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A real-space finite cluster approach to the correlated electron problem
Physiea C 235-240 (1994) 2251-2252
PHYSICA
North-Holland
A real-space finite cluster approach to the correlated electron problem J.tI.Jefferson, a and W . S t e p h a n , b
aDRA Electronics Sector, St.Andrews Road, Great Malvern, Worcestershire W R 1 4 3PS, England bDepartment of Physics, King's College London, Strand, London WC2R 2LS, England A new finite-cluster method for the correlated electron problem has been applied to Hubbard and Heisenberg models and shown to yield accurate solutions for the ground-state and low-lying excited states of finite systems. Iterating this method results in much improved estimates of ground-state energy densities in the thermodynamic limit compared with the usual numerical renormalisation group approach.
Renewed interest in the correlated electron problem, stimulated by the high-temperature cuprate superconductors, has led to a great deal of rece~-~t theoretical and numerical studies on charge-spin models, particularly the Hubbard model and the related Heisenberg and t - d models. Despite much progress, we still do not have reliable solutions of these models, except for small chlsters and special cases which are exactly sohlble. In this paper we report oa some recent calculations based on a new finite-cluster nlethod which give high-accuracy solutions when compared with other approximate methods, such as perturbation theory and the numerical renormalisation group. An outline of the method is as follows. (1) The correlated system is divided into cells, ~s in the numerical renormalisation group (NRG). (2) A cell is diagonalised explicitly and the Hamiltonian is expressed in terms of these cell eigenstates (this is exact). (3) A subset of states for eacl, cell is retained (the model subspace) and an effective Hamiltonian is derived for this smaller Hilbert space using exact solutions of groups of 2,3,4 ... CellS. "' (4) .1 l.i e .eltecT,1ve . . . . l -.l a n l l l"l , o n l "a n is l,' aI l e l l ( l'"l ~ t g o nalised explicitly for a large finite system. (5) For larger systems the process may be repeated, as in the NRG and greater accuracy may be achieved using a method of stepwise refinement ~Smilar to thai, used in the Kondo problem [I]. The method bears some similarity with earlier treatments of *Supported by the European Community under contract no. SCl*-0222-C(EDB)
the Hubbard model [2], but is much more accurate since it utilises exact solutions for groups of cells, which may be shown to be equivalent to a partial re-summation of the quasi-degenerate Rayleigh-Schr6diger perturbation expansion. A simple application is to the single-band ttubbard model in which a cell is chosen to be a site. For the half-filled band case tile model subspace has one electron at each site and it, is well-known that in tile strong-coupling regime (U > > t) tins reduces to a Ileisenberg model with antiferromagnetic exchange constant J = 412/U. This rapidly becomes a poor approximation when U/t is reduced but ::lay bc improved by calculating higher-order perturbation corrections or by using the finite-chlster method. In Fig. 1 we plot the percentage error ir~ the ground state energy vs U/t in one and two dimensions for 2, 3 and 4site finite chlster approximations, comparing this with the perturbation theory results [3]. In all cases the finite cluster method gives a substantial imt)rovement in accuracy. Similar accuracies are obtained for excited states and low-energy states of the original Itubbard mo¢ie 1,may be reproduced .... *l (~7' .l ..... wii.h all error 1~u~ u~an 1x+u u,_,~*. ,+O TT /-/ ---~ '} which includes the expecl.ed regime of tile nmtalinslllat.or transit ioD ; 2D. Similar accurac;es may be obtained for arbitrary band fillings, for which we obtain a chargespin model which is a generalisation of the t - J mode.1 with more accurate effective interact, ions of longer-range. The method has also been applied to the
0921-4534D4/$07.00 © 1994 - ElsevierScience B.V. All rights reserved.
SSDI 0921-4534(94)01691-7
. . . .
J H. Jefferson, W. Stephan/Physica C 235-240 (1994) 2251-2252
2252
ergy given by the fixed point. For example, in one dimension with cells of 9 spins, we get -0.4445 for the fixed point energy density compared with the exact Bethe Ansatz result of-0.443147 [5] and the usual NRG result of-0.4212 [6]. Similarly in two dimensions, with cells of 3 x 3 spins, the finitecluster NRG result is -0.6662 which again compares favourably with the quantum Monte-Carlo result of-0.6692 and is a considerable improvement on the usual NRG result of-0.5592 [7]. Preliminary results on the t - J model and on the test case of an uncorrelated tight-binding band show similar improvements in accuracy when compared with perturbative real-space expansions and demonstrate the wide applieabilty of the method. These results and further details of the method will be presented elsewhere.
