n The Bogoliubov dispersion relation depends on λ and ρ := L d only through λρ . It is therefore natural to set λ := ρ − 1 , which we will do in what follows. Thus the initial Hamiltonian becomes p a p + 1 ∞ � � ˆ p 2 a ∗ V (k) a ∗ p+k a ∗ H = n =0 H n = ⊕ q − k a q a p . 2 N p p , q , k The Bogoliubov Hamiltonian depends on L only through the choice of the lattice spacing 2 π L . We expect that the low energy part of the excitation spectra of H n and H Bog are close to one another for large n , hoping that then n − n 0 → 0 . We expect some kind of uniformity wrt L .
Note that formally we can even take the limit L → ∞ obtaining � H Bog − E Bog = (2 π ) − d a ∗ ω (p)˜ p ˜ a p dp , � P = (2 π ) − d a ∗ p˜ p ˜ a p dp .
(For finite L ) set β p � � � � a ∗ p a ∗ � U = exp − p − a p a − p . 2 p � =0 Then U is unitary and a p = U ∗ a p U, ˜ a ∗ p = U ∗ a ∗ ˜ p U, H Bog = E Bog + U ∗ � ω (p) a ∗ p a p U, p � =0 P = U ∗ � p a ∗ p a p U. p � =0
The excitation spectrum of H Bog is given by � � spec H Bog − E Bog , P \{ (0 , 0) } j j : p 1 , . . . , p j ∈ 2 π �� � � � � L Z d \{ 0 } , j = 1 , 2 , . . . = ω (p i ) , p i . i =1 i =1
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.
6 RIGOROUS RESULTS ON EXCITATION SPECTRUM OF INTERACTING BOSONS Jan Derezi´ nski and Marcin Napi´ orkowski: On the excitation spectrum of interacting bosons in the infinite-volume mean-field limit, Annales Henri Poincare, DOI: 10.1007/s00023-013-0302-4
Our main result says that for large n and not too large L the low energy part of the excitation spectrum of H n is well approximated by the low energy part of the excitation spectrum of H Bog . Note that we cannot make L go to infinity arbitrarily fast as n → ∞ . In particular, when we want to use arguments based on the weak coupling, we should assume λ − 1 = ρ = n L d → ∞ .
Before we describe our result let us introduce some notation. Let A be a bounded from below self-adjoint operator with only discrete 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 � � � � Bog (p) , . . . sp H Bog (p) − E Bog . The lowest eigenvalue of H Bog (0) − E Bog is obviously 0 . Set := − → 0 , K 1 Bog (0) , K 2 � � � � Bog (0) , . . . sp H Bog (0) − E Bog .
Besides the assumptions on V that we already mentioned ˆ V ≥ 0 , V ≥ 0 we add technical assumptions � V (x)dx < ∞ , ˆ V (p) ≤ C (1 + | p | ) − µ , µ > d.
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 v (0)( n − 1) + E Bog + K j E n + K j 2ˆ Bog (p) n (p) + L d � 3 / 2 . − Cn − 1 / 2 L d/ 2+3 � K j
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 v (0)( n − 1) + E Bog + K j E n + 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.
Basic tools of the proof: Consequence of the min-max principle: A ≤ B implies − → sp( A ) ≤ − → sp( B ) . Rayleigh-Ritz principle: − → sp( A ) ≤ − → � � � sp P K AP K . � � K
It is impossible to apply the Raileigh-Ritz principle directly, be- cause the physical Hamiltonian H n acts on the physical space H n and the Bogoliubov Hamiltonian H Bog acts on the Fock space l 2 � 2 π � �� L Z d \{ 0 } H Bog := Γ s . These spaces are incomparable – neither is contained in the other. Introduce the operator of the number of particles outside of the zeroth mode N > := � a ∗ p a p . p � =0 We want to use the fact that on low energy states N > is small.
