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Thermal phase transition for some spin-boson models
Physica A 392 (2013) 3765–3779
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Thermal phase transition for some spin-boson models M. Aparicio Alcalde ∗ , B.M. Pimentel Instituto de Física Teórica, UNESP - São Paulo State University, Caixa Postal 70532-2, 01156-970 São Paulo, SP, Brazil
highlights • • • •
We study the thermodynamics of the Dicke model considering spatial separation between atoms. We study the thermodynamics of the Thompson model which considers phonons in the Dicke model. Using the path integral approach, we compute the free energy and the collective spectrum of the models. We identify a Goldstone mode when the continuous symmetry is spontaneously broken in both models.
article
info
Article history: Received 12 December 2012 Received in revised form 7 March 2013 Available online 24 April 2013 Keywords: Dicke model Collective excitations Quantum phase transition
abstract In this work we study two different spin-boson models. Such models are generalizations of the Dicke model, it means they describe systems of N identical two-level atoms coupled to a single-mode quantized bosonic field, assuming the rotating wave approximation. In the first model, we consider the wavelength of the bosonic field to be of the order of the linear dimension of the material composed of the atoms, therefore we consider the spatial sinusoidal form of the bosonic field. The second model is the Thompson model, where we consider the presence of phonons in the material composed of the atoms. We study finite temperature properties of the models using the path integral approach and functional methods. In the thermodynamic limit, N → ∞, the systems exhibit phase transitions from normal to superradiant phase at some critical values of temperature and coupling constant. We find the asymptotic behavior of the partition functions and the collective spectrums of the systems in the normal and the superradiant phases. We observe that the collective spectrums have zero energy values in the superradiant phases, corresponding to the Goldstone mode associated to the continuous symmetry breaking of the models. Our analysis and results are valid in the limit of zero temperature β → ∞, where the models exhibit quantum phase transitions. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The study of spin-boson models, in particular the Dicke model and generalizations have several directions of interest. Such interest ranges from their use as physical implementation in quantum information and quantum computer to the studies of relations between concepts of entanglement and critical phenomena. In these cases, depending on the physical branch of study, we can distinguish between the different approaches to the analysis of the spin-boson models which could consider the system as a few-body system or a many-body system. An interesting case of many-body system corresponds to the thermodynamic limit, it means, a system with an infinite number of constituents. In such a case, important relations between concepts of entanglement and critical phenomena were obtained at zero and finite temperature. The Dicke model describes a system of N identical two-level atoms coupled to a single-mode radiation field, simplified according to the rotating wave approximation [1]. In this context, the superradiance is characterized as the coherent
∗
Corresponding author. Tel.: +55 11 23053238. E-mail addresses: [email protected], [email protected] (M. Aparicio Alcalde), [email protected] (B.M. Pimentel).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.04.003
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spontaneous radiation emission with intensity proportional to N 2 . Thermodynamic properties of the Dicke model and generalizations were studied in the thermodynamic limit N → ∞. It is found that the Dicke model exhibits a second order phase transition from normal to superradiant phase at certain critical values of temperature and the coupling constant between the atoms and the field [2,3]. The full Dicke model is defined when we consider the counter-rotating term in the Hamiltonian, using a coupling constant different from the coupling constant of the rotating term. It was found that the counter-rotating term contributes to the critical temperature, free energy and the collective spectrum of the system [4–7]. Another interesting generalization of the Dicke model includes the dipole–dipole interaction, where it is observed that such interaction contributes to the critical parameters at zero and finite temperature and also contributes to the free energy [8–10]. Experimental investigation of phase transition in the Dicke model and some generalizations are proposed. Dimer et al. [11], proposed a realization of the Dicke model in a optical cavity using two Raman resonances, and obtained an effective Hamiltonian equal to the full Dicke Hamiltonian. In such a realization, it is possible to control the parameters of the effective Hamiltonian, and it is also possible to operate in the phase transition regime. In recent works [12,13], a different experimental realization of the Dicke model was studied. It consist of a Bose–Einstein condensate coupled to a multimode optical cavity, where such a system undergoes a phase transition at a sufficient value of the intensity of a transverse pump laser. The thermal properties of the Dicke model with this Bose–Einstein condensate realization were studied [14], where the Bose–Einstein statistics for particles in a two-level system were considered. The path integral approach and functional methods were used for studying the Dicke model, some generalizations and other spin-boson problems, in the thermodynamic limit. In those model, the critical values of coupling constant and temperature were obtained, and asymptotic expressions for the free energy and the collective spectrum were found [5,6, 10,15–17]. In this paper and following the same integral functional approach, we study the critical behavior of two different spin-boson models. In the first model, we consider the wavelength of the bosonic field to be of the order of the linear dimension of the material composed of the atoms, therefore we consider the spatial sinusoidal form of the bosonic field [4,18]. The second model is the Thompson model [18,19] which considers the presence of phonons in the material composed of the atoms. In both cases, the systems exhibit a phase transition from normal to superradiant phase at some critical values of temperature and coupling constant. We find the asymptotic behavior of the partition function and the collective spectrum of the system in the normal and the superradiant phases. It is possible to observe that the collective spectrum has a zero energy value in the superradiant phase, corresponding to the Goldstone mode associated to the continuous symmetry breaking of the models. Our analysis and results are valid in the limit of zero temperature β → ∞, where the models exhibit a quantum phase transition. The two point correlation functions are fundamental tools to study critical phenomena in many-body systems [20]. Notwithstanding, recently studies of entanglement in many-body systems at zero and finite temperature show that concepts of entanglement of bipartite systems are good measures of correlations and have important relations with critical concepts [21]. At zero temperature, the concept of entanglement entropy is used to study quantum correlations in a bipartite system. At finite temperature, the concept of mutual information is an adequate quantity to measure the total correlations in a bipartite system. In several models, the concepts such as the entanglement entropy at zero temperature and the mutual information at finite temperature, show a maximum near the critical point and, in the thermodynamic limit, these quantities show relations with critical exponents [22–24]. The relationship between entanglement entropy and quantum phase transition in the Dicke model was also studied, where it is found that the atom–field entanglement entropy diverges at the critical point of the phase transition with the same critical exponent as the characteristic length [25,26]. This paper is organized as follows. In Section 2, we introduce the Hamiltonian and partition function of the models. In Section 3, the partition function in the path integral version is determined. In Section 4, using functional methods we study the phase transitions of the models, we find the critical temperature and asymptotic expressions for the partition function and collective spectrum in the thermodynamic limit. Finally, in Section 5 we state our summary. In this paper we use kB = h¯ = 1. 2. The Hamiltonian and the partition function of the models We will study two spin-boson models, which are generalizations of the Dicke model. Both models describe a system of N identical two-level atoms coupled to a single-mode quantized bosonic field, moreover the rotating wave approximation is assumed. The first model to be studied considers the spatial sinusoidal form of the bosonic field. Such consideration is valid when the wavelength λ of the bosonic field is of the order of the linear dimensions L in the material composed of the atoms. Therefore, the Hamiltonian of this model is given by HPos =
N Ω
2
i =1
g
σ(zi) + ω0 bĎ b + √
N
N
ξi b σ(+i) + bĎ σ(−i) ,
(1)
i =1
where we define the parameter ξi = sin(2 π n i/N ), with n = L/λ, the number of wavelengths of the bosonic field inside the material. In above equation we define the operators σ(±i) = 12 (σ(1i) ± i σ(2i) ), where the operators σ(1i) , σ(2i) and σ(zi) = σ(3i) , are the Pauli matrices. These matrices satisfy: [σ(+i) , σ(−i) ] = σ(zi) and [σ(zi) , σ(±i) ] = ± 2 σ(±i) . The b and bĎ are the boson annihilation and creation operators of mode excitations that satisfy the usual commutation relation rules.
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The second model to be studied is the Thompson model, which is the usual Dicke model with the presence of phonons in the material composed of the atoms. The Hamiltonian of the Thompson model can be written as HTh =
N Ω
2
g
σ(zi) + ω0 bĎ b + ϵ c Ď c + √
N
N i=1
i =1
b σ(+i) + bĎ σ(−i)
1 1 + √ (κ c + κ ∗ c Ď ) . N
(2)
where the operators σ(zi) , σ(±i) , b and bĎ , are the same as those in Eq. (1), and the new operators c and c Ď are boson annihilation and creation operators of the phonon excitations that satisfy the usual commutation relation rules. Let us define a fermion version for each model. For this purpose, let us define the raising and lowering Fermi operators αiĎ , αi , βiĎ and βi , that satisfy the anti-commutator relations αi αjĎ +αjĎ αi = δij and βi βjĎ +βjĎ βi = δij . In this analysis, we use a Ď
Ď
Ď
representation of the operators σ(zj) , σ(+j) and σ(−j) by the following bilinear combination of Fermi operators, αi αi − βi βi , αi βi and
βiĎ αi ,
the correspondence is given by
σ(zi) −→ αiĎ αi − βiĎ βi ,
(3)
σ(+i) −→ αiĎ βi ,
(4)
σ(−i) −→ βiĎ αi .
(5)
and
Using this representation, given by Eqs. (3)–(5), in the respective Hamiltonian of each model given by Eqs. (1) and (2), we define the Hamiltonian of the fermion models, so we have (F )
HPos = ω0 bĎ b +
N Ω
2 i=1
N g Ď ξi b αi βi + bĎ βiĎ αi . αiĎ αi − βiĎ βi + √
(6)
N i =1
and
N N g Ď 1 Ω Ď 1 + √ (κ c + κ ∗ c Ď ) . b αi βi + bĎ βi αi αiĎ αi − βiĎ βi + √
(F )
HTh = ω0 bĎ b + ϵ c Ď c +
2 i=1
N i =1
N
(7)
We are interested in studying thermodynamic properties of the systems, therefore we must compute the partition (F ) function in both cases. It is important to note that the Hamiltonians HPos and HPos , of the first model are defined in different (F )
spaces, and similar in the Thompson model, i.e., the Hamiltonians HTh and HTh are also defined in different spaces. Each operator σiα , appearing in the Hamiltonians HPos and HTh , acts on two-dimensional Hilbert space, notwithstanding, Fermi Ď
(F )
Ď
(F )
operators αi , αi , βi and βi , appearing in the Hamiltonians HPos and HTh , act on four-dimensional Fock space. The following properties relate the partition function of the models defined using the Pauli matrices, with the partition function of the fermion version of the models, we have
(F )
ZPos = Tr exp(−β HPos ) = iN Tr exp −β HPos −
iπ 2
N (F )
,
(8)
and
(F )
ZTh = Tr exp(−β HTh ) = iN Tr exp −β HTh −
iπ 2
N (F )
(F )
,
(9)
(F )
where HPos is given by Eq. (1), HPos by Eq. (6), HTh by Eq. (2), HTh by Eq. (7) and the operator N (F ) is defined by N (F ) =
N (αiĎ αi + βiĎ βi ).
(10)
i=1
The traces used in Eqs. (8) and (9) for each Hamiltonian are defined in their respective spaces. The relations given by Eqs. (8) and (9) allow us to express the partition function of the models we are studying, ZPos and ZTh , using the fermion Hamiltonians given by Eqs. (6) and (7). 3. The partition function using the path integral approach In this section we use path integral approach and functional methods [27,6] to compute the partition function in both models. Let us define the Euclidean action SPos of the first model in the following form β
SPos = 0
β
N ∗ ∗ dτ b (τ ) ∂τ b(τ ) + αi (τ ) ∂τ αi (τ ) + βi (τ ) ∂τ βi (τ ) − ∗
i=1
0
(F )
dτ HPos (τ ),
(11)
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(F )
where the Hamiltonian HPos (τ ) is obtained from the fermion Hamiltonian defined in Eq. (6), therefore we have (F )
HPos (τ ) = ω0 b ∗ (τ ) b(τ ) + g
+√
N
N
ξi
N Ω
2
α i∗ (τ ) α i (τ ) − β i∗ (τ )β i (τ )
i=1
α i (τ ) β i (τ ) b(τ ) + β i (τ ) α i (τ ) b (τ ) . ∗
∗
∗
(12)
i =1
For the Thompson model the Euclidean action STh , is given by β
STh =
dτ b∗ (τ ) ∂τ b(τ ) + c ∗ (τ ) ∂τ c (τ ) +
β
dτ
0
0
N
β
αi∗ (τ ) ∂τ αi (τ ) + βi∗ (τ ) ∂τ βi (τ ) −
(F )
dτ HTh (τ ), (13)
0
i =1
(F )
in this case, the Hamiltonian HTh (τ ) is obtained from the fermion Hamiltonian defined in Eq. (7), therefore we have (F )
HTh (τ ) = ω0 b ∗ (τ ) b(τ ) + ϵ c ∗ (τ ) c (τ ) + g
+√
N
N
N Ω
2
α i∗ (τ ) α i (τ ) − β i∗ (τ )β i (τ )
i =1
α i (τ ) β i (τ ) b(τ ) + β i (τ ) α i (τ ) b (τ ) ∗
∗
∗
i =1
1
1+ √ κ c (τ ) + κ c (τ ) N ∗ ∗
.
(14)
Since we are interested in studying thermodynamic properties of the systems, we must compute the partition function in both cases. Therefore we will study the following quantity Z Z (0)
β [dη] exp S − 2iπβ 0 n(τ )dτ , = β [dη] exp S0 − 2iπβ 0 n(τ )dτ
(15)
the function n(τ ) is defined by n(τ ) =
N
αi∗ (τ ) αi (τ ) + βi∗ (τ )βi (τ ) ,
(16)
i=1
where S is the Euclidean action given by Eq. (11) for the first model or Eq. (13) for the Thompson model. The action S0 is the free Euclidean action, corresponding to the case where there is no interaction between atoms and field, i.e., the expression of the complete action S setting g = 0. Finally, [dη] is the functional measure. Note that, the functional integrals involved in Eq. (15), are functional integrals with respect to the complex functions b∗ (τ ), b(τ ), and Fermi fields αi∗ (τ ), αi (τ ), βi∗ (τ ) and βi (τ ), for the first model. In the case of the Thompson model, the functional integrals involved in Eq. (15), are functional integrals with respect to the complex functions b∗ (τ ), b(τ ), c ∗ (τ ) and c (τ ), and Fermi fields αi∗ (τ ), αi (τ ), βi∗ (τ ) and βi (τ ). Since we are using thermal equilibrium boundary conditions, in the imaginary time formalism, the integration variables in Eq. (15) obey periodic boundary conditions for the Bose fields, i.e., b(β) = b(0) and c (β) = c (0) and anti-periodic boundary conditions for Fermi fields i.e., αi (β) = −αi (0) and βi (β) = −βi (0). Continuing with the calculation of the quantity (Z0) given by Eq. (15), let us use the following transformation Z
αi (τ ) → e
iπ t 2β
−
αi (τ ),
αi∗ (τ ) → e
βi (τ ) → e 2β βi (τ ),
βi∗ (τ ) → e
iπ
t
iπ t 2β
− 2iπβ t
αi∗ (τ ),
(17)
βi∗ (τ ).
With this last transformation, the term n(τ ) in Eq. (15) can be dropped. Therefore, applying the transformation given by Eq. (17) into the expression given by Eq. (15), we obtain that Z Z (0)
[dη] eS = . [dη] eS0
(18)
In Eq. (18), the Bose fields obey periodic boundary conditions, i.e., b(β) = b(0) and c (β) = c (0), and the Fermi fields obey the following boundary conditions:
αi (β) = i αi (0), βi (β) = i βi (0),
αi∗ (β) = −i αi∗ (0), βi∗ (β) = −i βi∗ (0).
