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(state)g(winding)e(n)m(um)m(b)s(er)i(to)f(the)h(Chern)g(n)m(um)m(b)s (ers)g(for)f(Harp)s(er's)h(equation)f(with)-118 4367 y(rational)20 b(\015ux)k([14)o(].)41 b(Akk)m(ermans,)26 b(Avron,)f(Narevic)m(h)e(and)g(Seiler)f(in)m(tro)s(duced)h(sp)s(ectral) f(b)s(oundary)i(conditions)-118 4487 y(giving)37 b(rise)i(to)g(a)f (linear)g(disp)s(ersion)g(relation)f(for)i(edge)g(states)h(and)f(a)g (natural)f(setting)h(for)f(the)h(Laughlin)-118 4607 y(w)m(a)m(v)m(e)32 b(function)d(as)h(a)g(man)m(y)g(b)s(o)s(dy)g(bulk)g(state)g([2].)43 b(The)31 b(stabilit)m(y)d(of)i(the)g(absolutely)g(con)m(tin)m(uous)g (sp)s(ectrum)-118 4728 y(asso)s(ciated)j(to)g(edge)h(states)g(under)g (the)f(p)s(erturbation)g(with)f(a)h(random)g(p)s(oten)m(tial)e(w)m(as)j (studied)g(b)m(y)g(sev)m(eral)-118 4848 y(authors)f(with)f(Mourre's)i (p)s(ositiv)m(e)e(comm)m(utator)f(estimates)h([16)o(,)h(8,)f(12].)28 5016 y(Our)48 b(\014rst)h(main)d(result)i(is)g(a)f(rigorous)g(pro)s(of) h(of)f(the)i(edge)f(curren)m(t)h(quan)m(tization)f(in)f(the)h(sense)i (of)-118 5136 y(Halp)s(erin)27 b(for)g(a)h(discrete)h(magnetic)e (half-plane)f(op)s(erator)i(con)m(taining)f(a)h(disordered)g(p)s(oten)m (tial,)g(notably)g(w)m(e)-118 5256 y(sho)m(w)40 b(quan)m(tization)e(of) g(what)h(w)m(e)h(call)e(the)h(edge)g(Hall)e(conductivit)m(y)-8 b(.)63 b(Our)39 b(second)h(mathematical)c(result)-118 5377 y(is)f(its)h(equalit)m(y)f(to)h(the)g(bulk)g(Hall)e(conductivit)m (y)j(as)f(calculated)f(b)m(y)i(the)f(Kub)s(o-Chern)g(form)m(ula)e([20,) i(6,)g(7].)-118 5497 y(The)42 b(pro)s(of)f(of)g(this)h(equalit)m(y)f (un)m(v)m(eals)i(a)e(deep)h(connection)g(b)s(et)m(w)m(een)i(the)e (plane)f(and)h(edge)g(geometry)f(as)-118 5618 y(it)46 b(is)g(based)i(on)f(Bott)g(p)s(erio)s(dicit)m(y)-8 b(,)48 b(the)g(heart)f(of)f(K-theory)h([9].)87 b(Using)46 b(these)j(results,)i (w)m(e)d(repro)s(duce)-118 5738 y(Halp)s(erin's)37 b(argumen)m(t)h (explaining)f(wh)m(y)j(the)f(Hall)d(conductance)k(of)e(a)g(Hall)f(bar)h (is)g(quan)m(tized)h(no)g(matter)-118 5858 y(what)44 b(prop)s(ortion)e(of)h(the)h(curren)m(t)h(is)e(actually)f(carried)h(b)m (y)h(the)g(edge)h(or)e(the)h(bulk)f(states)i(resp)s(ectiv)m(ely)-8 b(.)1924 6107 y(1)p eop %%Page: 2 2 2 1 bop -118 22 a Fj(Finally)27 b(w)m(e)k(presen)m(t)g(a)e(simple)f (theoretical)h(reasoning)g(sho)m(wing)h(that)f(in)g(a)h(t)m(ypical)f (exp)s(erimen)m(tal)f(situation)-118 142 y(at)k(most)g(10\045)g(of)h (the)g(curren)m(t)g(\015o)m(ws)h(b)m(y)f(edge)h(states.)28 310 y(F)-8 b(or)43 b(the)h(de\014nition)e(of)i(the)f(edge)i(Hall)c (conductivit)m(y)-8 b(,)47 b(w)m(e)d(consider)g(a)g(gas)f(of)g(c)m (harged)h(indep)s(enden)m(t)-118 430 y(particles)32 b(in)g(the)i (\(discrete\))f(upp)s(er)h(half)d(plane)i(\000)28 b(=)g Fi(f)p Fj(\()p Fh(x;)17 b(y)t Fj(\))28 b Fi(2)g Fg(Z)2383 394 y Fk(2)2423 430 y Fi(j)p Fh(y)j Fi(\025)e Fj(0)p Fi(g)k Fj(submitted)f(to)h(a)f(p)s(erp)s(endicular)-118 551 y(magnetic)j(\014eld)i Fh(B)5 b Fj(.)56 b(Let)885 526 y(^)860 551 y Fh(H)44 b Fj(denote)38 b(the)f(one-particle)e (Hamiltonian)e(acting)j(on)h Fh(`)3054 515 y Fk(2)3093 551 y Fj(\(\000\).)56 b(All)35 