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The existence of singular terms and their effects on the validity of Fermi liquid theory in two dimensions Beydaghyan, Gisia-Bano 1992

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THE EXISTENCE OF SINGULAR TERMS AND THEIR EFFECT ONTHE VALIDITY OF FERMI LIQUID THEORY IN TWO DIMENSIONSByGisia-Bano BeydaghyanB. Sc. (Physics) Dalhousie University, 1990.A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESPHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1992© Gisia-Bano Beydaghyan, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of ^The University of British ColumbiaVancouver, CanadaDate ^Ociohicr,^iq DE-6 (2/88)AbstractThe question of the breakdown of Fermi liquid theory in two dimensions is examined inthe context of perturbation theory for a dilute interacting Fermi gas. The quasiparticleinteraction function, fic;u;, is calculated for such a system. The interaction function,calculated to second order in terms of the dimensionless coupling constant, shows varioussingularities. The most divergent terms appear in the cross channel, but cancel outleaving a much weaker singularity in the limit of two momenta approaching each other( 0 —f 0 ). As in the case of the three dimensional Fermi gas, the Cooper channel containsa logarithmic singularity in the limit 0 —f 7r. This singularity can be summed and isknown to be harmless to the structure of Fermi liquid theory. A different feature in twodimensions is the existence of such a singularity for 0 —p 0. This feature needs furtherinvestigation.Calculations have also been extended to a polarized Fermi gas and the result is equiv-alent to the unpolarized case and does not show any additional features. In conclusion,the results do not indicate the presence of strong divergences which could cause thebreakdown of Fermi liquid theory in two dimensions for a dilute interacting Fermi gas.i iTable of ContentsAbstract^iiList of Figures^ ivAcknowledgement^ v1 Introduction^ 12 Theoretical background:Fermi Liquid Theory^ 52.1 The Foundation of Fermi Liquid Theory  ^52.2 The Dilute Interacting Fermi Gas  ^73 Quantum Mechanical Scattering in Two Dimensions in Free Space^103.1 General Formulations ^  103.2 Born Approximation  144 Methods of Calculations and Results^ 164.1 The Quasiparticle Interaction Function for a 2D Dilute Fermi Gas .^. 164.2 The Spin Polarized Case ^  265 Discussion^ 275.1 Recent Works on the Low Density Fermi Gas in 2D ^ 275.2 Discussion of Our Results ^  295.3 Discussion of the Phase Shift  30iiiBibliography^ 33ivList of Figures4.1 The phase space restrictions for the singular terms in / 2 + /3 . a) for poutside the Fermi sea, b) for p inside the Fermi sea. The shaded areabecomes narrower as p -f 0.   234.2 The function f (8). Note the singular behaviour at 0 = 0, 7r. The horizontalscale at it has been slightly magnified for the sake of clarity. ^ 25vAcknowledgementFirst and foremost, I wish to thank my supervisor, Dr. Philip Stamp for suggestingthis project and for his continuous guidance and support throughout the course of myresearch. The many discussions that I had with him were vital in my understanding ofthe subject and in the completion of this work. I also wish to thank Dr. Ian Affleck forexamining this thesis and providing valuable comments.I also thank the organizing committee of the School of Theoretical Physics at LesHouches and my supervisor for providing me with the opportunity to attend the school inAugust 1991 and to benefit from the outstanding lectures and the stimulating discussions.I specifically acknowledge useful discussions with Dr. Benoit Doucot.I appreciate the useful discussions with Dr. Nikolay Prokofiev during his stay at UBCand for his contribution to further investigation of the validity of Fermi liquid theory intwo dimensions.'Special thanks should also be given to Richard Szabo for reading the first draft andproviding helpful comments. I also appreciate the support of my friends and fellowstudents while I was writing. Finally, I appreciate the constant love and support thatmy family has given me.viChapter 1IntroductionThe motivation for this work partly arises from the recent emphasis on two dimentionalphysics. Until recently, physics in two dimensions was considered relatively trivial, stud-ied mainly as a matter of curiosity and occasionally to compare and contrast the resultsto properties in three dimensions. The current interest in two-dimensional physics partlystems from the success in producing thin films: for example, it has become possible' tomake monolayer films of 3He on superfluid 4He. The need to understand the physicsof surfaces, the two-dimensional electron gas and new high-transition-temperature su-perconductors also has put emphasis on understanding physics in two dimensions.Theexistence of anyons and their role in the mechanism of quantum Hall effect is just oneindication of the subtlety of the phenomena and features in two dimensions. 2With the recent discovery of high-T, copper-oxide superconductors 3 there has been anenormous effort to characterize and understand the physical behavior of these compoundsin the normal and superconducting states. 4 The measurements of electrical conductivityand electron tunneling in these materials show their anisotropic behaviour, indicatingthat the current carriers are confined to the layers of copper-oxide planes 4 . Despite theexperimental and theoretical efforts, the nature of the normal state and the mechanismof superconductivity in these materials are still open questions. Almost all candidatetheories, such as the Luttinger liquid theory of Anderson'', marginal Fermi liquid theoryof Varma, et al.' and the anyon mechanism' have concentrated on this two dimensionalbehaviour and almost all propose that electron (electron-pair) tunneling is responsible1Chapter 1. Introduction^ 2for conductivity (superconductivity) in the direction perpendicular to a-b plane (c-axis).The BCS theory of superconductivity' explains the nature and the onset of thesuperconductivity in normal three dimensional superconductors. The normal state of theelectrons in a metal is that of an interacting Fermi system: a Fermi liquid. Such a systemis reached by adiabatic continuation from a free Fermi gas and its low-lying energy levelshave the same structure as the non-interacting system. There are natural instabilitiesin a Fermi liquid which lead to the formation of Cooper pairs. 