arXiv:2108.10415

Honeycomb Hubbard Model at van Hove Filling II: Lower Bounds of the Self-Energy

Vincent Rivasseau, Zhituo Wang

Published Aug 23, 2021• 0 citations

Abstract

In this paper we complete the proof that the honeycomb Hubbard model with renormalized chemical potential $μ=1$ is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing the necessary lower bounds for second derivatives of the self-energy w.r.t. the external momentum.

DOI: 10.48550/arxiv.2108.10415

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arXiv:2108.10415

Honeycomb Hubbard Model at van Hove Filling II: Lower Bounds of the Self-Energy

Vincent Rivasseau, Zhituo Wang

Published Aug 23, 2021• 0 citations

arXiv HTML PDF

Abstract

In this paper we complete the proof that the honeycomb Hubbard model with renormalized chemical potential

μ=1

is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing the necessary lower bounds for second derivatives of the self-energy w.r.t. the external momentum.

DOI: 10.48550/arxiv.2108.10415

Related papers

Non-Fermi Liquid Behaviors in the Hubbard model on the Honeycomb lattice

Zhituo Wang

In this paper we study the low temperature ( $T\rightarrow 0$ ) behaviors of the weakly interacting Hubbard model on the honeycomb lattice with a bare chemical potential, which takes values in a small neighborhood of the renormalized chemical potential $μ$ , which is fixed to be $1$ . We prove that the two-point Schwinger's function is an analytic function of the coupling constant $λ$ in the domain $|λ|\cdot|\log T|^2<1$ . But the ground state of this model is not a Fermi liquid, because analytic properties of self-energy don't satisfy the Salmhofer's criterion on the Fermi liquids at finite temperature.In this paper we study the low temperature ($T\rightarrow 0$) behaviors of the weakly interacting Hubbard model on the honeycomb lattice with a bare chemical potential, which takes values in a small neighborhood of the renormalized chemical potential $μ$, which is fixed to be $1$. We prove that the two-point Schwinger's function is an analytic function of the coupling constant $λ$ in the domain $|λ|\cdot|\log T|^2<1$. But the ground state of this model is not a Fermi liquid, because analytic properties of self-energy don't satisfy the Salmhofer's criterion on the Fermi liquids at finite temperature.

arXiv:1907.12553Jul 29, 20190 citations

Etude du modèle de Hubbard bidimensionnel à demi remplissage par des méthodes constructives.

Stéphane Afchain

Jan 1, 20050 citations

Phase Transitions in the Hubbard Model on the Square Lattice

Zhituo Wang

We study the low temperature properties of the two-dimensional weakly interacting Hubbard model on $\ZZZ^2$ with renormalized chemical potential $μ=2-μ_0$ , $μ_0=10^{-10}$ fixed, in which case the Fermi surface is close to a perfect square. Using fermionic functional integrals, cluster expansions and rigorous renormalization group analysis, we prove that the perturbation series for the two-point Schwinger function is analytic in the coupling constant $ł$ in the domain $ł\in\RR_T=\{ł\in\RRR,\vertλ\log^2(μ_0T/C_1)|\le C_2\}$ for any fixed temperature $T>0$ , suggesting that there is a phase transition with critical temperature $T_c= \frac{C_1}{\m_0}\exp{(-C^{1/2}_2|λ|^{-1/2})}$. Here $C_1, C_2$ are positive constants independent of $T$ and $ł$ . We also prove that the second derivative of the momentum space self-energy function w.r.t. the external momentum is not uniformly bounded, suggesting that this model is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer. This result can be viewed as a first step towards rigorous study of the Fermi liquid-non Fermi liquid crossover phenomenon.We study the low temperature properties of the two-dimensional weakly interacting Hubbard model on $\ZZZ^2$ with renormalized chemical potential $μ=2-μ_0$, $μ_0=10^{-10}$ fixed, in which case the Fermi surface is close to a perfect square. Using fermionic functional integrals, cluster expansions and rigorous renormalization group analysis, we prove that the perturbation series for the two-point Schwinger function is analytic in the coupling constant $ł$ in the domain $ł\in\RR_T=\{ł\in\RRR,\vertλ\log^2(μ_0T/C_1)|\le C_2\}$ for any fixed temperature $T>0$, suggesting that there is a phase transition with critical temperature $T_c= \frac{C_1}{\m_0}\exp{(-C^{1/2}_2|λ|^{-1/2})}$. Here $C_1, C_2$ are positive constants independent of $T$ and $ł$. We also prove that the second derivative of the momentum space self-energy function w.r.t. the external momentum is not uniformly bounded, suggesting that this model is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer. This result can be viewed as a first step towards rigorous study of the Fermi liquid-non Fermi liquid crossover phenomenon.

