Eigenvalues for the massive Schwinger model from a finite-lattice Hamiltonian approach

Nuclear Physics B 170 [FS1] (1980) 353-368 © North-Holland Publishing Company

EIGENVALUES FOR THE MASSIVE SCHWINGER MODEL FROM A FINITE-LATTICE HAMILTONIAN APPROACH D.P. CREWTHER 1 and C.J. I-IAMER2 School of Physics, University of Melbourne, Parkville, Victoria, Australia 3052 Received 17 September 1979 (Final version received 6 February 1980) A "finite-lattice" hamiltonian method is applied to the problem of calculating eigenvalues for the massive Schwinger model. The vacuum energy per site is obtained to an accuracy of 0.1%. The masses of the first two excited states are found to within an accuracy of order 1%, and between 1 and 5%, respectively. These results are an order of magnitude more precise than has been achieved previously. 1. Introduction T h e S c h w i n g e r m o d e l is t h e s i m p l e s t of all g a u g e field t h e o r i e s , a n d can b e e x a c t l y s o l v e d in two o p p o s i t e limits, o n e w h e r e t h e f e r m i o n s a r e massless [1], a n d t h e o t h e r w h e r e t h e f e r m i o n s a r e v e r y h e a v y [2]. This m a k e s it a s t a n d a r d t e s t - b e d for lattice t e c h n i q u e s . L a t t i c e h a m i l t o n i a n m e t h o d s w e r e first a p p l i e d to t h e p r o b l e m b y B a n k s , K o g u t a n d S u s s k i n d [3], a n d t h e i r results h a v e b e e n e x t e n d e d in s e v e r a l s u b s e q u e n t p a p e r s [4, 2, 5, 6]. T h e t e c h n i q u e s u s e d w e r e b o r r o w e d f r o m statistical m e c h a n i c s : s t r o n g c o u p l i n g p e r t u r b a t i o n series w e r e c a l c u l a t e d for t h e h a m i l t o n i a n e i g e n v a l u e s , a n d w e r e t h e n e x t r a p o l a t e d to t h e c o n t i n u u m limit using P a d 6 a p p r o x i m a n t s . E s t i m a t e s w e r e o b t a i n e d for t h e m a s s e s of t h e first two e x c i t e d states w h i c h w e r e a c c u r a t e * to a b o u t 10 o r 20 p e r c e n t . N o w t h e use of P a d 6 a p p r o x i m a n t s carries a m a j o r d r a w b a c k [7], in t h a t t h e c o n t i n u u m limit of l a t t i c e h a m i l t o n i a n m o d e l s is u s u a l l y a s i n g u l a r p o i n t , a n d P a d 6 a p p r o x i m a n t s a r e p o o r l y c o n v e r g e n t in t h e n e i g h b o u r h o o d of a singularity. V a r i o u s special t e c h n i q u e s h a v e b e e n s u g g e s t e d to d e a l w i t h this p r o b l e m [ 7 - 9 ] , b u t n o n e of t h e m a p p e a r e n t i r e l y satisfactory. A n a l t e r n a t i v e p l a n of a t t a c k has r e c e n t l y b e e n s u g g e s t e d b y o n e of us [10], w h i c h w e shall r e f e r to as t h e " f i n i t e : l a t t i c e " a p p r o a c h . T h e first s t e p is to f o r m a t r u n c a t e d basis set of s t r o n g c o u p l i n g e i g e n s t a t e s , o n a lattice of finite size M, a n d to r e p r e s e n t t h e h a m i l t o n i a n at a n y finite c o u p l i n g b y a m a t r i x o p e r a t o r acting o n this finite set of states. N e x t , this m a t r i x can b e e x a c t l y d i a g o n a l i z e d b y s t a n d a r d m e t h o d s . In this w a y a s e q u e n c e of e s t i m a t e s of t h e h a m i l t o n i a n e i g e n v a l u e s is o b t a i n e d , w h i c h r a p i d l y i Present address: National Vision Research Institute of Australia, 386 Cardigan Street, Carlton, Victoria 3053, Australia. 2 Present address: Department of Theoretical Physics, School of Physical Sciences, Australian National University, Canberra City, A.C.T. 2601, Australia. * Some selected results were a good deal more accurate than this, but these successes may have been partly fortuitous. 353

