The Vandermonde polynomial / determinant |V| popped up in some operator algebra I was exploring along with the $n$-simplices, and as I was trying to come to a better understanding of these little fellows, I stumbled across some relations to Physics (as usual).
I) Coulomb interaction energy:
In "Notes on Fekete points on complex manifolds, " Julien Meyer states that minimizing the interaction energy for a system of equally charged particles in a plane is equivalent to maximizing the magnitude of |V| constructed from their mutual positions.
On page 10 of Zabrocki's "Matrix models and and growth processes: from viscous flow to the quantum hall effect," the log of |V| appears again in the Coloumb interaction energy of a Dyson gas.
II) The Fadeev-Popov determinant in quantum field theory:
Ryan Thorngren in the MathOverflow question "Wonderful applications of the Vandermonde determinant" describes the connection to |V|.
III) Wave functions:
Sabine Jansen in "Fermionic and bosonic Laughlin state on thick cylinders" presents a many-body wave function for particles on a cylinder that is the power of |V| times a Gaussian as a model in the fractional Hall effect. The use of the |V| in wave function expressions (and random matrix theory) seems to be closely related to |V|'s basic invariance under substitution of orthogonal polynomials in the standard definition of the Vandermonde matrix.
IV) Partition functions:
Powers of |V| occur in the Mehta integral, the partition function for a gas of point charges moving on a line and attracted to the origin. (Cf. also "Beta integrals" by Warnaar.)
(Edit 9/2015) See also The importance of the Selberg integral by Forrester and Warnaar and the Wronskian determinant associated with the Euler beta function integral and, therefore, the Veneziano amplitude in Dual N-point functions in PGL(N-2,C)-invariant formalism by Hanson (page 5).
Where else does the Vandermonde determinant appear in Physics?