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Crystallization and metal-insulator transition in an itinerant electron system
Volume 134, number 5
PHYSICS LETTERS A
9 January 1989
CRYSTALLIZATION AND METAL-INSULATOR TRANSITION IN AN ITINERANT ELECTRON SYSTEM J. J1~DRZEJEWSK1a,
j LACH b and R. LY~WAb Institute of Theoretical Physics, University of Wroclaw, Cybulskiego 36, 50-205 Wroclaw, Poland Institute of Low Temperatures and Structure Research, Polish Academy of Sciences, P1. Katedralny 1, 50-950 Wroclaw, Poland
Received 24 October 1988; accepted for publication 8 November 1988 Communicated by A.A. Maradudin
We study properties ofground states of an itinerant electron model beyond the hole—particle symmetry point. When the density ofband electrons is varied different sorts of periodic configurations oflocalized particles minimize the energy. The system can be a metal or an insulator accordingly.
Properties of interacting fermion systems are surely in the center of interest of contemporary solid state physics. A theory capable of describing these systems is an indispensable tool for studies of such fundamental phenomena as valence-change transitions, metal—insulator transitions, crystallization, superconductivity, itinerant magnetism etc. A common feature of the mentioned phenomena is that they are cooperative effects produced by strongly correlated fermion systems at sufficiently low temperatures. Despite numerous efforts, neither does a satisfactory theory of such systems exist nor is our understanding ofthesimplestmodelsinventeddeepenough [1] (an exception are mean-field models, like the BCS one). The most representative and most often explored models in the field are: the Hubbard model [2], the periodic Anderson model [1,3] and the Falicov— Kimball model [4]. Traditional approximate techniques when applied to these models yield results which cannot be trusted at least in some respects
ball model is defined by the following hamiltonian:
[1,5,61, since the results obtained are often contradictory. Apparently a number of rigorous results which could serve as reference points is strongly desirable. Quite recently, it has been demonstrated that the one-band, spinless version of the Falico—Kimball model is rather well suited for investigations. Brandt and Schmidt [7] and Kennedy and Lieb [8] obtamed several rigorous results for this model. The one-band, spinless version of the Falico—Kim-
The model (1) has several interesting interpretations. Let us mention a few of them. First, as indicated in refs. [1,8], it can be thought of as a model of a solid composed of electrons and ions, hence as a model of a crystallization. Second, it is a model of valence-change transitions and mixed-valence states in rare-earth compounds [1]. In this case the moving particles play the role of band s-electrons, the localized ones stand for f-electrons and E fixes the
H=
x,yeA
~
+E ~ W(x)
+2U~ n~W(x) .
(1)
The hamiltonian (1) describes a system composed of two sorts of Fermi particles on a lattice A: moving ones (we call them electrons) and localized ones (we call them ions). The first term of (1) represents the kinetic energy of electrons, c~is an annihilation operator of an electron at a sitex of A. The second term stands for a repulsion (U>0) or an attraction (U<0) of the electrons with the ions, ~ and W(x) denotes an occupation number operator of he ions at a site x (which can be regarded as a classical field assuming the value 1 if the site x is occupied and the value 0 if it is empty). The third term stands for an energy contribution due to a filling of the systern by N1 ions (N, ~xeA W( x)).
