Table Of ContentINTRODUCTION TO LATTICE GAUGE THEORIES
AND NUMERICAL METHODS
Carleton DeTar
Research Institute for Theoretical Phys&cs
University of Helsinki
Siltavuorenpenger 20 C
SF-00170 Helsinki 17
FINLAND
and
Department of Physics
University of Utah
Salt Lake City UT 84112
USA
i. Introduction
The lattice approximation has proved to be an important technique
for studying properties of continuum space-time gauge theories -
particularly for providing information about the ground state, the
low-lying excitations, and thermodynamic behavior. Numerical lattice
calculations have given the most convincing evidence to date that quarks
are confined in SU(2) and SU(3) Yang-Mills theories, and they promiee
to provide values for the masses of the low-lying mesons, baryons, and
"glueballs" and to give a thermodynamic description of the theory,
without the need for introducing extraneous assumptions about the
vacuum state. In principle all we should want to know about the structure
of the QCD vacuum and the low-lying states is given by the numerical
lattice calculations. We simply need a very large and fast computer.
However, most of us find it unsatisfying to characterize the physical
vacuum by a set of numbers that only modern mass storage devices can
record. Moreover, with present computer facilities it is inconceivable
that lattice gauge theories can give detailed information about hadron
wave functions, such as, e.g. the charge radius of the neutron. Our
ultimate goal should be to obtain approximate analytic models. Nevertheless,
the numerical calculations now provide a high standard of comparison
according to which any approximate description must be judged.
There is a distinguished history of numerical and analytical studies
of lattice field theories in condensed matter physics. Many of these
techniques have been adapted to high energy physics applications.
Wilson and Polyakov i and Kogut and Susskind 2 proposed lattice
versions of quantum chromodynamics in 1974. Creutz 3 demonstrated that
it was feasible to do numerical calculations with Wilson's lattice version
of the SU(2) Yang-Mills theory and gave evidence that the long range
confinement of the strong coupling lattice methods persists even in the
weak coupling regime, thereby "demonstrating" that the theory is con-
fining. His work opened the door for a host of subsequent numerical
studies. We now know a great deal about lattice gauge theories with
practically any gauge group. Of particular interest for QCD, the string
tension, a few glueball masses, and the deconfinement temperature have
been calculated for the pure Yang-Mills SU(2) and SU(3) gauge theories.
With fermions included, we have some preliminary evaluations of several
low-lying hadron masses in an approximation that neglects Feynman
diagrams with quark loops 4,5oEfforts to include quark loops are
beginning to succeed 6,7 .
Because of the considerable current interest in fermions and the
computational technology my lectures will dwell more heavily on these
topics. Lattice gauge theories have already been treated in several
reviews and summer school lectures ~,9,10 Therefore, I refer the
interested student to these sources for a more historically balanced
presentation. Nevertheless, I shall try to keep the discussion reasonably
self-contained for the benefit of the non-experts in this field.
In these lectures I emphasize the connection between the hamiltonian
and Euclidean action formulation of the gauge theories, since I believe
this gives more insight into the meaning of finite temperature corre-
lation functions. The massive Schwinger model on a finite lattice is
used to motivate the generalization to quantum chromodynamics in four
space-time dimensions. I begin by introducing the hamiltonian version
of the Schwinger model and proceed to derive the functional integral
expression for the partition function of this model. In the process a
number of related topics are treated, including the determination of
glueball and meson masses, the Grassmann calculus, and a brief discussion
of various popular fermion actions. I conclude with a discussion of
numerical methods.
2. Hamiltonian Formulation of a Simple Lattice Gauge Theory:
Massive Schwinger Model
Let us begin by discussing a finite lattice version of the massive
Schwinger model (quantum electrodyn~ics in one dimension). First, we
focus on the fe~ions and introduce the electromagnetic field later.
