# Theoretical Solid State Physics

Our group focuses on the study of strongly correlated many particle systems, where interactions play an important role. To do so we use and develop advanced theoretical tools, in particular tensor network and functional renormalization group methods.

## Current research highlights

## Symmetric improved estimators for multipoint vertex functions

Phys. Rev. B 109, 125138 (2024), J.-M. Lihm, J. Halbinger, J. Shim, J. von Delft, F. B. Kugler, S.-S. B. Lee

The two-particle vertex is a key quantity in quantum many-body physics, which describes how the effective interaction between particles depends on the particles’ energies. However, due to its complicated frequency dependence and many potential sources of numerical errors, the accurate calculation of the vertex, especially for strong interactions and low temperatures, remains extremely challenging. Here, we develop a new representation of vertex functions, “symmetric improved estimators,” that improve the accuracy and stability of their calculations.

classified The estimators naturally decompose a vertex into the sum of terms with different characteristic behaviors and large-frequency limits. As each term is evaluated separately, one thereby accurately resolves the multifaceted frequency dependence.

We use our method, in combination with the multipoint numerical renormalization group, to calculate the two-particle vertex of the Anderson impurity model, a prototypical many-body Hamiltonian. We demonstrate that using symmetric improved estimators drastically improves the quality of the vertex compared to using the asymmetric improved estimator. The accuracy of the vertex is demonstrated by multiple stringent tests, such as the agreement with perturbation theory in the weak-coupling regime, with Fermi-liquid theory at energy scales below the Kondo temperature, and with general fluctuation-dissipation relations.

Our finding opens the way for studying electronic correlation in systems or parameter regimes that had previously been beyond computational reach.

## Real-frequency quantum field theory applied to the single-impurity Anderson model

Phys. Rev. B 109, 115128 (2024)(1995), A. Ge, N. Ritz, E. Walter, S. Aguirre, J. v. Delft, F. B. Kugler

A major challenge in the field of correlated electrons is the computation of dynamical correlation functions. For comparisons with experiment, one is interested in their real-frequency dependence. This is difficult to compute, as imaginary-frequency data from the Matsubara formalism require analytic continuation, a numerically ill-posed problem. Here, we apply quantum field theory to the single-impurity Anderson model (AM), using the Keldysh instead of the Matsubara formalism with direct access to the self-energy and dynamical susceptibilities on the real-frequency axis. We presentresults from the functional renormalization group (fRG) at one-loop level and from solving the selfconsistent parquet equations in the parquet approximation. In contrast to previous Keldysh fRG works, we employ a parametrization of the four-point vertex which captures its full dependence on three real-frequency arguments.

classified We compare our results to benchmark data obtained with the numerical renormalization group and to second-order perturbation theory. We find that capturing the full frequency dependence of the four-point vertex significantly improves the fRG results compared to previous implementations, and that solving the parquet equations yields the best agreement with the NRG benchmark data, but is only feasible up to moderate interaction strengths. Our methodical advances pave the way for treating more complicated models in the future.

## Projective symmetry group classification of chiral Z2 spin liquids on the pyrochlore lattice: Application to the spin-1/2 XXZ Heisenberg model

Phys. Rev. B 105, 125122 (2022), B. Schneider, J. C. Halimeh, M. Punk

Simplified models for quantum spin-ice materials, such as the XXZ model on the pyrochlore lattice, may host interesting new spin-liquid ground states. In this work, we provide a complete classification of nearest-neighbor chiral Z2 spin-liquid states on the pyrochlore lattice using a projective symmetry group analysis. Moreover, the newly

classified states are applied to the XXZ model within a self-consistent mean-field scheme. We find a new class of spin-liquid states, the pi/3-states, that inherit their properties from fractionalization of the C3 symmetry of the pyrochlore lattice. Further, we discuss observable signatures that can be measured in neutron scattering experiments. Remarkably, the spin liquids preserve the SU(2) symmetry even in the presence of SU(2) symmetry breaking interactions.

## Multipoint Correlation Functions: Spectral Representation and Numerical Evaluation &

Computing Local Multipoint Correlators Using the Numerical Renormalization Group

Phys. Rev. X 11, 041007 (2021), S.-S. B. Lee ,F. B. Kugler, J. von Delft

We show how to compute multipoint correlation functions of quantum many-body systems with unprecedented accuracy. Multipoint functions describe correlations between multiple quantum particles at different space-time points. Such functions pervade many branches of physics and are relevant for interpreting numerous types of experimental measurements, like electrical conductivities and scattering spectra of photons or neutrons. However, multipoint functions are notoriously difficult to compute, especially for strong interactions and low temperatures of interest for understanding quantum systems such as high-temperature superconductors, heavy fermions, or Hund metals.