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arXiv:cond-mat/0107118Jul 5, 20014 citations

Low-energy effective theory at a quantum critical point of the two-dimensional Hubbard model: Mean-field analysis

Kambis Veschgini, Manfred Salmhofer B

We complement previous functional renormalization group (fRG) studies of the two-dimensional Hubbard model by mean-field calculations. The focus falls on Van Hove filling and the hopping amplitude ${t}^{\ensuremath{'}}/t=0.341$. The fRG data suggest a quantum critical point (QCP) in this region and in its vicinity a singular fermionic self-energy $\mathrm{Im}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})/\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}{\left|\ensuremath{\omega}\right|}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\approx}0.26$ [K-U. Giering and M. Salmhofer, Self-energy flows in the two-dimensional repulsive Hubbard model, Phys. Rev. B 86, 245122 (2012)]. Here we start a more detailed investigation of this QCP using a bosonic formulation for the effective action, where the bosons couple to the order parameter fields. To this end, we use the channel decomposition of the fermionic effective action developed in C. Husemann and M. Salmhofer, Efficient Parametrization of the Vertex Function, Omega-Scheme, and the (t, t')-Hubbard Model at Van Hove Filling, Phys. Rev. B 79, 195125 (2009), which allows us to perform Hubbard-Stratonovich transformations for all relevant order parameter fields at any given energy scale $\mathrm{\ensuremath{\Omega}}$. We stop the flow at a scale $\mathrm{\ensuremath{\Omega}}$ where the correlations of the order parameter field are already pronounced, but the flow is still regular, and derive the effective boson theory. It contains $d$ -wave superconducting, magnetic, and density-density interactions. We analyze the resulting phase diagram in the mean-field approximation. We show that the singular fermionic self-energy suppresses gap formation both in the superconducting and magnetic channel already at the mean-field level, thus rounding a first-order transition (without self-energy) to a quantum phase transition (with self-energy). We give a simple effective model that shows the generality of this effect. In the two-dimensional Hubbard model, the effective density-density interaction is peaked at a nonzero frequency, so that solving the mean-field equations already involves a functional equation instead of simply a matrix equation (on a technical level, similar to incommensurate phases). Within a certain approximation, we show that such an interaction leads to a short quasiparticle lifetime.We complement previous functional renormalization group (fRG) studies of the two-dimensional Hubbard model by mean-field calculations. The focus falls on Van Hove filling and the hopping amplitude ${t}^{\ensuremath{'}}/t=0.341$. The fRG data suggest a quantum critical point (QCP) in this region and in its vicinity a singular fermionic self-energy $\mathrm{Im}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})/\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}{\left|\ensuremath{\omega}\right|}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\approx}0.26$ [K-U. Giering and M. Salmhofer, Self-energy flows in the two-dimensional repulsive Hubbard model, Phys. Rev. B 86, 245122 (2012)]. Here we start a more detailed investigation of this QCP using a bosonic formulation for the effective action, where the bosons couple to the order parameter fields. To this end, we use the channel decomposition of the fermionic effective action developed in C. Husemann and M. Salmhofer, Efficient Parametrization of the Vertex Function, Omega-Scheme, and the (t, t')-Hubbard Model at Van Hove Filling, Phys. Rev. B 79, 195125 (2009), which allows us to perform Hubbard-Stratonovich transformations for all relevant order parameter fields at any given energy scale $\mathrm{\ensuremath{\Omega}}$. We stop the flow at a scale $\mathrm{\ensuremath{\Omega}}$ where the correlations of the order parameter field are already pronounced, but the flow is still regular, and derive the effective boson theory. It contains $d$-wave superconducting, magnetic, and density-density interactions. We analyze the resulting phase diagram in the mean-field approximation. We show that the singular fermionic self-energy suppresses gap formation both in the superconducting and magnetic channel already at the mean-field level, thus rounding a first-order transition (without self-energy) to a quantum phase transition (with self-energy). We give a simple effective model that shows the generality of this effect. In the two-dimensional Hubbard model, the effective density-density interaction is peaked at a nonzero frequency, so that solving the mean-field equations already involves a functional equation instead of simply a matrix equation (on a technical level, similar to incommensurate phases). Within a certain approximation, we show that such an interaction leads to a short quasiparticle lifetime.

arXiv:1806.08930Physical review. B./Physical review. BDec 13, 20184 citations

Second-order self-energy of the Hubbard Hamiltonian: Absence of quasiparticle excitations near half-filling

