Content-Type: multipart/mixed; boundary="-------------0104181459940" This is a multi-part message in MIME format. ---------------0104181459940 Content-Type: text/plain; name="01-145.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-145.comments" 4 pages ---------------0104181459940 Content-Type: text/plain; name="01-145.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-145.keywords" Chern-Simons invariant, Berry phase, fiber bundle ---------------0104181459940 Content-Type: application/postscript; name="berry1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="berry1.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.76 (MiKTeX 1.11) Copyright 1997 Radical Eye Software (www.radicaleye.com) %%Title: berry1.dvi %%CreationDate: Wed Apr 18 12:54:46 2001 %%Pages: 4 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\TEXMF\MIKTEX\BIN\DVIPS.EXE berry1 %DVIPSParameters: dpi=600 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Fq(\()p Fh(r)p Fq(\))24 b(=)e Fi(x\033)616 2739 y Fg(x)677 2727 y Fq(+)c Fi(y)s(\033)851 2739 y Fg(y)910 2727 y Fq(+)g Fi(z)t(\033)1083 2739 y Fg(z)1144 2727 y Ff(\021)23 b Fh(r)18 b Ff(\001)f Fi(~)-40 b(\033)s(;)382 b Fq(\(1\))-150 2918 y(where)31 b Fi(\033)141 2930 y Fg(x;y)r(;z)324 2918 y Fq(are)f(the)i(three)f(comp)r(onen)n(ts)g(of)g(the)h(P)n(auli)e (ma-)-150 3018 y(trices)i(and)h Fi(x;)14 b(y)s(;)g(z)36 b Fq(are)c(three)h(real)f(parameters.)51 b(F)-7 b(or)32 b(a)g(\014xed)-150 3117 y Fh(r)38 b Fq(=)f(\()p Fi(x;)14 b(y)s(;)g(z)t Fq(\))35 b(the)i(Hamiltonian)f(giv)n(en)f(in)i(Eq.)e (\(1\))i(has)e(t)n(w)n(o)-150 3217 y(eigen)n(v)-5 b(alues)27 b Fi(E)341 3229 y Fe(\006)421 3217 y Fq(=)c Ff(\006j)p Fh(r)p Ff(j)p Fq(.)37 b(Let)28 b(us)g(denote)g(the)g(corresp)r(onding) -150 3317 y(normalized)f(t)n(w)n(o-comp)r(onen)n(t)f(eigen)h(v)n (ectors)f(b)n(y)h Ff(j)p Fi( )1542 3329 y Fe(\006)1598 3317 y Fq(\()p Fh(r)p Fq(\))d Fi(>)p Fq(.)-67 3419 y(F)-7 b(or)43 b(the)h(rest)f(of)g(the)h(pap)r(er)f(let)h(us)f(fo)r(cus)g(on,) 48 b(sa)n(y)-7 b(,)46 b(the)-150 3518 y(p)r(ositiv)n(e-energy)31 b(eigen)i(v)n(ector)e Ff(j)p Fi( )977 3530 y Fd(+)1032 3518 y Fq(\()p Fh(r)p Fq(\))i Fi(>)p Fq(.)53 b(Since)33 b(the)g(trans-)-150 3618 y(formation)42 b Ff(j)p Fi( )323 3630 y Fd(+)378 3618 y Fq(\()p Fh(r)p Fq(\))48 b Fi(>)p Ff(!)g Fi(e)764 3588 y Fg(i\022)824 3618 y Ff(j)p Fi( )901 3630 y Fd(+)957 3618 y Fq(\()p Fh(r)p Fq(\))g Fi(>)42 b Fq(preserv)n(es)f(the)i(nor-)-150 3718 y(malization)c(and)g(the)h (eigen)g(nature)f(of)g Ff(j)p Fi( )1279 3730 y Fd(+)1335 3718 y Fq(\()p Fh(r)p Fq(\))44 b Fi(>)p Fq(,)e(there)e(is)-150 3817 y(a)30 b(con)n(tin)n(uum)h(of)g(t)n(w)n(o-comp)r(onen)n(t)e(v)n (ectors,)i(eac)n(h)f(lab)r(eled)h(b)n(y)-150 3917 y Fi(\022)r Fq(,)39 b(satisfying)d Fi(H)7 b Ff(j)p Fi( )485 3929 y Fd(+)p Fg(;\022)593 3917 y Fq(\()p Fh(r)p Fq(\))39 b Fi(>)p Fq(=)f Ff(j)p Fh(r)p Ff(jj)p Fi( )1065 3929 y Fd(+)p Fg(;\022)1173 3917 y Fq(\()p Fh(r)p Fq(\))h Fi(>)p Fq(.)64 b(F)-7 b(or)36 b(a)g(giv)n(en)-150 4017 y Fh(r)k Fq(the)g(in)n(ternal)e(space)h(formed)g(b)n(y)g(all)g Ff(j)p Fi( )1258 4029 y Fd(+)p Fg(;\022)1367 4017 y Fq(\()p Fh(r)p Fq(\))k Fi(>)c Fq(has)g(the)-150 4116 y(U\(1\))27 b(group)e(structure.)36 b(When)27 b Fh(r)f Fq(v)-5 b(aries)26 b(o)n(v)n(er)e(a)i(manifold)h Ff(M)-150 4216 y Fq(\(henceforth)c (referred)g(as)f(the)i(base)f(space\))g(the)g(in)n(ternal)g(space)-150 4316 y(sw)n(eeps)g(out)g(a)h(\\\014b)r(er)f(bundle".)35 b(Since)24 b(the)g(in)n(ternal)f(space)g(has)-150 4415 y(U\(1\))j(symmetry)f(this)h(\014b)r(er)f(bundle)h(is)g(called)f(a)g (U\(1\))h(bundle.)-150 4515 y(In)f(the)g(follo)n(wing)e(w)n(e)i(shall)f (refer)g(to)g(suc)n(h)g(\014b)r(er)h(bundle)g(as)f(the)-150 4614 y(eigen)j(bundle)h(of)g(Eq.)f(\(1\).)-67 4717 y(A)36 b(connection)f(can)h(b)r(e)g(de\014ned)g(on)f(the)h(eigen)f(bundle)i(b) n(y)-150 4816 y(\014rst)25 b(c)n(ho)r(osing)e(a)i(reference)e(eigen)i (v)n(ector)e Ff(j)p Fi( )1298 4828 y Fd(+)1354 4816 y Fq(\()p Fh(r)p Fq(\))h Fi(>)g Fq(at)h(eac)n(h)f Fh(r)-150 4916 y Fq(then)k(de\014ning)g(the)g(v)n(ector)e(\014eld)246 5138 y Fh(A)318 5150 y Fg(b)351 5138 y Fq(\()p Fh(r)p Fq(\))e(=)576 5082 y(1)p 576 5119 42 4 v 582 5195 a Fi(i)651 5138 y(<)e( )792 5150 y Fd(+)847 5138 y Fq(\()p Fh(r)p Fq(\))p Ff(jr)p Fi( )1096 5150 y Fd(+)1153 5138 y Fq(\()p Fh(r)p Fq(\))i Fi(>)e(:)396 b Fq(\(2\))-150 5352 y(W)-7 b(e)28 b(note)g(that)g(in)g(order)f(for)g Fh(A)872 5364 y Fg(b)934 5352 y Fq(to)g(b)r(e)i(w)n(ell)e(de\014ned,)i(the)f(ref-) -150 5452 y(erence)k(v)n(ector)g(m)n(ust)h(v)-5 b(ary)33 b(con)n(tin)n(uously)f(with)h Fh(r)p Fq(.)54 b(The)33 b(con-)-150 5552 y(nection)c Fh(A)214 5564 y Fg(b)276 5552 y Fq(de\014ned)g(ab)r(o)n(v)n(e)e(is)h(not)h(unique.)40 b(Indeed,)29 b(b)n(y)f(p)r(er-)-150 5651 y(forming)38 b(the)h(transformation)f