(19)
Let us define the bosonic modes free actions for the models as follows (B0)
SPos (b) =
β
0
dτ b∗ (τ ) ∂τ − ω0 b(τ )
(20)
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and (B0)
STh (b, c ) =
β
dτ
b (τ ) ∂τ − ω0 ∗
b(τ ) + c (τ ) ∂τ − ϵ c (τ ) . ∗
(21)
0
Then we can rewrite the actions SPos and STh given by Eqs. (11) and (13), using the bosonic modes free actions defined by Eqs. (20) and (21), plus some additional terms that can be expressed in matrix forms. Consequently, the total actions SPos and STh can be written as (B0)
SPos = SPos (b) +
β
dτ
N
0
ρiĎ (τ ) Mi (b∗ , b) ρi (τ ),
(22)
i =1
and β
(B0)
STh = STh (b, c ) +
dτ
N
0
ρiĎ (τ ) M (b∗ , b, c ∗ , c ) ρi (τ ),
(23)
i =1
Ď
where the column matrices ρ i (τ ) and ρi (τ ) are given in terms of Fermi field operators in the following way
ρ i (τ ) =
β i (τ ) , α i (τ )
ρ Ďi (τ ) = β ∗i (τ )
α ∗i (τ )
(24)
and the matrices Mi (b , b) and M (b , b, c , c ) are given by ∗
∗
Mi (b∗ , b) =
∗
g
L
− √ ξi b∗ (τ )
− √ ξi b (τ )
L∗
N
g
N
,
(25)
and M (b∗ , b, c ∗ , c )
g
− √ b∗ (τ )
L
N
= g 1 ∗ ∗ − √ b (τ ) 1 + √ κ c (τ ) + κ c (τ ) N
1 1+ √ κ c (τ ) + κ ∗ c ∗ (τ ) N L∗
N
.
(26)
The operators L and L∗ , in Eqs. (25) and (26), are defined by ∂τ + Ω /2 and ∂τ − Ω /2 respectively. In order to compute the partition function we use the expression given by Eq. (18). In such an equation, for each model, we substitute the actions SPos and STh given by Eqs. (22) and (23) respectively. For both models, the functional integrals are Gaussian in the Fermi fields. Therefore, let us begin the calculation integrating with respect these Fermi fields, so we obtain
N
(B0)
[dη(b)] eSPos
ZPos =
det Mi (b∗ , b) ,
(27)
i=1
and
ZTh =
(B0)
[dη(b, c )] eSTh
N
det M (b∗ , b, c ∗ , c )
.
(28)
In Eqs. (27) and (28) their respective functional measures [dη(b)] and [dη(b, c )] are functional measures only with respect of their respective bosonic fields. With the help of the following property for matrices with operator components
det
A C
B D
= det AD − ACA−1 B ,
(29)
and determinant properties, we have that
det Mi (b∗ , b) = det LL∗
1 −1 ∗ det 1 − N −1 g 2 ξi2 L− b , ∗ bL
(30)
and
1 −1/2 det M (b , b, c , c ) = det LL∗ det 1 − N −1 g 2 L− (κ c + κ ∗ c ∗ ) ∗ b 1+N
∗
∗
× L−1 b∗ 1 + N −1/2 (κ c + κ ∗ c ∗ ) .
(31)
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Substituting Eqs. (27) and (30) in Eq. (18), and substituting Eqs. (28) and (31) in Eq. (18), we have that ZPos (0)
ZPos
=
A ZPos (B0)
,
(32)
(B0)
,
(33)
[dη(b)] eSPos
and ZTh (0)
ZTh
=
A ZTh
[dη(b)] eSTh
A A with ZPos and ZTh defined by
A ZPos =
N (B0) 1 −1 ∗ [dη(b)] exp SPos + tr ln 1 − N −1 g 2 ξi2 L− b L b , ∗
(34)
i =1
and A ZTh
=
(B0)
1 −1/2 [dη(b, c )] exp STh + N tr ln 1 − N −1 g 2 L− (κ c + κ ∗ c ∗ ) ∗ b 1+N
× L−1 b∗ 1 + N −1/2 (κ c + κ ∗ c ∗ ) .
(35)
4. Phase transition and the partition function in the thermodynamic limit We are interested in knowing the asymptotic behavior of the quotients
ZPos (0) ZPos
and
ZTh (0) ZTh
N → ∞. With this intention, we analyze the asymptotic behavior of the expressions
√
(35). First, in such equations, let us scale the bosonic fields by b → therefore we get A ZPos
= APos (N )
N b, b → ∗
√
in the thermodynamic limit, i.e.,
A ZPos
A and ZTh given by Eqs. (34) and
N b ,c → ∗
√
N c and c ∗ →
[dη(b)] exp N ΦPos (b∗ , b) ,
√
N c∗,
(36)
and A ZTh = ATh (N )
[dη(b, c )] exp N ΦTh (b∗ , b, c ∗ , c ) ,
(37)
with the functions ΦPos (b∗ , b) and ΦTh (b∗ , b, c ∗ , c ) defined by (B0)
ΦPos (b∗ , b) = SPos +
N 1
N
1 −1 ∗ tr ln 1 − g 2 ξi2 L− b . ∗ bL
(38)
i=1
and
(B0) 1 ∗ ∗ −1 ∗ ∗ ∗ ΦTh (b∗ , b, c ∗ , c ) = STh + tr ln 1 − g 2 L− b 1 + κ c + κ c L b 1 + κ c + κ c . ∗
(39)
The terms APos (N ) and ATh (N ) from Eqs. (36) and (37) respectively, arise√from transforming the functional √ √ measures [dη(b)] √ and [dη(b, c )] under the scaling of bosonic fields by b → N b, b∗ → N b∗ , c → N c and c ∗ → N c ∗ . The asymptotic behavior of the functional integrals in Eqs. (36) and (37) when N → ∞, can be obtained by using the method of steepest descent [28]. According to this method, in the case of the first model we expand the function ΦPos (b∗ , b) around some critical point b(τ ) = b0 (τ ) and b∗ (τ ) = b∗0 (τ ) and in the Thompson model we expand the function ΦTh (b∗ , b, c ∗ , c ) around some critical point b(τ ) = b0 (τ ), b∗ (τ ) = b∗0 (τ ), c (τ ) = c0 (τ ) and c ∗ (τ ) = c0∗ (τ ). Such critical points can be of two kinds, one makes Re(ΦPos (b∗ , b)) or Re(ΦTh (b∗ , b, c ∗ , c )) maximums, and the other one is defined as a saddle point. To find some asymptotic expressions for the partition functions, we consider the first terms of the expansion, which are the leading terms for the value of the total functional integrals. We can find the maximum points, or saddle points, finding the δ ΦPos (b∗ ,b) (b∗ ,b) stationary points. In the first model, the stationary points are solution of the equations = 0 and δ ΦδPos = 0. δ b(τ ) b∗ (τ ) In the Thompson model, the stationary points are solution of the equations δ ΦTh (b∗ ,b,c ∗ ,c ) δ c (τ )
δ ΦTh (b∗ ,b,c ∗ ,c ) and δ c ∗ (τ ) ∗ ∗
δ ΦTh (b∗ ,b,c ∗ ,c ) δ b(τ )
= 0,
δ ΦTh (b∗ ,b,c ∗ ,c ) δ b∗ (τ )
= 0,
=0 = 0. As results of these equations, for both models, the stationary points are constant functions b(τ ) = b0 , b (τ ) = b0 , c (τ ) = c0 and c ∗ (τ ) = c0∗ . And, it is not difficult to show that for temperatures β −1 ≥ βc−1
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the stationary point is given by b0 = b∗0 = 0 in the first model and by b0 = b∗0 = c0 = c0∗ = 0 in the Thompson model, which are maximum points. Moreover, the critical temperature βc−1 in the first model is obtained by solving the following equation 2 ω0 Ω
= tanh
g2
βc Ω
2
,
(40)
and for the Thomson model we have
ω0 Ω g2
= tanh
βc Ω
2
.
(41)
In the first model, from Eq. (40), we can see that the critical temperature is independent of n, which appears in the definition of ξi = sin(2 π n i/N ). We also note that, it is possible to find some critical temperature when g 2 > 2 ω0 Ω . Similarly, in the Thomson model some value of critical temperature is found when g 2 > ω0 Ω . With these conditions the systems undergo a phase transition. When the systems have temperatures β −1 > βc−1 we say that the systems are in the normal phase. In the first model, for temperatures β −1 < βc−1 , the stationary points b(τ ) = b0 and b∗ (τ ) = b∗0 satisfy the following equation
ω0 g2
N 1
=
N
i =1
ξi2 β Ω 2 + 4 g 2 |b0 |2 ξi2 . tanh 2 Ω 2 + 4 g 2 |b0 |2 ξi2
(42)
Therefore, in the limit N → ∞, we can approximate the Eq. (42) by
ω0 g2
1
=
dx
sin2 (2 π n x)
0
Ω∆ ( x )
tanh
β
Ω∆ ( x ) ,
2
(43)
with Ω∆ (x) defined by
Ω∆ (x) =
Ω 2 + 4 g 2 |b0 |2 sin2 (2 π n x).
(44)
In the Thompson model, for temperatures β −1 < βc−1 , the stationary points b(τ ) = b0 , b∗ (τ ) = b∗0 , c (τ ) = c0 and c ∗ (τ ) = c0∗ satisfy the following equations
ω0 Ω∆ g2
= (1 + 2 κ c0 ) tanh 2
β Ω∆
2
,
(45)
with Ω∆ defined by
Ω∆ =
Ω 2 + 4 g 2 |b0 |2 (1 + 2 κ c0 )2 ,
(46)
and we find that κ c0 is real, satisfying
κ c0 (1 + 2 κ c0 ) =
2 ω0 |κ|2 |b0 |2
ϵ
.
(47)
Phase transition happens if it is possible to find some real solution for |b0 | ̸= 0 in Eq. (43) in the first model and some real solution for |b0 | ̸= 0 in Eq. (45) in the Thompson model. The maximum points are a continuous set of values given by the expression b0 = ρ ei φ and b∗0 = ρ e−i φ with φ ∈ [0, 2π ) and ρ = |b0 |, where |b0 | is a solution of Eq. (43) for the first model and |b0 | is a solution of Eq. (45) for the Thompson model. When the systems have temperatures β −1 < βc−1 we say that the systems are in the superradiant phase. Let us continue with the computation of the asymptotic behavior for the functional integrals appearing in Eqs. (36) and (37), in the thermodynamic limit, N → ∞. In the following steps, we shall find this asymptotic behavior when we only have one maximum point, defined by b0 = b∗0 in the first model and by b0 = b∗0 and κ c0 = κ c0∗ in the Thompson model. The resulting expressions, will be valid in the normal phases and useful for the superradiant phases, because in the superradiant phases, the calculation will follow the same steps for the modes different from zero when we use the Fourier representation of the fields. We consider the two first leading terms in the functional integrals appearing in Eqs. (36) and (37), arising from the expansions of ΦPos (b∗ , b) in the first model and of ΦTh (b∗ , b, c ∗ , c ) in the Thompson model around their maximum points, these expansions are given by
ΦPos (b∗ , b) = ΦPos (b∗0 , b0 ) +
1 2
β
0
dτ1 dτ2 (b∗ (τ1 ) − b∗0 , b(τ1 ) − b0 ) MΦPos (τ1 , τ2 )
b∗ (τ2 ) − b∗0 b(τ2 ) − b0
,
(48)
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M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
and 1
ΦTh (b∗ , b, c ∗ , c ) = ΦTh (b∗0 , b0 , c0∗ , c0 ) +
β
2
dτ1 dτ2 b∗ (τ1 ) − b∗0 , b(τ1 ) − b0 , c ∗ (τ1 ) − c0∗ , c (τ1 ) − c0
0
b∗ (τ2 ) − b∗0 b(τ ) − b0 , × MΦTh (τ1 , τ2 ) ∗ 2 c (τ2 ) − c0∗ c (τ2 ) − c0
(49)
the matrices MΦPos (τ1 , τ2 ) and MΦTh (τ1 , τ2 ), are given by
δ 2 ΦPos (b∗ , b) δ b∗ (τ1 ) δ b∗ (τ2 ) MΦPos (τ1 , τ2 ) = δ 2 ΦPos (b∗ , b) δ b(τ1 ) δ b∗ (τ2 )
δ 2 ΦPos (b∗ , b) δ b∗ (τ1 ) δ b(τ2 ) , 2 ∗ δ ΦPos (b , b) δ b(τ1 ) δ b(τ2 ) b∗ =b=b0
(50)
and MΦTh (τ1 , τ2 )
δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b∗ (τ1 ) δ b∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ b(τ1 ) δ b∗ (τ2 ) = δ 2 Φ (b∗ , b, c ∗ , c ) Th δ c ∗ (τ1 ) δ b∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c (τ1 ) δ b∗ (τ2 )
δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b∗ (τ1 ) δ b(τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ b(τ1 ) δ b(τ2 ) δ 2 ΦTh (b∗ , b, c ∗ , c ) δ c ∗ (τ1 ) δ b(τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c (τ1 ) δ b(τ2 )
δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b∗ (τ1 ) δ c ∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ b(τ1 ) δ c ∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c ∗ (τ1 ) δ c ∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c (τ1 ) δ c ∗ (τ2 )
δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b∗ (τ1 ) δ c (τ2 ) δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b(τ1 ) δ c (τ2 ) . 2 ∗ ∗ δ ΦTh (b , b, c , c ) δ c ∗ (τ1 ) δ c (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c (τ1 ) δ c (τ2 ) b∗ ,b ,c ∗ ,c 0 0 0 0
(51)
Substituting these expansions given by Eqs. (48) and (49) in Eqs. (36) and (37) respectively, we obtain A ZPos
=e
N ΦPos (b∗ ,b ) 0 0
[dη(b)] exp
β
1
2
dτ1 dτ2
0
∗ b (τ2 ) b (τ1 ), b(τ1 ) MΦPos (τ1 , τ2 ) , b(τ2 )
∗
(52)
and
A ZTh = eN ΦTh (b0 ,b0 ,c0 ,c0 ) ∗
∗
1 [dη(b, c )] exp
2
β
0
b∗ (τ2 ) b(τ ) b∗ (τ1 ), b(τ1 ), c ∗ (τ1 ), c (τ1 ) MΦTh (τ1 , τ2 ) ∗ 2 . (53) c (τ2 ) c (τ2 )
dτ1 dτ2
To obtain the last two expressions, we have applied the transformation b(τ ) →
√
b(τ ) + b0 / N, b∗ (τ ) →
b∗ (τ ) + √ √ √ b∗0 / N , c (τ ) → c (τ ) + c0 / N and c ∗ (τ ) → c ∗ (τ ) + c0∗ / N in the functional integrals involved. At this level, it is convenient to use Fourier representation of the fields b(τ ) and c (τ ) in the functional integrals in Eqs. (52) and (53). Therefore, we have that 1 b(ω)eiωτ , b(τ ) = √
β
ω
1 ∗ b∗ (τ ) = √ b (ω)e−iωτ ,
β ω 1 c (τ ) = √ c (ω)eiωτ , β ω 1 ∗ c ∗ (τ ) = √ c (ω)e−iωτ , β ω
(54)
since the bosonic fields b(τ ) and c (τ ) satisfy periodic boundary conditions, the parameter ω takes the values 2π n/β , with n being all the integers. These values correspond to the Matsubara frequencies for bosonic fields. Substituting this Fourier representations, Eq. (54) in Eqs. (52) and (53), we obtain that A ZPos
=e
N ΦPos (b∗ ,b ) 0 0
[dη(b)] exp
1 ∗ b∗ (ω2 ) b (ω1 ), b(ω1 ) MΦPos (ω1 , ω2 ) b(ω2 ) 2ω ω
1 2
,
(55)
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3773
and
∗ b (ω2 ) 1 b(ω ) [dη(b, c )] exp b∗ (ω1 ), b(ω1 ), c ∗ (ω1 ), c (ω1 ) MΦTh (ω1 , ω2 ) ∗ 2 , c (ω )
A ZTh = eN ΦTh (b0 ,b0 ,c0 ,c0 ) ∗
∗
2ω ω 1 2
(56)
2
c (ω2 )
with MΦPos (ω1 , ω2 ) and MΦTh (ω1 , ω2 ) being defined by
δ 2 ΦPos (b∗ , b) −i (ω1 τ1 +ω2 τ2 ) β δ b∗ (τ1 ) δ b∗ (τ2 ) e dτ1 dτ2 δ 2 ΦPos (b∗ , b) 0 ei (ω1 τ1 −ω2 τ2 ) δ b(τ1 ) δ b∗ (τ2 )
1
MΦPos (ω1 , ω2 ) =
β
δ 2 ΦPos (b∗ , b) −i (ω1 τ1 −ω2 τ2 ) e δ b∗ (τ1 ) δ b(τ2 ) , δ 2 ΦPos (b∗ , b) i (ω1 τ1 +ω2 τ2 ) e δ b(τ1 ) δ b(τ2 )
(57)
and MΦ (ω1 , ω2 ) =
1
β
β
dτ1 dτ2
0
δ 2 Φ (b∗ , b, c ∗ , c )
e−i (ω1 τ1 +ω2 τ2 )
δ b∗ (τ1 ) δ b∗ (τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) ei (ω1 τ1 −ω2 τ2 ) δ b(τ1 ) δ b∗ (τ2 ) × 2 ∗ ∗ δ Φ (b , b, c , c ) −i (ω τ +ω τ ) 1 1 2 2 δ c ∗ (τ ) δ b∗ (τ ) e 1 2 2 ∗ ∗ δ Φ (b , b, c , c ) i (ω1 τ1 −ω2 τ2 ) e δ c (τ1 ) δ b∗ (τ2 )
δ 2 Φ (b∗ , b, c ∗ , c ) −i (ω1 τ1 −ω2 τ2 ) e δ b∗ (τ1 ) δ b(τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) i (ω1 τ1 +ω2 τ2 ) e δ b(τ1 ) δ b(τ2 ) 2 ∗ ∗ δ Φ (b , b, c , c ) −i (ω1 τ1 −ω2 τ2 ) e δ c ∗ (τ1 ) δ b(τ2 ) 2 ∗ ∗ δ Φ (b , b, c , c ) i (ω1 τ1 +ω2 τ2 ) e δ c (τ1 ) δ b(τ2 )
δ 2 Φ (b∗ , b, c ∗ , c ) −i (ω1 τ1 +ω2 τ2 ) e δ b∗ (τ1 ) δ c ∗ (τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) i (ω1 τ1 −ω2 τ2 ) e δ b(τ1 ) δ c ∗ (τ2 ) 2 ∗ ∗ δ Φ (b , b, c , c ) −i (ω1 τ1 +ω2 τ2 ) e δ c ∗ (τ1 ) δ c ∗ (τ2 ) 2 ∗ ∗ δ Φ (b , b, c , c ) i (ω1 τ1 −ω2 τ2 ) e δ c (τ1 ) δ c ∗ (τ2 )
δ 2 Φ (b∗ , b, c ∗ , c ) −i (ω1 τ1 −ω2 τ2 ) e δ b∗ (τ1 ) δ c (τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) i (ω1 τ1 +ω2 τ2 ) e δ b(τ1 ) δ c (τ2 ) , δ 2 Φ (b∗ , b, c ∗ , c ) −i (ω1 τ1 −ω2 τ2 ) e δ c ∗ (τ1 ) δ c (τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) i (ω1 τ1 +ω2 τ2 ) e δ c (τ1 ) δ c (τ2 ) b∗ ,b ,c ∗ ,c 0 0 0 0
(58)
where in the last case Φ = ΦTh . In these Fourier representations of the functional integrals given by Eqs. (55) and (56), the integral measures [dη(b)] and [dη(b, c )] take the tractable forms ω db∗ (ω) db∗ (ω) and ω db∗ (ω) db(ω), dc ∗ (ω) dc (ω) respectively. In the first model, using the expression for ΦPos (b∗ , b) given by Eq. (38), we can calculate the matrix MΦPos (ω1 , ω2 ). Performing these calculations we obtain that MΦPos (ω1 , ω2 ) =
R(ω2 ) δω1 ,−ω2 S (ω2 ) δω1 , ω2
S (ω2 ) δω1 , ω2 R(ω2 ) δω1 ,−ω2
,
(59)
with δω1 , ω2 being the delta Kronecker and the functions R(ω) and S (ω) are given by sin4 (2 π n x)
1
β Ω∆ (x)
, 2 Ω∆ (x) ω2 + Ω∆2 (x) 1 sin2 (2 π n x) Ω∆ (x)2 + Ω 2 + 2 i ω Ω β Ω∆ (x) g2 dx tanh , S (ω) = i ω − ω0 + 2 0 2 Ω∆ (x) ω2 + Ω∆2 (x) R(ω) = −2 b20 g 4
dx
0
tanh
(60)
where the function Ω∆ (x) is defined in the Eq. (44). Substituting the matrix MΦ (ω1 , ω2 ), with components given by Eq. (59), A , given by Eq. (55), we obtain that in the functional integral appearing in ZPos A ZPos
=e
N ΦPos (b∗ ,b ) 0 0
[dη(b)] exp
S (ω) b(ω) b (ω) + ∗
ω
1 2
R(ω)
b(ω) b(−ω) + b (ω) b (−ω) ∗
∗
.
(61)
Performing this Gaussian functional integral, we finally obtain that
( 2 π i )2 2π i ∗ A . ZPos = eN ΦPos (b0 ,b0 ) 1/2 S (ω) S (−ω) − R2 (ω) S 2 (0) − R2 (0) ω≥1 In order to find the asymptotic behavior of
ZPos 0 ZPos
(62) (B0)
when N → ∞, we must calculate [dη(b)] eSPos appearing in Eq. (32). Using
the free bosonic action SB0 given by Eq. (20), we obtain that
B0
[dη(b)] eSPos =
2π i
ω
ω0 − i ω
.
(63)
Substituting Eqs. (62) and (63) in Eq. (32) we have that ZPos (0)
ZPos
= eN ΦPos (b0 ,b0 ) ∗
1
(H (0))
1/2
1
ω≥1
H (ω)
,
(64)
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M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
in this last equation, the function H (ω) is given by H (ω) =
S (ω) S (−ω) − R2 (ω)
ω2 + ω02
,
(65)
where the Eq. (60) gives the expressions for the functions S (ω) and R(ω). 4.1. Normal phases of the models, β −1 > βc−1 In the normal phase we have β −1 > βc−1 , for the first model we have b0 = b∗0 = 0, see Eq. (42). It implies that Ω∆ (x) = Ω . Substituting this equality in Eqs. (64) and (65), we obtain that ZPos (0)
=
ZPos
1
ω≥1
HI (ω)
1
(HI (0))1/2
,
(66)
where HI (ω) = 1 +
g4 4 (ω2 + Ω 2 ) (ω2 + ω02 )
tanh2
βΩ
g 2 (ω2 − Ω ω0 )
+
2
(ω2 + Ω 2 ) (ω2 + ω02 )
tanh
βΩ
2
.
(67)
For the Thompson model, from Eq. (45) we have that b0 = b∗0 = c0 = c0∗ = 0. Finding the matrix MΦTh (ω1 , ω2 ) given by Eq. (58) we obtain MΦTh (ω1 , ω2 )
0
i ω2 − ω0 + T (ω2 ) δω1 , ω2
0
0
0
0
0
0 0
0
(i ω2 − ϵ) δω1 , ω2
,
= i ω2 − ω0 + T (ω2 ) δω1 , ω2 0 0
(i ω2 − ϵ) δω1 , ω2
(68)
0
where T (ω) =
g 2 (Ω + i ω)
Ω 2 + ω2
tanh
βΩ 2
.
(69)
Therefore, performing the functional integral in Eq. (56) and substituting in Eq. (33), we have ZTh (0)
ZTh
=
1
ω≥1
KI (ω)
1
(KI (0))
1/2
,
(70)
where KI (ω) = 1 +
g4
2
(ω2 + Ω 2 ) (ω2 + ω02 )
tanh
βΩ
+
2
2 g 2 (ω2 − Ω ω0 )
(ω2 + Ω 2 ) (ω2 + ω02 )
tanh
βΩ 2
.
(71)
Making the analytic continuation (iω → E ) in HI (ω) and KI (ω), we solve the equations HI (−i E ) = 0 and KI (−i E ) = 0, such equations correspond to the collective spectrum equation of the first model and the Thompson model respectively. Solving such equations, the collective spectrum for the first model (EPos ) and Thompson model (ETh ), are given respectively by 2 EPos = ω0 + Ω ±
1/2 2 βΩ ω0 − Ω + 2 g 2 tanh , 2
(72)
and 2 ETh = ω0 + Ω ±
ω0 − Ω
2
2
+ 4 g tanh
βΩ 2
1/2
.
(73)
In this normal phase, the results for both models are the√same as the results for the usual Dicke model, only with the difference that for the first model we have to change g → 2 g. 4.2. Superradiant phases of the models, β −1 < βc−1 Let us study both models in the superradiant phase β −1 < βc−1 . We have seen that the maximum point, giving the maximum contribution in the functional integrals given by Eqs. (36) and (37), is not unique. Therefore, we have to perform the following calculations differently from the normal phase. First, we note that the functions ΦPos (b∗ , b) and
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3775
ΦTh (b∗ , b, c ∗ , c ) defined in Eqs. (38) and (39) respectively, are invariants by the transformation b(τ ) → exp(i θ ) b(τ ) and b∗ (τ ) → exp(−i θ ) b∗ (τ ), where θ is an arbitrary factor independent of τ . This continuous invariance is responsible for the appearing of Goldstone mode in the systems. In order to obtain asymptotic expressions for the functional integrals given by Eqs. (36) and (37), let us separate the functions b(τ ) and c (τ ) in the following form b (τ ) = bc + b′ (τ ), b∗ (τ ) = b∗c + b ′ ∗ (τ ), c (τ ) = cc + c ′ (τ ), c ∗ (τ ) = cc∗ + c ′ ∗ (τ ),
(74)
where bc and cc are constant functions, and the fields b (τ ), b (τ ), c (τ ) and c (τ ) satisfy the boundaries conditions b′ (0) = b′ (β) = 0, b ′ ∗ (0) = b ′ ∗ (β) = 0, c ′ (0) = c ′ (β) = 0 and c ′ ∗ (0) = c ′ ∗ (β) = 0. Using the representation bc = ρ ei φ and b∗c = ρ e−i φ in the functional integrals given by Eqs. (36) and (37), followed by the transformation b′ (τ ) → ei φ b′ (τ ) and b ′ ∗ (τ ) → e−i φ b ′ ∗ (τ ), we obtain that ′
A ZPos = 2 π i A(N )
∞
dρ 2
′∗
′
′∗
[dη(b′ )] exp N ΦPos (ρ, b ′ ∗ , b′ ) ,
(75)
0
and A ZTh = 2 π i A(N )
∞
dρ 2
dcc dcc∗
[dη(b′ , c ′ )] exp N ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) ,
(76)
0
where the functions ΦPos (ρ, b ′ ∗ , b′ ) and ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) are given by
ΦPos (ρ, b ′ ∗ , b′ ) =
β
dτ ρ + b ′ ∗ (τ )
∂τ − ω0
ρ + b′ (τ )
0
+
N 1
N i=1
1 tr ln 1 − g 2 ξi2 L− ρ + b′ L−1 ρ + b ′ ∗ ∗
,
(77)
and
ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) =
β
ρ + b ′ ∗ (τ ) ∂τ − ω0 ρ + b′ (τ ) 0 1 + cc∗ + c ′∗ (τ ) (∂τ − ϵ) cc + c ′ (τ ) + tr ln 1 − g 2 L− ρ + b′ ∗ × 1 + κ cc + κ ∗ cc∗ + κ c ′ + κ ∗ c ′∗ L−1 ρ + b ′ ∗ × 1 + κ cc + κ ∗ cc∗ + κ c ′ + κ ∗ c ′∗ .
