The book presents a thorough review of applications of functional-integral techniques to critical phenomena in the second-order phase transitions and of the modern state of the theory. The book originated from the courses on quantum field theory regularly delivered, since the early 1970s, by the first author, who is a well-known specialist in the theory of functional integrals. The second author attended in 1989 these courses and participated later in an ambitious program of calculations. The results of these calculations are among those presented in the book.
1. A phase transition is said to be of second order if it includes no latent heat and is characterized by continuous change of internal energy with temperature. Important materials undergoing such phase transitions are ferromagnets, superfluids and superconductors. Typical critical phenomena taking place in the second-order transitions are power behavior of the correlation length \(\xi\), specific heat \(C\), magnetization \(M\) and other characteristics of the system, \[ \xi\sim | T - T_c | ^{-\nu}, \quad C \sim | T - T_c | ^{-\alpha}, \quad | M | \sim | T - T_c | ^{-\beta}, \dots, \] near the critical temperature, \(T\to T_c\), as well as a power behavior of the correlation function, \[ G_c(x) \sim 1/r^{D-2+\eta}, \quad r = | x | , \] at the critical temperature, \(T = T_c\). The corresponding parameters (\(\nu\), \(\alpha\), \(\beta\), \(\eta\), etc.) are called critical exponents. The critical exponents satisfy the so-called scaling relations so that not all of them are independent. The reason is that the thermodynamic potentials and correlation functions have the form of generalized homogeneous functions near the critical temperature. For example, the correlation function has for \(T > T_c\) the form \[ G(x) = \frac{ x/| \tau | ^{-\nu}}{r^{D-2+ \eta}}, \quad \tau = \frac{T}{T_c} - 1. \] The theory of phase transitions has to explain qualitatively and quantitatively the critical behavior of the systems undergoing these transitions, particularly, to derive the values of critical exponents. The most efficient theory presented in the book deals with the so-called \(\Phi^4\) quantum field and makes use of the functional-integral technique.
2. The starting point and theoretical basis for the consideration in the book is the Ginzburg-Landau phenomenological theory of the second-order phase transitions and particularly the famous Ginzburg-Landau energy functional \[ E[\Phi] = \int d^Dx \left\{\frac{A_1}{2}\partial_i \Phi(x) \partial_i \Phi(x) + \frac{A_2}{2}\Phi^2(x) + \frac{A_4}{4!}\left[\Phi^2 (x) \right]^2 \right\} \tag{*} \] where \(D\) is the dimension of the model under consideration and \(\Phi(x)\) a (generally multi-component) order field. The latter may be for example an average of the localized magnetic moments of some lattice over a few lattice spacings. The first gradient term in the exponent of Eq. (*) distinguishes the Ginzburg-Landau energy functional from the form of free energy assumed in Landau theory (where \(\Phi\), actually not depending on \(x\), is called order parameter rather than order field). This form of the functional allows one to take into account, in the derivation of critical phenomena, the spatial fluctuations, very important near critical temperature. In quantum field theory (QFT), a system described by the Ginzburg-Landau functional (*) is called a \(\Phi^4\)-theory. This system has been extensively investigated in the context of both QFT and quantum statistics (obtained from QFT by ‘Wick rotation’ from real to imaginary time). Beginning from the seventies of the last century, \(\Phi^4\)-theory was systematically applied for exploring critical phenomena in phase transitions and proved to be a convenient mathematical tool for this. Making use of the field-theoretic techniques, it has been possible to satisfactorily understand second-order phase transitions of many magnetic systems (such as Ising model and Helium II) and their generalizations (e.g. diluted solutions of polymers and superfluid \(^4\)He at the transition to the normal phase). To underline the connection with QFT, the Ginzburg-Landau functional (*) is written as \(E = E_0 + E_{\text{int}}\) where \[ E_0[\Phi] = \int d^D x \, \frac{1}{2}\left\{[\partial_{x}\Phi(x)]^2 + m^2\Phi^2(x) \right\}, \quad E_{\text{int}} = \int d^D x \, \frac{\lambda}{4!} \Phi^4( {x}). \] The critical point where thermal fluctuations become violent is characterized by vanishing the mass parameter \(m\). All thermodynamic properties of the system are described by the partition function of the field \(\Phi(x)\) which is given by the functional integral \[ Z^{\text{phys}} = \int {\mathcal D}\Phi(x)\, e^{-E[\Phi]/k_B T}. \] The calculation of observable consequences of such a theory proceeds via perturbation theory: certain correlation functions are obtained as power series in the coupling strength \(\lambda\). Each coefficient in such a series is presented as a sum of Feynman integrals (multiple integrals in energy-momentum space) efficiently organized by means of Feynman diagrams. A useful classification is connected with the number of closed lines (loops) in the given diagram. Summing the diagrams containing not more than \(n\) loops is called \(n\)-loop approximation.
3. The book of Kleinert and Schulte-Frohlinde gives a review on the techniques of calculating the power series for the critical exponents. The generation of a great number of diagrams and their weight factors is made with the help of computer-algebraic calculations. Helpful for the counting process is a unique representation of the diagrams by a matrix with integer-valued elements and a simple method for extracting the multiplicity of the diagrams from this matrix. The computer programs applied by the authors allowed them to discover a counting error of the five-loop diagrams in the standard literature. The calculation of Feynman integrals corresponding to the various diagrams is performed in dimensional regularization of’t Hooft and Veltman making use of analytical continuation in the parameter \(D\) from the (complex) region where the relevant integrals converge to a wider area including the value \(D = 3\) necessary for applications. This way of regularization has many advantages, e.g. it preserves all symmetries of the theory and is applicable for the massless fields. The divergences of the theory are removed by counterterms defined in the so-called minimal subtraction scheme providing a simple form of the renormalization group functions. The renormalization constants are calculated with recursive subtracting the divergences of subdiagrams by means of the so-called Bogoliubov’s R-operation. The results of the integrals together with the symmetry factors yield the critical exponents of the systems as an expansion in \(\varepsilon = 4 - D\) up to \(\varepsilon^5\). For a comparison with experiments, the series for the critical exponents have to be evaluated at \(\varepsilon = 1\). The series being divergent, a special techniques for their ‘resummation’ is necessary. The most simple resummation method is Padé approximation: approximation of a divergent function by the rational function having the same lower coefficients in the power expansion. This method is the easiest to apply, since it does not require additional information on the series. More powerful methods are also reviewed and applied for calculating critical exponents. One of them uses the knowledge of the behavior of all power series at high orders in \(\lambda\). The series are re-expanded into functions with the same behavior at higher orders. This method can be applied to both, the series in \(\lambda\) and the series in \(\epsilon\). It has recently been employed by the authors of the book for the resummation of series which contain an additional interaction in cubic symmetry, thereby confirming the result from Padé approximations. The most efficient method for evaluating the perturbation expansions for the critical exponents proves to be variational perturbation theory which has been developed recently by Kleinert. This method leads to theoretical values for the critical exponent \(\alpha\) which are in complete agreement with the experimental numbers.
Concluding, the book of Kleinert and Schulte-Frohlinde gives a complete survey of the theory of critical phenomena including efficient schemes of calculating critical exponents and many examples of the results of such calculations. The book supplies an excellent introduction to the topic for those who wish to enter the area, but will also be of much help for specialists already working on the relevant subjects. Being well organized and written with a simple language, the book may be useful for students of last years and post-graduate students specializing in quantum field theory and quantum statistics as well as for the experts in these area.