Ff(j)p Fi( )977 5663 y Fd(+)1032 5651 y Fq(\()p Fh(r)p Fq(\))43 b Fi(>)p Ff(!)e Fi(e)1406 5621 y Fg(i\022)r Fd(\()p Fc(r)p Fd(\))1550 5651 y Ff(j)p Fi( )1627 5663 y Fd(+)1682 5651 y Fq(\()p Fh(r)p Fq(\))i Fi(>)-150 5751 y Fq(w)n(e)27 b(induce)h(a)f(\\gauge)f(transformation")g (on)h Fh(A)1376 5763 y Fg(b)1409 5751 y Fq(:)2722 2241 y Fh(A)2794 2253 y Fg(b)2851 2241 y Ff(!)c Fh(A)3029 2253 y Fg(b)3081 2241 y Fq(+)18 b Ff(r)p Fi(\022)r(:)681 b Fq(\(3\))2042 2412 y(The)28 b(Berry's)e(phase)h(caused)g(b)n(y)g(an)h (adiabatic)e(ev)n(olution)h(of)h Fh(r)2042 2511 y Fq(around)e(a)i (closed)e(lo)r(op)i Ff(C)k Fq(is)27 b(giv)n(en)g(b)n(y)2692 2720 y Fi(\015)g Fq(=)2850 2607 y Fj(I)2896 2796 y Fe(C)2953 2720 y Fi(d)p Fh(r)19 b Ff(\001)g Fh(A)3168 2732 y Fg(b)3201 2720 y Fq(\()p Fh(r)p Fq(\))p Fi(:)651 b Fq(\(4\))2042 2933 y(Ob)n(viously)26 b(the)i(Berry's)e(phase)h(is)h(gauge)e(in)n(v)-5 b(arian)n(t.)2125 3033 y(Let)36 b(us)f(\014rst)g(consider)g(t)n(w)n (o-dimensional)f(base)h(spaces)f Ff(M)2042 3132 y Fq(that)28 b(are)g(closed)f(surfaces.)38 b(It)29 b(turns)f(out)h(that)f(if)h Ff(M)f Fq(encloses)2042 3232 y(the)40 b(origin)f(\(for)h(example)g Ff(M)j Fq(=)g Fi(S)3262 3202 y Fd(2)3343 3232 y Fq(=)h Ff(f)p Fh(r)p Fq(;)54 b Ff(j)p Fh(r)p Ff(j)44 b Fq(=)f(1)p Ff(g)p Fq(\),)g(it)2042 3332 y(is)34 b(imp)r(ossible)h(to)f(c)n(ho)r (ose)f(a)h(gauge)f(in)i(whic)n(h)f Fh(A)3660 3344 y Fg(b)3694 3332 y Fq(\()p Fh(r)p Fq(\))h(is)g(non-)2042 3431 y(singular)26 b(ev)n(erywhere.)36 b(In)28 b(order)e(to)i(obtain)g(non-singular)e Fh(A)4051 3443 y Fg(b)2042 3531 y Fq(it)36 b(is)f(necessary)e(to)i (divide)h Ff(M)f Fq(in)n(to)g(a)g(n)n(um)n(b)r(er)g(of)g(\(o)n(v)n (erlap-)2042 3631 y(ping\))24 b(patc)n(hes)g(so)g(that)h(1\))p Fh(A)2972 3643 y Fg(b)3030 3631 y Fq(is)f(non-singular)f(in)i(eac)n(h)e (patc)n(h,)2042 3730 y(and)30 b(2\))h(in)g(the)g(o)n(v)n(erlapping)d (region)h(of)i(t)n(w)n(o)f(patc)n(hes)g(the)h(dif-)2042 3830 y(feren)n(t)i Fh(A)2355 3842 y Fg(b)2388 3830 y Fq('s)h(only)f(di\013er)g(b)n(y)g(a)g(gauge)f(transformation.)52 b(His-)2042 3929 y(torically)23 b(a)h(problem)f(lik)n(e)h(this)h(w)n (as)e(encoun)n(tered)g(when)i(Dirac)2042 4029 y(tried)g(to)g(write)f (do)n(wn)h(the)g(v)n(ector)f(p)r(oten)n(tial)h(in)g(the)h(surround-) 2042 4129 y(ing)k(of)g(a)g(magnetic)g(monop)r(ole.)g([2])g(It)h(turns)f (out)h(that)f(under)2042 4228 y(the)38 b(framew)n(ork)d(of)j(quan)n (tum)f(mec)n(hanics)g(the)h(condition)f(2\))2042 4328 y(stated)24 b(ab)r(o)n(v)n(e)g(requires)f(the)i(strength)f(of)h(the)g (monop)r(ole)f(to)h(b)r(e)2042 4428 y(quan)n(tized.)i([2,3])2125 4527 y(The)f(fact)g(that)g(it)h(is)f(imp)r(ossible)g(to)f(de\014ne)i (an)e(ev)n(erywhere-)2042 4627 y(nonsingular)45 b(connection)h(is)h (the)g(symptom)g(of)f(non-trivial)2042 4726 y(top)r(ology)-7 b(.)71 b(In)40 b(his)f(seminal)g(w)n(ork)f(Chern)i(disco)n(v)n(ered)d (a)i(set)2042 4826 y(of)d(in)n(v)-5 b(arian)n(ts)35 b(to)h(c)n (haracterize)e(suc)n(h)i(non-trivialit)n(y)-7 b(.)35 b([4])h(F)-7 b(or)2042 4926 y(the)29 b(simple)g(case)f(w)n(e)h(are)e (considering)h(the)h(in)n(v)-5 b(arian)n(ts)28 b(reduce)2042 5025 y(to)f(a)g(single)h(n)n(um)n(b)r(er,)f(the)h(Chern)f(n)n(um)n(b)r (er:)2659 5234 y Ff(C)h Fq(=)2854 5178 y(1)p 2829 5215 92 4 v 2829 5291 a(4)p Fi(\031)2944 5121 y Fj(Z)2990 5310 y Fe(M)3086 5234 y Fi(d)p Fh(a)20 b Ff(\001)e Fh(B)3304 5246 y Fg(b)3337 5234 y Fi(:)618 b Fq(\(5\))2042 5452 y(Here)26 b Fh(B)2305 5464 y Fg(b)2361 5452 y Fq(=)d Ff(r)16 b(\002)g Fh(A)2687 5464 y Fg(b)2747 5452 y Fq(is)26 b(the)h(curv)-5 b(ature)25 b(asso)r(ciated)g(with)i Fh(A)3995 5464 y Fg(b)4029 5452 y Fq(.\()2042 5552 y(Note)k(that)h(in)f(order)f (to)i(ha)n(v)n(e)e(a)h(w)n(ell)g(de\014ned)g Fh(B)3654 5564 y Fg(b)3719 5552 y Fq(w)n(e)g(need)h(a)2042 5651 y Fb(lo)l(c)l(al)t(ly)25 b Fq(di\013eren)n(tiable)f Fh(A)2855 5663 y Fg(b)2888 5651 y Fq(.\))36 b(F)-7 b(or)23 b(the)i(eigen)e (bundle)h(of)g(Eq.)f(\(1\))2042 5751 y(Berry)j(has)h(sho)n(wn)g(that)h ([1])1946 6000 y(1)p eop %%Page: 2 2 2 1 bop 547 302 a Fh(B)615 314 y Fg(b)648 302 y Fq(\()p Fh(r)p Fq(\))24 b(=)872 246 y(1)p 872 283 42 4 v 872 359 a(2)979 246 y(^)-46 b Fi(r)p 934 283 123 4 v 934 359 a Ff(j)p Fh(r)p Ff(j)1019 335 y Fd(2)1067 302 y Fi(:)696 b Fq(\(6\))-150 534 y(The)20 b(righ)n(t)g(hand)g(side)g(of)g(Eq.)f (\(6\))i(is)f(the)g(same)g(as)f(the)i(magnetic)-150 634 y(\014eld)26 b(pro)r(duced)g(b)n(y)g(a)f(monop)r(ole)g(sitting)h(at)g (the)g(origin.)36 b(Th)n(us)-150 733 y(if)31 b Ff(M)g Fq(encloses)f(the)h(origin)e(Eq.)i(\(5\))g(yields)f Ff(C)j Fq(=)28 b(1)p Fi(=)p Fq(2)i(indicat-)-150 833 y(ing)23 b(the)g(corresp)r(onding)e(eigen)h(bundle)i(is)e(non-trivial.)34 b(On)23 b(the)-150 932 y(other)k(hand)h(if)h Ff(M)f Fq(do)r(es)g(not)g (enclose)f(the)h(origin,)f(then)i Ff(C)f Fq(=)c(0)-150 