dτ
(78)
It is important to note that the integrals appearing in Eqs. (75) and (76), are combinations of integration of a real variable
ρ 2 , complex variables cc and cc∗ , and functional integration with respect to the fields b ′ ∗ , b′ , c ′ ∗ and c ′ . In the following, we use the steepest descent method in order to analyze the limit N → ∞. We find one stationary point with respect to the variable ρ 2 for the first model and one stationary point with respect to the variables ρ 2 , cc and cc∗ for the Thompson δΦ = 0 with b ′ ∗ (τ ) = b′ (τ ) = 0 for the first model model. These stationary points satisfy the following equations: δ (ρPos 2) ρ=ρ0 δΦ δΦ and δ (ρTh = 0, δδΦcTh = 0 and b ′ ∗ (τ ) = b′ (τ ) = c ′ ∗ (τ ) = c ′ (τ ) = 0 for the Thompson ∗ = 0, δ cTh ∗ 2) ∗ ∗ c ρ0 ,c0 ,c0
ρ0 ,c0 ,c0
c
ρ0 ,c0 ,c0
model. As results, the solutions for ρ0 in both models, are the same as |b0 | defined by Eq. (43) for the first model and (45) for the Thompson model. Moreover, for the Thompson model, the values of c0 and c0∗ obey the equation κ ∗ c0∗ = κ c0 and Eq. (47). Let us consider the two first leading terms in the functional integrals appearing in Eqs. (75) and (76), arising from the expansions of ΦPos (ρ, b ′ ∗ , b′ ) and ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) around the points defined by ρ0 and b ′ ∗ (τ ) = b′ (τ ) = 0 in the first case, and ρ0 , c0 , c0∗ and b ′ ∗ (τ ) = b′ (τ ) = c ′ ∗ (τ ) = c ′ (τ ) = 0 in the second case. Such points are maximum
points of Re ΦPos (ρ, b ′ ∗ , b′ ) and Re ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) respectively. Therefore, the expansions are given by
2 δ 2 Φ 2 2 ρ − ρ 0 2 δ(ρ 2 )2 ρ=ρ0 , b′ =b ′∗ =0 ′∗ 1 β b (τ2 ) dτ1 dτ2 b ′ ∗ (τ1 ), b′ (τ1 ) MΦPos (τ1 , τ2 ) , + ′ b (τ2 ) 2 0
ΦPos (ρ, b ′ ∗ , b′ ) = ΦPos (ρ0 , 0, 0) +
1
(79)
3776
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
and
ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ )
ρ 2 − ρ02 = ΦTh (ρ0 , c0∗ , c0 , 0, 0, 0, 0) + ρ 2 − ρ02 , cc∗ − c0∗ , cc − c0 M0 cc∗ − c0∗ 2 cc − c0 ′∗ b (τ2 ) ′ 1 β b (τ ) + dτ1 dτ2 b ′ ∗ (τ1 ), b′ (τ1 ), c ′ ∗ (τ1 ), c ′ (τ1 ) MΦTh (τ1 , τ2 ) ′ ∗ 2 , c (τ2 ) 2 0 c ′ (τ2 ) 1
(80)
where the matrices MΦPos (τ1 , τ2 ) and M0 are given by
δ 2 ΦPos δ 2 ΦPos δ b′∗ (τ1 ) δ b′∗ (τ2 ) δ b′∗ (τ1 ) δ b′ (τ2 ) MΦPos (τ1 , τ2 ) = , δ 2 ΦPos δ 2 ΦPos δ b′ (τ1 ) δ b′∗ (τ2 ) δ b′ (τ1 ) δ b′ (τ2 ) ρ=ρ0 , b′ =b ′∗ =0 2 δ ΦTh δ 2 ΦTh δ 2 ΦTh 2 ) δc ∗ 2 ) δc δ(ρ 2 )2 δ(ρ δ(ρ c c δ2 Φ δ 2 ΦTh δ 2 ΦTh Th , M0 = ∗ 2 ∗ 2 ∗ δ cc δ(ρ ) δ(cc ) δ cc δ cc δ 2 ΦTh δ 2 ΦTh δ 2 ΦTh 2 ∗ δ cc δ(ρ ) δ cc δ cc δ(cc )2 ρ0 ,c ∗ ,c0 , b′ =b ′∗ =c ′ =c ′∗ =0
(81)
(82)
0
and MΦTh (τ1 , τ2 ) is given by
δ 2 ΦTh δ b′∗ (τ1 ) δ b′∗ (τ2 ) δ 2 ΦTh δ b′ (τ1 ) δ b′∗ (τ2 ) δ 2 ΦTh ′∗ δ c (τ1 ) δ b′∗ (τ2 ) δ 2 ΦTh ′ δ c (τ1 ) δ b′∗ (τ2 )
δ 2 ΦTh δ b′∗ (τ1 ) δ b′ (τ2 ) δ 2 ΦTh ′ δ b (τ1 ) δ b′ (τ2 ) δ 2 ΦTh δ c ′∗ (τ1 ) δ b′ (τ2 ) δ 2 ΦTh ′ δ c (τ1 ) δ b′ (τ2 )
δ 2 ΦTh δ b′∗ (τ1 ) δ c ′∗ (τ2 ) δ 2 ΦTh ′ δ b (τ1 ) δ c ′∗ (τ2 ) δ 2 ΦTh δ c ′∗ (τ1 ) δ c ′∗ (τ2 ) δ 2 ΦTh ′ δ c (τ1 ) δ c ′∗ (τ2 )
δ 2 ΦTh δ b′∗ (τ1 ) δ c ′ (τ2 ) δ 2 ΦTh δ b′ (τ1 ) δ c ′ (τ2 ) . 2 δ ΦTh δ c ′∗ (τ1 ) δ c ′ (τ2 ) δ 2 ΦTh ′ ′ δ c (τ1 ) δ c (τ2 ) ρ0 ,c0∗ ,c0 , b′ =b ′∗ =c ′ =c ′∗ =0
(83)
The missing second order contributions in Eqs. (79) and (80) are zero. It happens since these terms are of the form, for 2 example, dτ1 C (τ1 ) (ρ 2 − ρ02 ) b′ (τ1 ), where C (τ1 ) = 21 δ(ρ 2δ) δΦb′ (τ ) , and it is possible to show that C (τ1 ) = C0 1
ρ=ρ0 , b′ =b ′∗ =0
and dτ1 b′ (τ1 ) = 0. Using the expansions given by Eqs. (79) and (80) to perform functional integrals given by Eqs. (75) and (76) respectively, we have that
A ZPos
= 2π i
√ Ne
N φPos
∞ √ − N ρ02
β
× exp
1 2
0
dy e
2 1 δ ΦPos 2 δ(ρ 2 )2
ρ=ρ0 , b′ =b ′∗ =0
y2
[dη(b′ )]
b ′ ∗ (τ2 ) dτ1 dτ2 b ′ ∗ (τ1 ), b′ (τ1 ) MΦPos (τ1 , τ2 ) b′ (τ2 )
,
(84)
and A ZTh
= 2π i
√ Ne
N φTh
2
∞ √ − N ρ02
1 × exp
β
0
dτ1 dτ2
dy
∗
dcc dcc exp
1 2
y, cc , cc M0 ∗
y cc∗ cc
[dη(b′ , c ′ )]
′∗ b (τ2 ) ′ b (τ ) b ′ ∗ (τ1 ), b′ (τ1 ), c ′ ∗ (τ1 ), c ′ (τ1 ) MΦTh (τ1 , τ2 ) ′ ∗ 2 , c (τ2 ) c ′ (τ2 )
(85)
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3777
√ N √ appearing in Eqs. (84) and (85) arise from the scaling ρ 2 → ρ 2 / N. For N → ∞, the integrals in Eqs. (84) and (85) are Gaussians. We represent the functions b′ (τ ), b ′ ∗ (τ ), c ′ (τ ) and c ′ ∗ (τ ) by Fourier series as follows 1 ′ ′ b′ (τ ) = √ b (ω)eiωτ , β ω 1 ′ ′∗ b (ω)e−iωτ , b ′∗ (τ ) = √ β ω 1 ′ ′ c ′ (τ ) = √ c (ω)eiωτ , β ω 1 ′ ′∗ c ′∗ (τ ) = √ c (ω)e−iωτ , (86) β ω where the expressions φPos = ΦPos (ρ0 , 0, 0) in Eq. (84) and φTh = ΦTh (ρ0 , c0∗ , c0 , 0, 0, 0, 0) in Eq. (85). The factors
since these functions satisfy the boundary conditions b′ (0) = b′ (β) = 0, b ′ ∗ (0) = b ′ ∗ (β) = 0, c ′ (0) = c ′ (β) = 0 and c ′ ∗ (0) = c ′ ∗ (β) = 0, the parameter ω takes the values 2π n/β , with n being all the integers different from zero. The prime after the summation symbols in Eq. (86) means that it is not considering ω = 0. Using the last Fourier representation in Eqs. (84) and (85), we obtain A ZPos = 2π i
√
N eN φPos
∞
dy e
2 1 δ ΦPos 2 δ(ρ 2 )2
ρ=ρ0 , b′ =b ′∗ =0
y2
−∞
[dη(b )] exp ′
×
′ 1 ′∗ b ′∗ (ω2 ) b (ω1 ), b′ (ω1 ) MΦPos (ω1 , ω2 ) b′ (ω2 ) 2ω ω
,
(87)
1 2
and A ZTh
= 2π i
√ Ne
N φTh
∞
√ − N ρ02
dy
∗
dcc dcc exp
1 y, cc∗ , cc M0 2
y cc∗ cc
[dη(b′ , c ′ )]
′∗ b (ω2 ) ′ b (ω ) 1 b ′ ∗ (ω1 ), b′ (ω1 ), c ′ ∗ (ω1 ), c ′ (ω1 ) MΦTh (ω1 , ω2 ) ′ ∗ 2 . (88) × exp c (ω2 ) 2 ω ,ω 1 2 ′ c (ω2 ) The matrix MΦPos (ω1 , ω2 ) has the same form as that appearing in Eq. (57) but changing ΦPos (b ∗ , b) by ΦPos (ρ, b ′ ∗ , b′ ) and evaluating in ρ0 and b ′ ∗ (0) = b ′ ∗ (β) = 0. The resulting matrix is the same as which appears in Eq. (59), but changing b0 by ρ0 . For the Thompson model, the matrix MΦTh (ω1 , ω2 ) has the same form as which appears in Eq. (58) but changing ΦTh (b ∗ , b, c ∗ , c ) by ΦTh (ρ, cc , cc∗ , b ′ ∗ , b′ , c ′ ∗ , c ′ ) and evaluating in ρ0 , c0∗ , c0 and b ′ ∗ (τ ) = b′ (τ ) = c ′ ∗ (τ ) = c ′ (τ ) = 0.
′
Performing some calculations, we arrive at MΦTh (ω1 , ω2 ) A1 (ω2 ) δω1 ,−ω2 A2 (ω2 ) δω1 , ω2 = κ ∗ A (−ω ) δ 4 2 ω1 ,−ω2 κ A4 (−ω2 ) δω1 , ω2
A2 (ω2 ) δω1 , ω2 A1 (ω2 ) δω1 ,−ω2 κ ∗ A4 (−ω2 ) δω1 , ω2 κ A4 (−ω2 ) δω1 ,−ω2
κ ∗ A4 (ω2 ) δω1 ,−ω2 κ ∗ A4 (−ω2 ) δω1 , ω2 κ ∗ 2 A3 (ω2 ) δω1 ,−ω2 (i ω2 − ϵ + |κ|2 A3 (ω2 )) δω1 , ω2
κ A4 (−ω2 ) δω1 , ω2 κ A4 (ω2 ) δω1 ,−ω2 (i ω2 − ϵ + |κ|2 A3 (ω2 )) δω1 , ω2 , (89) κ 2 A3 (ω2 ) δω1 ,−ω2
where the functions A1 (ω), A2 (ω), A3 (ω) and A4 (ω), are defined by
2 Ω∆2 − Ω 2 β Ω∆ , A1 (ω) = − tanh 2 2 8 ρ02 Ω∆ ω2 + Ω∆ Ω∆2 − Ω 2 Ω∆2 + Ω 2 + 2 i ω Ω β Ω∆ A2 (ω) = i ω − ω0 + tanh , 2 2 8 ρ02 Ω∆ ω2 + Ω∆ 2 g 2 Ω 2 ρ02 β Ω∆ , A3 (ω) = tanh 2 Ω∆ ω2 + Ω∆2
A4 (ω) =
2 g Ω∆ − Ω2
2 Ω∆
12 1+
Ω (Ω − i ω)
ω2 + Ω∆2
tanh
β Ω∆ 2
.
(90)
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Finally, performing the functional integrals given by Eqs. (87) and (88), and substituting in Eqs. (32) and (33) respectively, we obtain that ZPos (0)
√
N eN φPos
=
ZPos
1
1
DPos ω≥1 HII (ω)
,
(91)
and ZTh (0)
√
N eN φTh
=
ZTh
1
1
DTh ω≥1 KII (ω)
.
(92)
The functions φPos , φTh , DPos and DTh are given by
φPos = − ω0 β ρ02 +
1
cosh
cosh
φTh = − ω0 β ρ02 − ϵ β |c0 |2 + ln
DPos = √
g
π β ω0
β Ω∆ (x) 2
dx ln 0
2
βΩ
cosh
dx
β Ω∆ 2
0
,
(93)
2
cosh
βΩ
,
(94)
2
12
sin (2 π n x) β Ω∆ (x) tanh 1 − Ω∆3 (x) sinh β Ω∆ (x) 4
1
β Ω∆ (x) ,
2
(95)
and
DTh =
2 Ω∆2 − Ω 2 sech β Ω2 ∆ Ω∆ + 3 Ω 2 4 g 2 |κ|2 ρ02 sinh β Ω cosh β Ω∆ − 1 − β Ω − √ ∆ ∆ 3/2 2 2 2 ϵ Ω∆ Ω∆ − Ω 4 2 π β ω0 Ω∆ ρ0
+ 3 β Ω∆ tanh
β Ω∆ 2
21
.
(96)
The function HII (ω) is equal to the function in Eq. (65) setting b0 = ρ0 . For the Thompson model, the function KII (ω) is given by
2 Ω∆2 + 2 Ω 2 β Ω∆ KII (ω) = 1 + tanh3 2 3 4 ρ02 Ω∆ ω2 + ω02 ω2 + ϵ 2 ω2 + Ω∆2 2 2 2 2 Ω∆2 − Ω 2 2 Ω ω2 − ω0 (Ω∆2 + Ω 2 ) 4 g |κ| ϵ Ω ρ0 β Ω∆ − + tanh 2 4 ρ02 Ω∆ ω2 + ω02 ω2 + Ω∆2 Ω∆ ω2 + ϵ 2 ω2 + Ω∆2 2 2 Ω 2 Ω∆2 − Ω 2 g 2 |κ|2 ϵ Ω∆ − Ω 2 2 Ω ω2 − ω0 (Ω∆2 + Ω 2 + ω2 ) + + 2 4 2 2 2 2 2 2 2 2 2 2 2 16 ρ0 Ω∆ ω + ω0 Ω∆ ω + ω0 ω + ϵ ω + Ω∆ ω + Ω∆ β Ω∆ 2 × tanh . (97)
2 g 2 |κ|2 ϵ Ω∆ − Ω2
2
The collective spectrum of the systems are obtained by solving the equations HII (−i E ) = 0 and KII (−i E ) = 0. For the first model, it is possible to note that, in the limit where the linear dimension of the material composed of the atoms is very small compared with the wavelength of the bosons (L ≪ λ), we recover the results of the usual Dicke model. In this limit, we consider sin(2 π n x) ≈ 1. Moreover, it is possible to see that in the generalized case (L ≈ λ), the task of obtain some final expressions for the partition function and collective spectrum is a technical problem, since we have to solve some complicated integrations. Nevertheless, setting particular values of the parameters of the model we can use numerical techniques of integration to achieve the final results. Finally, in the general case, it is possible to verify the existence of the Goldstone mode (E Pos = 0), we get such result verifying that HII (0) = 0. For the Thompson model, we solve the equation KII (−i E ) = 0 using the relation given by Eq. (45), as a result, we arrive at the following solutions E1Th = 0,
(98)
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
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and 2
E2Th
2
2 64 g 2 |κ|2 ω ϵ (ω − Ω )2 ρ 2 12 0 0 0 2 2 2 = ω + ϵ + 2 ω0 Ω + Ω∆ ± ω0 − ϵ + 2 ω0 Ω + Ω∆ + . Ω∆2 − Ω 2 2 0
2
2
(99)
Setting κ = 0 in the last equation, we recover the collective spectrum of the usual Dicke model in the superradiant phase. Therefore, we can see that the phonons contribute to the collective spectrum through the term Ω∆ and in the last term in Eq. (99). 5. Summary In this paper we studied two different spin-boson models which are generalizations of the Dicke model. Both models describe a system of N identical two-level atoms coupled to a single-mode quantized bosonic field, where we use the rotating wave approximation. In the first model, we consider the wavelength of the bosonic field λ, to be of the order of the linear dimension of the material composed of the atoms L, therefore we consider the spatial sinusoidal form of the bosonic field. The second model is the Thompson model, where it was considered the presence of phonons in the material composed of the atoms. Using the path integral approach and functional methods, in the thermodynamic limit N → ∞, we found that the systems exhibit a phase transition from normal to superradiant phase, at some critical values of temperature and coupling constant. Moreover, we were able to compute the asymptotic behavior of the partition function and collective spectrum of the models in the normal and superradiant phase, respectively. In both models, the collective spectrum has a zero energy value in the superradiant phase, corresponding to the Goldstone mode associated to the continuous symmetry breaking. Since these models are generalizations of the Dicke model we are able to recover the free energy and the collective spectrum when in the first model we take the approximation L ≪ λ and in the Thompson model we set κ = 0. Beside that, in the first model, with the purpose to achieve some final result, we have to solve some complicated integrations, nevertheless it is possible to manage such integrations using numerical techniques of integration. Finally, in the Thompson model, we can identify the contribution due to the phonons in both the free energy and the collective spectrum. Acknowledgements The authors acknowledge the referee for valuable suggestions. MAA was supported by FAPESP, and BMP was partially supported by CNPq and CAPES. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
R.H. Dicke, Phys. Rev. 93 (1954) 99. K. Hepp, E.H. Lieb, Ann. Phys. 76 (1973) 360. Y.K. Wang, F.T. Hioe, Phys. Rev. A 7 (1973) 831. F.T. Hioe, Phys. Rev. A 8 (1973) 1440. M. Aparicio Alcalde, A.L.L. de Lemos, N.F. Svaiter, J. Phys. A 40 (2007) 11961. M. Aparicio Alcalde, B.M. Pimentel, Physica A 390 (2011) 3385. V.M. Bastidas, C. Emary, B. Regler, T. Brandes, Phys. Rev. Lett. 108 (2012) 043003. G. Chen, D. Zhao, Z. Chen, J. Phys. B 39 (2006) 3315. J. Nie, X.L. Huang, X.X. Yi, Opt. Commun. 282 (2009) 1478. M. Aparicio Alcalde, J. Stephany, N.F. Svaiter, J. Phys. A 44 (2011) 505301. F. Dimer, B. Estienne, A.S. Parkins, H.J. Carmichael, Phys. Rev. A 75 (2007) 013804. K. Baumann, C. Guerlin, F. Brennecke, T. Esslinger, Nature 464 (2010) 1301. D. Nagy, G. Kónya, G. Szirmai, P. Domokos, Phys. Rev. Lett. 104 (2010) 130401. M. Aparicio Alcalde, M. Bucher, C. Emary, T. Brandes, Phys. Rev. E 86 (2012) 012101. V.N. Popov, S.A. Fedotov, Theoret. Math. Phys. 51 (1982) 363. V.N. Popov, Functional Integrals and Collective Excitations, Cambridge University Press, Cambridge, 1987. M. Aparicio Alcalde, R. Kullock, N.F. Svaiter, J. Math. Phys. 50 (2009) 013511. B.M. Pimentel, A.H. Zimerman, Nuovo Cimento B 30 (1975) 43. B.V. Thompson, J. Phys. A 8 (1975) 126. J.J. Binney, N.J. Dowrick, A.J. Fisher, M.E.J. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group, Oxford University Press, Oxford, 1992. L. Amico, R. Fazio, A. Osterloh, V. Vedral, Rev. Modern Phys. 80 (2008) 517. G. Vidal, J.I. Latorre, E. Rico, A. Kitaev, Phys. Rev. Lett. 90 (2003) 227902. R.R.P. Singh, M.B. Hastings, A.B. Kallin, R.G. Melko, Phys. Rev. Lett. 106 (2011) 135701. J. Wilms, J. Vidal, F. Verstraete, S. Dusuel, J. Stat. Mech. P01023 (2012). N. Lambert, C. Emary, T. Brandes, Phys. Rev. Lett. 92 (2004) 073602. N. Lambert, C. Emary, T. Brandes, Phys. Rev. A 71 (2005) 053804. V.N. Popov, V.S. Yarunin, Collective Effects in Quantum Statistics of Radiation and Matter, Kluwer Academic Publishers, The Netherlands, 1988. D.J. Amit, V. Martin-Mayor, Field Theory, the Renormalization Group, and the Critical Phenomena, World Scientific Publishing, Singapore, 2005.