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
319
Volume 134, number 5
PHYSICS LETTERS A
position of a sharp f-electron energy level. Third, the model (1) can be regarded as an approximation to the Hubbard model. Then the two sorts of particles represent electrons with different spin, E = 0 [81. The rigorous results of refs. [7,8] can be summarized as follows. Let us assume, for definiteness, that A is a square lattice, the matrix elements t~.of the kinetic energy are equal to 1 when x, y are nearest neighbours and vanish otherwise (without any loss of generality t can be chosen positive) and U> 0. Under the conditions stated, if the total number Ne of electrons and the total number N~of ions are equal to ~ Al (lAl is the number of sites in A), then the system has two ground states. In the first state the ions occupy the even sublattice of A and in the second state they occupy the odd one, that is, in the ground state the ion configuration is periodic, a chessboard-like crystal [7,8]. The ground state is insulating [8]. These properties hold for periodic and free boundary conditions, they depend neither on the size and the dimension of A nor on the value of the unique parameter U/I of the system [8] (no metal— insulator transition). Moreover, the long-range order displayed in the ground state is preserved at sufficiently low temperatures. The system considered can be viewed as a classical system on a lattice (Ising-like configurations of the ions) with a complicated, many-body and temperature-dependent interaction. Usually it is expected that such an interaction can be well approximated by a pair interaction that is associated to it. For the model (1) the latter interaction has been determined by Litvin and Priezzhev [91. The complicated pair potential has been investigated numerically and ground state configurations ofthe ions have been found for a number of densities of the electrons and the ions. In this Letter we present the main results of our investigations of ground states of the model (1). We are interested in two problems primarily: (i) What sorts of periodic configurations (crystals) of the ions arise beyond the symmetry point Ne = N, = I Al, specifically for N, = I Al and Ne = IA I / 2~lwith n = 1, 2, (the reason of such a choice shall become clear later)? Contrary to ref. [9] we investigate the full potential between the ions. Therefore we are interested in comparing our results with corresponding predic...
320
9 January 1989
tionsof ref. [9] which are based on the pair potential. (ii) What is the character of the ground states beyond the symmetry point, metallic or insulating? In the literature on the subject it is ascertained that if Ne I Al —Ni, then the system should be a metal (see for instance ref. [11). Does there exist a metal—insulator transition driyen by the unique parameter UI 1 of the model (1)? For several reasons (see below and ref. [8]) it is convenient to introduce a new field s(x) = 2 W(x) 1. Then the operator —
HEN1 =
—
~
UNe
(2)
t~,c~c~+U ~ n~s(x) ,
x.yeA
xeA .
for a fixed configuration of the ions is the secondquantized version of the single particle hamiltonian .
.
h(S) = T+ US
.
.
(3)
where T is the matrix with elements ~ and S is a diagonal matrix whose matrix elements are s(x) = ±1. In this manner studies of the model (1) can be reduced to those of the spectrum of (3) for different configurations of the ions. It is this peculiarity of the model which makes it more suitable for investigations than the other above mentioned models. The condition N, = IA l~to which we always stick, is equivalent to Tr S=0 and as a result the spectrum of h(S) is symmetric with respect to zero (this property helps to control the results of numerical diagonalizations of h(S)). In our search for ground state configurations of the ions we restrict ourselves to periodic configurations only (in statistical mechanics there is a conjecture that, under some regularity conditions on the potential, non-periodic equilibrium states do not exist in two-dimensional systems [101). Thus since we investigate finite systems it is natural to impose penodic boundary conditions on the lattice A. To single out configurations of the ions that are candidates for the ground state configurations we divide A into 2~z, n = 1, 2, square sublattices. For each n we consider a set of the configurations that are obtained according to the following rules: (1) each one of the sublattices can be either empty or completely filled, (2) one half of the sublattices has to be empty ...,
Volume 134, number 5
PHYSICS LETTERS A
while the other half has to be completely occupied (to satisfy TrS=0). Of course the spectra of h(S) corresponding to configurations that are related by the symmetry transformations of the model are the same. For n = 1 there is only one equivalence class of the ion configurations (displayed in fig. 1 a). The corresponding spectrum can be found analytically. If Ne= ~ Al (the half-filled band), this spectrum gives the lowest energy of the Fermi sea of the one particle system defined by h(S) for any U/I, in agreement with ref. [8]. At the Fermi level there is a gap of width 2 U, hence the system is an insulator for any UI t. Also the case n = 2, where there are two equivalence classes (one of them coincides with that for n = 1 and the new one is displayed in fig. lb) can be solved analytically. By elementary inequalities between the spectra of the n = 1 and n = 2 cases it can be proven that if in the thermodynamic limit the
a
9 January 1989
density of electrons n~,= limA.~.NJ IA I = ~ then the Fermi sea energy of the new n=2 class is lower than that of the n = 1 class. Therefore the latter configuration cannot be the ground state one for fle = This conclusion is in contradiction with the predictions based on the pair potential [9]. One of the results of ref. [9] states that if N, = IA I then the chessboardlike configuration (fig. Ia) is the ground state one for any density of electrons. Thus contrary to cornmon expectations the many-body character of the interaction between the ions cannot be neglected. The cases n = 3 and n = 4 have been studied numerically only. If n = 3 (8 sublattices) there are 70 configurations which are admitted by the condition TrS=0. They split into 6 equivalence classes. We have calculated the spectra of all these configurations and also of several different configurations that are admissible in the 16 sublattices case (in the latter case the total number of admissible configurations ~.
