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Ground-state phase diagram of the generalized Falicov–Kimball model
ARTICLE IN PRESS
Physica B 378–380 (2006) 306–307 www.elsevier.com/locate/physb
Ground-state phase diagram of the generalized Falicov–Kimball model Hana Cˇencˇarikova´, Pavol Farkasˇ ovsky´ Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 043 53 Kosˇice, Slovakia
Abstract The extrapolation of small-cluster exact-diagonalization calculations is used to study the ground-state phase diagram of the spin-onehalf Falicov–Kimball model (FKM) extended by the spin-dependent on-site interaction between localized (f) and itinerant (d) electrons. Both the magnetic and charge ordering are analysed as functions of the spin-dependent on-site interaction (J) and the total number of itinerant electrons (N d ) at selected values of U (the spin-independent interaction between the localized and itinerant electrons) and N f (the total number of f-electrons). It is shown that the spin-dependent interaction (for N f ¼ L) stabilizes the ferromagnetic (FE) and ferrimagnetic (FI) state, while the stability region of antiferromagnetic (AF) phase is gradually reduced. The precisely opposite effect on the stability of FE, FI and AF phases has a reduction of N f . Moreover, the strong coupling between the f and d-electron subsystems is found for both N f ¼ L as well as N f oL. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Lp; 71.27.+a; 71.28.+d; 71.30.+h Keywords: Charge order; Magnetic order; Falicov–Kimball model
Recently Lemanski [1] proposed a simple model for a description of charge and magnetic order formation in systems having both localized and itinerant electrons. This model is based on a generalization of the spin-one-half Falicov–Kimball model (FKM) [2] with an anisotropic, spin-dependent local interaction that couples the localized and itinerant subsystems. The model Hamiltonian is X X X þ H¼ tij d þ fþ fþ is d js þ E is f is þ U is f is d is0 d is0 ijs
þJ
X
is
ðf þ is f is
iss0 þ fþ is f is Þd is d is ,
ð1Þ
is
where f þ is , f is are the creation and annihilation operators for an electron of spin s ¼"; # in the localized state at lattice site i and d þ is , d is are the creation and annihilation operators of the itinerant electrons in the d-band Wannier state at site i. The first term of (1) is the kinetic energy corresponding to quantum-mechanical hopping of d-electrons between sites i and j. The second term stands for the f-electrons Corresponding author. Tel.: +421 55 6336320; fax: +421 55 6336292.
E-mail address: [email protected] (P. Farkasˇ ovsky´). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.112
whose sharp energy level is E. The third term represents the on-site Coulomb interaction P between d-electrons with density nd ¼ N d =L ¼ ð1=LÞ Pis d þ is d is and f-electrons with density nf ¼ N f =L ¼ ð1=LÞ is f þ is f is , where L is the number of lattice sites. The last term is the spin-dependent (of the Ising type) local interaction between d and f electrons that reflects the Hund’s rule force. Since the felectron density operator f þ is f is commutes with the Hamiltonian (1), f þ f can be replaced by the classical is is variable wis ¼ 0; 1 and then the exact-diagonalization technique [3] can be used directly to study the ground states of (1). We start our study with the case N f ¼ L. In this case each lattice site is occupied by one (up or down) f-electron and thus only distributions over different spin configurations should be examined (we suppose that the interaction between f-electrons is infinite). In Fig. 1a, we summarize numerical results obtained by diagonalization calculations on the largest cluster that we were able to consider exactly ðL ¼ 32Þ in the form of N d J phase diagram. Various phases that enter intoP the phase diagram are classified according to S zf ¼ i w0i" w0i# (w0 is a configuration that minimizes the ground-state energy of (1)) and
ARTICLE IN PRESS H. Cˇencˇarikova´, P. Farkasˇovsky´ / Physica B 378–380 (2006) 306–307
32
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d−electrons L = 32 U=4
f−electrons L = 32 U=4
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Fig. 1. (a) The ground-state phase diagram of the generalized FKM in the N d J plane. (b) The ground-state phase diagram of the generalized FKM in the N f 2N d plane for d-electrons.