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Thermal phase transition for some spin-boson models M. Aparicio Alcalde ∗ , B.M. Pimentel Instituto de Física Teórica, UNESP - São Paulo State University, Caixa Postal 70532-2, 01156-970 São Paulo, SP, Brazil
highlights • • • •
We study the thermodynamics of the Dicke model considering spatial separation between atoms. We study the thermodynamics of the Thompson model which considers phonons in the Dicke model. Using the path integral approach, we compute the free energy and the collective spectrum of the models. We identify a Goldstone mode when the continuous symmetry is spontaneously broken in both models.
article
info
Article history: Received 12 December 2012 Received in revised form 7 March 2013 Available online 24 April 2013 Keywords: Dicke model Collective excitations Quantum phase transition
abstract In this work we study two different spin-boson models. Such models are generalizations of the Dicke model, it means they describe systems of N identical two-level atoms coupled to a single-mode quantized bosonic field, assuming the rotating wave approximation. In the first model, we consider the wavelength of the bosonic field to be of the order of the linear dimension of the material composed of the atoms, therefore we consider the spatial sinusoidal form of the bosonic field. The second model is the Thompson model, where we consider the presence of phonons in the material composed of the atoms. We study finite temperature properties of the models using the path integral approach and functional methods. In the thermodynamic limit, N → ∞, the systems exhibit phase transitions from normal to superradiant phase at some critical values of temperature and coupling constant. We find the asymptotic behavior of the partition functions and the collective spectrums of the systems in the normal and the superradiant phases. We observe that the collective spectrums have zero energy values in the superradiant phases, corresponding to the Goldstone mode associated to the continuous symmetry breaking of the models. Our analysis and results are valid in the limit of zero temperature β → ∞, where the models exhibit quantum phase transitions. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The study of spin-boson models, in particular the Dicke model and generalizations have several directions of interest. Such interest ranges from their use as physical implementation in quantum information and quantum computer to the studies of relations between concepts of entanglement and critical phenomena. In these cases, depending on the physical branch of study, we can distinguish between the different approaches to the analysis of the spin-boson models which could consider the system as a few-body system or a many-body system. An interesting case of many-body system corresponds to the thermodynamic limit, it means, a system with an infinite number of constituents. In such a case, important relations between concepts of entanglement and critical phenomena were obtained at zero and finite temperature. The Dicke model describes a system of N identical two-level atoms coupled to a single-mode radiation field, simplified according to the rotating wave approximation [1]. In this context, the superradiance is characterized as the coherent
∗
Corresponding author. Tel.: +55 11 23053238. E-mail addresses: [email protected], [email protected] (M. Aparicio Alcalde), [email protected] (B.M. Pimentel).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.04.003
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spontaneous radiation emission with intensity proportional to N 2 . Thermodynamic properties of the Dicke model and generalizations were studied in the thermodynamic limit N → ∞. It is found that the Dicke model exhibits a second order phase transition from normal to superradiant phase at certain critical values of temperature and the coupling constant between the atoms and the field [2,3]. The full Dicke model is defined when we consider the counter-rotating term in the Hamiltonian, using a coupling constant different from the coupling constant of the rotating term. It was found that the counter-rotating term contributes to the critical temperature, free energy and the collective spectrum of the system [4–7]. Another interesting generalization of the Dicke model includes the dipole–dipole interaction, where it is observed that such interaction contributes to the critical parameters at zero and finite temperature and also contributes to the free energy [8–10]. Experimental investigation of phase transition in the Dicke model and some generalizations are proposed. Dimer et al. [11], proposed a realization of the Dicke model in a optical cavity using two Raman resonances, and obtained an effective Hamiltonian equal to the full Dicke Hamiltonian. In such a realization, it is possible to control the parameters of the effective Hamiltonian, and it is also possible to operate in the phase transition regime. In recent works [12,13], a different experimental realization of the Dicke model was studied. It consist of a Bose–Einstein condensate coupled to a multimode optical cavity, where such a system undergoes a phase transition at a sufficient value of the intensity of a transverse pump laser. The thermal properties of the Dicke model with this Bose–Einstein condensate realization were studied [14], where the Bose–Einstein statistics for particles in a two-level system were considered. The path integral approach and functional methods were used for studying the Dicke model, some generalizations and other spin-boson problems, in the thermodynamic limit. In those model, the critical values of coupling constant and temperature were obtained, and asymptotic expressions for the free energy and the collective spectrum were found [5,6, 10,15–17]. In this paper and following the same integral functional approach, we study the critical behavior of two different spin-boson models. In the first model, we consider the wavelength of the bosonic field to be of the order of the linear dimension of the material composed of the atoms, therefore we consider the spatial sinusoidal form of the bosonic field [4,18]. The second model is the Thompson model [18,19] which considers the presence of phonons in the material composed of the atoms. In both cases, the systems exhibit a phase transition from normal to superradiant phase at some critical values of temperature and coupling constant. We find the asymptotic behavior of the partition function and the collective spectrum of the system in the normal and the superradiant phases. It is possible to observe that the collective spectrum has a zero energy value in the superradiant phase, corresponding to the Goldstone mode associated to the continuous symmetry breaking of the models. Our analysis and results are valid in the limit of zero temperature β → ∞, where the models exhibit a quantum phase transition. The two point correlation functions are fundamental tools to study critical phenomena in many-body systems [20]. Notwithstanding, recently studies of entanglement in many-body systems at zero and finite temperature show that concepts of entanglement of bipartite systems are good measures of correlations and have important relations with critical concepts [21]. At zero temperature, the concept of entanglement entropy is used to study quantum correlations in a bipartite system. At finite temperature, the concept of mutual information is an adequate quantity to measure the total correlations in a bipartite system. In several models, the concepts such as the entanglement entropy at zero temperature and the mutual information at finite temperature, show a maximum near the critical point and, in the thermodynamic limit, these quantities show relations with critical exponents [22–24]. The relationship between entanglement entropy and quantum phase transition in the Dicke model was also studied, where it is found that the atom–field entanglement entropy diverges at the critical point of the phase transition with the same critical exponent as the characteristic length [25,26]. This paper is organized as follows. In Section 2, we introduce the Hamiltonian and partition function of the models. In Section 3, the partition function in the path integral version is determined. In Section 4, using functional methods we study the phase transitions of the models, we find the critical temperature and asymptotic expressions for the partition function and collective spectrum in the thermodynamic limit. Finally, in Section 5 we state our summary. In this paper we use kB = h¯ = 1. 2. The Hamiltonian and the partition function of the models We will study two spin-boson models, which are generalizations of the Dicke model. Both models describe a system of N identical two-level atoms coupled to a single-mode quantized bosonic field, moreover the rotating wave approximation is assumed. The first model to be studied considers the spatial sinusoidal form of the bosonic field. Such consideration is valid when the wavelength λ of the bosonic field is of the order of the linear dimensions L in the material composed of the atoms. Therefore, the Hamiltonian of this model is given by HPos =
N Ω
2
i =1
g
σ(zi) + ω0 bĎ b + √
N
N
ξi b σ(+i) + bĎ σ(−i) ,
(1)
i =1
where we define the parameter ξi = sin(2 π n i/N ), with n = L/λ, the number of wavelengths of the bosonic field inside the material. In above equation we define the operators σ(±i) = 12 (σ(1i) ± i σ(2i) ), where the operators σ(1i) , σ(2i) and σ(zi) = σ(3i) , are the Pauli matrices. These matrices satisfy: [σ(+i) , σ(−i) ] = σ(zi) and [σ(zi) , σ(±i) ] = ± 2 σ(±i) . The b and bĎ are the boson annihilation and creation operators of mode excitations that satisfy the usual commutation relation rules.
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
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The second model to be studied is the Thompson model, which is the usual Dicke model with the presence of phonons in the material composed of the atoms. The Hamiltonian of the Thompson model can be written as HTh =
N Ω
2
g
σ(zi) + ω0 bĎ b + ϵ c Ď c + √
N
N i=1
i =1
b σ(+i) + bĎ σ(−i)
1 1 + √ (κ c + κ ∗ c Ď ) . N
(2)
where the operators σ(zi) , σ(±i) , b and bĎ , are the same as those in Eq. (1), and the new operators c and c Ď are boson annihilation and creation operators of the phonon excitations that satisfy the usual commutation relation rules. Let us define a fermion version for each model. For this purpose, let us define the raising and lowering Fermi operators αiĎ , αi , βiĎ and βi , that satisfy the anti-commutator relations αi αjĎ +αjĎ αi = δij and βi βjĎ +βjĎ βi = δij . In this analysis, we use a Ď
Ď
Ď
representation of the operators σ(zj) , σ(+j) and σ(−j) by the following bilinear combination of Fermi operators, αi αi − βi βi , αi βi and
βiĎ αi ,
the correspondence is given by
σ(zi) −→ αiĎ αi − βiĎ βi ,
(3)
σ(+i) −→ αiĎ βi ,
(4)
σ(−i) −→ βiĎ αi .
(5)
and
Using this representation, given by Eqs. (3)–(5), in the respective Hamiltonian of each model given by Eqs. (1) and (2), we define the Hamiltonian of the fermion models, so we have (F )
HPos = ω0 bĎ b +
N Ω
2 i=1
N g Ď ξi b αi βi + bĎ βiĎ αi . αiĎ αi − βiĎ βi + √
(6)
N i =1
and
N N g Ď 1 Ω Ď 1 + √ (κ c + κ ∗ c Ď ) . b αi βi + bĎ βi αi αiĎ αi − βiĎ βi + √
(F )
HTh = ω0 bĎ b + ϵ c Ď c +
2 i=1
N i =1
N
(7)
We are interested in studying thermodynamic properties of the systems, therefore we must compute the partition (F ) function in both cases. It is important to note that the Hamiltonians HPos and HPos , of the first model are defined in different (F )
spaces, and similar in the Thompson model, i.e., the Hamiltonians HTh and HTh are also defined in different spaces. Each operator σiα , appearing in the Hamiltonians HPos and HTh , acts on two-dimensional Hilbert space, notwithstanding, Fermi Ď
(F )
Ď
(F )
operators αi , αi , βi and βi , appearing in the Hamiltonians HPos and HTh , act on four-dimensional Fock space. The following properties relate the partition function of the models defined using the Pauli matrices, with the partition function of the fermion version of the models, we have
(F )
ZPos = Tr exp(−β HPos ) = iN Tr exp −β HPos −
iπ 2
N (F )
,
(8)
and
(F )
ZTh = Tr exp(−β HTh ) = iN Tr exp −β HTh −
iπ 2
N (F )
(F )
,
(9)
(F )
where HPos is given by Eq. (1), HPos by Eq. (6), HTh by Eq. (2), HTh by Eq. (7) and the operator N (F ) is defined by N (F ) =
N (αiĎ αi + βiĎ βi ).
(10)
i=1
The traces used in Eqs. (8) and (9) for each Hamiltonian are defined in their respective spaces. The relations given by Eqs. (8) and (9) allow us to express the partition function of the models we are studying, ZPos and ZTh , using the fermion Hamiltonians given by Eqs. (6) and (7). 3. The partition function using the path integral approach In this section we use path integral approach and functional methods [27,6] to compute the partition function in both models. Let us define the Euclidean action SPos of the first model in the following form β
SPos = 0
β
N ∗ ∗ dτ b (τ ) ∂τ b(τ ) + αi (τ ) ∂τ αi (τ ) + βi (τ ) ∂τ βi (τ ) − ∗
i=1
0
(F )
dτ HPos (τ ),
(11)
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(F )
where the Hamiltonian HPos (τ ) is obtained from the fermion Hamiltonian defined in Eq. (6), therefore we have (F )
HPos (τ ) = ω0 b ∗ (τ ) b(τ ) + g
+√
N
N
ξi
N Ω
2
α i∗ (τ ) α i (τ ) − β i∗ (τ )β i (τ )
i=1
α i (τ ) β i (τ ) b(τ ) + β i (τ ) α i (τ ) b (τ ) . ∗
∗
∗
(12)
i =1
For the Thompson model the Euclidean action STh , is given by β
STh =
dτ b∗ (τ ) ∂τ b(τ ) + c ∗ (τ ) ∂τ c (τ ) +
β
dτ
0
0
N
β
αi∗ (τ ) ∂τ αi (τ ) + βi∗ (τ ) ∂τ βi (τ ) −
(F )
dτ HTh (τ ), (13)
0
i =1
(F )
in this case, the Hamiltonian HTh (τ ) is obtained from the fermion Hamiltonian defined in Eq. (7), therefore we have (F )
HTh (τ ) = ω0 b ∗ (τ ) b(τ ) + ϵ c ∗ (τ ) c (τ ) + g
+√
N
N
N Ω
2
α i∗ (τ ) α i (τ ) − β i∗ (τ )β i (τ )
i =1
α i (τ ) β i (τ ) b(τ ) + β i (τ ) α i (τ ) b (τ ) ∗
∗
∗
i =1
1
1+ √ κ c (τ ) + κ c (τ ) N ∗ ∗
.
(14)
Since we are interested in studying thermodynamic properties of the systems, we must compute the partition function in both cases. Therefore we will study the following quantity Z Z (0)
β [dη] exp S − 2iπβ 0 n(τ )dτ , = β [dη] exp S0 − 2iπβ 0 n(τ )dτ
(15)
the function n(τ ) is defined by n(τ ) =
N
αi∗ (τ ) αi (τ ) + βi∗ (τ )βi (τ ) ,
(16)
i=1
where S is the Euclidean action given by Eq. (11) for the first model or Eq. (13) for the Thompson model. The action S0 is the free Euclidean action, corresponding to the case where there is no interaction between atoms and field, i.e., the expression of the complete action S setting g = 0. Finally, [dη] is the functional measure. Note that, the functional integrals involved in Eq. (15), are functional integrals with respect to the complex functions b∗ (τ ), b(τ ), and Fermi fields αi∗ (τ ), αi (τ ), βi∗ (τ ) and βi (τ ), for the first model. In the case of the Thompson model, the functional integrals involved in Eq. (15), are functional integrals with respect to the complex functions b∗ (τ ), b(τ ), c ∗ (τ ) and c (τ ), and Fermi fields αi∗ (τ ), αi (τ ), βi∗ (τ ) and βi (τ ). Since we are using thermal equilibrium boundary conditions, in the imaginary time formalism, the integration variables in Eq. (15) obey periodic boundary conditions for the Bose fields, i.e., b(β) = b(0) and c (β) = c (0) and anti-periodic boundary conditions for Fermi fields i.e., αi (β) = −αi (0) and βi (β) = −βi (0). Continuing with the calculation of the quantity (Z0) given by Eq. (15), let us use the following transformation Z
αi (τ ) → e
iπ t 2β
−
αi (τ ),
αi∗ (τ ) → e
βi (τ ) → e 2β βi (τ ),
βi∗ (τ ) → e
iπ
t
iπ t 2β
− 2iπβ t
αi∗ (τ ),
(17)
βi∗ (τ ).
