GROUND - STATE ENERGY OF CHARGED ANYON GASES

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1 GROUD - STATE EERGY OF CHARGED AYO GASES B. Abdullaev, Institute of Applied Physics, ational University of Uzbekistan APUAG FU Berlin 1

2 Contents Interacting anyons in D harmonic potential in the presence of external magnetic field: general setup Anyons: many-body Aharonov - Bohm effect on-interacting anyons in D harmonic well Coulomb-interacting anyons in D harmonic well on-interacting anyons in D harmonic well and magnetic field Coulomb-interacting anyons in D harmonic well and magnetic field Infinite Coulomb anyons gas Ground state energy of infinite Coulomb anyon gas Explicit derivation of ground state energy formulas by taking into account short range correlations in wave function Do anyons and fermions exist in the ground state of D in concept of anyons? Conclusion APUAG FU Berlin

3 Interacting anyons in D harmonic potential in the presence of external magnetic field: general setup (B. Abdullaev, et al., Phys. Rev. B 68, (003)) Hamiltonian H = 1 M k=1 p k A ν r k + ea ext r k /c + M ω 0 r k + 1 k,j k e r kj. Anyon vector potential A ν r k = ħν j k e z r kj r kj ; Magnetic field A ext r k = H r k Minimization of energy E = ψ R Hψ R dr ψ (R)ψ R dr with trail wave function ψ R = α π / k=1 exp α x k + y k Typically, E = ReE + iime, however, for Gaussian ψ R, one has ImE = APUAG FU Berlin 3

4 Anyons: many-body Aharonov - Bohm effect Inside tube H 0, outside H = 0 and A = ф/π φ, where ф is magnetic flux. If ф 0 = πħc/ e is elementary magnetic flux, then ψ r ψ r e iνφ ; ν = ф/ф 0 ;.. ψ r 1, r, ψ r 1, r, e iν φ ; e iν φ = z i z j ν i j ; A ν r k = ħν z i z j ν p i +. A ν r i j k e z r kj r kj ; for z = x + iy. Thus Schrödinger equation is: 1 M i=1 Ф r 1, r, = =EФ r 1, r, for bosonic representation of Ф r 1, r, APUAG FU Berlin 4

5 on-interacting anyons in D harmonic well Hamiltonian H = 1 M k=1 p k A ν r k + M ω 0 r k. Energy before minimization is E ħω 0 = α + 1 α, where = 1 + ν 1 ln 1 δ 31/ ln 4. When the 3 nearest distance between anyons δ 0 then. Origin of this divergence is three particle interaction term r kj r kl ψ R r kj r kl ψ R dr for k j, k l, j l. Minimization de dα = 0 gives α 0 = 1/, thus ground state energy is E 0 = 1/. Known from literature at ν 0 limit energy is E 0l + 1 ν/. Thus fitting at ν 0 E 0 to E 0l (regularization!) one obtains expression = 1 + ν 1 and expression for δ. Hence, ground state energy of non-interacting anyons in D harmonic well is E ħω 0 = 1 + ν 1 1/ APUAG FU Berlin 5

6 Coulomb-interacting anyons in D harmonic well Hamiltonian of system H = 1 M k=1 p k A ν r k + M ω 0 r k + 1 k,j k e r kj. Expression for energy is E = α ħω 0 α Mα1/ with M = π 1/ r0 1 and = 1 + ν 1. a B Minimization de = 0 gives equation dα X4 MX = 0 for X = 1/α 1/ with solution: X 0 = A + B 1/ + A + B + A AB + B 1/ 1/, where A = B = M M M M 18 1/ 1/3 1/ 1/3 and E 0 ħω 0 = X 0 + X 0 + M X 0. APUAG FU Berlin

