Abstract
In this chapter we consider the thermodynamics of integrable models. We assume that these models have a phase shift that can be computed exactly or approximately. For models with no bound states, by applying the thermodynamic limit to the corresponding Bethe equations, we obtain equations for the ground state. We then consider the case of finite temperature and derive the Yang-Yang equation describing the state of thermodynamic equilibrium. Further, using the example of a one-dimensional electron gas with delta-function interaction, we formulate the so-called string hypothesis and derive the corresponding Thermodynamic Bethe Ansatz equations, both canonical and simplified. We also exhibit solutions of these equations at weak and strong coupling.
This is a very controversial point of the thermodynamic Bethe-ansatz equations for soluble models, except for the repulsive boson case, which has no string solutions. But equations obtained using the string hypothesis seem to give the correct free energy and other thermodynamic quantities.
Minoru Takahashi
Thermodynamics of one-dimensional solvable models
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Notes
- 1.
For N even the ground state of a fermion system is doubly degenerate, see footnote 2 on Sect. 5.1.2.
- 2.
- 3.
These important properties of \(p(\eta )\) also follow from the variational principle, see [3].
- 4.
In physical units this density has the dimension \(1/[\hbar ]\).
- 5.
All physical parameters are in there, including the mass parameter \(\mu =mc\) (c is the speed of light), the length parameter \(\ell \) and the Planck constant \(\hbar \).
- 6.
We warn the reader that we use for pressure the same notation as for the total momentum of a microscopic system.
- 7.
This configuration is called p-\(\lambda \) string.
- 8.
References
Zamolodchikov, A.B.: Thermodynamic Bethe Ansatz in relativistic models. Scaling three state potts and Lee-yang models. Nucl. Phys. B 342, 695–720 (1990)
Yang, C.N., Yang, C.P.: Thermodynamics of one-dimensional system of bosons with repulsive delta function interaction. J. Math. Phys. 10, 1115–1122 (1969)
Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge Monographs on Mathematical Physics, Cambridge University Press (1997)
Gaudin, M.: Boundary energy of a Bose gas in one dimension. Phys. Rev. A 4, 386–394 (1971)
Sutherland, B.: Quantum many body problem in one-dimension. J. Math. Phys. 12, 251–256 (1971)
Takahashi, M.: One-dimensional electron gas with delta-function interaction at finite temperature. Prog. Theor. Phys. 46, 1388–1406 (1971)
Lai, C.K.: Thermodynamics of fermions in one dimension with a \(\delta \)-function interaction. Phys. Rev. Lett. 26, 1472–1475 (1971)
Lai, C.K.: Thermodynamics of a one-dimensions system of fermions with a repulsive \(\delta \)-function interaction. Phys. Rev. A 8, 2567–2573 (1973)
Takahashi, M.: Thermodynamics of One-Dimensional Solvable Models. Cambridge University Press (1999)
Essler, F.H.L., Frahm, H., Göhmann, F., Klümper, A., Korepin, V.E.: The One-Dimensional Hubbard Model. Cambridge University Press (2005)
Arutyunov, G., Frolov, S.: String hypothesis for the \({\rm AdS}_5\times {\rm S}^5\) mirror. JHEP 03, 152 (2009)
Arutyunov, G., Frolov, S.: Thermodynamic Bethe Ansatz for the \({\rm AdS}_5\times {\rm S}^5\) mirror model. JHEP 05, 068 (2009)
Arutyunov, G., de Leeuw, M., van Tongeren, S.J.: The quantum deformed mirror TBA I. JHEP 10, 090 (2012)
Arutyunov, G., Frolov, S.: Foundations of the \({\rm AdS}_5\times {\rm S}^5\) superstring. Part I. J. Phys. A 42, 254003 (2009)
Beisert, N., et al.: Review of AdS/CFT integrability: an overview. Lett. Math. Phys. 99, 3–32 (2012)
Yang, C.N.: Some exact results for the many body problems in one dimension with repulsive delta function interaction. Phys. Rev. Lett. 19, 1312–1314 (1967)
Šamaj, L., Bajnok, Z.: Introduction to the statistical physics of integrable many-body systems. Cambridge University Press (2013)
Landau, L.D., Lifshitz, E.M., Pitaevskij, L.P.: Statistical Physics, (Course of Theoretical Physics), 2 edn., vol. 5. Oxford (1980)
Kuniba, A., Nakanishi, T., Suzuki, J.: T-systems and Y-systems in integrable systems. J. Phys. A 44, 103001 (2011)
van Tongeren, S.J.: Introduction to the thermodynamic Bethe ansatz. J. Phys. A 49, 323005 (2016)
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Arutyunov, G. (2019). Integrable Thermodynamics. In: Elements of Classical and Quantum Integrable Systems . UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-24198-8_6
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