##### Analytic Continuation of Multipoint Correlation Functions

A. X. Ge, J. Halbinger, S. S. B. Lee, J. von Delft, F. B. Kugler

Annalen Der Physik 536 (7), 46 (2024).

Conceptually, the Matsubara formalism (MF), using imaginary frequencies, and the Keldysh formalism (KF), formulated in real frequencies, give equivalent results for systems in thermal equilibrium. The MF has less complexity and is thus more convenient than the KF. However, computing dynamical observables in the MF requires the analytic continuation from imaginary to real frequencies. The analytic continuation is well-known for two-point correlation functions (having one frequency argument), but, for multipoint correlators, a straightforward recipe for deducing all Keldysh components from the MF correlator had not been formulated yet. Recently, a representation of MF and KF correlators in terms of formalism-independent partial spectral functions and formalism-specific kernels was introduced by Kugler, Lee, and von Delft [Phys. Rev. X 11, 041006 (2021)]. This representation is used to formally elucidate the connection between both formalisms. How a multipoint MF correlator can be analytically continued to recover all partial spectral functions and yield all Keldysh components of its KF counterpart is shown. The procedure is illustrated for various correlators of the Hubbard atom. This article explains how multipoint correlators in the imaginary-frequency Matsubara formalism (MF) can be analytically continued to the real-frequency Keldysh formalism (KF). The physical information in both types of correlators is fully encoded in partial spectral functions (PSFs). We analytically extract PSFs from MF correlators and give explicit formulas for the MF-to-KF analytic continuation of two-, three-, and four-point correlators. image

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

A. X. Ge, N. Ritz, E. Walter, S. Aguirre, J. von Delft, F. B. Kugler

Physical Review B 109 (11), 115128 (2024).

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 because 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 using the Keldysh instead of the Matsubara formalism with direct access to the self-energy and dynamical susceptibilities on the real-frequency axis. We present results from the functional renormalization group (fRG) at the one-loop level and from solving the self-consistent parquet equations in the parquet approximation. In contrast with previous Keldysh fRG works, we employ a parametrization of the four-point vertex which captures its full dependence on three real-frequency arguments. 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 with previous implementations, and that solving the parquet equations yields the best agreement with the numerical renormalization group 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.

##### Finite-size subthermal regime in disordered SU(*N*)-symmetric Heisenberg chains

D. Saraidaris, J. W. Li, A. Weichselbaum, J. von Delft, D. A. Abanin

Physical Review B 109 (9), 94201 (2024).

SU(N) symmetry is incompatible with the many -body localized (MBL) phase, even when strong disorder is present. However, recent studies have shown that finite -size SU(2) systems exhibit nonergodic, subthermal behavior, characterized by the breakdown of the eigenstate thermalization hypothesis, and by the excited eigenstates entanglement entropy that is intermediate between area and volume law. In this paper, we extend previous studies of the SU(2 )-symmetric disordered Heisenberg model to larger systems, using the time-dependent density matrix renormalization group (tDMRG) method. We simulate quench dynamics from weakly entangled initial states up to long times, finding robust subthermal behavior at stronger disorder. Although we find an increased tendency towards thermalization at larger system sizes, the subthermal regime persists at intermediate time scales, nevertheless, and therefore should be accessible experimentally. At weaker disorder, we observe signatures of thermalization,. however, entanglement entropy exhibits slow sublinear growth, in contrast to conventional thermalizing systems. Furthermore, we study dynamics of the SU(3 )-symmetric disordered Heisenberg model. Similarly, strong disorder drives the system into subthermal regime, albeit thermalizing phase is broader compared to the SU(2) case. Our findings demonstrate the robustness of the subthermal regime in spin chains with non -Abelian continuous symmetry, and are consistent with eventual thermalization at large system sizes and long time scales, suggested by previous studies.

##### Symmetric improved estimators for multipoint vertex functions

J. M. Lihm, J. Halbinger, J. Shim, J. von Delft, F. B. Kugler, S. S. B. Lee

Physical Review B 109 (12), 125138 (2024).

