Abstract
The Quantum Heisenberg Ferromagnet can be naturally reformulated in terms of interacting bosons (called spin waves or magnons) as an expansion in the inverse spin size. We calculate the first order interaction correction to the free energy, as an upper bound in the limit where the spin size \(S \to \infty \) and ß S is fixed (ß being the inverse temperature). Our result is valid in two and three spatial dimensions. We extrapolate our result to compare with Dyson’s low-temperature expansion. While our first-order correction has the expected temperature dependence, in higher orders of the perturbation theory cancellations are necessary.
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References
Bloch, F.: Zur Theorie des Ferromagnetismus. Zeitschrift Für Physik 61, 206–219 (1930)
Conlon, J.G., Solovej, J.P.: On asymptotic limits for the quantum Heisenberg model. J. Phys. A 23, 3199–3213 (1990)
Conlon, J.G., Solovej, J.P.: Random walk representations of the Heisenberg model. J. Stat. Phys. 64, 251–270 (1991)
Conlon, J.G., Solovej, J.P.: Uniform convergence of the free energy of the classical Heisenberg model to that of the Gaussian model. J. Statist. Phys. 65, 235–245 (1991)
Conlon, J.G., Solovej, J.P.: Upper bound on the free energy of the spin 1/2 heisenberg ferromagnet. Lett. Math. Phys. 23, 223–231 (1991)
Correggi, M., Giuliani, A.: The free energy of the quantum Heisenberg ferromagnet at large spin. J. Stat. Physics 149(2), 234–245 (2012)
Correggi, M., Giuliani, A., Seiringer, R.: Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet. Comm. Math. Phys. 339, 279–307 (2015)
Dyson, F.: General theory of spin-wave interactions. Phys. Rev. 102, 1217–1230 (1956)
Dyson, F.: Thermodynamic behavior of an ideal ferromagnet. Phys. Rev. 102, 1230–1244 (1956)
Fannes, M., Vanheuverzwijn, P., Verbeure, A.: Quantum energy-entropy inequalities: A new method for proving the absence of symmetry breaking. J. Math. Phys. 25, 76–78 (1984)
Fröhlich, J., Simon, B., Spencer, T.: Phase transitions and continuous symmetry breaking. Phys. Rev. Lett. 36, 804–806 (1976)
Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Comm. Math. Phys. 50(1), 79–95 (1976)
Fröhlich, J., Pfister, C.: On the absence of spontaneous symmetry breaking and of crystalline order in two-dimensional systems. Commun. Math. Phys. 81, 277 (1981)
Hewson, A.C., Ter Haar, D.: On the T 3-term in the low-temperature expansion for the magnetisation of a spin-1/2 Heisenberg ferromagnet. Phys. Lett. 6(2), 136–137 (1963)
Hofmann, C.P.: Spontaneous magnetization of the 0(3) ferromagnet at low temperatures. Phys. Rev B 65, 094430 (2002)
Hofmann, C.P.: Spontaneous magnetization of an ideal ferromagnet: beyond Dyson’s analysis. Phys. Rev. B 84, 064414 (2011)
Hohenberg, P.C.: Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383–386 (1967)
Holstein, T., Primakoff, H.: Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev. 58, 1098–1113 (1940)
Kramers, H.A.: Zur Theorie des Ferromagnetismus. Commun. Kamerlingh Onnes Lab. Univ. Leiden 22(Suppl. 83), 67,68,72,75–81 (1936)
Lieb, E.H.: The classical limit of quantum spin systems. Comm. Math. Phys. 31, 327–340 (1973)
Loly, P.D., Doniach, S.: Removal of an apparent discrepancy between calculations of dyson and of oguchi for the heisenberg ferromagnet. Phys. Rev. 144, 319–321 (1966)
Mermin, D., Wagner, H.: Absence of ferromagnetism or anti ferromagnetism in one- or twodimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133–1136 (1966)
Oguchi, T.: Theory of spin-wave interactions in ferro- and antiferromagnetism. Phys. Rev. 117, 117–123 (1960)
Oguchi, T.: The equivalence of Hamiltonians of Holstein-Primakoff and Dyson in spin-wave theory in ferromagnetism. Prog. Theor. Phys. 25, 721–722 (1961)
Opechowski, W.: ÜBer die temperaturabhängigkeit der Magnetisierung eines Ferromagnetikums bei tiefen Temperaturen. Physica 4(8), 715–722 (1937)
Schafroth, M.R.: Self-consistent spin-wave theory for the ferromagnetic exchange problem. Proc. Phys. Soc. London, Sect. A 67(1), 33 (1954)
Theorem, M.-W., Wagner, H., Schollwoeck, U.: Scholarpedia 5(10), 9927 (2010). doi:10.4249/scholarpedia.9927. http://www.scholarpedia.org/article/Mermin-Wagner_Theorem
Toth, B.: Improved lower bound on the thermodynamic pressure of the spin 1/2 Heisenberg ferromagnet. Lett. Math. Phys. 28, 75–84 (1993)
van Hemmen, J.L., Brito, A.A.S., Wreszinski, W.F.: Spin waves in quantum ferromagnets. J. Stat. Phys. 37(1), 187–213 (1984)
van Kranendonk, J.: Theory of the low-temperature properties of ferromagnetic crystals. Physica 21(1), 81–82 (1954)
Zittartz, J.: On the spin wave problem in the Heisenberg model of ferromagnetism. Zeitschrift für Physik 184(5), 506–520 (1965)
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Benedikter, N. Interaction Corrections to Spin-Wave Theory in the Large-S Limit of the Quantum Heisenberg Ferromagnet. Math Phys Anal Geom 20, 5 (2017). https://doi.org/10.1007/s11040-016-9237-6
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DOI: https://doi.org/10.1007/s11040-016-9237-6
Keywords
- Ferromagnetic Heisenberg model
- Quantum Heisenberg Ferromagnet
- Magnon
- Spin-wave theory
- Interaction correction
- Holstein-Primakoff transformation
- Holstein-Primakov transformation
- Perturbation theory of the Heisenberg model