J. Galán, José Antonio Vergés, A. Martín‐Rodero

The electron self-energy Sigma(k,\ensuremath{\omega}) corresponding to the one-band Hubbard Hamiltonian on clusters of the square lattice has been obtained as a function of the band filling up to second order in the interaction parameter U. The k dependence of the self-energy is completely taken into account. As a consequence, the imaginary part of the self-energy shows a linear behavior for fillings close to one electron per site whereas for large doping rates the quadratic \ensuremath{\omega} dependence characterizing Fermi liquids is recovered. The origin of this behavior has been investigated analytically: linear terms associated with nesting are shown to exist in any finite-dimensional lattice although its numerical relevance decreases very rapidly for space dimensionality larger than two. Implications of these results on the renormalization factor Z have been analyzed. From a practical point of view, we conclude that a standard second-order perturbative treatment is quantitatively precise for U values of the on-site Coulomb interaction smaller than the bandwidth of the noninteracting spectrum.The electron self-energy Sigma(k,\ensuremath{\omega}) corresponding to the one-band Hubbard Hamiltonian on clusters of the square lattice has been obtained as a function of the band filling up to second order in the interaction parameter U. The k dependence of the self-energy is completely taken into account. As a consequence, the imaginary part of the self-energy shows a linear behavior for fillings close to one electron per site whereas for large doping rates the quadratic \ensuremath{\omega} dependence characterizing Fermi liquids is recovered. The origin of this behavior has been investigated analytically: linear terms associated with nesting are shown to exist in any finite-dimensional lattice although its numerical relevance decreases very rapidly for space dimensionality larger than two. Implications of these results on the renormalization factor Z have been analyzed. From a practical point of view, we conclude that a standard second-order perturbative treatment is quantitatively precise for U values of the on-site Coulomb interaction smaller than the bandwidth of the noninteracting spectrum.

Physical review. B, Condensed matterNov 1, 199315 citations

In this paper we complete the proof that the honeycomb Hubbard model with renormalized chemical potential

μ=1

is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing the necessary lower bounds for second derivatives of the self-energy w.r.t. the external momentum.

In this paper we study the low temperature (

T\rightarrow 0

) behaviors of the weakly interacting Hubbard model on the honeycomb lattice with a bare chemical potential, which takes values in a small neighborhood of the renormalized chemical potential

μ

, which is fixed to be

1

. We prove that the two-point Schwinger's function is an analytic function of the coupling constant

λ

in the domain

|λ|\cdot|\log T|^2<1

. But the ground state of this model is not a Fermi liquid, because analytic properties of self-energy don't satisfy the Salmhofer's criterion on the Fermi liquids at finite temperature.

We study the low temperature properties of the two-dimensional weakly interacting Hubbard model on $\ZZZ^2$ with renormalized chemical potential

μ=2-μ_0

μ_0=10^{-10}

fixed, in which case the Fermi surface is close to a perfect square. Using fermionic functional integrals, cluster expansions and rigorous renormalization group analysis, we prove that the perturbation series for the two-point Schwinger function is analytic in the coupling constant

ł

in the domain $ł\in\RR_T=\{ł\in\RRR,\vertλ\log^2(μ_0T/C_1)|\le C_2\}$ for any fixed temperature

T>0

, suggesting that there is a phase transition with critical temperature $T_c= \frac{C_1}{\m_0}\exp{(-C^{1/2}_2|λ|^{-1/2})}$. Here

C_1, C_2

are positive constants independent of

T

and

ł

. We also prove that the second derivative of the momentum space self-energy function w.r.t. the external momentum is not uniformly bounded, suggesting that this model is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer. This result can be viewed as a first step towards rigorous study of the Fermi liquid-non Fermi liquid crossover phenomenon.

We prove analyticity theorems in the coupling constant for the Hubbard model at half-filling. The model in a single renormalization group slice of index

i

is proved to be analytic in

λ

for

|λ| \le c/i

for some constant

c

, and the skeleton part of the model at temperature

T

(the sum of all graphs without two point insertions) is proved to be analytic in

λ

for

|λ| \le c/|\log T|^{2}

. These theorems are necessary steps towards proving that the Hubbard model at half-filling is {\it not} a Fermi liquid (in the mathematically precise sense of Salmhofer).

d

-wave superconducting, magnetic, and density-density interactions. We analyze the resulting phase diagram in the mean-field approximation. We show that the singular fermionic self-energy suppresses gap formation both in the superconducting and magnetic channel already at the mean-field level, thus rounding a first-order transition (without self-energy) to a quantum phase transition (with self-energy). We give a simple effective model that shows the generality of this effect. In the two-dimensional Hubbard model, the effective density-density interaction is peaked at a nonzero frequency, so that solving the mean-field equations already involves a functional equation instead of simply a matrix equation (on a technical level, similar to incommensurate phases). Within a certain approximation, we show that such an interaction leads to a short quasiparticle lifetime.