We study global anomalies of nonlocal effective theories proposed to describe symmetry-preserving Luttinger surfaces, i.e., the momentum-space manifolds of Green's function zeros (GFZs) at zero energy, in strongly interacting fermionic systems. In particular, we focus on simplest possible cases associated with a gapless Dirac zero, which is the counterpart of the gapless Dirac quasiparticle in weakly interacting systems. These theories may be derived by integrating out low-energy degrees of freedom that do not couple to the relevant gauge field. We discuss the global anomaly, the bulk-boundary correspondence, and the constraint on phases consistent with the anomaly, such as non-Fermi liquids and emergent gapless quasiparticles on Luttinger surfaces. Failing to avoid a spontaneous symmetry breaking in the thermodynamical limit inevitably leads unstable GFZs. We also provide some perspectives on why the related nonlocal fermionic effective theory studied recently is not a suitable starting point for a symmetrically gapped phase.
We study global anomalies of nonlocal effective theories proposed to describe symmetry-preserving Luttinger surfaces, i.e., the momentum-space manifolds of Green's function zeros (GFZs) at zero energy, in strongly interacting fermionic systems. In particular, we focus on simplest possible cases associated with a gapless Dirac zero, which is the counterpart of the gapless Dirac quasiparticle in weakly interacting systems. These theories may be derived by integrating out low-energy degrees of freedom that do not couple to the relevant gauge field. We discuss the global anomaly, the bulk-boundary correspondence, and the constraint on phases consistent with the anomaly, such as non-Fermi liquids and emergent gapless quasiparticles on Luttinger surfaces. Failing to avoid a spontaneous symmetry breaking in the thermodynamical limit inevitably leads unstable GFZs. We also provide some perspectives on why the related nonlocal fermionic effective theory studied recently is not a suitable starting point for a symmetrically gapped phase.
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 standard paradigm of topological phases posits that two phases with identical symm etries are separated by a bulk phase transition, while symmetry breaking provides a path in parameter space that allows adiabatic connection between the phases. Typically, if symmetry is broken only at the boundary, topological surface states become gapped, and single-particle surface properties no longer distinguish between the two phases. In this work, we challenge this expectation. We demonstrate that the single-particle surface Green's function contains zeros, or ``Luttinger surfaces,'' which maintain the same bulk-boundary correspondence as topological surface states. Remarkably, these Luttinger surfaces persist under symmetry-breaking perturbations that destroy the surface states. Moreover, we point out that low-energy and surface theories, often used synonymously in discussions of (gapped) topological matter, are actually different, with the difference captured by the Luttinger surfaces.
Global symmetries that define the number of low-energy degrees of freedom have profound consequences on universal properties near topological quantum critical points and in other gapless or nearly gapless states of emergent fermions. We take a Z2 global symmetry (such as time-reversal) as an example to study its effect on thermodynamic and transport properties. Although the thermal entropy density of Z2 symmetric systems is simply twice that of their counterparts without any global symmetries or the Z1 class, the temperature dependence of thermal conductivity $\ensuremath{\kappa}$ is distinctly and drastically different for different symmetries. For systems with dynamic exponent z=1, in the Z2 symmetric class, we have $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}(d\ensuremath{-}1)}$ in the quantum critical regime near weakly interacting fixed points, while for systems with no global symmetries (i.e., the Z1 class), we have $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}(d+3)}$, with d being the spatial dimension. Only near strong-coupling fixed points, both cases with or without Z2 global symmetries follow the same scaling function, $\ensuremath{\kappa}\ensuremath{\propto}{T}^{d\ensuremath{-}1}$. These distinct scalings of thermal conductivity can also appear in gapless surface Majorana states.
In this paper, we study gapless fermionic and bosonic systems in d-dimensional continuum space with U(1) particle-number-conservation and Rd translation symmetry. We present low-energy effective field theories for several gapless phases with the $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{R}}^{d}$ symmetry. The $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{R}}^{d}$ symmetry has a property that a U(1) symmetry twist will induce a nonzero momentum proportional to the U(1) charge density $\overline{\ensuremath{\rho}}$, which will be referred to as a mixed anomaly. The different effective field theories for different phases of the same system must have the same mixed anomaly. As a result, all the low-energy effective field theories must have fields with nonzero momenta of the order of ${\overline{\ensuremath{\rho}}}^{1/d}$. In particular, we present a low-energy effective field theory with infinite number of fields for Fermi liquid. We also present the Fermi-liquid effective field theory in the presence of a real space magnetic field and $\mathbit{k}$-space ``magnetic'' field, as well as in the presence of interaction described by Landau parameters. Our effective field theory correctly captures the mixed anomaly, which constrains the low-energy dynamics, such as determines the volume of the Fermi surface in terms of the mixed anomaly [i.e., in terms of the U(1) charge density]. This is another formulation of the Luttinger-Ward-Oshikawa theorem.
