- Email: [email protected]
Chern-Simons currents and chiral fermions on the lattice
Phystcs Letters B 301 (1993) 219-223 North-Holland
PHYSICS LETTERS B
Chern-Simons currents and chiral fermions on the lattice Maarten F.L. Golterman
I
Department of Phystcs, Washington Umverstty, St Louts, MO 63130-4899, USA
Karl
Jansen 2
and David B. Kaplan
3.4
Department of Physws, Umverstty of Cahforma, 9500 Gdman Drtve 0319, San Dwgo, La Jolla, CA 92093-0319, USA
Received 22 December 1992
We compute the Chern-Stmons current induced by latUce fermtons on a d( = 2n + I )-dtmens~onal lattice, usmg a topologacal mterpretat~on of the ferm~on propagator as a map from the torus to the sphere, Td--,S'~ Our techmques clarify the dependence of the current on short-distance phystcs. We show exphcltly thal for Wdson fermtons It changes dlscontmuously at d+ 1 different values for the mass m. Th~s result ~s relevant for a recently proposed model of choral latttce fermtons as zeromodes bound to a domam wall.
C h e r n - S i m o n s operators play an interesting role in gauge theories in o d d s p a c e - t i m e dimensions. Threed i m e n s i o n a l gauge theories may be relevant in hlgh t e m p e r a t u r e superconductivity [ 1 ] as well as the q u a n t u m Hall effect [2 ]. A n o t h e r place where o d d - d i m e n s i o n a l gauge s y m m e t r y and the C h e r n - S i m o n s o p e r a t o r play a big role is in a recently p r o p o s e d m e t h o d for simulating chiral lattice fermions [ 3 ]. In o d d - d i m e n s i o n a l gauge theories with charged fermions, heavy fermion modes can induce the C h e r n - S i m o n s o p e r a t o r as an effective o p e r a t o r in the low energy theory. The strength o f the o p e r a t o r does not decrease with the mass o f the heavy modes, and is in general sensitive to short distance physics. In this letter we show that the strength o f the induced o p e r a t o r has a simple topological interpretation, allowing one to easily d e t e r m i n e the way that short distance physics enters the low energy theory. The particular application o f the induced C h e r n - S i m o n s o p e r a t o r that is the focus o f this letter, is the model o f d o m a i n wall chiral fermions [ 3 ]. The idea is to i m p l e m e n t an o d d - d i m e n s i o n a l d - - 2 n + l theory o f Wilson fermions with a mass couphng to a d o m a i n wall. The low energy effective theory consists o f massless chiral fermlons b o u n d to the ( d - l ) - d i m e n s i o n a l d o m a i n wail, without there being " d o u b l e r " modes [4] from the Brillouin zone boundary. The a n o m a l o u s W a r d identities for these chirai fermions in the presence o f background gauge fields can be directly measured on a finite lattice; this has already been done numerically for d = 3 and the results are m agreement with the c o n t i n u u m a n o m a l y in l + ! d i m e n s i o n s [ 5 ]. In this system, the a n o m alous divergence o f the ( d - l )-dimensional z e r o m o d e flavor currents is due to charge flow in the direction normal to the d o m a i n wall, even though there ts a mass gap o f the wall - an effect discussed by Callan and Harvey [ 6 ] for c o n t i n u u m fermions coupled to a d o m a i n wall. They pointed out that the C h e r n - S i m o n s action induced by integrating out the heavy fermion m o d e s [ 7 ], being proportional to m~ ] m I, has opposite signs on the two sides o f the d o m a i n wall. This gives rise to a C h e r n - S i m o n s current in the presence o f background gauge fields with a nonzero divergence at the d o m a i n wall. Furthermore, the divergence exactly reproduces the even E-mad address: [email protected]. 2 E-mad [email protected]. a E-mad address, [email protected]. ( Sloan Fellow, NSF Presidential Young Investigator, and DOE Outstandmg Jumor InvesUgator. 0370-2693/93/$ 06.00 © 1993 ElsevmerScience Pubhshers B.V All rights reserved
219
Volume 301, number 2,3
PHYSICS LETTERS B
4 March 1993
