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Unitary transformation treatment of a single hole in an Ising antiferromagnet
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Physica B 194-196 (1994) 1563-1564 North-Holland
Unitary Transformation Treatment of a Single Hole in an Ising Antiferromagnet Heinz Barentzen M a x - Planck - Institut fiir FestkSrperforschung, 7000 S t u t t g a r t 80, F R G The motion of a single hole in a two-dimensional Ising antiferromagnet (t - Jz model) is studied in a representation, where the spins are treated in the linear spin-wave approximation and the hole is described as a spinless fermion. The formal similarity with FrShlich's polaron Hamiltonian suggests that the t - J, model can be approximately diagonalized by means of two successive unitary transformations, analogous to those used by Lee, Low, and Pines in their intermediate-coupling treatment of the polaron. The first one is the lattice version of the Jost transformation, and its effect on the Hamiltonian is that the latter becomes diagonal in the hole operators. The remaining pure boson part is then subject to a displaced-oscillator transformation to eliminate all terms linear in the boson operators. The resulting energy E(k) is a rigorous upper bound to the exact ground state energy and, for k = 0, compares well with analytic results based on the retraceable path approximation.
There is a widespread belief t h a t a detailed knowledge of the behavior of holes in q u a n t u m antiferromagnets is an i m p o r t a n t step towards an understanding of the mechanism underlying highTc superconductivity. This paper deals with the simplest possible case, the motion of a single hole in an Ising antiferromagnet on a square lattice. This is commonly referred to as the t - J~ model and described by the Hamiltonian
u=-t
F_,,a
s,'s/,
where C~. t and Ci~ are projected fermion operators taking into account that no double occupancy of sites is allowed because of the strong on-site Coulomb repulsion. In the Ising term, Si z denotes a local spin operator of q u a n t u m number S = 1/2, while the rest of the notation is standard. The t - Jz model has been studied by a variety of methods (for a review, see Ref.1). Here we follow an approach proposed by the authors of Ref. 2, where the spins are treated in the HolsteinPrimakoff boson representation and the holes are described as spinless fermions. In this way, the t - Jz model is transformed into an effective Hamiltonian representing a system of interacting bosons and fermions. If, in addition, the linear spinwave approximation is adopted, the effective Ha-
miltonian for a single hole can be written as [2]
H = wo Z i
b'tb' 4- 2t Z ( f ' t f i b ' + H.c.),
(1)
where w0 = zJz/2 and bil(fl t) creates a boson (spinless fermion) at site R/. The form (1) is reminiscent of FrShlich's polaron I-Iamiltonian and suggests t h a t it can be approximately diagonalized by means of two successive unitary transformations, analogous to those used by Lee, Low, and Pines (LLP) in their intermediate-coupling treatment of the polaron [3]. As in polaron theory, the total crystal moment u m P is a constant of the motion. This implies t h a t there exists a unitary transformation leading to the simultaneous diagonalization of b o t h P and H . Guided by the LLP approach [3], one m a y suspect that the unitary operator q . P~f,*f,),
U = exp(-i Z
(2a)
i where
Q = Z
qbqtbq
(2b)
q
is the m o m e n t u m of the boson field, has the required properties. Using Eqs. (2), the transformed Hamiltonian can be cMculated in a straightforward way [4]. The result is of the form
Ut H U = wo ~_~ H k f k t fk, k
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E1362-P
(3)
1564 which is manifestly diagonal in the hole operators. Each Hk describes a pure, interacting boson system and is given by Hk =
E bqlbq-]-- ~ q
E ( F k _ q b q + H.c.),
1
I
I
2
4
6
I
I
8
10
(4a) -2
q
where
E_. w0
Fk = z - 1 E c o s [ ( k -
Q ) - R a ] = r k I.
