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Extending the reach of strong coupling: An iterative technique for hamiltonian lattice models
Volume 138B, number 5,6
PHYSICS LETTERS
26 April 1984
EXTENDING THE REACH OF STRONG COUPLING: AN ITERATIVE TECHNIQUE FOR HAMILTONIAN LATTICE MODELS J. ALBERTY, J. GREENSITE and A. PATKOS 1 The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 1 7, DK-2100 Copenhagen O, Denmark Received 27 December 1983
We propose an iterative method for doing lattice strong-coupling-like calculations in a range of medium to weak couplings. The method is a modified Lanczos scheme, with greatly improved convergence properties. The technique is tested on the Mathieu equation and on a hamiltonian finite-chain XY model, with excellent results.
Strong-coupling calculations in lattice gauge theory can account for many qualitative features of the hadronic spectrum, such as confinement [1] and chiral symmetry breaking [2]. There is some reason to believe that the mass spectrum might attain realistic (i.e. continuum theory) values just inside the weak-coupling regime, since even at strong couplings the massratio calculations are not disastrously wrong [3]. This motivates a search for practical methods of doing strong-coupling-like spectrum calculations in regions of moderate to weak couplings. In this note we will outline such a method for hamiltonian lattice theories, and test it on some very simple models. Our hope is that the technique may be applicable to the more ambitious goal of analytical spectrum calculations in lattice gauge theory. Underlying our method is the Lanczos tridiagonalization scheme, which is a standard numerical technique for eigenvalue problems. To this scheme we add two new features: (1) coupling subdivision; and (2) subspace extension. We will discuss these in turn. (1) Coupling subdivision. This is a method for subdividing a large/3 problem into a series of small/3 problems. Consider the problem of finding the eigenvalues of the hamiltonian H=Ho+xV,
(1)
1 Permanent address: Department of Atomic Physics, E6tv6s University, Hungary. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where x is the perturbation parameter, and we imagine that the radius of convergence of perturbation theory is at x = 1. Suppose x 2> 1. Then consider solving, in N steps, the sequence of eigenvalue problems
H1 = u 0 + (x/N) v , Hn+ 1 = H n + (x/N) V ,
(2)
by Rayleigh-Schr6dinger perturbation theory (this is for purposes of explanation; in practice we will use the more efficient Lanczos algorithm). By diagonalizing Hn to sufficient accuracy at the nth step, Hn+l can be diagonalized to any required accuracy at the (n+ 1)th step. A f t e r N steps we have solved the original problem in eq. (1), although each step only requires perturbation in the small parameter x/N. (2) Subspace extension. In perturbation theory, the perturbed eigenvectors are expressed in terms of coefficients multiplying the unperturbed eigenstates. As the order of perturbation theory increases, the number of coefficients associated with each state increases explosively, especially in field theory calculations. To control this blow-up, we propose to solve the eigenvalue problem at the nth coupling step to arbitrary accuracy only within a truncated subspace ~ n of the full Hilbert space. At the next (n+ 1)th coupling step, the subspace is extended in a controlled way, so that a gradual increase in the total coupling Xn = n X x / N is correlated with a gradual increase in the subspace of states ~ n . 405
Volume 138B, number 5,6
PHYSICS LETTERS
To explain the mechanics of all this, we first review the Lanczos tridiagonalization procedure. The idea is to start with the ground state if0 of the zeroth order theory H0, and then construct a tridiagonal N × N matrix coefficients by operating successively with the full hamiltonian, i.e. H ~ 0 = cOCO+ b 0 ~ l , Ht~m = bm-1 ~ m - 1 + Cm ~m + bm ~m+l.
(3)
The matrix of coefficients, with Cm on the diagonal and bm off-diagonal, is a tridiagonal matrix by construction, which can then be diagonalized with great efficiency. This method has been applied to hamiltonian finite lattice spin chains by Romany et al. [4]. The main drawbacks of this method are, again, the explosive increase (as N increases) in the number o f H 0 eigenstates needed to represent the states ~m, as well as the relatively slow convergence rate (as compared to our method below). Our modified Lanczos procedure is as follows: At the first (n = 1) coupling step, we compute H1 ff~l) = Cl ff~l) + b~(1) , H 1 = H 0 + (x/N) V .
