Structure of large bosonic systems : the mean-field approximation and the quantum de Finetti theorem

Structure of large bosonic systems : the mean-field approximation and the quantum de Finetti theorem icolas Rougerie LPMMC CRS & Universite Grenoble 1 Mathematical horizons of quantum physics IMS, Singapore, September 2013 Joint work with Mathieu Lewin and Phan Tha nh am: arxiv:1303.0981

1. Introduction : bosons in the mean-field regime 2. Trapped systems and the quantum de Finetti theorem 3. The need for a weak de Finetti theorem 4. Quantitative de Finetti in finite dimension 5. Localization in Fock space and quantum de Finetti 6. Validity of Hartree s theory in the general case: conclusion

The mean-field approximation interacting particles in d space dimensions, 1 : too many degrees of freedom for an efficient description Look for relevant approximations: simplest one = assume independent, identically distributed particles Classical mechanics: probability measure µ on (R d ). Ansatz: with ρ probability on R d. µ(x 1,..., x ) = ρ(x i ) Quantum mechanics: wave function Ψ L 2 (R d ) L 2 (R d ). Ansatz: Ψ(x 1,..., x ) = i=1 u(x i ) = u (x 1,..., x ), u L 2 (R d ). i=1 Ok for bosons. For fermions, rather a Slater determinant (minimal correlations allowed by the Pauli exclusion principle) General question: is that a sensible approximation in the limit?

Bosons in the mean-field regime H := j=1 T j + 1 1 1 k<l w kl. Acting on H := s H, -fold symmetric tensor product of H a separable Hilbert space. Example: H = L 2 (R d ) H Ψ L 2 (R d ) satisfying Ψ(x 1,..., x i,..., x j,..., x ) = Ψ(x 1,..., x j,..., x i,..., x ) i, j One-body part: T self-adjoint operator on H. Example: T = + V, V : R d R T j = xj + V (x j ). Two-body part: pair interaction w, symmetric operator on H 2. Example: w kl = w ( x k x l ), w : R d R, w( x) = w(x) Adequate assumptions to have H bounded below. Scaling factor to have a well-defined limit problem. Simplest possibility : mean-field scaling. Mean-field approximation for the ground state energy: does E() := inf Ψ, H Ψ inf u, H u when? Ψ H, Ψ =1 u H, u =1

Motivation For non interacting systems (w = 0) inf Ψ, H Ψ = Ψ H, Ψ =1 inf u H, u =1 all the particles in the same quantum state. u, H u on interacting bosons with temperature: for T < T c, most particles in the same quantum state. Bose-Einstein condensation. Experimental motivation: Bose-Einstein condensates of cold dilute atomic gases (first in 1995, 2001 obel prize Cornell-Wieman-Ketterle): Many particles in the same quantum state macroscopic quantum effects. Superfluidity, quantized vortices... [Coddington-etal-Cornell, PRA 70 (2004)]

Hartree s theory Restrict states to purely uncorrelated ansätze Ψ = u Linear -body problem on-linear 1-body problem: E H [u] = u, Tu H + 1 2 u u, w u u H = 1 u, H 2 u ( Example = u 2 + V u 2) + 1 u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R d Hartree ground state energy e H = inf { E H [u], u H = 1 } when minimizer u H exists, satisfies Hartree s nonlinear equation. Example: ( ) + V + u H 2 w u H = ε H u H Trivial energy upper bound by the variational principle E() e H. Expect the lower bound to also hold for very general systems E() lim = e H Expect also Bose-Einstein condensation (BEC) on the Hartree ground state when it is unique.

Trapped bosonic gases bosons in the full space with a (say regular) external potential H = j=1 ( ) ( j + ia(x j )) 2 + V (x j ) + 1 1 1 k<l w(x k x l ), rotating trapped Bose gas, gases with magnetic fields: A : R d R d a (say regular) vector potential. Simplest case V (x) when x trapping potential. Seiringer, Grech-Seiringer 11/12 : validity of Hartree for ŵ 0, A 0. ŵ 0 allows to bound from below the two-body Hamiltonian by a one-body Hamiltonian. More relevant for cold atoms physics: Contact interactions. Scale the interaction potential w with so that w δ 0. Gross-Pitaevskii limit (more subtle, correlations play a role): Dyson 57, Lieb-Seiringer-Yngvason 00-13, for w 0.

