Using a variational Monte Carlo method, we study the competition of strong electron-electron and electron-phonon interactions in the ground state of the Holstein-Hubbard model on a square lattice. At half filling, an extended intermediate metallic or weakly superconducting (SC) phase emerges, sandwiched between antiferromagnetic and charge order (CO) insulating phases. By carrier doping into the CO insulator, the SC order dramatically increases for strong electron-phonon couplings, but is largely hampered by wide phase separation (PS) regions. Superconductivity is optimized at the border to the PS.
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TY - JOUR
T1 - Competition among Superconducting, Antiferromagnetic, and Charge Orders with Intervention by Phase Separation in the 2D Holstein-Hubbard Model
AU - Ohgoe, Takahiro
AU - Imada, Masatoshi
N1 - Funding Information: We now study the doped region. In Fig. 4 , we first present our ground-state phase diagram at U = 0 in the δ - λ plane for Ω = 8 t and Ω = t , because the U = 0 phase diagram captures an essential aspect. For U = 0 , the effective interaction U eff ( ω ) has negative parts for ω < Ω , which lead to s -wave SC states except for the gapped CO phase at half filling. In our phase diagram, the SC + CO phase is absent. Instead, the PS region appears adjacent to the CO phase at half filling. We find that for the smaller phonon frequency, the PS region is enlarged. In the Supplemental Material  , we present the phase diagram in the adiabatic limit as the extreme case. In Fig. 4 , we also plot S c ( π , π ) / N and the long-range part of the s -wave SC correlation function P s ∞ , which is defined by P s ∞ = ( 1 / M ) ∑ 2 L / 4 < | r | P s ( r ) . Here, r is the relative position vectors belonging to ( - L / 2 , L / 2 ] 2 , M is the number of vectors satisfying 2 L / 4 < | r | < 2 L / 2 , and the SC function P s ( r ) is defined by P s ( r ) = ( 1 / N ) ∑ r i ⟨ Δ s † ( r i ) Δ s ( r i + r ) ⟩ with the order parameter Δ s ( r i ) = c r i ↑ c r i ↓ . 4 10.1103/PhysRevLett.119.197001.f4 FIG. 4. Ground-state phase diagrams of the Holstein model in the δ - λ plane at (a) Ω = 8 t and (b) Ω = t . In the vertical axis, S c ( π , π ) / N (red squares) and P s ∞ (color plots) for L = 14 are plotted in the CO and SC phases, respectively. Black squares in the bottom plane represent boundaries between the PS and s -wave SC regions. White areas denote the PS regions. Thick red lines at δ = 0 indicate the CO phase. In Fig. 5(a) , we show physical quantities which were used to determine the phase diagrams in Fig. 4 in an example at ( Ω / t , U / t , λ ) = ( 8 , 0 , 0.3 ) . We also show an interacting case for ( Ω / t , U / t , λ ) = ( 8 , 8 , 1.3 ) in Fig. 5(b) for comparison. Since the model is mapped, in the antiadiabatic limit, to the standard Hubbard model with the on-site interaction U eff = U - W λ , the comparison between the interacting and noninteracting cases with the same U eff may provide us with insight for large Ω . The cases shown in Figs. 5(a) and 5(b) indeed have the same U eff = - 2.4 . The value of S c ( π , π ) / N decreases monotonically and the CO eventually disappears at δ ≃ 0.1 and 0.2 for U / t = 0 [Fig. 5(a) ] and U / t = 8 [Fig. 5(b) ], respectively. On the other hand, the value of P s ∞ increases as δ increases and we clearly observe the SC phase. For small δ , a CO order and an s -wave SC order coexist. By the Maxwell construction for the δ - μ curve, however, we find that the SC + CO phase is swallowed up by the PS region ( 0 < δ < 0.14 for U / t = 0 and 0 < δ < 0.37 for U / t = 8 ). Here, μ is the chemical potential which was calculated by μ ( N ¯ e ) = [ E ( N e ) - E ( N e ′ ) ] / ( N e - N e ′ ) . Here, E is the total energy, ( N e , N e ′ ) are the electron numbers, and we obtain the chemical potential at the mid filling N ¯ e = ( N e + N e ′ ) / 2 . Our Hamiltonian has the particle-hole symmetry at μ = - 8 λ - U / 2 = - 2.4 and - 6.4 for Figs. 5(a) and 5(b) , respectively. Since this value is above the line used for the Maxwell construction, there is a charge gap at half filling. For the interacting case Fig. 5(b) , the charge gap is even larger. We also present the negative inverse uniform charge susceptibility - χ c - 1 = d μ / d ρ in Fig. 5 . In our model, the spinodal point δ s , where the uniform charge susceptibility diverges ( χ c - 1 = 0 ), coincides with the critical point of the CO and, therefore, the PS is driven by the CO (see also the results for the adiabatic limit in the Supplemental Material  ). 