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Exact superconducting ground states of the extended Anderson model
Physica B 406 (2011) 3622–3624
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Exact superconducting ground states of the extended Anderson model L.G. Sarasua ´ blica, Montevideo, Uruguay Instituto de Fı´sica, Universidad de la Repu
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 April 2011 Received in revised form 8 June 2011 Accepted 22 June 2011 Available online 29 June 2011
We obtain exact ground states of an extended periodic Anderson model (EPAM) with non-local hybridization and Coulomb repulsion between f and c electrons (Falicov–Kimball term) in one dimension. We show that for a range of parameter values these ground states exhibit composite hole pairing and superconductivity that originate from purely electronic interactions. & 2011 Elsevier B.V. All rights reserved.
Keywords: Superconductivity Strongly correlated electron systems
1. Introduction In recent years there has been a great interest in the study of mechanisms for superconductivity that originate from electronic interactions. This is motivated by the fact that it is accepted that the electron pairing in heavy fermions or superconducting cuprates is caused by other than the usual BCS mechanism. Different theories have been proposed to explain the origin of superconductivity in strongly correlated electrons systems (SCES), based on distinct concepts like: spin fluctuations, the RVB state and correlated hopping, among others. In the study of these mechanisms serious drawbacks may emerge due to the fact that approximate schemes cannot give conclusive results when the interactions are strong. Thus, exact results for such systems are valuable since they are free of these inconveniences. The theoretical description of multi-band systems in which the electronic correlations are important are usually described by the Anderson model and its variants. The so called extended periodic Anderson model (EPAM) consists of the usual periodic Anderson model including a repulsive interaction between electrons of different bands, the so called Falicov–Kimball (FK) term. The FK model has been extensively used to study valence, metal-insulator transitions and excitonic phases in rare earths compounds and actinides [1,2]. Recently, it has been proposed that superconductivity may arise in the EPAM driven by a valence fluctuation mechanism [3]. In the present work, we obtain exact ground states of the EPAM for a limited region of the parameters space. We show that these ground states exhibit off diagonal long range order (ODLRO), which imply that they are superconducting, when
Tel.: 598 2525 8624; fax: þ598 2525 0580.
E-mail address: sarasua@fisica.edu.uy 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.055
certain conditions involving the local and interband Coulomb interaction, the energy level of the quasi-localized electrons and the hybridization is met.
2. Model hamiltonian We shall consider the extended Anderson model with on-site and intersite hybridizations in one-dimensional case. The model allows the hopping of f electrons between different sites. This situation is very adequate for many systems, in which the f electrons may be itinerant. In spite of this, we search for solutions with almost localized f electrons. The Hamiltonian model reads X y X y X f X f f X H ¼ tc cis cjs tf fis fjs þ U nim nik þ G nci nfi þ Ef ni /ijSs
þ V0
/ijSs
i
i
X y X y ðcis fis þ fiys cis Þ þ V1 ðcis fjs þ fiys cjs Þ is
i
ð1Þ
/ijSs
where ciys , fiys are the creation operators for c and f electrons at the site i with spin s, ncis ¼ ciys cis , nfis ¼ fiys fis , are the number operators, na ¼ nam þ nak (a¼f,c), tc and tf are the hopping integrals, V0 and V1 are the local and intersite hibridization, respectively, Ef measures the energy difference between the centers of the bare f and c bands, U is the on-site Coulomb repulsion and G is the Coulomb repulsion between f and c electrons at the same site. To obtain the ground state of the above Hamiltonian, we shall proceed in similar way as in Ref. [4]. The ground state is found showing that a wavefunction jcS minimizes the expectation value of the Hamiltonian. In doing so, we shall separate the Hamiltonian into two parts, one of them containing almost all the kinematical terms and the other containing local terms. To obtain a wavefunction that minimizes the expectation value of the first term we use the method of Brandt and Giesekus [5–8]. The minimization of the second term is performed diagonalizing the local
