Restoring the continuum limit in the time-dependent numerical renormalization group approach

The interplay between the Kondo screening of quantum impurities (by the electronic channels to which they couple) and the interimpurity RKKY interactions (mediated by the same channels) has been extensively studied. However, the effect of unidirectional channels (e.g., chiral or helical edge modes of 2D topological materials) which greatly restrict the mediated interimpurity interactions, has only more recently come under scrutiny, and it can drastically alter the physics. Here we take Wilson's numerical renormalization group (NRG), the most established numerical method for treating quantum impurity models, and extend it to systems consisting of two impurities coupled at different locations to unidirectional channel(s). This is challenging due to the incompatibility of unidirectionality with one of the main ingredients in NRG---the mapping of the channel(s) to a Wilson chain---a tight-binding chain with the impurity at one end and hopping amplitudes which decay exponentially with the distance. We bridge this gap by introducing a ``Wilson ladder'' consisting of two coupled Wilson chains, and demonstrate that this construction successfully captures the unidirectionality of the channel(s), as well as the distance between the two impurities. We use this mapping in order to study two Kondo impurities coupled to a single chiral channel, showing that all local properties and thermodynamic quantities are indifferent to the interimpurity distance, and correspond to two separate single-impurity models. Extensions to more impurities and/or helical channels are possible.

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.

The chemical master equation is a continuous time discrete space Markov model of chemical reactions. The chemical master equation is derived mathematically and it is shown that the corresponding initial value problem has a unique solution. Conditions are given under which this solution is a probability distribution. We present finite state and aggregation-disaggregation approximations and provide error bounds for the case of piecewise constant disaggregation. The aggregation-disaggregation approximation allows the solution of the chemical master equation for larger state spaces and is also an important tool for the solution of multidimensional problems. References Adam Arkin, John Ross, and Harley H. McAdams. {Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Ph age lambda-Infected Escherichia coli Cells}. Genetics, 149(4):1633--1648, 1998. http://www.genetics.org/cgi/content/abstract/149/4/1633 P. Deuflhard, W. Huisinga, T. Jahnke, and M. Wulkow. Adaptive discrete Galerkin methods applied to the chemical master equation. Technical Report ZIB-Report 07-04, Konrad--Zuse Zentrum fuer Informationstechnik Berlin, 2007. http://www.zib.de/Publications/Reports/ZR-07-04.pdf Stefan Engblom. Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput., 180(2):498--515, 2006. doi:10.1016/j.amc.2005.12.032 Lars Ferm and Per Loetstedt. Numerical method for coupling the macro and meso scales in stochastic chemical kinetics. BIT Numerical Mathematics, 47:735--762, 2007. doi:10.1007/s10543-007-0150-z Daniel T. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, 81(25):2340--2361, December 1977. Markus Hegland, Conrad Burden, Lucia Santoso, Shev MacNamara, and Hilary Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math., 205(2):708--724, 2007. doi:10.1016/j.cam.2006.02.053 Leo Huber and Markus Hegland. Dimension adaptive sparse grids for the chemical master equations. submitted, 2008. B. Munsky and M. Khammash. The finite state projection algorithm for the solution of the chemical master equation. The Journal of Chemical Physics, 124:044104--1--044104--1, 2006. doi:10.1063/1.2145882 Roger B. Sidje. EXPOKIT: Software package for computing matrix exponentials. ACM Transactions on Mathematical Software, 24(1):130--156, March 1998. doi:10.1145/285861.285868 T. Tian and K. Burrage. Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage-$\lambda $. Journal of Theoretical Biology, 227:229--237, 2004. doi:10.1016/j.jtbi.2003.11.003 N. G. van Kampen. Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam, the Netherlands, 1981.

We study the entanglement entropy of a region of length 2L with the remainder of an infinite one dimensional gapless quantum system in the case where the region is centered on a quantum impurity. The coupling to this impurity is not scale invariant, and the physics involves a crossover between weak and strong coupling regimes. While the impurity contribution to the entanglement has been computed numerically in the past, little is known analytically about it, since in particular the methods of conformal invariance cannot be applied because of the presence of a crossover length. We show in this paper that the small coupling expansion of the entanglement entropy in this problem is quite generally plagued by strong infrared divergences, implying a non-perturbative dependence on the coupling. The large coupling expansion turns out to be better behaved, thanks to powerful results from the boundary CFT formulation and, in some cases, the underlying integrability of the problem. However, it is clear that this expansion does not capture well the crossover physics. In the integrable case -- which includes problems such as an XXZ chain with a modified link, the interacting resonant level model or the anisotropic Kondo model -- a non perturbative approach is in principle possible using form-factors. We adapt in this paper the ideas of [1,2] to the gapless case and show that, in the rather simple case of the resonant level model, and after some additional renormalizations, the form factors approach yields remarkably accurate results for the entanglement all the way from short to large distances. This is confirmed by detailed comparison with numerical simulations. Both our form factor and numerical results are compatible with a non-perturbative form at short distance.