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REFErtENCES -10.0 -15.0
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.
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Figure 1. Plots of percentage error in grounds~ate energy for 2,3 and 4-site cluster approximations (solid lines) and for second and fourthorder Rayleigh-Schrodinger perturbation theory (dashed lines). (a) 9-site chain. (b) 3 x 3 cluster.
Heisenberg model, forming cells with both an even and odd number of sites (spins). For an even number of spins/cell, the ground-state energy in the thermodynamic limit may be estimated by retaining only the lowest (singlet) state in each cell. By considering different cell sizes ~tml various orders in tile finite-cluster expansion we estimate tile ground-state energy of the lleisenberg model in 2D to be -0.665 4-0.05, w!,ich is within tile limits of recent quantum Monte-Carlo simulations [4]. For cells with an o~I,I lltlilll~,'r of spins we have extended the usual Nil.(; tlu'thod to incorporate the finite cluster results :uld agai~l lind much improved e s t i m a t e s ~)f l l w gt'~lllld stal.t" Cll-
1. K.G. Wilson, Rev.Mod.Phys. 47, 773 (1975). 2. S.T.Chui and J.W.Bray, Phys.Rev.B 18, 2426 (1978); J.E.Hirsch and G.F.Mazenko, Phys.Rev.B 19, 2656 (1979). 3. M.Takahashi, J.Phys.C 10,1289 (1977). 4. N.Trivedi and D.M.Ceperley, Phys.Rev.B 41, 4552 (1990). 5. H.Bethe, Z.Phys. 71,205 (1931). 6. D.C.Mattis av,t C.Y.Pan in Magnetic Excita. tions and Fluctuations II, ed. S,Lovesey et.al., Springer Series in Solid State Physics 54, 88 (1984). 7. a.S.Yedidia, Phys.Rev.Lett. 61, 2278 (1988).
PHYSICA
North-Holland
A real-space finite cluster approach to the correlated electron problem J.tI.Jefferson, a and W . S t e p h a n , b
aDRA Electronics Sector, St.Andrews Road, Great Malvern, Worcestershire W R 1 4 3PS, England bDepartment of Physics, King's College London, Strand, London WC2R 2LS, England A new finite-cluster method for the correlated electron problem has been applied to Hubbard and Heisenberg models and shown to yield accurate solutions for the ground-state and low-lying excited states of finite systems. Iterating this method results in much improved estimates of ground-state energy densities in the thermodynamic limit compared with the usual numerical renormalisation group approach.
Renewed interest in the correlated electron problem, stimulated by the high-temperature cuprate superconductors, has led to a great deal of rece~-~t theoretical and numerical studies on charge-spin models, particularly the Hubbard model and the related Heisenberg and t - d models. Despite much progress, we still do not have reliable solutions of these models, except for small chlsters and special cases which are exactly sohlble. In this paper we report oa some recent calculations based on a new finite-cluster nlethod which give high-accuracy solutions when compared with other approximate methods, such as perturbation theory and the numerical renormalisation group. An outline of the method is as follows. (1) The correlated system is divided into cells, ~s in the numerical renormalisation group (NRG). (2) A cell is diagonalised explicitly and the Hamiltonian is expressed in terms of these cell eigenstates (this is exact). (3) A subset of states for eacl, cell is retained (the model subspace) and an effective Hamiltonian is derived for this smaller Hilbert space using exact solutions of groups of 2,3,4 ... CellS. "' (4) .1 l.i e .eltecT,1ve . . . . l -.l a n l l l"l , o n l "a n is l,' aI l e l l ( l'"l ~ t g o nalised explicitly for a large finite system. (5) For larger systems the process may be repeated, as in the NRG and greater accuracy may be achieved using a method of stepwise refinement ~Smilar to thai, used in the Kondo problem [I]. The method bears some similarity with earlier treatments of *Supported by the European Community under contract no. SCl*-0222-C(EDB)
