Study the phase transition in binary mixture

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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.

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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 λ λ2 + ψ0 ψ1 + φ0 φ1 (ψ12 + ψ22 ) + (ψ12 + ψ22 )2 2 4 8 λ 2 + (φ + φ22 )(ψ12 + ψ22 ), 8 1 4
Study the phase transition in binary mixture and the inverse propagators in the tree approximation are given by ~k 2 3λ1 2 ! 2mφ − µ1 + 2 φ0 + λ4 ψ02 −ω D0−1 (k) = ~k 2 λ1 2 ω 2mφ − µ1 + φ 2 0 + λ4 ψ02 ~k 2 3λ2 2 ! 2mψ − µ2 + 2 ψ0 + λ4 φ20 −ω G−1 0 (k) = ~k 2 λ2 2 . (2.6) ω 2mψ − µ2 + ψ 2 0 + λ4 φ20 Assuming the ansatz ~k 2 ! 2mφ + M12 −ω D −1 = ~k 2 ω 2mφ + M32 ~k 2 ! 2mψ + M22 −ω G−1 = ~k 2 ω 2mψ + M42 and following closely [6], the CJT effective potential VβCJT (φ0 , ψ0 , D, G) at finite temperature in the HF approximation is given by µ1 λ1 µ2 λ2 λ VβCJT (φ0 , ψ0 , D, G) = − φ20 + φ40 − ψ02 + ψ04 + φ20 ψ02 Z 2 8 2 8 8 1 −1 −1 −1 −1 + tr ln D (k) + ln G (k) + [D0 (k; φ0 , ψ0 )D] + [G0 (k; φ0 , ψ0 )G] − 211 2 β Z 2 Z 2 Z Z 3λ1 3λ1 λ1 + D11 (k) + D22 (k) + D11 (k) D22 (k) 8 β 8 β 4 β β Z 2 Z 2 Z Z 3λ2 3λ2 λ2 + G11 (k) + G22 (k) + G11 (k) G22 (k) 8 β 8 β 4 β β Z Z Z Z λ λ + D11 (k) G11 (k) + D11 (k) G22 (k) 8 β β 8 β β Z Z Z Z λ λ + D22 (k) G11 (k) + D22 (k) G22 (k) . (2.7) 8 β β 8 β β From Equation (2.7) we obtain the following equations: a - The gap equations λ1 2 λ 2 µ1 − φ0 − ψ0 − Σφ1 = 0 2 4 λ2 2 λ 2 µ2 − ψ0 − φ0 − Σψ1 = 0. (2.8) 2 4 b- The Schwinger-Dyson (SD) equations D −1 = D0−1 (k; φ0 , ψ0 ) + Σφ ; G−1 = G−1 ψ 0 (k; φ0 , ψ0 ) + Σ , (2.9) 5
Le Viet Hoa, To Manh Kien and Pham The Song where ! ! Σφ1 0 Σψ1 0 Σφ = ; Σψ = , (2.10) 0 Σφ2 0 Σψ2 and 3λ1 λ1 λ λ Z Z Z Z Σφ1 = D11 (k) + D22 (k) + G11 (k) + G22 (k), 2 β 2 β 4 β 4 β λ1 3λ1 λ λ Z Z Z Z Σφ2 = D11 (k) + D22 (k) + G11 (k) + G22 (k), 2 β 2 β 4 β 4 β 3λ2 λ2 λ λ Z Z Z Z Σψ1 = G11 (k) + G22 (k) + D11 (k) + D22 (k), 2 β 2 β 4 β 4 β λ2 3λ2 λ λ Z Z Z Z Σψ2 = G11 (k) + G22 (k) + D11 (k) + D22 (k), 2 β 2 β 4 β 4 β 3λ1 2 λ 2 3λ2 2 λ 2 M12 = −µ1 + φ0 + ψ0 + Σφ1 M22 = −µ2 + ψ0 + φ0 + Σψ1 . (2.11) 2 4 2 4 The explicit forms for propagators come out by combining (2.8) and (2.9), ~k 2 + λ4 ψ02 + Σφ1 3λ1 2 ! 2mφ − µ1 + 2 φ0 −ω D −1 = ~k 2 , ω 2mφ − µ1 + λ1 2 φ 2 0 + λ4 ψ02 + Σφ2 ~k 2 + λ4 φ20 + Σψ1 ! 3λ2 2 2mψ − µ2 + 2 ψ0 −ω G−1 = ~k 2 (2.12) ω 2mψ − µ2 + λ2 2 ψ 2 0 + λ4 φ20 + Σψ2 which clearly shows that the Goldstone theorem failed in the HF approximation. In order to restore it, we use the method developed in [5], which in our case is achieved by adding a correction ∆V to VβCJT , namely, V˜βCJT = VβCJT + ∆V, (2.13) with aλ1 bλ2 cλ ∆VβCJT = [2Pab Pba − Paa Pbb ] + [2Qab Qba − Qaa Qbb ] + Paa Qbb Z2 Z 2 2 Pab = Dab ; Qab = Gab . (2.14) β β It is easily checked that choosing a = b = −1/2 and c = 0 we are led to effective potential V˜βCJT obeying the requirements imposed in [5]. Indeed, substituting these 6
