Fermionic lattice models and electronic correlations: Magnetism and superconductivity

Journal of Non-Crystalline Solids 354 (2008) 4326–4329

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Fermionic lattice models and electronic correlations: Magnetism and superconductivity M. Matlak a,*, B. Grabiec b, S. Krawiec a a b

Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland Institute of Physics, University of Zielona Góra, Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland

a r t i c l e

i n f o

Article history: Available online 31 July 2008 PACS: 71.10.w 71.10.Ca 71.10.Fd 71.20.Be 71.27.+a 71.28.+d 73.21 La 78.67 Bf Keyword: Magnetic properties

a b s t r a c t We consider the dimer approach to the generalized Hubbard model. As a first step we solve the dimer eigenvalue problem exactly. We decompose the dimer Hamiltonian HD into a set of commuting partial ð aÞ ðaÞ Hamiltonians HD ða ¼ 1; 2; . . . ; 16Þ ascribed to each dimer energy level where each HD is represented in the second quantization. This procedure gives us a review of important two-site interactions, normally hidden in the original dimer Hamiltonian HD, several of them describing a competition between magnetism and superconductivity but belonging to different dimer energy levels. This feature is, however, a source of new problems discussed in the paper and connected with the practical use of the mean field approximation in the case of a real lattice. As a next step, we consider the decomposition of the real lattice into a set of interacting dimers to explicitly show that the competition between magnetism and superconductivity is a common feature of all electronic lattice models. This competition should be necessarily taken into account in practical calculations of the thermodynamics of such models. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction The aim of this paper is to show that electronic lattice models like the Hubbard model and its generalization (cf. e.g. Refs. [1– 28] and papers cited therein) contain as a rule such two-site electronic interactions which can lead to a competition between magnetism and superconductivity. This feature is explicitly demonstrated in the case of an exactly solvable model for a dimer (see Section 2) where we decompose a dimer Hamiltonian HD into ðaÞ a set of commuting partial Hamiltonians HD ¼ Ea jEa ihEa j represented in the second quantization and ascribed to each dimer energy level Ea (a = 1, 2, . . . , 16). This approach produces, however, quite new problems because the competitive two-site electronic interactions, describing magnetism and superconductivity are present only in several partial Hamiltonians. It means that they can be thermodynamically active only in the case when a given energy level is occupied. It is then evident that the problem how to introduce order parameters (using e.g. mean field approximation) is therefore very delicate. This procedure cannot be globally applied to the total Hamiltonian HD but only selectively, i.e. with respect to the mentioned partial Hamitonians and it should necessarily be correlated with the occupation of the corresponding * Corresponding author. E-mail address: [email protected] (M. Matlak). 0022-3093/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2008.06.045

levels. How to do it properly is an open question, especially in the case of a real lattice (see Section 3) where we decompose the lattice into a set of interacting dimers. We show in this way that the competition between magnetism and superconductivity is always present but practical calculations of the thermodynamical properties of the model seem to be very difficult. 2. The model A generalized one-band Hubbard model belongs to a class of fermionic lattice models widely used in solid state physics (magnetic and transport properties of solids, insulator–metal transitions, liquid 3He, fullerenes, high-TC superconductivity, etc.). In this paper we consider the extended Hubbard model supplemented by the intrasite Cooper pairs hopping (KPPK interaction). The Hamiltonian of this model has the following form:

H¼

X i6¼j;r

t i;j cþi;r cj;r þ U

X i

ni;" ni;# þ

1 X ð1Þ J ni;r nj;r 2 i6¼j;r ij

X 1 X ð2Þ þ J ni;r nj;r  V i;j cþi;" cþi;# cj;# cj;" : 2 i6¼j;r ij i6¼j

ð1Þ

The indices (i, j) enumerate the lattice points (Ri, Rj), ti,j is the hopping integral, U denotes the effective intrasite Coulomb interaction, J(1) and J(2) (generally, not necessary equal) describe

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effective intersite interactions, all of them resulting from the original intrasite and intersite Coulomb repulsion which can be modified by polaronic effects (see e.g. Ref. [9] for details) and therefore U,J(1,2) can be treated here as positive or negative parameters. The last term in (1) is responsible for the transport of intrasite Cooper pairs (Refs. [10,11]) with the coupling constant V. The model (1) cannot be solved exactly in a general case. We can, however, consider a special but nontrivial case of two interacting ions (a dimer problem) which has an exact, analytical solution. Thus, let us start with a dimer Hamiltonian, resulting from the expression (1). It has the following form:

