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Superfluid ground-state phase diagram of the $2d$ Hubbard Model in the emergent BCS regime

Fedor Šimkovic, Youjin Deng, Evgeny Kozik

Published Dec 30, 2019• 3 citations
OpenAlex

Abstract

In non-perturbative regimes, the superfluid instability in the $2d$ 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 $0 \leq t^{\prime} \leq 0.3t$, interaction strength $0 \leq U \leq 3t$, and lattice filling $0 \leq n \leq 2$. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and $t'$. At $t'=0.3$, the weak-coupling picture with the dominant triplet pairing sharply peaked at $n \approx 0.73$ due to the Van Hove singularity is replaced by a plateau of the singlet $d_{x^2-y^2}$ paring, while for $U \gtrsim 3t$ the effective couplings are consistent with the $d_{x^2-y^2}$ high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.

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Superfluid ground-state phase diagram of the 2d2d2d Hubbard Model in the emergent BCS regime
In non-perturbative regimes, the superfluid instability in the 2d2d2d 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≤t′≤0.3t0 \leq t^{\prime} \leq 0.3t0≤t′≤0.3t, interaction strength 0≤U≤3t0 \leq U \leq 3t0≤U≤3t, and lattice filling 0≤n≤20 \leq n \leq 20≤n≤2. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and t′t't′. At t′=0.3t'=0.3t′=0.3, the weak-coupling picture with the dominant triplet pairing sharply peaked at n≈0.73n \approx 0.73n≈0.73 due to the Van Hove singularity is replaced by a plateau of the singlet dx2−y2d_{x^2-y^2}dx2−y2​ paring, while for U≳3tU \gtrsim 3tU≳3t the effective couplings are consistent with the dx2−y2d_{x^2-y^2}dx2−y2​ high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.
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 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.
For over half a century, the Hubbard model has played a paradigmatic role in attempts to understand quantum phenomena exhibited by correlated electrons in solids. Despite a substantial effort and the apparent simplicity of the model, its behavior in many important regimes has remained unknown. Here we study superfluidity in the two-dimensional Hubbard model with controlled error bars up to the coupling strength U = 4 and filling factor . We show, by means of unbiased diagrammatic Monte Carlo simulations, that in this regime the superfluid transition is governed by Fermi liquid physics with an emergent weak BCS-type coupling driving the instability. The corresponding ground-state phase diagram in the plane describes the competition between the superfluid states of p - and d -wave symmetry. We also report dimensionless coupling constants in this effective BCS regime.
A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF) is formulated and used for the description of superconducting (SC) ground state in the two-dimensional Hubbard model with electron-transfer amplitudes ttt (and ${t}^{\ensuremath{'}}$) between nearest (and next-nearest) neighbors. The method is numerically very efficient and allows for a detailed analysis of the phase diagram as a function of all relevant parameters (UUU, $\ensuremath{\delta}$, ${t}^{\ensuremath{'}}$) and a determination of the kinetic-energy driven pairing region. SC states appear only for substantial interactions, $U/t\ensuremath{\gtrsim}3$, and for not too large hole doping, $\ensuremath{\delta}\ensuremath{\lesssim}0.32$ for ${t}^{\ensuremath{'}}=0.25t$; this upper critical doping value agrees well with experiment for the cuprate high-temperature superconductors. We also obtain other important features of the SC state: (i) the SC gap at the Fermi surface resembles ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave only around the optimal doping and the corrections to this state are shown to arise from the longer range of the pairing; (ii) the nodal Fermi velocity is almost constant as a function of doping and agrees quantitatively with the experimental results; (iii) the SC transition is driven by the kinetic-energy lowering for low doping and strong interactions.
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 (${t}^{\ensuremath{'}}$), 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 ${t}^{\ensuremath{'}}$ 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 ($n\ensuremath{\approx}0.85$) 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.
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.