Breakdown of Fermi liquid in correlated electron systems

PHYSICA EI~EVIER Physica A 263 (1999) 197-207

Breakdown of Fermi Liquid in correlated electron systems

Claudio Castellani and Carlo Di Castro

Istituto di Fisica della Materia e Dipartimento di Fisica, Universith di Roma "La 8apienza", Piazzale A. Moro 2, 00185 Roma, Italy

A b s t r a c t

The standard description of metals is based on the Landau theory of Fermi systems (Fermi Liquid theory). This picture breaks down in one dimensional systems, which are instead described by the Luttinger Liquid theory. Actually, experimental evidence indicates that Fermi Liquid theory breaks down in a variety of physical systems, including superconducting cuprates.

In the first part of this lecture we will consider the relevant problem of crossover from Luttinger to Fermi Liquid with increasing dimensionality, showing that the Fermi Liquid is stable with respect to residual scattering by regular (short range) interactions among quasiparticles in any d > 1. However singular interactions can modify these results and open the way to richer scenarios.

Relevant cases of singular interactions concern: i) gauge theories introduced to describe the half-filled Landau level and spin liquids considered in the context of high Te cuprates; ii) prox- imity to a quantum critical point which has been investigated as a source of non-Fermi Liquid behavior in both heavy fermion systems and cuprates.

Even though the theory of non-Fermi Liquid in d > 1 is hindered by the lack of exact non- perturbative methods in d ¢ 1, various progresses have been made. In particular the singular forward scattering has been approached with some success by both bosonization and renormal- ization group methods.

1 I n t r o d u c t i o n

Fermi Liquid theory has been the general framework for describing Fermion systems (like electrons in metals and He 3) for more than forty years since the work of Landau in the late fifties [1,2]. It was realized in the seventies that Fermi Liquid theory does not apply to ld systems which are instead described by a different theory: the Lutt inger Liquid theory [3,4]. However it is only in the late eighties tha t the problem Fermi Liquid versus non-Fermi Liquid theories becomes a very debated subject in the context of high Tc cuprate superconductors [5]. One of the terms of this debate is the question whether Lutt inger Liquid theory would also apply to 2d systems like the cuprates.

In this lecture we will review the main issues of this debate. A major message will be that Lutt inger Liquid crosses over to Fermi Liquid as soon as d > 1 for short range

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interactions [6]. A kind of Luttinger Liquid behavior at d > 1 survives only for systems with long range interactions [7]. In this respect, these systems with long range forces are the generalization of the ld Luttinger Liquid to higher dimensions. In the light of these results we shall finally discuss possible scenarios for the non-Fermi Liquid properties of cuprates.

Our presentation of the techniques introduced to approach the relevant problem of strongly correlated electron systems is far from being complete. We will mainly deal with the methods using Ward Identities to build non perturbative theories. Our aim is to convey the general ideas, while referring to the original works for all details.

2 Fermi Liquid t h e o r y

Fermi Liquid theory, which generically applies to 3d metals, relies on the existence of elementary excitations (quasiparticles) behaving as free particles but for residual Hartree- Fock corrections [2]. In terms of the modern renormalization group (RG) analysis this corresponds to the asymptotic freedom for quasiparticles near the Fermi surface. At the basis of this asymptotic freedom there are the kinematic restrictions for low energy de- caying as implied by the existence of a Fermi sphere and the Pauli exclusion principle. At zero temperature, the inverse lifetime for a quasiparticle of momentum k and excitation energy ek obeys the relation l I T k ,"-' ~ << £k, stating that quasiparticles are asymptotically stable excitations in the limit k -+ kF with ek --~ 0 . At T ~ 0, one gets 1/Tk N T 2.

The equilibrium properties of a Fermi Liquid are similar to those of the non interacting system. In particular c v / T , Xch,,-ge and Xs~n are constant in temperature at small T. Here cv is the specific heat, and )/~arge and )/8~n indicate charge and spin susceptibilities. The resistivity is quadratic in T at low T: p ~ T 2, plus a residual resistivity in the presence of weak disorder.

Fermi Liquid theory breaks down in a variety of physical systems even in the absence of symmetry breaking order. A large class includes disordered correlated systems like: i) metals with impurities having internal degrees of freedom, i.e. Kondo systems [8] (single channel Kondo systems at T > TK, TK being the Kondo temperature, and multichannel Kondo systems at any T); ii) 3d electronic systems near the metal-insulator transition in the presence of strong disorder [9]. The discussion of non-Fermi Liquid in disordered systems is beyond the scope of this review which concerns clean systems, where non-Fermi Liquid behavior, if any, derives from interactions. However we would like to mention the 2d disordered electronic gas, which is realized in MOSFET and heterostructures as a candidate for a non-Fermi Liquid behavior. Scaling theory predicts for this system a metallic phase with self-generated local moments and a non-Fermi Liquid behavior in Cv and X ~ [10,11]. The experimental verification of this prediction would be a major breakthrough.