(4)
Now ~O~1) is already an eigenstate, namely the ground state, of H0, while ~(1) can be decomposed into a set o f H 0 eigenstates. The subspace spanned by all these eigenstates will be denoted 9 @ We now do a sequence of substeps (m = 1, M) to determine the new ground state for H 1 within the restricted subspace c~ 1. This is done by repeated application of HI ff~l,m) ~_x Cl ff~l,m) + b~(1,m) + (neglected),
nl~(1,m ) ~1 bl]/~)l,m)+ c2~(1,m) + (neglected),
26 April 1984
Hn ~)n, 1) = ¢1 ~Dn' 1) + b-~(n, 1),
(6)
Hn = HO + Xn V .
But now the total set o f H 0 eigenstates, obtained in the decomposition of ~O~ n' 1) and ~(n, 1), defines an enlarged subspace 9gn. We then repeat the 2 × 2 Lanczos substeps n n ~),, rn) c~__nCl ~)n,m) + b~(n,m) ,
until convergence to the ground state o f H n in an extended ~gn subspace is obtained. At the Nth step, the original problem at coupling XN = x has been solved; the accuracy, of course, depends on the number of coupling subdivisions and Lanczos substeps that have been applied. Given the ground state at each n, excited states are constructed by an analogous procedure, with the subspace ~ n constrained to be orthogonal to ~0~n'm). This completes the algorithm. Its chief strength lies in the controlled extension of the solution subspace ~ n , which is only increased at each coupling step. This allows for the accurate determination of the best ground state within a given subspace, by large numbers of Lanczos substeps (in practice we have found M = 2 × N or 3 X N substeps is sufficient). In addition, it is not necessary at any stage to diagonalize a large (J X J, say) matrix, as in truncated-basis methods [5]. Instead, only manipulations on J component vectors are required. We will now show how this procedure works on some simple models. The first example is the Mathieu equation, Ht~ r = (-d2/d0 2 - x cos O) t]]r = Er~r ,
(5)
where the meaning of the symbol ~1 is that we omit, on the rhs of(5), all components which fall outside the subspace ~1- Each Lanczos substep m involves diagonalizing the 2 × 2 coefficient matrix generated by (5), to determine the improved ground state ~l,m+l) to be inserted into the (m+ 1)th Lanczos sub-step. The Lanczos sub-steps are repeated until convergence within the ~ 1 subspace is obtained. The next step, or in general the nth step, is to operate with H n on the ground state obtained at the ( n - 1 ) t h step ~O~ n'l)= ~(n-l,M),
(8)
which is equivalent to a one-dimensional XY model. The eigenfunctions and eigenvalues of (8) are tabulated (with slightly different notation) in ref. [6]. We have solved the eigenvalue problem using our method in the basis (1, X/2 cos(nO), ~ sin(n0)), treating separately the even and odd sectors. The hamiltonian of (8) is then represented by the tridiagonal matrix HKK = K 2 ,
HK,K+ 1 = HK+I, K = - x / 2
HKK = (K-- 1)2 ,
H12 = H21 = -x/X/c2
HK,K+ 1 = HK+I, K = - x / 2 .
406
(7)
Hn~(n,m ) ~__n bCDn,m) + C2-~(n,m) ,
(odd), (even) , (9)
Volume 138B, number 5,6
PHYSICS LETTERS
,o-3
which is equivalent to the hamiltonian version of the 2D XY-model in the L ~ oo limit. In the strong coupling basis exp (linK OK), the action of (10) on an arbitrary state is L
even (E o)
1oLJJ 10-5 Ld
26 April 1984
lO-S L
_ x ~ [~P[ml ... m/+l, m]+l--1 .... ] 2]=1
i0 -z
I 2'5
5
I 7'5 X
I0
125
+ qz[m I ... m ] - l , m/+l+ 1, ...]] .
