Luttinger Liquid Alexander Chudnovskiy Hamburg Unversity Luttinger - PowerPoint PPT Presentation

Luttinger Liquid Alexander Chudnovskiy Hamburg Unversity

Luttinger Liquid • Luttinger liquid concept • Bosonization in operator approach • Conformal field theory approach • Examples of correlation functions References: J. Voit, Rep. Prog. . Phys. 57 , 977-1116 (1994). A.O. Gogolin, A.A. Nersesyan, A.M. Tsvelik, “Bosonization and strongly correlated systems”, Cambridge University Press, 1998. T. Giamarchi, “Quantum physics in one dimension”, Oxford University Press, 2003. 2

Fermi liquid in 1D and in higher dimensions • quasiparticles are electrons and holes (fermions) ! T 2 • quasiparticles do not decay if close to the Fermi surface (decay rate )

Fermi liquid in 1D and in higher dimensions • quasiparticles are electrons and holes (fermions) ! T 2 • quasiparticles do not decay if close to the Fermi surface (decay rate ) Particle-hole excitations q : E p + q ! E p = 0 q : E p + q ! E p " ! F q the momentum does not fix the energy ph-excitations are not coherent

Fermi liquid in 1D and in higher dimensions • quasiparticles are electrons and holes (fermions) ! T 2 • quasiparticles do not decay if close to the Fermi surface (decay rate ) Particle-hole excitations Particle-hole excitations in 1D q : E p + q ! E p " ! F q q : E p + q ! E p = 0 q : E p + q ! E p " ! F q the momentum does not fix the energy the momentum fixes the energy ph-excitations are not coherent ph-extitations are coherent quasiparticles

Fermi liquid in 1D and in higher dimensions • quasiparticles are electrons and holes (fermions) ! T 2 • quasiparticles do not decay if close to the Fermi surface (decay rate ) Particle-hole excitations Particle-hole excitations in 1D q : E p + q ! E p " ! F q q : E p + q ! E p = 0 q : E p + q ! E p " ! F q the momentum does not fix the energy the momentum fixes the energy ph-excitations are not coherent ph-extitations are coherent quasiparticles

Fermi liquid in 1D: linearization of dispersion E k k ! = 2 !" F L Finite system: equidistant spectrum

Hamiltonian of 1D interacting electron system: Luttinger liquid model { } " " ! 0 = + + v F ( k ! k F ): c R , k ! c R , k ! : ! v F ( k + k F ): c L , k ! H c L , k ! : ! k ! ! O ! : = O ! : … : - normal ordering with respect to the Fermi see: : O FS . ! ! + + + Density fluctuations: ! r , " ( p ) = = c r , k , " " # p ,0 c r , k , " : c r , k + p , " c r , k , " : c r , k + p , " c r , k , " . k k

Hamiltonian of 1D interacting electron system: Luttinger liquid model { } " " ! 0 = + + v F ( k ! k F ): c R , k ! c R , k ! : ! v F ( k + k F ): c L , k ! H c L , k ! : ! k ! ! O ! : = O ! : … : - normal ordering with respect to the Fermi see: : O FS . ! ! + + + Density fluctuations: ! r , " ( p ) = = c r , k , " " # p ,0 c r , k , " : c r , k + p , " c r , k , " : c r , k + p , " c r , k , " . k k Interactions: take into account the small momentum transfer only ! 2 = 1 ' # % g 2 ! ( p ) ! " , ! ' + g 2 ! ( p ) ! " , " ! ' ! R " ( p ) ! L " ( " p ) H $ & L p , ! , ! '