( d - 1 )-dimensional anomaly for the single chiral fermion zeromode that is b o u n d to the d o m a m wall "~. This effect is a manifestation of the descent relations between the anomalies in odd and even dimensions [ 8 ]. In this letter, we show how to perform the C a l l a n - H a r v e y ( C H ) analysis for the lattice theory m euclidean space, where the zeromode spectrum is more complicated than m the c o n t i n u u m . It is far from obvious that the lamce theory should follow the CH c o n t i n u u m analysis; after all, the coefficient of the C h e r n - S l m o n s action gets O( 1 ) contributions from arbitrarily heavy fermlon modes, and the heavy spectrum on the lattice looks nothing like m the c o n t i n u u m . In fact, we know the induced C h e r n - S i m o n s operator must have a coefficient very different from the c o n t i n u u m result. While ref. [3 ] analysed the spectrum of the theory for a Wilson coupling r = 1 and a d o m a i n wall height 0 < mo< 2 and found a single chiral mode, a recent paper by Jansen and Schmaltz [ 9 ] analyses the same model for general parameters and shows that the spectrum b o u n d to the d o m a i n wall changes discontinuously with varying mo/r ~2. They find that for 2 k < I mo/rl < 2 k + 2 , where k is an integer in the range O<~k<~d- 1, there are (dZ~) choral modes bound to the domain wall with chirality ( - 1 ) k × s i g n ( m o ) ; there are no choral fermions for I mo/rl > 2d. This is qmte different than the c o n t i n u u m theory, for which there is a single chiral mode for any mo¢: 0. If the induced C h e r n - S i m o n s action on the lattice is to correctly account for the anomalous divergences of the chlral fermton currents on the d o m a i n wall, then evidently its coefficient must also depend discontinuously on mo/r in a very particular way. We show in this letter that that does indeed happen ~3. The abelian C h e r n - S i m o n s action in d = 2n + l c o n t m u o u s euchdean dimensions is given by
F(a) CS
~Otl
Ot2n+i
f d 2n+ Ix Aa, 0a2Aa3"" 0a,,Aa2,+t •
( 1)
When a massive fermion is integrated out of the theory it generates a c o n t n b u t m n to the effective action of the form S~fr=c, Fcs; absorbing the gauge coupling into the gauge field, Fcs is seen to be o f d t m e n s i o n d, and so the coefficient c, will be dimensionless and the operator will not decouple for large fermion mass. The coefficient c, can be computed by calculating the relevant portmn of the graph in fig. 1. This is true on the lattice as well in the weak field, long wavelength limit for the gauge fields. Denoting the fermion propagator and photon vertex as S ( p ) and iAu(p, p ' ) respectively, the graph of fig. 1 yields a value for c, which may be expressed as •
l ~ a l # I ct,,#notn+j
~
c . = (nT-i) ~ ) ! ×
"
" d2n+ Ip (2rt)2,+ , T r [ S ( p ) A ~ , ( p , p - q t ) S ( p - q ~ )
]
...A . . . . (P+q"+~'P)
q,=o"
(2)
BZ
4~ It should be pointed out that if the magmtude of the Chern-Slmons current is regular dependent, the graph needs to be regulated. A regulator cannot change the divergenceof the current, however. We thank M. Lfischerfor this comment. 42 All dlmenslonful parameters are gdven m lattice units. By a domam wall of height mo we mean a spatially dependent mass term re(s) --. +mo as s-, + oo, where s is the coordinate transverse to the domain wall. 43 The dependence of the reduced Chern-Slmons actmn on the Wdson couphng • has been previouslydiscussed for three d~mensmnsm the continuum hmlt (spatmlly constant rn--,0) in ref. [ 10] and for Iml < 1 m ref. [ I 1] Some of the techmques used m this letter are samdar to those found m the latter work.
---> ql 220
\
/
~
<--
qn+1
Fig. 1. The Feynman diagram m 2n+ I dtmensmns conmbutmg to the induced Chern-Simons acuon for abehan gauge fields, Y 7=+1 ~q, = 0. Graphs with mulUplephoton vertmespecuhar to the latuce do not conmbute, as each A field from such a vertex has the same Lorentz index and the contnbutlonvanishes by the anusymmetry of the ~tensor
Volume 301, number 2,3
PHYSICS LETTERS B
4 March 1993
The factor o f n + 1 m eq. ( 2 ) is due to the s y m m e t r y factor o f the graph m fig. !; the factor o f i is the product o f i "+ ~ from the photon vertices and ( - ~)" from relating the d e r i v a u v e s in eq. ( l ) to powers o f momenta. The pintegration is over the Brilloum zone o f a hypercubic lattice w~th lattice spacing a = I. The integral ( 2 ) looks very difficult to c o m p u t e on the lattice for arbitrary n, as both S(p) and Au(p, p ' ) are in general rather c o m p l i c a t e d functions. It is m a d e quite tractable, however, by exposing its topological properUes. Gauge invariance implies that the p h o t o n coupling satisfies the W a r d i d e n u t y '~
Au(p,P)=-i~p u
(3)
(P) ,
allowing c, m eq. ( 2 ) to be reexpressed as " f d2~+lp ( - - i ) ¢u, u2,., j T r { [ S ( p ) Ou, S ( p ) -~1 [ S ( p ) 0 u 2 , ÷ , S ( p ) - ' ] : c,= (n+l)(2n+l)! ( 2 n ) 2"+' . . . .
