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Constraint quantization of slave-particle theories
ELSEVIER
Physica B 230-232 (1997) 451-453
Constraint quantization of slave-particle theories C h r i s t i a n H e l m *, J o a c h i m K e l l e r lnstitut fiir Theoretische Physik der Universitiit Regensburg, 93040 Regensburg, Germany
Abstract
We start from the Barnes~Coleman slave-particle description, where the Hubbard operators X are decomposed into a product of fermionic (f~) and bosonic (b) operators. The quantum mechanical constraint btb + ~-~ f t f ~ = 1 is treated within the framework of Dirac's method for the quantization of classical constrained systems. This leads to modified algebraic properties of the fundamental operators: bbtb = b, f~f~f~ = 6,~fy and f,b t = 0. Thereby the algebra of the X-operators is preserved exactly on the operator level. Matrix representations of the above algebra are constructed and a resolvent-like perturbation theory for the single-impurity Anderson model is developed.
Keywords." Slave-boson theory
In systems like the single-impurity Anderson model
X~/~ = ~ ,
N
" = Z c oc,o +
Zxoo
k~r
(r=l N
+
v (xo c . +X ock ),
(1)
ka=l
the doubly occupied state of the strongly correlated f-electron (N: spin degeneracy) is forbidden due to a strong local Coulomb repulsion. This is described by the so-called Hubbard operators X~/~ =1 ct) (fl I (~, fl = 0 . . . . . N), which fulfill a projector-like algebra and a completeness relation N
X~#X,:6 = 6~X~6
ZX~
= 17%h,s-
(2)
0c=O
In the standard slave-boson approach [1], the Xoperators are decomposed into a product of fermionic * Corresponding author. 0921-4526/97/$17.00
(f~) and bosonic (b) degrees of freedom: (3)
which are intuitively interpreted as creators and annihilators of empty (b) and occupied ( f a ) states. In order to eliminate unphysical degrees of freedom, an additional constraint has to be implemented in the functional integral:
(btb + ~ - - ~ f t f ~ _
1)[ p h y s ) = 0.
(4)
In the usual treatment of slave-boson theories this constraint is only fulfilled on a mean-field level and it is difficult to control the contribution of unphysical parts of the Hilbert space. Motivated by this we aim at a decomposition X~13(t ) = ~t(t)~b#(t) of X-operators into a product of two operators with own dynamics while fulfilling the algebra and completeness relation of Eq. (2) exactly on the operator level. The constraint Equation (4) can be treated exactly by Dirac's method [2] for the quantization of classical
Copyright © 1997 Elsevier Science B.V. All rights reserved
PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 6 1 1 - 4
~9~t = (bt,fGt),
452
C Helm, J. Keller/ Physica B 230-232 (1997) 451-453
constrained systems, where the standard (anti)commutation relations of the constrained degrees of freedom are modified in order to incorporate the constraint on the operator level. This procedure has also been carried out (with inconsistent final results) in [3]. The corrected result reads as
which are functions of ~ q $ only, are independent of the choice of the representation and coincide with the physical result tr~(e-#HA) tr~phys(e-#/4A) (A(~k~))7-t- trn(e-~/~) - tr~/phy~(e-#~)
(8)
= (a(X~))~h. Note that these algebraic properties are equal for all ~O~,which are neither bosons nor fermions, but that Eq. (2) is reproduced exactly. One possibility to fulfill this algebra is to enlarge the (N + 1)-dimensional physical Hilbert space '~phys by an unphysical "vacuum" state I vac) of the slaveparticle operators: [ 0) = b f [ vac) and [ tr) = f~f I vac). Then on the space 7-/-- ~t'~phys ~) 7-~ ~t := (0 ..... e~) ~ C (N+2)×(N+2)
(6)
With the help of these auxiliary operators the Laplace transform of the f-electron Green's function G~#(t) := -i([Xo~(t),X3o]+)O(t)
(9)
can be wriRen as
G~(z)
=/
1 cf~),f~b]+), bt)(z+z1+
C
(10) with (e~), := 6n,~, n = 0 ..... N. Thereby the usual resolvent theory is recovered, as the dynamics on the unphysical space 7-[ is reduced to the identity: ~ ( t ) = l~q~e -ira. In addition to this we constructed all possible matrix representations of the algebra of Eq. (5), which can serve as a starting point of alternative perturbational approaches beyond the usual resolvent expansion. In this paper we present for simplicity only representations on Hilbert spaces ~ := 7~ ® '~phys, which are tensor product of an arbitrary separable Hilbert space ~ and the physical Hilbert space 7-/phys and which turn out to show interesting new features. In this case ~t can be represented as
~ . . . (0,. .