Multipoint vertex functions, and the four-point vertex in particular, are crucial ingredients in many-body theory. Recent years have seen significant algorithmic progress toward numerically computing their dependence on multiple frequency arguments. However, such computations remain challenging and are prone to suffer from numerical artifacts, especially in the real-frequency domain. Here, we derive estimators for multipoint vertices that are numerically more robust than those previously available. We show that the two central steps for extracting vertices from correlators, namely, the subtraction of disconnected contributions and the amputation of external legs, can be achieved accurately through repeated application of equations of motion, in a manner that is symmetric with respect to all frequency arguments and involves only fully renormalized objects. The symmetric estimators express the core part of the vertex and all asymptotic contributions through separate expressions that can be computed independently, without subtracting the large-frequency limits of various terms with different asymptotic behaviors. Our strategy is general and applies equally to the Matsubara formalism, the real-frequency zero-temperature formalism, and the Keldysh formalism. We demonstrate the advantages of the symmetric improved estimators by computing the Keldysh four-point vertex of the single-impurity Anderson model using the numerical renormalization group.

##### Quantics Tensor Cross Interpolation for High-Resolution Parsimonious Representations of Multivariate Functions

M. K. Ritter, Y. N. Fernández, M. Wallerberger, J. von Delft, H. Shinaoka, X. Waintal

Physical Review Letters 132 (5), 56501 (2024).

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, which combines the advantages of both schemes. We illustrate its potential with an application from condensed matter physics: the computation of Brillouin zone integrals.

##### Controlled Bond Expansion for Density Matrix Renormalization Group Ground State Search at Single-Site Costs

A. Gleis, J. W. Li, J. von Delft

Physical Review Letters 130 (24), 246402 (2023).

DMRG ground state search algorithms employing symmetries must be able to expand virtual bond spaces by adding or changing symmetry sectors if these lower the energy. Traditional single-site DMRG does not allow bond expansion,. two-site DMRG does, but at much higher computational costs. We present a controlled bond expansion (CBE) algorithm that yields two-site accuracy and convergence per sweep, at single-site costs. Given a matrix product state 'I' defining a variational space, CBE identifies parts of the orthogonal space carrying significant weight in H'I' and expands bonds to include only these. CBE-DMRG uses no mixing parameters and is fully variational. Using CBE-DMRG, we show that the KondoHeisenberg model on a width 4 cylinder features two distinct phases differing in their Fermi surface volumes.

##### U(1)-symmetric Gaussian fermionic projected entangled paired states and their Gutzwiller projection

J. W. Li, J. von Delft, H. H. Tu

Physical Review B 107 (8), 85148 (2023).

We develop a formalism for constructing particle-number-conserving Gaussian fermionic projected entangledpair states [U(1)-GfPEPSs] and show that these states can describe ground states of band insulators and gaplessfermions with band touching points. When using them as variationalAnsatzefor two Dirac fermion systems(the pi-flux model on the square lattice and the [0,pi]-flux model on the kagome lattice), we find that the U(1)-GfPEPSs, even with a relatively small bond dimension, can accurately approximate the Dirac Fermi sea groundstates. By applying Gutzwiller projectors on top of these U(1)-GfPEPSs, we obtain a PEPS representation ofU(1)-Dirac spin liquid states for spin-1/2 systems. With state-of-the-art tensor network numerics, the criticalexponent in the spin-spin correlation function of the Gutzwiller-projected pi-flux state is estimated to be eta approximate to 1.7.

##### Projector formalism for kept and discarded spaces of matrix product states

A. Gleis, J. W. Li, J. von Delft

Physical Review B 106 (19), 195138 (2022).

Any matrix product state |psi > has a set of associated kept and discarded spaces, needed for the description of |psi >, and changes thereof, respectively. These induce a partition of the full Hilbert space of the system into mutually orthogonal spaces of irreducible n-site variations of |psi >. Here, we introduce a convenient projector formalism and diagrammatic notation to characterize these n-site spaces explicitly. This greatly facilitates the formulation ofMPS algorithms that explicitly or implicitly employ discarded spaces. As an illustration, we derive an explicit expression for the n-site energy variance and evaluate it numerically for a model with long-range hopping. We also describe an efficient algorithm for computing low-lying n-site excitations above a finite MPS ground state.

