This thesis is concerned with recent progress made for the antiferromagnetic quantum critical metal in two space dimensions. Firstly, we develop a field-theoretic functional renormalization group formalism for the low-energy effective field theories for non-Fermi liquids by using renormalizable field theories for metals. The formalism is applied to the antiferromagnetic quantum critical metal in two space dimensions. In the space of coupling functions, we identify the interacting fixed point with a vanishing nesting angle. For theories with non-zero nesting angles, the coupling functions acquire universal momentum dependent profiles controlled by the bare nesting angle before flowing towards a superconducting state in the low-energy limit. The superconducting instability is inevitable because “lukewarm” electrons that are coherent enough to be susceptible to pairing are subject to a universal attractive interaction mediated by the critical spin fluctuations. Despite the superconducting instability being unavoidable, theories with repulsive or weakly attractive four-fermion interaction at a UV scale must flow through a “bottleneck” where there is a slow RG flow due to the proximity to non-Hermitian fixed points. This bottleneck of the RG flow controls the scaling behaviour of the normal state and the “quasi-universal” pathway to superconductivity. Secondly, we show that momentum-dependent quantum corrections dynamically give rise to a curved momentum-spacetimes for quasiparticles. In the antiferromagnetic quantum critical metal, the curved momentum-spacetime geometry arises from a momentum-dependent red shift. With increasing nesting angle, the red shift near the hot spots becomes stronger while the hot region on the Fermi surface with the strong red shift shrinks. This creates the possibility of realizing a momentum-space black hole horizon where electrons are perpetually slowed down as they approach the hot spots. However, the singularity in the momentum-dependent red shift is cut off at finite temperatures above the superconducting transition temperature. Finally, we study a magnetic impurity immersed in the antiferromagnetic quantum critical metal. Critical spin fluctuations represented by bosonic fields compete with the conduction electrons to couple with the impurity spin. In the low-energy limit, the electron-impurity (Kondo) coupling dominates over the boson-impurity coupling. However, the Kondo screening is suppressed by the boson with an increasing severity when the hot spots connected by the antiferromagnetic ordering wavevector are better nested. The origin of this suppression of Kondo screening lies in the ultraviolet/infrared (UV/IR) mixing: bosons that carry large momenta up to the UV cutoff actively suppress the Kondo screening at low energies.
We analyze quantum fluctuation effects at the onset of charge- or spin-density wave order with a
2kF wave vector
Q in two-dimensional metals, for the special case where
Q connects a pair of hot spots situated at high symmetry points of the Fermi surface with a vanishing Fermi surface curvature. We compute the order parameter susceptibility and the fermion self-energy in one-loop approximation. The susceptibility has a pronounced peak at
Q, and the self-energy displays non-Fermi liquid behavior at the hot spots, with a linear frequency dependence of its imaginary part. The real part of the one-loop self-energy exhibits logarithmic divergencies with universal prefactors as a function of both frequency and momentum, which may be interpreted as perturbative signatures of power laws with universal anomalous dimensions. As a result, one obtains a non-Fermi liquid metal with a vanishing quasiparticle weight at the hot spots, and a renormalized dispersion relation with anomalous algebraic momentum dependencies near the hot spots.
We study non-Fermi-liquid states that arise at the quantum critical points associated with the spin density wave (SDW) and charge density wave (CDW) transitions in metals with twofold rotational symmetry. We use the dimensional regularization scheme, where a one-dimensional Fermi surface is embedded in $(3\ensuremath{-}\ensuremath{\epsilon})$-dimensional momentum space. In three dimensions, quasilocal marginal Fermi liquids arise both at the SDW and CDW critical points: the speed of the collective mode along the ordering wave vector is logarithmically renormalized to zero compared to that of Fermi velocity. Below three dimensions, however, the SDW and CDW critical points exhibit drastically different behaviors. At the SDW critical point, a stable anisotropic non-Fermi-liquid state is realized for small $\ensuremath{\epsilon}$, where not only time but also different spatial coordinates develop distinct anomalous dimensions. The non-Fermi liquid exhibits an emergent algebraic nesting as the patches of Fermi surface are deformed into a universal power-law shape near the hot spots. Due to the anisotropic scaling, the energy of incoherent spin fluctuations disperse with different power laws in different momentum directions. At the CDW critical point, on the other hand, the perturbative expansion breaks down immediately below three dimensions as the interaction renormalizes the speed of charge fluctuations to zero within a finite renormalization group scale through a two-loop effect. The difference originates from the fact that the vertex correction antiscreens the coupling at the SDW critical point whereas it screens at the CDW critical point.
