Structure of the Excitation Spectrum for Many-Body Quantum Systems Robert Seiringer IST Austria Variational Problems in Physics Fields Institute, Toronto, October 2, 2014 R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 1
Introduction First realization of Bose-Einstein Condensation (BEC) in cold atomic gases in 1995: In these experiments, a large num- ber of (bosonic) atoms is confined to a trap and cooled to very low tem- peratures. Below a critical tem- perature condensation of a large fraction of particles into the same one-particle state occurs. Interesting quantum phenomena arise, like the appearance of quantized vortices and superfluidity. The latter is related to the low-energy excitation spectrum of the system. BEC was predicted by Einstein in 1924 from considerations of the non-interacting Bose gas. The presence of particle interactions represents a major difficulty for a rigorous derivation of this phenomenon. R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 2
R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 3
Quantum Mechanics 101 At low temperature , quantum mechanics determines the motion of the particles. Allowed quantum states ψ j determined by Schr¨ odinger’s equation − ∆ ψ j ( x ) + V ( x ) ψ j ( x ) = E j ψ j ( x ) ( ∂/∂x ( i ) ) 2 . Mathematically with ∆ = ∑ 3 i =1 extremely well understood. Explicit solutions for some potentials V ( x ) , e.g., harmonic os- cillator V ( x ) = | x | 2 . R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 4
Bosons and Fermions Indistinguishable particles in nature come in two types: bosons (fermions) have permutation-(anti-)symmetric wavefunctions Ψ( x 1 , . . . , x i , . . . , x j , . . . , x N ) = ( − 1) Ψ( x 1 , . . . , x j , . . . , x i , . . . , x N ) ���� for fermions If one neglects interactions among the particles, Ψ( x 1 , . . . , x N ) is just an (anti-) symmetrized product of functions ψ k 1 ( x 1 ) ψ k 2 ( x 2 ) · · · ψ k N ( x N ) with ψ k appearing n k times, say. For fermions, n k ∈ { 0 , 1 } ( Pauli exclusion principle ), for bosons n k ∈ { 0 , 1 , . . . , N } . Bosons at zero temperature display complete Bose-Einstein condensation . R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 5
The Bose Gas: A Quantum Many-Body Problem Quantum-mechanical description in terms of the Hamiltonian for a gas of N bosons with pair-interaction potential v ( x ) . In appropriate units, N ∑ ∑ H N = − ∆ i + v ( x i − x j ) i =1 1 ≤ i<j ≤ N The kinetic energy is described by the ∆ , the Laplacian on a box [0 , L ] 3 , with periodic boundary conditions. As appropriate for bosons , H acts on permutation-symmetric wave functions Ψ( x 1 , . . . , x N ) in ⊗ N L 2 ([0 , L ] 3 ) . The interaction v is assumed to be repulsive and of short range . Example: hard spheres, v ( x ) = ∞ for | x | ≤ a , 0 for | x | > a . R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 6
Quantities of Interest • Ground state energy E 0 ( N, L ) = inf spec H N In particular, energy density in the thermodynamic limit N → ∞ , L → ∞ with N/L 3 = ϱ fixed, i.e., E 0 ( ϱL 3 , L ) e ( ϱ ) = lim L 3 L →∞ • At positive temperature T = β − 1 > 0 , one looks at the free energy F ( N, L, T ) = − 1 β ln Tr exp( − βH N ) and the corresponding energy density in the thermodynamic limit F ( ϱL 3 , L, T ) f ( ϱ, T ) = lim L 3 L →∞ R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 7
• The one-particle density matrix of the ground state Ψ 0 (or any other state) is given by the integral kernel ∫ γ 0 ( x, x ′ ) = N R 3( N − 1) Ψ 0 ( x, x 2 , . . . , x N )Ψ ∗ 0 ( x ′ , x 2 , . . . , x N ) dx 2 · · · dx N It satisfies 0 ≤ γ 0 ≤ N as an operator, and Tr γ 0 = N . Bose-Einstein condensation in a state means that the one-particle density ma- trix γ 0 has an eigenvalue of order N , i.e., that ∥ γ 0 ∥ ∞ = O ( N ) . The corresponding eigenfunction is called the condensate wave function . For Gibbs states of translation invariant systems ∫ ∥ γ 0 ∥ ∞ = 1 [0 ,L ] 6 γ 0 ( x, x ′ ) dx dx ′ L 3 and this being order N = ϱL 3 means that γ 0 ( x, x ′ ) does not decay as | x − x ′ | → ∞ , which is also termed long range order . BEC is expected to occur below a critical temperature . R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 8