##### Nonequilibrium spintronic transport through Kondo impurities

A. Manaparambil, A. Weichselbaum, J. von Delft, I. Weymann

Physical Review B 106 (12), 125413 (2022).

In this work we analyze the nonequilibrium transport through a quantum impurity (quantum dot or molecule) attached to ferromagnetic leads by using a hybrid numerical renormalization group-time-dependent density matrix renormalization group thermofield quench approach. For this, we study the bias dependence of the differential conductance through the system, which shows a finite zero-bias peak, characteristic of the Kondo resonance and reminiscent of the equilibrium local density of states. In the nonequilibrium settings, the resonance in the differential conductance is also found to decrease with increasing the lead spin polarization. The latter induces an effective exchange field that lifts the spin degeneracy of the dot level. Therefore, as we demonstrate, the Kondo resonance can be restored by counteracting the exchange field with a finite external magnetic field applied to the system. Finally, we investigate the influence of temperature on the nonequilibrium conductance, focusing on the split Kondo resonance. Our work thus provides an accurate quantitative description of the spin-resolved transport properties relevant for quantum dots and molecules embedded in magnetic tunnel junctions.

##### Benchmark calculations of multiloop pseudofermion fRG

M. K. Ritter, D. Kiese, T. Muller, F. B. Kugler, R. Thomale, S. Trebst, J. von Delft

European Physical Journal B 95 (7), 102 (2022).

The pseudofermion functional renormalization group (pffRG) is a computational method for determining zero-temperature phase diagrams of frustrated quantum magnets. In a recent methodological advance, the commonly employed Katanin truncation of the flow equations was extended to include multiloop corrections, thereby capturing additional contributions from the three-particle vertex (Thoenniss et al. https://arxiv.org/abs/2011.01268,. Kiese et al. https://arxiv.org/abs/2011.01269) . This development has also stimulated significant progress in the numerical implementation of pffRG, allowing one to track the evolution of pseudofermion vertices under the renormalization group flow with unprecedented accuracy. However, cutting-edge solvers differ in their integration algorithms, heuristics to discretize Matsubara frequency grids, and more. To lend confidence in the numerical robustness of state-of-the-art multiloop pffRG codes, we present and compare results produced with two independently developed and algorithmically distinct solvers for Heisenberg models on three-dimensional lattice geometries. Using the cubic lattice Heisenberg (anti)ferromagnet with nearest and next-nearest neighbor interactions as a generic benchmark model, we find the two codes to quantitatively agree, often up to several orders of magnitude in digital precision, both on the level of spin-spin correlation functions and renormalized fermionic vertices for varying loop orders. These benchmark calculations further substantiate the usage of multiloop pffRG solvers to tackle unconventional forms of quantum magnetism.

##### Multiloop flow equations for single-boson exchange fRG

M. Gievers, E. Walter, A. X. Ge, J. von Delft, F. B. Kugler

European Physical Journal B 95 (7), 108 (2022).

The recently introduced single-boson exchange (SBE) decomposition of the four-point vertex of interacting fermionic many-body systems is a conceptually and computationally appealing parametrization of the vertex. It relies on the notion of reducibility of vertex diagrams with respect to the bare interaction U, instead of a classification based on two-particle reducibility within the widely used parquet decomposition. Here, we re-derive the SBE decomposition in a generalized framework (suitable for extensions to, e.g., inhomogeneous systems or real-frequency treatments) following from the parquet equations. We then derive multiloop functional renormalization group (mfRG) flow equations for the ingredients of this SBE decomposition, both in the parquet approximation, where the fully two-particle irreducible vertex is treated as an input, and in the more restrictive SBE approximation, where this role is taken by the fully U-irreducible vertex. Moreover, we give mfRG flow equations for the popular parametrization of the vertex in terms of asymptotic classes of the two-particle reducible vertices. Since the parquet and SBE decompositions are closely related, their mfRG flow equations are very similar in structure.

##### Fulfillment of sum rules and Ward identities in the multiloop functional renormalization group solution of the Anderson impurity model

P. Chalupa-Gantner, F. B. Kugler, C. Hille, J. von Delft, S. Andergassen, A. Toschi

Physical Review Research 4 (2), 23050 (2022).

