We study the quantum critical behavior in an isotropic Fermi liquid in the vicinity of a zero-temperature density-wave transition at a finite wave vector
qc. We show that, near the transition, the Landau damping of the soft bosonic mode yields a crossover in the fermionic self-energy from $\ensuremath{\Sigma}(k,\ensuremath{\omega})\ensuremath{\approx}\ensuremath{\Sigma}(k)$ to $\ensuremath{\Sigma}(k,\ensuremath{\omega})\ensuremath{\approx}\ensuremath{\Sigma}(\ensuremath{\omega})$, where
k and $\ensuremath{\omega}$ are momentum and frequency. Because of this self-generated locality, the fermionic effective mass diverges right at the quantum critical point, not before; i.e., the Fermi liquid survives up to the critical point.
We consider the effect of the coupling between two-dimensional (2D) quantum rotors near an XY ferromagnetic quantum critical point and spins of itinerant fermions. We analyze how this coupling affects the dynamics of rotors and the self-energy of fermions. A common belief is that near a
q=0 ferromagnetic transition, fermions induce an $\mathrm{\ensuremath{\Omega}}/q$ Landau damping of rotors (i.e., the dynamical critical exponent is
z=3) and Landau overdamped rotors give rise to non-Fermi liquid fermionic self-energy $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}$. This behavior has been confirmed in previous quantum Monte Carlo (QMC) studies. Here we show that for the XY case the behavior is different. We report the results of large-scale quantum Monte Carlo simulations, which show that at small frequencies
z=2 and $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{1/2}$. We argue that the new behavior is associated with the fact that a fermionic spin is by itself not a conserved quantity due to spin-spin coupling to rotors, and a combination of self-energy and vertex corrections replaces
1/q in the Landau damping by a constant. We discuss the implication of these results to experiments.
We revisit Stoner instability, an old problem but in a modern point of view. An idea is to extract out dynamics of directional fluctuations of spins explicitly, resorting to the CP
1 representation and integrating over their amplitude fluctuations. As a result, we derive an effective field theory for ferromagnetic quantum phase transitions in terms of bosonic spinons and fermionic holons. We show that this effective field theory reproduces overdamped spin dynamics in a paramagnetic Fermi liquid and magnon spectrum in a ferromagnetic Fermi liquid. An interesting observation is that the velocity of spinons becomes zero, approaching the ferromagnetic quantum critical point, which implies emergence of local quantum criticality. Based on this scenario, we predict the
ω/T scaling behavior near ferromagnetic quantum criticality beyond the conventional scenario of the weak-coupling approach.
We consider the problem of fermions interacting with gapless long-wavelength collective bosonic modes. The theory describes, among other cases, a ferromagnetic quantum-critical point (QCP) and a QCP towards nematic ordering. We construct a controllable expansion at the QCP in two steps: we first create a non-Fermi-liquid ``zero-order'' Eliashberg-type theory, and then demonstrate that the residual interaction effects are small. We prove that this approach is justified under two conditions: the interaction should be smaller than the fermionic bandwidth, and either the band mass
mB should be much smaller than
m=kF∕vF, or the number of fermionic flavors
N should be large. For an SU(2) symmetric ferromagnetic QCP, we find that the Eliashberg theory itself includes a set of singular renormalizations which can be understood as a consequence of an effective long-range dynamic interaction between quasiparticles, generated by the Landau damping term. These singular renormalizations give rise to a negative nonanalytic
q3∕2 correction to the static spin susceptibility, and destroy a ferromagnetic QCP. We demonstrate that this effect can be understood in the framework of the ${\ensuremath{\phi}}^{4}$ theory of quantum criticality. We also show that the nonanalytic
q3∕2 correction to the bosonic propagator is specific to the SU(2) symmetric case. For systems with a scalar order parameter, the
q3∕2 contributions from individual diagrams cancel out in the full expression of the susceptibility, and the QCP remains stable.
We study the problem of disorder-free metals in the vicinity of a quantum critical point in
d=3+1 dimensions. We begin with perturbation theory in the Yukawa coupling between the electrons and undamped bosons (order-parameter fluctuations) and show that the perturbation expansion breaks down below energy scales where the bosons get substantially Landau damped. Above this scale, however, we find a regime in which low-energy fermions obtain an imaginary self-energy that varies linearly with frequency, realizing the ``marginal Fermi liquid'' phenomenology [Varma et al., Phys. Rev. Lett. 63, 1996 (1989)]. We discuss a large-
N theory in which the marginal Fermi liquid behavior is enhanced while the role of Landau damping is suppressed, and show that quasiparticles obtain a decay rate parametrically larger than their energy.
We present the expression for the quasiparticle vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ (proportional to the Landau interaction function) in a 2D Fermi liquid (FL) near an instability towards antiferromagnetism. This function is relevant in many ways in the context of metallic quantum criticality. Previous studies have found that near a quantum critical point, the system enters into a regime in which the fermionic self-energy is large near hot spots on the Fermi surface [points on the Fermi surface connected by the antiferromagnetic ordering vector ${q}_{\ensuremath{\pi}}=(\ensuremath{\pi},\ensuremath{\pi})$] and has much stronger dependence on frequency than on momentum. We show that in this regime, which we termed a critical FL, the conventional random-phase-approximation- (RPA) type approach breaks down, and to properly calculate the vertex function one has to sum up an infinite series of terms which were explicitly excluded in the conventional treatment. Besides, we show that, to properly describe the spin component of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ even in an ordinary FL, one has to add Aslamazov-Larkin (AL) terms to the RPA vertex. We show that the total ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is larger in a critical FL than in an ordinary FL, roughly by an extra power of magnetic correlation length $\ensuremath{\xi}$, which diverges at the quantum critical point. However, the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ is highly nonuniform: It holds only when, for one of the two momentum variables, the distance from a hot spot along the Fermi surface is much larger than for the other one. This fact renders our case different from quantum criticality at small momentum, where the enhancement of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ was found to be homogeneous. We show that the charge and spin components of the total vertex function satisfy the universal relations following from the Ward identities related to the conservation of the particle number and the total spin. We show that in a critical FL, the Ward identity involves ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ taken between particles on the FS. We find that the charge and spin components of ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ are identical to leading order in the magnetic correlation length. We use our results for ${\ensuremath{\Gamma}}^{\ensuremath{\omega}}({K}_{F},{P}_{F})$ and for the quasiparticle residue to derive the Landau parameters
Fcl=0=Fsl=0, the density of states, and the uniform (
q=0) charge and spin susceptibilities ${\ensuremath{\chi}}_{c}^{l=0}={\ensuremath{\chi}}_{s}^{l=0}$. We show that the density of states
NF diverges as $log\ensuremath{\xi}$; however,
Fc,sl=0 also diverge as $log\ensuremath{\xi}$, such that the total ${\ensuremath{\chi}}_{c,s}^{(l=0)}\ensuremath{\propto}{N}_{F}/(1+{F}_{c}^{l=0})$ remain finite at $\ensuremath{\xi}=\ensuremath{\infty}$. We show that at weak coupling these susceptibilities are parametrically smaller than for free fermions.