11 The prerequisite forthis mechanism is the existence of an effective attractive two-body interaction which ina metal is provided by mediating phonons. Although the phonon-mediated mechanismis not completely ruled out in the cuprate superconductors, the evidence points againstit . 12,13The normal state properties of the cuprate superconductors are also quite puzzling.These materials show metallic behaviour near their transition temperatures; however,unlike normal metals, they do not obey the predictions of Fermi liquid theory in regardto their decay rate, resistivity and tunneling properties ( for a review on this matter referto ref. [4-7}). Varma, et al. 8 ' 14 have suggested that Fermi liquid theory is obeyed in a"marginal" sense and have proposed a phenomenological postulate from which severalfeatures of the normal state could be derived. However, others have claimed that Fermiliquid theory is irrelevant with regard to the ground state of these materials. SpecificallyAnderson has claimed"' that in two dimensions, Fermi liquid theory breaks down dueto the presence of strong singular interaction terms.The question of the relevance of Fermiliquid theory to two dimensional systems is obviously important. Perhaps the crucial testwould come from the experiments on the liquid 'He films on the substrate of superfluid4He. This has been made possible recentlyl, and it could be the closest system to a twodimensional Fermi liquid, if there is one.Chapter 1. Introduction^ 3The fundamental assumptions and the consequences of Fermi liquid theory are ex-plained in the next chapter. We only mention here that a Fermi liquid is the state of aninteracting system of fermions reached by adiabatic continuation from a non-interactingFermi gas. In a Fermi liquid, the role of particles is taken up by quasiparticles which areparticlelike excitations of the interacting system. The overlap between the wavefunctionsof the bare particle and the quasiparticle is given by the renormalization factor z:zk =< 14 1 0 >Anderson's argument for the breakdown of the Fermi liquid theory begins by suggest-ing the existence of singular terms of the form11 • (1)^Pi ) fppi^ (1.2)IP P1 2in the quasiparticle interaction function, and that these interactions produce a finite phaseshift for quasiparticles with p p' = 2pF . (Note that this form is actually asymmetric.One way of justifying this would be to assume that one excitation is above and the other isbelow the Fermi surface.) He argues that due to these singularities, the renormalizationconstant z vanishes and it is no longer correct to speak of continuing from the non-interacting system to the interacting system.Anderson further draws analogy from the problem of a quantum impurity in a staticpotential that this is also related to the overlap of the initial and final many-body wave-functions:< 0 I V > exP(—(7r0)2 (1.3)where 10 > and IV > are the wavefuctions for the free and interacting systems, respec-tively, and 8 is the forward scattering phase shift. He argues that the scattering ofquasiparticles in the vicinity of the Fermi surface results in a finite phase shift whichmakes this overlap zero and therefore adiabatic continuation is impossible.Chapter 1. Introduction^ 4Faced with such propositions, there are questions which should be dealt with. Thefirst question is whether there are such singularities and and if so what their originsare. Also one must show that they indeed lead to the breakdown of Fermi liquid theory,since it is well known that many singularities, such as the Cooper pairing, only lead toa modification of Fermi liquid theory.' Assuming that these tasks are accomplished, westill have to find an alternative quantum liquid to describe the interacting ground state.This work mainly deals with the first question and examines the validity of Fermi liquidtheory in the context of perturbation theory.Chapter 2Theoretical background:Fermi Liquid Theory2.1 The Foundation of Fermi Liquid TheoryFermi liquid theory is a microscopic theory of systems of interacting fermions at tem-peratures where quantum effects dominate, i.e. below the degeneracy temperature ofthe system. Initially proposed in 1957 by Lev Landau l' 19 to be applied to the prob-lem of liquid 3He, Fermi liquid theory has also been successfully applied to the electronliquid in a metal. The theory assumes that the low level excitations of the liquid havefermionic statistics and consequently a necessary but not sufficient condition for Fermiliquid theory to be applicable is that the interacting particles be fermions. Fermi liquidtheory has been applied to the dilute solutions of 3He in superfluid 'He' and it best suc-ceeds in explaining the quantum behaviour of liquid 'Ile.' Fermi liquid theory predictslow temperature properties of the electron liquid in a metal and provides a conceptualunderstanding of the success of the free electron approximation.In constructing his theory, Landau assumed that it is possible to turn on the inter-action slowly so that there is an adiabatic continuation from a non-interacting Fermigas to the interacting Fermi liquid. An adiabatic continuation means that there is anunambiguous and one-to-one correspondence between the low-lying levels of the inter-acting system and those of a non-interacting Fermi gas. Therefore one is able to labelsuch states of the Fermi fluid by the levels of the initial Fermi gas. The justification forapplying this procedure is subtle, and depends on further assumptions appropriate to a5Chapter 2. Theoretical background:Fermi Liquid Theory^ 6Fermi system.Even in a strongly interacting Fermi system, the exclusion principle dramaticallyreduces the phase space