On the infimum of the excitation spectrum of a homogeneous Bose gas H.D. Cornean, J. Derezi´ nski, P. Zi´ n. 1
Homogeneous Bose gas n bosons on R d interacting with a 2-body potential v are described by the Hilbert space L 2 � ( R d ) n � and the s Hamiltonian 1 � � H n = − 2∆ i + v ( x i − x j ) i =1 1 ≤ i<j ≤ n commuting with total momentum n P n := � − i ∇ x i . i =1 We assume that the potential v decays fast at infinity, v ( x ) = v ( − x ) and ˆ v ( k ) ≥ 0. 2
System in a box I We are interested in the properties at fixed density ρ . Therefore, we enclose the system in Λ = [0 , L ] d . We assume that the number of particles equals n = ρV , where V = L d is the volume. The Hilbert space is L 2 s (Λ n ). We replace the potential by its periodization v L ( x ) = 1 e i k · x ˆ � v ( k ) . V k ∈ 2 π L Z d 3
System in a box II The Hamiltonian, with periodic boundary conditions, equals n 1 � � H L,n 2∆ L v L ( x i − x j ) = − i + i =1 1 ≤ i<j ≤ n and has the ground state energy E L,n := inf sp H L,n . The total momentum n P L,n := � − i ∇ L x i i =1 has the spectrum sp P L,n = 2 π L Z d . 4
Infimum of the excitation spectrum L Z d we set For k ∈ 2 π ǫ L,n ( k ) := inf sp H L,n ( k ) − E L,n . Let k ∈ R d . For L → ∞ , keeping n V = ρ > 0, we set (informally) ǫ ρ ( k ) := lim L →∞ ǫ L,n ( k ) . 5
Boost operator Define the Boost operator in the direction of the first coordinate: � � n i2 π � U 1 := exp . x i, 1 L i =1 We easily compute 1 P L,n P L,n U ∗ + 2 πρL d − 1 , U 1 = 1 1 � � 1 1 H L,n − H L,n − 2 ρL d ( P L,n 2 ρL d ( P L,n U ∗ ) 2 ) 2 . U 1 = 1 1 1 Therefore, in particular e 1 ) = sp H L,n (0) + m 2 (2 π ) 2 ρL d − 2 sp H L,n ( m 2 πρL d − 1 ˆ . 2 6
Excitation spectrum of free Bose gas in finite volume L ( k ) k n 2 n 2 - L L 7
Excitation spectrum of interacting Bose gas in finite volume L ( k ) k n 2 n 2 - L L 8
1 -dimensional case Theorem. In dimension d = 1 we have ǫ ( k + 2 πρ ) = ǫ ( k ). Proof. If Φ ( H L,n − E )Φ = 0 , ( P L,n − k )Φ = 0 , then 1 ( H L,n − E ) U Φ L (2 π k + 2 π 2 ρ ) U Φ , = ( P L,n − k − 2 πρ ) U Φ = 0 . Then we let L → ∞ . 9
Excitation spectrum of 1-dimensional interacting Bose gas ( k ) k 2 -2 0 10
Conjecture about the infimum of the excitation spectrum Conjecture. (1) The map R d ∋ k �→ ǫ ρ ( k ) ∈ R + is continuous. (2) Let k ∈ R d . If L j → ∞ , n j j → ρ , k s ∈ 2 π L j Z d , then we L d have that ǫ L j ,n j ( k j ) → ǫ ρ ( k ). (3) If d ≥ 2, then there exists c cr > 0 such that ǫ ρ ( k ) > c cr | k | . (4) For some c ph > 0, ǫ ρ ( k ) ≈ c ph | k | for small k . (5) k �→ ǫ ρ ( k ) is subadditive, that is ǫ ρ ( k 1 ) + ǫ ρ ( k 2 ) ≥ ǫ ρ ( k 1 + k 2 ). 11
(1) and (2) can be interpreted a kind of a spectral thermodynamic limit. If (1) and (2) are true around k = 0, then there is no gap in the excitation spectrum. (It is easy to show that ǫ (0) = 0). (3) implies the superfluidity of the Bose gas. More precisely, a drop of Bose gas will travel without friction as long as its speed is less than c cr . (4) suggests that speed of sound for at low energies is well defined and equals c ph . If one can superimpose (almost) independent elementary excitations, then (5) is true. 12
The above conjecture has important physical consequences. In particular, it implies the superfluidity of the Bose gas at zero temperature. The above conjecture seems plausible. It is suggested by the arguments going back to Bogoliubov, Hugenholz, Pines, Bieliaev, as well as Bijls and Feynman. Nobody has an idea how to prove it rigorously. 13
Subadditive functions We say that R d ∋ k �→ ǫ ( k ) ∈ R is subadditive iff k 1 , k 2 ∈ R d . ǫ ( k 1 + k 2 ) ≤ ǫ ( k 1 ) + ǫ ( k 2 ) , Let R d ∋ k �→ ω ( k ) ∈ R be another function. We define the subbadditive hull of ω to be ǫ ( k ) := inf { ω ( k 1 )+ · · · + ω ( k n ) : k 1 + · · · + k n = k , n = 1 , 2 , . . . } . Clearly, the subadditive hull is always subadditive. 14
Quadratic Hamiltonians Consider the Hamiltonian � R d ω ( k ) a ∗ H = k a k d k , and the total momentum � R d k a ∗ P = k a k d k . We will call the function ω appearing in H the elementary excitation spectrum. Clearly, the infimum of the energy-momentum spectrum of H is equal to the subadditive hull of ω . 15
Excitation spectrum of free Bose gas energy ( k ) ( k ) k 16
