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 2 N ) via φ jk = arg x j − x k � e iαφ jk ˜ Ψ( x 1 , . . . , x N ) = Ψ( x 1 , . . . , x N ) , | x j − x k | . j<k Magnetic Gauge sym ( R 2 N ) the Schrödinger operator � ( − ∆ j + V ( x j )) is mapped to On L 2 N ( − i ∇ j + α A j ) 2 + V ( x j ) � � � H N = j =1 ( x j − x k ) ⊥ � with Aharonov-Bohm magnetic potentials A j = A ( x j ) := | x j − x k | 2 . k � = j 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., R 2 d y ( x − y ) ⊥ � α A j ≃ ( Nα ) | x − y | 2 ρ ( y ) , with ρ the one-particle density associated to Φ ∈ L 2 sym ( R 2 N ) ; We should then expect that 1 N � Φ | H N | Φ � ≃ E af Nα [ u ] , for some u ∈ L 2 ( R 2 ) such that | u | 2 ( x ) = ρ ( x ) (self-consistency). AF Functional � � 2 + V | u | 2 � �� E af − i ∇ + β A [ | u | 2 ] � �� � β [ u ] = R 2 d x u 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 β � � 2 + V | u | 2 � �� A [ ρ ] = ∇ ⊥ ( w 0 ∗ ρ ) − i ∇ + β A [ | u | 2 ] � E af �� � β [ u ] = R 2 d x u , → u ∗ , − β , we can assume β � 0 ; Thanks to the symmetry u, β − The domain of E af β is D [ E af ] = H 1 ( R 2 ) , since by 3-body Hardy inequality � � 2 | u | 2 � C � u � 4 L 2 ( R 2 ) �∇| u |� 2 � A [ | u | 2 ] � � R 2 d x L 2 ( R 2 ) . Proposition (Minimization [Lundholm, Rougerie ’15]) For any β � 0 , there exists a minimizer u af β ∈ D [ E af ] of the functional E af β : � u � 2 =1 E af β [ u ] = E af β [ u af E af β := inf β ] . 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 β α = N − 1 for some β ∈ R , i.e., in the almost-bosonic limit; 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 ) � u � 2 =1 E af lim = inf β [ u ] N N →∞ 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 β → 0 , the anyonic gas behaves as a Bose gas. More interesting is the regime β → ∞ , i.e., “less-bosonic” anyons: what is the energy asymptotics of E af β ? β | 2 almost constant in the homogeneous case, i.e., for V = 0 and is | u af confinement to a bounded region? how does the inhomogeneity of V modify the density | u af β | 2 ? what is u af β 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 � � 2 . − i ∇ + β A [ | u | 2 ] � �� � � E N / D (Ω , β, M ) := inf d x u u ∈ H 1 0 (Ω) , � u � 2 = M Ω We want to study the limit β → ∞ of E N / D (Ω , β, M ) /β ; The above limit is equivalent to the TD limit ( β, ρ ∈ R + fixed) E N / D ( L Ω , β, ρL 2 | Ω | ) lim . L 2 | Ω | L →∞ Lemma (Scaling Laws) � � 1 β For any λ, µ ∈ R + , E N / D (Ω , β, M ) = λ 2 µ 2 , λ 2 µ 2 M λ 2 E N / D µ Ω , . M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 8 / 16
Main Results – Homogeneous Gas Heuristics ( β ≫ 1 ) β | 2 for Numerical simulations by R. Duboscq (Toulouse): plot of | u af β = 25 , 50 , 200 . β | 2 can be constant only in a very weak In the homogeneous case, | u af sense (say in L p , p < ∞ not too large); The phase of u af β should contain vortices (with # ∼ β ) almost 1 uniformly distributed with average distance ∼ √ β (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 β, ρ ∈ R + , the limits E N / D (Ω , ˜ E N / D ( L Ω , β, ρL 2 | Ω | ) β, ρ ) e ( β, ρ ) := lim = β | Ω | lim ˜ L 2 | Ω | L →∞ ˜ β β →∞ exist, coincide and are independent of Ω . Moreover e ( β, ρ ) = βρ 2 e (1 , 1) e (1 , 1) is a finite quantity satisfying the lower bound e (1 , 1) � 2 π which follows from the inequality for u ∈ H 1 0 � 2 L 2 (Ω) � 2 π | β | � u � 4 � �� − i ∇ + β A [ | u | 2 ] � � u L 4 (Ω) . M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 10 / 16
Main Results – Trapped Gas Trapped Anyons � � 2 + V | u | 2 � �� A [ ρ ] = ∇ ⊥ ( w 0 ∗ ρ ) − i ∇ + β A [ | u | 2 ] � E af �� � β [ u ] = R 2 d x u , Let V ( x ) be a smooth homogenous potential of degree s � 1 , i.e., V ( λ x ) = λ s V ( x ) , V ∈ C ∞ ( R 2 ) , and such that min | x | � R V ( x ) − − R →∞ + ∞ (trapping potential). − − → We consider the minimization problem for β ≫ 1 E af u ∈ D [ E af ] , � u � 2 =1 E af β = inf β [ u ] , V | u | 2 ∈ L 1 ( R 2 ) with D [ E af ] = H 1 ( R 2 ) ∩ and u af � � β any minimizer. Since B ( x ) = β curl A [ ρ ] = 2 πβρ ( x ) , if one could minimize the magnetic energy alone, the effective functional for β ≫ 1 should be � � 2 πβρ 2 + V ( x ) ρ � � R 2 d x [ B ( x ) + V ( x )] ρ = R 2 d x . 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 β is � E TF e (1 , 1) βρ 2 ( x ) + V ( x ) ρ ( x ) � � β [ ρ ] := R 2 d x with ground state energy E TF := inf � ρ � 1 =1 E TF β [ ρ ] and minimizer ρ TF β ( x ) . β Under the hypothesis we made on V , we have s 2 � 1 � β ( x ) = β − β − E TF s +2 E TF ρ TF s +2 ρ TF s +2 x = β 1 , . 1 β � 2 Given the chemical potential µ TF := E TF � � ρ TF � + e (1 , 1) 2 , we have 1 1 1 ρ TF 1 µ TF � � 1 ( x ) = − V ( x ) + . 1 2 e (1 , 1) M. Correggi (Roma 3) Almost-bosonic Anyon Gas Atlanta 8/10/2016 12 / 16
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