(4)
where the d i f f e r e n t i a u o n 0, is with respect to p,. The fermlon propagator may be written in the genetic form
S -~ (p) =a(p) + i b ( p ) . y = N ( p ) [ c o s
IO(P)l+i~(P)'~' sm [0(P) I 1 - N ( p ) V ( p ) ,
(5)
where N ( p ) -= a 2 ~ , 0 ( p ) = ~ arctan ( Ib l / a ), and V(p) is seen to be a 2" X 2" unitary matrix. P r o v i d e d that S - z (p) does not vanish for any p, eq. ( 4 ) ts i n d e p e n d e n t o f N ( p ) , allowing S - ~(p) and S(p) to be replaced everywhere by V(p) and Vt(p) respectively. This matrix V(p) is seen to describe a m a p p i n g from m o m e n t u m space - with on the hypercub~c latuce is the torus T a - onto the sphere S d defined by the vector O(p). The h o m o t o p y classes o f such m a p s are ~dentified by integers, and so the integral in eq. ( 4 ) has a simple topological interpretation; It is, to a n o r m a l i z a u o n constant, the winding n u m b e r o f the m a p described by the ferm~on propagator. (As an example o f a n o n m v a l m a p from T2--*S2, picture the torus as a square with oppostte edges identified, and m a p the square onto a sphere so that the center o f the square is m a p p e d to the N o r t h pole o f S 2, whtle the edge o f the square are all m a p p e d to the South pole. ) We now proceed to c o m p u t e th~s winding n u m b e r for the Wilson fermion propagator. Although we are ultimately interested in the effective action for lattice fermlons in the presence o f a d o m a i n wall, we can compute the effective action far from the mass defect on e~ther side by treating the fermion mass as a constant rn. Thus we can use the s t a n d a r d Wilson propagator d
S-l(p)=
~ ju=l
d
iTusmpu+m+r ~ ( c o s p u _ l ) .
(6)
,u=l
Continuous changes in the mass and Wilson couphng, m and r, cannot change the value o f the winding n u m b e r except at points for which S - t has a zero for some m o m e n t u m p. Such singular points o f the m a p p i n g occur only at m o m e n t a corresponding to the corners o f the Brillouin zone, and then only for m/r=O, 2, ..., 2d. The C h e r n - S i m o n s coefficient c, as a function o f m / r must therefore be piecewise constant, changing only at these crmcai values. Furthermore, for fixed r, V(p) -~ _+ 1 as rn-~ + oo, so we may deduce that
c,(m,r)=O
form/r2d.
(7)
To c o m p u t e c, for 0 < m / r < 2d, we need only evaluate the d e n v a u v e o f the integral in eq. ( 4 ) with respect to m across the critical values m/r=O, 2 ..... 2d. This task is simphfied by the fact that c , ( m + d m , r) is unchanged as one deforms d m in a p - d e p e n d e n t way so that d i n ( p ) vamshes for all p except in the vicinity o f the Brilloum zone corners; these points are d e n o t e d by p = ¢ ~ k ) , the a = 1, .., (~) vectors with k n o n v a n i s h m g c o m p o n e n t s We expand the lamce gauge field as Uu(x) = 1- L4u(X)+ 221
Volume 301, number 2,3
PHYSICS LETTERS B
4 March 1993
equal to zt. Therefore we need only evaluate the mtegrals m infimtesimal regions near the Brillouin zone corners ~k). After some algebra this yields ~5
dcn(m,r) dm
-
i(-l)n2 ~ )k d ( n + 1 ) /,.a )-" ( - I dm
l(--1) "
a
= (n+l)(2zr).n!k~=o(--l)k
(d)
f
d2~+'p
m-2rk
(27t)xn+l [IPl 2.+. (m_2rk)2]
~+l
8(m-2rk),
(8)
where the integration d ~ is over the mfimtesimal region ]P] < e ~ 0. This may be trivially integrated with respect to m, given the boundary values (7), to yield the Chern-Stmons coefficient for Wilson fermions:
1, 1,. c~(m, r ) = 2 ( n ~ - i ) (2n)nn !
k=o
( - 1 )k
m-2rk Im-2rkl
(Wilson fermions) .
(9)
This is to be compared to the continuum result for a fermlon of mass m, computed by inserting the continuum free propagator in eq. (2) and integrating over continuum m o m e n t u m space:
cn(m) =
i(-l)
n
(continuum result) .