b(op)s(erators)i(on)-118 671 y(the)42 b(half-plane)e(space)j(carry)f(a)f(hat)h(from)e(no)m(w)j (on.)71 b(T)m(ypically)2443 646 y(^)2418 671 y Fh(H)49 b Fj(is)41 b(the)h(pro)5 b(jection)42 b(on)m(to)g Fh(`)3575 635 y Fk(2)3614 671 y Fj(\(\000\))g(of)f(an)-118 792 y(op)s(erator)29 b Fh(H)35 b Fj(=)28 b Fh(H)573 807 y Ff(H)656 792 y Fj(+)16 b Fh(V)51 b Fj(acting)29 b(on)h Fh(`)1320 755 y Fk(2)1359 792 y Fj(\()p Fg(Z)1465 755 y Fk(2)1505 792 y Fj(\))g(where)g Fh(H)1932 807 y Ff(H)2029 792 y Fj(is)f(Harp)s(er's)i(magnetic)d(Hamiltonian)e(and)k Fh(V)51 b Fj(is)29 b(the)-118 912 y(sum)37 b(of)g(a)g(p)s(erio)s(dic)f (and)h(a)g(disordered)h(p)s(oten)m(tial.)56 b(As)38 b(the)g(edge)g(of)f (the)h(plane)f(in)m(tercepts)h(the)g(cyclotron)-118 1032 y(orbits,)k(there)g(ma)m(y)f(b)s(e)g(a)f(net)i(electric)e(curren)m(t)i (along)e(the)h(edge.)69 b(In)41 b(order)g(to)g(calculate)f(it,)i(let)e Fh(J)50 b Fj(b)s(e)41 b(a)-118 1153 y(sp)s(ectral)36 b(in)m(terv)-5 b(al)35 b(lying)g(in)g(a)i(gap)f(of)f(the)i(plane)f (Hamiltonian)d Fh(H)8 b Fj(.)54 b(Suc)m(h)37 b(an)g(in)m(terv)-5 b(al)35 b(t)m(ypically)g(con)m(tains)-118 1273 y(extended)g(edge)g (states)f(of)936 1248 y(^)911 1273 y Fh(H)41 b Fj([14],)34 b(ev)m(en)h(in)e(presence)i(of)e(a)g(w)m(eak)i(disordered)f(p)s(oten)m (tial)e([13,)h(16,)g(8,)h(12)o(].)47 b(If)-96 1368 y(^)-118 1393 y Fh(P)-55 1408 y Ff(J)21 1393 y Fj(is)27 b(the)h(sp)s(ectral)g (pro)5 b(jection)27 b(of)1229 1368 y(^)1203 1393 y Fh(H)36 b Fj(on)27 b Fh(J)9 b Fj(,)29 b(then)f(the)g(electric)f(edge)h(curren)m (t)h(in)e Fh(x)p Fj(-direction)g(carried)g(b)m(y)h(the)-118 1514 y(eigenstates)k(in)e Fh(J)40 b Fj(is)30 b(equal)h(to)g Fh(q)1116 1489 y Fj(^)1098 1514 y Fi(T)25 b Fj(\()1237 1489 y(^)1215 1514 y Fh(P)1278 1529 y Ff(J)1327 1514 y Fi(r)1410 1529 y Ff(x)1454 1514 y Fj(\()1517 1489 y(^)1492 1514 y Fh(H)7 b Fj(\)\).)43 b(Here)32 b Fh(q)i Fj(is)d(the)g(c)m(harge) h(of)e(the)i(particles,)e Fi(r)3375 1529 y Ff(x)3419 1514 y Fj(\()3482 1489 y(^)3457 1514 y Fh(H)8 b Fj(\))27 b(=)h Fh({)p Fj([)p Fh(X)r(;)3925 1489 y Fj(^)3900 1514 y Fh(H)8 b Fj(])-118 1634 y(is)30 b(the)i(curren)m(t)g(op)s(erator)e (giv)m(en)h(b)m(y)h(the)f(comm)m(utator)f(of)g(the)h(Hamiltonian)d(and) j(the)g Fh(X)8 b Fj(-p)s(osition)28 b(op)s(erator)-118 1755 y(and)h(\014nally)e(the)i(trace)782 1729 y(^)763 1755 y Fi(T)54 b Fj(=)27 b Fi(T)1028 1770 y Ff(x)1072 1755 y Fj(T)-8 b(r)1173 1770 y Ff(y)1243 1755 y Fj(is)28 b(the)h(trace)g(p)s(er)g(unit)f(v)m(olume)g([6,)h(7])f(in)g(the)h Fh(x)p Fj(-direction)f(and)h(the)g(usual)-118 1875 y(trace)k(in)f Fh(y)t Fj(-direction.)41 b(No)m(w)33 b(giv)m(en)f(an)h(energy)h Fh(E)k Fj(in)32 b(a)g(gap)h(of)f(extended)i(states)g(of)e Fh(H)8 b Fj(,)32 b(w)m(e)i(de\014ne)1114 2150 y Fh(\033)1173 2109 y Ff(e)1169 2175 y Fe(?)1229 2150 y Fj(\()p Fh(E)6 b Fj(\))55 b(=)1579 2083 y Fh(q)1626 2046 y Fk(2)p 1579 2127 87 4 v 1595 2218 a Fj(\026)-50 b Fh(h)1745 2150 y Fj(lim)1692 2214 y Ff(J)6 b Fe(!f)p Ff(E)t Fe(g)2012 2083 y Fj(1)p 1977 2127 119 4 v 1977 2218 a Fi(j)p Fh(J)j Fi(j)2140 2125 y Fj(^)2122 2150 y Fi(T)25 b Fj(\()2261 2125 y(^)2239 2150 y Fh(P)2302 2165 y Ff(J)2351 2150 y Fi(r)2434 