available for scattering processes. At T ,---- 0, the volume ofavailable phase space for scattering of a particle on the Fermi surface is zero, and thereforeits life-time is infinite. A particle with a momentum k > kF, has a volume proportional to(k—kF ) 2 available for scattering. At low temperatures, a particle's energy is proportionalto the absolute temperature and its decay rate is proportional to the square of theabsolute temperature. Therefore, at sufficiently low temperatures it is possible to turnon the interaction in a time that is less than the lifetime of a particle, and one can thenspeak of one particle states which are approximate eigenstates of the interacting system.Furthermore it is obvious that there are no single particle stationary states of theinteracting system. In the theory of Fermi liquids the role of particles is taken up byquasiparticles which are the particle-like excitations of the liquid and which obey Fermi-Dirac statistics. A quasiparticle can be thought of as a particle in the self-consistentfield of all other particles, and it carries the same charge and momentum as the actualparticles.' With this notion, the problem of interacting particles is replaced by inter-acting quasiparticles whose number is always equal to the number of actual particles.One should note that the energy of the particle depends on the state of the surroundingparticles, and therefore the total energy is no longer the sum of the energies of the in-dividual particles; it is a functional of the distribution function. The energy of a singlequasiparticle is defined as the functional derivative of the total energy with respect tothe distribution function.Now assuming that the state of the system remains of the same symmetry, we ask thequestion: what happens if the distribution fuction,nk, varies slightly? The energies ofthe quasiparticles are no longer independent, and by varying nk, the energy of any otherquasiparticle changes. The total energy of the system is a functional of the distributionChapter 2. Theoretical background:Fermi Liquid Theory^ 7function and has a perturbation expansion whose first few terms are :E = E0 E Srik€(1), + -1 E E f(k,11(5nOnk,^(2.4)2 kE li)( is the energy of the quasiparticle of wavevector k and f (k, k') is the second functionalderivative of the E. The great practical advantage of Landau's theory is that this ex-pansion to second order is sufficient for obtaining the low temperature properties of thesystem. These properties turn out to depend on some integral of the function f.There are many reviews of Fermi liquid theory. For further discussions and applica-tions to physical systems see ref.[22-24].2.2 The Dilute Interacting Fermi GasIn order to introduce the methods presented in this thesis, we must mention the caseof a three dimensional low-density degenerate Fermi gas. This was studied by Lee andYang', Abrikosov and Khalatnikov26, and others'.Lee and Yang considered the cases of Fermi, Bose and Classical Boltzmann gases ofhard spheres of diameter a. The method involved finding a two-body pseudopotential toreplace the hard sphere potential. An expansion was obtained in terms of the diameter a- which also coincides with the s-wave scattering length - for the ground state of a Fermigas at a finite density p and infinite volume. The energy per particle of such a systemwas found to be (at T = 0):E^3p2N— = ( 1 ) + Eirap J(2J 1) - ' [1 + 6(1 - 21n 2) 357r + 0(n2 a2)]^(2.5)^5^ .t-F /where J is the spin of the particles. To the orders specified, this expansion is exact.Abrikosov and Khalatnikov approached this problem from a different angle. Theyalso considered a dilute Fermi gas with two-body interactions, and assumed that theinteraction range is much smaller than the distance between two particles.This allowsChapter 2. Theoretical background:Fermi Liquid Theory^ 8the expansion of the energy in terms of the small parameter (kFa) where kF is the Fermiwavevector and a is the s-wave scattering length. Their method was the first to renormal-ize the potential in terms of a physical quantity such as the s-wave scattering length. Asopposed to Lee and Yang, they did not calculate the ground state energy directly, ratherthey used the quasiparticle approximation and calculated the quasiparticle interactionfunction for the system. The advantage of their method is that one is able to obtainformulas for thermodynamic and transport properties of the system without further in-tegration, and that the energy can be obtained from the chemical potential. The detailsof this method are explained in the next two chapters. Here we give a brief outline:Consider a gas of fermions with two-body interactions as explained. As long asthe interaction range and the momentum exchange are small, one can approximate theinteraction to be independent of the momenta of the two particles. One can also ignorethree-body collisions if one is only interested in the first few terms of the series expansion.This is because such collisions only affect terms of fourth order in (kFa) and higher. Withthis method, they obtained the following expression for the quasiparticle interactionfunction:^27ra^3^cos 0^1 + sin(0/ 2) f (0) = ^h2[ 1 +2( 7r NN3 (2 +m^ 2 sin(0/2) in 1 — sin(0/2) )18irah 2^3^sin(0/2)^1 + sin(0/2) \ -1 ( 6)m (0-10-2)^+ 2(7r Pa/(1 —^2^In 1 — sin(0/2)where 0 is the angle between the two momenta. The energy is obtained from the relation:E = f ,a dN^ (2.7)and coincides with that of Lee and Yang.From the eq. (2.6), we observe that for angles near r, the function f has a logarithmicsingularity and that strictly speaking, the series is no longer meaningful. However, thisdilemma is resolved by summing the divergent terms to infinite order. This gives aChapter 2. Theoretical background:Fermi Liquid Theory^ 9non-singular result for a > 0. However, when a < 0, that is in the case of attractiveinteractions, the scattering amplitude has a pole at a small imaginary value of E whereE = p2 + p '2 - 2pF. This pole corresponds to the instability of the Fermi liquid groundstate to formation of Cooper pairs and is the cause of (s-wave ) superfluidity in a Fermiliquid.With regard to the recent developements it is worthwhile doing a similar calculation intwo dimensions. Not only could this lead us to the formation of bound states as in threedimensions, but it can also reveal divergences which signal the breakdown of validity ofthe Fermi liquid theory. The latter is actually what happens for a one dimensional gasof fermions and it is well known that the properties of a Fermi gas in one dimension arefundamentally different from that in three dimension. 