2(n+ l)(2n)"n![m[
(10)
Our computation is exact in the limit of small and adiabatic external gauge fields, even for m o f O ( l ) m lattice units; it is readily generalized to non-abehan gauge fields. So far we have only considered lattice fermions with a constant mass; our primary interest though is in studying current flow m the presence of a domain wall mass term for the fermion. Following Callan and Harvey, we assume that sufficiently far away from the domain wall on either stde, we can treat the mass as constant and compute the induced current by varying the Chern-Simons action, juts = ~iSefr
n+ 1
6A~ - ~2
c,(m(s), r)¢~,,~ 2 ,2,÷,F,,~2...F,~2._t~2.,
(11)
where the factor of 2" arises from replacing 0.4 by the field strength F. Since c, depends on the mass re(s) which is space dependent, Jucs has a nonzero divergence corresponding to current flow normal to the d o m a m wall. In their continuum analysis, Callan and Harvey pomted out that with c, given by eq. (10), there was equal current flow toward the wall from each side that exactly compensated for the anomalous divergence of the chtral fermlon current along the wall's surface. For lattice fermlons in the presence of the domain wall, we must use the value (9) for c,(, r). Thus on the side o f the domain wall for which m(s)/r is negative, the Chern-Simons current vanishes. On the other side, where m (s)/r--, Imo/rl, the current either vanishes if Imp~r[ > 2 ( d + 1 ), or else, for 2l< [rap~r l < 2 ( l + 1 ), the current is given by J~(lattice) =--(--l) J~(continuum) k=o
k
-
E (-l) k k=/+l
=2(-1)'
dll
(12)
For example, for 1=0 (a d o m a m wall height satisfymg 0 < [mo/r[ < 2 ) the lattice current vanishes on one side o f the wall, while having twice the continuum magnitude on the other stde; in this case the total divergence of the continuum and lattice currents are the same. In general the lattice Chern-S~mons current vantshes on the side o f the domain wall for which rn(s)/r is negative, while on the other side has the correct magnitude to compensate for the anomaly of not one positive chirahty zeromode at the d o m a m wall, but rather (aT ~) zeromodes with chirality ( - 1 )t× sgn (rap/r). Thts result agrees precisely with the zeromode spectrum bound to the lattice domain wall found by Jansen and Schmaltz [9]. It also agrees wtth a numerical computation of the .5 We work in d = 2n + 1 dimensions, and our gamma matrix conventions are { 7., y~} = 2 3 ~ , Yu = 7~, ( 7 , . Ya) = i"
222
Volume 301, number 2,3
PHYSICS LETTERS B
4 March 1993
C h e r n - S i m o n s c u r r e n t o n a d = 3 fimte lattice that we have p e r f o r m e d (see ref. [ 5 ] ), which exhibits the b e h a v ior peculiar to the lattice that the C h e r n - S l m o n s c u r r e n t flows o n only one side o f the d o m a i n wall, as well as the d i s c o n t i n u o u s d e p e n d e n c e o f its m a g n i t u d e o n mo/r. O u r analysis c o n f i r m s that the m o d e l o f ref. [ 3 ] correctly reproduces the c o n t i n u u m a n o m a l y for chiral ferm~ons o n a finite lattice in the presence o f weak gauge fields, e v e n for a d o m a i n wall height t o o = O ( 1 ) m lattice units. We w o u l d like to t h a n k J. Kutl, A. M a n o h a r , G. Moore, J. R a b i n a n d M. Schmaltz for useful c o n v e r s a t i o n s . M.G. w o u l d hke to t h a n k the T8 D i w s i o n o f L A N L a n d the Physics D e p a r t m e n t s o f U C San Diego a n d U C S a n t a Barbara for hospitality. M.G. is s u p p o r t e d m part by the D e p a r t m e n t o f Energy u n d e r grant # D E - 2 F G 0 2 9 1 E R 4 0 6 2 8 . K.J. a n d D.K. are s u p p o r t e d in part by the D e p a r t m e n t o f Energy u n d e r grant # D E - F G 0 3 9 0 E R 4 0 5 4 6 , a n d D.K. by the N S F u n d e r c o n t r a c t P H Y - 9 0 5 7 1 3 5 , a n d by a fellowship from the Alfred P. Sloan Foundation.
References [ i ] G Baskaran and P.W Wdson, Phys. Rev. B 37 (1988) 580; P.B. Wlegrnann, Phys. Rev Lett. 60 (1988) 821, I. Dzyaloshinskn, A M. Polyakov and P.B. Wlegmann, Phys. Lett A 127 ( 1988 ) I 12. [ 21 See e.g E Fradkm, Phys. Rev B 42 (1990) 570. [3] D B. Kaplan, Phys. Lett. B 288 (1992) 342, [4] K G. Wilson, m: New phenomena m subnuclear physics, (Ertce, 1975), ed, A. Zichtcht (Plenum, New York, 1977), L.H Karsten and J. Smll, Nucl. Phys. B 183 ( 1981 ) 103 [5] K. Jansen, Phys. Lett B 288 (1992) 348 [ 6 ] C,G. Callan Jr. and J A. Harvey, Nucl Phys. B 250 ( 1985 ) 427. [7] J Goldstone and F. Wdczek, Phys. Rev Left. 47 ( 1981 ) 986, S Deser, R Jacklw and S Templeton, Ann. Phys 140 (1982) 372, 185 ( 1988 ) 406 (E), N. Redhch, Phys. Rev. Lett 52 (1984) 18, A J. Nleml and G Semenoff, Phys. Rev. Lett. 51 (1984) 2077 [ 81 B. Zummo, lectures at Les Houches Summer School ( 1983 ), R Stora, Carg/:se lectures (1983) [9] K. Jansen and M. Schmaltz, Phys. Left. B 296 (1992) 374. [ 10] H. So, Prog Theor. Phys 73 (1985) 528, 74 (1985) 585. [ I I ] A. Coste and M Liischer, Nucl Phys. B 323 (1989) 631.