2165 y Ff(x)2478 2150 y Fj(\()2541 2125 y(^)2516 2150 y Fh(H)7 b Fj(\)\))33 b(.)1151 b(\(1\))-118 2375 y(Because)42 b(an)e(in\014nite)f(half-plane)f(is)h(a)h(t)m(ypical)f(mo) s(del)g(for)g(a)h(mesoscopic)g(v)m(olume)g(with)g(a)f(b)s(oundary)-8 b(,)43 b(w)m(e)-118 2495 y(call)e Fh(\033)130 2459 y Ff(e)126 2520 y Fe(?)185 2495 y Fj(\()p Fh(E)6 b Fj(\))43 b(the)g(edge)g(Hall)d(conductivit)m(y)j(rather)g(than)f(the)h(edge)g (Hall)e(conductance)j(just)f(as)g(the)g(bulk)-118 2616 y(Hall)33 b(conductivit)m(y)j(is)f(calculated)g(with)g(an)g(in\014nite) f(planar)h(mo)s(del)e(for)i(a)h(mesoscopic)f(v)m(olume,)h(while)e(the) -118 2736 y(conductance)44 b(is)e(alw)m(a)m(ys)i(asso)s(ciated)e(to)h (a)f(\014nite)h(macroscopic)f(sample.)73 b(Both)42 b(the)h(edge)h(and)f (the)g(bulk)-118 2856 y(Hall)35 b(conductivit)m(y)j(are)g(idealized)e (quan)m(tities)h(for)g(whic)m(h)h(clear)f(mathematical)e(results)j(can) f(b)s(e)h(obtained.)-118 2977 y(F)-8 b(urther)38 b(w)m(e)h(note)f(that) g(one)g(could)g(de\014ne)h(the)g(edge)f(Hall)e(conductivit)m(y)j(for)e (a)h(strip)g(geometry)-8 b(,)39 b(but)f(this)-118 3097 y(w)m(ould)e(not)g(lead)f(to)h(quan)m(tization)f(b)s(ecause)i(of)f(bac) m(kscattering,)h(that)f(is)g(tunneling)f(from)f(upp)s(er)j(to)f(lo)m(w) m(er)-118 3217 y(edge)d(states)h([10)o(].)28 3385 y(Before)j(starting)e (the)i(more)f(mathematical)d(analysis,)k(let)f(us)h(consider)g(the)g (Harp)s(er)f(Hamiltonian)d Fh(H)3949 3400 y Ff(H)-118 3505 y Fj(on)f Fh(`)58 3469 y Fk(2)97 3505 y Fj(\()p Fg(Z)203 3469 y Fk(2)243 3505 y Fj(\))g(in)g(order)g(to)g(familiarize) 27 b(ourselv)m(es)34 b(with)e(the)g(notion)f(of)h(edge)h(Hall)d (conductivit)m(y)-8 b(.)43 b(It)33 b(is)e(de\014ned)-118 3626 y(b)m(y)38 b(the)g(\014nite)f(di\013erence)i(equation)e(\()p Fh(H)1413 3641 y Ff(H)1480 3626 y Fh( )t Fj(\))1585 3641 y Ff(n;m)1750 3626 y Fj(=)f Fh( )1925 3641 y Ff(n)p Fk(+1)p Ff(;m)2170 3626 y Fj(+)25 b Fh( )2334 3641 y Ff(n)p Fe(\000)p Fk(1)p Ff(;m)2579 3626 y Fj(+)h Fh(e)2726 3590 y Fk(2)p Ff(\031)r({')2877 3626 y Fh( )2940 3641 y Ff(n;m)p Fk(+1)3186 3626 y Fj(+)f Fh(e)3332 3590 y Fe(\000)p Fk(2)p Ff(\031)r({')3539 3626 y Fh( )3602 3641 y Ff(n;m)p Fe(\000)p Fk(1)3859 3626 y Fj(and)-118 3746 y(w)m(e)37 b(supp)s(ose)g(here)g(that)f(the)g (magnetic)f(\015ux)i(p)s(er)f(unit)g(cell)f(is)g(rational)f Fh(')f Fj(=)h Fh(p=q)t Fj(.)53 b(Then)37 b(the)g(sp)s(ectrum)f(of)-118 3867 y Fh(H)-37 3882 y Ff(H)60 3867 y Fj(is)30 b(kno)m(wn)i(to)f(b)s(e) f(a)h(band)f(sp)s(ectrum.)44 b(T)-8 b(o)30 b(analyse)h(the)g(half)e (plane)h(op)s(erator)3010 3841 y(^)2985 3867 y Fh(H)3066 3882 y Ff(H)3164 3867 y Fj(on)g Fh(`)3338 3830 y Fk(2)3377 3867 y Fj(\(\000\),)h(w)m(e)h(use)f(the)-118 3987 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21600 2250 l 21600 1350 l 11730 1350 l cp gs col7 1.00 shd ef gr % Polyline n 11280 10350 m 21600 10350 l 21600 8550 l 11280 8550 l cp gs col7 1.00 shd ef gr % Polyline n 11700 9450 m 10350 9450 l 10350 1350 l 11700 1350 l cp gs col7 1.00 shd ef gr /Symbol ff 540.00 scf sf 20220 9090 m gs 1 -1 sc (+p) col0 sh gr /Symbol ff 540.00 scf sf 11579 8955 m gs 1 -1 sc (-p) col0 sh gr /Times-Roman-iso