EXCITATION SPECTRUM OF INTERACTING BOSONS IN THE MEAN FIELD INFINITE - PowerPoint PPT Presentation

EXCITATION SPECTRUM OF INTERACTING BOSONS IN THE MEAN FIELD INFINITE VOLUME LIMIT JAN DEREZI´ NSKI Dept. of Math. Methods in Phys., Faculty of Physics, University of Warsaw Joint work with MARCIN NAPI´ ORKOWSKI

We show that low lying excitation spectrum of N -body bosonic Schr¨ odinger Hamiltonians with repulsive interaction is approximately given by the Bogoliubov approximation. We consider the limit N → ∞ , weak coupling and large density. We allow for an ar- bitrarily large size of a box provided that it does not grow too fast with N .

We start with a potential that is a real function v on R d such that v (x) = v ( − x) and v ∈ L 1 ( R d ) , v ∈ L 1 ( R d ) , ˆ v (x) ≥ 0 , x ∈ R d , v (p) ≥ 0 , p ∈ R d . ˆ Then we replace the original v by the periodized potential v L (x) = 1 � e ipx ˆ v (p) , L d p ∈ (2 π/L ) Z d which is well defined on the torus [ − L/ 2 , L/ 2[ d .

We use the symmetric N -particle Hilbert space �� [ − L/ 2 , L/ 2[ d � N � L 2 s and the periodic boundary conditions indicated by L . Momentum N � P L i ∂ L N := − x i . i =1 Hamiltonian N i + L d � � H L ∆ L v L (x i − x j ) . N = − N i =1 1 ≤ i<j ≤ N In the sequel, we drop the superscript L .

L Z d and H N P N = P N H N . Hence Note that spec P N = 2 π H N = ⊕ H N (k) . k ∈ spec P N We can define the energy-momentum spectrum spec ( H N , P N ) . We will denote by E N the ground state energy of H N . By the excitation spectrum we will mean spec ( H N − E N , P N ) .

We introduce the Bogoliubov energy E Bog := − 1 � | p | 2 + ˆ � | p | 2 + 2ˆ � � v (p) − | p | v (p) 2 p ∈ 2 π L Z d \{ 0 } and the Bogoliubov dispersion relation | p | 2 + 2ˆ � e p = | p | v (p) .

Bogoliubov Hamiltonian � e p a † H Bog := E Bog + p a p , p � =0 Bogoliubov momentum � p a † P Bog := p a p , p � =0 Clearly, H Bog P Bog = P Bog H Bog . Above, a † p and a p are bosonic creation/annihilation operators on � �� L 2 � the bosonic Fock space Γ s spec ( P N ) \ 0 .

We would like to show that the excitation spectrum of H N is well approximated by the excitation spectrum of the Bogoliubov Hamiltonian. In the examples below we ilustrate that the latter has a special shape involving a positive critical velocity, which according to the Landau criterion is responsible for superfluidity.

v 1 (p) = e − p 2 / 5 ˆ 10

Excitation spectrum of 1-dimensional homogeneous Bose gas with potential v 1 in the Bogoliubov approximation.

v 2 (p) = 15e − p 2 / 2 ˆ 2

Excitation spectrum of 1-dimensional homogeneous Bose gas with potential v 2 in the Bogoliubov approximation.

Let A be a bounded from below self-adjoint operator with only dis- crete spectrum. We define − → sp( A ) := ( a 1 , a 2 , . . . ) , where a 1 , a 2 , . . . are the eigenvalues of A in the increasing order. If dim H = n , then we set a n +1 = a n +2 = · · · = ∞ .

Excitation energies of the N -body Hamiltonian. If p ∈ 2 π L Z d \{ 0 } , set := − → K 1 N (p) , K 2 � � � � N (p) , . . . sp H N (p) − E N . The lowest eigenvalue of H N (0) − E N is 0 by general arguments. Set := − → 0 , K 1 N (0) , K 2 � � � � N (0) , . . . sp H N (0) − E N .

