# A Modern Approach to Critical Phenomena by Igor Herbut

By Igor Herbut

Best thermodynamics and statistical mechanics books

Order, disorder and criticality: advanced problems of phase transition

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Dynamical Processes in Condensed Matter

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20) The measure of the functional integral in Eq. 19) should be understood as the “sum” over all complex functions α (τ ) that satisfy the boundary condition α (0) = α (β). The quantum number α can be anything that labels the states in the single-particle basis: momentum, position, lattice site in a system on a discrete lattice, etc. For example, choosing α = x with x as the particle’s continuous coordinate, the action for the system of bosons of mass m interacting via V (x − y) becomes β S= dτ ∗ dx (x, τ ) −∂τ − μ − 0 + ∇2 2m 2 dy| (x, τ )|2 V (x − y)| (y, τ )|2 .

Since the finite value of 0 signals the ordered phase, it can be taken to be the order parameter for the superfluid phase transition. In general, the order parameter is defined as the average (r ) , which in the saddle-point approximation coincides with 0 . Note that only the absolute value of 0 is fixed by the condition in Eq. 38), whereas its phase is left arbitrary. For convenience, it may be chosen to be zero. 36). 4 Landau’s mean-field theory 37 disordered (normal) phase. The U (1) symmetry here is an analog of the familiar rotational symmetry that becomes spontaneously broken by the direction of the finite magnetization in the example of ferromagnetic phase transition.

Assume μ < 0 and expand the action around the trivial saddle point, in the normal phase. The result is again Eq. 36). In the first approximation we may set λ = 0, so that the action becomes quadratic in the fluctuating fields. Z [ j] can then be straightforwardly computed by simply completing the square in the exponent. 44) where the constant is independent of the source field. The mean-field susceptibility in the normal phase, which equals the susceptibility for the 38 Ginzburg–Landau–Wilson theory non-interacting system, is then dk (2π )d χ0 (r ) = kB T eik·r 2k2 2m −μ .