ff 540.00 scf sf 18346 9195 m gs 1 -1 sc (x) col0 sh gr % Polyline 45.000 slw n 11700 2250 m 11700 8550 l 20700 8550 l 20700 2250 l gs col0 s gr % Polyline gs clippath 16080 1404 m 16200 828 l 16320 1404 l 16320 660 l 16080 660 l cp clip n 16200 9000 m 16200 720 l gs col0 s gr gr % arrowhead n 16080 1404 m 16200 828 l 16320 1404 l 16200 1308 l 16080 1404 l cp gs 0.00 setgray ef gr col0 s /Times-Roman-iso ff 900.00 scf sf 16500 1665 m gs 1 -1 sc (E) col0 sh gr /Times-Roman-iso ff 720.00 scf sf 18015 9090 m gs 1 -1 sc (k) col0 sh gr $F2psEnd rs %%EndDocument endTexFig -118 1243 a Fj(Figure)35 b(1:)51 b Fb(Sc)m(hematic)36 b(represen)m(tation)i(of)e(the)g(sp)s(ectrum)h(of)2286 1218 y Fj(^)2260 1243 y Fh(H)2341 1258 y Ff(H)2445 1243 y Fb(in)e(a)h(giv)m(en)h(gap)f(of)g Fh(H)3290 1258 y Ff(H)3357 1243 y Fb(,)i(the)f(solid)e(lines)-118 1363 y(are)d(the)h(Diric)m(hlet)e(bands)i(and)g(the)g(shaded)h(regions)e (are)g(the)h(Blo)s(c)m(h)f(bands)i(\(tak)m(en)f(from)e Fj([14])p Fb(\).)-118 1636 y Fj(Hence)k(w)m(e)g(obtain)e Fh(\033)682 1600 y Ff(e)678 1661 y Fe(?)737 1636 y Fj(\()p Fh(E)6 b Fj(\))30 b(=)g Fh(\033)1086 1600 y Ff(b)1082 1661 y Fe(?)1141 1636 y Fj(\()p Fh(E)6 b Fj(\))34 b(for)f(all)f (energies)i(in)f(the)h(gaps)g(of)g Fh(H)2688 1651 y Ff(H)2755 1636 y Fj(,)g(whic)m(h)g(is)g(a)f(particular)f(case)j(of)-118 1757 y(Theorem)c(2)g(b)s(elo)m(w.)42 b(Note)31 b(that)g(the)g(equiv)-5 b(alen)m(t)31 b(result)f(for)h(the)g(Landau)g(Hamiltonian)c(simply)i (states)j(that)-118 1877 y(there)h(are)g Fh(n)g Fj(edge)g(c)m(hannels)g (in)f(the)h(gap)f(b)s(et)m(w)m(een)j(the)e Fh(n)p Fj(th)g(and)f(\()p Fh(n)23 b Fj(+)f(1\)th)32 b(Landau)h(bands)g([13].)28 2045 y(No)m(w)43 b(w)m(e)g(w)m(ould)g(lik)m(e)e(to)h(add)h(a)f (disordered)g(p)s(oten)m(tial)f Fh(V)22 b Fj(.)72 b(First)42 b(of)f(all,)i(if)e Fh(V)64 b Fj(is)42 b(su\016cien)m(tly)h(small,)-118 2165 y(su\016cien)m(tly)j(large)e(gaps)h(of)g Fh(H)1074 2180 y Ff(H)1186 2165 y Fj(remain)f(op)s(en)i(for)e Fh(H)57 b Fj(=)49 b Fh(H)2278 2180 y Ff(H)2376 2165 y Fj(+)30 b Fh(V)22 b Fj(.)81 b(It)46 b(follo)m(ws)e(further)h(from)f(Mourre)-118 2286 y(estimates)38 b(on)g(the)h(curren)m(t)h(op)s(erator)e(that)g(the) h(sp)s(ectrum)g(remains)e(absolutely)h(con)m(tin)m(uous)h(in)f(the)h (gaps)-118 2406 y(of)d Fh(H)44 b Fj(for)36 b(a)h(w)m(eak)h(p)s(oten)m (tial)c(whenev)m(er)40 b(the)d(curren)m(t)g(of)g(the)g(edge)g(states)g (of)2907 2381 y(^)2882 2406 y Fh(H)2963 2421 y Ff(H)3066 2406 y Fj(has)g(a)g(de\014nite)g(sign)f([8].)-118 2526 y(Whereas)h(the)f(latter)f(condition)f(is)i(alw)m(a)m(ys)g(satis\014ed) g(for)f(the)i(Landau)e(Hamiltonian,)e(it)i(ma)m(y)h(not)f(hold)g(in) -118 2647 y(the)j(discrete)g(case)g(\()p Fb(cf.)58 b Fj(Fig.)f(1)37 b(and)h(the)g(n)m(umerical)e(studies)i(in)f([14)o(])h (where)g(edge)h(c)m(hannels)f(ha)m(ving)f(edge)-118 2767 y(states)f(with)f(group)g(v)m(elo)s(cit)m(y)g(b)s(oth)g(to)g(the)g (left)g(and)g(to)g(the)h(righ)m(t)e(are)h(exhibited\).)51 b(In)36 b(this)f(situation,)f(the)-118 2888 y(p)s(ositiv)m(e)d(comm)m (utator)f(metho)s(ds)i(cannot)g(b)s(e)g(applied.)42 b(Nev)m(ertheless,) 35 b(w)m(e)e(shall)d(b)s(e)i(able)f(to)h(sho)m(w)h(that)e(the)-118 