the Hubbard model [2], but is much more accurate since it utilises exact solutions for groups of cells, which may be shown to be equivalent to a partial re-summation of the quasi-degenerate Rayleigh-Schr6diger perturbation expansion. A simple application is to the single-band ttubbard model in which a cell is chosen to be a site. For the half-filled band case tile model subspace has one electron at each site and it, is well-known that in tile strong-coupling regime (U > > t) tins reduces to a Ileisenberg model with antiferromagnetic exchange constant J = 412/U. This rapidly becomes a poor approximation when U/t is reduced but ::lay bc improved by calculating higher-order perturbation corrections or by using the finite-chlster method. In Fig. 1 we plot the percentage error ir~ the ground state energy vs U/t in one and two dimensions for 2, 3 and 4site finite chlster approximations, comparing this with the perturbation theory results [3]. In all cases the finite cluster method gives a substantial imt)rovement in accuracy. Similar accuracies are obtained for excited states and low-energy states of the original Itubbard mo¢ie 1,may be reproduced .... *l (~7' .l ..... wii.h all error 1~u~ u~an 1x+u u,_,~*. ,+O TT /-/ ---~ '} which includes the expecl.ed regime of tile nmtalinslllat.or transit ioD ; 2D. Similar accurac;es may be obtained for arbitrary band fillings, for which we obtain a chargespin model which is a generalisation of the t - J mode.1 with more accurate effective interact, ions of longer-range. The method has also been applied to the
0921-4534D4/$07.00 © 1994 - ElsevierScience B.V. All rights reserved.
SSDI 0921-4534(94)01691-7
. . . .
J H. Jefferson, W. Stephan/Physica C 235-240 (1994) 2251-2252
2252
ergy given by the fixed point. For example, in one dimension with cells of 9 spins, we get -0.4445 for the fixed point energy density compared with the exact Bethe Ansatz result of-0.443147 [5] and the usual NRG result of-0.4212 [6]. Similarly in two dimensions, with cells of 3 x 3 spins, the finitecluster NRG result is -0.6662 which again compares favourably with the quantum Monte-Carlo result of-0.6692 and is a considerable improvement on the usual NRG result of-0.5592 [7]. Preliminary results on the t - J model and on the test case of an uncorrelated tight-binding band show similar improvements in accuracy when compared with perturbative real-space expansions and demonstrate the wide applieabilty of the method. These results and further details of the method will be presented elsewhere.
5.0
/
h4
0,0 h= =~' -5.0
(a)
S S
, ¢,
t-.
/
:
~-10.0
;
•
,,
/ u¢
:
.
,
W eL-
o
y
0.0
u.l -5.0
.3
REFErtENCES -10.0 -15.0
/
I...-';,
0.0
2.0
,
.
4.0
,
6.0
,:
,
8.0
0.0
U/t
Figure 1. Plots of percentage error in grounds~ate energy for 2,3 and 4-site cluster approximations (solid lines) and for second and fourthorder Rayleigh-Schrodinger perturbation theory (dashed lines). (a) 9-site chain. (b) 3 x 3 cluster.
Heisenberg model, forming cells with both an even and odd number of sites (spins). For an even number of spins/cell, the ground-state energy in the thermodynamic limit may be estimated by retaining only the lowest (singlet) state in each cell. By considering different cell sizes ~tml various orders in tile finite-cluster expansion we estimate tile ground-state energy of the lleisenberg model in 2D to be -0.665 4-0.05, w!,ich is within tile limits of recent quantum Monte-Carlo simulations [4]. For cells with an o~I,I lltlilll~,'r of spins we have extended the usual Nil.(; tlu'thod to incorporate the finite cluster results :uld agai~l lind much improved e s t i m a t e s ~)f l l w gt'~lllld stal.t" Cll-
1. K.G. Wilson, Rev.Mod.Phys. 47, 773 (1975). 2. S.T.Chui and J.W.Bray, Phys.Rev.B 18, 2426 (1978); J.E.Hirsch and G.F.Mazenko, Phys.Rev.B 19, 2656 (1979). 3. M.Takahashi, J.Phys.C 10,1289 (1977). 4. N.Trivedi and D.M.Ceperley, Phys.Rev.B 41, 4552 (1990). 5. H.Bethe, Z.Phys. 71,205 (1931). 6. D.C.Mattis av,t C.Y.Pan in Magnetic Excita. tions and Fluctuations II, ed. S,Lovesey et.al., Springer Series in Solid State Physics 54, 88 (1984). 7. a.S.Yedidia, Phys.Rev.Lett. 61, 2278 (1988).