3 Luttinger Liquid theory

In clean systems Fermi Liquid theory is known to breakdown in ld systems where the problem of interacting fermions is exactly solvable under quite general conditions [12,3]

C. Castellani et al. / Physica A 263 (1999) 197-207 199

and by various methods like bosonization [13], Dyson equation plus Ward Identities (later referred to as WI approach) [14] , renormalization group [3,15].

The resulting theory has been called Luttinger Liquid theory [16]. Notice that phase space arguments for vanishing scattering at kF do not apply in ld: there is no asymptotic freedom and no quasiparticles in ld. The renormalization group analysis shows indeed a line of fixed points with finite fixed-point interactions. The single particle Green function G(x, t) shows an anomalous dimension ~?. While the non interacting G°(x, t) = 1 / ( x - VFt) indicates that particles only propagate with the Fermi velocity VF in the x direction, in the interacting case the result is

G(x, t) : 1/(x - Vc~)I/2-1-'/2(X -{- Vc~) ' /2(X - - Vst) 1/2 (1)

The particles appear as complex objects propagating with the velocities of the two (charge and spin) sound modes and with vanishing low energy spectral weight. This can be indeed revealed in photoemission experiments in quasi-ld systems like Bechgaard salts, blue bronzes, spin ladders [17]. The collective properties at small q and w are instead those of a standard metal (i.e. Fermi Liquid) despite the absence of Fermi Liquid quasiparticles.

4 D imens iona l c r o s s o v e r

Contrary to the ld case, Fermi Liquid is found to be a quite robust theory in d > 1. Here a major difficulty to assess a non-Fermi Liquid behavior is the lack of exact methods in d > 1 and, at the same time, the intrinsic inadequacy of perturbation theory to answer this issue.

On the experimental side the most important example of non-Fermi Liquid behavior is given by the normal phase (T > To) of superconducting cuprates which are 2d systems from the electronic point of view [18]. It is worth noting that non-Fermi Liquid properties are also found in various heavy Fermion systems in specific points of their phase diagram [19].

Theoretically, the problem has been tackled by extending the ld approaches (specifically WI and bosonization) to d > 1 and taking into account forward (i.e. small q) scattering. In ld systems forward scattering is indeed the generic mechanism for Luttinger Liquid theory and one can show that this is also the only relevant scattering "near" dimension one ("near" in the sense explained below), but for the Cooper scattering giving rise to superconductivity [7].

The method based on the Dyson equation and Ward Identities is the most suitable for analyzing the dimensional crossover to assess the stability of Luttinger Liquid for d > 1. Here the issue is to solve an interacting system in dimension d, with d being a real num- ber (not necessarily an integer) approaching one. While bosonization techniques require a given integer dimension, the Dyson equation and Ward Identities can be considered at arbitrary d by generalizing in a standard way integrals in continuous dimensions. More- over, by taking 1 < d < 2, the various steps which allow for the exact solution in d -- 1 are still asymptotically valid in the low energy and momentum regime.

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The WI approach starts from the Dyson equation with self-energy given by

E(k) = i f D(q)G(k - q)A(k - q/2; q) (2) q

where here k : (e, k), q : (w, q) and fq is a notation for (2~r) -(d+U f dw f dq. D(q) is the effective dynamical interaction between particles with parallel spin. Interaction has been assumed to be only forward (corresponding in ld to the so called g2 - g4-model). A is the density component of the three point irreducible vertex (A, A) with the density and the current insertions defined by (1, Vk) with Vk --~ VFt¢.

The exact solution of the ld problem relies on the fact tha t A in Eq. (2) can be expressed in terms of the Green functions themselves:

A(k,q) = G - l ( k + q / 2 ) - G - l ( k - q / 2 )

o ) - - q V k (3)

Two properties which hold in d = 1 are used to derive this equation: i) the density-current vertex satisfies the Ward Identity

wA(k,q) - qA(k, q) = G - l ( k + q/2) - G - l ( k - q/2) (4)

which follows from continuity equation. ii) All vectors are parallel or antiparallel in d = 1 so that the current vertex A(k, q) is

simply given by

A(k, q) = vkA(k, q) (5)

and can be inserted in Eq. (4) to get Eq. (3) [20]. Notice that because of Eq. (3) there is an exact cancellation between vertex and wave function corrections in the polarization bubbles and the effective dynamical interaction D(q) is given by its random phase ap- proximation (RPA) expression with bare bubbles. The Dyson equation is therefore closed, and solving Eq. (2) by the use of Eq. (3) one gets [14] the exact Green function in Eq. (1).

