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

Stéphane Afchain

Published Jan 1, 2005• 0 citations

Abstract

We prove that the Hubbard model at finite temperature T and half-filling is analytic in its coupling constant λ for |λ| ≤ c/| log T |2, where c is some numerical constant. We also bound the self-energy and prove that the Hubbard model at half-filling is not a Fermi liquid (in the mathematically precise sense of Salmhofer), modulo a simple lower bound on the first non-trivial self-energy graph, which will be published in a companion paper.

Related papers

The two dimensional Hubbard Model at half-filling: I. Convergent Contributions

Vincent Rivasseau

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

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

Zhituo Wang

arXiv:1907.12553Jul 29, 20190 citations

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

Vincent Rivasseau, Zhituo Wang

arXiv:2108.10415Aug 23, 20210 citations

Phase Transitions in the Hubbard Model on the Square Lattice

Zhituo Wang

arXiv:2303.13628Mar 23, 20230 citations

Fermi liquid behavior in the 2D Hubbard model at low temperatures

G. Benfatto, Alessandro Giuliani, Vieri Mastropietro

Jan 1, 200523 citations

Fermi Liquid Behavior in the 2D Hubbard Model at Low Temperatures

G. Benfatto, Alessandro Giuliani, V. Mastropietro

arXiv:cond-mat/0507686Annales Henri PoincaréAug 1, 200638 citations

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

Stéphane Afchain

Published Jan 1, 2005• 0 citations

OpenAlex

Abstract

Related papers

The two dimensional Hubbard Model at half-filling: I. Convergent Contributions

Vincent Rivasseau

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).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).

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

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

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

Vincent Rivasseau, Zhituo Wang

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 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.

arXiv:2108.10415Aug 23, 20210 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.

arXiv:2303.13628Mar 23, 20230 citations

Fermi liquid behavior in the 2D Hubbard model at low temperatures

G. Benfatto, Alessandro Giuliani, Vieri Mastropietro

Abstract. We prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures. In particular we show that the wave function renormalization is an order 1 constant and essentially temperature independent in the considered range of temperatures and that the interacting Fermi surface is a regular convex curve. This result is obtained by deriving a convergent expansion (which is not a power series) for the two point Schwinger function by Renormalization Group methods and proving at each order suitable power counting improvements due to the convexity of the interacting Fermi surface. Convergence follows from determinant bounds for the fermionic expectations.Abstract. We prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures. In particular we show that the wave function renormalization is an order 1 constant and essentially temperature independent in the considered range of temperatures and that the interacting Fermi surface is a regular convex curve. This result is obtained by deriving a convergent expansion (which is not a power series) for the two point Schwinger function by Renormalization Group methods and proving at each order suitable power counting improvements due to the convexity of the interacting Fermi surface. Convergence follows from determinant bounds for the fermionic expectations.

Jan 1, 200523 citations

Fermi Liquid Behavior in the 2D Hubbard Model at Low Temperatures

G. Benfatto, Alessandro Giuliani, V. Mastropietro

We prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures. In particular we show that the wave function renormalization is an order 1 constant and essentially temperature independent in the considered range of temperatures and that the interacting Fermi surface is a regular convex curve. This result is obtained by deriving a convergent expansion (which is not a power series) for the two point Schwinger function by Renormalization Group methods and proving at each order suitable power counting improvements due to the convexity of the interacting Fermi surface. Convergence follows from determinant bounds for the fermionic expectations.We prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures. In particular we show that the wave function renormalization is an order 1 constant and essentially temperature independent in the considered range of temperatures and that the interacting Fermi surface is a regular convex curve. This result is obtained by deriving a convergent expansion (which is not a power series) for the two point Schwinger function by Renormalization Group methods and proving at each order suitable power counting improvements due to the convexity of the interacting Fermi surface. Convergence follows from determinant bounds for the fermionic expectations.

arXiv:cond-mat/0507686Annales Henri PoincaréAug 1, 200638 citations

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).

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 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.

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.