THE SHIFTED COUPLED CLUSTER METHOD: A NEW APPROACH

arXiv:hep-lat/9212025v1 18 Dec 1992OUTP-92-42PCPT-92/PE.2840December 1992THE SHIFTED COUPLED CLUSTER METHOD: A NEW APPROACHTO HAMILTONIAN LATTICE GAUGE THEORIESC H Llewellyn SmithDepartment of Physics, Theoretical Physics, 1 Keble Road,Oxford OX1 3NP, England.N J WatsonCentre de Physique Th´eorique, C.N.R.S. - Luminy, Case 907,F-13288 Marseille Cedex 9, France.AbstractIt is shown how to adapt the non-perturbative coupled cluster method of many-bodytheory so that it may be successfully applied to Hamiltonian lattice SU(N) gauge theories.The procedure involves first writing the wavefunctions for the vacuum and excited statesin terms of linked clusters of gauge invariant excitations of the strong coupling vacuum.The fundamental approximation scheme then consists of i) a truncation of the infinite setof clusters in the wavefunctions according to their geometric size, with all larger clustersappearing in the Schr¨odinger equations simply discarded, ii) an expansion of the truncatedwavefunctions in terms of the remaining clusters rearranged, or “shifted”, to describe gaugeinvariant fluctuations about their vacuum expectation values.

The resulting non-lineartruncated Schr¨odinger equations are then solved self-consistently and exactly. Results arepresented for the case of SU(2) in d = 3 space-time dimensions.

This letter reports the first results of a new approach to Hamiltonian lattice gaugetheories. The goal is to find approximate non-perturbative solutions of the Schr¨odingerequation for both vacuum and excited states.

Space is treated as a discrete lattice in orderto i) reduce field theory to n →∞body quantum mechanics, ii) render the theory ultra-violet finite, and iii) make trivial the imposition of Gauss’s law, which must be imposedas a subsidiary condition on the states in an A0 = 0 gauge, which is the natural choicewith a Hamiltonian approach1.The basic variables are the SU(N) matrices U(L) defined on each link L. Conjugate“chromoelectric” fields Ea(L), a = 1 . .

. N 2 −1, are given by2Ea(L) = −T aU(L)ij∂∂U(L)ij+U †(L)T aij∂∂U †(L)ij(1)in Schr¨odinger representation, where T a are the group generators.

The Hamiltonian forSU(N) in d space-time dimensions, without quarks, may be written [2]H(x) = a−1x−1/2W(x)(2)where the dimensionless variable x is related to the coupling g by x = 4g−4a2d−8, andW(x) =XLEa(L)2 −xXpRe Up(3)where Up is the trace of the product of Us and U †s around the four sides of a plaquettep: Up = TrU(1)U(2)U †(3)U †(4).The Schr¨odinger equation for the “reduced” energyw = ax1/2E is thenWΨ = wΨ(4)The continuum limit occurs at x →∞.We consider first perturbation theory about the strong coupling x = 0 vacuum, butemphasize that the method we shall develop applies at all couplings 0 ≤x < xc, where1 To avoid ghosts, a non-covariant gauge is necessary, which must be A0 = 0 to retain rotationalinvariance. In contrast to QED, Gauss’s law cannot be solved explicitly [1].2 This operation, in which the elements of the unitary matrices U(L) are treated as if they wereindependent, and with the indices i, j summed over, correctly reproduces the fundamental commutationrelations for Ea(L) with U(L′), U †(L′) and Ea′(L′).1

xc is the lowest x at which there is a phase transition. For SU(N), it is expected thatxc = ∞i.e.

the continuum limit. At x ≪1, the vacuum wavefunction Ψ0 of, for example,the SU(2) theory in d = 3 isΨ0 = 1 + x13Xs+ x2118XsXs′−172Xs−1117Xs++8351Xs++ x31162XsXs′Xs′′−1216XsXs′+ .

. .

−88163180Xs+ . .

. (5)where the sums are over all sites s on the lattice.

