Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Constructive analysis of two dimensional Fermi systems at finite temperature

Gui‐Lu Long

Published Jan 1, 2013• 0 citations
Publisher

Abstract

We consider a dilute Fermion system in continuum two spatial dimensions with short-range interaction. We prove nonperturbatively that at low temperature the renormalized perturbation expansion has non-zero radius of convergence. The convergence radius shrinks when the energy scale goes to the infrared cutoff. The shrinking rate of the convergence radius is established to be dependent of the sign of the coupling constant by a detailed analysis of the so-called ladder contributions. We prove further that the self-energy of the model is uniformly of C1, in the analytic domain of the theory. The proofs are based on renormalization of the Fermi surface and multiscale analysis employing mathematical renormalization group technique. Tree expansion is introduced to reorganize perturbation expansion nicely. Finally we apply these techniques to construct a half-filled Hubbard model on honeycomb bilayer lattice with local interaction.

DOI
10.11588/heidok.00014947

Related papers

Rigorous Proof of Fermi Liquid Behavior for Jellium Two-Dimensional Interacting Fermions

Margherita Disertori, Vincent Rivasseau

Using the method of continuous constructive renormalization group around the Fermi surface, we prove that a jellium two-dimensional interacting system of fermions at low temperature T is analytic in the coupling constant lambda for |lambda| |logT|</=K, where K is some numerical constant and T is the temperature. In that range of parameters, the first and second derivatives of the self-energy remain bounded, a behavior seen in Fermi liquids but, in particular, not in Luttinger liquids. Our results also prove that in 2D any transition temperature must be nonperturbative in the coupling constant, a result expected on physical grounds. The proof exploits specific momentum conservation rules in two dimensions.

arXiv:cond-mat/9912368•Physical Review Letters•Jul 10, 2000•21 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.

arXiv:cond-mat/0507686•Annales Henri Poincaré•Aug 1, 2006•38 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.

arXiv:2303.13628•Mar 23, 2023•0 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.

•Jan 1, 2005•23 citations

Low temperature analysis of two dimensional Fermi systems with symmetric Fermi surface

G. Benfatto, Alessandro Giuliani, Vieri Mastropietro

We prove the convergence of the perturbative expansion, based on Renormalization Group, of the two point Schwinger function of a system of weakly interacting fermions in d = 2, with symmetric Fermi surface and up to exponentially small temperatures, close to the expected onset of superconductivity.

•Jan 1, 2002•12 citations

Renormalization group analysis of multi-band many-electron systems at half-filling

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]).

arXiv:1405.0655•May 4, 2014•3 citations
We consider a dilute Fermion system in continuum two spatial dimensions with short-range interaction. We prove nonperturbatively that at low temperature the renormalized perturbation expansion has non-zero radius of convergence. The convergence radius shrinks when the energy scale goes to the infrared cutoff. The shrinking rate of the convergence radius is established to be dependent of the sign of the coupling constant by a detailed analysis of the so-called ladder contributions. We prove further that the self-energy of the model is uniformly of C1, in the analytic domain of the theory. The proofs are based on renormalization of the Fermi surface and multiscale analysis employing mathematical renormalization group technique. Tree expansion is introduced to reorganize perturbation expansion nicely. Finally we apply these techniques to construct a half-filled Hubbard model on honeycomb bilayer lattice with local interaction.
Using the method of continuous constructive renormalization group around the Fermi surface, we prove that a jellium two-dimensional interacting system of fermions at low temperature T is analytic in the coupling constant lambda for |lambda| |logT|</=K, where K is some numerical constant and T is the temperature. In that range of parameters, the first and second derivatives of the self-energy remain bounded, a behavior seen in Fermi liquids but, in particular, not in Luttinger liquids. Our results also prove that in 2D any transition temperature must be nonperturbative in the coupling constant, a result expected on physical grounds. The proof exploits specific momentum conservation rules in two dimensions.
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 study the low temperature properties of the two-dimensional weakly interacting Hubbard model on $\ZZZ^2$ with renormalized chemical potential μ=2−μ0μ=2-μ_0μ=2−μ0​, μ0=10−10μ_0=10^{-10}μ0​=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 łłł 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>0T>0T>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 C1,C2C_1, C_2C1​,C2​ are positive constants independent of TTT and łłł. 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.
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.
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]).