In one dimension Schwinger described relativistic fermions with two-
component spinors. We introduce
1 c
\~2x
where x = 0,1...,N-I labels the lattice points. The lattice Hamiltonian
is 2
-i N-I N-I
- Z ~x ~i (~x+l - ~x-i ) + Z m ~x~x. (2.2)
HF 2a x= 0 x= 0
where a is the lattice spacing, and we use periodic boundary conditions
~0=~N . With the anticommutation relations
{ t
~x~'~x'~ }' ~' ~xx' (2.3)
=
it is easily verified that the lattice version of the Dirac equation
i~x = ~x HF -iY071
' - 2a (~x+l - ~x-l) + mY0~ x (2.4)
is obtained. Notice that the hermiticity of H is associated with the
appearance of the central difference in (2.4). The equation of motion
is brought to 2x2 block diagonal form in momentum space:
N-I
1 eikx2~/N
k = 0,1,...,N-I .
~k = N~ ~ ~x
x=0
The eigenfrequencies are the zeros of the determinant
m-~ --1 sln--~- _ 2~k
a
' )5.2(
1 _ 2zk
- sln-~ 0L-m-
i.e.
m 1 . . 2~k. 2
= +- +--2 "-~--nls~ )L~ (2.6)
a
The continuum limit of the theory is defined by a + 0 with• Na fixed.
For m to remain finite, it is necessary that k ~ 0 or N or k~N/2
F (2.7)
p =~2wk k>0; 2~(k-N) k<N .
17 aN
(k-N/2)
t aN k ~ N/2
Therefore, there are two orthogonal sets of low-lying excitations with
the physical characteristics of two types of free particles of mass m
and their antiparticles. This species doubling is a persistent difficulty
for lattice fermion theories. There is a simple remedy in the one
dimensional case. Let N be even and define
~x = (YOYI)x ~x ' " (2.8)
Then
HF -i N-I N-I ~ ,
- 2a Z ~x% (~x+l - ~x-1 ) + ~ m~x~ x (_)x, (2.9)
x=0 x=0
which is the sum of two identical hamiltonians, one from the upper and
one from the lower components. If we write ~x = ~x 1 ' the upper
component, then we have the one-component hamiltonian
N-I N-I
, -i ~ # x
H F - ~-~ Z ~x (~x+l - %x-1 ) + ~ m~x ~x )-( (2.10)
x=0 x=0
The Fourier transform
N-I
_ 1 Z eikx2w/N #x 2( Ii)
~k N
X=O
gives
'FH = k=0Z 4( ~ +N/2) ~--2'nis ~ m
-i
m ~-- sin
(2.12)
This expression is easily diagonalized and the dispersion relation (2.6)
is the same. However, k now covers only half the range 0~k<N/2-1,
giving a single fermion species in the continuum limit. In Susskind's
language, the transformation from ~ to ¢ corresponds to keeping only
the upper component of ~x on the even sites and lower on the odd sites
2 .
Although the species doubling problem can be solved by this device
the continuous chiral symmetry in the massless theory is then lost ii
When m=0 the Hamiltonian is invariant under the global transformations
~x ~ eie/2~x
(2.13)
~x + eiY5e/2~x
where Y5 = Y071" The first symmetry is associated with charge
conservation. It is valid also for the massive theory, and is unaffected
by the reduction ~ ÷ ¢ The second is obviously lost in the reduction
÷ ¢ , since it involves both upper and lower components.
An alternative solution to the species doubling problem involves
introducing a non-local difference operator. See 12 The non-locality
has discouraged widespread use of this form.
