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Open systems in contact with a ferromagnetic Fermi sea: quantum particle-number fluctuations
19December1994 PHYSICS LETTERS A ELSEVIER
Physics Letters A 196 (1994) 87-96
Open systems in contact with a ferromagnetic Fernti sea: quantum particle-number fluctuations Changfeng Chen Department of Physics, University of Nevada, Las Vegas, NV 89154, USA Received 25 August 1994; revised manuscript received 28 October 1994; accepted for publication 31 October 1994 Communicated by L.J. Sham
Abstract
We study the effects of the particle-number fluctuation in small fermionic quantum systems in contact with a completely ferromagnetic Fermi sea in an exact diagonalization approach based on a formalism previously developed by Artacho and Falicov [Phys. Rev. B 47 (1993) 1190]. A symmetry associated with the particle-number fluctuation is identified and used, together with other symmetries of the system, in the projection of the eigenstates and the block diagonalization of the Hamiltonian matrix. A two-site Hubbard model is used as an example to illustrate the implementation of the theory. Physical consequences of the quantum particle-number fluctuation and further applications of the theoretical approach are discussed.
1. Introduction
The small-cluster approach, used in the quantum Monte Carlo simulation [ 1 ], the exact diagonalization [ 2 ], and other schemes, has become one of the most powerful techniques in solving many-body problems. Compared with perturbative and variational methods, the small-cluster approach has the distinct advantage in being able to provide exact, albeit numerical, solutions to many-body problems. It is particularly useful in describing properties dominated by uniform and/or short-range strong correlations. However, it does have a limitation in that only very small systems can be studied in this approach, due to the rapid growth of the number of many-body states with the size of the system. On the other hand, the small-cluster approach is a natural choice in the study of physically small systems, which are isolated or, as in most cases in condensed matter physics, in contact with its environment. In many cases, small systems are used with Elsevier Science B.V.
SSDI0375-9601 (94)00896-5
or without periodic boundary conditions to model a macroscopic system (sampling some high symmetry points in the Brillouin zone for periodic systems). It has proved to be successful in describing "local" quantities of strongly correlated model systems [3,4] and real materials [5,6]. Finite-size scaling is often applied to the extent allowed by available computational power. Meanwhile, various "modified boundary condition" schemes have been developed [7] to extract information on properties that are relevant in the thermodynamic limit. In general, the proper choice and treatment of boundary conditions is of crucial importance in the application of the small-cluster approach and the interpretation of the calculated results. However, in almost all previous small-cluster calculations, there is one very important aspect that is not considered, i.e., the quantum particle-number fluctuation. In other words, calculations are performed for a fixed number of particles in the system, and then they are repeated
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C, Chen / Physics Letters A 196 (1994) 87-96
for another fixed number of particles, and so on. There is no particle exchange between the system and the environment to which it is connected. Some previous papers have addressed this issue in perturbative [8] and variational [9] approaches with applications to model systems. More recently, a general thermodynamic description of elementary quanta in open systems [10] has been proposed. A set of constraints, in particular the chemical potential of the "environment" has been imposed to enforce the influence the system. Nevertheless, the formalism does not explicitly allow the exchange of particles between the system and its environment. In the small-cluster approach, some "pseudo grand canonical ensemble" schemes have been proposed [ 11] to connect the results with different fixed-particle-number calculations. However, those are not true quantum mechanical treatments. Recently, Artacho and Falicov (AF) proposed [12] a new, fully quantum mechanical scheme to treat the particle-number fluctuation in small systems. It has been applied to several noninteracting and interacting electron systems and has been demonstrated to be a powerful and accurate method. In this work, we adopt the approach proposed by AF and extend it to the case of small clusters in contact with a completely ferromagnetic Fermi sea. Systems in contact with fully polarized ferromagnetic environments have been the subject of some recent experimental studies [ 13]. We attempt to construct a manybody theoretical scheme that later can be easily applied to large-scale numerical studies of such systems. An application to a two-site Hubbard cluster will be presented, with the intention of illustrating features of the algorithms rather than giving a detailed study of the model systems. There is a symmetry associated with the particle-number fluctuation in the ferromagnetic environment. This symmetry will prove to be very useful in reducing the size of the Hilbert space when dealing with larger systems. This is a very interesting case because a wide range of interacting electron systems can be described by the Hubbard model and its variants; also itinerant ferromagnetic systems and/or environment, to which impurities and adsorbates are connected, have received considerable attention for fundamental physics and practical applications associated with them. However, it should be emphasized that the present formalism is not restricted in any way to the Hubbard model only, as will be discussed below.
The proposed scheme can be used either on isolated systems (but connected to the environment or reservoir) or on extended systems with proper boundary conditions, particularly low-dimensional systems with broken spatial symmetries. Applications can be found in magnetic surfaces, interfaces, and multilayers, impurities/defects in magnetic materials, and adsorbates or artificially fabricated nanostructures on magnetic surfaces. In this contribution, we will focus on the development of the formalism, with an illustrative application to a two-site Hubbard model in contact with a ferromagnetic Fermi sea. The problem can be solved analytically so that the theoretical procedure involved can be fully demonstrated. Symmetries inherent in the system will be discussed in detail; their application to the block diagonalization of the Hamiltonian matrix will be presented. The scheme can be extended lor larger and more complicated/realistic systems in a straightforward manner.
2. F o r m a l i s m
First, we give a brief review of the formalism proposed by Artacho and Falicov [ 12]. The exact Hamiltonian for a fermion c o m p l e x consisting of a small system connected to a reservoir has the general form H = H~ + Hr ÷ Hint.
(2.1)
Here the three terms on the r.h.s, include operators that refer to the system, the reservoir, and the interactions and transfer of particles between the system and the reservoir. The aim here is to restrict the formalism to the system only and replace the reservoir with a simple set of outside parameters while guaranteeing fermionic consistency, i.e., the states of the reservoir must retain fermion character: they must satisfy antisymmetry properties with the fermionic states of the system and among themselves. To achieve this goal, an approximation is proposed to project the Hilbert space of the states in the reservoir onto the much smaller one of the system with a one-to-one correspondence between the eigenstates of Hs and the relevant (retained) states of H of the complex. The approximation essentially means that the transfer to and from the reservoir takes place only in the neighborhood of the Fermi level, which is reasonable for a "large and almost shapeless" reservoir.
c Chen / Physics Letters A 196 (1994) 87-96 With the above mapping, Hr and Hint in Eq. (2.1) can now be replaced by an effective Hamiltonian Heft=
Z(T;o.C~o.Rio--~-Tio-e[o.Cio-),
(2.2)
i,o
where c[~ (ci~) are creation and annihilation operators within the system, R ~ (Ri~r) are the operators that connect the states in the system to the corresponding states in the reservoir. The system-reservoir "hopping" matrix elements ri~ are taken here to be parameters that define a priori the properties of the complex, in particular the exchange of particles between the system and the reservoir. With this approximation, an effective Hamiltonian for the open system reads Hos = H~ + Heft.