Fig. 1. Plot of the difference in energy values Eexact-E between energies determined by our method (a [3,4] iterated Lanczos), and the exact values. Results are shown for both the ground state (e0) and first excited state (el), in a range coupling that lies well within the weak-coupling regime. Starting from the x = 0 eigenstates $0 --- 1, ~1 = X/2 sin(0), our method generates extremely accurate eigenvalues after two or three iterations in the weakcoupling region x = 2 . 5 - 1 2 . 5 (the strong-coupling series diverges after x = 0.7345). The results, compared to the exact values, are shown on fig. 1, which plots the difference (E - Eexact) between our values for the energy and the exact values, for both the ground and first excited states. We have tried a number of IN, M] schemes, where N = number of subdivisions and M = number of Lanczos substeps, but the differences between, e.g., the [3,4] and [2,6] values cannot even be resolved on fig. 1. The maximum dimension of the subspace (in the [3,4] scheme) used to produce these results was D = 11. A variant of the [N, M] procedure above is to fix all x n = x, i.e. treat the subdivisions simply as steps where the subspace is extended, but without any corresponding change in the coupling. This will be denoted [N, 3/]-fix. This variant also gives accurate eigenvalues. However, in testing the stability of the method over a long series of iterations, we observed an instability in [N, M]-fax after 12 iterations (similar instabilities occur in the standard Lanczos algorithm [7]). This instability was not observed after a similar number of iterations with the [N, M] method. The second model we discuss is a generalization of the Mathieu equation to a chain of L coupled quantum spins, L H = ~ [ - d 2 / d 0 k - x CoS(0K+ 1 -- OK)l , K=I
(10)
(11)
Each strong-coupling basis state can be represented by a string of {inK}. Starting at the x = 0 ground state tI'0 [0 ... 0] = 1, the Lanczos procedure stays within the EKmK = 0 sector of Hllbert space. Even so, the space spanned by the basis vectors expands explosively as the number of(ordinary) Lanczos steps (or the order of perturbation theory) is increased. This is the problem that subspace extansion, by freezing the subspace dimensionality over M Lanczos substeps, is intended to alleviate. In table 1 we compare results obtained for the ground state energy at L = 3 and x = 5, which is deep within the weak-coupling regime, by several variants of our method. Method I is a simple iteration of 2 × 2 Lancz0s steps, with no coupling subdivision and no freezing of the subspace, which is allowed to increase freely. Method II is the IN, M]-fix procedure, with no coupling subdivision, and method III is the full [N,M] scheme. For methods II and III we have used N = 5 (M= 2 - 3 XN). All three methods converge to the right answer, but the convergence rate of method I is rather slow compared to the other two methods. The convergence rates of methods II and III, i.e. IN, M] procedures with and without fixed coupling, turn out to be comparable. It can be seen from the table that the [N, M]-fLx method has converged to 1% accuracy already at the third iteration, using an eleven-dimensional subspace ~ 3 . This accuracy agrees with what is found by directly diagonalizing the hamiltonian in a truncated basis of the same dimensionality, which is the non-iterative procedure carried out in ref. [5]. The exact results were obtained by establishing six-place agreement between the results obtained in direct truncated-basis diagonalization in ~ 6 and ~ 7 subspaces. A side result of the [N, M] method, method III, is the determina407
Volume 138B, number 5,6
PHYSICS LETTERS
26 April 1984
Table 1 The convergence of three versions of the [N, M] method to the ground state energy at x = 5, as the step number n (associated with the subspace ~ n ) is increased. The methods are explained in the text. Method III gives an estimate for the ground state energy at each intermediate coupling Xn, while the corresponding exact values are shown in the rightmost column. n(C~n)
