Fluctuation effects at the onset of $\bf 2k_{F}$ density wave order
with two pairs of hot spots in two-dimensional metals

Quantum critical fluctuations at the onset of charge or spin density wave order in two-dimensional metals destroy Fermi liquid behavior and lead to unconventional dependences on temperature and other control parameters. loehneysen07 The theory of such systems is difficult due to a complex interplay of the critical order parameter fluctuations and the gapless excitations of the electrons. A purely bosonic order parameter theory as developed by Hertz hertz76 and Millis millis93 may describe some features, but it is not generally applicable since the electronic excitations lead to singular interactions of the order parameter fluctuations. abanov04 Secondary instabilities generated by the fluctuations, especially pairing instabilities, are almost unavoidable and further complicate the analysis.

The most intensively studied case is the quantum critical point at the onset of Néel-type antiferromagnetic order. abanov03 Non-Fermi liquid behavior is obtained already at leading order in perturbation theory, but new features appear at higher orders. metlitski10_af1 Impressive analytical lee18 and numerical berg12 efforts have led to substantial progress, but a complete theory of this important universality class has not yet been constructed.

A special and particularly intricate situation arises when the density wave vector is a nesting vector of the Fermi surface, that is, when it connects Fermi points with antiparallel Fermi velocities. footnote_perfnest Charge and spin correlation functions exhibit a singularity at such wave vectors due to an enhanced phase space for low-energy particle-hole excitations. In inversion symmetric crystals with a valence band dispersion $\epsilon_{\bf k}$, nesting vectors ${\bf Q}$ are determined by the condition $\epsilon_{({\bf Q}+{\bf G})/2}=\epsilon_{F}$, where ${\bf G}$ is zero or a reciprocal lattice vector, and $\epsilon_{F}$ is the Fermi energy. Since nesting vectors in isotropic continuum systems are determined by the Fermi surface radius $k_{F}$ via the simple relation $|{\bf Q}|=2k_{F}$, we refer to nesting vectors also as “$2k_{F}$-vectors”.

Nesting singularities are particularly pronounced in metals with reduced spatial dimensionality. Charge and spin susceptibilities in low-dimensional metals exhibit peaks at nesting vectors, so that these vectors are natural charge or spin density wave vectors. For example, the ground state of the two-dimensional Hubbard model undergoes a continuous quantum phase transition into a spin-density wave state with a wave vector satisfying the nesting condition, at least within mean-field theory. schulz90 ; igoshev10 Also $d$-wave bond charge order generated by antiferromagnetic fluctuations in spin-fermion models for cuprates metlitski10_af2 ; sachdev13 occurs naturally with nesting wave vectors. holder12 ; punk15

Quantum criticality in a two-dimensional metal at the onset of density wave order with a nesting wave vector was first analyzed by Altshuler et al. altshuler95 They obtained non-Fermi liquid power laws for the electron self-energy and the susceptibility for the special case where the wave vector ${\bf Q}$ is not only a nesting vector, but also half a reciprocal lattice vector. Later it was shown that due to additional umklapp processes Landau quasi-particles actually survive, albeit with an enhanced decay rate, for the particular case where ${\bf Q}$ is the antiferromagnetic wave vector $(\pi,\pi)$. bergeron12 ; wang13 For generic, that is, incommensurate footnote_incomm nesting vectors instead, Altshuler et al. altshuler95 found strong infrared divergences in two-loop contributions to the susceptibility and concluded that the order-parameter fluctuations destroy the quantum critical point, replacing the second-order by a first-order phase transition.

The analysis of non-Fermi liquid behavior at the onset of charge and spin-density wave order with an incommensurate nesting vector in two-dimensional metals was initiated more recently. The fermion self-energy was calculated to first order (one loop) in the order parameter fluctuation propagator. holder14 The fluctuation propagator was computed from a one-loop approximation with bare particle-hole bubbles. A breakdown of Landau Fermi liquid theory was obtained at hot spots on the Fermi surface, that is, Fermi points connected by the ordering wave vector ${\bf Q}$. Two qualitatively distinct cases arise according to the number of hot spots connected by ${\bf Q}$.