(2.3)
The average value of any observable has the form (A) = T r ( A p ) ,
(2.4)
89
by the Fermi-Dirac statistics. These eigenstates are labeled by various quantum numbers, such as energy, momentum, and spin. The proposed projection scheme selects states with E = Ef and k = kf, and replaces the "bare" system-reservoir hopping matrix elements with effective ones. When the number of exchanged particles is small compared to that in the reservoir, this is a reasonable approximation. This leaves the spin index as the only remaining dynamical variable. In a ferromagnetic reservoir, all the spins are aligned in a single well-defined direction. In this case, the operators R / R t lose the dynamical degree of freedom and become "slave-fermion operators", whose only function is to ensure the already enforced (as outlined above) fermionic order. Then the effective Hamiltonian (2.2) can be further effectively written as i H e tf =
Z(T;C]o. -~ Tio-Cio-),
(2.7)
i,o"
i.e., without explicitly including the R operators. This Hamiltonian can be rewritten as
with the density matrix p given by
Heft = ~-~ ( Jl~c~8 + aiacia), p = exp[-(nos - tzNs)/kuT]/Z,
(2.5)
where Z = Tr{exp[ - (nos - # N s ) / k B T ] }
(2.6)
and Ns is the number of particles in the system and /x the chemical potential. At zero temperature, (A) in Eq. (2.4) reduces to the expectation value of A evaluated in the ground state of Ho~ - / x N s . Now the open system has a Hilbert space of the same size as that of the original system and the normal procedure in the small-cluster approach can be applied. Quantum particle-number fluctuations are described by the parameters/, and r. In this paper we extend the AF formalism to a special yet, at the same time, quite general and interesting case, i.e., the case of a completely ferromagnetic reservoir. Many physical systems can be described by this formalism, as discussed in the Introduction. We identify a symmetry associated with the particle-number fluctuation in this case. Its application to reducing the size of the Hamiltonian matrix will be illustrated in the next section. The key in the AF formalism is the recognition of the order of the eigenstates in the reservoir imposed
(2.8)
i
where 6 indicates a well-defined direction of the spin in the reservoir, i.e., the direction of the magnetization, which can be explicitly related to the "up" and "down" z-component states ] 1") and [ 1) as 16) = cos(½0)] T ) + sin(10) e-ix ] ~),
(2.9)
with 0 ~< 0 ~< rr and 0 ~< X ~< 2rr. By varying 0 and rr, one can align the spin in any desired direction. In this representation, the system-reservoir "hopping" matrix elements in the Hamiltonian (2.7) can be written as
riT = Ai cos(½0),
(2.10)
ri+ = hi sin(½0) e -ix,
(2.11)
In Ref. [ 121, this case is discarded because the effective Hamiltonian introduces artificial broken-spin (ferromagnetic) symmetry in the reservoir. In the present work, it is the ferromagnetic reservoir that we are interested in. Here the particle exchange between the system and the reservoir occurs only in a single spin direction defined by the magnetization of the reservoir. Notice, however, the spin states in the system may or may not align with the direction defined by the reservoir. In the latter case the spin states may be projected onto this direction as will be shown in the text.
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C. Chen / Physics Letters A 196 (1994) 8 7 - 9 6
where Ai is the effective "hopping" matrix element between the system and the reservoir magnetized in the 18) direction. Due to the particle-number fluctuation, both the total number of particles and the total spin of the system are no longer good quantum numbers. The loss of these symmetries is partially compensated by a new symmetry defined by the operator
Q = N - 2S(O,x),
S( O, x ) = ~-~, ( Six sin O cos X + Siy sin O sin X i
(2.13
where
Si,. = ~1 Z(C~TCi~ + c]LCiT),
2.14
i
Si,. = ( 1 / 2 i ) Z ( c ] r c i ~ - c~tciT),
2.15)
i
s,:
= 1
Z(cJF,T
(2161
i
It is straightlorward to prove that [ Ho~, Q] = 0 ,
N
0
I
2
3
4
Np
0 0
0 .'~
0, 2 4, -~
0. 2 6. 4
0, 2..4 8, 6. 4
Np
(2.12)
where N is the total particle number operator and the total spin operator is defined as
+ Se cos 0),
Table 1 All possible eigenvalues of the operator Q (Np and 19p) for a two-site Hubbard model with zero to four (N) electrons
(2.17)
where Hos is the sum of the system Hamiltonian and the effective Hamiltonian (2.7), i.e., Q is a good quantum number. The physics associated with this new symmetry is easy to understand: when the exchange of particles only occurs for a single well-defined spin orientation, the changes in particle number and spin take place in a "synchronized" manner such that their difference remains the same in the fluctuation process. We notice that the particle states in the system can be projected onto the new spin axis 16) and I - 8 ) , even though the exchange of particles between the system and reservoir occurs only in the 16) direction. Once the states are expressed in this representation, the possible eigenvalues of the operator S(O, X) range tYom - N to N in the subspace with the fixed number of N particles. When the net spin of the system is aligned in the direction of 18), the eigenvalues of the operator Q are determined (see Eq. ( 2 . 1 2 ) ) by the number of paired particles (with spin 8 and - 3 ) Np in
the system; when the net spin of the system is aligned along I - 8), the eigenvalues of Q are given by /Vp = 2N - Np. Therefore, the eigenvalues of Q are always either zero or even integers. As an example, Table I lists all possible eigenvalues of Q for a two-site Hubbard model with up to four electrons. However, there is another system-size dependent constraint on the allowed eigenvalues of Q imposed by the Pauli principle. For example, in the two-site Hubbard model case, when N = 3, the allowed values of Q are 2 (Np) and 4 (Elp ) ; Q = 0 (No) and 6 ( ~'p ) are prohibited by the Pauli principle. For a larger system with three or more sites all values (0, 2, 4, 6) are allowed for the fourelectron system. This rule applies to general cases with a different number of sites and particles in the system. Since Q is a conserved quantity, states in the system with a different number of particles but with the same value of Q can mix; those with different values of Q cannot. States can be in general classified by Q, space symmetry, and any other symmetry the system may possess. It should be emphasized that the Q-symmetry is not particularly associated with the Hubbard model. As long as the Hamiltonian of the system commutes with the total particle number and total spin operators, Q will be a good quantum number. In other words, it is essentially determined by the property (e.g., ferromagnetism) of the reservoir. In the next section, we apply the above formalism to a two-site Hubbard model to illustrate the implementation of the theory and discuss some physical consequences.
3. Application to a two-site H u b b a r d model
We study the Hubbard model which is defined by the Hamiltonian, in the standard notation
C. Chen / Physics Letters A 196 (1994) 87- 96
i,j, tr -[- U ~
i,o"
nit ni~ ,
(3.1)
i
E(N,t) = E ( 4 - N , - t ) + ( N - 2 ) U + (2N - 4)e
simplify the derivation, but not lose generality 2. To be specific, we choose 0 = ½~" and X = 0. In this case, the effective Hamiltonian can be written as n e f f = ~ ~-~(c~T --[-c~[ -.}-CiT -]- cij.) i
where - t is the nearest-neighbor hopping integral, U the on-site Coulomb interaction,/~ the chemical potential, and E the orbital energy. In a closed system with a fixed number of particles e simply gives a uniform shift in energy and can be omitted. However, in the open system it must be explicitly included in the calculation. The total Hamiltonian of the open system now is the sum of Eqs. (3.1) and (2.7) with the system-reservoir "hopping" matrix elements given by Eqs. (2.10) and (2.11). Here we choose to study a two-site Hubbard model to illustrate the application of some symmetry projection procedures that one will encounter in more complicated situations. For this case, the summation of the site index in Eq. (3.1) runs from one to two only. For the purpose of comparison, we first solve the Hamiltonian (3.1) for 0 - 4 electrons in the c l o s e d system in the "standard" approach. There are in total 16 many-body electron states. For the two-site cluster, there is an inversion symmetry with even and odd parities, denoted as 2?+ and 2 - . The electron states are projected according to the particle-number, spin, and inversion symmetry into various subspaces; the Hamiltonian is solved in the subspaces since there is no nonzero matrix element between states with different symmetries. The energy equations with corresponding symmetry and degeneracy are obtained as shown in Table 2. There is a particle-hole symmetry in the energy equations, i.e.,
(3.2)
for N ~> 3. Notice that the orbital energy term has been explicitly included here. We now turn our attention to the case with the quantum particle-number fluctuation. We choose a special direction of the spin orientation 18) in the following calculation. We also assume that the system-reservoir "hopping" matrix elements for different sites are the same, i.e., the fluctuation is uniform. Doing so will
91
(3.3)
and the total spin operator (2.13) reduces to S(O, X ) = Sx.