The exponential property of Fock spaces says Γ s ( Z 1 ⊕ Z 2 ) ≃ Γ s ( Z 1 ) ⊗ Γ s ( Z 2 ) . We have l 2 � 2 π ≃ C ⊕ l 2 � 2 π L Z d � � L Z d \{ 0 } Thus H ≃ Γ s ( C ) ⊗H Bog . 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 H ext := l 2 ( Z ) ⊗ H Bog
The operator N 0 extends to an operator N ext 0 . Similarly, N ex- tends to N ext = N ext + N > . The space H sits in H ext : 0 l [0 , ∞ [ ( N ext 0 ) H ext , H = 1 l n ( N ext )1 l [0 , ∞ [ ( N ext 0 ) H ext . H n = 1 For any value of n there is a copy of H Bog in H ext : H Bog ≃ H ext l n ( N ext ) H ext . := 1 n
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 ∗ k satisfy the same CCR as a p and a ∗ k .
Let us repeat Bogoliubov’s heuristic argument: ˆ � ˆ V (0) p 2 + N 0 � �� � V (p) + ˆ 2 N a ∗ 0 a ∗ a ∗ H ≃ 0 a 0 a 0 + V (0) p a p N p � =0 1 � � � ˆ a ∗ 0 a ∗ 0 a p a − p + a 0 a 0 a ∗ p a ∗ + V (p) − p 2 N p � =0 ˆ � ˆ V (0) p 2 + N 0 � �� � V (p) + ˆ b ∗ = 2 N N 0 ( N 0 − 1) + V (0) p b p N p � =0 1 �� � � ˆ N 0 ( N 0 − 1) b p b − p + b ∗ p b ∗ � + V (p) N 0 ( N 0 − 1) − p 2 N p � =0 ˆ V (0) p 2 + ˆ � b ∗ � � ≃ ( N − 1) + V (p) p b p 2 p � =0 ˆ V (p) � b ∗ p b ∗ � � + − p + b p b − p 2 p � =0
In the actual proof we use an estimating Hamiltonian on H n H n,ǫ := 1 | p | 2 + ˆ ˆ � a ∗ � � V (0)( n − 1) + V (p) p a p 2 p � =0 + 1 � � � ˆ a ∗ 0 a ∗ 0 a p a − p + a ∗ p a ∗ V (p) − p a 0 a 0 2 n p � =0 ˆ ˆ � ˆ − 1 V (0) V (0) p a p N > + � a ∗ 2 n N > � V (p) + n 2 p � =0 � ˆ + ǫ p a p N 0 + +(1 + ǫ − 1 ) 1 2 nV (0) L d N > ( N > − 1) � V (p) + ˆ a ∗ � 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 ˆ � H ext b ∗ � � V (0)( n − 1) + V (p) p b p 2 p � =0 �� ( N ext − 1) N ext +1 � � ˆ 0 0 V (p) b p b − p + hc 2 n p � =0 ˆ ˆ � ˆ − 1 V (0) V (0) p b p N > + � b ∗ 2 n N > � V (p) + n 2 p � =0 � ˆ + ǫ � V (p) + ˆ b ∗ p b p N ext � V (0) 0 n p � =0 +(1 + ǫ − 1 ) 1 2 nV (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 � � � � ˆ b ∗ b p b − p + b ∗ p b ∗ � � V (p) V (p) . − p 2 p � =0 p � =0 preserves H ext Its restriction to H ext n . will be denoted H Bog ,n . n Clearly, H Bog ,n is unitarily equivalent to H Bog .
