- Email: [email protected]
Ladder approximation for the AC conductivity in the generalized two-dimensional Hubbard model
Solid State Communications 258 (2017) 21–24
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
Ladder approximation for the AC conductivity in the generalized two-dimensional Hubbard model
MARK
L.S. Lima Departamento de Física e Matemática, Centro Federal de Educação Tecnológica de Minas Gerais, 30510-000 Belo Horizonte, MG, Brazil
A BS T RAC T We calculate the optical conductivity of the generalized two-dimensional Hubbard model including vertex corrections, by using a ladder approximation in the diagrammatic expansion. We have obtained a superconductor behavior for this system at low temperature. Employing the ladder approach, we have included the influence of the electron-electron interaction on a previous result obtained in the mean field approximation. We have obtained the behavior of the optical conductivity, employing the ladder approximation, tending to zero in the DC limit. Since this approximation is better than the mean field approximation (it include information about the electron-electron interaction) then we have a better description to the behavior of the AC optical conductivity for the two-dimensional Hubbard model.
1. Introduction Is well known that the conductivity of the two-dimensional Hubbard model is particularly relevant for high-temperature superconductors. Into this framework, vertex corrections are expected to be important because of strongly momentum-dependent self-energies [1]. Moreover, the superconductivity in hole-doped high-temperature superconductors is studied with emphasis on the connections among the Luttinger theorem, topological quantum field theories and critical theories involving change in the size of the Fermi surface [2]. The pseudogap phase have been described as a Higgs phase in a SU(2) gauge theory, where the Higgs field represents the local antiferromagnetism. Usually the superconductivity has been studied employing besides of the two-dimensional Hubbard model, the projected version of it, the t-J model [3–7]. However, recently, the utility of such modeling for high-temperature superconductors is highly questionable. In such system the gap is d-wave, i.e. there are regions in momentum space where the gap is zero. This has a strongly impact on the shape of the optical conductivity. On the other hand the superconductivity at high-temperature can be studied also using disordered spin systems modeled by the spin-1/2 two-dimensional Heisenberg antiferromagnet (AF) [8,9]. In general, is well known that the fact the electric conductivity tending to infinity, is not taken as the true definition of superconductivity. A material is superconducting if it presents the Meissner-Ochsenfeld effect. This effect is the fact that metals in the superconductor state are perfect diamagnets and hence expels a weak external magnetic field. Is well known still that the type-II superconductivity cannot be explained using the standard BCS theory because this theory is valid only when the coupling constant between the par of electrons is small
E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.ssc.2017.04.012 Received 1 March 2017; Received in revised form 5 April 2017; Accepted 17 April 2017 Available online 18 April 2017 0038-1098/ © 2017 Elsevier Ltd. All rights reserved.
[2,10,11]. However, the Ginzburg-Landau theory was derived from the BCS theory by Gorkov long time ago. In materials such as 3He there are two types of superfluidity depending on its pressure and its temperature. In a superconductor, electrons (fermions) form Cooper pairs in order to form bosons, which enables Bose-Einstein condensation. In 3He, there is a similar situation: the 3He atoms (which are fermions) pair up and form bosons too. However, in that case, they are not atoms vibrations that are responsible for the formation of pairs, but rather the fact that the atom magnetization become parallel to one another. The total magnetization of a pair of 3He atoms taking part in superfluidity is in that case not equal to zero, contrary to the magnetization of a Cooper pair in the BCS theory. One of the most important challenges in the study of high temperature superconductivity is to understand the relation between antiferromagnetism 2D with the superconductivity. There are many different families which include the iron-pnictides, electron-doped cuprates and heavy-fermion superconductors that are in close connection with the AFM phase [10]. Recently, there are a large number of measurements reported in iron-pnictide family as a function of the concentration of holes x [12,13]. The plan of this paper is the following. In Section 2 we describe the model, in Section 3 we describe the techniques to calculate the transport coefficients. In Section 4, we present our conclusions and final remarks. 2. The model In this paper, we calculate the conductivity considering the influence of the electron-electron interaction on the superconductivity
Solid State Communications 258 (2017) 21–24
L.S. Lima
given by [14].
of the generalized two-dimensional Hubbard model. The model is given by the following Hamiltonian [14].