(3.4)
So the good quantum number now is Q = N - 2Sx.
(3.5)
The 16 many-body electron states in the system first split according the inversion symmetry, with eight states in each subspace, as summarized in Table 3 in the "old" basis. For systems with more complicated structures electron states split according to irreducible representations of the space group [5]. Then these spatially symmetrized states are further projected according to the eigenvalues of the Q operator, as shown in Table 4. It is seen that only Q = 0, 2, and 4 are allowed for the two-site Hubbard model, although in principle Q = 6 and 8 are also possible for a fourelectron system. This is due to the constraint imposed by the Pauli principle as discussed above. There is no nonzero Hamiltonian matrix element between states with different symmetry indices. Therefore, the original 16 x 16 Hamiltonian matrix is now block diagonalized into four 2 × 2 and two 4 × 4 matrices, according to symmetries ~ . The resulting Hamiltonian matrices are shown in Tables 5 and 6. Direct diagonalization of these matrices provides all eigenvalues and eigenstates. Since the particle number in the system is not conserved, the average occupation number per site (N) or, equivalently the band filling in the case of extended systems, is measured by the grand canonical ensemble average (see Eq. (2.4)) and is controlled by the chemical potential #. For the case of (N) = 1, i.e., o n avera g e two electrons in the two-site cluster, the chemical 2 These assumptions are valid for the case of a single ferromagnetic Fermi sea. The energy spectrum does not depend on the direction of 16). If the two sites are in contact with two different ferromagnetic Fermi seas, as at the interface of two different ferromagnetic materials, the fluctuation at different sites must be treated differently.
C. Chen / Physics Letters A 196 (1994) 87-96
92
Table 2 The energy equations with the corresponding symmetry and degeneracy for the two-site Hubbard model; the left superscripts in the symmetry symbols indicate the spin degeneracy (2S + 1 ) with S being the total spin of the system Number of electrons
Symmetry
Energy equation
Degeneracy
N = 0 N=I
t X+ 2v+~ 2x .... 15' ~ 12"32'22'+ 2X-
E = 0 E - • -t=0 E e -'I'-/ = 0 E 2 - ( U + 4e) E + 4+ 2 + 2+U - 4t 2 = 0 E- 2e-U=0 E 2e = 0 E 3e-t-U=0
I ,~~' "~ I I 3 2
E -3e+t-U=0 E 4e-2U=0
2 1
N =2
N=3
N=4
12.+
Table 3 The 16 eigenstates in the old (closed system) basis projected according to particle number ( N ) and inversion symmetry: the vacuum state J0) contains no particle in the system Symmetry
Eigenstate
Symmetry
N = 0 N= I
I ~'+ 2X+
i/.t1 = 10) ~2 = (I/v/2)(C~T + c{r)]0>
22'-
N:2
I~'+
t//4 +,,
(I/v%(,m:+.r,.[t+,.+,tr,.~t)lO)
Eigeustute
g'lo = ( 1/x~-2)(Ctl - c~+)lO) ,.+.-
~bll = I. I/'~-)(C~fC~l - - c 'r21.~21 "'} )1())
'"
~,,e = ~Y, ¢~r )Ira
2X -
g't4 = (I/'f2-)(cl;-c21 - c2rcij )]0) / • .t .t ,t .t .t .t I//15 { I., V/2~)((II( 11(2[ -- {2t(2~.(i] )] 0} t t t t t t tO e, I/~"2-)(cIt¢'ll c25 -- C2~C21 eli )10)
t
9
N:4
~v+
~8
.t
.t
.t
,t
.t
1
t
t
,
,t c t .t .t ( e l i 11~2T(21)10)
Table 4 The 16 eigenstates in the new (open system) basis projected according to the good quantum number Q and expressed in terms of states in the old basis (see Table 3); notice that states with symmetries 2 .+ 0/n to g'8) and X - (~,) to ~Pt6 ) do not mix as they form new states
Eigenstate
4,2 = ( l/x/2~) ~02 ~3 = ( I / ' / 2 ) ( ~ 2 4)4 = 64 45 =¢,s q56 = ( 1/~v~) (,I//6 4,7 = ( 1/x/2) (¢6 r~8 = ~8
+ 03) -63)
-k //J7 ) - 07 )
Q
Eigenstatc
Q
0 2 2 2 2 4 4
4, m ~,l ~2 ,b~3 4) 14 4hs 616
2 e 2 0 4 2 4
= ( I/,/2_) (J,9 - ,P.+ ) =4,~, = ¢ 1/,/5-')C~2 - ~,~ I = 4(6~2 + 6 ~ 3 + ~--6,4 / = ~ (/ill2 Jr- ,I//13 -- "v/2-ff;14 } = ( 1/C2-) (¢,Ls + g,,~ ) = (I/x/2-)(~15 - &lr)
C. Chen / Physics Letters A 196 (1994) 87-96
93
Table 5 The Hamiltonian matrices for states with symmetries 27~; the symbol ~ represents E - #
- t
0 2)t 2,~ g- - t
-v~a
-v"5,~
-x/2A 2~+U -v~,~ -2t 0 v'2,~
0
-2t x/~A 2~ -x/2a -x/2,~ 3 ~ + t + U
3~'+t+U -2,~
-23. 4~ + 2U
Table 6 The Hamiltonian matrices for states with symmetries 2?a ; the symbol ~ represents • - / ~
g+ t g + t -2a -2,~ 2~
-V'2A
0
0
- v ' S a 2~ + u
0
v'5,~
0 0
0 v/2h
2g - x/2,~ -x/2A 3~-t+U
potential is/z = • ~ t + ½U for states with .,~+ symmetries. In the following we examine the ground-state energy and the magnetization of the system with IN) = 1 to study some physical consequences of quantum particle-number fluctuation and also to show a practical way of determining the system-reservoir "hopping" parameter from known properties of the system. We first study the ground-state energy of the systems as a function of interaction parameters. In the case of the closed system with two electrons (also two holes) in the system, it is given by E = 2e + ½U - ½x,/U 2 + 16t 2.
