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.
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.
For weakly interacting many-body quantum systems one can apply the extensive framework of perturbative quantum field theory. On the other hand, strong interactions can not be treated practically in perturbation theory because of the absence of a small control parameter. In this context variational methods are a powerful tool to obtain reliable informations regarding the properties of the ground state of the strongly correlated electron system. In this thesis we present novel formulations of two widely used methods: the Gutzwiller wave function and the slave-spin theory. These techniques share a similar philosophy and the extensions are based on the common attitude towards a more detailed description of the high-energy incoherent excitations characterizing interacting electron systems together with low-energy quasiparticles excitations, captured by standard approaches. Our task is not only crucial for a faithful description of the insulating phase but also improves the variational characterization of the low-energy quasiparticle excitations. We apply the novel ghost-Gutzwiller wave function technique to tackle an intriguing phenomenon, namely the exciton Mott transition in photoexcited semiconductors. Despite being a quite old topic that goes back to the 1970's, the nature of the exciton Mott transition still defies a complete understanding. By considering an idealised model of photoexcited semiconductors we unveil the important role of the exciton binding energy in determining the nature of the transition. Moreover, our results uncover rather anomalous electron-hole liquid phase next to the transition, which still sustains excitonic excitations although being a degenerate Fermi liquid of heavy mass quasiparticles. By means of the Dirac-Frenkel variational principle, we generalize the ghost-Gutzwiller wave function to study the out of equilibrium dynamics of strongly correlated electron systems. Numerical results on the single-band Hubbard model show a remarkable agreement with those obtained with time-dependent dynamical mean field theory. We believe that the method opens the way to several promising developments and future applications. Concerning the slave-particle approach, we consider Anderson impurity models and we show that, within our formulation, the constraint on the slave-spin variable can be removed, leaving thermal average free by any projection on the "physical" subspace of the enlarged Hilbert space. To the best of our knowledge, this represents an exception and shows that our formulation is more convenient than other slave-particle approaches, where the constraint have to be implemented explicitly. The method, suited to deal with impurity models, finds direct application in studying time-dependent transport across an interacting quantum dot. To this aim we formulate, by means of the Keldysh Green's function, the slave-spin mean-field theory in the out of equilibrium framework and we apply the method to the dynamics of a driven quantum dot. Finally we study the current-voltage characteristic of an interacting quantum dot tunnel-coupled to the edge of a superconductive nanowire. The appearance of the topological Majorana edge mode is signalized by distinctive features in the charge transport properties of the junction. Apart from the well known half-integer zero bias conductance we find that the topological region is characterized by a vanishing Fano factor, that can be measured in experiments to detect the presence of a Majorana zero mode hybridized with the quantum dot.
Recent theoretical and experimental advances have opened new perspectives on the characterization of strongly correlated phases of matter, adding a new layer of understanding based on concepts and tools borrowed from quantum information theory. We join the effort by investigating, within dynamical mean-field theory (DMFT) and its cluster extension, the local and quasilocal (i.e. short-range) quantum information content of salient landmarks in the phase diagram of the two-dimensional Hubbard model, the cornerstone model for the description of the physics of cuprate materials, and of a closely related model for interacting topological states on the honeycomb lattice. In the first part of the thesis, we focus on the DMFT solution of the Kane-Mele-Hubbard model, at zero temperature. In the thermodynamic limit we find evidence of an interaction-driven discontinuous quantum phase transition between the Z2 quantum spin-Hall insulator and anisotropic antiferromagnetic solutions, with an intermediate regime of coexistence of nontrivial topology and magnetic order. A clear-cut interpretation of these results is given in terms of a rigorous quantification of quantum and classical correlations contained in a single local orbital (which we refer to as intra-orbital mutual information), providing a notion of "statistical distance" from the Hartree-Fock description of the system. The resulting analysis complements the established Green's function based understanding of the relationship between dynamical and static mean-field theories. In particular, we find the magnetic solutions of dynamical mean-field theory to asymptotically approach the corresponding uncorrelated Hartree-Fock states, in the strong coupling limit, in stark contrast with the paramagnetic Mott-Hubbard solution, which in turn reveals a maximally correlated local spin-orbital pair at large interaction strength. Furthermore, these findings cast some light on the relationship between Mott