3008 y(curren)m(t)39 b(remains)f(constan)m(t.)61 b(Ho)m(w)m(ev)m(er,)43 b(but)38 b(w)m(e)i(cannot)e(deduce)i(that)e(the)h(sp)s(ectrum)g(is)e (still)f(absolutely)-118 3128 y(con)m(tin)m(uous)d(once)g(a)g(small)d (p)s(erturbation)i(is)g(added.)28 3249 y(In)e(order)g(to)g(treat)g(the) g(situation)e(with)i(brok)m(en)h(translation)d(in)m(v)-5 b(ariance,)29 b(w)m(e)i(parallel)c(Bellissard's)i(non-)-118 3369 y(comm)m(utativ)m(e)j(generalization)f(of)h(the)i(TKN)1588 3384 y Fk(2)1661 3369 y Fj(w)m(ork)g([6,)f(7].)45 b(No)33 b(particular)e(structure)j(of)f(the)g(Hamiltonian)-118 3489 y Fh(H)42 b Fj(on)35 b Fh(`)184 3453 y Fk(2)223 3489 y Fj(\()p Fg(Z)329 3453 y Fk(2)369 3489 y Fj(\))f(is)h(needed)h (except)g(for)e(its)g(homogen)m(uit)m(y)h(in)e(the)j(sense)g(of)e([6,)h (7].)49 b(The)36 b(main)d(mathematical)-118 3610 y(to)s(ol)i(in)h([6,)h (7])g(is)f(the)i(C)800 3574 y Fe(\003)839 3610 y Fj(-algebra)e Fi(A)g Fj(of)h(homogeneous)g(observ)-5 b(ables)37 b(in)f(the)i(plane.) 56 b(It)37 b(has)g(the)g(structure)-118 3730 y(of)e(a)h(crossed)i(pro)s (duct)e(algebra)f Fi(A)e Fj(=)h Fh(C)7 b Fj(\(\012\))24 b Fi(\002)h Fg(Z)1776 3745 y Ff(x)1845 3730 y Fi(\002)g Fg(Z)2015 3745 y Ff(y)2093 3730 y Fj(asso)s(ciated)36 b(to)g(the)g(dynamical)f(system)i(giv)m(en)f(b)m(y)-118 3851 y(the)d(magnetic)f(translations)f Fg(Z)1068 3866 y Ff(x)1145 3851 y Fj(and)i Fg(Z)1403 3866 y Ff(y)1478 3851 y Fj(in)f(the)i Fh(x)f Fj(and)g Fh(y)t Fj(-direction)d(resp)s (ectiv)m(ely)k(acting)e(on)h(the)g(compact)-118 3971 y(space)g(of)e(disorder)h(con\014gurations)g(\012)g(whic)m(h)g(is)g (the)g(h)m(ull)f(of)g Fh(H)8 b Fj(.)43 b(Eac)m(h)33 b(suc)m(h)g (con\014guration)e Fh(!)g Fi(2)e Fj(\012)j(induces)-118 4091 y(a)e(represen)m(tation)i Fh(\031)652 4106 y Ff(!)733 4091 y Fj(of)f(the)g(observ)-5 b(able)31 b(algebra)e Fi(A)i Fj(on)f(ph)m(ysical)h(Hilb)s(ert)e(space)j Fh(`)3072 4055 y Fk(2)3112 4091 y Fj(\()p Fg(Z)3218 4055 y Fk(2)3257 4091 y Fj(\).)43 b(There)32 b(exists)g(an)-118 4212 y Fh(H)37 b Fi(2)30 b(A)j Fj(suc)m(h)i(that)f Fh(\031)698 4227 y Ff(!)748 4212 y Fj(\()p Fh(H)8 b Fj(\))33 b(is)g(precisely)h (the)g(Hamilton)d(op)s(erator)i(with)h(disordered)g(con\014guration)e Fh(!)h Fi(2)d Fj(\012.)-118 4332 y(Ph)m(ysical)45 b(quan)m(tities)f(as) h(the)g(Hall)e(conductivities)h(hence)j(carry)e(a)f(supplemen)m(tary)i (index)f Fh(!)t Fj(.)79 b(W)-8 b(e)45 b(no)m(w)-118 4453 y(consider)c(the)h(T)-8 b(o)s(eplitz)40 b(extension)i Fh(T)14 b Fj(\()p Fi(A)p Fj(\))40 b(with)h(resp)s(ect)h(to)f(the)h (crossed)g(pro)s(duct)g(structure)g(of)f Fg(Z)3754 4468 y Ff(y)3837 4453 y Fj([18].)-118 4573 y(Its)33 b(ph)m(ysical)f (represen)m(tations)h(giv)m(e)f(op)s(erators)g(in)g(the)g(half)f (plane.)43 b(This)33 b(naturally)d(giv)m(es)j(rise)f(to)g(an)g(exact) -118 4693 y(sequence)j(of)d Fh(C)474 4657 y Fe(\003)514 4693 y Fj(-algebras)f([18])1244 4913 y(0)55 b Fi(!)g(E)1659 4860 y Fc(i)1620 4913 y Fi(!)g Fh(T)14 b Fj(\()p Fi(A)p Fj(\))2088 4860 y Fc(\031)2057 4913 y Fi(!)55 b(A)g(!)g Fj(0)32 b(.)1281 b(\(2\))-118 5133 y(Here)34 b Fi(E)42 b Fj(is)33 b(the)h(C)546 5097 y Fe(\003)586 5133 y Fj(-algebra)e(of)h (observ)-5 b(ables)34 b(lo)s(calized)e(near)h(the)h(edge)g Fh(y)f Fj(=)c(0;)34 b(it)e(is)h(isomorphic)f(to)h(the)h(C)3943 5097 y Fe(\003)3983 5133 y Fj(-)-118 5254 y(tensor)e(pro)s(duct)g(of)f Fh(C)7 b Fj(\(\012\))20 b Fi(\002)h Fg(Z)1058 5269 y Ff(x)1134 5254 y Fj(with)31 b(the)h(compact)f(op)s(erators)h Fh(K)7 b Fj(.)43 b(The)33 b(exact)f(sequence)i(\(2\))e(induces)g(t)m(w) m(o)-118 5374 y(six-term)25 b(exact)i(sequences,)k(one)26 b(for)g(K-theory)g(groups)g([18,)g(9])g(and)g(one)g(for)g(the)g(cyclic) g(cohomology)e(groups)-118 5494 y([17],)30 b(and)g(w)m(e)h(shall)e(use) i(their)e(dualit)m(y)g([17])h(to)g(pro)m(v)m(e)h(the)f(equalit)m(y)g (of)f(bulk)h(and)g(edge)h(Hall)d(conductivities.)28 5615 y(Let)34 b(us)g(illustrate)d(these)k(notions)e(for)g(the)h(Harp)s(er)f (Hamiltonian)d(with)j(arbitrary)f(\015ux)j Fh(')p Fj(,)e(but)h(without) -118 5735 y(a)29 b(further)h(p)s(oten)m(tial.)41 b(The)31 b(C)1002 5699 y Fe(\003)1041 5735 y Fj(-algebra)d Fi(A)i Fj(is)f(then)h(the)g(rotation)e(algebra)h(generated)h(b)m(y)h(the)f(t)m (w)m(o)g(magnetic)-118 5856 y(translations)i Fh(U)481 5871 y Ff(x)559 5856 y Fj(and)h Fh(U)815 5871 y Ff(y)890 5856 y Fj(satisfying)g(the)h(comm)m(utation)d(relation)h Fh(U)2514 5871 y Ff(x)2558 5856 y Fh(U)2624 5871 y Ff(y)2695 5856 y Fj(=)d Fh(e)2845 5819 y Fk(2)p Ff(\031)r({')2997 5856 y Fh(U)3063 5871 y Ff(y)3104 5856 y Fh(U)3170 5871 y Ff(x)3215 5856 y Fj(.)46 b(Th)m(us)35 b(in)e(this)g(case)1924 6107 y(3)p eop %%Page: 4 4 4 3 bop -118 47 a Fh(C)7 b Fj(\(\012\))138 20 y Fi(\030)139 51 y Fj(=)248 47 y Fg(C)p Fj(.)53 b(The)37 b(T)-8 b(o)s(eplitz)34 b(extension)j(is)e(generated)h(b)m(y)2133 22 y(^)2117 47 y Fh(U)2183 62 y Ff(x)2263 47 y Fj(and)2472 22 y(^)2455 47 y Fh(U)2521 62 y Ff(y)2599 47 y Fj(satisfying)e(the)i(same)g(comm)m (utation)-118 178 y(relation,)41 b(but,)j(while)774 153 y(^)757 178 y Fh(U)823 193 y Ff(x)908 178 y Fj(remains)c(unitary)-8 b(,)1670 153 y(^)1654 178 y Fh(U)1720 193 y Ff(y)1802 178 y Fj(is)41 b(no)m(w)h(only)e(an)h(isometry)f(satisfying)3354 153 y(^)3337 178 y Fh(U)3413 142 y Fe(\003)3403 202 y Ff(y)3470 153 y Fj(^)3454 178 y Fh(U)3520 193 y Ff(y)3603 178 y Fj(=)i(1)28 b Fi(\000)g Fj(\005)3976 193 y Fk(0)-118 298 y Fj(where)j(\005)234 313 y Fk(0)302 298 y Fj(is)e(the)h(pro)5 b(jection)29 b(on)h(the)g(states)g(supp)s(orted)g(b)m(y)h(the)e(b)s (oundary)h(of)f(\000.)42 b(Finally)-8 b(,)28 b Fi(E)38 b Fj(is)29 b(isomorphic)-118 419 y(to)37 b(the)i(tensor)f(pro)s(duct)g (of)g(C)1036 382 y Fe(\003)1076 419 y Fj(\()1130 393 y(^)1114 419 y Fh(U)1180 434 y Ff(x)1224 419 y Fj(\))1298 391 y Fi(\030)1299 423 y Fj(=)1413 419 y Fh(C)7 b Fj(\()p Fh(S)1594 382 y Fk(1)1633 419 y Fj(\))38 b(with)f Fh(K)7 b Fj(.)60 b(The)38 b(maps)g(in)f(\(2\))h(are)g(the)g(inclusion)e Fh(i)i Fj(and)g(the)-118 539 y(pro)5 b(jection)32 b Fh(\031)37 b Fj(giv)m(en)32 b(b)m(y)i Fh(\031)t Fj(\()940 514 y(^)924 539 y Fh(U)990 554 y Ff(x;y)1091 539 y Fj(\))27 b(=)h Fh(U)1326 554 y Ff(x;y)1427 539 y Fj(.)28 659 y(The)f(traces)f Fi(T)551 674 y Ff(x;y)678 659 y Fj(of)f(ph)m(ysical)g(represen)m (tations)i Fh(\031)1873 674 y Ff(!)1949 659 y Fj(of)f(an)f(observ)-5 