1032 y(and)f(the)g(corresp)r(onding)e(eigen)h(bundle)h(is)g(trivial.)35 b(The)23 b(Chern)-150 1132 y(n)n(um)n(b)r(er)g(\(Eq.)f(\(5\)\))h(has)g (a)f(simple)h(ph)n(ysical)f(in)n(terpretation)f(-)i(it)-150 1231 y(is)28 b(the)h(total)g(\015ux)f(pro)r(duced)h(b)n(y)f(the)h (bundle)g(curv)-5 b(ature.)39 b(In)29 b(a)-150 1331 y(pro)r(of)23 b(similar)f(to)h(that)h(giv)n(en)e(in)i(Ref.)f([3],)h(Chern)f(sho)n(w)n (ed)f(that)-150 1431 y Ff(C)34 b Fq(should)c(b)r(e)g(quan)n(tized)g(to) f(v)-5 b(alues)30 b Fi(n=)p Fq(2)f(where)g Fi(n)h Fq(=)f(in)n(teger.) -150 1530 y([4])-67 1630 y(Since)j(the)f(Chern)h(n)n(um)n(b)r(er)f(can) g(b)r(e)h(larger)d(than)j(1)p Fi(=)p Fq(2,)f(it)h(is)-150 1730 y(in)n(teresting)19 b(to)i(kno)n(w)e(what)h(kind)g(of)h (Hamiltonian)f(will)g(exhibit)-150 1829 y Ff(C)28 b Fq(=)22 b Fi(n=)p Fq(2)27 b(\()p Fi(n)c(>)g Fq(1\))k(eigen)g(bundles.)-67 1929 y(One)f(answ)n(er)g(is)g(giv)n(en)g(b)n(y)h(the)g(follo)n(wing)e (\()p Fi(n)17 b Fq(+)g(1\))f Ff(\002)g Fq(\()p Fi(n)h Fq(+)g(1\))-150 2029 y(matrix)585 2212 y Fi(H)7 b Fq(\()p Fh(r)p Fq(\))24 b(=)f Fh(r)18 b Ff(\001)h Fh(S)p Fi(;)735 b Fq(\(7\))-150 2395 y(where)27 b Fh(S)c Fq(=)g(\()p Fi(S)337 2407 y Fg(x)379 2395 y Fi(;)14 b(S)467 2407 y Fg(y)507 2395 y Fi(;)g(S)595 2407 y Fg(z)633 2395 y Fq(\))28 b(are)f(the)h(matrices)f(represen)n(ting)f(the)-150 2495 y(three)32 b(generators)f(of)h(SU\(2\))i(in)f(the)g(higher)f(spin) h(\()p Fi(S)j Fq(=)31 b Fi(n=)p Fq(2\))-150 2594 y(represen)n(tation.)k (F)-7 b(or)27 b(example)g(for)g Fi(n)c Fq(=)g(2)k(w)n(e)g(ha)n(v)n(e) -37 2872 y Fi(S)14 2884 y Fg(x)79 2872 y Fq(=)211 2816 y(1)p 176 2853 111 4 v 176 2870 a Ff(p)p 245 2870 42 4 v 68 x Fq(2)311 2705 y Fj(0)311 2855 y(@)397 2773 y Fq(0)83 b(1)f(0)397 2872 y(1)h(0)f(1)397 2972 y(0)h(1)f(0)702 2705 y Fj(1)702 2855 y(A)788 2872 y Fi(;)42 b(S)904 2884 y Fg(y)967 2872 y Fq(=)1099 2816 y(1)p 1064 2853 111 4 v 1064 2870 a Ff(p)p 1133 2870 42 4 v 68 x Fq(2)1199 2705 y Fj(0)1199 2855 y(@)1285 2773 y Fq(0)83 b Ff(\000)p Fi(i)108 b Fq(0)1292 2872 y Fi(i)115 b Fq(0)108 b Ff(\000)p Fi(i)1285 2972 y Fq(0)115 b Fi(i)141 b Fq(0)1693 2705 y Fj(1)1693 2855 y(A)-37 3204 y Fi(S)14 3216 y Fg(z)75 3204 y Fq(=)163 3037 y Fj(0)163 3187 y(@)249 3105 y Fq(1)83 b(0)115 b(0)249 3204 y(0)83 b(0)115 b(0)249 3304 y(0)83 b(0)f Ff(\000)p Fq(1)618 3037 y Fj(1)618 3187 y(A)705 3204 y Fi(:)1058 b Fq(\(8\))-67 3487 y(There)39 b(is)h(another)f(mo)r (di\014cation)h(of)f(Eq.)h(\(1\))g(whic)n(h)g(also)-150 3587 y(leads)23 b(to)g Ff(C)k Fq(=)c Fi(n=)p Fq(2)f(eigen)h(bundle.)36 b(In)n(terestingly)22 b(this)i(time)g(w)n(e)-150 3687 y(do)h(not)f(need)h(to)g(enlarge)f(the)h(dimension)g(of)f(the)i (Hamiltonian)-150 3786 y(matrix.)36 b(Consider)27 b(the)h(follo)n(wing) e(2)18 b Ff(\002)g Fq(2)28 b(matrix)530 3969 y Fi(H)7 b Fq(\()p Fh(r)p Fq(\))24 b(=)822 3947 y(^)821 3969 y Fi(h)p Fq(\()p Fh(r)p Fq(\))19 b Ff(\001)d Fi(~)-40 b(\033)t(;)680 b Fq(\(9\))-150 4161 y(where)90 4139 y(^)89 4161 y Fi(h)p Fq(\()p Fh(r)p Fq(\))28 b(is)f(a)f(suitably)h(c)n(hosen)f(unit)i(v)n (ector)d(\014eld)i(that)h(de-)-150 4261 y(\014nes)i(a)g(mapping)f(from) h Ff(M)g Fq(\(a)g(closed)f(t)n(w)n(o-dimensional)f(sur-)-150 4361 y(face\))e(to)g Fi(S)205 4330 y Fd(2)242 4361 y Fq(.)36 b(It)26 b(is)g(kno)n(wn)f(that)i(suc)n(h)e(mappings)h(can)f(b)r (e)h(clas-)-150 4460 y(si\014ed)i(in)n(to)f(homotop)n(y)f(classes)h (eac)n(h)g(lab)r(eled)g(b)n(y)g(an)h(in)n(teger)546 4681 y Ff(P)i Fq(=)721 4568 y Fj(Z)768 4757 y Fd(\000)827 4681 y Fi(d)p Fh(a)19 b Ff(\001)f Fh(J)p Fi(:)697 b Fq(\(10\))-150 4906 y(Here)27 b Fh(J)p Fq(,)h(the)g(dual)g(of)f(the)h(P)n(on)n(try)n (agin)d(form,)i(is)h(giv)n(en)f(b)n(y)277 5124 y Fi(J)331 5089 y Fg(\026)398 5124 y Fq(=)521 5068 y(1)p 496 5105 92 4 v 496 5181 a(8)p Fi(\031)598 5124 y(\017)632 5089 y Fg(\026\027)t(\025)753 5102 y Fq(^)752 5124 y Fi(h)19 b Ff(\001)f Fq(\()p Fi(@)936 5136 y Fg(\027)979 5102 y Fq(^)978 5124 y Fi(h)g Ff(\002)g Fi(@)1171 5136 y Fg(\025)1216 5102 y Fq(^)1215 5124 y Fi(h)p Fq(\))p Fi(:)427 b Fq(\(11\))-150 5352 y(W)-7 b(e)24 b(will)f(sho)n(w)f(later)h(that)g(b)n(y)g(c)n(ho)r (osing)f(a)1216 5330 y(^)1215 5352 y Fi(h)p Fq(\()p Fh(r)p Fq(\))i(with)g Ff(P)29 b Fq(=)23 b Fi(n)g Fq(in)-150 5452 y(Eq.)29 b(\(9\))g(the)g(corresp)r(onding)f(eigen)g(bundle)i(has)f (Chern)g(n)n(um-)-150 5551 y(b)r(er)f Ff(C)f Fq(=)c Fi(n=)p Fq(2.)