223
PHYSICS LETTERS B
Chern-Simons currents and chiral fermions on the lattice Maarten F.L. Golterman
I
Department of Phystcs, Washington Umverstty, St Louts, MO 63130-4899, USA
Karl
Jansen 2
and David B. Kaplan
3.4
Department of Physws, Umverstty of Cahforma, 9500 Gdman Drtve 0319, San Dwgo, La Jolla, CA 92093-0319, USA
Received 22 December 1992
We compute the Chern-Stmons current induced by latUce fermtons on a d( = 2n + I )-dtmens~onal lattice, usmg a topologacal mterpretat~on of the ferm~on propagator as a map from the torus to the sphere, Td--,S'~ Our techmques clarify the dependence of the current on short-distance phystcs. We show exphcltly thal for Wdson fermtons It changes dlscontmuously at d+ 1 different values for the mass m. Th~s result ~s relevant for a recently proposed model of choral latttce fermtons as zeromodes bound to a domam wall.
C h e r n - S i m o n s operators play an interesting role in gauge theories in o d d s p a c e - t i m e dimensions. Threed i m e n s i o n a l gauge theories may be relevant in hlgh t e m p e r a t u r e superconductivity [ 1 ] as well as the q u a n t u m Hall effect [2 ]. A n o t h e r place where o d d - d i m e n s i o n a l gauge s y m m e t r y and the C h e r n - S i m o n s o p e r a t o r play a big role is in a recently p r o p o s e d m e t h o d for simulating chiral lattice fermions [ 3 ]. In o d d - d i m e n s i o n a l gauge theories with charged fermions, heavy fermion modes can induce the C h e r n - S i m o n s o p e r a t o r as an effective o p e r a t o r in the low energy theory. The strength o f the o p e r a t o r does not decrease with the mass o f the heavy modes, and is in general sensitive to short distance physics. In this letter we show that the strength o f the induced o p e r a t o r has a simple topological interpretation, allowing one to easily d e t e r m i n e the way that short distance physics enters the low energy theory. The particular application o f the induced C h e r n - S i m o n s o p e r a t o r that is the focus o f this letter, is the model o f d o m a i n wall chiral fermions [ 3 ]. The idea is to i m p l e m e n t an o d d - d i m e n s i o n a l d - - 2 n + l theory o f Wilson fermions with a mass couphng to a d o m a i n wall. The low energy effective theory consists o f massless chiral fermlons b o u n d to the ( d - l ) - d i m e n s i o n a l d o m a i n wail, without there being " d o u b l e r " modes [4] from the Brillouin zone boundary. The a n o m a l o u s W a r d identities for these chirai fermions in the presence o f background gauge fields can be directly measured on a finite lattice; this has already been done numerically for d = 3 and the results are m agreement with the c o n t i n u u m a n o m a l y in l + ! d i m e n s i o n s [ 5 ]. In this system, the a n o m alous divergence o f the ( d - l )-dimensional z e r o m o d e flavor currents is due to charge flow in the direction normal to the d o m a i n wall, even though there ts a mass gap o f the wall - an effect discussed by Callan and Harvey [ 6 ] for c o n t i n u u m fermions coupled to a d o m a i n wall. They pointed out that the C h e r n - S i m o n s action induced by integrating out the heavy fermion m o d e s [ 7 ], being proportional to m~ ] m I, has opposite signs on the two sides o f the d o m a i n wall. This gives rise to a C h e r n - S i m o n s current in the presence o f background gauge fields with a nonzero divergence at the d o m a i n wall. Furthermore, the divergence exactly reproduces the even E-mad address: [email protected]. 2 E-mad [email protected]. a E-mad address, [email protected]. ( Sloan Fellow, NSF Presidential Young Investigator, and DOE Outstandmg Jumor InvesUgator. 0370-2693/93/$ 06.00 © 1993 ElsevmerScience Pubhshers B.V All rights reserved
219
Volume 301, number 2,3
PHYSICS LETTERS B
4 March 1993