We investigate several fundamental characteristics of the multiloop functional renormalization group (mfRG) flow by hands of its application to a prototypical many-electron system: the Anderson impurity model (AIM). We first analyze the convergence of the algorithm in the different parameter regions of the AIM. As no additional approximation is made, the multiloop series for the local self-energy and response functions converge perfectly to the corresponding results of the parquet approximation (PA) in the weak- to intermediate-coupling regime. Small oscillations of the mfRG solution as a function of the loop order gradually increase with the interaction, hindering a full convergence to the PA in the strong-coupling regime, where perturbative resummation schemes are no longer reliable. By exploiting the converged results, we inspect the fulfillment of (i) sum rules associated to the Pauli principle and (ii) Ward identities related to conservation laws. For the Pauli principle, we observe a systematic improvement by increasing the loop order and including the multiloop corrections to the self-energy. This is consistent with the preservation of crossing symmetries and two-particle self-consistency in the PA. For the Ward identities, we numerically confirm a visible improvement by means of the Katanin substitution. At weak coupling, violations of the Ward identity are further reduced by increasing the loop order in mfRG. In this regime, we also determine the precise scaling of the deviations of the Ward identity as a function of the electronic interaction. For larger interaction values, the overall behavior becomes more complex, and the benefits of the higher-loop terms are mostly present in the contributions at large frequencies.

##### Abelian SU(N)(1 )chiral spin liquids on the square lattice

J. Y. Chen, J. W. Li, P. Nataf, S. Capponi, M. Mambrini, K. Totsuka, H. H. Tu, A. Weichselbaum, J. von Delft, D. Poilblanc

Physical Review B 104 (23), 235104 (2021).

In the physics of the fractional quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin like systems, we propose a family of SU(N)-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary N of an SU(3) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of SU(2)1 and SU(3)1 chiral spin liquids (CSL) on similar models we search for topological SU(N)1 CSL in some range of the Hamiltonian parameters via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small (periodic and open) clusters up to N = 10 and an innovative SU(N)-symmetric version of iDMRG to compute entanglement spectra on (infinitely long) cylinders in all topological sectors provide unambiguous signatures of the SU(N)1 character of the chiral liquids. An SU(4)-symmetric chiral PEPS, constructed in a manner similar to its SU(2) and SU(3) analogs, is shown to give a good variational ansatz of the N = 4 ground state, with chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups where the possibility of varying N provides a tuning parameter similar to the magnetic field in the physics of the FQH effect.

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

F. B. Kugler, S. S. B. Lee, J. von Delft

Physical Review X 11 (4), 41006 (2021).

The many-body problem is usually approached from one of two perspectives: the first originates from an action and is based on Feynman diagrams, the second is centered around a Hamiltonian and deals with quantum states and operators. The connection between results obtained in either way is made through spectral (or Lehmann) representations, well known for two-point correlation functions. Here, we complete this picture by deriving generalized spectral representations for multipoint correlation functions that apply in all of the commonly used many-body frameworks: the imaginary-frequency Matsubara and the realfrequency zero-temperature and Keldysh formalisms. Our approach separates spectral from time-ordering properties and thereby elucidates the relation between the three formalisms. The spectral representations of multipoint correlation functions consist of partial spectral functions and convolution kernels. The former are formalism independent but system specific,. the latter are system independent but formalism specific. Using a numerical renormalization group method described in the accompanying paper, we present numerical results for selected quantum impurity models. We focus on the four-point vertex (effective interaction) obtained for the single-impurity Anderson model and for the dynamical mean-field theory solution of the one-band Hubbard model. In the Matsubara formalism, we analyze the evolution of the vertex down to very low temperatures and describe the crossover from strongly interacting particles to weakly interacting quasiparticles. In the Keldysh formalism, we first benchmark our results at weak and infinitely strong interaction and then reveal the rich real-frequency structure of the dynamical mean-field theory vertex in the coexistence regime of a metallic and insulating solution.

##### Computing Local Multipoint Correlators Using the Numerical Renormalization Group

S. S. B. Lee, F. B. Kugler, J. von Delft

Physical Review X 11 (4), 41007 (2021).

