Landau theory of Fermi liquids is the foundational concept behind our understanding of conventional metals. Its basic postulate is that the low-energy physics can be described by weakly interacting quasiparticles --- a dressed but effectively free version of electrons formally represented by poles of the single-particle Green's function. However, this concept notoriously fails for strongly interacting systems such as high-${T}_{c}$ cuprate superconductors, and our understanding of properties of these ``non-Fermi'' quantum liquids is far from complete. One well-known candidate theory for non-Fermi liquids is Hertz-Millis theory: a finite-density fermion system coupled to a massless critical boson. In (2+1) dimensions, this theory is strongly interacting at low energies, but as of yet there is no controlled insight in this regime; conventional perturbation theory fails. In this article, the authors show that in the so-called quenched approximation the fermionic spectrum can be computed exactly to all orders in the coupling constant. They show extensive detailed analysis that, in this quenched limit, the nonperturbative low-energy dynamics are indeed that of a non-Fermi liquid. Instead of coherent quasiparticles one finds the emergence of critical fermionic excitations described by branch cuts in the single particle Green's function.
Landau theory of Fermi liquids is the foundational concept behind our understanding of conventional metals. Its basic postulate is that the low-energy physics can be described by weakly interacting quasiparticles --- a dressed but effectively free version of electrons formally represented by poles of the single-particle Green's function. However, this concept notoriously fails for strongly interacting systems such as high-Tc cuprate superconductors, and our understanding of properties of these ``non-Fermi'' quantum liquids is far from complete. One well-known candidate theory for non-Fermi liquids is Hertz-Millis theory: a finite-density fermion system coupled to a massless critical boson. In (2+1) dimensions, this theory is strongly interacting at low energies, but as of yet there is no controlled insight in this regime; conventional perturbation theory fails. In this article, the authors show that in the so-called quenched approximation the fermionic spectrum can be computed exactly to all orders in the coupling constant. They show extensive detailed analysis that, in this quenched limit, the nonperturbative low-energy dynamics are indeed that of a non-Fermi liquid. Instead of coherent quasiparticles one finds the emergence of critical fermionic excitations described by branch cuts in the single particle Green's function.
Non-Fermi liquids are unconventional metals whose physical properties deviate qualitatively from those of noninteracting fermions due to strong quantum fluctuations near Fermi surfaces. They arise when metals are subject to singular interactions mediated by soft collective modes. In the absence of well-defined quasiparticles, universal physics of non-Fermi liquids is captured by interacting field theories which replace Landau Fermi liquid theory. However, it has been difficult to understand their universal low-energy physics due to a lack of theoretical methods that take into account strong quantum fluctuations in the presence of abundant low-energy degrees of freedom. In this review, we discuss two approaches that have been recently developed for non-Fermi liquid theory with emphasis on two space dimensions. The first is a perturbative scheme based on a dimensional regularization, which achieves a controlled access to the low-energy physics by tuning the codimension of Fermi surface. The second is a nonperturbative approach which treats the interaction ahead of the kinetic term through a non-Gaussian scaling called interaction-driven scaling. Examples of strongly coupled non-Fermi liquids amenable to exact treatments through the interaction-driven scaling are discussed.
We study the problem of disorder-free metals near a continuous quantum critical point. We depart from the standard paradigm [Hertz, Phys. Rev. B 14, 1165 (1976) and Millis, Phys. Rev. B 48, 7183 (1993)] and treat both fermions and bosons (i.e., order parameter fields) on equal footing. We construct a Wilsonian effective field theory that integrates out only high-energy boson and fermion modes. Below the upper critical dimension of the theory (d=3 spatial dimensions), we find new fixed points in which the bosons are described by the Wilson-Fisher fixed point and are coupled to a non-Fermi-liquid metal. We describe subtleties with the renormalization group flow of four-Fermi interactions, which can be surmounted in a controlled large-N limit. In this limit, we find that the theory has no superconducting instability.
Non-Fermi liquids arise when strong interactions destroy stable fermionic quasiparticles. The simplest models featuring this phenomenon involve a Fermi surface coupled to fluctuating gapless bosonic order parameter fields, broadly referred to as ``Hertz-Millis'' models. We revisit a controlled approach to Hertz-Millis models that combines an expansion in the inverse number (N) of fermion species with an expansion in the deviation of the boson dynamical critical exponent z from 2. The structure of the expansion is found to be qualitatively different in the quantum critical regime $\mathrm{\ensuremath{\Omega}}\ensuremath{\ll}q$ and in the transport regime $\mathrm{\ensuremath{\Omega}}\ensuremath{\gg}q$. In particular, correlation functions in the transport regime involve infinitely many diagrams at each order in perturbation theory. We provide an explicit and tractable recipe to classify and resum these diagrams. For the simplest Hertz-Millis models, we show that this recipe is consistent with nonperturbative anomaly arguments and correctly captures the fixed point optical conductivity as well as leading corrections from irrelevant operators. We comment on potential applications of this expansion to transport in more complicated Hertz-Millis models as well as certain beyond-Landau metallic quantum critical points.