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D.P. Crewther, C.J. Hamer / Massive $chwinger model

converges as the lattice size M and the basis set of eigenstates is increased. So far, the approach is equivalent to a variational method of calculating the eigenvalues. The final step in the calculation is extrapolation of the eigenvalues for finite coupling to the continuum limit, which is simply carried out using linear or quadratic extrapolants. We expect this method to be more reliable than the Pad6 approximant approach, particularly when the continuum limit is singular. It has been tested successfully for the simple case of a non-relativistic harmonic oscillator on a lattice [ 10]; the purpose of the present work was to try it out on a fully relativistic field theory, i.e., the Schwinger model. The results may be summarized as follows: a remarkably accurate estimate for the vacuum energy per site within 0.1% of the exact value [11, 4] was found. For the first ("vector") excited state, the estimates obtained were accurate to 1 or 2% in both the zero (fermion) mass and non-relativistic limits, and interpolate smoothly between these limits for intermediate values of the fermion mass. For the second ("scalar") excited state, the estimates appear to achieve about 1-5% accuracy. The accuracy of the estimates is expected to decrease as one gets to higher excited states, because of the extra structure and complexity of these states. An order-of-magnitude increase in accuracy has thus been achieved, compared with previous lattice studies. This demonstrates that satisfactory quantitative results are now within reach, by lattice methods, for simple two-dimensional field theories. There remains much work to be done, however, before a similar statement can be made regarding four-dimensional QCD. A disappointing feature of the results is that we have found it necessary to go to high orders in perturbation theory in order to achieve the quoted accuracy. The convergence appears very slow for fermion models, in the "staggered lattice" formulation of Kogut and Susskind [12]. Unfortunately, we have not been able to think of a better way of doing things. The layout of the rest of the paper is as follows. In sect. 2, a brief review of the hamiltonian lattice formulation developed for the Schwinger model by Banks et al. [3] is given. In sect. 3, we indicate how a set of strong coupling eigenstates may be obtained on a finite lattice, together with the hamiltonian matrix connecting them. Sect. 4 illustrates the method of extrapolation to the continuum limit, using the case of the massless Schwinger model as an example. Sect. 5 contains results for the massive case, and in sect. 6 our conclusions are summarized.

2. The lattice theory We use the hamiltonian lattice formulation of Kogut and Susskind [12, 3]. The field theory is defined on a spatial lattice, of spacing a, with the time dimension continuous. At each site n of the lattice, a single component fermion field &(n) is defined, obeying the anticommutation relations {&(n), ¢(m)} = O,

{&+(n), ¢ (rn)} = 6 . . . .

(1)

D.P. Crewther, C.J. H a m e r / Massive Schwinger model

355

A two-component field tp can then be defined as

(2)

~0= ~ , o ' where toe = ¢(n),

n even,

~o = & (n),

n odd.

This results in a staggered lattice, where the fermions and antifermions can only reside on alternate sites, and a "unit cell' of the lattice contains two sites. The gauge field,

U(n, n + 1)

=

e iagA(n) ~

e i°(n) ,

(3)

is defined on the links between sites. H e r e A(n) is the spatial component of the abelian "vector" potential (we work in the class of gauges A ° = 0). The operator L(n) which generates cyclic translations in O(n) is just a multiple of the electric field, L(n ) = (1/ g)E(n ). The commutation relations between O(n ) and L(n ), [0(n), L(m)] = i6 . . . .

(4)

mean that these operators can be represented on a ladder space {[l}} such that

Lit) tit),

l = o, +1, ±2 . . . . .

