Third-order strong-coupling impurity solver for real-frequency DMFT: Accurate spectral functions for antiferromagnetic and photo-doped states

TL;DR

This paper introduces a real-frequency, third-order strong-coupling impurity solver for DMFT that leverages the QTCI framework to efficiently evaluate high-dimensional diagram weights. By applying it to the single-band Hubbard model on the Bethe lattice, the authors benchmark NCA, OCA, and TOA against QMC and IF-MPS references across paramagnetic, antiferromagnetic, and photodoped steady states, demonstrating improved spectral resolution and convergence in many regimes. However, they also reveal limitations at weak coupling and near AFM phase boundaries, where unphysical features or slow convergence can occur, highlighting the need for careful integration schemes and potential vertex resummations. The work establishes a viable pathway toward high-order, real-frequency impurity solvers for both equilibrium and non-equilibrium DMFT, with promising extensions to multi-band and electron-boson coupled systems.

Abstract

We present a real-frequency third-order strong-coupling impurity solver which employs quantics tensor cross interpolation (QTCI) for an efficient evaluation of the diagram weights. Applying the method to dynamical mean-field theory (DMFT) calculations of the single-band Hubbard model on the Bethe lattice, we clarify the interaction and temperature range in which the third-order approach yields accurate results. Since the calculations are implemented on the real-time/frequency axis, the detailed structure of spectral functions can be obtained without analytical continuation, as we demonstrate with examples for paramagnetic, antiferromagnetic and photo-doped states. Our work establishes a viable path toward high-order, real-frequency impurity solvers for both equilibrium and non-equilibrium DMFT studies.

Figures (26)

• Figure 1: Kadanoff-Baym contour with forward, backward, and imaginary-time branches.
• Figure 2: Topologies of the skeleton diagrams for the pseudo-particle self-energy $\Sigma_\text{pp}(t',t)$ at the NCA, OCA, and TOA level. Arrows indicate the direction along the Keldysh contour. Red squares represent the first operator at time $t$, while blue circles denote the last operator at time $t'$. Dashed lines represent hybridization functions $\Delta$, which can have two possible directions. The two ends of a dashed line correspond to a pair of impurity creation and annihilation operators. The solid lines connecting $t$ and $t'$ are projected pseudo-particle Green's functions $\mathcal{G}$, which always follow the direction of the Keldysh contour.
• Figure 3: Topologies of the skeleton diagrams for the physical Green's function $G(t,t')$ within the NCA, OCA, and TOA approximations. Here, red squares correspond to annihilation operators, while blue circles correspond to creation operators.
• Figure 4: The ring-shaped Keldysh contour employed in this study. The branch-cut point connects the end of the backward branch to the beginning of the forward branch. If the time arguments of a Green's function, self-energy, or hybridization function cross this branch-cut point, the corresponding component is identified as the lesser component; otherwise, it is classified as the greater component.
• Figure 5: Illustration of three different QTCI-based encoding schemes for a two-variable function, where each variable $v_i$ ($i=1,2$) is represented by its binary digits $v_{ij}$.