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Fig. 1. Ground state configurations of the ions: (a) N, = A, (b) N,=~(A~,(c) N,=IIA! and U/t>a, (d) N,=~AI, (e) N, = A and U/t
321
Volume 134, numberS
PHYSICS LETTERS A
amounts to 12870). Then the Fermi sea energies of these spectra were compared. It follows from our analysis that the ground state properties of the considered system depend strongly on the band filling and that in contradistinction to the half-filled band case they may depend on the value of U/i. Specifically we have found that if Ne IAI/2~with n=l, 2, 4 then the ground state configurations of the ions are those displayed in figs. Ia, lb and ld respectively, that is, those corresponding to 2’~,n = 1, 2, 4, sublattices. As mentioned above for n = 1 the system is known to be an insulator. For n = 2 and n = 4 it turns out to be a metal. In the former case we have been able to prove this property in the thermodynamic limit using the exact spectrum. No dependence on the value of U/i has been observed. However, the size of the system could not be too small. The case n=3 appeared to be more involved. For Ne = IA I 2~and for U/i larger than some critical value a, 0< a < 1, the Fermi sea energy is minimized by the configuration displayed in fig. ic which can be constructed if A is splitted at least into 2~sublattices. There is a gap on the Fermi level in the corresponding spectrum. This gap grows when U/i increases and tends to a limit whose estimated value amounts to 2. On the other hand if U/I approaches a from above the gap tends to zero. Simultaneously the Fermi sea energy (NeflAI of another configuration, which is displayed in fig. le, approaches the energy of the considered one and for
322
9 January 1989
sufficiently small U/I it becomes the ground state configuration. If U/i is below ci both configurations have no gap on the Fermi level. Thus in the n = 3 case a metal—insulator transition takes place. Most likely it is accompanied by a rearrangement of the ion configuration. The results obtained incline us to the hypothesis that when Ne IAl/2~,with n odd, the system is an insulator for U/i greaten than some critical value and it is a metal for U/i smaller than this value. On the other hand if n is even then the system is metallic irrespective ofthe value of U/ t. More details and additional results will be published elsewhere [11]. References [I] DI. Khomskii, in: Quantum theory of solids, ed. I.M. Lifshits (Mir, Moscow, i982). [2] J. Hubbard, Proc. R. Soc. A 276 (1963) 238. [31P.W. Anderson, Phys. Rev. 124 (1961) 41. [4] L.M. Falicov and J.C. Kimball, Phys. Rev. Lelt. 22 (1969) 997. [5 ID. Vollhardt, Rev. Mod. Phys. 56(1984)99. [6] G. Czycholl, Phys. Rep. 143 (1986) 277. [7]U. Brandt and R. Schmidt, Z. Phys. B 63 (1986) 45. [8] T. Kennedy amd E.H. Lieb, Physica A 138 (1986) 320. [9] A.A. Litvin and V.B. Priezzhev, J1NR preprint P17-87-635, Dubna (1987). 110] R.L. Dobroushin and S.B. Shlosman, Preprint, Institute for Problems ofInformation Transmission, Moscow (1984) [in Russian]. [111 J. Jçdrzejewski, J. Lach and R. Ly±wa,Physica A, to be published.
