❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❯❜❡r❧â♥❞✐❛ ✲ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛
❉❛❧s♦♥ ❊❧♦②
❊♠❛r❛♥❤❛♠❡♥t♦ ❡ ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s
❡♠ ❝❛❞❡✐❛s ❞❡ s♣✐♥ ❡①❛t❛♠❡♥t❡ s♦❧ú✈❡✐s
❖r✐❡♥t❛❞♦r✿ ❏♦sé ❈â♥❞✐❞♦ ❳❛✈✐❡r
❯❜❡r❧â♥❞✐❛ ✲ ✷✵✶✸
❉❛❧s♦♥ ❊❧♦② ❆❧♠❡✐❞❛
❊♠❛r❛♥❤❛♠❡♥t♦ ❡ ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s
❡♠ ❝❛❞❡✐❛s ❞❡ s♣✐♥ ❡①❛t❛♠❡♥t❡ s♦❧ú✈❡✐s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡✲ ❞❡r❛❧ ❞❡ ❯❜❡r❧â♥❞✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❋ís✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❋ís✐❝❛ ❞❛ ▼❛tér✐❛ ❈♦♥❞❡♥s❛❞❛✴❋ís✐❝❛ ❊st❛tíst✐❝❛✳
❖r✐❡♥t❛❞♦r✿ ❏♦sé ❈â♥❞✐❞♦ ❳❛✈✐❡r
❯❜❡r❧â♥❞✐❛
❴
Dados Internacionais de Catalogação na Publicação (CIP) Sistema de Bibliotecas da UFU , MG, Brasil
A447e 2013
Almeida, Dalson Eloy, 1989-
Emaranhamento e propriedades críticas em cadeias de spin exata-mente solúveis / Dalson Eloy Almeida. - 2013.
96 f. : il.
Orientador: José Cândido Xavier.
Dissertação (mestrado) – Universidade Federal de Uberlândia, Pro- grama de Pós-Graduação em Física.
Inclui bibliografia.
1. 1. Física - Teses. 2. Matéria condensada - Teses. 3. Problema de muitos corpos - Teses. I. Xavier, José Cândido. II. Universidade Fede-ral de Uberlândia. Programa de Pós-Graduação em Física. III. Título.
❉❛❧s♦♥ ❊❧♦② ❆❧♠❡✐❞❛
❊♠❛r❛♥❤❛♠❡♥t♦ ❡ ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s
❡♠ ❝❛❞❡✐❛s ❞❡ s♣✐♥ ❡①❛t❛♠❡♥t❡ s♦❧ú✈❡✐s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡✲ ❞❡r❛❧ ❞❡ ❯❜❡r❧â♥❞✐❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❋ís✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❋ís✐❝❛ ❞❛ ▼❛tér✐❛ ❈♦♥❞❡♥s❛❞❛✴❋ís✐❝❛ ❊st❛tíst✐❝❛✳
❉❛t❛ ❞❡ ❛♣r♦✈❛çã♦✿ ✶✾ ❞❡ ❢❡✈❡r❡✐r♦ ❞❡ ✷✵✶✸
❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ❉r✳ ❏♦sé ❈â♥❞✐❞♦ ❳❛✈✐❡r ✲ ❖r✐❡♥t❛❞♦r ■◆❋■❙✴❯❋❯
Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♥♦ ❞✐ ▲♦r❡♥③♦ ■◆❋■❙✴❯❋❯
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ✐♥✈❡st✐❣❛♠♦s ♦ ♠♦❞❡❧♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❳❨✱ ❜❡♠ ❝♦♠♦ ✉♠❛ ❡s❝❛❞❛ ❞❡ s♣✐♥s ❞❡ ❞✉❛s✲♣❡r♥❛s ❝♦♠ ✐♥t❡r❛çã♦ ❞❡ três✲s♣✐♥s ❡①❛t❛♠❡♥t❡ s♦❧ú✈❡❧✳ ❖❜t❡♠♦s ❛♥❛❧✐t✐❝❛♠❡♥t❡ ❛s ❝♦rr❡✲ çõ❡s ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ❞❛s ❡♥❡r❣✐❛s ❡ ❞❡t❡r♠✐♥❛♠♦s ❛ ❝❛r❣❛ ❝❡♥tr❛❧✱ ❜❡♠ ❝♦♠♦ ❛s ❞✐♠❡♥sõ❡s ❞❡ ❡s❝❛❧❛✳ ❯t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦✱ t❛♠❜é♠ ❡st✉❞❛♠♦s ❛s ❝♦rr❡çõ❡s ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ❞❛ ❡♥tr♦♣✐❛ ❞❡ ❘é♥②✐ ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❞♦s ❡st❛❞♦s ❡①❝✐t❛❞♦s✳ ◆♦ss♦s r❡s✉❧t❛❞♦s ❡stã♦ ❡♠ ❛❝♦r❞♦ ❝♦♠ ❛s ♣r❡❞✐çõ❡s ❞❛ t❡♦r✐❛ ❝♦♥❢♦r♠❡ ❞❡ ❝❛♠♣♦s✳ P♦r ú❧t✐♠♦✱ ♠❛s ♥ã♦ ♠❡♥♦s ✐♠♣♦rt❛♥t❡✱ ♥♦ ❆♣ê♥❞✐❝❡ ❆✱ r❡s✉♠✐♠♦s ♦s ♣r✐♥❝✐♣❛✐s ♣❛ss♦s ❞❛ ❞✐❛❣♦♥❛❧✐③❛çã♦ ❞❛ ❤❛✲ ♠✐❧t♦♥✐❛♥❛ ❜✐q✉❛❞rát✐❝❛ ❣❡r❛❧ ❡♠ t❡r♠♦s ❞❡ ♦♣❡r❛❞♦r❡s ❢❡r♠✐ô♥✐❝♦s ❡ ✉❧t❡r✐♦r♠❡♥t❡✱ ❞✐s❝✉t✐♠♦s ♦ ♠ét♦❞♦ ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❡♥tr♦♣✐❛ ❞❡ ❘é♥②✐✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❙✐st❡♠❛s ❞❡ ♠✉✐t♦s ❝♦r♣♦s✱ ▼♦❞❡❧♦s ❡①❛t❛♠❡♥t❡ s♦❧ú✈❡✐s✱ ❈❛❞❡✐❛ ❳❨✱ ■♥t❡✲ r❛çã♦ ♠✉❧t✐s♣✐♥✱ ❋❡r♠✐♦♥✐③❛çã♦✱ Pr♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s ❡♠ s✐st❡♠❛s ❞❡ s♣✐♥s✱ ❊♥tr♦♣✐❛ ❞❡ ❡♠❛✲ r❛♥❤❛♠❡♥t♦✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❳❨ ♠♦❞❡❧✱ ❛s ✇❡❧❧ ❛s ❛♥ ❡①❛❝t❧② s♦❧✈❛❜❧❡ t✇♦✲❧❡❣ s♣✐♥ ❧❛❞❞❡r ✇✐t❤ t❤r❡❡✲s♣✐♥ ✐♥t❡r❛❝t✐♦♥s✳ ❲❡ ♦❜t❛✐♥ ❛♥❛❧②t✐❝❛❧❧② t❤❡ ✜♥✐t❡✲s✐③❡ ❝♦rr❡❝t✐♦♥s ♦❢ t❤❡ ❧♦✇✲❧②✐♥❣ ❡♥❡r❣✐❡s ❛♥❞ ❞❡t❡r♠✐♥❡ t❤❡ ❝❡♥tr❛❧ ❝❤❛r❣❡ ❛s ✇❡❧❧ ❛s t❤❡ s❝❛❧✐♥❣ ❞✐♠❡♥s✐♦♥s✳ ❇② ✉s✐♥❣ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ♠❡t❤♦❞✱ ✇❡ ❛❧s♦ st✉❞② t❤❡ ✜♥✐t❡✲s✐③❡ ❝♦rr❡❝t✐♦♥s ♦❢ t❤❡ ❘é♥②✐ ❡♥tr♦♣② ♦❢ t❤❡ ❣r♦✉♥❞ st❛t❡ ❛♥❞ ♦❢ t❤❡ ❡①❝✐t❡❞ st❛t❡s✳ ❖✉r r❡s✉❧ts ❛r❡ ✐♥ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡ ♣r❡❞✐❝t✐♦♥s ♦❢ t❤❡ ❝♦♥❢♦r♠❛❧ ✜❡❧❞ t❤❡♦r②✳ ▲❛st ❜✉t ♥♦t ❧❡❛st✱ ✐♥ t❤❡ ❛♣♣❡♥❞✐① ✭❆♣ê♥❞✐❝❡ ❆✮✱ ✇❡ s✉♠♠❛r✐③❡ t❤❡ ♠❛✐♥ st❡♣s ♦❢ t❤❡ ❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❣❡♥❡r❛❧ ❜✐q✉❛❞r❛t✐❝ ❤❛♠✐❧t♦♥✐❛♥s ✐♥ t❡r♠ ♦❢ ❋❡r♠✐ ♦♣❡r❛t♦rs✱ ❛♥❞ t❤❡r❡❛❢t❡r✱ ✇❡ ❛❧s♦ ❞✐s❝✉ss t❤❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ♠❡t❤♦❞ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❘é♥②✐ ❡♥tr♦♣②✳
❑❡②✇♦r❞s✿ ▼❛♥② ❜♦❞② s②st❡♠s✱ ❊①❛❝t❧② s♦❧✈❛❜❧❡ ♠♦❞❡❧s✱ ❳❨ ❝❤❛✐♥✱ ▼✉❧t✐s♣✐♥ ✐♥t❡r❛❝t✐♦♥✱ ❋❡r♠✐♦♥✐③❛t✐♦♥✱ ❈r✐t✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s♣✐♥ s②st❡♠s✱ ❊♥t❛♥❣❧❡♠❡♥t ❡♥tr♦♣②✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❖ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦ ❢♦✐ ❡①❡❝✉t❛❞♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♠❡✉ ♦r✐❡♥t❛❞♦r ❏✳ ❈â♥✲ ❞✐❞♦ ❳❛✈✐❡r✱ ❝✉❥♦s ❡stí♠✉❧♦s✱ ❞✐s♣♦s✐çã♦✱ ❝rít✐❝❛s✱ ❜♦❛s ✐❞❡✐❛s✱ ✐♥str✉çõ❡s ❡ ❝✉✐❞❛❞♦s❛s r❡✈✐sõ❡s ❞❡st❡ ♠❛♥✉s❝r✐t♦ sã♦ r❡❝♦♥❤❡❝✐❞♦s ❝♦♠ ♣r♦❢✉♥❞❛ ❣r❛t✐❞ã♦✳
❆❣r❛❞❡ç♦ t❛♠❜é♠ à ❛❣ê♥❝✐❛ ❋❆P❊▼■● ♣❡❧♦ ❛♠♣❛r♦ ✜♥❛♥❝❡✐r♦✳
❈♦♥t❡ú❞♦
■♥tr♦❞✉çã♦ ✶
✶ ❋❡♥ô♠❡♥♦s ❝rít✐❝♦s ❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ✻
✶✳✶ ❋❡♥ô♠❡♥♦s ❝rít✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❈♦rr❡çõ❡s ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ❞❛ ❡♥❡r❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ ❆ ❡♥tr♦♣✐❛ ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ❡♠ s✐st❡♠❛s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷ ❙♦❧✉çã♦ ❡①❛t❛ ❡ ♣r♦♣r✐❡❞❛❞❡ ❝rít✐❝❛s ❞♦ ♠♦❞❡❧♦ ❳❨ ✷✻
✷✳✶ ❉✐❛❣♦♥❛❧✐③❛çã♦ ❡①❛t❛ ❡ ❞✐❛❣r❛♠❛ ❞❡ ❢❛s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ Pr♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✶ ▼♦❞❡❧♦ ❳❳ (γ = 0) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷✳✷ ❈❛♠♣♦ ♠❛❣♥ét✐❝♦ ❝rít✐❝♦ hc = 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸ ❊♠❛r❛♥❤❛♠❡♥t♦ ♥♦ ♠♦❞❡❧♦ ❳❨ ✸✾
✸✳✶ ▼♦❞❡❧♦ ❞❡ ■s✐♥❣(γ = 1) ❡ ❝❛♠♣♦ ❝rít✐❝♦ hc = 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✸✳✷ ▼♦❞❡❧♦ ❳❳ (γ = 0) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✷✳✶ ❊st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✷✳✷ ❊st❛❞♦s ❡①❝✐t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✹ ❊s❝❛❞❛ ❞❡ ❞✉❛s✲♣❡r♥❛s ❝♦♠ ✐♥t❡r❛çã♦ ❞❡ três✲s♣✐♥s ✺✻
