Tex
AMP
2020/21
-Benjamin
Schlein
University
of
Zurich
Correlation
energy
of
weakly
interacting
fermions
#
Joint
with
N. Benedikter, RT. Nam , M . Porta , R. Seir
ingen
Inmany
-body
QM, two
types
of
particles
a)
hoses
: describedby
symmetric
wave-functions
, ie .
NIN
( Xp , , .. . , XAN)
=NIN
(x, i -- -, XN)
tf
ite SNa)
fs
: describedby
antisymmetric
wave-functions
Bosowicsyste.ms#
I
cinmriiadriynminnmtiaahaeiu
:
:
ripe
.Io
; er)
'Hamakw
:Hn
= - Ox ; tNt
¥
; V ki-xp
acting
on L's
Inn
) . .Goatee
:compute
ground
- stateenergy
andexcitations
. BEC : mostparticle
in state 4. ( x) = Ifxe
ABogolivbovthe.org
Describe Bose gas onFockspaa=
Fs
÷¥
. L's
Inn)
with
creation and annihilation ops .apt
, ap , p e Z? We have
CCI
:lap
,aof
)
-Spg
,lap
,ad
.-Tapi
, a;]
= 0apt
ap = XXparticles
with
momentum p. In
particular
:O
lap
.We write HN =
Fez
, P ' aTap
tTN
Efg
, r e z ,✓
( r)aftra
;
ay
rapE
substitute
again
rn
' n I =I
a. , ai)
Hn
elN-ii
2 +Tpo
(p'tVtpllapap
tftp.ovlplfapa.pt
tap
aNeglect
cubic,
quartic
terms__
: we
find
Hn
eInitial
+
p&o
(p'ttlpllapap
+ftp.ovlpl/ap*a-ptapa-p
)
Diagonalize
quadratic
Hamiltonian :#
with
1- =
exp
I
z'¥
.rep
lap
't
a
:p
-apa
-pl
)
,
we
bind
tap
T = ashEp
.apt
t sink2ps
. aChoosing
Ep
sit . :tanh
kept
=THI
p't
Jlp
) we obtain T 'Hn
telNmzV
-ftp.olrittlpl-fptzptvpi
]
+pzofpfepilpT.apap-a.gr#e
energy
t ' low -energyspectrum
, up to*
Rigorous
: Seiringer
, Grech - Seiringer
,Lewin- Nam-
Serf
ally
-Solovej
, Berezinski
-
Napierkowski
,
Pizzo
,
Bossmann- Perra t- Sei
ringer
, ... .E
. Baccara - Brennecke- Cena
tiempo
- SMean-fieldfermious-F.fm
, consider Nfermions
,trapped
in rows n .Because
of
statistics
, kineticenergy
isof
order N 513 .We consider
Menefee
theHamiltonian
HN
=¥
. - E ' Ox ; ttf
§
; V ki-xp
ontalk
)
Here , we set E - N -" 3 .→ Mean -
field
limit is linkedwith
semiclassical
,
with E = N
-"3
Hartree-Fock
For{
Life
.... .. an ONS onLYN
, considerStared
.Nllxi
.-- xn )-f
!
fj
Hi . - -xn)
-ftp.dltlfilxjllisi.jsn
Wedefine
Eiht
-inf
{
of
, Hn 47 :f-
th
!
Lj
}
"
.÷÷÷÷
::
:*
:÷÷±÷÷÷:i÷÷÷it÷÷
.If
I
> o and Fermiball
fined
, wefind
↳⇐ se . N "'
EFF
- life, Hn Ye > even
if
Ho
§
see Hartllet
En
=inf
{
CNI
, HnY
)
:Ye Ra
IM
}
Interested incorrelate
: E arr = En -EYE
s O .theorem
AssumeI
>o,
compactly
supported
, smallenough
. Then :E.re Ea
-Eee
,IN
17g
't
timeWH
-H
-eardrop
Hall
)
dh-Eating
Remache
: • to second order in V , we find : +OLE
't'T
,
Eary
q = Ahh- log 2)-Fez
,1h11
that
' . ( it ONT)
,
as
previously
shownby
Hai¥¥e
.• result consistent with
formula
derived inphysics
through
random- phase
To
estimate
correlation
energy
,it's convenient
to
factor
Fermi sea andfocus
onexcitations
.top
: wedefine
fermionic
Foch spaceFa
-ng
.ha
IN
Creation and
annihilation
operators
satisfy
CAI
On
Fa
, wefind
aunity
sit . • Rr =put
↳af
r =Te
(
A-fr
, go , ..-3
is -Vacuum on Fa)
and •Raf
R!
{
af
if
Ipl
> k,Ap
if
Ipl
E keAfter
conjugation
with
R, r
represents
Fermi sea ,apt
creates anexit
with
momentum p a •( a
particle
if Ipl
> Ke, a hole
if
lple
Kt)
.o
With R
, we can
define
excitation HamiltonianL
- RHn
R . -Wecompute
: z - Z 2R
p?
Ep
'apt
ap R 't = , ↳Ep
' apapt
t?p
,weep
'apiap
=Fps
↳Ep
'-Fpu
.eep
Ep
' ap .apt
Ep
, >neap
'apiap
=⇐
¥Ep
' t¥
, IEp
' - E 'KE
I
apt
ap -tastes
with kinetic energy kin-energy
of
* holes = #particles
)
Similarly
, we cancompute
RIn
Four
.#TH
aItr
af
a g.rap R-This
generates
ms
.Some
of
them, like
I
TN
Fg
, q, > ↳UH
CY , ap:
a;
ag.rapY
)/
Iptr l , lqtrl>KI EIN
ETH
N aprray
YA
N a g.rap
4N
E4N
.ANY
n ' aresmell
,on stares
with
few
excitations .We
find
L =
Eth
+ thatQB
t smallwith
tho =¥
, lEp
' - E ' KEI
apt
apQB
-TN
fry
:b
,¥k¥
""Iain
a.gain
a.pt
h
.)
+Nt Fpm
, Igm > k,VH
-Ipw
A&p
Ag
agar
Ipl, I ql E UF
↳ terms involve two
particles
and two holes-refine
pie
iaagga.gg
bit
= I a:*
a;
Ipl 's KF lpsrl > heThen
:QB
-f
?
#V'
Ir)
fbibrtzlbikitbrb
-RD
Furthermore :[
br ,bn
]
-[
bit
,but
]
= ofr
, he 23[
br ,but
]
I C.Snu
, on stareswith
Problem
:Ttr
-¥
.-2
lip
' -hit
ap.apafma.gr
lqkhtlqtiel
> he =-2
E'
fatal
'-of
]
aol.ie
air
lqlekf Ht ht> he ← 2E '-2
g.
kagin
a;
r INE KF lat H> Ut↳
cannot be Tosolve
problem
, we need toexpressed
inHii
i
,We
decompose
a thin shell around/ ,
, , i
Fermi
sphene
inpatches
{ Balan
,... ,m ,for
M -- N
'
We
define
corresponding
oi¥pai
operators
We conclude that - HF
L
-Ew
=ZEKE
,!H
In
,I
Dlhltwlhllag
bait
by
,u-tf
Wlhlag
(
bimba
:
tba.nbg.ie
)
)
+ small
Nene
:leading
contributions
to correlationenergy
arise
from
aqvadraticttamiltonianin
approximately
fields
bin
,