Fork Tensor-Product States: Efficient Multiorbital Real-Time DMFT Solver

We present a tensor network especially suited for multi-orbital Anderson impurity models and as an impurity solver for multi-orbital dynamical mean-field theory (DMFT). The solver works directly on the real-frequency axis and yields high spectral resolution at all frequencies. We use a large number $\mathbf{(}\mathcal{O}(100)\mathbf{)}$ of bath sites and therefore achieve an accurate representation of the bath. The solver can treat full rotationally invariant interactions with reasonable numerical effort. We show the efficiency and accuracy of the method by a benchmark for the three-orbital test-bed material ${\mathrm{SrVO}}_3$. There we observe multiplet structures in the high-energy spectrum, which are almost impossible to resolve by other multi-orbital methods. The resulting structure of the Hubbard bands can be described as a broadened atomic spectrum with rescaled interaction parameters. Additional features emerge when $U$ is increased. Finally, we show that our solver can be applied even to models with five orbitals. This impurity solver offers a new route to the calculation of precise real-frequency spectral functions of correlated materials.

Popular Summary

Strongly correlated materials—characterized by strong electron-electron repulsive forces—exhibit a variety of fascinating properties such as metal-insulator transitions and unconventional superconductivity. However, theoretically describing such materials is challenging because of the exponential complexity of the quantum many-body problem: The most successful theory proposed to date is a combination of density-functional theory and dynamical mean-field theory. Although a variety of approaches exist that are, in principle, exact, so far none of them has been able to provide exact data directly about the real-frequency axis for multiorbital problems, with sufficient enough resolution at all energies to be comparable to experimental results. Here, we present a method that can solve this important problem for many relevant cases.

We formulate a multiorbital quantum impurity solver directly on the real-frequency axis and focus on the cubic crystal ${\mathrm{SrVO}}_3$. Our solver is based on a tensor-product representation of the many-body ground state—with separated tensors for every orbital-spin combination—that is particularly suited for impurity problems. We use real-time evolution to calculate the interacting Green’s function. Our method does not require an analytic continuation procedure, which is prone to energy-resolution problems. This algorithm allows us, for the first time, to solve a realistic three-orbital problem directly on the real-frequency axis with excellent energy resolution at all energy scales. Comparing our results with findings from continuous-time quantum Monte Carlo methods, we find very good agreement on the imaginary frequency axis. For real frequencies, our approach shows a three-peaked spectrum, where each of those peaks corresponds to an atomic excitation. Furthermore, we show that even five-band models are within reach of this approach.

We expect that this advance will allow many problems in strongly correlated materials to be addressed and compared with experimental data with much higher accuracy than previously possible.

Figure 1

(a) Graphical representation of a MPS. Every circle corresponds to a tensor $A_i^{s_i}$ and each line to an index of this tensor. In this picture, the physical indices are the vertical lines, while the horizontal lines show the bond indices. Connected lines mean that the corresponding index is summed over. Fixing all the physical indices $s_i$ for each site results in a tensor of rank zero with the value of the coefficient $c_{s_1,\dots ,s_N}$. (b) Graphical representation of a MPO. The difference from a MPS is that a MPO has incoming indices $s_i$ and outgoing indices $s_i^{\prime }$ corresponding to the bra- and ket vectors of the operator.

Figure 2

Graphical representation of a FTPS for multi-orbital AIM. Separating bath degrees of freedom leads to a forklike structure. In this picture, a two-orbital model is shown, with four bath sites in each orbital. Orbitals are labeled $A$ and $B$, and the arrows denote the spin. Each spin-orbital combination has its own bath sticking out to the right. As in Fig. 1, the vertical lines are the physical degrees of freedom [all of dimension two, for empty (respectively, occupied) bath sites]. All other lines are bond indices, and like in the MPS, they are summed over. As mentioned in the text, the bath is represented in star geometry because of the smaller bond dimensions needed. The bath sites are ordered according to their on-site energies. Two example hoppings $V_1$ and $V_2$ are drawn.

Figure 3

Spectral functions $A(\omega )$ for density-density interactions (DD) only (blue line), and with spin-flip and pair-hopping terms included (red line). In both calculations, we used $U=4.0\mathrm{eV}$ and $J=0.6\mathrm{eV}$. Both spectra show a three-peak structure in the upper Hubbard band and additional features at high energies (around 8 eV).