Local three- and four-point correlators yield important insight into strongly correlated systems and have many applications. However, the nonperturbative, accurate computation of multipoint correlators is challenging, particularly in the real-frequency domain for systems at low temperatures. In the accompanying paper, we introduce generalized spectral representations for multipoint correlators. Here, we develop a numerical renormalization group approach, capable of efficiently evaluating these spectral representations, to compute local three- and four-point correlators of quantum impurity models. The key objects in our scheme are partial spectral functions, encoding the system's dynamical information. Their computation via numerical renormalization group allows us to simultaneously resolve various multiparticle excitations down to the lowest energies. By subsequently convolving the partial spectral functions with appropriate kernels, we obtain multipoint correlators in the imaginary-frequency Matsubara, the realfrequency zero-temperature, and the real-frequency Keldysh formalisms. We present exemplary results for the connected four-point correlators of the Anderson impurity model, and for resonant inelastic x-ray scattering spectra of related impurity models. Our method can treat temperatures and frequenciesimaginary or real-of all magnitudes, from large to arbitrarily small ones.

##### Differentiating Hund from Mott physics in a three-band Hubbard-Hund model: Temperature dependence of spectral, transport, and thermodynamic properties

K. M. Stadler, G. Kotliar, S. S. B. Lee, A. Weichselbaum, J. von Delft

Physical Review B 104 (11), 115107 (2021).

We study the interplay between Mott physics, driven by Coulomb repulsion U, and Hund physics, driven by Hund's coupling J, for a minimal model for Hund metals, the orbital-symmetric three-band Hubbard-Hund model (3HHM) for a lattice filling of 1/3. Hund-correlated metals are characterized by spin-orbital separation (SOS), a Hund's-rule-induced two-stage Kondo-type screening process, in which spin screening occurs at much lower energy scales than orbital screening. By contrast, in Mott-correlated metals, lying close to the phase boundary of a metal-insulator transition, the SOS window becomes negligibly small and the Hubbard bands are well separated. Using dynamical mean-field theory and the numerical renormalization group as real-frequency impurity solver, we identify numerous fingerprints distinguishing Hundness from Mottness in the temperature dependence of various physical quantities. These include ARPES-type spectra, the local self-energy, static local orbital and spin susceptibilities, resistivity, thermopower, and lattice and impurity entropies. Our detailed description of the behavior of these quantities within the context of a simple model Hamiltonian will be helpful for distinguishing Hundness from Mottness in experimental and theoretical studies of real materials.

##### Study of spin symmetry in the doped t-J model using infinite projected entangled pair states

J. W. Li, B. Bruognolo, A. Weichselbaum, J. von Delft

Physical Review B 103 (7), 75127 (2021).

We study the two-dimensional t-J model on a square lattice using infinite projected entangled pair states (iPEPS). At small doping, multiple orders, such as antiferromagnetic order, stripe order and superconducting order, are intertwined or compete with each other. We demonstrate the role of spin symmetry at small doping by either imposing SU(2) spin symmetry or its U(1) subgroup in the iPEPS ansatz, thereby excluding or allowing spontaneous spin-symmetry breaking, respectively, in the thermodynamic limit. From a detailed comparison of our simulations, we provide evidence that stripe order is pinned by long-range antiferromagnetic order. We also find SU(2) iPEPS, enforcing a spin-singlet state, yields a uniform charge distribution and favors d-wave singlet pairing.

##### Quantum many-body simulations of the two-dimensional Fermi-Hubbard model in ultracold optical lattices

B. B. Chen, C. Chen, Z. Y. Chen, J. Cui, Y. Y. Zhai, A. Weichselbaum, J. von Delft, Z. Y. Meng, W. Li

Physical Review B 103 (4), L041107 (2021).

