arXiv:cond-mat/9706143 · Physical review. B, Condensed matter

Spectral function of two-dimensional Fermi liquids

Christoph J. Halboth, Walter Metzner

Published Apr 15, 1998• 17 citations

Abstract

We show that the spectral function for single-particle excitations in a two-dimensional Fermi liquid has a Lorentzian shape in the low-energy limit. Landau quasiparticles have a uniquely defined spectral weight and a decay rate which is much smaller than the quasiparticle energy. By contrast, perturbation theory and the $T$-matrix approximation yield spurious deviations from Fermi-liquid behavior, which are particularly pronounced for a linearized dispersion relation.

DOI: 10.1103/physrevb.57.8873

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arXiv:cond-mat/9706143 · Physical review. B, Condensed matter

Spectral function of two-dimensional Fermi liquids

Christoph J. Halboth, Walter Metzner

Published Apr 15, 1998• 17 citations

arXiv HTML PDF

Abstract

T

-matrix approximation yield spurious deviations from Fermi-liquid behavior, which are particularly pronounced for a linearized dispersion relation.

DOI: 10.1103/physrevb.57.8873

Related papers

Decay of Fermionic Quasiparticles in One-Dimensional Quantum Liquids

К. А. Матвеев, Akira Furusaki

The low-energy properties of one-dimensional quantum liquids are commonly described in terms of the Tomonaga-Luttinger liquid theory, in which the elementary excitations are free bosons. To this approximation, the theory can be alternatively recast in terms of free fermions. In both approaches, small perturbations give rise to finite lifetimes of excitations. We evaluate the decay rate of fermionic excitations and show that it scales as the eighth power of energy, in contrast to the much faster decay of bosonic excitations. Our results can be tested experimentally by measuring the broadening of power-law features in the density structure factor or spectral functions.The low-energy properties of one-dimensional quantum liquids are commonly described in terms of the Tomonaga-Luttinger liquid theory, in which the elementary excitations are free bosons. To this approximation, the theory can be alternatively recast in terms of free fermions. In both approaches, small perturbations give rise to finite lifetimes of excitations. We evaluate the decay rate of fermionic excitations and show that it scales as the eighth power of energy, in contrast to the much faster decay of bosonic excitations. Our results can be tested experimentally by measuring the broadening of power-law features in the density structure factor or spectral functions.

arXiv:1308.3523Physical Review LettersDec 18, 201334 citations

Quasi-Fermi liquid behavior in a one-dimensional system of interacting spinless fermions

Joshua D. Baktay, A. V. Rozhkov, Adrian E. Feiguin +1 more

We present numerical evidence for a paradigm in one-dimensional interacting fermion systems, whose phenomenology has traits of both Luttinger liquids and Fermi liquids. This state, dubbed a quasi-Fermi liquid, possesses a discontinuity in its fermion occupation number at the Fermi momentum. The excitation spectrum presents particlelike quasiparticles and absence of holelike quasiparticles, giving rise instead to edge singularities. Such a state is realized in a one-dimensional spinless fermion lattice Hamiltonian by fine-tuning the interactions to a regime where they become irrelevant in the renormalization group sense. We show, using uniform infinite matrix products states and finite-entanglement scaling analysis, that the system ground state is characterized by a Luttinger parameter $K=1$ and a discontinuous jump in the fermion occupation number. We support the characterization with calculations of the spectral function that show a particle-hole asymmetry reflected in the existence of well-defined Landau quasiparticles above the Fermi level and edge singularities without the associated quasiparticles below. These results indicate that the quasi-Fermi liquid paradigm can be realized beyond the low-energy perturbative realm.We present numerical evidence for a paradigm in one-dimensional interacting fermion systems, whose phenomenology has traits of both Luttinger liquids and Fermi liquids. This state, dubbed a quasi-Fermi liquid, possesses a discontinuity in its fermion occupation number at the Fermi momentum. The excitation spectrum presents particlelike quasiparticles and absence of holelike quasiparticles, giving rise instead to edge singularities. Such a state is realized in a one-dimensional spinless fermion lattice Hamiltonian by fine-tuning the interactions to a regime where they become irrelevant in the renormalization group sense. We show, using uniform infinite matrix products states and finite-entanglement scaling analysis, that the system ground state is characterized by a Luttinger parameter $K=1$ and a discontinuous jump in the fermion occupation number. We support the characterization with calculations of the spectral function that show a particle-hole asymmetry reflected in the existence of well-defined Landau quasiparticles above the Fermi level and edge singularities without the associated quasiparticles below. These results indicate that the quasi-Fermi liquid paradigm can be realized beyond the low-energy perturbative realm.