28 The situation in two dimensionis far from clear and we hope to shed some light on it.Chapter 3Quantum Mechanical Scattering in Two Dimensions in Free Space3.1 General FormulationsWe are interested in calculating the quasiparticle interaction function and from that thethermodynamical properties of a dilute Fermi gas. The Hamiltonian includes the kineticenergy and a second term for pair interaction of the particles. However, the interactionenergy increases at short distances (typical interatomic distances), and perturbation the-ory is no longer valid. We can overcome this problem by renormalizing the potential interms of a physical quantity such as the scattering length. That is, we consistently re-place the potential with one which has the same scattering amplitude at low energies andis well behaved at short distances. As long as the energies are low, and the calculatedquantity includes the interaction only in terms of the scattering amplitude, the resultwould be the same as the one which uses the actual, non-renormalized potential.The scattering length is defined as:a = — lim f (0)^ (3.8)k--40where f(0) is the scattering amplitude. In three dimensions a has the formMttoa= ^47h 2where uo = I d3r V(r)^(3.9)and V(r) is the interaction potential. However, as we shall soon see, in two dimensionsall scattering amplitudes diverge as (k) -1 at low energies. The problem is easily fixed byintroducing a well behaved dimensionless quantity as is done in the next section.10a2Tao2a , aR,^2^in2VC - U(r) - )R = 0r ar ar r2m 2T = 0 (3.13)(3.14)Chapter 3. Quantum Mechanical Scattering in Two Dimensions in Free Space^11The following is a calculation of the scattering amplitudes in two dimensions. Theyare included for completeness. The derivation and notations are mainly followed fromLapidus. 29 . For further discussion see ref. [30-32].We begin by writing the SchrOdinger equation in two dimensions V 20 + V (r, 0)0 = (3.10)2mwhich we can write asV2 + (k 2 — U(r, 0))/ = 0^ (3.11)where k 2 = 2mEh2 and U(r, 0) = V(r , 0). Furthermore, we assume that the potentialis central which means U(r, 0) = U(r). Now in polar coordinates, the equation has theforma a^a2 ( ?75-r (rw-r) r2a92 + (k 2 — U(r))0 = 0 (3.12)This equation can be separated into radial and angular parts, so we take 0(r, 0) =R(r)T(0). The two equations areIf we take the x-axis to be along the direction of the incident beam, it would be an axisof the symmetry of the system. Therefore the probability distribution must be symmetricabout this axis. That means 'TOW IT(-6)1 2 . Then the normalized angular part ofthe solution would have the formT(0) = \Fir cos me^ (3.15)m must be an integer so that T(0) is periodic in 0(single-valued).Chapter 3. Quantum Mechanical Scattering in Two Dimensions in Free Space^12Now, let's look at the asymptotic form of the radial equation. Consider a potentialthat vanishes at sufficiently large values of r. An example is a potential of the formV(r) =The radial equation has the asymptotic form2 d2 R^dRp^p_V(r)0(p 2r < ar > ani2) R 0(3.16)(3.17)dp2^dpwhere p = kr. Eq. (3.17) is in the form of the Bessel differential equation, solutionsof which are Bessel and Neumann functions of the first kind. At large values of r theirleading asymptotic terms areJrn (kr) --4 ( rkr )1 cos(kr — (m 2)2)^(3.18).^1 7rNm (kr)^( irkr ) 2 sm(kr — (m + —2 )-2 ) (3.19)and therefore the radial solution has the asymptotic form2^1 7rRm (kr)^ (m+2 + 8,)^(3.20)The quatity Sni is called the phase shift of the mth partial wave.h2 7.2Now consider a free particle of a fixed energy E^4 . The incoming particle has2mthe wavefunction:ikxOinc = e^ (3.21)For a steady state configuration, conservation of energy requires that the scattered wavehave the asymptotic dependence ofeikrI,„(r, 0) r.,^f (0)\fr(3.22)where 1(0) contains the angular dependence of the scattered wave and has dimension ofChapter 3. Quantum Mechanical Scattering in Two Dimensions in Free Space^13Therefore the asymptotic form of the steady state wavefunction must have the form :e ikr0(r, 0) = eikr go) (3.23)Comparison of the above equation with the previously obtained asymptotic form of thewavefunction (eq. 3.20) gives the scattering amplitude and phase shifts. We proceed byfirst expanding the incident wavefunction in terms of Bessel functions. The Jacobi-Angerrelation (ref. [33], p585) gives00e = eiks^ikr cos()^E im Jni (kOe imem=-0000^= Jo (kr) + 2 E im Jrn (kr) cos(m0)^(3.24)m=1Then the following equality must be satisfied:oo^ eikr^co^2^7rJo(kr)+2 E imJm (kr)cos(mo)+f(6)—,_ E Am (  ,r cos(kr—(m+1)--2-+ ,5,) cos(m0)^v r^m.0^iricm=1(3.25)Writing this equation in terms of e ikr and e —ikr and putting the coefficient of each to zerogives the following two equations:Em im cos(m0)^Am ism\/2lrk^z/Ue^(3.26)1(0)^(emim cos(m0)^Am e iSm)e—i=m=o^f2irk^ViTcwhere Em = 2, m 0 and co = 1. These equations give:^Am = 2Em im(27r)e i6rn^ (3.28)^1(0) =^1^E^cos(m0)(e2ism — 1)^(3.29)(27rik) m=oFor m^0, the phase shift vanishes as km and therefore ensures that f(0) remainsregular. However, in the case of m = 0, we have ( C i is a fixed constant ):itSo ^, (ln Clka ) -12^2(3.30)0^(3.27)m=0P = —k E €77, sing (5m, (3.32)Chapter 3. Quantum Mechanical Scattering in Two Dimensions in Free Space^14which is not enough to make the scattering amplitude finite.30 The divergence of thescattering amplitude at small energies also appears in the Born approximation.The analogous quantity to the three dimensional scattering cross section is a lengthin two dimensions. We define it asFor a central potential:P = i' 27r If (6)1 2 de(3CO(3.31)3.2 Born ApproximationIn cases where the scattering potential is weak and the phase shifts are small, one maytreat the scattered wave as a perturbation to the incident wavefunction. This is theessence of the Born approximation. (see discussion in ref.