In the simplest case, ${\bf Q}$ connects only a single pair of hot spots, and points typically in a high symmetry direction (axial or diagonal). The frequency dependence of the one-loop fermion self-energy at the hot spots follows a power law with an exponent $\frac{2}{3}$ in this case. holder14 The decay rate of fermionic excitations is thus larger than their excitation energy in the low energy limit. The momentum dependence yields a singular renormalization of the Fermi velocity and a flattening of the Fermi surface near the hot spots. sykora18 An important issue is the self-consistency of these results, or whether the non-Fermi liquid self-energy destroys the $2k_{F}$ quantum critical point, since it has a tendency to wipe out the $2k_{F}$ peak in the density wave susceptibility. It was shown that the $2k_{F}$ quantum critical point can survive only with the assistance of a sufficiently strong Fermi surface flattening around the hot spots. sykora18 This scenario was subsequently supported by a systematic $\epsilon$-expansion around the critical dimension $d_{c}=5/2$, at least to leading order in $\epsilon$. halbinger19

In the second case, the density wave vector ${\bf Q}$ connects two pairs of hot spots. The Fermi velocities are collinear within each pair, but not between the pairs. In Fig. 1 we show various possible geometries, with Fermi surfaces and hot spot pairs on the left, and the lines of all nesting vectors ${\bf Q}$ satisfying the condition $\epsilon_{({\bf Q}+{\bf G})/2}=\epsilon_{F}$ on the right. These lines are obtained by backfolding the line consisting of all doubled Fermi wave vectors $2{\bf k}_{F}$ into the first Brillouin zone. Wave vectors ${\bf Q}$ connecting two pairs of hot spots correspond to crossing points of the lines of nesting vectors. The geometry in the top row of Fig. 1 is realized by the spin-density wave quantum critical point in the two-dimensional Hubbard model, while the $d$-wave bond charge order instability mentioned above provides an example for the geometry in the second row. For two hot spot pairs, the imaginary part of the one-loop self-energy at the hot spots was found to be a linear function of frequency, with a distinct prefactor for positive and negative frequencies. holder14 Hence, the decay rate of fermionic excitations is proportional to their excitation energy, implying a marginal breakdown of Fermi liquid theory.

In this work we extend the analysis of fluctuation effects at the onset of $2k_{F}$ density wave order with two pairs of hot spots. We compute the frequency dependence of the one-loop self-energy on the imaginary frequency axis and show that the results are consistent with the real frequency self-energy obtained previously. holder14 We also compute the momentum dependence near the hot spots. Unlike the case of a single hot spot pair, sykora18 the momentum dependence of the self-energy entails only a finite renormalization of the Fermi velocity, and the renormalized Fermi surface maintains a finite curvature at the hot spots.

The perturbative one-loop results are not self-consistent. Incorporating the one-loop self-energy in the fermion propagator can lead to a shift of the peak in the susceptibility away from the nesting vector, which spoils the $2k_{F}$ quantum critical point. To include the self-energy feedback self-consistently, we explore two distinct routes. First, we compute the fluctuation propagator and the fermion self-energy from a renormalization group flow. Due to a peculiar cutoff dependence of the fermion self-energy, the self-energy feedback is incomplete in this approach and the shift of the susceptibility peak persists. Second, we obtain a fully self-consistent solution by solving the coupled integral equations for the particle-hole bubble and the fermion self-energy with self-energy feedback. Here the peak at the nesting vector survives.

The paper is structured as follows. In Sec. II we derive the order parameter susceptibility and effective interaction at the quantum critical point in one-loop approximation with bare fermion propagators. In Sec. III we evaluate the frequency and momentum dependence of the one-loop fermion self-energy near the hot spots. In Sec. IV we show that the perturbative one-loop results are not self-consistent. In Sec. V we discuss our renormalization group analysis of the quantum critical point, and in Sec. VI we present the self-consistent solution of the one-loop equations with self-energy feedback. A conclusion in Sec. VII closes the article.