L.G. Sarasua / Physica B 406 (2011) 3622–3624
Hamiltonian. In this part the method is similar to the optimal ground state approach proposed in Ref. [9]. The local hibridization will be separated into two parts V0 ¼ V 0 þ V, and each part is included in separated terms. Thus, we write H ¼ Hk þH0 , with X
Hk ¼ tc
X
ciys cjs tf
/ijSs
/ijSs
fiys fjs þ V 0
X y X y ðcis fis þ fiys cis Þ þ V1 ðcis fjs þ fiys cjs Þ /ijSs
is
ð2Þ We introduce now the operators ayis ¼ aðciys þ cjys Þ þ bðfiys þfjys Þ
ð3Þ
being a and b real constants. With the use of them, we can write Hk in the form X X X ais ayis 2tc ðnci þnfi Þ þ2ðtf tc Þ nfi ð4Þ Hk ¼ is
i 2
2
i
if tc ¼ a , tf ¼ b , V1 ¼ V ¼ ab. This imply the following relations: tc tf ¼ V12 ,
0
V1 ¼ V 0
ð5Þ
Assuming that these conditions hold, the Hamiltonian can be written as X X ais ayis 2tc ðnci þnfi Þ þHl ð6Þ H¼ is
i
with Hl ¼
3623
The wavefunction jcS may be the ground state if jjl S coincides with the eigenstates of Hl, which in turn must satisfy the relations (14). Although this could seem in principle difficult to accomplish, we found that for the condition U ¼ 2G, Eq. (14) is satisfied for any value of V0 and E~ f . It is to be noticed that the relations (5) restrict the values of V 0 , but not those of V (we recall that V0 ¼ V þ V 0 Þ. Then we can obtain ground states of Eq. (1) varying the values of V and E~ f . However, the values of A, B, C will change in order to satisfy Eq. (14) which in turn modifies the ratio a=b and also the value of tf =tc . Hence, after fixing the values of U and G, we can find the ground states for arbitrary values of tc, V0 and Ef, but the values of tf and V1 become determined by Eqs. (5) and (14). In all the numerical results that are presented in this work, the value of tf is small in comparison with tc, with tf =tc o 0:1. For instance, for U¼1, G¼0.5, tc ¼ 1, V¼0.1, E~ f ¼ 1, the ground state of Hl has the coefficients A¼0.9956, B¼ .0661, C¼0.00438, which satisfy the condition (5). These values imply that tf ¼ 0:0044. For these values of the parameters, jcS is the ground state of Eq. (1), with a ¼ 1, b ¼ 0:0664. The corresponding state for the totally filled band case is given by Y y y y y 0 cim cik fim fik j0S ð15Þ jc S ¼ i 0
The expectation value /HS obtained with jc S equals the lower bound (9) for n¼4.
P
i Hli , and
Hli ¼ E~ f nfi þ Unfim nfik þGnci nfi þV
X y ðcis fis þ fiys cis Þ
ð7Þ
3. Superconducting ground states
s
where we defined E~ f ¼ Ef þ 2ðtf tc Þ. Denoting the total number of particles as N, H can be written as ^ H ¼ P2t ð8Þ c N þ Hl P y where P^ ¼ is ais ais . To obtain the ground states, we need a lower bound for the expectation value of H. We note that P^ is semidefinite positive, so that the minimal value of its expectation value is zero. Thus a lower bound for /HS is
El ¼ 2tc N þ /Hl Smin
ð9Þ
The value of /Hl Smin is obtained diagonalizing the local Hamiltonian Hli. Consider now the wavefunction Y y y aim aik j0S ð10Þ jcS ¼ i
This state locates two electron per site and thus corresponds to ^ cS ¼ 0 since the half filled band case. It is easy to verify that /cjPj ay2 ¼ 0, due to the properties of the fermionic operators. Thus, in is virtue of Ritz’s theorem, jcS is the ground state if /cjHl jcS ¼ /Hl Smin since in this case /cjHjcS equals the lower bound El . It can be verified that the expression of jcS simplifies to Y y y y y ðacim þ bfim Þðacik þ bfik Þj0S jcS ¼
The wavefunction jcS found in the previous section is a ground state of Eq. (1) under the already mentioned conditions but it is not superconducting. A system is superconducing if it exhibits the off diagonal long range order (ODLRO) property [10,11]. A state jFS has this property if lim
y y /Fjclm cmk cpm cqk jFSa 0
where ri denotes the position of the site i. Here l and p are at the neighborhood of m and q, respectively, i.e. rl rm ,rp rq . The wavefunction jcS locates two electrons in each site and consequently does not have the property (16). In order to have ODLRO, the wavefunction must contain pairs of electrons or holes that can move along the lattice. From now on, we shall focus on the three quarter filled band case, i.e. 1 hole per site. We will show that a state of hole pairs has lower energy than one of unpaired holes. In doing so, we shall obtain the minimal values of e ¼ /Hli S=ni for different values of ni, through the diagonalization of Hli. We shall denote as e0,p the minimal value of e for ni ¼ p. The results are shown in Fig. 1 for various values of E~ f . From this we can see that