-67 5651 y(The)33 b(Chern)g(n)n(um)n(b)r(er)g(records)f(the)i (highest)f(lev)n(el)g(of)g(top)r(o-)-150 5751 y(logical)c (non-trivialit)n(y)-7 b(.)42 b(When)30 b(the)h(Chern)e(n)n(um)n(b)r(er) h(v)-5 b(anishes)2042 271 y(the)26 b(eigen)g(bundle)g(can)g(still)g(b)r (e)g(non-trivial)f(at)h(a)g(more)f(subtle)2042 371 y(lev)n(el.)44 b(Let)31 b(us)f(consider)f(a)h(three-dimensional)g(base)f(space)h(in) 2042 471 y(whic)n(h)c Ff(C)i Fq(=)22 b(0)k(for)g(all)g(closed)f (surfaces,)h(implying)g(the)h(absence)2042 570 y(of)36 b(\\)p Fh(B)2255 582 y Fg(b)2288 570 y Fq(-monop)r(ole".)60 b(In)37 b(that)f(case)f(the)i(\\\015ux)e(lines")h(of)g Fh(B)4051 582 y Fg(b)2042 670 y Fq(form)28 b(closed)h(lo)r(ops.)40 b(There)29 b(is)g(a)g(top)r(ological)e(in)n(teresting)h(sit-)2042 769 y(uation)36 b(in)g(whic)n(h)h(these)f(\015ux)h(lines)f(link)g(with) h(one)f(another.)2042 869 y(It)e(is)g(clear)f(that)h(this)g(class)f(of) h(eigen)f(bundles)i(are)e(top)r(ologi-)2042 969 y(cally)g(distinct)h (from)g(those)f(whose)g(curv)-5 b(ature)33 b(\015ux)h(lo)r(ops)f(do) 2042 1068 y(not)27 b(link.)2125 1170 y(In)c(1974)e(Chern)h(and)h (Simons)g(disco)n(v)n(ered)e(an)h(in)n(v)-5 b(arian)n(t,)23 b(the)2042 1270 y(Chern-Simons)36 b(in)n(v)-5 b(arian)n(t,)38 b(that)f(quan)n(ti\014es)g(this)g(more)f(sub-)2042 1369 y(tle)28 b(top)r(ological)f(non-trivialit)n(y)-7 b(.)27 b([5])h(F)-7 b(or)28 b(a)f(three-dimensional)2042 1469 y(manifold)g Ff(M)h Fq(the)g(Chern-Simons)e(in)n(v)-5 b(arian)n(t)27 b(is)g(giv)n(en)g(b)n(y)2455 1694 y Ff(C)5 b(S)29 b Fq(=)2706 1638 y(1)p 2681 1675 V 2681 1751 a(4)p Fi(\031)2796 1581 y Fj(Z)2842 1769 y Fe(M)2938 1694 y Fi(d)2981 1659 y Fd(3)3019 1694 y Fi(r)r Fh(A)3130 1706 y Fg(b)3183 1694 y Ff(\001)18 b(r)h(\002)f Fh(A)3467 1706 y Fg(b)3500 1694 y Fi(:)414 b Fq(\(12\))2042 1925 y(In)23 b(order)f(for)h Ff(C)5 b(S)29 b Fq(to)23 b(b)r(e)h(gauge)e(in)n (v)-5 b(arian)n(t)22 b(w)n(e)h(require)f Ff(M)h Fq(to)g(b)r(e)2042 2024 y(free)32 b(of)g(b)r(oundary)-7 b(.)50 b(The)32 b(top)r(ological)f(information)h(recorded)2042 2124 y(b)n(y)c Ff(C)5 b(S)35 b Fq(is)29 b(precisely)f(the)h(linking)g(b)r(et)n(w)n (een)g(the)g Fh(B)3672 2136 y Fg(b)3734 2124 y Fq(\015ux)g(lines.)2042 2224 y(The)e(fact)g(that)g(linking)g(is)g(only)g(de\014ned)g(in)h (three)f(dimensions)2042 2323 y(explains)20 b(wh)n(y)h(the)g (Chern-Simons)f(in)n(v)-5 b(arian)n(t)19 b(requires)h(a)g(three)2042 2423 y(dimensional)27 b(base)g(space.)2125 2525 y(After)22 b(realizing)e(that)i(there)f(is)h(another)e(lev)n(el)h(of)h(top)r (ological)2042 2624 y(non-trivialit)n(y)29 b(it)h(is)g(natural)g(to)g (ask)f(whether)h(one)g(can)f(mo)r(d-)2042 2724 y(ify)23 b(Eq.)f(\(1\))h(so)f(that)h(the)g(eigen)g(bundle)g(exhibits)g(non-zero) e Ff(C)5 b(S)h Fq(.)2042 2824 y(W)-7 b(e)20 b(shall)g(pro)n(v)n(e)e (that)j(the)f(Hamiltonian)g(giv)n(en)g(b)n(y)f(Eq.)h(\(9\))g(also)2042 2932 y(w)n(orks)26 b(so)h(long)g(as)2662 2910 y(^)2661 2932 y Fi(h)p Fq(\()p Fh(r)p Fq(\))i(is)e(c)n(hosen)g(appropriately)-7 b(.)2125 3034 y(No)n(w)21 b(let)i(us)f(restrict)f(ourselv)n(es)f(to)i (the)h(case)e(where)g(the)i(base)2042 3134 y(space)40 b Ff(M)h Fq(is)f Fi(R)2578 3104 y Fd(3)2629 3071 y Fj(S)2698 3134 y Ff(f1g)p Fq(,)k(the)d(stereographic)d(pro)5 b(jection)40 b(of)2042 3233 y Fi(S)2098 3203 y Fd(3)2161 3233 y Ff(\021)25 b(f)p Fq(\()p Fi(x;)14 b(y)s(;)g(z)t(;)g(w)r Fq(\);)g Fi(x)2747 3203 y Fd(2)2804 3233 y Fq(+)19 b Fi(y)2932 3203 y Fd(2)2989 3233 y Fq(+)g Fi(z)3116 3203 y Fd(2)3172 3233 y Fq(+)g Fi(w)3317 3203 y Fd(2)3381 3233 y Fq(=)26 b(1)p Ff(g)i Fq(.)42 b(In)29 b(that)h(case)2043 3320 y(^)2042 3342 y Fi(h)p Fq(\()p Fh(r)p Fq(\))41 b(is)f(a)f(mapping)h (from)g Fi(R)3038 3312 y Fd(3)3089 3280 y Fj(S)3158 3342 y Ff(f1g)f Fq(to)h Fi(S)3534 3312 y Fd(2)3571 3342 y Fq(.)74 b(Due)41 b(to)f(the)2042 3442 y(w)n(ork)26 b(of)h(Hopf)h(it)g (is)g(kno)n(wn)f(that)h(suc)n(h)f(mapping)g(can)g(also)g(b)r(e)2042 3541 y(classi\014ed)f(in)n(to)h(homotop)n(y)e(classes.)36 b(There)26 b(also)g(exists)h(an)f(in-)2042 3641 y(teger,)34 b(the)g(Hopf)g(in)n(v)-5 b(arian)n(t,)34 b(that)g(c)n(haracterizes)d (eac)n(h)i(class.)2042 3740 y(The)k(meaning)g(of)g(a)f(non-trivial)g (Hopf)i(map)f(is)g(rev)n(ealed)e(b)n(y)2042 3840 y(the)28 b(dual)f(of)h(its)f(P)n(on)n(try)n(agin)e(form)i(\(Eq.)g(\(11\)\).)37 b(F)-7 b(or)27 b(b)r(oth)h(the)2042 3940 y(trivial)j(and)g(non-trivial) g(Hopf)h(maps)f Ff(r)21 b(\001)g Fh(J)31 b Fq(=)e(0)i(ev)n(erywhere) 2042 4039 y(hence)22 b(the)h Fh(J)g Fq(\015ux)f(lines)h(form)f(closed)f (lo)r(ops.)35 b(F)-7 b(or)21 b(a)h(non-trivial)2042 4139 y(map)i(the)g(closed)g(\015ux)g(lo)r(ops)g(link)g(with)g(one)g(another) f(while)i(for)2042 4239 y(a)35 b(trivial)h(map)f(they)i(don't.)62 b(No)n(w)35 b(w)n(e)h(explain)f(what)h(is)g(the)2042 4338 y(Hopf)h(in)n(v)-5 b(arian)n(t.)63 b(Since)37 b Ff(r)25 b(\001)g Fh(J)38 b Fq(=)g(0,)h(there)e(exists)f(a)g(v)n(ector) 2042 4438 y(\014eld)28 b Fh(A)2294 4450 y Fg(h)2365 4438 y Fq(so)e(that)2699 4656 y Fh(J)d Fq(=)2894 4599 y(1)p 2869 4636 V 2869 4713 a(2)p Fi(\031)2971 4656 y Ff(r)18 b(\002)g Fh(A)3213 4668 y Fg(h)3256 4656 y Fi(:)658 b Fq(\(13\))2042 4868 y(The)23 b(Hopf)h(in)n(v)-5 b(arian)n(t)23 b(is)g(simply)h(the)g(Chern-Simons)e(in)n(v)-5 