Ginzburg-Landau Theory of Type II Superconductors This interesting application of quantum field theory is reviewed by B. Rosenstein and D. Li, Ginzburg-Landau theory of type II superconductors in magnetic field, Rev. Mod. Phys. 82, 109 168 (2010). G[Ψ, A] = G n [A] + d 2 r dz ( ) 2π D = + i A, Φ 0 [ ] 2 + 2 D 2m DΨ 2 2m z Ψ 2 + a (T ) Ψ 2 + b (T ) c 2 Ψ 4 + d 3 (B H)2 r 8π Φ 0 = 2π c, e = 2e, a (T ) α (T e c2 T ), b (T ) const > 0, where H = Hẑ is the applied magnetic field, B = A is the magentic induction of the photon field, and Φ 0 is the Magnetic flux quantum of the superconducting vortices. PHY 510 1 10/25/2013,
Feynman Diagrams for High-Temperature Perturbation Series The nature of transition between the normal (high-temperature) and mixed phases was studied by G.J. Ruggeri and D.J. Thouless, Perturbation series for the critical behavior of type II superconductors near H c2, J. Phys. F: Metal Phys. 6, 2063 2079 (1976). They evaluated the free energy to sixth order, computing 1 + 2 + 5 + 13 + 59 = 80 Feynman diagrams in orders n = 2, 3, 4, 5, 6. The free energy was evaluated by others up to order n = 13. The 2010 URGE to Compute team, R.A. Dygert, M. Heavner, D. Pandey and M. Skvarch, counted n = 14, 15, see R.A. Dygert Honors Thesis. Order n = 7 8 9 10 11 12 13 14 15 Diagrams 285 1987 16057 149430 1551863 17747299 221015026 2975850329 43060406372 PHY 510 2 10/25/2013
Exponentiation of Vacuum Diagrams Anharmonic oscillator ground state energy diagrams on the left. In QED and the Standard Model, the PHY 510 3 10/25/2013
vacuum diagrams contribute to the Casimir effect and the cosmological constant. In high-energy applications they exponentiate and do not contribute to scattering amplitudes as shown in Peskin-Schroeder 4.4 Feynman Diagrams. In condensed matter field theory, the vacuum (Fermi sea) diagrams are fundamentally important, as discussed in Altland-Simons 5.2 Ground state energy of the interacting electron gas: PHY 510 4 10/25/2013
Feynman Diagrams for Scalar Field Theory There are many software packages available to generate Feynman diagrams and compute probability amplitudes, see the Karlsruhe University Institut für Theoretische Teilchenphysik (TTP) Links to algebraic programs. These packages are based on special algorithms to handle the combinatorics of large numbers of diagrams. A good example of an algorithm to generate scalar field theory diagrams is given in the article by H. Kleinert, A. Pelster, B. Kastening, and M. Bachmann, Recursive graphical construction of Feynman diagrams and their multiplicities in φ 4 and φ 2 A theory, Phys. Rev. E 62, 1537 1559 (2000). The important equations from this article are summarized below. Definitions and Notation Consider a self-interacting scalar field φ with N components in d Euclidean dimensions whose thermal fluctuations are controlled by the energy functional E[φ] = 1 G 1 12 2 φ 1φ 2 + g V 1234 φ 1 φ 2 φ 3 φ 4 4! 12 with some coupling constant g. In this short-hand notation, the spatial and tensorial arguments of the field φ, the bilocal kernel G 1, and the quartic interaction V are indicated by simple number indices, for example 1234 PHY 510 5 10/25/2013
the index 1 x 1, α 1, and d d x 1, φ 1 φ α1 (x 1 ), G 1 12 G 1 α 1,α 2 (x 1, x 2 ), V 1234 V α1,α 2,α 3,α 4 (x 1, x 2, x 3, x 4 ). 1 α 1 The scalar field propagator, also called the kernel, is a functional matrix G 1, while V is a functional tensor, both being symmetric in their indices. The energy functional describes generically d-dimensional Euclidean φ 4 -theories. These are models for a family of universality classes of continuous phase transitions, such as the O(N)- symmetric φ 4 -theory which serves to derive the critical phenomena in dilute polymer solutions (N = 0), Ising- and Heisenberg-like magnets (N = 1, 3), and superfluids (N = 2). In all these cases, the energy functional is specified by G 1 α 1,α 2 (x 1, x 2 ) = δ α1,α 2 ( 2 x1 + m 2) δ(x 1 x 2 ), V α1,α 2,α 3,α 4 (x 1, x 2, x 3, x 4 ) = 1 3 δ α 1,α 2 δ α3,α 4 + δ α1,α 3 δ α2,α 4 + δ α1,α 4 δ α2,α 3 δ(x 1 x 2 )δ(x 1 x 3 )δ(x 1 x 4 ), where the mass m 2 is proportional to the temperature distance from the critical point. In the following G 1 and V are completely general, except for the symmetry with respect to their indices. By using natural units in which the Boltzmann constant k B times the temperature T equals unity, the PHY 510 6 10/25/2013
