Quantum and classical phase transitions in double-layer quantum Hall ferromagnets

We consider the problem of quantum and classical phase transitions in double-layer quantum Hall systems at ν = 1/m (m odd integers) from a long-wavelength statistical mechanics viewpoint. We derive an explicit mapping of the long-wavelength Lagrangian of the quantum Hall system into that of a three-dimensional isotropic classical XY model whose coupling constant depends on the quantum fluctuation in the original quantum Hall Hamiltonian. Universal properties of the quantum phase transition at the critical layer separation are completely determined by this mapping. The dependence of the Kosterlitz-Thouless transition temperature on layer separation, including quantum fluctuation effects, is approximately obtained by simple finite-size scaling analyses. 73.40.Hm, 73.20.Dx, 75.30.Kz Typeset using REVTEX 1 Low-dimensional electron systems exhibit a richer variety of physical properties than their higher-dimensional counterparts due to enhanced interaction effects. For a two-dimensional electron gas in a perpendicular magnetic field, the interaction effects are especially important because of Landau level quantization. When electrons are entirely restricted to the lowest Landau level by a strong magnetic field, electron-electron interaction completely dominates the properties of the system as the electron kinetic energy is quenched to an unimportant constant. One of the most interesting phenomena in these strongly correlated electron systems is the quantum Hall effect (QHE), which has attracted a great deal of experimental and theoretical interest. In recent years, a lot of attention has been directed to quantum Hall systems in double-layer structures where electrons are confined to two parallel planes separated by a distance comparable to the in-plane inter-electron distance. With the introduction of this layer degree of freedom, many qualitatively new effects due entirely to interlayer electron correlations appear.2–9 These new features include QHE phases with various spontaneously-broken symmetries, such as the interlayer coherent state at ν = 1/m (m odd integers) and the canted antiferromagnetic state at ν = 2, where interesting phase transitions both at zero and finite temperatures may occur. Thus, multi-component quantum Hall systems provide a suitable platform for studying various quantum phase transitions and their crossover behaviors. In this paper, we consider the quantum phase transition at d = dc and the Kosterlitz-Thouless transition at d < dc in double-layer systems at ν = 1/m, where d is the layer separation. Our consideration is based on an explicit mapping of the long-wavelength Lagrangian of the quantum Hall system into that of a three-dimensional (3D) isotropic classical XY model whose coupling constant g depends on the quantum fluctuation terms of the original Hamiltonian. The mapping shows unambiguously that the quantum phase transition at dc is in the same universality class as that of a 3D XY -model transition at its critical coupling constant gc. The dependence of the Kosterlitz-Thouless transition temperature on layer separation is approximately obtained by a straightforward finite-size scaling analysis around the quantum critical point. In this way, both quantum and classical phase transitions in this problem are described in terms of the known properties of 2 a 3D classical XY model. To be specific, we restrict ourselves to ν = 1 (i.e. m = 1), where various energy scales can be determined in the Hartree-Fock approximation. Our results, however, apply qualitatively to the general case of ν = 1/m with m an odd integer. There has been a lot of work on the ν = 1 quantum Hall system.2–4,8,9 At large layer separations, where the interlayer Coulomb interaction is negligible, the double-layer system is effectively a pair of decoupled half-filled single layers which exhibit no QHE. At small layer separations, the interlayer Coulomb interaction is almost as important as the intralayer interaction. All electrons are in the symmetric state where interlayer and intralayer electron correlations are treated on an equal footing. At ν = 1, the electrons form a filled band and exhibit the QHE. The QHE phase at small d and the non-QHE phase at large d are separated by a continuous transition at d = dc. For convenience, we introduce a pseudospin variable S to describe the layer degree of freedom, where Sz = ±1/2 represent electron occupation of the right or left layers, respectively, and Sx = ±1/2 represent electron occupation of the symmetric or antisymmetric subbands, respectively. The transition between the QHE and non-QHE phases at d = dc may be viewed as a magnetization transition: The non-QHE phase at d > dc corresponds to a pseudospin disordered phase and the QHE phase at d < dc corresponds to a pseudospin magnetization in x̂-direction. (Note that we assume that physical spins of the electrons are completely polarized by the applied magnetic field and are not relevant variables at all.) Even though, there has been a lot of work on the ν = 1 system, most theoretical efforts, however, were directed towards the understanding of the properties in the QHE phase. These studies usually ignore quantum fluctuations, and hence shed no light on the nature of the quantum phase transition at dc. In fact, many quantities obtained in this way, such as the pseudospin stiffness, the magnetization, and the susceptibility, do not show any sign of a phase transition at all. The present work is concerned solely with the phase transition, so it is essential that we include quantum fluctuations. Our goal is accomplished by a mapping of the low energy physics of the ν = 1 system into that of a 3D classical XY model. Although, the basic ideas involved here have largely been known 3 in the literature, we think it is still very useful to explicitly carry out the derivations and put these ideas into a concrete basis so that more sophisticated calculations may start from here. For the purpose of discussing the spontaneous pseudospin magnetization, we prohibit interlayer tunneling. (The tunneling acts like a Zeeman term in the pseudospin space.) This is not an unreasonable restriction, as the interlayer tunneling can be made very small in real semiconductor samples. The Hamiltonian of the ν = 1 double-layer quantum Hall system is