As shown originally by Hubbard [3] inthe case of continuum field theory, and observed by Greensite for lattice theories [4], theproducts of sums of independent excitations exponentiate:Ψ0 = expS(6)For all couplings 0 ≤x < xc, and for general SU(N) and d, the exponential form (6)holds, where the function S is a single sum over the sites s of the lattice of all possiblelinearly independent “linked clusters” Ci,s of SU(N) Wilson loops:S =XsXiaiCi,s(7)By “linked” is meant products of Wilson loops at fixed relative separation and orientation.The “amplitudes” ai are then the coefficients of these clusters, summed to all ordersin strong coupling perturbation theory. The representation (6),(7) holds, and is exact,provided the true vacuum is not orthogonal to the strong coupling vacuum Ψ0(x = 0)i.e.

up to the lowest x at which there is a phase transition. Thus, for any N and d, theSchr¨odinger equation for the vacuum energy is equivalent toXLEa(L)2S + Ea(L)S · Ea(L)S−xXpRe Up =Xsw0(8)The wavefunctions of excited states (glueballs) with zero momentum can be writtenΨg = G expS(9)2

whereG =Xsb0 +XibiCi,s(10)i.e. G is a similar sum3 over the lattice of linked clusters of SU(N) Wilson loops.

Thecorresponding Schr¨odinger equation is thenXLEa(L)2G + 2Ea(L)S · Ea(L)G= (wg −w0)G(11)The effect of the electric field operators on a given link L in (8), (11) is to recombinethe clusters in S and G containing U(L) or U †(L) according to the relation for SU(N)generators T aαβT aγδ = 12δαδδβγ −12N δαβδγδ. Some examples for general SU(N) areEa(L)2= N2−12Nr r ✻✻Ea(L)· Ea(L)= 12−12Nr r✻r r❄✦❛✻❄✻❄where the dots indicate the ends of the link L.The exponential form (6) for the vacuum wavefunction is in fact a very general ex-pression for the fully interacting ground state of a many-body system, with S a sum oflinked clusters each of which describes linked excitations of a bare or unperturbed stateΨ0(x=0).

The exponentiation then gives the correct statistical weighting for unlinked ex-citations consisting of products of linked components. Furthermore, it guarantees that theSchr¨odinger equation for the vacuum and for excited states is a single sum of linked termsonly, so that there are then no size extensivity problems with approximation schemes.

Theexpression (6) is the starting point for the many-body theory coupled cluster method4.This is an intrinsically non-perturbative approach, involving the amplitudes ai, bi directlyrather than their perturbative expansions.In order to apply the coupled cluster method to lattice SU(N) gauge theories, thefundamental approximation in general consists of two steps:3 Excited states of a given spin and parity are constructed through the appropriate dependence of thebi on the clusters’ rotation/reflection orientations, left implicit in (10).4For an introductory review of this method, which has been widely used in nuclear and condensedmatter physics and quantum chemistry, see ref. [5].

Application to lattice gauge theory was suggested byGreensite [4], but this suggestion was not followed up.3

I) A truncation of the sums of clusters in S and G according to the clusters’ geometricalsize. All clusters then formed in the Schr¨odinger equations with the truncated S and Gwhich are larger than the given cut-offare simply discarded.II) A Taylor expansion of S and G in the remaining clusters, rearranged, or “shifted”, todescribe fluctuations of the individual Wilson loops about their vacuum expectation values.These expectation values are calculated self-consistently from the vacuum wavefunction.The first step was originally suggested by Greensite [4], but in order for the methodto provide a tractable, valid and efficient approximation scheme beyond the perturbativerange of couplings, the second step is in general essential.The physical idea is thatamplitudes in S involving fluctuations on scales larger than the vacuum correlation lengthξ, and in G larger than the size ρ of the glueball, are expected to be greatly reducedrelative to those for fluctuations smaller than ξ and ρ, and to decrease very rapidly withincreasing size.

For fixed lattice spacing, the results should therefore converge rapidly tothe exact result as the dimensions of the cut-offcluster size are increased beyond ξ andρ.As the lattice spacing a →0, ξ/a and ρ/a diverge and the size in lattice units ofclusters required to describe the theory must diverge correspondingly. The success of themethod therefore depends on the results obtained at successsively increasing values of x(i.e.

decreasing a) showing a sufficiently rapid approach to the expected scaling behaviourthat the continuum limit can be extrapolated.We consider first the case of SU(2) with a one-plaquette cut-off, for which step I isexactly soluble. For SU(2), the identity TrA TrB = TrAB + TrAB−1 for SU(2) matricesA, B gives linear relations between clusters e.g.=+☎☎❉❉= 2 +☎☎❉❉Thus, the one-plaquette clusters may be written as “powers” of the one-plaquette singleloop C1,p.

