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arXiv:1208.1645

Wilson's numerical renormalization group and AdS_3 geometry

Kouichi Okunishi

Published Aug 8, 2012• 3 citations
arXivHTMLPDF

Abstract

We discuss the relation between the Wilson's numerical renormalization group(NRG) for the Kondo impurity problem and a field theory in the background AdS_3 space time, where the radial coordinate plays a role of the controlling parameter of the effective mass scale. We find that the Wilson NRG can be described by the boundary Rindler field and then the cutoff parameter λof the Wilson NRG is related to the AdS radius L through λ= 2k_F/ωL, where k_F is the effective Fermi wave number. It is also found that the Rindler space is discretized with the lattice space of a=π/k_F.

DOI
10.48550/arxiv.1208.1645

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We discuss the relation between the Wilson's numerical renormalization group(NRG) for the Kondo impurity problem and a field theory in the background AdS_3 space time, where the radial coordinate plays a role of the controlling parameter of the effective mass scale. We find that the Wilson NRG can be described by the boundary Rindler field and then the cutoff parameter λof the Wilson NRG is related to the AdS radius L through λ= 2k_F/ωL, where k_F is the effective Fermi wave number. It is also found that the Rindler space is discretized with the lattice space of a=π/k_F.
We discuss the scale-free property of Wilson's numerical renormalization group (NRG) for the Kondo-impurity problem. The single-particle state of the effective Hamiltonian with a cutoff $\ensuremath{\Lambda}$ is described by the wave-packet basis having the scale-free property. The energy scale of the system can be controlled by the lattice translation of the wave-packet basis with no reference of rescaling of the lattice space. We also analyze the role of the Kondo interaction in the context of wave-packet basis and then discuss the scaling and renormalization of the Kondo coupling. In addition, we clarify the role of the edge state in the lowest energy scale of Wilson NRG.
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Wilson's Numerical Renormalization Group (NRG) is so far the only nonperturbative technique that can reliably access low-energy properties of quantum impurity systems. We present a recent extension of the method, the DM-NRG, which yields highly accurate results for dynamical quantities at arbitrary frequencies and temperatures. As an application, we determine the spectrum of a quantum dot in an external magnetic field. Furthermore, we discuss magnetic impurities with orbital degeneracy, which have been inferred in recent experiments on quantum dots in an Aharonov-Bohm geometry. It is demonstrated that for spinless electrons, interference between neighbouring levels sets the low-energy scale of the system. Switching on an external field leads to a remarkable crossover into a regime dominated by orbital Kondo screening. We predict that the broadening-induced level splitting should be clearly visible in measurements of the optical absorption power. A more general model including the electron spin is studied within an extended two-band NRG procedure. We observe competition between interference and Kondo screening, similar to the situation in two-impurity models (RKKY).
Quantum impurity models describe interactions between some local degrees of freedom and a continuum of non-interacting fermionic or bosonic states. The investigation of quantum impurity models is a starting point towards the understanding of more complex strongly correlated systems, but quantum impurity models also provide the description of various correlated mesoscopic structures, biological and chemical processes, atomic physics and describe phenomena such as dissipation or dephasing. Prototypes of these models are the Anderson impurity model, or the single- and multi-channel Kondo models. The numerical renormalization group method (NRG) proposed by Wilson in mid 70's has been used in its original form for a longtime as one of the most accurate and powerful methods to deal with quatum impurity problems. Recently, a number of new developments took place: First, a spectral sum-conserving density matrix NRG approach (DM-NRG) has been developed, which has also been generalized for non-Abelian symmetries. In this manual we introduce some of the basic concepts of the NRG method and present recently developed Flexible DM-NRG code. This code uses user-defined non-Abelian symmetries dynamically, computes spectral functions, expectation values of local operators for user-defined impurity models. The code can also use a uniform density of states as well as a user-defined density of states. The current version of the code assumes fermionic bath's and it uses any number of U(1), SU(2) charge SU(2) or Z(2) symmetries. The Flexible DM-NRG code can be downloaded from http://www.phy.bme.hu/~dmnrg