( d - 1 )-dimensional anomaly for the single chiral fermion zeromode that is b o u n d to the d o m a m wall "~. This effect is a manifestation of the descent relations between the anomalies in odd and even dimensions [ 8 ]. In this letter, we show how to perform the C a l l a n - H a r v e y ( C H ) analysis for the lattice theory m euclidean space, where the zeromode spectrum is more complicated than m the c o n t i n u u m . It is far from obvious that the lamce theory should follow the CH c o n t i n u u m analysis; after all, the coefficient of the C h e r n - S l m o n s action gets O( 1 ) contributions from arbitrarily heavy fermlon modes, and the heavy spectrum on the lattice looks nothing like m the c o n t i n u u m . In fact, we know the induced C h e r n - S i m o n s operator must have a coefficient very different from the c o n t i n u u m result. While ref. [3 ] analysed the spectrum of the theory for a Wilson coupling r = 1 and a d o m a i n wall height 0 < mo< 2 and found a single chiral mode, a recent paper by Jansen and Schmaltz [ 9 ] analyses the same model for general parameters and shows that the spectrum b o u n d to the d o m a i n wall changes discontinuously with varying mo/r ~2. They find that for 2 k < I mo/rl < 2 k + 2 , where k is an integer in the range O<~k<~d- 1, there are (dZ~) choral modes bound to the domain wall with chirality ( - 1 ) k × s i g n ( m o ) ; there are no choral fermions for I mo/rl > 2d. This is qmte different than the c o n t i n u u m theory, for which there is a single chiral mode for any mo¢: 0. If the induced C h e r n - S i m o n s action on the lattice is to correctly account for the anomalous divergences of the chlral fermton currents on the d o m a i n wall, then evidently its coefficient must also depend discontinuously on mo/r in a very particular way. We show in this letter that that does indeed happen ~3. The abelian C h e r n - S i m o n s action in d = 2n + l c o n t m u o u s euchdean dimensions is given by
F(a) CS
~Otl
Ot2n+i
f d 2n+ Ix Aa, 0a2Aa3"" 0a,,Aa2,+t •
( 1)
When a massive fermion is integrated out of the theory it generates a c o n t n b u t m n to the effective action of the form S~fr=c, Fcs; absorbing the gauge coupling into the gauge field, Fcs is seen to be o f d t m e n s i o n d, and so the coefficient c, will be dimensionless and the operator will not decouple for large fermion mass. The coefficient c, can be computed by calculating the relevant portmn of the graph in fig. 1. This is true on the lattice as well in the weak field, long wavelength limit for the gauge fields. Denoting the fermion propagator and photon vertex as S ( p ) and iAu(p, p ' ) respectively, the graph of fig. 1 yields a value for c, which may be expressed as •
l ~ a l # I ct,,#notn+j
~
c . = (nT-i) ~ ) ! ×
"
" d2n+ Ip (2rt)2,+ , T r [ S ( p ) A ~ , ( p , p - q t ) S ( p - q ~ )
]
...A . . . . (P+q"+~'P)
q,=o"
(2)
BZ
4~ It should be pointed out that if the magmtude of the Chern-Slmons current is regular dependent, the graph needs to be regulated. A regulator cannot change the divergenceof the current, however. We thank M. Lfischerfor this comment. 42 All dlmenslonful parameters are gdven m lattice units. By a domam wall of height mo we mean a spatially dependent mass term re(s) --. +mo as s-, + oo, where s is the coordinate transverse to the domain wall. 43 The dependence of the reduced Chern-Slmons actmn on the Wdson couphng • has been previouslydiscussed for three d~mensmnsm the continuum hmlt (spatmlly constant rn--,0) in ref. [ 10] and for Iml < 1 m ref. [ I 1] Some of the techmques used m this letter are samdar to those found m the latter work.
---> ql 220
\
/
~
<--
qn+1
Fig. 1. The Feynman diagram m 2n+ I dtmensmns conmbutmg to the induced Chern-Simons acuon for abehan gauge fields, Y 7=+1 ~q, = 0. Graphs with mulUplephoton vertmespecuhar to the latuce do not conmbute, as each A field from such a vertex has the same Lorentz index and the contnbutlonvanishes by the anusymmetry of the ~tensor
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The factor o f n + 1 m eq. ( 2 ) is due to the s y m m e t r y factor o f the graph m fig. !; the factor o f i is the product o f i "+ ~ from the photon vertices and ( - ~)" from relating the d e r i v a u v e s in eq. ( l ) to powers o f momenta. The pintegration is over the Brilloum zone o f a hypercubic lattice w~th lattice spacing a = I. The integral ( 2 ) looks very difficult to c o m p u t e on the lattice for arbitrary n, as both S(p) and Au(p, p ' ) are in general rather c o m p l i c a t e d functions. It is m a d e quite tractable, however, by exposing its topological properUes. Gauge invariance implies that the p h o t o n coupling satisfies the W a r d i d e n u t y '~
Au(p,P)=-i~p u
(3)
(P) ,
allowing c, m eq. ( 2 ) to be reexpressed as " f d2~+lp ( - - i ) ¢u, u2,., j T r { [ S ( p ) Ou, S ( p ) -~1 [ S ( p ) 0 u 2 , ÷ , S ( p ) - ' ] : c,= (n+l)(2n+l)! ( 2 n ) 2"+' . . . .
