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arXiv:1308.5946 · Physical Review B

Evolution of the superconductivity dome in the two-dimensional Hubbard model

K.-S. Chen, Zi Yang Meng, Shuxiang Yang, Thomas Pruschke, Juana Moreno, Mark Jarrell

Published Dec 10, 2013• 33 citations
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Abstract

In a recent publication [Chen et al., Phys. Rev. B 86, 165136 (2012)], we identified a line of Lifshitz transition points separating the Fermi liquid and pseudogap regions in the hole-doped two-dimensional Hubbard model. Here, we extend the study to further determine the superconducting transition temperature in the phase diagram. By means of large-scale dynamical cluster quantum Monte Carlo simulations, we are able to identify the evolution of the $d$-wave superconducting dome in the hole-dope side of the phase diagram, with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. To obtain the superconducting transition temperature ${T}_{c}$, we employ two-particle measurements of the pairing susceptibilities. As ${t}^{\ensuremath{'}}$ goes from positive to negative values, we find the $d$-wave projected irreducible pairing vertex function is enhanced, and the curvature of its doping dependence changes from convex to concave, which fixes the position of the maximum superconducting temperature at the same filling ($n\ensuremath{\approx}0.85$) and constraints the dome from precisely following the Lifshitz line. We furthermore decompose the irreducible vertex function into fully irreducible, charge and spin components via the parquet equations, and consistently find that the spin component dominates the pairing vertex function in the doping range where the dome is located. Our investigations deepen the understanding of the phase diagram of the two-dimensional Hubbard model and, more importantly, pose new questions to the field. For example, we found as ${t}^{\ensuremath{'}}$ goes from positive to negative values, the curvature of the pairing strength as a function of doping changes from convex to concave, and the nature of the dominant fluctuations changes from charge degree of freedom to spin degree of freedom. The study of these issues will lead to further understanding of the phase diagram of the two-dimensional Hubbard model and also the physics of the hole-doped cuprate high-temperature superconductors.