Satyendra Nath Bose Albert Einstein (1894–1974) (1879–1955) R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 9
• The structure of the excitation spectrum , i.e., the spectrum of H N above the ground state energy E 0 ( N ) , and the relation of the corresponding eigenstates to the ground state. For translation invariant systems, H N commutes with the total momentum N ∑ P = − i ∇ j j =1 and hence one can look at their joint spectrum . Of particular relevance is the infimum E q ( N, L ) = inf spec H N ↾ P = q and one can investigate the limit ( ) E q ( ϱL 3 , L ) − E 0 ( ϱL 3 , L ) e q ( ϱ ) = lim for fixed ϱ and q L →∞ For interacting systems, one expects a linear behavior of e q ( ϱ ) for small q . R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 10
The Ideal Bose Gas For non-interacting bosons ( v ≡ 0 ), the free energy can be calculated explicitly: [ ] ∫ ( ) 1 1 − exp( − β ( p 2 − µ )) f 0 ( ϱ, T ) = sup µϱ + R 3 ln dp (2 π ) 3 β µ< 0 If ( T ) 3 / 2 ∫ 1 1 ϱ ≥ ϱ c ( β ) ≡ e βp 2 − 1 dp = ζ (3 / 2) (2 π ) 3 4 π R 3 the supremum is achieved at µ = 0 and hence ∂f 0 /∂ϱ = 0 for ϱ ≥ ϱ c . In other words, the critical temperature equals 4 π T (0) ζ (3 / 2) 2 / 3 ϱ 2 / 3 ( ϱ ) = c The one-particle density matrix for the ideal Bose gas is given by ∑ e βµ ϱ n (4 πβn ) 3 / 2 e −| x − y | 2 / (4 βn ) γ 0 ( x, y ) = [ ϱ − ϱ c ( β )] + + n ≥ 0 R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 11
The spectrum of the Laplacian on [0 , L ] 3 with periodic boundary conditions is { ) 3 } ( 2 π | p | 2 : p ∈ σ ( − ∆) = L Z with corresponding eigenfunctions the plane waves φ p ( x ) = L − 3 / 2 e ip · x . Hence the spectrum of the ideal gas Hamiltonian N ∑ H (0) N = − ∆ i i =1 is simply ∑ ∑ σ ( H (0) | p | 2 n p : n p ∈ N 0 , N ) = n p = N p p ∈ ( 2 π L Z ) 3 and the corresponding eigenfunctions are symmetrized tensor products of the φ p ’s. R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 12
Second Quantization on Fock space In the following, it will be convenient to regard ⊗ N sym L 2 ([0 , L ] 3 ) as a subspace of the bosonic Fock space ∞ n ⊕ ⊗ L 2 ([0 , L ] 3 ) F = sym n =0 A basis of L 2 ([0 , L ] 3 ) is given by the plane waves L − 3 / 2 e ipx for p ∈ ( 2 π L Z ) 3 , and we introduce the corresponding creation and annihilation operators , satisfying the CCR [ ] [ ] [ ] a † p , a † a p , a † a p , a q = = 0 , = δ p,q q q The Hamiltonian H N is equal to the restriction to the subspace ⊗ N sym L 2 ([0 , L ] 3 ) of ∑ ∑ ∑ 1 a † q + p a † | p | 2 a † H = p a p + � v ( p ) k − p a k a q 2 L 3 p p q,k ∫ where [0 ,L ] 3 v ( x ) e − ipx dx � v ( p ) = denotes the Fourier transform of v . R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 13
The Bogoliubov Approximation At low energy and for weak interactions, one expects Bose-Einstein condensation, meaning that a † 0 a 0 ∼ N . Hence p = 0 plays a special role. The Bogoliubov approximation consists of • dropping all terms higher than quadratic in a † p and a p for p ̸ = 0 . √ • replacing a † 0 and a 0 by N The resulting Hamiltonian is quadratic in the a † p and a p , and equals (( ( )) ∑ ) H Bog = N ( N − 1) | p | 2 + ϱ � p a † a † a † p a p + 1 v (0) + � v ( p ) 2 ϱ � v ( p ) − p + a p a − p 2 L 3 p ̸ =0 with ϱ = N/L 3 . It can be diagonalized via a Bogoliubov transformation . R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 14
Bogoliubov Transformation Let b p = cosh( α p ) a p + sinh( α p ) a † − p , with √ tanh( α p ) = | p | 2 + ϱ � | p | 4 + 2 | p | 2 ϱ � v ( p ) − v ( p ) ϱ � v ( p ) Here, we have to assume that | p | 2 + 2 ϱ � v ( p ) ≥ 0 for all p . The b p and b † p again satisfy CCR . A simple calculation yields ∑ H Bog = E Bog e p b † + p b p 0 p ̸ =0 where ( ) ∑ √ = N ( N − 1) v (0) − 1 | p | 2 + ϱ � E Bog | p | 4 + 2 | p | 2 ϱ � � v ( p ) − v ( p ) 0 2 L 3 2 p ̸ =0 and √ | p | 4 + 2 | p | 2 ϱ � e p = v ( p ) R. Seiringer – Excitation Spectrum for Many-Body Quantum Systems – October 2, 2014 Nr. 15
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