Abstract This chapter discusses effective theories of quantum fermionic liquids. In the low-energy limit, the type of the effective theory depends on the structure of the quasiparticle spectrum, which in turn is determined by topology in momentum space (p-space topology). The p-space topology distinguishes three main generic classes of the stable fermionic spectrum in the quantum vacuum of a 3+1 fermionic system: vacua with Fermi surfaces, vacua with Fermi points, and vacua with a fully gapped fermionic spectrum. Fermi surface is stable because it represents the topological object — the vortex in momentum space. As a result, Fermi liquids with Fermi surface share the properties of their simplest representative: weakly interacting Fermi gas. The low-energy physics of the interacting particles in a Fermi liquid is equivalent to the physics of a gas of quasiparticles moving in collective Bose fields produced by all other particles. Another topological object in p-space is the hedgehog, which is responsible for stability of Fermi points. Example of vacuum of the Fermi-point universality class is provided by 3He-A. Near the Fermi point the effective relativistic field theory emerges with emerging Weyl fermions, gauge fields, and gravity. The chapter discusses p-space and r-space topology, topological invariant for Fermi surface and Fermi points in terms of Green's function, Landau and non-Landau Fermi liquids, collective modes of the Fermi surface, volume of the Fermi surface as invariant of adiabatic deformations, collective modes of vacuum with Fermi points (electromagnetic and gravitational fields), and manifolds of zeros in p-space of higher dimensions.
States with gapless degrees of freedom are typically more complicated and less well- understood than systems possessing a gap. In this thesis, we study strongly-correlated systems described by gapless fermions. In most of the systems we consider, the fermionic excitations are emergent, i.e. they are not adiabatically connected to the electrons which constitute the fundamental building blocks of the system in question.\nWe start by proposing several novel states of matter which we claim are relevant to strongly-correlated systems. The rst state we present is a fractionalized Fermi liquid on the surface of a topological Kondo insulator. A Kondo insulator is a material which becomes insulating at low temperatures as a result of strong electron-spin interactions. When spin- orbit coupling is present, this insulator might be topological and, consequently, host robust gapless surface modes. Given the strong interactions and the decreased dimensionality of the surface, we propose that the spins and electrons there may decouple, resulting in the formation of a fractionalized Fermi liquid.\nWe next argue that quantum electrodynamics in 2+1 dimensions (QED3) with Nf = 4 fermion avours may describe a continuous, decon ned phase transition connecting the 120◦ coplanar Néel phase and the √12 × √12 valence bond solid phase of the triangular lattice antiferromagnet (AF). In addition to being a critical point, QED3 is also believed to describe a critical phase of matter called the Dirac spin liquid. Regardless of whether QED3 is manifest as a phase or a critical point, impurities and imperfections are always present in the real world, and it is therefore important to understand what e ects this may have. We show that when QED3 is perturbed by weak disorder, under certain circumstances, it ows to a new critical point in which both interactions and disorder are present.\nConclusively identifying a system described by QED3 has proven to be a di cult task, not only experimentally, but in numerical studies as well. Since QED3 is a critical theory, its spectrum on a torus is a universal quantity dependent only on low-energy degrees of freedom, such as the torus area. It follows that this data is accessible via exact diagonalization and can serve as an identifying signature of the theory. To this end, we calculate the QED3 torus spectrum using path integral methods.\nWe conclude this thesis with a comprehensive study of gapped Z2 spin liquids of the square lattice antiferrogmagnet, using a critical theory of Dirac fermions coupled to an SU(2) gauge eld as a starting point. There are many di erent Z2 spin liquid groundstates which preserve the same symmetry group. Even if one knows that a system has a Z2 spin liquid groundstate, it is by no means obvious which Z2 spin liquid is being realized. By starting from a gapless theory, the set of possibilities can be reduce considerably. Further, using certain recently proposed dualities, comparisons with theories formulated using bosons can be made.