Colloquium: Chaotic quantum dots with strongly correlated electrons

Quantum dots pose a problem where one must confront three obstacles: randomness, interactions, and finite size. Yet it is this confluence that allows one to make some theoretical advances by invoking three theoretical tools: random matrix theory, the renormalization group, and the $1∕N$ expansion. Here the reader is introduced to these techniques and shown how they may be combined to answer a set of questions pertaining to quantum dots.

Figure 1

A typical situation where leads bring in electrons which tunnel into and out of a dot (opposite to the current shown by arrows). A gate voltage $V_g$ is the control parameter and $V$ is the vanishingly small source-drain voltage difference.

Figure 2

A caricature of conductance $G$ vs gate voltage $V_g$. The peaks have varying heights, widths, and spacings.

Figure 3

The wheel-of-fortune (WOF) states within a band of energy $E_T$ concentric with the Fermi circle. There are roughly $g$ such states of mean momenta $\mathbf{k}$ centered on $g$ equally spaced points on the Fermi circle. The WOF states are obtained by chopping off plane waves of the desired mean momentum at the edges of the dot. These states are very nearly orthonormal.

Figure 4

The low-energy region for nonrelativistic fermions lies within the annulus concentric with the Fermi circle. It extends $\Lambda$ in momentum and $E_L$ in energy from the Fermi circle. In units where the Fermi velocity $v_F$ is chosen to be unity, the two are equal.

Figure 5

Kinematical representation why momenta are individually conserved up to a permutation.

Figure 6

Schematic of scattering amplitude in ${\varphi }^4$ theory. The solid dot is the full answer and the empty one is the coupling constant that goes in. The loop is the first of many that enter.

Figure 7

The flow of the coupling $u_m$ for any $m\ne 0$. The origin of this flow is the universal Hamiltonian where every $u_m=0$ except $u_0$ which can have any value. All points to the right of $u^{*}$ flow to this. We ask what happens if we begin to the left.

Figure 8

The phase diagram in the $u-\frac{1}{g}$ plane. The actual phase transition takes place on the line $1∕g=0$. However, it can be perceived at finite $g$ if one is within the $V$-shaped quantum critical regime. In typical cases, the $1∕g$ axis is replaced by the $T$ axis, where $T$ is the temperature. There a phase transition of quantum-mechanical origin at $T=0$ influences the quantum critical regime at finite $T$.