=

(5)

The hamiltonian for the lattice Schwinger model is then

H

=-iy 2a

[d~÷(n)ei°(")~(n+l)-h.c.]+mE(-1)n~*(n)4~(n)+½gEa~L2(n).

(6)

Banks et al. [3] have demonstrated that this hamiltonian is gauge invariant, and reproduces the Dirac equation and Q E D correctly in the continuum limit. If one performs a Jordan-Wigner transformation [13]: ~ ( n ) = 1-[ [io'3(1)]~r-(n), '<"

(7)

c~+(n) = F[ [-io'3(l)]o'+(n), l
where the o-i(n) are Pauli spin matrices at each site, then the hamiltonian can be represented by H = ~ l z [~r+(n) ei°(")o'-(n + 1) +h.c.]+½m • (-1)"o'3(n)+½g2a EL2(n). za n , n Now define the dimensionless operator

W=

22H= Wo-xV, ag

(8)

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D.P. Crewther, C..L Hamer / Massive Schwingermodel

where

Wo=EL2(n)+½~ E (-1)"~3(n), "

"

(9)

V = E [~r+(n) ei°~")er-(n + 1) +h.c.], rt

tz-

2m 2 , ga

x-

1 2 2ga

(10)

For x << i one can employ strong coupling perturbation theory on this model, treating the operator V as the perturbation.

3. Strong coupling expansion Our next objective is to obtain the unperturbed strong' coupling eigenstates of the operator W0, and calculate matrix elements of the hamiltonian between them. The unperturbed ground state, 10), is an eigenstate of

o'3(n) and L(n), with o ' 3 ( n ) = - ( - 1 ) " , and L ( n ) = 0.

(11)

This will be represented diagrammatically by an empty lattice (e.g., fig. la). We will then represent sites containing "flipped" spins, relative to the ground state, by vertical lines with arrows denoting the spin direction (i.e., whether the excitation is a fermion or an antifermion). Gauge field excitations will be denoted by wiggly lines at the appropriate link, with the n u m b e r of lines denoting IL(n)l, and an arrow denoting the sign of L(n).

G

t-- t

-t

t

b

÷t ¢

Fig. 1. Diagrammatic representation of unperturbed strong-coupling eigenstates on a 2-site lattice with periodic boundary conditions. (a) Ground state; (b) "vector" state; (c) "scalar" state.

D.P. Crewther, C.J. Hamer / Massive Schwinger model

357

The first two excited states in the strong coupling limit, then, are the "vector" state

I1-) = ~ 1 Z [o'+(n) ei°(")o'-( n + 1) - h.c.]10),

(12)

where M is the number of sites in the lattice, and periodic boundary conditions are assumed, and the "scalar" state,

12+)= ~ 1 E [o'+(n) eiO(,)cr_(n

+ 1) + h.c.][0).

(13)

These may be represented diagrammatically as in figs. lb, c. The unperturbed energy of the ground state is defined to be zero: relative to this, the vector and scalar states have unperturbed "energy" (i.e., eigenvalue of W0) equal to 1 + 2ix. Our general method of calculation is to assume a lattice with some fixed number of sites M, start with the ground state 10), and then generate a series of new states by applying the perturbation operator V successively up to order N. The set of states { V" [0), n = 0, 1 . . . . . N} is then orthogonalized by a Gram-Schmidt procedure, and the result is a set of orthonormal, even parity, translationally invariant* eigenstates of W0. The matrix elements of V between these states may be computed simultaneously. For a 2-site lattice at order N = 1, for example, we have only two even-parity states generated, the ground state[0) and the scalar state 12+) illustrated in figs. la, c. The hamiltonian matrix connecting them is

o W+=

_x/~ x

-4 x] l+2g

j,

(14)

and the ground state energy per site, in this approximation, is M°J°= ½(1 + 21z - 4(12+ 2tz)2 + 8x2) "

(15)

