##### 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, 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.

##### 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, 075127 (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. Chen, J. Cui, Y. Zhai, A. Weichselbaum, J. von Delft, Z.Y. Meng, W. Li

Physical Review B 103, 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, 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, 043052 (2021).

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, 033372 (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, 031052 (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. Liu, H.-H. Zhao, H.-J. Liao, L. Wang, T. Xiang, W. Li, Z.Y. Xie

Physical Review B 101, 220409(R) (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, 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, 085110 (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, 016401 (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.

##### 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), 045110 (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, 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.