/ = −2t ∑
(ci† cj
1i (→ n , t ) = −it
+ h . c . ) + U ∑ n i ↑ nj ↓ ,
〈i, j〉
(1)
i
2U 3
→
∑ ( Si )2 +
(cn ) is a creation (annihilation) electron operator. We use the where fermion model of two-dimensional Hubbard model [14] to determine the regular part of the spin conductivity (AC conductivity) or continuum conductivity and for the DC conductivity. The Superconductivity is the ability of fermions to form a persistent or non-dissipative current without an external field [13]. This does not happens in conventional metals where an electric current appears as ⎯→ ⎯ ⎯→ ⎯ response of the system to electric field, given by Ohm law, 1 = σ E . ⎯→ ⎯ Where E is an external electric field and σ is the electric conductivity. The linearly growing of the optical conductivity for large ω cannot satisfy in any way the f-sum rule [1], since the method used is accurate in the range of low ω. In the Kubo formalism [16–20] the optical conductivity is given by the real part of the conductivity σ (ω), σ ′(ω), being written in a standard form as [20].
Ne U . 2
(2) 〈i, j〉 i → = † where Si = 2 cα σαα ′ cα ′ with α = x, y, z and σαα′ are the three Pauli matrices. Therefore
→
∑ ( Si )
2
=
∑ Siα Siα ′.
i
(3)
i
By expanding the components and making use of the SU(2) identity
∑
α α σβδ σμν = 2δβν δδμ − δβδ δμν
(4)
α =1,2,3
one gets
∑
σ ′(ω) = DS δ (ω) + σ reg (ω),
→2 ( Si ) =
∑
i
i
⎛3 ⎞ 3 ⎜ ni − ni ↑ nj ↓⎟ ⎝4 ⎠ 2
→ → DS = π [〈2 〉 + Λ′( k = 0, ω → 0 )],
Thus, we can write
∑
(ciα† cjα + h . c. ) + D ∑
i, j, α =↑,↓
i Λ (→ q , ω) = =N
→ ( Si )2 .
σ reg (ω),
(7)
i
We have calculated the vertex corrections for the conductivity in 2U → r . The r ) is the total spin of the band in → this model, where D = − 3 . S (→ system is SU(2) invariant on the spin rotations. For large values of U, i.e. U / t → ∞, the local spin becomes as large as possible. The single-particle fermionic spectrum is given as
/ =
∑ ∫ α =↑,↓
d 2k † ζk ckα ckα , (2π )2
where Δ is the energy gap given as Δ = relation of free fermion
σ reg (ω) =
(9)
dteiωt 〈[1 (→ q , t ), 1 (−→ q , 0)]〉.
(14)
Λ″(→ q = 0, ω) , ω
(15)
→ (gμB )2 Λ″( k = 0, ω) = ω =
π
∫0 ∫0
π
d 2k cos k (1 − e ζk / T ) f (ζk ) δ (ω − 2ζk ). (2π )2 ζk
(16)
U /3 |ϕ| and εk is the dispersion
εk = −2t (cos k1 + cos k2 ).
∞
where Λ″ is the imaginary part of the current-current correlation function. It represents the continuum contribution to the conductivity or AC conductivity. In the Eqs. (13) and (23), Λ′ and Λ″ stand for the real and imaginary part of Λ. After a long calculation we obtain the AC electric conductivity, neglecting electron-electron interactions, as
(8)
εk2 + Δ2 ,
∫0
the regular part of σ ′(ω), is given by [19].
σ reg (ω) =
and thus the dispersion relation of the free Hamiltonian is
ζk =
(13)
where it is given by two terms. The first term is given by the kinetic energy of the particles and the second term from Λ′(k = 0, ω → 0) that is the real part of the current-current correlation function defined as
(6)
The last term is an additive constant. Thus, we have the model Eq. (1) is reduced to the model
/ = −2t
(12)
where the Dirac delta term represents the DC contribution, where DS is the Drude's weight
(5)
2U → 2 NU /1 = U ∑ ni ↑ nj ↓ = − ( Si ) + e . 3 2 i
(11)
cn†,
where 〈i , j〉 stands for the sum over nearest-neighbors. Such model can be written in the form
/ = −2t ∑ (ci† cj + h . c. ) −
(cα† (→ n , t ) cα (→ n +→ xi , t ) − h . c.)
∑ α =↑,↓
where f (ζk ) =
(10)
(eζk / T
+
1)−1
is the Fermi-Dirac distribution.
3.2. Ladder approximation
From the solution of the analog BCS gap equation [15], we turn out that a number of different possible states can occur. The most important ones are the Anderson-Brinkman-Morrel state (ABM) and the Balain-Werthamer state (BW). The BW state has a gap, which has a constant magnitude over all the Fermi surface, rather like a BCS superconductor, while the ABM state has a gap which vanishes at the → two points on the Fermi surface, k = (0, 0, ± k f ). This difference leads to different physical properties. The ABM state is identified with 3HeA-phase, while the BW state corresponds to the 3He-B-phase. The Bphase is generally the more stable one, except in the high pressure region near to Tc [15].