Then, with the one-plaquette cut-offS, writing the infinite sum of one-plaquetteclusters on each p as a function Sp(C1,p)S =XpSp(C1,p)(12)4

Schr¨odinger’s equation for the vacuum becomes the differential equationXp(dSpdC1,p3C1,p + d2SpdC21,p+ dSpdC1,p2! C21,p −4!−xC1,p)=Xsw0(13)where terms involving clusters spanning adjacent plaquettes p, p′dSpdC1,pdSp′dC1,p′ EaC1,p · EaC1,p′(14)have been discarded.

The solution is5Sp = ln 1 −14C21,p−1/2se2−12cos−1−12C1,p! (15)where se2 is a Mathieu function [10].

The associated vacuum energy per site w0 for d = 3is plotted in fig. 1, together with Pad´e approximants obtained from the strong couplingseries to O(x14) [11] with which it shows very remarkable agreement at all x. Furthermore,using the Feynman-Hellman theorem, the vacuum expectation value ⟨C1⟩= −dw0/dx, hasthe correct weak coupling behaviour⟨C1⟩= 2 −O(x−1/2)x ≫1(16)The first excited state involves the Mathieu function se4.

The mass gap ∆wg = wg−w0for d = 3 is shown plotted against x−1/2 in fig. 2, together with Pad´e approximants fromO(x10) series [11].

In d = 3, the SU(2) theory is superrenormalizable and so the true(reduced) mass gap ∆wg is expected to go to a constant at the continuum limit x →∞.At weak coupling, the Mathieu mass gap diverges as ∆wg = 4x1/2 −54 + O(x−1/2). Thisx1/2 behaviour is as expected from the uncertainty principle with a wavefunction involvingexcitations spanning only a finite number of lattice spacings.In general, increasing the cut-offsize beyond one plaquette or going beyond SU(2),the resulting partial differential equation for Sp is intractable.

We therefore introduce the5 Eqn. (15), which is the exact solution for an SU(2) “one plaquette universe”, has been derived bymany authors e.g.

refs. [6], [7], and also as the d = 3 variational solution if Ψ is taken to be a productof functions of one plaquette variables [8].

However, the fact that it reproduces so well the results of ref. [11] for d = 3 has not been noticed previously.

For d = 4 (for which a variational solution is not available)eqn. (15) still results in a vacuum energy which compares well with that obtained from Pad´e approximantsderived from strong coupling series [9].5

second step of the approximation scheme and rearrange the sums of clusters in the givencut-offS and G in terms of “shifted” linked clusters C′i,sS=XsXia′iC′i,s(17)G=Xsb′0 +Xib′iC′i,s(18)These shifted linked clusters consist of products of Wilson loops each minus their vacuumexpectation values, still at fixed relative separation and orientation e.g. for general SU(N)′✲❄with✲′=✲−⟨⟩,✲❄′=❄−⟨⟩❄To illustrate the method, we consider again the one-plaquette cut-offand include in Sand G shifted one-plaquette clusters up to the n’th power of the shifted one-plaquette loopC′1,p = C1,p −⟨C1⟩.

Substituting into (8) and (11), the resulting clusters spanning twoplaquettes are discarded (step I) together with all “higher order” fluctuations (C′1,p)m,m > n, (step II) and coefficients of the constant term and each shifted cluster (C′1,p)i,i = 1 to n, are equated. For the vacuum, this gives a set of n non-linear equations for theamplitudes a′i and an expression for w0, all in terms of x and ⟨C1⟩.

For the glueball, theprocedure gives an n × n matrix equation for the b′i with eigenvalues ∆wg, together withan equation for b′0, all in terms of the a′i and ⟨C1⟩. These coupled cluster equations arethen solved using numerical library routines, with ⟨C1⟩calculated self-consistentlyThe results for w0 and ∆wg obtained [12] with the one plaquette cut-offand theexpansions of S and G truncated at n = 1, 2, 3 .

. .

converge rapidly to the Mathieu resultsat all x. Each successive power of C′1,p generates the next term in the series expansionsin x for x ≪1 and x−1/2 for x ≫1 of the Mathieu eigenvectors and eigenvalues.