A similar procedure may be carried out for the negative-parity states, starting from the vector state [1-). We have used a computer for the calculations, although for this simple model one may proceed to quite high orders by hand. This same method has previously been used on the two-dimensional Ising model by Barber and Hamer [14]. There, a lattice of finite size M can contain only a finite set of states, so that by going to a sufficiently high order of perturbation N one may obtain the exact energy eigenvalues on the finite lattice. The Schwinger model, by contrast, is "open ended", because one may proceed indefinitely to generate new

Fig. 2. Diagram for an (even-parity) strong-coupling eigenstate containing a constant "background" fietd of strength [L(n)[ = 2. *" A "translation" corresponds to displacement by an even number of sites, on the staggered lattice.

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D.P. Crewther, C.J. Hamer / Massive Schwinger model

states which contain higher and higher values of a constant, " b a c k g r o u n d " gauge field. An example is shown in fig. 2. It is therefore not clear a priori what sequence of values (N, M ) will give the most useful set of approximations to the eigenvalues: the question must be settled by trial and error. We proceed to a discussion of this matter, and of the method of extrapolation to the continuum limit, in sect. 4.

4. Extrapolation to the continuum limit: the massless Schwinger model By the methods outlined above, one obtains a finite hamiltonian matrix corresponding to any given pair of parameters (N, M ) ; this may be diagonalized by standard methods to give its exact eigenvalues for any value of the inverse coupling p a r a m e t e r x. Our next object is to extrapolate these results, in some fashion, to the continuum limit (M -~ oo, a ~ 0, x -~ oo). To illustrate the procedure, we shall discuss the case of the massless Schwinger model (/z = 0), where exact results are known [1]. 4.1. GROUND-STATE ENERGY In the continuum limit, x = 1/g2a2~oo, the hamiltonian is dominated by the " p e r t u r b a t i o n " term xV. The operator V is just the hamiltonian of the X - Y antiferromagnetic spin lattice [3], for which the ground-state energy is known exactly [11]. Hence one expects lim ~Oo/2Mx = - l / r r

=-0.3183,

(exact).

(16)

x ..-~ ¢x3, M~oo

We therefore plot values for the quantity

fo(X ) = ~Oo/2Mx

(17)

versus 1/~/x (cf. discussion below), for various pairs of values (N, M ) (see fig. 3). Now for a fixed lattice size M, t h e approximations obtained as one raises the order of perturbation N are found to " m a p out" the exact eigenvalue to successively larger values of x, within any given level of accuracy. In turn, as one now raises the lattice size M, the eigenvalue for the infinite lattice is m a p p e d out to successively larger values of x. For our present purposes, we are not interested in following the eigenvalue for a given M beyond the point where it diverges from the infinite lattice result. We find that an order of perturbation N = M - 2 is generally sufficient to carry us to this point. Fig. 3 shows the approximations obtained to fo(X) for N = M - 2, M = 4, 6 and 8. It can be seen that the approximations converge with r e m a r k a b l e rapidity, so that the 6-site and 8-site results are indistinguishable even as far as 1/~/x = 0.05. A linear extrapolation of this curve to the continuum limit gives an estimate lim COo = - 0 . 3 1 8 6 , x-,oo, 2 M x M

(lattice estimate),

-~ o o

which agrees with the exact result (16) within 0.1%.

(18)

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D.P. Crewther, C.J. Hamer / Massive Schwinger model i

-0-1

-02 0

t 2 / . ~ I

I

I

I

I

I

l

I

I

I

Fig. 3. Successive approximations to the vacuum energy function [o(X) of the massless Schwinger model as a function of 1/#x, for different perturbation orders N and lattice sizes M. The curves are labelled IN, M]. The [4, 6], [6, 8] approximations are indistinguishable from the bulk limit (solid line), right in to 1/#x = 0.05.