2.0
i
Y 2 y y y y y y y y ða2 cim cik þ abðcim fik þ fim cik Þ þ b fim fik Þj0S ¼
ð16Þ
jrl rp j-1
1.5
ð11Þ
i
e
Thus, jcS is formed by local states of the form jjl S ¼ Aj20S þBjmkSþ BjkmSþ Cj02S
ð12Þ
0.5
with y y cik j0S, j20S ¼ cim
y y j02S ¼ fim fik j0S
y y fik j0S, jmkS ¼ cim
y y jkmS ¼ fim cik j0S 2
0.0 ð13Þ 2
4
and A ¼ a2 =D, B ¼ ab=D, C ¼ b =D, where D ¼ a4 þ 2a2 b þ b , which imply that the coefficients verify the condition AC=B2 ¼ 1:
1.0
ð14Þ
0
1
2 ni
3
4
Fig. 1. The minimal values of /Hli =ni S as a function of ni for U ¼ 1, G ¼0.5, V¼ 0.1 and E~ f ¼ 0 (diamonds), E~ f ¼ 0:5 (circles), E~ f ¼ 1 (triangles) and E~ f ¼ 1:5 (squares).
3624
L.G. Sarasua / Physica B 406 (2011) 3622–3624
1.5
m nor l. Using Eq. (20), it can be verified that 00
2
y y y y y y y y cmm cmk clk clm jc S ¼ b fmm fmk cmm cmk flm flk jcr S 00
e
1.0
0.5
0.0 0
1
2 ni
3
4
Fig. 2. In this figure a graphical comparison between the values of /Hli =ni S for ni ¼3 and the average of the values for ni ¼2 and ni ¼ 4 is performed. This shows that for n¼ 3, the value of /Hl S is minimal when there are two electrons in half of the lattice sites and four electrons in the remainder sites.
when E~ f is above U/2, the curve representing e0,p as a function of ni has negative curvature around ni ¼3. This means that for the three quarter filled band case (n ¼3) the value of /Hl S is minimized when there are two electrons in some sites and four electrons in the remainder sites (see Fig. 2). In other terms, we have that for n¼ 3, /Hl Smin ¼
N ðe þ el0,4 Þ 2 l0,2
ð17Þ
The corresponding curves for other values of V are similar to the ones shown in Fig. 1. For any value of V, the curvature curve becomes negative if E~ f 4U=2. Consider the wavefunction XY y Y y 00 jc S ¼ b2i b4i j0S ð18Þ C iAC
j= 2C
where we defined the operators 2
y y y y y y y y cik þ abðcim fik þ fim cik Þ þ b fim fik by2i ¼ a2 cim y y y y by4i ¼ cim cik fim fik
ð19Þ
and C is a set of L=2 odd natural numbers qs, such that qs o N=2. L denotes the number of lattice points. For conveniency, we define 00 the local states jji S ¼ by2i j0S and jfi S ¼ by4i j0S. The state jc S corresponds to n¼ 3, since it locates two electrons in half of the lattice points and four electrons in the remainder. Thus, the expectation value /Hl S gets the minimal value with this wavefunction, since the local states jji S and jfi S get the minimal values e0,2 and e0,4 , respectively, as shown in Section 2. On the 00 ^ other hand, /c jPj c00 S ¼ 0. Thus, /c00 jHjc00 S equals the lower 00 bound (9). This implies that jc S is a ground state of Eq. (1). It can 00 also be verified that jc S has ODLRO. To do this, we express the 00 ground state jc S as 00
jc S ¼ by2m by4 l jcr Sþ by2l by4m jcr S
ð20Þ
where jcr S ¼
XY 0
C iAC
0
by2i
Y y b4j j0S
ð21Þ
j= 2C 0
being C0 a set of natural numbers qs such that qs oN=21 and qs am,qs a l. Thus, cr does not contain electrons on the sites
00 y y jcmm cmk clk clm jc S ¼ 00
ð22Þ 4
and as a consequence /c b for any value of l and m. This shows that jc S exhibits ODLRO, and thus it is superconducting Now we discuss the relation of the obtained results with real systems. Recently, it has been pointed out that the superconductivity in some heavy fermions like PuCoGa5 and NpPd5 Al2 cannot be explained with the usual spin fluctuations theory. It was proposed that the superconducting state in these systems is a condensate of composite pairs involving localized and conduction electrons. The emergence of such a kind of condensate was studied in a two-channel Kondo model by using a large N approximation [12]. The obtained results in the present work demonstrate that the same type of pairing can occur in the EPAM. The binding energy is the difference between the energies of the paired and unpaired holes. We can estimate this value with the difference Eb ¼ 2e0,3 ðe0,2 þ e0,4 Þ. For the parameters values U¼1, G ¼0.5, V¼0.1 and Ef ¼1, we obtain that Eb ¼ 0:087. The values for U in real systems range between 2 and 20 eV. Taking U¼5 eV gives Eb ¼ 0:44 eV, which corresponds to a very high temperature.