~f ..... a ~ ~f ) E C k(N+l)xk(N+l) 0,al~
(7)
with ai~a)~ = 6ij6~#, a~ C C k(u+l) for i,j : 1. . . . . k and ~,fl = 0 ..... N. Without loss of generality we choose the special orthonormal basis (ai~)p := 6p,i(N+l)+~+l. Consequently, ~ b ; = 6~#Pg, is not the identity 1~ (as claimed in [3]), but the projector Pq, on the k-dimensional image of qJ in the k ( N + 1)dimensional space 7-/. Thereby physical combinations ~ f ~ = lk ® D~# with (D:cfl)O
£ being the Liouville operator and the contour C enclosing all eigenvalues of H. The expansion of G~# in orders of V using the formula
1 ..~0 OCFz____~O)~ z - Co1 - ~CF __ z -Z2o
(11)
can be translated into diagrams similar to those used in the usual resolvent theory. Thereby the application of E on ~(t) creates unusual non-vanishing products of four and more slave operators like ~t ~b;~ . These unphysical terms cancel out in any representation in any finite order in V, reproducing the correct results of a direct X-operator approach order by order, which shows the consistency of the approach. But when summing up an infinite number of diagrams, these unusual terms can remain and the results can depend on the choice of representation. The approximation 1/(z - £ ) = 1/(z - go - E) with E = £ v [ 1 / ( z - Z;0)]Z:v shows Kondo-like resonances at the energy To = D e x p ( - I ef [ / [ ( N - 1) NoV2]) - consistent with the physical intuitiononly for spin degeneracy N > 1. Here the inclusion of vertex corrections seems to be crucial for the reproduction of the correct Kondo scale for N # 1, c~. This and the comparison with the usual resolvent
C. Helm, J. Keller/ Physica B 230-232 (1997) 451-453
reveal the physical significance of the used subclass of diagrams, are currently being investigated. To conclude, we constructed new slave-particle decompositions of the Hubbard operators, which are e x a c t on the operator level, and developed a formal concept for a perturbation theory of these auxiliary operators. This will allow to study the above theory in (e.g. self-consistent) approximations, which go beyond the results presented above. One of the authors (C.H.) of this work has been supported by grants from the Studienstiftung des Deutschen Volkes.
453
References [1] P. Coleman, Phys. Rev. B 28 (1983) 5255; P. Coleman, in: Theory of Heavy Fermions and Valence Fluctuations, eds. T. Kasuya and T. Saso (Springer, New York, 1985) p.163; L. Yu, in: Recent Progress in Many-body Theories, Vol. 3, eds. T.L. Ainsworth et al. (Plenum, New York, 1992) p.157, and references therein. [2] P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York (1964). [3] J.C. Le Guillon and E. Ragoney, Phys. Rev. B 52 (1995) 2403.
Physica B 230-232 (1997) 451-453
Constraint quantization of slave-particle theories C h r i s t i a n H e l m *, J o a c h i m K e l l e r lnstitut fiir Theoretische Physik der Universitiit Regensburg, 93040 Regensburg, Germany
Abstract
We start from the Barnes~Coleman slave-particle description, where the Hubbard operators X are decomposed into a product of fermionic (f~) and bosonic (b) operators. The quantum mechanical constraint btb + ~-~ f t f ~ = 1 is treated within the framework of Dirac's method for the quantization of classical constrained systems. This leads to modified algebraic properties of the fundamental operators: bbtb = b, f~f~f~ = 6,~fy and f,b t = 0. Thereby the algebra of the X-operators is preserved exactly on the operator level. Matrix representations of the above algebra are constructed and a resolvent-like perturbation theory for the single-impurity Anderson model is developed.
Keywords." Slave-boson theory
In systems like the single-impurity Anderson model
X~/~ = ~ ,
N
" = Z c oc,o +
Zxoo
k~r
(r=l N
+
v (xo c . +X ock ),
(1)
ka=l
the doubly occupied state of the strongly correlated f-electron (N: spin degeneracy) is forbidden due to a strong local Coulomb repulsion. This is described by the so-called Hubbard operators X~/~ =1 ct) (fl I (~, fl = 0 . . . . . N), which fulfill a projector-like algebra and a completeness relation N
X~#X,:6 = 6~X~6
ZX~
= 17%h,s-
(2)
0c=O
In the standard slave-boson approach [1], the Xoperators are decomposed into a product of fermionic * Corresponding author. 0921-4526/97/$17.00
(f~) and bosonic (b) degrees of freedom: (3)
which are intuitively interpreted as creators and annihilators of empty (b) and occupied ( f a ) states. In order to eliminate unphysical degrees of freedom, an additional constraint has to be implemented in the functional integral:
(btb + ~ - - ~ f t f ~ _
1)[ p h y s ) = 0.
