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

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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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Nội dung Text: 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 λ yielding u = − . At minimum the potential energy U reads U = m2 u2 + u4 . Let λ 2 us next decompose φ and φ∗ 1 1 φ = √ (u + φ1 + iφ2 ) , φ∗ = √ (u + φ1 − iφ2 ) . (2.2) 2 2 Inserting (2.2) into (1.2) we get the matrix D representing the interaction betweenφ1 , φ2 139
Tran Huu Phat and Nguyen Thi Tham iD−1 = ∥Aik ∥ , A11 = ω 2 + ⃗k 2 + 2λu2 , A22 = ω 2 + ⃗k 2 , A12 = A21 = 0. The partition function is established as ∫ Z = Dφ∗ Dφ exp [−SE ] In a one-loop approximation we have that ∫ [ ∫ ] −V LU −1 Z=e Dφ1 Dφ2 exp − dxϕ iD ϕ + where ϕ+ = (φ1 , φ2 ) The effective potential is defined as ln Z Ω=− = Ωs + U, (2.3) VL in which ∞ ∫ T ∑ ∑ ∞ dk⊥ { [ 2 2 ] [ 2 2 ]} ΩS = 2 ln ωm + E1n + ln ωm + E2n . (2.4) 2L m=−∞ n=−∞ (2π) with √ √ E1n 2 = k⊥ 2 + k3n + M 2, E2n 2 = k⊥ 2 + k3n , M 2 = 2λu2 (2.5) and π k3n = (2n + 1) , n = 0, ±1, ±2, .... (2.6) L for anti-periodic boundary conditions, and π k3n = 2n , n = ±1, ±2, ... (2.7) L for periodic boundary conditions. Note that En in (2.5) is exactly the gapless spectrum of the Goldston boson in the broken phase and there is a similarity between L appearing in (2.6) and (2.7) with T in the Matsubara formula β = 1/T , and therefore it is convenient to impose a = 1/L for later use. Parameter a has the dimension of energy. Making use of the formula: ∑∞ ( ) ( ) T ln ω 2 + E 2 = E + T ln 1 − e−E/T , n=−∞ and taking into account (2.5), (2.6) and (2.7), we arrive at the expression for ΩS ∑∞ ∫ dk⊥ { [ −βE1n ] [ ]} ΩS = −a 2 E1n + E2n + T ln 1 − e + T ln 1 − e−βE2n . n=−∞ (2π) (2.8) 140
Phase transition and the casimir effect in a complex scalar field... The two first terms under the integral in (2.8) are exactly the energy of an electromagnetic vacuum restricted between two plates which gives rise to the Casimir energy. Next let us study the phase transition of the complex scalar field without the Casimir effect at various values of a. The effective potential (2.3) is rewritten in the form Ω = ΩS (T ) − ΩS (T = 0) + U ∑∞ ∫ dk⊥ { ( −βE1n ) ( )} = −aT 2 T ln 1 − e + T ln 1 − e−βE2n + U (2.9) n=−∞ (2π) ∂Ω From (2.9) we derive the gap equation = 0, or ∂u ∫ ∞ aλ ∑ n k 1 2 m + λu + 2 dk √ = 0. (2.10) π n=−∞ 2 k 2 + k3n + 2λu2 e n − 1 βE 0 In order to numerically study the evolution of u versus T at several values of a, the model parameters chosen are those associated with pions and sigma mesons in the linear sigma model: 3m2π − m2σ m2 − m2 m2 = , λ = σ 2 π , mσ = 500M eV, mπ = 138M eV, fπ = 93M eV. 2 fπ where mσ , mπ are respectively the mass of sigma mesons and pions, and fπ is the pion decay constant. Starting from this parameter set and the gap equation (2.10), we get the behaviors of u as a function of temperature at several values of a, given in Figures 1a and 1b for periodic and anti-periodic cases. It is clear that the phase transitions in both cases are of the second order. (1a) (1b) Figure 1. The evolution of u versus T at several values of a which correspond to periodic (1a) and anti-periodic (1b) boundary conditions 141
Tran Huu Phat and Nguyen Thi Tham 2.2. Casimir effect Let us mention that the vacuum energy caused by the electromagnetic field restricted between two parallel planar plates is of the form ∫ dk⊥ ∑ ∞ E(a) = −a En S, (2.11) (2π)2 n=−∞ in which S is the area of planar plate, . It is evident that E(a) diverges. So we try to renormalize it by introducing a rapid damping factor ∫ dk⊥ ∑ ∞ ER (a) = −a 3 En e−δEn S, (2.12) (2π) n=−∞ and the Casimir energy then reads EC (a) = limER (a). (2.13) δ→0 Applying the Abel-Plana formula [8, 9] ∑ ∞ ∫∞ ∫∞ 1 F (it) − F (−it) F (n) − F (t)dt = F (0) + i dt , n=0 2 e2πt − 1 0 0 for periodic conditions and ∑ ∞ ∫∞ ∫∞ ∑ F (it) − F (−it) 1 F (n + ) − 2 F (t)dt = i dt λ . n=0 2 0 0 λ=±1 e2π(t+i 2 ) − 1 for anti-periodic conditions to the calculation of (2.12) and (2.13), we derive the expressions for the Casimir energy ∫∞ ∫∞ √ 16π 2 t2 − b2 ECP (a) = 3 ydy dt 2πt , L e −1 0 b 2.14a which correspond to periodic conditions and ∫∞ ∫∞ √ 16π 2 t2 − b2 ECA (a) = − 3 ydy dt 2πt . (2.14) L e +1 0 b which correspond to anti-periodic conditions. The parameters appearing in 2.14a and 2.14 are defined as |k⊥ | 2 M2 y=L , b = y 2 + M∗2 , M∗2 = L2 2π (2π)2 . 142
