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Bulletin of the American Physical Society

Finite-temperature superconducting correlations in the square lattice Hubbard model

Ehsan Khatami, Richard T. Scalettar, Rajiv R. P. Singh

Published Mar 5, 2014• 0 citations
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Superconductivity and magnetism in strongly correlated electronic systems

M. J. Sampaio

In this thesis, symmetry and related transformation laws under a suitable unitary transformation are used in the study of the superconducting and magnetic properties of strongly correlated electronic systems. Such a study profits from a recently found global SO(3)×SO(3)×U(1) = [SO(4)×U(1)]/Z2 symmetry in the Hubbard model on a bipartite lattice. The introduction of the square-lattice quantum liquid relies on a general description of the Hubbard model consistent with its exact global SO(3) × SO(3) × U(1) symmetry. Such a quantum liquid is expected to play the same role for many-electron problems with short-range interactions on a square lattice, as a Fermi liquid for three-dimensional isotropic metals. It refers to the Hubbard model on the square lattice in the subspace spanned by the ground state, and the excited states generated by application onto it of one- and two-electron operators. For such a square-lattice quantum liquid only the charge c fermions and spin-neutral two-spinon s1 fermions play an active role. Part of the studies of this thesis use an effective superconductivity theory according to which, for zero spin density m = 0 and the range 0 < x < x∗ of finite hole concentrations of a short-range spin ordered phase, the ground state of the square-lattice quantum liquid of c and s1 fermions with residual interactions is superconducting. (For U/4t > ui ≈ 3/4 the critical hole concentration x∗ increases continuously upon increasing the U/4t value from x∗ = 1/√2 ≈ 0.225 at U/4t = ui to x∗ = 1/ ≈ 0.318 for U/4t → ∞.) Superconductivity emerges naturally from the effects of the residual interactions of the c and s1 fermions and as a side product of the short-range spin correlations. The phase of the c fermion pairs reads = 0 + 1 where the fluctuations of the phases 0 and 1 become large for x → 0 and x → x∗, respectively. For 0 < x < x∗ and x = 0, the symmetry of the action for the phase is a global superconducting U(1) symmetry and a local compact gauge symmetry, respectively. The former U(1) symmetry is identified with the U(1) symmetry of the Hubbard-model global SO(3) × SO(3) × U(1) = [SO(4) × U(1)]/Z2 symmetry. The amplitudes g = g0 g1, g0 = |heiθ0i|, and g1 = |heiθ1i|, their zero-temperature values ˘g ≈ (x/x∗)(1−x/x∗), ˘g0 ≈ (x/x∗), and ˘g1 ≈ (1−x/x∗) and the fluctuations of the phase , play an important role in the physics of the square-lattice quantum liquid. Together with the energy scale 2 0, which gives the maximum magnitude of the s1 fermion spinon pairing energy, they fully control the order parameters of both the short-range spin and long-range superconducting correlations. Indeed, for temperatures T < Tc, the short-range spin ordered phase has both a macroscopic quantum phase-coherent pair superconducting order so that |heiθi| > 0, and a short-range spin order, whereas for Tc < T < T∗ only the latter order prevails yet there remain c fermion pairing correlations such that |heiθ1i| > 0 but heiθ0i = 0. Here the critical temperature Tc and the crossover pseudogap temperature T∗ are fully controlled by the absolute value of the order parameters of the two above orders, respectively. However, the square-lattice quantum liquid alone does not describe quantitatively the unusual properties of the hole-doped cuprates. To fulfill such a goal we profit from a suitable theory according to which the interplay of electronic correlations described by the square-lattice quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions with intrinsic disorder, is behind the unusual properties of the holedoped hight-temperature cuprate superconductors. By profiting from such an interplay, an excellent quantitative agreement with experiments for the physical quantities of several families of hole-doped cuprates is reached. That includes universal properties such as a dome-like critical-temperature hole-concentration dependence and normal-state linear-T resistivity near optimal doping, extending over a wide temperature window. The results confirm a coexisting two-gap scenario associated with an incoherent two-spinon pairing and a coherent twoc- fermion pairing, respectively. While most of our studies focus on the superconducting and magnetic properties of the square-lattice quantum liquid associated with the Hubbard model on the square lattice, the magnetic properties of a related model – the periodic Anderson model – are also addressed, on the cubic lattice.

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Rodrigo A. Fontenele, Nathan Vasconcelos, Natanael C. Costa +2 more

The emergence of superconductivity (SC) in lattice models, such as the attractive Hubbard one, has renewed interest since the realization of cold-atom experiments. However, reducing the temperature in these experiments is a bottleneck; therefore, investigating how to increase the energy scale for SC is crucial to cold atoms. In view of this, we examine the effects of next-nearest-neighbor hoppings (t′) on the pairing properties of the attractive Hubbard model in a square lattice. To this end, we analyze the model through unbiased Quantum Monte Carlo simulations for fixed density n=0.87, and perform finite-size scaling analysis to the thermodynamic limit. As our main result, we notice that the existence of further hopping channels leads to an enhancement of the pairing correlations, which, in turn, increases the ground-state order parameter Δ. Finally, at finite temperatures, for t′/t≠0, this enhancement of pairing correlations leads to an increase in the critical temperature Tc. That is, the fine-tuning of second-neighbor hoppings increases the energy scales for SC, and may be a route by which cold-atom experiments can achieve such a phase and to help us further understand the nature of this phenomenon.