HD ¼ t

(

ð5Þ

HD

X ðcþ1;r c2;r þ cþ2;r c1;r Þ þ Uðn1;" n1;# þ n2;" n2;# Þ r

þ J ð1Þ

X

n1;r n2;r þ J ð2Þ

r

X

ð6Þ

HD

n1;r n2;r

r

 Vðcþ1;" cþ1;# c2;# c2;" þ cþ2;" cþ2;# c1;# c1;" Þ; ð1;2Þ

ð2Þ

ð1;2Þ

where t 1;2 ¼ t 2;1 ¼ t; J1;2 ¼ J2;1 ¼ J ð1;2Þ and V1,2 = V2,1 = V. We start from the Fock’s basis jn1,", n1,;; n2,", n2, ;i(ni,r = 0, 1; i = 1, 2; r = ", ;) and we find the exact solution of the dimer eigenvalue problem (HDjEai = EajEai). We obtain in this way 16 energy levels (some of them degenerated) with corresponding 16 eigenvectors (see the formulae (A.1) and (A.2) in the Appendix). In the next step we use the equivalent form for the dimer Hamiltonian (2)

HD ¼

16 X

Ea Pa ;

ð3Þ

ð7Þ

HD

a¼1

" #)  ! ! J ð2Þ J ð2Þ ðU  V  J ð2Þ Þ 2t 2 na na S1  S2  1 2  ¼ E8 P8 ¼  þ 4C 2 C 4 " ! # ð2Þ ðU  VÞ UV J t2 þ þ 1þ þ ½d1 d2 þ d2 d1  þ 4 2C C " ! # ðU  VÞ U  V  J ð2Þ t2 þ 1þ þ 8 2C 2C      nb nb  nb1 1  na2  2 þ nb2 1  na1  1 2 2 " # ð2Þ X X 2 t UV þJ þ 1þ ½aþi;r bi;r þ bi;r ai;r ; ð12Þ  2 2C r i¼1 ( " #)  ! ! J ð2Þ J ð2Þ ðU  V  J ð2Þ Þ 2t 2 na na S1  S2  1 2  ¼ E9 P9 ¼   4C 2 C 4 " ! # ð2Þ 2 ðU  VÞ UV J t þ þ þ 1  ½d1 d2 þ d2 d1  4 2C C " ! # ðU  VÞ U  V  J ð2Þ t2 þ  1 2C 2C 8      b n nb  nb1 1  na2  2 þ nb2 1  na1  1 2 2 " # 2 t U  V þ J ð2Þ X X þ  ½aþi;r bi;r þ bi;r ai;r ; ð13Þ 1 2 2C r i¼1   na na ¼ E10 P 10 þ E11 P11 ¼ 2Jð1Þ Sz1  Sz2 þ 1 2 ; ð14Þ 4

ð8Þ

where Pa = jEaihEaj. Each product EaPa in the formula (3) can be rewritten in the second quantization with the use of the formulae (A.1) in the Appendix and the definition of the Hubbard and spin operators:

ai;r ¼ ci;r ð1  ni;r Þ; bi;r ¼ ci;r ni;r ; 1 Szi ¼ ðnai;"  nai;# Þ; nai;r ¼ aþi;r ai;r ; 2 Sþi ¼ cþi;" ci;# ¼ aþi;" ai;# ; Si ¼ cþi;# ci;" ¼ aþi;# ai;" :

ð4Þ

ð9Þ

ð6Þ ð10Þ

HD ¼

10 X

ðiÞ

HD ;

ð7Þ

i¼1

where ð1Þ

HD ¼ E2 P2 þ E4 P 4 ¼ 

     t a nb nb n1 1  na2  2 þ na2 1  na1  1 2 2 2

t X þ ½a a2;r þ aþ2;r a1;r ; 2 r 1;r      t a nb nb n1 1  na2  2 þ na2 1  na1  1 ¼ E3 P3 þ E5 P 5 ¼ 2 2 2 i t Xh þ a1;r a2;r þ aþ2;r a1;r ;  2 r   na na ¼ E6 P6 ¼ J ð2Þ Sz1  Sz2  1 2 4 

ð2Þ

HD

ð3Þ

HD

J ð2Þ þ  ðS  S þ S1  Sþ2 Þ; 2 1 2      ðU þ VÞ b nb nb ¼ E7 P7 ¼ n1 1  na2  2 þ nb2 1  na1  1 4 2 2 ðU þ VÞ þ þ  ½d1 d2 þ d2 d1 ; 2 þ

ð4Þ

HD

ð8Þ

ð9Þ

ð10Þ

ð11Þ

ðt þ U þ J ð1Þ þ J ð2Þ Þ a b ½n1 n2 þ na2 nb1  4 ðt þ U þ J ð1Þ þ J ð2Þ Þ X þ þ ½b1;r b2;r þ b2;r b1;r ; þ 2 r

ð15Þ

HD ¼ E13 P 13 þ E15 P15 ¼

ð5Þ

Taking into account the degeneration of the dimer energy levels (cf. (A.1) in the Appendix) we obtain