We consider a one-band system of interacting fermions with a bare dispersion relation $\epsilon_{\bf k}$. Our calculations are based on the standard quantum many-body formalism with an imaginary frequency representation of dynamical quantities.negele87 The bare fermion propagator is given by

where $k_{0}$ is the frequency variable, and $\xi_{\bf k}=\epsilon_{\bf k}-\mu$ is the single-particle energy relative to the chemical potential $\mu$. At zero temperature the Matsubara frequency $k_{0}$ is a continuous variable.

We assume that, in mean-field theory, the system undergoes a charge or spin density-wave instability with an incommensurate and nested ($2k_{F}$) modulation vector ${\bf Q}$ at a QCP which can be reached by tuning a suitable parameter such as electron density or interaction strength. Approaching the QCP from the normal metallic regime, the instability is signalled by a diverging RPA susceptibility

where $g<0$ is the coupling constant parametrizing the bare interaction in the instability channel, and $\chi_{0}$ is the bare susceptibility

$N$ is the spin-multiplicity ($N=2$ for spin-$\frac{1}{2}$ fermions), and $f_{\bf p}$ is a form factor related to the internal structure of the density-wave order parameter. For an order parameter with $s$-wave symmetry, $f_{\bf p}$ is symmetric under rotations and reflections. In the following we assume $f_{\bf p}=1$ for definiteness. A generalization to form factors with other symmetries such as $d$-wave symmetry is straightforward. Eq. (2) holds also for the spin susceptibility in the normal (symmetric) phase of spin-rotation invariant systems, where all components of the spin susceptibility are equal.

The bare susceptibility $\chi_{0}({\bf q},iq_{0})$ exhibits a square-root singularity on the $2k_{F}$ line defined by all $2k_{F}$ vectors in ${\bf q}$-space. To parametrize momenta near a $2k_{F}$ momentum ${\bf Q}$, we introduce relative coordinates normal and tangential to the $2k_{F}$ line at ${\bf Q}$, which we denote by $q_{r}$ and $q_{t}$, respectively (see Fig. 2). For momenta near ${\bf Q}$ and low frequencies, the bare susceptibility can be expanded as altshuler95 ; holder14

where $a$ and $b$ are positive constants, $c$ is a real constant, and

The energy-momentum relation $e_{\bf q}$ is given by

where $v_{F}$ is the Fermi velocity at the hot spots on the Fermi surface $\pm{\bf k}_{H}$ connected by ${\bf Q}$, and $m$ parametrizes the Fermi surface curvature at these points ($mv_{F}$ is the radius of curvature). Hence, $e_{\bf q}/v_{F}$ is the oriented distance of ${\bf q}$ from the $2k_{F}$-line. The prefactor of the square-root term is determined by the Fermi velocity and curvature near $\pm{\bf k}_{H}$ as

while the other constants $b$ and $c$ receive contributions from everywhere. For fermions with a parabolic dispersion in the continuum and a constant form factor, $b$ and $c$ vanish, as can be seen from Stern’s exact analytic formula for $\chi_{0}$ in this case. stern67 The constant $c$ vanishes also at high symmetry points for lattice fermions, especially if ${\bf k}_{H}$ points along an axis or a diagonal in the Brillouin zone. In this case a correction of order $q_{t}^{2}$ may become relevant. sykora18

Throughout this article we consider the case where the ordering wave vector ${\bf Q}$ connects two pairs of hot spots $\pm{\bf k}_{H}$ and $\pm{\bf k}^{\prime}_{H}$ on the Fermi surface. The wave vector ${\bf Q}$ is then a crossing point of two $2k_{F}$ lines, as shown in Fig. 1 (right panel). Let $q_{r}$ and $q_{t}$ be the normal and tangential coordinates relative to the first of the $2k_{F}$ lines as introduced above. The normal and tangential coordinates relative to the second $2k_{F}$ line, $q^{\prime}_{r}$ and $q_{t}^{\prime}$, respectively, are related to the former by