The extension of the WI approach to 1 _< d < 2 is based on the observation that for systems dominated by forward scattering and "near" dimension one both steps i) and ii) are still allowed in analyzing asymptotic behaviors. This leads to a non perturbat ive evaluation of G(x, t) which is asymptotically exact in the limit x -+ c~ or t -+ oc and reduces to the ld expression (1) at d = 1.

The result of this analysis is that Luttinger Liquid is strictly confined in d = 1, and the system crosses over to a Fermi Liquid as soon as d (real number) is larger than one [6]. A finite quasiparticle weight at kF and a corresponding discontinuity in the momentum distribution are obtained, which tend to zero exponentially as d --+ 1. This crossover to a Fermi Liquid is valid for regular (short range) forward interactions and has been con- firmed by RG analysis [21] and by bosonization [22]. Non-Fermi Liquid requires singular forward scattering, e.g. V(q) ~ 1/q ~ with a _> 2d - 2 [23]. Indeed, in this case, the phase space reduction for the scattering in d > 1 is overcome by the singular momentum de- pendent interaction. Notice that, at d = 2, the value a = 1 appropriate to the physical

C. Castellani et al. / Physica A 263 (1999) 197-207 201

2d Coulomb force is not singular enough. We shall consider in the following sections two relevant physical situations with singular interactions, i.e. the gauge theories (GT) and the proximity to a quantum critical point (QCP).

Before ending this discussion on dimensional crossover, a specific mention should be given to the so called "loop cancellation" which states that the sum of all possible fermion loops with n insertions vanishes for n > 2 [7]. This cancellation is exact in d = 1 (and is strictly related to the non-dressing of polarization bubbles). For d > 1 the cancellation is only approximate in the sense that the cancellation concerns the leading singularities [24]. As discussed below, this "loop cancellation" appears also in many models with singular forward scattering and provides Fermi Liquid collective properties (RPA-like) for w, q ~ 0 even when the single particle properties are not Fermi Liquid-like.

5 G a u g e theo r i e s

The physics of a Fermi system coupled to a transverse gauge field has recently become important in two different contexts. The first case concerns the gauge theory introduced to implement Anderson's idea of the Resonating Valence Bond spin liquid phase for cuprates [25]. The second case is the description of the half-filled Landau level [26]. In both cases the coupling reads:

HI = g f ¢ + ¢ v F " A (6)

and the transverse component of gauge field propagator < A , A z > in the Random Phase Approximation is described by the following propagator:

D G T ( q , w ) _ i 7 ~ _ 1 x q 2-~ (7)

Here the exponent 0 < ff < 1 and the coefficients "y and X are determined by the underlying microscopic model, ff = 0 in the spin liquid case and ff = 1 for the half-filled Landau level.

The GT problem has been studied with various methods [27-29]. The RG approach appears to be particularly suitable to deal with the singularities introduced by the singular forward scattering mediated by the gauge propagator (7). The results of this analysis give a non-Fermi Liquid for d < dc = 3 - ~. The non-Fermi Liquid behavior appears in the vanishing of the quasiparticle spectral weight Z: one gets Z ~ w (de d)/(3-¢) for d < dc and Z ,-~ 1 / l logw [ at d = de.

As for the ld Luttinger Liquid, the collective properties at small q and w are instead Fermi Liquid like and are well described by the RPA expressions, but with finite correc- tions of the coefficients entering the RPA expressions. This is an outcome of the "loop cancellation" in the present problem. The "loop cancellation" is also acting on transport properties in the sense that the inverse transport time is much less than the inverse life- time of the excitations: for instance in d = 2 and for ~ = 0 they are proportional to w 4/3 and w ~/3 respectively [30]. Non-Fermi liquid collective properties instead appear at increasing q and w [30].