localization and the possible development of nonlocal entanglement. In the second part of the thesis, we provide an alternative view of both the interaction-driven and density-driven paramagnetic Mott transitions in the two-dimensional Hubbard model, in terms of rigorous measures of entanglement and correlation between two spatially separated electronic orbitals, with no contribution from their environment. A space-resolved analysis of cluster dynamical mean-field theory results elucidates the prominent role of the nearest-neighbor entanglement in probing Mott localization: two traditional upper bounds and two recently introduced lower bounds for its magnitude sharply increase at the metal-insulator transition, in contrast with the moderate variation found at all interaction strengths that are sufficiently far from the transition point. At half-filling, the two-site entanglement beyond nearest neighbors is shown to be quickly damped as the inter-site distance is increased within the cluster, suggesting that Mott-Hubbard insulators may follow an area law. However, the size of the simulated clusters does not allow a quantitative analysis of the decay, so that a precise classification of the spatial entanglement properties of the system is left for future study. In the presence of hole-doping, we show how the pseudogap metal separating the Mott-Hubbard insulator from the hole-dominated Fermi liquid features quasilocal entanglement properties that are strikingly similar to the localized Mott phase, while it is separated from the low-entangled Fermi liquid by a discontinuous jump in all the computed entanglement and correlation measures. All the presented results ultimately resolve a conundrum of previous analyses based on the single-site von Neumann entropy, which has been found to monotonically decrease when the interaction is increased, defeating the purpose of capturing and understanding strong electronic correlations with the aid of quantum information concepts. Both the intra-orbital mutual information and the quasilocal two-site entanglement, on the other hand, recover instead the distinctive character of Mott insulators and pseudogap metals as strongly correlated many-body states, demonstrating its central role in future advancements in the field of quantum materials.
Though most fermionic Mott insulators order at low temperatures, ordering is ancillary to their insulating behavior. Our emphasis here is on disentangling ordering from the intrinsic strongly correlated physics of a doped half-filled band. To this end, we focus on the two-dimensional Hubbard model. Because the charge gap arises from on-site correlations, we have been refining the nonperturbative approach of Matsumoto and Mancini [Phys. Rev. B 55, 2095 (1997)] which incorporates local physics. Crucial to this method is a self-consistent two-site dynamical cluster expansion which builds in the nearest-neighbor energy scale J. At half-filling, we find that the spectral function possesses a gap of order U and is devoid of any coherent quasiparticle peaks although ordering or charge fractionalization are absent. At low temperatures, local antiferromagnetic correlations emerge. In the doped case, we find that the Fermi surface exceeds the Luttinger volume. The breakdown of Luttinger's theorem in the underdoped regime is traced both to the dynamically generated Mott gap as well to a nonvanishing of the imaginary part of the self-energy at the Fermi level. Spectral weight transfer across the Mott gap also emerges as a ubiquitous feature of a doped Mott insulator and suggests that high- and low-energy scales are inseparable. Additionally in the underdoped regime, we find that a pseudogap exists in the single-particle density of states as well as in the heat capacity. The pseudogap (which is set by the energy scale t2/U) is argued to be a ubiquitous feature of a lightly doped Mott state and simply represents the fact that hole transport involves double occupancy. In analogy with the Mott gap and antiferromagnetism, we propose that ordering may also accompany the formation of a pseudogap. We suggest a current pattern within a one-band model that preserves translational but breaks time-reversal symmetry along the canonical x and y axes but not along $x=\ifmmode\pm\else\textpm\fi{}y$ that is consistent with the experimental observations. Finally, we show that the Hall coefficient in a doped Mott insulator must change sign at a doping level x<1/3. The sign change is tied to a termination of strong correlation physics in the doped Mott state.
Underdoped Mott insulators provide us with a challenge of many-body physics. Recent renewed understanding is discussed in terms of the evolution of pole and zero structure of the single-particle Green's function. Pseudogap as well as Fermi arc/pocket structure in the underdoped cuprates is well reproduced from the recent cluster extension of the dynamical mean-field theory. Emergent coexisting zeros and poles set the underdoped Mott insulator apart from the Fermi liquid, separated by topological transitions. Cofermion proposed as a generalization of exciton in the slave-boson framework accounts for the origin of the zero surface formation. The cofermion-quasiparticle hybridization gap offers a natural understanding of the pseudogap and various unusual Mottness. Furthermore the cofermion offers a novel pairing mechanism, where the cofermion has two roles: It reinforces the Cooper pair as a pair partner of the quasiparticle and acts as a glue as well. It provides a strong insight for solving the puzzle found in the dichotomy of the gap structure.
