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arXiv:2303.13628

Phase Transitions in the Hubbard Model on the Square Lattice

Zhituo Wang

Published Mar 23, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2303.13628

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We study the low temperature properties of the two-dimensional weakly interacting Hubbard model on \ZZZ2\ZZZ^2\ZZZ2 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∈\RRT={ł∈\RRR,∣λlog⁡2(μ0T/C1)∣≤C2}ł\in\RR_T=\{ł\in\RRR,\vertλ\log^2(μ_0T/C_1)|\le C_2\}ł∈\RRT​={ł∈\RRR,∣λlog2(μ0​T/C1​)∣≤C2​} for any fixed temperature T>0T>0T>0, suggesting that there is a phase transition with critical temperature Tc=C1\m0exp⁡(−C21/2∣λ∣−1/2)T_c= \frac{C_1}{\m_0}\exp{(-C^{1/2}_2|λ|^{-1/2})}Tc​=\m0​C1​​exp(−C21/2​∣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.
In this paper we study the low temperature (T→0T\rightarrow 0T→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 μμμ, which is fixed to be 111. We prove that the two-point Schwinger's function is an analytic function of the coupling constant λλλ in the domain ∣λ∣⋅∣log⁡T∣2<1|λ|\cdot|\log T|^2<1∣λ∣⋅∣logT∣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.
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.
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 analyticity theorems in the coupling constant for the Hubbard model at half-filling. The model in a single renormalization group slice of index iii is proved to be analytic in λλλ for ∣λ∣≤c/i|λ| \le c/i∣λ∣≤c/i for some constant ccc, and the skeleton part of the model at temperature TTT (the sum of all graphs without two point insertions) is proved to be analytic in λλλ for ∣λ∣≤c/∣log⁡T∣2|λ| \le c/|\log T|^{2}∣λ∣≤c/∣logT∣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).
forbidding momenta close to the angles of the square shaped Fermi surface. By Renormalization Group methods we find a convergent expansion for the Schwinger function up to exponentially small temperatures. We prove that the system is not a Fermi liquid, but on the contrary it behaves like a Marginal Fermi liquid, a behaviour observed in the normal phase of high Tc superconductors. 1. Main results 1.1 Motivations. The notion of Fermi liquids, introduced by Landau, refers to a wide class of interacting fermionic systems whose thermodynamic properties (like the specific heat or the resistivity) are qualitatively the same of a gas of non interacting fermions. While there is an enormous number of metals having Fermi liquid behaviour, in recent times new materials has been found whose properties are qualitatively different. In particular the high-temperature superconducting materials (so anisotropic to be considered essentially bidimensional) in their normal phase have a non Fermi liquid behaviour, in striking contrast with previously known superconductors, which are Fermi liquids above the critical temperature.
We analyze the particle-hole symmetric two-dimensional Hubbard model on a square lattice starting from weak-to-moderate couplings by means of the field-theoretical renormalization group approach up to two-loop order. This method is essential in order to evaluate the effect of the momentum-resolved anomalous dimension $\ensuremath{\eta}(\mathbf{p})$ which arises in the normal phase of this model on the corresponding low-energy single-particle excitations. As a result, we find important indications pointing to the existence of a non-Fermi-liquid (NFL) regime at temperature $T\ensuremath{\rightarrow}0$ displaying a truncated Fermi surface (FS) for a doping range exactly in between the well-known antiferromagnetic insulating and the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave singlet superconducting phases. This NFL evolves as a function of doping into a correlated metal with a large FS before the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing susceptibility finally produces the dominant instability in the low-energy limit.