Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics

Transcription

1 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Douglas Lundholm IHES / IHP / CRMIN joint work with Jan Philip Solovej, University of Copenhagen Variational & Spectral Methods in Quantum Mechanics Paris, June 20, 2013 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 1/25

2 Outline of Talk 1 Identical particles and statistics in lower dimensions 2 Local exclusion principle 3 New Lieb-Thirring type inequalities 4 pplications 5 Outlook Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 2/25

3 Identical particles and statistics: standard theory N-particle wavefunction ψ L 2 ((R d ) N ) ψ(x 1,...,x j,...,x k,...,x N )=±ψ(x 1,...,x k,...,x j,...,x N ) +: bosons: ψ N sym L2 (R d ) : fermions: ψ N L 2 (R d ) Pauli s exclusion principle: ϕ ϕ =0, ϕ L 2 (R d ) This classification is valid for elementary particles in d =3spatial dimensions Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 3/25

4 Identical particles and statistics: standard theory N-particle wavefunction ψ L 2 ((R d ) N ) ψ(x 1,...,x j,...,x k,...,x N )=±ψ(x 1,...,x k,...,x j,...,x N ) +: bosons: ψ N sym L2 (R d ) : fermions: ψ N L 2 (R d ) Pauli s exclusion principle: ϕ ϕ =0, ϕ L 2 (R d ) This classification is valid for elementary particles in d =3spatial dimensions Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 3/25

5 Identical particles and statistics: generalized theory Streater, Wilde, 1970 (QFT)... Leinaas, Myrheim, 1977; Goldin, Menikoff, Sharp, 1981; Wilczek, Fractional statistics quasiparticles in d =2 anyons: ψ(...,x j,...,x k,...)=e iαπ ψ(...,x k,...,x j,...), e iαπ U(1) Note: continuous interchange, π 1 (R 2N / \ SN )=B N Intermediate/fractional statistics in d =1: x 1 <x 2 <...<x N Particle collision b.c. for ψ at r = x j+1 x j 0 r ψ = ηψ or ψ(r) r α Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 4/25

6 Identical particles and statistics: generalized theory Streater, Wilde, 1970 (QFT)... Leinaas, Myrheim, 1977; Goldin, Menikoff, Sharp, 1981; Wilczek, Fractional statistics quasiparticles in d =2 anyons: ψ(...,x j,...,x k,...)=e iαπ ψ(...,x k,...,x j,...), e iαπ U(1) Note: continuous interchange, π 1 (R 2N / \ SN )=B N Intermediate/fractional statistics in d =1: x 1 <x 2 <...<x N Particle collision b.c. for ψ at r = x j+1 x j 0 r ψ = ηψ or ψ(r) r α Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 4/25

7 Identical particles and statistics: generalized theory Streater, Wilde, 1970 (QFT)... Leinaas, Myrheim, 1977; Goldin, Menikoff, Sharp, 1981; Wilczek, Fractional statistics quasiparticles in d =2 anyons: ψ(...,x j,...,x k,...)=e iαπ ψ(...,x k,...,x j,...), e iαπ U(1) Note: continuous interchange, π 1 (R 2N / \ SN )=B N Intermediate/fractional statistics in d =1: x 1 <x 2 <...<x N Particle collision b.c. for ψ at r = x j+1 x j 0 r ψ = ηψ or ψ(r) r α Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 4/25

8 Modelling anyons p p e i2pαπ e i(2p+1)απ Bosons (ψ L 2 sym) inr 2 with haronov-bohm magnetic interactions: ˆT := 1 2m N j=1 D 2 j, D j = i j + j, j (x) =α k j (x j x k )I x j x k 2 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 5/25

9 Modelling anyons Morally, free kinetic energy ˆT = 1 2m j j acting on multivalued ψ = U α ψ, U := j<k e iφ jk = j<k z j z k z j z k. Precise definition in magnetic gauge: (DL, Solovej, 2013) D : L 2 sym(r 2N ) D (R 2N \ ; C 2N ) ˆT (α R) := 1 2m (D min) D min = 1 2m (D max) D max (α=2n) Dom( ˆT ) = U 2n Hsym(R 2 2N ) (α=2n+1) Dom( ˆT ) = U (2n+1) Hasym(R 2 2N ) Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 6/25

10 Modelling 1D intermediate statistics Bosons on R with pairwise interaction potential, singular at the diagonals (boundary) Schrödinger-type quantization (η) Lieb-Liniger ˆT LL := 1 2 N 2 x 2 j=1 j +2η j<k δ(x k x j ) Heisenberg-type quantization (α) Calogero-Sutherland ˆT CS := 1 2 N 2 x 2 j=1 j + j<k α(α 1) (x k x j ) 2 = 1 2 Q αq α Leinaas, Myrheim, 1977; 1988; 1993; Polychronakos, 1989; neziris, Balachandran, Sen, 1991; Isakov, 1992; Myrheim, 1999 (review); DL, Solovej, 2013 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 7/25

