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b(the)h(absence)e(of)h(an)n(y)f(lo)r(cal)g(b)r(ene\014t)i(for)f (the)g(electrons)f(in)n(v)n(olv)n(ed,)h(the)g(remo)n(v)-5 b(al)33 b(of)-150 3279 y(doubly)c(o)r(ccupied)g(states)g(has)g(no)g Fk(a)h(priori)38 b Fn(classical)28 b(limit,)i(but)g(remains)e(a)h (dynamical)g(pro)r(cess)f(with)i(short)e(time)i(and)f(length)-150 3378 y(scales.)37 b(Then)28 b(the)g(simplest)g(w)n(a)n(y)f(to)g (imagine)h(the)g(long-time)f(b)r(eha)n(vior,)g(whic)n(h)g(dominates)h (bulk)g(prop)r(erties,)f(is)h(that)g(it)g(is)g(due)-150 3478 y(to)35 b(electrons)g(whic)n(h)g(ha)n(v)n(e)g(managed)f(to)h(a)n (v)n(oid)f(eac)n(h)h(other,)i Fk(i.e.)p Fn(,)i(ha)n(v)n(e)c(built)h(up) g(w)n(a)n(v)n(e)d(functions)j(with)g(double)g(o)r(ccupation)-150 3578 y(suppressed.)48 b(All)32 b(of)f(the)h(`arti\014cial)f(limits')h (considered)e(ab)r(o)n(v)n(e)h(try)g(to)g(capture)g(this)h(regime)f(of) g Fk(avoiding)38 b Fn(double)32 b(o)r(ccupancy)-7 b(,)-150 3677 y(additionally)35 b(simplifying)i(the)f(construction)f(of)h(the)h 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Fh(X)1449 456 y Fg()1510 504 y(\033)1631 369 y Fn(\()p Fi(n)1713 381 y Ff(i;)p Fe(\000)p Ff(\033)1871 369 y Fd(\000)c Fi(n)2004 381 y Ff(j;)p Fe(\000)p Ff(\033)2147 369 y Fn(\))2193 277 y Fh(\020)2243 369 y Fi(a)2287 329 y Fe(y)2287 392 y Ff(i\033)2355 369 y Fi(a)2399 392 y Ff(j)s(\033)2493 369 y Fd(\000)g Fi(a)2620 329 y Fe(y)2620 392 y Ff(j)s(\033)2695 369 y Fi(a)2739 392 y Ff(i\033)2808 277 y Fh(\021)2871 369 y Fi(;)1050 b Fn(\(1\))-150 644 y(with)25 b Fi(t)g Fn(the)f(hopping)h(o)n(v)n(erlap,)e(and)h Fi(U)33 b Fn(the)25 b(on-site)f(repulsion.)35 b(T)-7 b(o)24 b(`smear)g(out')g(the)h(other)f(spin,)h(replace)f(the)h(n)n(um)n (b)r(er)f(op)r(erators)-150 744 y(b)n(y)k(their)h(exp)r(ectation)g(v)-5 b(alues,)28 b(assuming)g(they)h(are)f(ev)n(erywhere)e(the)j(same.)40 b(This)29 b(mak)n(es)e(the)i(comm)n(utator)f(zero,)g(and)g(that)h(is) -150 843 y(the)f(formal)f(expression)f(of)h(Gutzwiller's)h(v)-5 b(ariational)26 b(assumption.)-25 975 y(If)32 b(the)g(comm)n(utator)e 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b(direct)g(w)n(a)n(y)f(to)h(impro)n(v)n(e)f(on)h(this)g(is)g(to)g (resum)g(some)f(of)i(the)f(neglected)g(comm)n(utators,)g(obtaining)f(a) h(b)r(etter)h(expression)-150 2273 y(for)e(the)g(kinetic)h(op)r(erator) d(in)j(the)f(righ)n(t-hand)f(side)h(of)h(\(2\),)g(instead)f(of)g Fi(K)36 b Fn(itself.)45 b(In)30 b(other)g(w)n(ords,)f(a)h(b)r(etter)h (disen)n(tanglemen)n(t)-150 2372 y(form)n(ula)d(than)h(\(2\))g(is)g (needed.)41 b(If)30 b(one)e(included)i(some)e(additional)h(comm)n (utators)e(to)i(a)g(\014nite)g(order)f(in)h Fi(\014)t Fn(,)h(this)f(w)n(ould)g(amoun)n(t)-150 2472 y(to)c(a)g(high-temp)r (erature)g(expansion;)g(I)h(prop)r(ose)e(a)h(di\013eren)n(t)h(line)g (of)f(attac)n(k,)g(and)h(this)f(is)h(to)f(sum)h(all)f(comm)n(utators)f (of)i(a)f(certain)-150 2572 y(class.)38 b(The)29 b(ob)n(vious)e(ones)h (to)g(include)h(in)g(the)g(case)e(of)i(a)f(large)f Fi(U)37 b Fn(are)27 b(those)h(with)h(only)f(one)g Fi(K)6 b Fn(,)29 b(and)f(man)n(y)g Fi(V)19 b Fn('s,)28 b(since)h(to)f(eac)n(h)-150 2671 y(order)f(in)i Fi(\014)t Fn(,)h(all)e(others)g(are)g(smaller)f(b)n (y)i(at)f(least)h Fi(t=U)9 b Fn(.)39 b(The)29 b(relev)-5 b(an)n(t)28 b(disen)n(tanglemen)n(t)g(form)n(ula)g(app)r(ears)f(in)i (Ref.)g(16.)40 b(There,)-150 2771 y(comm)n(utators)26 b(with)i(only)f(one)h Fi(K)6 b Fn(,)27 b Fk(i.e.)38 b Fn(of)28 b(the)g(t)n(yp)r(e)1505 2937 y Fi(\014)1556 2903 y Ff(n)1615 2937 y Fn([)p Fi(V)5 b(;)14 b Fn([)p Fi(V)5 b(;)14 b(:)g(:)g(:)g(;)g Fn([)p Fi(V)5 b(;)1615 2993 y Fh(|)p 1652 2993 162 10 v 162 w({z)p 1888 2993 V 162 w(})1676 3075 y Fm(\()p Ff(n)p Fe(\000)p Fm(1\))19 b(times)2102 2937 y Fi(K)6 b Fn(]])14 b Fi(:)g(:)g(:)o Fn(])g Fi(;)1549 b Fn(\(3\))-150 3230 y(are)27 b(summed)h(to)f(all)g (orders)f(in)i Fi(\014)t Fn(,)g(giving)f(an)g(impro)n(v)n(ed)g(v)n (ersion)f(of)h Fi(K)6 b Fn(:)892 3503 y Fi(K)969 3468 y Fe(0)1015 3503 y Fn(=)22 b Fi(t)1175 3424 y Fh(X)1156 3590 y Fg()1217 3637 y(\033)1348 3416 y Fn(sinh)1510 3324 y Fh(\020)1569 3379 y Ff(\014)s(U)p 1569 3397 93 4 v 1599 3445 a Fm(2)1685 3416 y Fn(\()p Fi(n)1767 3428 y Ff(i;)p Fe(\000)p Ff(\033)1926 3416 y Fd(\000)c Fi(n)2059 3428 y Ff(j;)p Fe(\000)p Ff(\033)2202 3416 y Fn(\))2234 3324 y Fh(\021)p 1348 3484 936 4 v 1488 3534 a Ff(\014)s(U)p 1488 3552 93 4 v 1518 3600 a Fm(2)1604 3571 y Fn(\()p Fi(n)1686 3583 y Ff(i;)p Fe(\000)p Ff(\033)1845 3571 y Fd(\000)g Fi(n)1978 3583 y Ff(j;)p Fe(\000)p Ff(\033)2121 3571 y Fn(\))2308 3411 y Fh(\020)2357 3503 y Fi(a)2401 3463 y Fe(y)2401 3526 y Ff(i\033)2469 3503 y Fi(a)2513 3526 y Ff(j)s(\033)2607 3503 y Fn(+)g Fi(a)2734 3463 y Fe(y)2734 3526 y Ff(j)s(\033)2810 3503 y Fi(a)2854 3526 y Ff(i\033)2922 3411 y Fh(\021)2985 3503 y Fi(:)936 b Fn(\(4\))-150 3778 y(The)36 b(fraction)f(in)n(v)n(olving)f(n)n(um)n (b)r(er)h(op)r(erators)f(should)h(b)r(e)h(understo)r(o)r(d)f(in)h (terms)f(of)h(its)g(T)-7 b(a)n(ylor)33 b(expansion.)60 b(This)36 b(is)f(a)g(most)-150 3878 y(surprising)26 b(result,)h(since)g (it)g(runs)g(directly)g(coun)n(ter)f(to)h(classical)f(in)n(tuition,)i 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Fn(in)27 b(\(4\),)g(and)g(taking)f(the)h(limit)-150 4407 y(of)h(long)e(times,)i Fi(U)9 b(T)34 b Fd(!)23 b(1)p Fn(,)28 b(the)g(fraction)f(with)h(the)g (h)n(yp)r(erb)r(olic)f(sine)h(easily)e(b)r(ecomes)677 4573 y(1)18 b Fd(\000)g Fi(n)870 4585 y Ff(i;)p Fe(\000)p Ff(\033)1028 4573 y Fd(\000)g