(3.6)
The ground state is of symmetry i f + (see Table 2), i.e., it is always a spin singlet state. This is in agreement with the conjecture [4] that there is a two-hole extension to Nagaoka's theorem [ 14] which yields a spin-singlet ground state. However, we will see below that this will be changed, as expected, in general for open systems, where the "boundary conditions" imposed by the surrounding environment will strongly influence or even dominate the properties of the system. In the strong interaction limit, i.e., as U approaches to infinity, the energy reduces to E ~ = 2e. In the case of an open system the ground-state energy is determined by numerically solving the 4 x 4 matrices in Tables 5 and 6. In the strong interaction limit,
2g-2,~ -2,~ 3 , ~ - t + U
the energy approaches to 2(e - tz). Here we take t to be positive; the ground state is then of symmetry ~ - . Fig. 1 shows the energy of the system relative to its value at infinite U for both the closed and the open system as a function of U/t. It is seen that the energies for the open system have larger slopes, i.e., they are more sensitive to the interaction term. The physics involved is easy to understand: local fluctuations decrease the band energy but increase the interaction energy; the compromise of the two opposite effects yield the calculated results. It has been shown [ 12] that the full quantum mechanical treatment of the system-reservoir hopping always yields lower total energy, thus a better description, for the system than a mean-field treatment. We now turn to the behavior of the magnitude of the magnetic moment of the system defined by
m = Z (C[~rS°'cr'Ci°-')'
(3.7)
i , o - , o -I
where S ~ , are the standard Pauli matrices and ( ) means the ground-state average. As shown above the ground state of the closed system with two electrons in the cluster is always a spin singlet, i.e., there is no net magnetic moment. When the system is in contact with a ferromagnetic Fermi sea, it is expected to pick up a finite magnetic moment. We express the ground
C. Chen / Physics Letters A 196 (1994) 87-96
94
1
I
!
l
-0.5 -1
(E
-1.5 -
E~)/t -2 -2.5 -3 -3.5
1
I
I
I
2
4
6
8
10
U/t Fig. I . The total energy of the closed and open two-site Hubbard cluster as a function of U/t for average occupation number per site
0.3
i
I
I
'l
0.25
0.2
rt~
/s-
/
0.15 0.1 0.05
0
~
/
/
~
,, , ,
/ *
0.2
/
'
~"-
1,
0.4
/
~
I
I
0.6
0.8
,\/t Fig. 2. The magnetic moment in the ground state of the open system as a function of A/t for U/t -= 0.0, 2.0.4.0, 6.0, 8.0 and 10.0 (top to bottom curves). The average occupation number per site (N) = I
state in terms of the combination of all the states with E4 symmetry as
m , = 0,
(3.10)
m= = 0.
(3.11)
6 'P = ~ alibi, l=3
(3.8)
where q51 are given in Table 4 and 0/~ are coefficients. The magnetization of the open system evaluated in the ground state is obtained as mx = (0/6) 2 - (0/3) 2,
(3.9)
It is seen that the system now may have non-zero magnetization along the x-direction. This particular magnetic "polarization" in the system is due to the special choice of the effective Hamiltonian (3.3) that connects the system and the reservoir. In general, all three components of the magnetization of the system may be non-zero when spin fluctuations in these di-
C. Chen / Physics Letters A 196 (1994)87-96 rections are allowed. However, the physics remains the same. Fig. 2 shows the magnitude of the groundstate magnetic moment as a function of the systemreservoir "hopping" matrix element for various values of the interaction strength. It is seen that the magnetization of the system rises smoothly with increasing system-reservoir exchange for all values of the interaction strength. However, one notices that the value of induced magnetization in the system decreases drastically as the interaction becomes strong. This is due to the more localized nature of the particles in the system for large values of U/t. The properties of the system become less dependent on the "boundary conditions" as the value of U/t grows. The above calculation of the magnetization of the open system as a function of the system-reservoir "hopping" parameter provides a practical way to determine the value of A in the Hamiltonian for real materials. For example, if the induced ground-state magnetization of a system, which is in contact with a ferromagnetic reservoir but otherwise is less or non-magnetic, is known, then ,~ can be obtained by fitting the calculated magnetization of the system to the known value. This scheme also works for extended systems such as magnetic surfaces where, however, the magnetization is intrinsic rather than induced; the task is to reproduce the correct ground-state magnetization. In the case of extended systems the parameter A may also be obtained by applying the open-cluster model to bulk material and fitting the result of open-cluster calculations to known properties, such as the density of states at the Fermi level, of the bulk material. These known properties may be the result of either bulk electronic structure calculations or experimental measurements. Application of the present formalism to more complicated/realistic systems is technically quite straightforward. We have recently developed a manybody approach [5,6] that uses and goes beyond the results of ab initio band structure calculations. Material-specific Hamiltonian parameters can be obtained. Interaction parameters beyond conventional local-density-approximation (LDA) results may be obtained from constrained LDA calculations [15] which take into consideration the correlation effects in the system. One crucial new ingredient here is the system-reservoir "hopping" matrix element, which may be determined by fitting the calculated results to known properties as discussed above.
95
In our previous work [5,6], surface effects have been studied by use of free-standing thin films or by including a substrate layer in the cluster and then considering only crystal-filed contribution from the rest of the substrate. While these schemes may be well justified for some cases, inclusion of quantum particle-number fluctuation should provide a better framework for such studies. More importantly, for physically small systems in contact with a reservoir, such as small adsorbed atomic/molecular clusters at transition-metal surfaces, the quantum particle-number fluctuation is expected to have much stronger effects on the properties of the systems and must be built-in from the beginning.
4. Summary In summary, we have studied the properties of small fermionic quantum systems in contact with a ferromagnetic Fermi sea. Quantum particle-number fluctuation in the system is explicitly included. We have identified a symmetry associated with the particlenumber fluctuation in a ferromagnetic environment. It is used, together with other symmetries of the system, in block diagonalization of the Hamiltonian matrix. Application to a two-site Hubbard model has been used as an example to illustrate the implementation of the theory. It is obvious that the present formalism has many applications in the study of magnetic surfaces, interfaces, and multilayers, as well as adsorbates in bulk and surface magnetic environment. The work reported in this paper has laid the foundation for its application to larger systems in numerical form. One crucial task there, especially for real materials, will be to properly determine the effective system-reservoir "hopping" parameter, which may be obtained by fitting the calculated results to known properties, such as the ground-state magnetization of the system. It may also be determined through more laborious fitting of the results of open-cluster calculations to ab initio electronic structures of extended systems in some cases.
Acknowledgement The author acknowledges interesting discussions with Dr. X.R. Wang and Dr. T.K. Ng, and the hospi-
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C Chen/Physics LettersA 196 (1994) 87-96
tality of the Theory Group at the Physics Department, Hong Kong University of Science and Technology where part of this work was performed. This work was supported in part by the National Science Foundation under the EPSCoR program.