(4)
In the usual treatment of slave-boson theories this constraint is only fulfilled on a mean-field level and it is difficult to control the contribution of unphysical parts of the Hilbert space. Motivated by this we aim at a decomposition X~13(t ) = ~t(t)~b#(t) of X-operators into a product of two operators with own dynamics while fulfilling the algebra and completeness relation of Eq. (2) exactly on the operator level. The constraint Equation (4) can be treated exactly by Dirac's method [2] for the quantization of classical
Copyright © 1997 Elsevier Science B.V. All rights reserved
PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 6 1 1 - 4
~9~t = (bt,fGt),
452
C Helm, J. Keller/ Physica B 230-232 (1997) 451-453
constrained systems, where the standard (anti)commutation relations of the constrained degrees of freedom are modified in order to incorporate the constraint on the operator level. This procedure has also been carried out (with inconsistent final results) in [3]. The corrected result reads as
which are functions of ~ q $ only, are independent of the choice of the representation and coincide with the physical result tr~(e-#HA) tr~phys(e-#/4A) (A(~k~))7-t- trn(e-~/~) - tr~/phy~(e-#~)
(8)
= (a(X~))~h. Note that these algebraic properties are equal for all ~O~,which are neither bosons nor fermions, but that Eq. (2) is reproduced exactly. One possibility to fulfill this algebra is to enlarge the (N + 1)-dimensional physical Hilbert space '~phys by an unphysical "vacuum" state I vac) of the slaveparticle operators: [ 0) = b f [ vac) and [ tr) = f~f I vac). Then on the space 7-/-- ~t'~phys ~) 7-~ ~t := (0 ..... e~) ~ C (N+2)×(N+2)
(6)
With the help of these auxiliary operators the Laplace transform of the f-electron Green's function G~#(t) := -i([Xo~(t),X3o]+)O(t)
(9)
can be wriRen as
G~(z)
=/
1 cf~),f~b]+), bt)(z+z1+
C
(10) with (e~), := 6n,~, n = 0 ..... N. Thereby the usual resolvent theory is recovered, as the dynamics on the unphysical space 7-[ is reduced to the identity: ~ ( t ) = l~q~e -ira. In addition to this we constructed all possible matrix representations of the algebra of Eq. (5), which can serve as a starting point of alternative perturbational approaches beyond the usual resolvent expansion. In this paper we present for simplicity only representations on Hilbert spaces ~ := 7~ ® '~phys, which are tensor product of an arbitrary separable Hilbert space ~ and the physical Hilbert space 7-/phys and which turn out to show interesting new features. In this case ~t can be represented as
~ . . . (0,. .
~f ..... a ~ ~f ) E C k(N+l)xk(N+l) 0,al~
(7)
with ai~a)~ = 6ij6~#, a~ C C k(u+l) for i,j : 1. . . . . k and ~,fl = 0 ..... N. Without loss of generality we choose the special orthonormal basis (ai~)p := 6p,i(N+l)+~+l. Consequently, ~ b ; = 6~#Pg, is not the identity 1~ (as claimed in [3]), but the projector Pq, on the k-dimensional image of qJ in the k ( N + 1)dimensional space 7-/. Thereby physical combinations ~ f ~ = lk ® D~# with (D:cfl)O
£ being the Liouville operator and the contour C enclosing all eigenvalues of H. The expansion of G~# in orders of V using the formula
1 ..~0 OCFz____~O)~ z - Co1 - ~CF __ z -Z2o
(11)
can be translated into diagrams similar to those used in the usual resolvent theory. Thereby the application of E on ~(t) creates unusual non-vanishing products of four and more slave operators like ~t ~b;~ . These unphysical terms cancel out in any representation in any finite order in V, reproducing the correct results of a direct X-operator approach order by order, which shows the consistency of the approach. But when summing up an infinite number of diagrams, these unusual terms can remain and the results can depend on the choice of representation. The approximation 1/(z - £ ) = 1/(z - go - E) with E = £ v [ 1 / ( z - Z;0)]Z:v shows Kondo-like resonances at the energy To = D e x p ( - I ef [ / [ ( N - 1) NoV2]) - consistent with the physical intuitiononly for spin degeneracy N > 1. Here the inclusion of vertex corrections seems to be crucial for the reproduction of the correct Kondo scale for N # 1, c~. This and the comparison with the usual resolvent
C. Helm, J. Keller/ Physica B 230-232 (1997) 451-453
reveal the physical significance of the used subclass of diagrams, are currently being investigated. To conclude, we constructed new slave-particle decompositions of the Hubbard operators, which are e x a c t on the operator level, and developed a formal concept for a perturbation theory of these auxiliary operators. This will allow to study the above theory in (e.g. self-consistent) approximations, which go beyond the results presented above. One of the authors (C.H.) of this work has been supported by grants from the Studienstiftung des Deutschen Volkes.
453
References [1] P. Coleman, Phys. Rev. B 28 (1983) 5255; P. Coleman, in: Theory of Heavy Fermions and Valence Fluctuations, eds. T. Kasuya and T. Saso (Springer, New York, 1985) p.163; L. Yu, in: Recent Progress in Many-body Theories, Vol. 3, eds. T.L. Ainsworth et al. (Plenum, New York, 1992) p.157, and references therein. [2] P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York (1964). [3] J.C. Le Guillon and E. Ragoney, Phys. Rev. B 52 (1995) 2403.