b(arian)n(t)2042 4968 y(for)27 b Fh(A)2241 4980 y Fg(h)2284 4968 y Fq(,)h([6])f(i.e.,) 2462 5197 y Ff(H)d Fq(=)2679 5141 y(1)p 2654 5178 V 2654 5254 a(4)p Fi(\031)2770 5084 y Fj(Z)2816 5273 y Fe(M)2912 5197 y Fi(d)2955 5163 y Fd(3)2992 5197 y Fi(r)r Fh(A)3103 5209 y Fg(h)3166 5197 y Ff(\001)18 b(r)h(\002)f Fh(A)3450 5209 y Fg(h)3493 5197 y Fi(:)421 b Fq(\(14\))2125 5430 y(In)28 b(the)g(rest)f(of)g(the)h(pap)r(er)f(w)n(e)h(pro)n(v)n(e)e(the) i(follo)n(wing.)2078 5617 y(1)p Fi(:)f Fq(F)-7 b(or)27 b(base)g(space)g Ff(M)c Fq(=)f Fi(S)2992 5583 y Fd(2)3057 5617 y Fq(the)28 b(eigen)f(bundle)h(of)34 b Fi(E)5 b(q)s(:)28 b Fq(\(9\))2142 5751 y(has)f Ff(C)h Fq(=)23 b Fi(n=)p Fq(2)j(if)2694 5729 y(^)2693 5751 y Fi(h)p Fq(\()p Fh(r)p Fq(\))i(has)f Ff(P)j Fq(=)23 b Fi(n:)668 b Fq(\(15\))1946 6000 y(2)p eop %%Page: 3 3 3 2 bop -123 275 a Fq(2)p Fi(:)28 b Fq(F)-7 b(or)27 b(base)g(space)f Ff(M)d Fq(=)g Fi(R)800 241 y Fd(3)851 197 y Fj([)943 275 y Ff(f1g)k Fq(the)h(eigen)f(bundle)h(of)-123 434 y Fi(E)5 b(q)s(:)28 b Fq(\(9\))g(has)f Ff(C)5 b(S)29 b Fq(=)23 b Fi(n)k Fq(if)693 412 y(^)692 434 y Fi(h)p Fq(\()p Fh(r)p Fq(\))h(has)f Ff(H)d Fq(=)f Fi(n:)471 b Fq(\(16\))-67 640 y(The)28 b(pro)r(of)f(rests)g(on)g(the)h(iden)n (tit)n(y)g(that)609 835 y Fh(B)677 847 y Fg(b)734 835 y Fq(=)22 b(2)p Fi(\031)s Fh(J)p Fi(:)760 b Fq(\(17\))-150 1036 y(whic)n(h)28 b(w)n(e)f(no)n(w)g(sho)n(w.)36 b(The)27 b(Berry)g(curv)-5 b(ature)26 b(is)i(giv)n(en)f(b)n(y)316 1265 y Fh(B)384 1277 y Fg(b)441 1265 y Fq(=)538 1208 y(1)p 538 1246 42 4 v 544 1322 a Fi(i)590 1265 y Ff(r\002)22 b Fi(<)h( )888 1277 y Fd(+)943 1265 y Ff(jr)p Fi( )1089 1277 y Fd(+)1168 1265 y Fi(>)f(:)467 b Fq(\(18\))-150 1493 y(Since)43 b Fi(<)48 b( )249 1505 y Fd(+)304 1493 y Ff(j)p Fi( )381 1505 y Fd(+)485 1493 y Fi(>)p Fq(=)g(1,)e(it)d(follo) n(ws)f(that)h Fi(<)49 b Ff(r)p Fi( )1592 1505 y Fd(+)1647 1493 y Ff(j)p Fi( )1724 1505 y Fd(+)1828 1493 y Fi(>)-150 1593 y Fq(+)23 b Fi(<)f( )79 1605 y Fd(+)134 1593 y Ff(jr)p Fi( )280 1605 y Fd(+)359 1593 y Fi(>)p Fq(=)g(0)p Fi(:)i Fq(F)-7 b(urthermore)23 b(since)g Fi(<)g Ff(r)p Fi( )1488 1605 y Fd(+)1543 1593 y Ff(j)p Fi( )1620 1605 y Fd(+)1699 1593 y Fi(>)p Fq(=)p Fi(<)-150 1692 y( )-96 1704 y Fd(+)-41 1692 y Ff(jr)p Fi( )105 1704 y Fd(+)184 1692 y Fi(>)249 1662 y Fe(\003)306 1692 y Fq(w)n(e)d(conclude)f(that)h Fi(<)j( )1068 1704 y Fd(+)1123 1692 y Ff(jr)p Fi( )1269 1704 y Fd(+)1348 1692 y Fi(>)c Fq(is)h(pure)f(imag-)-150 1792 y(inary)-7 b(.)36 b(Th)n(us)239 1987 y Fh(A)311 1999 y Fg(b)367 1987 y Fq(=)23 b Fi(I)7 b(m)14 b Fq([)p Fi(<)22 b( )749 1999 y Fd(+)805 1987 y Ff(jr)p Fi( )951 1999 y Fd(+)1029 1987 y Fi(>)p Fq(])239 2111 y Fh(B)307 2123 y Fg(b)363 2111 y Fq(=)h Fi(I)7 b(m)14 b Fq([)p Ff(r\002)22 b Fi(<)h( )902 2123 y Fd(+)957 2111 y Ff(jr)p Fi( )1103 2123 y Fd(+)1182 2111 y Fi(>)p Fq(])13 b Fi(:)439 b Fq(\(19\))-150 2312 y(In)28 b(comp)r(onen)n(t)f(form)g(the)h(second)f (of)h(Eq.)f(\(19\))g(read)-150 2506 y Fi(B)-83 2472 y Fg(\026)-38 2506 y Fq(=)22 b Fi(\017)83 2472 y Fg(\026\027)t(\025)204 2506 y Fi(I)7 b(m)14 b Fq([)p Fi(<)22 b(@)488 2518 y Fg(\027)530 2506 y Fi( )584 2518 y Fd(+)639 2506 y Ff(j)p Fi(@)706 2518 y Fg(\025)750 2506 y Fi( )804 2518 y Fd(+)882 2506 y Fi(>)p Fq(])-38 2687 y(=)59 2631 y Fi(\017)93 2600 y Fg(\026\027)t(\025)p 59 2668 155 4 v 116 2744 a Fq(2)224 2687 y Fi(I)7 b(m)14 b Fq([)p Fi(<)22 b(@)508 2699 y Fg(\027)550 2687 y Fi( )604 2699 y Fd(+)659 2687 y Ff(j)p Fi(@)726 2699 y Fg(\025)770 2687 y Fi( )824 2699 y Fd(+)902 2687 y Fi(>)h Ff(\000)f Fi(<)h(@)1209 2699 y Fg(\025)1252 2687 y Fi( )1306 2699 y Fd(+)1362 2687 y Ff(j)p Fi(@)1429 2699 y Fg(\027)1470 2687 y Fi( )1524 2699 y Fd(+)1602 2687 y Fi(>)p Fq(])14 b Fi(:)50 b Fq(\(20\))-150 2915 y(T)-7 b(o)31 b(compute)g Fi(<)d(@)453 2927 y Fg(\027)494 2915 y Fi( )548 2927 y Fd(+)603 2915 y Ff(j)p Fi(@)670 2927 y Fg(\025)714 2915 y Fi( )768 2927 y Fd(+)852 2915 y Fi(>)g Ff(\000)g Fi(<)g(@)1175 2927 y Fg(\025)1219 2915 y Fi( )1273 2927 y Fd(+)1328 2915 y Ff(j)p Fi(@)1395 2927 y Fg(\027)1437 2915 y Fi( )1491 2927 y Fd(+)1574 2915 y Fi(>)j Fq(w)n(e)f(in-)-150 3015 y(sert)k(a)g(complete)h(set)g (of)f(states)h(\()p Fi(I)42 b Fq(=)1144 2953 y Fj(P)1232 3040 y Fg(n)p Fd(=)p Fe(\006)1394 3015 y Ff(j)p Fi( )1471 3027 y Fg(n)1551 3015 y Fi(><)34 b( )1769 3027 y Fg(n)1814 3015 y Ff(j)p Fq(\),)-150 3114 y(and)27 b(that)h(giv)n(es)46 3309 y Fi(<)22 b(@)177 3321 y Fg(\027)219 3309 y Fi( )273 3321 y Fd(+)328 3309 y Ff(j)p Fi(@)395 3321 y Fg(\025)439 3309 y Fi( )493 3321 y Fd(+)571 3309 y Fi(>)g Ff(\000)h Fi(<)g(@)878 3321 y Fg(\025)921 3309 y Fi( )975 3321 y Fd(+)1031 3309 y Ff(j)p Fi(@)1098 3321 y Fg(\027)1139 3309 y Fi( )1193 3321 y Fd(+)1271 3309 y Fi(>)46 3450 y Fq(=)145 3371 y Fj(X)133 3546 y Fg(n)p Fd(=)p Fe(\006)300 3450 y