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arXiv:1308.1903 · Physical Review B

Generalized Wilson chain for solving multichannel quantum impurity problems

Andrew K. Mitchell, Martin R. Galpin, Samuel Wilson-Fletcher, David E. Logan, R. Bulla

Published Mar 12, 2014• 65 citations
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Abstract

The numerical renormalization group is used to solve quantum impurity problems, which describe magnetic impurities in metals, nanodevices, and correlated materials within dynamical mean field theory. Here we present a simple generalization of the Wilson chain, which improves the scaling of computational cost with the number of conduction bands, bringing more complex problems within reach. The method is applied to calculate the $t$ matrix of the three-channel Kondo model at $T=0$, which shows universal crossovers near non-Fermi-liquid critical points. A nonintegrable three-impurity problem with three bands is also studied, revealing a rich phase diagram and novel screening and overscreening mechanisms.

DOI
10.1103/physrevb.89.121105

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The numerical renormalization group is used to solve quantum impurity problems, which describe magnetic impurities in metals, nanodevices, and correlated materials within dynamical mean field theory. Here we present a simple generalization of the Wilson chain, which improves the scaling of computational cost with the number of conduction bands, bringing more complex problems within reach. The method is applied to calculate the ttt matrix of the three-channel Kondo model at T=0T=0T=0, which shows universal crossovers near non-Fermi-liquid critical points. A nonintegrable three-impurity problem with three bands is also studied, revealing a rich phase diagram and novel screening and overscreening mechanisms.
In the early 1970s, Wilson developed the concept of a fully nonperturbative renormalization group transformation. When applied to the Kondo problem, this numerical renormalization group (NRG) method gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG method was later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method, including some guidelines for calculating physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi-liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean-field theory.
A multichannel Kondo model, where two or more equivalent but independent channels of electrons compete to screen a spin-12\frac{1}{2}21​ impurity, shows overcompensation of the impurity spin, leading to the non-Fermi-liquid behavior in various thermodynamic and transport properties. However, when the channel symmetry is broken, an impurity quantum phase transition can occur at zero temperature. Identification of an order parameter describing the impurity quantum phase transition is very difficult since it is beyond the conventional Landau-Ginzburg-Wilson theory. By employing the natural orbitals renormalization group method, we study both two-channel and three-channel Kondo models, from the perspective of spin correlation between the impurity and electrons in electronic channels. Here we demonstrate that by introducing the spin-correlation ratio as an order parameter we can characterize impurity quantum phase transitions driven by channel asymmetry. In particular, the universal critical exponents $\ensuremath{\beta}$ of the spin-correlation ratio and $\ensuremath{\nu}$ of the correlation length are explicitly determined by finite-size-scaling analysis, namely, $\ensuremath{\beta}=0.10(1)$ and $\ensuremath{\nu}=2.0(1)$, and $\ensuremath{\beta}=0.10(1)$ and $\ensuremath{\nu}=2.5(1)$ for the two-channel and three-channel Kondo models, respectively.
The Kondo problem, which describes the interaction of a spin sss magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in 1/s1/s1/s with the renormalization group. We study both the case of an impurity interacting with a single channel K=1K=1K=1 of fermions in the s→∞s \rightarrow \inftys→∞ limit and the case with KKK channels in the double-scaling limit K→∞K \rightarrow \inftyK→∞, s→∞s \to \inftys→∞, K/sK/sK/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large sss and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime K>2sK > 2sK>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.
Quantum impurity problems can be solved using the numerical renormalization group (NRG), which involves discretizing the free conduction electron system and mapping to a ``Wilson chain.'' It was shown recently that Wilson chains for different electronic species can be interleaved by use of a modified discretization, dramatically increasing the numerical efficiency of the RG scheme [Phys. Rev. B 89, 121105(R) (2014)]. Here we systematically examine the accuracy and efficiency of the ``interleaved'' NRG (iNRG) method in the context of the single impurity Anderson model, the two-channel Kondo model, and a three-channel Anderson-Hund model. The performance of iNRG is explicitly compared with ``standard'' NRG (sNRG): when the average number of states kept per iteration is the same in both calculations, the accuracy of iNRG is equivalent to that of sNRG but the computational costs are significantly lower in iNRG when the same symmetries are exploited. Although iNRG weakly breaks SU(NNN) channel symmetry (if present), both accuracy and numerical cost are entirely competitive with sNRG exploiting full symmetries. iNRG is therefore shown to be a viable and technically simple alternative to sNRG for high-symmetry models. Moreover, iNRG can be used to solve a range of lower-symmetry multiband problems that are inaccessible to sNRG.
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. The new non-Fermi-liquid phase shows frustrated behavior involving alternating overscreenings in spin and orbital sectors, with universal power laws in the spin (ω^{-1/5}), orbital (ω^{1/5}) and spin-orbital (ω^{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.
We present a unified framework for renormalization group methods, including Wilson's numerical renormalization group (NRG), and White's density-matrix renormalization group (DMRG), within the language of matrix-product-states. This allows improvements over Wilson's NRG for quantum impurity models, as we illustrate for the one-channel Kondo model. Moreover, we use a variational method for evaluating Green's functions. The proposed method is more flexible in its description of spectral properties at finite frequencies, opening the way to time-dependent, out-of-equilibrium impurity problems. It also substantially improves computational efficiency for one-channel impurity problems, suggesting potentially linear scaling of complexity for nnn-channel problems.