Fi(n)1161 4585 y Ff(j;)p Fe(\000)p Ff(\033)1323 4573 y Fn(+)g(2)p Fi(n)1498 4585 y Ff(i;)p Fe(\000)p Ff(\033)1637 4573 y Fi(n)1687 4585 y Ff(j;)p Fe(\000)p Ff(\033)1853 4573 y Fn(=)23 b(\(1)18 b Fd(\000)g Fi(n)2166 4585 y Ff(i;)p Fe(\000)p Ff(\033)2306 4573 y Fn(\)\(1)g Fd(\000)g Fi(n)2563 4585 y Ff(j;)p Fe(\000)p Ff(\033)2707 4573 y Fn(\))g(+)g Fi(n)2890 4585 y Ff(i;)p Fe(\000)p Ff(\033)3030 4573 y Fi(n)3080 4585 y Ff(j;)p Fe(\000)p Ff(\033)3944 4573 y Fn(\(5\))-150 4739 y(and)27 b(so)g(one)h(obtains)f(pro)5 b(jected)27 b(hopping,)g(with)h(particle-hole)e(symmetry)h(preserv)n(ed:)300 4926 y Fi(K)371 4938 y Fm(e\013)471 4926 y Fn(=)c Fi(t)632 4847 y Fh(X)613 5013 y Fg()674 5060 y(\033)794 4833 y Fh(h)834 4926 y Fn(\(1)18 b Fd(\000)g Fi(n)1059 4938 y Ff(i;)p Fe(\000)p Ff(\033)1198 4926 y Fn(\))p Fi(a)1274 4886 y Fe(y)1274 4949 y Ff(i\033)1343 4926 y Fi(a)1387 4949 y Ff(j)s(\033)1462 4926 y Fn(\(1)h Fd(\000)f Fi(n)1688 4938 y Ff(j;)p Fe(\000)p Ff(\033)1831 4926 y Fn(\))h(+)f(\()p Fk(h)6 b Fi(:)p Fk(c)f Fi(:)p Fn(\))2167 4833 y Fh(i)2225 4926 y Fn(+)41 b Fi(t)2404 4847 y Fh(X)2385 5013 y Fg()2446 5060 y(\033)2566 4833 y Fh(h)2606 4926 y Fi(n)2656 4938 y Ff(i;)p Fe(\000)p Ff(\033)2795 4926 y Fi(a)2839 4886 y Fe(y)2839 4949 y Ff(i\033)2907 4926 y Fi(a)2951 4949 y Ff(j)s(\033)3027 4926 y Fi(n)3077 4938 y Ff(j;)p Fe(\000)p Ff(\033)3239 4926 y Fn(+)18 b(\()p Fk(h)6 b Fi(:)p Fk(c)f Fi(:)p Fn(\))3524 4833 y Fh(i)3577 4926 y Fi(:)344 b Fn(\(6\))-150 5201 y(The)22 b(same)f(hops,)h(whic)n(h)f(w)n(ere)g(singularly)f(enhanced)i(on)f(the) h(imaginary)e(time)i(axis,)g(are)e(no)n(w)h(strictly)h(forbidden!)34 b(This)22 b(op)r(erator)-150 5300 y(allo)n(ws)d(only)i(those)f(hops,)i (whic)n(h)f(do)f(not)h(c)n(hange)f(the)h(n)n(um)n(b)r(er)f(of)h(doubly) g(o)r(ccupied)f(sites,)i(either)f(b)n(y)g(ha)n(ving)e(b)r(oth)i(sites,) i(in)n(v)n(olv)n(ed)-150 5400 y(in)28 b(a)f(hop,)h(uno)r(ccupied)f(b)n (y)h(an)f(opp)r(osite)g(spin)h(\(\014rst)g(term\),)g(or)e(b)n(y)i(ha)n (ving)e(them)i(b)r(oth)g(o)r(ccupied)g(\(second)f(term\).)p eop %%Page: 4 4 4 3 bop -25 83 a Fn(T)-7 b(o)27 b(summarize,)h(after)g(a)g(non-p)r (erturbativ)n(e)f(resummation)h(has)f(included)i(dominan)n(t)f (additional)g(comm)n(utators,)f(the)i(e\013ect)-150 183 y(of)c(a)g(large)f(con)n(tact)h(p)r(oten)n(tial)g(is)g(v)n(ery)f (di\013eren)n(t)h(along)g(the)g(real)g(and)g(imaginary)e(time)j(axes.) 35 b(Con)n(trast)24 b(this)i(with)g(the)f(situation)-150 282 y(near)20 b(a)g(mean-\014eld)h(solution:)33 b(in)20 b(imaginary)g(time,)i(deviations)e(from)h(it)g(b)n(y)f(energies)g Fi(!)j Fn(are)d(exp)r(onen)n(tially)g(suppressed)g(b)n(y)g(factors)-150 382 y(exp\()p Fd(\000)p Fi(\014)t(!)s Fn(\),)33 b(while)f(in)g(real)f (time,)i(they)f(pro)r(duce)f(rapid)h(oscillations)e(b)n(y)i(exp)o(\()p Fd(\000)p Fi(i!)s(T)12 b Fn(\),)32 b(whic)n(h)g(giv)n(e)e(an)i (analogous)d(suppression)-150 482 y(through)h(cancellation.)45 b(In)30 b(other)g(w)n(ords,)h(a)f(mean)g(\014eld)h(giv)n(es)e(the)i Fk(same)38 b Fn(ph)n(ysics)30 b(along)f(b)r(oth)i(axes,)f(while)h (here,)g(going)e(from)-150 581 y(one)e(to)h(the)g(other)f(in)n(v)n(olv) n(es)1524 652 y(sinh)14 b Fi(\014)t(U)t(=)p Fn(2)p 1524 689 358 4 v 1605 765 a Fi(\014)t(U)t(=)p Fn(2)1915 708 y Fd(!)2031 652 y Fn(sin)f Fi(U)c(T)g(=)p Fn(2)p 2031 689 323 4 v 2088 765 a Fi(U)g(T)g(=)p Fn(2)2363 708 y Fi(;)1558 b Fn(\(7\))-150 901 y(replacing)20 b(in\014nit)n(y)h(b)n(y)g (zero)e(in)j(the)f(exp)r(onen)n(t,)h(when)f Fi(U)30 b Fn(is)21 b(large,)f(whic)n(h)h(implies)g(t)n(w)n(o)f Fk(di\013er)l(ent)29 b Fn(ph)n(ysical)20 b(regimes.)33 b(The)21 b(one)g(with)-150 1001 y(pro)5 b(jected)21 b(hopping)h(is,)h (ob)n(viously)-7 b(,)21 b(asso)r(ciated)g(with)h Fk(avoiding)29 b Fn(double)22 b(o)r(ccupation;)h(but)g(since)e(the)i(o)n(v)n(erlap)c (is)j(zero)f(b)r(oth)h(for)g(the)-150 1101 y(creation)i(and)h (destruction)g(of)h(a)f(doubly)g(o)r(ccupied)g(site,)h(it)g(has)f(the)h (curious)e(prop)r(ert)n(y)g(that)i(an)f(initially)g(doubly)h(o)r (ccupied)f(site,)-150 1200 y(ho)n(w)n(ev)n(er)g(small)i(its)h (probabilit)n(y)e(b)n(y)h(the)h(side)f(factors)g(in)g(\(2\),)h(cannot)f (ev)n(er)f(b)r(e)i(destro)n(y)n(ed.)35 b(The)27 b(regime)g(of)g (enhanced)g(o)n(v)n(erlaps,)-150 1300 y(Eq.)19 b(\(4\),)i(w)n(ould)e(b) n(y)h(con)n(trast)e(destro)n(y)g(it)i(v)n(ery)e(quic)n(kly)-7 b(,)21 b(so)e(it)h(is)f(asso)r(ciated)f(with)i(the)g(dynamics)f(of)g Fk(r)l(emoving)27 b Fn(double)20 b(o)r(ccupation,)-150 1399 y(as)32 b(discussed)h(in)g(the)g(In)n(tro)r(duction.)52 b(The)33 b(fact)g(that)g(b)r(oth)g(are)f(symmetric)g(with)i(resp)r(ect) e(to)h(creation)f(and)g(destruction)h(of)f(a)-150 1499 y(doubly)27 b(o)r(ccupied)h(site)g(is)f(a)g(consequence)g(of)h(the)g (requiremen)n(t)e(that)i(the)g(e\013ectiv)n(e)g(kinetic)f(op)r(erator)f (b)r(e)i(Hermitian.)-25 1634 y(No)n(w)f(I)h(c)n(ho)r(ose)e(to)i(study)g (only)f(one)g(of)h(these)g(regimes,)e(whic)n(h)i(giv)n(es)f(a)g(formal) g(impro)n(v)n(emen)n(t)f(to)i(Gutzwiller's)g(assumption,)-150 1734 y(but)23 b(also)e(falls)h(in)h(line)f(with)h(those)f(previous)f (approac)n(hes)f(to)i(the)h(sub)5 b(ject,)23 b(whic)n(h)f(assumed)g (the)h(dominan)n(t)f(b)r(eha)n(vior)f(w)n(as)g(similar)-150 1834 y(for)29 b(long)h(times,)g(and)g(lo)n(w)f(temp)r(eratures.)43 b(Since)31 b(the)f(aim)g(is)f(to)h(in)n(v)n(estigate)f(the)h(thermo)r (dynamics)f(of)h(the)g(normal)f(state,)i(the)-150 1933 y(op)r(erator)25 b(\(6\),)i(obtained)g(in)g(the)g(limit)h(of)f(long)f (times,)h(is)g(preferable,)f(esp)r(ecially)h(since)f(it)i(ob)n(viously) d(allo)n(ws)h(for)g(the)h(creation)f(of)-150 2033 y(long-w)n(a)n(v)n (elength)20 b(mo)r(des)i(b)n(y)g(delo)r(calization)g(o)n(v)n(er)f 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b(c)n(hoice)f(to)h(use)g(\(6\))g(instead)g(of)g Fi(K)30 b Fn(in)24 b(\(2\))g(amoun)n(ts)g(to)g(a)f(v)-5 b(ariational)23 b(assumption,)h(and)g(the)g(thermal)g(form)n(ulation)f (allo)n(ws)g(in)-150 2898 y(principle)29 b(for)f(an)h(in)n(v)n (estigation)f(of)g(the)i(asso)r(ciated)e(thermo)r(dynamics.)40 b(A)n(t)29 b(the)h(presen)n(t)e(state)h(of)g(the)g(art,)g(sev)n(eral)e (limitations)-150 2998 y(m)n(ust)h(b)r(e)g(accepted,)f(in)h(order)e(to) i(carry)e(out)h(this)h(program.)