PHYSICS LETTERS A
9 January 1989
CRYSTALLIZATION AND METAL-INSULATOR TRANSITION IN AN ITINERANT ELECTRON SYSTEM J. J1~DRZEJEWSK1a,
j LACH b and R. LY~WAb Institute of Theoretical Physics, University of Wroclaw, Cybulskiego 36, 50-205 Wroclaw, Poland Institute of Low Temperatures and Structure Research, Polish Academy of Sciences, P1. Katedralny 1, 50-950 Wroclaw, Poland
Received 24 October 1988; accepted for publication 8 November 1988 Communicated by A.A. Maradudin
We study properties ofground states of an itinerant electron model beyond the hole—particle symmetry point. When the density ofband electrons is varied different sorts of periodic configurations oflocalized particles minimize the energy. The system can be a metal or an insulator accordingly.
Properties of interacting fermion systems are surely in the center of interest of contemporary solid state physics. A theory capable of describing these systems is an indispensable tool for studies of such fundamental phenomena as valence-change transitions, metal—insulator transitions, crystallization, superconductivity, itinerant magnetism etc. A common feature of the mentioned phenomena is that they are cooperative effects produced by strongly correlated fermion systems at sufficiently low temperatures. Despite numerous efforts, neither does a satisfactory theory of such systems exist nor is our understanding ofthesimplestmodelsinventeddeepenough [1] (an exception are mean-field models, like the BCS one). The most representative and most often explored models in the field are: the Hubbard model [2], the periodic Anderson model [1,3] and the Falicov— Kimball model [4]. Traditional approximate techniques when applied to these models yield results which cannot be trusted at least in some respects
ball model is defined by the following hamiltonian:
[1,5,61, since the results obtained are often contradictory. Apparently a number of rigorous results which could serve as reference points is strongly desirable. Quite recently, it has been demonstrated that the one-band, spinless version of the Falico—Kimball model is rather well suited for investigations. Brandt and Schmidt [7] and Kennedy and Lieb [8] obtamed several rigorous results for this model. The one-band, spinless version of the Falico—Kim-
The model (1) has several interesting interpretations. Let us mention a few of them. First, as indicated in refs. [1,8], it can be thought of as a model of a solid composed of electrons and ions, hence as a model of a crystallization. Second, it is a model of valence-change transitions and mixed-valence states in rare-earth compounds [1]. In this case the moving particles play the role of band s-electrons, the localized ones stand for f-electrons and E fixes the
H=
x,yeA
~
+E ~ W(x)
+2U~ n~W(x) .
(1)
The hamiltonian (1) describes a system composed of two sorts of Fermi particles on a lattice A: moving ones (we call them electrons) and localized ones (we call them ions). The first term of (1) represents the kinetic energy of electrons, c~is an annihilation operator of an electron at a sitex of A. The second term stands for a repulsion (U>0) or an attraction (U<0) of the electrons with the ions, ~ and W(x) denotes an occupation number operator of he ions at a site x (which can be regarded as a classical field assuming the value 1 if the site x is occupied and the value 0 if it is empty). The third term stands for an energy contribution due to a filling of the systern by N1 ions (N, ~xeA W( x)).