ðt þ U þ Jð1Þ þ J ð2Þ Þ a b ½n1 n2 þ na2 nb1  4 ðt þ U þ J ð1Þ þ Jð2Þ Þ X þ þ ½b1;r b2;r þ b2;r b1;r ;  2 r

HD ¼ E12 P 12 þ E14 P14 ¼

HD

¼ E16 P16 ¼

ðU þ J ð1Þ þ J ð2Þ Þ b b n1 n2 2 þ

ð16Þ ð17Þ

a;b a;b and nia;b ¼ ni;" þ ni;# ; nbi;r ¼ bi;r bi;r ¼ ni;r ni;r (i = 1, 2), d1(2) = a1(2),   ;b1(2)," = c1(2), ;c1(2),",i ¼ 1 if i = 2 and i ¼ 2 if i = 1. The constant C in the formulae (12) and (13) is given by the formula (A.2) in the Appendix. The partial Hamiltonians (8)–(17) belong to the subspaces of N = 1, 2, 3 and 4, respectively. The expressions (8)–(17) are exact and having summed them up (see (7)) we obtain again the original dimer Hamiltonian (2). It is easy to see that the dimer Hamiltonian decomposition (8)–(17) visualizes a competition between different magnetic interactions (ferromagnetic, antiferromagnetic – it depends on the sign of the corresponding coupling constants), scattered between different dimer energy levels. The Ising type interactions are present in the formulae (10) and (14). The formula (10) contains also a transverse interaction between spins. The Heisenberg type magnetic interactions (or generalized t–J interactions) can be seen in the first terms of the formulae (12) and (13). It is interesting to note that if J(2) = V = 0 the first terms in the formulae (12) and (13) describe ferromagnetic or antiferromagnetic interactions, similar to the well-known t–J model (see e.g. Refs. [12–14]) 2 2 because the coefficient 2tC  4t for large jUj and is exactly the same jUj as in the case of a real lattice. Let us also note that such products þ þ þ þ þ þ þ as d1 d2 ¼ b1;" aþ 1;# a2;# b2;" ¼ c 1;" c 1;# c2;# c 2;" and d2 d1 ¼ b2;" a2;# a1;# b1;" ¼ þ þ c2;" c2;# c1;# c1;" are present in the expressions (11)–(13). They describe the hopping of intrasite Cooper pairs, typical for the KPPK superconductivity models (cf. e.g. Refs. [10,11]) with positive or negative coupling constants and this type of interactions are present within

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the model (2) as hidden interactions even in the case of V = 0. Such   þ products as Sþ 1 S2 ðS1 S2 Þ, present in the formulae (10), (12) and (13), describe the intersite Cooper pairs (cf. e.g. Ref. [9] and papers cited   þ þ þ þ þ therein) because Sþ 1 S2 ¼ c 1;" c2;# c 1;# c 2;" ðS1 S2 ¼ c2;" c 1;# c 2;# c 1;" Þ. Alternatively, the application of the resonating valence bond approach (cf. Refs. [7,8]) allows us also to treat the terms like ! ! na na ðS1  S2  14 2 ) in (12) and (13) when introducing the pairing operator f2;1 ¼ p1ffiffi2 ðc2;# c1;"  c2;" c1;# Þ. For example,! when we restrict ourselves ! þ to subspace na1 ¼ na2 ¼ 1 we obtain ðS1  S2  14Þ ¼ f2;1 f2;1 and such terms lead to superconductivity (cf. Refs. [7,8]). It is also interesting to note that even in the case when J(1) = J(2) = U = V = 0 (see formulae (8)–(17)) the main competitive interactions (ferromagnetic, antiferromagnetic and superconducting) are always present (t 6¼ 0). It is interesting to note that the formulae (7)–(17) vizualize the important fact that with an increase in the average number of electrons n = hNi the system will pass through different phases, depending on the result of the competition between different thermodynamically activated two-site interactions and this feature explains the origin of the model’s phasediagram in qualitative terms.

Appendix

3. Conclusions

jE9 i ¼ a ðj1; 1; 0; 0i þ j0; 0; 1; 1iÞ þ aþ ðj1; 0; 0; 1i  j0; 1; 1; 0iÞ;

Let us consider the case of a real lattice described by the Hamiltonian (1). We assume that the number of lattice points i(j) is equal to N (an even number) and we decompose the lattice into a set of M ¼ N2 dimers described by the dimer index I, a (J, b) where I(J) = 1, 2, . . . , M and a(b) = 1, 2. The Hamiltonian (1) can be thus replaced by the equivalent form

