Local Density Approximation for the Almost-bosonic Anyon Gas Michele Correggi Università degli Studi Roma Tre www.cond-math.it QMATH13 Many-body Systems and Statistical Mechanics joint work with D. Lundholm (Stockholm) and N. Rougerie (Grenoble) M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 0 / 16
Outline 1 Introduction: Fractional statistics and anyons; Almost-bosonic limit for extended anyons and the Average Field (AF) functional [LR]; Minimization of the AF functional. 2 Main results [CLR]: Existence of the Thermodynamic Limit (TL) for homogeneous anyons; Local density approximation of the AF functional in terms of a Thomas-Fermi (TF) effective energy. Main References [LR] D. Lundholm, N. Rougerie, J. Stat. Phys. 161 (2015). [CLR] MC, D. Lundholm, N. Rougerie, in preparation. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 1 / 16
Introduction Anyons The wave function Ψ(x 1,..., x N ) of identical particles must satisfy Ψ(..., x j,..., x k,...) 2 = Ψ(..., x k,..., x j,...) 2 ; In 3D there are only 2 possible choices: Ψ(..., x j,..., x k,...) = ± Ψ(..., x k,..., x j,...); In 2D there are other options, related to the way the particles are exchanged (braid group). Fractional Statistics (Anyons) For any α [ 1, 1] (statistics parameter), it might be Ψ(..., x j,..., x k,...) = e iπα Ψ(..., x k,..., x j,...) α = 0 = bosons and α = 1 = fermions; anyonic quasi-particle are expected to describe effective excitations in the fractional quantum Hall effect [physics; Lundholm, Rougerie 16]. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 2 / 16
Introduction Anyons One can work on wave functions Ψ satisfying the anyonic condition (anyonic gauge): complicate because Ψ is not in general single-valued. Equivalently one can associate to any anyonic wave function Ψ a bosonic (resp. fermionic) one Ψ L 2 sym(r 2N ) via Ψ(x 1,..., x N ) = j<k Magnetic Gauge e iαφ jk Ψ(x1,..., x N ), φ jk = arg x j x k x j x k. On L 2 sym(r 2N ) the Schrödinger operator ( j + V (x j )) is mapped to H N = N j=1 [ ] ( i j + αa j ) 2 + V (x j ) with Aharonov-Bohm magnetic potentials A j = A(x j ) := k j (x j x k ) x j x k 2. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 3 / 16
Introduction AF Approximation If the number of anyons is larger, i.e., N but at the same time α N 1, then one expects a mean-field behavior, i.e., R2 (x y) αa j (Nα) dy x y 2 ρ(y), with ρ the one-particle density associated to Φ L 2 sym(r 2N ); We should then expect that 1 N Φ H N Φ ENα[u], af for some u L 2 (R 2 ) such that u 2 (x) = ρ(x) (self-consistency). AF Functional E af [u] = R 2 dx { ( i + A[ u 2 ] ) u 2 + V u 2} with A[ρ] = (w 0 ρ) and w 0 (x) := log x. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 4 / 16
Introduction Minimization of E af { E R af[u] = ( dx i + A[ u 2 ] ) u 2 + V u 2}, A[ρ] = (w 0 ρ) 2 Thanks to the symmetry u, u,, we can assume 0; The domain of E af is D[Eaf ] = H 1 (R 2 ), since by 3-body Hardy inequality dx A[ u 2 ] 2 u 2 C u 4 L 2 (R 2 ) u 2 L 2 (R 2 ). R 2 Proposition (Minimization [Lundholm, Rougerie 15]) For any 0, there exists a minimizer u af D[Eaf ] of the functional E af : E af := inf u 2 =1 Eaf [u] = Eaf [uaf ]. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 5 / 16
Introduction Almost-bosonic Limit Consider N non-interacting anyons with statistics parameter α = for some R, i.e., in the almost-bosonic limit; N 1 Assume that the anyons are extended, i.e., the fluxes are smeared over a disc of radius R = N γ. Theorem (AF Approximation [Lundholm, Rougerie 15]) Under the above hypothesis and assuming that V is trapping and γ γ 0, inf σ(h N,R ) lim = inf N N u 2 =1 Eaf [u] and the one-particle reduced density matrix of any sequence of ground states of H N,R converges to a convex combination of projectors onto AF minimizers. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 6 / 16
Introduction Motivations The AF approximation is used heavily in physics literature, but typically the nonlinearity is resolved by picking a given ρ (usually the constant density); As expected, when 0, the anyonic gas behaves as a Bose gas. More interesting is the regime, i.e., less-bosonic anyons: what is the energy asymptotics of E af? is u af 2 almost constant in the homogeneous case, i.e., for V = 0 and confinement to a bounded region? how does the inhomogeneity of V modify the density u af 2? what is u af like? in particular how does its phase behave? The AF functional is not the usual mean-field-type energy (e.g., Hartree or Gross-Pitaevskii), since the nonlinearity depends on the density but acts on the phase of u via a magnetic field. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 7 / 16
Main Results Homogeneous Gas Homogeneous Gas Let Ω R 2 be a bounded and simply connected set with Lipschitz boundary; We consider the following two minimization problems E N/D (Ω,, M) := inf dx ( i + A[ u 2 ] ) u 2. u H0 1(Ω), u 2 =M We want to study the limit of E N/D (Ω,, M)/; The above limit is equivalent to the TD limit (, ρ R + fixed) lim L Lemma (Scaling Laws) Ω E N/D (LΩ,, ρl 2 Ω ) L 2. Ω For any λ, µ R +, E N/D (Ω,, M) = 1 λ 2 E N/D (µω, ), λ 2 µ 2 M. λ 2 µ 2 M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 8 / 16