Figure 4

(a) We take the bath spectral function $\mathrm{\Delta }(\omega )$ from the DMFT self-consistent solution for $N_b=109$ and represent it using various numbers of bath sites. It is obvious that $N_b=9$ is too small to represent the bath well. (b) Converged DMFT spectral function using the AIM with different numbers of bath sites. Only the smallest bath shows a noticeable difference. This is mostly due to the fact that, in this case, a higher broadening of ${\eta }_{\mathrm{FT}}=0.1\mathrm{eV}$ had to be used in the Fourier transform and time evolution was only possible for $t=14{\mathrm{eV}}^{-1}$. The additional small structure at $\omega =0$ for $N_b=59$ bath sites is most likely a linear-prediction artifact.

Figure 5

DMFT spectral functions $A(\omega )$ from $\mathrm{CTQMC}+\mathrm{MaxEnt}$ (blue line) at $\beta =200{\mathrm{eV}}^{-1}$, and from FTPS (red line). The FTPS result shows a distinctive three-peak structure in the upper Hubbard band.

Figure 6

Comparison of imaginary-time Green’s functions $G(\tau )$ from CTQMC (${\mathrm{G}}_{\mathrm{CTQMC}}$, blue line) and FTPS using Eq. (15) (${\mathrm{G}}_{\mathrm{A}(\omega )}$, red squares). The agreement is also equally good at $\beta =100{\mathrm{eV}}^{-1}$ and $\beta =400{\mathrm{eV}}^{-1}$ (not shown). The difference between the two Green’s functions is shown in the bottom panel. Note that on both ends, ${\mathrm{G}}_{\mathrm{A}(\omega )}$ is smaller than ${\mathrm{G}}_{\mathrm{CTQMC}}$. The normalization of the spectral function demands that $G(\tau =0)+G(\tau =\beta )=-1$. The CTQMC data deviate in the order of ${10}^{-2}$ from this constraint due to statistical noise, while FTPS gives (by construction) the correct result to a precision of ${10}^{-8}$. This explains the bigger differences of the Green’s functions around $\tau =0$ and $\tau =\beta$. For better visibility of the $\tau >0$ data, the value of $9\times {10}^{-3}$ at $\tau =0$ is not shown.

Figure 7

Spectral functions from analytically continued imaginary-time Green’s functions $G(\tau )$ calculated by CTQMC (blue line) and by FTPS (red line). Clearly, the analytic continuation cannot resolve the peak structure in the upper Hubbard band.

Figure 8

(a) Close-up of the three-peak structure for various values of $J$. Additionally, we show vertical lines for the $J=0.6\mathrm{eV}$ spectrum at energies ${\omega }_M$ (position of the middle peak) and at ${\omega }_M+2J$ and ${\omega }_M-J$. We see that the width of the upper Hubbard band is close to $3J$. (b) Close-up of the small spectral peaks at high energies. These correspond to excitations into the $N=3$ sector of the atomic model (see Table 1). The height of each peak can be estimated by the degeneracy of the atomic states. Effective parameters $\overline{J}$ are 0.53 eV ($J=0.5\mathrm{eV}$), 0.59 eV ($J=0.6\mathrm{eV}$), and 0.68 eV ($J=0.7\mathrm{eV}$). They are obtained from the difference between the two peaks that are highest in energy.

Figure 9

(a) Increasing $U$ results in a slimmer central peak and a shift of the upper Hubbard band. Also, the three-peak structure gets smeared out. (b) Close-up of the upper Hubbard band. As in Fig. 8, additional vertical lines are plotted at ${\omega }_M$ (position of the middle peak) and at ${\omega }_M+2J$ and ${\omega }_M-J$ as a rough guide to where the atomic peaks would be located. With the help of these lines, one can discern a three-peak structure again, but it is extended by a feature at the low energy side of the Hubbard band.

Figure 10

Comparison of imaginary-time Green’s functions $G(\tau )$ from CTQMC (${\mathrm{G}}_{\mathrm{CTQMC}}$, blue line) and FTPS using Eq. (15) (${\mathrm{G}}_{\mathrm{A}(\omega )}$, red squares). As in Fig. 6, they compare very well.

Figure 11

Spectrum $A(\omega )$ using different linear prediction parameters for a calculation without spin-flip and pair-hopping terms. The calculations with linear prediction were performed with a broadening of ${\eta }_{\mathrm{FT}}=0.02\mathrm{eV}$. Except for small changes around $\omega =0$, the effect of the various linear prediction parameters is minor. The blue line lies directly below the red and green lines. We also show a DMFT calculation without any linear prediction. The main features are still present.

Figure 12

Different truncation values $t_w$ in the time evolution do not influence the shape of the spectrum $A(\omega )$.