Study the phase transition in binary mixture values of a, b and c into (2.13) and (2.14) it is found that µ1 λ1 µ2 λ2 λ V˜βCJT (φ0 , ψ0 , D, G) = − φ20 + φ40 − ψ02 + ψ04 + φ20 ψ02 Z 2 8 2 8 8 1 −1 −1 −1 −1 + tr ln D (k) + ln G (k) + [D0 (k; φ0 , ψ0 )D] + [G0 (k; φ0 , ψ0 )G] − 211 2 β Z 2 Z 2 Z Z λ1 λ1 3λ1 + D11 (k) + D22 (k) + D11 (k) D22 (k) 8 β 8 β 4 β β Z 2 Z 2 Z Z λ2 λ2 3λ2 + G11 (k) + G22 (k) + G11 (k) G22 (k) 8 β 8 β 4 β β Z Z Z Z λ λ + D11 (k) G11 (k) + D11 (k) G22 (k) 8 β β 8 β β Z Z Z Z λ λ + D22 (k) G11 (k) + D22 (k) G22 (k) . (2.15) 8 β β 8 β β Since V˜βCJT contains divergent integrals corresponding to zero temperature contributions we must proceed to the regularization. To this end, we make use of the dimensional regularization by performing momentum integration in d = 3 − dimensions and then taking → 0, the regularized integrals turn out to be finite. By this way, we obtain the effective potential consisting only finite terms. From (2.15), instead of (2.8), (2.11) and (2.12), we immediately deduce the following equations a- The gap equations λ1 2 λ 2 −µ1 + φ + ψ + Σφ2 = 0, 2 0 4 0 λ λ2 −µ2 + φ20 + ψ02 + Σψ2 = 0. (2.16) 4 2 At critical temperatures (see Section 4) we have φ0 = ψ0 = 0, and Equation (2.16) give µ1 = Σφ2 , µ2 = Σψ2 , which manifest exactly the Hugenholz-Pines theorem [7] extended to binary mixture. b- The SD equations D −1 = D0−1 (k) + Σφ ; G−1 = G−1 ψ 0 (k) + Σ , (2.17) 7
Le Viet Hoa, To Manh Kien and Pham The Song where ! ! Σφ1 0 Σ ψ 0 Σφ = ; Σψ = 1 , 0 Σφ2 0 Σψ2 λ1 3λ1 λ λ Z Z Z Z Σφ1 = D11 (k) + D22 (k) + G11 (k) + G22 (k), 2 β 2 β 4 β 4 β 3λ1 λ1 λ λ Z Z Z Z Σφ2 = D11 (k) + D22 (k) + G11 (k) + G22 (k), 2 β 2 β 4 β 4 β λ2 3λ2 λ λ Z Z Z Z Σψ1 = G11 (k) + G22 (k) + D11 (k) + D22 (k), 2 β 2 β 4 β 4 β 3λ2 λ2 λ λ Z Z Z Z Σψ2 = G11 (k) + G22 (k) + D11 (k) + D22 (k), (2.18) 2 β 2 β 4 β 4 β and 3λ1 2 λ 2 M12 = −µ1 + φ + ψ + Σφ1 , 2 0 4 0 3λ2 2 λ 2 M22 = −µ2 + ψ0 + φ0 + Σψ1 . 2 4 Combining (2.16) and (2.17) we get the forms for inverse propagators ~k 2 2 ! + M1 −ω 3λ1 2 λ 2 D −1 = 2mφ ~k 2 ; M12 = −µ1 + φ0 + ψ0 + Σφ1 , ω 2mφ 2 4 ~k 2 2 ! + M 2 −ω 3λ2 2 λ 2 G−1 = 2mψ ~k 2 ; M22 = −µ2 + ψ0 + φ0 + Σψ1 . (2.19) ω 2m 2 4 ψ Ultimately the one-particle-irreducible effective potential V˜βCJT (φ0 , ψ0 ) is Z µ 1 λ 1 µ 2 λ 2 λ 1 V˜β (φ0 , ψ0 ) = − φ0 + φ0 − ψ0 + ψ0 + φ0 ψ0 + CJT 2 4 2 4 2 2 −1 −1 tr ln D (k) + ln G (k) 2 8 2 8 8 2 β Z 2 Z 2 Z Z λ1 λ1 3λ1 − D11 (k) − D22 (k) − D11 (k) D22 (k) 8 β 8 β 4 β β Z 2 Z 2 Z Z λ2 λ2 3λ2 − G11 (k) − G22 (k) − G11 (k) G22 (k) 8 β 8 β 4 β β Z Z Z Z λ λ − D11 (k) G11 (k) − D11 (k) G22 (k) 8 β β 8 β β Z Z Z Z λ λ − D22 (k) G11 (k) − D22 (k) G22 (k) (2.20) 8 β β 8 β β 8
Study the phase transition in binary mixture 3. Physical Properties 3.1. Equations of state Let us now consider equations of state. We begin with the pressure defined
˜ CJT P = − Vβ (φ0 , ψ0 , D, G)
(3.1) at minimum values from which the total particle densities are determined by ∂P ρi = , i = 1, 2. ∂µi Taking into account the fact that derivatives of V˜βCJT (φ0 , ψ0 , D, G) with respect to its arguments vanish at minimum values we get: ∂VβCJT φ2 1 1 Z Z ρ1 = − = 0+ D11 + D22 , ∂µ1 2 2 β 2 β ∂VβCJT ψ02 1 1 Z Z ρ2 = − = + G11 + G22 . (3.2) ∂µ2 2 2 β 2 β Hence, the gap Equation (2.16) becomes λ Z µ1 = λ1 ρ1 + ρ2 + λ1 D11 , 2 Zβ λ µ2 = λ2 ρ2 + ρ1 + λ2 G11 , (3.3) 2 β and the particle densities in condensates are φ20 1 1 Z Z = ρ1 − D11 − D22 , 2 2 β 2 β ψ02 1 1 Z Z = ρ2 − G11 − G22 . (3.4) 2 2 β 2 β Combining Equations (2.18), (3.1) and (3.2) together produces the following expres- sion for the pressure λ1 2 λ2 2 λ 1 Z ˜ P = −V = ρ + ρ2 + ρ1 ρ2 − tr{ln D −1 (k) + ln G−1 (k)} − 2 1 2 2 2 β Z 2 Z 2 λ1 λ2 Z Z − D11 − G11 + λ1 ρ1 D11 + λ2 ρ2 G11 . (3.5) 2 β 2 β β β 9
