Application of quantum Monte-Carlo method for fractional quantum Hall systems

Sei Suzuki, Tatsuya Nakajima

Ting-Tung Wang, Ha Quang Trung, Qianhui Xu +3 more

We develop a hybrid Monte Carlo (HMC) method to efficiently compute the physical observables from the samplings of the Laughlin and the Moore-Read (MR) wave functions of fractional quantum Hall (FQH) systems. With the advancements in methodology, including global updates and double stereographic projection on spherical geometry, our HMC simulation is significantly faster than the widely used Metropolis Monte Carlo scheme. As a result, we can readily simulate systems with electron numbersN>1000on both disk and sphere geometries. We apply this method to investigate the topological shift obtained from the edge dipole moment, computed from the wave function density on the disk. We also numerically computed the non-Abelian braiding matrices for different braiding schemes of the MR quasiholes on the sphere. Results with much better quality compared with previous works have been achieved. With the thermodynamic limit results obtained at ease, we also discuss the future usage of our method to clarify the questions on the instability of FQH states in an ideal Chern band setting or under quantum decoherence.

D. Yoshioka

Jie Wang, Scott Geraedts, Edward H. Rezayi +1 more

The Monte Carlo method is very useful for studying various model states proposed for the fractional quantum Hall systems. In this paper, we introduce a lattice Monte Carlo method based on an exact lattice formalism for quantum Hall problems defined on a torus. This ``lattice representation'' is applied to study the many body Berry phase of the composite-Fermi liquid phase in a half-filled Landau level. The Monte Carlo result agrees with the exact numerical result found on small systems sizes and is consistent with the prediction from the Dirac fermion low-energy effective theory. Many other aspects of other model states, including the structure factor, Coulomb energy and particle-hole symmetry, are studied and discussed.

John J. Quinn, Arkadiusz Wójs, Jennifer J. Quinn +1 more

Stephanie Jane Huntington

The microscopic picture for fractional quantum Hall effect (FQHE) is difficult to work with analytically for a large number of electrons. Therefore to make predictions and attempt to describe experimental measurements on quantum Hall systems, effective theories are usually employed such as the chiral Luttinger liquid system. In this thesis the Monte Carlo method is used for Laughlin-type quantum Hall systems to compute microscopic observables. In particular such computations are carried out for the large system size expansion of the free energy. This work was motivated by some disagreement in the literature about the form of the free energy expansion and is still an ongoing project. Tunnelling in the FQHE is an interesting problem since the tunnelling operators are derived from an effective theory which has not yet been checked microscopically. To perform a test for the effective tunnelling Hamiltonian, microscopic calculations were performed numerically for charges tunnelling across the bulk states of a FQH device. To compute these matrix elements, two methods were found to overcome a phase problem encountered in the Monte Carlo simulations. The Monte Carlo results were compared to the matrix elements predicted by the effective tunnelling Hamiltonian and there was a good match between the data. Performing this comparison enabled the operator ordering in the effective tunnelling Hamiltonian to be deduced and the data also showed that the quasiparticle tunnelling processes were more relevant than the electron tunnelling processes for all system sizes, supporting the idea that when tunnelling is considered at a weak barrier, the electron tunnelling process can be neglected.