1/Nexpansion for two-dimensional quantum ferromagnets

Abstract

We present 1/N Schwinger boson and quantum Monte Carlo calculations of the magnetization and NMR relaxation rate for the two-dimensional ferromagnetic Heisenberg model representing a quantum Hall system at filling factor ν = 1. Comparing the analytic and numerical calculations, we find that the SU(N ) version of Schwinger boson theory gives accurate results for the magnetization at low temperatures, whereas the O(N ) model works well at higher temperatures.PACS numbers: 73.40.Hm,75.40.Mg,76.50.+g,75.30.Ds Two-dimensional quantum magnets have received particular attention in recent years because of advances in materials synthesis associated with high-temperature superconductivity, thin films and surfaces, and semiconductor quantum wells. It has recently come to be appreciated that two-dimensional electron gases in quantum wells subjected to strong magnetic fields in the quantum Hall regime are novel itinerant ferromagnets. The strong external magnetic field quenches the orbital kinetic energy but couples only very weakly to the spin degrees of freedom allowing low energy spin fluctuations to survive.Two-dimensional ferromagnets exhibit novel topological defects referred to as skyrmions by analogy with the corresponding objects in the Skyrme model in nuclear physics. What is unique about quantum Hall ferromagnets [1,2] is that these defects carry fermion charge and hence their ground state density can be controlled by moving the filling factor away from ν = 1. The combination of low energy spin fluctuations and these topological defects dramatically alters the NMR spectrum [3,2] and the specific heat [4].With the advent of NMR and magnetoabsorption measurements in quantum Hall systems, it is now possible to measure the temperature dependence of the electron magnetization. Recent theoretical work [5] has evaluated the magnetization of this quantum critical system at ν = 1 using SU(N ) and O(N ) formulations of mean field theory (with N = ∞). In this paper we present analytic results for the 1/N corrections to the magnetization and compare them with extensive quantum Monte Carlo simulations. We also present numerical results for the NMR relaxation rate 1/T 1 . In addition to their relevance to 2D quantum ferromagnets in general and QHE magnets in particular, these results provide new information on the level of accuracy of large N expansion methods and show a highly non-trivial difference in the behavior of the SU(N ) and O(N ) models.Low energy spin fluctuations in quantum Hall ferromagnets at ν = 1 are expected to be well described by the Heisenberg model [5]. The large N approach to this model is a systematic expansion around a mean field theory for N = ∞ and has the advantage of being equally valid at all temperatures T . Furthermore, even at the mean field level this approach correctly captures the fact that arbitrarily small thermal fluctuations destroy the long range order in two dimensions. An alternative microscopic approach which includes spin-wave corrections to the electronic self-energy has also recen...

Monte Carlo study of a two-dimensional quantum ferromagnet

“…We notice a fairly good agreement between the analytic and numerical results and, as in the case for the magnetization, 8 time the numerical results do not lie between the O(N ) and SU(N ) solutions, which was the case for the magnetization. At zero temperature the relaxation rate is zero, since no spins flip in the ordered ferromagnet and therefore the nuclear spins cannot relax.…”

“…Some results have previously been compared to measurements of the ν = 1 quantum Hall state and analytical Schwinger Boson calculations. 8 The high relative accuracy (10 −4 ) of the Monte Carlo results made it possible to make statements about the relative merits of the SU(N ) and O(N ) solutions. We have, in addition, calculated the spin-lattice relaxation rate 1/T 1 using a maximum entropy method, and compared also these results with the Schwinger boson results.…”

“…For a more comprehensive discussion of the Schwinger boson results and the experimental data, we refer to our previous paper. 8 …”

“…[3][4][5][6] The system at filling factor ν = 1 should be well described by a two-dimensional Heisenberg ferromagnet 7 , and in a recent publication we presented analytical Schwinger boson and numerical Monte Carlo results for this model. 8 The ferromagnetic Heisenberg model has been much less studied than the antiferromagnetic model, probably because the ground state and lowest excitations are known exactly. At finite temperatures there is, however, no exact analytic solution, and in our first paper we gave a detailed discussion of the calculation of the leading 1/N correction to the Schwinger boson mean-field theory.…”

Theory for the reduction of products of spin operators

“…For instance, the Holstein-Primakoff approach has been applied frequently [13,14], which is valid for low tem-2 peratures. On the other hand, the Green's function method [15][16][17] and the Schwinger-Boson approximations [18,19] can be used also at higher temperatures, since they consider partly interactions between spin-waves. The Green's function method is a hierachical approach, i.e.…”