[34])We write:(r) = 00(0 + OM^(3.33)where 7b0 (r), OH and OM denote the incident, the scattered and the total wavefunctionrespectively. They must satisfy :^(V 2 + k 2 )0(r) = U(r)0(r)^ (3.34)Or(V 2 + k2 )0(r) = Uibo(r)^(3.35)The Green's function of the Helmholtz equation in two dimensions has the form':^G (r, r') = Ti le ) (kir —71)^ (3.36)where le ) is the zeroth order Hankel function of the first kind, defined by^HO1) (kP) = Jo(kP) + iNo(kP)^ (3.37)1^ei(kir—r'l+ U (r')e ikx 1 dr' (3.39)= 8rkir — r'lChapter 3. Quantum Mechanical Scattering in Two Dimensions in Free Space^15Then the scattered wave must be of the form— J G(r,r')U(r')00 (r')dr'= --4 f Ho (klr — r'1)U(r')00 (r')dr'^(3.38)Using the asymptotic form of the Hankel function' :where q is the momentum transter, i.e. q = k' — k and k' is the wavevector of modulusk in the direction r'. In the limit of large r and using equation (3.23) for the scatteringamplitude, we obtain:f(0) = ^1 ^I e i g .r i U(r')d2 r'^ (3.40)rkand in the limit q^0Uo^2muo AO) = V8rk — h2V8rkwhere00uo^V(r) d2 r^(3.41)The analogous quantity to the three dimensional scattering length has the dimension of(L) and is equal to :2muoa =  ^ (3.42)h 2 \/8 kThis relation can be used for the scattering of two particles by replacing the wavevectorwith the relative wavevector and the mass by the effective mass.Chapter 4Methods of Calculations and Results4.1 The Quasiparticle Interaction Function for a 2D Dilute Fermi GasIn the previous chapter, we described two particle scattering in free space. Now, we con-sider the scattering of quasiparticles in a dilute Fermi gas. As mentioned in section(2.2),this is a generalization of the method of Abrikosov and Khalatnikov to two dimensions.We consider a short-range two body interaction that is independent of particle momenta.The interaction is then renormalized in terms of its low energy scattering amplitude.This procedure removes the difficulty of having to deal with strong interactions at shortdistances.The Hamiltonian of a system of particles with pair interaction is :H = ckunk, +EEEE<Aai,Aa2IVIPicti,p2a2 >^alp-a2ap2c,, api , ik,cr^Pl>al P2,cg2(4.43)where the summation is over all four momenta and four spin indices. Now, one can write:<^f v(r)e-g-ind2r^(4.44)where q = p'1 — pl = — (p'2 — p2 ), and fl is the area of the system.Conservation ofmomentum is implicitly assumed in this expression. Spin indices are suppressed becausethe interaction is assumed to be independent of spin. Assuming the momentum exchangeof the particles to be small, and the interaction to be short ranged one can replace theintegral in the above equation with its value at q = 0 and the matrix element with 10-.16UpA(1)En <n 2C2^4,(74--,a a2 , — (T a l,cr I On° >Pi ,P2U0 = —29^E < Own I (m1,, —^— 1), (n2 , ,_, + 1),^+ 1) >a 1,2 1',2'—^\71 — nv ,, (4.51)Chapter 4. Methods of Calculations and Results^ 17We now consider the approximate HamiltonianH = Ecp,n2„, + "±-0EEEEcit,a+, a a4S2^P1CX1 P2% P2 a2 Pi a ].P,cr^P1,a1 P2 ,a2 14,c4 P1, 04(4.45)We restrict ourselves to s-wave scattering which is dominant in the limit of slow collisions.Then only particles of antiparallel spin can scatter each other, and for particles of spin2, the Hamiltonian simplifies to :uoH = E Epu np, + E EP,6^1,2 1',2'where + and - represent the two possible spin states of the particles.We consider this Hamiltonian as(4.46)^H = H0 + OH^ (4.47)and find its ground state energy to second order of perturbative theory. The first andsecond order corrections to the energy level E n are:0(1)E,,, = (OH)nn =< q OH 1 en, >^(4.48)\--• I (All )nm 12^I< C6Cn1 I 'Ali 1^>1 2 (4.49)A (2) En^Eo Eo^nth^7E9, —mOn n^mwhere En° and 0°„ represent the eigenvalues and eigenvectors of H0 . The ground stateenergy of the non-interacting system gives the zeroth order contribution:Er) = E p ncr PgP,crand the first order correction is given by eq. 4.48(4.50)Chapter 4. Methods of Calculations and Results^ 18The matrix elements will not vanish only ifnil =^—1n2, = n 2 — 1^ (4.52)so thatAME = U0 N:Nr-N2_, 711,0-n2,—n^2S-2 L-1a 1,2(4.53)This term represent the shift of the energy levels due to the interaction. Similarly<^AH 0913, >=---< 0 1-• E E^>1,2 1',2'2Quo—EEE <o f;)„ (n i , + 1), (n2, + 1), (n2 — 1), (n i — 1) >211^- 1,2 1',2'— 111/0 n2,07,2TT1Matrix elements will not vanish only if= — 1m2 = n 2 — 1mt , = ni , + 1m2, = n 2 , + 1So now we have„,2A (2) En^160^E ^—cr (1 —^,a)(1 ^21/ 2^EVcra 1,2 1',2'(4.54)(4.55)(4.56)with implicit conservation of momentum. This term represents the energy correction dueto pair collisions. It is proportional to the occupation number of the initial states andthe number of unoccupied final states.Uo u2+  0  E E ^1 ^uo= —21^4122 a 1 , 2' 1 °.E + E2,- cf — EV,a — E2/,_a 212(4.58)Up _.,2lChapter 4. Methods of Calculations and Results^ 19We now recall from the previous section that in two dimensions the "scattering length"diverges as (k)* . However, we can renormalize the potential in terms of the dimension-less coupling parameter A defined by:A = acd87rk = mu° (4.57)h 2For the sake of consistency, we should also include the next term in eq. (3.41) or eq.(4.57) which we have thus far ignored. Fortunately, this is easily done, as the secondorder Born approximation is obtained by simply changing':u2 1Up = u0 — -'s E E211 a 1/ ,2/ E1,a + E2,--, — Ev,, — E2 , ,-,L2 A,^h4 V^1m^m2S1 E E Ei , + E2,-, —Elsa — E2 1 ,-.7= — a l',2 1^/h^h2—m2 A(1 ^ A E Emm521. 1,2E 1,a + E2,-a — Eli,a — E2',-a )=Equations (4.50), (4.53) and (4.56) give the ground state energy to second order in :UpE = E ,p,np, + —252 E E n i ,,n2,,P,cr^0- 1,2,2+ L.0 v. E v. n1,,n2,_,(1 — ni,,,)(1 — 122/ 7 _a )2S2 2 L'a, 1,2 1',2' Ei,a + E2,-a — E1',a — E2 , ,-,and finally using the renormalized value of the potential from eq.