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arXiv:1002.0591 · Physical Review B

Superconductivity in the repulsive Hubbard model: An asymptotically exact weak-coupling solution

S. Raghu, Steven Allan Kivelson, Douglas J. Scalapino

Published Jun 4, 2010• 282 citations
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Abstract

We study the phase diagram of the Hubbard model in the limit where $U$, the onsite repulsive interaction, is much smaller than the bandwidth. We present an asymptotically exact expression for ${T}_{c}$, the superconducting transition temperature, in terms of the correlation functions of the noninteracting system which is valid for arbitrary densities so long as the interactions are sufficiently small. Our strategy for computing ${T}_{c}$ involves first integrating out all degrees of freedom having energy higher than an unphysical initial cutoff ${\ensuremath{\Omega}}_{0}$. Then, the renormalization group (RG) flows of the resulting effective action are computed and ${T}_{c}$ is obtained by determining the scale below which the RG flows in the Cooper channel diverge. We prove that ${T}_{c}$ is independent of ${\ensuremath{\Omega}}_{0}$. Using this method, we find a variety of unconventional superconducting ground states in two- and three-dimensional lattice systems, and present explicit results for ${T}_{c}$ and pairing symmetries as a function of the electron concentration.

DOI
10.1103/physrevb.81.224505

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We study the phase diagram of the Hubbard model in the limit where UUU, the onsite repulsive interaction, is much smaller than the bandwidth. We present an asymptotically exact expression for Tc{T}_{c}Tc​, the superconducting transition temperature, in terms of the correlation functions of the noninteracting system which is valid for arbitrary densities so long as the interactions are sufficiently small. Our strategy for computing Tc{T}_{c}Tc​ involves first integrating out all degrees of freedom having energy higher than an unphysical initial cutoff \ensuremathΩ0{\ensuremath{\Omega}}_{0}\ensuremathΩ0​. Then, the renormalization group (RG) flows of the resulting effective action are computed and Tc{T}_{c}Tc​ is obtained by determining the scale below which the RG flows in the Cooper channel diverge. We prove that Tc{T}_{c}Tc​ is independent of \ensuremathΩ0{\ensuremath{\Omega}}_{0}\ensuremathΩ0​. Using this method, we find a variety of unconventional superconducting ground states in two- and three-dimensional lattice systems, and present explicit results for Tc{T}_{c}Tc​ and pairing symmetries as a function of the electron concentration.
In superconducting phases, repulsive interactions between the electrons usually lead to higher-angular momentum Cooper pairing. Here, the authors employ the weak-coupling renormalization group approach to study unconventional superconducting phases emerging in the extended, repulsive Hubbard model on paradigmatic two-dimensional lattices. Among their findings are extended regions of chiral topological superconductivity on the square, triangular, and honeycomb lattices. They provide detailed superconducting phase diagrams as well as the coupling strengths that quantify the corresponding critical temperatures depending on the band-structure parameters, band filling, and interaction parameters. The importance of the resolution and accuracy of the numerical integration for the method is discussed and a way to higher numerical efficiency provided.
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We formulate the exact Wilsonian renormalization group for a system of interacting fermions on a lattice. The flow equations for all vertices of the Wilson effective action are expressed in form of the Polchinski equation. We apply this method to the Hubbard model on a square lattice using both zero- and finite- temperature methods. Truncating the effective action at the sixth term in fermionic variables we obtain the one-loop functional renormalization equations for the effective interaction. We find the temperature of the instability Tc^{RG} as function of doping. We calculate furthermore the renormalization of the angle-resolved correlation functions for the superconductivity (SC) and for the antiferromagnetism (AF). The dominant component of the SC correlations is of the type d while the AF fluctuations are of the type s Following the strength of both SC and AF fluctuation along the instability line we obtain the phase diagram. The temperature Tc^{RG} can be identified with the crossover temperature T{co} found in the underdoped regime of the high-temperature superconductors, while in the overdoped regime Tc^{RG} corresponds to the superconducting critical temperature.
We present a functional renormalization group analysis of superconductivity in the ground state of the attractive Hubbard model on a square lattice. Spontaneous symmetry breaking is treated in a purely fermionic setting via anomalous propagators and anomalous effective interactions. In addition to the anomalous interactions arising already in the reduced BCS model, effective interactions with three incoming legs and one outgoing leg (and vice versa) occur. We accomplish their integration into the usual diagrammatic formalism by introducing a Nambu matrix for the effective interactions. From a random-phase approximation generalized through use of this matrix we conclude that the impact of the 3+1 effective interactions is limited, especially considering the effective interactions which are important for the determination of the order parameter. The exact hierarchy of flow equations for one-particle irreducible vertex functions is truncated on the two-particle level, with higher-order self-energy corrections included in a scheme proposed by Katanin (2004 Phys. Rev. B 70 115109). Using a parametrization of effective interactions by patches in momentum space, the flow equations can be integrated numerically to the lowest scales without encountering divergences. Momentum-shell as well as interaction-flow cutoff functions are used, including a small external field or a large external field and a counterterm, respectively. Both approaches produce momentum-resolved order parameter values directly from the microscopic model. The size of the superconducting gap is in reasonable agreement with expectations from other studies.
We obtain a complete and numerically exact in the weak-coupling limit $(U\ensuremath{\rightarrow}0)$ ground-state phase diagram of the repulsive fermionic Hubbard model on the square lattice for filling factors 0<n<20<n<20<n<2 and next-nearest-neighbor hopping amplitudes $0\ensuremath{\le}{t}^{\ensuremath{'}}\ensuremath{\le}0.5$. Phases are distinguished by the symmetry and the number of nodes of the superfluid order parameter. The phase diagram is richer than may be expected and typically features states with a high---higher than that of the fundamental mode of the corresponding irreducible representation---number of nodes. The effective coupling strength in the Cooper channel $\ensuremath{\lambda}$, which determines the critical temperature Tc{T}_{c}Tc​ of the superfluid transition, is calculated in the whole parameter space and regions with high values of $\ensuremath{\lambda}$ are identified. It is shown that besides the expected increase of $\ensuremath{\lambda}$ near the Van Hove singularity line, joining the ferromagnetic and antiferromagnetic points, another region with high values of $\ensuremath{\lambda}$ can be found at quarter filling and ${t}^{\ensuremath{'}}=0.5$ due to the presence of a line of nesting at ${t}^{\ensuremath{'}}\ensuremath{\ge}0.5$. The results can serve as benchmarks for controlled nonperturbative methods and guide the ongoing search for high-Tc{T}_{c}Tc​ superconductivity in the Hubbard model.
The repulsive two-dimensional Hubbard model is analysed within a functional renormalisation group (RG) approach. The flow equation for the effective interactions is evaluated on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyse the nature of the instabilities signalled by the diverging interactions, we compute the flow of the superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find spin density wave instabilities or d-wave superconductivity as leading instability close to half-filling. This calculation establishes the existence of d-wave superconductivity in the Hubbard model beyond doubt.