We show how spectral functions for quantum impurity models can be calculated very accurately using a complete set of "discarded" numerical renormalization group eigenstates, recently introduced by Anders and Schiller. The only approximation is to judiciously exploit energy scale separation. Our derivation avoids both the overcounting ambiguities and the single-shell approximation for the equilibrium density matrix prevalent in current methods, ensuring that relevant sum rules hold rigorously and spectral features at energies below the temperature can be described accurately. PACS numbers: 71.27.+a, 75.20.Hr, 73.21.La Quantum impurity models describe a quantum system with a small number of discrete states, the "impurity", coupled to a continuous bath of fermionic or bosonic excitations. Such models are relevant for describing transport through quantum dots, for the treatment of correlated lattice models using dynamical mean field theory, or for the modelling of the decoherence of qubits.The impurity's dynamics in thermal equilibrium can be characterized by spectral functions of the type A BC (ω) = dt 2π e iωt B (t)Ĉ T . Their Lehmann representation readswith Z = a e −βEa and E ba = E b − E a , which implies the sum rule dωA BC (ω) = BĈ T . In this Letter, we describe a strategy for numerically calculating A BC (ω) that, in contrast to previous methods, rigorously satisfies this sum rule and accurately describes both high and low frequencies, including ω T , which we test by checking our results against exact Fermi liquid relations. Our work builds on Wilson's numerical renormalization group (NRG) method [1]. Wilson discretized the environmental spectrum on a logarithmic grid of energies Λ −n , (with Λ > 1, 1 ≤ n ≤ N → ∞), with exponentially high resolution of low-energy excitations, and mapped the impurity model onto a "Wilson tight-binding chain", with hopping matrix elements that decrease exponentially as Λ −n/2 with site index n. Because of this separation of energy scales, the Hamiltonian can be diagonalized iteratively: adding one site at a time, a new "shell" of eigenstates is constructed from the new site's states and the M K lowest-lying eigenstates of the previous shell (the so-called "kept" states), while "discarding" the rest.Subsequent authors [2,3,4,5,6,7,8,9,10] have shown that spectral functions such as A BC (ω) can be calculated via the Lehmann sum, using NRG states (kept and discarded) of those shells n for which ω ∼ Λ −n/2 . Though plausible on heuristic grounds, this strategy entails double-counting ambiguities [5] about how to combine data from successive shells. Patching schemes [9] for addressing such ambiguities involve arbitrariness. As a result, the relevant sum rule is not satisfied rigorously, with typical errors of a few percent. Also, the density matrix (DM)ρ = e −βĤ /Z has hitherto been represented rather crudely using only the single N T -th shell for which, with a chain of length N = N T , resulting in inaccurate spectral information for ω T . In this Letter we avoid these probl...

A general framework for non-abelian symmetries is presented for matrix-product and tensornetwork states in the presence of well-defined orthonormal local as well as effective basis sets. The two crucial ingredients, the Clebsch-Gordan algebra for multiplet spaces as well as the Wigner-Eckart theorem for operators, are accounted for in a natural, well-organized, and computationally straightforward way. The unifying tensor-representation for quantum symmetry spaces, dubbed QSpace, is particularly suitable to deal with standard renormalization group algorithms such as the numerical renormalization group (NRG), the density matrix renormalization group (DMRG), or also more general tensor networks such as the multi-scale entanglement renormalization ansatz (MERA). In this paper, the focus is on the application of the non-abelian framework within the NRG. A detailed analysis is presented for a fully screened spin-3/2 three-channel Anderson impurity model in the presence of conservation of total spin, particle-hole symmetry, and SU(3) channel symmetry. The same system is analyzed using several alternative symmetry scenarios. This includes the more traditional symmetry setting SU(2)spin ⊗ SU(2) ⊗3 charge , the larger symmetry SU(2)spin ⊗ U(1) charge ⊗ SU(3) channel , and their much larger enveloping symplectic symmetry SU(2)spin ⊗ Sp(6). These are compared in detail, including their respective dramatic gain in numerical efficiency. In the appendix, finally, an extensive introduction to non-abelian symmetries is given for practical applications, together with simple self-contained numerical procedures to obtain Clebsch-Gordan coefficients and irreducible operators sets. The resulting QSpace tensors can deal with any set of abelian symmetries together with arbitrary non-abelian symmetries with compact, i.e. finitedimensional, semi-simple Lie algebras.

For phase transitions in dissipative quantum impurity models, the existence of a quantum-toclassical correspondence has been discussed extensively. We introduce a variational matrix product state approach involving an optimized boson basis, rendering possible high-accuracy numerical studies across the entire phase diagram. For the sub-ohmic spin-boson model with a power-law bath spectrum ∝ ω s , we confirm classical mean-field behavior for s < 1/2, correcting earlier numerical renormalization-group results. We also provide the first results for an XY-symmetric model of a spin coupled to two competing bosonic baths, where we find a rich phase diagram, including both critical and strong-coupling phases for s < 1, different from that of classical spin chains. This illustrates that symmetries are decisive for whether or not a quantum-to-classical correspondence exists.

We show that recursively generated Chebyshev expansions offer numerically efficient representations for calculating zero-temperature spectral functions of one-dimensional lattice models using matrix product state (MPS) methods. The main features of this Chebychev matrix product state (CheMPS) approach are: (i) it achieves uniform resolution over the spectral function's entire spectral width; (ii) it can exploit the fact that the latter can be much smaller than the model's many-body bandwidth; (iii) it offers a well-controlled broadening scheme that allows finite-size effects to be either resolved or smeared out, as desired; (iv) it is based on using MPS tools to recursively calculate a succession of Chebychev vectors |tn , (v) whose entanglement entropies were found to remain bounded with increasing recursion order n for all cases analyzed here; (vi) it distributes the total entanglement entropy that accumulates with increasing n over the set of Chebyshev vectors |tn , which need not be combined into a single vector. In this way, the growth in entanglement entropy that usually limits density matrix renormalization group (DMRG) approaches is packaged into conveniently manageable units. We present zero-temperature CheMPS results for the structure factor of spin-antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to three benchmark methods, we find that CheMPS (1) yields results comparable in quality to those of correction vector DMRG, at dramatically reduced numerical cost; (2) agrees well with Bethe Ansatz results for an infinite system, within the limitations expected for numerics on finite systems; (3) can also be applied in the time domain, where it has potential to serve as a viable alternative to time-dependent DMRG (in particular at finite temperatures). Finally, we present a detailed error analysis of CheMPS for the case of the noninteracting resonant level model.