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Register a new userFractional quantum Hall effect in the Hofstadter model of interacting fermions
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Authors
Igor N.Karnaukhov
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Applying a unified approach, we study integer quantum Hall effect (IQHE) and fractional quantum Hall effect (FQHE) in the Hofstadter model with short range interaction between fermions. An effective field, that takes into account the interaction, is determined by both the amplitude and phase. Its amplitude is proportional to the interaction strength, the phase corresponds to the minimum energy. In fact the problem is reduced to the Harper equation with two different scales: the first is a magnetic scale (cell size corresponding to a unit quantum magnetic flux), the second scale (determines the inhomogeneity of the effective field) forms the steady fine structure of the Hofstadter spectrum and leads to the realization of fractional quantum Hall states. In a sample of finite sizes with open boundary conditions, the fine structure of the Hofstadter spectrum also includes the fine structure of the edge chiral modes. The subbands in a fine structure of the Hofstadter band (HB) are separated extremely small quasigaps. The Chern number of a topological HB is conserved during the formation of its fine structure. Edge modes are formed into HB, they connect the nearest-neighbor subbands and determine the fractional conductance for the fractional filling at the Fermi energies corresponding to these quasigaps.
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We study how the stability of the fractional quantum Hall effect (FQHE) is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria have a dominant effect on the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.
The nature of the fractional quantum Hall effect at $ u=1/2$ observed in wide quantum wells almost three decades ago is still under debate. Previous studies have investigated it by the variational Monte Carlo method, which makes the assumption that the transverse wave function and the gap between the symmetric and antisymmetric subbands obtained in a local density approximation at zero magnetic field remain valid even at high perpendicular magnetic fields; this method also ignores the effect of Landau level mixing. We develop in this work a three-dimensional fixed phase Monte Carlo method, which gives, in a single framework, the total energies of various candidate states in a finite width quantum well, including Landau level mixing, directly in a large magnetic field. This method can be applied to one-component states, as well two-component states in the limit where the symmetric and antisymmetric bands are nearly degenerate. Our three-dimensional fixed-phase diffusion Monte Carlo calculations suggest that the observed 1/2 fractional quantum Hall state in wide quantum wells is likely to be the one-component Pfaffian state supporting non-Abelian excitations. We hope that this will motivate further experimental studies of this state.
We construct model wavefunctions for the collective modes of fractional quantum Hall systems. The wavefunctions are expressed in terms of symmetric polynomials characterized by a root partition and a squeezed basis, and show excellent agreement with exact diagonalization results for finite systems. In the long wavelength limit, the model wavefunctions reduce to those predicted by the single-mode approximation, and remain accurate at energies above the continuum of roton pairs.
A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M>N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1(2m+1). Within this model the complimentary wavefunction for v = 1-1/(2m + 1) is found explicitly and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.
At small momenta, the Girvin-MacDonald-Platzman (GMP) mode in the fractional quantum Hall (FQH) effect can be identified with gapped nematic fluctuations in the isotropic FQH liquid. This correspondence would be exact as the GMP mode softens upon approach to the putative point of a quantum phase transition to a FQH nematic. Motivated by these considerations as well as by suggestive evidence of an FQH nematic in tilted field experiments, we have sought evidence of such a nematic FQHE in a microscopic model of interacting electrons in the lowest Landau level at filling factor 1/3. Using a family of anisotropic Laughlin states as trial wave functions, we find a continuous quantum phase transition between the isotropic Laughlin liquid and the FQH nematic. Results of numerical exact diagonalization also suggest that rotational symmetry is spontaneously broken, and that the phase diagram of the model contains both a nematic and a stripe phase.
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