S zd ¼ N d" N d# . Three different phases depicted in Fig. 1a as (+) (the ferromagnetic (FE) phase), () (the ferrimagnetic (FI) phase) and () (the anti-ferromagnetic (AF) phase) are characterized by jS zf j ¼ N f ; jSzd j ¼ N d (+), 0ojS zf joN f , 0ojS zd joN d () and jSzf j ¼ 0, jSzd j ¼ 0 (). Comparing numerical results obtained for jSzf j and jSzd j (see inset) one can find nice correspondence between the magnetic phase diagrams of localized and itinerant subsystems. Indeed, with the exception of several isolated points at J ¼ 0:05, the corresponding FE, FI, and AF phases perfectly coincide over the remaining part of diagrams showing on the strong coupling between the magnetic subsystems of d and f electrons for nonzero values of J. It is interesting that the d-electrons are fully polarized in the region where the f-electrons are in the FE state. Instead, one would rather expect only a partial polarization of the d-electrons (FI phase) as they would have then more freedom to move through the lattice than the same number of fully polarized d-electrons (due to the Pauli principle). A simple justification follows directly from Eq. (1). Indeed, when the f-electrons are in the FE state then Eq. (1) yields for the up and down spin d-electrons the same electronic spectra that are shifted one against other by factor 2J. Thus up to some critical d-electron filling the fully polarized state of d-electrons has the lowest energy. Above this critical value the partially polarized state is stabilized. This confirms the supposition that the spin-
307
dependent interaction J plays an important role in description of ground-state properties of the generalized FKM. In general, the spin-dependent interaction stabilizes the FE and FI phases while the AF phase is gradually suppressed with increasing J. Moreover, we have observed that within the AF phase the ground states (for given N d ) do not change when J increases, while within the FI phase very strong effects of J on ground states were found. For example, the transition from the AF to FE phase at N f ¼ 12 realizes through the following sequence of FI phases: ½"2 #6 4 ! "14 ½#2 "2 4 #2 ! "26 ½#2 "2 1 #2 ! "30 #2 , where the lower index denotes the number of consecutive sites occupied by up or down spin f-electrons, or the number of repetitions of the block ½:::. In the AF region one can find different examples of periodic and non-periodic configurations, but the most interesting example represent configurations formed by antiparallel FE domains (½"n #n k ), that clearly demonstrate the cooperative effects of spin-dependent interaction J between the localized and itinerant electrons. Let us now briefly discuss to the case N f aL. In this case one has to minimize the ground state energy not only over all different spin configurations but also over all different felectron distributions. Due to we were able to investigate exactly only the clusters up to L ¼ 24 for N f aL. In Fig. 1b, we present the phase diagram of the generalized FKM in the N f 2N d plane obtained by exact-diagonalization calculations for L ¼ 24, J ¼ 0:5 and U ¼ 4. Again, the stability regions of FE, FI, and AF phase are marked by +, and . Here we display the numerical results only for the itinerant subsystem since the analysis of jSzd j and jS zf j showed that the FE, FI, and AF phases corresponding to localized and itinerant subsystems coincide also for N f oL. The most striking feature of the N f 2N d phase diagram is that with decreasing N f the AF phase is stabilized, while the FE and FI phases are suppressed. It is interesting that this effect is strongly asymmetric. For small N d the FE phase is stable only at N f ¼ L and once N f aL, the FE phase is replaced by FI or AF phases. The FI phase survives along the main diagonal in the narrow band and fully disappears near the point N f ¼ L=2; N d ¼ L. In the opposite limit (large N d ) the FE phase persists also for N f oL and with decreasing N f disappears gradually. The same behaviour exhibits also the FI phase. Acknowledgement This work was supported by the Slovak Grant Agency VEGA (no. 2/4060/04) and the Science and Technology Assistance Agency (APVT-20-021602). References [1] R. Lemanski, Phys. Rev. B 71 (2005) 035107. [2] L.M. Falicov, et al., Phys. Rev. Lett. 22 (1969) 997. [3] P. Farkasˇ ovsky´, Phys. Rev. B 20 (1995) 1507.