(4.59):(4.60)h2^h21 v.„^2mE = E fp,np,, + 2mf2 A(1mS2 /` La „2 + F, F2 „,.12^i2) E E n l,a n2 , -aP'cr^ 1',2' Fl I 2 — l — P2^a 1,2h4^(1 — ni , ,,)(1 — 7121,_,)+ ^ A2^ni un2,E E    2m 2 52 2 a 1,2 1',2' El,a + E2,-a — El , ,a — E2 , ,-ah 2= E cp,u np,, + 2m11 A E E ni,, n2,-0-p,o^ a 1,2(4.61)(4.59)E = E 2n2S2p,a^ a 1,24^2,—cr (n1',cr^n2/,—o)m12A2711,crn^ E E ^+^P? — P122PI 492 pl ,p2c nP,crh2A E EChapter 4. Methods of Calculations and Results^ 20h4 — nv , 0.)(1 —^— 1]—92 A2 E E  ^P1,P2^ +^P? — P/22h 2E fp,unp,, + 2rn9 A E Ep,cr^ a 1,2h4^Az^E nl,an2,—cr^— (71,v,o.,d2MEP a 1,2^,t2Fl I F2 — Fl^F2The third term is antisymmetric with respect to the transformation P1, P2(4.62)(4.63)Fl F2while the summation is over all four momenta, and therefore the summation of this termvanishes:(4.64)The spin dependence of denominator has vanished because in the first approximationE±, = E po-.B^(4.65)Let's rewrite E as :0E^nk,a + 2h 2 A^v,= E •--1c,c7"'^2mSZnl , al n2,0-2 Pal a2k,a^ pl ,Q1 112, 0.22h4A2 6031 + P2 — P3 — P4) mf22 E E E 2_,^ n4,„)Pcri,2 4446)Pi111 ,c7 1. P2 '0'2 pa 1 0-3 P410.4 p22 p32 p42 n1,0.1 n2,(72 (n3,Cr3 —where Po- 1 0-2 is the spin exchange operator.The energy per quasiparticle cp,, is the functional derivative of the total energy E:^SE^0^^= E^o"^P,0"SnP, 4h4 A2nify E EP2,a22h 2 API , (7 1( 8(pi + P2 — P3 p) 2^ni ,0.1 n2 , 0.2 P0. 10.2 P,„p3^+733^p2mIZ E ni Pcra ia(p + Pl — P2 — P3) (4.67)P2 + P1 — P2 —n i ai (n2,2 + n3,0 -3)Pcri , P0-20-3)Chapter 4. Methods of Calculations and Results^ 21and the quasiparticle interaction function fPP'" is the second functional derivative of thetotal energy with respect to the quasiparticle distribution function n. So we havefo- odPP'Scp,,„ 2h 2 A n^2h4A2 2 2(nk+-racr' _,Trif2^mn2 k P2 + Pf2 k2^—n ic ,_a lp2 1 2 k2 (p pl + 102 + 1 2 p2 k2 (pl p k)2 )^(4.68)Note the slight change of notation, namely initial momenta are now denoted by p and p'.For the case of an unpolarized liquid n k , cr = n 11 ,— c and the interaction function becomes:2h 2 A^2h4X2^4P„,=. m^St P°"''^ms22 k p2 pI2 k2 (p pt 1^02p2 k2 pl2 (p pl +^p/2 k2 p2 (p/ p^)nk (4.69)Except for the prefactor, this expression is identical to its form for the 3 — d case, seeref.[26].At this point, we could simply assume that p p' = pF , since important scatteringprocesses are on the Fermi surface. But note that we are also interested in expressionsof the form (1.2). Therefore we first evaluate the integrals for general values of p and p',and at the end take the momenta to be on the Fermi surface. The following three termsshould be evaluated:1(4.70)Lk p2 + p'2 — k 2 —^p' — k) 21(4.71)(4.72)(4.73)k 132 + k 2 — p'2 — (p — p' k) 21^112 k2 p2^P k) 2We can use the following geometric identities:09 +^= p2 112 + 2p^2k • (p p') k 2fac'P131Chapter 4. Methods of Calculations and Results^ 22(p — p k) 2 = p2 + pi2 — 2p • p + 2k • (p — p) k 2^(4.74)(p — p k) 2 = p2 p'2 — 2p • p/ — 2k • (p — p) k 2^(4.75)Let's first evaluate the last two integrals:52^fpF^kdOdk= ^^2(27rh) 2^Jo p' • (p — p') — k(p — p') cos 0and with a change of variables:=^ PF ^11 27  kdOdk122(27rh) 2 I p — p' j Jo Jo S — k cos°(4.76)(4.77)where S is defined by S — p1F^The principle value of the integral over the angle can be evaluated by contour inte-gration in the complex plane, and in general:2ir dco 27rP ^ sgn(a) 0(a 2 — b2 )Jo a — b cos co^/a2 — b2SO:/2 =42 P- P'iF ISI — e(ISI — 1)V S2 — 1) sgn(S)7rh where 0 is the step function. Similarly:(4.78)(4.79)/3 = ^h2 IpP—p'l (IS'l —0(IS/ 1-1)N/592 — 1) sgn(S')^(4.80)where S' is defined by S' = 1 PPi)IP-p'lIt is interesting to notice that /2 and /3 both contain terms of the same form as (1.2).But we then see that these two terms actually add up to a constant:1/ ^p'(p'— p) P'1 2= —1Ip --^-- (4.81)Chapter 4. Methods of Calculations and Results^ 23Figure 4.1: The phase space restrictions for the singular terms in 1 -2 +1-3 . a) for p outsidethe Fermi sea, b) for p inside the Fermi sea. The shaded area becomes narrower as p 0.Therefore, these terms are harmless! Yet, there are other singularities as well, but thestep function puts severe phase space restrictions on them ; see Fig. (4.1).Finally, the first integral is:V  fpF^ kdkdOIl = (271-h) 2 Jo Jo (k 2 p • /1) — klp III cos 0This integral is equal to:(4. 82)1-1if2Joir kk2ipw;c16it kc2o+sp.0p, \(4.83)= (27C/02 foPF kipk +dkplk^sgnl kip+pil ) ^0[(k 2 + P • II ) 2 — 1]^(4.84)9Jodk^klp + pi= 27^(2irh) 2 o^kIP + /II \ I ( k2ip+47 /1 ) 2^10{(k2 + p • p)2 — (14 +111)fj1.85)f2 IPF=  ^27rk sgn(k 2 + P • P') ^dk  27rh 2 0^Ok2 + p . /39 2 — (kiP +71 1) 2Let's define:Q(k)..(k 2 4- 10. 1/) 2 --(k11)-k ii)2^(4. 86)The roots of Q(k) 0 are given by :ki = 71 (1.13 +^+ - 11 1) 2^(4.87)Chapter 4. Methods of Calculations and Results^ 24114 = 4(113 + 11 1 — 1P — P'1) 2 (4.88)and the k values that make Q(k) positive are 0 < k 2 < /4 and ki2 < k 2 < p2 or0 < k 2 < min(14,p2F). By a simple argument we can see that k 2 + p • p' is positive whenki2 < k2 < pF , otherwise sgn(k 2 + p • p') = sgn(p • p').After taking care of this small detail, we would like to evaluate the integral in eq.(4.84).With a change of variable to y = k 2 , we have :f  kdk^1 i  dy ^ — log[2VQ(y) + 2y — ( 2 + p' 2 )]^(4.89)Now Ii can be written as:f2^P + 11Ii =^{log( p — )27r i2^p'+ (0 [P2F — kfl + O[ki — P2F]sgn(P ' P')) x {log2 V14 — (P2 + P'2 )P2F + (P ' p') 2 + 2P2F — (P2 + P'2 ) i}2 1P • P'1 — (P 2 + 112 )(4.90)Note that the most important contribution in the above expression is the first term.When p and p' are on the Fermi surface, it is this term that describes their interaction.These equations give the following form for the quasiparticle interaction function intwo dimensions:h2^h2^P + 11 = 2A( rnf2 )P,,,, + A 2 ( 27rTnn ){2P„, [log( p — p,)+ ( 19 [P2F — ki] + O[kZ — P2F]sgn(P '71) x (log 204 — (192 + P'2 )P2F + (P • p') 2 + 2132F — (132 + 13'2) )]21P • P'1 — (132 +112 )+ [1 + ^111IP PF^(0 (ISI - 1)\/S2 - 1 + 0 (1,5" 1 - 1)1/912 - 1)]}^(4.91)- \/Q(k) 2 OWfC17 1i PP /The cross channel (the last part of the above ) has singularities as p —> p' which arestrong, but only appear if p' > pF , while the Cooper channel (i.e. the part proportionalChapter 4. Methods of Calculations and Results^ 25Figure 4.2: The function f(0). Note the singular behaviour at 0 ,---  0, r. The horizontalscale at r has been slightly magnified for the sake of clarity.to P e ) has a logarithmic singularity at that limit. In the case when both momenta areon the Fermi surface the latter singularity remains, but the former disappears. In thatcase the interaction function simplifies to :1 + cos°17 , .