✹✳✶ ❉✐❛❣r❛♠❛ ❞❡ ❢❛s❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❝rít✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✹✳✷ ❊♥tr♦♣✐❛ ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✷✳✶ ❊st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✷✳✷ ❊st❛❞♦s ❡①❝✐t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
❈♦♥❝❧✉sõ❡s ❡ ❝♦♠❡♥tár✐♦s ✜♥❛✐s ✼✵
❆ ❍❛♠✐❧t♦♥✐❛♥❛ ❜✐q✉❛❞rát✐❝❛ ✼✹
❆✳■ ❉✐❛❣♦♥❛❧✐③❛çã♦ ❡①❛t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹
❈❖◆❚❊Ú❉❖ ① ❆✳■■ ❈♦rr❡❧❛çã♦ ❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
■♥tr♦❞✉çã♦
❆s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞♦s ❝♦♠♣♦♥❡♥t❡s ❢✉♥❞❛♠❡♥t❛✐s ❞♦ ♥♦ss♦ ♠✉♥❞♦✱ ❢♦r♠❛❞♦ ♣♦r ♠❛tér✐❛ ❡ r❛❞✐❛çã♦✱ sã♦ ❞❡s❝r✐t❛s ♣❡❧❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ ❚❛✐s ❝♦♠♣♦♥❡♥t❡s sã♦ ♣❛rtí❝✉❧❛s q✉❡ r❛r❛♠❡♥t❡ sã♦ ❡♥❝♦♥tr❛❞❛s ✐s♦❧❛❞❛s ❡ q✉❡ ✉s✉❛❧♠❡♥t❡ ✐♥t❡r❛❣❡♠ ❝♦♠ ♦ ❛♠❜✐❡♥t❡✳ ❊ ❛✐♥❞❛✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛rtí❝✉❧❛s sã♦ ❡♠ ❣❡r❛❧ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞✐❢❡r❡♥t❡s ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ✐♥❞✐✈✐❞✉❛❧ ❞❡st❛s ♣❛rtí❝✉❧❛s✱ ❞❡✈✐❞♦ à ✐♥t❡r❛çã♦ ❡♥tr❡ ❡❧❛s✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱ é ♣♦ssí✈❡❧ ❞❡s❝r❡✲ ✈❡r ✉♠ s✐st❡♠❛ ❝♦♥t❡♥❞♦ ♠✉✐t❛s ♣❛rtí❝✉❧❛s ✐♥t❡r❛❣❡♥t❡s s✐♠♣❧❡s♠❡♥t❡ ❞❡s♣r❡③❛♥❞♦ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ❛s ♣❛rtí❝✉❧❛s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡♠ ❝❛s♦s ♠❛✐s ❣❡r❛✐s✱ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s ❣r❛✉s ❞❡ ❧✐❜❡r✲ ❞❛❞❡ ♠✐❝r♦s❝ó♣✐❝♦s sã♦ ❛ ❝❤❛✈❡ ♣❛r❛ ♦s ❢❡♥ô♠❡♥♦s q✉❡ ♦❜s❡r✈❛♠♦s✳ P♦r ❡①❡♠♣❧♦✱ ❛♦ t❡♥t❛r♠♦s ❞❡s❝r❡✈❡r ♦s ❡❧étr♦♥s ❡♠ ✉♠ só❧✐❞♦✱ ❛ ❝❤❛♠❛❞❛ t❡♦r✐❛ ❞❡ ❜❛♥❞❛s é ❝❡rt❛♠❡♥t❡ ❛q✉❡❧❛ ❞❡ ♠❛✐♦r s✉❝❡ss♦✳ ❈♦♥t✉❞♦✱ s❡ ❡st❡s ❡❧étr♦♥s ✐♥t❡r❛❣❡♠ ❢♦rt❡♠❡♥t❡ ❡♥tr❡ s✐✱ ♥ã♦ ♣♦❞❡r❡♠♦s tr❛t❛r ♣❡r✲ t✉r❜❛t✐✈❛♠❡♥t❡ s✉❛s ✐♥t❡r❛çõ❡s✱ ❡ ❡♥tã♦ ❛ t❡♦r✐❛ ❞❡ ❜❛♥❞❛s é ❢❛❧❤❛✳
❈♦♠♦ ❥á ❢♦✐ ❞✐t♦✱ ❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞❡ ✉♠ s✐st❡♠❛ ♣♦❞❡♠ ✈❛r✐❛r ♥❛ ♣r❡s❡♥ç❛ ❞❡ ❛❣❡♥t❡s ❡①t❡r♥♦s ✭❡✳❣✳✱ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✮✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱ t❛✐s ❛❣❡♥t❡s ❡①t❡r♥♦s ❛❧t❡r❛♠ tã♦ ❞r❛st✐✲ ❝❛♠❡♥t❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ s✐st❡♠❛ q✉❡ ❡❧❛s sã♦ ✐❞❡♥t✐✜❝❛❞❛s ❝♦♠♦ ♦✉tr❛ ❢❛s❡✳ ◆❡st❡s ❝❛s♦s✱ ❞✐③❡♠♦s q✉❡ ♦ s✐st❡♠❛ s♦❢r❡✉ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡✳ ❈❡rt❛s tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ sã♦ ❝❛r❛❝t❡r✐✲ ③❛❞❛s ♣♦r ♣♦♥t♦s ❝rít✐❝♦s✱ ♥♦s q✉❛✐s ❛s ❢❛s❡s t♦r♥❛♠✲s❡ ✐♥❞✐st✐♥❣✉í✈❡✐s✳ ■st♦ ♦❝♦rr❡ ❞❡✈✐❞♦ ❛♦s ❝♦♥st✐t✉✐♥t❡s ❞♦ s✐st❡♠❛ ❡st❛r❡♠ ❝♦rr❡❧❛❝✐♦♥❛❞❛s ❡♠ t♦❞❛s ❛s ❡s❝❛❧❛s ❞❡ ❞✐stâ♥❝✐❛ ♥♦ ♠❛t❡r✐❛❧✱ ❞❡ ♠♦❞♦ q✉❡ ❞✐③❡♠♦s q✉❡ ♦ s✐st❡♠❛ é ❝rít✐❝♦ ❡ ❛ ❡s❝❛❧❛ ❡s♣❛❝✐❛❧ é ♣❡r❞✐❞❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡♠ s✐st❡♠❛s ❞✐t♦s ♥ã♦ ❝rít✐❝♦s✱ ❡st❡s ❝♦♥st✐t✉✐♥t❡s ✐♥t❡r❛❣❡♠✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❝♦♠ s❡✉s ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐♠♦s✳ ❆ss✐♠✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ξ✱ ❞✐stâ♥❝✐❛ s♦❜r❡ ❛ q✉❛❧ ❛s ♣❛rtí❝✉❧❛s ❡stã♦ ❡❢❡t✐✈❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞❛s✱ é ✐♥✜♥✐t♦ ✭✜♥✐t♦✮ ❡♠ s✐st❡♠❛s ❝rít✐❝♦s ✭♥ã♦ ❝rít✐❝♦s✮✳