(4)
where the d i f f e r e n t i a u o n 0, is with respect to p,. The fermlon propagator may be written in the genetic form
S -~ (p) =a(p) + i b ( p ) . y = N ( p ) [ c o s
IO(P)l+i~(P)'~' sm [0(P) I 1 - N ( p ) V ( p ) ,
(5)
where N ( p ) -= a 2 ~ , 0 ( p ) = ~ arctan ( Ib l / a ), and V(p) is seen to be a 2" X 2" unitary matrix. P r o v i d e d that S - z (p) does not vanish for any p, eq. ( 4 ) ts i n d e p e n d e n t o f N ( p ) , allowing S - ~(p) and S(p) to be replaced everywhere by V(p) and Vt(p) respectively. This matrix V(p) is seen to describe a m a p p i n g from m o m e n t u m space - with on the hypercub~c latuce is the torus T a - onto the sphere S d defined by the vector O(p). The h o m o t o p y classes o f such m a p s are ~dentified by integers, and so the integral in eq. ( 4 ) has a simple topological interpretation; It is, to a n o r m a l i z a u o n constant, the winding n u m b e r o f the m a p described by the ferm~on propagator. (As an example o f a n o n m v a l m a p from T2--*S2, picture the torus as a square with oppostte edges identified, and m a p the square onto a sphere so that the center o f the square is m a p p e d to the N o r t h pole o f S 2, whtle the edge o f the square are all m a p p e d to the South pole. ) We now proceed to c o m p u t e th~s winding n u m b e r for the Wilson fermion propagator. Although we are ultimately interested in the effective action for lattice fermlons in the presence o f a d o m a i n wall, we can compute the effective action far from the mass defect on e~ther side by treating the fermion mass as a constant rn. Thus we can use the s t a n d a r d Wilson propagator d
S-l(p)=
~ ju=l
d
iTusmpu+m+r ~ ( c o s p u _ l ) .
(6)
,u=l
Continuous changes in the mass and Wilson couphng, m and r, cannot change the value o f the winding n u m b e r except at points for which S - t has a zero for some m o m e n t u m p. Such singular points o f the m a p p i n g occur only at m o m e n t a corresponding to the corners o f the Brillouin zone, and then only for m/r=O, 2, ..., 2d. The C h e r n - S i m o n s coefficient c, as a function o f m / r must therefore be piecewise constant, changing only at these crmcai values. Furthermore, for fixed r, V(p) -~ _+ 1 as rn-~ + oo, so we may deduce that
c,(m,r)=O
form/r
(7)
To c o m p u t e c, for 0 < m / r < 2d, we need only evaluate the d e n v a u v e o f the integral in eq. ( 4 ) with respect to m across the critical values m/r=O, 2 ..... 2d. This task is simphfied by the fact that c , ( m + d m , r) is unchanged as one deforms d m in a p - d e p e n d e n t way so that d i n ( p ) vamshes for all p except in the vicinity o f the Brilloum zone corners; these points are d e n o t e d by p = ¢ ~ k ) , the a = 1, .., (~) vectors with k n o n v a n i s h m g c o m p o n e n t s We expand the lamce gauge field as Uu(x) = 1- L4u(X)+ 221
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equal to zt. Therefore we need only evaluate the mtegrals m infimtesimal regions near the Brillouin zone corners ~k). After some algebra this yields ~5
dcn(m,r) dm
-
i(-l)n2 ~ )k d ( n + 1 ) /,.a )-" ( - I dm
l(--1) "
a
= (n+l)(2zr).n!k~=o(--l)k
(d)
f
d2~+'p
m-2rk
(27t)xn+l [IPl 2.+. (m_2rk)2]
~+l
8(m-2rk),
(8)
where the integration d ~ is over the mfimtesimal region ]P] < e ~ 0. This may be trivially integrated with respect to m, given the boundary values (7), to yield the Chern-Stmons coefficient for Wilson fermions:
1, 1,. c~(m, r ) = 2 ( n ~ - i ) (2n)nn !
k=o
( - 1 )k
m-2rk Im-2rkl
(Wilson fermions) .
(9)
This is to be compared to the continuum result for a fermlon of mass m, computed by inserting the continuum free propagator in eq. (2) and integrating over continuum m o m e n t u m space:
cn(m) =
i(-l)
n
(continuum result) .