The Landau theory of phase and phase transition, which describes the state of matter by the spontaneous symmetry breaking, and the Fermi liquid theory of interacting Fermi systems are the two corner stones of contemporary condensed matter physics. They act as our starting points to understand the structure, excitations and phase transitions of any condensed matter system. According to such a Landau paradigm, an interacting Fermi system will behave just as non-interacting Fermi gas system at sufficient low energy, if the interaction is not strong enough to induce a symmetry breaking phase transition in the system. In such a case, the perturbation expansion in the interaction strength is assumed to converge and the low energy excitation can be connected to that of a Fermi gas system adiabatically. In strongly correlated electron systems, such as the high temperature superconductors and quantum antiferromagnets, a pertubative picture on the interaction effect usually fails. The correlation effect between the electrons can induce new forms of order that are beyond the Landau symmetry breaking paradigm. Non-Fermi liquid behavior can emerge in the fully symmetric correlated phases. The Mott insulating phase and the quantum spin liquid phase are the most prominent example of such exotic state of matter. After about three decades of intensive studies, people find that the quantum non-locality effect is at the heart of such new forms of order. The quantum non-locality effect manifests itself in the emergence of gauge degree of freedom in the low energy physics of such strongly correlated system. Such emergent gauge filed is not only important for the characterization and classification of the ground state structure, but is also directly responsible for the non-Fermi liquid behavior of the system. The Landau paradigm should be extended to accommodate such new forms of order and the related new form of excitations and new universal class of phase transitions. Such an extension will elevate our understanding of the structure of matter to the level of quantum wave function. Unlike the conventional symmetry breaking order, which is characterized by the correlation functions between local physical observables, the new form of orders, which are commonly called quantum order (with the topological order as one of its typical example), can be better characterized by the entanglement property in the quantum wave function. From a mathematical point of view, the new form of order is closely related to the projective representation theory of the symmetry group, while the conventional symmetry breaking order is well described by the linear representation theory of the symmetry group.
Many strongly correlated electron systems fall outside the conventional quasiparticle paradigm. Their low-energy fermionic excitations need not be adiabatically connected to the bare electrons or appear as poles of the physical electron single-particle Green's function. This thesis investigates to what extent an effective quasiparticle description can nevertheless survive, and which emergent degrees of freedom provide the appropriate low-energy structure. First, the interaction-driven evolution from a quantum spin Hall insulator to a Mott insulator is studied using the dynamical cluster approximation. Dispersive zeros of the single-particle Green's function emerge near the Mott transition and retain a nontrivial topological character. When symmetry breaking is allowed, a nontopological excitonic insulator is found to intrude between quantum spin Hall and Mott insulators. Encoding both poles and zeros within a low-energy quasiparticle Hamiltonian provides a continuous one-body description of the three phases. On the Mott side, its gap evolution supports interpreting the soft excitonic mode as a bound state of quasiparticle and quasihole excitations tied to the Green's function zeros, rather than to the widely separated Hubbard bands. Second, the intrinsic anomalous Hall conductivity of a topological metal is analyzed within a multiband extension of Landau Fermi-liquid theory. Residual interactions among quasiparticles at the Fermi surface dress the dynamic current vertex and generate corrections to the Berry-curvature contribution obtained from the quasiparticle bands alone. Our result supports recent claims that the correct expressions for topological observables include vertex corrections besides the topological invariants built just upon the Green's function. It also demonstrates that such corrections are naturally accounted for by Landau Fermi-liquid theory. Extracting the relevant ingredients for an effective description of material-relevant correlated systems requires computational tools that remain reliable at low temperatures. To this end, a deterministic strong-coupling impurity solver for dynamical mean-field theory is discussed. By providing static, dynamical, and thermodynamic observables without stochastic sampling, it offers a promising route toward multiorbital and spin-orbit coupled problems where conventional Monte Carlo methods currently struggle. Finally, the auxiliary quasiparticle framework underlying the ghost-Gutzwiller Ansatz is investigated. On a model for correlated quantum spin Hall insulators, we show that the topology of the interacting state is encoded in the auxiliary one-body Hamiltonian. The resulting band-structure reveals topological Hubbard bands whose topological character can be tuned through a finite magnetization. When applied to the single-band t-J model, the same framework yields a fractionalized Fermi liquid (FL*). In this phase, dispersive neutral spinons, with vanishing physical-electron spectral weight, coexist with conventional quasiparticles forming a small hole-like Fermi surface, thus violating Luttinger's theorem. The corresponding temperature--doping phase diagram contains a low-doping FL*, a d-wave superconducting dome, and an overdoped conventional Fermi liquid, thereby reproducing key qualitative features of cuprate phenomenology.
The paradigm of Landau Fermi liquid theory has successfully described the low-energy behavior of numerous metallic systems. However, a growing number of experimental observations in strongly correlated electron materials, such as high-temperature superconductors and heavy fermion compounds, defy this conventional description, exhibiting what is known as non-Fermi liquid (NFL) behavior. These exotic states lack well-defined quasiparticle excitations, displaying anomalous scaling laws and transport properties. Ultracold atomic gases have emerged as highly controllable quantum simulators for strongly correlated matter, offering unprecedented access to tunable interaction strengths, dimensionality, and lattice geometries. This paper explores the potential of ultracold fermionic gases to unveil and characterize NFL behavior. We review the theoretical underpinnings of Landau Fermi liquid theory and its breakdown, survey prominent theoretical models for NFL states, and highlight experimental techniques in ultracold atomic systems pertinent to their detection. Specific attention is given to phenomena such as the BCS-BEC crossover, quantum criticality, and pseudogap phases in ultracold fermions, where strong interactions are expected to lead to deviations from quasiparticle descriptions. We discuss experimental signatures like anomalous spectral functions, linear-in-temperature resistivity analogs, and deviations in thermodynamic properties, proposing how these could be observed and analyzed in current and future ultracold atom platforms. This research aims to leverage the purity and tunability of atomic gases to gain fundamental insights into the nature of NFL states, contributing to a broader understanding of quantum many-body physics beyond the conventional quasiparticle picture.