-25 3133 y(First,)21 b(it)f(is)g(easier)f(to)g(treat)h(the)g(three-band)f(Emery)g(mo)r(del,) 1906 3103 y Fm(17)1997 3133 y Fn(with)h(repulsion)f(only)g(on)h(the)g (copp)r(er)g(sites,)h(than)f(the)g(Hubbard)-150 3233 y(mo)r(del,)28 b(b)r(ecause)f(in)h(the)g(latter,)f(neigh)n(b)r(oring)f (sites)h(are)g(correlated,)f(whic)n(h)h(e\013ectiv)n(ely)h(in)n(tro)r (duces)f(a)g(b)r(ond)h(degree)e(of)h(freedom)-150 3333 y(in)i(statistical)f(sums.)40 b(So)29 b(from)f(no)n(w)g(on,)h(only)f 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y(is)e(driv)n(en)g(b)n(y)h(c)n(harge)e(order.)778 3936 y Fm(3,4)893 3966 y Fn(In)i(La)1091 3978 y Fm(2)p Fe(\000)p Ff(x)1217 3966 y Fn(Sr)1296 3978 y Ff(x)1337 3966 y Fn(CuO)1508 3978 y Fm(4)1545 3966 y Fn(,)g(magnetic)f(p)r(eaks)g (in)h(neutron)f(di\013raction)h(disapp)r(ear)e(ab)r(o)n(v)n(e)g Fd(\030)d Fn(100)j(K.)3927 3936 y Fm(4)-25 4102 y Fn(Third)k(and)g (most)g(imp)r(ortan)n(t,)g(the)h(Gutzwiller)f(appro)n(ximation)e(is)i (implemen)n(ted:)43 b(up-spins)30 b(see)g(do)n(wn-spins)f(as)g (in\014nitely)-150 4201 y(hea)n(vy)-7 b(,)30 b(and)g Fk(vic)l(e)k(versa)p Fn(.)46 b(This)30 b(is)h(an)f(`arti\014cial)f (limit',)j(as)e(noted)g(in)h(the)g(In)n(tro)r(duction,)g(and)f(reduces) g(the)h(problem)e(to)i(a)f(one-)-150 4301 y(particle)24 b(one,)g(but)h(of)f(a)g(sto)r(c)n(hastic)f(nature.)36 b(In)24 b(its)h(simplest)f(implemen)n(tation,)h(discussed)f(here,)g (one)g(puts)h(the)f(hopping)g(o)n(v)n(erlap)-150 4400 y(for)i(spins)g(of)g(one)g(kind)h(\(sa)n(y)-7 b(,)26 b(do)n(wn-spins\))f(to)i(zero)e(`b)n(y)h(hand',)h(and)f(so)f(obtains)h (a)g(three-band)g(v)-5 b(arian)n(t)25 b(of)h(the)h(F)-7 b(alico)n(v-Kim)n(ball)-150 4500 y(mo)r(del.)92 4470 y Fm(13)186 4500 y Fn(Then)24 b(the)g(other)g(kind,)g(up-spins,)h(see)e (a)h(plane,)g(randomly)f(tiled)h(with)g(CuO)2653 4512 y Fm(4)2714 4500 y Fn(`molecules',)g(o)n(v)n(er)e(whic)n(h)i(they)g (can)f(hop,)-150 4600 y(wherev)n(er)j(there)h(is)h(not)f(a)h(do)n (wn-spin,)f(whic)n(h)g(blo)r(c)n(ks)g(hopping)g(to)h(the)g(site)f(whic) n(h)h(it)g(o)r(ccupies.)-25 4735 y(In)n(terestingly)23 b(enough,)h(the)h(free)f(energy)f(of)h(this)h(random-tiling)e(problem)h (can)g(b)r(e)g(obtained)g(without)h(mean-\014eld)f(a)n(v)n(eraging)-150 4835 y(of)31 b(the)g(fermion)f(o)r(ccupation)h(n)n(um)n(b)r(er.)46 b(The)31 b(tric)n(k)f(is)g(to)h(in)n(tro)r(duce)f(a)h(com)n(binatorial) e(transform)g(of)i(the)g(sto)r(c)n(hastic)f(ev)n(olution)-150 4934 y(op)r(erator,)c(whic)n(h)h(factors)g(the)h(asso)r(ciated)e (generating)g(function)j(\(grand)d(canonical)h(sum\))h Fk(exactly)p Fn(.)-25 5070 y(V)-7 b(ery)27 b(sc)n(hematically)-7 b(,)27 b(this)h(is)g(ho)n(w)f(it)h(w)n(orks.)36 b(If)28 b(I)g(ha)n(v)n(e)f Fk(any)35 b Fn(list)28 b(of)f(ob)5 b(jects,)28 b Fi(U)2567 5082 y Ff(LM)2686 5070 y Fn(,)g(with)g Fi(M)k Fn(=)23 b(0)14 b Fi(:)g(:)g(:)f(L)p Fn(,)28 b(the)g(corresp)r (onding)-150 5169 y(generating)e(function)i(can)f(b)r(e)h(written)1294 5202 y Fh(X)1319 5380 y Ff(M)1428 5281 y Fi(x)1475 5247 y Ff(M)1549 5281 y Fi(U)1606 5293 y Ff(LM)1748 5281 y Fn(=)23 b(\(1)18 b(+)g Fi(x)p Fn(\))2090 5247 y Ff(L)2154 5202 y Fh(X)2197 5376 y Ff(p)2288 5281 y Fi(y)2332 5247 y Ff(p)2370 5281 y Fi(U)2427 5293 y Ff(L)2476 5281 y Fn(\()p Fi(p)p Fn(\))p Fi(;)1339 b Fn(\(8\))p eop %%Page: 5 5 5 4 bop 1387 1382 a Fc(r)1387 1146 y(r)1387 910 y(r)1387 674 y(r)1387 437 y(r)1387 201 y(r)1269 1382 y(e)1269 1146 y(e)1269 910 y(e)1269 674 y(e)1269 437 y(e)1269 201 y(e)1387 1500 y(e)1387 1264 y(e)1387 1028 y(e)1387 792 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y(The)c(third)i(prop)r(ert)n(y)d(of)i(\(8\),)g(most)g(imp)r(ortan) n(t)g(for)f(practical)g(calculation,)g(is)h(that)g(the)g Fi(U)2905 4330 y Ff(L)2955 4318 y Fn(\()p Fi(p)p Fn(\),)g(view)n(ed)f (as)h(functions)g(of)g(the)-150 4418 y Fi(A)-88 4430 y Ff(i)-60 4418 y Fn(,)35 b(ha)n(v)n(e)e(a)g(smo)r(oth)g(thermo)r (dynamic)h(limit,)i Fi(L)c Fd(!)h(1)p Fn(.)56 b(As)33 b(new)h(sites)g(are)e(added)i(to)f(the)h(lattice,)i(only)d(new)h(terms) f(app)r(ear)-150 4518 y(in)c(the)g Fi(U)149 4530 y Ff(L)199 4518 y Fn(\()p Fi(p)p Fn(\),)g(while)g(terms)g(referring)e(to)i(`old')g (sites)g(remain)f(unc)n(hanged.)40 b(This)28 b(means)h(c)n(hemical)f(p) r(oten)n(tials)h(can)f(b)r(e)h(used)g(to)-150 4617 y(extract)23 b(the)i(con)n(tribution)e(corresp)r(onding)f(to)i(a)g(giv)n(en)f (concen)n(tration,)h Fi(M)t(=L)p Fn(,)g(of)g(randomly)f(tiled)h (molecules.)35 b(By)24 b(con)n(trast,)g(the)-150 4717 y Fi(U)-93 4729 y Ff(LM)56 4717 y Fn(ha)n(v)n(e)29 b(no)i(thermo)r (dynamic)e(limit)j(at)e(all:)42 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b(op)r(erator,)g(has)g(a)h(one-particle)f(part,)h (whic)n(h)-150 5350 y(amoun)n(ts)j(to)h(a)f(tigh)n(t-binding)h (Hamiltonian,)g(with)g(hopping)g(to)f(as)g(man)n(y)h(nearest)f(neigh)n (b)r(ors,)f(as)h(there)h(are)f(terms)g(included)i(in)p eop %%Page: 6 6 6 5 bop -150 83 a Fn(the)33 b(expansion)e(in)i Fi(p)p Fn(.)50 b(The)33 b(diagonalization)d(of)i(this)h(Hamiltonian)f(giv)n (es)f(stationary)g(states)h(for)f(the)i(up-spins,)g(whic)n(h)g(retain) -150 183 y(information)j(ab)r(out)g(the)h(original)e(disorder,)i (displa)n(ying,)h(for)e(instance,)i(band)f(renormalization)d(as)i(a)g (function)h(of)f(molecule)-150 282 y(concen)n(tration,)23 b(since)f(their)h(e\013ectiv)n(e)g(parameters)f(dep)r(end)h(on)g(the)g (c)n(hemical)g(p)r(oten)n(tial)g Fi(\027)28 b Fn(of)23 b(the)h(do)n(wn-spins,)f(putting)g Fi(x)h Fn(=)e Fi(e)3968 252 y Ff(\014)s(\027)-150 382 y Fn(in)28 b(Eq.)f(\(8\).)37 b(In)28 b(particular,)e(a)h(Mott-Hubbard)h(transition)e(is)i(found.)