11 Modelling 1D intermediate statistics Bosons on R with pairwise interaction potential, singular at the diagonals (boundary) Schrödinger-type quantization (η) Lieb-Liniger ˆT LL := 1 2 N 2 x 2 j=1 j +2η j<k δ(x k x j ) Heisenberg-type quantization (α) Calogero-Sutherland ˆT CS := 1 2 N 2 x 2 j=1 j + j<k α(α 1) (x k x j ) 2 = 1 2 Q αq α Leinaas, Myrheim, 1977; 1988; 1993; Polychronakos, 1989; neziris, Balachandran, Sen, 1991; Isakov, 1992; Myrheim, 1999 (review); DL, Solovej, 2013 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 7/25

12 Modelling 1D intermediate statistics Bosons on R with pairwise interaction potential, singular at the diagonals (boundary) Schrödinger-type quantization (η) Lieb-Liniger ˆT LL := 1 2 N 2 x 2 j=1 j +2η j<k δ(x k x j ) Heisenberg-type quantization (α) Calogero-Sutherland ˆT CS := 1 2 N 2 x 2 j=1 j + j<k α(α 1) (x k x j ) 2 = 1 2 Q αq α Leinaas, Myrheim, 1977; 1988; 1993; Polychronakos, 1989; neziris, Balachandran, Sen, 1991; Isakov, 1992; Myrheim, 1999 (review); DL, Solovej, 2013 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 7/25

13 Pauli exclusion and energy inequalities Pauli exclusion: say q N particles allowed in each one-particle state of Ĥ1 = 1 2 R d + V (x) Lieb-Thirring inequality: (Lieb, Thirring, 1975) Ĥ N = ˆT + ˆV N ( = 1 ) 2 j + V (x j ) j=1 q λ k q C d V (x) 1+ d 2 dx R d kinetic energy inequality: T = 1 2 N R dn j=1 k=0 Bosons: q = N trivial bounds j ψ 2 dx C d q 2/d ρ(x) 1+ 2 d dx R d Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 8/25

14 Local exclusion for fermions cp. Dyson, Lenard, 1967 Lemma (Local exclusion for fermions in d =3) Let ψ n L 2 (R 3 ) be a wavefunction of n fermions and let Ω be a ball of radius l. Then n j ψ 2 dx (n 1) ξ2 l 2 ψ 2 dx, Ω n Ω n j=1 where ξ is the smallest positive root of the equation d 2 dx 2 sin x x =0. Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 9/25

15 Local exclusion for anyons Lemma (Local exclusion for anyons) Let ψ be a wavefunction of n anyons with α R and let Ω R 2 be either a disk or a square, with area Ω. Then n Ω n j=1 D j ψ 2 dx (n 1) c ΩC 2 α,n Ω Ω n ψ 2 dx, where c Ω is a constant which satisfies c Ω for the disk and c Ω for the square, and C α,n := min min (2p + 1)α 2q. p {0,1,...,n 2} q Z Idea of proof: pairwise relative magnetic Hardy inequality Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 10/25

16 Local exclusion for anyons C α,n := min { 1 ν, if α = µ ν N p {0,1,...,N 2},q Z (2p + 1)α 2q is a reduced fraction with µ odd, 0 otherwise. Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 11/25

17 Local exclusion for 1D intermediate statistics Lemma (Local exclusion in 1D) Let ψ be a symmetric wavefunction of n particles on R and let Ω be an interval of length l. Thenforη 0 resp. α 1 Ω n 1 2 n j ψ 2 + V LL/CS (x j x k ) ψ 2 dx j<k (n 1) ξ2 LL/CS l 2 ψ 2 dx, Ω n j=1 where ξ LL/CS = ξ LL (ηl) resp. ξ CS (α) is the smallest positive root of the equation d x tan x = ηl resp. xjα 1 (x) =0. dx 2 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 12/25

18 Local exclusion for 1D intermediate statistics ξ LL (ηl) resp. ξ CS (α) as a function of ηl resp. α Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 13/25

19 Lieb-Thirring inequalities for anyons Theorem (Kinetic energy inequality for anyons) Let ψ be an N-anyon wavefunction on R 2 with any α R. Then for a constant 10 4 C π. T C 2 α,n C R 2 ρ(x) 2 dx, Corollary (Lieb-Thirring inequality for anyons) Let V be a real-valued potential on R 2.Then ˆT + V C 2 α,n C V (x) 2 dx, R 2 for a positive constant C. Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 14/25

