-99-5884-9_9
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