H¼

X

HD;I þ

X

t I;a;J;a cþI;a;r cJ;a;r þ

I6¼J;a;r

I

þ

X

1 X ð1Þ 1 J nI;a;r nJ;a;r þ 2 I6¼J;a;r I;a;J;a 2

t I;a;J;b cþI;a;r cJ;b;r

E1 ¼ 0;

jE1 i ¼ j0; 0; 0; 0i; 1 E2 ¼ t; jE2 i ¼ pffiffiffi ðj1; 0; 0; 0i þ j0; 0; 1; 0iÞ; 2 1 E3 ¼ t; jE3 i ¼ pffiffiffi ðj1; 0; 0; 0i  j0; 0; 1; 0iÞ; 2 1 E4 ¼ t; jE4 i ¼ pffiffiffi ðj0; 1; 0; 0i þ j0; 0; 0; 1iÞ; 2 1 E5 ¼ t; jE5 i ¼ pffiffiffi ðj0; 1; 0; 0i  j0; 0; 0; 1iÞ; 2 1 ð2Þ E6 ¼ J ; jE6 i ¼ pffiffiffi ðj1; 0; 0; 1i þ j0; 1; 1; 0iÞ; 2 1 E7 ¼ U þ V; jE7 i ¼ pffiffiffi ðj1; 1; 0; 0i  j0; 0; 1; 1iÞ; 2 U  V þ J ð2Þ U  V þ J ð2Þ E8 ¼ C þ ; E9 ¼ C þ ; 2 2 jE8 i ¼ aþ ðj1; 1; 0; 0i þ j0; 0; 1; 1iÞ  a ðj1; 0; 0; 1i  j0; 1; 1; 0iÞ; E10 ¼ J ð1Þ ; ð1Þ

E11 ¼ J ;

X

E13 E14 E15

ð1Þ

J I;a;J;b nI;a;r nJ;b;r

I6¼J;a6¼b;r

jE10 i ¼ j1; 0; 1; 0i; jE11 i ¼ j0; 1; 0; 1i;

1 jE12 i ¼ pffiffiffi ðj0; 1; 1; 1i þ j1; 1; 0; 1iÞ; 2 1 ð1Þ ð2Þ ¼ t þ U þ J þ J ; jE13 i ¼ pffiffiffi ðj0; 1; 1; 1i  j1; 1; 0; 1iÞ; 2 1 ð1Þ ð2Þ ¼ t þ U þ J þ J ; jE14 i ¼ pffiffiffi ðj1; 0; 1; 1i þ j1; 1; 1; 0iÞ; 2 1 ð1Þ ð2Þ ¼ t þ U þ J þ J ; jE15 i ¼ pffiffiffi ðj1; 0; 1; 1i  j1; 1; 1; 0iÞ; 2 ¼ 2ðU þ J ð1Þ þ J ð2Þ Þ; jE16 i ¼ j1; 1; 1; 1i;

E12 ¼ t þ U þ J ð1Þ þ J ð2Þ ;

I6¼J;a6¼b;r

E16

1 X ð2Þ 1 X ð2Þ þ J nI;a;r nJ;a;r þ J nI;a;r nJ;b;r 2 I6¼J;a;r I;a;J;a 2 I6¼J;a6¼b;r I;a;J;b X X  V I;a;J;a cþI;a;" cþI;a;# cJ;a;# cJ;a;"  V I;a;J;b cþI;a;" cþI;a;# cJ;b;# cJ;b;" ; I6¼J;a

The eigenvalues and eigenvectors of the dimer (2) are

ðA:1Þ where

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u U  V  J ð2Þ 2 t C¼ þ 4t2 ; 2

I6¼J;a6¼b

1 a ¼ 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU  V  J ð2Þ Þ 1 : 2C

ðA:2Þ

ð18Þ where HD,I is the dimer Hamiltonian given by the expression (2) where the lower dimer index I in the operators should be introduced (as e.g. c1,r ? cI,1,r, etc.). The dimer Hamiltonian HD,I in (18) can, however, be replaced by an equivalent expression (7) where the partial dimer Hamiltonians are given by the expressions (8)– (17). The Hamiltonian (18), equivalent to (1), describes now ‘‘free” dimers (first term in (18)) and their interactions (the next eight terms in (18)). The main difference between Hamiltonians (1) and (18), both describing the same physics, lies in the fact that the mentioned competitive magnetic and superconducting interactions are hidden in the Hamiltonian (1) whereas in the Hamiltonian (18) they appear in a direct way. It, however, means that to find thermodynamical properties of the model we should simultaneously introduce order parameters in four competitive channels (ferromagnetic, antiferromagnetic and superconducting (intrasite and intersite pairing)) where special attention should be paid to the nontrivial fact that all competitive interactions are additionally scattered between different energy levels and therefore their activities have to be correlated with the occupation of these levels. These conclusions seem to be also important in the case of such small systems like quantum dots and nanosystems (cf. e.g. Refs. [29–32]).

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