Bosonic atoms: Bosonic atoms and Boson stars Classical nucleus of charge Z, electrons without Pauli principle. Large almost neutral atom : Z large t = ( 1)/Z fixed. ( H := i 1 ) + 1 1 t x i 1 x k x l. j=1 1 k<l Validity of Hartree: Benguria-Lieb 83, using (formally) ŵ 0 via a Lieb-Oxford inequality. See also Kiessling 12. Boson stars: White dwarf with local neutrality and no Pauli principle (Lieb-Thirring 84). H = j=1 with 0 < κ < 2/π. (( 1 xj ) 1/2 1 ) κ 1 1 k<l Validity of Hartree: Lieb-Yau 87, uses that (formally) ŵ 0. 1 x k x l, This talk : Validity of Hartree s theory can be seen as a consequence of the structure of the set of bosonic states.

Related literature Fannes-Spohn-Verbeure 80, Raggio-Werner 89, Petz-Raggio-Verbeure 89... similar in spirit to this talk, but very abstract. Hepp-Lieb 74: Dicke maser model. Gross-Pitaevskii limit (more subtle, correlations play a role): Lieb-Seiringer-Yngvason 00-13, for w 0. BECs in particular geometries, for high rotation: Lieb-Seiringer-Yngvason, Schnee-Yngvason, Bru-Correggi-Pickl-Yngvason... Beyond Hartree (Bogoliubov s theory): Lieb-Solovej 01-06, Cornean-Dereziński-Ziń 09, Seiringer 11, Grech-Seiringer 12, Lewin-am-Serfaty-Solovej 12, Dereziński-apiórkowski 13 Dynamics (-body Schrödinger flow preserves factorization). Hepp, Ginibre-Velo, Spohn, Erdös-Schlein-Yau, Bardos-Golse-Mauser, Rodniansky-Schlein, Pickl, Knowles-Pick, Fröhlich-Knowles-Schwarz, Machedon-Grillakis-Margetis, Chen-Pavlovic, Ammari-ier, Lewin-am-Schlein... and others.

H := Setting the stage j=1 T j + 1 1 1 k<l w kl. Assume T has compact resolvent. Example on a compact domain. (General) mixed states = positive, self-adjoint, trace-class operators on H Pure states = wave functions rank-one projectors To any Ψ H associate a density matrix, projector onto Ψ γ Ψ = Ψ Ψ S 1 (H ) Reduced density matrices (identify operators and kernels) γ (k) Ψ = Tr k+1 Ψ Ψ S 1 (H k ) i.e. for any bounded operators b 1,..., b k on H [ ] Tr H k γ (k) Ψ b1... b k = Tr H [ Ψ Ψ b 1... b k 1 k] Energy only depends on γ (2) Ψ : 1 Ψ, H Ψ = Tr H 1[T γ (1) Ψ ] + 1 2 Tr H 2[wγ(2) Ψ ] = 1 2 Tr H 2[H2γ(2) Ψ ] Is the infimum attained for γ (2) Ψ = u 2 u 2?

Representability of mixed states Spectral theorem: mixed states = convex combinations of pure states without loss E() = 1 { ( 2 inf Tr H 2 H 2γ (2)) }, γ (2) P (2) with the set of -representable 2-body mixed states { } P (2) = γ (2) S 1 (H 2 ) : 0 G S 1 (H ), Tr H G = 1, γ (2) = Tr 3 G How does P (2) look like? Representability Problem. Minimize a fixed energy functional on a -dependent set. Does the set have a non trivial limit? With a leap of faith (OK in the confined case), hope that E() lim = 1 { ( 2 inf Tr H 2 H 2γ (2)) }, γ (2) lim P(2) A decreasing sequence of sets P (2) +1 P(2), define the set of -representable 2-body mixed states P (2) := 2 P (2) = lim P(2) (for the strong S1 topology)