Fi(<)g(@)432 3462 y Fg(\027)473 3450 y Fi( )527 3462 y Fd(+)583 3450 y Ff(j)p Fi( )660 3462 y Fg(n)728 3450 y Fi(><)f( )934 3462 y Fg(n)979 3450 y Ff(j)p Fi(@)1046 3462 y Fg(\025)1090 3450 y Fi( )1144 3462 y Fd(+)1222 3450 y Fi(>)h Ff(\000)p Fq([)p Fi(\027)28 b Ff($)23 b Fi(\025)p Fq(])46 3656 y(=)p Fi(<)f(@)242 3668 y Fg(\027)283 3656 y Fi( )337 3668 y Fd(+)392 3656 y Ff(j)p Fi( )469 3668 y Fe(\000)549 3656 y Fi(><)g( )755 3668 y Fe(\000)811 3656 y Ff(j)p Fi(@)878 3668 y Fg(\025)922 3656 y Fi( )976 3668 y Fd(+)1054 3656 y Fi(>)h Ff(\000)p Fq([)p Fi(\027)28 b Ff($)23 b Fi(\025)p Fq(])p Fi(:)246 b Fq(\(21\))-150 3857 y(In)47 b(reac)n(hing)e(the)i(last)f(line)h(w)n(e)f(ha)n(v)n(e)f (used)i(the)f(fact)h(that)-150 3957 y Fi(<)23 b(@)-18 3969 y Fg(\027)23 3957 y Fi( )77 3969 y Fd(+)132 3957 y Ff(j)p Fi( )209 3969 y Fd(+)288 3957 y Fi(><)f( )494 3969 y Fd(+)549 3957 y Ff(j)p Fi(@)616 3969 y Fg(\025)660 3957 y Fi( )714 3969 y Fd(+)792 3957 y Fi(>)h Ff(\000)p Fq([)p Fi(\027)k Ff($)d Fi(\025)p Fq(])f(=)g(0)p Fi(:)-67 4062 y Fq(T)-7 b(o)30 b(compute)g Fi(<)c( )542 4074 y Fe(\000)598 4062 y Ff(j)p Fi(@)665 4074 y Fg(j)700 4062 y Fi( )754 4074 y Fd(+)836 4062 y Fi(>)k Fq(in)g(Eq.)f(\(21\))h(w)n(e)f (express)g(the)-150 4162 y(eigen)h(v)n(ector)f(of)h Fi(H)7 b Fq(\()p Fh(r)561 4132 y Fe(0)585 4162 y Fq(\))28 b(=)f Fi(H)7 b Fq(\()p Fh(r)20 b Fq(+)g Fi(\016)s Fh(r)p Fq(\))28 b(=)f Fi(H)7 b Fq(\()p Fh(r)p Fq(\))21 b(+)f Fi(\016)s Fh(r)h Ff(\001)f(r)p Fi(H)37 b Fq(in)-150 4261 y(terms)20 b(of)h(those)f(of)g Fi(H)7 b Fq(\()p Fh(r)p Fq(\))22 b(via)e(\014rst)g(order)f(p)r(erturbation)h(theory)-7 b(.)-150 4361 y(T)g(o)27 b(the)h(\014rst)g(order)e(in)i Ff(j)p Fi(\016)s Fh(r)p Ff(j)g Fq(w)n(e)f(obtain)-150 4605 y Ff(j)p Fi( )-73 4617 y Fd(+)-18 4605 y Fq(\()p Fh(r)53 4571 y Fe(0)77 4605 y Fq(\))d Fi(>)e Fq(=)308 4488 y Fj(\024)352 4605 y Ff(j)p Fi( )429 4617 y Fd(+)507 4605 y Fi(>)h Fq(+)670 4549 y Fi(<)f( )811 4561 y Fe(\000)867 4549 y Ff(j)p Fi(\016)s Fh(r)d Ff(\001)f(r)p Fi(H)7 b Ff(j)p Fi( )1251 4561 y Fd(+)1329 4549 y Fi(>)p 670 4586 725 4 v 864 4662 a(E)925 4674 y Fd(+)999 4662 y Ff(\000)18 b Fi(E)1143 4674 y Fe(\000)1404 4605 y Ff(j)p Fi( )1481 4617 y Fe(\000)1560 4605 y Fi(>)1625 4488 y Fj(\025)220 4863 y Fq(=)308 4721 y Fj(")356 4863 y Ff(j)p Fi( )433 4875 y Fd(+)512 4863 y Fi(>)k Fq(+)674 4807 y Fi(<)g( )815 4819 y Fe(\000)871 4807 y Ff(j)p Fi(\016)s(r)971 4819 y Fg(\026)1017 4807 y Fi(@)1061 4819 y Fg(\026)1106 4785 y Fq(^)1105 4807 y Fi(h)c Ff(\001)f Fi(~)-40 b(\033)s Ff(j)p Fi( )1340 4819 y Fd(+)1419 4807 y Fi(>)p 674 4844 810 4 v 1058 4920 a Fq(2)1493 4863 y Ff(j)p Fi( )1570 4875 y Fe(\000)1649 4863 y Fi(>)1714 4721 y Fj(#)1776 4863 y Fi(:)50 b Fq(\(22\))-150 5138 y(Eq.)27 b(\(22\))g(implies)h (that)201 5398 y Fi(<)22 b( )342 5410 y Fe(\000)398 5398 y Ff(j)p Fi(@)465 5410 y Fg(\027)507 5398 y Fi( )561 5410 y Fd(+)639 5398 y Fi(>)h Fq(=)824 5342 y Fi(<)g( )966 5354 y Fe(\000)1022 5342 y Ff(j)p Fi(@)1089 5354 y Fg(\027)1131 5320 y Fq(^)1131 5342 y Fi(h)18 b Ff(\001)e Fi(~)-40 b(\033)t Ff(j)p Fi( )1366 5354 y Fd(+)1444 5342 y Fi(>)p 824 5379 685 4 v 1146 5455 a Fq(2)1519 5398 y Fi(:)1745 5550 y Fq(\(23\))-150 5751 y(As)28 b(the)g(result)f(w)n(e)g(ha)n(v)n(e) 2045 271 y Fi(<)22 b(@)2176 283 y Fg(\027)2218 271 y Fi( )2272 283 y Fd(+)2327 271 y Ff(j)p Fi( )2404 283 y Fe(\000)2483 271 y Fi(><)g( )2689 283 y Fe(\000)2745 271 y Ff(j)p Fi(@)2812 283 y Fg(\025)2856 271 y Fi( )2910 283 y Fd(+)2988 271 y Fi(>)h Ff(\000)p Fq([)p Fi(\027)28 b Ff($)23 b Fi(\025)p Fq(])2045 435 y(=)2142 379 y(1)p 2142 416 42 4 v 2142 492 a(4)2194 435 y([)p Fi(<)f( )2358 447 y Fd(+)2414 435 y Ff(j)p Fi(@)2481 447 y Fg(\027)2523 413 y Fq(^)2522 435 y Fi(h)c Ff(\001)f Fi(~)-40 b(\033)s Ff(j)p Fi( )2757 447 y Fe(\000)2837 435 y Fi(><)22 b( )3043 447 y Fe(\000)3099 435 y Ff(j)p Fi(@)3166 447 y Fg(\025)3210 413 y Fq(^)3209 435 y Fi(h)d Ff(\001)d Fi(~)-40 b(\033)t Ff(j)p Fi( )3445 447 y Fd(+)3523 435 y Fi(>)23 b Ff(\000)p Fq([)p Fi(\027)28 b Ff($)23 b Fi(\025)p Fq(])2045 635 y(=)2142 579 y(1)p 2142 616 V 2142 692 a(4)2220 556 y Fj(X)2208 730 y Fg(n)p Fd(=)p Fe(\006)2351 635 y Fq([)p Fi(<)g( )2516 647 y Fd(+)2571 635 y Ff(j)p Fi(@)2638 647 y Fg(\027)2681 613 y Fq(^)2680 635 y Fi(h)18 b Ff(\001)f Fi(~)-40 b(\033)s Ff(j)p Fi( )2915 647 y Fg(n)2983 635 y Fi(><)23 b( )3190 647 y Fg(n)3235 635 y Ff(j)p Fi(@)3302 647 y Fg(\025)3346 613 y Fq(^)3346 635 y Fi(h)18 b Ff(\001)e Fi(~)-40 b(\033)t Ff(j)p Fi( )3581 647 y Fd(+)3659 635 y Fi(>)23 b Ff(\000)p Fq([)p Fi(\027)28 b Ff($)23 b Fi(\025)p Fq(])2045 888 y(=)2142 832 y(1)p 2142 869 V 2142 945 a(4)2194 888 y([)p Fi(<)f( )2358 900 y Fd(+)2414 888 y Ff(j)p Fq([)p Fi(@)2504 900 y Fg(\027)2546 866 y Fq(^)2545 888 y Fi(h)c Ff(\001)f Fi(~)-40 b(\033)s(;)14 b(@)2784 900 y