partition function is determined as a functional integral over the Boltzmann weight e E[φ] Z = Dφ e E[φ] and may be evaluated perturbatively as a power series in the coupling constant g. From this we obtain the negative free energy W = ln Z as an expansion ( ) p 1 g W = W (p). p! 4! p=0 The coefficients W (p) may be displayed as connected vacuum diagrams constructed from lines and vertices. Each line represents a free correlation function G 12, which is the functional inverse of the kernel G 1 in the energy functional, defined by G 12 G 1 23 = δ 13. The vertices represent an integral over the interaction 2 1234 V 1234. PHY 510 7 10/25/2013
Multiplicity of Vacuum Diagrams To construct all connected vacuum diagrams contributing to W (p) to each order p in perturbation theory, one connects p vertices with 4p legs in all possible ways according to Feynman s rules which follow from Wick s expansion of correlation functions into a sum of all pair contractions. This yields an increasing number of Feynman diagrams, each with a certain multiplicity which follows from combinatorics. In total there are 4! p p! ways of ordering the 4p legs of the p vertices. This number is reduced by permutations of the legs and the vertices which leave a vacuum diagram invariant. Denoting the number of self-, double, triple and fourfold connections with S, D, T, F, there are 2! S, 2! D, 3! T, 4! F leg permutations. An additional reduction arises from the number N of identical vertex permutations where the vertices remain attached to the lines emerging from them in the same way as before. The resulting multiplicity of a connected vacuum diagram in the φ 4 -theory is therefore given by the formula M E=0 φ 4 = 4! p p! 2! S+D 3! T 4! F N. The superscript E records that the number of external legs of the connected vacuum diagrams is zero. The diagrammatic representation of the coefficients W (p) in the expansion of the negative free energy W is displayed Table 1 up to five loops, and Table 2 lists diagrams and coefficients for corrections to the scalar field propagator. PHY 510 8 10/25/2013
PHY 510 9 10/25/2013
PHY 510 10 10/25/2013
Source Function Approach In the path integral method, a source function J(x) is coupled to the scalar field and modifies the free energy functional E[φ, J] = E[φ] J 1 φ 1. The functional integral for the generating functional Z[J] = Dφ e E[φ,J] is first explicitly calculated for a vanishing coupling constant g, yielding Z (0) [J] = exp 1 2 Tr ln G 1 + 1 G 12 J 1 J 2, 2 where the trace of the logarithm of the kernel is defined by the series Tr ln G 1 ( 1) n+1 = G 1 12 n δ 12 G 1 n1 δ n1. n=1 1...n For non-vanishing coupling constant g, the generating functional Z[J] is expanded in powers of the quartic interaction V. Functional derivatives with respect to the current J then give the correlation functions of the theory. The original partition function can thus be obtained from the free generating functional by the formula Z = exp g 4! 1234 1 12 δ 4 V 1234 δj 1 δj 2 δj 3 δj 4 Z (0) [J]. J=0 PHY 510 11 10/25/2013
Expanding the exponential in a power series, we arrive at the perturbation expansion Z = 1 + g δ 4 V 1234 4! 1234 δj 1 δj 2 δj 3 δj 4 + 1 ( ) 2 g δ 8 V 1234 V 5678 +... Z (0) [J], 2 4! δj 1 δj 2 δj 3 δj 4 δj 5 δj 6 δj 7 δj 8 J=0 12345678 in which the pth order contribution for the partition function requires the evaluation of 4p functional derivatives with respect to the current J. Kernel Approach The derivation of the perturbation expansion simplifies, if we use functional derivatives with respect to the kernel G 1 in the energy functional rather than with respect to the current J. This allows us to substitute the previous expression for the partition function by Z = exp g 6 1234 V 1234 δ 2 δg 1 12 δg 1 34 e W (0), where the zeroth order of the negative free energy has the diagrammatic representation W (0) = 1 2 Tr ln G 1 1 2. PHY 510 12 10/25/2013
Expanding again the exponential in a power series, we obtain Z = 1 + g δ 2 V 1234 6 1234 δg 1 + 1 ( ) 2 g δ 4 V 1234 V 5678 12 δg 1 34 2 6 12345678 δg 1 12 δg 1 Thus we need only half as many functional derivatives and gives δw (0) = 1 2 G δ 2 W (0) 12, = 1 4 G 13G 24 + G 14 G 23, δg 1 12 δg 1 12 δg 1 34 such that the partition function Z becomes Z = 1 + g 3 V 1234 G 12 G 34 + 1 ( ) 2 g V 1234 V 5678 4! 1234 2 4! 12345678 ] [9 G 12 G 34 G 56 G 78 + 24 G 15 G 26 G 37 G 48 + 72 G 12 G 35 G 46 G 78 This has the diagrammatic representation Z = 1 + g 3 + 1 ( ) [ 2 g 9 + 24 + 72 4! 2 4! 34 δg 1 56 δg 1 78 +... ] +... e W (0). +... e W (0). e W (0). All diagrams in this expansion follow directly by successively cutting lines of the basic one-loop vacuum diagram. By going to the logarithm of the partition function Z, we find a diagrammatic expansion for the negative free energy W W = 1 2 + g 4! 3 + 1 2 ( ) 2 g 24 + 72 4! +.... PHY 510 13 10/25/2013