Main Results Homogeneous Gas Heuristics ( 1) Numerical simulations by R. Duboscq (Toulouse): plot of u af 2 for = 25, 50, 200. In the homogeneous case, u af 2 can be constant only in a very weak sense (say in L p, p < not too large); The phase of u af should contain vortices (with # ) almost uniformly distributed with average distance 1 (Abrikosov lattice). M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 9 / 16
Main Results Homogeneous Gas TD Limit Theorem ( TD Limit [MC, Lundholm, Rougerie 16]) Under the above hypothesis on Ω and for any, ρ R +, the limits E N/D (LΩ,, ρl 2 Ω ) E N/D (Ω,, ρ) e(, ρ) := lim L L 2 = Ω lim Ω exist, coincide and are independent of Ω. Moreover e(, ρ) = ρ 2 e(1, 1) e(1, 1) is a finite quantity satisfying the lower bound e(1, 1) 2π which follows from the inequality for u H0 1 ( i + A[ u 2 ] ) u 2 L 2 (Ω) 2π u 4 L 4 (Ω). M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 10 / 16
Main Results Trapped Gas Trapped Anyons { E R af[u] = ( dx i + A[ u 2 ] ) u 2 + V u 2}, A[ρ] = (w 0 ρ) 2 Let V (x) be a smooth homogenous potential of degree s 1, i.e., V (λx) = λ s V (x), V C (R 2 ), and such that min x R V (x) + (trapping potential). R We consider the minimization problem for 1 E af = inf E u D[E af af [u], ], u 2 =1 with D[E af ] = H 1 (R 2 ) { V u 2 L 1 (R 2 ) } and u af any minimizer. Since B(x) = curla[ρ] = 2πρ(x), if one could minimize the magnetic energy alone, the effective functional for 1 should be dx [B(x) + V (x)] ρ = dx [ 2πρ 2 + V (x)ρ ]. R 2 R 2 M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 11 / 16
Main Results Trapped Gas TF Approximation TF Functional The limiting functional for E af is E TF[ρ] := dx [ e(1, 1)ρ 2 (x) + V (x)ρ(x) ] R 2 with ground state energy E TF := inf ρ 1 =1 E TF [ρ] and minimizer ρtf(x). Under the hypothesis we made on V, we have ) E TF = s s+2 E TF 1, ρ TF 2 (x) = s+2 ρ TF 1 ( 1 s+2 x. Given the chemical potential µ TF 1 := E1 TF + e(1, 1) ρ TF 2 1, we have 2 ρ TF 1 (x) = 1 [ 2e(1,1) µ TF 1 V (x) ] +. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 12 / 16
Main Results Trapped Gas Local Density Approximation Theorem (TF Approx. [MC, Lundholm, Rougerie 16]) Under the hypothesis on V, lim E af s s+2 E TF 1 = 1, 2 s+2 u af ( ) 2 1 W1 s+2 x ρtf 1 (x) in the space W 1 of probability measure on R 2 with the Wasserstein distance. The result applies to more general potentials, e.g., asymptotically homogeneous potentials; The homogeneous case (confinement to Ω, V = 0) is included: we recover the asymptotics E N/D (Ω,, 1)/ e(1, 1)/ Ω and u af 2 W (x) 1 ρtf 1 (x) Ω 1/2. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 13 / 16
Main Results Trapped Gas Local Density Approximation Theorem (LDA [MC, Lundholm, Rougerie 16]) Under the hypothesis on V, for any x 0 R 2 and any 0 η < s+1 sup φ C 0 (R 2 ), L(φ) 1 where ρ af 2 (x) := [ dx φ ( η (x x 0 )) R 2 s+2 u af ( 2 1 s+2 x ). ρ af 2(s+2), (x) ρtf 1 (x)] 0 ρ af is well approximated in weak sense by ρtf 1 on any scale up to η (in the homogeneous case 1/ ); One can not presumably go beyond that scale because that is the mean distance between vortices. M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 14 / 16
Perspectives AF functional: Obtain more information about e(1, 1); Investigate the vortex structure of u af, which should be given by some Abrikosov lattice (which in turn is expected to provide info on e(1, 1)). Anyon gas: Recover the behavior at the many-body level, in a scaling limit N, α = α(n) and find out the parameter region where it emerges; Prove the existence of the thermodynamic limit in the same setting. Thank you for the attention! M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 15 / 16
Extended Anyons The operator H N is too singular to be defined on L 2 sym(r 2N ), but one can remove the planes x j = x k ; Equivalently one can consider extended anyons, i.e., smear the Aharonov-Bohm fluxes over disc of radius R: A j A j,r = ) ( w R (x j x k ), k j with w R (x) := 1 πr 2 ( log 1BR (0)) (x). Extended Anyons For any R > 0 the operator H N = N j=1 [ ] ( i j + αa j,r ) 2 + V (x j ) with magnetic potentials A j,r is self-adjoint on D(H N ) L 2 sym(r 2N ). M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 15 / 16
Proof (Homogeneous Gas) 1 of TD limit when Ω is a unit square with Dirichlet b.c.; 2 E D (LQ,, ρl 2 ) E N (LQ,, ρl 2 ) = o(l 2 ) for squares (IMS); 3 Prove of TD for general domains Ω by localizing into squares. Key observation for 1 & 3: the magnetic field generated by a bounded region can be gauged away outside (Newton s theorem); Pick a smooth and radial f with supp(f) B δ (0) and N points so that x j x k > 2δ: consider then the trial state N u(x) = f(x x j )e iφ j, u 2 2 = N f 2 2 ; j=1 In { x x j δ} the magnetic field generated by the other discs is arg(x x k ) =: φ j. k j ( w 0 f(x x k ) 2) = f 2 2 k j M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 16 / 16