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We examine the validity of Fermi-liquid theory in the low-density two-dimensional Hubbard model. From the ground-state energy we obtain the quasiparticle energy and the Landau interaction function. Then we prove that the quasiparticle-interaction function is finite and that the quasiparticle residue is nonzero for repulsive interactions. We also find, as a function of the excitation energy \ensuremath{\epsilon}, that the quasiparticle lifetime \ensuremath{\sim}1/${\mathrm{\ensuremath{\epsilon}}}^{2}$ln\ensuremath{\epsilon}. Hence, we conclude that Fermi-liquid theory holds in our model.We examine the validity of Fermi-liquid theory in the low-density two-dimensional Hubbard model. From the ground-state energy we obtain the quasiparticle energy and the Landau interaction function. Then we prove that the quasiparticle-interaction function is finite and that the quasiparticle residue is nonzero for repulsive interactions. We also find, as a function of the excitation energy \ensuremath{\epsilon}, that the quasiparticle lifetime \ensuremath{\sim}1/${\mathrm{\ensuremath{\epsilon}}}^{2}$ln\ensuremath{\epsilon}. Hence, we conclude that Fermi-liquid theory holds in our model.

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We study the dilute-gas expansion for a two-dimensional Fermi system with arbitrary short-range repulsive interactions. In contrast with the three-dimensional case, we find an unusual pole in the vertex part in the particle-particle channel, for all center-of-mass momenta q2 ${\mathit{k}}_{\mathit{F}}$ . We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2 ${\mathit{k}}_{\mathit{F}}$ . We study the effect of these bound states on the single-particle self-energy and find that the quasiparticles are well defined. Thus in the low-density regime, there is no breakdown of Fermi-liquid theory. In the Appendix we discuss the two-particle phase shift in the dilute Fermi gas, its connection with bound states, and the analogy with the potential scattering phase shift.We study the dilute-gas expansion for a two-dimensional Fermi system with arbitrary short-range repulsive interactions. In contrast with the three-dimensional case, we find an unusual pole in the vertex part in the particle-particle channel, for all center-of-mass momenta q2${\mathit{k}}_{\mathit{F}}$. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2${\mathit{k}}_{\mathit{F}}$. We study the effect of these bound states on the single-particle self-energy and find that the quasiparticles are well defined. Thus in the low-density regime, there is no breakdown of Fermi-liquid theory. In the Appendix we discuss the two-particle phase shift in the dilute Fermi gas, its connection with bound states, and the analogy with the potential scattering phase shift.

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Spectral properties of the two-dimensional Hubbard model

V. Zlatić, K. D. Schotte, Gudrun Schliecker

The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function ${\mathit{A}}_{\mathbf{p}}$ (\ensuremath{\omega}) is momentum dependent. Using ${\mathit{A}}_{\mathbf{p}}$ (\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function ${\mathit{A}}_{\mathbf{p}}$(\ensuremath{\omega}) is momentum dependent. Using ${\mathit{A}}_{\mathbf{p}}$(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.

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Extremely correlated Fermi liquids: Self-consistent solution of the second-order theory

