Excitation spectrum of interacting bosons in the mean-field - PowerPoint PPT Presentation

Excitation spectrum of interacting bosons in the mean-field infinite-volume limit Marcin Napi´ orkowski Jan Derezi´ nski Faculty of Physics, University of Warsaw 21 III 2014 Warwick EPSRC Symposium on Statistical Mechanics: Many-Body Quantum Systems Marcin Napi´ orkowski Excitation spectrum of interacting bosons

Model Model We consider an interacting, homogeneous Bose gas. The Hamiltonian of such an N -particle system is given by N � � H N = − ∆ i + λ v ( x i − x j ) i =1 1 ≤ i<j ≤ N defined on the Hilbert space L 2 sym (( R d ) N ) . λ ≥ 0 is a coupling constant. We assume v is a real and symmetric function such that v ∈ L 1 ( R d ) , v ∈ L 1 ( R d ) ˆ v ( x ) ≥ 0 , x ∈ R d , ˆ v ( p ) ≥ 0 , p ∈ R d . Marcin Napi´ orkowski Excitation spectrum of interacting bosons

Model We want to describe a physical system of positive density in a large volume limit. To this end we replace R d by the torus ( − L/ 2 , L/ 2] d and the potential by its periodized version v L ( x ) = 1 � v ( p )e i px . ˆ L d p ∈ (2 π/L ) Z d The Hamiltonian in the box reads N � � H L ∆ L v L ( x i − x j ) . N = − i + λ i =1 1 ≤ i<j ≤ N The total momentum operator N P L � − i ∂ L N = x i . i =1 Marcin Napi´ orkowski Excitation spectrum of interacting bosons

The excitation spectrum The excitation spectrum H N and P N commute, thus we can consider the joint energy-momentum spectrum sp( H N , P N ) ⊂ R d +1 . Let E N denote the ground state energy of H N . Then Excitation spectrum := sp( H N − E N , P N ) . Marcin Napi´ orkowski Excitation spectrum of interacting bosons

The excitation spectrum Bogoliubov excitation spectrum The diagonalised Bogoliubov Hamiltonian � e ( p ) b † H Bog = E Bog + p b p p � =0 p 4 + 2 λρ ˆ v ( p ) p 2 and � with e ( p ) = E Bog := − 1 | p | 2 + ˆ � � � | p | 4 + 2 λρ ˆ � v ( p ) | p | 2 v ( p ) − . 2 p ∈ 2 π L Z d \{ 0 } Marcin Napi´ orkowski Excitation spectrum of interacting bosons

The excitation spectrum Bogoliubov excitation spectrum The diagonalised Bogoliubov Hamiltonian � e ( p ) b † H Bog = E Bog + p b p p � =0 p 4 + 2 λρ ˆ v ( p ) p 2 and � with e ( p ) = E Bog := − 1 | p | 2 + ˆ � � � | p | 4 + 2 λρ ˆ � v ( p ) | p | 2 v ( p ) − . 2 p ∈ 2 π L Z d \{ 0 } ⇒ Choice of mean-field scaling λ = 1 /ρ. Marcin Napi´ orkowski Excitation spectrum of interacting bosons

The excitation spectrum L Z d define the Bogoliubov elementary excitation For p ∈ 2 π p 4 + 2ˆ � spectrum e ( p ) := v ( p ) p 2 . L Z d we consider the Bogoliubov excitation energies For any p ∈ 2 π with total momentum p : � j � e ( k i ) : k 1 , . . . , k j ∈ 2 π � L Z d , k 1 + . . . + k j = p , j = 1 , 2 , . . . i =1 Marcin Napi´ orkowski Excitation spectrum of interacting bosons

The excitation spectrum L Z d define the Bogoliubov elementary excitation For p ∈ 2 π p 4 + 2ˆ � spectrum e ( p ) := v ( p ) p 2 . L Z d we consider the Bogoliubov excitation energies For any p ∈ 2 π with total momentum p : � j � e ( k i ) : k 1 , . . . , k j ∈ 2 π � L Z d , k 1 + . . . + k j = p , j = 1 , 2 , . . . i =1 Let K 1 Bog ( p ) , K 2 Bog ( p ) , . . . be these energies in the increasing order. Similarly, let K 1 N ( p ) , K 2 N ( p ) , . . . be the corresponding excitation energies of H N , that is, the eigenvalues of H N − E N of total momentum p in the increasing order. Marcin Napi´ orkowski Excitation spectrum of interacting bosons

Theorem Theorem Lower bound Let c > 0 . Then there exists C such that 1 if L 2 d +2 ≤ cN, then E N ≥ 1 v ( 0 )( N − 1) + E Bog − CN − 1 / 2 L 2 d +3 ; 2ˆ 2 if in addition K j N ( p ) ≤ cNL − d − 2 , then 1 E N + K j v ( 0 )( N − 1) + E Bog + K j N ( p ) ≥ 2ˆ Bog ( p ) K j N ( p ) + L d � 3 / 2 . − CN − 1 / 2 L d/ 2+3 � Marcin Napi´ orkowski Excitation spectrum of interacting bosons

Theorem Upper bound Let c > 0 . Then there exists c 1 > 0 and C such that 1 if L 2 d +1 ≤ cN L d +1 ≤ c 1 N, and then E N ≤ 1 v ( 0 )( N − 1) + E Bog + CN − 1 / 2 L 2 d +3 / 2 ; 2ˆ 2 if in addition K j cNL − d − 2 Bog ( p ) ≤ K j c 1 NL − 2 , and Bog ( p ) ≤ then 1 E N + K j v ( 0 )( N − 1) + E Bog + K j N ( p ) ≤ 2ˆ Bog ( p ) + CN − 1 / 2 L d/ 2+3 ( K j Bog ( p ) + L d − 1 ) 3 / 2 . Marcin Napi´ orkowski Excitation spectrum of interacting bosons

Theorem By the exponential property of Fock spaces we have the identification � l 2 (2 π � � C ⊕ l 2 (2 π � L Z d \ { 0 } ) L Z d ) H = Γ s = Γ s l 2 � 2 π � �� L Z d \{ 0 } ≃ Γ s ( C ) ⊗ Γ s . We embed the space of the zeroth mode Γ s ( C ) = l 2 ( { 0 , 1 , . . . } ) in a larger space l 2 ( Z ) . The extended space l 2 � 2 π H ext := l 2 ( Z ) ⊗ Γ s � �� L Z d \{ 0 } . We have also a unitary operator U | n 0 � ⊗ Ψ > = | n 0 − 1 � ⊗ Ψ > . For p � = 0 we define b p := a p U † (on H ext ). Marcin Napi´ orkowski Excitation spectrum of interacting bosons

Theorem Marcin Napi´ orkowski Excitation spectrum of interacting bosons

Theorem Thank you for your attention! Marcin Napi´ orkowski Excitation spectrum of interacting bosons