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arXiv:2103.05545 · Physical review. B./Physical review. B

Tree tensor-network real-time multiorbital impurity solver: Spin-orbit coupling and correlation functions in Sr2RuO4

Xiaodong Cao, Yi Lu, P. Hansmann, M. W. Haverkort

Published Sep 9, 2021• 22 citations
arXivHTMLPDF

Abstract

We present a tree tensor-network impurity solver suited for general multiorbital systems. The network is constructed to efficiently capture the entanglement structure and symmetry of an impurity problem. The solver works directly on the real-time/frequency axis and generates spectral functions with energy-independent resolution of the order of one percent of the correlated bandwidth. Combined with an optimized representation of the impurity bath, it efficiently solves self-consistent dynamical mean-field equations and calculates various dynamical correlation functions for systems with off-diagonal Green's functions. For the archetypal correlated Hund's metal ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$, we show that both the low-energy quasiparticle spectra related to the van Hove singularity and the high-energy atomic multiplet excitations can be faithfully resolved. In particular, we show that while the spin-orbit coupling has only minor effects on the orbital-diagonal one-particle spectral functions, it has a more profound impact on the low-energy spin and orbital response functions.

DOI
10.1103/physrevb.104.115119

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We present a tree tensor-network impurity solver suited for general multiorbital systems. The network is constructed to efficiently capture the entanglement structure and symmetry of an impurity problem. The solver works directly on the real-time/frequency axis and generates spectral functions with energy-independent resolution of the order of one percent of the correlated bandwidth. Combined with an optimized representation of the impurity bath, it efficiently solves self-consistent dynamical mean-field equations and calculates various dynamical correlation functions for systems with off-diagonal Green's functions. For the archetypal correlated Hund's metal Sr2RuO4{\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}Sr2​RuO4​, we show that both the low-energy quasiparticle spectra related to the van Hove singularity and the high-energy atomic multiplet excitations can be faithfully resolved. In particular, we show that while the spin-orbit coupling has only minor effects on the orbital-diagonal one-particle spectral functions, it has a more profound impact on the low-energy spin and orbital response functions.
We introduce a tree tensor network (TTN) impurity solver that enables highly efficient and accurate real-time simulations of quantum impurity models. By decomposing a noninteracting bath Hamiltonian into a Cayley tree, the method provides a tensor network representation that naturally captures the multiscale entanglement structure intrinsic to impurity-bath systems. This geometry differs from conventional chain-based mappings and yields a substantial reduction of entanglement, allowing accurate ground-state properties and long-time dynamics to be captured at significantly lower bond dimensions. Benchmark calculations for the single-impurity Anderson model demonstrate that the TTN solver achieves markedly enhanced resolution of real-frequency spectral functions, without invoking analytic continuation. This impurity solver provides a balanced, scale-uniform description of impurity physics and offers a versatile approach for real-time dynamical mean-field theory and related applications involving quantum impurity models.
Using an imaginary-time matrix-product state (MPS) based quantum impurity solver we perform a realistic dynamical mean-field theory (DMFT) calculation combined with density functional theory (DFT) for Sr2RuO4. We take the full Hubbard-Kanamori interactions and spin-orbit coupling (SOC) into account. The MPS impurity solver works at essentially zero temperature in the presence of SOC, a regime of parameters currently inaccessible to continuous-time quantum Monte Carlo methods, due to a severe sign problem. We show that earlier results obtained at high temperature, namely, that the diagonal self-energies are nearly unaffected by SOC and that interactions lead to an effective enhancement of the SOC, hold even at low temperature. We observe that realism makes the numerical solution of the impurity model with MPS much more demanding in comparison to earlier works on Bethe lattice models, requiring several algorithmic improvements.
We present a minimally entangled typical thermal state quantum impurity solver for general multiorbital systems at finite temperatures. We introduce an improved estimator for the single-particle Green's function that strongly reduces the large fluctuations at long imaginary time and low temperature, which were a severe limitation of the original algorithm. In combination with the fork tensor product states Ansatz, we obtain a dynamical mean field theory (DMFT) quantum impurity solver, which we benchmark for single and three-band models down to low temperatures, including the effect of spin-orbit coupling in a realistic DMFT computation for the Hund's metal Sr2RuO4{\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}Sr2​RuO4​ down to low temperatures.
The full frequency dependence of electronic interactions is central to dynamical correlations, but its numerical cost restricts multiorbital calculations. We formulate the multiloop functional renormalization group directly in the quantics tensor train representation, retaining all three independent vertex frequencies over broad ranges without an explicit high-frequency parametrization. Analytical tensor operators perform frequency transformations and internal summations without dense vertex intermediates. Multiloop corrections improve agreement of vertices and responses with independent references and reproduce the low-energy numerical renormalization group self-energy. Lower-temperature calculations resolve a quadratic frequency contribution to the analytically continued scattering rate. A degenerate two-orbital impurity exhibits distinct spin, orbital, and charge responses at fixed Hund coupling. In a compensated altermagnetic environment, an orbital-level splitting induces a local spin response despite zero magnetization of the unperturbed impurity. Increasing Hund coupling suppresses the orbital response while increasing its spin polarization. Renormalized one-particle propagation alone does not reproduce this strong polarization dependence. These results extend compressed multiloop calculations to dynamical correlations in multiorbital environments.
We present the combination of a complex-time tensor-network impurity solver with an analytic continuation scheme based on exponential fitting as an efficient framework for single and multiorbital dynamical mean-field calculations. By performing time evolution along a complex-time contour, the approach balances computational cost with the difficulty of spectral recovery, offering greater flexibility than methods confined to the real or imaginary axis. By complementing the complex-time evolution with an exponential fitting scheme, we faithfully extract real-time information at negligible cost. The resulting method obtains high-resolution spectra at a significantly lower computational cost than real-time evolution, offering a promising tool for studies of strongly correlated materials.
We implement an efficient numerical method to calculate response functions of complex impurities based on the Density Matrix Renormalization Group (DMRG) and use it as the impurity-solver of the Dynamical Mean Field Theory (DMFT). This method uses the correction vector to obtain precise Green's functions on the real frequency axis at zero temperature. By using a self-consistent bath configuration with very low entanglement, we take full advantage of the DMRG to calculate dynamical response functions paving the way to treat large effective impurities such as those corresponding to multi-orbital interacting models and multi-site or multi-momenta clusters. This method leads to reliable calculations of non-local self energies at arbitrary dopings and interactions and at any energy scale.