Ground state energy (Lsite = 3, x = 5) I
straight iteration M= 1 1 2 3 4 5 6 7
-7.80476 -9.43860 -9.72003 -9.76628 -9.77841 -9.78210 -9.78354
II IN, M] fix
III [N, M] (coupling division) (Ndiv = 5)
-7.80476 -9.56297 -9.77468 -9.78467 -9.78486
-0.822876 -2.79802 -5.0234 -7.36486 -9.78486
tion of the ground state energy and eigenstates at a set of intermediate couplings (in this case Xn = n). The exact intermediate energies are also displayed in table 1. The superior convergence of our method to the standard Lanczos scheme shows up even at strong couplings. Table 2 is a comparison of the [iV, M] method with standard Lanczos (method IV) on an L = 5 chain at coupling x = 1 (where we are able to compare with detailed published results [8]). Both methods converge; ours converges faster. Since the main problem one faces in calculations at moderate-to-weak x is the blow up in the size of the space of states, this enhanced convergence is crucial. Table 2 Ordinary Lanczos versus the iterated [N, M] method at strong coupling x = 1 on an L = 5 site chain, for several values of N. Ordinary Lanczos, at a given N, diagonalizes an (N + 1) × (N + 1) tridiagonal matrix; the eigenstates belong to an ~ N subspace. N
1 2 3 4 5 6
408
Ground state energy (Lsite = 5, x = 1) IV ordinary Lanczos
III [N, M]
-0.87083 -1.12826 -1.24072 -1.27391 -1.28732 -1.28816
-0.87083 -1.18628 -1.2707 -1.28349
xn
Exact ground state energy at x = x n
1 2 3 4 5
-0.8861 -2.81345 -5.0244 -7.36498 -9.78487
Our algorithm is also easily adapted to mass gap calculations, which, in the XY spin system, is the energy difference between the lowest states in the E m K = 1 and ~ , m K = 0 sectors. The exploratory run for L = 3 reproduced the results reported in fig. 4 of ref.
[4]. The most systematic investigation of the original Lanczos method, as applied to the 0 ( 2 ) and 0 ( 3 ) finite spin chains, has been made by Romany et al. [4]. They have analyzed 0 ( 2 ) chains of length up to L = 7, truncating the m at rn = 3, and 0 ( 3 ) chains up to L --- 5. Using 10 Lanczos steps they reached convergence on 9000- and 15 000-dimensional subspaces, respectively. We believe that the present improved method will allow the investigation of longer chains, which, if supplemented by finite size scaling analysis, should allow a better determination of the critical properties via an accurate determination of the mass gap. We also feel that the advantages of our procedure, which include the controlled expansion of the subspace, the correspondingly reduced storage requirements, and the fact that in our method there is no need to diagonalize large matrices, may allow its application to higher dimensional lattice systems, and to lattice gauge theory in particular. One of us (A.P.) thanks Pal Rujan for discussions on the Lanczos method, and would also like to express his gratitude to the Niels Bohr Institute for its continued hospitality.
Volume 138B, number 5,6
PHYSICS LETTERS
References [1] K. Wilson, Phys. Rev. D10 (1974) 2445; J. Kogut and L. Susskind, Phys. Rev. D l l (1975) 395. [2] S. Drell, tt. Quinn, B. Svetitsky and M. Weinstein, Phys. Rev. D22 (1980) 490; J. Greensite and J. Primack, Nucl. Phys. B180 [FS2] (1981) 170; J. Smit, Nucl. Phys. B175 (1980) 307; J-M. Blairon, R. Brout, F. Englert and J. Greensite, Nucl. Phys. B180 [FS2] (1981) 439; N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100; H. Kluberg-Stern, A. Morel, O. Napoly and B. Peterson, Nucl. Phys. B190 [FS3] (1981) 504.
26 April 1984
[3] T. Banks et al., Phys. Rev. D15 (1977) 1111; J. Shigemitsu, Phys. Rev. D18 (1978) 1709; G. Munster, Nucl. Phys. B190 [FS3] (1981) 1709; H. Kluberg-Stern, A. Morel and B. Peterson, Phys. Lett. l14B (1982) 152. [4] H.H. Romany and H.W. Wyld, Phys. Rev. D21 (1980) 3341. [5] C.J. Hamer, Phys. Lett. 82B (1979) 75. [6] M. Abramowitz and J. Stegun, eds., Handbook of Mathematical Functions (US Bureau of National Standards, Washington, DC, 1964). [7] J.H. Wilkinson, The algebraic eigenvalue problem (Clarendon, Oxford, 1965). [8] C.J. Hamer and M.N. Barber, J. Phys. A14 (1981) 259.