The electron-electron interaction can be represented by the vertex function Πkk′ which satisfies the Bethe-Salpeter equation [21]. ∞
Πkk ′ = i
→
→
′ Gαα ( k , ω + ω′) Gαα ( k , ω′) Γkk ′ (ω, ω′), ∫−∞ dω 2π
(17)
and Γkk ′ (ω , ω′) = δkk ′ −
2it =N
∑∫ k1
∞
−∞
→ → dω1 αα Vkk1k1k (ω′, ω1) Gαα ( k1, ω + ω1) Gαα ( k1, ω1) Γkk ′ (ω , ω1). 2π
(18) The matrix Vkkαα1k1k (ω′, ω1) is the sum of all irreducible interaction parts. We take into account the electron-electron interaction to lowest order approximating V αα by its first-order irreducible interaction part [18,21].
3. Calculus of the transport coefficients 3.1. Kubo formalism of transport
Vkkαα1k1k (ω′, ω1) = Vkk1k1k ,
The current operator for the two-dimensional Hubbard model is
where we neglect all the contributions to 22
(19)
V αα
where two or more of the
Solid State Communications 258 (2017) 21–24
L.S. Lima
Fig. 1. Behavior of the AC conductivity σ reg (ω) for the value of the gap Δ = 1.5. The AC conductivity, tends to zero at ω = 2 4 − Δ2 (left panel). We have gotten the inclusion of the electron-electron interaction into the ladder approximation changes this behavior previously obtained where now the AC conductivity is non-zero for all ω. The inset shows the noisy behavior of the conductivity near ω = 0 which is due a stronger scattering of electrons for this range of ω.
bare interactions V are involved. Then the propagators can be replaced by the bare expressions G 0 (ω), since all the first-order diagrams for the electron self-energy vanish at T=0. In terms of electron propagators and the vertex function, Gj (ω) is given by
Gj (ω) =
i (2tα (S ))2
=N
∞
→
r (m ) =
Ir (m) =
→
(20)
∑ k′
(21)
sin(kx ) γk gk = 0, sin(kx ) γk − k1 gk =
k
sin(k1, x ) 2
∑
sin2 (kx ) gk ,
k
(22)
which hold for any function gk which has the symmetry of the lattice. By virtue of the identities Eq. (22) and from the Green's function Eq. (20), the analytical steps can be performed to obtain the continuum contribution for the electric conductivity given as
⎤ r (2) − (r (1) r (1) − r (0) r (0) ) 1 ⎡ I⎢ ⎥, ω ⎣ 1 + (r (0) + r (2) ) − (r (1) r (1) − r (0) r (2) ) ⎦ 1 ⎛ BC − AD ⎞ σ reg (ω) = ⎜ 2 ⎟, ω ⎝ C + D2 ⎠
r (m )
(25) can be evaluated analytically as
(26)
∫0
π
dk
sin2 k 1 . ζkm ω − ζk
(27)
In Fig. 1, we present the behavior of the AC conductivity in the mean field approach that neglects the electron-electron interaction (left panel) and with the effect of electron-electron (right panel). We have gotten the AC conductivity falling to zero at ω = 4 − Δ2 when the effect of electron-electron scattering is not considered. With the inclusion of the electron-electron scattering the behavior of the AC conductivity changes to non-zero for all ω values (right panel). The optical conductivity decreases with ω tending to zero in ω → 0 limit. Therefore, we have obtained that the inclusion of the electron-electron scattering in the mean field approximation to AC conductivity changes the behavior of the AC conductivity from zero at ω = 4 − Δ2 , as showed in left panel, to non-zero for all ω as showed in the right panel. Hence, the optical conductivity increases with ω and tends to zero in low frequency ω. However, Near ω = 0 , we have gotten a noisy behavior for the conductivity that might be due a stronger scattering of the electrons for this range of ω. In cases where one has a good handle on the single-particle Green's function, the difficulty of calculating transport in the 2D Hubbard model stems from the fact that one cannot neglect the effect of vertex corrections when strong momentum-dependent correlations are present. These vertex corrections are the analog of the self-energy, but for the two-particle response functions. When vertex corrections are not included, conservation laws can be violated and the results gotten inaccurate. In the case of small finite systems tractable by exact diagonalization or quantum Monte Carlo calculations, the correlation function is directly evaluated and vertex corrections are not an issue [1].