b. Hamiltonian with Gauge Fields
Kogut and Susskind 2 showed how to construct a lattice hamiltonian
invariant under local gauge transformations. For a U(1) theory in
A 0 = 0 gauge, it is
N-I
1 N-I ~2 i iex - ~x+l e-leX~l~x
H = ~ g2a ~ - ~ ~x e Yl~x+l
x=0 x 2a x=0
N-I
+ Z m~x~ x (2.14)
x=0
where the guage link angle e e 0,27), the conjugate flux variable ~x'
x
satisfying
e x,£x, = i6xx, , (2.15)
and the gauge coupling constant g have been introduced. The familiar
continuum limit is obtained, if
e x = g a A 1 (ax)
£x = E1 (ax)/g (2.16)
~x = /~ X (ax)
and the limit ga ÷ 0 is taken. Then
1
H ~ /dx { ~ El 2 )x( + --.~~(x)71-i~x+gAl(X) X )x( +m~ )x( X )x( } •
(2.17)
Notice that finite values of the vector potential and electric field
correspond respectively to small 8 x and large £x"
The Hamiltonian is invariant under the time independent gauge trans-
formations
~x ÷ eilx ~x (2.18)
8 x ÷ 8 x - Ix+ 1 + I x •
The specification of the model is completed by requiring that the
physical states satisfy Gauss' law:
£x - £x-I = Px -:~x: " (2.19)
where : : is a suitably chosen normal ordering discussed below, Because
of the assumed periodicity of 8 x and £x under x + x+N, Gauss' law
requires that the physical states be neutral:
N-I
0 = ~N-I - £-i = 7 Px' (2.20)
x=0
Usually one arranges for the vacuum to be neutral and studies neutral
excitations thereof. In perturbation theory the vacuum in lowest order
has all free negative energy fermion states filled. From (2.6) we see
that there are N such states in the two-component theory. Thus we obtain
% %
: ~x ~x : = ~x ~x -i , (2.21)
which corresponds to the introduction on each site of a "heavy ion" with
charge opposite to the fermion's charge. On the other hand, the one-
component theory has 1/2 particle per site in the vacuum state. The
charge must be quantized in whole units of g since £x is quantized.
At the expense of changing the lattice theory, but presumably not the
continuum limit , one can define
f*
°x ~xO~ x -1
= x even (2.22)
I*%x~x X odd ,
which corresponds to the introduction of heavy ions on the even sites
,31 Another possibility involves doubling the species, but this time
adding a particle of negative charge. Then
= #%~xx c%c
Px *~x ~x
is a suitable choice. This solution is typical of fermion actions with
symmetric time derivatives and does not require the introduction of
heavy ions.
3. The Pure Gauge Theory
a. Feynman Path Integral for the Partition Function
From the hamiltonian
N-I
-_ 1
H G ~ g2a 7 £ 2 3( )i.
k=0 x
with £x - £x-i = 0 for physical states and £0 = £N ' we shall calculate
the partition function
ZG(8) = Tr exp(-SH G) (3.2)
The problem is a trivial one with the solution
ZG(8) = ~ exp(-~2aS£2N) , (3.3)
but we shall take a devious approach to obtain the functional integral
representation on the Wilson lattice, which is useful for non-trivial
cases in higher dimensions and with non-Abelian gauge groups. To this end
end we write
ZG(8)= Tr ~ (3.4)
where
T = exp (-HGdT) (3.5)
and 8 = MAT. T is the transfer matrix, an important operator in its own
right. It is the evolution operator for an imaginary time interval.
The functional integral is usually written on the group manifold -
in this case, over the variables 8x, conjugate to £x" We therefore
use the basis in which 8 x is diagonal to calculate the trace. However
the general state vector l{Sx}> does not satisfy Coulomb's law, so it
is necessary to introduce the projection operator
N-I 2~ d~ x
P = n I ~ exp i(£ - 3( )6
x=0 0 x ~x-l)~x ' "
whereupon the trace over physical states may be obtained by integrating
over all 8 as follows:
X
N-I
Z(6) = Tr exp(-SH G) = / H d8 x <{Sx}lexp(-6HG) Pl{Sx}> . (3.7)
x=0
Now the projection P commutes with H, so
Z(6) = Tr (TP) M (3.8)
and on the 8 x basis the matrix product is
~dex,0<{ex0}ITPl{eX,M-1}><{ex,M-1 }
Z(6) = f xHdSx,M-I~dOx,M-2"'"
>}oxe{lPTI}lxe{<
TpI{Sx,M_2} ~ ... (3.9)
To evaluate the transfer matrix element, we first find the matrix ele-
ment
N-I N-1
J = <{8'} x exp ( -~¢ ~2 x) exp ~ £~x(~x - ~x-i ) l{Sx}>
x=0 x=0
3( .i0
with e= g2aAT and then integrate over all ~x" Since
eim£ul{Sx}> = I{e 0 ..... 8U_I,Su-~,SU+I,---SN_I }> 3( .Ii
we find
N-I
J = <{8 x I} exp(-~ e Z Zx 2 ) I{8 ,x , }> (3.12)
x=0
where
x"O = 8x - ~x + ~x+l (3.13)
The expression (3.12) is readily evaluated for small e as follows:
,, dSei£xS-82/2¢I
<8' x I exp(-elx2) 18 x > = ~ <e' x I Z~ e" x
2 -~
!