Fg(\025)2829 866 y Fq(^)2828 888 y Fi(h)k Ff(\001)f Fi(~)-40 b(\033)s Fq(])p Ff(j)p Fi( )3086 900 y Fd(+)3164 888 y Fi(>)p Fq(])2045 1089 y(=)2149 1033 y Fi(i)p 2142 1070 V 2142 1146 a Fq(2)2194 1089 y Fi(\017)2228 1101 y Fg(abc)2326 1089 y Fq(\()p Fi(@)2402 1101 y Fg(\027)2445 1068 y Fq(^)2444 1089 y Fi(h)2492 1101 y Fg(a)2532 1089 y Fq(\)\()p Fi(@)2640 1101 y Fg(\025)2685 1068 y Fq(^)2684 1089 y Fi(h)2732 1101 y Fg(b)2765 1089 y Fq(\)[)p Fi(<)23 b( )2962 1101 y Fd(+)3018 1089 y Ff(j)p Fi(\033)3088 1101 y Fg(c)3122 1089 y Ff(j)p Fi( )3199 1101 y Fd(+)3277 1089 y Fi(>)p Fq(])2045 1290 y(=)2149 1234 y Fi(i)p 2142 1271 V 2142 1347 a Fq(2)2194 1290 y Fi(\017)2228 1302 y Fg(abc)2326 1290 y Fq(\()p Fi(@)2402 1302 y Fg(\027)2445 1269 y Fq(^)2444 1290 y Fi(h)2492 1302 y Fg(a)2532 1290 y Fq(\)\()p Fi(@)2640 1302 y Fg(\025)2685 1269 y Fq(^)2684 1290 y Fi(h)2732 1302 y Fg(b)2765 1290 y Fq(\))2798 1269 y(^)2797 1290 y Fi(h)2845 1302 y Fg(c)2902 1290 y Fq(=)3007 1234 y Fi(i)p 3000 1271 V 3000 1347 a Fq(2)3052 1269 y(^)3052 1290 y Fi(h)18 b Ff(\001)g Fq(\()p Fi(@)3235 1302 y Fg(\027)3278 1269 y Fq(^)3277 1290 y Fi(h)g Ff(\002)g Fi(@)3470 1302 y Fg(\025)3515 1269 y Fq(^)3514 1290 y Fi(h)p Fq(\))p Fi(:)320 b Fq(\(24\))2042 1482 y(In)26 b(going)g(from)g(the)g(second)g(to)g(the)h(third)g(line)f(of)h(Eq.)f (\(24\))g(w)n(e)2042 1590 y(ha)n(v)n(e)34 b(used)i(the)f(fact)h(that)g Fi(<)g( )3104 1602 y Fd(+)3159 1590 y Ff(j)p Fi(@)3226 1602 y Fg(\027)3268 1568 y Fq(^)3267 1590 y Fi(h)24 b Ff(\001)e Fi(~)-41 b(\033)t Ff(j)p Fi( )3513 1602 y Fd(+)3604 1590 y Fi(><)36 b( )3824 1602 y Fd(+)3879 1590 y Ff(j)p Fi(@)3946 1602 y Fg(\025)3990 1568 y Fq(^)3990 1590 y Fi(h)23 b Ff(\001)2040 1690 y Fi(~)-40 b(\033)s Ff(j)p Fi( )2169 1702 y Fd(+)2275 1690 y Fi(>)49 b Ff(\000)p Fq([)p Fi(\027)55 b Ff($)50 b Fi(\025)p Fq(])h(=)f(0.)85 b(Substituting)45 b(Eq.)e(\(24\))g(in)n(to)2042 1789 y(Eq.)27 b(\(20\))g(w)n(e)g(obtain)2107 1992 y Fi(B)2174 1958 y Fg(\026)2218 1992 y Fq(=)c Fi(\017)2340 1958 y Fg(\026\027)t(\025)2461 1992 y Fi(I)7 b(m)p Fq([)p Fi(<)22 b(@)2731 2004 y Fg(\027)2773 1992 y Fi( )2827 2004 y Fe(\006)2883 1992 y Ff(j)p Fi(@)2950 2004 y Fg(\025)2994 1992 y Fi( )3048 2004 y Fe(\006)3127 1992 y Fi(>)p Fq(])g(=)3335 1936 y(1)p 3335 1973 V 3335 2049 a(4)3386 1992 y Fi(\017)3420 1958 y Fg(\026\027)t(\025)3542 1970 y Fq(^)3541 1992 y Fi(h)d Ff(\001)f Fi(@)3693 2004 y Fg(\027)3735 1970 y Fq(^)3734 1992 y Fi(h)h Ff(\002)f Fi(@)3928 2004 y Fg(\025)3972 1970 y Fq(^)3971 1992 y Fi(h)2218 2144 y Fq(=)23 b(2)p Fi(\031)s(J)2452 2110 y Fg(\026)2496 2144 y Fi(:)1418 b Fq(\(25\))2125 2307 y(After)35 b(establishing)f(Eq.)g (\(17\))h(it)g(is)g(simple)g(to)f(pro)n(v)n(e)g(\(15\))2042 2407 y(and)27 b(\(16\).)37 b(F)-7 b(or)27 b(\(15\))g(the)h(Chern)f(n)n (um)n(b)r(er)h(is)f(giv)n(en)g(b)n(y)2279 2613 y Ff(C)h Fq(=)2473 2557 y(1)p 2448 2594 92 4 v 2448 2670 a(4)p Fi(\031)2564 2500 y Fj(Z)2610 2688 y Fe(S)2673 2613 y Fi(d)p Fh(a)19 b Ff(\001)g Fh(B)2891 2625 y Fg(b)2947 2613 y Fq(=)3045 2557 y(1)p 3045 2594 42 4 v 3045 2670 a(2)3110 2500 y Fj(Z)3156 2688 y Fe(S)3219 2613 y Fi(d)p Fh(a)g Ff(\001)g Fh(J)k Fq(=)g Ff(P)7 b Fi(=)p Fq(2)p Fi(:)236 b Fq(\(26\))2042 2823 y(As)30 b(the)h(result)f Ff(P)k Fq(=)27 b Fi(n)j Fq(implies)g Ff(C)j Fq(=)27 b Fi(n=)p Fq(2.)43 b(No)n(w)30 b(let)h(us)f(pro)n(v)n(e)2042 2922 y(\(16\).)36 b(Eq.)27 b(\(17\))h(implies)f(that)2539 3125 y Ff(r)19 b(\002)f Fh(A)2782 3137 y Fg(b)2839 3125 y Fq(=)k(2)p Fi(\031)3053 3068 y Fq(1)p 3028 3105 92 4 v 3028 3181 a(2)p Fi(\031)3130 3125 y Ff(r)d(\002)f Fh(A)3373 3137 y Fg(h)3416 3125 y Fi(:)498 b Fq(\(27\))2042 3311 y(As)27 b(the)h(result)g Fh(A)2611 3323 y Fg(b)2672 3311 y Fq(and)f Fh(A)2905 3323 y Fg(h)2976 3311 y Fq(di\013er)h(b)n(y)f (a)g(pure)h(gauge)e(at)i(most)2702 3474 y Fh(A)2774 3486 y Fg(b)2830 3474 y Fq(=)23 b Fh(A)2990 3486 y Fg(h)3052 3474 y Fq(+)18 b Ff(r)p Fi(\036:)661 b Fq(\(28\))2042 3637 y(Since)31 b(Eq.)f(\(12\))g(is)h(gauge)e(in)n(v)-5 b(arian)n(t)30 b(when)g Ff(M)e Fq(=)g Fi(R)3775 3607 y Fd(3)3826 3575 y Fj(S)3895 3637 y Ff(f1g)p Fq(,)2042 3737 y(w)n(e)f(conclude)2074 3922 y Ff(C)5 b(S)h Fq(=)2301 3866 y(1)p 2276 3903 V 2276 3979 a(4)p Fi(\031)2392 3809 y Fj(Z)2438 3998 y Fg(S)2482 3981 y Fa(3)2532 3922 y Fi(d)2575 3888 y Fd(3)2613 3922 y Fi(r)r Fh(A)2724 3934 y Fg(b)2777 3922 y Ff(\001)18 b(r)h(\002)f Fh(A)3061 3934 y Fg(b)3117 3922 y Fq(=)3240 3866 y(1)p 3215 3903 V 3215 3979 a(4)p Fi(\031)3331 3809 y Fj(Z)3377 3998 y Fg(S)3421 3981 y Fa(3)3471 3922 y Fi(d)3514 3888 y Fd(3)3551 3922 y Fi(r)r Fh(A)3662 3934 y Fg(h)3725 3922 y Ff(\001)g(r)h(\002)f Fh(A)4009 3934 y Fg(h)2179 4093 y Fq(=)k Ff(H)q Fi(:)1577 b Fq(\(29\))2042 4256 y(As)27 b(the)h(result)g Ff(H)c Fq(=)e Fi(n)28 b Fq(implies)g Ff(C)5 b(S)29 b Fq(=)22 b Fi(n)p