Ground-state of a two dimensional Heisenberg antiferromagnet

Solid State Communications,

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GROUND-STATE OF A TWO DIMENSIONAL HEISENBERG ANTIFERROMAGNET Jay D. Mancini,P’* William J. Massano,b Yu Zhou’ and Peter F. Meier’ “Physic8 Institute, University of Zurich, CH-8057, Zurich, Switzerland bScience Department, SUNY Maritime College, For Schuyler, NY 10465, U.S.A. ‘Departmen of Physics and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN 55455, U.S.A. (Received 21 November

1997; accepted in revised form 4 March 1998 by S.G. Louie)

The Lanczos tridiagonal formalism is employed in the study of the

ground-state of the s = 3 anisotropic Heisenberg antiferromagnet for the honeycomb, simple cubic and body-centered cubic lattices. Using the connected moments of Lee and Lo, we are able to generate, in a straightforward manner, an equivalent 6 X 6 Lanczos matrix and in turn, evaluate the ground-state energy and the singlet-triplet energy for all values of the anisotropy parameter. 0 1998 Elsevier Science Ltd. All rights reserved

Since the discovery of high T, superconductors by Bednorz and Muller [l] over ten years ago, there has been a great deal of interest both experimentally and theoretically in the antiferromagnetic undoped insulator La2Cu04, the oxygen deficient YBa2Cu06 and other undoped copper-oxide materials [2-51. The nature of the ground-state of these systems continued to be a hotly contested topic. This interest was precipitated by Anderson [6], as well as others, who hold that the mechanism for high-T, systems is magnetic in nature. Anderson has conjectured that a full disordered resonatingvalence bond (RVB) state [7] may be the ground-state of the two-dimensional Heisenberg model. Recently, a number of techniques have been applied to the Heisenberg antiferromagnetic spin 1 square-lattice which indicate that Neel order may reign as the ground-state. Furthermore, introduction of frustration in the form of next-nearest neighbor interactions suggests that no RVB state is required to describe its properties. A great deal of the physics literature has been devoted to the study of low-dimensional systems [8]. Particular attention has been paid to quantum spin models with the Heisenberg Hamiltonian as a prototype. Many years ago, Bethe [9] completely solved the onedimensional spin- 1 ferromagnetic Heisenberg model

* Permanent address: Physics Department, Fordham University, Bronx, NY 10458, U.S.A.

with isotropic coupling using a method which has come to be known as the “Bethe Ansatz” (BA). Unfortunately, the BA becomes intractable for dimensions greater than one, thus necessitating the use of other schemes. In this study, we wish to apply the Lanczos [9] variational scheme [lo-131 to study the ground-state of the spinhalf anisotropic Heisenberg antiferromagnet for a honeycomb, simple cubic and body-centered lattices. This represents an extension of earlier work wherein the lanczos algorithm was applied with great success to the one-dimensional frustrated anisotropic Heisenberg Model [14-171. Our results will be compared to those of Lee and Lo [ 171 who have recently applied the Connected Moments Expansion (CMX) [18-211 to a number of anisotropic Heisenberg antiferromagnetic lattices. The Lanczos (tridiagonal) scheme is generated according to the algorithm AI&I) = en,n-~I~,-~)+en.nl~~)+en,n+lI~n+l)r

(1)

where Dirac notation is invoked. Here an initial trial ket ]$o) is chosen both for calculation efficacy as well as by the physics of the Hamiltonian. That is a necessary condition is for the vector I$,-,) to be in the Hilbert space of the model under study. That is (&]$a) # 0. Any further a priori knowledge of the ground-state, such as total momentum and spin, may be incorporated into the choice of I#&

779

780

GROUND-STATE

OF A HEISENBERG

Note that a tridiagonal energy matrix I? with matrix elements en,m = ($n]k]$m) is also being generated. A variational upper-bound on the ground-state energy is obtained by direct diagonalization of any submatrix of k. Furthermore, by MacDonald’s theorem [22] an estimate of the excited state energies may also be obtained from the eigenvalues of k. The eigenstate I+,) is given recursively by the relation [ 121 ,$ >= (~-e~-,.~-,)l~n-,)-en-,,n-21~,-2) n en,m- 2

=

(~oIB,(ri)AB,(A)l~o)).