Bogoliubov excitation energies. If p ∈ 2 π L Z d \{ 0 } , set := − → K 1 Bog (p) , K 2 H L Bog (p) − E L � � � � Bog (p) , . . . sp . Bog The lowest eigenvalue of H Bog (0) − E Bog is obviously 0 . Set := − → 0 , K 1 Bog (0) , K 2 H L Bog (0) − E L � � � � Bog (0) , . . . sp . Bog

L Z d the Bogoliubov excitation energies are given by For any p ∈ 2 π j e k i : k 1 , . . . , k j ∈ 2 π � � � L Z d \{ 0 } , k 1 + · · · +k j = p , j = 1 , 2 , . . . , i =1 in the increasing order.

Upper bound Let c > 0 . Then there exists C such that if L 2 d +2 ≤ cN, then E N ≥ 1 v (0)( N − 1) + E Bog − CN − 1 / 2 L 2 d +3 ; 2ˆ If in addition K j N (p) ≤ cNL − d − 2 , then N (p) ≥ 1 E N + K j v (0)( N − 1) + E Bog + K j 2ˆ Bog (p) N (p) + L d � 3 / 2 . K j − CN − 1 / 2 L d/ 2+3 �

Lower bound. Let c > 0 . Then there exists c 1 > 0 and C such that if L 2 d +1 ≤ cN , L d +1 ≤ c 1 N, then E N ≤ 1 v (0)( N − 1) + E Bog + CN − 1 / 2 L 2 d +3 / 2 ; 2ˆ Bog (p) ≤ cNL − d − 2 and K j If in addition K j Bog (p) ≤ c 1 NL − 2 , then N (p) ≤ 1 E N + K j v (0)( N − 1) + E Bog + K j 2ˆ Bog (p) + CN − 1 / 2 L d/ 2+3 ( K j Bog (p) + L d − 1 ) 3 / 2 .

Special case of this theorem with L = 1 was proven by R. Seiringer. Mimicking his proof gives big error terms for large L : they are of the order N − 1 / 2 exp( L d/ 2 ) . To get better error estimates we need to use additional ideas.

Bosonic Fock space l 2 � 2 π ∞ � L Z d �� H := N =0 H N = Γ s ⊕ . Hamiltonian in second quantized notation p a p + 1 ∞ v (k) a † p+k a † � � p 2 a † H := N =0 H N = ⊕ ˆ q − k a q a p . 2 N p p , q , k Number of particles in condensate N 0 = a † 0 a 0 . Number of particles outside of condensate N > = � a † p a p . p � =0

The exponential property of Fock spaces gives l 2 � 2 π � �� L Z d \{ 0 } H ≃ Γ s ( C ) ⊗ Γ s . Embed the space of zero modes Γ s ( C ) = l 2 ( { 0 , 1 , . . . } ) in a larger space l 2 ( Z ) . Thus we obtain the extended Hilbert space l 2 � 2 π H ext := l 2 ( Z ) ⊗ Γ s � �� L Z d \{ 0 } .

The operator N 0 extends to an operator N ext satisfying 0 l [0 , ∞ [ ( N ext H = Ran1 0 ) . N for the subspace of H ext corresponding If N ∈ Z , we will write H ext to N > + N ext = N . 0

We have also a unitary operator U | n 0 � ⊗ Ψ > = | n 0 − 1 � ⊗ Ψ > . We now define for p � = 0 the following operator on H ext : b p := a p U † . Operators b p and b † q satisfy the same CCR as a p and a † q .

Estimating Hamiltonian on H N H N,ǫ := 1 | p | 2 + ˆ � a † � � 2ˆ v (0)( N − 1) + v (p) p a p p � =0 + 1 � � � a † 0 a † p a † 0 a p a − p + a † v (p) ˆ − p a 0 a 0 2 N p � =0 − 1 v (p) + ˆ v (0) p a p N > + ˆ v (0) � a † 2 N N > � � ˆ N 2 p � =0 + ǫ p a p N 0 + +(1 + ǫ − 1 ) 1 2 N v (0) L d N > ( N > − 1) � a † � � v (p) + ˆ ˆ v (0) N p � =0 H N ≥ H N, − ǫ , 0 < ǫ ≤ 1; H N ≤ H N,ǫ , 0 < ǫ.

