Fermi liquid features of the one-dimensional Luttinger liquid

a r X i v :c o n d -m a t /9712300v 2 [c o n d -m a t .s t r -e l ] 28 A p r 1999Fermi-liquid features of the one-dimensional Luttinger liquidYupeng Wang 1,21Laboratory of Ultra-Low Temperature Physics,Chinese Academy of Sciences,P.O.Box 2711,Beijing 100080,China2Institut f¨u r Physik,Universit¨a t Augsburg,86135Augusburg,GermanyWe show that the one-dimensional (1D)electron systemscan also be described by Landau’s phenomenological Fermi-liquid theory.Most of the known results derived from the Luttinger-liquid theory can be retrieved from the 1D Fermi-liquid theory.Exact correspondence between the Landau pa-rameters and Haldane parameters is established.The expo-nents of the dynamical correlation functions and the impu-rity problem are also discussed based on the finite size cor-rections of elementary excitations with the predictions of the conformal field theory,which provides a bridge between the 1D Fermi-liquid and the Luttinger liquid.71.10.Pm,71.10.Ay,72.15.NjLandau’s Fermi-liquid theory (FLT)[1]is a fundamen-tal paradigm in condensed matter physics.However,as suggested by many authors,the situation in reduced di-mensions may be different.Especially in 1D,it is gen-erally believed that the systems are marginal [2–4]and the low temperature properties of the 1D gapless quan-tum systems can be described by the Luttinger-liquid theory (LLT)[5,6].Even in 2D,Anderson showed that a strongly correlated electron system may exhibit the main characters of a Luttinger liquid [7].A Luttinger liquid generally has the following characteristics:(i).Its low en-ergy excitations can be completely described by the col-lective density waves,i.e.,the sound waves (charge den-sity fluctuation)and the spin density waves characterized by two velocities v ρand v σrespectively;(ii).Various op-erators exhibit anomalous dimensions determined by two stiffness constants K ρand K σ,which lead to the nonuni-versal power-law decay of correlation functions;(iii).The momentum distribution of the physical particles is con-tinuous at the Fermi surface.In fact,the low temperature properties of a 1D gap-less electron system are uniquely determined by four con-stants v ρ,v σ,K ρand K σand the Haldane relations hold:v N γv J γ=v γ2,v J γ=K γv γ,γ=ρ,σ,(1)where v N γand v J γ(γ=ρ,σ)are the velocities of excita-tions induced by additional charges and currents,respec-tively [5,6].Some important thermodynamic quantities such as the specific heat C ,the susceptibility χand the compressibility κare given by C =ππv N σ,κ=2that if we use the Green’s function of the quasiparticlesrather than that of the real particles,the imaginary part of the self energy may be very small compared to thereal part,as long as the creation and annihilation oper-ators of the quasiparticles are well-defined.This can beseen in the following discussion for the integrable mod-els.For the second case,we note that the relaxation time is nothing but the response time of a quasiparticleto an external perturbation.Without external perturba-tion,the system must be in equilibrium and the collision integral is exactly zero[9].Therefore,the quasiparti-cle relaxation time makes no sense to the equilibrium properties.To clarify this,we consider the pure states(eigenstates)of a1D system.For afinite non-interacting1D system,the quantum states are described by a se-quence of integers I j.Each I j corresponds to a singleparticle state with the momentum2πI j/L.If we turn onthe interaction continuously,the distribution{I j}must not be changed since I j are discrete while the interac-tion is continuous.Therefore,an eigenstate of the in-teracting system corresponds exactly to one of the non-interacting system with the same quantum number{I j}and each allowed I j may describe a quasiparticle state. Without external perturbation,the distribution{I j}istime independent.Therefore,in a pure quantum statewe can say a quasiparticle has an infinite life-time and is thus well defined.In the ground state,only N(parti-cle number)states of I j with the smallest absolute val-ues are occupied.These occupied I j states form a per-fect Fermi sphere.In such a sense,the Luttinger theo-rem is satisfied.In addition,the particle-hole excitations can be constructed by