-25 517 y(The)j(prescription)f(results)h(in)h(a)f(real)f(sp)r(ectrum,)i(b)r (ecause)f(the)h(free)f(energy)f(is)i(real,)f(and)g(since)g(Eq.)g(\(9\)) h(is)f(translationally)-150 617 y(in)n(v)-5 b(arian)n(t,)28 b(crystal)g(momen)n(tum)i(is)f(a)f(go)r(o)r(d)h(quan)n(tum)g(n)n(um)n (b)r(er,)g(and)g(there)g(is)g(a)g(Brillouin)f(zone.)41 b(Quan)n(tum)29 b(correlations)e(o)n(v)n(er)-150 717 y(larger)h(clusters)h(of)h(sites)g(are)f(systematically)g(tak)n(en)g (in)n(to)h(accoun)n(t,)g(but)h(here)e(I)h(stop)g(at)g(the)g(\014rst)g (non-trivial)f(order,)g Fi(p)e Fn(=)f(1)k(in)-150 816 y(Eq.)i(\(8\).)52 b(This)32 b(already)f(includes)h(the)h(most)g(imp)r (ortan)n(t)f(nearest-neigh)n(b)r(or)e(correlaton)g(b)r(ey)n(ond)i(the)h (Gutzwiller)g(assumption)-150 916 y(in)n(to)h(accoun)n(t,)h(b)r(ecause) e(hopping)h(is)g(forbidden)g(precisely)f(to)h(those)g(copp)r(er)f (sites,)j(on)e(whic)n(h)g(there)f(are)g(do)n(wn-spins,)i(rather)-150 1015 y(than)28 b(to)f(some)g(a)n(v)n(erage)e(n)n(um)n(b)r(er)i(of)h (sites,)f(giv)n(en)g(b)n(y)g(the)h(concen)n(tration)e(of)i(do)n (wn-spins.)600 1240 y Fo(4.)52 b(THERMOD)m(YNAMICS)34 b(OF)g(THE)g(MODEL)h(SPECTR)m(UM)-150 1383 y Fn(Once)28 b(an)h(e\013ectiv)n(e)f(one-particle)g(sp)r(ectrum)g(is)h(obtained,)g (it)g(is)f(easy)g(to)g(explore)g(the)h(asso)r(ciated)e(thermo)r (dynamics.)40 b(Since)29 b(the)-150 1482 y(basis,)h(diagonalizing)e (the)i(free)f(energy)g(op)r(erator,)g(is)g(a)h(unitary)f (transformation)f(of)i(the)g(initial)g(site)g(represen)n(tation,)f(it)h (could,)-150 1582 y(in)f(principle,)h(b)r(e)g(used)f(as)f(suc)n(h,)i (to)f(analyse)f(an)n(y)g(mo)r(del,)i(with)g(the)f(whole)g(presen)n(t)g (w)n(ork)e(pro)n(viding)h(only)h(the)h(motiv)-5 b(ation)29 b(for)-150 1682 y(its)j(particular)f(c)n(hoice.)49 b(Instead,)32 b(let)h(me)f(discuss)f(the)h(basis,)h(deriv)n(ed)e(from)g(\(8\),)j(as)d (a)g(mo)r(del)h(in)h(its)f(o)n(wn)f(righ)n(t.)49 b(I)32 b(call)f(it)i(the)-150 1781 y(`mo)r(del)28 b(of)f(annealed)g (disorder'.)-25 1917 y(A)j(Mott-Hubbard)h(transition)f(is)g(found,)i (as)e(a)g(function)h(of)f(concen)n(tration.)45 b(Bey)n(ond)29 b(the)i(transition,)g(when)g(the)g(c)n(hemical)-150 2016 y(p)r(oten)n(tial)38 b(is)g(in)g(a)f(narro)n(w)f(band)i(in)g(the)g(c)n (harge-transfer)d(gap,)40 b(there)e(is)f(a)h(quan)n(titativ)n(e)f (argumen)n(t)g(that)h(higher)f(terms)h(in)-150 2116 y(\(8\))f(cannot)g (c)n(hange)f(the)h(basic)g(picture)g(established)g(b)n(y)g(the)g(lo)n (w)n(er)f(ones,)j(except)e(for)g(suc)n(h)f(qualitativ)n(e)h(e\013ects,) j(as)c(app)r(ear)-150 2215 y(only)31 b(with)h(higher-order)d(hopping.) 47 b(Namely)-7 b(,)32 b(for)f(the)h(in-gap)e(band)h Fi(y)h Fd(\030)d Fn(exp\()p Fd(\000)p Fi(\014)t Fn(\001)2647 2227 y Ff(pd)2720 2215 y Fn(\),)k(with)f(\001)3070 2227 y Ff(pd)3174 2215 y Fn(the)g(bare)e(c)n(harge-transfer)-150 2315 y(gap)38 b(\(di\013erence)h(b)r(et)n(w)n(een)g(bare)f(copp)r(er)h (and)g(o)n(xygen)e(lev)n(els\),)42 b(p)r(ositiv)n(e)c(in)h(the)h(hole)e (picture.)71 b(On)39 b(the)h(other)e(hand,)k(eac)n(h)-150 2415 y Fi(U)-93 2427 y Ff(L)-44 2415 y Fn(\()p Fi(p)p Fn(\))27 b Fd(\030)f Fn(exp\()p Fi(\014)t Fd(j)p Fi(E)474 2427 y Ff(p)513 2415 y Fd(j)p Fn(\),)31 b(with)f Fi(E)874 2427 y Ff(p)942 2415 y Fn(the)g(\(negativ)n(e\))f(delo)r(calization)g (energy)-7 b(,)29 b(asso)r(ciated)f(with)i(h)n(ybridization)f(o)n(v)n (er)f(a)h(molecule)g(of)-150 2514 y Fi(p)f Fn(sites.)40 b(Since)28 b(the)h(latter)f(gain)g(saturates)f(with)i(the)g(size)f(of)h (the)g(molecule,)f(its)h(limit)g(b)r(eing)g(the)g(delo)r(calization)e (energy)g(of)i(the)-150 2614 y(whole)e(crystal,)g(the)h(exp)r(ense)f (of)h Fi(p)f Fn(times)h(the)g(bare)f(gap)g(so)r(on)g(cuts)g(o\013)h (the)g(higher)f(terms)g(in)h(\(8\),)g(unless)f(the)h(hopping)f(in)n (tegral)-150 2714 y Fi(t)h Fn(is)f(greater)f(than)i(\001)536 2726 y Ff(pd)609 2714 y Fn(,)g(whic)n(h)f(is)h(not)f(the)h(case)f(in)h (real)e(materials.)-25 2849 y(The)i(sp)r(ectrum)g(is)g(of)f(a)h(F)-7 b(ermi)28 b(liquid)g(b)n(y)g(construction,)f(since)h(only)f(the)i (one-particle)d(part)i(of)g(the)g(free)g(energy)f(w)n(as)g(tak)n(en.) -150 2949 y(Th)n(us)h(it)g(cannot)g(break)f(do)n(wn)h(formally)-7 b(,)27 b(but)i(one)f(can)f(lo)r(ok)h(for)f(signs)h(of)g(instabilit)n(y) -7 b(,)28 b(or)f(departures)g(from)h(free)g(b)r(eha)n(vior.)37 b(F)-7 b(or)-150 3048 y(one,)31 b(no)f(signals)g(of)h(phase)f (separation)f(ha)n(v)n(e)g(b)r(een)i(found:)44 b(c)n(hemical)30 b(p)r(oten)n(tials)g(are)g(monotonically)f(increasing)g(functions)i(of) -150 3148 y(concen)n(tration.)k(Ho)n(w)n(ev)n(er,)23 b(the)i(mo)r(del)g(b)r(eha)n(v)n(es)f(di\013eren)n(tly)h(in)g(the)g(lo) n(w)n(er)f(Hubbard)g(band,)i(whic)n(h)f(app)r(ears)e(for)i(concen)n (trations)-150 3247 y(b)r(elo)n(w)k(the)h(Mott-Hubbard)f(transition,)g (and)g(in)g(the)h(in-gap)e(band,)i(whic)n(h)f(lies)g(in)h(the)f(c)n (harge-transfer)d(gap,)j(separated)f(b)n(y)h(an)-150 3347 y(energy)130 3326 y(~)117 3347 y(\001)186 3359 y Ff(pd)284 3347 y Fn(=)373 3278 y Fh(p)p 456 3278 538 4 v 69 x Fn(\(\001)557 3359 y Ff(pd)631 3347 y Fi(=)p Fn(2\))747 3323 y Fm(2)802 3347 y Fn(+)18 b(4)p Fi(t)957 3323 y Fm(2)1013 3347 y Fd(\000)h Fn(\001)1166 3359 y Ff(pd)1239 3347 y Fi(=)p Fn(2)27 b(from)i(the)g(p)r(osition)g(of)f(the) h(bare)f(o)n(xygen)g(lev)n(el.)39 b(The)29 b(F)-7 b(ermi)29 b(liquid)g(of)g(the)g(in-gap)-150 3447 y(band)35 b(is)h(e\013ectiv)n (ely)f(free,)i Fk(i.e.)p Fn(,)i(it)d(is)f(a)g(F)-7 b(ermi)36 b(gas,)g(while)g(in)f(the)h(lo)n(w)n(er)e(Hubbard)h(band,)i(it)f(is)g (a)f(liquid,)i(judging)f(from)f(the)-150 3546 y(substan)n(tial)27 b(con)n(tribution)g(of)h(the)f(in)n(teraction)g(to)h(thermo)r(dynamic)f (functions.)-25 3682 y(This)32 b(is)g(not)h(to)f(sa)n(y)f(that)i(the)g (parameters)e(of)h(said)g(F)-7 b(ermi)33 b(gas)e(are)h(similar)f(to)i (the)g(bare)e(ones.)51 b(Indeed,)34 b(with)f(resp)r(ect)f(to)-150 3781 y(quan)n(titativ)n(e)f(v)-5 b(alues,)32 b(the)g(similarit)n(y)e (to)i(the)g(non-in)n(teracting)e(case)g(is)i(precisely)e(rev)n(ersed:) 43 b(the)32 b(lo)n(w)n(er)e(Hubbard)h(band)h(has)e(a)-150 3881 y(bandwidth)d(and)f(e\013ectiv)n(e)h(mass)e(whic)n(h)i(di\013er)f (but)h(little)g(from)g(the)f(bare)g(v)-5 b(alues)26 b(\(although)g (\014nitely)h(little\),)h(while)e(the)h(free)f(gas)-150 3981 y(in)j(the)f(in-gap)g(band)g(can)g(consist)g(of)g(particles)g(so)g (hea)n(vy)-7 b(,)27 b(that)i(the)g(bandwidth)f(b)r(ecomes)g(less)g (than)h(the)g(temp)r(erature,)f(lik)n(e)g(in)-150 4080 y(a)f(Brinkmann-Rice)g(transition,)g(with)h Fi(t=)p Fn(\001)1272 4092 y Ff(pd)1372 4080 y Fn(the)g(con)n(trol)e(parameter.)-25 4216 y(The)39 b(source)g(of)g(suc)n(h)h(statemen)n(ts)f(is)h(the)g(en)n (trop)n(y)e(of)i(the)g(system,)i(whic)n(h)e(can)f(b)r(e)h(analytically) f(separated)f(in)n(to)h(three)-150 4315 y(con)n(tributions,)d(the)g(en) n(trop)n(y)e(of)h(mobile)g(spins)g Fi(S)1493 4327 y Fe(")1531 4315 y Fn(,)i(of)e(static)g(spins)g Fi(S)2194 4327 y Fe(#)2232 4315 y Fn(,)i(and)e(the)h(con)n(tribution)e(to)h(the)h(en)n (trop)n(y)d(due)j(to)f(the)-150 4415 y(in)n(teraction)h(b)r(et)n(w)n (een)h(the)h(t)n(w)n(o,)g Fi(S)1002 4427 y Fm(in)n(t)1086 4415 y Fn(.)65 b(The)37 b(last)g(part)g(is)g(calculated)f(from)h(the)g (temp)r(erature)g(dep)r(endence)h(of)f(the)g(e\013ectiv)n(e)-150 4514 y(sp)r(ectrum,)1574 4653 y Fi(S)1625 4665 y Fm(in)n(t)1732 4653 y Fd(/)1819 4540 y Fh(Z)1866 4728 y Ff(B)s(Z)1986 4653 y Fi(f)2027 4665 y Ff(\026)2071 4653 y Fn(\()p Fi(")p Fn(\))2195 4597 y Fi(@)5 b(")p 2184 4634 109 4 v 2184 4710 a(@)g(T)2303 4653 y(;)1576 b Fn(\(10\))-150 4854 y(and)29 b(m)n(ust)h(b)r(e)f(negativ)n(e,)g(since)g(in)n(teractions)f (order)g(the)i(system.)1980 4824 y Fm(18)2079 4854 y Fn(It)g(turns)f(out)g(that)h(it)g(giv)n(es)e(a)g(substan)n(tial)h(con)n (tribution)g(in)-150 4954 y(the)e(lo)n(w)n(er)e(Hubbard)h(band,)h(of)g (the)g(order)e(of)i Fd(\000)p Fn(0)p Fi(:)p Fn(2)e Fi(k)1571 4966 y Ff(B)1655 4954 y Fn(p)r(er)h(site,)h(and)g(easily)f(greater)e (than)j Fi(S)2887 4966 y Fe(")2925 4954 y Fn(,)g(so)f(that)h(only)f (the)h(total)f(en)n(trop)n(y)-150 5053 y Fi(S)-99 5065 y Fe(")-47 5053 y Fn(+)14 b Fi(S)83 5065 y Fe(#)136 5053 y Fn(+)g Fi(S)266 5065 y Fm(in)n(t)375 5053 y Fn(is)25 b(p)r(ositiv)n(e.)36 b(On)25 b(the)h(other)f(hand,)h(in)g(the)g(in-gap) e(band)i Fi(S)2260 5065 y Fm(in)n(t)2369 5053 y Fn(is)g(negligible,)f (so)g(the)h(mobile)f(spins)h(are)e(decoupled)-150 5153 y(from)j(the)h(static)g(ones,)f(hence)g(a)h(`gas'.)p eop %%Page: 7 7 7 6 bop -25 83 a Fn(A)27 b(similar)g(conclusion)g(ma)n(y)g(b)r(e)h(dra) n(wn)f(from)g(the)h(linear)f(sp)r(eci\014c)g(heat)h(co)r(e\016cien)n (t,)1606 302 y Fi(\015)5 b Fn(\()p Fi(T)12 b Fn(\))22 b(=)1889 185 y Fh(\022)1960 246 y Fi(@)5 b(S)2060 258 y Fe(")p 1960 283 138 4 v 1974 359 a Fi(@)g(T)2108 185 y Fh(\023)2169 385 y Ff(V)t(;N)3902 302 y Fn(\(11\))-150 541 y(whic)n(h)25 b(is)f(a)h(bulk)g(measure)f(of)g(the)i(e\013ectiv)n (e)e(mass)g(of)h(the)g(mobile)g(spins)g(\()p Fi(S)2273 553 y Fe(#)2336 541 y Fn(and)g Fi(S)2546 553 y Fm(in)n(t)2654 541 y Fn(giv)n(e)f(negligible)g(con)n(tributions)g(to)h Fi(\015)5 b Fn(\).)36 b(F)-7 b(or)-150 641 y(a)27 b(non-in)n(teracting) f(system,)i(a)f(simple)h(scaling)e(holds,)i(whenev)n(er)e(the)i(bare)f (hopping)g(o)n(v)n(erlap)f Fi(t)h Fn(is)h(c)n(hanged)e(to)i Fi(t)3530 611 y Fe(0)3553 641 y Fn(:)1620 815 y Fi(\015)1663 827 y Ff(t)1688 810 y Fa(0)1714 815 y Fn(\()p Fi(T)12 b Fn(\))23 b(=)g Fi(c\015)2029 827 y Ff(t)2058 815 y Fn(\()p Fi(T)9 b(=c)p Fn(\))p Fi(;)1621 b Fn(\(12\))-150 988 y(where)27 b Fi(c)h Fn(is)f(a)g(dimensionless)g(n)n(um)n(b)r(er.)37 b(The)28 b(bulk)g(e\013ectiv)n(e)f(mass)g(of)h(the)g(annealed)f (disorder)f(mo)r(del)i(ob)r(eys)f(the)h(same)f(form)n(ula)-150 1088 y(in)35 b(the)f(in-gap)g(band,)i(while)e(it)h(fails)f(to)g(do)g (so)g(in)g(the)h(lo)n(w)n(er)e(Hubbard)h(band.)57 b(Quan)n(titativ)n (ely)-7 b(,)35 b(the)f(similarit)n(y)g(to)g(the)h(non-)-150 1188 y(in)n(teracting)h(case)g(is)g(again)g(rev)n(ersed:)53 b(the)37 b(actual)f Fk(values)46 b Fn(of)36 b Fi(\015)5 b Fn(\()p Fi(T)12 b Fn(\))36 b(in)h(the)g(lo)n(w)n(er)f(Hubbard)g(band) h(are)f(v)n(ery)f(similar)h(to)h(the)-150 1287 y(non-in)n(teracting)29 b(ones,)i(while)f(the)h(in-gap)f(band)g(sho)n(ws)g(a)g(signi\014can)n (tly)f(enhanced)i(sensitivit)n(y)f(to)g(the)h(bare)f(parameters.)44 b(F)-7 b(or)-150 1387 y(instance,)29 b(if)g(\001)344 1399 y Ff(pd)442 1387 y Fn(=)c(3)j(eV,)h(a)f(c)n(hange)g(from)g Fi(t)d Fn(=)f(1)29 b(eV)g(to)f Fi(t)1771 1357 y Fe(0)1820 1387 y Fn(=)c(0)p Fi(:)p Fn(2)k(eV)h(giv)n(es)e(an)i(enhancemen)n(t)f Fi(c)d Fd(\030)g Fn(10)j(in)g(the)i(non-in)n(teracting)-150 1486 y(case,)d(while)h Fi(c)23 b Fd(\030)f Fn(230)k(\(!