In [F. B. Kugler, S.-S. B. Lee, and J. von Delft, Phys. Rev. X 11, 041006 (2021)], we develop a theoretical framework ("spectral representations") expressing general multipoint functions through the system's quantum states, revealing their structure with great clarity. In [S.-S. B. Lee, F. B. Kugler, and J. von Delft, Phys. Rev. X 11, 041007 (2021)], we devise a powerful method for numerically evaluating the spectral representations of local multipoint functions. With this, we compute effective two-particle interactions ("vertex functions") and, as an example, spectra for resonant inelastic x-ray scattering (RIXS), a promising experimental tool in solid-state physics. Our novel approach allows us to accurately treat energies and temperatures ranging over many orders of magnitude, beyond the reach of other numerical methods. This makes it a valuable addition to the toolbox of theoretical techniques for studying quantum matters.

## Uncovering Non-Fermi-Liquid Behavior in Hund Metals: Conformal Field Theory Analysis of an SU(2)×SU(3) Spin-Orbital Kondo Model,

Phys. Rev. X 10, 031052 (2020), A. M. Tsvelik 1 and O. M. Yevtushenko

Hund metals have attracted attention in recent years due to their unconventional superconductivity, which supposedly originates from non-Fermi-liquid (NFL) properties of the normal state. When studying Hund metals using dynamical mean-field theory (DMFT), one arrives at a self-consistent "Hund impurity problem" involving a multiorbital quantum impurity with nonzero Hund coupling interacting with a metallic bath. If its spin and orbital degrees of freedom are screened at different energy scales, the intermediate energy window is governed by a novel NFL fixed point, whose nature had not yet been clarified. We resolve this problem by providing an analytical solution of a paradigmatic example of a Hund impurity problem, involving two spin and three orbital degrees of freedom. To this end, we combine a state-of-the-art implementation of the numerical renormalization group, capable of exploiting non-Abelian symmetries, with a generalization of Affleck and Ludwig’s conformal field theory (CFT) approach for multichannel Kondo models.

Multivariate functions of continuous variables arise in countless branches of science. Numerical computations with such functions typically involve a compromise between two contrary desiderata: accurate resolution of the functional dependence, versus parsimonious memory usage. Recently, two promising strategies have emerged for satisfying both requirements: (i) The quantics representation, which expresses functions as multi-index tensors, with each index representing one bit of a binary encoding of one of the variables; and (ii) tensor cross interpolation (TCI), which, if applicable, yields parsimonious interpolations for multi-index tensors. Here, we present a strategy, quantics TCI (QTCI), which combines the advantages of both schemes. We illustrate its potential with an application from condensed matter physics: the computation of Brillouin zone integrals.

Many correlated metallic materials are described by Landau Fermi-liquid theory at low energies, but for Hund metals the Fermi-liquid coherence scale T_FL is found to be surprisingly small. In this Letter, we study the simplest impurity model relevant for Hund metals, the three-channel spin-orbital Kondo model, using the numerical renormalization group (NRG) method and compute its global phase diagram. In this framework, T_FL becomes arbitrarily small close to two new quantum critical points that we identify by tuning the spin or spin-orbital Kondo couplings into the ferromagnetic regimes. We find quantum phase transitions to a singular Fermi-liquid or a novel non-Fermi-liquid phase. We compute dynamical susceptibilities in these phases, which reveal universal power laws. These can be fully understood using conformal field theory arguments, which also clarify the nature of the non-Fermi-liquid phase.

The crossover from fluctuating atomic constituents to a collective state as one lowers temperature or energy is at the heart of the dynamical mean-field theory description of the solid state. We demonstrate that the numerical renormalization group is a viable tool to monitor this crossover in a real-materials setting. The renormalization group flow from high to arbitrarily small energy scales clearly reveals the emergence of the Fermi-liquid state of Sr2RuO4. We find a two-stage screening process, where orbital fluctuations are screened at much higher energies than spin fluctuations, and Fermi-liquid behavior, concomitant with spin coherence, below a temperature of 25 K. Our work demonstrates the potential of DFT+DMFT+NRG as a new computational paradigm for real-material systems to (i) directly access real-frequency properties at arbitrarily low temperatures and (ii) reveal and elucidate the intricate renormalization process that occurs during the dressing of atomic excitations by their solid-state environment.

We demonstrate that low dimensional Kondo-Heisenberg systems, consisting of itinerant electrons and localized magnetic moments (Kondo impurities), can be used as a principally new platform to realize scalar chiral spin order. The underlying physics is governed by a competition of the Ruderman-Kittel-Kosuya-Yosida (RKKY) indirect exchange interaction between the local moments with the direct Heisenberg one. When the direct exchange is weak and RKKY dominates, the isotropic system is in the disordered phase. A moderately large direct exchange leads to an Ising-type phase transition to the phase with chiral spin order. Our finding paves the way towards pioneering experimental realizations of the chiral spin liquid in systems with spontaneously broken time-reversal symmetry.