20 Lieb-Thirring inequalities for 1D statistics Theorem (Kinetic energy inequality for 1D Lieb-Liniger) Let ψ be a symmetric N-particle wavefunction on R. Thenfor η 0 C LL ξ LL (2η/ ρ(x)) 2 ρ(x) 3 dx, T LL for a constant C LL 2/3. R Theorem (Kinetic energy inequality for 1D Calogero-Sutherland) Let ψ be a symmetric N-particle wavefunction on R. Thenfor α 1 ξ CS (α) 2 C CS ρ(x) 3 dx, T CS for a constant 1/32 C CS 2/3. ρ is a local approximation of ρ: ρ Q := R Q ρ Q. Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 15/25

21 Proofs using splitting algorithm Q 0 B B B B B B Split recursively until each box contains approximately 2 particles (B) or 0 particles (). pply local uncertainty on every box with non-constant density. pply local exclusion on B s, which also cover for s with constant density. Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 16/25

22 pplication: Ground state energy of an ideal gas Energy per unit area of a gas of free anyons with odd numerator fractional statistics parameter α = µ ν, confined to an area L2,is bounded below by T L 2 C ρ 2 ν 2, with density ρ := N/L2. In the 1D cases, with density ρ := N/L: resp. T LL L C LL ξ LL (2η/(γ ρ)) 2 ρ 3, if ρ(x) γ ρ, T CS L C CS ξ CS (α) 2 ρ 3. For N cp. Lieb, Liniger, 1963; resp. Sutherland, 1971 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 17/25

23 pplication: Ground state energy of an ideal gas Energy per unit area of a gas of free anyons with odd numerator fractional statistics parameter α = µ ν, confined to an area L2,is bounded below by T L 2 C ρ 2 ν 2, with density ρ := N/L2. In the 1D cases, with density ρ := N/L: resp. T LL L C LL ξ LL (2η/(γ ρ)) 2 ρ 3, if ρ(x) γ ρ, T CS L C CS ξ CS (α) 2 ρ 3. For N cp. Lieb, Liniger, 1963; resp. Sutherland, 1971 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 17/25

24 pplication: External potentials Harmonic oscillator Hamiltonian: Ĥ = ˆT + ˆV ext = ˆT + N j=1 ω 2 2 x j 2. By minimization of the corresponding quadratic forms we obtain Ĥ 1 8C 3 π C α,n ωn resp. Ĥ CS 8π ξ CS(α) ωn 2 cp. with fermions in 2D: E 0 3 ωn 3 2 as N resp. exact C-S ground state: E 0 = 1 2ωN(1 + α(n 1)) 8 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 18/25

25 Outlook: nyon trial states What is the true ground state energy E 0 as a function of α? Is it even true that E 0 (α) E 0 (α = 1)? General bound for anyons in a harmonic oscillator: (Chitra, Sen, 1992) ( ) Ĥ ω N + 1) L + αn(n 2 angular momentum for ground states ψ s.t. E 0 N 3/2 must be L = α ( N 2 ) + O(N 3/2 ) Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 19/25

26 Outlook: nyon trial states N =2: Leinaas, Myrheim, 1977 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 20/25

27 Outlook: nyon trial states N =3: Murthy, Law, Brack, Bhaduri, 1991; Sporre, Verbaarschot, Zahed, 1991 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 21/25

28 Outlook: nyon trial states N =4: Sporre, Verbaarschot, Zahed, 1992 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 22/25

29 Outlook: nyon trial states N : Chitra, Sen, 1992 Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 23/25

30 Outlook: nyon trial states Consider N = Kν particles arranged into ν disjoint complete graphs (V q,e q ).Forα = µ/ν with even numerator: ψ α (z) := ν N z jk α S ( z jk ) µ ϕ 0 (z k ) j<k q=1 (j,k) E q k=1 For α = µ/ν with odd numerator: ψ α (z) := ν z jk α S ( z jk ) µ j<k q=1 (j,k) E q K 1 k=0 ϕ k (z j Vq ) These have L = α ( N 2 ) + O(N) and yield cancellations in Dj ψ Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 24/25

31 References D. L., J.P. Solovej, Hardy and Lieb-Thirring inequalities for anyons, arxiv: , toappearincmp D. L., J.P. Solovej, Local exclusion for intermediate and fractional statistics, arxiv: D. L., J.P. Solovej, Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics, arxiv: , to appear in HP See also R.L. Frank, R. Seiringer, Lieb-Thirring Inequality for a Model of Particles with Point Interactions, JMP 53, (2012). Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics Slide 25/25