We present exact results for the susceptibility of the interacting resonant\nlevel model in equilibrium. Detailed simulations using both the Numerical\nRenormalization Group and Density Matrix Renormalization Group were performed\nin order to compare with closed analytical expressions. By first bosonizing the\nmodel and then utilizing the integrability of the resulting boundary\nsine-Gordon model, one finds an analytic expression for the relevant energy\nscale $T_K$ with excellent agreement to the numerical results. On the other\nhand, direct application of the Bethe ansatz of the interacting resonant level\nmodel does not correctly reproduce $T_K$ - however if the bare parameters in\nthe model are renormalised, then quantities obtained via the direct Bethe\nansatz such as the occupation of the resonant level as a function of the local\nchemical potential do match the numerical results. The case of one lead is\nstudied in the most detail, with many results also extending to multiple leads,\nalthough there still remain open questions in this case.\n

Using numerical renormalization group method the temperature dependences of the magnetic susceptibility $\chi(T)$ and specific heat $C(T)$ are obtained in the single-impurity Anderson model with inclusion of the $d-f$ Coulomb interaction. It is shown that the exciton effects owing to this interaction can change considerably the dependence $C(T)$ in comparison with the standard Anderson model at not too low temperatures, whereas the dependence $\chi (T)$ remains universal. The renormalization of the effective hybridization parameter and $f$-level position, which is connected with the $d-f$ interaction, is calculated, a satisfactory agreement with the Hartree-Fock approximation being derived.

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.

The use of quantum mechanical approximation methods in the solution of master equations

Solution of the master equation for unimolecular reactions

We develop a quantum trajectories technique for the unraveling of the quantum adiabatic master equation in Lindblad form. By evolving a complex state vector of dimension $N$ instead of a complex density matrix of dimension $N^2$, simulations of larger system sizes become feasible. The cost of running many trajectories, which is required to recover the master equation evolution, can be minimized by running the trajectories in parallel, making this method suitable for high performance computing clusters. In general, the trajectories method can provide up to a factor $N$ advantage over directly solving the master equation. In special cases where only the expectation values of certain observables are desired, an advantage of up to a factor $N^2$ is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for $8$-qubit quantum annealing examples. We also apply the quantum trajectories method to a $16$-qubit example originally introduced to demonstrate the role of tunneling in quantum annealing, which is significantly more time consuming to solve directly using the master equation. The quantum trajectories method provides insight into individual quantum jump trajectories and their statistics, thus shedding light on open system quantum adiabatic evolution beyond the master equation.

A system of N spin-1/2 particles interacting with a thermal reservoir is used as a pedagogical example for advanced undergraduate and graduate students. We introduce and illustrate some methods, approximations, and phenomena related to dissipation and nonlinearity in many-particle physics. We start our analysis from the dynamical Sudarshan-Lindblad quantum master equation for the density operator of a system S interacting with a thermal reservoir R. We derive the quantum version of the so-called Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equations such that the master equation can be decomposed in a hierarchical set of N - 1 equations (N > 1). The hierarchy is broken by introducing the mean-field approximation and reducing the problem to a nonlinear single particle system. In this scenario, the Hamiltonian is nonlinear (i.e., it depends on the state of <S), although the superoperator responsible for the dissipation and decoherence of S remains unaffected. To provide a useful tool to students: (1) we discuss the physical approximations involved, (2) we derive the analytical solution to the mean values equations of motion resulting from the Hamiltonian, (3) we solve analytically the master equation in the stationary regime, (4) we obtain and discuss the solution of the nonlinear master equation, numerically, and finally, (5) we discuss the master equation beyond the mean-field approximation and show how to introduce higher order quantum correlations that have been previously neglected.

