Excitation spectrum of interacting bosons in the
mean-field infinite-volume limit
Marcin Napiórkowski
Jan Dereziński
Faculty of Physics, University of Warsaw
21 III 2014
Warwick EPSRC Symposium on Statistical Mechanics:
Many-Body Quantum Systems
Marcin Napiórkowski
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
HN = −
N
X
∆i + λ
i=1
X
v(xi − xj )
1≤i<j≤N
defined on the Hilbert space L2sym ((Rd )N ).
λ ≥ 0 is a coupling constant.
We assume v is a real and symmetric function such that
v ∈ L1 (Rd ), v̂ ∈ L1 (Rd )
v(x) ≥ 0, x ∈ Rd , v̂(p) ≥ 0, p ∈ Rd .
Marcin Napiórkowski
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 Rd by the torus (−L/2, L/2]d and the
potential by its periodized version
v L (x) =
1
Ld
X
v̂(p)eipx .
p∈(2π/L)Zd
The Hamiltonian in the box reads
L
HN
=−
N
X
X
∆L
i +λ
i=1
v L (xi − xj ).
1≤i<j≤N
The total momentum operator
PNL =
N
X
−i∂xLi .
i=1
Marcin Napiórkowski
Excitation spectrum of interacting bosons
The excitation spectrum
The excitation spectrum
HN and PN commute, thus we can consider the joint
energy-momentum spectrum sp(HN , PN )⊂ Rd+1 .
Let EN denote the ground state energy of HN . Then
Excitation spectrum := sp(HN − EN , PN ).
Marcin Napiórkowski
Excitation spectrum of interacting bosons
The excitation spectrum
Bogoliubov excitation spectrum
The diagonalised Bogoliubov Hamiltonian
X
HBog = EBog +
e(p)b†p bp
p6=0
with e(p) =
EBog := −
p
p4 + 2λρv̂(p)p2 and
1
2
X
|p|2 + v̂(p) −
p
|p|4 + 2λρv̂(p)|p|2 .
Zd \{0}
p∈ 2π
L
Marcin Napiórkowski
Excitation spectrum of interacting bosons
The excitation spectrum
Bogoliubov excitation spectrum
The diagonalised Bogoliubov Hamiltonian
X
HBog = EBog +
e(p)b†p bp
p6=0
with e(p) =
EBog := −
p
p4 + 2λρv̂(p)p2 and
1
2
X
|p|2 + v̂(p) −
p
|p|4 + 2λρv̂(p)|p|2 .
Zd \{0}
p∈ 2π
L
⇒ Choice of mean-field scaling λ = 1/ρ.
Marcin Napiórkowski
Excitation spectrum of interacting bosons
The excitation spectrum
d
For p ∈ 2π
the Bogoliubov elementary excitation
L Z define
p
spectrum e(p) := p4 + 2v̂(p)p2 .
d
For any p ∈ 2π
L Z we consider the Bogoliubov excitation energies
with total momentum p:
( j
)
X
2π d
e(ki ) : k1 , . . . , kj ∈
Z , k1 + . . . + kj = p, j = 1, 2, . . .
L
i=1
Marcin Napiórkowski
Excitation spectrum of interacting bosons
The excitation spectrum
d
For p ∈ 2π
the Bogoliubov elementary excitation
L Z define
p
spectrum e(p) := p4 + 2v̂(p)p2 .
d
For any p ∈ 2π
L Z we consider the Bogoliubov excitation energies
with total momentum p:
( j
)
X
2π d
e(ki ) : k1 , . . . , kj ∈
Z , k1 + . . . + kj = p, j = 1, 2, . . .
L
i=1
1 (p), K 2 (p), . . . be these energies in the increasing order.
Let KBog
Bog
1 (p), K 2 (p), . . . be the corresponding excitation
Similarly, let KN
N
energies of HN , that is, the eigenvalues of HN − EN of total
momentum p in the increasing order.
Marcin Napiórkowski
Excitation spectrum of interacting bosons
Theorem
Theorem
Lower bound
Let c > 0. Then there exists C such that
1 if
L2d+2 ≤ cN,
then
1
EN ≥ v̂(0)(N − 1) + EBog − CN −1/2 L2d+3 ;
2
2
if in addition
j
KN
(p) ≤ cN L−d−2 ,
then
j
EN + KN
(p) ≥
1
j
(p)
v̂(0)(N − 1) + EBog + KBog
2
3/2
j
−CN −1/2 Ld/2+3 KN
(p) + Ld
.
Marcin Napiórkowski
Excitation spectrum of interacting bosons
Theorem
Upper bound
Let c > 0. Then there exists c1 > 0 and C such that
1 if
L2d+1 ≤ cN
and Ld+1 ≤ c1 N,
then
2
1
EN ≤ v̂(0)(N − 1) + EBog + CN −1/2 L2d+3/2 ;
2
if in addition
j
KBog
(p) ≤ cN L−d−2
and
j
KBog
(p) ≤ c1 N L−2 ,
then
j
EN + KN
(p) ≤
1
j
v̂(0)(N − 1) + EBog + KBog
(p)
2
j
+CN −1/2 Ld/2+3 (KBog
(p) + Ld−1 )3/2 .
Marcin Napiórkowski
Excitation spectrum of interacting bosons
Theorem
By the exponential property of Fock spaces we have the
identification
2 2π d
2 2π d
H = Γs l ( Z ) = Γs C ⊕ l ( Z \ {0})
L
L
2π
' Γs (C) ⊗ Γs l2
Zd \{0} .
L
We embed the space of the zeroth mode
Γs (C) = l2 ({0, 1, . . . }) in a larger space l2 (Z). The extended
space
2π
Hext := l2 (Z) ⊗ Γs l2
Zd \{0} .
L
We have also a unitary operator U |n0 i ⊗ Ψ> = |n0 − 1i ⊗ Ψ> .
For p 6= 0 we define bp := ap U † (on Hext ).
Marcin Napiórkowski
Excitation spectrum of interacting bosons
Theorem
Marcin Napiórkowski
Excitation spectrum of interacting bosons
Theorem
Thank you for your attention!
Marcin Napiórkowski
Excitation spectrum of interacting bosons