Understanding quantum many-body states of correlated electrons is one main theme in modern condensedmatter physics. Given that the Fermi-Hubbard model, the prototype of correlated electrons, was recently realized in ultracold optical lattices, it is highly desirable to have controlled numerical methodology to provide precise finite-temperature results upon doping to directly compare with experiments. Here, we demonstrate the exponential tensor renormalization group (XTRG) algorithm [Chen et al., Plrys. Rev. X 8. 031082 (2018)], complemented by independent determinant quantum Monte Carlo, offers a powerful combination of tools for this purpose. XTRG provides full and accurate access to the density matrix and thus various spin and charge correlations, down to an unprecedented low temperature of a few percent of the tunneling energy. We observe excellent agreement with ultracold fermion measurements at both half filling and finite doping, including the sign-reversal behavior in spin correlations due to formation of magnetic polarons, and the attractive hole-doublon and repulsive hole-hole pairs that are responsible for the peculiar bunching and antibunching behaviors of the antimoments.

##### New signatures of the spin gap in quantum point contacts

K. L. Hudson, A. Srinivasan, O. Goulko, J. Adam, Q. Wang, L. A. Yeoh, O. Klochan, I. Farrer, D. A. Ritchie, A. Ludwig, A. D. Wieck, J. von Delft, A. R. Hamilton

Nature Communications 12 (1), 5 (2021).

One dimensional semiconductor systems with strong spin-orbit interaction are both of fundamental interest and have potential applications to topological quantum computing. Applying a magnetic field can open a spin gap, a pre-requisite for Majorana zero modes. The spin gap is predicted to manifest as a field dependent dip on the first 1D conductance plateau. However, disorder and interaction effects make identifying spin gap signatures challenging. Here we study experimentally and numerically the 1D channel in a series of low disorder p-type GaAs quantum point contacts, where spin-orbit and hole-hole interactions are strong. We demonstrate an alternative signature for probing spin gaps, which is insensitive to disorder, based on the linear and non-linear response to the orientation of the applied magnetic field, and extract a spin-orbit gap Delta E approximate to 500 mu eV. This approach could enable one-dimensional hole systems to be developed as a scalable and reproducible platform for topological quantum applications. In one-dimensional systems, the combination of a strong spin-orbit interaction and an applied magnetic field can give rise to a spin-gap, however experimental identification is difficult. Here, the authors present new signatures for the spin-gap, and verify these experimentally in hole QPCs.

##### Renormalized Lindblad driving: A numerically exact nonequilibrium quantum impurity solver

M. Lotem, A. Weichselbaum, J. von Delft, M. Goldstein

Physical Review Research 2 (4), 43052 (2020).

"The accurate characterization of nonequilibrium strongly correlated quantum systems has been a longstanding challenge in many-body physics. Notable among them are quantum impurity models, which appear in various nanoelectronic and quantum computing applications. Despite their seeming simplicity, they feature correlated phenomena, including small emergent energy scales and non-Fermi-liquid physics, requiring renormalization group treatment. This has typically been at odds with the description of their nonequilibrium steady state under finite bias, which exposes their nature as open quantum systems. We present a numerically exact method for obtaining the nonequilibrium state of a general quantum impurity coupled to metallic leads at arbitrary voltage or temperature bias, which we call ""RL-NESS"" (renormalized Lindblad-driven nonequilibrium steady state). It is based on coherently coupling the impurity to discretized leads which are treated exactly. These leads are furthermore weakly coupled to reservoirs described by Lindblad dynamics which impose voltage or temperature bias. Going beyond previous attempts, we exploit a hybrid discretization scheme for the leads together with Wilson's numerical renormalization group, in order to probe exponentially small energy scales. The steady state is then found by evolving a matrix-product density operator via real-time Lindblad dynamics, employing a dissipative generalization of the time-dependent density matrix renormalization group. In the long-time limit, this procedure successfully converges to the steady state at finite bond dimension due to the introduced dissipation, which bounds the growth of entanglement. We thoroughly test the method against the exact solution of the noninteracting resonant level model. We then demonstrate its power using an interacting two-level model, for which it correctly reproduces the known limits, and gives the full I-V curve between them."

##### Quantitative functional renormalization group description of the two-dimensional Hubbard model

C. Hille, F. B. Kugler, C. J. Eckhardt, Y. Y. He, A. Kauch, C. Honerkamp, A. Toschi, S. Andergassen

Physical Review Research 2 (3), 33372 (2020).

