arXiv:cond-mat/0107118

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

Vincent Rivasseau

Published Jul 5, 2001• 4 citations

Abstract

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).

DOI: 10.48550/arxiv.cond-mat/0107118

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arXiv:cond-mat/0107118

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

Vincent Rivasseau

Published Jul 5, 2001• 4 citations

arXiv HTML PDF

Abstract

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).

DOI: 10.48550/arxiv.cond-mat/0107118

Related papers

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

Stéphane Afchain

Jan 1, 20050 citations

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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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Yohei Kashima

Renormalization group analysis for multi-band many-electron systems at half-filling at positive temperature is presented. The analysis includes the Matsubara ultra-violet integration and the infrared integration around the zero set of the dispersion relation. The multi-scale integration schemes are implemented in a finite-dimensional Grassmann algebra indexed by discrete position-time variables. In order that the multi-scale integrations are justified inductively, various scale-dependent estimates on Grassmann polynomials are established. We apply these theories in practice to prove that for the half-filled Hubbard model with nearest-neighbor hopping on a square lattice the infinite-volume, zero-temperature limit of the free energy density exists as an analytic function of the coupling constant in a neighborhood of the origin if the system contains the magnetic flux pi (mod 2 pi) per plaquette and 0 (mod 2 pi) through the large circles around the periodic lattice. Combined with Lieb's result on the flux phase problem ([Lieb, E. H., Phys. Rev. Lett. 73 (1994), 2158]), this theorem implies that the minimum free energy density of the flux phase problem converges to an analytic function of the coupling constant in the infinite-volume, zero-temperature limit. The proof of the theorem is based on a four-band formulation of the model Hamiltonian and an extension of Giuliani-Mastropietro's renormalization designed for the half-filled Hubbard model on the honeycomb lattice ([Giuliani, A. and V. Mastropietro, Commun. Math. Phys. 293 (2010), 301-346]).Renormalization group analysis for multi-band many-electron systems at half-filling at positive temperature is presented. The analysis includes the Matsubara ultra-violet integration and the infrared integration around the zero set of the dispersion relation. The multi-scale integration schemes are implemented in a finite-dimensional Grassmann algebra indexed by discrete position-time variables. In order that the multi-scale integrations are justified inductively, various scale-dependent estimates on Grassmann polynomials are established. We apply these theories in practice to prove that for the half-filled Hubbard model with nearest-neighbor hopping on a square lattice the infinite-volume, zero-temperature limit of the free energy density exists as an analytic function of the coupling constant in a neighborhood of the origin if the system contains the magnetic flux pi (mod 2 pi) per plaquette and 0 (mod 2 pi) through the large circles around the periodic lattice. Combined with Lieb's result on the flux phase problem ([Lieb, E. H., Phys. Rev. Lett. 73 (1994), 2158]), this theorem implies that the minimum free energy density of the flux phase problem converges to an analytic function of the coupling constant in the infinite-volume, zero-temperature limit. The proof of the theorem is based on a four-band formulation of the model Hamiltonian and an extension of Giuliani-Mastropietro's renormalization designed for the half-filled Hubbard model on the honeycomb lattice ([Giuliani, A. and V. Mastropietro, Commun. Math. Phys. 293 (2010), 301-346]).

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C. Bourbonnais, Hugues Nélisse, A. Reid +1 more

Recent computer simulations performed on the one-dimensional Hubbard model with a new algorithm developed by Sorella et al. confirm the absence of a Fermi surface in this model at half filling. However, these same simulations suggest the existence of a Fermi surface at zero temperature away from half filling, contrary to expectations based on renormalization-group analysis. The results found here, using a more standard algorithm and a simple low-temperature extrapolation, appear consistent with the above-mentioned numerical results in the half-filled and quarter-filled band cases. However, it is argued that the discontinuity at the Fermi level found in all these simulations is likely to come from the zero-temperature finite-size dependence of the quasiparticle weight Z, which is also discussed here.Recent computer simulations performed on the one-dimensional Hubbard model with a new algorithm developed by Sorella et al. confirm the absence of a Fermi surface in this model at half filling. However, these same simulations suggest the existence of a Fermi surface at zero temperature away from half filling, contrary to expectations based on renormalization-group analysis. The results found here, using a more standard algorithm and a simple low-temperature extrapolation, appear consistent with the above-mentioned numerical results in the half-filled and quarter-filled band cases. However, it is argued that the discontinuity at the Fermi level found in all these simulations is likely to come from the zero-temperature finite-size dependence of the quasiparticle weight Z, which is also discussed here.

Physical review. B, Condensed matterAug 1, 198919 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).

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.