❯♠ ❡①❡♠♣❧♦ ❞❡ s✐st❡♠❛ q✉❡ ❛♣r❡s❡♥t❛ ✉♠ ♣♦♥t♦ ❝rít✐❝♦ sã♦ ♦s ❢❡rr♦♠❛❣♥❡t♦s✳ ❆s♣✐r❛♥❞♦ ❡st✉❞❛r✴❡♥t❡♥❞❡r t❛✐s s✐st❡♠❛s ▲❡♥③ ♣r♦♣ôs✱ ❡♠ ✶✾✷✵✱ ✉♠ s✐♠♣❧❡s ♠♦❞❡❧♦ ♣❛r❛ ■s✐♥❣✳ ❍♦❥❡ ❡st❡ ♠♦❞❡❧♦ é ❝❤❛♠❛❞♦ ❞❡ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣✳ ❆ ✈❡rsã♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❞❡st❡ t❡✈❡ ❛ s✉❛ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ♣✉❜❧✐❝❛❞❛ ♣♦r ■s✐♥❣ ❡♠ ✶✾✷✺ ❬✶❪✱ ♣❡❧♦ ♠ét♦❞♦ ❞❛ ♠❛tr✐③ ❞❡ tr❛♥s❢❡rê♥❝✐❛✱✶ ♦ q✉❡
❞❡♠♦♥str♦✉ q✉❡ ♦ ♠♦❞❡❧♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❡r❛ ✐♥s✉✜❝✐❡♥t❡ ♣❛r❛ s❡ ❡①♣❧✐❝❛r ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ✶❱❛❧❡ s❛❧✐❡♥t❛r q✉❡ t❛♠❜é♠ ✐♥✈❡st✐❣❛♠♦s ❛s tr❛♥s✐çõ❡s ♠❛❣♥ét✐❝❛s✱ ❞❡ ♠♦❞♦ ❡①❛t♦✱ ❡♠ ✉♠ ♠♦❞❡❧♦ ❢❡rr♦✲
♠❛❣♥ét✐❝♦ ❞❡ s♣✐♥s ■s✐♥❣ ❣❡r❛✐s ✭S, S′✮ ❞✐♠❡r✐③❛❞♦ ❡ ❝♦♠ ❛♥✐s♦tr♦♣✐❛ ❞❡ í♦♥s✲s✐♠♣❧❡s ❬✷❪✳
■◆❚❘❖❉❯➬➹❖ ✷ ❢❡rr♦♠❛❣♥ét✐❝❛ à t❡♠♣❡r❛t✉r❛ ✜♥✐t❛✳ ❈♦♥t✉❞♦✱ ♣❛r❛ ♦ ❝❛s♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ t❛❧ tr❛♥s✐çã♦ ♦❝♦rr❡ à t❡♠♣❡r❛t✉r❛ ♥ã♦ ♥✉❧❛✱ ✈✐♥❞♦ ❛ s❡r ♣r♦✈❛❞♦ ❝♦♠ ❛ s♦❧✉çã♦ ❡①❛t❛ ♦❜t✐❞❛ ♣♦r ❖♥s❛❣❡r ❡♠ ✶✾✹✹ ❬✸✱ ✹❪✳ ❯♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❢♦✐ ♣r♦♣♦st❛ ❛✐♥❞❛ ♥♦ ✐♥í❝✐♦ ❞♦ s✉r❣✐♠❡♥t♦ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ♦ ♠♦❞❡❧♦ ❍❡✐s❡♥❜❡r❣✱ q✉❡ t❛♠❜é♠ é ❢✉♥❞❛♠❡♥t❛❧ ❡♠ ♠❡❝â♥✐❝❛ ❡st❛tíst✐❝❛ ❡ t❡♠ s✐❞♦ ❡st✉❞❛❞♦ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡✳ ❋✐♥❛❧♠❡♥t❡✱ ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ♣♦❞❡ t❛♠❜é♠ s❡r ❣❡♥❡r❛❧✐③❛❞♦ ♣♦r ♠♦❞❡❧♦s t❛✐s ❝♦♠♦ ♦ ♠♦❞❡❧♦ ❞❡ ❍✉❜❜❛r❞ ❬✺❪ ❡ ♦ ♠♦❞❡❧♦ ❞❡ ❑♦♥❞♦ ♥❛ r❡❞❡ ❢❡rr♦♠❛❣♥ét✐❝❛ ❬✻❪✳ ❊st❡ ú❧t✐♠♦ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❡①♣❧✐❝❛r ❛ ♠❛❣♥❡t♦✲r❡s✐stê♥❝✐❛ ❝♦❧♦ss❛❧ ❞❛s ♠❛♥❣❛♥✐t❛s ❬✻❪✳ ❖ ♣rê♠✐♦ ◆♦❜❡❧ ❞❛ ❢ís✐❝❛ ❞❡ ✷✵✵✼ ❢♦✐ ❡①❛t❛♠❡♥t❡ ❞❡✈✐❞♦ à ❞❡s❝♦❜❡rt❛ ❞❡st❡ ❢❡♥ô♠❡♥♦✳✷
❆✐♥❞❛ q✉❡ ♦ ♦❜❥❡t✐✈♦ ❣❡r❛❧ ❡♠ ❋ís✐❝❛ ❞❛ ▼❛tér✐❛ ❈♦♥❞❡♥s❛❞❛ é ❞❡ s❡ ❞❡s❝r❡✈❡r ✉♠ só❧✐❞♦✱ q✉❡ ❝♦♥s✐st❡ ❞❡ í♦♥s ❡ ❡❧étr♦♥s ❡♠ ✉♠❛ ❡str✉t✉r❛ ❝r✐st❛❧✐♥❛ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ❛ s♦❧✉çã♦ ❞❡ t❛❧ ♣r♦❜❧❡♠❛ é ✐♥✈✐á✈❡❧✱ ❡♠ ❣❡r❛❧✱ t❛♥t♦ ♥✉♠ér✐❝❛✱ ❝♦♠♦ ❛♥❛❧✐t✐❝❛♠❡♥t❡✱ ♦ q✉❡ ❥✉st✐✜❝❛ ❛ ❝r✐❛çã♦ ❞❡ ♠♦❞❡❧♦s ❝♦♠ ♣❛râ♠❡tr♦s ❞❡ ✐♥t❡r❛çã♦ s✐♠♣❧✐✜❝❛❞♦s✱ ❝♦♠♦ ♦s ❞♦s ❡①❡♠♣❧♦s ❝✐t❛❞♦s ♥♦ ♣❛✲ rá❣r❛❢♦ ❛♥t❡r✐♦r✳ ❉❡ ❢❛t♦✱ ✉♠ ❡❧❡♠❡♥t♦ ❝❤❛✈❡ ❞❛ ❢ís✐❝❛ t❡ór✐❝❛ t❡♠ s✐❞♦ ❛ ❝♦♥❝❡♣t✉❛❧✐③❛çã♦ ❞❡ ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s ❛tr❛✈és ❞❡ ♠♦❞❡❧♦s✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ✐♥✈❡st✐❣❛çã♦ ❞❡ t❛✐s ♠♦❞❡❧♦s ❡stá s✉❥❡✐t❛ às ❢❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s✴❝♦♠♣✉t❛❝✐♦♥❛✐s ❞✐s♣♦♥í✈❡✐s✱ ❡✱ ❛té ♠❡s♠♦ ♣❛r❛ ♠♦❞❡❧♦s r❡❧❛t✐✈❛✲ ♠❡♥t❡ s✐♠♣❧❡s✱ ♥♦r♠❛❧♠❡♥t❡✱ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❡♠ três ❞✐♠❡♥sõ❡s é ✉♠❛ t❛r❡❢❛ ❡①tr❡♠❛♠❡♥t❡ ❝♦♠♣❧✐❝❛❞❛✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ s✐st❡♠❛s ❡♠ ♠❡♥♦r❡s ❞✐♠❡♥sõ❡s sã♦ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ♠❛✐s s✐♠✲ ♣❧❡s ❞❡ s❡r❡♠ ❛❜♦r❞❛❞♦s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✳ ❆q✉✐✱ ❡st❛♠♦s ♣r❡♦❝✉♣❛❞♦s ❝♦♠ ♠♦❞❡❧♦s q✉❛s✐✲✉♥✐❞✐♠❡♥s✐♦♥❛✐s q✉❡ sã♦ ❡①❛t❛♠❡♥t❡ s♦❧ú✈❡✐s✱ ✐✳❡✳✱ ❛q✉❡❧❡s ❝✉❥❛s ♣r♦♣r✐❡❞❛❞❡s ♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞❛s ❞❡ ♠♦❞♦ ❛♥❛❧ít✐❝♦✳ ❱❛❧❡ s❛❧✐❡♥t❛r q✉❡ ♠♦❞❡❧♦s t❡ór✐❝♦s q✉❛s✐✲✉♥✐❞✐♠❡♥s✐♦♥❛✐s sã♦ ♣r❡❝✐s♦s ♥❛ ❞❡s❝r✐çã♦ ❞❡ ❝❡rt♦s ❡①♣❡r✐♠❡♥t♦s ❬✾❪✱ s❡♥❞♦ ♣♦rt❛♥t♦ ♠❛✐s ❞♦ q✉❡ ❛♣❡♥❛s ❡s♣❡❝✉❧❛✲ çõ❡s ♠❛t❡♠át✐❝❛s✳ ❆ ✐♠♣♦rtâ♥❝✐❛ ♣r✐♥❝✐♣❛❧ ❞❛ ♦❜t❡♥çã♦ ❞❡ s♦❧✉çõ❡s ❡①❛t❛s r❡s✐❞❡ ♥♦ ❢❛t♦ ❞❡ q✉❡ ❡st❛s ♥♦s ♣❡r♠✐t❡♠ ❡♥t❡♥❞❡r ❡ ❝❛r❛❝t❡r✐③❛r ❝♦rr❡t❛♠❡♥t❡ ❞✐✈❡rs♦s ❢❡♥ô♠❡♥♦s ❢ís✐❝♦s✱ s❡♥❞♦ ❛ss✐♠ sã♦ ✐♥❞✐s❝✉t✐✈❡❧♠❡♥t❡ ✉♠ ❛ss✉♥t♦ ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❡✱ ❡♠ ❣❡r❛❧✱ ♠❛✐s ❢á❝❡✐s ❞❡ s❡r❡♠ ❡st✉❞❛❞♦s✳
❆ ❤✐stór✐❛ ❞❛ s♦❧✉çã♦ ❡①❛t❛ ❞❡ s✐st❡♠❛s q✉â♥t✐❝♦s ❞❡ ♠✉✐t♦s ❝♦r♣♦s r❡♠♦♥t❛ ❛♦ ❛rt✐❣♦ ❞❡ ✶✾✷✽ ❞❡ ❏♦r❞❛♥ ❡ ❲✐❣♥❡r ❬✶✵❪✱ ♥♦ q✉❛❧ ❡♥❝♦♥tr❛r❛♠ ❛ r❡❧❛çã♦ ❡①❛t❛ ❡♥tr❡ ♦♣❡r❛❞♦r❡s ❞❡ ❧❡✈❛♥✲ t❛♠❡♥t♦✴❛❜❛✐①❛♠❡♥t♦ ❞❡ s♣✐♥✲1/2 ❡ ♦♣❡r❛❞♦r❡s ❢❡r♠✐ô♥✐❝♦s ❞❡ ❝r✐❛çã♦✴❛♥✐q✉✐❧❛çã♦✳ ❉❡ ♠♦❞♦ q✉❡ t❛❧ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ♣❡r♠✐t❡ ❝♦♥✈❡rt❡r s✐st❡♠❛s ❞❡ s♣✐♥✲1/2 ❡♠ ❢ér♠✐♦♥s ✐♥t❡r❛❣❡♥t❡s s❡♠ s♣✐♥✳ ❖ ♣ró①✐♠♦ ♣❛ss♦ ❞❡ss❛ ❤✐stór✐❛ ❢♦✐ ❛ s♦❧✉çã♦ ❞❡ ❇❡t❤❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ✉♥✐✲ ❞✐♠❡♥s✐♦♥❛❧ ❞❡ s♣✐♥✲1/2 ❡♠ ✶✾✸✶ ❬✶✶❪✱ ♦ ❝❤❛♠❛❞♦ ❛♥s❛t③ ❞❡ ❇❡t❤❡✳ ❖✉tr♦ ✐♠♣♦rt❛♥t❡ ♠❛r❝♦ ♦❝♦rr❡✉ ❡♠ ✶✾✹✹✱ ❝♦♠ ❛ ❥á ❝✐t❛❞❛ s♦❧✉çã♦ ❡①❛t❛ ♣❛r❛ ❛ ❢✉♥çã♦ ❞❡ ♣❛rt✐çã♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❜✐❞✐♠❡♥s✐♦♥❛❧ ♦❜t✐❞❛ ♣♦r ❖♥s❛❣❡r✳ ❱❛❧❡ ❛❝❡♥t✉❛r t❛♠❜é♠ q✉❡✱ ❡♠ ✶✾✻✶✱ ▲✐❡❜ ❡t ❛❧✳ ❬✶✷❪ ❡st✉✲ ❞❛r❛♠ ✉♠ s✐st❡♠❛ ❞❡ L❢ér♠✐♦♥s ❣♦✈❡r♥❛❞♦ ♣♦r ❤❛♠✐❧t♦♥✐❛♥❛s ❜✐q✉❛❞rát✐❝❛s✱ ❡ ♠♦str❛r❛♠ q✉❡
■◆❚❘❖❉❯➬➹❖ ✸ ❛♣❡s❛r ❞♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ❞❛ ❤❛♠✐❧t♦♥✐❛♥❛ t❡r t❛♠❛♥❤♦ 2L ❛ ❞✐❛❣♦♥❛❧✐③❛çã♦ ❞❡st❛ é r❡❞✉③✐❞❛
à s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ ✈❛❧♦r❡s ♣ró♣r✐♦s ❞❡ ❞✉❛s ♠❛tr✐③❡s ❞❡ ♦r❞❡♠ L✳ ❆✐♥❞❛ ♥♦s ❛♥♦s ✶✾✻✵ ♥♦✈❛s ❛♣❧✐❝❛çõ❡s ♣❛r❛ ♦ ❛♥s❛t③ ❇❡t❤❡ ❢♦r❛♠ ❞❡s❝♦❜❡rt❛s✱ ♣♦r ❡①❡♠♣❧♦✿ ♦ ❡st✉❞♦ ❞♦ ❣ás ❞❡ ❜ós♦♥s ❝♦♠ ✐♥t❡r❛çã♦ ❞❡ ❢✉♥çã♦ ❞❡❧t❛ ❬✶✸✱ ✶✹❪❀ ❡ ♦ ♠♦❞❡❧♦ ❞❡ ❍✉❜❜❛r❞ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❬✶✺✱ ✶✻❪✳✸
❯♠ ♥♦✈♦ ❝❛♣ít✉❧♦ ♥❛ ❛♥á❧✐s❡ ❞❡ ♠♦❞❡❧♦s ❡①❛t❛♠❡♥t❡ s♦❧ú✈❡✐s ❢♦✐ ❛❜❡rt♦ ❝♦♠ ♦ ❛❞✈❡♥t♦ ❞❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦s ❝♦♥❢♦r♠❡ ✭❈❋❚✮ ❬✶✽❪✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✈ár✐❛s r❡str✐çõ❡s ❛♣❛r❡❝❡♠ s❡ ❛ss✉♠✐r♠♦s q✉❡ ♦s s✐st❡♠❛s ❝rít✐❝♦s sã♦ ✐♥✈❛r✐❛♥t❡s ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❝♦♥❢♦r♠❡s✳ ❆❧é♠ ❞✐ss♦✱ ❛s ♣♦ssí✈❡✐s ❝❧❛ss❡s ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ ❞❡ s✐st❡♠❛s q✉â♥t✐❝♦s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s ✜❝❛♠ ✐♥❞❡①❛❞♦s ♣❡❧❛ ❛♥♦♠❛❧✐❛ ❝♦♥❢♦r♠❡ ✭❝❛r❣❛ ❝❡♥tr❛❧✮ c✳ ❊♠ ♠❡❛❞♦s ❞♦s ❛♥♦s ♦✐t❡♥t❛✱ ❞❡♠♦♥str♦✉✲s❡ ❛tr❛✈és ❞❛ ❈❋❚✱ ❝♦♠♦ ❛ ❝❛r❣❛ ❝❡♥tr❛❧ c♣♦❞❡ s❡r ❡①tr❛í❞❛ ❛tr❛✈és ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❡♥❡r❣✐❛ ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ s✐st❡♠❛s ❣r❛♥❞❡s✱ ♣♦ré♠ ✜♥✐t♦s✳ ❊ q✉❡ ❛❧é♠ ❞✐ss♦✱ ♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s✱ q✉❡ ❣♦✈❡r♥❛♠ ♦ ❞❡❝❛✐♠❡♥t♦ ❡♠ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❞❛s ❢✉♥çõ❡s ❞❡ ❝♦rr❡❧❛çã♦✱ sã♦ t❛♠❜é♠ ♣r❡❝✐s❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ♦s ♥í✈❡✐s ❞❡ ❡♥❡r❣✐❛ ❞❡ s✐st❡♠❛s ♣ró①✐♠♦s ❞♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦✳
❊st❛s ❞❡s❝♦❜❡rt❛s r❡s✉❧t❛r❛♠ ❡♠ ✉♠ ❡st✉❞♦ s✐st❡♠át✐❝♦ ❞❛s ❝♦rr❡çõ❡s ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ♥♦s ❡s♣❡❝tr♦s ❞❡ ❞✐✈❡rs♦s ♠♦❞❡❧♦s✳ Pr♦✈❛✈❡❧♠❡♥t❡✱ ❡st❡ é ✉♠ ❞♦s ♠♦❞♦s ♠❛✐s ♣r❡❝✐s♦s ❞❡ s❡ ❞❡✲ t❡r♠✐♥❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ ❡♠ s✐st❡♠❛s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s✱ ❞❡ ♠♦❞♦ q✉❡✱ ♠❡r❡❝✐❞❛♠❡♥t❡ é ✉♠ ❞♦s ♠❛✐s ♣♦♣✉❧❛r❡s✳ ❆t✉❛❧♠❡♥t❡✱ ❞❡✈✐❞♦ à ❝♦♥❡①ã♦ ❡♥tr❡ ❛s ❝♦rr❡❧❛çõ❡s q✉â♥t✐❝❛s ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦✱ é ♣♦ssí✈❡❧ ❛ ❧♦❝❛❧✐③❛çã♦ ❞❡ ♣♦♥t♦s ❝rít✐❝♦s ❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❝❧❛ss❡ ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ ✭❝❛r❣❛ ❝❡♥tr❛❧c✮ ❞❡❧❡s ❛tr❛✈és ❞❡ ❝♦♥❝❡✐t♦s ❞❛ ✐♥❢♦r♠❛çã♦ q✉â♥t✐❝❛ ❬✶✾✕✷✺❪✳ ❊st❛ é ✉♠❛ ❞❛s r❛③õ❡s ♣❡❧❛ q✉❛❧✱ ❡♠ ❛♥♦s r❡❝❡♥t❡s✱ ❛ ❋ís✐❝❛ ❞❛ ▼❛tér✐❛ ❈♦♥❞❡♥s❛❞❛ ❡ ❛ ❚❡♦r✐❛ ❞❡ ■♥❢♦r♠❛çã♦ tê♠ ♠♦str❛❞♦ ✐♥t❡r❡ss❡s ❝♦♠✉♥s✱ ❡ ❝♦♥s❡q✉❡♥✲ t❡♠❡♥t❡✱ ❢❡rr❛♠❡♥t❛s ❡ ✐❞❡✐❛s ❞❡s❡♥✈♦❧✈✐❞❛s ❡♠ ✉♠ ❝❛♠♣♦ tê♠ ❞❛❞♦ ♥♦✈❛s ❝♦♠♣r❡❡♥sõ❡s ❛♦ ♦✉tr♦✳✹
❙❛❜❡✲s❡ q✉❡ ♦ ❡♠❛r❛♥❤❛♠❡♥t♦ é ✉♠❛ ❞❛s ❛ss✐♥❛t✉r❛s ♠❛✐s ♣❡❝✉❧✐❛r❡s ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ♥ã♦ ♣♦ss✉✐♥❞♦ ♥❡♥❤✉♠ ❛♥á❧♦❣♦ ❝❧áss✐❝♦✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ ✏♣❛rtí❝✉❧❛✑ q✉❡ ❝❧❛ss✐❝❛♠❡♥t❡ ♣♦ss✉✐ ❛♣❡♥❛s ❞♦✐s ♣♦ssí✈❡✐s ❡st❛❞♦s✱ ♣❡❧❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ♣♦❞❡r✐❛ ❡st❛r s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥♦s ❞♦✐s ❡st❛❞♦s ❜✐♥ár✐♦s✳ ◗✉❛♥❞♦ t❡♠♦s ✉♠ s✐st❡♠❛ ❝♦♠ ♠❛✐s ❞❡ ✉♠❛ ✏♣❛rtí❝✉❧❛✑✱ ❝❛❞❛ ✉♠❛ ❞❡❧❛s ♣♦❞❡ ❡st❛r ❡♠ ✉♠ ❡st❛❞♦ ❞❡ s✉♣❡r♣♦s✐çã♦✱ ♦ q✉❡ r❡s✉❧t❛r✐❛ ♥✉♠ ❡st❛❞♦ ❣❧♦❜❛❧ ❡♠❛r❛♥❤❛❞♦✳ ◆♦t❛❞❛♠❡♥t❡✱ ♦ ♣rê♠✐♦ ◆♦❜❡❧ ❞❛ ❢ís✐❝❛ ❞❡st❡ ❛♥♦ ❢♦✐ ❝♦♥s❛❣r❛❞♦ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠ét♦❞♦s ♣❛r❛ ❛ ♠❡❞✐❞❛ ❡ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛rtí❝✉❧❛s ✐♥❞✐✈✐❞✉❛✐s s❡♠ ❞❡str✉✐r s✉❛ ♥❛t✉r❡③❛ q✉â♥t✐❝❛ ✭❡st❛❞♦ ❞❡ s✉♣❡r♣♦s✐çã♦✮✳ ❆❜❛✐①♦✱ r❡t♦♠❛r❡♠♦s✱ ❛✐♥❞❛ q✉❛❧✐t❛t✐✈❛♠❡♥t❡✱ ♦ ❝♦♥❝❡✐t♦ ❞❛ ❝♦♥❡①ã♦ ❡♥tr❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❢❛s❡s q✉â♥t✐❝❛s✳
P♦❞❡♠♦s✱ ❜❛s✐❝❛♠❡♥t❡✱ r♦t✉❧❛r ❛s ❤❛♠✐❧t♦♥✐❛♥❛s ❞❡ s✐st❡♠❛s ❞❡ ♠✉✐t♦s ❝♦r♣♦s ❡♠ ❞✉❛s ❝❧❛s✲ ✸❱❡❥❛ ❛ ❘❡❢✳ ❬✶✼❪ ♣❛r❛ ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❢❛s❝✐♥❛♥t❡ ❛ss✉♥t♦ ❞❡ s✐st❡♠❛s q✉❡ ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞♦s ❡①❛t❛♠❡♥t❡✳ ✹❖✉tr♦ ❣r❛♥❞❡ ♠♦t✐✈❛❞♦r ♣❛r❛ ♦ ❝♦♠♣❛rt✐❧❤❛♠❡♥t♦ ❞❡ ✐♥t❡r❡ss❡s ❡ ❡s❢♦rç♦s ❞❡st❛s ár❡❛s ❡stá ♥❛ ♠❛✐s ❛✉❞❛✲
■◆❚❘❖❉❯➬➹❖ ✹ s❡s✳ ❆q✉❡❧❛s ❝✉❥♦s ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛✱ ♥♦ ❧✐♠✐t❡ t❡r♠♦❞✐♥â♠✐❝♦✱ ❛♣r❡s❡♥t❛ ✉♠ ❣❛♣ ✜♥✐t♦ ✭♥✉❧♦✮ sã♦ ❞✐t❛s ♥ã♦ ❝rít✐❝❛s ✭❝rít✐❝❛s✮✳✺ ❊♠ ❣❡r❛❧✱ ♦s ❛✉t♦❡st❛❞♦s ❞❡st❛s ❤❛♠✐❧t♦♥✐❛♥❛s sã♦ ❝♦♠♣♦st♦s