Physica B 378–380 (2006) 306–307 www.elsevier.com/locate/physb
Ground-state phase diagram of the generalized Falicov–Kimball model Hana Cˇencˇarikova´, Pavol Farkasˇ ovsky´ Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 043 53 Kosˇice, Slovakia
Abstract The extrapolation of small-cluster exact-diagonalization calculations is used to study the ground-state phase diagram of the spin-onehalf Falicov–Kimball model (FKM) extended by the spin-dependent on-site interaction between localized (f) and itinerant (d) electrons. Both the magnetic and charge ordering are analysed as functions of the spin-dependent on-site interaction (J) and the total number of itinerant electrons (N d ) at selected values of U (the spin-independent interaction between the localized and itinerant electrons) and N f (the total number of f-electrons). It is shown that the spin-dependent interaction (for N f ¼ L) stabilizes the ferromagnetic (FE) and ferrimagnetic (FI) state, while the stability region of antiferromagnetic (AF) phase is gradually reduced. The precisely opposite effect on the stability of FE, FI and AF phases has a reduction of N f . Moreover, the strong coupling between the f and d-electron subsystems is found for both N f ¼ L as well as N f oL. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Lp; 71.27.+a; 71.28.+d; 71.30.+h Keywords: Charge order; Magnetic order; Falicov–Kimball model
Recently Lemanski [1] proposed a simple model for a description of charge and magnetic order formation in systems having both localized and itinerant electrons. This model is based on a generalization of the spin-one-half Falicov–Kimball model (FKM) [2] with an anisotropic, spin-dependent local interaction that couples the localized and itinerant subsystems. The model Hamiltonian is X X X þ H¼ tij d þ fþ fþ is d js þ E is f is þ U is f is d is0 d is0 ijs
þJ
X
is
ðf þ is f is
iss0 þ fþ is f is Þd is d is ,
ð1Þ
is
where f þ is , f is are the creation and annihilation operators for an electron of spin s ¼"; # in the localized state at lattice site i and d þ is , d is are the creation and annihilation operators of the itinerant electrons in the d-band Wannier state at site i. The first term of (1) is the kinetic energy corresponding to quantum-mechanical hopping of d-electrons between sites i and j. The second term stands for the f-electrons Corresponding author. Tel.: +421 55 6336320; fax: +421 55 6336292.
E-mail address: [email protected] (P. Farkasˇ ovsky´). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.112
whose sharp energy level is E. The third term represents the on-site Coulomb interaction P between d-electrons with density nd ¼ N d =L ¼ ð1=LÞ Pis d þ is d is and f-electrons with density nf ¼ N f =L ¼ ð1=LÞ is f þ is f is , where L is the number of lattice sites. The last term is the spin-dependent (of the Ising type) local interaction between d and f electrons that reflects the Hund’s rule force. Since the felectron density operator f þ is f is commutes with the Hamiltonian (1), f þ f can be replaced by the classical is is variable wis ¼ 0; 1 and then the exact-diagonalization technique [3] can be used directly to study the ground states of (1). We start our study with the case N f ¼ L. In this case each lattice site is occupied by one (up or down) f-electron and thus only distributions over different spin configurations should be examined (we suppose that the interaction between f-electrons is infinite). In Fig. 1a, we summarize numerical results obtained by diagonalization calculations on the largest cluster that we were able to consider exactly ðL ¼ 32Þ in the form of N d J phase diagram. Various phases that enter intoP the phase diagram are classified according to S zf ¼ i w0i" w0i# (w0 is a configuration that minimizes the ground-state energy of (1)) and
ARTICLE IN PRESS H. Cˇencˇarikova´, P. Farkasˇovsky´ / Physica B 378–380 (2006) 306–307
32
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32
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24
d−electrons L = 32 U=4
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Fig. 1. (a) The ground-state phase diagram of the generalized FKM in the N d J plane. (b) The ground-state phase diagram of the generalized FKM in the N f 2N d plane for d-electrons.