With this last transformation, the term n(τ ) in Eq. (15) can be dropped. Therefore, applying the transformation given by Eq. (17) into the expression given by Eq. (15), we obtain that Z Z (0)
[dη] eS = . [dη] eS0
(18)
In Eq. (18), the Bose fields obey periodic boundary conditions, i.e., b(β) = b(0) and c (β) = c (0), and the Fermi fields obey the following boundary conditions:
αi (β) = i αi (0), βi (β) = i βi (0),
αi∗ (β) = −i αi∗ (0), βi∗ (β) = −i βi∗ (0).
(19)
Let us define the bosonic modes free actions for the models as follows (B0)
SPos (b) =
β
0
dτ b∗ (τ ) ∂τ − ω0 b(τ )
(20)
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3769
and (B0)
STh (b, c ) =
β
dτ
b (τ ) ∂τ − ω0 ∗
b(τ ) + c (τ ) ∂τ − ϵ c (τ ) . ∗
(21)
0
Then we can rewrite the actions SPos and STh given by Eqs. (11) and (13), using the bosonic modes free actions defined by Eqs. (20) and (21), plus some additional terms that can be expressed in matrix forms. Consequently, the total actions SPos and STh can be written as (B0)
SPos = SPos (b) +
β
dτ
N
0
ρiĎ (τ ) Mi (b∗ , b) ρi (τ ),
(22)
i =1
and β
(B0)
STh = STh (b, c ) +
dτ
N
0
ρiĎ (τ ) M (b∗ , b, c ∗ , c ) ρi (τ ),
(23)
i =1
Ď
where the column matrices ρ i (τ ) and ρi (τ ) are given in terms of Fermi field operators in the following way
ρ i (τ ) =
β i (τ ) , α i (τ )
ρ Ďi (τ ) = β ∗i (τ )
α ∗i (τ )
(24)
and the matrices Mi (b , b) and M (b , b, c , c ) are given by ∗
∗
Mi (b∗ , b) =
∗
g
L
− √ ξi b∗ (τ )
− √ ξi b (τ )
L∗
N
g
N
,
(25)
and M (b∗ , b, c ∗ , c )
g
− √ b∗ (τ )
L
N
= g 1 ∗ ∗ − √ b (τ ) 1 + √ κ c (τ ) + κ c (τ ) N
1 1+ √ κ c (τ ) + κ ∗ c ∗ (τ ) N L∗
N
.
(26)
The operators L and L∗ , in Eqs. (25) and (26), are defined by ∂τ + Ω /2 and ∂τ − Ω /2 respectively. In order to compute the partition function we use the expression given by Eq. (18). In such an equation, for each model, we substitute the actions SPos and STh given by Eqs. (22) and (23) respectively. For both models, the functional integrals are Gaussian in the Fermi fields. Therefore, let us begin the calculation integrating with respect these Fermi fields, so we obtain
N
(B0)
[dη(b)] eSPos
ZPos =
det Mi (b∗ , b) ,
(27)
i=1
and
ZTh =
(B0)
[dη(b, c )] eSTh
N
det M (b∗ , b, c ∗ , c )
.
(28)
In Eqs. (27) and (28) their respective functional measures [dη(b)] and [dη(b, c )] are functional measures only with respect of their respective bosonic fields. With the help of the following property for matrices with operator components
det
A C
B D
= det AD − ACA−1 B ,
(29)
and determinant properties, we have that
det Mi (b∗ , b) = det LL∗
1 −1 ∗ det 1 − N −1 g 2 ξi2 L− b , ∗ bL
(30)
and
1 −1/2 det M (b , b, c , c ) = det LL∗ det 1 − N −1 g 2 L− (κ c + κ ∗ c ∗ ) ∗ b 1+N
∗
∗
× L−1 b∗ 1 + N −1/2 (κ c + κ ∗ c ∗ ) .
(31)
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M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
Substituting Eqs. (27) and (30) in Eq. (18), and substituting Eqs. (28) and (31) in Eq. (18), we have that ZPos (0)
ZPos
=
A ZPos (B0)
,
(32)
(B0)
,
(33)
[dη(b)] eSPos
and ZTh (0)
ZTh
=
A ZTh
[dη(b)] eSTh
A A with ZPos and ZTh defined by
A ZPos =
N (B0) 1 −1 ∗ [dη(b)] exp SPos + tr ln 1 − N −1 g 2 ξi2 L− b L b , ∗
(34)
i =1
and A ZTh
=
(B0)
1 −1/2 [dη(b, c )] exp STh + N tr ln 1 − N −1 g 2 L− (κ c + κ ∗ c ∗ ) ∗ b 1+N
× L−1 b∗ 1 + N −1/2 (κ c + κ ∗ c ∗ ) .
(35)
4. Phase transition and the partition function in the thermodynamic limit We are interested in knowing the asymptotic behavior of the quotients
ZPos (0) ZPos
and
ZTh (0) ZTh
N → ∞. With this intention, we analyze the asymptotic behavior of the expressions
√
(35). First, in such equations, let us scale the bosonic fields by b → therefore we get A ZPos
= APos (N )
N b, b → ∗
√
in the thermodynamic limit, i.e.,
A ZPos
A and ZTh given by Eqs. (34) and
N b ,c → ∗
√
N c and c ∗ →
[dη(b)] exp N ΦPos (b∗ , b) ,
√
N c∗,
(36)
and A ZTh = ATh (N )
[dη(b, c )] exp N ΦTh (b∗ , b, c ∗ , c ) ,
(37)
with the functions ΦPos (b∗ , b) and ΦTh (b∗ , b, c ∗ , c ) defined by (B0)
ΦPos (b∗ , b) = SPos +
N 1
N
1 −1 ∗ tr ln 1 − g 2 ξi2 L− b . ∗ bL
(38)
i=1
and
(B0) 1 ∗ ∗ −1 ∗ ∗ ∗ ΦTh (b∗ , b, c ∗ , c ) = STh + tr ln 1 − g 2 L− b 1 + κ c + κ c L b 1 + κ c + κ c . ∗
(39)
The terms APos (N ) and ATh (N ) from Eqs. (36) and (37) respectively, arise√from transforming the functional √ √ measures [dη(b)] √ and [dη(b, c )] under the scaling of bosonic fields by b → N b, b∗ → N b∗ , c → N c and c ∗ → N c ∗ . The asymptotic behavior of the functional integrals in Eqs. (36) and (37) when N → ∞, can be obtained by using the method of steepest descent [28]. According to this method, in the case of the first model we expand the function ΦPos (b∗ , b) around some critical point b(τ ) = b0 (τ ) and b∗ (τ ) = b∗0 (τ ) and in the Thompson model we expand the function ΦTh (b∗ , b, c ∗ , c ) around some critical point b(τ ) = b0 (τ ), b∗ (τ ) = b∗0 (τ ), c (τ ) = c0 (τ ) and c ∗ (τ ) = c0∗ (τ ). Such critical points can be of two kinds, one makes Re(ΦPos (b∗ , b)) or Re(ΦTh (b∗ , b, c ∗ , c )) maximums, and the other one is defined as a saddle point. To find some asymptotic expressions for the partition functions, we consider the first terms of the expansion, which are the leading terms for the value of the total functional integrals. We can find the maximum points, or saddle points, finding the δ ΦPos (b∗ ,b) (b∗ ,b) stationary points. In the first model, the stationary points are solution of the equations = 0 and δ ΦδPos = 0. δ b(τ ) b∗ (τ ) In the Thompson model, the stationary points are solution of the equations δ ΦTh (b∗ ,b,c ∗ ,c ) δ c (τ )
δ ΦTh (b∗ ,b,c ∗ ,c ) and δ c ∗ (τ ) ∗ ∗
δ ΦTh (b∗ ,b,c ∗ ,c ) δ b(τ )
= 0,
δ ΦTh (b∗ ,b,c ∗ ,c ) δ b∗ (τ )
= 0,
=0 = 0. As results of these equations, for both models, the stationary points are constant functions b(τ ) = b0 , b (τ ) = b0 , c (τ ) = c0 and c ∗ (τ ) = c0∗ . And, it is not difficult to show that for temperatures β −1 ≥ βc−1
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3771
the stationary point is given by b0 = b∗0 = 0 in the first model and by b0 = b∗0 = c0 = c0∗ = 0 in the Thompson model, which are maximum points. Moreover, the critical temperature βc−1 in the first model is obtained by solving the following equation 2 ω0 Ω
= tanh
g2
βc Ω
2
,
(40)
and for the Thomson model we have
ω0 Ω g2
= tanh
βc Ω
2
.
(41)
In the first model, from Eq. (40), we can see that the critical temperature is independent of n, which appears in the definition of ξi = sin(2 π n i/N ). We also note that, it is possible to find some critical temperature when g 2 > 2 ω0 Ω . Similarly, in the Thomson model some value of critical temperature is found when g 2 > ω0 Ω . With these conditions the systems undergo a phase transition. When the systems have temperatures β −1 > βc−1 we say that the systems are in the normal phase. In the first model, for temperatures β −1 < βc−1 , the stationary points b(τ ) = b0 and b∗ (τ ) = b∗0 satisfy the following equation
ω0 g2
N 1
=
N
i =1
ξi2 β Ω 2 + 4 g 2 |b0 |2 ξi2 . tanh 2 Ω 2 + 4 g 2 |b0 |2 ξi2
(42)
Therefore, in the limit N → ∞, we can approximate the Eq. (42) by
ω0 g2
1
=
dx
sin2 (2 π n x)
0
Ω∆ ( x )
tanh
β
Ω∆ ( x ) ,
2
(43)
with Ω∆ (x) defined by
Ω∆ (x) =
Ω 2 + 4 g 2 |b0 |2 sin2 (2 π n x).
(44)
In the Thompson model, for temperatures β −1 < βc−1 , the stationary points b(τ ) = b0 , b∗ (τ ) = b∗0 , c (τ ) = c0 and c ∗ (τ ) = c0∗ satisfy the following equations
ω0 Ω∆ g2
= (1 + 2 κ c0 ) tanh 2
β Ω∆
2
,
(45)
with Ω∆ defined by
Ω∆ =
Ω 2 + 4 g 2 |b0 |2 (1 + 2 κ c0 )2 ,
(46)
and we find that κ c0 is real, satisfying
κ c0 (1 + 2 κ c0 ) =
2 ω0 |κ|2 |b0 |2
ϵ
.
(47)
Phase transition happens if it is possible to find some real solution for |b0 | ̸= 0 in Eq. (43) in the first model and some real solution for |b0 | ̸= 0 in Eq. (45) in the Thompson model. The maximum points are a continuous set of values given by the expression b0 = ρ ei φ and b∗0 = ρ e−i φ with φ ∈ [0, 2π ) and ρ = |b0 |, where |b0 | is a solution of Eq. (43) for the first model and |b0 | is a solution of Eq. (45) for the Thompson model. When the systems have temperatures β −1 < βc−1 we say that the systems are in the superradiant phase. Let us continue with the computation of the asymptotic behavior for the functional integrals appearing in Eqs. (36) and (37), in the thermodynamic limit, N → ∞. In the following steps, we shall find this asymptotic behavior when we only have one maximum point, defined by b0 = b∗0 in the first model and by b0 = b∗0 and κ c0 = κ c0∗ in the Thompson model. The resulting expressions, will be valid in the normal phases and useful for the superradiant phases, because in the superradiant phases, the calculation will follow the same steps for the modes different from zero when we use the Fourier representation of the fields. We consider the two first leading terms in the functional integrals appearing in Eqs. (36) and (37), arising from the expansions of ΦPos (b∗ , b) in the first model and of ΦTh (b∗ , b, c ∗ , c ) in the Thompson model around their maximum points, these expansions are given by
ΦPos (b∗ , b) = ΦPos (b∗0 , b0 ) +
1 2
β
0
dτ1 dτ2 (b∗ (τ1 ) − b∗0 , b(τ1 ) − b0 ) MΦPos (τ1 , τ2 )
b∗ (τ2 ) − b∗0 b(τ2 ) − b0
,
(48)
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M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
and 1
ΦTh (b∗ , b, c ∗ , c ) = ΦTh (b∗0 , b0 , c0∗ , c0 ) +
β
2
dτ1 dτ2 b∗ (τ1 ) − b∗0 , b(τ1 ) − b0 , c ∗ (τ1 ) − c0∗ , c (τ1 ) − c0
0
b∗ (τ2 ) − b∗0 b(τ ) − b0 , × MΦTh (τ1 , τ2 ) ∗ 2 c (τ2 ) − c0∗ c (τ2 ) − c0
(49)
the matrices MΦPos (τ1 , τ2 ) and MΦTh (τ1 , τ2 ), are given by
δ 2 ΦPos (b∗ , b) δ b∗ (τ1 ) δ b∗ (τ2 ) MΦPos (τ1 , τ2 ) = δ 2 ΦPos (b∗ , b) δ b(τ1 ) δ b∗ (τ2 )
δ 2 ΦPos (b∗ , b) δ b∗ (τ1 ) δ b(τ2 ) , 2 ∗ δ ΦPos (b , b) δ b(τ1 ) δ b(τ2 ) b∗ =b=b0
(50)
and MΦTh (τ1 , τ2 )
δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b∗ (τ1 ) δ b∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ b(τ1 ) δ b∗ (τ2 ) = δ 2 Φ (b∗ , b, c ∗ , c ) Th δ c ∗ (τ1 ) δ b∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c (τ1 ) δ b∗ (τ2 )
δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b∗ (τ1 ) δ b(τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ b(τ1 ) δ b(τ2 ) δ 2 ΦTh (b∗ , b, c ∗ , c ) δ c ∗ (τ1 ) δ b(τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c (τ1 ) δ b(τ2 )
δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b∗ (τ1 ) δ c ∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ b(τ1 ) δ c ∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c ∗ (τ1 ) δ c ∗ (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c (τ1 ) δ c ∗ (τ2 )
δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b∗ (τ1 ) δ c (τ2 ) δ 2 ΦTh (b∗ , b, c ∗ , c ) δ b(τ1 ) δ c (τ2 ) . 2 ∗ ∗ δ ΦTh (b , b, c , c ) δ c ∗ (τ1 ) δ c (τ2 ) 2 δ ΦTh (b∗ , b, c ∗ , c ) δ c (τ1 ) δ c (τ2 ) b∗ ,b ,c ∗ ,c 0 0 0 0
(51)
Substituting these expansions given by Eqs. (48) and (49) in Eqs. (36) and (37) respectively, we obtain A ZPos
=e
N ΦPos (b∗ ,b ) 0 0
[dη(b)] exp
β
1
2
dτ1 dτ2
0
∗ b (τ2 ) b (τ1 ), b(τ1 ) MΦPos (τ1 , τ2 ) , b(τ2 )
∗
(52)
and
A ZTh = eN ΦTh (b0 ,b0 ,c0 ,c0 ) ∗
∗
1 [dη(b, c )] exp
2
β
0
b∗ (τ2 ) b(τ ) b∗ (τ1 ), b(τ1 ), c ∗ (τ1 ), c (τ1 ) MΦTh (τ1 , τ2 ) ∗ 2 . (53) c (τ2 ) c (τ2 )
dτ1 dτ2
To obtain the last two expressions, we have applied the transformation b(τ ) →
√
b(τ ) + b0 / N, b∗ (τ ) →
b∗ (τ ) + √ √ √ b∗0 / N , c (τ ) → c (τ ) + c0 / N and c ∗ (τ ) → c ∗ (τ ) + c0∗ / N in the functional integrals involved. At this level, it is convenient to use Fourier representation of the fields b(τ ) and c (τ ) in the functional integrals in Eqs. (52) and (53). Therefore, we have that 1 b(ω)eiωτ , b(τ ) = √
β
ω
1 ∗ b∗ (τ ) = √ b (ω)e−iωτ ,
β ω 1 c (τ ) = √ c (ω)eiωτ , β ω 1 ∗ c ∗ (τ ) = √ c (ω)e−iωτ , β ω
(54)
since the bosonic fields b(τ ) and c (τ ) satisfy periodic boundary conditions, the parameter ω takes the values 2π n/β , with n being all the integers. These values correspond to the Matsubara frequencies for bosonic fields. Substituting this Fourier representations, Eq. (54) in Eqs. (52) and (53), we obtain that A ZPos
=e
N ΦPos (b∗ ,b ) 0 0
[dη(b)] exp
1 ∗ b∗ (ω2 ) b (ω1 ), b(ω1 ) MΦPos (ω1 , ω2 ) b(ω2 ) 2ω ω
1 2
,
(55)
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3773
and
∗ b (ω2 ) 1 b(ω ) [dη(b, c )] exp b∗ (ω1 ), b(ω1 ), c ∗ (ω1 ), c (ω1 ) MΦTh (ω1 , ω2 ) ∗ 2 , c (ω )
A ZTh = eN ΦTh (b0 ,b0 ,c0 ,c0 ) ∗
∗
2ω ω 1 2
(56)
2
c (ω2 )
with MΦPos (ω1 , ω2 ) and MΦTh (ω1 , ω2 ) being defined by
δ 2 ΦPos (b∗ , b) −i (ω1 τ1 +ω2 τ2 ) β δ b∗ (τ1 ) δ b∗ (τ2 ) e dτ1 dτ2 δ 2 ΦPos (b∗ , b) 0 ei (ω1 τ1 −ω2 τ2 ) δ b(τ1 ) δ b∗ (τ2 )
1
MΦPos (ω1 , ω2 ) =
β
δ 2 ΦPos (b∗ , b) −i (ω1 τ1 −ω2 τ2 ) e δ b∗ (τ1 ) δ b(τ2 ) , δ 2 ΦPos (b∗ , b) i (ω1 τ1 +ω2 τ2 ) e δ b(τ1 ) δ b(τ2 )
(57)
and MΦ (ω1 , ω2 ) =
1
β
β
dτ1 dτ2
0
δ 2 Φ (b∗ , b, c ∗ , c )
e−i (ω1 τ1 +ω2 τ2 )
δ b∗ (τ1 ) δ b∗ (τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) ei (ω1 τ1 −ω2 τ2 ) δ b(τ1 ) δ b∗ (τ2 ) × 2 ∗ ∗ δ Φ (b , b, c , c ) −i (ω τ +ω τ ) 1 1 2 2 δ c ∗ (τ ) δ b∗ (τ ) e 1 2 2 ∗ ∗ δ Φ (b , b, c , c ) i (ω1 τ1 −ω2 τ2 ) e δ c (τ1 ) δ b∗ (τ2 )
δ 2 Φ (b∗ , b, c ∗ , c ) −i (ω1 τ1 −ω2 τ2 ) e δ b∗ (τ1 ) δ b(τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) i (ω1 τ1 +ω2 τ2 ) e δ b(τ1 ) δ b(τ2 ) 2 ∗ ∗ δ Φ (b , b, c , c ) −i (ω1 τ1 −ω2 τ2 ) e δ c ∗ (τ1 ) δ b(τ2 ) 2 ∗ ∗ δ Φ (b , b, c , c ) i (ω1 τ1 +ω2 τ2 ) e δ c (τ1 ) δ b(τ2 )
δ 2 Φ (b∗ , b, c ∗ , c ) −i (ω1 τ1 +ω2 τ2 ) e δ b∗ (τ1 ) δ c ∗ (τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) i (ω1 τ1 −ω2 τ2 ) e δ b(τ1 ) δ c ∗ (τ2 ) 2 ∗ ∗ δ Φ (b , b, c , c ) −i (ω1 τ1 +ω2 τ2 ) e δ c ∗ (τ1 ) δ c ∗ (τ2 ) 2 ∗ ∗ δ Φ (b , b, c , c ) i (ω1 τ1 −ω2 τ2 ) e δ c (τ1 ) δ c ∗ (τ2 )
δ 2 Φ (b∗ , b, c ∗ , c ) −i (ω1 τ1 −ω2 τ2 ) e δ b∗ (τ1 ) δ c (τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) i (ω1 τ1 +ω2 τ2 ) e δ b(τ1 ) δ c (τ2 ) , δ 2 Φ (b∗ , b, c ∗ , c ) −i (ω1 τ1 −ω2 τ2 ) e δ c ∗ (τ1 ) δ c (τ2 ) δ 2 Φ (b∗ , b, c ∗ , c ) i (ω1 τ1 +ω2 τ2 ) e δ c (τ1 ) δ c (τ2 ) b∗ ,b ,c ∗ ,c 0 0 0 0
(58)
where in the last case Φ = ΦTh . In these Fourier representations of the functional integrals given by Eqs. (55) and (56), the integral measures [dη(b)] and [dη(b, c )] take the tractable forms ω db∗ (ω) db∗ (ω) and ω db∗ (ω) db(ω), dc ∗ (ω) dc (ω) respectively. In the first model, using the expression for ΦPos (b∗ , b) given by Eq. (38), we can calculate the matrix MΦPos (ω1 , ω2 ). Performing these calculations we obtain that MΦPos (ω1 , ω2 ) =
R(ω2 ) δω1 ,−ω2 S (ω2 ) δω1 , ω2
S (ω2 ) δω1 , ω2 R(ω2 ) δω1 ,−ω2
,
(59)
with δω1 , ω2 being the delta Kronecker and the functions R(ω) and S (ω) are given by sin4 (2 π n x)
1
β Ω∆ (x)
, 2 Ω∆ (x) ω2 + Ω∆2 (x) 1 sin2 (2 π n x) Ω∆ (x)2 + Ω 2 + 2 i ω Ω β Ω∆ (x) g2 dx tanh , S (ω) = i ω − ω0 + 2 0 2 Ω∆ (x) ω2 + Ω∆2 (x) R(ω) = −2 b20 g 4
dx
0
tanh
(60)
where the function Ω∆ (x) is defined in the Eq. (44). Substituting the matrix MΦ (ω1 , ω2 ), with components given by Eq. (59), A , given by Eq. (55), we obtain that in the functional integral appearing in ZPos A ZPos
=e
N ΦPos (b∗ ,b ) 0 0
[dη(b)] exp
S (ω) b(ω) b (ω) + ∗
ω
1 2
R(ω)
b(ω) b(−ω) + b (ω) b (−ω) ∗
∗
.
(61)
Performing this Gaussian functional integral, we finally obtain that
( 2 π i )2 2π i ∗ A . ZPos = eN ΦPos (b0 ,b0 ) 1/2 S (ω) S (−ω) − R2 (ω) S 2 (0) − R2 (0) ω≥1 In order to find the asymptotic behavior of
ZPos 0 ZPos
(62) (B0)
when N → ∞, we must calculate [dη(b)] eSPos appearing in Eq. (32). Using
the free bosonic action SB0 given by Eq. (20), we obtain that
B0
[dη(b)] eSPos =
2π i
ω
ω0 − i ω
.
(63)
Substituting Eqs. (62) and (63) in Eq. (32) we have that ZPos (0)
ZPos
= eN ΦPos (b0 ,b0 ) ∗
1
(H (0))
1/2
1
ω≥1
H (ω)
,
(64)
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M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
in this last equation, the function H (ω) is given by H (ω) =
S (ω) S (−ω) − R2 (ω)
ω2 + ω02
,
(65)
where the Eq. (60) gives the expressions for the functions S (ω) and R(ω). 4.1. Normal phases of the models, β −1 > βc−1 In the normal phase we have β −1 > βc−1 , for the first model we have b0 = b∗0 = 0, see Eq. (42). It implies that Ω∆ (x) = Ω . Substituting this equality in Eqs. (64) and (65), we obtain that ZPos (0)
=
ZPos
1
ω≥1
HI (ω)
1
(HI (0))1/2
,
(66)
where HI (ω) = 1 +
g4 4 (ω2 + Ω 2 ) (ω2 + ω02 )
tanh2
βΩ
g 2 (ω2 − Ω ω0 )
+
2
(ω2 + Ω 2 ) (ω2 + ω02 )
tanh
βΩ
2
.
(67)
For the Thompson model, from Eq. (45) we have that b0 = b∗0 = c0 = c0∗ = 0. Finding the matrix MΦTh (ω1 , ω2 ) given by Eq. (58) we obtain MΦTh (ω1 , ω2 )
0
i ω2 − ω0 + T (ω2 ) δω1 , ω2
0
0
0
0
0
0 0
0
(i ω2 − ϵ) δω1 , ω2
,
= i ω2 − ω0 + T (ω2 ) δω1 , ω2 0 0
(i ω2 − ϵ) δω1 , ω2
(68)
0
where T (ω) =
g 2 (Ω + i ω)
Ω 2 + ω2
tanh
βΩ 2
.
(69)
Therefore, performing the functional integral in Eq. (56) and substituting in Eq. (33), we have ZTh (0)
ZTh
=
1
ω≥1
KI (ω)
1
(KI (0))
1/2
,
(70)
where KI (ω) = 1 +
g4
2
(ω2 + Ω 2 ) (ω2 + ω02 )
tanh
βΩ
+
2
2 g 2 (ω2 − Ω ω0 )
(ω2 + Ω 2 ) (ω2 + ω02 )
tanh
βΩ 2
.
(71)
Making the analytic continuation (iω → E ) in HI (ω) and KI (ω), we solve the equations HI (−i E ) = 0 and KI (−i E ) = 0, such equations correspond to the collective spectrum equation of the first model and the Thompson model respectively. Solving such equations, the collective spectrum for the first model (EPos ) and Thompson model (ETh ), are given respectively by 2 EPos = ω0 + Ω ±
1/2 2 βΩ ω0 − Ω + 2 g 2 tanh , 2
(72)
and 2 ETh = ω0 + Ω ±
ω0 − Ω
2
2
+ 4 g tanh
βΩ 2
1/2
.
(73)
In this normal phase, the results for both models are the√same as the results for the usual Dicke model, only with the difference that for the first model we have to change g → 2 g. 4.2. Superradiant phases of the models, β −1 < βc−1 Let us study both models in the superradiant phase β −1 < βc−1 . We have seen that the maximum point, giving the maximum contribution in the functional integrals given by Eqs. (36) and (37), is not unique. Therefore, we have to perform the following calculations differently from the normal phase. First, we note that the functions ΦPos (b∗ , b) and
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3775
ΦTh (b∗ , b, c ∗ , c ) defined in Eqs. (38) and (39) respectively, are invariants by the transformation b(τ ) → exp(i θ ) b(τ ) and b∗ (τ ) → exp(−i θ ) b∗ (τ ), where θ is an arbitrary factor independent of τ . This continuous invariance is responsible for the appearing of Goldstone mode in the systems. In order to obtain asymptotic expressions for the functional integrals given by Eqs. (36) and (37), let us separate the functions b(τ ) and c (τ ) in the following form b (τ ) = bc + b′ (τ ), b∗ (τ ) = b∗c + b ′ ∗ (τ ), c (τ ) = cc + c ′ (τ ), c ∗ (τ ) = cc∗ + c ′ ∗ (τ ),
(74)
where bc and cc are constant functions, and the fields b (τ ), b (τ ), c (τ ) and c (τ ) satisfy the boundaries conditions b′ (0) = b′ (β) = 0, b ′ ∗ (0) = b ′ ∗ (β) = 0, c ′ (0) = c ′ (β) = 0 and c ′ ∗ (0) = c ′ ∗ (β) = 0. Using the representation bc = ρ ei φ and b∗c = ρ e−i φ in the functional integrals given by Eqs. (36) and (37), followed by the transformation b′ (τ ) → ei φ b′ (τ ) and b ′ ∗ (τ ) → e−i φ b ′ ∗ (τ ), we obtain that ′
A ZPos = 2 π i A(N )
∞
dρ 2
′∗
′
′∗
[dη(b′ )] exp N ΦPos (ρ, b ′ ∗ , b′ ) ,
(75)
0
and A ZTh = 2 π i A(N )
∞
dρ 2
dcc dcc∗
[dη(b′ , c ′ )] exp N ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) ,
(76)
0
where the functions ΦPos (ρ, b ′ ∗ , b′ ) and ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) are given by
ΦPos (ρ, b ′ ∗ , b′ ) =
β
dτ ρ + b ′ ∗ (τ )
∂τ − ω0
ρ + b′ (τ )
0
+
N 1
N i=1
1 tr ln 1 − g 2 ξi2 L− ρ + b′ L−1 ρ + b ′ ∗ ∗
,
(77)
and
ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) =
β
ρ + b ′ ∗ (τ ) ∂τ − ω0 ρ + b′ (τ ) 0 1 + cc∗ + c ′∗ (τ ) (∂τ − ϵ) cc + c ′ (τ ) + tr ln 1 − g 2 L− ρ + b′ ∗ × 1 + κ cc + κ ∗ cc∗ + κ c ′ + κ ∗ c ′∗ L−1 ρ + b ′ ∗ × 1 + κ cc + κ ∗ cc∗ + κ c ′ + κ ∗ c ′∗ .