Daniel Hansen, B. Sriram Shastry

We present detailed results from a recent microscopic theory of extremely correlated Fermi liquids, applied to the $t$ - $J$ model in two dimensions, developed recently by Shastry [Phys. Rev. Lett. 107, 056403 (2011); Phys. Rev. B 87, 125124 (2013)]. The second-order theory in the parameter $\ensuremath{\lambda}$, related to the density, is argued to be quantitatively valid in the overdoped regime for $0\ensuremath{\le}n\ensuremath{\lesssim}0.75$, with $n$ denoting the particle density. The calculation involves the self-consistent solution of equations for an auxiliary Fermi liquid Green's function and an adaptive spectral weight. We present numerical results at low as well as high $T$ , at various low to intermediate densities in the normal phase, using a minimal set of band parameters relevant to the cuprate superconductors. We display the momentum space occupation function ${m}_{k}$ , energy dispersion curves locating the peaks of spectral functions, the optical conductivity, relaxation rates for quasiparticles, and the electronic spectral functions on an absolute scale. The line shapes have an asymmetric shape and a broad background that is also seen in experiments, and our calculations validate approximate recent versions of the theory. The results also display the experimentally noted high-energy kink and provide an in-depth understanding of its origin and dependence on band parameters.We present detailed results from a recent microscopic theory of extremely correlated Fermi liquids, applied to the $t$-$J$ model in two dimensions, developed recently by Shastry [Phys. Rev. Lett. 107, 056403 (2011); Phys. Rev. B 87, 125124 (2013)]. The second-order theory in the parameter $\ensuremath{\lambda}$, related to the density, is argued to be quantitatively valid in the overdoped regime for $0\ensuremath{\le}n\ensuremath{\lesssim}0.75$, with $n$ denoting the particle density. The calculation involves the self-consistent solution of equations for an auxiliary Fermi liquid Green's function and an adaptive spectral weight. We present numerical results at low as well as high $T$, at various low to intermediate densities in the normal phase, using a minimal set of band parameters relevant to the cuprate superconductors. We display the momentum space occupation function ${m}_{k}$, energy dispersion curves locating the peaks of spectral functions, the optical conductivity, relaxation rates for quasiparticles, and the electronic spectral functions on an absolute scale. The line shapes have an asymmetric shape and a broad background that is also seen in experiments, and our calculations validate approximate recent versions of the theory. The results also display the experimentally noted high-energy kink and provide an in-depth understanding of its origin and dependence on band parameters.

arXiv:1211.0594Physical Review BJun 6, 201315 citations

T

-matrix approximation yield spurious deviations from Fermi-liquid behavior, which are particularly pronounced for a linearized dispersion relation.

K=1

and a discontinuous jump in the fermion occupation number. We support the characterization with calculations of the spectral function that show a particle-hole asymmetry reflected in the existence of well-defined Landau quasiparticles above the Fermi level and edge singularities without the associated quasiparticles below. These results indicate that the quasi-Fermi liquid paradigm can be realized beyond the low-energy perturbative realm.

{\mathit{k}}_{\mathit{F}}

. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2

{\mathit{k}}_{\mathit{F}}

. We study the effect of these bound states on the single-particle self-energy and find that the quasiparticles are well defined. Thus in the low-density regime, there is no breakdown of Fermi-liquid theory. In the Appendix we discuss the two-particle phase shift in the dilute Fermi gas, its connection with bound states, and the analogy with the potential scattering phase shift.

{\mathit{A}}_{\mathbf{p}}

(\ensuremath{\omega}) is momentum dependent. Using

{\mathit{A}}_{\mathbf{p}}

(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.

We present detailed results from a recent microscopic theory of extremely correlated Fermi liquids, applied to the

t

J

model in two dimensions, developed recently by Shastry [Phys. Rev. Lett. 107, 056403 (2011); Phys. Rev. B 87, 125124 (2013)]. The second-order theory in the parameter $\ensuremath{\lambda}$, related to the density, is argued to be quantitatively valid in the overdoped regime for $0\ensuremath{\le}n\ensuremath{\lesssim}0.75$, with

n

denoting the particle density. The calculation involves the self-consistent solution of equations for an auxiliary Fermi liquid Green's function and an adaptive spectral weight. We present numerical results at low as well as high

T

, at various low to intermediate densities in the normal phase, using a minimal set of band parameters relevant to the cuprate superconductors. We display the momentum space occupation function

{m}_{k}

, energy dispersion curves locating the peaks of spectral functions, the optical conductivity, relaxation rates for quasiparticles, and the electronic spectral functions on an absolute scale. The line shapes have an asymmetric shape and a broad background that is also seen in experiments, and our calculations validate approximate recent versions of the theory. The results also display the experimentally noted high-energy kink and provide an in-depth understanding of its origin and dependence on band parameters.