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.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.

arXiv:2105.12528May 26, 20210 citations

Spin-Liquid Insulators Can Be Landau’s Fermi Liquids

Michele Fabrizio

The long search for insulating materials that possess low-energy quasiparticles carrying electron's quantum numbers except charge-inspired by the neutral spin-1/2 excitations, the so-called spinons, exhibited by Anderson's resonating-valence-bond state-seems to have reached a turning point after the discovery of several Mott insulators displaying the same thermal and magnetic properties as metals, including quantum oscillations in a magnetic field. Here, we show that such anomalous behavior is not inconsistent with Landau's Fermi liquid theory of quasiparticles at a Luttinger surface. That is the manifold of zeros within the Brillouin zone of the single-particle Green's function at zero frequency, and which thus defines the spinon Fermi surface conjectured by Anderson.The long search for insulating materials that possess low-energy quasiparticles carrying electron's quantum numbers except charge-inspired by the neutral spin-1/2 excitations, the so-called spinons, exhibited by Anderson's resonating-valence-bond state-seems to have reached a turning point after the discovery of several Mott insulators displaying the same thermal and magnetic properties as metals, including quantum oscillations in a magnetic field. Here, we show that such anomalous behavior is not inconsistent with Landau's Fermi liquid theory of quasiparticles at a Luttinger surface. That is the manifold of zeros within the Brillouin zone of the single-particle Green's function at zero frequency, and which thus defines the spinon Fermi surface conjectured by Anderson.

arXiv:2211.16296Physical Review LettersApr 12, 202326 citations

Theory of the Luttinger surface in doped Mott insulators

Tudor D. Stanescu, Philip Phillips, Ting-Pong Choy

We prove that the Mott insulating state is characterized by a divergence of the electron self-energy at well-defined values of momenta in the first Brillouin zone. When particle-hole symmetry is present, the divergence obtains at the momenta of the Fermi surface for the corresponding noninteracting system. Such a divergence gives rise to a surface of zeros (the Luttinger surface) of the single-particle Green function and offers a single unifying principle of Mottness from which pseudogap phenomena, spectral weight transfer, and broad spectral features emerge in doped Mott insulators. We also show that only when particle-hole symmetry is present does the volume of the zero surface equal the particle density. We identify that the general breakdown of Luttinger's theorem in a Mott insulator arises from the breakdown of a perturbative expansion for the self-energy in the single-particle Green function around the noninteracting limit. A modified version of Luttinger's theorem is derived for special cases.We prove that the Mott insulating state is characterized by a divergence of the electron self-energy at well-defined values of momenta in the first Brillouin zone. When particle-hole symmetry is present, the divergence obtains at the momenta of the Fermi surface for the corresponding noninteracting system. Such a divergence gives rise to a surface of zeros (the Luttinger surface) of the single-particle Green function and offers a single unifying principle of Mottness from which pseudogap phenomena, spectral weight transfer, and broad spectral features emerge in doped Mott insulators. We also show that only when particle-hole symmetry is present does the volume of the zero surface equal the particle density. We identify that the general breakdown of Luttinger's theorem in a Mott insulator arises from the breakdown of a perturbative expansion for the self-energy in the single-particle Green function around the noninteracting limit. A modified version of Luttinger's theorem is derived for special cases.

arXiv:cond-mat/0602280Physical Review BMar 6, 200769 citations

The Many Guises of Quasiparticles in Correlated Electron Systems

Ivan Pasqua

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.

Sep 14, 20260 citations

Luttinger many-fermion model

Ladislav Šamaj, Zoltán Bajnok

Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.

Cambridge University Press eBooksMay 16, 20130 citations

Beyond Particles: Unparticles in Strongly Correlated Electron Matter

Philip Phillips

I am concerned in these lectures with the breakdown of the particle concept in strongly correlated electron matter. I first show that the standard procedure for counting particles, namely Luttinger's theorem, breaks down anytime pole-like excitations are replaced by ones that have a divergent self-energy. Such a breakdown obtains in electronic systems whose pole-like excitations do not extend to the edge of the Brillouin zone, as in Fermi arcs in the cuprates. Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, unparticles are the natural candidate to explain the presence of charged degrees of freedom that have no particle content. The continuous mass formulation of unparticles is recast as an action in anti de Sitter space. Such an action serves as the generating functional for the propagator. This mapping fixes the scaling dimension of the unparticle to be $d_U=d/2+\sqrt{d^2+4}/2$ and ensures that the corresponding propagator has zeros with $d$ the spacetime dimension of the unparticle field. The general {\it dynamical} mechanism by which bulk operators, such as the Pauli term, couple to the scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energy is reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicates that the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.I am concerned in these lectures with the breakdown of the particle concept in strongly correlated electron matter. I first show that the standard procedure for counting particles, namely Luttinger's theorem, breaks down anytime pole-like excitations are replaced by ones that have a divergent self-energy. Such a breakdown obtains in electronic systems whose pole-like excitations do not extend to the edge of the Brillouin zone, as in Fermi arcs in the cuprates. Since any non-trivial infrared dynamics in strongly correlated electron matter must be controlled by a critical fixed point, unparticles are the natural candidate to explain the presence of charged degrees of freedom that have no particle content. The continuous mass formulation of unparticles is recast as an action in anti de Sitter space. Such an action serves as the generating functional for the propagator. This mapping fixes the scaling dimension of the unparticle to be $d_U=d/2+\sqrt{d^2+4}/2$ and ensures that the corresponding propagator has zeros with $d$ the spacetime dimension of the unparticle field. The general {\it dynamical} mechanism by which bulk operators, such as the Pauli term, couple to the scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energy is reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicates that the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.

arXiv:1412.1098Jul 22, 20152 citations

d_U=d/2+\sqrt{d^2+4}/2

and ensures that the corresponding propagator has zeros with

d

the spacetime dimension of the unparticle field. The general {\it dynamical} mechanism by which bulk operators, such as the Pauli term, couple to the scaling dimension of the boundary operator and thereby lead to a vanishing of the spectral weight at zero energy is reviewed in the context of unparticles and zeros. The analogue of the BCS gap equations with unparticles indicates that the transition temperature increases as the attractive interaction strength decreases, indicating that unparticles are highly susceptible to a superconducting instability.