where $\theta$ is the angle between the Fermi surface normal vectors at ${\bf k}_{H}$ and ${\bf k}^{\prime}_{H}$, which is also the angle between the two $2k_{F}$ lines crossing at ${\bf Q}$ (see Fig. 2). The $2k_{F}$ singularities on these two $2k_{F}$ lines need to be added, such that holder14

where $e_{\bf q}$ and $q_{t}$ are defined relative to the first $2k_{F}$ line, while $e^{\prime}_{\bf q}=v^{\prime}_{F}q^{\prime}_{r}+\frac{1}{4m^{\prime}}{q^{\prime}_{t}}^{2}$ is defined relative to the second $2k_{F}$-line. The last two terms in Eq. (9) describe the leading regular momentum dependence near ${\bf Q}$. We have chosen the variables $e_{\bf q}$ and $q_{t}$ to parametrize this dependence. footnote_chi_0 The hot spots are often related by a point group symmetry of the lattice, as in the first two examples in Fig. 1, so that $v_{F}=v^{\prime}_{F}$, $m=m^{\prime}$, and $a=a^{\prime}$. The bare susceptibility exhibits a peak at ${\bf Q}$ if and only if $c\,\theta>0$. In the following we assume that this is the case.

At the QCP one has $g\chi_{0}({\bf Q},0)=-1$, so that the RPA susceptibility assumes the singular form

To deal with the critical order parameter fluctuations, the perturbation expansion has to be organized in powers of a dynamical effective interaction. This arises naturally as a boson propagator by decoupling the bare interaction in the instability channel via a Hubbard-Stratonovich transformation.hertz76 ; millis93 Alternatively it can be obtained by an RPA resummation of particle-hole bubbles or ladders. In the simplest case of a charge-density wave instability in a spinless fermion system, the RPA effective interaction can be written as

For a charge-density wave instability in a spin-$\frac{1}{2}$ fermion system, the effective interaction has the diagonal spin structure

where $\sigma_{1},\sigma_{2}$ ($\sigma^{\prime}_{1},\sigma^{\prime}_{2}$) are the spin-indices of the ingoing (outgoing) fermions. For a spin-density wave, the effective interaction acquires a non-diagonal spin structure. In a spin-rotation invariant system of spin-$\frac{1}{2}$ fermions, it can be written as

where ${\boldsymbol{\tau}}=(\tau^{x},\tau^{y},\tau^{z})$ is the vector formed by the three Pauli matrices $\tau^{x},\tau^{y},\tau^{z}$. footnote3

Expanding the bare susceptibility as in Eq. (9), the effective interaction at the QCP assumes the asymptotic form

It thus features the same singularity as the RPA susceptibility. In case that the coupling $g$ has a (regular) momentum dependence for ${\bf q}$ near ${\bf Q}$, the coefficients $b$ and $c$ are not determined by $\chi_{0}({\bf q},0)$ only, but receive additional contributions from the expansion of $g^{-1}({\bf q})$ around ${\bf Q}$.

To leading order in the effective interaction, the fermion self-energy is given by the one-loop expression

with $M=1$ for a charge density and $M=3$ for a spin density instability. We evaluate $\Sigma({\bf k},ik_{0})$ for low frequencies $k_{0}$ and momenta ${\bf k}$ near one of the hot spots, say ${\bf k}_{H}$. The dominant contributions come from momentum transfers ${\bf q}$ near ${\bf Q}$, such that ${\bf k}-{\bf q}$ is situated near the antipodal hot spot $-{\bf k}_{H}$.

In our derivations we assume that the Fermi surface is convex at the hot spots. Results for the case of a concave Fermi surface follow from a simple particle-hole transformation. Using normal and tangential coordinates for ${\bf k}$ near ${\bf k}_{H}$, we expand the dispersion relation to leading order as $\xi_{\bf k}=v_{F}k_{r}+\frac{1}{2m}k_{t}^{2}$, and $\xi_{{\bf k}-{\bf q}}=-v_{F}(k_{r}-q_{r})+\frac{1}{2m}(k_{t}-q_{t})^{2}$ for ${\bf q}$ near ${\bf Q}$.