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We want to illustrate briefly how the above results can be obtained and how they strongly rely on the implementation of Ward Identies within the RG procedure [28,31]. The first consideration is that "loop cancellation" prevents higher order corrections to the RPA expression of the gauge propagator (7). There are no anomalous dimensions for the boson field. The only renormalizations which are needed are those of the three point current vertex and of the fermion self-energy (wave function renormalization Z -- (1 - or.~-i Y-dJ and Fermi velocity renormalization Zvs - Z(1 + ~-~-k~)). Actually, these renormalizations are related by Ward Identities. By indicating wi ts Asi~a the singular part of the current vertex one gets

0 ~ d-Zt~ Asin9 ~- --VF-'~W "" VFW 3-¢ (8)

in the "dynamical limit" q / w ~ 0 and

^ 0 E Asia9 -~ k ~ - ,,~ 0 (9)

in the "static" limit w / q ~ O.

In writing the above equations one takes into account that E(q, w) for gauge theories is weakly dependent on momentum, while it has a strong non-Fermi Liquid w dependence already at perturbative level. The choice of the kinematic regime for the renormalization of the current vertex is a central issue. A is a function of the transferred momentum q and energy w, and of the momentum k and energy e present in the incoming ( k - q / 2 , e - w / 2 )

and outcoming (k + q/2, e + w / 2 ) fermion lines. Let us decompose q into its transverse and parallel component to k: q =- qt + qll" The most singular contribution to the self- energy involves the regime qt ~ w 1/(3-¢) and qll ~ wd/(3-¢) [32]. In this region, considering the Ward Identities at finite momentum and energy and assuming the generalization of ld relation (3) we get A -~ 1. This means that no vertex renormalization is needed. Therefore, the fermion-boson coupling g renormalizes into Z g , just because of the wave function renormalization ¢ -~ Z1/2¢R. The full effective coupling in the problem comes out to be u ,,~ g 2 . D O S where D O S indicates the quasiparticle density of states which scales as 1 / Z . (This last property is a consequence of a self-energy depending critically only on energy E --- ~(w)). u in terms of the bare u0 is therefore given by

Z u o u = (10)

where # is an infrared momentum scale and x~ = 3 - ( - d -= dr - d. From the perturbative expression of Z (at first order in u) we obtain the one-loop renormalization equation for Z and u

2 = - u Z + .... (11)

2 iL = x~,u + u-~ = x , ,u - u 2 + .... (12)

du where 2--- - # ~ and u = - ~..

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A nonzero universal fixed point exists (u* -~ xu) for d < d~ while u* = 0 for d > d~ and u ~ 1/[log#l for d = d~. The fixed point u* depends on the perturbative order at which Z (i.e. the self-energy) is evaluated. At the fixed point the scaling behavior of the wave function renormalization Z can be evaluated from Eq. (11). Actually, given Eq. (10), the only existence of a fixed point at d < de, independently of its value, allows to derive the non-Fermi Liquid result

Z ~ #xu ~ o j ( 3 - ( - d ) / ( 3 - ~ ) (13)

3 - ~ being the dimension of w in terms of the momentum scale #. At d = d~, u vanishes logarithmically and

1 Z ~ - - (14)

[logw[

indicating a marginal Fermi Liquid behavior for the single particle spectral weight at the marginal dimension dc. The results (13) and (14) only rely on the absence of renormal- ization for the current vertex and within this assumption they are exact scaling relations.

6 P r o x i m i t y t o a q u a n t u m cr i t i ca l p o i n t

A further relevant case in which Fermi Liquid breaks down because of singular interac- tions is in the proximity of a QCP, i.e. in the proximity of a second order phase transition at zero temperature [33].

A simple RPA analysis shows that the critical fluctuations mediate an effective singular interaction

1 V(q,w) ~ (15)

iT[w[- (q - q~)2 _ ~-2

Here we are assuming that the transition is described at Gaussian level with a dynamical index z = 2. qc is the wave vector at which the instability occurs, ~ is the correlation length which diverges at the transition and '7 is a constant which fixes the characteristic energy-momentum scale. In the 6 - T plane the QCP is specified by T = 0 and ~ = 5~,

being a given tuning parameter (for instance the doping in the Cu02 planes for the models described below for the cuprates).

A non-Fermi Liquid behavior is expected in the so called quantum critical region which lies above the QCP and it characterized by having the correlation length only depend- ing on temperature: ~ --- ~(T) [33]. In this region there is no finite energy scale, but the temperature itself, in the scattering processes. The evaluation of the lowest order contri- bution to the self-energy leads to a strong non-Fermi Liquid lifetime for the k-points on the Fermi surface which are connected by the scattering processes with wave vector q~ ("hot" points). When qc --- 0 all Fermi surface points are "hot" points and the problem presents a strict resemblance with the gauge theories discussed in the previous section.