409
PHYSICS LETTERS
26 April 1984
EXTENDING THE REACH OF STRONG COUPLING: AN ITERATIVE TECHNIQUE FOR HAMILTONIAN LATTICE MODELS J. ALBERTY, J. GREENSITE and A. PATKOS 1 The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 1 7, DK-2100 Copenhagen O, Denmark Received 27 December 1983
We propose an iterative method for doing lattice strong-coupling-like calculations in a range of medium to weak couplings. The method is a modified Lanczos scheme, with greatly improved convergence properties. The technique is tested on the Mathieu equation and on a hamiltonian finite-chain XY model, with excellent results.
Strong-coupling calculations in lattice gauge theory can account for many qualitative features of the hadronic spectrum, such as confinement [1] and chiral symmetry breaking [2]. There is some reason to believe that the mass spectrum might attain realistic (i.e. continuum theory) values just inside the weak-coupling regime, since even at strong couplings the massratio calculations are not disastrously wrong [3]. This motivates a search for practical methods of doing strong-coupling-like spectrum calculations in regions of moderate to weak couplings. In this note we will outline such a method for hamiltonian lattice theories, and test it on some very simple models. Our hope is that the technique may be applicable to the more ambitious goal of analytical spectrum calculations in lattice gauge theory. Underlying our method is the Lanczos tridiagonalization scheme, which is a standard numerical technique for eigenvalue problems. To this scheme we add two new features: (1) coupling subdivision; and (2) subspace extension. We will discuss these in turn. (1) Coupling subdivision. This is a method for subdividing a large/3 problem into a series of small/3 problems. Consider the problem of finding the eigenvalues of the hamiltonian H=Ho+xV,
(1)
1 Permanent address: Department of Atomic Physics, E6tv6s University, Hungary. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where x is the perturbation parameter, and we imagine that the radius of convergence of perturbation theory is at x = 1. Suppose x 2> 1. Then consider solving, in N steps, the sequence of eigenvalue problems
H1 = u 0 + (x/N) v , Hn+ 1 = H n + (x/N) V ,
(2)
by Rayleigh-Schr6dinger perturbation theory (this is for purposes of explanation; in practice we will use the more efficient Lanczos algorithm). By diagonalizing Hn to sufficient accuracy at the nth step, Hn+l can be diagonalized to any required accuracy at the (n+ 1)th step. A f t e r N steps we have solved the original problem in eq. (1), although each step only requires perturbation in the small parameter x/N. (2) Subspace extension. In perturbation theory, the perturbed eigenvectors are expressed in terms of coefficients multiplying the unperturbed eigenstates. As the order of perturbation theory increases, the number of coefficients associated with each state increases explosively, especially in field theory calculations. To control this blow-up, we propose to solve the eigenvalue problem at the nth coupling step to arbitrary accuracy only within a truncated subspace ~ n of the full Hilbert space. At the next (n+ 1)th coupling step, the subspace is extended in a controlled way, so that a gradual increase in the total coupling Xn = n X x / N is correlated with a gradual increase in the subspace of states ~ n . 405
Volume 138B, number 5,6
PHYSICS LETTERS
To explain the mechanics of all this, we first review the Lanczos tridiagonalization procedure. The idea is to start with the ground state if0 of the zeroth order theory H0, and then construct a tridiagonal N × N matrix coefficients by operating successively with the full hamiltonian, i.e. H ~ 0 = cOCO+ b 0 ~ l , Ht~m = bm-1 ~ m - 1 + Cm ~m + bm ~m+l.