Hypothethic excitation spectrum of interacting Bose gas with no “rotons” energy ( k ) ( k ) k 17
Hypothethic excitation spectrum of interacting Bose gas with “rotons” energy ( k ) ( k ) k 18
Criterion for subadditivity Theorem. (1) Let f be an increasing concave function with f (0) ≥ 0. Then f ( | k | ) is subadditive. (2) Let ǫ 0 be subadditive and ǫ 0 ≤ ω . Let ǫ be the subadditive hull of ω . Then ǫ 0 ≤ ǫ . It often happens that the subadditive hull of a function is equal to zero everywhere. This is the case e.g. when ω ( k ) = c k 2 , which corresponds to the free Bose gas. But not for superfluid systems: 19
Subadditive hulls with phononic shape and positive critical velocity Corollary. Suppose that c cr , c s > 0 and ω satisfies 1. ω ( k ) ≥ c cr | k | ; ω ( k ) 2. lim k → 0 | k | = c s . Let ǫ be the subadditive hull of ω . Then ǫ also satisfies 1. ǫ ( k ) ≥ c cr | k | ; ǫ ( k ) 2. lim k → 0 | k | = c s . 20
Landau’s argument for the superfluidity I We add to H the perturbation u travelling at a speed w: n i d � � � d t Ψ t = H + λ u ( x i − w t ) Ψ t . i =1 We go to the moving frame: Ψ w t ( x 1 , . . . , x n ) := Ψ t ( x 1 − w t, . . . , x n − w t ). We obtain a Schr¨ odinger equation with a time-independent Hamiltonian n i d � d t Ψ w Ψ w � � t = H − w P + λ u ( x i ) t . i =1 Is the ground state H Ψ gr = E Ψ gr stable against the travelling perturbation? 21
Bose gas travelling slower than critical velocity energy k 22
Bose gas travelling faster than critical velocity energy k 23
Travelling Bose gas in finite volume energy k n 2 n 2 - L L 24
Stability Define the global critical velocity ǫ L,n ( k ) c L,n := inf cr | k | | k | cr , then the ground state of H L,n remains the If | w | < c L,n ground state of the “tilted Hamiltonian”, hence it is stable. = π For the free Bose gas we have c L,n L > 0 . In general, cr c L,n ≤ π L . Hence the global critical velocity is very small cr and vanishes in the thermodynamic limit. 25
Metastability 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 , V = ρ, exists and, in dimension d ≥ 2, R →∞ c ρ lim inf cr ,R > 0 . We expect metastability against a travelling perturbation travelling at a smaller speed. 26
2nd quantized grand-canonical approach I Consider the symmetric Fock space Γ s ( L 2 (Λ)). For a chemical potential µ > 0, we define the grand-canonical Hamiltonian x ( − 1 � a ∗ H L = 2∆ x − µ ) a x d x +1 � � a ∗ x a ∗ y v L ( x − y ) a y a x d x d y , 2 n =0 ( H n,L − µn ) . ⊕ ∞ = 27
2nd quantized grand-canonical approach II In the momentum representation it equals (1 H L = 2 k 2 − µ ) a ∗ � k a k k + 1 � v ( k 2 − k 3 ) a ∗ k 1 a ∗ δ ( k 1 + k 2 − k 3 − k 4 )ˆ k 2 a k 3 a k 4 . 2 V k 1 , k 2 , k 3 , k 4 The momentum operator equals P L := � k k a ∗ k a k . If E L is the ground state energy of H L , then one can get the corresponding density by ∂ µ E L = − V ρ. 28
Infimum of the excitation spectrum – a rigorous definition I We define the ground state energy in the box E L = inf sp H L . For k ∈ 2 π L Z d , we define the infimum of the excitation spectrum in the box ǫ L ( k ) := inf sp H L ( k ) − E L . 29
Infimum of the excitation spectrum – a rigorous definition II For k ∈ R d we define the infimum of the excitation spectrum in the thermodynamic limit � � �� L | <δ ǫ L ( k ′ ǫ ( k ) := sup lim inf inf L ) . L →∞ L ∈ 2 π L Z d , | k − k ′ k ′ δ> 0 Proposition. At zero total momentum, the excitation spectrum has a global minimum where it equals zero: ǫ L (0) = ǫ (0) = 0. 30
Conjecture about the infimum of the excitation spectrum Conjecture. (1) The map R d ∋ k �→ ǫ ( k ) ∈ R + is continuous. (2) Let k ∈ R d . If L j → ∞ , k j ∈ 2 π L j Z d , k j → k , then ǫ L j ( k j ) → ǫ ( k ). (3) If d ≥ 2, then there exists c cr > 0 such that ǫ ( k ) > c cr | k | . (4) For some c ph > 0 such that ǫ ( k ) ≈ c ph | k | for small k . (5) k �→ ǫ ( k ) is subadditive. 31
Minimization among coherent states For α ∈ C , we define the displacement or Weyl operator of the zeroth mode: W α := e − αa ∗ 0 + αa 0 . Set Ω α := W α Ω. Note that P L Ω α = 0. The expectation of the Hamiltonian in those coherent states equals (Ω α | H L Ω α ) = − µ | α | 2 + ˆ v (0) 2 V | α | 4 , √ V µ and is minimized for α = e i τ √ v (0) . ˆ 32
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