Inparticular, even at the very simplest n = 1 approximation the method gives the correctweak coupling leading behaviour (16) for the vacuum expectation value ⟨C1⟩.Proceeding to larger cut-offs with SU(2) in d = 3, we have included in S and G allthe shifted clusters which, unshifted, occur in 2nd, 3rd and 4th order strong couplingperturbation theory (which is used solely as a guide to which terms to include - theresulting coupled cluster equations are all solved non-perturbatively). The complete set6

xorder 1order 2order 3order 4Pad´e [11]0.5-0.106298-0.0843647-0.0827613-0.0822829-0.0821561.0-0.386012-0.332963-0.322072-0.319032-0.315871.5-0.779896-0.707363-0.685324-0.681253-0.671152.0-1.25000-1.16553-1.13371-1.13132-1.1162 (2)2.5-1.77435-1.68175-1.64112-1.64197-1.6263.0-2.33949-2.24077-2.19185-2.19667-2.183 (3)3.5-2.93665-2.83296-2.77600-2.78509-2.774.0-3.55974-3.45180-3.38692-3.40037-3.40 (2)4.5-4.20439-4.09265-4.01992-4.03771-5.0-4.86730-4.75207-4.67154-4.69358-10.0-12.1254-11.9822-11.8255-11.8828-20.0-28.2197-28.0321-27.7362-27.8358-Table 1. Vacuum energies E0 for the first four orders of approximation in the shifted coupled clustermethod, together with the results of Pad´e approximants derived from the O(x14) strong coupling seriestaken from [11] (estimated errors in the last figure are given in brackets).of 69 linearly independent clusters occuring at 4th order are shown in the appendix.

Theloop expectation values are calculated self-consistently from the wavefunction using theFeynman-Hellman theorem by making the changeW →W +XsXloops∈S,GǫiCi,s(19)so that the a′i become functions of the ǫi and x. Then⟨Ci⟩= ∂w0∂ǫi{ǫi=0}(20)These are then simple quantities to calculate numerically, avoiding any group integrals.The results, obtained using a computer to carry out all the calculations [12], aregiven in table 1 for w0 and shown plotted in fig. 3 for ∆wg.

Assuming that the Pad´eapproximants are a good guide to the exact results, the method gives excellent resultsfor the vacuum energy w0 at strong and intermediate couplings and the correct form7

w0 = −2x + O(x1/2) at weak coupling. All the Wilson loop vacuum expectation valueshave the correct weak coupling form, as in (16).

For the mass gap ∆wg, reasonable resultsare obtained at intermediate couplings, before the inevitable x1/2 scaling sets in at weakcoupling.In addition to the eigenvalues, the method also provides the wavefunctionsand the cluster vacuum expectation values at all couplings.The amplitudes ai, bi ofthe most dominant of the 69 shifted clusters occuring in the wavefunctions at 4th orderare shown in figs. 4 and 5 respectively.

At weak coupling all the amplitudes behave asa′i = a′i(1/2)x1/2 +O(const. ), b′i = b′i(0) +O(x−1/2), with, for the 69 clusters occuring at 4thorder, the a′i(1/2) and b′i(0) ranging over six orders of magnitude.

For the vacuum, it is foundthat, at weak coupling, simple shifted clusters spanning few plaquettes and/or containingfew Wilson loops dominate over those for complicated multi-loop clusters. This can beunderstood as a combinatorical effect in the coupled cluster equations.

For the lowest(scalar) glueball, shifted clusters consisting of single Wilson loops dominate at intermediateand weak coupling. With the glueball calculation involving the diagonalization of a (large)matrix, this effect is harder to understand.

No evidence is found of a phase transition atfinite x.Returning to general SU(N), it can be shown [13] that the shifting procedure guar-antees that, for any N and d, the approximation scheme gives the correct weak couplingvacuum expectation value N + O(x−1/2) for a Wilson loop in S. Thus, at x →∞, theeffect of the electric field operators coupling together e.g. a pair of shifted single loopsbecomesEa(L)· Ea(L)= 12−−−12N+ const.

term O(x−1/2)r r✻′r r❄′✦❛✻❄′✻′❄′✻❄′′It is straightforward to show that, in general, in the weak coupling, large N limit, onlyshifted single loops6 survive in S and G. Why this appears to be a good approximationat N = 2 remains unclear.6For SU(N), a single Wilson loop can “wrap around” itself up to N −1 times before it becomes linearlydependent on clusters of simpler loops.8

ConclusionsIn general, the application of the coupled cluster method to lattice gauge theoriesdepends crucially on the rearrangement, or shifting, of the clusters in S and G to describegauge invariant fluctuations. At x ≪1 the method matches on to perturbation theory7,while at x ≫1, it is always found that small, simple shifted clusters dominate in thewavefunctions over large, complicated shifted clusters due to combinatorical effects in theSchr¨odinger equations.Furthermore, the interpretation of the discarded terms in theSchr¨odinger equations as higher order fluctuations enables otherwise formidable problemsinvolving linear dependences among clusters to be largely circumvented.In summary, the shifted coupled cluster method provides a non-perturbative semi-analytic approach to lattice gauge theories, involving a direct physical approximationscheme, and gives non-perturbative information on both eigenvalues and eigenfunctionsof the Schr¨odinger equations.