[81 [91 101 11]
References I l l M.H. Kalos, ed., NATO Advanced Study Institute Series C, Vol. 125. Monte Carlo methods in quantum problems ( Reidel, Dordrecht, 1984). 121 L.M. Falicov, in: Recent progress in many-body theories, Vol. 1, eds. A.J. Kallio, E. Pajanne and R.E Bishop (Plenum, New York, 1988) p. 275; J. Callaway, Physics B 149 11988) 17. 131 E. Dagotto, Int. J. Mod. Phys. B 5 ( 1991 ) 77, and references therein. 141 J. Freericks and L.M. Falicov, Phys. Rev. B 42 ( 19901 4960; 44 (1991) 1458. [5] Changfeng Chen, Int. J. Mod. Phys. B 5 (1991) 1147, and references therein. [ 6 ] Changfeng Chen, Phys. Rev. B 45 (1992) 13811 ; 48 ( 1993 ) 1318; J. Phys. Condens. Matter 4 (19921 9855. 17] R. Jullien and R.M. Martin, Phys. Rev. B 26 119821 6173: D.K. Campbell, J.T. Gammel and E.Y. Loh Jr., Int. J. Mod. Phys. B 3 (1989) 2131; D. Poilblanc, Phys. Rev. B 44 (19911 9562; R. Valenti, C. Gros, RJ. Hirschfeld and W. Stephan, Phys. Rev. B 44 (1991) 13203:
121 131
14l
15]
S.R. White and R.M. Noack+ Phys. Rev. Lett. 68 (1992) 3487; S.R. White, Phys. Rev. B 48 11993) 10345. C.M. Varma and Y. Yafet, Phys. Rev. B 13 119761 2950. K. Sch6nhammer, Phys. Rev. B 13 (19761 4336. A. Elci and D. Depatie, Phys. Len. A 176 (19931 393. A. Reich and L.M. Falicov, Phys. Rev. B 34 (1986) 6752: L. Milans del Bosch and L.M. Falicov, Phys. Rev. B 37 (1988) 6073. E. Artacho and L.M. Falicov, Phys. Rev. B 47 (1993) 1190. M.G. Samant et al., Phys. Rev. Lett. 72 (1994) I112; S. Bode et al., Phys. Rev. Lett. 72 ( 19941 1072: B.A. Gurney et al., Phys. Rev. Lett. 71 11994) 41123: O. Rader et al., Phys. Rev. Lett. 72 ( 19941 2247. Y. Nagaoka, Solid State Commun. 3 (1965) 409; D+J. Thouless, Proc. Phys. Soc. London 86 (19651 893: Y. Nagaoka, Phys. Rev. 147 (1966) 392; H. Tasaki, Phys. Rev. B 40 (1989) 9192. M.S. Hybertsen, M. Schltiter and N.E. Christensen, Phys. Rev. B 39 (1989) 9028: AK+ McMahan, R.M. Martin and S. Satpathy, Phys. Rev. B 38 (1989) 6650; A.K. McMahan, J.E Annett and R.M. Martin, Phys. Rev. B 42 (19901 6268; J.B. Grant and A.K. McMahan, Phys. Rev. Lett. 66 11991 ) 488: J. Zaanen, O. Jepsen, O. Gunnarsson, A.T. Paxton and O.K. Anderson, Physica C 153-155 119881 1636; M.S. Hybertsen, E.B. Stechel, W . M . C . Foulkes and M. Schliiter, Phys. Rev. B 45 (1992) 10032.
Physics Letters A 196 (1994) 87-96
Open systems in contact with a ferromagnetic Fernti sea: quantum particle-number fluctuations Changfeng Chen Department of Physics, University of Nevada, Las Vegas, NV 89154, USA Received 25 August 1994; revised manuscript received 28 October 1994; accepted for publication 31 October 1994 Communicated by L.J. Sham
Abstract
We study the effects of the particle-number fluctuation in small fermionic quantum systems in contact with a completely ferromagnetic Fermi sea in an exact diagonalization approach based on a formalism previously developed by Artacho and Falicov [Phys. Rev. B 47 (1993) 1190]. A symmetry associated with the particle-number fluctuation is identified and used, together with other symmetries of the system, in the projection of the eigenstates and the block diagonalization of the Hamiltonian matrix. A two-site Hubbard model is used as an example to illustrate the implementation of the theory. Physical consequences of the quantum particle-number fluctuation and further applications of the theoretical approach are discussed.
1. Introduction
The small-cluster approach, used in the quantum Monte Carlo simulation [ 1 ], the exact diagonalization [ 2 ], and other schemes, has become one of the most powerful techniques in solving many-body problems. Compared with perturbative and variational methods, the small-cluster approach has the distinct advantage in being able to provide exact, albeit numerical, solutions to many-body problems. It is particularly useful in describing properties dominated by uniform and/or short-range strong correlations. However, it does have a limitation in that only very small systems can be studied in this approach, due to the rapid growth of the number of many-body states with the size of the system. On the other hand, the small-cluster approach is a natural choice in the study of physically small systems, which are isolated or, as in most cases in condensed matter physics, in contact with its environment. In many cases, small systems are used with Elsevier Science B.V.
SSDI0375-9601 (94)00896-5
or without periodic boundary conditions to model a macroscopic system (sampling some high symmetry points in the Brillouin zone for periodic systems). It has proved to be successful in describing "local" quantities of strongly correlated model systems [3,4] and real materials [5,6]. Finite-size scaling is often applied to the extent allowed by available computational power. Meanwhile, various "modified boundary condition" schemes have been developed [7] to extract information on properties that are relevant in the thermodynamic limit. In general, the proper choice and treatment of boundary conditions is of crucial importance in the application of the small-cluster approach and the interpretation of the calculated results. However, in almost all previous small-cluster calculations, there is one very important aspect that is not considered, i.e., the quantum particle-number fluctuation. In other words, calculations are performed for a fixed number of particles in the system, and then they are repeated
88
C, Chen / Physics Letters A 196 (1994) 87-96
for another fixed number of particles, and so on. There is no particle exchange between the system and the environment to which it is connected. Some previous papers have addressed this issue in perturbative [8] and variational [9] approaches with applications to model systems. More recently, a general thermodynamic description of elementary quanta in open systems [10] has been proposed. A set of constraints, in particular the chemical potential of the "environment" has been imposed to enforce the influence the system. Nevertheless, the formalism does not explicitly allow the exchange of particles between the system and its environment. In the small-cluster approach, some "pseudo grand canonical ensemble" schemes have been proposed [ 11] to connect the results with different fixed-particle-number calculations. However, those are not true quantum mechanical treatments. Recently, Artacho and Falicov (AF) proposed [12] a new, fully quantum mechanical scheme to treat the particle-number fluctuation in small systems. It has been applied to several noninteracting and interacting electron systems and has been demonstrated to be a powerful and accurate method. In this work, we adopt the approach proposed by AF and extend it to the case of small clusters in contact with a completely ferromagnetic Fermi sea. Systems in contact with fully polarized ferromagnetic environments have been the subject of some recent experimental studies [ 13]. We attempt to construct a manybody theoretical scheme that later can be easily applied to large-scale numerical studies of such systems. An application to a two-site Hubbard cluster will be presented, with the intention of illustrating features of the algorithms rather than giving a detailed study of the model systems. There is a symmetry associated with the particle-number fluctuation in the ferromagnetic environment. This symmetry will prove to be very useful in reducing the size of the Hilbert space when dealing with larger systems. This is a very interesting case because a wide range of interacting electron systems can be described by the Hubbard model and its variants; also itinerant ferromagnetic systems and/or environment, to which impurities and adsorbates are connected, have received considerable attention for fundamental physics and practical applications associated with them. However, it should be emphasized that the present formalism is not restricted in any way to the Hubbard model only, as will be discussed below.
The proposed scheme can be used either on isolated systems (but connected to the environment or reservoir) or on extended systems with proper boundary conditions, particularly low-dimensional systems with broken spatial symmetries. Applications can be found in magnetic surfaces, interfaces, and multilayers, impurities/defects in magnetic materials, and adsorbates or artificially fabricated nanostructures on magnetic surfaces. In this contribution, we will focus on the development of the formalism, with an illustrative application to a two-site Hubbard model in contact with a ferromagnetic Fermi sea. The problem can be solved analytically so that the theoretical procedure involved can be fully demonstrated. Symmetries inherent in the system will be discussed in detail; their application to the block diagonalization of the Hamiltonian matrix will be presented. The scheme can be extended lor larger and more complicated/realistic systems in a straightforward manner.
2. F o r m a l i s m
First, we give a brief review of the formalism proposed by Artacho and Falicov [ 12]. The exact Hamiltonian for a fermion c o m p l e x consisting of a small system connected to a reservoir has the general form H = H~ + Hr ÷ Hint.