Le Viet Hoa, To Manh Kien and Pham The Song The free energy follows from the Legendre transform E = µ1 ρ1 + µ2 ρ2 − P, and reads λ1 2 λ2 2 λ E = ρ + ρ2 + ρ1 ρ2 + 2 1 2 2 Z 2 Z 2 1 λ1 λ2 Z −1 −1 + tr{ln D (k) + ln G (k)} + D11 + G11 . (3.6) 2 β 2 β 2 β Equations (3.5) and (3.6) constitute the equations of state governing all ther- modynamical processes, in particular, phase transitions of the binary mixture, which is a two-component system with two conserved charges. To proceed further it is interesting to consider the high temperature regime, T /µi 1, associating with symmetry restoration/nonrestoration (SR/SNR) and inverse symmetry breaking (ISB), which are the main subject of the next section. Using the high temperature expansions of all quantities we find the pressure to first order in λ1 , λ2 and λ for temperature just below the critical temperature 3/2 3/2 λ1 ρ21 + λ2 ρ22 + λρ1 ρ2 (mφ + mψ )ζ(5/2) 5/2 (m3φ λ1 + m3ψ λ2 )[ζ(3/2)]2 3 P = + √ T + T 2 2 2π 3/2 16π 3 which reduces to the well-known result of Lee and Yang for single component Bose gas [8] without invoking the double counting subtraction as done in [9]. Based on the formula: ∂ E=− [βP (µ)]µ, β = 1/T, ∂β the high temperature behaviour of the free energy density is also derived in the same approximation 3/2 3/2 1 3(mφ λ1 ρ1 + mψ λ2 ρ2 )ζ(3/2) 3/2 E = = − (λ1 ρ21 + λ2 ρ22 + λρ1 ρ2 ) − √ T + 2 4 2π 3/2 3/2 3/2 3(mφ + mψ )ζ(5/2) 5/2 (m3φ λ1 + m3ψ λ2 )[ζ(3/2)]2 3 + √ T + T . 4 2π 3/2 8π 3 Let us remark that the preceding expression for E does not reduce to the cor- responding one given in [9] for single component Bose gas because the approximation taken there is different from ours. 10
Study the phase transition in binary mixture Next the low temperature regime, T /µi 1, is concerned. Using the low temperature expansions of all quantities, we are able to write the low temperature behaviours of the equations for M12 and M22 as follows √ 3/2 √ 2 2 2M13 mφ λ1 2 2m3φ λ1 π 2 4 M1 = 2λ1 ρ1 − − T 3π 2 15M15 √ 3 3/2 √ 3 2 2 2M 2 m ψ λ 2 2 2mψ λ2 π 2 4 M2 = 2λ2 ρ2 − − T 3π 2 15M25 which require a self-consistent solution for M12 and M22 as functions of densities and temperature. 3.2. Symmetry non restoration and inverse symmetry break- ing Introducing the effective chemical potentials µ1 = µ1 − Σφ2 µ2 = µ2 − Σψ2 the gap Equation (2.16) can be rewritten as λ1 2 λ 2 φ + ψ = µ1 2 0 4 0 λ 2 λ2 2 φ + ψ0 = µ2 4 0 2 which yield φ20 2µ λ2 − µ2 λ ψ02 2µ λ1 − µ1 λ =2 1 ; =2 2 . (3.7) 2 4λ1 λ2 − λ2 2 4λ1 λ2 − λ2 Equations (3.7) resemble (2.4) with µi replaced by µi . It is evident that the symmetry breaking in φ sector is restored at T = Tc1 if φ20 = 0 or 2λ2 µ1 (Tc1 ) − λµ2 (Tc1 ) = 0. (3.8) A similar process occurs in ψ sector at T = Tc2 if ψ02 = 0 11
