Correlation Diagrams: An Intuitive Approach to Interactions in Quantum Hall Systems

Abstract

In this chapter, we study correlations resulting from Coulomb interactions in fractional quantum Hall systems. Our objective is to use correlation diagrams to gain new insights into correlations in strongly interacting many-body systems. We introduce correlation diagrams to guide in the selection of the correlation functions correlation function caused by interactions. Electrons are represented by points located at positions $z_{i}$ in the complex plane, and there are correlation lines connecting pairs of electrons. A correlation line connecting particles i and j represents a correlation factor (cf) $z_{ij}=z_{i}-z_{j}$ .

Appendices

Problems

17.1

Consider a system of N fermions and prove an identity given by

$$ {\hat{L}}^2+N(N-2)\hat{l}^2-\sum _{\left\langle i, j\right\rangle }\hat{L}_{ij}^2=0. $$

Here $\hat{L}$ is the total angular momentum operator, $\hat{L}_{ij}=\hat{l}_i+\hat{l}_j$, and the sum is over all pairs. Hint: One can write out the definitions of ${\hat{L}}^2$ and $\sum _{\left\langle i, j\right\rangle }{\hat{L}_{ij}}^2$ and eliminate $\hat{l}_i \cdot \hat{l}_j$ from the pair of equations.

17.2

Demonstrate that the expectation value of square of the pair angular momentum $L_{ij}$ summed over all pairs is totally independent of the multiplicity $\alpha$ and depends only on the total angular momentum L.

17.3

Derive the two sum rules involving $\mathcal {P}_{L\alpha }(L_{12})$, i.e., the probability that the multiplet $|l^N;N\alpha \rangle$ contains pairs having pair angular momentum $L_{12}$:

$$ \frac{1}{2}N(N-1)\sum _{L_{12}} L_{12}(L_{12}+1)\mathcal {P}_{L\alpha }(L_{12}) = L(L+1)+N(N-2)l(l+1) $$

and

$$ \sum _{L_{12}} \mathcal {P}_{L\alpha }(L_{12}) = 1. $$

17.4

Show that the energy of the multiplet $|l^N;L\alpha \rangle$ is given, for harmonic pseudopotential $V_\mathrm{H}(L_{12})$, by

$$ E_\alpha (L) = N\left[ \frac{1}{2}(N-1)A+B(N-2)l(l+1)\right] +BL(L+1). $$

Summary

Here we study correlations resulting from Coulomb interactions in fractional quantum Hall systems, and correlation diagrams are introduced to guide in the selection of the correlation functions caused by interactions.

It is established that a harmonic pseudopotential does not cause correlations (i.e. it does not lift the degeneracy of different multiplets with the same value of the total angular momentum L). The pseudopotential $V_{n}(L_{12})$ for electrons in a partially filled $n{\text {th}}$ Landau level (LLn) is evaluated and the interaction energies of quasielectrons and of quasiholes are determined. The use of partitions and permutation symmetry is introduced to construct correlation diagrams and correlation functions causing the most important correlations. Comparison of the energy spectra of an (N, 2l) electron system with the corresponding $(N_{\text {QE}}, 2l_{\text {QE}})$ quasielectron system shows that the latter system gives a very reasonable approximation to the low energy states of the former. Because of the form $V_{1}(L_{12})$ for the LL0 there can be no Laughlin correlated states for $2/5>\nu >1/3$, and that states like $\nu =4/11$ must involve pairing of the electrons and a much weaker interaction between these pairs.

Conditions on the correlation function $\mathcal {G}\{z_{ij}\}$ are imposed in terms of the number of pairs $n_{j}$ in the correlation diagram containing j correlation factors, which greatly limit the allowed choices of the total number $K_{\mathcal {G}}$ of correlation factors in $\mathcal {G}\{z_{ij}\}$. In the numerical diagonalizations, the intuitive model wave function is in reasonable qualitative agreement with numerical experiment. This demonstrates that the novel intuitive approach to fermion correlations does give new insight into understanding many fermion interactions in fractional quantum Hall effect – the paradigm for strongly interacting systems.