(2.1)
Here the three terms on the r.h.s, include operators that refer to the system, the reservoir, and the interactions and transfer of particles between the system and the reservoir. The aim here is to restrict the formalism to the system only and replace the reservoir with a simple set of outside parameters while guaranteeing fermionic consistency, i.e., the states of the reservoir must retain fermion character: they must satisfy antisymmetry properties with the fermionic states of the system and among themselves. To achieve this goal, an approximation is proposed to project the Hilbert space of the states in the reservoir onto the much smaller one of the system with a one-to-one correspondence between the eigenstates of Hs and the relevant (retained) states of H of the complex. The approximation essentially means that the transfer to and from the reservoir takes place only in the neighborhood of the Fermi level, which is reasonable for a "large and almost shapeless" reservoir.
c Chen / Physics Letters A 196 (1994) 87-96 With the above mapping, Hr and Hint in Eq. (2.1) can now be replaced by an effective Hamiltonian Heft=
Z(T;o.C~o.Rio--~-Tio-e[o.Cio-),
(2.2)
i,o
where c[~ (ci~) are creation and annihilation operators within the system, R ~ (Ri~r) are the operators that connect the states in the system to the corresponding states in the reservoir. The system-reservoir "hopping" matrix elements ri~ are taken here to be parameters that define a priori the properties of the complex, in particular the exchange of particles between the system and the reservoir. With this approximation, an effective Hamiltonian for the open system reads Hos = H~ + Heft.
(2.3)
The average value of any observable has the form (A) = T r ( A p ) ,
(2.4)
89
by the Fermi-Dirac statistics. These eigenstates are labeled by various quantum numbers, such as energy, momentum, and spin. The proposed projection scheme selects states with E = Ef and k = kf, and replaces the "bare" system-reservoir hopping matrix elements with effective ones. When the number of exchanged particles is small compared to that in the reservoir, this is a reasonable approximation. This leaves the spin index as the only remaining dynamical variable. In a ferromagnetic reservoir, all the spins are aligned in a single well-defined direction. In this case, the operators R / R t lose the dynamical degree of freedom and become "slave-fermion operators", whose only function is to ensure the already enforced (as outlined above) fermionic order. Then the effective Hamiltonian (2.2) can be further effectively written as i H e tf =
Z(T;C]o. -~ Tio-Cio-),
(2.7)
i,o"
i.e., without explicitly including the R operators. This Hamiltonian can be rewritten as
with the density matrix p given by
Heft = ~-~ ( Jl~c~8 + aiacia), p = exp[-(nos - tzNs)/kuT]/Z,
(2.5)
where Z = Tr{exp[ - (nos - # N s ) / k B T ] }
(2.6)
and Ns is the number of particles in the system and /x the chemical potential. At zero temperature, (A) in Eq. (2.4) reduces to the expectation value of A evaluated in the ground state of Ho~ - / x N s . Now the open system has a Hilbert space of the same size as that of the original system and the normal procedure in the small-cluster approach can be applied. Quantum particle-number fluctuations are described by the parameters/, and r. In this paper we extend the AF formalism to a special yet, at the same time, quite general and interesting case, i.e., the case of a completely ferromagnetic reservoir. Many physical systems can be described by this formalism, as discussed in the Introduction. We identify a symmetry associated with the particle-number fluctuation in this case. Its application to reducing the size of the Hamiltonian matrix will be illustrated in the next section. The key in the AF formalism is the recognition of the order of the eigenstates in the reservoir imposed
(2.8)
i
where 6 indicates a well-defined direction of the spin in the reservoir, i.e., the direction of the magnetization, which can be explicitly related to the "up" and "down" z-component states ] 1") and [ 1) as 16) = cos(½0)] T ) + sin(10) e-ix ] ~),
(2.9)
with 0 ~< 0 ~< rr and 0 ~< X ~< 2rr. By varying 0 and rr, one can align the spin in any desired direction. In this representation, the system-reservoir "hopping" matrix elements in the Hamiltonian (2.7) can be written as
riT = Ai cos(½0),
(2.10)
ri+ = hi sin(½0) e -ix,
(2.11)
In Ref. [ 121, this case is discarded because the effective Hamiltonian introduces artificial broken-spin (ferromagnetic) symmetry in the reservoir. In the present work, it is the ferromagnetic reservoir that we are interested in. Here the particle exchange between the system and the reservoir occurs only in a single spin direction defined by the magnetization of the reservoir. Notice, however, the spin states in the system may or may not align with the direction defined by the reservoir. In the latter case the spin states may be projected onto this direction as will be shown in the text.
90
C. Chen / Physics Letters A 196 (1994) 8 7 - 9 6
where Ai is the effective "hopping" matrix element between the system and the reservoir magnetized in the 18) direction. Due to the particle-number fluctuation, both the total number of particles and the total spin of the system are no longer good quantum numbers. The loss of these symmetries is partially compensated by a new symmetry defined by the operator
Q = N - 2S(O,x),
S( O, x ) = ~-~, ( Six sin O cos X + Siy sin O sin X i
(2.13
where
Si,. = ~1 Z(C~TCi~ + c]LCiT),
2.14
i
Si,. = ( 1 / 2 i ) Z ( c ] r c i ~ - c~tciT),
2.15)
i
s,:
= 1
Z(cJF,T
(2161
i
It is straightlorward to prove that [ Ho~, Q] = 0 ,
N
0
I
2
3
4
Np
0 0
0 .'~
0, 2 4, -~
0. 2 6. 4
0, 2..4 8, 6. 4
Np
(2.12)
where N is the total particle number operator and the total spin operator is defined as
+ Se cos 0),
Table 1 All possible eigenvalues of the operator Q (Np and 19p) for a two-site Hubbard model with zero to four (N) electrons
(2.17)
where Hos is the sum of the system Hamiltonian and the effective Hamiltonian (2.7), i.e., Q is a good quantum number. The physics associated with this new symmetry is easy to understand: when the exchange of particles only occurs for a single well-defined spin orientation, the changes in particle number and spin take place in a "synchronized" manner such that their difference remains the same in the fluctuation process. We notice that the particle states in the system can be projected onto the new spin axis 16) and I - 8 ) , even though the exchange of particles between the system and reservoir occurs only in the 16) direction. Once the states are expressed in this representation, the possible eigenvalues of the operator S(O, X) range tYom - N to N in the subspace with the fixed number of N particles. When the net spin of the system is aligned in the direction of 18), the eigenvalues of the operator Q are determined (see Eq. ( 2 . 1 2 ) ) by the number of paired particles (with spin 8 and - 3 ) Np in
the system; when the net spin of the system is aligned along I - 8), the eigenvalues of Q are given by /Vp = 2N - Np. Therefore, the eigenvalues of Q are always either zero or even integers. As an example, Table I lists all possible eigenvalues of Q for a two-site Hubbard model with up to four electrons. However, there is another system-size dependent constraint on the allowed eigenvalues of Q imposed by the Pauli principle. For example, in the two-site Hubbard model case, when N = 3, the allowed values of Q are 2 (Np) and 4 (Elp ) ; Q = 0 (No) and 6 ( ~'p ) are prohibited by the Pauli principle. For a larger system with three or more sites all values (0, 2, 4, 6) are allowed for the fourelectron system. This rule applies to general cases with a different number of sites and particles in the system. Since Q is a conserved quantity, states in the system with a different number of particles but with the same value of Q can mix; those with different values of Q cannot. States can be in general classified by Q, space symmetry, and any other symmetry the system may possess. It should be emphasized that the Q-symmetry is not particularly associated with the Hubbard model. As long as the Hamiltonian of the system commutes with the total particle number and total spin operators, Q will be a good quantum number. In other words, it is essentially determined by the property (e.g., ferromagnetism) of the reservoir. In the next section, we apply the above formalism to a two-site Hubbard model to illustrate the implementation of the theory and discuss some physical consequences.