The Fermi liquid theory is a central concept in modern condensed matter physics used to describe conventional metals. A state of this universality class consists of well-defined quasiparticles, which occupy a finite number of states within a Fermi volume defined in momentum space. According to the Luttinger’s theorem, such a state encloses a Fermi volume proportional to the electron density modulo the filled bands if no symmetry is broken. In the past decades strong deviations from Fermi liquid theory have been observed in the scaling behavior of thermodynamic observables in different materials, among which are for instance cuprates and iron pnictides. This thesis is about two prototypical systems that can not be described by Fermi liquid theory. The first part of this work investigates two-dimensional effective models, which are in a Mott insulating state at half-filling and have a finite conductivity when doped with holes. In such a Mott insulating state, the charge carriers are strongly localized due to the Coulomb repulsion. Hence, strong correlations are assumed to have a strong impact on the formation of the ground state, also in the regime of small hole concentration. In a first project we have used a so-called spinon dopon mean-field theory to represent the two-dimensional Fermi-Hubbard model with strong on-site repulsive interaction in effective degrees of freedom, in which the holes can be embedded into a quantum spin liquid. The corresponding SU(2) invariant ground state belongs to the class of fractionalized Fermi liquids. In a second project we investigate a quantum dimer model, an effective model based on a Hilbert space spanned by short range singlets and bound states of holes and spins. The focus here is on the calculation of the hole-part of the electron spectral function by using exact diagonalization and its comparison with two analytic methods, a diagrammatic computation based on the Bethe-Salpeter equation and a so-called two-mode approximation. The electron spectral function shows a similar analytic form in momentum space between nodal and antinodal point when compared to results from photoemission spectroscopy experiments on cuprates. Furthermore, in a subsequent work we calculate the exact ground state of the quantum dimer model along a certain parameter line. In order to analyze the behavior of the elctron spectral function when increasing the density of holes, we investigate the Fermi-Hubbard model in a current project using a dynamical mean-field approach. To solve the 4-site cluster impurity problem, we use a numerical renormalization group approach. However, numerical limitations force us to restrict the analysis to spin-polarized baths. According to the spectral data, the system is similar to the SU(2) invariant case at half-filling in a Mott insulating state and posseses a momentum-selective energy gap at finite doping. The topology of the Fermi surface shows a Lifshitz transition when increasing the hole concentration. Here, the curvature of the Fermi surface changes from electron- to hole-like. For comparison we apply the dynamical mean-field theory also to the quantum dimer model and observe that the electron spectral functions at finite doping are qualitatively similar to that of the two-dimensional Fermi-Hubbard model. The second half of the work is about Tomonaga-Luttinger liquid theory, which is used to describe the low-energy effective degrees of freedom of one-dimensional systems. Here, we first discuss a conceptional extension of the operator-based bosonization theory for one-dimensional systems. This extension is especially suited for inhomogeneous one-dimensional systems. First, we investigate a one-dimensional system with a local interation potential and compute an exact solution of the single particle propagator at T = 0. The critical exponent of the single particle propagator has an unconventional form as a function of the microscopic Tomonaga-Luttinger parameters, which is not covered by the original Luttinger paradigm postulated by F. Duncan M. Haldane. In a second project on one-dimensional systems, we study the impact of scattering processes among bosonic low-energy excitations on the thermalization process. Such scattering processes are irrelevant on large length scales, however strongly affect the dynamics. In our analytic analysis we focus on a experimental setup, where a one-dimensional Bose gas is instantaneously splitted in two identical, however strongly correlated, halves of one-dimensional electronic systems. In the following, the corresponding non-equilibrium state runs through multiple regimes in time, such as a metastable prethermalization regime. However, above a certain threshold in time such scattering processes cause an effective thermalization of the system. In order to demonstrate this, we compute the kinetic equation in the Keldysh field integral formalism from a diagrammatic expansion based on a self-consistent Born approximation.
In periodic systems of interacting electrons, Fermi and Luttinger surfaces refer to the locations within the Brillouin zone of poles and zeros, respectively, of the single-particle Green's function at zero energy and temperature. Such difference in analytic properties underlies the emergence of well-defined quasiparticles close to a Fermi surface, in contrast to their supposed non-existence close to a Luttinger surface, where the single-particle density-of-states vanishes at zero energy. We here show that, contrary to such common belief, coherent `quasiparticles' do exist also approaching a Luttinger surface in compressible interacting electron systems. Thermodynamic and dynamic properties of such 'quasiparticles' are just those of conventional ones. For instance, they yield well defined quantum oscillations in Luttinger's surface and linear in temperature specific heat, which is striking given the vanishing density of states of physical electrons, but actually not uncommon in strongly correlated materials.