= /4~ Z exp 8(-- x 0-- II +2zn)2/2e (3.14)
x
/4-~e exp{ ~1 c°S(Sx | - ~ x II )-i}
In the second step we used (3.11) and
co
<e I l e"x +e> = E 6 (e x -e x !w -e+2"rrn) (3..15)
n_~-~
Putting everything together, we find that up to an irrelevant constant
factor the matrix element is
N-I
i
J ~ exp{ E cos(8 x -8x+mx-~x+l)-l} (3.16)
g~aA~ x=0
for small g2aAT. Since the projection operator P appears M times in
(3.8) it is necessary tO integrate over M sets of ~x"
Thus
M-I N-I
(3.17)
Z(6) ~ / I II dSx d~x~ exp(-S G)
T=0 x=0
N-I M-I
l-coS(Sx~ f ex~+~x~-~x+l,~) (3,18)
g2aA-r O=x "r=O
where 8xMHSx0 . This is the Feynman path integral with the Wilson action
for the U(1) gauge theory. The action can be described by constructing
a space-time lattice shown in Fig.3.1. The angles 8xT are the space-
like link variables and mxT' the time-like variables. The argument of
the cosine is the plaquette angle 8p formed from the "vectorial" sum of
the angles around a square. Gauge invariance on the space-time lattice
takes the following form: For an
802 812 ~ > arbitrary set of angles IxT , let
T=2
8xT 8xT lx+I,T + lXT (3.19)
~0~
ii°~ ~12
801 811~ ÷ +l - l (3.20)
~xT ~xT x~ x,Y+l ,
i='
The action S G is invariant under this
(0m ~i0 w20
transformation.
OO8' 810
T=0 The continuum limit of the action
x=0 x=l x=2
S G is expressed in the familiar form by
putting
Fig.3.1
10
8xT = gaAl(ax, T~) (3.21)
axT = gATA0(ax' TAT). (3.22)
Then in the limit ATg + 0 and ag÷ 0 with A 1 and A 0 fixed,
1 2
S G ~ /dx ~SdT ~ F (3.23)
0 Hu
with F = ~A - ~A
The generalization of the Wilson action to higher dimensions and
other gauge groups is straightforward. For SU(3) in four space-time
dimensions on a symmetric hypercubical lattice of lattice constant a ,
each site is labeled by a four vector x and has eight links to its
nearest neighbors in the directions ±e , where e is a unit lattice
vector. To each of the four positive directions an SU(3) link matrix
Ux~ is assigned. Each site x also has 24 associated plaquettes,
PxHu' six with H>u>0. The plaquette product around the border ~px~u p
U % U %
U(~Pxuu) = Uxu Ux+eH,~ x+eu,u x,u ~ (3.24)
appears in Wilson's Yang-Mills action:
SG = 1 Z Z Tr 2-U(~Px~u) - U%(~pxHu) (3.25)
6g 2 x ~>~>0
The partition function is
Z(8) = I H dUx~ exp(-SG), (3.26)
x,~
where dUx~ is the invariant Haar measure for SU(3). Putting
Ux~ = exp(iga A cl )c (3.27)
where ic are the eight generators of SU(3) and taking a~0 gives the
Euclidean SU(3) Yang-Mills action
YM
SG ~0r8 dT/d3x ¼ ( F ~c H )2 (3.28)
where
H~ Ac~ - HCA~ - gfabcAa~A~" (3.29)
u~CF =