= 2Ah 2 pero., + A2h2 1^ ( + P„, log^JPP^mf/^27rnill^1 — cos 0 )(4.92)where 0 is the angle between the two momenta. The above can be separated into thesymmetric(parallel spin) and antisymmetric (antiparallel spin) parts:andA 2 n 2fpp, _ ^271-rat(4.93)fa = )h 2^A2h2^(1 + 1 log 1 + cos 0 \Pp,^+mfg 27mQ^2^1 — cos 0 )(4.94)Recall that A is dimensionless and 77- 1 -2h 2 has the dimension of energy. The behaviour off as a function of angle is shown in Fig. (4.2).Chapter 4. Methods of Calculations and Results^ 264.2 The Spin Polarized CaseComparing equations (4.59) and (4.60), we observe that the form of the interactionfunction for a spin polarized system is very similar to the unpolarized case. Beforeproceeding to write the result, we note a few points.The degree of polarization in the system is given by :N.+. — N_ p2F+ — p2F _a =^ =  ^ (4.95)N+ + N_ 4+ + p2F_As the degree of polarization increases in the system, s-wave scatterng becomes lessimportant. This is due to the fact mentioned earlier, that in the limit of s-wave scatteringonly particles of antiparallel spin can scatter each other. By increasing polarization,the amount of available momentum space for s-wave scattering decreases, and in highmagnetic fields, the p-wave scattering dominates. So our result is only valid for weakmagnetic fields.For a polarized Fermi liquid , equation(4.59) gives the result :h2^2  h 2^P + Pi' = 2)4—Finz )P„, + A( 27rrn2 {2P„, [log( p _ 13,)+ [O[p2F+ — 4] + 0[14 — pF+ ]sgn(p - p')]x (log 204+ — (p2 + P'2 )P2F+ + (p • 130 2 + 213F+ — (p2 + p12)  )12 113 • PI I — (132 + 13'2 )+ (pF+ --* pF+)+ [1+ IP 1 Pii 09 (1 ,9 1 - (PF,-,0)\152 - (PF,--,0 2 + 0 0 51- (PF,-,))\15'2 - (PF,--M31)When the two momenta are on the respective Fermi surfaces, this expression reduces to :f;;T: = 2P,„, [1( 2 ) + X2(2^g(P+ 13 ^)(lo^) + 1)1 (4.97)n1Q 27m12^p — p'Equation (4.95) does not indicate any pecularities due to polarization. However, thepresence of two Fermi surfaces may put restrictions on the scattering events. 36 Therefore,this case needs further investigation.fa 03 pp'Chapter 5Discussion5.1 Recent Works on the Low Density Fermi Gas in 2DFor a long time, it was implicitly assumed that the structure of Fermi liquid theorywas consistent in two dimensions as well as three dimensions. Numerous papers havebeen devoted to different aspects of the two dimensional Fermi liquids ( especially inconnection with the experiments on films of 'He ), without questioning its validity. 36-40However, it is well known that interacting Fermions in one dimension do not form a Fermiliquid, but form a state which has been called the Luttinger liquid.' It is characterizedby separation of charge and spin degrees of freedom and its low-energy excitations havebosonic character.In connection with new high-T, superconductors and their anamolous normal stateproperties, 4 Anderson''' has claimed that the normal state of these new materialsis a new quantum liquid. To support his claim, Anderson asserts that there are twoknown fundamentally different fixed points ( in the context of renormalization group )for systems of interacting Fermions in any dimension and consequently, such systemsexhibit Fermi liquid behaviour or Luttinger liquid behaviour. Anderson argues that dueto singular terms in the quasiparticle interaction function, Fermi liquid theory breaksdown in two dimensions and a system of interacting Fermions shows non-Fermi liquidbehaviour. The breakdown occurs because as a result of the singular interactions, thequasiparticle wavefunction's renormalization factor vanishes. The last point was explored274 In 2z(w = 0) = 1 ha2(pa2) (5.99)Chapter 5. Discussion^ 28in a paper by P. Stamp. 41In his paper,' Stamp showed that if such singularities as (1.2) are present, theyindeed lead to a breakdown of Fermi liquid theory. The proof was based on examin-ing the perturbation series and separating graphs that contained such singularies. Themost divergent terms were self-consistently summed, and it was shown that they give aquasiparticle pole vanishing as2'["—] 2 f6Oz(w) 1-1-Fo h N(0)w0 (5.98)where So is the phase shift, w = E — 2EF and coo is some upper energy cutoff. p representsthe number density, No is the renormalized density of states and fro is the renormalizedzeroth order Fermi liquid parameter: coming from non-singular graphs. This form isvalid in the vicinity of Fermi surface ( —f 0 ) and signals the breakdown of Fermiliquid theory. The subdominant (less divergent) terms were ignored in this calculation,but later they were summed42 ( by eikonal expansion ) and were found to preserve theessential structure of (5.98). It must be emphasized that Stamp's paper did not give ajustification for the existence of such singularities, but merely assumed it. Neither did itgive any clues to the relevant ground state.Another work that has examined the validity of Fermi liquid theory in two dimensionsis by Fabrizio, et al.'. They calculated a nonperturbative, exact solution for the groundstate of a finite number of particles in the low density limit. What is of particular interestto us is the Migdal discontinuity which they found to beThis indicates that z is well defined and non-zero in the low-density limit. Another workwhich came to the same conclusion was by Fukuyama et al.' This work used t-matrixapproximation to study the Hubbard model in the limit of low density and found thatChapter 5. Discussion^ 29it behaves like a Fermi liquid. However, both these results are shown by Stamp to beconsistent with formula (5.98), since z is nonzero in the limit of low density.Furthermore, one should mention several papers by Engelbrecht and Randeria 45-49on this subject. Their work" on the low density repulsive Fermi gas in 2D is of particularinterest to us. Their method was similar to ours in that they extended the method ofref.