In this thesis, we study the low-energy effective theory for the antiferromagnetic quantum critical metal in two dimensions. The theory has been the subject of intense study for more than twenty years, due to the novel physics of non-Fermi liquid metals and its potential relevance to high-temperature superconductors and heavy-fermion compounds. In the first part of the thesis, we present the perturbative study of the theory in 3 minus epsilon space dimensions by extending the earlier one-loop analysis to higher-loop orders. We show that the expansion is not organized by the standard loop expansion, and a two-loop graph becomes as important as one-loop graphs even in the small epsilon limit due to an infrared singularity caused by an emergent quasilocality. This qualitatively changes the nature of the infrared fixed point, and the epsilon expansion is controlled only after the two-loop effect is taken into account. Furthermore, we show that a ratio between velocities emerges as a small parameter, which suppresses a large class of diagrams. We show that the critical exponents do not receive quantum corrections beyond the linear order in epsilon in the limit that the ratio of velocities vanishes. In the second part of the thesis, we present a nonperturbative solution to the theory in two dimensions based on an ansatz that is inspired by the perturbative analysis. Being a strongly coupled theory, it can still be solved reliably in the low-energy limit as quantum fluctuations are organized by the ratio of velocities that dynamically flows to zero in the low-energy limit. We predict the exact critical exponents that govern the universal scaling of physical observables at low temperatures.
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 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 antiferromagnetic quantum critical metal in $3\ensuremath{-}\ensuremath{\epsilon}$ space dimensions by extending the earlier one-loop analysis [Sur and Lee, Phys. Rev. B 91, 125136 (2015)] to higher-loop orders. We show that the $\ensuremath{\epsilon}$ expansion is not organized by the standard loop expansion, and a two-loop graph becomes as important as one-loop graphs due to an infrared singularity caused by an emergent quasilocality. This qualitatively changes the nature of the infrared fixed point, and the $\ensuremath{\epsilon}$ expansion is controlled only after the two-loop effect is taken into account. Furthermore, we show that a ratio between velocities emerges as a small parameter, which suppresses a large class of diagrams. We show that the critical exponents do not receive corrections beyond the linear order in $\ensuremath{\epsilon}$ in the limit that the ratio of velocities vanishes. The $\ensuremath{\epsilon}$ expansion gives critical exponents which are consistent with the exact solution obtained in $0<\ensuremath{\epsilon}\ensuremath{\le}1$.
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.
Properties of the two-dimensional Hubbard model with nearest-neighbor hopping are under scrutiny in cold-atom experiments and diagrammatic quantum Monte Carlo. Given the controlled nature of these approaches, it is timely to make predictions about strange-metal behavior and its relation to quantum-critical properties. Even for interaction strengths below the Mott transition, several peculiarities occur at the quantum critical point separating Fermi liquid and incommensurate spin-density wave order. Here, we predict, using the non-perturbative improved two-particle self-consistent approach, that for correlation lengths ranging from about one to one hundred lattice spacings, Kohn anomalies on the underlying Fermi surface lead to unconventional critical exponents. The temperature dependence of the single-particle self-energy acquires strong momentum dependence along the Fermi surface with no clear evidence of Landau quasiparticles. Nevertheless, we observe strange-metal behavior, namely, resistivity that scales linearly with temperature as
T→0. Our work shows that a linear temperature dependence of resistivity arises without a linear-in-temperature self-energy along the Fermi surface, as is often assumed.