\))38 b(in)28 b(the)g(in-gap)e(band)i(of)g(the)g(annealed)e(disorder)h(mo)r (del.)-25 1620 y(The)36 b(Mott-Hubbard)g(transition)g(o)r(ccurs)f(at)i (an)f(incommensurate)f(\014lling,)k(near,)f(but)f(not)f(quite)h(at,)i (half-\014lling.)3782 1589 y Fm(19)3888 1620 y Fn(This)-150 1719 y(is)33 b(p)r(ossible)f(b)r(ecause)g(particle-hole)f(symmetry)h (is)h(brok)n(en)e(b)n(y)i(a)f(large)f Fi(U)42 b Fn(in)32 b(the)h(presence)f(of)h(an)f(o)n(xygen)f(band.)52 b(The)33 b(mo)r(del)-150 1819 y(predicts)23 b(this)h(in)f(a)g(v)n(ery)f(robust)h (w)n(a)n(y)-7 b(,)23 b(dep)r(ending)h(only)f(on)g(the)h(form)f(of)g (the)h(equations)e(for)h(the)h(c)n(hemical)e(p)r(oten)n(tial,)i(and)g (on)f(the)-150 1918 y(fact)k(that)g(the)g(band-width)h(of)e(the)i (in-gap)e(band)g(narro)n(ws,)f(as)h(the)i(concen)n(tration)d(of)i(do)n (wn-spins)f(increases)f(\()p Fk(i.e.)p Fn(,)k(the)e(n)n(um)n(b)r(er) -150 2018 y(of)h(molecules)f(o)n(v)n(er)e(whic)n(h)j(the)g(up-spins)f (hop)h(decreases\).)-25 2151 y(Another)h(robust)f(e\013ect,)i(this)g (time)f(of)h(the)f(Gutzwiller)g(appro)n(ximation,)f(is)h(that)h(the)f (width)h(of)f(the)h(in-gap)e(band)h(do)r(es)g(not)-150 2251 y(c)n(hange)h(with)i(the)g(doping.)49 b(This)31 b(is)h(v)n(ery)e(simple)i(to)f(understand:)45 b(if)32 b(the)g(disp)r(ersion)f(of)g(up-spins)g(dep)r(ends)h(on)g(the)g(c)n (hemical)-150 2350 y(p)r(oten)n(tial)26 b(of)g(do)n(wn-spins,)f Fi(")e Fn(=)g Fi(")p Fn(\()p Fi(\027)5 b Fn(\),)27 b(then)f(it)h(will)f (dep)r(end)h(on)e(the)i(concen)n(tration)d(insofar)h(as)h(the)g(c)n (hemical)g(p)r(oten)n(tial)f(c)n(hanges)-150 2450 y(with)33 b(concen)n(tration.)51 b(But)33 b(in)g(the)g(Gutzwiller)g(appro)n (ximation,)f(the)h(`other')f(spins)h(o)r(ccup)n(y)f(a)g(band)h(of)g (width)g(zero,)g(so)f(their)-150 2550 y(c)n(hemical)27 b(p)r(oten)n(tial)h(do)r(es)f(not)g(c)n(hange)g(with)h(the)g(doping)f (throughout)g(the)h(in-gap)f(band.)-25 2683 y(Finally)-7 b(,)29 b(the)g(Mott-Hubbard)g(transition)f(itself)h(is)g(\014rst-order) e(in)i(the)h(doping.)40 b(It)29 b(is)g(accompanied)f(b)n(y)h(a)f(jump)i (in)f(en)n(trop)n(y)-7 b(,)-150 2782 y(\001)p Fi(S)28 b Fn(=)23 b(\001)p Fi(S)206 2794 y Fe(")244 2782 y Fn(,)h(wholly)e(due) i(to)f(the)g(mobile)g(spins.)35 b(They)23 b(seem)g(to)g(su\013er)f(a)h (kind)g(of)g(melting)g(transition,)g(accompanied)f(b)n(y)h(a)g(rapid) -150 2882 y(disapp)r(earance)e(of)i Fi(S)524 2894 y Fm(in)n(t)608 2882 y Fn(,)h(the)f(con)n(tribution)f(of)h(the)g(in)n(teraction.)35 b(The)23 b(same)f(transition)g(cannot)h(b)r(e)g(crossed)e(in)j(temp)r (erature:)34 b(the)-150 2982 y(discon)n(tin)n(uities)26 b(in)h(the)g(en)n(trop)n(y)e(and)h(c)n(hemical)g(p)r(oten)n(tial)h(are) e(parallel)h(to)g(the)h(temp)r(erature)f(axis)g(in)g(the)h(temp)r (erature-doping)-150 3081 y(plane.)36 b(The)27 b(bulk)g(e\013ectiv)n(e) g(mass)f(sho)n(ws)g(no)h(sign)f(of)h(div)n(erging,)e(as)h(the)i (transition)e(is)g(approac)n(hed.)35 b(This)27 b(is)g(easy)f(to)g (establish,)-150 3181 y(b)r(ecause)h(it)i(o)r(ccurs)e(sligh)n(tly)g(a)n (w)n(a)n(y)f(from)h(half-\014lling,)h(and)f(so)h(the)g(div)n(ergence)e (of)i Fi(\015)5 b Fn(,)27 b(due)h(to)g(the)g(v)-5 b(an)28 b(Ho)n(v)n(e)f(singularit)n(y)-7 b(,)26 b(is)i(k)n(ept)-150 3281 y(distinct.)45 b(It)31 b(is)f(in)h(k)n(eeping)e(with)i(the)g (nature)e(of)i(a)e(\014rst-order)g(transition,)h(where)g(an)g(exc)n (hange)e(of)j(minima)f(do)r(es)g(not)g(require)-150 3380 y(the)e(mo)r(des)f(around)g(them)h(to)g(soften.)1558 3599 y Fo(5.)53 b(DISCUSSION)-150 3739 y Fn(The)31 b(ph)n(ysical)g (assumption,)h(on)f(whic)n(h)g(this)h(w)n(ork)d(is)j(based,)g(is)f (that)g(a)n(w)n(a)n(y)f(from)h(half-\014lling,)h(c)n(harge)d (correlations)g(dominate)-150 3839 y(spin)35 b(correlations)e(in)i(the) h(cuprates.)58 b(This)35 b(seems)f(to)h(b)r(e)h(b)r(orne)e(out)h(in)h (the)f(normal)f(state,)2939 3809 y Fm(4)3011 3839 y Fn(and)h(is)g(also) f(indicated)h(b)n(y)g(the)-150 3938 y(decomp)r(osition)i(\(2\),)j(to)e (whic)n(h)g(the)g(dominan)n(t)f(correction)f(is)h(pro)5 b(jected)37 b(hopping,)j Fk(i.e.)69 b Fn(a)37 b(c)n(harge)f (correlation,)i(while)g(spin)-150 4038 y(correlations)24 b(are)h(due)h(to)g(comm)n(utators)f(in)h(whic)n(h)g(the)h(kinetic)f(op) r(erator)e(app)r(ears)h(t)n(wice,)i(so)e(they)h(are)g(smaller)f(b)n(y)g Fi(t=U)35 b Fn(at)26 b(eac)n(h)-150 4138 y(order)g(in)i Fi(\014)t Fn(.)-25 4271 y(The)d(attempt)h(at)f(a)g(complete,)g (self-con)n(tained)g(calculation)f(failed)h(on)g(t)n(w)n(o)g(coun)n (ts,)g(one)g(of)g(them)h(wholly)e(unexp)r(ected.)37 b(This)-150 4370 y(w)n(as)30 b(the)h(essen)n(tially)f(non-p)r(erturbativ)n(e)f(c)n (haracter)g(of)h(the)h(transition)g(amplitude)g(\(2\),)g(once)g(the)g (dominan)n(t)f(corrections)f(w)n(ere)-150 4470 y(resummed)e(to)g (in\014nite)h(order)e(in)h(the)h(in)n(v)n(erse)d(temp)r(erature.)37 b(The)27 b(c)n(hoice)f(of)h(lo)r(oking)g(at)g(the)g(pro)5 b(jected)27 b(hopping)g(regime)f(alone)-150 4570 y(m)n(ust)36 b(b)r(e)g(regarded)e(as)h(uncon)n(trolled,)i(though)e(it)h(migh)n(t)g (b)r(e)g(argued)e(that)i(the)g(limit)h Fi(U)9 b(T)47 b Fd(!)37 b(1)p Fn(,)g(in)f(whic)n(h)g(it)g(w)n(as)f(deriv)n(ed,)-150 4669 y(includes)j(all)f(but)h(the)g(shortest)f(times,)k(if)d