A Modern Approach to Critical Phenomena by Igor Herbut

By Igor Herbut

Serious phenomena is likely one of the most enjoyable components of contemporary physics. This 2007 publication presents an intensive yet financial creation into the rules and strategies of the idea of severe phenomena and the renormalization staff, from the point of view of contemporary condensed topic physics. Assuming simple wisdom of quantum and statistical mechanics, the ebook discusses section transitions in magnets, superfluids, superconductors, and gauge box theories. specific cognizance is given to themes comparable to gauge box fluctuations in superconductors, the Kosterlitz-Thouless transition, duality ameliorations, and quantum section transitions - all of that are on the leading edge of physics examine. This ebook comprises a number of difficulties of various levels of hassle, with suggestions. those difficulties offer readers with a wealth of fabric to check their knowing of the topic. it really is excellent for graduate scholars and more matured researchers within the fields of condensed topic physics, statistical physics, and many-body physics.

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N=0 |n n| = 1. 11) where β = 1/(kB T ) is the inverse temperature, and μ the chemical potential. 12) and Nˆ the particle number operator, aˆ †α aˆ α . 13) Using Eq. 9) we first rewrite the partition function as d Z= ∗ αd α − e 2πi α ∗ α α |e−β( H −μ N ) | ˆ α ˆ . 14) Dividing the imaginary time interval β into M pieces and inserting the unity operator in Eq. 15) 1 the requisite matrix elements where 0 = M = , and = β/M. For may be approximated as k−1 |e M α,k − ( Hˆ −μ Nˆ ) | k = e− k−1 | H −μ N | ˆ ˆ k + O( 2 ).

4 Landau’s mean-field theory the Hubbard–Stratonovich transformation (Appendix A) λ e− 2N dr | (r )|4 = Dχ(r )e− N 2 dr [ 2λ χ (r )+iχ (r )| (r )|2 ] , N | i (r )|2 . After this transformation the action becomes where | (r )|2 = i=1 quadratic in i , which can therefore in principle be integrated out. Anticipating the condensation transition, we integrate out all the components i , 2 ≤ i ≤ N , except the first. Since N can be pulled out in front of the action in the exponent, in the limit N → ∞ the saddle-point approximation to the partition function becomes exact.

612 . 29) 2 (2π)d e 2mkB2TkBEC h2 −1 Since at all temperatures below TBEC the chemical potential of the noninteracting system stays zero, the value of the integral for total number of N = V 30 Ginzburg–Landau–Wilson theory particles in Eq. 28) is in fact at its finite maximum at T = TBEC . If the number of particles is fixed, this means that at T < TBEC some of the singleparticle states must be occupied by a finite fraction of the total number of particles to make up for the difference. This singular contribution has been missed by approximating the discrete sum over wavevectors with the integral, and needs to be added separately.

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