dτ
(78)
It is important to note that the integrals appearing in Eqs. (75) and (76), are combinations of integration of a real variable
ρ 2 , complex variables cc and cc∗ , and functional integration with respect to the fields b ′ ∗ , b′ , c ′ ∗ and c ′ . In the following, we use the steepest descent method in order to analyze the limit N → ∞. We find one stationary point with respect to the variable ρ 2 for the first model and one stationary point with respect to the variables ρ 2 , cc and cc∗ for the Thompson δΦ = 0 with b ′ ∗ (τ ) = b′ (τ ) = 0 for the first model model. These stationary points satisfy the following equations: δ (ρPos 2) ρ=ρ0 δΦ δΦ and δ (ρTh = 0, δδΦcTh = 0 and b ′ ∗ (τ ) = b′ (τ ) = c ′ ∗ (τ ) = c ′ (τ ) = 0 for the Thompson ∗ = 0, δ cTh ∗ 2) ∗ ∗ c ρ0 ,c0 ,c0
ρ0 ,c0 ,c0
c
ρ0 ,c0 ,c0
model. As results, the solutions for ρ0 in both models, are the same as |b0 | defined by Eq. (43) for the first model and (45) for the Thompson model. Moreover, for the Thompson model, the values of c0 and c0∗ obey the equation κ ∗ c0∗ = κ c0 and Eq. (47). Let us consider the two first leading terms in the functional integrals appearing in Eqs. (75) and (76), arising from the expansions of ΦPos (ρ, b ′ ∗ , b′ ) and ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) around the points defined by ρ0 and b ′ ∗ (τ ) = b′ (τ ) = 0 in the first case, and ρ0 , c0 , c0∗ and b ′ ∗ (τ ) = b′ (τ ) = c ′ ∗ (τ ) = c ′ (τ ) = 0 in the second case. Such points are maximum
points of Re ΦPos (ρ, b ′ ∗ , b′ ) and Re ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ ) respectively. Therefore, the expansions are given by
2 δ 2 Φ 2 2 ρ − ρ 0 2 δ(ρ 2 )2 ρ=ρ0 , b′ =b ′∗ =0 ′∗ 1 β b (τ2 ) dτ1 dτ2 b ′ ∗ (τ1 ), b′ (τ1 ) MΦPos (τ1 , τ2 ) , + ′ b (τ2 ) 2 0
ΦPos (ρ, b ′ ∗ , b′ ) = ΦPos (ρ0 , 0, 0) +
1
(79)
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M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
and
ΦTh (ρ, cc∗ , cc , b ′ ∗ , b′ , c ′∗ , c ′ )
ρ 2 − ρ02 = ΦTh (ρ0 , c0∗ , c0 , 0, 0, 0, 0) + ρ 2 − ρ02 , cc∗ − c0∗ , cc − c0 M0 cc∗ − c0∗ 2 cc − c0 ′∗ b (τ2 ) ′ 1 β b (τ ) + dτ1 dτ2 b ′ ∗ (τ1 ), b′ (τ1 ), c ′ ∗ (τ1 ), c ′ (τ1 ) MΦTh (τ1 , τ2 ) ′ ∗ 2 , c (τ2 ) 2 0 c ′ (τ2 ) 1
(80)
where the matrices MΦPos (τ1 , τ2 ) and M0 are given by
δ 2 ΦPos δ 2 ΦPos δ b′∗ (τ1 ) δ b′∗ (τ2 ) δ b′∗ (τ1 ) δ b′ (τ2 ) MΦPos (τ1 , τ2 ) = , δ 2 ΦPos δ 2 ΦPos δ b′ (τ1 ) δ b′∗ (τ2 ) δ b′ (τ1 ) δ b′ (τ2 ) ρ=ρ0 , b′ =b ′∗ =0 2 δ ΦTh δ 2 ΦTh δ 2 ΦTh 2 ) δc ∗ 2 ) δc δ(ρ 2 )2 δ(ρ δ(ρ c c δ2 Φ δ 2 ΦTh δ 2 ΦTh Th , M0 = ∗ 2 ∗ 2 ∗ δ cc δ(ρ ) δ(cc ) δ cc δ cc δ 2 ΦTh δ 2 ΦTh δ 2 ΦTh 2 ∗ δ cc δ(ρ ) δ cc δ cc δ(cc )2 ρ0 ,c ∗ ,c0 , b′ =b ′∗ =c ′ =c ′∗ =0
(81)
(82)
0
and MΦTh (τ1 , τ2 ) is given by
δ 2 ΦTh δ b′∗ (τ1 ) δ b′∗ (τ2 ) δ 2 ΦTh δ b′ (τ1 ) δ b′∗ (τ2 ) δ 2 ΦTh ′∗ δ c (τ1 ) δ b′∗ (τ2 ) δ 2 ΦTh ′ δ c (τ1 ) δ b′∗ (τ2 )
δ 2 ΦTh δ b′∗ (τ1 ) δ b′ (τ2 ) δ 2 ΦTh ′ δ b (τ1 ) δ b′ (τ2 ) δ 2 ΦTh δ c ′∗ (τ1 ) δ b′ (τ2 ) δ 2 ΦTh ′ δ c (τ1 ) δ b′ (τ2 )
δ 2 ΦTh δ b′∗ (τ1 ) δ c ′∗ (τ2 ) δ 2 ΦTh ′ δ b (τ1 ) δ c ′∗ (τ2 ) δ 2 ΦTh δ c ′∗ (τ1 ) δ c ′∗ (τ2 ) δ 2 ΦTh ′ δ c (τ1 ) δ c ′∗ (τ2 )
δ 2 ΦTh δ b′∗ (τ1 ) δ c ′ (τ2 ) δ 2 ΦTh δ b′ (τ1 ) δ c ′ (τ2 ) . 2 δ ΦTh δ c ′∗ (τ1 ) δ c ′ (τ2 ) δ 2 ΦTh ′ ′ δ c (τ1 ) δ c (τ2 ) ρ0 ,c0∗ ,c0 , b′ =b ′∗ =c ′ =c ′∗ =0
(83)
The missing second order contributions in Eqs. (79) and (80) are zero. It happens since these terms are of the form, for 2 example, dτ1 C (τ1 ) (ρ 2 − ρ02 ) b′ (τ1 ), where C (τ1 ) = 21 δ(ρ 2δ) δΦb′ (τ ) , and it is possible to show that C (τ1 ) = C0 1
ρ=ρ0 , b′ =b ′∗ =0
and dτ1 b′ (τ1 ) = 0. Using the expansions given by Eqs. (79) and (80) to perform functional integrals given by Eqs. (75) and (76) respectively, we have that
A ZPos
= 2π i
√ Ne
N φPos
∞ √ − N ρ02
β
× exp
1 2
0
dy e
2 1 δ ΦPos 2 δ(ρ 2 )2
ρ=ρ0 , b′ =b ′∗ =0
y2
[dη(b′ )]
b ′ ∗ (τ2 ) dτ1 dτ2 b ′ ∗ (τ1 ), b′ (τ1 ) MΦPos (τ1 , τ2 ) b′ (τ2 )
,
(84)
and A ZTh
= 2π i
√ Ne
N φTh
2
∞ √ − N ρ02
1 × exp
β
0
dτ1 dτ2
dy
∗
dcc dcc exp
1 2
y, cc , cc M0 ∗
y cc∗ cc
[dη(b′ , c ′ )]
′∗ b (τ2 ) ′ b (τ ) b ′ ∗ (τ1 ), b′ (τ1 ), c ′ ∗ (τ1 ), c ′ (τ1 ) MΦTh (τ1 , τ2 ) ′ ∗ 2 , c (τ2 ) c ′ (τ2 )
(85)
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3777
√ N √ appearing in Eqs. (84) and (85) arise from the scaling ρ 2 → ρ 2 / N. For N → ∞, the integrals in Eqs. (84) and (85) are Gaussians. We represent the functions b′ (τ ), b ′ ∗ (τ ), c ′ (τ ) and c ′ ∗ (τ ) by Fourier series as follows 1 ′ ′ b′ (τ ) = √ b (ω)eiωτ , β ω 1 ′ ′∗ b (ω)e−iωτ , b ′∗ (τ ) = √ β ω 1 ′ ′ c ′ (τ ) = √ c (ω)eiωτ , β ω 1 ′ ′∗ c ′∗ (τ ) = √ c (ω)e−iωτ , (86) β ω where the expressions φPos = ΦPos (ρ0 , 0, 0) in Eq. (84) and φTh = ΦTh (ρ0 , c0∗ , c0 , 0, 0, 0, 0) in Eq. (85). The factors
since these functions satisfy the boundary conditions b′ (0) = b′ (β) = 0, b ′ ∗ (0) = b ′ ∗ (β) = 0, c ′ (0) = c ′ (β) = 0 and c ′ ∗ (0) = c ′ ∗ (β) = 0, the parameter ω takes the values 2π n/β , with n being all the integers different from zero. The prime after the summation symbols in Eq. (86) means that it is not considering ω = 0. Using the last Fourier representation in Eqs. (84) and (85), we obtain A ZPos = 2π i
√
N eN φPos
∞
dy e
2 1 δ ΦPos 2 δ(ρ 2 )2
ρ=ρ0 , b′ =b ′∗ =0
y2
−∞
[dη(b )] exp ′
×
′ 1 ′∗ b ′∗ (ω2 ) b (ω1 ), b′ (ω1 ) MΦPos (ω1 , ω2 ) b′ (ω2 ) 2ω ω
,
(87)
1 2
and A ZTh
= 2π i
√ Ne
N φTh
∞
√ − N ρ02
dy
∗
dcc dcc exp
1 y, cc∗ , cc M0 2
y cc∗ cc
[dη(b′ , c ′ )]
′∗ b (ω2 ) ′ b (ω ) 1 b ′ ∗ (ω1 ), b′ (ω1 ), c ′ ∗ (ω1 ), c ′ (ω1 ) MΦTh (ω1 , ω2 ) ′ ∗ 2 . (88) × exp c (ω2 ) 2 ω ,ω 1 2 ′ c (ω2 ) The matrix MΦPos (ω1 , ω2 ) has the same form as that appearing in Eq. (57) but changing ΦPos (b ∗ , b) by ΦPos (ρ, b ′ ∗ , b′ ) and evaluating in ρ0 and b ′ ∗ (0) = b ′ ∗ (β) = 0. The resulting matrix is the same as which appears in Eq. (59), but changing b0 by ρ0 . For the Thompson model, the matrix MΦTh (ω1 , ω2 ) has the same form as which appears in Eq. (58) but changing ΦTh (b ∗ , b, c ∗ , c ) by ΦTh (ρ, cc , cc∗ , b ′ ∗ , b′ , c ′ ∗ , c ′ ) and evaluating in ρ0 , c0∗ , c0 and b ′ ∗ (τ ) = b′ (τ ) = c ′ ∗ (τ ) = c ′ (τ ) = 0.
′
Performing some calculations, we arrive at MΦTh (ω1 , ω2 ) A1 (ω2 ) δω1 ,−ω2 A2 (ω2 ) δω1 , ω2 = κ ∗ A (−ω ) δ 4 2 ω1 ,−ω2 κ A4 (−ω2 ) δω1 , ω2
A2 (ω2 ) δω1 , ω2 A1 (ω2 ) δω1 ,−ω2 κ ∗ A4 (−ω2 ) δω1 , ω2 κ A4 (−ω2 ) δω1 ,−ω2
κ ∗ A4 (ω2 ) δω1 ,−ω2 κ ∗ A4 (−ω2 ) δω1 , ω2 κ ∗ 2 A3 (ω2 ) δω1 ,−ω2 (i ω2 − ϵ + |κ|2 A3 (ω2 )) δω1 , ω2
κ A4 (−ω2 ) δω1 , ω2 κ A4 (ω2 ) δω1 ,−ω2 (i ω2 − ϵ + |κ|2 A3 (ω2 )) δω1 , ω2 , (89) κ 2 A3 (ω2 ) δω1 ,−ω2
where the functions A1 (ω), A2 (ω), A3 (ω) and A4 (ω), are defined by
2 Ω∆2 − Ω 2 β Ω∆ , A1 (ω) = − tanh 2 2 8 ρ02 Ω∆ ω2 + Ω∆ Ω∆2 − Ω 2 Ω∆2 + Ω 2 + 2 i ω Ω β Ω∆ A2 (ω) = i ω − ω0 + tanh , 2 2 8 ρ02 Ω∆ ω2 + Ω∆ 2 g 2 Ω 2 ρ02 β Ω∆ , A3 (ω) = tanh 2 Ω∆ ω2 + Ω∆2
A4 (ω) =
2 g Ω∆ − Ω2
2 Ω∆
12 1+
Ω (Ω − i ω)
ω2 + Ω∆2
tanh
β Ω∆ 2
.
(90)
3778
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
Finally, performing the functional integrals given by Eqs. (87) and (88), and substituting in Eqs. (32) and (33) respectively, we obtain that ZPos (0)
√
N eN φPos
=
ZPos
1
1
DPos ω≥1 HII (ω)
,
(91)
and ZTh (0)
√
N eN φTh
=
ZTh
1
1
DTh ω≥1 KII (ω)
.
(92)
The functions φPos , φTh , DPos and DTh are given by
φPos = − ω0 β ρ02 +
1
cosh
cosh
φTh = − ω0 β ρ02 − ϵ β |c0 |2 + ln
DPos = √
g
π β ω0
β Ω∆ (x) 2
dx ln 0
2
βΩ
cosh
dx
β Ω∆ 2
0
,
(93)
2
cosh
βΩ
,
(94)
2
12
sin (2 π n x) β Ω∆ (x) tanh 1 − Ω∆3 (x) sinh β Ω∆ (x) 4
1
β Ω∆ (x) ,
2
(95)
and
DTh =
2 Ω∆2 − Ω 2 sech β Ω2 ∆ Ω∆ + 3 Ω 2 4 g 2 |κ|2 ρ02 sinh β Ω cosh β Ω∆ − 1 − β Ω − √ ∆ ∆ 3/2 2 2 2 ϵ Ω∆ Ω∆ − Ω 4 2 π β ω0 Ω∆ ρ0
+ 3 β Ω∆ tanh
β Ω∆ 2
21
.
(96)
The function HII (ω) is equal to the function in Eq. (65) setting b0 = ρ0 . For the Thompson model, the function KII (ω) is given by
2 Ω∆2 + 2 Ω 2 β Ω∆ KII (ω) = 1 + tanh3 2 3 4 ρ02 Ω∆ ω2 + ω02 ω2 + ϵ 2 ω2 + Ω∆2 2 2 2 2 Ω∆2 − Ω 2 2 Ω ω2 − ω0 (Ω∆2 + Ω 2 ) 4 g |κ| ϵ Ω ρ0 β Ω∆ − + tanh 2 4 ρ02 Ω∆ ω2 + ω02 ω2 + Ω∆2 Ω∆ ω2 + ϵ 2 ω2 + Ω∆2 2 2 Ω 2 Ω∆2 − Ω 2 g 2 |κ|2 ϵ Ω∆ − Ω 2 2 Ω ω2 − ω0 (Ω∆2 + Ω 2 + ω2 ) + + 2 4 2 2 2 2 2 2 2 2 2 2 2 16 ρ0 Ω∆ ω + ω0 Ω∆ ω + ω0 ω + ϵ ω + Ω∆ ω + Ω∆ β Ω∆ 2 × tanh . (97)
2 g 2 |κ|2 ϵ Ω∆ − Ω2
2
The collective spectrum of the systems are obtained by solving the equations HII (−i E ) = 0 and KII (−i E ) = 0. For the first model, it is possible to note that, in the limit where the linear dimension of the material composed of the atoms is very small compared with the wavelength of the bosons (L ≪ λ), we recover the results of the usual Dicke model. In this limit, we consider sin(2 π n x) ≈ 1. Moreover, it is possible to see that in the generalized case (L ≈ λ), the task of obtain some final expressions for the partition function and collective spectrum is a technical problem, since we have to solve some complicated integrations. Nevertheless, setting particular values of the parameters of the model we can use numerical techniques of integration to achieve the final results. Finally, in the general case, it is possible to verify the existence of the Goldstone mode (E Pos = 0), we get such result verifying that HII (0) = 0. For the Thompson model, we solve the equation KII (−i E ) = 0 using the relation given by Eq. (45), as a result, we arrive at the following solutions E1Th = 0,
(98)
M. Aparicio Alcalde, B.M. Pimentel / Physica A 392 (2013) 3765–3779
3779
and 2
E2Th
2
2 64 g 2 |κ|2 ω ϵ (ω − Ω )2 ρ 2 12 0 0 0 2 2 2 = ω + ϵ + 2 ω0 Ω + Ω∆ ± ω0 − ϵ + 2 ω0 Ω + Ω∆ + . Ω∆2 − Ω 2 2 0
2
2
(99)
Setting κ = 0 in the last equation, we recover the collective spectrum of the usual Dicke model in the superradiant phase. Therefore, we can see that the phonons contribute to the collective spectrum through the term Ω∆ and in the last term in Eq. (99). 5. Summary In this paper we studied two different spin-boson models which are generalizations of the Dicke model. Both models describe a system of N identical two-level atoms coupled to a single-mode quantized bosonic field, where we use the rotating wave approximation. In the first model, we consider the wavelength of the bosonic field λ, to be of the order of the linear dimension of the material composed of the atoms L, therefore we consider the spatial sinusoidal form of the bosonic field. The second model is the Thompson model, where it was considered the presence of phonons in the material composed of the atoms. Using the path integral approach and functional methods, in the thermodynamic limit N → ∞, we found that the systems exhibit a phase transition from normal to superradiant phase, at some critical values of temperature and coupling constant. Moreover, we were able to compute the asymptotic behavior of the partition function and collective spectrum of the models in the normal and superradiant phase, respectively. In both models, the collective spectrum has a zero energy value in the superradiant phase, corresponding to the Goldstone mode associated to the continuous symmetry breaking. Since these models are generalizations of the Dicke model we are able to recover the free energy and the collective spectrum when in the first model we take the approximation L ≪ λ and in the Thompson model we set κ = 0. Beside that, in the first model, with the purpose to achieve some final result, we have to solve some complicated integrations, nevertheless it is possible to manage such integrations using numerical techniques of integration. Finally, in the Thompson model, we can identify the contribution due to the phonons in both the free energy and the collective spectrum. Acknowledgements The authors acknowledge the referee for valuable suggestions. MAA was supported by FAPESP, and BMP was partially supported by CNPq and CAPES. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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