4. Conclusions In the present work we obtained exactly ground states of the EPAM containing the f c Coulomb repulsion and inter-site hibridization. With the method that we employed, the major part of the parameters values may be chosen arbitrarily, while two of them become determined by the others for the solution to be valid. The values of the parameter for which the ground states can be obtained are not in principle unphysical. In spite of the restrictions involving the system parameters, we demonstrated that the EPAM containing purely repulsive electronic interactions can exhibit hole pairing and superconductivity. The pairs are composite type since they involve both conduction and valence electrons on the same site. We considered here the one-dimensional case, but the same kind of ground states can be obtained for higher dimensions. The main difference is that the restrictions involving the parameters increase with the dimension.
Acknowledgements We would like to thank PEDECIBA and Facultad de CienciasUruguay for partial financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
L.M. Falicov, J.C. Kimball, Phys. Rev. Lett. 22 (1969) 997. J.K. Freericks, V. Zlatic, Rev. Mod. Phys. 75 (2003) 1333. K. Miyake, J. Phys. Soc. Jpn. 71 (2002) 1007. L.G. Sarasua, Phys. Rev. B 75 (2007) 054504. U. Brandt, A. Giesekus, Phys. Rev. Lett. 68 (1992) 2648. R. Strack, Phys. Rev. Lett. 70 (1993) 833. I. Orlik, Z. Gulacsi, Phil. Mag. B 81 (2001) 1587; P. Gurin, Z. Gulacsi, Phys. Rev. B 64 (2001) 45118. L.G. Sarasua, M.A. Continentino, Phys. Rev. B 65 (2002) 233107. J. de Boer, A. Schadschneider, Phys. Rev. B 75 (1995) 4298. C.N. Yang, Rev. Mod. Phys. 34 (1962) 694. G.L. Sewell, J. Stat. Phys. 61 (1990) 415. R. Flint, M. Dzero, P. Coleman, Nat. Phys. 4 (2008) 643.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Exact superconducting ground states of the extended Anderson model L.G. Sarasua ´ blica, Montevideo, Uruguay Instituto de Fı´sica, Universidad de la Repu
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 April 2011 Received in revised form 8 June 2011 Accepted 22 June 2011 Available online 29 June 2011
We obtain exact ground states of an extended periodic Anderson model (EPAM) with non-local hybridization and Coulomb repulsion between f and c electrons (Falicov–Kimball term) in one dimension. We show that for a range of parameter values these ground states exhibit composite hole pairing and superconductivity that originate from purely electronic interactions. & 2011 Elsevier B.V. All rights reserved.
Keywords: Superconductivity Strongly correlated electron systems
1. Introduction In recent years there has been a great interest in the study of mechanisms for superconductivity that originate from electronic interactions. This is motivated by the fact that it is accepted that the electron pairing in heavy fermions or superconducting cuprates is caused by other than the usual BCS mechanism. Different theories have been proposed to explain the origin of superconductivity in strongly correlated electrons systems (SCES), based on distinct concepts like: spin fluctuations, the RVB state and correlated hopping, among others. In the study of these mechanisms serious drawbacks may emerge due to the fact that approximate schemes cannot give conclusive results when the interactions are strong. Thus, exact results for such systems are valuable since they are free of these inconveniences. The theoretical description of multi-band systems in which the electronic correlations are important are usually described by the Anderson model and its variants. The so called extended periodic Anderson model (EPAM) consists of the usual periodic Anderson model including a repulsive interaction between electrons of different bands, the so called Falicov–Kimball (FK) term. The FK model has been extensively used to study valence, metal-insulator transitions and excitonic phases in rare earths compounds and actinides [1,2]. Recently, it has been proposed that superconductivity may arise in the EPAM driven by a valence fluctuation mechanism [3]. In the present work, we obtain exact ground states of the EPAM for a limited region of the parameters space. We show that these ground states exhibit off diagonal long range order (ODLRO), which imply that they are superconducting, when
Tel.: 598 2525 8624; fax: þ598 2525 0580.