Phase transition and the casimir effect in a complex scalar field... Next, based on equations 2.14a, 2.14 and the gap equation (2.10), let us consider the evolution of Casimir energy versus a = 1/L at several values of T and versus T at several values of a. In Figure 2 is shown a dependence of Casimir energy that corresponds to periodic P
A
(2a) and anti-periodic conditions (2b). It is easily recognized that EC (a) and conditions
E (a)
are negligible as L is large enough, while they increases rapidly in the opposite C case. Corresponding to periodic conditions and anti-periodic conditions respectively, we
the T dependence of Casimir energy at L = 6.5f m. It is seen show in Figures
3a and 3b that ECP (a) and
ECA (a)
decrease as T increases. (2a) (2b) Figure 2. The behavior of Casimir energy as a function of L at several values of T. Figure 2a and (2b) shows the periodic (anti-periodic) condition (3a) (3b) Figure 3. The behavior of Casimir energy as a function of T at L = 6.5f m. Figure 3a (3b) shows the periodic (anti-periodic) condition 143
Tran Huu Phat and Nguyen Thi Tham Finally, the Casimir forces FCP (L) and FCA (L) acting on two parallel plates are concerned for both cases of periodic and anti-periodic conditions. They are determined by ∂ECP,A (L) FC (L) = − P,A , (2.15) ∂L Inserting 2.14 into 2.15 we find immediately that ∫∞ ∫∞ 8λu2 dt 4 P FCP (L) = ydy √ + EC (L), L2 (e2πt − 1) t2 − b2 3L 0 b (2.16) ∫∞ ∫∞ 8λu2 dt 4 A FCA (L) = − 2 ydy √ + EC (L) . L (e2πt + 1) t2 − b2 3L 0 b Combining equations 2.14a, 2.14 and 2.16 together leads to the graphs representing the L dependence of Casimir forces at T = 200M eV depicted in Figure 4 and the graphs representing the T dependence of Casimir forces at L = 6.5 fm plotted in Figure 5. Figures 4 and 5 indicate that - The Casimir force is repulsive for periodic boundary conditions and becomes attractive when boundary conditions are anti-periodic. - The strength of the Casimir force decreases quickly as the distance L between the two plates increases. - Casimir forces depend considerably on temperature T for T < 300M eV. (4a) (4b) Figure 4. The evolution of Casimir forces versus L at T = 200M eV 144
Phase transition and the casimir effect in a complex scalar field... (5a) (5b) Figure 5. The evolution of Casimir forces versus T at L = 6.5f m 3. Conclusion In the preceding sections we systematically studied the phase transition and Casimir effect in a complex scalar field embedded in compactified space-time. We obtained the following results: i) The U(1) restoration phase transition at high temperature is of the second order for both boundary conditions and the critical temperature depends on L. ii) Based on the calculated Casimir energies and Casimir forces for periodic and anti-periodic boundary conditions, we investigated numerically their T dependences at several values of L, plotted in Figures 3 and 5, and their L dependences at several values of T , plotted in Figures 2 and 4. iii) The physical content of periodic and anti-periodic boundary conditions is cleared up in Figure 4 which shows that the Casimir force is repulsive for periodic conditions while it is attractive when boundary conditions are anti-periodic. Evidently this statement is very interesting for those working in nanophysics and nanotechnology. The revelation that the critical temperature depends on the compactified length L reveals a new direction for the investigation of high temperature superconductors and Bose-Einstein condensations in space with (2D + ε) dimensions. Acknowledgments. This paper is financially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 103.01-2011.05. 145
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