I ~ LI[II q f_,1 : i 'j -" L'd~'] [ I N :]

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 34 (1994) 756-758 North-Holland

A New Method for Solving Lattice QFT with Dynamical Fermions John Lawson * Department of Physics, Brown University, Providence, RI 02912

We present a new numerical approach for solving Lattice Quantum Field Theory. The Source Galerkin Method is fundamentally different from Monte Carlo methods. Because it treats fermions and bosons in an equivalent manner, solutions to problems involving dynamical fermions are possible with existing computers.

1. I n t r o d u c t i o n

We present a new and powerful method to at- tack problems in quantum field theory [1]. This method is a numerical scheme which is not based on any statistical methods. Calculations are done in a fraction of the time as that of Monte Carlo and to very high degree of accuracy. In particu- lar, it is symmetric in its treatment of boson and fermion fields.

Our approach is based on QFT in the pres- ence of a source. On a lattice, the functional relations become a set of coupled linear differ- ential equations for Z in the discretized sources. After constructing a power series solution and using the Galerkin procedure, we can solve for the lattice Green's functions of a theory. We call this approach the Source Galerkin Method. We illustrate it for simple models with dynamical fermions

2. O u t l i n e o f t h e M e t h o d

2.1. S o u r c e F o r m u l a t i o n We study the dynamics of a field in the pres-

ence of an external source. The vacuum persis- tence amplitude Z[J] ~-g (OlO)j is the generating functional for the Green's functions and obeys the relation

2 6Z 6aZ (D + M )6--~(x) + g-~-j-(x) 3 - J ( x )Z (1)

After solving for Z[J], the Green's functions can be extracted by functional differentiation.

*Research supported in part by DOE Grant DE-FG02- 91ER40688- Task D

On a Euclidean lattice, the functional relation becomes a set of coupled differential equations

M2, 0Z 0Z aaZ (2+ )-yg - E + = (2) n n

where the sum is over nearest neighbors. There is one equation per site.

2.2. P o w e r Ser ies a n d B o u n d a r y C o n d i - t ions

For finite lattices with N sites, we will con- struct Z as a power series in the N source vari- ables Ji. The number of independent coefficients can be reduced by exploiting the symmetries of the lattice. Thus Z is constructed to be invariant under the symmetry group of the lattice.

In order to solve the differential equations, we must specify boundary conditions. Normalization of the vacuum amplitude implies

Z [ J : O ] = 1 (3)

and we will exclude symmetry breaking, so that

dZ b : 0 = (¢d : 0 (4)

We cannot specify the second derivatives of Z because they correspond to the two-point func- tions. Instead we will truncate Z at some finite order. The truncation order is dictated by com- putational constraints. As a boundary condition, truncation guarantees that as g ~ 0 in the in- teraction theory, the solution reduces to the free field. This is an implicit limit of all path integral formulations [3].

Truncation in this sense not only sets a bound- ary condition, but also introduces an approxima- tion scheme. This scheme be made systematic by

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J Lawson/14 new method for solving lattice QFT with dynamical fermions 757

t runcat ing at successively higher orders and then taking the limit as M goes to infinity.

2.3. G a l e r k i n P r o c e d u r e The truncated polynomial is an approximate

solution to the differential equations where the error due to truncation is called the residual R. In order to minimize R, we define an inner product in the source space

(g, f ) = g ( J ~ . . . J N ) I ( J I . . . JN) [dY] (5) £

where the integration is over all Ji, and e is con- sidered small. Requiring the inner product of R with linearly independent test functions Tk to vanish

(R, T1) = 0

(6) = 0

generates a set of linear algebraic equations. We can generate as many of these equations as we like as long as the Tk are linearly independent. We will construct as many equations as there are independent unknowns in our power series solu- tion These equations can be solved by a single mat r ix inversion where the resulting coefficients are the lattice Green's functions.

3. D y n a m i c a l F e r m i o n s

For fermions, the functional formulation is identical except that the sources become Grass- mann variables. In a lattice 4-fermion theory with dynamical staggered fermions, the functional re- lation becomes a set of coupled Grassman differ- ential equations

5Z 53Z Ds ~ + g arJ~O~+lrli+l = ritz (7)

where / ) s is the Susskind operator for staggered fermions, and rli , Oi are staggered Grassmann source variables.

As with the boson case, we will solve these equations by power series in the source variables.

Z E -k l kl -k-N kN = a{~iki}~ll ql ' ' ' q N ~N (8)

{ l~iki }

One bonus is that due to Fermi statistics, the polynomial will self-terminate. This means that for small lattices, exact solutions are possible. For realistic systems, though, you will have to trun- cate the series artificially because of the rapid in- crease in the number of terms.

On two sites, the exact power series is

Z = 1 + al(Oir h + r/2r/2)

+a2(01/12 + 712~]1) +a3~l r]27]i ~2 (9)

Putting this into the lattice functional equations and equating like powers of the sources gives sim- ple algebraic equations for the coefficients. These can be solved trivially to give

al = M / ( M 2 + g + l)

a2 = 1 / (M 2 + g + 1) (10)

a3 = 1 / ( M 2 + g + l)

Since this represents an exact solution to the Grassman equations, we do not need to use the Galerkin procedure in this case.

For realistic systems, a Galerkin procedure must be devised. A major obstacle is that the Galerkin method gives a weak solution to the differential equations. Grassmann integration, though, has no measure theoretic interpretation. This problem can be resolved by defining a mod- ified inner product in the Grassmann function space. After this modification, the Galerkin pro- cedure is identical to that used for bosons. Calcu- lating with fermions is as easy as calculating with bosons. Details of this procedure, in addition to a numerical solution of the lattice Gross-Neveu model, will be given elsewhere [2].

4. D i s c u s s i o n

We have presented a new method to numeri- cally at tack problems in lattice QFT. It does not depend on any statistical method and possesses extreme flexibility. In many ways, it resembles a variational principle, but for interacting theories, it is definitely NOT a variational method. Sys- tematic improvements of the approximations are possible, forcing convergence to the exact solu- tion.

758 J Lawson/A new method for solving lattice QFT with dynamical fermions

Calculations can be done in a fraction of the t ime of Monte Carlo and to a very high de- gree of accuracy. The integrations involved in the Galerkin procedure are trivial and the linear equations for the coefficients of Z can be solved numerically. The matrices to be inverted become large, but unlike Monte Carlo, there is only one inversion for a given set of parameters.

Since our method is deterministic, it does not suffer from "minus sign" problem. In addition, a determinant is never calculated. This means that systems with dynamical fermions and at nonzero chemical potential might be accessible to numer- ical methods. Among these is included the Hub- bard model, popular as a model for high Tc super- conducivity. We believe that the Source Galerkin Method will be a powerful tool for studying fermionic systems.

5. A c k n o w l e d g e m e n t s

This work was developed at Brown University in collaboration with G.S. Guralnik and S. Gar- cia.

R E F E R E N C E S

1. S. Garcia, G.S. Guralnik and J. Lawson, sub- mit ted to Phys. Lett B

2. J. Lawson and G.S. Guralnik, to be published 3. S. Garcia and G.S. Guralnik, to be published 4. C.A.J. Fletcher, Computat ional

Galerkin Methods 1984