4.2. VECTOR AND SCALAR PARTICLE MASSES W e e x p e c t t h e p h y s i c a l p a r t i c l e m a s s e s to b e finite in this s u p e r r e n o r m a l i z a b l e t h e o r y [3], i.e., we e x p e c t m i = l i m oJi - w0 g x-,oo ~

(19)

M~oo

to b e finite. T h e e x a c t s o l u t i o n [1] for t h e m a s s l e s s S c h w i n g e r m o d e l is, in fact*, m1 T = ~ = 0.564,

(vector), (20)

m+ 2 --~-- = ~--~ = 1 . 1 2 8 ,

(scalar).

T h e a p p e a r a n c e of a f a c t o r ~/x in eq. (19) l e a d s us to e x p e c t a s q u a r e - r o o t s i n g u l a r i t y (at least) in t h e b e h a v i o u r of t h e e i g e n v a l u e s at t h e c o n t i n u u m limit, x -->~ . T h i s is t h e r e a s o n for t h e p o o r c o n v e r g e n c e of P a d ~ a p p r o x i m a n t e s t i m a t e s [7, 2, 8, 5]. *

In the massless limit, the Schwinger model [1] reduces to the theory of a non-interactingmassive boson (the vector particle). The scalar state is then merely a pair of these bosons.

360

D.P. Crewther, C.J. Hamer / Massive Schwinger model i

.t °, \

tl0.~2j \

rnlg: 0

I

I

I

o

I

I

i

i

I

i

I

Fig. 4. Successive approximations, labelled by [N, M], to the excited state masses [i(x) for the massless Schwinger model. The lower curves are for the vector state, the upper ones for the scalar. The solid lines are the bulk limits, as mapped out by these successive approximations.

W e are led to plot the quantity

(,oi-,oo) ~(x) = 2---~x

(21)

versus 1/x/x. Fig. 4 shows the approximations o b t a i n e d to [_(x), f+(x) for p a r a m e t e r s N = M-2, M = 4, 6, 8, 10 and 12. O n c e again, it can be seen that the a p p r o x i m a tions m a p out the infinite lattice result (solid line) to successively larger values of x, although the c o n v e r g e n c e is not nearly so spectacular as for the g r o u n d - s t a t e energy. A similar plot for the ratio f+(x)/[_(x) is shown in fig. 5. O u r task n o w is to extrapolate these curves to the c o n t i n u u m limit, i.e., the axis 1/x/x = 0. F o r this purpose, we have calculated straight line and parabolic fits to the [N, M ] curves at each value of x, and calculated the intercepts of these fits on the axis 1/x/x = 0: these are the " l i n e a r " and " q u a d r a t i c extrapolants", respectively. Fig. 6 shows these extrapolants G_, G+ for the vector and scalar states, respectively, as a function of 1/x/x. O n c e again, it can be seen that the [N, M ] approximations m a p out the infinite lattice results (solid lines) as one goes to higher orders. It can also be seen that the linear and quadratic extrapolants for the vector state converge and flatten out very nicely as o n e approaches 1/x/x = 0.5, which is about as far as these results will take us. Thus one m a y be confident that [_(x) can be well a p p r o x i m a t e d by a straight line over the range 1/x/x = 0 to 0.5 (weak coupling regime); and that its

D.P. Crewther, CJ. Hamer / Massive Schwinger model

361

~.8]

[8JO~\ \\

m/g=0

I

I

I

I

I

I

'

1

0

I

I

1/~

Fig. 5. Successive approximations to the mass ratio f÷(x)/f_(x) versus 1/Vx for the massless Schwinger model. Notation as for fig. 4.

intercept is accurately determined by the linear extrapolant at 1/x/x = 0.5. Hence we estimate m-

= lim {/_(x)}= 0.560.

g

(22)

x~OO M---, oo

The linear extrapolant is generally more stable than the quadratic. Inspection of fig. 6 shows that the best estimates for the continuum eigenvalues at a given order

[8,~] t6#,

ro~] ~,

//~

[6~BL

,,.

quod

m/g = 0 i

1

1/¢'~ Fig. 6. Extrapolants for the vector and scalar states of the massless Schwinger model, as a function of 1/x/x. Successive approximations to the linear extrapolants are labelled [N, M]I, and to the quadratic, [N, M]2. The solid lines show the bulk limits for these quantities, as far as they can be m a p p e d out.