Using a leading algorithmic implementation of the functional renormalization group (fRG) for interacting fermions on two-dimensional lattices, we provide a detailed analysis of its quantitative reliability for the Hubbard model. In particular, we show that the recently introduced multiloop extension of the fRG flow equations for the self-energy and two-particle vertex allows for a precise match with the parquet approximation also for two-dimensional lattice problems. The refinement with respect to previous fRG-based computation schemes relies on an accurate treatment of the frequency and momentum dependences of the two-particle vertex, which combines a proper inclusion of the high-frequency asymptotics with the so-called truncated unity fRG for the momentum dependence. The adoption of the latter scheme requires, as an essential step, a consistent modification of the flow equation of the self-energy. We quantitatively compare our fRG results for the self-energy and momentum-dependent susceptibilities and the corresponding solution of the parquet approximation to determinant quantum Monte Carlo data, demonstrating that the fRG is remarkably accurate up to moderate interaction strengths. The presented methodological improvements illustrate how fRG flows can be brought to a quantitative level for two-dimensional problems, providing a solid basis for the application to more general systems.

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

E. Walter, K. M. Stadler, S. S. B. Lee, Y. Wang, G. Kotliar, A. Weichselbaum, J. von Delft

Physical Review X 10 (3), 31052 (2020).

"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, 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, T-sp < T-orb, 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-ofthe-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. We characterize the NFL fixed point of Hund metals in detail for a Kondo model with an impurity forming an SU(2) x SU(3) spin-orbital multiplet, tuned such that the NFL energy window is very wide. The impurity's spin and orbital susceptibilities then exhibit striking power-law behavior, which we explain using CFT arguments. We find excellent agreement between CFT predictions and numerical renormalization group results. Our main physical conclusion is that the regime of spin-orbital separation, where orbital degrees of freedom have been screened but spin degrees of freedom have not, features anomalously strong local spin fluctuations: the impurity susceptibility increases as chi(imp)(sp) similar to omega(-gamma), with gamma > 1."

##### Automatic differentiation for second renormalization of tensor networks

B. B. Chen, Y. Gao, Y. B. Guo, Y. Z. Liu, H. H. Zhao, H. J. Liao, L. Wang, T. Xiang, W. Li, Z. Y. Xie

Physical Review B 101 (22), 220409 (2020).

"Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models. Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable TRG (partial derivative TRG) that can be applied to improve various TRG methods, in an automatic fashion. partial derivative TRG systematically extends the essential concept of second renormalization [Phys. Rev. Lett. 103. 160601 (2009)] where the tensor environment is computed recursively in the backward iteration. Given the forward TRG process, partial derivative TRG automatically finds the gradient of local tensors through backpropagation, with which one can deeply ""train"" the tensor networks. We benchmark partial derivative TRG in solving the square-lattice Ising model, and we demonstrate its power by simulating one- and two-dimensional quantum systems at finite temperature. The global optimization as well as GPU acceleration renders partial derivative TRG a highly efficient and accurate many-body computation approach."

##### Global Phase Diagram of a Spin-Orbital Kondo Impurity Model and the Suppression of Fermi-Liquid Scale

Y. Wang, E. Walter, S. S. B. Lee, K. M. Stadler, J. von Delft, A. Weichselbaum, G. Kotliar

Physical Review Letters 124 (13), 136406 (2020).

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, TFL 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. The new non-Fermi-liquid phase shows frustrated behavior involving alternating overscreenings in spin and orbital sectors, with universal power laws in the spin (omega(-1/5)), orbital (omega(1/5)) and spin-orbital (omega(1)) dynamical susceptibilities. These power laws, and the NRG eigenlevel spectra, can be fully understood using conformal field theory arguments, which also clarify the nature of the non-Fermi-liquid phase.

##### Non-Fermi-liquid Kondo screening under Rabi driving

S. S. B. Lee, J. von Delft, M. Goldstein

Physical Review B 101 (8), 85110 (2020).

We investigate a Rabi-Kondo model describing an optically driven two-channel quantum dot device featuring a non-Fermi-liquid Kondo effect. Optically induced Rabi oscillation between the valence and conduction levels of the dot gives rise to a two-stage Kondo effect: Primary screening of the local spin is followed by secondary nonequilibrium screening of the local orbital degree of freedom. Using bosonization arguments and the numerical renormalization group, we compute the dot emission spectrum and residual entropy. Remarkably, both exhibit two-stage Kondo screening with non-Fermi-liquid properties at both stages.