❞❡ s♦❜r❡♣♦s✐çõ❡s ❞❡ ❡st❛❞♦s ♣r♦❞✉t♦s✱ ✐✳❡✳✱ s❡ ♦ s✐st❡♠❛ ❞❡s❝r✐t♦ ♣❡❧❛ ❤❛♠✐❧t♦♥✐❛♥❛ é ❞✐✈✐❞♦ ❡♠ ❞✉❛s ♣❛rt❡s✱ ❡❧❛s✱ ❡♠ ❣❡r❛❧✱ sã♦ ❡♠❛r❛♥❤❛❞❛s ❡ ♣♦r ✐ss♦ ❝♦♠♣❛rt✐❧❤❛♠ ✐♥❢♦r♠❛çã♦✳ ❯♠❛ ♣♦ssí✈❡❧ ♠❡❞✐❞❛ ❞❡st❛ ✐♥❢♦r♠❛çã♦ ❝♦♠♣❛rt✐❧❤❛❞❛ é ✐♥❢❡r✐❞❛ ♣❡❧❛ ❡♥tr♦♣✐❛ ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦✳ ■♥❣❡♥✉❛✲ ♠❡♥t❡✱ ♣♦❞❡rí❛♠♦s ❡s♣❡r❛r q✉❡ ❡st❛ ❡♥tr♦♣✐❛ ❢♦ss❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❡①t❡♥s✐✈❛✱ ✐✳❡✳✱ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ✈♦❧✉♠❡ ❞♦ s✉❜s✐st❡♠❛✱ ♠❛s ✐st♦ ♥ã♦ ♦❝♦rr❡✳ ▲❡♠❜r❛♥❞♦ q✉❡ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ♥❛ ✐♥✲ ❢♦r♠❛çã♦ ♠✉t✉❛♠❡♥t❡ ❝♦♠♣❛rt✐❧❤❛❞❛✱ ♥ã♦ ❡♠ ✉♠❛ ✐♥❢♦r♠❛çã♦ ❝♦♥t✐❞❛ ❡♠ ✉♠❛ ❞❛❞❛ ♣❛rt❡✱ ❡ q✉❡ ❛s ❞✉❛s ♣❛rt❡s ❝♦♠♣❛rt✐❧❤❛♠ ❛ s✉♣❡r❢í❝✐❡ q✉❡ ❛s s❡♣❛r❛♠✱ ❞❡✈❡rí❛♠♦s ❡s♣❡r❛r ❡♥tã♦ q✉❡ ❛ ❡♥tr♦♣✐❛ s❡❥❛ ♣r♦♣♦r❝✐♦♥❛❧ ❛ ❡st❛ ár❡❛✳ ■st♦ é ❞❡ ❢❛t♦ ❡s♣❡r❛❞♦ ♣❛r❛ s✐st❡♠❛s ♥ã♦ ❝rít✐❝♦s✱ ❝✉❥♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ ξ é ✜♥✐t♦✱ ❡ é ❜❡♠ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ▲❡✐ ❡♥tró♣✐❝❛ ❞❛ ➪r❡❛ ❬✷✼❪✳ ❊♠ ❝♦♥tr❛♣❛rt✐❞❛✱ ❡♠ s✐st❡♠❛s ❝rít✐❝♦s ✭ξ → ∞✮✱ ❛s ♣❛rt❡s ❞♦ s✐st❡♠❛ ❝♦♠♣❛rt✐❧❤❛♠ ✐♥❢♦r♠❛çã♦ ♠✉✐t♦ ❛❧é♠ ❞♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❡♠ t♦r♥♦ ❞❛ s✉♣❡r❢í❝✐❡ q✉❡ ♦s s❡♣❛r❛✳ ❉❡ ♠♦❞♦ q✉❡ ♦ ❝♦♥❤❡✲ ❝✐♠❡♥t♦ ❞❡ ❛♣❡♥❛s ✉♠❛ ❞❛s ♣❛rt❡s ✐♠♣❧✐❝❛ ❡♠ ✉♠❛ ♣❡r❞❛ ❞❛ ✐♥❢♦r♠❛çã♦ ✭❛✉♠❡♥t♦ ❞❛ ❡♥tr♦♣✐❛✮✳ P♦rt❛♥t♦ ✉♠❛ ✈✐♦❧❛çã♦ ❞❛ ▲❡✐ ❡♥tró♣✐❝❛ ❞❛ ➪r❡❛ é ❡s♣❡r❛❞❛ ❬✷✼❪ ❬✈❡r t❛♠❜é♠ ❛ ❋✐❣✉r❛ ✶✳✸✳✶✭❞✮❪✳ ◆❡st❛ ❞✐ss❡rt❛çã♦✱ ♥♦ ❈❛♣ít✉❧♦ ✶✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛ ♠❛✐s ✉s✉❛❧ ♠❡❞✐❞❛ ❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çã♦ ❝♦♠♣❛rt✐❧❤❛❞❛ ❡♥tr❡ ❞✉❛s ♣❛rt❡s ❞❡ ✉♠ s✐st❡♠❛✱ ❛ ❡♥tr♦♣✐❛ ❞❡ ❘é♥②✐✳ ❉✐s❝✉t✐r❡♠♦s t❛♠❜é♠ ❝♦♠♦ ❡♥❝♦♥tr❛r ❛ ❝❛r❣❛ ❝❡♥tr❛❧ ♣♦r ♠❡✐♦ ❞❡st❛ ❡♥tr♦♣✐❛ ✭❝♦♠♣♦rt❛♠❡♥t♦ ❣❡r❛❧ ❞❛ ✈✐♦❧❛çã♦ ❞❛ ▲❡✐ ❞❛ ➪r❡❛✮✳
◆♦s ❈❛♣ít✉❧♦s s❡❣✉✐♥t❡s✱ ❡st✉❞❛r❡♠♦s ❛❧❣✉♥s s✐st❡♠❛s ❞❡ s♣✐♥s q✉❛s✐✲✉♥✐❞✐♠❡♥s✐♦♥❛✐s ❡①❛✲ t❛♠❡♥t❡ s♦❧ú✈❡✐s ✉t✐❧✐③❛♥❞♦ ♦s ♠ét♦❞♦s ♠❡♥❝✐♦♥❛❞♦s ❛❝✐♠❛✳ ❆ s❛❜❡r✱ ♥♦ ❈❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥✲ t❛r❡♠♦s ❛ s♦❧✉çã♦ ❡①❛t❛ ❞♦ ♠♦❞❡❧♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❳❨ ❡♠ ✉♠ ❝❛♠♣♦ tr❛♥s✈❡rs♦✱ ❞✐s❝✉t✐♥❞♦ s❡✉ ❡s♣❡❝tr♦ ❞❡ ❡♥❡r❣✐❛s ❡ ❝❛❧❝✉❧❛♥❞♦ ❞❡ ♠♦❞♦ ❡①❛t♦ s✉❛ ❝❛r❣❛ ❝❡♥tr❛❧❀ ❝♦♠♦ ✈❡r❡♠♦s c = 1 ❡ c= 1/2 ♣❛r❛ ♦s ❝❛s♦s ✐s♦tró♣✐❝♦ ❡ ❝❛♠♣♦ ❡①t❡r♥♦ ❝rít✐❝♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦r ✜♠✱ ♦ ❈❛♣í✲ t✉❧♦ ✸✱ s❡rá ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ ❳❨✱ ♥♦ q✉❛❧ ❝❛❧❝✉❧❛r❡♠♦s ❛s ❡♥tr♦♣✐❛s ❞❡ ❘é♥②✐ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❛s ❝♦rr❡❧❛çõ❡s✱ ❝✉❥♦s r❡s✉❧t❛❞♦s ❝♦♠✲ ♣❛r❛r❡♠♦s ❝♦♠ ❛s ♣r❡✈✐sõ❡s ❞❛ ❈❋❚✳ ◆❛ ✈❡r❞❛❞❡✱ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❳❳✱ ❡st❡s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞♦s ❞❡ ♠♦❞♦ ❡①❛t♦✱ ❡♠❜♦r❛ ❡st❡ ♥ã♦ s❡rá ♦ ♥♦ss♦ ❢♦❝♦✳ ◗✉❛♥t♦ à ❡♥tr♦♣✐❛ ❞❡ ❘é♥②✐ ❞❡ ❡st❛❞♦s ❡①❝✐t❛❞♦s✱ ♣♦r ✉♠ ❧❛❞♦✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡s♣❡r❛❞♦s ❞❡ tr❛❜❛❧❤♦s ❛♥t❡r✐♦r♠❡♥t❡ s❡rã♦ ❝♦♥✜r♠❛❞♦s ❬✷✽✱ ✷✾❪✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ❡♥tr♦♣✐❛s ❞❡ ❘é♥②✐ ♣❛r❛ ❡①❝✐t❛çõ❡s ❞✐t❛s ♥ã♦✲❝♦♠♣❛❝t❛s s❡rã♦ ✐♥t❡r♣r❡t❛❞♦s✴❞✐s❝✉t✐❞♦s ❞❡ ✉♠ ♠♦❞♦ ❞✐❢❡r❡♥t❡ ❞❛ ♣r♦♣♦st❛ ♦r✐❣✐♥❛❧ ❬✷✽✱ ✷✾❪✳
◆♦ ❈❛♣ít✉❧♦ ✹ ❝♦♥s✐❞❡r❛r❡♠♦s ✉♠ ♠♦❞❡❧♦✱ ❡①❛t❛♠❡♥t❡ s♦❧ú✈❡❧✱ ❡♠ ❣❡♦♠❡tr✐❛ ❞❡ ❡s❝❛❞❛s ❞❡ ❞✉❛s ♣❡r♥❛s ❡ ❝♦♠ ✐♥t❡r❛çã♦ ❞❡ três✲s♣✐♥s✳ ❊st❡ ♠♦❞❡❧♦ ♣♦ss✉✐ ✉♠ ✐♥t❡r❡ss❛♥t❡ ❞✐❛❣r❛♠❛ ❞❡ ❢❛s❡s ❝♦♠ r❡❣✐õ❡s ❝♦♠ ❣❛♣ ✜♥✐t♦ ✭r❡❣✐õ❡s ♥ã♦ ❝rít✐❝❛✮ ❡ r❡❣✐õ❡s ❝♦♠ ✉♠ ❡ ❞♦✐s ♠♦❞♦s ❞❡ ❣❛♣✲