(4b) -4
a
In (4a), the coupling parameter a is defined by = 2t/J~, while the sum in (45) runs over the z nearest neighbors of a given site. The wave vectors k appearing in Eqs. (3) and (4) are just the eigenvalues of P in the transformed system. To accomplish the approximate diagonalization of the operators (4a), we again follow LLP [3] by subjecting Hk to a displaced-oscillator transformation Vk defined by Yk = exp[N-1/2 E )~q(k)(bq - bqt)], q
(5)
where the real parameters Aq(k) are determined from the condition that VktHkVk does not contain terms linear in the boson operators. Since this is equivalent to a variational treatment of Hk [3], the resulting energy E(k) will be an upper bound to the exact ground state energy. Here the k-dependence is solely due to those trajectories, where the hole moves along closed loops [5,6]. If the latter are excluded by setting k = 0, our result for the energy may be compared with that of the retraceable path approximation (rpa) [5,7]. For k = 0, all relevant quantities like Aq(0) and E(0) -- E can be determined analytically and evaluated as functions of the coupling parameter a. The result for E/wo is plotted in Fig. 1 and compared with the result of the rpa [7]. Qualitatively, both curves are very similar, the quantitative agreement being still quite reasonable in the physically relevant coupling range (~r .~ 6). Further improvement of the results can probably be achieved by applying a Bogoliubov transformation to diagonalize the terms quadratic in the boson operators. Details of the LLP approach as well as the dispersion of E(k) will be reported elsewhere [4].
-6
12
O*
Figure 1. Ground state energy of a single hole for k = 0 on the square lattice as function of the coupling parameter ~r = 2t/Jz. The upper curve shows the result of the present theory, the lower one the result obtained by means of the rpa. REFERENCES
1. Yu.A. Izyumov, Sov.Phys.Usp. 34, 935 (1991) [Usp.Fiz.Nauk 161, 1 (1991)] 2. S. Schmitt-Rink, C.M. Varma, A.E. Ruckenstein, Phys.Rev.Lett. 60, 2793 (1988) 3. D. Pines, in Polarons and Excitons, edited by C.G. Kuper and G.D. Whitfield (Oliver and Boyd, Edinburgh, 1962) 4. H. Barentzen, to be published 5. W.F. Brinkman, T.M. Rice, Phys.Rev.B 2,
1324 (1970) 6. S.A. Trugman, Phys.Rev.B 37, 1597 (1988) 7.
M.M. Mohan, J.Phys.: Condens. Matter 3, 4307 (1991)
Physica B 194-196 (1994) 1563-1564 North-Holland
Unitary Transformation Treatment of a Single Hole in an Ising Antiferromagnet Heinz Barentzen M a x - Planck - Institut fiir FestkSrperforschung, 7000 S t u t t g a r t 80, F R G The motion of a single hole in a two-dimensional Ising antiferromagnet (t - Jz model) is studied in a representation, where the spins are treated in the linear spin-wave approximation and the hole is described as a spinless fermion. The formal similarity with FrShlich's polaron Hamiltonian suggests that the t - J, model can be approximately diagonalized by means of two successive unitary transformations, analogous to those used by Lee, Low, and Pines in their intermediate-coupling treatment of the polaron. The first one is the lattice version of the Jost transformation, and its effect on the Hamiltonian is that the latter becomes diagonal in the hole operators. The remaining pure boson part is then subject to a displaced-oscillator transformation to eliminate all terms linear in the boson operators. The resulting energy E(k) is a rigorous upper bound to the exact ground state energy and, for k = 0, compares well with analytic results based on the retraceable path approximation.