362

D.P. Crewther, C.J. Hamer / Massive $chwinger model

(N, M ) are the values of the linear extrapolant at its first turning point* for the vector state, and at its second turning point for the scalar state. We can then obtain a crude idea of the errors in these estimates by looking at the discrepancy between the linear and quadratic extrapolants at these points. A sequence of such estimates for the two states is given in table 1. It can be seen that the final error estimate for the scalar state, 5%, is a good deal larger than for the vector. As can be seen from fig. 4, this is because the higher excited state has more "structure" at large x and also the ( N , M ) approximations converge more slowly at any given x; thus the extrapolation to the axis is inevitably more uncertain. The discrepancy from the exact value turns out to be less than 1%, in fact: but this may be fortuitous. TABLE 1 Estimates for vector and scalar masses in the massless Schwinger model, obtained for a series of lattice sizes M, at orders of perturbation N = M - 2

(N, M)

m_/g

m+/g

(4, 6) (6, 8) (8, 10) (10, 12) Exact

0.49 + . 0 7 0.56 ± . 0 2 0.555 +.01 0.560+.01 0.564

1.44±.2 1.22±.12 1.15 + .08 1.12+.05 1.128

4.3. F I N I T E - S I Z E S C A L I N G E S T I M A T E S

A second approach to the continuum limit may be made using the "finite-size scaling" method of Fisher and Barber [15, 14]. The idea here is to take a sequence of eigenvalues for lattices of different size M at a fixed value of x, and extrapolate to the limit M-~ oo using finite-size scaling. In the present case, since no phase transition occurs in the model, we expect an exponential approach to the bulk limit:

f~(x)-fl~(x) ~

~ e -BM ,

(23)

where fM (x) denotes the eigenvalue for lattice size M. Now for small lattices it is easy to go to high orders N, and map out the finite-lattice eigenvalues to relatively large values of x: we have computed up to values (N, M ) = (26, 4), (26, 6), and (20, 8). Assuming that the scaling relation (23) holds at such small values of M, one can deduce the bulk limit f~(x). The results of this procedure are shown in fig. 7. The dotted lines show the finite-lattice eigenvalues, and the solid line the estimated bulk limits. A problem arises in the region where the finite-lattice curves are crossing one another: the scaling hypothesis, eq. (23), is clearly inapplicable in this region. * The turning point for the vector state is hardly visible in fig. 6.

D.P. Crewther, CJ. Hamer / Massive Schwinger model

363

i

fi(x)

1"5

tms]~--:-- _ - - - ~ - - - ~ - ~ - - -

.

1.0

G5 *

i

I

i

I

I

I

I

I

I

V~

Fig. 7. Finite-size scaling results for the massless Schwinger model. T h e dashed curves, labelled [N, M ] are essentially exact results for the vector and scalar masses on small, finite lattices. T h e solid lines are estimates of the bulk limit, using the finite-size scaling prediction, eq. (23).

Nevertheless, one can use the finite-size scaling technique to m a p the bulk limit out to about the same values of x as for our first technique; and one can even apply it beyond the crossover region, to obtain what appear to be sensible estimates of the eigenvalues very close to the continuum limit (fig. 7).

5. Results for the massive Schwinger model Because chiral symmetry is broken in the (abelian) massless Schwinger model, the unperturbed vacuum state turns out to be the same in the massive case as in the massless case [3]*. H e n c e we would expect our finite-lattice techniques to work just as well for the massive fermion case as for the massless case; i.e., the ground-state energy is again dominated by the x V term in the hamiltonian, and the continuum limit is expected to be the same as for the massless case. Our estimates of the vacuum energy verify this expectation to within the same level of accuracy as eq. (18). Fig. 8 shows the "binding energies" (~oi - OJo) 2 m fi(x) = ~ g

(24)

* This is in contrast to non-abelian gauge models where a phase transition is expected as the quark mass goes to zero.

