Identifying Particle Correlations in Quantum Hall Regime

Abstract

We introduce a method that allows the disclosure of correlations between particle positions in an arbitrary many-body system. The method is based on a well-known simulated annealing algorithm and the proposed artificial distribution technique. Additionally, we investigate correlations in quantum Hall liquids (we consider many-body wave functions that have been recently determined via the cyclotron subgroup model) and present three-dimensional plots of configuration probability distributions that have been established from numerical simulations. We demonstrate that the preferred simultaneous positions of particles (configurations of positions, which correspond to large values of a system's probability distribution, | N | 2 ) tend to form complicated geometric structures, which are equivalent to classical Wigner crystals only for Laughlin states. Furthermore, we claim that quantum Hall liquids attributed to non-Laughlin fillings are correlated on subdomains rather than on a whole particle domain (due to a quantizing magnetic field, which modifies the topology of a system's dynamics). Finally, we characterize Hall-like internal orders in terms of statistical correlations (one-dimensional unitary representations of cyclotron subgroups). Our conclusions concerning the stability of many-body states agree with transport measurements and various numerical studies.

Many-body wave functions for correlated systems in magnetic fields: Monte Carlo simulations in the lowest Landau level

“…Exemplary snapshots are presented in figure 8 (parameters of Monte Carlo calculations are included in table 3). It is worth mentioning that typical configurations of point particles / wave function arguments have already been investigated for CF and cyclotron-subgroup states [56]. Furthermore, interesting geometric structures, which resemble x shifted classical Wigner crystals comprising xth neighbours, were discovered.…”

“…Typical configurations of point particles (snapshots taken during Monte Carlo simulations after the thermalization is obtained) can be implemented to characterize many-body wave functions. Typical configurations of point particles / wave function arguments have crystal-like character for Laughlin's liquids [56]. This, together with the oscillatory behavior of pair distribution functions, is a manifestation of a solid-like local order.…”

“…Simultaneously, we postulate that the shoulder appears not due to the formation of quasi-particle clusters, but due to the division of a wave function into Jastrow factors with different exponents corresponding to different sets of N a ζ neighbours/arguments. This shoulder is a direct consequence of an additional characteristic distance between neighbouring elements of the most probable configuration of arguments; in the global maximum of |Ψ N | 2 [56].…”

Topological Classification of Correlations in 2D Electron Systems in Magnetic or Berry Fields

“…In the gas case, the finite averaged separation of electrons (i.e., the mean separation of electron densities acc. to the wave function ( A11 )) is caused by fermionic ’repulsion’ and can be called Pauli virtual crystallization in a Fermi gas [ 42 ], although no other homotopic class exists in the gaseous system. In the gas, any commensurability cannot hold because noninteracting particles can be arbitrarily distributed.…”

Homotopy quantum phase transitions

“…3 (right) are the same for interacting and noninteracting systems because the two-particle wave function has the same form in both systems. In the gas case, the finite separation of electrons is caused by fermionic 'repulsion' and is called Pauli virtual crystallization 45 , although no other homotopic class exists in the gaseous system. In gas any commensurability does not hold and does not impose any restrictions on a full braid group.…”