The results presented here show (table 1) that the methodconverges very rapidly for the vacuum energy (assuming that the Pad´e results are a goodguide) and should be capable of providing excellent approximations to the vacuum wave-function. Further work is needed to establish how well it works for excited states; througha more judicious choice of clusters in S and G, it should be possible to improve signifi-cantly upon the results presented here for the mass gap at intermediate couplings so thatit can be reliably extrapolated to the weak coupling limit.

The method therefore promisesto provide much valuable analytic information on the behaviour of lattice gauge theories.Fuller accounts of this work will appear elsewhere [13].AcknowledgementsN.J.W. acknowledges the financial support of an SERC/NATO Postdoctoral Fellow-ship.7 The method, without shifting, in fact provides an efficient way, avoiding any group integrals, ofcalculating perturbative expansions for both eigenvalues and eigenvectors due to the linked property of theSchr¨odinger equations.9

Appendix: Fourth Order (Unshifted) ClustersC1,s =C2,s =C3,s =C4,s =C5,s =C6,s =C7,s =C8,s =C9,s =C10,s =C11,s =C12,s =C13,s =C14,s =C15,s =C16,s =C17,s =C18,s =C19,s =C20,s =C21,s =C22,s =C23,s =C24,s =C25,s =C26,s =C27,s =C28,s =C29,s =C30,s =C31,s =C32,s =C33,s =C34,s =C35,s =C36,s =C37,s =C38,s =C39,s =C40,s =C41,s =C42,s =C43,s =C44,s =C45,s =C46,s =C47,s =C48,s =C49,s =C50,s =C51,s =C52,s =C53,s =C54,s =C55,s =C56,s =C57,s =C58,s =C59,s =C60,s =C61,s =C62,s =C63,s =C64,s =C65,s =C66,s =C67,s =C68,s =C69,s =10

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B139:1, 1978. [2] J. Kogut and L. Susskind, Phys.

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[4] J.P. Greensite, Nucl. Phys.

B166:113, 1980. [5] R. Bishop and H. Kummel, Physics Today p.82, March 1987.

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Phys. 70:229, 1983.

[7] D. Robson and D. Webber, Z. Phys. C7:53, 1980.

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[12] N.J. Watson, D.Phil. thesis, Oxford University, 1992.

[13] C.H. Llewellyn Smith and N.J. Watson, in preparation.11

Figure CaptionsFig. 1.

Graph of the vacuum energy w0 vs. x for the exact Mathieu solution of theone-plaquette-cutoffcoupled cluster equation (13), together with i) the Pad´e approximantsderived from the O(x14) strong coupling perturbation theory series taken from [11], andii) the results from O(x2) and O(x4) strong coupling perturbation theory, which show thatthe Mathieu result is non-trivial.Fig.2.Graph of the mass gap ∆wg vs.x−1/2 for the exact Mathieu solution ofthe one-plaquette-cutoffcoupled cluster equation, together with i) the Pad´e approximantsderived from the O(x10) strong coupling perturbation theory series taken from [11], andii) the estimated bounds, shown dotted, for their extrapolation to the continuum limitx →∞, also from [11].Fig. 3.

Graph of the mass gap ∆wg vs. x−1/2 for the shifted coupled cluster wave-functions Ψg = G expS with S, G including the sets of clusters which, unshifted, occurat the first four orders of strong coupling perturbation theory. Also shown are the Pad´eapproximants and continuum extrapolations derived from the O(x10) strong coupling per-turbation theory series taken from [11].Fig.

4. Graph of the absolute values |a′i/a′1| vs. x for the amplitudes a′i of the mostdominant of the 69 shifted clusters occuring in the vacuum wavefunction Ψ0 = expS atthe 4th order approximation, where a′1 is the amplitude of the shifted single plaquette loopC′1.

The cluster labels i are shown on the right side of the figure.Fig. 5.

Graph of the ampitudes b′i vs. x for the most dominant of the 69 shifted clustersoccuring in the scalar glueball wavefunction Ψg = G expS at the 4th order approximation.The cluster labels i are shown on the right side of the figure.12

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