##### Strongly Correlated Materials from a Numerical Renormalization Group Perspective: How the Fermi-Liquid State of Sr2RuO4 Emerges

F. B. Kugler, M. Zingl, H. U. R. Strand, S. S. B. Lee, J. von Delft, A. Georges

Physical Review Letters 124 (1), 16401 (2020).

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. By computing real-frequency correlation functions, we directly observe this spin-orbital scale separation and show that the van Hove singularity drives strong orbital differentiation. We extract quasiparticle interaction parameters from the low-energy spectrum and find an effective attraction in the spin-triplet sector.

##### Orbital differentiation in Hund metals

F. B. Kugler, S. S. B. Lee, A. Weichselbaum, G. Kotliar, J. von Delft

Physical Review B 100 (11), 115159 (2019).

Orbital differentiation is a common theme in multiorbital systems, yet a complete understanding of it is still missing. Here, we consider a minimal model for orbital differentiation in Hund metals with a highly accurate method: We use the numerical renormalization group as a real-frequency impurity solver for a dynamical mean-field study of three-orbital Hubbard models, where a crystal field shifts one orbital in energy. The individual phases are characterized with dynamic correlation functions and their relation to diverse Kondo temperatures. Upon approaching the orbital-selective Mott transition, we find a strongly suppressed spin coherence scale and uncover the emergence of a singular Fermi liquid and interband doublon-holon excitations. Our theory describes the diverse polarization-driven phenomena in the t(2g) bands of materials such as ruthenates and iron-based superconductors, and our methodological advances pave the way toward real-frequency analyses of strongly correlated materials.

##### Thermal tensor renormalization group simulations of square-lattice quantum spin models

H. Li, B. B. Chen, Z. Y. Chen, J. von Delft, A. R. A. Weichselbaum, W. Li

Physical Review B 100 (4), 45110 (2019).

In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a finite thermal correlation length xi, such that for system sizes L >> xi finite-size calculations actually simulate the thermodynamic limit. In this paper, we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width W = 10. We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc. are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH, we obtain an energy per site u*(g) similar or equal to -0.6694(4) and a spontaneous magnetization m*(S) similar or equal to 0.30(1) already consistent with the ground-state properties, which is obtained from extrapolated low-T thermal data on W <= 8 cylinders and W <= 10 open strips, respectively. We extract an exponential divergence versus T of the structure factor S(M), as well as the correlation length xi, at the ordering wave vector M = (pi, pi), which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analyzing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature T-c.

##### Two-temperature scales in the triangular-lattice Heisenberg antiferromagnet

L. Chen, D. W. Qu, H. Li, B. B. Chen, S. S. Gong, J. von Delft, A. Weichselbaum, W. Li

Physical Review B 99 (14), 140404 (2019).

"The anomalous thermodynamic properties of the paradigmatic frustrated spin-1/2 triangular-lattice Heisenberg antiferromagnet (TLH) has remained an open topic of research over decades, both experimentally and theoretically. Here, we further the theoretical understanding based on the recently developed, powerful exponential tensor renormalization group method on cylinders and stripes in a quasi-one-dimensional (1D) setup, as well as a tensor product operator approach directly in 2D. The observed thermal properties of the TLH are in excellent agreement with two recent experimental measurements on the virtually ideal TLH material Ba8CoNb6O24. Remarkably, our numerical simulations reveal two crossover temperature scales, at T-l/J similar to 0.20 and T-h/J similar to 0.55, with J the Heisenberg exchange coupling, which are also confirmed by a more careful inspection of the experimental data. We propose that in the intermediate regime between the low-temperature scale T-l and the higher one T-h, the ""rotonlike"" excitations are activated with a strong chiral component and a large contribution to thermal entropies. Bearing remarkable resemblance to the renowned roton thermodynamics in liquid helium, these gapped excitations suppress the incipient 120 degrees order that emerges for temperatures below T-l."