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arXiv:1805.10011 · Journal of the Physical Society of Japan

Edge-induced Strongly Correlated Electronic States in Two-dimensional Hubbard Model: Enhancement of Magnetic Correlations and Self-energy Effects

Shun Matsubara, Youichi Yamakawa, Hiroshi Kontani

Published Jun 15, 2018• 4 citations
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Abstract

To understand nontrivial edge electronic states in strongly-correlated metals such as cuprate superconductors, we study the two-dimensional Hubbard models with open edge boundary. The position-dependences of the spin susceptibility and the self-energy are carefully analyzed self-consistently, by using the fluctuation-exchange (FLEX) approximation. It is found that spin susceptibilities are strongly enlarged near the (1,1) open edge when the system is near the half-filling. The enhancement is large even if the negative feedback from the self-energy is considered in the FLEX approximation. The present study predicts the emergence of nontrivial spin-fluctuation-driven phenomena near the edge, like the quantum criticality, edge superconductivity, and the bond-density-wave order.

DOI
10.7566/jpsj.87.073705

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To understand nontrivial edge electronic states in strongly-correlated metals such as cuprate superconductors, we study the two-dimensional Hubbard models with open edge boundary. The position-dependences of the spin susceptibility and the self-energy are carefully analyzed self-consistently, by using the fluctuation-exchange (FLEX) approximation. It is found that spin susceptibilities are strongly enlarged near the (1,1) open edge when the system is near the half-filling. The enhancement is large even if the negative feedback from the self-energy is considered in the FLEX approximation. The present study predicts the emergence of nontrivial spin-fluctuation-driven phenomena near the edge, like the quantum criticality, edge superconductivity, and the bond-density-wave order.
Since the two-dimensional (2D) Hubbard model is one of the simplest model having features of layered cuprates which is widely known as high-Tc superconductor, many researchers pay their attention to this model. One of the purposes of this thesis is to discuss, on the basis of the fluctuation exchange (FLEX) approximation which satisfies the conservation law and takes account of the spin fluctuations, how the shape of the Fermi surface (FS) is modified and how the charge susceptibility behaves in the 2D t−t′−U Hubbard model near half filling as strength of the onsite Coulomb interaction U is increased. The antiferromagnetic (AF) correlation length is shown to be enhanced as the Coulomb interaction get closer to the critical value Uc for the critical point to onset AF order. At the same time, the shape of the renormalized FS is deformed showing the tendency of nesting and the charge susceptibility shows a pronounced enhancement near Uc. This result is consistent with experimental observations in under-doped cuprates LSCO. The enhancement of the charge susceptibility is so large that it cannot be explained by the van Hove singularity of the density of states (DOS) due to the nesting of FS. Indeed, the van Hove singularity of DOS is easily cancelled by the RPA type contribution. Another purpose of this thesis is to investigate an origin of the divergent charge susceptibility. It is expected that the effect of the AF spin fluctuations is the main origin for this divergence of the charge susceptibility from the fact that the anomaly is observed near the antiferromagnetic insulating (AFI) phase. In order to verify this physical picture, we calculate the Aslamasov-Larkin (AL) type contribution to the charge susceptibility, and compare its value with that calculated by the FLEX approximation. The result is that the AL-type contribution nearly exhaust the value of the enhanced charge susceptibility where the charge susceptibility is enhanced enough as
It is discussed on the fluctuation-exchange (FLEX) approximation that the charge susceptibility χ c increases divergently in the 2D t – t ' Hubbard model near the half-filling as the strength of the onsite Coulomb interaction U is approached to the critical value U cr for the critical point to onset of the antiferromagnetic (AF) order. It turns out that the main origin of the divergence of χ c is the Aslamazov–Larkin (AL) type contribution which can be taken into account by the FLEX approximation and is enhanced by the strong AF correlations because two AF-fluctuation modes and the vertex corretions work cooperatively to enhance χ c . Indeed, AL-type contribution almost exhausts the enhanced part of χ c in the region U ≃ U cr . This result is consistent with experimental observations in under-doped cuprates LSCO and quantum Monte Carlo simulations by Furukawa and Imada [J. Phys. Soc. Jpn. 62 (1993) 2557].
As the surface Andreev bound state (ABS) forms at the open (1,1) edge of a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductor, the local density of states increases. Therefore, a strong electron correlation and drastic phenomena may occur. However, a theoretical study on the effects of the ABS on the electron correlation has not been performed yet. To understand these effects, we study the large-cluster Hubbard model with an open (1,1) edge in the presence of a bulk ddd-wave gap. We calculate the site-dependent spin susceptibility by performing the random-phase approximation and modified fluctuation-exchange approximation in real space. We find that near the (1,1) edge, drastic ferromagnetic (FM) fluctuations occur owing to the ABS. In addition, as the temperature decreases, the system rapidly approaches a magnetic-order phase slightly below the transition temperature of the bulk ddd-wave superconductivity (SC). In this case, the FM fluctuations are expected to induce interesting phenomena such as edge-induced triplet SC and quantum critical phenomena.