(3)

At this point, it is worthwhile to note the intimate relationship between the CMX method of Lee and Lo [ 171 and the variational scheme [ 121. The CMX expression for the ground-state energy is given, to third order, by the expression [ 181

(4) Here the Zk are defined Hamiltonian k-2

I,=-

x i=O

k-1

[

i

as connected

Vol. 106, No. 12

purposes, we give the 3 X 3 results: Qo = II 7

(6)

(2)

and thus I$,,) is related to I$,,) through a polynomial of the Hamiltonian operator 1#a) = ~&)]$a>. Therefore, the energy matrix elements en,m may be written as a groundstate moment expansion of the Hamiltonian e n.m

ANTIFERROMAGNET

moments

of the

1

Furthermore, the ground-state energy obtained by a 2 X 2 truncation of the Lanczos matrix E(o2x2) reproduces the second-order CMX expression eMX(*) in the regime where ]&$] < 0.25 [12]. Lee and Lo [17] have calculated up to the eleventh connected moment, I,], for the spin i anisotropic Heisenberg model for the honeycomb, square, simple cubic and body-centered cubic lattices. We may therefore use these calculated moments Zk to find explicitly each of the matrix elements in the corresponding (tridiagonal) Lanczos matrix. This is accomplished straightforwardly by solving equation (5) for (H ) in terms of the Ik and lower-order ground-state matrix elements of the Hamiltonian. Then, the Lanczos matrix elements en,m given by equation (3) may be evaluated explicitly. The eigenvalues of this matrix are then easily evaluated. It may be shown that for diagonal matrix elements en,n, to leading order, one must calculate (k2n- ‘). That is en,n = (p

where we have used the notation (Hk> = ($u]fik ]$a). It has been shown elsewhere [ 121 that the Lanczos matrix elements en.m may be expressed explicitly in terms of the connected moments Ik. For illustrative

- ‘) + (lower order terms).

(7)

Thus, the evaluation of Ill in the CMX scheme corresponds to a 6 X 6 Lanczos matrix. It is this matrix which we diagonalize in order to evaluate the energy

Table 1. Lanczos results for the ground-state energy of the honeycomb lattice for each truncation size, as a function of the anisotropy size, as a function of the anisotropy parameter R. The last column represents the results of Lee and Lo [I71 R

Eo(2 x 2)

Eo(3 X 3)

Eo(4 X 4)

Eo(5 X 5)

Eo(6 X 6)

Eo(CMX)

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

-0.37687 -0.38247 -0.39173 -0.40456 -0.42083 -0.44036 -0.463 -0.48855 -0.51682 -0.5476

-0.37687 -0.38243 -0.39153 -0.40394 -0.41936 -0.43745 -0.45786 -0.48024 -0.50424 -0.52956

-0.37687 -0.38245 -0.39163 -0.4042 1 -0.41994 -0.43853 -0.4597 -0.48315 -0.50862 -0.53589

-0.376872 -0.382457 -0.39 167 -0.404456 -0.452905 -2.587363 -6.530795 - 14.69708 -38.2836 -552.9481

-0.37687 -0.38246 -0.39166 -0.40434 -0.4203 -0.43927 -0.4609 -0.48483 -0.5 1072 -0.583 1

-3.076872 -0.382457 -0.391662 -0.404341 -0.420288 -0.439222 -0.460806 -0.484725 -0.5 10709 -0.538521

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GROUND-STATE

OF .4 HEISENBERG

ANTIFERROMAGNET

Table 2. Lanczos results for the ground-state energy of the simple cubic lattice for each truncation of the anisotropy parameter R. The last column represents the results of Lee and Lo [ 173

781 size, as a function

R

Eo(2x 2)

Eo(3 x 3)

E’o(4 x 4)

Eo(5 x 5)

&(6 x 6)

Eo(CMX)

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.oo

-0.7515 -0.75599 -0.76346 -0.77389 -0.78722 -0.80343 -0.82245 -0.84422 -0.86868 -0.89857

-0.7515 -0.75599 -0.76344 -0.77382 -0.78707 -0.803 12 -0.82189 -0.84329 -0.8672 1 -0.89355

-0.7515 -0.756 -0.76349 -0.77398 --0.78743 - 0.80382 -0.8231 - 0.84522 -0.87011 -0.89768

-0.75 15 -0.756 -0.7635 -0.77399 -0.7875 -0.80403 -0.82364 -0.84642 -0.8726 1 -0.90273

-0.75 15 -0.756 -0.7635 -0.77399 -0.78749 -0.804 -0.82352 -0.84609 -0.87173 -0.90047

-0.75 1500 -0.755999 -0.763497 -0.773993 -0.787490 -0.803995 -0.823520 -0.84608 1 -0.871700 -0.900393

eigenstates. We consider the Hamiltonian of the spin i heisenberg antiferromagnet with anisotropic exchange interaction [14-161

We wish to point out that within this basis, the Hamiltonian fro is just that for the ferromagnetic Ising, model with then well-known ground-state given by a system with all spins in the “up” state.