[26] to 2D. Their dimensionless parameter, although defined differently, is equivalentto ours. They also found the logorithmic singularity that appears in the Cooper channeland signals the superconducting instability for 0 = 7r. However, their calculations weredone with the momenta on the Fermi surface , and hence is unsuited to detect anysingularities of the form (1.2). Note that (1.2) actually is finite (= 1,-) for k and k' onthe Fermi surface.5.2 Discussion of Our ResultsIn this work, we have studied a two dimensional dilute interacting Fermi gas. Thequasiparticle interaction function has been calculated to second order in terms of thedimensionless coupling parameter, A, which characterizes the renormalized potential.The interaction function shows a number of interesting features which are :I. The appearance of singular terms of the form proposed in equation(1.2). This showsthat in essence such terms can exist in the interaction function. However, in our case,the two most divergent terms add up to a harmless constant. What remains is a muchweaker singularity which is described below.II. From eq.(4.78), (4.79) and (4.91), we have other singular terms which are of theform:A PF,-, ^IP - PI 1( P • (11 — 13) ) 2 — 1PF IP — liChapter 5. Discussion^ 30= 1^IP — P'I v/p2 cos 2 7 — p2F, \fipF  t ip' — PF )1IP - PI I l PF jAs can be seen from the aforementioned equations, these terms only appear when por p' or both of them are above the Fermi surface. Even then, they have additional phasespace restrictions due to the presence of the step function, ( see Fig. 4.1). and they arefurther weakend by a factor of (IPIPF' )1. Therefore, it is unlikely that they could leadto a breakdown of the Fermi liquid ground state. The work by Stamp and Prokofiev 5°shows that this conclusion is indeed correct. We shall look at their work in a bit moredetail later.III.Finally, our result agrees with those of Randeria,et al.' that there is a logorith-mic singularity in the Cooper channel This feature is not unique to two dimensionalsystems and has previously been noted in 3D as well. It can be shown' that despite thissingularity, the interaction function remains regular for a > 0.This is done by summingthe ladder of such singular terms to infinite order. However, attractive interactions a < 0lead to the appearance of a pole in the scattering amplitude, which is indicative of theinstability of the ground state to Cooper pairing. It marks the onset of superfluidityin a Fermi liquid. However, our result also shows the presence of such a singularity for0 = 0. This singularity has the peculiar feature that it is of exactly the same form as0 = 7r divergence, but with a sign difference. This feature is definitely worthy of furtherinvestigation.5.3 Discussion of the Phase ShiftWe have already looked at one or two aspects of Anderson's argument for the breakdownof Fermi liquid theory. So far we have looked at the question of existence of such termsChapter 5. Discussion^ 31and their effect on the validity of Fermi liquid theory in two dimensions. Anderson arguesin analogy to the problem of a impurity with a static potential. He has argued that in thepresence of the Fermi sea, the two particle relative scattering phase shift (q , (.4.7) (whereq is the center of mass momentum and w = E — 2€F, ) is finite at q = 2kF, w = 0, and thatit leads to an orthogonality catastrophe:< 0IV >f"4 eXP(—(7r-60 ) 2 ln (5.100)where < 01 V > represents the overlap of the non-interacting state with the state in thepresence of the potential. In the case of a finite phase shift this overlap vanishes. Thisleads to an orthogonality catastrophe which means we can no longer consider the freeFermi gas as the relavant fixed point of the problem.Now, one relevant question is whether there is a finite phase shift. Randeria andEngelbrecht have claimed that near the Fermi surfaceNAT) 8(2kF,^21n(koa)(5. 10 1)which goes to zero as w -4 0. A work that has studied this problem in detail is that ofStamp and Prokofiev.' In their paper, starting from the two-body SchrOdinger equationfor two particles, they calculated the scattered wave function and the phase shift forsuch scattering in the presence of a Fermi sea. They found the behavior of (q , co) at thelimit of q 2kF and w -4 0 to be peculiar. This was also noted by Fukuyama et al.'This dilemna is resolved using the Lippmann-Schwinger equation and applying boundaryconditions of a finite box. They noted that a) the answer should be independent of theshape of the box in the thermodynamic limit (L oo) and b) that the direction ofQ, which is the momentum exchange, should be arbitrary. By angular averaging overall possible directions, one derives an expression for the phase shift that properly andunambiguiously vanishes in the limit of low energy. Therefore they concluded that withinChapter 5. Discussion^ 32the context of perturbation theory and validity of the Lippmann-Schwinger equation thephase shift remains zero, and Fermi liquid description remains valid.In conclusion, our calculations show the presence of weak singularities which arelimited to a very narrow region of the phase space, and so there is no hint of a breakdownof Fermi liquid theory. This finding is in agreement with other works on the subject,especially that of Stamp and Prokofiev.' The conclusion is that perturbation theoryis consistent with a two dimensional Fermi liquid and if such strong singularities existwhich lead to the breakdown of the adiabatic continuity from a free Fermi gas, they musthave a non-perturbative origin.Bibliography[1] J. M. Valles et al., Phys. Rev. Lett. 60, 428 (1988); R. H. Higley et al., Phys. Rev.Lett. 63, 2570 (1989); and D. T. Spragne et al., Phys. Rev. B 44, 9776 (1991).[2] R. E. Prange and S. M. 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