Inserting Eq. (14) into Eq. (15), the one-loop self-energy is thus given by the integral

Note that we have not yet introduced an ultraviolet cutoff for the momentum integral. We will do so at a later stage of the evaluation, whenever needed.

The self-energy obtained from the perturbative one-loop calculation modifies the fermion propagator substantially at the hot spots. Since the effective interaction $D$ and the fermion self-energy were computed with bare fermion propagators, we need to check whether the results remain qualitatively the same when the bare propagator is replaced by the interacting propagator with the non-Fermi liquid form

for low frequencies $k_{0}$ and momenta ${\bf k}$ near a hot spot. From the Dyson equation $G^{-1}=G_{0}^{-1}-\Sigma$ and Eqs. (17) and (19) one can read off the renormalization factors

The dispersion relation $\bar{\xi}_{\bf k}$ in Eq. (27) has a renormalized Fermi velocity following from the momentum dependence of the self-energy in Eq. (25),

with a distinct renormalization for momenta inside and outside the Fermi surface, and a renormalized mass $\bar{m}$.

Eq. (27) describes the frequency dependence for $k_{0}\to 0$ only for hot spot momenta. For momenta near a hot spot the frequency dependence is the same only above a momentum-dependent crossover scale, below which Fermi liquid behavior with a large finite $Z$ and a vanishing $Z^{\prime}$ is recovered. The crossover scale is of order $k_{r}$ in the normal direction, and of order $k_{t}^{2}$ in the tangential direction. Approximating the fermion propagator by Eq. (27) in quantities that involve momentum integrals of $G({\bf k},ik_{0})$ will lead to quantitative errors, but it is not expected to affect the qualitative behavior.

In a self-consistent one-loop calculation, the effective interaction $D$ is still given by the RPA form

as in Eq. (11), where the particle-hole bubble

is now being computed with the interacting propagator $G$ instead of the bare one. Vertex corrections do not play an important role here, since there are no singular one-loop vertex corrections for incommensurate density waves. altshuler95 For ordering wave vectors distinct from half a reciprocal lattice vector, there is no coalescence of divergences of propagators in the vertex correction.

To extract the leading momentum and frequency dependence of $\Pi({\bf q},iq_{0})$ for momenta near ${\bf Q}$ and low frequencies, we expand the dispersion $\bar{\xi}_{\bf k}$ to linear order in $k_{r}$ and to quadratic order in $k_{t}$ as previously. We first discard the (finite) renormalizations of the dispersion and replace $\bar{\xi}_{\bf k}$ by the bare dispersion $\xi_{\bf k}$, focusing thus on the changes imposed by the frequency dependence of the self-energy. The momentum integrals in Eq. (32) can then be computed analytically. In Appendix B we derive the result

where $\{k_{0}\}=Z(k_{0})ik_{0}+Z^{\prime}|k_{0}|$. For a numerical evaluation of the frequency integral in Eq. (33), we extend Eq. (27) to large frequencies by the ansatz $Z(k_{0})=1+\bar{Z}\ln\big{(}1+\Lambda/|k_{0}|\big{)}$ and $Z^{\prime}=\bar{Z}^{\prime}/(1+|k_{0}|/\Lambda)$, where $\bar{Z}$ and $\bar{Z}^{\prime}$ are constants, and $\Lambda$ is an arbitrary ultraviolet cutoff. In this way the propagator has the correct large frequency asymptotics $G\sim(ik_{0})^{-1}$ for $k_{0}\to\infty$. Eqs. (28) and (29) imply that the ratio $\bar{Z}^{\prime}/\bar{Z}$ is given by $R(\alpha)$.

