Phase transition and the casimir effect in a complex scalar field with one compactified spatial dimension

Phase transition and the Casimir effect are studied in the complex scalar field with one spatial dimension to be compactified. It is shown that the phase transition is of the second order and the Casimir effect behaves quite differently depending on whether it’s under periodic or anti-periodic boundary conditions.
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Phase transition and the casimir effect in a complex scalar field with one compactified spatial dimension JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 138-146 This paper is available online at http://stdb.hnue.edu.vn PHASE TRANSITION AND THE CASIMIR EFFECT IN A COMPLEX SCALAR FIELD WITH ONE COMPACTIFIED SPATIAL DIMENSION Tran Huu Phat1 and Nguyen Thi Tham2 1 Vietnam Atomic Energy Institute, 2 Faculty of Physics, Hanoi University of Education No. 2, Xuan Hoa, Vinh Phuc Abstract. Phase transition and the Casimir effect are studied in the complex scalar field with one spatial dimension to be compactified. It is shown that the phase transition is of the second order and the Casimir effect behaves quite differently depending on whether it’s under periodic or anti-periodic boundary conditions. Keywords: Phase transition, Casimir effect, complex scalar field, compactified spatial. 1. Introduction It is well known that a characteristic of quantum field theory in space-time with nontrivial topology is the existence of non-equivalent types of fields with the same spin [1, 2, 3]. For a scalar system in space-time which is locally flat but with topolog, that is a Minkowskian space with one of the spatial dimensions compactified in a circle of finite radius L, the non-trivial topology is transferred into periodic (sign +) and anti-periodic (sign -) boundary conditions: φ(t, x, y, z) = ±φ(t, x, y, z + L) (1.1) The seminal discovery in this direction is the so-called Casimir effect [4, 5], where the Casimir force generated by the electromagnetic field that exists in the area between two parallel planar plates was found to be π 2 ~cS F =− , 240L4 here S is the area of the parallel plates and L is the distance between two plates fulfilling the condition L2 ≪ S. The Casimir effect was first written about in 1948 [4], but since the Received June 30, 2013. Accepted August 27, 2013. Contact Nguyen Thi Tham, e-mail address: nguyenthamhn@gmail.com 138 Phase transition and the casimir effect in a complex scalar field... 1970s this effect has received increasing attention of scientists. Newer and more precise experiments demonstrating the Casimir force have been performed and more are under way. Recently, the Casimir effect has become a hot topic in various domains of science and technology, ranging from cosmology to nanophysics [5, 6, 7]. Calculations of the Casimir effect do not exist below zero degrees. It has been seen that the strength of the Casimir force decreases as the distance between two plates increases. At this time it is not possible to predict the repulsive or attractive force for different objects and there is no indication that the Casimir force is dependent on distance at finite temperatures. The Lagrangian we consider is of the form λ L = ∂µ φ∗ ∂φ − U, U = m2 φ∗ φ + (φ∗ φ)2 . (1.2) 2 in which φ is a complex scalar field and m and λ are coupling constants. In the present article, we calculate the effective potential in a complex scalar field and, based on this, the phase transition in compactified space-time is derived. We then study the Casimir effect at a finite temperature and calculate Casimir energies and Casimir forces that correspond to both boundary conditions (periodic and anti-periodic). 2. Content 2.1. Effective potential and phase transition The space is compactified along the oz axis with length L. Then the Euclidian Action is defined as ∫L ∫ SE = i dz LE dτ dx⊥ , dx⊥ = dxdy, (2.1) 0 where LE is the Euclidian form of the Lagrangian (1.2) and t = iτ . Assume that when λ > 0 and m2 < 0, the field operator φ develops vacuum expectation value ⟨φ⟩ = ⟨φ∗ ⟩ = u, then the U (1) symmetry of the complex scalar field given in (1.2) is spontaneously broken. ( ) ∂U In the tree approximation u corresponds to the minimum of U , = 0, ∂φ φ=φ∗ =0 √ m2 ...