E-mail address: sarasua@fisica.edu.uy 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.055
certain conditions involving the local and interband Coulomb interaction, the energy level of the quasi-localized electrons and the hybridization is met.
2. Model hamiltonian We shall consider the extended Anderson model with on-site and intersite hybridizations in one-dimensional case. The model allows the hopping of f electrons between different sites. This situation is very adequate for many systems, in which the f electrons may be itinerant. In spite of this, we search for solutions with almost localized f electrons. The Hamiltonian model reads X y X y X f X f f X H ¼ tc cis cjs tf fis fjs þ U nim nik þ G nci nfi þ Ef ni /ijSs
þ V0
/ijSs
i
i
X y X y ðcis fis þ fiys cis Þ þ V1 ðcis fjs þ fiys cjs Þ is
i
ð1Þ
/ijSs
where ciys , fiys are the creation operators for c and f electrons at the site i with spin s, ncis ¼ ciys cis , nfis ¼ fiys fis , are the number operators, na ¼ nam þ nak (a¼f,c), tc and tf are the hopping integrals, V0 and V1 are the local and intersite hibridization, respectively, Ef measures the energy difference between the centers of the bare f and c bands, U is the on-site Coulomb repulsion and G is the Coulomb repulsion between f and c electrons at the same site. To obtain the ground state of the above Hamiltonian, we shall proceed in similar way as in Ref. [4]. The ground state is found showing that a wavefunction jcS minimizes the expectation value of the Hamiltonian. In doing so, we shall separate the Hamiltonian into two parts, one of them containing almost all the kinematical terms and the other containing local terms. To obtain a wavefunction that minimizes the expectation value of the first term we use the method of Brandt and Giesekus [5–8]. The minimization of the second term is performed diagonalizing the local
L.G. Sarasua / Physica B 406 (2011) 3622–3624
Hamiltonian. In this part the method is similar to the optimal ground state approach proposed in Ref. [9]. The local hibridization will be separated into two parts V0 ¼ V 0 þ V, and each part is included in separated terms. Thus, we write H ¼ Hk þH0 , with X
Hk ¼ tc
X
ciys cjs tf
/ijSs
/ijSs
fiys fjs þ V 0
X y X y ðcis fis þ fiys cis Þ þ V1 ðcis fjs þ fiys cjs Þ /ijSs
is
ð2Þ We introduce now the operators ayis ¼ aðciys þ cjys Þ þ bðfiys þfjys Þ
ð3Þ
being a and b real constants. With the use of them, we can write Hk in the form X X X ais ayis 2tc ðnci þnfi Þ þ2ðtf tc Þ nfi ð4Þ Hk ¼ is
i 2
2
i
if tc ¼ a , tf ¼ b , V1 ¼ V ¼ ab. This imply the following relations: tc tf ¼ V12 ,
0
V1 ¼ V 0
ð5Þ
Assuming that these conditions hold, the Hamiltonian can be written as X X ais ayis 2tc ðnci þnfi Þ þHl ð6Þ H¼ is
i
with Hl ¼
3623
The wavefunction jcS may be the ground state if jjl S coincides with the eigenstates of Hl, which in turn must satisfy the relations (14). Although this could seem in principle difficult to accomplish, we found that for the condition U ¼ 2G, Eq. (14) is satisfied for any value of V0 and E~ f . It is to be noticed that the relations (5) restrict the values of V 0 , but not those of V (we recall that V0 ¼ V þ V 0 Þ. Then we can obtain ground states of Eq. (1) varying the values of V and E~ f . However, the values of A, B, C will change in order to satisfy Eq. (14) which in turn modifies the ratio a=b and also the value of tf =tc . Hence, after fixing the values of U and G, we can find the ground states for arbitrary values of tc, V0 and Ef, but the values of tf and V1 become determined by Eqs. (5) and (14). In all the numerical results that are presented in this work, the value of tf is small in comparison with tc, with tf =tc o 0:1. For instance, for U¼1, G¼0.5, tc ¼ 1, V¼0.1, E~ f ¼ 1, the ground state of Hl has the coefficients A¼0.9956, B¼ .0661, C¼0.00438, which satisfy the condition (5). These values imply that tf ¼ 0:0044. For these values of the parameters, jcS is the ground state of Eq. (1), with a ¼ 1, b ¼ 0:0664. The corresponding state for the totally filled band case is given by Y y y y y 0 cim cik fim fik j0S ð15Þ jc S ¼ i 0
The expectation value /HS obtained with jc S equals the lower bound (9) for n¼4.