In Fig. 5 we plot the static ($q_{0}=0$) particle-hole bubble $\delta\Pi({\bf q},0)$ as a function of $e_{\bf q}$ for $\bar{Z}=2/\pi$ and various choices of $\bar{Z}^{\prime}$. One can see that $\delta\Pi({\bf q},0)$ exhibits a peak at ${\bf q}={\bf Q}$ only if $\bar{Z}^{\prime}>1$. Since the behavior for small $e_{\bf q}$ is determined by low frequency contributions, the criterion for a peak is independent of the choice of $\Lambda$ and depends only on the ratio $\bar{Z}^{\prime}/\bar{Z}$. Hence, a peak is obtained if and only if this ratio is larger than $\pi/2$. However, the perturbative result yields $\bar{Z}^{\prime}/\bar{Z}=R(\alpha)$ which is smaller than one. The one-loop fermion self-energy computed with bare propagators thus removes the peak in the particle-hole bubble at the nesting vector, and thus the corresponding peak in the susceptibility and in the effective interaction $D$. The $2k_{F}$ quantum critical point seems thus spoiled by the self-energy feedback, and the perturbative calculation is not even qualitatively consistent.

We now analyze the effect of the Fermi velocity renormalization (30) generated by the momentum dependence of the self-energy. Although this is a finite renormalization, it leads to a qualitative change since the renormalization factors differ for particles inside and outside the Fermi sea. We compute $\delta\Pi({\bf q},iq_{0})$ for momenta near ${\bf Q}$ and low frequencies as previously, but now with the renormalized Fermi velocity as obtained from the one-loop self-energy. We still ignore the finite renormalization of the mass. In this case only the $k_{t}$ integration in Eq. (32) can be performed analytically by residues, but the remaining two integrals can be easily carried out numerically.

In Fig. 6 we show the resulting static particle-hole bubble $\delta\Pi({\bf q},0)$ as a function of $q_{r}$ for various choices of the parameter $\alpha$. The $Z$-factors in the fermion propagators assume the $\alpha$-dependent values described by Eqs. (28) and (29) at low frequencies. The renormalized Fermi velocity is given by Eq. (30) for all momenta. One can see that that $\Pi({\bf q},0)$ now exhibits a sharp peak at ${\bf Q}$ for sufficiently large values of $\alpha$. The peak at ${\bf Q}$ is again destroyed for small $\alpha$, but this time due to a change of slope for positive $q_{r}$. The critical value for the stability of the peak at ${\bf Q}$ is $\alpha_{c}\approx 0.03$. Note that this value depends on the choice of $M$ and $N$. For $M=1$ and $N=2$ we find $\alpha_{c}\approx 1$. In any case, the Fermi velocity renormalization with a larger renormalization factor for particles inside the Fermi sea helps in stabilizing the QCP with a nested wave vector. Indeed, taking only the Fermi velocity renormalization into account, and ignoring the frequency dependence of the self-energy, we found that the peak at ${\bf Q}$ is stabilized for any choice of $\alpha$.

While the QCP seems stabilized at least for sufficiently large $\alpha$, the one-loop self-energy feedback leads to a steeper, non-linear momentum dependence of $\Pi({\bf q},0)$ for negative $q_{r}$. This, in turn, reduces the singular contributions to the self-energy, so that the perturbative one-loop result with bare propagators is not self-consistent anyway.

In the following two sections we try to incorporate the self-energy feedback self-consistently, first by a renormalization group flow, and then by a self-consistent solution of the coupled integral equations for the particle-hole bubble and the fermion self-energy. In both sections we focus on the frequency dependence of the self-energy, discarding its momentum dependence for simplicity.

A systematic way of dealing with low energy singularities and the corresponding divergences in perturbation theory is provided by the renormalization group, where self-consistency can in principle be achieved by solving a set of flow equations, that is, ordinary differential equations, instead of solving non-linear integral equations. In this section we derive and solve a flow equation for the fermion self-energy with self-energy feedback on the right hand side, and we compare the results to those from perturbation theory. To access the full frequency dependence of the self-energy, we use a functional renormalization group (fRG) framework. metzner12 Flow equations based on the fRG have already been applied to other quantum critical points in two-dimensional interacting fermion systems, for example, at the onset of nematic drukier12 and (non-nested) antiferromagnetic maier16 order in metals, and at the onset of superfluidity in semi-metals. obert11