Three identical quantum dots sit at the corners of an equilateral triangle

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations, Schur's lemma and characters

Problem

Three identical quantum dots sit at the corners of an equilateral triangle. Each dot carries one electronic orbital of on-site energy

\alpha

(in eV), and any two dots are coupled by an identical tunnelling matrix element

\beta

(in eV, real). In the basis of the three localised orbitals

|1\rangle,|2\rangle,|3\rangle

the single-particle Hamiltonian is

H=\begin{pmatrix}\alpha&\beta&\beta\\ \beta&\alpha&\beta\\ \beta&\beta&\alpha\end{pmatrix}.

The triangle's symmetry group is

C_{3v}

, represented on the three orbitals by the permutation matrices

D(g)

that relabel the dots exactly as

g

permutes the triangle's vertices (this is the same representation used for the three hydrogens of ammonia, decomposing as

A_1\oplus E

, with

A_1

spanned by

(1,1,1)

). ((a)) State and prove: if two diagonalisable matrices

A,B

on the same finite-dimensional space commute, they possess a common eigenbasis, even when an eigenvalue of

A

is degenerate. ((b)) Explain, without appealing to the averaging argument, why the permutation matrices

D(g)

already form a unitary representation of

C_{3v}

. ((c)) Show that

H

commutes with every

D(g)

. Using part (a) and the

A_1\oplus E

splitting, together with Schur's lemma (state and prove the part you need), explain why the two eigenvalues of

H

associated with the

E

subspace must be exactly equal for any real

\alpha,\beta

. Then find both eigenvalues of

H

(one non-degenerate, one doubly degenerate) for

\alpha=-6.0\,\mathrm{eV}

\beta=-2.5\,\mathrm{eV}