k
∑
sin2 k 1 . ζkm ω − ζk
σ reg (ω)
The equations above must be solved in the calculation of the Green's function Gj (ω). When the electron-electron interactions are treated within the ladder approximation, the algebraic solution of the coupled integral → → equations above is based on the decoupling of the sums over k and k1 by means of the identities
∑
d 2k
4π ⎡ (Δ2 − ω 2 ) ⎤ ⎥, ⎢1 − ⎦ ωm ⎣ 4t 2
Rr (m) = 4℘
where α (S ) is an Oguchi correction factor [21] and the reduced vertex function Γk is defined by
Γk (ω, ω′) =
π
and the real part is given as
k
sin(k′x ) Γkk ′ (ω, ω′). ζk ′
∫0
The imaginary part of
′ ∑ sin(kx ) Gαα ( k , ω + ω′) Gαα ( k , ω′) ∫−∞ dω 2π ζk
Γk (ω, ω′),
2 π
σ reg (ω) =
(23)
where
A = Rr (2) − (Rr (1) )2 − (Ir (1) )2 − (Rr (0) )2 − (Ir (0) )2 , B = Ir (2) + 2Rr (1) Ir (2) + 2Rr (0) Ir (0), C = 1 + Rr (0) + Rr (2) − (Rr (1) )2 + (Ir (1) )2 + Rr (0) Rr (2) − Ir (0) Ir (2), D = Ir (0) + Ir (2) − 2Rr (1) Ir (1) + Rr (0) Ir (2) + Ir (0) Rr (2),
4. Conclusions and final remarks In summary, we have calculated the AC conductivity for the generalized two-dimensional Hubbard model employing the ladder approximation for the Bethe Salpeter Equation. A connection of the two-dimensional Hubbard model with superconductivity at high tem-
(24) and 23
Solid State Communications 258 (2017) 21–24
L.S. Lima
[4] D.J. Scalapino, J.R. Schrieffer, J.S. Brooks (Eds.), Handbook of High-Temperature Superconductivity, Springer, New York, 2007. [5] M.R. Norman, J. Supercond. Nov. Magn. 25 (2012) 2131. [6] M. Ogata, H. Fukuyama, Rep. Prog. Phys. 71 (2008) 036501. [7] P.A. Lee, N. Nagaosa, X.G. Wen, Rev. Mod. Phys. 78 (2006) 17. [8] L.S. Lima, A.S.T. Pires, B.V. Costa, Phys. A 438 (2015) 579. [9] L.S. Lima, A.S.T. Pires, B.V. Costa, J. Magn. Magn. Mater. 371 (2014) 89. [10] D.J. Scalapino, Rev. Mod. Phys. 84 (2012) 1383. [11] S. Sachdev, Science 336 (2012) 1510. [12] D. Chowdhury, J. Orenstein, S. Sachdev, T. Senthil, arXiv:1502.04122v1, 2015. [13] Saurabh Maiti, Andrey V. Chubukov, AIP Conference Proceedings, 1550, 3, 2013. [14] E. Fradkin, Field Theories of Condensed Matter Physics, Second edition, Cambridge, U.K, 2013. [15] F. James, Superconductivity, Superfluids and Condensates, Oxford Masters series in Condensed Matter Physics, Oxford University Press, London, 2004. [16] L.S. Lima, Phys. C 527 (2016) 33. [17] A.S.T. Pires, L.S. Lima, Phys. Rev. B 79 (2009) 064401. [18] A.S.T. Pires, L.S. Lima, J. Phys.:Condens. Matter 21 (2009) 245502. [19] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Springer-Verlag, New York, 1985. [20] G.D. Mahan, Many Particles Physics, Plenum, New York, 1990. [21] M. Sentef, M. Kollar, A.P. Kampf, Phys. Rev. B 75 (2007) 214403. [22] Tianni Lua, Cuiping Zhangb, Shengwu Guoa, Yifang Wua, Chengshan Lib, Lian Zhoua, Phys. C 519 (2015) 24. [23] Zhang Degan, Phys. C 519 (2015) 43. [24] Tetsuya Kajitaa, Takayuki Kawamatab, Takashi Nojib, Takehiro Hatakedab, Masatsune Katob, Yoji Koikeb, Takashi Itoha, Phys. C 519 (2015) 104.
perature can be made which is a subject of intense research in the actuality [14]. With relation to force of the coupling that generates values higher of Tc, we can not simply expect that a stronger coupling constant will lead to higher value of Tc because increasing the constant of coupling leads to several new effects that are not included in the standard BCS theory, some of which are detrimental to superconductivity [11]. We can make the calculations also with adding of nonmagnetic impurities or holes in the lattice to verify the influence on the behavior of the conductivity [22–24]. This can be made in a future work. Acknowledgment The Brazilian Agencies CNPq, CAPES and FAPEMIG. References [1] Dominic Bergeron, Vasyl Hankevych, Bumsoo Kyung, A.-M.S. Tremblay, Phys. Rev. B 84 (2011) 085128. [2] S. Sachdev, D. Chowdhury, Progress. Theor. Exp. Phys. (2016) (arXiv:1605. 03579v1). [3] J. Kaczmarczyk, J. Spalek, T. Schickling, J. Bunemann, Phys. Rev. B 88 (2013) 115127.
