Elsevier

Nuclear Physics B

Volume 795, Issue 3, 1 June 2008, Pages 578-595
Nuclear Physics B

Disordered loops in the two-dimensional antiferromagnetic spin–fermion model

https://doi.org/10.1016/j.nuclphysb.2007.11.015Get rights and content

Abstract

The spin–fermion model has long been used to describe the quantum-critical behavior of 2d electron systems near an antiferromagnetic (AFM) instability. Recently, the standard procedure to integrate out the fermions and obtain an effective action for spin waves has been questioned in the clean case. We show that, in the presence of disorder, the single fermion loops display two crossover scales: upon lowering the energy, the singularities of the clean fermionic loops are first cut off, but below a second scale new singularities arise that lead again to marginal scaling. In addition, impurity lines between different fermion loops generate new relevant couplings which dominate at low energies. We outline a non-linear σ model formulation of the single-loop problem, which allows to control the higher singularities and provides an effective model in terms of low-energy diffusive as well as spin modes.

Introduction

The spin–fermion model is a low-energy effective model describing the interaction of conductance electrons (fermions) with spin waves (bosons). It has been used, e.g., to describe the quantum critical behavior of an electron system near an antiferromagnetic instability [1], [2], [3]. An important example where this might be realized experimentally is in itinerant heavy-fermion materials [4]. By integrating out the fermions completely, a purely bosonic effective action for spin waves is obtained. This action is written in terms of a bare spin propagator and bare bosonic vertices with any even number of spin lines. The value of each bosonic vertex is given by a fermionic loop with spin–vertex insertions: in general, these are complicated functions of all external bosonic frequencies and momenta. Hertz [1] and Millis [2] considered only the static limit of these vertices, i.e., setting all frequencies to zero at finite momenta. In this limit the 4-point vertex and all higher vertices vanish for a linear dispersion relation, while they are constants proportional to a power of the inverse bandwidth if the band curvature is taken into account. For the AFM the dynamic critical exponent is z=2 due to Landau damping of spin modes by particle–hole pairs. The scaling of these vertices in d=2 under an RG flow toward low energy scales is marginal for the 4-point vertex while all higher vertices are irrelevant (d+z=4 is the upper-critical dimension). Thus, a well-defined bosonic action with only quadratic and quartic parts in the spin field is obtained.

Recently, Lercher and Wheatley [5] as well as Abanov et al. [6] considered for the 2d case not only the static limit of the 4-point vertex but the full functional dependence on frequencies and momenta. Surprisingly, they found that in the dynamic limit, setting the momenta to the ordering wave vector at finite frequency, the 4-point vertex is strongly divergent as the external frequencies tend to zero, implying an effective spin interaction nonlocal in time. The higher bosonic vertices display an even stronger singularity [7].

To assess the relevance of the singular bosonic vertices, Ref. [7] considered the scaling limit ωq2 with z=2. In this limit, the bosonic vertices are less singular than in the dynamic limit but the related couplings are still marginal, i.e., they cannot be neglected in the effective bosonic action, in apparent contradiction to Hertz and Millis. Employing an expansion in a large number of hot spots N or fermion flavors, Ref. [6] argues that vertex corrections are resummed to yield a spin propagator with an anomalous dimension η=2/N=1/4 (for N=8). At the same time, z=2 remains unchanged up to two-loop order, i.e., the frequency dimension is given by xω=2(1η/2).

The infinite number of marginal vertices renders the purely bosonic theory difficult to use for perturbative calculations. A relevant question is whether the above difficulty persists upon the inclusion of a weak static disorder potential present in real materials. To address this issue we insert disorder corrections into single fermionic loops and find two different crossover scales: at frequencies ω1/τ (i.e., much larger than the impurity scattering rate) and momenta q1/ (with mean free path =vFτ, where vF is the Fermi velocity), the fermionic loops resemble the clean case, while below this scale the singularity is cut off by self-energy corrections and the loops saturate. However, at yet lower frequencies, a second crossover scale ω1/(τkF), q1/(kF) appears where the loops acquire a diffusive form due to impurity ladder corrections and the related couplings again scale marginally, as in the clean case. Therefore, in an intermediate energy range disorder regularizes the singular vertices and appears to restore Hertz and Millis theory, while ultimately at the lowest scales the disordered loops are as singular as the clean ones, albeit with a different functional form: the linear dispersion of the electrons is replaced by a diffusive form. We outline a non-linear σ model formulation of the disordered single-loop problem which allows us to identify all disorder corrections which exhibit the maximum singularity, and provides an action for spin modes coupled to low-energy diffusive electronic modes, instead of the original electrons.