We study the dynamics of critical spin fluctuations and hot electrons at the metallic antiferromagnetic quantum critical points with
Z2 and O(2) spin symmetries, building on earlier works on the O(3) symmetric theory. The interacting theories in
2+1 dimensions are approached from
3+1-dimensional theories in the $\ensuremath{\epsilon}$ expansion that tunes the codimension of Fermi surface as a control parameter. The low-energy physics of the
Z2 and O(2) theories qualitatively differ from each other and also from that of the O(3) theory. The difference is caused by higher-order quantum corrections beyond the one-loop order that are important even to the leading order in $\ensuremath{\epsilon}$. The naive loop expansion breaks down due to dynamical quenching of kinetic energy: the speed of the collective mode (
c) and the Fermi velocity perpendicular to the magnetic ordering vector (
v) become vanishingly small at low energies. What sets the three theories apart is the hierarchy that emerges between the quenched kinetic terms. At the infrared fixed point,
c/v becomes 0, 1, and $\ensuremath{\infty}$ in the
Z2, O(2), and O(3) theories, respectively. At intermediate energy scales, the slow renormalization group (RG) flows of
c and
v toward their fixed point values create approximate scale invariance controlled by approximate marginal parameters. The manifold of those quasifixed points and the RG flow therein determines crossovers from scaling behaviors with transient critical exponents at intermediate energy scales to the universal scaling in the low-energy limit. If the symmetry group is viewed as a tuning parameter, then the O(2) theory corresponds to a multicritical point which has one additional quasimarginal parameter than the other two theories.
We study the electronic spectral function of a metal in the vicinity of an antiferromagnetic (AFM) quantum critical point, focusing on a situation where the bare bandwidth of the spin fluctuations is significantly smaller than the Fermi energy. In this limit, we identify a range of energies where the fermionic quasiparticles near the ``hot spots'' on the Fermi surface are strongly scattered by the quantum critical fluctuations, whereas the damping of the AFM fluctuations by the electrons is negligible. Within a one-loop approximation, there is a parameter range where the
T=0 spectral function at the hot spots has a ``precursor gap'' feature, with a local maximum at a finite frequency. However, we find that the ratio of the bare spin-wave velocity to the Fermi velocity required to obtain a precursor gap is smaller by about an order of magnitude than the value in the electron-doped cuprate superconductors. Hence, our results cannot easily account for the recent angle-resolved photoemission experiments in this system [He et al., Proc. Natl. Acad. Sci. USA 116, 3449 (2019)]. At lower frequencies, the Landau damping of the AFM fluctuations becomes important, and the electronic spectral function has the familiar ${\ensuremath{\omega}}^{\ensuremath{-}1/2}$ singularity. Our one-loop perturbative results are supported by a numerical Monte Carlo simulation of electrons coupled to an undamped, nearly critical AFM mode.
One of the key factors that determine the fates of quantum many-body systems in the zero temperature limit is the competition between kinetic energy that delocalizes particles in space and interaction that promotes localization. While one dominates over the other in conventional metals and insulators, exotic states can arise at quantum critical points where none of them clearly wins. Here we present a class of metallic states that emerge at an antiferromagnetic (AF) quantum critical point in the presence of one-dimensional Fermi surfaces embedded in space dimensions three and below. At the critical point, interactions between particles are screened to zero in the low-energy limit and at the same time the kinetic energy is suppressed in certain spatial directions to the leading order in a perturbative expansion that becomes asymptotically exact in three dimensions. The resulting dispersionless and interactionless state exhibits distinct quasilocal strange metallic behaviors due to a subtle dynamical balance between screening and infrared singularity caused by spontaneous reduction of effective dimensionality. The strange metal, which is stable near three dimensions, shows enhanced fluctuations of bond density waves,
d-wave pairing, and pair density waves.