Disordered loops in the two-dimensional antiferromagnetic spin–fermion model
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 due to Landau damping of spin modes by particle–hole pairs. The scaling of these vertices in under an RG flow toward low energy scales is marginal for the 4-point vertex while all higher vertices are irrelevant ( 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 with . 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 (for ). At the same time, remains unchanged up to two-loop order, i.e., the frequency dimension is given by .
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 (i.e., much larger than the impurity scattering rate) and momenta (with mean free path , where 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 , 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 . We find that these diagrams may dominate the single-loop contributions below , 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 action for a fermionic field ψ, and a bosonic spin field ϕ.1 The inverse fermionic propagator is in terms of the Matsubara frequency iϵ and a dispersion relation with a roughly circular Fermi surface (FS), which we approximate by a quadratic dispersion with electron mass , chemical potential μ, Fermi momentum , and constant density of states . is the bare spin propagator.
We assume that the above model describes an AFM quantum critical point at finite . The Fermi surface has so-called hot regions connected by exchange of , and cold regions where scattering off spin waves is weak. Here we shall assume an underlying lattice and a commensurate , which is equivalent (up to a reciprocal lattice vector) to .
When computing fermionic loops with only spin–vertex insertions, the momentum integration can be reduced to the region around two hot spots separated by , which we shall denote by α and (Fig. 1). The fermionic dispersion relation is linearized around any hot spot α at momentum as [6] where denotes the distance from the hot spot. The components () of the Fermi velocity parallel (perpendicular) to at a given hot spot α are related by and , with the angle between hot spots α and as seen from the center of the circular Fermi surface. The case () corresponds to perfect nesting, but here we consider a generic without nesting. For a pair of hot spots, the momentum integration can be written as where and are two independent momentum directions at hot spot α ( coincides with the radial direction at hot spot ), 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 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 [8], with the dimensionless strength of the spin fluctuations [6] The inverse spin propagator resummed in the random-phase approximation has dynamical exponent , where the mass term m measures the distance from the quantum critical point and . Near criticality , the momentum exchanged by scattering off a spin wave is peaked near . From here on, we shall denote by q the deviation from . 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: where we have labeled the three independent external frequencies and momenta as shown in Fig. 2. This is a nonanalytic function whose value for , depends on the order of the limits: it vanishes in the static limit ( first) while it diverges as in the dynamic limit ( 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 form 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 and .
To estimate the relevance of the vertices in the scaling limit , where q dominates ω in the denominator for , consider the term in the effective action [7]: where is the coupling strength related to the vertex function . Using the scaling dimension of the field (in frequency and momentum space) in two dimensions, The scaling dimension of all 2n-point functions is zero for , 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 [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 the Matsubara frequency ϵ is cut off as at the scale of the impurity scattering rate . Although the particle–hole bubble with small momentum transfer q is cut
Smallness of ladder corrections and second crossover scale
As for , 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
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 , the singularities of the clean (marginal) vertices are cut off at scale and the 4-point vertex and beyond become irrelevant. Below , 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.
References (20)
Quantum critical phenomena
Phys. Rev. B
(1976)Effect of non-zero temperature on quantum critical points in itinerant fermion systems
Phys. Rev. B
(1993)Quantum Phase Transitions
(1999)Non-Fermi-liquid behavior in d- and f-electron metals
Rev. Mod. Phys.
(2001)- et al.
Breakdown of mode–mode coupling expansion for commensurate itinerant antiferromagnetism in two dimensions
Phys. Rev. B
(2000) - et al.
Quantum-critical theory of the spin–fermion model and its application to cuprates: Normal state analysis
Adv. Phys.
(2003) - et al.
Anomalous scaling at the quantum critical point in itinerant antiferromagnets
Phys. Rev. Lett.
(2004) Polarizability of a two-dimensional electron gas
Phys. Rev. Lett.
(1967)- et al.
Fermion loops, loop cancellation, and density correlations in two-dimensional Fermi systems
Phys. Rev. B
(1998) - et al.
Methods of Quantum Field Theory in Statistical Physics
(1975)