moving some particles from theFermi sphere(thus leaving some I j holes)up to higher I jstates.This can be exactly shown in the integrable mod-els[10].In these models,the excitation energy is exactlygiven by∆E= pǫ(I p)− hǫ(I h),whereǫ(I p)(ǫ(I h)) is the dressed energy of the quasiparticles(holes)[10]. Also,the exact creation(annihilation)operator of thequasiparticles R†(p)(R(p))for the integrable models can be constructed in the framework of the algebraic Betheansatz[11].Though these operators satisfy an nonlocalcommutation relation R†(p)R†(q)=S(p,q)R†(q)R†(p), the Pauli exclusion principle holds for these quasiparti-cles since the scattering matrix S(p,q)takes the value-1when p=q.In fact,R†(p)is the exact creation opera-tor of the eigenstates and has the perfect time evolution form R†(p,t)=exp(iǫ(p))R†(p,0).Based on the above discussions,we can see that at leastfor the equilibrium state,Landau’s Fermi liquid theoryis applicable in1D.Caution should be taken only for the transport properties and other non-equilibrium proper-ties.We note a different approach,the so-called Landau-Luttinger-liquid theory[12]based on the exact solution of the Hubbard model was established by Carmelo et al..The quasi-particles are defined in the charge-and spin-sectors respectively rather than in the conventional way.This is the poineering work approaching to the 1D quantum systems along the line of the generalized Fermi-liquid theory and obtained the related quantities in a microscopic way[13,14].The main conclusion of the present letter is that almost all the known results derived from the LLT can be retrieved from Landau’s phenomenological FLT.The one-to-one correspondence between the Landau parameters and the Haldane param-eters can be established.The present results strongly suggests that the Luttinger-liquid,the Landau-Luttinger liquid and Landau’s conventional Fermi liquid are essen-tially paralell in1D.A microscopic state of the Fermi liquid is described by the quasiparticle distribution{n ps}.In an equilibrium state,n ps takes the well known Fermi-Dirac form n ps= {exp[(ǫp−µ)/T]+1}−1,whereµis the chemical potential. Notice that the quasiparticle energyǫp is no longer a c-number but a functional of{n p′s′},ǫp=ǫ0p+12 r=± s′f p F s,rp F s′,f s1=1v F=1+F s1.(5)In another hand,the density of states per unit length of the quasiparticles at the Fermi surface isN(0)=1πv F,(6)where v F is the Fermi velocity.Notice that we have taken the particle mass m as our unit.To derive some im-portant quantities such as the compressibilityκand the Drude weight D,we considerδn ps induced by the exter-nal pressure andfluxΦδn ps=∂n ps|p|δΦ).(7)At low temperatures,with eq.(3)wefind the following relations hold∂n1+F s0,1∂Φ=2N(0)πn2C Nρ=21+F s0,D≡2πv F1∂H =2N(0)L∂Jσ1+F a1,(10)and the magnetic susceptibilityχand the spin Drude weight Dσare given byχ≡1πv F1πC Jσ=11+F a1,(11)where M and Jσare the magnetization and the spin current induced by the magneticfield H and the spin fluxΦσrespectively.Notice above four new parametersC Nρ,C Jρ,C Nσ,C Jσare defined.They are nothing butthe zero sounds in a1D Fermi liquid.As the tempera-ture is lowered,the mean quasiparticle scattering timeτincreases and,forfixed frequency,ωτincreases.Asωτnears one,the quasiparticles no longer have time to relax in one period of the sound;the liquid then no longer re-mains in local thermodynamic equilibrium,and the char-acter of the sound propagation begins to change.If thereis no quasiparticle collision,f ps,p′s′,then in the collision regimeωτ>>1,the excess quasiparticles in afluid ele-ment with increased density simply diffuse away,withoutdriving the neighboring elements.However,as Landau observed,if there are quasiparticle interactions,then a local increase in the density of afluid element can drive neighboring elements via the modification of the effec-tivefield p′s′f ps,p′s′δn p′s′(x,t).Such restoring forces between neighboring elements can give rise to a sound-like collective modes of oscillation of thefluid,called zero sound.For the liquid in nonequilibrium and inhomogeneous situations that differ slightly from the equilibrium state of the homogeneous liquid,the state of the liquid can be specified by the quasiparticle distribution n ps(x,t)as a function of position and time,which satisfies the Landau kinetic equation[1,9]∂n ps(x,t)∂pǫp(x,t)∂∂xǫp(x,t)∂∂t+v p∂∂ǫp)v p∂L p′s′f ps,p′s′δn p′s′(x,t).