P
i Hli , and
Hli ¼ E~ f nfi þ Unfim nfik þGnci nfi þV
X y ðcis fis þ fiys cis Þ
ð7Þ
3. Superconducting ground states
s
where we defined E~ f ¼ Ef þ 2ðtf tc Þ. Denoting the total number of particles as N, H can be written as ^ H ¼ P2t ð8Þ c N þ Hl P y where P^ ¼ is ais ais . To obtain the ground states, we need a lower bound for the expectation value of H. We note that P^ is semidefinite positive, so that the minimal value of its expectation value is zero. Thus a lower bound for /HS is
El ¼ 2tc N þ /Hl Smin
ð9Þ
The value of /Hl Smin is obtained diagonalizing the local Hamiltonian Hli. Consider now the wavefunction Y y y aim aik j0S ð10Þ jcS ¼ i
This state locates two electron per site and thus corresponds to ^ cS ¼ 0 since the half filled band case. It is easy to verify that /cjPj ay2 ¼ 0, due to the properties of the fermionic operators. Thus, in is virtue of Ritz’s theorem, jcS is the ground state if /cjHl jcS ¼ /Hl Smin since in this case /cjHjcS equals the lower bound El . It can be verified that the expression of jcS simplifies to Y y y y y ðacim þ bfim Þðacik þ bfik Þj0S jcS ¼
The wavefunction jcS found in the previous section is a ground state of Eq. (1) under the already mentioned conditions but it is not superconducting. A system is superconducing if it exhibits the off diagonal long range order (ODLRO) property [10,11]. A state jFS has this property if lim
y y /Fjclm cmk cpm cqk jFSa 0
where ri denotes the position of the site i. Here l and p are at the neighborhood of m and q, respectively, i.e. rl rm ,rp rq . The wavefunction jcS locates two electrons in each site and consequently does not have the property (16). In order to have ODLRO, the wavefunction must contain pairs of electrons or holes that can move along the lattice. From now on, we shall focus on the three quarter filled band case, i.e. 1 hole per site. We will show that a state of hole pairs has lower energy than one of unpaired holes. In doing so, we shall obtain the minimal values of e ¼ /Hli S=ni for different values of ni, through the diagonalization of Hli. We shall denote as e0,p the minimal value of e for ni ¼ p. The results are shown in Fig. 1 for various values of E~ f . From this we can see that
2.0
i
Y 2 y y y y y y y y ða2 cim cik þ abðcim fik þ fim cik Þ þ b fim fik Þj0S ¼
ð16Þ
jrl rp j-1
1.5
ð11Þ
i
e
Thus, jcS is formed by local states of the form jjl S ¼ Aj20S þBjmkSþ BjkmSþ Cj02S
ð12Þ
0.5
with y y cik j0S, j20S ¼ cim
y y j02S ¼ fim fik j0S
y y fik j0S, jmkS ¼ cim
y y jkmS ¼ fim cik j0S 2
0.0 ð13Þ 2
4
and A ¼ a2 =D, B ¼ ab=D, C ¼ b =D, where D ¼ a4 þ 2a2 b þ b , which imply that the coefficients verify the condition AC=B2 ¼ 1:
1.0
ð14Þ
0
1
2 ni
3
4
Fig. 1. The minimal values of /Hli =ni S as a function of ni for U ¼ 1, G ¼0.5, V¼ 0.1 and E~ f ¼ 0 (diamonds), E~ f ¼ 0:5 (circles), E~ f ¼ 1 (triangles) and E~ f ¼ 1:5 (squares).