Fq(.)2125 4356 y(In)40 b(ph)n(ysics)f(one)g(often)h(encoun)n(ters)f(Berry's)f(phase)h(when) 2042 4456 y(a)30 b(system)h(p)r(osses's)f(\\fast")g(and)h(\\slo)n(w")e (dynamic)i(degrees)e(of)2042 4555 y(freedom.)67 b(When)39 b(the)f(fast)g(degrees)f(of)g(freedom)h(are)f(\\in)n(te-)2042 4655 y(grated)i(out")h(it)h(often)f(pro)r(duces,)j(as)d(part)f(of)i (the)f(e\013ectiv)n(e)2042 4754 y(action)31 b(of)h(the)g(slo)n(w)f(v)-5 b(ariables,)32 b(a)f(term)h(\(the)g(Berry's)e(phase\))2042 4854 y(that)22 b(is)h(non-zero)e(ev)n(en)g(when)i(the)g(slo)n(w)e(v)-5 b(ariables)21 b(c)n(hange)h(adi-)2042 4954 y(abatically)i(with)i(time.) 37 b(Suc)n(h)26 b(term)f(can)g(fundamen)n(tally)h(alter)2042 5053 y(the)i(b)r(eha)n(vior)e(of)i(the)g(slo)n(w)e(v)-5 b(ariables.)2125 5153 y(In)27 b(the)h(follo)n(wing)e(w)n(e)g(presen)n (t)h(an)g(example)f(where)h(the)g(fast)2042 5253 y(degrees)20 b(of)h(freedom)g(generate)f(an)i(e\013ectiv)n(e)f(action)g(that)h(is)f (the)2042 5352 y(Hopf)k(in)n(v)-5 b(arian)n(t)23 b(of)i(the)g(slo)n(w)e (v)-5 b(ariables.)35 b(The)25 b(mo)r(del)f(is)h(a)f(\014eld)2042 5452 y(theory)18 b(in)i(t)n(w)n(o)f(space)f(and)h(one)g(\(Euclidean\))h (time)f(dimensions.)2042 5552 y(It)32 b(consists)e(of)i(t)n(w)n(o)e (\014elds:)45 b(1\))31 b(a)g(fermion)h(\014eld)f Fi( )3679 5564 y Fg(\033)3724 5552 y Fq(\()p Fh(r)p Fi(;)14 b(t)p Fq(\),)33 b(and)2042 5651 y(2\))26 b(an)h(unit)g(v)n(ector)f(\014eld)31 b(^)-46 b Fi(n)o Fq(\()p Fh(r)p Fi(;)14 b(t)p Fq(\).)38 b(The)27 b(Lagrangian)d(densit)n(y)i(is)2042 5751 y(giv)n(en)h(as)1946 6000 y(3)p eop %%Page: 4 4 4 3 bop 135 271 a Ff(L)p Fq(=)23 b Ff(L)337 283 y Fg( )406 271 y Fq(+)18 b Ff(L)546 283 y Fg(n)610 271 y Ff(\000)g Fi(g)7 b Fq(^)-46 b Fi(n)18 b Ff(\001)862 249 y Fq(\026)845 271 y Fi( )899 283 y Fg(\013)945 271 y Fi(~)-40 b(\033)994 283 y Fg(\013\014)1082 271 y Fi( )1136 283 y Fg(\014)85 438 y Ff(L)142 450 y Fg( )192 438 y Fq(=)297 416 y(\026)280 438 y Fi( )334 450 y Fg(\013)381 438 y Fq(\()p Fi(@)457 450 y Fg(t)505 438 y Ff(\000)19 b Fi(\026)p Fq(\))p Fi( )725 450 y Fg(\013)791 438 y Ff(\000)920 382 y Fq(1)p 884 419 115 4 v 884 495 a(2)p Fi(m)1025 416 y Fq(\026)1008 438 y Fi( )1062 450 y Fg(\013)1109 438 y Fq(\()p Ff(r)g(\000)f Fi(i)p Fh(A)1413 450 y Fg(ex)1486 438 y Fq(\))1518 404 y Fd(2)1556 438 y Fi( )1610 450 y Fg(\013)90 641 y Ff(L)147 653 y Fg(n)192 641 y Fq(=)23 b Fi(i)k Fq(\012[)t(^)-46 b Fi(n)p Fq(])19 b(+)604 585 y Fi(K)p 604 622 77 4 v 621 698 a Fq(2)690 641 y Ff(jr)t Fq(^)-46 b Fi(n)p 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b(that)h(condition)f (in)n(tegrating)f(out)h(the)g(electrons)2042 271 y(pro)r(duces)23 b(a)g(term)2675 239 y Fg(k)p 2656 253 75 4 v 2656 300 a Fd(4)p Fg(\031)2754 204 y Fj(R)2823 271 y Fi(d)2866 241 y Fd(2)2904 271 y Fi(xdt)p Fh(A)3096 283 y Fg(h)3150 271 y Ff(\001)10 b(r)g(\002)g Fh(A)3409 283 y Fg(h)3453 271 y Fq(,)25 b(whic)n(h)e(is)h(prop)r(or-)2042 371 y(tional)h(to)h (the)h(Hopf)f(in)n(v)-5 b(arian)n(t)25 b(of)h(the)k(^)-46 b Fi(n)p Fq(\()p Fh(r)p Fi(;)14 b(t)p Fq(\).)37 b(This)26 b(term)g(has)2042 471 y(the)j(e\013ect)g(of)g(c)n(hanging)e(the)i (spins)g(and)f(statistics)g(of)h(solitons)2042 570 y(\(the)f (skyrmions\))f(in)g(the)33 b(^)-47 b Fi(n)p Fq(\()p Fh(r)p Fi(;)14 b(t)p Fq(\))29 b(\014eld.)f([6])2125 781 y(Ac)n(kno)n (wledgemen)n(ts:)55 b(DHL)38 b(is)f(in)h(debt)g(to)g(Geo\013rey)e(Lee) 2042 881 y(for)d(helping)i(him)f(to)g(visualize)g(the)g(dual)g(of)h (the)f(P)n(on)n(try)n(agin)2042 980 y(form)28 b(of)g(the)g(non-trivial) f(Hopf)i(map.)38 b(W)-7 b(e)29 b(thank)f(Qiang-Hua)2042 1080 y(W)-7 b(ang)41 b(for)g(helping)h(plotting)g(the)g(dual)f(of)h (the)g(P)n(on)n(try)n(agin)2042 1179 y(form.)36 b(DHL)29 b(is)e(supp)r(orted)g(b)n(y)h(NSF)g(gran)n(t)e(DMR)j(99-71503.)p 2042 1398 2043 4 v 2042 1832 a Fo([1])34 b(M.)26 b(Berry)-6 b(,)26 b(Pro)r(c.)h(R.)e(So)r(c.)i(London,)e(A)h Fm(392)p Fo(,)g(45)g(\(1984\).)2042 1923 y([2])34 b(P)-6 b(.A.M.)26 b(Dirac,)h(Pro)r(c.)g(Ro)n(y)-6 b(.)25 b(So)r(c.)h(A)f Fm(133)p Fo(,)i(60)f(\(1931\).)2042 2015 y([3])34 b(C.N.)26 b(Y)-6 b(ang,)26 b(Ann.)f(of)h(N.Y.)g(Acad.)f(of)i(Sci.)f Fm(294)p Fo(,)g(86)h(\(1977\).)2042 2106 y([4])34 b(S.S.)26 b(Chern,)g(Ann.)e(of)j(Math.)f Fm(47)p Fo(,)g(85,)h(\(1946\).)2042 2197 y([5])34 b(S.S.)26 b(Chern)f(and)h(J.)g(Simoms)e(Ann.)h(Math.)h Fm(99)p Fo(,)h(48)f(\(1974\).)2042 2289 y([6])34 b(F.)26 b(Wilczek)g(and)f(A.)g(Zee,)h(Ph)n(ys.)g(Rev.)f(Lett.)g Fm(52)p Fo(,)h(2111)h(\(1984\).)2042 2380 y([7])34 b(F)-6 b(or)24 b(a)h(review)g(see,)g(e.g.,)h(\\The)f(Quan)n(tum)d(Hall)j (E\013ect",)g(edited)2156 2471 y(b)n(y)31 b(R.E.)h(Prange)g(and)f(S.M.) h(Girvin,)i(\(Springer-V)-6 b(erlag,)33 b(New)2156 2563 y(Y)-6 b(ork)25 b(1986.\))1946 6000 y Fq(4)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0104181459940--