0375-9601/89/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
319
Volume 134, number 5
PHYSICS LETTERS A
position of a sharp f-electron energy level. Third, the model (1) can be regarded as an approximation to the Hubbard model. Then the two sorts of particles represent electrons with different spin, E = 0 [81. The rigorous results of refs. [7,8] can be summarized as follows. Let us assume, for definiteness, that A is a square lattice, the matrix elements t~.of the kinetic energy are equal to 1 when x, y are nearest neighbours and vanish otherwise (without any loss of generality t can be chosen positive) and U> 0. Under the conditions stated, if the total number Ne of electrons and the total number N~of ions are equal to ~ Al (lAl is the number of sites in A), then the system has two ground states. In the first state the ions occupy the even sublattice of A and in the second state they occupy the odd one, that is, in the ground state the ion configuration is periodic, a chessboard-like crystal [7,8]. The ground state is insulating [8]. These properties hold for periodic and free boundary conditions, they depend neither on the size and the dimension of A nor on the value of the unique parameter U/I of the system [8] (no metal— insulator transition). Moreover, the long-range order displayed in the ground state is preserved at sufficiently low temperatures. The system considered can be viewed as a classical system on a lattice (Ising-like configurations of the ions) with a complicated, many-body and temperature-dependent interaction. Usually it is expected that such an interaction can be well approximated by a pair interaction that is associated to it. For the model (1) the latter interaction has been determined by Litvin and Priezzhev [91. The complicated pair potential has been investigated numerically and ground state configurations ofthe ions have been found for a number of densities of the electrons and the ions. In this Letter we present the main results of our investigations of ground states of the model (1). We are interested in two problems primarily: (i) What sorts of periodic configurations (crystals) of the ions arise beyond the symmetry point Ne = N, = I Al, specifically for N, = I Al and Ne = IA I / 2~lwith n = 1, 2, (the reason of such a choice shall become clear later)? Contrary to ref. [9] we investigate the full potential between the ions. Therefore we are interested in comparing our results with corresponding predic...
320
9 January 1989
tionsof ref. [9] which are based on the pair potential. (ii) What is the character of the ground states beyond the symmetry point, metallic or insulating? In the literature on the subject it is ascertained that if Ne I Al —Ni, then the system should be a metal (see for instance ref. [11). Does there exist a metal—insulator transition driyen by the unique parameter UI 1 of the model (1)? For several reasons (see below and ref. [8]) it is convenient to introduce a new field s(x) = 2 W(x) 1. Then the operator —
HEN1 =
—
~
UNe
(2)
t~,c~c~+U ~ n~s(x) ,
x.yeA
xeA .
for a fixed configuration of the ions is the secondquantized version of the single particle hamiltonian .
.
h(S) = T+ US
.
.
(3)
where T is the matrix with elements ~ and S is a diagonal matrix whose matrix elements are s(x) = ±1. In this manner studies of the model (1) can be reduced to those of the spectrum of (3) for different configurations of the ions. It is this peculiarity of the model which makes it more suitable for investigations than the other above mentioned models. The condition N, = IA l~to which we always stick, is equivalent to Tr S=0 and as a result the spectrum of h(S) is symmetric with respect to zero (this property helps to control the results of numerical diagonalizations of h(S)). In our search for ground state configurations of the ions we restrict ourselves to periodic configurations only (in statistical mechanics there is a conjecture that, under some regularity conditions on the potential, non-periodic equilibrium states do not exist in two-dimensional systems [101). Thus since we investigate finite systems it is natural to impose penodic boundary conditions on the lattice A. To single out configurations of the ions that are candidates for the ground state configurations we divide A into 2~z, n = 1, 2, square sublattices. For each n we consider a set of the configurations that are obtained according to the following rules: (1) each one of the sublattices can be either empty or completely filled, (2) one half of the sublattices has to be empty ...,
Volume 134, number 5
PHYSICS LETTERS A
while the other half has to be completely occupied (to satisfy TrS=0). Of course the spectra of h(S) corresponding to configurations that are related by the symmetry transformations of the model are the same. For n = 1 there is only one equivalence class of the ion configurations (displayed in fig. 1 a). The corresponding spectrum can be found analytically. If Ne= ~ Al (the half-filled band), this spectrum gives the lowest energy of the Fermi sea of the one particle system defined by h(S) for any U/I, in agreement with ref. [8]. At the Fermi level there is a gap of width 2 U, hence the system is an insulator for any UI t. Also the case n = 2, where there are two equivalence classes (one of them coincides with that for n = 1 and the new one is displayed in fig. lb) can be solved analytically. By elementary inequalities between the spectra of the n = 1 and n = 2 cases it can be proven that if in the thermodynamic limit the
a
9 January 1989
density of electrons n~,= limA.~.NJ IA I = ~ then the Fermi sea energy of the new n=2 class is lower than that of the n = 1 class. Therefore the latter configuration cannot be the ground state one for fle = This conclusion is in contradiction with the predictions based on the pair potential [9]. One of the results of ref. [9] states that if N, = IA I then the chessboardlike configuration (fig. Ia) is the ground state one for any density of electrons. Thus contrary to cornmon expectations the many-body character of the interaction between the ions cannot be neglected. The cases n = 3 and n = 4 have been studied numerically only. If n = 3 (8 sublattices) there are 70 configurations which are admitted by the condition TrS=0. They split into 6 equivalence classes. We have calculated the spectra of all these configurations and also of several different configurations that are admissible in the 16 sublattices case (in the latter case the total number of admissible configurations ~.