Representability and quantum de Finetti theorem Quantum analogue of the Hewitt-Savage theorem for symmetric probability measures. Theorem (Størmer 69, Hudson-Moody 75) An infinite hierarchy {γ (k) } k=0 of states, γ (k) acts on H k. Consistency condition Tr k+1 γ (k+1) = γ (k) γ (0) = 1 (implies Tr H k γ (k) = 1 for all k 0) Then there Exists a unique Borel probability measure µ on the sphere SH of H, invariant under the group action of S 1, such that γ (k) = SH u k u k dµ(u) for all k 0. Representability problem is solved for P (2) : { } P (2) = u 2 u 2 dµ(u), µ Borel probability measure SH Take a -body state, with large. Look at the k-body density matrix, for k small. It is almost a convex combination of Hartree states.

Validity of Hartree for confined systems lim 1 E() = 1 { 2 inf Tr H 2 (H 2γ (2)), γ (2) P (2)} = 1 ( ) 2 inf Tr H 2 H µ 2 u 2 u 2 dµ(u) = inf E H [u]dµ(u) e H SH µ SH Justify the exchange of lim and inf: T has compact resolvent strong S 1 convergence of reduced density matrices for a sequence of ground states Theorem (Lewin-am-R 13) Energy: lim 1 E() = e H. If (Ψ ) is any sequence of (approximate) minimizers, there is a probability measure µ on the set M of minimizers of e H (modulo a phase), such that lim j γ(k) Ψ j = dµ(u) u k u k strongly in S 1 (H k ) M Bose-Einstein condensation: if e H has a unique minimizer u H (up to phase) lim γ(k) Ψ = u k H u k H strongly in S 1 (H k ).

Examples Strategy related to earlier works Fannes-Spohn-Verbeure 80, Petz-Raggio-Verbeure 89, Raggio-Werner 89 cf the use of the classical de Finetti theorem in statistical mechanics: Messer-Spohn 82, Kiessling 93, Caglioti-Lions-Marchioro-Pulvirenti 93, Kiessling-Spohn 99 Homogeneous Bose gas: bosons in a periodic box, H = L 2 per([0, 1] d ) H := j=1 j + 1 1 1 k<l previously dealt with by Seiringer 11, with ŵ 0 w(x k x l ), Trapped Bose gas: bosons in the full space with a (say regular) trapping potential V (x) when x ( ) H = ( j + ia(x j )) 2 + V (x j ) + 1 1 j=1 1 k<l w(x k x l ), rotating trapped Bose gas, gases with magnetic fields: A : R d R d a (say regular) vector potential (Grech-Seiringer 12, with ŵ 0 and A = 0)

on confined case One-body potentials going to 0 at (cf bosonic atoms and boson stars) Typical example H V ( = xj + V (x j ) ) + 1 1 on L 2 (R d ) with j=1 1 k<l w(x k x l ) V, w L p + L 0, max(1, d/2) < p <, w(x) = w( x) V, w 0 at Can add magnetic fields, fractional Laplacians... only need stability. Hartree s functionals. Possible loss of mass at infinity ( EH V [u] = u 2 + V u 2) + 1 u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R d EH[u] 0 = u 2 + 1 u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R { d eh V (λ) = inf EH V [u] } u 2 = λ. R d Large binding inequalities: for all 0 λ 1: e V H (1) e V H (λ) + e 0 H(1 λ)

Loss of compactness γ density matrices of (approximate) ground state, reduced density matrices γ (k) Weak- compactness in S 1 + diagonal argument: γ (k) γ(k) Tr H k [γ (k) B k] Tr H k [γ (k) B k ] for any compact operator B k on H k The trace is not weak- continuous in infinite dimension (i.e. the identity is not compact), the limiting hierarchy is not consistent. Only γ (k) Tr k+1 γ (k+1) Quantum de Finetti requires equality, hence strong S 1 convergence. Try to describe exhaustively the possible ways of losing compactness, i.e. in this context loss of mass at infinity One-body problems: concentration-compactness principle of Lions, 80 s (see also Lieb, Struwe, Brézis-Coron...) Many-body problem: geometric methods, cf Simon, Enss, Sigal... late 70 s. Main novelty: combine quantum de Finetti and geometric localization.

