LUTTINGER LIQUID Speaker Iryna Kulagina T. Giamarchi “Quantum Physics in One Dimension” (Oxford,...
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Transcript of LUTTINGER LIQUID Speaker Iryna Kulagina T. Giamarchi “Quantum Physics in One Dimension” (Oxford,...
LUTTINGER LIQUIDLUTTINGER LIQUID
Speaker Iryna Kulagina
T. Giamarchi “Quantum Physics in One Dimension” (Oxford, 2003)
J. Voit “One-Dimensional Fermi Liquids” arXiv:cond-mat/9510014
Nichols T. Bronn “Luttinger Liquids”G. F. Giuliani and G. Vignale “Quantum Theory of the
Electron Liquids” (Cambridge, 2005)
Fermi GasFermi GasEnergy for single particle
Hamiltonian ( )
Elementary excitation:
1) addition of a particle at wavevector k (δnk=1),
energy
2) destruction of a particle at wavevector k (δnk=-1),
energy
2
2
2m k
k
2
2F
F
kE
m ( 0)FE T
k kk
H n k k
Fk k
0k k
Fk k0k k
Landau’s Fermi Liquid Landau’s Fermi Liquid Theory Theory
Adding a particle
Destructing a particle
Ground state quasiparticle distribution
Changing of quasiparticle occupation number
Energy change
Expansion of energy
Energy of quasiparticle added to the system
3
, 1 0,pp N a N
, 1 0,pp N a N
0
1,( )
0,F
F
k kn k
k k
0 0( ) ( ) ( )n k n k n k
0
'
1( ) ( , ') ( ) ( ')
2kk kk
E n k f k k n k n kL
0*F
k F
kk k
m
0
'
1( , ') ( ')k k
k
f k k n kL
Luttinger Liquid.Luttinger Liquid.Noninteracting problemNoninteracting problem
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Hamiltonian
Linear spectrum
New Hamiltonian
Spectrum
k k kk
H a a
( )k F F Fv k k for k k
( )k F F Fv k k for k k
, 1
( )F F kr krk r
H v rk k a a
, ( ) ( )R k F F FE q v k q v k v q
BozonizationBozonizationFourier components of the particle density operators
Commutation relations for operators
Commutation relations for Hamiltonian
Hamiltonian
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, ,( ) k q kk
q a a
' '( ), ( ')2qq
qLq q
0 , ( ) ( )FH q v q q
00,
2( ) ( )F
q
vH q q
L
Interacting HamiltonianInteracting HamiltonianInteracting Hamiltonian
Excitation spectrum
Field operators
Commutation relations
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int 2 4,
1( ) ( ) ( ) ( ) ( ) ( )
2 q
H g q q q g q q qL
2 2
4 2( ) ( )( )
2 2F
g q g qq q v
/2
0
1( ) ( ) ( )q iqx
q
i xx e q q N
L q L
/2
0
1 1( ) ( ) ( )q iqx
q
Jx e q q
L q L
, , ( 0)N N N J N N N q
( ), ( ) ( )x y i x y
Interacting HamiltonianInteracting HamiltonianFull Hamiltonian
Parameters
Current density
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220 int ( )
2 2 x
uK uH H H dx x
K
2 2
4 4 4 2
4 2
2 2,
2 2F
FF
g g v g gu v K
v g g
( ) ( )j x uK x
Model with spinModel with spinKinetic energy
Where
Interacting Hamiltonians
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int,2 2 , , 4 , ,, , ,
1( ) ( ) ( ) ( ) ( ) ( )
2 s t s tq s t
H g q q q g q q qL
0 , , , , , , , ,,
, ,0, ,
( ) ( )
2( ) ( )
F F k s k s F k s k sk s
Fs s
q s
H v k k a a k k a a
vq q
L
, , , , ,( )s k q s k sk
q a a
int,1 1 , , , , , 2 , , 2 ,, , , ,
1F Fk s p t p k q t k k q s
k p q s t
H g a a a aL
Model with spinModel with spinFull Hamiltonian
Where
with
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12
2cos( 8 )
2
gH H H dx
22
2 2 x
u K uH dx
K
2 2
4, 4, 4,
4,
2 2,
2 2F
FF
g g v g gu v K
v g g
1 2 1 4, 4 4,, , 2 , , , 0g g g g g g g g
Physical propertiesPhysical propertiesThe Specific heat
The specific heat coefficient
Spin susceptibility
Compressibility
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( )C T T
0
1
2F Fv v
u u
0
Fv
u
0
Fv K
u
ConclusionsConclusionsThe important properties of ID liquides:a continuous momentum distribution function n(k), varying with as |k−kF|α with an interaction-dependent exponent α, and a pseudogap in the single-particle density of states ∝|ω|α, consequences of the non-existence of fermionic quasi-particlessimilar power-law behaviour in all correlation functions, specifically in those for superconducting and spin or charge density wave fluctuations, with universal scaling relations between the different nonuniversal exponents, which depend only on one effective coupling constant per degree of freedomfinite spin and charge response at small wavevectors, and a finite Drude weight in the conductivitycharge-spin separationpersistent currents quantized in units of 2kF
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