n,ǫ = 1 ˆ H ext V (0)( n − 1) + H Bog ,n + R n,ǫ , 2 ��� ( N ext − 1) N ext R n,ǫ := 1 � � ˆ � 0 0 V (p) − 1 b p b − p + hc 2 n p � =0 ˆ ˆ � ˆ − 1 V (0) V (0) p b p N > + � b ∗ 2 n N > � V (p) + n 2 p � =0 � ˆ + ǫ + (1 + ǫ − 1 ) 1 2 nV (0) L d N > ( N > − 1) . � V (p) + ˆ b ∗ p b p N ext � V (0) 0 n p � =0
Proof of lower bound. We use the inclusion H n ⊂ H ext n . For brevity set l n 1 κ := 1 l [0 ,κ ] ( H n − E n ) . For 0 < ǫ ≤ 1 , � 1 � ˆ l n l n l n l n 1 κ H n 1 κ ≥ 1 V (0)( n − 1) + H Bog ,n + R n, − ǫ 1 κ . κ 2 Hence, ≥ 1 − → V (0)( n − 1) + − → � � � � ˆ l n l n sp 1 κ H n 1 sp H Bog − � R n, − ǫ � . κ 2
Proof of upper bound. Let G ∈ C ∞ ([0 , ∞ [) , G ≥ 0 , 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 Z κ is a partial isometry with initial space Ran( G ( N > /n )1 ) ⊂ H κ l Bog ) ⊂ H ext and final space Ran(1 n . κ
� � � � − → sp H n ≤ − → � = − → � Z ∗ κ Z κ H n Z ∗ Z κ H n Z ∗ sp κ Z κ sp . � � κ l Bog � � Ran Z ∗ Ran1 κ κ Z κ H n Z ∗ κ ≤ Z κ H n,ǫ Z ∗ κ = 1 ˆ l Bog l Bog V (0)( n − 1)1 + H Bog 1 κ κ 2 + Z κ ( H Bog − E Bog ) Z ∗ l Bog κ − ( H Bog − E Bog )1 κ + Z κ R n,ǫ Z ∗ κ .
Therefore, → − sp( H n ) ≤ − → Z κ H n,ǫ Z ∗ � � sp κ = 1 V (0)( n − 1) + − → � � ˆ l Bog sp H Bog 1 κ 2 � � � Z κ ( H Bog − E Bog ) Z ∗ l Bog + κ − ( H Bog − E Bog )1 � � κ � � � � Z κ R n,ǫ Z ∗ + � . � � κ
7 FINITE VOLUME EFFECTS
L Z d we define the boost operator in the direction of w : For w ∈ 2 π n � � � U (w) := exp i x i w . i =1 We easily compute U ∗ (w) P n U (w) = P n + w n, � � H n − 1 U (w) = H n − 1 U ∗ (w) n ( P n ) 2 n ( P n ) 2 Hence spec H (p + n w) − (p + n w) 2 = spec H (p) − p 2 n . n
Excitation spectrum of free Bose gas in finite volume L ( k ) k n 2 n 2 - L L
Excitation spectrum of interacting Bose gas in finite volume L ( k ) k n 2 n 2 - L L
In dimension d = 1 in the limit L → ∞ we have ǫ (k + 2 πρ ) = ǫ (k) , because ( H L,n − E )Φ = 0 , ( P L,n − k)Φ = 0 , with U = U ( 2 π L ) , implies ( H L,n − E ) U Φ = 1 L (2 π k + 2 π 2 ρ ) U Φ → 0 , ( P L,n − k − 2 πρ ) U Φ = 0 .
Excitation spectrum of 1-dimensional interacting Bose gas ( k ) k 2 -2 0
In Landau’s argument we gave the following picture of the tilted Hamiltonian: energy k In finite volume it is incorrect.
Travelling Bose gas in finite volume energy k 2 n 2 n - L L
Define the global critical velocity ǫ L,n (k) c L,n cr := inf | k | | k | cr , then the ground state of H L,n remains the ground If | w | < c L,n state of the “tilted Hamiltonian”, hence it is stable. cr = π cr ≤ π For the free Bose gas we have c L,n L > 0 . In general, c L,n L . Hence the global critical velocity is very small and vanishes in the thermodynamic limit.
Define the restricted critical velocity below the momentum R as � ǫ L,n (k) � c L,n cr ,R := inf k � = 0 , | k | < R . | k | We expect that for repulsive potentials n c ρ L →∞ c L,n cr ,R := lim cr ,R , L d = ρ, R →∞ c ρ exists and, in dimension d ≥ 2 , we have c ρ cr := lim inf cr ,R > 0 . This may imply the metastability against travelling perturbations travelling at a speed smaller than c ρ cr .
8 GRAND-CANONICAL APPROACH
� L 2 ([ L/ 2 , L/ 2] d ) � Consider the symmetric Fock space Γ s and the (canonical) Hamiltonian H with λ = 1 . For a chemical potential µ > 0 , we define the grand-canonical Hamiltonian H µ := H − µN (p 2 − µ ) a ∗ � = p a p p + 1 � ˆ V (k) a ∗ p+k a ∗ q − k a q a p . 2 L d p , q , k
If E µ is the ground state energy of H µ , then it is realized in the sector n satisfying ∂ µ E µ = − n. In what follows we drop the subscript µ .