A weak quantum de Finetti theorem Theorem (Lewin-am-R 13) Let (γ ) be a sequence of mixed states, γ S 1 (H ) such that, for all k 1 γ (k) γ(k) in S 1 (H k ). Then there exists a unique Borel probability measure µ on the unit ball BH of H, invariant under the group action of S 1, such that for all k 0. γ (k) = BH dµ(u) u k u k Similar in spirit to results of Ammari-ier 08-13 Sufficient to deal with weakly lower semi continuous systems General case: more is needed (see later)

Validity of Hartree for systems with no bound states at infinity Assume that the two-body potential has no bound states + w 0. energy is a weakly lower semi-continuous function of the density matrices and e 0 H = 0 Theorem (Lewin-am-R 13) Energy: lim 1 E V () = e V H (1). Take Ψ a sequence of approximate ground states in H. There exists a Borel probability measure µ supported on the set { M V = u H, u 1 } E V H [u] = eh V ( u 2 ) = eh V (1), such that γ (k) Ψ j M V u k u k dµ(u). If e V H (1) < e V H (λ), then µ is supported on SH and the limit is strong. If in addition e V H (1) admits a unique minimizer u H, there is complete Bose-Einstein condensation.

Quantitative versions of the de Finetti theorem? Pick a -body state Γ S(H ) for some Hilbert space H Try to construct a Γ of the form Γ = u u dµ (u) SH with µ a probability measure, such that Γ (k) Γ (k) C(, k) S 1 (H k ) C(, k) should at least satisfy C(, k) 0 when with k fixed Clearly implies the Størmer-Hudson-Moody theorem, with a (hopefully) constructive proof. Answer unknown for general Hilbert spaces. Maybe true, maybe not... Progress in finite dimension, mostly König-Renner 05 / Christandl-König-Mitchison-Renner 07. Sufficient to discuss pure states Γ = Ψ Ψ, Ψ H.

A coherent state decomposition by Schur s lemma By rotational invariance of the normalized uniform measure du on the unit sphere SH dim H du u u = 1 H where d = dim H and SH ( ) dim H + d 1 =. d 1 Generalized coherent state representation: continous partition of the identity Lower symbol of Ψ H in this representation: dµ low (u) := dim H u, Ψ 2 du. Looking for a de Finetti measure looking for an upper symbol, i.e. dµ up such that Ψ Ψ = u u dµ up (u). SH Lieb 73, Simon 80 : upper symbols (when they exist!) have a tendency to turn into lower symbols in semi-classical limits.

Making a guess for the upper symbol To get an idea, pretend that you do have an upper symbol to begin with Ψ Ψ = u u dµ up (u). How to obtain dµ up (u) as a function of Ψ? Compute the lower symbol of the upper symbol When SH dµ low (u) = dim H u, Ψ 2 du ( ) = dim H u, v 2 dµ up (v) du SH dµ low (u) dµ up (u) reasonable guess : for any state Ψ, pick the lower symbol as a candidate for the de Finetti measure: Γ := dim H SH u u u, Ψ 2 du.

The CKMR estimate Reduced density matrices of the original state Ψ H γ (k) := Tr k+1 Ψ Ψ Reduced density matrices of the de Finetti state = dµ (u) u k u k γ (k) SH dµ (u) = dim H u, Ψ 2 du. Theorem (Christandl-König-Mitchison-Renner 07) For every k = 1, 2,..., it holds Tr H k γ (k) γ(k) 4kd. has the desired property C(k, ) = kd/ 0 for and d, k fixed. efficient when d, i.e. much more particles than degrees of freedom. rather short proof, I present a variant (Lewin-am-R 13)

Alternative representation of the CKMR state Theorem (Lewin-am-R 13) γ (k) with the convention that γ (l) ( ( + d 1)! k = ( + k + d 1)! l=0 l )( k l ) γ (l) s 1 H k l s 1 H k l = σ S k (γ (l) ) σ(1),...,σ(l) (1 H k l) σ(l+1),...,σ(k). Then γ (k) with A, B 0 and γ(k) Obviously Tr A = Tr B. = (C(d, k, ) 1)γ(k) + B = A + B ( + d 1)!! C(d, k, ) = ( + k + d 1)! ( k)! < 1. Triangle inequality Tr γ (k) γ(k) TrA + TrB = 2 Tr A = 2(1 C(d, k, )) ot as good as before, but not so bad... 2k(d + 2k).

