VOLUME 50 NUMBER 18 PHYSICAL REVIEW LETTERS 2MAY1983 bserve that the condition that the electrons lie TABLE I. Projection of variational three-body wave in the lowest Landau level is that f(z) be poly functionsψ in the manner伸mlm)/({钟nlψm)(4n nomial in z. The antisymmetry of p requires that xom)2.4m is the lowest-energy eigenstate of angu f be odd. Conservation of angular momentum re nomial of degree M, where M is the total angular ctronic potential of either 1//,-In(r), or exp(-r2/ quires that Il f(2j-zkbe a homogeneous poly- momentum. We have, therefore, f(2)=2, with m odd. To determine which m minimizes the en- m 1/y exp ergy, I write m2={<k(x1-2k)}exp(-∑1|z1|2)2 0.99673 0.99966 0.99468 0.99195 0.99939 0.99476 0.99981 0.99573 99437 99999 where B=1/m and is a classical poter ntial ener 11 0.99652 0.99542 0,99996 gy given by 0.99708 0.99985 ∑<2m21n|2-k|+mD1|21|2.(8) g describes a system of N identical particles of within a few percent by the ion disk energy charge Q=m, interacting via logarithmic poten tials and embedded in a uniform neutralizing r127a12dr2 background of charge density o=(2ao)". This =(4/3-1)2e2/R, is the classical one-component plasma(OCP),a where the integration domain is a disk of radius system which has been studied in great detai R=(om)-/2. At r=2 we have the exact result5 Monte Carlo calculations have indicated that the that g(r)=1-exp[-(r/R)I, giving Utot OCP is a hexagonal crystal when the dimension- R. At m=3 and m=5 I have reproduced the monte less plasma parameter r=2BQ2=2m is greater Carlo g(r)of Caillol et aL. 3 using the modified than 140 and a fluid otherwise. m 2 describes a hypernetted chain technique described by them. system uniformly expanded to a density of a I obtain Utot=(0.4156+0.0012)e2/a, and U tor (5) m"(2Tao 2). It minimizes the energy when om(-0 3340*0.0028 )e2/ao The corresponding values equals the charge density generating V. for the charge-density wave are -0 389e2/ao In Table I, I list the projection of fm for three -0322e2/an Utot is a smooth function of r. I particles onto the lowest-energy eigenstate of interpolate it crudely in the manner angular momentum 3m calculated numericall These are all nearly 1. This supports my asser Utot(m)0.814/0.230 (11) tion that a wave function of the form of Eq.(6)has adequate variational freedom. I have done a sim- This interpolation converges to the CDw energy ar calculation for four particles with Coulombic repulsions and find projections of 0. 979 and 0.947 not be determined from that of the ocp since the for the m=3 and m=5 states CDW has a lower energy than the crystal de w m has a total energy per particle which for scribed byψ for m>71 small m is more negative than that of a charge I generate the elementary excitations of *m by ity wave(CDw). It is given in terms of the piercing the fluid at zo with an infinitely thin ial distribution function g(r) of the ocp b solenoid and passing through it a flux quantum △φ=hc/ e adiabatically.. The effect of this opera Utot =fre[g(r)-1]rdr. tion on the single-body wave functions is (2-2)exp(-是|z12)-(z-20)m“exp(-划1212.(12) In the limit of large r, Utot is approximated Let us take as approximate representations of these excited states +磊0=Ag mxp(-An|212)江(x4-20)江<(2-2), …A(,江,-4列