b(able)26 b(are)f(almost)f(surely)i(indep)s(enden)m(t)h(of)-118 780 y Fh(!)33 b Fj(with)d(resp)s(ect)h(to)e(an)m(y)i(giv)m(en)f(in)m(v) -5 b(arian)m(t)29 b(and)h(ergo)s(dic)f(measure)h Fg(P)g Fj(on)f(\012)i([7].)42 b(Hence)32 b(they)e(allo)m(w)f(to)g(de\014ne) -118 900 y(traces)35 b(on)g(the)g(observ)-5 b(able)35 b(algebras)g Fi(A)f Fj(and)h Fi(E)9 b Fj(.)50 b(No)m(w)35 b(the)g(de\014nition)f(\(1\))g(of)h(the)g(edge)g(Hall)e(conductivit)m (y)-118 1020 y(remains)g(v)-5 b(alid)31 b(as)j(long)e(as)i(the)g(pro)5 b(jections)1631 995 y(^)1609 1020 y Fh(P)1672 1035 y Ff(J)1754 1020 y Fj(are)34 b(in)f(the)h(Sc)m(hatten)g(ideal)f(of)g (traceclass)h(op)s(erators)f(with)-118 1141 y(resp)s(ect)k(to)361 1116 y(^)342 1141 y Fi(T)62 b Fj(for)36 b Fh(J)45 b Fj(su\016cien)m (tly)37 b(close)g(to)f Fi(f)p Fh(E)6 b Fi(g)p Fj(.)54 b(This)36 b(is)g(p)s(ossible)g(ev)m(en)i(though)3103 1116 y(^)3081 1141 y Fh(P)3144 1156 y Ff(J)3229 1141 y Fj(is)e(only)g(an)g(elemen)m(t)-118 1261 y(of)j(the)g(bicomm)m(utan)m (t)f Fi(E)839 1225 y Fe(00)881 1261 y Fj(,)i(the)g(en)m(v)m(elopping)f (v)m(on)h(Neumann)f(algebra.)62 b(No)m(w)40 b(the)g(crucial)e(observ)-5 b(ation)38 b(is)-118 1382 y(that)f(the)g(curren)m(t)i(of)d(the)i(edge)g (states)g(in)e(an)h(in)m(terv)-5 b(al)36 b Fh(J)47 b Fj(lying)35 b(in)i(a)g(gap)g(of)f Fh(H)45 b Fj(can)37 b(b)s(e)h(calculated)e(using)-118 1502 y(Duhamel's)31 b(form)m(ula)g(and)i(taking)f(in)m(to)f(accoun)m(t)j(the)f(discon)m (tin)m(uit)m(y)f(of)2651 1477 y(^)2629 1502 y Fh(P)2692 1517 y Ff(J)2741 1502 y Fj(:)1083 1748 y(^)1065 1773 y Fi(T)25 b Fj(\()1204 1748 y(^)1182 1773 y Fh(P)1245 1788 y Ff(J)1294 1773 y Fi(r)1377 1788 y Ff(x)1421 1773 y Fj(\()1484 1748 y(^)1459 1773 y Fh(H)8 b Fj(\)\))55 b(=)1830 1705 y Fi(j)p Fh(J)9 b Fi(j)p 1820 1750 139 4 v 1820 1841 a Fj(2)p Fh(\031)t({)2015 1748 y Fj(^)1996 1773 y Fi(T)26 b Fj(\()2134 1748 y(^)2114 1773 y Fi(U)10 b Fj(\()p Fh(J)f Fj(\))2325 1732 y Fe(\003)2365 1773 y Fi(r)2448 1788 y Ff(x)2512 1748 y Fj(^)2491 1773 y Fi(U)i Fj(\()p Fh(J)e Fj(\)\))32 b(,)1091 b(\(3\))-118 1971 y(where)882 2116 y(^)862 2141 y Fi(U)10 b Fj(\()p Fh(J)f Fj(\))56 b(=)f(exp)1426 1995 y Fd( )1491 2141 y Fj(2)p Fh(\031)t({)1652 2116 y Fj(^)1630 2141 y Fh(P)1693 2156 y Ff(J)1777 2049 y Fj(^)1752 2074 y Fh(H)29 b Fi(\000)23 b Fh(E)2040 2038 y Fe(0)p 1752 2118 312 4 v 1848 2210 a Fi(j)p Fh(J)9 b Fi(j)2073 1995 y Fd(!)2188 2141 y Fj(,)228 b Fh(E)2521 2100 y Fe(0)2572 2141 y Fj(=)28 b(inf)6 b(\()p Fh(J)j Fj(\))32 b(.)899 b(\(4\))-118 2396 y(Although)337 2371 y(^)315 2396 y Fh(P)378 2411 y Ff(J)461 2396 y Fj(con)m(tains)34 b(discon)m(tin)m(uities,)h(the)f(construction)h(assures)h(that)f(the)g (unitary)f(op)s(erator)3724 2371 y(^)3704 2396 y Fi(U)11 b Fj(\()p Fh(J)e Fj(\))34 b(is)-118 2517 y(an)c(elemen)m(t)f(of)g(the)i (unitization)1153 2491 y(~)1136 2517 y Fi(E)38 b Fj(of)29 b(the)h(C)1570 2480 y Fe(\003)1610 2517 y Fj(-algebra)e Fi(E)9 b Fj(.)42 b(F)-8 b(urther)30 b Fi(r)2546 2532 y Ff(x)2610 2491 y Fj(^)2590 2517 y Fi(U)10 b Fj(\()p Fh(J)f Fj(\))30 b(is)2944 2491 y(^)2926 2517 y Fi(T)c Fj(-traceclass)k(if)3584 2491 y(^)3562 2517 y Fh(P)3625 2532 y Ff(J)3704 