Quantum Field Theory in Condensed Matter Physics
Alexei M. Tsvelik
Language: English
Pages: 380
ISBN: 0521529808
Format: PDF / Kindle (mobi) / ePub
This course in modern quantum field theory for condensed matter physics includes a derivation of the path integral representation, Feynman diagrams and elements of the theory of metals. Alexei Tsvelik also covers Landau Fermi liquid theory and gradually turns to more advanced methods used in the theory of strongly correlated systems. The book contains a thorough exposition of such non-perturbative techniques, as 1/N-expansion, bosonization (Abelian and non-Abelian), conformal field theory and theory of integrable systems. First edition Hb (1995): 0-521-45467-0 First edition Pb (1996): 0-521-58989-4
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(1) = −G 0 [2i p∇r − = G (2) 2iG 30 p∇r V r ]G 0 ( p, r ) + G0 = −G 0 [2i p∇r − r G0 r ]G (4.17) (1) ≈ 4G 0 p∇r G 30 p∇r V Here we have used the fact that ∇G 0 = −G 20 ∇V . The first correction to TrG is given by TrG (1) = dr =− d D+1 p 2iG 30 p∇r V + G 0 (2π) D+1 (∇V )2 d D+1 p dr (2π) D+1 ( p 2 + V + m 2 )4 =− r G0 d D+1 p (∇G 0 )2 (2π ) D+1 dr (4.18) The second correction is equal to TrG (2) = d D+1 p 4G 0 p∇r G 30 p∇r V = 4 (2π) D+1 dr d D+1 p 5 G ( p∇V )2 (4.19) (2π) D+1
(Vladimir Solovyev). The history of science strongly supports this belief: all great physical theories are at the same time beautiful. Einstein, for example, openly admitted that ideas of beauty played a very important role in his formulation of the theory of general relativity, for which any experimental support had remained minimal for many years. Einstein is by no xii Preface means alone; the reader is advised to read the philosophical essays of Werner Heisenberg, whose authority in the
(N´eel) order. Substituting expression (16.12) into the expression for the energy and expanding the latter in gradients of n and powers of L, we get 1 1 2 JS N(r j + ae)N(r j ) ≈ (16.13) d D x[ρs (∂µ n)2 + χ⊥ S 2 L2 ] 2 2 e where the spin stiffness ρs and the transverse magnetic susceptibility χ⊥ in this particular model are given by ρs = J S 2 a 2−D χ⊥ = 2D J a D (16.14) The latter expression is model dependent; therefore I shall treat ρs and χ⊥ as formal parameters. Substituting expression
(the S = 1/2 model is discussed in great detail in Chapters 29 and 31). If this is indeed the case, the pair correlation function of vector fields decays at large distances, ξ 2 , as the correlation function of staggered magnetization in the S = 1/2 x 2 + c2 τ 2 isotropic Heisenberg chain: n(x, τ )n(0, 0) ∼ ξ2 2 x + c2 τ 2 1/2 (16.28) Our confidence in Haldane’s conjecture is supported by numerical studies on finite (but large) chains (Moreo, 1987; Ziman and Shultz, 1987). From this equivalence
1/T ; at small temperatures the system is described by the standard O(3) nonlinear sigma model with renormalized ρs and . Now recall that the phase transition we are studying occurs at finite temperatures when the average staggered magnetization is zero. Therefore its existence is not related to the fact that the corresponding Heisenberg model has an ordered ground state. In their paper Chandra et al. (1990) came up with the suggestion that one can have a magnetic phase transition without a
