By Altland A., Simons B.D.

Sleek experimental advancements in condensed topic and ultracold atom physics current ambitious demanding situations to theorists. This ebook offers a pedagogical creation to quantum box concept in many-particle physics, emphasizing the applicability of the formalism to concrete difficulties. This moment variation comprises new chapters constructing course vital techniques to classical and quantum nonequilibrium phenomena. different chapters disguise a number of subject matters, from the creation of many-body recommendations and practical integration, to renormalization staff tools, the idea of reaction features, and topology. Conceptual features and formal technique are emphasised, however the dialogue specializes in useful experimental purposes drawn mostly from condensed topic physics and neighboring fields. prolonged and not easy issues of totally labored suggestions supply a bridge among formal manipulations and research-oriented considering. aimed toward raising graduate scholars to a degree the place they could have interaction in autonomous study, this booklet enhances graduate point classes on many-particle concept

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**Example text**

Diﬀerentiation with respect to T then leads to the relation cv = ∂T u ∼ T . However, for temperatures T > vΛ higher than the highest frequencies stored in the phonon modes, the procedure above no longer makes sense (formally, due to the now non-negligible temperature dependence of the boundaries). Yet in this regime we may expand eβv|k| − 1 βv|k|, which brings us back to the classical result cv = const. d Consider now a d-dimensional solid with “isotropic” exchange coupling k2s i=1 (φR+ei − 2 φR ) (ei : unit vector in the i-direction).

1. Convince yourx self that the continuity equation of the system assumes the form ∂t x12 dxρ(x, t) = ∂t (φ(x2 , t) − φ(x1 , t)), where ρ(x, t) is the local density of the medium. Use this result to show that ρ = ∂x φ. Show that the momentum dx (particle density) × (velocity) = dxρ∂t φ carried by the system coincides with the Noether momentum. EXERCISE Compute the energy-momentum tensor for the Lagrangian of the free electromagnetic ﬁeld. Derive the corresponding energy–momentum vector and convince yourself that you obtain results familiar from electrodynamics.

8 Phonon spectra of the transition metal oxide Sr2 RuO4 measured along diﬀerent axes in momentum space. Notice the approximate linearity of the low-energy branches (acoustic phonons) at small momenta. Superimposed at high frequencies are various branches of optical phonons. (Source: Courtesy of M. Braden, II. ) Expressing the operators (φˆk , π ˆk ) in terms of (ˆ ak , a ˆ†k ), it is now straightforward to bring the Hamiltonian into the quasi-particle oscillator form (exercise) ˆ†k a ωk a ˆk + ˆ = H k 1 2 .