(10)

where Src- = s” t iSy defines the spin raising and lowering operators. Here J > 0 represents the antiferromagnetic exchange interaction and 0 5 R 5 1 is the anisotropy parameter. Following Lee and Lo [ 171 and in anticipation of antiferromagnetism, a rotation of the spin quantization axis is performed at each site of one sublattice (“down”) into the direction of the local mean field. We have then

Lee and Lo then use (10) as the initial vector in the evaluation of the connected moments Ik given by equation (5). Though this is a tedious operation, it is straightforwardly done. Once the connected moments are generated, they may be inserted into the CMX series (4) for the ground-state energy while the corresponding Lanczos matrix elements may also be constructed. In Table 1 we give our results for the ground-state energy of the honeycomb Lattice for each Lanczos truncation as a function of the anisotropy parameter R. The last column represents the CMX results of Lee and Lo [ 161. We note that each truncation yields reasonable results in comparison to those of CMX. However, it should be pointed out that for Eo(5 X 5), as the value of R 2 0.6, the

Table 3. Lanczos results for the ground-state energy of the body centered cubic lattice for each truncation function of the anisotropy parameter R. The last column represents the results of Lee and Lo [ 171

size, as a

R

Eo(2x 2)

Eo(3 x 3)

Eo(4 x 4)

Eo(5 x 5)

Eo(6 x 6)

Eo(CMX)

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

- 1.001428 -1.00571 -1.012834 - 1.022783 - 1.035534 -1.051056 -1.069314 - 1.090265 -1.113862 -1.140055

- 1.090000 - 1.090002 - 1.090004 - 1.090007 -1.09001 I -1.090016 - 1.09002 1 - 1.090028 - 1.090035 -1.090043

- 1.001429 - 1.005722 - 1.012897 - 1.022983 - 1.036022 - 1.05207 -1.071193 - 1..093475 -1.11901 -1.14791

-1.001429 - 1.005722 -1.012898 - 1.022989 - 1.036049 -1.052152 -1.071413 - 1.093996 -1.12015 -1.150255

- 1.001429 - 1JO5722 -1.012898 - 1.022989 - 1.036047 - 1.052 146 -1.071395 - 1.093948 -1.12002 -1.149908

-1.001429 - 1.005722 -1.012898 - 1.022989 - 1.036047 - 11.052 147 -1.071396 - 1.093950 -1.120026 -1.149919

Vol. 106, No. 12

GROUND-STATE OF A HEISENBERG ANTIFERROMAGNET

782

Heisenberg honeycomb 20

18

16

14

12

s ii

10

--4-

2x2

8

--c

3x3

6

-A-4x4 . ..x... 5x5

:. :’ ,’

4

::

x’ :’

0

6x6

:. 8

2-

0

0.I

0.2

0.3

0.4

0.5

0.6

I

0.7

0.8

Fig. 1. The singlet-triplet energy is plotted as a function of the anisotropy parameter R for the honeycomb lattice.

Heisenberg simple cubic

??

3

4==

----

A

---___*______~______*_______L--_-----*--------__

‘.

‘m

d .\

*.

-+-

.. ‘w

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4x4

-.*w.. 5x5 -_t

0

3x3

6x6

I .o

R

Fig. 2. The singlet-triplet energy is plotted as a function of the anisotropy parameter R for the simple cubic lattice.