There is a widespread belief t h a t a detailed knowledge of the behavior of holes in q u a n t u m antiferromagnets is an i m p o r t a n t step towards an understanding of the mechanism underlying highTc superconductivity. This paper deals with the simplest possible case, the motion of a single hole in an Ising antiferromagnet on a square lattice. This is commonly referred to as the t - J~ model and described by the Hamiltonian
u=-t
F_,
s,'s/,
where C~. t and Ci~ are projected fermion operators taking into account that no double occupancy of sites is allowed because of the strong on-site Coulomb repulsion. In the Ising term, Si z denotes a local spin operator of q u a n t u m number S = 1/2, while the rest of the notation is standard. The t - Jz model has been studied by a variety of methods (for a review, see Ref.1). Here we follow an approach proposed by the authors of Ref. 2, where the spins are treated in the HolsteinPrimakoff boson representation and the holes are described as spinless fermions. In this way, the t - Jz model is transformed into an effective Hamiltonian representing a system of interacting bosons and fermions. If, in addition, the linear spinwave approximation is adopted, the effective Ha-
miltonian for a single hole can be written as [2]
H = wo Z i
b'tb' 4- 2t Z ( f ' t f i b ' + H.c.),
(1)
where w0 = zJz/2 and bil(fl t) creates a boson (spinless fermion) at site R/. The form (1) is reminiscent of FrShlich's polaron I-Iamiltonian and suggests t h a t it can be approximately diagonalized by means of two successive unitary transformations, analogous to those used by Lee, Low, and Pines (LLP) in their intermediate-coupling treatment of the polaron [3]. As in polaron theory, the total crystal moment u m P is a constant of the motion. This implies t h a t there exists a unitary transformation leading to the simultaneous diagonalization of b o t h P and H . Guided by the LLP approach [3], one m a y suspect that the unitary operator q . P~f,*f,),
U = exp(-i Z
(2a)
i where
Q = Z
qbqtbq
(2b)
q
is the m o m e n t u m of the boson field, has the required properties. Using Eqs. (2), the transformed Hamiltonian can be cMculated in a straightforward way [4]. The result is of the form
Ut H U = wo ~_~ H k f k t fk, k
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E1362-P
(3)
1564 which is manifestly diagonal in the hole operators. Each Hk describes a pure, interacting boson system and is given by Hk =
E bqlbq-]-- ~ q
E ( F k _ q b q + H.c.),
1
I
I
2
4
6
I
I
8
10
(4a) -2
q
where
E_. w0
Fk = z - 1 E c o s [ ( k -
Q ) - R a ] = r k I.
(4b) -4
a
In (4a), the coupling parameter a is defined by = 2t/J~, while the sum in (45) runs over the z nearest neighbors of a given site. The wave vectors k appearing in Eqs. (3) and (4) are just the eigenvalues of P in the transformed system. To accomplish the approximate diagonalization of the operators (4a), we again follow LLP [3] by subjecting Hk to a displaced-oscillator transformation Vk defined by Yk = exp[N-1/2 E )~q(k)(bq - bqt)], q
(5)
where the real parameters Aq(k) are determined from the condition that VktHkVk does not contain terms linear in the boson operators. Since this is equivalent to a variational treatment of Hk [3], the resulting energy E(k) will be an upper bound to the exact ground state energy. Here the k-dependence is solely due to those trajectories, where the hole moves along closed loops [5,6]. If the latter are excluded by setting k = 0, our result for the energy may be compared with that of the retraceable path approximation (rpa) [5,7]. For k = 0, all relevant quantities like Aq(0) and E(0) -- E can be determined analytically and evaluated as functions of the coupling parameter a. The result for E/wo is plotted in Fig. 1 and compared with the result of the rpa [7]. Qualitatively, both curves are very similar, the quantitative agreement being still quite reasonable in the physically relevant coupling range (~r .~ 6). Further improvement of the results can probably be achieved by applying a Bogoliubov transformation to diagonalize the terms quadratic in the boson operators. Details of the LLP approach as well as the dispersion of E(k) will be reported elsewhere [4].
-6
12
O*
Figure 1. Ground state energy of a single hole for k = 0 on the square lattice as function of the coupling parameter ~r = 2t/Jz. The upper curve shows the result of the present theory, the lower one the result obtained by means of the rpa. REFERENCES
1. Yu.A. Izyumov, Sov.Phys.Usp. 34, 935 (1991) [Usp.Fiz.Nauk 161, 1 (1991)] 2. S. Schmitt-Rink, C.M. Varma, A.E. Ruckenstein, Phys.Rev.Lett. 60, 2793 (1988) 3. D. Pines, in Polarons and Excitons, edited by C.G. Kuper and G.D. Whitfield (Oliver and Boyd, Edinburgh, 1962) 4. H. Barentzen, to be published 5. W.F. Brinkman, T.M. Rice, Phys.Rev.B 2,
1324 (1970) 6. S.A. Trugman, Phys.Rev.B 37, 1597 (1988) 7.
M.M. Mohan, J.Phys.: Condens. Matter 3, 4307 (1991)