S zd ¼ N d" N d# . Three different phases depicted in Fig. 1a as (+) (the ferromagnetic (FE) phase), () (the ferrimagnetic (FI) phase) and () (the anti-ferromagnetic (AF) phase) are characterized by jS zf j ¼ N f ; jSzd j ¼ N d (+), 0ojS zf joN f , 0ojS zd joN d () and jSzf j ¼ 0, jSzd j ¼ 0 (). Comparing numerical results obtained for jSzf j and jSzd j (see inset) one can find nice correspondence between the magnetic phase diagrams of localized and itinerant subsystems. Indeed, with the exception of several isolated points at J ¼ 0:05, the corresponding FE, FI, and AF phases perfectly coincide over the remaining part of diagrams showing on the strong coupling between the magnetic subsystems of d and f electrons for nonzero values of J. It is interesting that the d-electrons are fully polarized in the region where the f-electrons are in the FE state. Instead, one would rather expect only a partial polarization of the d-electrons (FI phase) as they would have then more freedom to move through the lattice than the same number of fully polarized d-electrons (due to the Pauli principle). A simple justification follows directly from Eq. (1). Indeed, when the f-electrons are in the FE state then Eq. (1) yields for the up and down spin d-electrons the same electronic spectra that are shifted one against other by factor 2J. Thus up to some critical d-electron filling the fully polarized state of d-electrons has the lowest energy. Above this critical value the partially polarized state is stabilized. This confirms the supposition that the spin-
307
dependent interaction J plays an important role in description of ground-state properties of the generalized FKM. In general, the spin-dependent interaction stabilizes the FE and FI phases while the AF phase is gradually suppressed with increasing J. Moreover, we have observed that within the AF phase the ground states (for given N d ) do not change when J increases, while within the FI phase very strong effects of J on ground states were found. For example, the transition from the AF to FE phase at N f ¼ 12 realizes through the following sequence of FI phases: ½"2 #6 4 ! "14 ½#2 "2 4 #2 ! "26 ½#2 "2 1 #2 ! "30 #2 , where the lower index denotes the number of consecutive sites occupied by up or down spin f-electrons, or the number of repetitions of the block ½:::. In the AF region one can find different examples of periodic and non-periodic configurations, but the most interesting example represent configurations formed by antiparallel FE domains (½"n #n k ), that clearly demonstrate the cooperative effects of spin-dependent interaction J between the localized and itinerant electrons. Let us now briefly discuss to the case N f aL. In this case one has to minimize the ground state energy not only over all different spin configurations but also over all different felectron distributions. Due to we were able to investigate exactly only the clusters up to L ¼ 24 for N f aL. In Fig. 1b, we present the phase diagram of the generalized FKM in the N f 2N d plane obtained by exact-diagonalization calculations for L ¼ 24, J ¼ 0:5 and U ¼ 4. Again, the stability regions of FE, FI, and AF phase are marked by +, and . Here we display the numerical results only for the itinerant subsystem since the analysis of jSzd j and jS zf j showed that the FE, FI, and AF phases corresponding to localized and itinerant subsystems coincide also for N f oL. The most striking feature of the N f 2N d phase diagram is that with decreasing N f the AF phase is stabilized, while the FE and FI phases are suppressed. It is interesting that this effect is strongly asymmetric. For small N d the FE phase is stable only at N f ¼ L and once N f aL, the FE phase is replaced by FI or AF phases. The FI phase survives along the main diagonal in the narrow band and fully disappears near the point N f ¼ L=2; N d ¼ L. In the opposite limit (large N d ) the FE phase persists also for N f oL and with decreasing N f disappears gradually. The same behaviour exhibits also the FI phase. Acknowledgement This work was supported by the Slovak Grant Agency VEGA (no. 2/4060/04) and the Science and Technology Assistance Agency (APVT-20-021602). References [1] R. Lemanski, Phys. Rev. B 71 (2005) 035107. [2] L.M. Falicov, et al., Phys. Rev. Lett. 22 (1969) 997. [3] P. Farkasˇ ovsky´, Phys. Rev. B 20 (1995) 1507.