Expectations in Hartree vectors determine the state Let s prove the explicit formula. γ (k) = ( + d 1)! ( + k + d 1)! ( )( ) k k l l l=0 γ (l) First: Lemma If a trace class self-adjoint operator γ (k) on H k satisfies then γ (k) 0. u k, γ (k) u k = 0 for all u H, s 1 H k l. States are fully determined by their lower symbol in a coherent state representation. Well-known in very general settings : Klauder 64, Simon 80. to prove the formula, only need to compute v k, γ (k) v k.

Wick quantization Creation/annihilation operators: a (f ) : H k 1 H k a (f ) (f 1... f k ) = (k) 1/2 f f 1... f k (+ symetrization). a(f ) : H k+1 H k adjoint of a (f ) (+symetrization) a(f ) (f 1... f k+1 ) = (k + 1) 1/2 f, f 1 f 2... f k+1 Canonical Commutation Relations [a(f ), a(g)] = 0, [a (f ), a (g)] = 0, [a(f ), a (g)] = f, g H. Characterization of the density matrices: v k, γ (k) v k = ( k)! Ψ a (v) k a(v) k Ψ! a.k.a. normal order: creation on the left, annihilation on the right.

Anti-Wick quantization v k, γ (k) v k = dim H du u, Ψ 2 u k, v k 2 SH! = dim H du u (+k), a (v) k Ψ 2 ( + k)!! dim H a by Schur s lemma = (v) k 2 Ψ ( + k)! dim H +k SH! dim H = ( + k)! dim H Ψ, a(v) k a (v) k Ψ +k a.k.a. anti-normal order: creation on the right, annihilation on the left. Only need to commute creation and annihilation using the CCR repeatedly. Linked to ideas of Ammari-ier 07-11. a(v) k and a (v) k almost commute for large, small k, fixed d. Lemma For any v SH and n a(v) n a (v) n = ( ) n n n! k k! a (v) k a(v) k. k=0

Theorem (Lewin-am-R 13) If for any k Then for any k. Simple to see that Weak quantum de Finetti again γ (k) γ(k) in S 1 (H k ), γ (k) = BH dµ(u) u k u k ow we will argue that weak def strong def finite dimension def weak def We use localization methods in Fock space to obtain an (almost) constructive proof. Størmer-Hudson-Moody prove a stronger def for general (not necessarily normal) states with stronger def weak def But there s more to the constructive proof using localization: description of particles lost at infinity, i.e. defect of compactness.

Geometric localization (1) Main Idea (Derezinski-Gérard 99, Ammari 04, Hainzl-Lewin-Solovej 09, Lewin 11...) Take a sequence of (mixed) -body states γ S 1 (H ), w.l.o.g. assume weak- convergence of reduced density matrices Take a compact self-adjoint 1-body operator A to convert weak convergence into stong convergence (example A = 1 B(0,R) ) Define a state G A by localizing the reduced density matrices of γ : ( ) (n) G A = A n γ (n) A n Actually, one needs to go to the truncated bosonic Fock space (localization can mean loss of particles): F = C H H 2... H State on F : positive trace-class self-adjoint operator on F.

Lemma (Lewin 11) Geometric localization (2) For any -body state γ S 1 (H ) and bounded self adjoint operator A on H with 0 A 2 1, there exists a unique state G A on F such that where G A,k S 1 (H k ) and ( ) (n) G A = A n γ (n) A n = ( ) 1 ( ) Tr n+1 k G,k A n k=n k n G A = G A,0 G A,1... G A, Fundamental relation (follows from an explicit formula) Tr H k G,k A 1 A = Tr H k G 2, k Think of A = 1 B(0,R) The probability of having k particles inside the ball B(0, R) equals the probability of having k particles outside the ball B(0, R).