(3)
The matrix of coefficients, with Cm on the diagonal and bm off-diagonal, is a tridiagonal matrix by construction, which can then be diagonalized with great efficiency. This method has been applied to hamiltonian finite lattice spin chains by Romany et al. [4]. The main drawbacks of this method are, again, the explosive increase (as N increases) in the number o f H 0 eigenstates needed to represent the states ~m, as well as the relatively slow convergence rate (as compared to our method below). Our modified Lanczos procedure is as follows: At the first (n = 1) coupling step, we compute H1 ff~l) = Cl ff~l) + b~(1) , H 1 = H 0 + (x/N) V .
(4)
Now ~O~1) is already an eigenstate, namely the ground state, of H0, while ~(1) can be decomposed into a set o f H 0 eigenstates. The subspace spanned by all these eigenstates will be denoted 9 @ We now do a sequence of substeps (m = 1, M) to determine the new ground state for H 1 within the restricted subspace c~ 1. This is done by repeated application of HI ff~l,m) ~_x Cl ff~l,m) + b~(1,m) + (neglected),
nl~(1,m ) ~1 bl]/~)l,m)+ c2~(1,m) + (neglected),
26 April 1984
Hn ~)n, 1) = ¢1 ~Dn' 1) + b-~(n, 1),
(6)
Hn = HO + Xn V .
But now the total set o f H 0 eigenstates, obtained in the decomposition of ~O~ n' 1) and ~(n, 1), defines an enlarged subspace 9gn. We then repeat the 2 × 2 Lanczos substeps n n ~),, rn) c~__nCl ~)n,m) + b~(n,m) ,
until convergence to the ground state o f H n in an extended ~gn subspace is obtained. At the Nth step, the original problem at coupling XN = x has been solved; the accuracy, of course, depends on the number of coupling subdivisions and Lanczos substeps that have been applied. Given the ground state at each n, excited states are constructed by an analogous procedure, with the subspace ~ n constrained to be orthogonal to ~0~n'm). This completes the algorithm. Its chief strength lies in the controlled extension of the solution subspace ~ n , which is only increased at each coupling step. This allows for the accurate determination of the best ground state within a given subspace, by large numbers of Lanczos substeps (in practice we have found M = 2 × N or 3 X N substeps is sufficient). In addition, it is not necessary at any stage to diagonalize a large (J X J, say) matrix, as in truncated-basis methods [5]. Instead, only manipulations on J component vectors are required. We will now show how this procedure works on some simple models. The first example is the Mathieu equation, Ht~ r = (-d2/d0 2 - x cos O) t]]r = Er~r ,
(5)
where the meaning of the symbol ~1 is that we omit, on the rhs of(5), all components which fall outside the subspace ~1- Each Lanczos substep m involves diagonalizing the 2 × 2 coefficient matrix generated by (5), to determine the improved ground state ~l,m+l) to be inserted into the (m+ 1)th Lanczos sub-step. The Lanczos sub-steps are repeated until convergence within the ~ 1 subspace is obtained. The next step, or in general the nth step, is to operate with H n on the ground state obtained at the ( n - 1 ) t h step ~O~ n'l)= ~(n-l,M),
(8)
which is equivalent to a one-dimensional XY model. The eigenfunctions and eigenvalues of (8) are tabulated (with slightly different notation) in ref. [6]. We have solved the eigenvalue problem using our method in the basis (1, X/2 cos(nO), ~ sin(n0)), treating separately the even and odd sectors. The hamiltonian of (8) is then represented by the tridiagonal matrix HKK = K 2 ,
HK,K+ 1 = HK+I, K = - x / 2
HKK = (K-- 1)2 ,
H12 = H21 = -x/X/c2
HK,K+ 1 = HK+I, K = - x / 2 .