24
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
Ladder approximation for the AC conductivity in the generalized two-dimensional Hubbard model
MARK
L.S. Lima Departamento de Física e Matemática, Centro Federal de Educação Tecnológica de Minas Gerais, 30510-000 Belo Horizonte, MG, Brazil
A BS T RAC T We calculate the optical conductivity of the generalized two-dimensional Hubbard model including vertex corrections, by using a ladder approximation in the diagrammatic expansion. We have obtained a superconductor behavior for this system at low temperature. Employing the ladder approach, we have included the influence of the electron-electron interaction on a previous result obtained in the mean field approximation. We have obtained the behavior of the optical conductivity, employing the ladder approximation, tending to zero in the DC limit. Since this approximation is better than the mean field approximation (it include information about the electron-electron interaction) then we have a better description to the behavior of the AC optical conductivity for the two-dimensional Hubbard model.
1. Introduction Is well known that the conductivity of the two-dimensional Hubbard model is particularly relevant for high-temperature superconductors. Into this framework, vertex corrections are expected to be important because of strongly momentum-dependent self-energies [1]. Moreover, the superconductivity in hole-doped high-temperature superconductors is studied with emphasis on the connections among the Luttinger theorem, topological quantum field theories and critical theories involving change in the size of the Fermi surface [2]. The pseudogap phase have been described as a Higgs phase in a SU(2) gauge theory, where the Higgs field represents the local antiferromagnetism. Usually the superconductivity has been studied employing besides of the two-dimensional Hubbard model, the projected version of it, the t-J model [3–7]. However, recently, the utility of such modeling for high-temperature superconductors is highly questionable. In such system the gap is d-wave, i.e. there are regions in momentum space where the gap is zero. This has a strongly impact on the shape of the optical conductivity. On the other hand the superconductivity at high-temperature can be studied also using disordered spin systems modeled by the spin-1/2 two-dimensional Heisenberg antiferromagnet (AF) [8,9]. In general, is well known that the fact the electric conductivity tending to infinity, is not taken as the true definition of superconductivity. A material is superconducting if it presents the Meissner-Ochsenfeld effect. This effect is the fact that metals in the superconductor state are perfect diamagnets and hence expels a weak external magnetic field. Is well known still that the type-II superconductivity cannot be explained using the standard BCS theory because this theory is valid only when the coupling constant between the par of electrons is small
E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.ssc.2017.04.012 Received 1 March 2017; Received in revised form 5 April 2017; Accepted 17 April 2017 Available online 18 April 2017 0038-1098/ © 2017 Elsevier Ltd. All rights reserved.
[2,10,11]. However, the Ginzburg-Landau theory was derived from the BCS theory by Gorkov long time ago. In materials such as 3He there are two types of superfluidity depending on its pressure and its temperature. In a superconductor, electrons (fermions) form Cooper pairs in order to form bosons, which enables Bose-Einstein condensation. In 3He, there is a similar situation: the 3He atoms (which are fermions) pair up and form bosons too. However, in that case, they are not atoms vibrations that are responsible for the formation of pairs, but rather the fact that the atom magnetization become parallel to one another. The total magnetization of a pair of 3He atoms taking part in superfluidity is in that case not equal to zero, contrary to the magnetization of a Cooper pair in the BCS theory. One of the most important challenges in the study of high temperature superconductivity is to understand the relation between antiferromagnetism 2D with the superconductivity. There are many different families which include the iron-pnictides, electron-doped cuprates and heavy-fermion superconductors that are in close connection with the AFM phase [10]. Recently, there are a large number of measurements reported in iron-pnictide family as a function of the concentration of holes x [12,13]. The plan of this paper is the following. In Section 2 we describe the model, in Section 3 we describe the techniques to calculate the transport coefficients. In Section 4, we present our conclusions and final remarks. 2. The model In this paper, we calculate the conductivity considering the influence of the electron-electron interaction on the superconductivity
Solid State Communications 258 (2017) 21–24
L.S. Lima
given by [14].
of the generalized two-dimensional Hubbard model. The model is given by the following Hamiltonian [14].
/ = −2t ∑
(ci† cj
1i (→ n , t ) = −it
+ h . c . ) + U ∑ n i ↑ nj ↓ ,
〈i, j〉
(1)
i
2U 3
→
∑ ( Si )2 +
(cn ) is a creation (annihilation) electron operator. We use the where fermion model of two-dimensional Hubbard model [14] to determine the regular part of the spin conductivity (AC conductivity) or continuum conductivity and for the DC conductivity. The Superconductivity is the ability of fermions to form a persistent or non-dissipative current without an external field [13]. This does not happens in conventional metals where an electric current appears as ⎯→ ⎯ ⎯→ ⎯ response of the system to electric field, given by Ohm law, 1 = σ E . ⎯→ ⎯ Where E is an external electric field and σ is the electric conductivity. The linearly growing of the optical conductivity for large ω cannot satisfy in any way the f-sum rule [1], since the method used is accurate in the range of low ω. In the Kubo formalism [16–20] the optical conductivity is given by the real part of the conductivity σ (ω), σ ′(ω), being written in a standard form as [20].
