Charge and spin correlations in metals are singular at wave vectors that connect points on the Fermi surface with antiparallel Fermi velocities. The singularity is caused by an enhanced phase space for low-energy particle-hole excitations near such wave vectors. It leads, among other effects, to the Kohn anomaly in phonon spectra and to the RKKY interaction between magnetic impurities in metals. For isotropic Fermi surfaces the singularity is located at wave vectors with modulus 2*k*_{F}, where *k*_{F} is the radius of the Fermi surface. In inversion symmetric crystalline solids, singular wave vectors are given by the condition ε_{(Q+G)/2} = 0, where ε** _{k}** is the single-particle excitation energy, and

**G**is a reciprocal lattice vector. This is the lattice generalization of the condition |

**Q**| = 2

*k*

_{F}for isotropic systems. Hence, wave vectors satisfying that condition are referred to as 2

*k*

_{F}vectors.

2*k*_{F} singularities are stronger in systems with reduced dimensionality. Charge and spin correlations in low dimensional systems are often peaked at 2*k*_{F} vectors, such that they are privileged wave vectors for charge and spin density wave instabilities. 2*k*_{F} instabilities are ubiquitous in (quasi) one-dimensional electron systems. 2*k*_{F} instabilities also play an important role in two-dimensional systems. In particular, the ground state of the two-dimensional Hubbard model, the most intensively studied model for cuprate high temperature superconductors, exhibits a spin density wave instability at a 2*k*_{F} vector, at least at weak coupling. Furthermore, *d*-wave bond charge order triggered by antiferromagnetic fluctuations in models for cuprate superconductors occurs preferably at *k*_{F} vectors [1].

Here we discuss consequences of quantum criticality at the onset of *incommensurate* 2*k*_{F} charge or spin density wave order in the ground state, in cases where the phase transition (e.g., as a function of electron density) is continuous. The momentum and energy dependences of the 2*k*_{F} density fluctuation propagator differs strongly from that for generic incommensurate wave vectors, and also from the one for commensurate (π,π) charge or spin density wave instabilities, for which quantum critical properties have been extensively studied in the past. The quantum critical behavior at 2*k*_{F} density wave transitions in two dimensional metals was addressed already by Altshuler *et al. *[2], who computed several properties for the case that the 2*k*_{F} vector is half a reciprocal lattice vector. For incommensurate 2*k*_{F} vectors, Altshuler *et al.* found strong infrared divergencies and concluded that fluctuations destroy the quantum critical point (QCP), such that the phase transition is ultimately discontinuous.

## Non-Fermi liquid behavior

We have analyzed the influence of incommensurate 2*k*_{F} quantum critical fluctuations on single-particle excitations by computing the electronic self-energy at 2*k*_{F} hot spots on the Fermi surface to first order in the fluctuation propagator [3]. Hot spots are points on the Fermi surface that are connected by the ordering wave vector **Q**. The momentum and frequency dependence of the self-energy can be measured by angular resolved photoemission spectroscopy. One needs to distinguish the case where the 2*k*_{F} vector connects only one pair of hot spots from cases where it connects two hot spot pairs (**Fig. 1**). 2*k*_{F} instabilities with a single pair of hot spots ±**k**_{F} occur naturally at high symmetry points, that is, with **k**_{F} and **Q** in axial or diagonal direction. That case was considered in Ref. [2], but the self-energy was computed only for a commensurate wave vector. In any case the quasi-particle decay rate obeys non-Fermi liquid power laws as a function of energy.

**Fig. 1:**Axial 2

*k*

_{F}wave vector of the form (

*Q*,0) (left) and 2

*k*

_{F}wave vector of the form (π,

*Q*) (right), and the corresponding hot spots on the Fermi surface.

To compute the electronic self-energy, one first needs to derive the momentum and energy dependence of the effective interaction (fluctuation propagator) at the QCP. At leading order, the latter is given by the RPA expression *D*(**q**,ω) = *g* [1 – *g* Π_{0}(**q**,ω)]^{–1}, where *g* is the coupling parametrizing the bare interaction in the instability channel, and Π_{0} is the bare polarization function of the system. The RPA effective interaction and the one-loop self-energy are not affected qualitatively by details such as the spin structure and form factors (*d*-wave, etc.) in Π_{0}(**q**,ω). Finite renormalizations from non-critical fluctuations could be incorporated by a renormalized coupling and a reduced quasi-particle weight. Density wave instabilities are driven by the particle-hole channel such that the RPA is a suitable starting point for the analysis. At the onset of density wave order, *g* Π_{0}(**Q**,0) is equal to one such that the effective interaction diverges. The momentum and energy dependence of *D*(**q**,ω) at that point is obtained by expanding Π_{0}(**q**,ω) for **q** near **Q **and small energies ω. Here the *k*_{F}-singularity comes into play. If **Q** is a 2*k*_{F}-vector, Π_{0}(**q**,ω) and hence *D*(**q**,ω) exhibit a peculiar square root singularity at **q** = **Q** and ω = 0.

**Fig. 2:**Decay rate of single-particle excitations at hot spots as a function of the excitation energy for the case of two hot-spot pairs.

Non-Fermi liquid behavior of electronic excitations near the hot spots of the Fermi surface emerges already from the self-energy computed to first order in *D*(**q**,ω). The asymptotic low-energy behavior can be computed analytically. For the case of a single hot-spot pair at high symmetry points, we have obtained a single-particle decay rate of the form *C*|ε|^{2/3} at low excitation energies ε, where *C* is a constant. In case of two hot-spot pairs, the single-particle decay rate scales as *C*_{±}|ε|, with distinct constants *C*_{+} and *C*_{–} for positive and negative excitation energies, respectively. It is thus *linear* in energy, with a pronounced *particle-hole asymmetry* (**Fig. 2**). Hence, in both cases there are no Fermi liquid quasi-particles at the hot spots.

For momenta away from the hot spots, the self-energy obeys Fermi liquid behavior in the low energy limit. Close to the hot spots, there is a crossover between the non-Fermi liquid power-laws at intermediate energies, and Fermi liquid behavior at very low energies. We have computed the self-energy only to one-loop order. This suffices to detect the breakdown of Fermi liquid theory, but higher orders may modify the power-laws. Higher order terms may also destroy the QCP at low energy scales, and replace it by a first order transition [2], or trigger another instability such as pairing.

In future work one should analyze higher order contributions in a suitable renormalization group framework. It will also be very interesting to extend the present analysis to the quantum critical regime at finite temperature, study transport properties, and relate the results to correlated electron compounds with 2*k*_{F} instabilities. Cuprates are natural candidates, but putative 2*k*_{F} QCPs seem to be suppressed by superconductivity in those compounds. Nevertheless, the strange metal behavior in the normal phase might be caused at least partially by quantum critical fluctuations related to a closeby 2*k*_{F} instability. Quasi-two-dimensional incommensurate spin density wave order has recently been observed in nickel oxide heterostructures. Exploiting the various options to manipulate oxide heterostructures, one should be able to tune such systems to a QCP.