Proximity to a QCP has been found to account for non-Fermi Liquid properties in various heavy fermions near a magnetic instability [19].

204 C. Castellani et al. / Physica A 263 (1999) 197-207

In the cuprates the source of the non-Fermi Liquid behavior is still a debated issue. The relevance of proximity to a QCP has been suggested by various groups. A main difference is the nature of the instability and therefore of the critical fluctuations which induce the singular scattering. Pines and coworkers [34] assume antiferromagnetic fluctuations as the relevant source of scattering. In making this assumption they are relying on the existence of an antiferromagnetic phase at very low doping. However the doping at which antiferromagnetism disappears (i.e (~cAF) is far from the "optimal" doping at which the "best" non-Fermi Liquid behavior is observed.

The group in Rome suggests [35,36] that the QCP should be instead near optimal doping. The same suggestion has been given by Varma [37] who relates the QCP to an excitonic-like instability. The proposal of Rome is that the QCP at optimal doping marks the onset of an incommensurate charge-spin density wave in the form of stripes at lower doping. The ordered phase associated to the QCP would consist of charged ld domain walls [38] in an antiferromagnetic environment, as discussed at length by the Brookhaven group [39,40]. The presence of a superconducting phase at low temperature (induced indeed by stripe-fluctuations) prevents the direct access to the QCP, which, however, could be revealed when destroying superconductivity by a large magnetic field [41]. According to the stripe-QCP scenario the fluctuations associated to proximity to the stripe phase govern the non-Fermi Liquid physics of cuprates for temperature above the superconducting transition [35,36,42].

While this scenario provides a good qualitative description of cuprates, it comes out to be quite difficult to build a fully consistent quantitative theory. The main difficulty is to go beyond the Gaussian approximation for the fluctuations and the related singular interaction, and beyond the lowest order perturbative evaluation of their effect on the fermions. Unfortunately the presence of a finite qc in Eq. (15) does not allow to apply the RG procedure for singular forward scattering described in the previous section. The solution to this open problem is a main challenge for the theory.

7 Conclus ion

At the end of this short review we like to add some general remarks on the use of the renormalization group in stable liquid phases that has been applied to obtain some of the results reported in this paper. The most successful use of RG is in critical phenomena where RG sums singular perturbative terms to provide the power law behavior (charac- teristic of the scaling theory) for quantities like the order parameter, the specific heat and the relevant susceptibility (e.g. the spin susceptibility in a magnetic transition).

By contrast, in stable (i.e. non-critical) liquid phases any singular terms of the perturba- tion theory (like in interacting Fermions in d = 1 or Bosons in the presence of condensate below 3d) must cancel exactly in the response functions which remain finite to implement the absence of criticality. Therefore, a singular perturbation theory and a stable phase im- ply additional symmetries to guarantee these cancellations. The implementation of these symmetries via Ward Identities leads in general to relations between different skeleton structures and to a reduction of the required renormalizations. In some cases this results into a closure of the RG equations and provides the exact solution of the problem, as for instance in the case of the interacting Bose system in the presence of a condensate [43].

c. Castellani etal. / Physica A 263 (1999) 197-207 205

In the case of interacting Fermions with linear spectrum around the Fermi momentum and short range interactions, perturbation theory is logarithmically divergent in d = 1. For forward scattering, in addition to the total charge (and spin) conservation leading to the ordinary WI in Eq. (4), charge (and spin) is conserved separately at each Fermi point (+kF), due to the absence of large momentum transfer. The corresponding additional WI implies the relation (5) (with Vk = +vF) between the charge and the current vertex, the closure of the Dyson equation and standard (Fermi Liquid-like) finite results for the collective modes and the thermodynamic properties like specific heat, compressibility and spin susceptibility, as from loop cancellation. The signature of the singular perturbation theory remains in the anomalous dimensions y of the Green function, which controls the vanishing of the spectral weight at the Fermi surface.

Even in the presence of singular interactions, as in the case of long range forces and gauge theories, a stable phase results into (or is implied by) WI and loop cancellation. Fermi Liquid collective properties at small q and ~ coexist with non-Fermi Liquid single particle properties. Again, WI can lead to exact relations like the expressions (13) and (14) for the renormalization parameter Z characterizing the non-Fermi liquid behavior. In the proximity of the QCP of section 6, the physical picture is much more involved. The presence of a critical behavior for the fluctuations mediating the interaction among parti- cles and the presence of an additional finite relevant scale qc prevent all the simplifications of the RG approach for stable phases, leaving this problem open.

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