Finally, while all disorder corrections to a single fermion loop lead to couplings which scale at most marginally, impurity lines connecting different fermion loops are a relevant perturbation in d=2. We find that these diagrams may dominate the single-loop contributions below ω1/τ, depending on the typical values of the bosonic momenta.

We proceed as follows: in the remaining part of this section, we introduce the model and the scaling arguments for the clean case. We then insert disorder corrections into a single fermion loop and discuss a class of most singular diagrams in Section 2. Their scaling behavior and the emergence of two crossover scales is the subject of Section 3. The multi-loop diagrams are discussed in Section 4. Appendix A contains the non-linear σ model for the disordered single-loop case.

The 2d spin–fermion model is defined by the actionS[ψ,ψ¯,ϕ]=(ψ¯,G0−1ψ)+(ϕ,χ0−1ϕ)+gϕψ¯ψ for a fermionic field ψ, ψ¯ and a bosonic spin field ϕ.1 The inverse fermionic propagator is G0−1(iϵ,p)=iϵξp in terms of the Matsubara frequency iϵ and a dispersion relation ξp with a roughly circular Fermi surface (FS), which we approximate by a quadratic dispersion ξp=|p|22meμ with electron mass me, chemical potential μ, Fermi momentum kF=2meμ, and constant density of states 2πρ0=ϵF/vF2. χ0(q) is the bare spin propagator.

We assume that the above model describes an AFM quantum critical point at finite q=qc. The Fermi surface has so-called hot regions connected by exchange of qc, and cold regions where scattering off spin waves is weak. Here we shall assume an underlying lattice and a commensurate qc=(π,π), which is equivalent (up to a reciprocal lattice vector) to qc.

When computing fermionic loops with only spin–vertex insertions, the momentum integration can be reduced to the region around two hot spots separated by qc, which we shall denote by α and α¯ (Fig. 1). The fermionic dispersion relation ξp is linearized around any hot spot α at momentum phsα as [6]ξp=vF(pphsα)=vxαp¯x+vyαp¯yξp¯α, where p¯ denotes the distance from the hot spot. The components vx (vy) of the Fermi velocity vF parallel (perpendicular) to qc at a given hot spot α are related by vF2=vx2+vy2 and vx/vy=tan(ϕ0/2), with ϕ0 the angle between hot spots α and α¯ as seen from the center of the circular Fermi surface. The case ϕ0=π (vy=0) corresponds to perfect nesting, but here we consider a generic ϕ0 without nesting. For a pair of hot spots, the momentum integration can be written asd2p(2π)2=Jαdξαdξα¯, where ξα and ξα¯ are two independent momentum directions at hot spot α (ξα¯ coincides with the radial direction at hot spot α¯), J−1=4π2vF2sinϕ0 is the corresponding Jacobian which depends on the shape of the Fermi surface and the filling, and one still has to perform the summation over all N=8 hot spots.

The fermionic loops with 2n spin–vertex insertions—i.e., the 2n-point functions—are in general complicated functions of the external frequencies and momenta. The loop with two spin insertions contributes to the self-energy for the spin propagator and has the well-known Landau damping form for small frequencies ω and momenta near qc [8],Σ(iω,qqc)=γ|ω|, with the dimensionless strength of the spin fluctuations [6]γ2πJNg2=g2N2πvF2sinϕ0=g2N4πvxvy. The inverse spin propagator resummed in the random-phase approximation has dynamical exponent z=2,χ−1(iω,q)=m+γ|ω|+ν|qqc|2, where the mass term m measures the distance from the quantum critical point and νg2/ϵF. Near criticality m0, the momentum exchanged by scattering off a spin wave is peaked near qc. From here on, we shall denote by q the deviation from qc. In the commensurate case there are logarithmic singularities in the clean bosonic self-energy (from contracting two spin lines diagonally in Fig. 2) that may lead to an anomalous dimension of the spin propagator [6].