364

D.Po Crewther, C.J. Hamer / Massive Schwinger model i

~)

[6.s]

m / g = 10

8,1q.1;"",-: / ,

I

,

i

0

I

,

1

,

,

i

1/,/'~

Fig. 8. Binding energies f/(x) versus 1/~/x, for m/g = 10. Notation as in fig. 4.

versus 1/x/x for the vector and scalar states, at m / g = 10 (which is in the nonrelativistic regime). It is clear that the finite-lattice estimates do not depart from the bulk limit until much higher values of x in this case; but the structure of the graphs is also pushed to higher values of x. The linear and quadratic extrapolants Gi for the vector and scalar states at m / g = 8 are shown in fig. 9. They behave in a similar fashion to the massless case, and the resulting errors attached to our estimates are also very similar; although the extrapolants do become somewhat less stable with increasing mass (cf. figs. 6 and 9). Estimates for the continuum limit of the vector and scalar binding energies are plotted against m / g in fig. 10, over a range 0 . 0 1 < m / g < 100. These may be compared with known exact results:

E_

m_ g

2m 0.564-0.219(re~g), g m/g-~o

E+

m+ g

2m 1.128 + 1 . 5 6 2 ( m / g ) , g m/g-,O

(25)

in the small mass limit [3, 4], and

E_ m / g ~ 0"642(re~g)-1~3 (26)

E+

~ m/g~C~

1.473(m/g) -1/3 ,

D.P. Crewther, C.J. Hamer / Massive $chwinger model

1

365

[6,8k

[8,1012',,

'

i

\

i

i

i

I

0

i

i

i

i

1 1/~-~

Fig. 9. Linear and quadratic extrapolants for the vector and scalar masses, for m/g = 8. Notation as in fig. 6.

I

I' ~ f

SCALAR

i\

\l

\l \1 e

® b e

VECTOR

0 •01

I 0.1

I 1

I 10

I 100

m/g Fig. 10. Estimates for the c o n t i n u u m binding energies of the vector and scalar states of the massive Schwinger model, as a function of m/g. T h e points are the finite-lattice estimates; the solid lines show the expected behaviour in the large- and small-mass limits; and the dotted lines are merely to guide the eye.

D.P. Crewther, C.J. Hamer / Massive Schwinger model

366

in the large mass limit [2]. These results are shown as solid lines in fig. 10. It can be seen that the lattice estimates generally agree with the known limiting results to within the expected systematic errors, except for the vector state in the nonrelativistic limit, where the estimates fall a little low. We therefore expect that our estimates can be trusted to a similar level of accuracy for intermediate values of m/g, where no exact answers are known. The vector state changes smoothly from the ultra-relativistic to the non-relativistic regimes, with a transition region centred around m / g ~-0.4; while the scalar state appears to develop a peak in the transition region, at around m / g = 1. Lastly, our estimates for the ratio E + / E _ are plotted against m / g in fig. 11. Again, they seem to agree within the expected errors with the known behaviour in the large [2] and small [3, 4] mass limits: E+ E _ mlg~O

2 + 3.55(re~g) m/~- ~ 2.295.

(27)

Again, a peak seems to develop in the transition region near m / g = 1.

6. Conclusions Application of the finite-lattice approach to the massive Schwinger model has b e e n shown to produce a reliable and nicely convergent sequence of estimates for the continuum eigenvalues. The ground-state energy per site has been estimated, and agrees with the exact result [4, 11] to within 0.1%. The binding energies of the vector and scalar excited states have been estimated to an accuracy of order 1%, and 1-5 %, respectively. It thus appears that satisfactory quantitative results are now within reach by lattice methods for reasonably simple two-dimensional models.