3. Application to a two-site H u b b a r d model
We study the Hubbard model which is defined by the Hamiltonian, in the standard notation
C. Chen / Physics Letters A 196 (1994) 87- 96
i,j, tr -[- U ~
i,o"
nit ni~ ,
(3.1)
i
E(N,t) = E ( 4 - N , - t ) + ( N - 2 ) U + (2N - 4)e
simplify the derivation, but not lose generality 2. To be specific, we choose 0 = ½~" and X = 0. In this case, the effective Hamiltonian can be written as n e f f = ~ ~-~(c~T --[-c~[ -.}-CiT -]- cij.) i
where - t is the nearest-neighbor hopping integral, U the on-site Coulomb interaction,/~ the chemical potential, and E the orbital energy. In a closed system with a fixed number of particles e simply gives a uniform shift in energy and can be omitted. However, in the open system it must be explicitly included in the calculation. The total Hamiltonian of the open system now is the sum of Eqs. (3.1) and (2.7) with the system-reservoir "hopping" matrix elements given by Eqs. (2.10) and (2.11). Here we choose to study a two-site Hubbard model to illustrate the application of some symmetry projection procedures that one will encounter in more complicated situations. For this case, the summation of the site index in Eq. (3.1) runs from one to two only. For the purpose of comparison, we first solve the Hamiltonian (3.1) for 0 - 4 electrons in the c l o s e d system in the "standard" approach. There are in total 16 many-body electron states. For the two-site cluster, there is an inversion symmetry with even and odd parities, denoted as 2?+ and 2 - . The electron states are projected according to the particle-number, spin, and inversion symmetry into various subspaces; the Hamiltonian is solved in the subspaces since there is no nonzero matrix element between states with different symmetries. The energy equations with corresponding symmetry and degeneracy are obtained as shown in Table 2. There is a particle-hole symmetry in the energy equations, i.e.,
(3.2)
for N ~> 3. Notice that the orbital energy term has been explicitly included here. We now turn our attention to the case with the quantum particle-number fluctuation. We choose a special direction of the spin orientation 18) in the following calculation. We also assume that the system-reservoir "hopping" matrix elements for different sites are the same, i.e., the fluctuation is uniform. Doing so will
91
(3.3)
and the total spin operator (2.13) reduces to S(O, X ) = Sx.
(3.4)
So the good quantum number now is Q = N - 2Sx.
(3.5)
The 16 many-body electron states in the system first split according the inversion symmetry, with eight states in each subspace, as summarized in Table 3 in the "old" basis. For systems with more complicated structures electron states split according to irreducible representations of the space group [5]. Then these spatially symmetrized states are further projected according to the eigenvalues of the Q operator, as shown in Table 4. It is seen that only Q = 0, 2, and 4 are allowed for the two-site Hubbard model, although in principle Q = 6 and 8 are also possible for a fourelectron system. This is due to the constraint imposed by the Pauli principle as discussed above. There is no nonzero Hamiltonian matrix element between states with different symmetry indices. Therefore, the original 16 x 16 Hamiltonian matrix is now block diagonalized into four 2 × 2 and two 4 × 4 matrices, according to symmetries ~ . The resulting Hamiltonian matrices are shown in Tables 5 and 6. Direct diagonalization of these matrices provides all eigenvalues and eigenstates. Since the particle number in the system is not conserved, the average occupation number per site (N) or, equivalently the band filling in the case of extended systems, is measured by the grand canonical ensemble average (see Eq. (2.4)) and is controlled by the chemical potential #. For the case of (N) = 1, i.e., o n avera g e two electrons in the two-site cluster, the chemical 2 These assumptions are valid for the case of a single ferromagnetic Fermi sea. The energy spectrum does not depend on the direction of 16). If the two sites are in contact with two different ferromagnetic Fermi seas, as at the interface of two different ferromagnetic materials, the fluctuation at different sites must be treated differently.
C. Chen / Physics Letters A 196 (1994) 87-96
92
Table 2 The energy equations with the corresponding symmetry and degeneracy for the two-site Hubbard model; the left superscripts in the symmetry symbols indicate the spin degeneracy (2S + 1 ) with S being the total spin of the system Number of electrons
Symmetry
Energy equation
Degeneracy
N = 0 N=I
t X+ 2v+~ 2x .... 15' ~ 12"32'22'+ 2X-
E = 0 E - • -t=0 E e -'I'-/ = 0 E 2 - ( U + 4e) E + 4+ 2 + 2+U - 4t 2 = 0 E- 2e-U=0 E 2e = 0 E 3e-t-U=0
I ,~~' "~ I I 3 2
E -3e+t-U=0 E 4e-2U=0
2 1
N =2
N=3
N=4
12.+
Table 3 The 16 eigenstates in the old (closed system) basis projected according to particle number ( N ) and inversion symmetry: the vacuum state J0) contains no particle in the system Symmetry
Eigenstate
Symmetry
N = 0 N= I
I ~'+ 2X+
i/.t1 = 10) ~2 = (I/v/2)(C~T + c{r)]0>
22'-
N:2
I~'+
t//4 +,,
(I/v%(,m:+.r,.[t+,.+,tr,.~t)lO)
Eigeustute
g'lo = ( 1/x~-2)(Ctl - c~+)lO) ,.+.-
~bll = I. I/'~-)(C~fC~l - - c 'r21.~21 "'} )1())
'"
~,,e = ~Y, ¢~r )Ira
2X -
g't4 = (I/'f2-)(cl;-c21 - c2rcij )]0) / • .t .t ,t .t .t .t I//15 { I., V/2~)((II( 11(2[ -- {2t(2~.(i] )] 0} t t t t t t tO e, I/~"2-)(cIt¢'ll c25 -- C2~C21 eli )10)
t
9
N:4
~v+
~8
.t
.t
.t
,t
.t
1
t
t
,
,t c t .t .t ( e l i 11~2T(21)10)
Table 4 The 16 eigenstates in the new (open system) basis projected according to the good quantum number Q and expressed in terms of states in the old basis (see Table 3); notice that states with symmetries 2 .+ 0/n to g'8) and X - (~,) to ~Pt6 ) do not mix as they form new states
Eigenstate
4,2 = ( l/x/2~) ~02 ~3 = ( I / ' / 2 ) ( ~ 2 4)4 = 64 45 =¢,s q56 = ( 1/~v~) (,I//6 4,7 = ( 1/x/2) (¢6 r~8 = ~8
+ 03) -63)
-k //J7 ) - 07 )
Q
Eigenstatc
Q
0 2 2 2 2 4 4
4, m ~,l ~2 ,b~3 4) 14 4hs 616
2 e 2 0 4 2 4
= ( I/,/2_) (J,9 - ,P.+ ) =4,~, = ¢ 1/,/5-')C~2 - ~,~ I = 4(6~2 + 6 ~ 3 + ~--6,4 / = ~ (/ill2 Jr- ,I//13 -- "v/2-ff;14 } = ( 1/C2-) (¢,Ls + g,,~ ) = (I/x/2-)(~15 - &lr)
C. Chen / Physics Letters A 196 (1994) 87-96
93
Table 5 The Hamiltonian matrices for states with symmetries 27~; the symbol ~ represents E - #
- t
0 2)t 2,~ g- - t
-v~a
-v"5,~
-x/2A 2~+U -v~,~ -2t 0 v'2,~
0
-2t x/~A 2~ -x/2a -x/2,~ 3 ~ + t + U
3~'+t+U -2,~
-23. 4~ + 2U
Table 6 The Hamiltonian matrices for states with symmetries 2?a ; the symbol ~ represents • - / ~
g+ t g + t -2a -2,~ 2~
-V'2A
0
0
- v ' S a 2~ + u
0
v'5,~
0 0
0 v/2h
2g - x/2,~ -x/2A 3~-t+U
potential is/z = • ~ t + ½U for states with .,~+ symmetries. In the following we examine the ground-state energy and the magnetization of the system with IN) = 1 to study some physical consequences of quantum particle-number fluctuation and also to show a practical way of determining the system-reservoir "hopping" parameter from known properties of the system. We first study the ground-state energy of the systems as a function of interaction parameters. In the case of the closed system with two electrons (also two holes) in the system, it is given by E = 2e + ½U - ½x,/U 2 + 16t 2.