2(n+ l)(2n)"n![m[
(10)
Our computation is exact in the limit of small and adiabatic external gauge fields, even for m o f O ( l ) m lattice units; it is readily generalized to non-abehan gauge fields. So far we have only considered lattice fermions with a constant mass; our primary interest though is in studying current flow m the presence of a domain wall mass term for the fermion. Following Callan and Harvey, we assume that sufficiently far away from the domain wall on either stde, we can treat the mass as constant and compute the induced current by varying the Chern-Simons action, juts = ~iSefr
n+ 1
6A~ - ~2
c,(m(s), r)¢~,,~ 2 ,2,÷,F,,~2...F,~2._t~2.,
(11)
where the factor of 2" arises from replacing 0.4 by the field strength F. Since c, depends on the mass re(s) which is space dependent, Jucs has a nonzero divergence corresponding to current flow normal to the d o m a m wall. In their continuum analysis, Callan and Harvey pomted out that with c, given by eq. (10), there was equal current flow toward the wall from each side that exactly compensated for the anomalous divergence of the chtral fermlon current along the wall's surface. For lattice fermlons in the presence of the domain wall, we must use the value (9) for c,(, r). Thus on the side o f the domain wall for which m(s)/r is negative, the Chern-Simons current vanishes. On the other side, where m (s)/r--, Imo/rl, the current either vanishes if Imp~r[ > 2 ( d + 1 ), or else, for 2l< [rap~r l < 2 ( l + 1 ), the current is given by J~(lattice) =--(--l) J~(continuum) k=o
k
-
E (-l) k k=/+l
=2(-1)'
dll
(12)
For example, for 1=0 (a d o m a m wall height satisfymg 0 < [mo/r[ < 2 ) the lattice current vanishes on one side o f the wall, while having twice the continuum magnitude on the other stde; in this case the total divergence of the continuum and lattice currents are the same. In general the lattice Chern-S~mons current vantshes on the side o f the domain wall for which rn(s)/r is negative, while on the other side has the correct magnitude to compensate for the anomaly of not one positive chirahty zeromode at the d o m a m wall, but rather (aT ~) zeromodes with chirality ( - 1 )t× sgn (rap/r). Thts result agrees precisely with the zeromode spectrum bound to the lattice domain wall found by Jansen and Schmaltz [9]. It also agrees wtth a numerical computation of the .5 We work in d = 2n + 1 dimensions, and our gamma matrix conventions are { 7., y~} = 2 3 ~ , Yu = 7~, ( 7 , . Ya) = i"
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Volume 301, number 2,3
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C h e r n - S i m o n s c u r r e n t o n a d = 3 fimte lattice that we have p e r f o r m e d (see ref. [ 5 ] ), which exhibits the b e h a v ior peculiar to the lattice that the C h e r n - S l m o n s c u r r e n t flows o n only one side o f the d o m a i n wall, as well as the d i s c o n t i n u o u s d e p e n d e n c e o f its m a g n i t u d e o n mo/r. O u r analysis c o n f i r m s that the m o d e l o f ref. [ 3 ] correctly reproduces the c o n t i n u u m a n o m a l y for chiral ferm~ons o n a finite lattice in the presence o f weak gauge fields, e v e n for a d o m a i n wall height t o o = O ( 1 ) m lattice units. We w o u l d like to t h a n k J. Kutl, A. M a n o h a r , G. Moore, J. R a b i n a n d M. Schmaltz for useful c o n v e r s a t i o n s . M.G. w o u l d hke to t h a n k the T8 D i w s i o n o f L A N L a n d the Physics D e p a r t m e n t s o f U C San Diego a n d U C S a n t a Barbara for hospitality. M.G. is s u p p o r t e d m part by the D e p a r t m e n t o f Energy u n d e r grant # D E - 2 F G 0 2 9 1 E R 4 0 6 2 8 . K.J. a n d D.K. are s u p p o r t e d in part by the D e p a r t m e n t o f Energy u n d e r grant # D E - F G 0 3 9 0 E R 4 0 5 4 6 , a n d D.K. by the N S F u n d e r c o n t r a c t P H Y - 9 0 5 7 1 3 5 , a n d by a fellowship from the Alfred P. Sloan Foundation.
References [ i ] G Baskaran and P.W Wdson, Phys. Rev. B 37 (1988) 580; P.B. Wlegrnann, Phys. Rev Lett. 60 (1988) 821, I. Dzyaloshinskn, A M. Polyakov and P.B. Wlegmann, Phys. Lett A 127 ( 1988 ) I 12. [ 21 See e.g E Fradkm, Phys. Rev B 42 (1990) 570. [3] D B. Kaplan, Phys. Lett. B 288 (1992) 342, [4] K G. Wilson, m: New phenomena m subnuclear physics, (Ertce, 1975), ed, A. Zichtcht (Plenum, New York, 1977), L.H Karsten and J. Smll, Nucl. Phys. B 183 ( 1981 ) 103 [5] K. Jansen, Phys. Lett B 288 (1992) 348 [ 6 ] C,G. Callan Jr. and J A. Harvey, Nucl Phys. B 250 ( 1985 ) 427. [7] J Goldstone and F. Wdczek, Phys. Rev Left. 47 ( 1981 ) 986, S Deser, R Jacklw and S Templeton, Ann. Phys 140 (1982) 372, 185 ( 1988 ) 406 (E), N. Redhch, Phys. Rev. Lett 52 (1984) 18, A J. Nleml and G Semenoff, Phys. Rev. Lett. 51 (1984) 2077 [ 81 B. Zummo, lectures at Les Houches Summer School ( 1983 ), R Stora, Carg/:se lectures (1983) [9] K. Jansen and M. Schmaltz, Phys. Left. B 296 (1992) 374. [ 10] H. So, Prog Theor. Phys 73 (1985) 528, 74 (1985) 585. [ I I ] A. Coste and M Liischer, Nucl Phys. B 323 (1989) 631.
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