Ne U . 2
(2) 〈i, j〉 i → = † where Si = 2 cα σαα ′ cα ′ with α = x, y, z and σαα′ are the three Pauli matrices. Therefore
→
∑ ( Si )
2
=
∑ Siα Siα ′.
i
(3)
i
By expanding the components and making use of the SU(2) identity
∑
α α σβδ σμν = 2δβν δδμ − δβδ δμν
(4)
α =1,2,3
one gets
∑
σ ′(ω) = DS δ (ω) + σ reg (ω),
→2 ( Si ) =
∑
i
i
⎛3 ⎞ 3 ⎜ ni − ni ↑ nj ↓⎟ ⎝4 ⎠ 2
→ → DS = π [〈2 〉 + Λ′( k = 0, ω → 0 )],
Thus, we can write
∑
(ciα† cjα + h . c. ) + D ∑
i, j, α =↑,↓
i Λ (→ q , ω) = =N
→ ( Si )2 .
σ reg (ω),
(7)
i
We have calculated the vertex corrections for the conductivity in 2U → r . The r ) is the total spin of the band in → this model, where D = − 3 . S (→ system is SU(2) invariant on the spin rotations. For large values of U, i.e. U / t → ∞, the local spin becomes as large as possible. The single-particle fermionic spectrum is given as
/ =
∑ ∫ α =↑,↓
d 2k † ζk ckα ckα , (2π )2
where Δ is the energy gap given as Δ = relation of free fermion
σ reg (ω) =
(9)
dteiωt 〈[1 (→ q , t ), 1 (−→ q , 0)]〉.
(14)
Λ″(→ q = 0, ω) , ω
(15)
→ (gμB )2 Λ″( k = 0, ω) = ω =
π
∫0 ∫0
π
d 2k cos k (1 − e ζk / T ) f (ζk ) δ (ω − 2ζk ). (2π )2 ζk
(16)
U /3 |ϕ| and εk is the dispersion
εk = −2t (cos k1 + cos k2 ).
∞
where Λ″ is the imaginary part of the current-current correlation function. It represents the continuum contribution to the conductivity or AC conductivity. In the Eqs. (13) and (23), Λ′ and Λ″ stand for the real and imaginary part of Λ. After a long calculation we obtain the AC electric conductivity, neglecting electron-electron interactions, as
(8)
εk2 + Δ2 ,
∫0
the regular part of σ ′(ω), is given by [19].
σ reg (ω) =
and thus the dispersion relation of the free Hamiltonian is
ζk =
(13)
where it is given by two terms. The first term is given by the kinetic energy of the particles and the second term from Λ′(k = 0, ω → 0) that is the real part of the current-current correlation function defined as
(6)
The last term is an additive constant. Thus, we have the model Eq. (1) is reduced to the model
/ = −2t
(12)
where the Dirac delta term represents the DC contribution, where DS is the Drude's weight
(5)
2U → 2 NU /1 = U ∑ ni ↑ nj ↓ = − ( Si ) + e . 3 2 i
(11)
cn†,
where 〈i , j〉 stands for the sum over nearest-neighbors. Such model can be written in the form
/ = −2t ∑ (ci† cj + h . c. ) −
(cα† (→ n , t ) cα (→ n +→ xi , t ) − h . c.)
∑ α =↑,↓
where f (ζk ) =
(10)
(eζk / T
+
1)−1
is the Fermi-Dirac distribution.
3.2. Ladder approximation
From the solution of the analog BCS gap equation [15], we turn out that a number of different possible states can occur. The most important ones are the Anderson-Brinkman-Morrel state (ABM) and the Balain-Werthamer state (BW). The BW state has a gap, which has a constant magnitude over all the Fermi surface, rather like a BCS superconductor, while the ABM state has a gap which vanishes at the → two points on the Fermi surface, k = (0, 0, ± k f ). This difference leads to different physical properties. The ABM state is identified with 3HeA-phase, while the BW state corresponds to the 3He-B-phase. The Bphase is generally the more stable one, except in the high pressure region near to Tc [15].