Idea of proof for weak quantum de Finetti (1) A sequence of -body states ( large) γ with γ (n) γ(n), γ (n) Tr n+1γ (n+1) Want to convert weak convergence into strong convergence obtain a consistent hierarchy and apply strong quantum de Finetti Pick a finite dimensional projector P and localize the state, (in the very end, take P 1). Look at the localized reduced density matrices ( ) 1 ( ) P n γ (n) k P n = Tr n+1 k G,k P n n k=n k=n ( ) n k Tr n+1 kg,k P Small k: ( k ) n goes to 0 rather fast ignore these terms Large k: up to normalization G P,k is a k-body state on PH apply quantum de Finetti in the finite dimensional space PH (no compactness issue) when k.

Idea of proof for weak quantum de Finetti (2) P n γ (n) P n k ) n Tr n+1 kg P,k ( k k=n ( k k ( ) n k ) n Tr n+1 kg P,k u n u n dµ k (u) S(PH) In the limit, the sum becomes an integral over the unit ball of H, λ k/ P n γ (n) P n ) n k 1 0 λ n ( k S(PH) u n u n dµ k (u) S(PH) u n u n dµ λ (u) Measure on the ball in spherical coordinates. Add a δ at the origin to obtain a probability. Ultimately, remove localization P by taking P l 1.

Localization and the de Finetti Theorem By-product of the proof: information about particles lost at infinity Smooth partition of unity, χ 2 R + η 2 R = 1, χ R 1 BR localization in a ball For any continuous function f, with µ the de Finetti measure lim f k=0 ( ) k Tr H k G χ R,k = dµ(u) f ( χ R u 2 ) BH Interpretation: µ ({ u 2 = λ }) = the probability that a fraction λ of the particles does not escape to infinity Using the fundamental property of the geometric localization ( ) k lim f Tr H k G η R,k = dµ(u) f (1 χ R u 2 ) BH k=0 gives information on particles lost at infinity. Crucial for non weakly lower semi continuous systems (bound states at infinity) Use with (essentially) f ground state energy in the k-particle sector

Typical setting One-body potentials going to 0 at (cf bosonic atoms and boson stars) Typical example H V ( = xj + V (x j ) ) + 1 1 on L 2 (R d ) with j=1 1 k<l w(x k x l ) V, w L p + L 0, max(1, d/2) < p <, w(x) = w( x) V, w 0 at Can add magnetic fields, fractional Laplacians... Hartree s functionals. Possible loss of mass at infinity ( EH V [u] = u 2 + V u 2) + 1 u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R d EH[u] 0 = u 2 + 1 u(x) 2 w(x y) u(y) 2 dxdy R d 2 R d R { d eh V (λ) = inf EH V [u] } u 2 = λ. R d Large binding inequalities: for all 0 λ 1: e V H (1) e V H (λ) + e 0 H(1 λ)

Validity of Hartree for general systems Theorem (Lewin-am-R 13) Energy: lim 1 E V () = e V H (1). Take Ψ a sequence of approximate ground states in H. There exists a Borel probability measure µ supported on the set { M V = u H, u 1 } EH V [u] = eh V ( u 2 ) = eh V (1) eh(1 0 u 2 ), such that γ (k) Ψ j M V u k u k dµ(u). If strict binding holds for Hartree: e V H (1) < e V H (λ) + e 0 H(1 λ). Then µ is supported on SH and the limit is strong. If in addition e V H (1) admits a unique minimizer u H, there is complete Bose-Einstein condensation γ (k) Ψ j u k H u k H strongly in S 1 (H k ).

Conclusion The validity of Hartree s theory in the mean-field scaling is a very general fact and follows from the structure of bosonic states. Main structure property: quantum de Finetti theorem, solving the representability problem in the limit. Explicit construction and estimates in finite dimension. General case using localization in Fock space. Outlook Quantitative estimates for the validity of Hartree s theory by this method? Extend the method to derive (static) non-linear Schrödinger equations by scaling the interaction potential? Discuss attractive interactions and many-body collapse? Positive (large) temperature?