Low‐Temperature Properties of a Heisenberg Antiferromagnet

It is shown how the propagator formalism can be used to obtain the low‐temperature expansion of the free energy of an isotropic Heisenberg antiferromagnet. The lowest‐order terms in such an expansion can be calculated using the proper self‐energy evaluated at zero temperature. The analytic properties of this quantity are investigated by expressing it in terms of time ordered diagrams. The low‐temperature expansion of the free energy is shown to be of the form AT 4+BT 4+CT 8, where A, B, and C are given by Oguchi correctly to order 1/S. For spin 1⁄2 the term in 1/S 2 gives a 2% reduction in A for a body‐centered lattice. Disciplines Physics | Quantum Physics Comments At the time of publication, author A. Brooks Harris was affiliated with Duke University. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania. This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/431 JOURNAL OF APPLIED PHYSICS VOLUME 35, NO.3 (TWO PARTS-PART 2) MARCH 1964 Low-Temperature Properties of a Heisenberg Antiferromagnet* A. BROOKS HARRIS Duke University, Durham, North Carolina It is shown how the propagator formalism can be used to obtain the low-temperature expansion of the free energy of an isotropic Heisenberg antiferromagnet. The lowest-order terms in such an expansion can be calculated using the proper self-energy evaluated at zero temperature. The analytic properties of this quantity are investigated by expressing it in terms of time ordered diagrams. The low-temperature expansion of the free energy is shown to be of the form AT4+BT4+Crs, where A, B, and C are given by Oguchi correctly to order 1/ S. For spin! the term in 1/ S2 gives a 2% reduction in A for a body-centered lattice. THE low-temperature properties of magnetic systems governed by a Heisenberg Hamiltonian have been intensively studied. For the ferromagnet, Dyson' has shown how to obtain the virial series by systematic use of perturbation theory. More recently attempts have been made to reproduce his results using Green's function methods. These calculations have not been altogether satisfactory because: (a) there is some ambiguity in the decoupling procedure, and (b) it is difficult to obtain the low-temperature expansion of the free energy by isolating terms with a given temperature dependence. For ferrimagnets and antiferromagnets the progress has been less substantial, as Dyson's method of calculation does not seem feasible in this case. Thus no systematic application of manybody perturbation theory has yet been attempted. However, the propagator formalism of Luttinger and Ward2 is well suited to this problem. By using such a formalism one performs a partial summation of the perturbation series. Also the low-temperature expansion of the free energy can be obtained conveniently. In addition, long wavelength divergences can be explicitly avoided. Finally, the use of diagrammatic techniques allows one to examine terms of arbitrarily high order in the perturbation. Our treatment is incomplete in that we assume that the kinematic interaction can be neglected and that the perturbation series is a useful one. We treat the case of an isotropic body-centered cubic Heisenberg antiferromagnet whose Hamiltonian is Je=2JEsi,Sj (1) ( iJ/ in the usual notation. Using Oguchi's transformation3 one can express this in terms of the boson operators ak and bk as Je= E+Jeo+ V, where E= -JVzJ S2 and, Jeo= 16J SEak+ak+bk+bk+'Ykak+b_k++'Ykakb_k, V = -16 J N-' E[2'Yk-k,ak +ak,bk" + bk-k'+k" (2) +l'k,ak +bk,ak"ak-k'-k"+l'kak+bk,+bk,,+bk+k'+k" J, (3) * Supported by a contract with the U.S. Office of Naval Research. 1 F. J. Dyson, Phys. Rev. 102, 1217 (1956). 2 J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 (1960); J. M. Luttinger, Phys. Rev. 119, 1153 (1960); 121, 942 (1961). 3 T. Oguchi, Progr. Theoret. Phys. (Kyoto) 22, 721 (1961). where 'Yk= cos (kxa/2) cos(kya/2) cos (k.a/2) . To diagonalize Jeo one introduces boson operators Ck and dk which are given as d_k= -qkak++pkb_k, (4) qk2= [(1-'Yk2)-L 1J/2. (5) In this cd representation Jeo takes the simple form ek=e(k) = 16J S(1-'Yk2)!.