3624
L.G. Sarasua / Physica B 406 (2011) 3622–3624
1.5
m nor l. Using Eq. (20), it can be verified that 00
2
y y y y y y y y cmm cmk clk clm jc S ¼ b fmm fmk cmm cmk flm flk jcr S 00
e
1.0
0.5
0.0 0
1
2 ni
3
4
Fig. 2. In this figure a graphical comparison between the values of /Hli =ni S for ni ¼3 and the average of the values for ni ¼2 and ni ¼ 4 is performed. This shows that for n¼ 3, the value of /Hl S is minimal when there are two electrons in half of the lattice sites and four electrons in the remainder sites.
when E~ f is above U/2, the curve representing e0,p as a function of ni has negative curvature around ni ¼3. This means that for the three quarter filled band case (n ¼3) the value of /Hl S is minimized when there are two electrons in some sites and four electrons in the remainder sites (see Fig. 2). In other terms, we have that for n¼ 3, /Hl Smin ¼
N ðe þ el0,4 Þ 2 l0,2
ð17Þ
The corresponding curves for other values of V are similar to the ones shown in Fig. 1. For any value of V, the curvature curve becomes negative if E~ f 4U=2. Consider the wavefunction XY y Y y 00 jc S ¼ b2i b4i j0S ð18Þ C iAC
j= 2C
where we defined the operators 2
y y y y y y y y cik þ abðcim fik þ fim cik Þ þ b fim fik by2i ¼ a2 cim y y y y by4i ¼ cim cik fim fik
ð19Þ
and C is a set of L=2 odd natural numbers qs, such that qs o N=2. L denotes the number of lattice points. For conveniency, we define 00 the local states jji S ¼ by2i j0S and jfi S ¼ by4i j0S. The state jc S corresponds to n¼ 3, since it locates two electrons in half of the lattice points and four electrons in the remainder. Thus, the expectation value /Hl S gets the minimal value with this wavefunction, since the local states jji S and jfi S get the minimal values e0,2 and e0,4 , respectively, as shown in Section 2. On the 00 ^ other hand, /c jPj c00 S ¼ 0. Thus, /c00 jHjc00 S equals the lower 00 bound (9). This implies that jc S is a ground state of Eq. (1). It can 00 also be verified that jc S has ODLRO. To do this, we express the 00 ground state jc S as 00
jc S ¼ by2m by4 l jcr Sþ by2l by4m jcr S
ð20Þ
where jcr S ¼
XY 0
C iAC
0
by2i
Y y b4j j0S
ð21Þ
j= 2C 0
being C0 a set of natural numbers qs such that qs oN=21 and qs am,qs a l. Thus, cr does not contain electrons on the sites
00 y y jcmm cmk clk clm jc S ¼ 00
ð22Þ 4
and as a consequence /c b for any value of l and m. This shows that jc S exhibits ODLRO, and thus it is superconducting Now we discuss the relation of the obtained results with real systems. Recently, it has been pointed out that the superconductivity in some heavy fermions like PuCoGa5 and NpPd5 Al2 cannot be explained with the usual spin fluctuations theory. It was proposed that the superconducting state in these systems is a condensate of composite pairs involving localized and conduction electrons. The emergence of such a kind of condensate was studied in a two-channel Kondo model by using a large N approximation [12]. The obtained results in the present work demonstrate that the same type of pairing can occur in the EPAM. The binding energy is the difference between the energies of the paired and unpaired holes. We can estimate this value with the difference Eb ¼ 2e0,3 ðe0,2 þ e0,4 Þ. For the parameters values U¼1, G ¼0.5, V¼0.1 and Ef ¼1, we obtain that Eb ¼ 0:087. The values for U in real systems range between 2 and 20 eV. Taking U¼5 eV gives Eb ¼ 0:44 eV, which corresponds to a very high temperature.
4. Conclusions In the present work we obtained exactly ground states of the EPAM containing the f c Coulomb repulsion and inter-site hibridization. With the method that we employed, the major part of the parameters values may be chosen arbitrarily, while two of them become determined by the others for the solution to be valid. The values of the parameter for which the ground states can be obtained are not in principle unphysical. In spite of the restrictions involving the system parameters, we demonstrated that the EPAM containing purely repulsive electronic interactions can exhibit hole pairing and superconductivity. The pairs are composite type since they involve both conduction and valence electrons on the same site. We considered here the one-dimensional case, but the same kind of ground states can be obtained for higher dimensions. The main difference is that the restrictions involving the parameters increase with the dimension.
Acknowledgements We would like to thank PEDECIBA and Facultad de CienciasUruguay for partial financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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