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

Particle-Hole ExcitationsParticle-Hole Excitations

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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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Thank you for

attention

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