Fi(U)46 b Fn(is)38 b(large.)65 b(Also,)40 b(the)e(large)e(enhancemen)n (t)i(of)f(the)h(hopping)g(o)n(v)n(erlap,)g(Eq.)f(\(4\),)-150 4769 y(requires)30 b(double)h(o)r(ccupation,)h(so)f(there)g(is)h Fk(a)g(priori)40 b Fn(a)31 b(small)g(probabilit)n(y)f(that)i(it)g(will) g(b)r(e)g(realized,)f(if)h Fi(U)40 b Fn(is)31 b(repulsiv)n(e;)i(while) -150 4868 y(the)e(pro)5 b(jected)30 b(hopping)g(\(6\))h(easily)e (acquires)h(some)g(`a)n(v)n(erage')d(v)-5 b(alue.)46 b(Nonetheless,)31 b(the)g(qualitativ)n(e)f(di\013erence)g(b)r(et)n(w)n (een)h(the)-150 4968 y(b)r(eha)n(vior)j(on)h(the)h(real)f(and)g (imaginary)f(time)i(axes)e(remains)h(a)g(c)n(hallenge,)h(esp)r(ecially) f(since)g(mean-\014eld)g(and)h(diagrammatic)-150 5068 y(approac)n(hes)29 b(necessarily)h(giv)n(e)g(the)i(same)f(ph)n(ysics)g (along)f(b)r(oth)h(axes:)44 b(for)31 b(them,)i(long-time)e(and)g(lo)n (w-temp)r(erature)f(b)r(eha)n(viors)-150 5167 y(are)d(similar)f(b)n(y)i (construction.)-25 5300 y(The)g(other)f(uncon)n(trolled)g(step)h(is)g (the)h(Gutzwiller)f(appro)n(ximation,)e(whic)n(h)i(is)g(more)f(con)n(v) n(en)n(tional.)37 b(In)28 b(the)h(presen)n(t)e(form)n(u-)-150 5400 y(lation,)f(it)f(reduces)g(the)h(ph)n(ysics)f(of)g(electron)g (transp)r(ort)f(in)i(the)g(presence)e(of)i(a)f(con)n(tact)f(repulsion)h (to)g(a)g(random-tiling)g(problem.)p eop %%Page: 8 8 8 7 bop -150 83 a Fn(As)31 b(discussed)g(in)g(the)g(In)n(tro)r (duction,)h(`slo)n(w)e(mo)r(des',)i(arti\014cially)e(imp)r(osed,)i(are) e(ubiquitous)h(in)h(computationally)e(feasible)h(ap-)-150 183 y(proac)n(hes)d(to)i(the)g(problem.)43 b(A)31 b(conclusiv)n(e)d (demonstration)h(that)h(they)h(exist,)f(or)f(do)h(not)g(exist,)g(w)n (ould)f(b)r(e)i(an)e(imp)r(ortan)n(t)h(step)-150 282 y(forw)n(ard)c(in)i(understanding)f(strong)f(lo)r(cal)h(repulsion)g (sim)n(ultaneously)g(in)g(space)g(and)h(time.)-25 418 y(On)j(the)h(brigh)n(t)f(side,)i(the)f(random)f(tiling)h(giv)n(es)e(a)i (transparen)n(t)e(in)n(terpretation)h(of)g(the)h(appro)n(ximation)e(in) i(real)f(space.)49 b(It)-150 517 y(amoun)n(ts)22 b(to)h(sa)n(ying)f (that)h(the)h(t)n(w)n(o)e(kinds)h(of)g(fermions)f(ha)n(v)n(e)g (established)h(a)f(p)r(ercolation)g(net)n(w)n(ork)g(in)h(order)f(to)h (a)n(v)n(oid)e(eac)n(h)i(other,)-150 617 y(suc)n(h)28 b(that)h(the)g(net)n(w)n(ork)e(mo)n(v)n(es)g(m)n(uc)n(h)h(more)g(slo)n (wly)f(than)i(it)g(tak)n(es)e(an)h(individual)h(electron)f(to)g(cross)f (it.)40 b(The)28 b(Mott-Hubbard)-150 717 y(transition)h(is)h(then)g(an) f(order-disorder)e(transition)i(in)h(this)g(net)n(w)n(ork,)e(to)i(a)n (v)n(oid)e(the)i(tendency)g(to)g(crystallize,)f(as)g(half-\014lling)g (is)-150 816 y(approac)n(hed)c(in)j(the)g(lo)n(w)n(er)e(Hubbard)h (band.)37 b(Since)27 b(all)g(this)h(is)f(happ)r(ening)g(in)h(the)g (clean)f(limit,)h(the)f(`disorder')f(b)r(eing)i(the)g(other)-150 916 y(electrons,)k(there)f(is)h(a)f(Brillouin)g(zone,)h(enforced)f(b)n (y)g(the)h(Blo)r(c)n(h)f(theorem,)i(but)f(the)g(ph)n(ysics)f(of)g(a)n (v)n(oiding)f(double)i(o)r(ccupation)-150 1015 y(is)38 b(in)g(real)f(space.)67 b(Actual)38 b(crystallization,)g Fk(i.e.)70 b Fn(static)37 b(order)g(at)h(lo)n(w)f(temp)r(erature,)j (requires)c(an)i(additional)f(correlation,)-150 1115 y(and)32 b(ob)n(viously)e(magnetic)h(in)n(teractions)g(are)g(go)r(o)r (d)g(candidates)g(for)g(pro)n(viding)f(it.)50 b(It)32 b(is)g(also)f(in)n(teresting)g(that)h(a)f(fairly)g(direct)-150 1215 y(exp)r(erimen)n(tal)f(consequence)f(of)h(the)g(Gutzwiller)g (appro)n(ximation)e(has)i(b)r(een)g(deduced,)h(namely)f(that)g(the)h (width)g(of)f(the)g(in-gap)-150 1314 y(band)e(should)f(b)r(e)h(indep)r (enden)n(t)g(of)g(doping.)-25 1450 y(The)h(e\013ectiv)n(e)h(fermion)f (mo)r(del,)h(or)f(basis,)g(obtained)h(in)f(this)h(w)n(ork,)f(has)g(no)g (spurious)g(degrees)f(of)i(freedom.)42 b(It)30 b(emphasizes)-150 1549 y(short-range)23 b(c)n(harge)g(correlations,)h(rather)g(lik)n(e)h (the)g Fk(unc)l(ondense)l(d)33 b Fn(sla)n(v)n(e)24 b(b)r(oson.)2471 1519 y Fm(20)2566 1549 y Fn(I)h(ha)n(v)n(e)f(k)n(ept)h(only)g(the)h (shortest,)f(at)g(the)h(lev)n(el)-150 1649 y(of)d(one)f(molecule,)h (but)h(an)e(extension)g(to)h(longer)e(ranges,)h(b)n(y)h(including)g (larger)d(molecules,)k(is)e(straigh)n(tforw)n(ard.)32 b(It)23 b(w)n(ould)g(include)-150 1749 y(additional)g(e\013ects)h(of)f (quan)n(tum)h(delo)r(calization,)f(so)g(the)h(presen)n(t)f(form)n (ulation)f(is)i(exp)r(ected)g(to)f(o)n(v)n(eremphasize)e(the)j (detrimen)n(tal)-150 1848 y(e\013ects)j(of)g(c)n(harge)e(correlations)g (on)h(hopping.)37 b(F)-7 b(or)26 b(instance,)h(if)g Fi(t)c Fn(=)g(1)j(eV)h(and)g(\001)2502 1860 y Ff(pd)2598 1848 y Fn(=)c(3)j(eV,)i(one)e(obtains)g(an)h(e\013ectiv)n(e)g(copp)r(er-) -150 1948 y(o)n(xygen)d(o)n(v)n(erlap)f Fi(t)443 1960 y Ff(pd)539 1948 y Fn(=)g(0)p Fi(:)p Fn(2)h(eV)i(in)g(the)f(in-gap)g (band,)h(while)f(a)g Fk(c)l(ondense)l(d)33 b Fn(sla)n(v)n(e)24 b(b)r(oson)h(calculation)f(giv)n(es)h Fi(t)3359 1960 y Ff(pd)3455 1948 y Fn(=)d(0)p Fi(:)p Fn(5)j(eV)h(at)f(10\045)-150 2047 y(hole)f(doping.)290 2017 y Fm(21)385 2047 y Fn(On)g(the)h(other)e (hand,)j(the)e(latter)g(is)h(exp)r(ected)g(to)f(underemphasize)g(the)h (same)e(correlations,)g(as)h(it)h(only)f(includes)-150 2147 y(them)31 b(through)e(a)h(global)g(constrain)n(t.)43 b(The)31 b(e\013ectiv)n(e)f(bandwidths)h(in)f(the)h(presen)n(t)f(metho) r(d,)h(suc)n(h)f(as)g Fd(\030)d Fn(0)p Fi(:)p Fn(4)j(eV)g(in)h(the)f (ab)r(o)n(v)n(e)-150 2247 y(case,)g(are)g(ab)r(out)g(the)h(observ)n(ed) e(v)-5 b(alues.)1168 2217 y Fm(22)1269 2247 y Fn(Note)30 b(that)h(these)g(bandwidths)f(ha)n(v)n(e)g(nothing)g(to)g(do)h(with)g (what)f(the)h(electron)f(sees)-150 2346 y(lo)r(cally)-7 b(,)33 b(where)e(it)h(alw)n(a)n(ys)e(h)n(ybridizes)h(b)n(y)h(the)h (full)f(bare)f(hopping)h(o)n(v)n(erlap,)f(if)h(an)g(electron)f(of)h (the)h(other)e(kind)h(is)g(absen)n(t;)i(the)-150 2446 y(e\013ectiv)n(e)28 b(o)n(v)n(erlaps)d(asso)r(ciated)h(with)i(the)g (in-gap)f(band)g(are)g(the)h(outcome)f(of)h(the)g(random)e(tiling)i(of) g(the)g(lo)r(cal)f(molecules.)