Study the phase transition in binary mixture

Basing on the Cornwall-Jackiw-Tomboulis (CJT) effective action approach, a theoretical formalism is established to study the Phase Transition in a binary mixture. The effective potential, which preserves the Goldstone theorem, is found in the Hartree-Fock (HF) approximation. This quantity is then used to consider the equation of state and the phase transition of the system.
Nội dung trích xuất từ tài liệu:
Study the phase transition in binary mixture JOURNAL OF SCIENCE OF HNUE Natural Sci., 2010, Vol. 55, No. 6, pp. 3-13 STUDY THE PHASE TRANSITION IN BINARY MIXTURE Le Viet Hoa(∗) Hanoi National University of Education To Manh Kien Xuan Mai High School, Chuong My, Hanoi Pham The Song Tay Bac University (∗) E-mail: hoalv@yahoo.com Abstract. Basing on the Cornwall-Jackiw-Tomboulis (CJT) effective ac- tion approach, a theoretical formalism is established to study the Phase Transition in a binary mixture. The effective potential, which preserves the Goldstone theorem, is found in the Hartree-Fock (HF) approximation. This quantity is then used to consider the equation of state and the phase transition of the system. Keywords: Phase Transition, binary, equation of state. 1. Introduction In recent years, there have been a lot of experimental works dealing with phase transition of systems composed of two distinct species of atoms [1-3]. The typical experiments were performed with atoms of 87 Rb in two different hyperfine states |F = 1, mf = −1i and |F = 2, mf = 1i, which behave as two completely distinguishable species [1] because the hyperfine splitting is much larger than any other relevant energy scale in the system. The multicomponent phase transition is not a simple extension of the single component phase transition. There arise many novel phenomena such as the quantum tunnelling of spin domain [2], vortex configuration[1], phase segregation of binary mixture [3] and so on. In this article, a theoretical formalism for studying phase transition in the global U(1) × U(1) model is formulated by means of the CJT effective action [4] combining with the gapless HF resummation [5]. We then have obtained the effective potential in the HF approximation, which respects the Goldstone theorem. 3 Le Viet Hoa, To Manh Kien and Pham The Song The paper is organized as follows. In Section 2 we derive the desired effective potential. Section 3 is devoted to the physical property study of binary mixture. The conclusion and outlook are presented in Section 4. 2. Effective potential in HF Approximation Let us begin with the idealized binary mixture given by the Lagrangian ∇2 ∇2     ∗ ∂ ∗ ∂ £ = φ −i − φ + ψ −i − ψ ∂t 2mφ ∂t 2mψ λ1 λ2 λ − µ1 φ∗ φ + (φ∗ φ)2 − µ2 ψ ∗ ψ + (ψ ∗ ψ)2 + (φ∗ φ)(ψ ∗ ψ), (2.1) 2 2 2 where µi (i = 1, 2) represents the chemical potential of the field φ (ψ), mi (i = 1, 2) is the mass of φ atom (ψ atom), and λi (i = 1, 2) and λ are the coupling constants. The boundedness of the potential requires that 4λ1 λ2 − λ2 > 0, (2.2) for repulsive self-interactions, λ1 > 0, λ2 > 0. The constraint (2.2) ensures the stability for the mixture of condensates in experimental realization. In the tree approximation, the condensate densities φ20 and ψ02 correspond to local minimum of the potential. They fulfill λ1 λ −µ1 φ0 + φ30 + φ0 ψ02 = 0 2 4 λ2 3 λ 2 −µ2 ψ0 + ψ0 + φ0 ψ0 = 0, (2.3) 2 4 yield φ20 2µ1 λ2 − µ2 λ ψ02 2µ2 λ1 − µ1 λ =2 ; =2 . (2.4) 2 4λ1 λ2 − λ2 2 4λ1 λ2 − λ2 Now let us focus on the calculation of effective potential in HF approximation. At first order the fields φ and ψ are decomposed as 1 1 φ = √ (φ0 + φ1 + iφ2 ), ψ = √ (ψ0 + ψ1 + iψ2 ). (2.5) 2 2 Insert (2.5) into (2.1) we get   λ1 λ λ1 £int = φ0 φ1 + ψ0 ψ1 (φ21 + φ22 ) + (φ21 + φ22 )2 2 4 8   λ2 ...