406
(7)
Hn~(n,m ) ~__n bCDn,m) + C2-~(n,m) ,
(odd), (even) , (9)
Volume 138B, number 5,6
PHYSICS LETTERS
,o-3
which is equivalent to the hamiltonian version of the 2D XY-model in the L ~ oo limit. In the strong coupling basis exp (linK OK), the action of (10) on an arbitrary state is L
even (E o)
1oLJJ 10-5 Ld
26 April 1984
lO-S L
_ x ~ [~P[ml ... m/+l, m]+l--1 .... ] 2]=1
i0 -z
I 2'5
5
I 7'5 X
I0
125
+ qz[m I ... m ] - l , m/+l+ 1, ...]] .
Fig. 1. Plot of the difference in energy values Eexact-E between energies determined by our method (a [3,4] iterated Lanczos), and the exact values. Results are shown for both the ground state (e0) and first excited state (el), in a range coupling that lies well within the weak-coupling regime. Starting from the x = 0 eigenstates $0 --- 1, ~1 = X/2 sin(0), our method generates extremely accurate eigenvalues after two or three iterations in the weakcoupling region x = 2 . 5 - 1 2 . 5 (the strong-coupling series diverges after x = 0.7345). The results, compared to the exact values, are shown on fig. 1, which plots the difference (E - Eexact) between our values for the energy and the exact values, for both the ground and first excited states. We have tried a number of IN, M] schemes, where N = number of subdivisions and M = number of Lanczos substeps, but the differences between, e.g., the [3,4] and [2,6] values cannot even be resolved on fig. 1. The maximum dimension of the subspace (in the [3,4] scheme) used to produce these results was D = 11. A variant of the [N, M] procedure above is to fix all x n = x, i.e. treat the subdivisions simply as steps where the subspace is extended, but without any corresponding change in the coupling. This will be denoted [N, 3/]-fix. This variant also gives accurate eigenvalues. However, in testing the stability of the method over a long series of iterations, we observed an instability in [N, M]-fax after 12 iterations (similar instabilities occur in the standard Lanczos algorithm [7]). This instability was not observed after a similar number of iterations with the [N, M] method. The second model we discuss is a generalization of the Mathieu equation to a chain of L coupled quantum spins, L H = ~ [ - d 2 / d 0 k - x CoS(0K+ 1 -- OK)l , K=I
(10)
(11)
Each strong-coupling basis state can be represented by a string of {inK}. Starting at the x = 0 ground state tI'0 [0 ... 0] = 1, the Lanczos procedure stays within the EKmK = 0 sector of Hllbert space. Even so, the space spanned by the basis vectors expands explosively as the number of(ordinary) Lanczos steps (or the order of perturbation theory) is increased. This is the problem that subspace extansion, by freezing the subspace dimensionality over M Lanczos substeps, is intended to alleviate. In table 1 we compare results obtained for the ground state energy at L = 3 and x = 5, which is deep within the weak-coupling regime, by several variants of our method. Method I is a simple iteration of 2 × 2 Lancz0s steps, with no coupling subdivision and no freezing of the subspace, which is allowed to increase freely. Method II is the IN, M]-fix procedure, with no coupling subdivision, and method III is the full [N,M] scheme. For methods II and III we have used N = 5 (M= 2 - 3 XN). All three methods converge to the right answer, but the convergence rate of method I is rather slow compared to the other two methods. The convergence rates of methods II and III, i.e. IN, M] procedures with and without fixed coupling, turn out to be comparable. It can be seen from the table that the [N, M]-fLx method has converged to 1% accuracy already at the third iteration, using an eleven-dimensional subspace ~ 3 . This accuracy agrees with what is found by directly diagonalizing the hamiltonian in a truncated basis of the same dimensionality, which is the non-iterative procedure carried out in ref. [5]. The exact results were obtained by establishing six-place agreement between the results obtained in direct truncated-basis diagonalization in ~ 6 and ~ 7 subspaces. A side result of the [N, M] method, method III, is the determina407