2517 y Fj(is)f(\(that)-118 2637 y(\(3\))37 b(is)g(\014nite)g(also)f(follo)m(ws)g(from)g(the)i(exp)s(onen)m(tial)e (map)h(\(5\))g(and)g(the)h(dualit)m(y)e(theorem)h(b)s(elo)m(w\).)57 b(Finally)-8 b(,)-118 2757 y(w)m(e)33 b(note)g(that)473 2732 y(^)453 2757 y Fi(U)10 b Fj(\()p Fh(J)f Fj(\))32 b(remains)g(in)f(the)i(same)f(homotop)m(y)g(class)h(of)f(unitaries)f (in)2912 2732 y(~)2894 2757 y Fi(E)41 b Fj(as)33 b(the)f(b)s(orders)h (of)f Fh(J)41 b Fj(v)-5 b(ary)-118 2878 y(in)32 b(a)g(gap)g Fh(G)h Fj(of)f Fh(H)8 b Fj(.)43 b(Th)m(us)34 b(\(4\))e(asso)s(ciates)h (to)f Fh(G)g Fj(a)h(class)f(in)g(the)h(K-group)f Fh(K)2782 2893 y Fk(1)2821 2878 y Fj(\()p Fi(E)9 b Fj(\).)28 2998 y(It)35 b(no)m(w)g(follo)m(ws)e(from)g(Connes')j(non-comm)m(utativ)m(e) d(geometry)h([11])g(that)2910 2959 y Fk(1)p 2877 2975 102 4 v 2877 3032 a(2)p Ff(\031)r({)3006 2973 y Fj(^)2988 2998 y Fi(T)26 b Fj(\()3125 2973 y(^)3106 2998 y Fi(U)3178 2962 y Fe(\003)3217 2998 y Fi(r)3300 3013 y Ff(x)3364 2973 y Fj(^)3344 2998 y Fi(U)10 b Fj(\))35 b(is)e(an)i(in)m(teger)-118 3135 y(for)g(an)m(y)i(unitary)590 3110 y(^)570 3135 y Fi(U)46 b Fj(in)35 b(\(a)h(suitable)f(subalgebra)h(of)7 b(\))1954 3110 y(~)1937 3135 y Fi(E)h Fj(.)54 b(Actually)35 b Fh(\020)2521 3150 y Fk(1)2560 3135 y Fj(\()2624 3110 y(^)2598 3135 y Fh(A;)2738 3110 y Fj(^)2715 3135 y Fh(B)5 b Fj(\))34 b(=)3018 3096 y Fk(1)p 2985 3112 V 2985 3169 a(2)p Ff(\031)r({)3115 3110 y Fj(^)3097 3135 y Fi(T)25 b Fj(\()3240 3110 y(^)3214 3135 y Fh(A)p Fi(r)3370 3150 y Ff(x)3414 3135 y Fj(\()3475 3110 y(^)3452 3135 y Fh(B)5 b Fj(\)\))36 b(de\014nes)i(a)-118 3272 y(1-co)s(cycle)f(on)g Fi(E)45 b Fj(b)s(ecause)926 3247 y(^)908 3272 y Fi(T)62 b Fj(is)37 b(in)m(v)-5 b(arian)m(t)36 b(under)i Fi(r)1905 3287 y Ff(x)1949 3272 y Fj(.)57 b(With)36 b(some)h(calculatory)f (e\013ort,)i(this)f(co)s(cycle)g(can)-118 3393 y(b)s(e)c(link)m(ed)g (to)g(the)h(standard)g(1-co)s(cyle)f(of)g(the)g(F)-8 b(redholm)32 b(mo)s(dule)g(\()p Fh(C)2544 3408 y Fk(1)2606 3393 y Fi(\012)23 b(E)2759 3408 y Fk(0)2798 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(follo)m(ws)d(that)i Fh(\033)1358 5702 y Ff(e)1354 5763 y Fe(?)1414 5738 y Fj(\()p Fh(E)6 b Fj(\))34 b(actually)e(results)j (from)e(a)g(pairing)f(b)s(et)m(w)m(een)k([)3358 5713 y(^)3337 5738 y Fi(U)11 b Fj(\()p Fh(G)p Fj(\)])3590 5753 y Fk(1)3659 5738 y Fi(2)31 b Fh(K)3839 5753 y Fk(1)3878 5738 y Fj(\()p Fi(E)9 b Fj(\))-118 5858 y(with)31 b(the)i(o)s(dd)f (cyclic)f(cohomology)g(class)h(de\014ned)h(b)m(y)g(the)f(1-co)s(cycle)g Fh(\020)2589 5873 y Fk(1)2660 5858 y Fj(giv)m(en)g(ab)s(o)m(v)m(e.)44 b(Similarly)-8 b(,)28 b(the)33 b(bulk)1924 6107 y(4)p eop %%Page: 5 5 5 4 bop -118 35 a Fj(Hall)43 b(conductivit)m(y)j Fh(\033)736 -1 y Ff(b)732 60 y Fe(?)792 35 y Fj(\()p Fh(\026)p Fj(\))f(for)f(a)i(F) -8 b(ermi)43 b(lev)m(el)i Fh(\026)g Fj(in)g(a)g(gap)g(of)g Fh(H)53 b Fj(comes)46 b(from)e(a)h(pairing)f(of)h(the)h(class)-118 156 y(of)c(the)i(F)-8 b(ermi)41 b(pro)5 b(jection)42 b([)p Fh(P)1033 171 y Ff(\026)1080 156 y Fj(])1107 171 y Fk(0)1191 156 y Fi(2)k Fh(K)1386 171 y Fk(0)1425 156 y 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