For α ∈ C , we define the displacement or Weyl operator of the zeroth mode: W α := e − αa ∗ 0 + αa 0 . Let Ω α := W α Ω be the corre- sponding coherent vector. Note that P Ω α = 0 . The expectation of the Hamiltonian in Ω α is ˆ V (0) (Ω α | H Ω α ) = − µ | α | 2 + 2 L d | α | 4 . √ L d µ It is minimized for α = e i τ √ V (0) , where τ is an arbitrary phase. ˆ
We apply the Bogoliubov translation to the zero mode of H by W ( α ) . This means making the substitution a 0 + α, a ∗ a ∗ a 0 = ˜ 0 = ˜ 0 + α, a ∗ a ∗ a k = ˜ a k , k = ˜ k , k � = 0 . Note that a k = W ∗ a ∗ k = W ∗ α a ∗ ˜ α a k W α , ˜ k W α , and thus the operators with and without tildes satisfy the same commutation relations. We drop the tildes.
Translated Hamiltonian µ 2 H := − L d 2 ˆ V (0) � � 1 V (k) µ 2k 2 + ˆ � a ∗ + k a k ˆ V (0) k µ � ˆ e − i2 τ a k a − k + e i2 τ a ∗ k a ∗ � � + V (k) − k 2 ˆ V (0) k V (k) √ µ ˆ � (e − i τ a ∗ k+k ′ a k a k ′ + e i τ a ∗ k a ∗ + k ′ a k+k ′ ) � ˆ V (0) L d k , k ′ ˆ V (k 2 − k 3 ) � a ∗ k 1 a ∗ + k 2 a k 3 a k 4 . 2 L d k 1 +k 2 =k 3 +k 4
If we (temporarily) replace the potential V (x) with λV (x) , where λ is a (small) positive constant, the translated Hamiltonian can be rewritten as √ H λ = λ − 1 H − 1 + H 0 + λH 1 2 + λH 1 . Thus the 3rd and 4th terms are in some sense small, which sug- gests dropping them.
Thus µ 2 0 + e − i τ a 0 ) 2 + H Bog , H ≈ − L d + µ (e i τ a ∗ 2 ˆ V (0) where � 1 V (k) µ 2k 2 + ˆ � � a ∗ H Bog = k a k ˆ V (0) k � =0 µ � ˆ e − i2 τ a k a − k + e i2 τ a ∗ k a ∗ � � + V (k) − k 2 ˆ V (0) k � =0
Then we proceed as before with the Bogoliubov energy � ˆ ˆ E Bog := − 1 V (p) V (p) | p | 2 + µ | p | 2 + 2 µ � − | p | ˆ ˆ 2 V (0) V (0) p � =0 and the Bogoliubov dispersion relation � ˆ V (p) | p | 2 + 2 µ ω (p) = | p | . ˆ V (0)
Note that the grand-canonical Hamiltonian H µ is invariant wrt the U (1) symmetry e i τN . The parameter α has an arbitrary phase. Thus we broke the symmetry when translating the Hamiltonian. The zero mode is not a harmonic oscillator – it has continuous spectrum and it can be interpreted as a kind of a Goldstone mode.
9 IMPROVING BOGOLIUBOV APPROXIMATION
L Z d ∋ k �→ θ k ∈ C be a sequence with θ k = θ − k . Let α ∈ C and 2 π Set e − 1 2 θ k a ∗ k a ∗ − k + 1 � 2 θ k a k a − k U θ := k Recall that W α := e − αa ∗ 0 + αa 0 . Then U α,θ := U θ W α is the general form of a Bogoliubov transformation commuting with momentum.
Let Ω denote the vacuum vector. Ψ α,θ := U ∗ α,θ Ω is the general form of a squeezed vector of zero momentum. We are looking for α, θ such that (Ψ α,θ | H Ψ α,θ ) ( ∗ ) attains the minimum. ( ∗ ) is equal to (Ω | U α,θ HU ∗ α,θ Ω) . Therefore, to find ( ∗ ) it is enough to compute the Bogoliubov- rotated Hamiltonian U α,θ HU ∗ α,θ and transform it to the Wick or- dered form.