5 10.1103/PhysRevLett.119.197001.f5 FIG. 5. Physical quantities S c ( π , π ) / N , P s ∞ , μ , and - χ c - 1 as functions of doping δ at (a) ( Ω / t , U / t , λ ) = ( 8 , 0 , 0.3 ) and (b) (8, 8, 1.3), respectively. The shaded area denotes the PS region, which was determined by the Maxwell construction. The dashed horizontal line in the middle panel is used for the Maxwell construction. The curves of - χ c - 1 were derived from the derivative of the μ - δ curves (black curves) which were obtained by the seventh-order polynomial fit. The spinodal points δ s are indicated as the arrows. Comparisons between Figs. 5(a) and 5(b) show a quantitative difference that the CO (SC) orders are enhanced (suppressed) for large U / t . However, we find a universal common feature both in Figs. 5(a) and 5(b) ; a clear one-to-one correspondence among the peak of the SC order, the spinodal point, and the border of the CO phase thus indicates tight connections of the mechanism of the SC, CO, and uniform charge instability. The strong effective attractive interaction of carriers is certainly the key, because it causes all of these three properties. The strong attraction is caused by the electron-phonon interaction here, while the resultant charge fluctuations may also work as additional glue of the Cooper pair. The same trend between the enhancement of the s -wave SC and the uniform charge susceptibility has been reported for d -wave SC in the Hubbard model  and extended s -wave SC in the ab initio effective Hamiltonian for LaFeAsO  as well. To summarize, by studying the ground states of the Holstein-Hubbard model on a square lattice, we have clarified where the s -wave SC is enhanced in the phase diagram. At half filling, we have found an intermediate metallic or weakly SC region sandwiched by the CO and AF phases. In the doped case, the SC is dramatically enhanced, but a wide PS region triggered by the CO largely hinders the SC and completely preempts the SC + CO phase. We have revealed that the SC is optimized at the border of the PS. These findings have been obtained by the VMC method extended for electron-phonon coupled systems. Our method is quite flexible, and therefore it will be also useful to study more complicated systems such as ab initio Hamiltonians of high- T c cuprates, where several different phonon modes are present. We thank Kota Ido for useful discussions. T. O. also thanks Yuta Murakami for discussions. The code was developed based on the open-source software m vmc  . This work is financially supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) High Performance Computing Infrastructure (HPCI) Strategic Programs for Innovative Research (SPIRE), the Computational Materials Science Initiative (CMSI), and Creation of New Functional Devices and High-Performance Materials to Support Next Generation Industries (CDMSI). 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PY - 2017/11/8
Y1 - 2017/11/8
N2 - Using a variational Monte Carlo method, we study the competition of strong electron-electron and electron-phonon interactions in the ground state of the Holstein-Hubbard model on a square lattice. At half filling, an extended intermediate metallic or weakly superconducting (SC) phase emerges, sandwiched between antiferromagnetic and charge order (CO) insulating phases. By carrier doping into the CO insulator, the SC order dramatically increases for strong electron-phonon couplings, but is largely hampered by wide phase separation (PS) regions. Superconductivity is optimized at the border to the PS.
AB - Using a variational Monte Carlo method, we study the competition of strong electron-electron and electron-phonon interactions in the ground state of the Holstein-Hubbard model on a square lattice. At half filling, an extended intermediate metallic or weakly superconducting (SC) phase emerges, sandwiched between antiferromagnetic and charge order (CO) insulating phases. By carrier doping into the CO insulator, the SC order dramatically increases for strong electron-phonon couplings, but is largely hampered by wide phase separation (PS) regions. Superconductivity is optimized at the border to the PS.
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U2 - 10.1103/PhysRevLett.119.197001
DO - 10.1103/PhysRevLett.119.197001
M3 - Article
C2 - 29219494
AN - SCOPUS:85033572533
VL - 119
JO - Physical Review Letters
JF - Physical Review Letters
SN - 0031-9007
IS - 19
M1 - 197001