(3.6)
The ground state is of symmetry i f + (see Table 2), i.e., it is always a spin singlet state. This is in agreement with the conjecture [4] that there is a two-hole extension to Nagaoka's theorem [ 14] which yields a spin-singlet ground state. However, we will see below that this will be changed, as expected, in general for open systems, where the "boundary conditions" imposed by the surrounding environment will strongly influence or even dominate the properties of the system. In the strong interaction limit, i.e., as U approaches to infinity, the energy reduces to E ~ = 2e. In the case of an open system the ground-state energy is determined by numerically solving the 4 x 4 matrices in Tables 5 and 6. In the strong interaction limit,
2g-2,~ -2,~ 3 , ~ - t + U
the energy approaches to 2(e - tz). Here we take t to be positive; the ground state is then of symmetry ~ - . Fig. 1 shows the energy of the system relative to its value at infinite U for both the closed and the open system as a function of U/t. It is seen that the energies for the open system have larger slopes, i.e., they are more sensitive to the interaction term. The physics involved is easy to understand: local fluctuations decrease the band energy but increase the interaction energy; the compromise of the two opposite effects yield the calculated results. It has been shown [ 12] that the full quantum mechanical treatment of the system-reservoir hopping always yields lower total energy, thus a better description, for the system than a mean-field treatment. We now turn to the behavior of the magnitude of the magnetic moment of the system defined by
m = Z (C[~rS°'cr'Ci°-')'
(3.7)
i , o - , o -I
where S ~ , are the standard Pauli matrices and ( ) means the ground-state average. As shown above the ground state of the closed system with two electrons in the cluster is always a spin singlet, i.e., there is no net magnetic moment. When the system is in contact with a ferromagnetic Fermi sea, it is expected to pick up a finite magnetic moment. We express the ground
C. Chen / Physics Letters A 196 (1994) 87-96
94
1
I
!
l
-0.5 -1
(E
-1.5 -
E~)/t -2 -2.5 -3 -3.5
1
I
I
I
2
4
6
8
10
U/t Fig. I . The total energy of the closed and open two-site Hubbard cluster as a function of U/t for average occupation number per site
0.3
i
I
I
'l
0.25
0.2
rt~
/s-
/
0.15 0.1 0.05
0
~
/
/
~
,, , ,
/ *
0.2
/
'
~"-
1,
0.4
/
~
I
I
0.6
0.8
,\/t Fig. 2. The magnetic moment in the ground state of the open system as a function of A/t for U/t -= 0.0, 2.0.4.0, 6.0, 8.0 and 10.0 (top to bottom curves). The average occupation number per site (N) = I
state in terms of the combination of all the states with E4 symmetry as
m , = 0,
(3.10)
m= = 0.
(3.11)
6 'P = ~ alibi, l=3
(3.8)
where q51 are given in Table 4 and 0/~ are coefficients. The magnetization of the open system evaluated in the ground state is obtained as mx = (0/6) 2 - (0/3) 2,
(3.9)
It is seen that the system now may have non-zero magnetization along the x-direction. This particular magnetic "polarization" in the system is due to the special choice of the effective Hamiltonian (3.3) that connects the system and the reservoir. In general, all three components of the magnetization of the system may be non-zero when spin fluctuations in these di-
C. Chen / Physics Letters A 196 (1994)87-96 rections are allowed. However, the physics remains the same. Fig. 2 shows the magnitude of the groundstate magnetic moment as a function of the systemreservoir "hopping" matrix element for various values of the interaction strength. It is seen that the magnetization of the system rises smoothly with increasing system-reservoir exchange for all values of the interaction strength. However, one notices that the value of induced magnetization in the system decreases drastically as the interaction becomes strong. This is due to the more localized nature of the particles in the system for large values of U/t. The properties of the system become less dependent on the "boundary conditions" as the value of U/t grows. The above calculation of the magnetization of the open system as a function of the system-reservoir "hopping" parameter provides a practical way to determine the value of A in the Hamiltonian for real materials. For example, if the induced ground-state magnetization of a system, which is in contact with a ferromagnetic reservoir but otherwise is less or non-magnetic, is known, then ,~ can be obtained by fitting the calculated magnetization of the system to the known value. This scheme also works for extended systems such as magnetic surfaces where, however, the magnetization is intrinsic rather than induced; the task is to reproduce the correct ground-state magnetization. In the case of extended systems the parameter A may also be obtained by applying the open-cluster model to bulk material and fitting the result of open-cluster calculations to known properties, such as the density of states at the Fermi level, of the bulk material. These known properties may be the result of either bulk electronic structure calculations or experimental measurements. Application of the present formalism to more complicated/realistic systems is technically quite straightforward. We have recently developed a manybody approach [5,6] that uses and goes beyond the results of ab initio band structure calculations. Material-specific Hamiltonian parameters can be obtained. Interaction parameters beyond conventional local-density-approximation (LDA) results may be obtained from constrained LDA calculations [15] which take into consideration the correlation effects in the system. One crucial new ingredient here is the system-reservoir "hopping" matrix element, which may be determined by fitting the calculated results to known properties as discussed above.
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In our previous work [5,6], surface effects have been studied by use of free-standing thin films or by including a substrate layer in the cluster and then considering only crystal-filed contribution from the rest of the substrate. While these schemes may be well justified for some cases, inclusion of quantum particle-number fluctuation should provide a better framework for such studies. More importantly, for physically small systems in contact with a reservoir, such as small adsorbed atomic/molecular clusters at transition-metal surfaces, the quantum particle-number fluctuation is expected to have much stronger effects on the properties of the systems and must be built-in from the beginning.
4. Summary In summary, we have studied the properties of small fermionic quantum systems in contact with a ferromagnetic Fermi sea. Quantum particle-number fluctuation in the system is explicitly included. We have identified a symmetry associated with the particlenumber fluctuation in a ferromagnetic environment. It is used, together with other symmetries of the system, in block diagonalization of the Hamiltonian matrix. Application to a two-site Hubbard model has been used as an example to illustrate the implementation of the theory. It is obvious that the present formalism has many applications in the study of magnetic surfaces, interfaces, and multilayers, as well as adsorbates in bulk and surface magnetic environment. The work reported in this paper has laid the foundation for its application to larger systems in numerical form. One crucial task there, especially for real materials, will be to properly determine the effective system-reservoir "hopping" parameter, which may be obtained by fitting the calculated results to known properties, such as the ground-state magnetization of the system. It may also be determined through more laborious fitting of the results of open-cluster calculations to ab initio electronic structures of extended systems in some cases.
Acknowledgement The author acknowledges interesting discussions with Dr. X.R. Wang and Dr. T.K. Ng, and the hospi-
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tality of the Theory Group at the Physics Department, Hong Kong University of Science and Technology where part of this work was performed. This work was supported in part by the National Science Foundation under the EPSCoR program.
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