Extended estimating Hamiltonian on H ext N N,ǫ := 1 | p | 2 + ˆ � H ext b † � � 2ˆ v (0)( N − 1) + v (p) p b p p � =0 �� ( N ext − 1) N ext +1 � � 0 0 v (p) ˆ b p b − p + hc 2 N p � =0 − 1 v (p) + ˆ v (0) p b p N > + ˆ v (0) � b † 2 N N > � � ˆ N 2 p � =0 + ǫ � b † p b p N ext � � v (p) + ˆ ˆ v (0) 0 N p � =0 +(1 + ǫ − 1 ) 1 2 N v (0) L d N > ( N > − 1) . H ext N,ǫ preserves H N and restricted to H N coincides with H N,ǫ .

p b p + 1 | p | 2 + ˆ � � � � p b † b † b p b − p + b † � � v (p) v (p) ˆ . − p 2 p � =0 p � =0 preserves H ext N . Its restriction to H ext N will be denoted H Bog ,N . Ap- plying an appropriate Bogoliubov transformation we see that H Bog ,N is unitarily equivalent to H Bog , which we introduced before.

N,ǫ = 1 H ext 2ˆ v (0)( N − 1) + H Bog ,N + R N,ǫ , ��� ( N ext − 1) N ext R N,ǫ := 1 � � � 0 0 v (p) ˆ − 1 b p b − p + hc 2 N p � =0 − 1 v (p) + ˆ v (0) p b p N > + ˆ v (0) � b † 2 N N > � � ˆ N 2 p � =0 + ǫ + (1 + ǫ − 1 ) 1 2 N v (0) L d N > ( N > − 1) . � b † p b p N ext � � v (p) + ˆ ˆ v (0) 0 N p � =0

Consequence of the min-max principle: A ≤ B implies − → sp( A ) ≤ − → sp( B ) . Rayleigh-Ritz principle: � − → sp( A ) ≤ − → � � sp P K AP K . � � K

Proof of lower bound For brevity set l N 1 κ := 1 l [0 ,κ ] ( H N − E N ) . For 0 < ǫ ≤ 1 , � 1 � l N l N l N l N 1 κ H N 1 κ ≥ 1 2ˆ v (0)( N − 1) + H Bog ,N + R N, − ǫ 1 κ . κ Hence, ≥ 1 − → v (0)( N − 1) + − → � � � � l N l N sp 1 κ H N 1 2ˆ sp H Bog − � R N, − ǫ � . κ

Suppose now that G is a smooth nonnegative function on [0 , ∞ [ such that  if s ∈ [0 , 1  1 , 3 ]   G ( s ) =  0 , if s ∈ [1 , ∞ [ .   l Bog For brevity, we set 1 := 1 l [0 ,κ ] ( H Bog ,N − E Bog ) . We define κ � − 1 / 2 1 l Bog G ( N > /N ) 2 1 l Bog l Bog G ( N > /N ) . � Z κ := 1 κ κ κ l Bog Clearly, Z κ is a partial isometry with initial space Ran( G ( N > /N )1 ) κ l Bog and final space Ran(1 ) . κ

� � � � → − sp H N ≤ − → � = − → � Z † κ Z κ H N Z † Z κ H N Z † sp κ Z κ sp . � � κ Ran Z † l Bog � � Ran1 κ κ Z κ H N Z † κ ≤ Z κ H N,ǫ Z † κ = 1 l Bog l Bog 2ˆ v (0)( N − 1)1 + H Bog 1 κ κ l Bog + Z κ ( H Bog − E Bog ) Z † κ − ( H Bog − E Bog )1 κ + Z κ R N,ǫ Z † κ .

Therefore, − → sp( H N ) ≤ Z κ H N,ǫ Z † κ = 1 v (0)( N − 1) + − → � � l Bog 2ˆ sp H Bog 1 κ � � � Z κ ( H Bog − E Bog ) Z † l Bog + κ − ( H Bog − E Bog )1 � � κ � � � � Z κ R N,ǫ Z † + � . � � κ