Approximate solutions for multistep master equations describing the time evolution of product formation in multiphoton or thermal unimolecular reactions are investigated. In particular, a method based on fitting the first few moments of the passage time distribution associated with the given stochastic process to proposed simple expressions for the product yield function is studied. It is shown that reasonable agreement with the exact numerical solution of the corresponding master equation is obtained with a two parameter fit (using two passage time moments) and an excellent agreement is obtained with a three parameter fit (using three passage time moments). In no case studied does a need arise for more than a three-moment description and the quality of available experimental results makes the simpler two-moment description sufficient in most cases. Analytical solutions for the first and second passage time moments are obtained for simple discrete and continuous master equation models. Expressions for the incubation time and the reaction rate are obtained in terms of these solutions. The validity of discretizing a continuous master equation (which is an important simplifying step in evaluating the time evolution associated with multiphoton dissociations in the presence of collisions, or with thermal unimolecular reactions involving large molecules) is studied using both the approximate two-moment solutions and exact numerical solutions. It is concluded that a proper discretization of a continuous master equation may be carried out provided ε≪kBT, where ε is the discretization energy step, kB the Boltzmann constant, and T the effective (density of states weighted) temperature. A larger discretization step can be used if only the incubation time is required. Using the approximately discretized master equation, we next calculate the effect of collisions on the incubation time and the rate of multiphoton dissociation using a model constructed to correspond to the unimolecular dissociation of tetramethyldioxethane. Incubation times are found to be less sensitive to collisions then the reaction rates. Finally, we investigate the applicability of the passage time moments method to describe the time evolution of product formation in a system whose dynamics is determined by a quantum mechanical Liouville equation. Again the two-moment description provides a reasonable and the three-moment approximation a good approximation to the exact solution. The three-moment approximations, however, cannot be used when the pressure (i.e., the dephasing rate) is too low.

Numerical renormalization group (NRG) calculations of quantum impurity models, based on a logarithmic discretization in energy of electronic or bosonic Hamiltonians, provide a powerful tool to describe physics involving widely separated energy scales, as typically encountered in nanostructures and strongly correlated materials. This main advantage of the NRG was however considered a drawback for resolving sharp spectral features at finite energy, such as dissipative atomic peaks. Surprisingly, we find a bunching of many-body levels in NRG spectra near dissipative resonances, and exploit this by combining the widely-used Oliveira's $z$-trick, using an averaging over {\it few} discrete NRG spectra, with an optimized {\it frequency-dependent} broadening parameter $b(\w)$. This strategy offers a tremendous gain in computational power and extracts all the needed information from the raw NRG data without {\it a priori} knowledge of the various energy scales at play. As an application we investigate with high precision the crossover from coherent to incoherent dynamics in the spin boson model.

Quantum spin impurities coupled to superconductors are under intense investigation for their relevance to fundamental research as well as the prospects to engineer novel quantum phases of matter. Here we develop a large-NN mean-field theory of a strongly coupled spin-\tfrac{1}{2}12 quantum impurity in a conventional ss-wave superconductor. The approach is benchmarked against Wilson’s numerical renormalization group (NRG). While the large-NN method is not applicable in the weak-coupling regime where the Kondo temperature T_KTK is smaller than the superconducting gap \DeltaΔ, it performs very well in the strong coupling regime where T_K \gtrsim \DeltaTK≳Δ, thus allowing us to obtain a reasonably accurate description of experimentally relevant quantities. The latter include the energy of the Yu-Shiba-Rusinov subgap states, their spectral weight, as well as the local density of continuum states. The method provides a reliable analytical tool that complements other perturbative and non-perturbative methods, and can be extended to more complex impurity models for which NRG may not be easily applicable.

Local three- and four-point correlators yield important insight into strongly\ncorrelated systems and have many applications. However, the nonperturbative,\naccurate computation of multipoint correlators is challenging, particularly in\nthe real-frequency domain for systems at low temperatures. In the accompanying\npaper, we introduce generalized spectral representations for multipoint\ncorrelators. Here, we develop a numerical renormalization group (NRG) approach,\ncapable of efficiently evaluating these spectral representations, to compute\nlocal three- and four-point correlators of quantum impurity models. The key\nobjects in our scheme are partial spectral functions, encoding the system's\ndynamical information. Their computation via NRG allows us to simultaneously\nresolve various multiparticle excitations down to the lowest energies. By\nsubsequently convolving the partial spectral functions with appropriate\nkernels, we obtain multipoint correlators in the imaginary-frequency Matsubara,\nthe real-frequency zero-temperature, and the real-frequency Keldysh formalisms.\nWe present exemplary results for the connected four-point correlators of the\nAnderson impurity model, and for resonant inelastic x-ray scattering (RIXS)\nspectra of related impurity models. Our method can treat temperatures and\nfrequencies -- imaginary or real -- of all magnitudes, from large to\narbitrarily small ones.\n