Vol. 106, No. 12

GROUND-STATE OF A HEISENBERG ANTIFERROMAGNET

values for the energy become unreasonable. This may be understood by looking at equation (2) and noting that it is possible that poles may appear in the Lanczos expressions. Indeed such poles are common occurances in expansions which rely on moments of the Hamiltonian [20,23]. In Table 2 the results for the simple cubic lattice are given. Once again the last column represents the CMX results of Lee and Lo. Here we note that each Lanczos truncation is well-behaved throughout parameter space and that the energies calculated are comparable to those of CMX. Table 3 represents our results for the body-centered cubic lattice. We note that once again anomalous results appear, this time in Es(3 X 3). However, each of the other truncations yield excellent agreement with CMX. Overall then, we may conclude that the Lanczos scheme, is a viable method calculating ground-state energies for these systems. In particular it should be pointed out that even the Lanczos 2 X 2 result, which requires only the evaluation (to highest order) of (Hi ‘). Obviously however, the downside of the Lanczos method is that, at least for the model Hamiltonians chosen in this work, certain truncations may lead to anomalous results in certain regions of parameter space. In Figs l-3 we have plotted the singlet-triplet energy A = El - EO for the honeycomb, simple cubic

783

and body-centered cubic lattices, respectively. For each lattice we note that as R - 0, A - Ac,, a fixed value for each configuration. In particular & = 2 (honeycomb), & = 5 (simple cubic) and At, = 7 (body-centered cubic). For the honeycomb, Fig. 1, we see that A remains fairly fixed for all values of R for each truncation except the 5 X 5 which once again yields spurious results. For the simple cubic, Fig. 2, we note that as R - 1 both the 5 X 5 and 6 X 6 truncations become concave downward while the lower-order truncations (more moderately) become concave upward. Thus, the energy gap becomes smaller (as R - 1) for increasing truncation size. Finally, for the body-centered cubic, Fig. 3, we see behavior similar to that of the simple cubic. However, once again the 5 X 5 truncation appears to give a slightly spurious result since its gap dips below that for the 6 X 6 truncation. In this brief work, we have employed the Lanczos tridiagonal formalism to study the ground-state of the s = 3 anisotropic Heisenberg antiferromagnet for the honeycomb, simple cubic and body-centered cubic. Both the ground-state energy and the singlet-triplet energy are calculated as a function of the anisotropy parameter. There is good agreement with the work of Lee and Lo who used a connected moments scheme, although our 5 X 5 Lanczos truncation seems to yield results

Heisenberg body centered cubic 8 r

6

3-

2-

I-

0

I 0.1

I 0.2

I 0.3

I 0.4

I 0.5

I 0.6

I 0.7

I 0.8

I 0.9

I 1.0

R

Fig. 3. The singlet-triplet energy is plotted as a function of the anisotropy parameter R for the body-centered cubic lattice.

784

GROUND-STATE

OF A HEISENBERG

which are spurious, at best. Overall the method works well and could be applied with confidence to similar many-body Hamiltonian systems.

10.

ANTIFERROMAGNET Mancini, 1983,606l;

11.

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J.D. and Mattis, D.C., Phys. Rev., B28, B29, 1984,6988;

B31, 1985,744O.

Mancini, J.D. and Potter, C.D., Nuovo Cimento, D9, 1987,481.

.n

Acknowledgements-One of the authors (JDM) would like to thank the University of Zurich for their hospitality while this work was completed and also the Swiss National Science Foundation for financial support. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

9.

Bednorz, J.G. and Miiller, K.A., Z. Phys., B64, 1986, 188. Vaknin, D. et al., Phys. Rev. Lett., 58, 1987, 2802. Mitsuda, S. et al., Phys. Rev., B36, 1987, 882. Gotlieb, D., Lagos, M., Hallberg, K. and Balseiro, C., Phys. Rev., B43, 1991, 13688. Hallberg, K., Gagliano, E.R. and Balseiro, C., Phys. Rev., B41, 1990, 9474. Anderson, P.W., Science, 235, 1987, 1196. Anderson, P.W., Mater. Res. Bull., 8, 1973, 153. Lieb, E.H. and Mattis, D.C., Mathematical Physics in One Dimension. Academic Press, New York, 1965. Lanczos, J., Res. Nat. Bur. Stand,, 45, 1950, 222.

Prie, J.D., Schwall, D., Mancini, J.D., Kraus, D. and “’ Massano, W.J., Nuovo Cimento, D16, 1994,433. 13 For an excellent review of the Lanczos Method, see ’ Dagotto, E., Rev. Mod. Phys., 66(3), 1994,763 (and references therein). 14. Massano, W.J., Prie, J.D. and Mancini, J.D., Phys. Rev., B46, 1992, 11133. 15. Mancini, J.D., Prie, J.D. and Massano, W.J., Solid State Commun.,

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