Volume 138B, number 5,6
PHYSICS LETTERS
26 April 1984
Table 1 The convergence of three versions of the [N, M] method to the ground state energy at x = 5, as the step number n (associated with the subspace ~ n ) is increased. The methods are explained in the text. Method III gives an estimate for the ground state energy at each intermediate coupling Xn, while the corresponding exact values are shown in the rightmost column. n(C~n)
Ground state energy (Lsite = 3, x = 5) I
straight iteration M= 1 1 2 3 4 5 6 7
-7.80476 -9.43860 -9.72003 -9.76628 -9.77841 -9.78210 -9.78354
II IN, M] fix
III [N, M] (coupling division) (Ndiv = 5)
-7.80476 -9.56297 -9.77468 -9.78467 -9.78486
-0.822876 -2.79802 -5.0234 -7.36486 -9.78486
tion of the ground state energy and eigenstates at a set of intermediate couplings (in this case Xn = n). The exact intermediate energies are also displayed in table 1. The superior convergence of our method to the standard Lanczos scheme shows up even at strong couplings. Table 2 is a comparison of the [iV, M] method with standard Lanczos (method IV) on an L = 5 chain at coupling x = 1 (where we are able to compare with detailed published results [8]). Both methods converge; ours converges faster. Since the main problem one faces in calculations at moderate-to-weak x is the blow up in the size of the space of states, this enhanced convergence is crucial. Table 2 Ordinary Lanczos versus the iterated [N, M] method at strong coupling x = 1 on an L = 5 site chain, for several values of N. Ordinary Lanczos, at a given N, diagonalizes an (N + 1) × (N + 1) tridiagonal matrix; the eigenstates belong to an ~ N subspace. N
1 2 3 4 5 6
408
Ground state energy (Lsite = 5, x = 1) IV ordinary Lanczos
III [N, M]
-0.87083 -1.12826 -1.24072 -1.27391 -1.28732 -1.28816
-0.87083 -1.18628 -1.2707 -1.28349
xn
Exact ground state energy at x = x n
1 2 3 4 5
-0.8861 -2.81345 -5.0244 -7.36498 -9.78487
Our algorithm is also easily adapted to mass gap calculations, which, in the XY spin system, is the energy difference between the lowest states in the E m K = 1 and ~ , m K = 0 sectors. The exploratory run for L = 3 reproduced the results reported in fig. 4 of ref.
[4]. The most systematic investigation of the original Lanczos method, as applied to the 0 ( 2 ) and 0 ( 3 ) finite spin chains, has been made by Romany et al. [4]. They have analyzed 0 ( 2 ) chains of length up to L = 7, truncating the m at rn = 3, and 0 ( 3 ) chains up to L --- 5. Using 10 Lanczos steps they reached convergence on 9000- and 15 000-dimensional subspaces, respectively. We believe that the present improved method will allow the investigation of longer chains, which, if supplemented by finite size scaling analysis, should allow a better determination of the critical properties via an accurate determination of the mass gap. We also feel that the advantages of our procedure, which include the controlled expansion of the subspace, the correspondingly reduced storage requirements, and the fact that in our method there is no need to diagonalize large matrices, may allow its application to higher dimensional lattice systems, and to lattice gauge theory in particular. One of us (A.P.) thanks Pal Rujan for discussions on the Lanczos method, and would also like to express his gratitude to the Niels Bohr Institute for its continued hospitality.
Volume 138B, number 5,6
PHYSICS LETTERS
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26 April 1984
[3] T. Banks et al., Phys. Rev. D15 (1977) 1111; J. Shigemitsu, Phys. Rev. D18 (1978) 1709; G. Munster, Nucl. Phys. B190 [FS3] (1981) 1709; H. Kluberg-Stern, A. Morel and B. Peterson, Phys. Lett. l14B (1982) 152. [4] H.H. Romany and H.W. Wyld, Phys. Rev. D21 (1980) 3341. [5] C.J. Hamer, Phys. Lett. 82B (1979) 75. [6] M. Abramowitz and J. Stegun, eds., Handbook of Mathematical Functions (US Bureau of National Standards, Washington, DC, 1964). [7] J.H. Wilkinson, The algebraic eigenvalue problem (Clarendon, Oxford, 1965). [8] C.J. Hamer and M.N. Barber, J. Phys. A14 (1981) 259.
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