The entanglement entropy of the ν=1/3 and ν=5/2 quantum Hall states in the presence of short-range random disorder has been calculated using direct diagonalization. A microscopic model of electron–electron interaction is used and spin-polarized electrons are confined to a single Landau level and interact with long-range Coulomb interaction. In the case of very weak disorder, the values of topological entanglement entropy are roughly consistent with the expected theoretical results. By considering a broad range of disorder strengths, entanglement entropy was studied in an effort to detect quantum phase transitions. In particular, there is a signature of the transition as the function of the disorder strength for the ν=5/2 state. Prospects of using the density matrix renormalization group to compute the entanglement entropy for larger system sizes are discussed.
The entanglement entropy of
ν=1/2 and
ν=9/2 quantum Hall states in the presence of short range disorder has been calculated by direct diagonalization. Spin polarized electrons are confined to a single Landau level and interact with long range Coulomb interaction. For
ν=1/2 the entanglement entropy is a smooth monotonic function of disorder strength. For
ν=9/2 the entanglement entropy is non monotonic suggestive of a solid-liquid phase transition. As a model of the transition at
ν=1/2 free fermions with disorder in 2 dimensions were studied. Numerical evidence suggests the entanglement entropy scales as
L rather than the
LlnL as in the disorder free case.
We present a detailed study of the ground-state entanglement in disordered fractional quantum Hall liquids. We consider electrons at various filling fractions
f in the lowest Landau level, with Coulomb interactions. At
f=1/3,1/5, and
2/5 where an incompressible ground-state manifold exists at zero disorder, we observe a pronounced minimum in the derivative of entanglement entropy with respect to disorder. At each filling, the position of this minimum is stable against increasing system size, but its magnitude grows monotonically and appears to diverge in the thermodynamic limit. We consider this behavior of the entropy derivative as a compelling signal of the expected disorder-driven phase transition from a topological fractional quantum Hall phase to an insulating phase. On the contrary, at
f=1/2 where a compressible composite fermion sea is present at zero disorder, the entropy derivative exhibits much greater, almost chaotic, finite-size effects, without a clear phase transition signal for system sizes within our exact diagonalization limit. However, the dependence of entanglement entropy with system size changes with increasing disorder, consistent with the expectation of a phase transition from a composite fermion sea to an insulator. Finally, we consider
f=1/7 where compressible Wigner crystals are quite competitive at zero disorder, and analyze the level statistics of entanglement spectrum at
f=1/3.
The fractional quantum Hall (FQH) effect at the filling number ν=5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. We calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν=1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian-anti-Pfaffian puddle state.
We study the behavior of two-dimensional electron gas in the fractional quantum Hall regime in the presence of finite layer thickness and correlated disordered potential. Generalizing the Chern number calculation to many-body systems, we determine the mobility gaps of fractional quantum Hall states based on the distribution of Chern numbers in a microscopic model. We find excellent agreement between experimentally measured activation gaps and our calculated mobility gaps, when combining the effects of both disordered potential and layer thickness. We clarify the difference between mobility gap and spectral gap of fractional quantum Hall states and explain the disorder-driven collapse of the gap and the subsequent transitions from the fractional quantum Hall states to the insulator.
We investigate the disorder-driven phase transitions in bosonic fractional quantum Hall liquids at filling factors
f=1/2 and
f=1 in the lowest Landau level. We use the evolution of ground-state entanglement entropy, fidelity susceptibility, and Hall conductance with increasing disorder strength to identify the underlying phase transitions. The critical disorder strengths obtained from these different quantities are consistent with each other, validating the reliability of our numerical calculations based on exact diagonalization. At
f=1/2, we observe a clear transition from the bosonic Laughlin state to a trivial insulating phase. At
f=1, we identify a direct phase transition from the non-Abelian bosonic Moore-Read state to a trivial insulating phase, although some signs of a disorder-induced intermediate fractional quantum Hall phase were recently reported for the
f=5/2 fermionic cousin.