This can be done by noting that U α,θ a ∗ k U ∗ α,θ = c k a ∗ k − s k a − k + δ 0 , k α, U α,θ a k U ∗ α,θ = c k a k − s k a ∗ − k + δ 0 , k α, where c k := cosh | θ k | , s k := − θ k | θ k | sinh | θ k | . and inserting this into H .
This is usually presented in a different but equivalent way: one introduces b k := U ∗ b ∗ k := U ∗ α,θ a ∗ α,θ a k U α,θ , k U α,θ , and one inserts k − s k b − k + δ 0 , k α, a k = c k b k − s k b ∗ a ∗ k = c k b ∗ − k + δ 0 , k α, into the expression for the Hamiltonian.
H = B + Cb ∗ 0 + Cb 0 + 1 − k + 1 � � � O (k) b ∗ k b ∗ D (k) b ∗ O (k) b k b − k + k b k 2 2 k k k + terms higher order in b ’s .
Clearly we have bound E ≤ (Ψ α,θ | H Ψ α,θ ) = B, Vectors Ψ α,θ, k := U ∗ α,θ a ∗ k Ω have momentum k , that means ( P − k)Ψ α,θ, k = 0 . We can use Ψ α,θ, k to obtain a variational upper bound for the infi- mum of energy-momentum spectrum: E + ǫ (k) ≤ (Ψ α,θ, k | H Ψ α,θ, k ) = B + D (k) .
Recall that we look for the infimum of (Ψ α,θ | H Ψ α,θ ) = B , Computing the derivatives with respect to α and α we obtain C = c 0 ∂ α B − s 0 ∂ α B so that the condition ∂ α B = ∂ α B = 0 entails C = 0 .
Computing the derivatives with respect to s and s we obtain − 2 c k + | s k | 2 ∂ s k B − s 2 � � k O (k) = ∂ s k B. c k c k Thus ∂ s k B = ∂ s k B = 0 entails O (k) = 0 .
Instead of s k , c k , it is more convenient to use functions S k := 2 s k c k , C k := c 2 k + | s k | 2 . We will keep α = | α | e i τ instead of µ as the parameter of the theory. We can later on express µ in terms of α 2 : ˆ V (0) + ˆ ˆ ˆ V (k ′ ) V (k ′ ) V (0) L d | α | 2 + � ( C k ′ − 1) − e i2 τ � µ = 2 L d S k ′ , 2 L d k ′ k ′ ρ = | α | 2 + � k | s k | 2 . L d
We obtain a fixed point equation � f 2 k − | g k | 2 , D (k) = g k S k = D (k) , C k = f k , D k 2 + | α | 2 ˆ f k : = k 2 V (k) L d v (k ′ − k) − ˆ ˆ V (k ′ ) V (k ′ ) ˆ � � 2 L d e i2 τ S k ′ , ( C k ′ − 1) + + 2 L d k ′ k ′ V (k ′ − k) g k : = | α | 2 e i2 τ ˆ ˆ V (k) � L d − S k ′ . 2 V k ′
√ L d κ , where κ has In the limit L → ∞ one should take α = the interpretation of the density of the condensate. Then one could expect that S k will converge to a function depending on k ∈ R d in 1 1 � � a reasonable class and we can replace by dk . L d (2 π ) d k In particular, � � � ˆ ˆ ˆ � V (0) V (k) V (0) κ � 2 L d α 2 � ˆ � D (0) = L d S k → V (k) S k dk . 2(2 π ) d k
Thus we expect that D (0) > 0 , which would mean that we have an energy gap in this approximation. It is believed that this is an artefact of the approach and that the true excitation spectrum of the Bose gas has no energy gap. Thus while we improved the approximation quantitatively, we made it worse qualitatively.
10 HOMOGENEOUS FERMI GAS
We consider fermions with spin 1 2 described by the Hilbert space H n := ⊗ n L 2 ( R d , C 2 ) � � . a We use the chemical potential from the beginning and we do not to assume the locality of interaction, so that the Hamiltonian is n � � � � H n = − ∆ i − µ + λ v ij . i =1 1 ≤ i<j ≤ n
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