## Exact Methods in Low-dimensional Statistical Physics and by Jesper Jacobsen, Stephane Ouvry, Vincent Pasquier, Didina

By Jesper Jacobsen, Stephane Ouvry, Vincent Pasquier, Didina Serban, Leticia Cugliandolo

Contemporary years have proven vital and extraordinary convergences among innovations regularly utilized in theoretical physics and techniques rising from glossy arithmetic (combinatorics, chance idea, topology, algebraic geometry, etc). those strategies, and particularly these of low-dimensional statistical versions, are instrumental in enhancing our realizing of rising fields, resembling quantum computing and cryptography, advanced structures, and quantum fluids. This e-book units those matters right into a greater and extra coherent theoretical context than is at present on hand. for example, realizing the most important ideas of quantum entanglement (a degree of knowledge density) necessitates a radical wisdom of quantum and topological box thought, and integrable types. to accomplish this objective, the lectures got via foreign leaders within the fields of precisely solvable types in low dimensional condensed topic and statistical physics.

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**Additional info for Exact Methods in Low-dimensional Statistical Physics and Quantum Computing: Lecture Notes of the Les Houches Summer School: Volume 89, July 2008**

**Example text**

E. 14) is modiﬁed to ψR (r) = −ψL (−r) (r > 0) . e. perturbation theory in 1/TK . This will be discussed in detail later, but basic features follow from power counting. An important result is for the impurity magnetic susceptibility. 33) where ST is the total spin operator including both impurity and electron spin operators. The impurity susceptibility is deﬁned, motivated by experiments, as the difference in susceptibilities of samples with and without the impurity. In practice, for a ﬁnite density of impurities, it is the term in the virial expansion of the susceptibility of ﬁrst order in the impurity density ni : χ = χ0 + ni χimp + .

This latter quantity, in the k = 2 case, was recently measured in quantum dot experiments, as I discuss in the next lecture. Using the Kubo formula, these quantities can be expressed in terms of the single-electron Green’s function. Owing to the δfunction nature of the Kondo interaction, the exact retarded Green’s function (in one, two, or three dimensions) with a single impurity at r = 0 can be written as G(r, r ; ω) = G0 (|r − r |, ω) + G0 (r, ω)T(ω)G0 (r , ω) . 85) Here G0 is the noninteracting Green’s function.

In the more modern language of the RG, this simply means that the renormalized coupling constant, λ(E), increases as the characteristic energy scale, E, is lowered. The “problem” is how to understand the low-energy behavior given this failure of perturbation theory, a failure which occurs despite the fact that the original coupling constant λ 1. The β-function may be calculated using Feynman diagram methods; the ﬁrst few diagrams are shown in Fig. 1. The dotted line represents the impurity spin.