(14) Assuming the externalfield U(x,t)=U exp i(qx−ωt), we haveδn ps(x,t)=δn ps(q,ω)exp i(qx−ωt),and ∂n0psL p′s′f psp′s′δn ps(q,ω))+(ω−qv p)δn ps(q,ω)=0.(15) If we writeδn ps in the formδn ps=−∂n0ps/∂ǫpνps,then theνps obeyνps−qv pω−qv pU.(16)Notice above p=±p F and v±pF=±v F.By solving the four coupled linear equations,we easily obtain the velocities of the four branch zero soundsC0ρ=v F(1+F s0),C0Jρ=v F(1+F s1),C0σ=v F(1+F a0),C0Jσ=v F(1+F a1).(17) These quantities are exactly the same given in(9)and (11).The former two correspond to the zero sounds of charges and the latter two correspond to those of the spins,respectively.In fact,the zero sounds describe the motions of the whole Fermi sphere’s deformation in the extreme collisionless limitωτ>>1.For each sector(spin and charge),there are only two such modes in1D.One is the breath mode of the Fermi sphere and the other is the oscillation of the whole Fermi sphere around the equilibrium position.The situation is very different from that of the Fermi liquid in3D,where there are an infinite number of zero sounds.At frequenciesωsufficiently small thatωτ<<1,sound in Fermi liquids takes the form of ordinary hydrody-namic;orfirst sound.The sound velocity Cρcan be given byC ρ=(1+F s 0)(1+F s 1).(18)In fact,in the hydrodynamic limit,the density fluctua-tion δn (x,t )satisfy the following equation∂2δn∂x 2δP =0,(19)where δP is the variation of the pressure and can beapproximated by n (∂µ/∂n )T =0δn =C ρ2δn .If we put δn (x,t )as a quantum field rather than a semiclassical quantity,with the canonical quantization[δn (x,t ),∂t δn (x ′,t )]=iδ(x −x ′),(20)the density fluctuations in 1D Fermi liquid can be simply “bosonized”asH b =[(∂δn∂x )2]dx.(21)Such an effective Hamiltonian is exactly the same derived in the Luttinger liquid theory.The spin density fluctu-ations can be quantized in a similar way with the spin wave sound velocityC σ=v F3(1C σ)T.(23)By comparing the measurable quantities derived from FLT and those from LLT,we can easily deduce that an exactly one-to-one correspondence exists between the pa-rameters of the 1D Fermi liquid and those of the Lut-tinger liquid:C N γ=v N γ,C J γ=v J γ,C γ=v γ,γ=σ,ρ.(24)In addition,the Haldane relations are also satisfied in the Fermi liquid descriptionC N γC J γ=C 2γ,C J γ=K γC γ,(25)where the stiffness constants in the Luttinger liquid the-ory can be expressed by the Fermi liquid parameters as K ρ,σ=1+F s,a 0.(26)An interesting fact is that all the quantities of the forward scattering g-ology model (spin-1/2Luttinger model)ob-tainedfromthebosonization method [5,6]are exactly thesame to those derived from FLT with the notation cor-respondence:g 4ss ′=f p F s,p F s ′,g 2ss ′=f p F s,−p F s ′.Such a correspondence has also been pointed out in a recent review [15].The correlation functions can not be derived from FLT in a microscopic way since the theory itself is phenomeno-logical.However,we know that 1D quantum systems are conformally invariant at zero temperature [16]and the exponents of a variety of correlation functions are directly related to the finite size corrections of the corresponding elementary excitations [17]which can be easily derived from the present theory.In a 1D Fermi liquid,there are only three types of elementary excitations in each sector,i.e.,the small momentum particle-holes,the 2k F particle-holes or the currents and the additional charges.The finite size correction for the excitation energy of a 1D Fermi liquid can be derived from (8),(10)and (21)as ∆F =γ=ρ,σ2πC γ8K γ+14K γ],(28)where I γare non-negative integers characterizing the particle-hole excitations.Suppose the impurity is sited at the origin.The tunneling current through it can be written asJ (x,y )∼−i [C †σ(x )C σ(y )−h.c ],x ∼0+,y ∼0−.(29)¿From(28)we obtain the asymptotic time correlation function<[J(x,y|τ),J(x,y|0)]>∼τ−K−1ρ−K−1σ.(30) Therefore,the tunneling resistivity for the spin-rotational invariant systems(Kσ=1)has the asymptotic formρi(T)∼T1−K−1ρ,(31) which also coincides with the result derived from LLT [18,21,22].The author acknowledges thefinancial supports of China National Foundation of Natural Science and Alexander von Humboldt-Stiftung.。