The electron-electron interaction can be represented by the vertex function Πkk′ which satisfies the Bethe-Salpeter equation [21]. ∞
Πkk ′ = i
→
→
′ Gαα ( k , ω + ω′) Gαα ( k , ω′) Γkk ′ (ω, ω′), ∫−∞ dω 2π
(17)
and Γkk ′ (ω , ω′) = δkk ′ −
2it =N
∑∫ k1
∞
−∞
→ → dω1 αα Vkk1k1k (ω′, ω1) Gαα ( k1, ω + ω1) Gαα ( k1, ω1) Γkk ′ (ω , ω1). 2π
(18) The matrix Vkkαα1k1k (ω′, ω1) is the sum of all irreducible interaction parts. We take into account the electron-electron interaction to lowest order approximating V αα by its first-order irreducible interaction part [18,21].
3. Calculus of the transport coefficients 3.1. Kubo formalism of transport
Vkkαα1k1k (ω′, ω1) = Vkk1k1k ,
The current operator for the two-dimensional Hubbard model is
where we neglect all the contributions to 22
(19)
V αα
where two or more of the
Solid State Communications 258 (2017) 21–24
L.S. Lima
Fig. 1. Behavior of the AC conductivity σ reg (ω) for the value of the gap Δ = 1.5. The AC conductivity, tends to zero at ω = 2 4 − Δ2 (left panel). We have gotten the inclusion of the electron-electron interaction into the ladder approximation changes this behavior previously obtained where now the AC conductivity is non-zero for all ω. The inset shows the noisy behavior of the conductivity near ω = 0 which is due a stronger scattering of electrons for this range of ω.
bare interactions V are involved. Then the propagators can be replaced by the bare expressions G 0 (ω), since all the first-order diagrams for the electron self-energy vanish at T=0. In terms of electron propagators and the vertex function, Gj (ω) is given by
Gj (ω) =
i (2tα (S ))2
=N
∞
→
r (m ) =
Ir (m) =
→
(20)
∑ k′
(21)
sin(kx ) γk gk = 0, sin(kx ) γk − k1 gk =
k
sin(k1, x ) 2
∑
sin2 (kx ) gk ,
k
(22)
which hold for any function gk which has the symmetry of the lattice. By virtue of the identities Eq. (22) and from the Green's function Eq. (20), the analytical steps can be performed to obtain the continuum contribution for the electric conductivity given as
⎤ r (2) − (r (1) r (1) − r (0) r (0) ) 1 ⎡ I⎢ ⎥, ω ⎣ 1 + (r (0) + r (2) ) − (r (1) r (1) − r (0) r (2) ) ⎦ 1 ⎛ BC − AD ⎞ σ reg (ω) = ⎜ 2 ⎟, ω ⎝ C + D2 ⎠
r (m )
(25) can be evaluated analytically as
(26)
∫0
π
dk
sin2 k 1 . ζkm ω − ζk
(27)
In Fig. 1, we present the behavior of the AC conductivity in the mean field approach that neglects the electron-electron interaction (left panel) and with the effect of electron-electron (right panel). We have gotten the AC conductivity falling to zero at ω = 4 − Δ2 when the effect of electron-electron scattering is not considered. With the inclusion of the electron-electron scattering the behavior of the AC conductivity changes to non-zero for all ω values (right panel). The optical conductivity decreases with ω tending to zero in ω → 0 limit. Therefore, we have obtained that the inclusion of the electron-electron scattering in the mean field approximation to AC conductivity changes the behavior of the AC conductivity from zero at ω = 4 − Δ2 , as showed in left panel, to non-zero for all ω as showed in the right panel. Hence, the optical conductivity increases with ω and tends to zero in low frequency ω. However, Near ω = 0 , we have gotten a noisy behavior for the conductivity that might be due a stronger scattering of the electrons for this range of ω. In cases where one has a good handle on the single-particle Green's function, the difficulty of calculating transport in the 2D Hubbard model stems from the fact that one cannot neglect the effect of vertex corrections when strong momentum-dependent correlations are present. These vertex corrections are the analog of the self-energy, but for the two-particle response functions. When vertex corrections are not included, conservation laws can be violated and the results gotten inaccurate. In the case of small finite systems tractable by exact diagonalization or quantum Monte Carlo calculations, the correlation function is directly evaluated and vertex corrections are not an issue [1].
