Structure of the Excitation Spectrum for Many-Body Quantum Systems - - PowerPoint PPT Presentation
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
i=1
N
i=1
1≤i<j≤N
L→∞
L→∞
R3(N−1) Ψ0(x, x2, . . . , xN)Ψ∗ 0(x′, x2, . . . , xN) dx2 · · · dxN
[0,L]6 γ0(x, x′)dx dx′
N
j=1
L→∞
µ<0
R3 ln
R3
c
n≥0
N = − N
i=1
N ) =
p∈( 2π
L Z)3
p
sym L2([0, L]3) as a subspace of the
∞
n=0 n
sym
L Z)3, and we
p, a† q
q
sym L2([0, L]3) of
p
pap +
p
q,k
q+pa† k−pakaq
[0,L]3 v(x)e−ipxdx
0a0 ∼ N. Hence p = 0 plays a special role.
p and ap for p ̸= 0.
0 and a0 by
p and ap, and equals
p̸=0
pap + 1 2ϱ
pa† −p + apa−p
−p, with
p again satisfy
p̸=0
pbp
p̸=0
R3
R3
R3
1 2(2π)3
R3 | v(p)|2 |p|2 dp are the first two terms in the Born series for 8πa, the
p
pn · · · b† p1Ψ0
p = cosh(αp)a† p + sinh(αp)a−p.
L→∞
q
∑
p pnp=q
p
0 and a0 (or any other mode) by a c-number
N
j=1
j
1≤i<j≤N
sym L2([0, L]).
N
i=1
1≤i<j≤N
p∈(2πZ)3
2N
p̸=0
p∈(2πZ)3\{0}
{np}, ∑
p p np=P
p̸=0
p
j=1 xj), q ∈ (2πZ)3,
j=1 ∇j denotes the total momentum operator. Hence our results
R3
R3×R3 |φ(x)|2v(x − y)|φ(y)|2dx dy
p∈(2πZ)3
pap +
p
q,k
q+pa† k−pakaq
0 and a0 by
p and ap, p ̸= 0.
2
p̸=0
pap + 1 2
pa† −p + apa−p
−p,
p̸=0
pbp
p → c† p := a† pa0
p and cp, and hence particle number conserv-
−p, analyze the spectrum of
p̸=0
pdp
0a0 is close to N.
0a0 ≤ const. [1 + HN − E0(N)]
p and ap, p ̸= 0, in the Hamiltonian are negligible,
0a0
p∈(2πZ)3\{0}
j=1
1≤i<j≤N
N
i=1
i=1 ∆i ≥ (2π)2(N − a† 0a0).
p̸=0
pcp + c† −pc−p
pc† −p + cpc−p
pcp + c† −pc−p
pc† −p + cpc−p
p − B2 p
p + βpc−p
−p
p
−p + βpcp
p
p
p − B2 p
p] + [c−p, c† −p]
p − B2 p
−p
p̸=0
pa† −p − apa−p
q] = δp,q. Our operators cp = apa† 0/
q
q
dp
−p + Error
p replaced by cp and c† p, respectively. Moreover, the
0a0)2 ≪ N 2.