Le Viet Hoa, To Manh Kien and Pham The Song or 2λ1 µ2 (Tc2 ) − λµ1 (Tc2 ) = 0. (3.9) Taking into account the high temperature expansions for µ1 and µ2 , Equations (3.8) and (3.9) provide the approximate formular for the critical temperatures Tc1 and Tc2 " #2/3 2(λµ2 − 2λ2 µ1 ) Tc1 = 2π 3/2 3/2 3/2 (mφ λ2 + mψ λλ2 − 8mφ λ1 λ2 )ζ(3/2) " #2/3 2(λµ1 − 2λ1 µ2 ) Tc2 = 2π 3/2 3/2 3/2 (3.10) (mψ λ2 + mφ λλ1 − 8mψ λ1 λ2 )ζ(3/2) which suggest several scenarios for symmetry restoration (SR), symmetry non restora- tion(SNR) and inverse symmetry breaking (ISB) in our model. It is known that in comparison with single component systems phase transition in two-component one is much more involved. The fact is that, in addition to the phase transition caused by the mechanical instability taking place in one-component systems, there exist in binary mixture the diffusive instabilities. In order to deter- mine the state of two-component bodies it is necessary to specify three quantities, for instance, P , T and the concentration fraction y which is defined as y = ρ1 /ρ, ρ = ρ1 + ρ2 . For symmetrical reason, we need to consider only 0 < y < 0.5. Then the condition for mechanical stability states that ∂P ρ ≥0 (3.11) ∂ρ T, y and the constraints for diffusive stabilities read ∂µ1 ∂µ2 ≥ 0 or ≤ 0. (3.12) ∂y T, P ∂y T, P 4. Conclusion and Outlook Due to growing interest in binary mixture we studied a non-relativistic model of two-component complex field. Our main goal is to formulate a theoretical for- malism for this physical system. To this end, with the aid of the CJT approach we 12
Study the phase transition in binary mixture established the finite CJT effective potential, which preserves the Goldstone theorem in broken phase. This is our major success. The expression for the pressure, which depends on particle densities, was derived by means of the fact that the pressure is determined by the effective potential at minimum. As a consequence, the free energy was obtained straightforwardly. The equations of state at low and high temperatures were considered. In par- ticular, the critical temperatures were determined, which generated various scenarios for SR, SNR/ISB with some constraints on coupling constants. In order to under- stand better the specific properties of phase transition patterns in two-component systems further study would be carried out by means of numerical computation. REFERENCES [1] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999). [2] H. -J. Miesner, D. M. Stemper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur, and W. Ketterle, Phys. Rev. Lett. 82 (1999) 2228. [3] J. Stenger, S. Inouye, D. M. Stemper-Kurn, H. -J. Miesner, A. P. Chikkatur, and W. Ketterle, Nature (London). 396, 345 (1998). [4] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D10, 2428 (1974). G. Amelino-Camelia, and S.Y.Pi, Phys. Rev. D47, 2356 (1993). [5] Yu. B. Ivanov, F. Riek, and J. Knoll, Phys. Rev. D71, 105016 (2005). [6] Tran Huu Phat, Nguyen Tuan Anh, and Le Viet Hoa, Eur. Phys. J. A19, 359 (2004); Tran Huu Phat, Le Viet Hoa, Nguyen Tuan Anh, and Nguyen Van Long, Phys. Rev. D76, 125027 (2007). [7] N. M. Hugenholz, and D. Pines, Phys. Rev. 116, 489 (1958). [8] T. D. Lee, and C. N. Yang, Phys. Rev. 112, 1419 (1958); Phys. Rev. 117, 897 (1960). [9] T. Haugset, H. Haugerud, and F. Ravndal, Ann. Phys. (NY) 266, 27 (1998). 13