The 4-point function is given by the fermion loop with four spin insertions, which provides the bare two spin interaction:b4=πJg4α|ω1+ω|+|ω1ω||ω2+ω||ω2ω|[i(ω1+ω2)ξq1+q2α][i(ω1ω2)ξq1q2α¯], where we have labeled the three independent external frequencies and momenta as shown in Fig. 2. This is a nonanalytic function whose value for ω0, q0 depends on the order of the limits: it vanishes in the static limit (ω0 first) while it diverges as 1/ω in the dynamic limit (q0 first). There is an important difference from the forward-scattering loop with small external momenta [9]: here, symmetrization of the external lines does not lead to loop cancellation, i.e., a reduction of the leading singularity. Instead, symmetrization only modifies the prefactors but does not change the scaling dimension.

We recall the scaling behavior of the 2n-point functions [7]. Since no cancellation of the leading singularity occurs, we only need to consider the power of external frequency and momentum and not the particular linear combinations of frequencies and momenta involved. Introducing a symbolic notation where ω denotes a positive linear combination of external frequencies and q a linear combination of momenta, the 2n-point functions have the scaling formb2ng2nvF2ω(ω+ivFq)2(n1), where an average over hot spots is understood, which leads to a real positive function of the frequencies and momenta represented by ω and q. For the purpose of scaling, we have substituted J1/vF2 and γg2/vF2.

To estimate the relevance of the vertices in the scaling limit ω2/zq20, where q dominates ω in the denominator for z>1, consider the ϕ2n term in the effective action [7]:g2n(d2qdω)2n1ω(vFq)2(n1)ϕ2n, where g2n is the coupling strength related to the vertex function b2n. Using the scaling dimension of the field [ϕ2]=(d+z+2) (in frequency and momentum space) in two dimensions,[g2n]=(2n1)(2+z)[z2(n1)]n(4z)=(2z)n. The scaling dimension of all 2n-point functions is zero for z=2, i.e., all bosonic vertices are marginal in the scaling limit, and it is not clear how to perform calculations with such an action. Our aim is to see if and how the disorder which is present in real systems changes the scaling dimension of the fermionic loops.

Section snippets

Disorder corrections to a single fermion loop

We consider static impurities modeled by a random local potential with mean squared amplitude u2 [10]. As long as no spin–vertex insertions appear between impurity scatterings, the disorder corrections in the Born approximation have the standard form. In the fermionic propagator G(iϵ,p)=(iϵ˜ξp)−1 the Matsubara frequency ϵ is cut off as ϵ˜ϵ+sgn(ϵ)/(2τ) at the scale of the impurity scattering rate 1/τ=2πρ0u2. Although the particle–hole bubble B(iϵ+iω,iϵ,q) with small momentum transfer q is cut

Smallness of ladder corrections and second crossover scale

As for b4,dirty, the disordered loops (beyond the 2-point function) feature two crossover scales, one where disorder corrections in the self-energy of the fermion lines cut off the vertices, and another where ladder corrections lead again to marginal scaling. The existence of these two scales can be traced back to the presence of hot spots.

For comparison, consider the case of forward-scattering bosonic vertices. Self-energy disorder corrections become important and cut off the fermionic

Disorder corrections to multiple fermion loops

The multi-loop disorder corrections arise from impurity lines connecting different fermionic loops. In the simplest case, one takes n static particle–hole bubbles with two spin insertions each (mass terms) and connects them with single impurity lines. As the impurity lines do not carry frequency, there are only n independent frequencies in this 2n-point spin vertex. The corresponding coupling in the action has therefore a different scaling dimension than the single-loop contribution (with 2n1

Discussion and conclusions

We have shown that the fermionic loops with large momentum transfer, which are relevant to describe an AFM transition, exhibit two crossover energy scales if disorder corrections are added. In the scaling limit set by charge diffusion ωDq2, the singularities of the clean (marginal) vertices g2n are cut off at scale ω1/τ and the 4-point vertex and beyond become irrelevant. Below ω1/(τkF), however, diffusive ladders lead again to marginal scaling, albeit with a diffusive functional form of

Acknowledgements

T.E. wishes to thank A. Chubukov, A. Rosch and M. Salmhofer for fruitful discussions. We thank the Alexander von Humboldt foundation (T.E. and C.D.C.), and the Italian Ministero dell'Università e della Ricerca (PRIN 2005, prot. 2005022492) for financial support.

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