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Weakly correlated electrons on a square lattice: a renormalization group theory

Dražen Zanchi, Heinz J. Schulz

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.

arXiv:cond-mat/9812303•Dec 17, 1998•0 citations
In this thesis, symmetry and related transformation laws under a suitable unitary transformation are used in the study of the superconducting and magnetic properties of strongly correlated electronic systems. Such a study profits from a recently found global SO(3)×SO(3)×U(1) = [SO(4)×U(1)]/Z2 symmetry in the Hubbard model on a bipartite lattice. The introduction of the square-lattice quantum liquid relies on a general description of the Hubbard model consistent with its exact global SO(3) × SO(3) × U(1) symmetry. Such a quantum liquid is expected to play the same role for many-electron problems with short-range interactions on a square lattice, as a Fermi liquid for three-dimensional isotropic metals. It refers to the Hubbard model on the square lattice in the subspace spanned by the ground state, and the excited states generated by application onto it of one- and two-electron operators. For such a square-lattice quantum liquid only the charge c fermions and spin-neutral two-spinon s1 fermions play an active role. Part of the studies of this thesis use an effective superconductivity theory according to which, for zero spin density m = 0 and the range 0 < x < x∗ of finite hole concentrations of a short-range spin ordered phase, the ground state of the square-lattice quantum liquid of c and s1 fermions with residual interactions is superconducting. (For U/4t > ui ≈ 3/4 the critical hole concentration x∗ increases continuously upon increasing the U/4t value from x∗ = 1/√2 ≈ 0.225 at U/4t = ui to x∗ = 1/ ≈ 0.318 for U/4t → ∞.) Superconductivity emerges naturally from the effects of the residual interactions of the c and s1 fermions and as a side product of the short-range spin correlations. The phase of the c fermion pairs reads = 0 + 1 where the fluctuations of the phases 0 and 1 become large for x → 0 and x → x∗, respectively. For 0 < x < x∗ and x = 0, the symmetry of the action for the phase is a global superconducting U(1) symmetry and a local compact gauge symmetry, respectively. The former U(1) symmetry is identified with the U(1) symmetry of the Hubbard-model global SO(3) × SO(3) × U(1) = [SO(4) × U(1)]/Z2 symmetry. The amplitudes g = g0 g1, g0 = |heiθ0i|, and g1 = |heiθ1i|, their zero-temperature values ˘g ≈ (x/x∗)(1−x/x∗), ˘g0 ≈ (x/x∗), and ˘g1 ≈ (1−x/x∗) and the fluctuations of the phase , play an important role in the physics of the square-lattice quantum liquid. Together with the energy scale 2 0, which gives the maximum magnitude of the s1 fermion spinon pairing energy, they fully control the order parameters of both the short-range spin and long-range superconducting correlations. Indeed, for temperatures T < Tc, the short-range spin ordered phase has both a macroscopic quantum phase-coherent pair superconducting order so that |heiθi| > 0, and a short-range spin order, whereas for Tc < T < T∗ only the latter order prevails yet there remain c fermion pairing correlations such that |heiθ1i| > 0 but heiθ0i = 0. Here the critical temperature Tc and the crossover pseudogap temperature T∗ are fully controlled by the absolute value of the order parameters of the two above orders, respectively. However, the square-lattice quantum liquid alone does not describe quantitatively the unusual properties of the hole-doped cuprates. To fulfill such a goal we profit from a suitable theory according to which the interplay of electronic correlations described by the square-lattice quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions with intrinsic disorder, is behind the unusual properties of the holedoped hight-temperature cuprate superconductors. By profiting from such an interplay, an excellent quantitative agreement with experiments for the physical quantities of several families of hole-doped cuprates is reached. That includes universal properties such as a dome-like critical-temperature hole-concentration dependence and normal-state linear-T resistivity near optimal doping, extending over a wide temperature window. The results confirm a coexisting two-gap scenario associated with an incoherent two-spinon pairing and a coherent twoc- fermion pairing, respectively. While most of our studies focus on the superconducting and magnetic properties of the square-lattice quantum liquid associated with the Hubbard model on the square lattice, the magnetic properties of a related model – the periodic Anderson model – are also addressed, on the cubic lattice.
The emergence of superconductivity (SC) in lattice models, such as the attractive Hubbard one, has renewed interest since the realization of cold-atom experiments. However, reducing the temperature in these experiments is a bottleneck; therefore, investigating how to increase the energy scale for SC is crucial to cold atoms. In view of this, we examine the effects of next-nearest-neighbor hoppings (t′) on the pairing properties of the attractive Hubbard model in a square lattice. To this end, we analyze the model through unbiased Quantum Monte Carlo simulations for fixed density n=0.87, and perform finite-size scaling analysis to the thermodynamic limit. As our main result, we notice that the existence of further hopping channels leads to an enhancement of the pairing correlations, which, in turn, increases the ground-state order parameter Δ. Finally, at finite temperatures, for t′/t≠0, this enhancement of pairing correlations leads to an increase in the critical temperature Tc. That is, the fine-tuning of second-neighbor hoppings increases the energy scales for SC, and may be a route by which cold-atom experiments can achieve such a phase and to help us further understand the nature of this phenomenon.
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.