Characterizing the three-orbital Hubbard model with determinant quantum Monte Carlo

We characterize the three-orbital Hubbard model using state-of-the-art determinant quantum Monte Carlo (DQMC) simulations with parameters relevant to the cuprate high-temperature superconductors. The simulations find that doped holes preferentially reside on oxygen orbitals and that the $(\pi ,\pi )$ antiferromagnetic ordering vector dominates in the vicinity of the undoped system, as known from experiments. The orbitally-resolved spectral functions agree well with photoemission spectroscopy studies and enable identification of orbital content in the bands. A comparison of DQMC results with exact diagonalization and cluster perturbation theory studies elucidates how these different numerical techniques complement one another to produce a more complete understanding of the model and the cuprates. Interestingly, our DQMC simulations predict a charge-transfer gap that is significantly smaller than the direct (optical) gap measured in experiment. Most likely, it corresponds to the indirect gap that has recently been suggested to be on the order of 0.8 eV, and demonstrates the subtlety in identifying charge gaps.

Figure 1

A copper $d_{x^2-y^2}$ orbital and its surrounding oxygen $p_x$ or $p_y$ orbitals are shown. The colors indicate the phase factors (blue for positive, red for negative).

Figure 2

The average fermion sign is plotted versus filling for $N=4, N=16$, and $N=36$ at $\beta =8{\mathrm{eV}}^{-1}$ and $U_{pp}=4.1$ eV. In general, a larger system size results in a more severe sign problem.

Figure 3

The average fermion sign is plotted versus filling for different values of $U_{dd}$ and $U_{pp}$ at $\beta =8{\mathrm{eV}}^{-1}$ and $N=36$. Reducing the interactions improves the sign problem and, in the case of $U_{pp}$, can even flip which side of diagram has the more severe problem.

Figure 4

The average fermion sign is plotted versus filling for a range of temperatures for the (a) $N=16$ and (b) $N=36$ systems, with $U_{pp}=0$ to access lower temperatures. Decreasing the temperature significantly decreases the average sign but preserves the direction of the particle-hole asymmetry. The partially protected fillings in the $N=16$ system become more pronounced with decreasing temperature.

Figure 5

The average fermion sign is plotted versus filling for different values of $t_{pp}$ to demonstrate the effect of decreasing the number of hopping pathways. Again, the particle-hole asymmetry is evident, as $t_{pp}=0$ invariably enhances the average sign for hole doping more than it does for electron doping.

Figure 6

The double occupancy ${\mathrm{D}}_{\alpha }$ versus filling on the copper and oxygen orbitals is shown for different temperatures, with $N=36$ and $U_{pp}=4.1$ eV. It exhibits no significant system size or temperature dependence.

Figure 7

Kinetic energy of the holes versus filling for different temperatures in $N=16$ and $N=36$ systems, with $U_{pp}=4.1$ and 0 eV. The non-interacting kinetic energy (solid black line) is shown for comparison.

Figure 8

(a) The total filling curve shows a gap opening with decreasing temperature, with $U_{pp}=0$ eV and $N=16$ to access the lowest possible temperatures. The inset shows the total filling for $N=36, U_{pp}=4.1$ eV, and $\beta =10{\mathrm{eV}}^{-1}$ and has the same axes. (b) The orbitally resolved fillings are shown for the same parameters as the inset in (a) and demonstrate that doped holes preferentially reside on oxygen atoms. As we are using the larger system size, the fermion sign is too small at certain chemical potentials to determine the filling; however, the solid lines indicate the trend that is consistent with results in smaller systems. A dashed line indicates the chemical potential corresponding to the undoped system.

Figure 9

The spin-spin correlation function is plotted versus filling for four different possible ordering vectors on the copper and oxygen (inset) orbitals. On copper, $(\pi ,\pi )$ antiferromagnetism dominates in the undoped system, decreasing with doping. The oxygen orbitals do not show signs of any particular spin order. Parameters used here are $N=16, U_{pp}=4.1$ eV, and $\beta =10{\mathrm{eV}}^{-1}$.

Figure 10

The copper spin-spin correlation function shows that $\mathbf{q}=(\pi ,\pi )$ order is strengthened by increasing system size and decreasing temperature ($U_{pp}=0$). The inset shows the finite size scaling at $0\%$ doping and $\beta =8{\mathrm{eV}}^{-1}$.

Figure 11

Nearest $\left[{\mathrm{S}}_{\mathrm{Cu}}(1,0)\right]$ and next-nearest $\left[{\mathrm{S}}_{\mathrm{Cu}}(1,1)\right]$ neighbor Cu-Cu spin-spin correlation function versus filling, with $U_{pp}=4.1$ eV and $\beta =8{\mathrm{eV}}^{-1}$, showing the crossover from AFM to short-ranged FM correlations for $N=16$ and 36.

Figure 12

Density-density correlation function versus filling on the (a) copper and (b) oxygen orbitals for different ordering vectors with $N=16$ and $U_{pp}=0$ (qualitatively similar results are obtained for $U_{pp}=4.1$ eV). The system shows a slight tendency to $(\pi ,\pi )$ charge ordering on the electron-doped side on copper.

Figure 13

Density-density correlations between the copper and oxygen orbitals, and between the oxygen orbitals for different ordering vectors in a $N=16, U_{pp}=0$ system (qualitatively similar results are obtained for $U_{pp}=4.1$ eV).

Figure 14

The orbitally-resolved spectral functions at (a) $0\%$, (b) $12.5\%$, (c) $25\%$, and (d) $37.5\%$ hole doping illustrate the doping evolution of the lower Hubbard band (LHB), Zhang-Rice triplet (ZRT) band, nonbonding (NB) band, Zhang-Rice singlet (ZRS) band, and upper Hubbard band (UHB) in the $N=16$ system, where $\beta =8{\mathrm{eV}}^{-1}, t_{pp}=0.49$ eV, and $U_{pp}=4.1$ eV. The insets show the density of states for each doping level, with the same frequency axis as the spectral functions.

Figure 15

The orbitally-resolved spectral functions at $12.5\%$ hole doping are shown for high-symmetry cuts in the first Brillouin zone in the $N=36$ system, where $\beta =8{\mathrm{eV}}^{-1}, t_{pp}=0.49$ eV, and $U_{pp}=4.1$ eV.

Figure 16

The (a) copper and (b) oxygen spectral functions are calculated in the undoped system using DQMC, ED, and CPT. Despite the different system sizes and temperatures, the peak positions show reasonable agreement. With fine momentum resolution, CPT resolves the (c) band dispersion and (d) DOS in detail. In (c), the noninteracting bands are overlaid as dashed lines on the CPT band structure. In (d), the shading in the DOS indicates orbital content (red for copper, blue for oxygen) in the filled bands. For example, the color gradient in the ZRS band indicates the increasing copper content away from $E_F$.

Figure 17

The filling versus chemical potential curves are shown on an $N=4$ cluster in order to access low temperatures to highlight the opening of a charge transfer gap. The inset shows an extrapolation of the gap size $\mathrm{\Delta }$ to zero temperature.

Figure 18

Comparison of the (a) orbitally-resolved filling and (b) local moments on copper and oxygen, showing good quantitative agreement between DQMC and ED. The DQMC simulations are performed with $N=16$ and $\beta =8{\mathrm{eV}}^{-1}$.