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Fig. 1. Ground state configurations of the ions: (a) N, = A, (b) N,=~(A~,(c) N,=IIA! and U/t>a, (d) N,=~AI, (e) N, = A and U/t
321
Volume 134, numberS
PHYSICS LETTERS A
amounts to 12870). Then the Fermi sea energies of these spectra were compared. It follows from our analysis that the ground state properties of the considered system depend strongly on the band filling and that in contradistinction to the half-filled band case they may depend on the value of U/i. Specifically we have found that if Ne IAI/2~with n=l, 2, 4 then the ground state configurations of the ions are those displayed in figs. Ia, lb and ld respectively, that is, those corresponding to 2’~,n = 1, 2, 4, sublattices. As mentioned above for n = 1 the system is known to be an insulator. For n = 2 and n = 4 it turns out to be a metal. In the former case we have been able to prove this property in the thermodynamic limit using the exact spectrum. No dependence on the value of U/i has been observed. However, the size of the system could not be too small. The case n=3 appeared to be more involved. For Ne = IA I 2~and for U/i larger than some critical value a, 0< a < 1, the Fermi sea energy is minimized by the configuration displayed in fig. ic which can be constructed if A is splitted at least into 2~sublattices. There is a gap on the Fermi level in the corresponding spectrum. This gap grows when U/i increases and tends to a limit whose estimated value amounts to 2. On the other hand if U/I approaches a from above the gap tends to zero. Simultaneously the Fermi sea energy (NeflAI of another configuration, which is displayed in fig. le, approaches the energy of the considered one and for
322
9 January 1989
sufficiently small U/I it becomes the ground state configuration. If U/i is below ci both configurations have no gap on the Fermi level. Thus in the n = 3 case a metal—insulator transition takes place. Most likely it is accompanied by a rearrangement of the ion configuration. The results obtained incline us to the hypothesis that when Ne IAl/2~,with n odd, the system is an insulator for U/i greaten than some critical value and it is a metal for U/i smaller than this value. On the other hand if n is even then the system is metallic irrespective ofthe value of U/ t. More details and additional results will be published elsewhere [11]. References [I] DI. Khomskii, in: Quantum theory of solids, ed. I.M. Lifshits (Mir, Moscow, i982). [2] J. Hubbard, Proc. R. Soc. A 276 (1963) 238. [31P.W. Anderson, Phys. Rev. 124 (1961) 41. [4] L.M. Falicov and J.C. Kimball, Phys. Rev. Lelt. 22 (1969) 997. [5 ID. Vollhardt, Rev. Mod. Phys. 56(1984)99. [6] G. Czycholl, Phys. Rep. 143 (1986) 277. [7]U. Brandt and R. Schmidt, Z. Phys. B 63 (1986) 45. [8] T. Kennedy amd E.H. Lieb, Physica A 138 (1986) 320. [9] A.A. Litvin and V.B. Priezzhev, J1NR preprint P17-87-635, Dubna (1987). 110] R.L. Dobroushin and S.B. Shlosman, Preprint, Institute for Problems ofInformation Transmission, Moscow (1984) [in Russian]. [111 J. Jçdrzejewski, J. Lach and R. Ly±wa,Physica A, to be published.