Electron-electron and electron-phonon interactions are responsible for the formation of spin, charge, and superconducting correlations in layered quantum materials. A paradigmatic model for such materials that captures both kinds of interactions is the two-dimensional Hubbard-Holstein model with a dispersionless Einstein phonon. In this work, we provide a detailed analysis of the magnetic, density, and superconducting fluctuations at and away from half filling. To that end, we employ the functional renormalization group using the recently introduced extension of the single-boson exchange formulation. More precisely, we go beyond previous approaches to the model by resolving the full frequency dependence of the two-particle vertex and taking into account the feedback from the electronic self-energy. We perform broad parameter scans in the space of Hubbard repulsion, electron-phonon coupling strength, and phonon frequency to explore the leading magnetic, density, and superconducting susceptibilities from the adiabatic to the anti-adiabatic regime. Our numerical data reveal that self-energy effects lead to an enhancement of the d -wave superconducting susceptibility toward larger phonon frequencies, in contrast to earlier isotope-effect studies. At small phonon frequencies, large density contributions to the s -wave superconducting susceptibility change sign and eventually lead to a reduction of s -wave superconductivity with increasing electron-phonon coupling, signaling the breakdown of Migdal-Eliashberg theory. We analyze our findings systematically, employing detailed diagnostics of the intertwined fluctuations and pinning down the various positive and negative isotope effects of the physical susceptibilities. Our results pave the way for a more comprehensive understanding of the interplay of electron-electron and electron-phonon interactions in correlated quantum materials.
We have calculated within the fluctuation exchange approximation the full momentum and real-frequency dependence of the quasiparticle self-energy, the gap function, and the dynamic spin susceptibility for the two-dimensional Hubbard model near half filling. The spin-density-wave (SDW) instability is reached only asymptotically in the limit of infinite U or zero temperature. Near the SDW instability we find approximately marginal Fermi-liquid behavior, a pseudogap in the density of states, and a singlet dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 pairing instability. Below Tc{\mathit{T}}_{\mathit{c}}Tc​, gaps open rapidly in the frequency spectra of the gap function, the quasiparticle scattering rate, and the density of states. The spectral density of the dynamic spin susceptibility agrees qualitatively with the neutron-scattering data on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​. The quasiparticle scattering rate and spin-lattice relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$ decrease rapidly below Tc{\mathit{T}}_{\mathit{c}}Tc​. The photoemission intensities along the antinode and node of the gap are in good agreement with the angle-resolved photoemission spectroscopy data on Bi 2212. Including next-nearest-neighbor hopping we obtain for sufficiently large coupling (U/t\ensuremath{\gtrsim}7) a broad commensurate neutron-scattering peak like the observed one in YBa2{\mathrm{YBa}}_{2}YBa2​Cu3{\mathrm{Cu}}_{3}Cu3​O6+x{\mathrm{O}}_{6+\mathit{x}}O6+x​.
The influence of spin and charge fluctuations on spectra of the two-dimensional fermionic Hubbard model is considered using the strong coupling diagram technique. Infinite sequences of diagrams containing ladder inserts, which describe the interaction of electrons with these fluctuations, are summed, and obtained equations are self-consistently solved for the ranges of Hubbard repulsions [Formula: see text], temperatures [Formula: see text] and electron concentrations [Formula: see text] with t the intersite hopping constant. For all considered U the system exhibits a transition to the long-range antiferromagnetic order at [Formula: see text]. At the same time no indication of charge ordering is observed. Obtained solutions agree satisfactorily with results of other approaches and obey moments sum rules. In the considered region of the U-T plane, the curve separating metallic solutions passes from [Formula: see text] at the highest temperatures to U = 2t at [Formula: see text] for half-filling. If only short-range fluctuations are allowed for the remaining part of this region is occupied by insulating solutions. Taking into account long-range fluctuations leads to strengthening of maxima tails, which transform a part of insulating solutions into bad-metal states. For low T, obtained results allow us to trace the gradual transition from the regime of strong correlations with the pronounced four-band structure and well-defined Mott gap for [Formula: see text] to the Slater regime of weak correlations with the spectral intensity having a dip along the boundary of the magnetic Brillouin zone due to an antiferromagnetic ordering for [Formula: see text]. For [Formula: see text] and [Formula: see text] doping leads to the occurrence of a pseudogap near the Fermi level, which is a consequence of the splitting out of a narrow band from a Hubbard subband. Obtained spectra feature waterfalls and Fermi arcs, which are similar to those observed in hole-doped cuprates.