k
∑
sin2 k 1 . ζkm ω − ζk
σ reg (ω)
The equations above must be solved in the calculation of the Green's function Gj (ω). When the electron-electron interactions are treated within the ladder approximation, the algebraic solution of the coupled integral → → equations above is based on the decoupling of the sums over k and k1 by means of the identities
∑
d 2k
4π ⎡ (Δ2 − ω 2 ) ⎤ ⎥, ⎢1 − ⎦ ωm ⎣ 4t 2
Rr (m) = 4℘
where α (S ) is an Oguchi correction factor [21] and the reduced vertex function Γk is defined by
Γk (ω, ω′) =
π
and the real part is given as
k
sin(k′x ) Γkk ′ (ω, ω′). ζk ′
∫0
The imaginary part of
′ ∑ sin(kx ) Gαα ( k , ω + ω′) Gαα ( k , ω′) ∫−∞ dω 2π ζk
Γk (ω, ω′),
2 π
σ reg (ω) =
(23)
where
A = Rr (2) − (Rr (1) )2 − (Ir (1) )2 − (Rr (0) )2 − (Ir (0) )2 , B = Ir (2) + 2Rr (1) Ir (2) + 2Rr (0) Ir (0), C = 1 + Rr (0) + Rr (2) − (Rr (1) )2 + (Ir (1) )2 + Rr (0) Rr (2) − Ir (0) Ir (2), D = Ir (0) + Ir (2) − 2Rr (1) Ir (1) + Rr (0) Ir (2) + Ir (0) Rr (2),
4. Conclusions and final remarks In summary, we have calculated the AC conductivity for the generalized two-dimensional Hubbard model employing the ladder approximation for the Bethe Salpeter Equation. A connection of the two-dimensional Hubbard model with superconductivity at high tem-
(24) and 23
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L.S. Lima
[4] D.J. Scalapino, J.R. Schrieffer, J.S. Brooks (Eds.), Handbook of High-Temperature Superconductivity, Springer, New York, 2007. [5] M.R. Norman, J. Supercond. Nov. Magn. 25 (2012) 2131. [6] M. Ogata, H. Fukuyama, Rep. Prog. Phys. 71 (2008) 036501. [7] P.A. Lee, N. Nagaosa, X.G. Wen, Rev. Mod. Phys. 78 (2006) 17. [8] L.S. Lima, A.S.T. Pires, B.V. Costa, Phys. A 438 (2015) 579. [9] L.S. Lima, A.S.T. Pires, B.V. Costa, J. Magn. Magn. Mater. 371 (2014) 89. [10] D.J. Scalapino, Rev. Mod. Phys. 84 (2012) 1383. [11] S. Sachdev, Science 336 (2012) 1510. [12] D. Chowdhury, J. Orenstein, S. Sachdev, T. Senthil, arXiv:1502.04122v1, 2015. [13] Saurabh Maiti, Andrey V. Chubukov, AIP Conference Proceedings, 1550, 3, 2013. [14] E. Fradkin, Field Theories of Condensed Matter Physics, Second edition, Cambridge, U.K, 2013. [15] F. James, Superconductivity, Superfluids and Condensates, Oxford Masters series in Condensed Matter Physics, Oxford University Press, London, 2004. [16] L.S. Lima, Phys. C 527 (2016) 33. [17] A.S.T. Pires, L.S. Lima, Phys. Rev. B 79 (2009) 064401. [18] A.S.T. Pires, L.S. Lima, J. Phys.:Condens. Matter 21 (2009) 245502. [19] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Springer-Verlag, New York, 1985. [20] G.D. Mahan, Many Particles Physics, Plenum, New York, 1990. [21] M. Sentef, M. Kollar, A.P. Kampf, Phys. Rev. B 75 (2007) 214403. [22] Tianni Lua, Cuiping Zhangb, Shengwu Guoa, Yifang Wua, Chengshan Lib, Lian Zhoua, Phys. C 519 (2015) 24. [23] Zhang Degan, Phys. C 519 (2015) 43. [24] Tetsuya Kajitaa, Takayuki Kawamatab, Takashi Nojib, Takehiro Hatakedab, Masatsune Katob, Yoji Koikeb, Takashi Itoha, Phys. C 519 (2015) 104.
perature can be made which is a subject of intense research in the actuality [14]. With relation to force of the coupling that generates values higher of Tc, we can not simply expect that a stronger coupling constant will lead to higher value of Tc because increasing the constant of coupling leads to several new effects that are not included in the standard BCS theory, some of which are detrimental to superconductivity [11]. We can make the calculations also with adding of nonmagnetic impurities or holes in the lattice to verify the influence on the behavior of the conductivity [22–24]. This can be made in a future work. Acknowledgment The Brazilian Agencies CNPq, CAPES and FAPEMIG. References [1] Dominic Bergeron, Vasyl Hankevych, Bumsoo Kyung, A.-M.S. Tremblay, Phys. Rev. B 84 (2011) 085128. [2] S. Sachdev, D. Chowdhury, Progress. Theor. Exp. Phys. (2016) (arXiv:1605. 03579v1). [3] J. Kaczmarczyk, J. Spalek, T. Schickling, J. Bunemann, Phys. Rev. B 88 (2013) 115127.
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