DOI
10.1103/physrevb.88.245110

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In a recent publication [Chen et al., Phys. Rev. B 86, 165136 (2012)], we identified a line of Lifshitz transition points separating the Fermi liquid and pseudogap regions in the hole-doped two-dimensional Hubbard model. Here, we extend the study to further determine the superconducting transition temperature in the phase diagram. By means of large-scale dynamical cluster quantum Monte Carlo simulations, we are able to identify the evolution of the ddd-wave superconducting dome in the hole-dope side of the phase diagram, with next-nearest-neighbor hopping (t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′), chemical potential, and temperature as control parameters. To obtain the superconducting transition temperature Tc{T}_{c}Tc​, we employ two-particle measurements of the pairing susceptibilities. As t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ goes from positive to negative values, we find the ddd-wave projected irreducible pairing vertex function is enhanced, and the curvature of its doping dependence changes from convex to concave, which fixes the position of the maximum superconducting temperature at the same filling (n\ensuremath≈0.85n\ensuremath{\approx}0.85n\ensuremath≈0.85) and constraints the dome from precisely following the Lifshitz line. We furthermore decompose the irreducible vertex function into fully irreducible, charge and spin components via the parquet equations, and consistently find that the spin component dominates the pairing vertex function in the doping range where the dome is located. Our investigations deepen the understanding of the phase diagram of the two-dimensional Hubbard model and, more importantly, pose new questions to the field. For example, we found as t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ goes from positive to negative values, the curvature of the pairing strength as a function of doping changes from convex to concave, and the nature of the dominant fluctuations changes from charge degree of freedom to spin degree of freedom. The study of these issues will lead to further understanding of the phase diagram of the two-dimensional Hubbard model and also the physics of the hole-doped cuprate high-temperature superconductors.
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Using large-scale dynamical cluster quantum Monte Carlo simulations, we study the Lifshitz transition of the two-dimensional Hubbard model with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. At ${t}^{\ensuremath{'}}\ensuremath{\le}0$, we identify a line of Lifshitz transition points associated with a change in the Fermi surface topology at zero temperature. In the overdoped region, the Fermi surface is complete and electron-like; across the Lifshitz transition, the Fermi surface becomes hole-like and develops a pseudogap. At (or very close to) the Lifshitz transition points, a van Hove singularity in the density of states crosses the Fermi level. The van Hove singularity occurs at finite doping due to correlation effects and becomes more singular when ${t}^{\ensuremath{'}}$ becomes more negative. The resulting temperature dependence on the bare ddd-wave pairing susceptibility close to the Lifshitz points is significantly different from that found in the traditional van Hove scenarios. Such unambiguous numerical observation of the Lifshitz transition at ${t}^{\ensuremath{'}}\ensuremath{\le}0$ extends our understanding of the quantum critical region in the phase diagram and shines lights on future investigations of the nature of the quantum critical point in the two-dimensional Hubbard model.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.
Many properties of the two-dimensional Hubbard model have been explored for the model parameters appropriate for strongly correlated electronic systems, especially cuprate superconductors.Most of the calculations are done using a well-established dynamical cluster quantum Monte Carlo method.Using this method, we investigate the effect of long-range hoppings on superconductivity with and without the presence of phonons on small clusters.The superconducting transition temperature, T c , is found to generally decrease with a negative next nearest neighbor hopping, t ′ .In the presence of the Holstein phonons, a finite t ′ enhances T c in the under-doped region for the hole-doped system, consistent with band structure calculations and experiment.The validity of the spin-susceptibility-mediated pairing in this model is studied and found to yield symmetries other than d-wave when a finite t ′ is considered.A new numerical algorithm for solving the embedded cluster problems is introduced, and used to calculate the thermodynamic properties of the model on larger clusters, especially those associated with the quantum critical behavior at finite doping.Our results suggest that the quantum critical point (QCP) which separates the Fermi liquid and pseudogap regions, is the second order terminus of the line of first order phase separation transition in the limit when a positive t ′ goes to zero.For small t ′ , the superconducting dome is centered at the QCP, suggesting that charge fluctuations might have a role in the pairing mechanism in this model.I am very thankful to University of Cincinnati for providing me with the opportunity to pursue my education and the support I received during my program.Part of this thesis is a result of my work as a researcher at Louisiana State University.I am also thankful to them for their support.I would like to thank my adviser, Mark Jarrell, who taught me how to be an effective researcher, provided me with the opportunities to interact with other professionals and fully supported me financially for most of the duration of the
In nonperturbative regimes, the superfluid instability in the two-dimensional (2D) Hubbard model can be described by an emergent BCS theory with small effective pairing constants. We compute the effective couplings using a controlled bold-line diagrammatic Monte Carlo approach, which stochastically sums all skeleton Feynman diagrams dressed in the one- and two-particle channels to high expansion orders, and map out the resulting superfluid ground-state phase diagram in a range of next-nearest-neighbor hopping $0\ensuremath{\le}{t}^{\ensuremath{'}}\ensuremath{\le}0.3t$, interaction strength $0\ensuremath{\le}U\ensuremath{\le}3t$, and lattice filling $0\ensuremath{\le}n\ensuremath{\le}2$. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and ${t}^{\ensuremath{'}}$. At ${t}^{\ensuremath{'}}=0.3$, the weak-coupling picture with the dominant triplet pairing sharply peaked at $n\ensuremath{\approx}0.73$ due to the Van Hove singularity is replaced by a plateau of the singlet ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ paring, while for $U\ensuremath{\gtrsim}3t$ the effective couplings are consistent with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.
We compute the two-particle quantities relevant for superconducting correlations in the two-dimensional Hubbard model within the dynamical cluster approximation. In the normal state we identify the parameter regime in density, interaction, and second-nearest-neighbor hopping strength that maximizes the d_{x^{2}-y^{2}} superconducting transition temperature. We find in all cases that the optimal transition temperature occurs at intermediate coupling strength, and is suppressed at strong and weak interaction strengths. Similarly, superconducting fluctuations are strongest at intermediate doping and suppressed towards large doping and half filling. We find a change in sign of the vertex contributions to d_{xy} superconductivity from repulsive near half filling to attractive at large doping. p-wave superconductivity is not found at the parameters we study, and s-wave contributions are always repulsive. For negative second-nearest-neighbor hopping the optimal transition temperature shifts towards the electron-doped side in opposition to the van Hove singularity, which moves towards hole doping. We surmise that an increase of the local interaction of the electron-doped compounds would increase T_{c}.