-25 2581 y(A)h(\014nal)g(prop)r(ert)n(y)f (of)h(the)g(e\013ectiv)n(e)g(mo)r(del)g(is)g(that)g(its)g(deriv)-5 b(ation)28 b(is)f(completely)h(constrained.)37 b(F)-7 b(or)28 b(instance,)f(all)h(e\013ectiv)n(e)-150 2681 y(hopping)i(o)n(v)n(erlaps,)g(up)h(to)f(an)n(y)g(n)n(um)n(b)r(er)h(of)g (nearest)e(neigh)n(b)r(ors,)i(are)f(\014xed)g(in)h(adv)-5 b(ance)30 b(b)n(y)h(the)g(form)f(\(8\).)47 b(This)31 b(is)f(imp)r(ortan)n(t)-150 2781 y(in)e(ligh)n(t)f(of)g(the)h(fact)g (that)f(e\013ectiv)n(e)h(mo)r(dels)f(with)h(second,)f(and)g(third,)h (neigh)n(b)r(or)e(hopping)h(seem)g(necessary)f(to)h(mo)r(del)h (realistic)-150 2880 y(disp)r(ersions.)270 2850 y Fm(23)377 2880 y Fn(V)-7 b(arious)36 b(generalizations,)i(suc)n(h)f(as)g (treating)f(b)r(oth)i(kinds)g(of)f(spins)g(symmetrically)-7 b(,)40 b(or)c(allo)n(wing)g(for)h(lattice)-150 2980 y(e\013ects,)k(all) d(ha)n(v)n(e)f(their)h(place)g(within)h(the)f(same)g(framew)n(ork,)h (again)d(de\014ning)j(the)f(corresp)r(onding)e(mo)r(dels)i(precisely)-7 b(,)40 b(at)e(a)-150 3079 y(comparable)27 b(lev)n(el)g(of)h(tec)n (hnical)g(di\016cult)n(y)-7 b(.)39 b(Suc)n(h)28 b(formal)f(rigidit)n(y) h(is,)g(in)g(principle,)h(desirable,)e(as)g(long)h(as)f(it)i(do)r(es)f (not)g(in)n(v)n(olv)n(e)-150 3179 y(to)r(o)22 b(great)g(a)g(compromise) f(in)i(computational)f(\015exibilit)n(y)-7 b(.)35 b(So)22 b(far,)i(exp)r(erience)e(with)h(the)g(basis,)g(or)e(mo)r(del,)j(of)f (annealed)e(disorder,)-150 3279 y(indicates)27 b(it)h(to)g(b)r(e)g(a)f (viable)g(to)r(ol)h(for)f(in)n(v)n(estigating)f(the)i(thermo)r (dynamics)f(of)g(strong)g(lo)r(cal)g(repulsion.)1363 3503 y Fo(A)m(CKNO)m(WLEDGMENTS)-150 3646 y Fn(The)f(author)f(w)n(ould) g(lik)n(e)h(to)f(thank)h(S.)g(Bari)l(\024)-37 b(si)n(\023)e(c)22 b(and)k(E.)f(T)-7 b(uti)l(\024)-37 b(s)26 b(for)f(in)n(teresting)g (discussions.)35 b(This)26 b(w)n(ork)e(has)i(b)r(een)g(supp)r(orted)f (b)n(y)-150 3746 y(the)j(Croatian)e(Go)n(v)n(ernmen)n(t)g(through)h (Pro)5 b(ject)27 b(No.)g(119204.)1594 3970 y Fo(REFERENCES)-85 4106 y Fn(1.)41 b(Z.)d(X.)g(Shen,)k(W.)c(E.)g(Spicer,)j(D.)d(M.)h (King,)h(D.)f(S.)f(Dessau,)j(and)d(B.)g(O.)g(W)-7 b(ells,)41 b(\\Photo)r(emission)36 b(studies)j(of)f(high-T)4017 4118 y Ff(c)21 4205 y Fn(sup)r(erconductors:)d(the)28 b(sup)r(erconducting)f(gap,")g Fk(Scienc)l(e)h Fl(267)p Fn(,)e(pp.)i(343{50,)d(1995.)-85 4313 y(2.)41 b(K.)h(Y)-7 b(amamoto,)45 b(T.)e(Katsufuji,)j(T.)d(T)-7 b(anab)r(e,)46 b(and)c(Y.)h(T)-7 b(okura,)45 b(\\Raman)c(scattering)h(of)g(the)h(c)n (harge-spin)e(strip)r(es)h(in)21 4413 y(La)115 4425 y Fm(1)p Ff(:)p Fm(67)237 4413 y Fn(Sr)316 4425 y Fm(0)p Ff(:)p Fm(33)439 4413 y Fn(NiO)589 4425 y Fm(4)626 4413 y Fn(,")27 b Fk(Phys.)k(R)l(ev.)e(L)l(ett.)g Fl(80)p Fn(,)e(pp.)h(1493{6,)d(1998.)-85 4521 y(3.)41 b(G.)f(Blum)n(b)r(erg,)i (M.)e(V.)h(Klein,)i(and)c(S.)h(W.)h(Cheong,)h(\\Charge)c(and)i(spin)g (dynamics)f(of)h(an)g(ordered)e(strip)r(e)i(phase)g(in)21 4620 y(La)115 4641 y Fm(1)157 4618 y Fb(2)p 157 4627 29 4 v 157 4661 a(3)199 4620 y Fn(Sr)288 4618 y Fb(1)p 288 4627 V 288 4661 a(3)330 4620 y Fn(NiO)480 4632 y Fm(4)545 4620 y Fn(in)n(v)n(estigated)27 b(b)n(y)g(Raman)g(sp)r (ectroscop)n(y)-7 b(,")25 b Fk(Phys.)31 b(R)l(ev.)f(L)l(ett.)e Fl(80)p Fn(,)f(pp.)h(564{7,)d(1998.)-85 4728 y(4.)41 b(J.)51 b(M.)h(T)-7 b(ranquada,)56 b(\\Charge)49 b(strip)r(es)j(and)f (an)n(tiferromagnetism)f(in)i(insulating)f(nic)n(k)n(elates)f(and)i (sup)r(erconducting)21 4828 y(cuprates,")26 b Fk(c)l(ond-mat)g Fl(9802043)p Fn(,)g(1998.)-85 4935 y(5.)41 b(I.)30 b(Isaac)f(and)h(J.)g (P)-7 b(.)30 b(F)-7 b(ranc)n(k,)30 b(\\Oxygen-isotop)r(e)e(e\013ect)j (on)f(the)g(c)n(harge-ordering)d(transition)i(of)h(La)3262 4947 y Fm(1)p Fe(\000)p Ff(x)3389 4935 y Fn(Ca)3491 4947 y Ff(x)3532 4935 y Fn(MnO)3719 4947 y Fm(3)3756 4935 y Fn(,")h Fk(Phys.)21 5035 y(R)l(ev.)c Fl(B57)p Fn(,)h(pp.)g(R5602{5,)d (1998.)-85 5143 y(6.)41 b(J.)28 b(A.)g(F)-7 b(ernandez-Baca,)27 b(P)-7 b(.)27 b(Dai,)i(H.)g(Y.)f(Hw)n(ang,)g(C.)g(Klo)r(c,)g(and)g (S.-W.)g(Cheong,)g(\\Ev)n(olution)f(of)h(the)g(lo)n(w-frequency)f(spin) 21 5242 y(dynamics)g(in)h(ferromagnetic)e(manganites,")g Fk(Phys.)k(R)l(ev.)g(L)l(ett.)e Fl(80)p Fn(,)f(pp.)h(4012{5,)d(1998.)p eop %%Page: 9 9 9 8 bop -85 83 a Fn(7.)41 b(N.)33 b(A.)h(Babushkina,)g(L.)f(M.)h(Belo)n (v)-5 b(a,)33 b(V.)h(I.)g(Ozhogin,)f(O.)g(Y.)h(Gorb)r(enk)n(o,)f(A.)h (R.)g(Kaul,)g(A.)g(A.)f(Bosak,)h(D.)f(I.)h(Khomskii,)21 183 y(and)29 b(K.)g(I.)g(Kugel,)h(\\Metal-insulator)d(transition)i (induced)h(b)n(y)2085 153 y Fm(16)2155 183 y Fn(O{)2262 153 y Fm(18)2331 183 y Fn(O)f(o)n(xygen)f(isotop)r(e)h(exc)n(hange)f (in)i(colossal)d(negativ)n(e)21 282 y(magnetoresistance)e(manganites,") h Fk(c)l(ond-mat)g Fl(9805315)p Fn(,)g(1998.)-85 390 y(8.)41 b(G.)24 b(A.)g(Levin,)h(T.)f(Stein,)i(C.)e(C.)g(Almasan,)g(S.)h (H.)f(Han,)h(D.)f(A.)h(Ga)5 b(jewski,)24 b(and)g(M.)g(B.)g(Maple,)h (\\Tw)n(o-dimensional)d(lo)r(caliza-)21 490 y(tion)31 b(and)g(sup)r(erconductor-insulator)d(transition)i(in)i(strongly)d 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