Hamiltonian of 1D interacting electron system: Luttinger liquid model { } " " ! 0 = + + v F ( k ! k F ): c R , k ! c R , k ! : ! v F ( k + k F ): c L , k ! H c L , k ! : ! k ! ! O ! : = O ! : … : - normal ordering with respect to the Fermi see: : O FS . ! ! + + + Density fluctuations: ! r , " ( p ) = = c r , k , " " # p ,0 c r , k , " : c r , k + p , " c r , k , " : c r , k + p , " c r , k , " . k k Interactions: take into account the small momentum transfer only ! 4 = 1 ' ' # % g 4 ! ( p ) ! " , ! ' + g 4 ! ( p ) ! " , " ! ' : ! r " ( p ) ! r " ' ( " p ): H $ & 2 L r = R , L p , ! , ! '

Symmetries of the Hamiltonian The Hamiltonian commutes with the total number of particles in each branch and with a given spin-projection ! ! N r ! = + ! # $ = 0. c r , k , ! c r , k , ! : N r ! , H " k ! # N r ! , ! r , p , ! $ = 0; " % % + c k + ( ! k ', k + p + c k + p + c k ' ) c k & + ( ! kk ' + c k ' ! # + c k ' , c k + p = + c k ) c k ' ) = c k ' ( c k ' c k + p " $ k , k ' k , k ' ( ) % + c k + c k ' + c k ' c k & c k + p + c k & c k + p + c k ' + c k + p + c k c k ' = 0 c k + p k For the election wave function: ! r " ( x ) ! e i # r " ! r " ( x )

Symmetries of the Hamiltonian ! r " ( x ) ! e i # ! r " ( x ) U(1) SYMMETRY # ! R " ( x ) ! e i # ! R " ( x ) % $ CHIRAL SYMMETRY ! L " ( x ) ! e " i # ! L " ( x ) % & SU(2) SPIN SYMMETRY, IF g 2, ! = g 2, ! ( ) ! " # e i ! " " ! r " ( x ) ! ! r " ' ( x ) " ' "" '

Boson solution of the Luttinger model Map the fermion problem with interactions to the problem of noninteracting bosons • Density operators obey bosonic commutation relations: relation to canonical bosonic creation/annihilation operators � • Hamiltonian can be represented as a bilinear of the density operators � • Explicit construction of a fermion through the bosonic operators

Commutator of the density operators ( ) ( ) & & ! ! + c k c k ' ! p ' + c k + c k + p ' ! c k + p ! p ' " $ + + + ! p , ! % = c k ' ! c k ' ! p ' = c k + p c k ' c k + p c k + p c k Right-movers: # ! p ' k , k ' k ( ) & + c k + p ' : ! : c k + p ! p ' + c k + p ' = + c k : + c k + p 0 ! c k + p ! p ' + : c k + p c k 0 k ( ) = ! pL & + c k + p + c k = " pp ' 0 ! c k 2 # " pp ' c k + p 0 k ! k = 2 " + c k + p 0 = 0 c k + p L + c k 0 = 1 c k

Commutator of the density operators U(1) Kac-Moody algebra % = ! pL % = pL ! ! ! ! " $ " $ ! p , ! 2 " # pp ' ! L , p , ! 2 " # pp ' Right-movers: Left-movers: # # ! p ' L , ! p ' $ = i $ = % i ! ! ! ! ! # ! # ! R ( x ), ! 2 " % x # ( x & x ') ! L ( x ), ! 2 " & x # ( x % x ') R ( x ') L ( x ') " " Relation between the density operators and canonical bosons 2 " 2 " % = pL + ( p ), (p>0). ! ! ! ! L ( p ), ! ! ! L " $ ! L , p , ! 2 " # pp ' & ! L , p = L , ! p = pL b pL b # L , ! p ' 2 " 2 " + ( p ), ! % = ! pL ! ! ! ! R ! ! R ( p ), (p>0). " $ ! R , p , ! 2 " # pp ' & ! R , p = R , ! p = pL b pL b # R , ! p '