E~

E 3

i~-!-.l I i ~

-01

0.1

1 m/g

10

100

Fig. 11. Estimates of the continuum ratio of binding energies between the scalar and vector states, versus m/g. Notation as in fig. 10.

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Fig. 12. A perturbation diagram for the translation of a fermion-antifermion state from one lattice cell to the next. Diagram conventions as in Banks et al. [3]. This diagram is fourth order in x. Two methods were considered for mapping out the "bulk limit" ( M ~ oo) of an eigenvalue as a function of the inverse coupling p a r a m e t e r x. (a) Increase the order N and lattice size M in tandem, with N = M - 2 . The resulting sequence of curves, as M increases, will rapidly converge at any finite x, and m a p out the bulk limit, with a high degree of accuracy, to larger and larger values of x; (b) Fix the lattice size M and increase the order N as far as possible. By this means, the exact eigenvalue for the finite lattice can be m a p p e d out to very large values of x. Next, obtain a series of such curves for different lattice sizes M, and use finite-size scaling [15, 14] to predict the bulk limit at a given value of x. The final step in both methods is to extrapolate to the continuum limit (x ~ o0) using linear a n d / o r quadratic extrapolants. Results of a similar accuracy seem to be obtainable by either method. A somewhat disturbing feature of the results is that we have had to go to quite high orders, e.g. (N, M ) = (10, 12), to achieve the quoted accuracy. This is equivalent to a perturbation series expansion for the eigenvalues up to twentieth order in x. It appears that this slow convergence may be attributed, in part, to the "staggered lattice" formulation [12] for fermions: e.g., the motion of a state from one point to another on the lattice only occurs via a relatively complicated and high-order diagram [3, 4] in this formulation - see fig. 12, for example. We have not been able to think of a better arrangement. If one is satisfied with a lower level of accuracy however, lower-order calculations will suffice. A glance at table 1 shows that quite reasonable estimates of the eigenvalues can be obtained a t ( N , M ) = (6, 8), for instance. Nevertheless, it is clear that a great deal of work, and m o r e efficient calculational methods, will be required before reliable results can be obtained for four-dimensional Q C D . We would like to thank Dr. M.N. Barber for advice regarding finite-size scaling.

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References [1] J. Schwinger, Phys. Rev. 128 (1962) 2425; J. Lowenstein and J. Swieca, Ann. of Phys. 68 (1971) 172 [2] C.J. Hamer, Nucl. Phys. B121 (1977) 159; B132 (1978) 542 [3] T. Banks, L. Susskind and J. Kogut, Phys. Rev. D13 (1976) 1043 [4] A. Carroll, J. Kogut, D.K. Sinclair and L. Susskind, Phys. Rev. D13 (1976) 2270 [5] R.D. Kenway and C.J. Hamer, Nucl. Phys. B139 (1978) 85 [6] D.R.T. Jones, R.D. Kenway, J.B. Kogut and D.K. Sinclair, to be published [7] A. Carroll, G.A. Baker Jr. and J.L. Gammel, Nucl. Phys. B129 (1977) 361 [8] J. Jurkiewicz and J. Wosiek, Nucl. Phys. B135 (1978) 416; B145 (1978) 445 [9] A. Carroll and J.B. Kogut, Illinois preprint ILL-(TH)-78-52, Phys. Rev. D, to be published [10] C.J. Hamer, Phys. Lett. 82B (1979) 75 [11] D. Mattis, The theory of magnetism (Harper and Row, New York, 1975) [12] J. Kogut and L. Susskind, Phys. Rev. D l l (1975) 395 [13] P. Jordan and E.P. Wigner, Z. Phys. 47 (1928) 631 [14] M.N. Barber and C.J. Hamer, Melbourne preprint UM-P-79-21, J. Phys. A, submitted [15] M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28 (1972) 1516