7 on-interacting anyons in D harmonic well and magnetic field Hamiltonian H = 1 M Fock-Darwin spectrum E nl H. k=1 Single electron Fock Darwin spectrum E nl = P n + l lr, where n and l are radial and angular quantum numbers, P = ħ p k A ν r k + ea ext r k /c + M ω 0 r k. ω 0 + ω c / 1/, R = ħω c /, ω c = e H/mc with magnetic field H = H. Filling these states by electrons one obtains ground state energy for lowest Landau levels at ω c ω 0 / 1 1/ : E = P + 1 R 1. Calculation for ground state energy for anyons gives: E 0 = P 1/ νr 1. For ω c it should be E 0 R for fermions ν = 1 and bosons ν = 0. For arbitrary large ω c E 0 P for bosons. Thus 1/ = 1 + ν APUAG FU Berlin 7

8 Coulomb-interacting anyons in D harmonic well and magnetic field Hamiltonian H = 1 M k=1 p k A ν r k + ea ext r k /c + M ω 0 r k + 1 k,j k e r kj. Calculated energy is E 0 ħω 0 = X ω c ω 0 Expression for X 0 is the same but replacing 1 + X 0 νω c ω M X 0. ω c ω 0 1 and M M 1 + ω c ω APUAG FU Berlin 8

9 Infinite Coulomb anyons gas Hamiltonian H = 1 M k=1 p k + A ν +M ω 0 r k + 1 k,j k e r kj + V r k with V r k = ρ e d r. r k r Let us consider no interacting case and ν = 1 then at E 0 = ħω 0 3/ since E 0 = ħω 0 1/ and = 1 + ν 1. Ground state energy of D electron gas with no interaction is E 0eg = ħ /m r 0, where r 0 is mean distance between electrons. From E 0 = E 0eg one gets ħω 0 = ħ / m r 0 1/ (harmonic potential regularization with vanishing confinement at!). From E 0 = + X ħω 0 X 0 + M 0 X 0 units, where a B is Bohr radius): one obtains energy per particle (in Rydberg Ry = e /a B E 0 = f ν, r s r s ν K + K X X K K X K X = K A + K B 1/ + K A + K B + K A K A K B + K B 1/ 1/ APUAG FU Berlin 9

10 Ground state energy of infinite Coulomb anyon gas K A = K + ν K 18 1/ 1/3 K B = K ν K 18 1/ 1/3, where one used = ν, M = 3/4 K and K = c WC r s /f 1/ ν, r s with c /3 WC =.1 taken from classical Wigner crystal energy. Ground state energy as function of Coulomb density parameter. From B. Abdullaev, U. Roessler, M. Musakhanov, Phys. Rev. B 76, (007) APUAG FU Berlin 10

11 Explicit derivation of ground state energy formulas by taking into account short range correlations in wave function Replacing trial wave function ψ R i j r ij ν ψ R logarithmic divergence regularization procedure! ):, one derives explicitly (with no E = 1/ with = 1 + ν 1 (Abdullaev, C.-H. Park, and M. M. Musakhanov, ħω 0 Physica C 471, 486 (011)) ; E 0 = ħω 0 X 0 + X 0 + M X 0 with = 1 + ν 1 (unpublished); E 0 = P 1/ νr 1 with = 1 + ν 1 (unpublished); E 0 = ħω 0 X ω c ω 0 X 0 νω c ω M X 0 with = 1 + ν 1 (unpublished) APUAG FU Berlin 11

12 Do anyons and fermions exist in the ground state of D in concept of anyons? Introducing the Zeeman term one obtains for Schrödinger equation with b k 1 term connected with statistics m sˆ k 1 with anyon (statistical) magnetic field: [( pk Ak ) sˆ bk ] ( r1, r,...) E( r1, r m k 1 m () and the Zeeman term ( r k r j ).,...) APUAG FU Berlin 1 () e ( r r ) and / z j( k j) ( r1, r,...) rij ( r1, r,...) i j k m j( k j) b k j m j( k j) () s z ( r k r j )

13 Conclusion 1. Approximate expression for ground state energy of Coulomb interacting anyons in D harmonic potential in the presence of external magnetic field has derived;. Approximate expression for ground state energy of Coulomb interacting infinite anyon gas has derived; 3. Exact cancellation of statistics and Zeeman terms in the anyon Hamiltonian has found APUAG FU Berlin 13

14 Thanks for attention APUAG FU Berlin 14