Equivalence of the free fermion Hamiltonian to the Hamiltonian quadratic in density operators { } " " ! 0 = + + ! F ( k ! k F ): c R , k ! c R , k ! : ! ! F ( k + k F ): c L , k ! H c L , k ! : ! k % = pL ! 0 , ! R , ! ( p ) ! # ! ! " $ $ = ! F p " R , ! ( p ) ! R , " , ! p , ! Easy to check: H cf. " # R , " , p 2 # % = pL ! 0 , ! L , ! ( p ) ! # ! ! " $ $ = % ! F p " L , ! ( p ) cf. ! L , " , ! p , ! H " # L , " , p 2 # ! 0 = !" F " : ! r , ! ( p ) " r , ! ( # p ): + const. H L p ! 0, r = ( R , L ), ! GS = ! F " p = 2 !" F GS ! E N E N + 1 “const.” is the ground state energy: L � It changes with the total particle number: ! 0 = !" F " r , ! ( # p ): + !" F 2 + N L ! " " 2 ) : ! r , ! ( p ) H ( N R ! L L p ! 0, r = ( R , L ), ! !

Luttinger liquid Hamiltonian ! 4 = !" F + "# F 2 + N L ! ! = H ! ! ! 0 + H ! 2 + H 2 ) : ! r , ! ( p ) " r , ! ( # p ): H ( N R ! L L ! p " 0, r = ( R , L ), ! + 1 ! % ' g 2 ! ( p ) ! " , ! ' + g 2 $ ( p ) ! " , # ! ' : ! R " ( p ) ! L " ' ( # p ): & ( L p , ! , ! ' + 1 ! ! % ' g 4 ! ( p ) ! " , ! ' + g 4 $ ( p ) ! " , # ! ' : ! r " ( p ) ! r " ' ( # p ): & ( 2 L r = R , L p , ! , ! '

Separation of spin and charge Charge and spin variables ! r ( p ) = 1 & N r ! = 1 # ! r ! ( p ) + ! r " ( p ) % # N r ! + N r " ( p ) % & $ $ 2 2 " r ( p ) = 1 & N r " = 1 # % # % ! r ! ( p ) ' ! r " ( p ) N r ! ' N r " ( p ) & , r = R , L $ $ 2 2 ( ) , g i ! = 1 ( ) , i = 2,4. Interaction constants: g i ! = 1 2 g i ! + g i ! 2 g i ! " g i ! The Hamiltonian ! = H ! ! + H ! ! . H ! ! = H ! 0 + H ! 2 + H ! 4 , ! = " , ! . H same structure with new interaction constants

Diagonalization of LL-Hamiltonian by a Bogoliubov transformation [ ] + ! L ( p )sinh " ( p ) [ ] ! ! R ( p ) = ! R ( p )cosh " ( p ) [ ] + ! R ( p )sinh " ( p ) [ ] ! ! L ( p ) = ! L ( p )cosh " ( p ) !" F + g 4 ( p ) " g 2 ( p ) K ( p ) ! e 2 ! ( p ) = ! ( p ) !" F + g 4 ( p ) + g 2 ( p ). Choice of : dq ! ! For a typical interaction: H int = V ( x " x ')ˆ n ( x ') = n ( " q ) n ( x )ˆ n ( q ) V ( q )ˆ ˆ dxdx ' 2 ! g 4 ! ! V ( q = 0), g 2 ! ! V ( q = 2 k F ). Repulsive interactions: K < 1. K > 1. Attractive interactions:

The diagonalized Hamiltonian ! = ! ! ( # p ): + ! 2 L ! N ( p )( N R + N L ) 2 + ! J ( p )( N R # N L ) 2 " ! ( p ) " ( p ): ! ! $ & H % ' L p ! 0 !" F + g 4 ( p ) " g 2 ( p ) 2 2 ! $ ! $ K ( p ) ! e 2 ! ( p ) = ! F + g 4 ( p ) ' g 2 ( p ) ! ( p ) = !" F + g 4 ( p ) + g 2 ( p ). . # & # & " " " % " % ! N = ! / K = ! F + ( g 4 + g 2 ) / " , ! J = ! K = ! F + ( g 4 ! g 2 ) / " , ! N ! J = ! 2 . g 2 , g 4 ! ! , K ! ! N , ! J