■◆❚❘❖❉❯➬➹❖ ✺ ♥✉❧♦ ✭r❡❣✐õ❡s ❝rít✐❝❛s✮✳ ❖❜t❡r❡♠♦s ❛♥❛❧✐t✐❝❛♠❡♥t❡ ❛s ❝♦rr❡çõ❡s ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ❞❛s ❡♥❡r❣✐❛s ♣❛r❛ ❛s r❡❣✐õ❡s ❞❡ ❣❛♣ ♥✉❧♦✱ ❡ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ s✉❛ ❝❛r❣❛ ❝❡♥tr❛❧❀ ❛ s❛❜❡r✱ ❡st❛ ❡s❝❛❞❛ ♣♦ss✉✐ ❛ ♠❡s♠❛ ❝❧❛ss❡ ❞❡ ✉♥✐✈❡rs❛❧✐❞❛❞❡ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦ ❞♦ ♠♦❞❡❧♦ ❳❳✱ c = 1✳ ❈♦♠♦ ✉s✉❛❧✱ ✉t✐❧✐③❛r❡♠♦s ♦ ♠ét♦❞♦ ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛s ❡♥tr♦♣✐❛s ❞❡ ❘é♥②✐✳ ❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡st❛s ❡♥tr♦♣✐❛s ❢♦✐ ❝♦♠♣❧❡t❛♠❡♥t❡ ❡♥t❡♥❞✐❞♦ ♥❛s ❞✉❛s r❡❣✐õ❡s ♥ã♦ ❝rít✐❝❛s✱ ✐♥❝❧✉s✐✈❡ ✉♠ r❡s✉❧t❛❞♦ ♥ã♦ tr✐✈✐❛❧ ❢♦✐ ❡♥❝♦♥tr❛❞♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ s✐st❡♠❛ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♣❡r✐ó❞✐❝❛✱ ♥❛ r❡❣✐ã♦ ❝rít✐❝❛ ❝♦♠ ❞♦✐s ♠♦❞♦s ❞❡ ❣❛♣✲♥✉❧♦✱ ❛ ❡♥tr♦♣✐❛ ❞❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡①✐❜❡ ❢♦rt❡s ♦s❝✐❧❛çõ❡s✳
❋✐♥❛❧✐③❛♥❞♦✱ ❛♣r❡s❡♥t❛♠♦s ♥♦ ❆♣ê♥❞✐❝❡ ❆✱ ♣❛r❛ ✉♠❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❜✐q✉❛❞rát✐❝❛ ❣❡r❛❧ ❡♠ ♦♣❡r❛❞♦r❡s ❢❡r♠✐ô♥✐❝♦s✿ ✭✐✮ s✉❛ ❞✐❛❣♦♥❛❧✐③❛çã♦ ❡①❛t❛ ❛tr❛✈és ❞❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❇♦❣♦❧✐✉❜♦✈❀ ✭✐✐✮ ❛ ❡♥tr♦♣✐❛ ❞❡ ❘é♥②✐ ❡♠ ❢✉♥çã♦ ❞♦s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ ❞❡ ❝♦rr❡❧❛çã♦✳
❈❛♣ít✉❧♦ ✶
❋❡♥ô♠❡♥♦s ❝rít✐❝♦s ❡ ❡♠❛r❛♥❤❛♠❡♥t♦
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❞✐✈❡rs♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s q✉❡ s❡rã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛ ❝♦♠♣r❡✲ ❡♥sã♦ ❞♦ r❡st❛♥t❡ ❞❡st❡ t❡①t♦✳ ❚❛♠❜é♠ ❧✐st❛r❡♠♦s ❞✐✈❡rs♦s r❡s✉❧t❛❞♦s ❣❡r❛✐s q✉❡ s❡rã♦ ❛♠♣❧❛✲ ♠❡♥t❡ ✉t✐❧✐③❛❞♦s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳ ❏á ♥❛ ♣ró①✐♠❛ s❡çã♦✱ ✐♥tr♦❞✉③✐r❡♠♦s ❛❧❣✉♠❛s ♥♦çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ❢❡♥ô♠❡♥♦s ❝rít✐❝♦s✳ ❈♦♠ ❝❡rt❡③❛✱ ✉♠❛ ❞❛s ❝❛r❛❝t❡ríst✐❝❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♦❜s❡r✈❛❞❛s ♥♦ ❡st✉❞♦ ❞❡ ❢❡♥ô♠❡♥♦s ❝rít✐❝♦s é ❛ ✉♥✐✈❡rs❛❧✐❞❛❞❡✿ ❞✐❢❡r❡♥t❡s s✐st❡♠❛s✶ ❛♣r❡s❡♥t❛♠ ❝♦♠♣♦rt❛♠❡♥t♦ ✐❞ê♥t✐❝♦s q✉❛♥❞♦ sã♦ r❡❡s❝❛❧❛❞♦s ♣♦r ❛❧❣✉♠ ♣❛râ♠❡tr♦
❝rít✐❝♦✳ ❚❛❧ ❝♦♥❝❡✐t♦ s❡rá ♠❡❧❤♦r ❡①♣❧✐❝❛❞♦ ♥♦ ❞❡❝♦rr❡r ❞❡st❡ ❝❛♣ít✉❧♦✳ ❆ ✜♠ ❞❡ ❡♥t❡♥❞❡r q✉❛❧✐✲ t❛t✐✈❛♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❝rít✐❝♦✱ ✈❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ✐♥tr♦❞✉③✐r ❛s ❞❡✜♥✐çõ❡s ❞❡ ♣♦♥t♦ ❝rít✐❝♦✱ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠✱ ❡①♣♦❡♥t❡s ❝rít✐❝♦s✱ ❡t❝✳ ❋✐♥❛❧♠❡♥t❡✱ ♥❛s s❡çõ❡s s❡❣✉✐♥✲ t❡s✱ r❡s✉♠✐r❡♠♦s ❛s ♣r❡❞✐çõ❡s ❞❛ ❈❋❚ ♣❛r❛ ❛s ❝♦rr❡çõ❡s ❞❡ t❛♠❛♥❤♦ ✜♥✐t♦ ❞❛s ❡♥❡r❣✐❛s ❡ ❞❛s ❡♥tr♦♣✐❛s ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦✳
✶✳✶ ❋❡♥ô♠❡♥♦s ❝rít✐❝♦s
❈♦♥s✐❞❡r❡♠♦s ✉♠ ♣❡❞❛ç♦ ❞❡ ❛❧❣✉♠ ♠❛t❡r✐❛❧✱ ♠❡❞✐♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛❝r♦s❝ó✲ ♣✐❝❛s✱ ♣♦r ❡①❡♠♣❧♦ s✉❛ ❞❡♥s✐❞❛❞❡✱ ❝♦♠♣r❡ss✐❜✐❧✐❞❛❞❡✱ ♠❛❣♥❡t✐③❛çã♦ ♦✉ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡✳ ❉✐✈✐❞❡✲ s❡ ❡♥tã♦ ❡st❡ ♠❛t❡r✐❛❧ ❡♠ ❞✉❛s ♣❛rt❡s✱ ♠❛♥t❡♥❞♦ ❛s ✈❛r✐á✈❡✐s ❡①t❡r♥❛s ✭♣r❡ssã♦✱ ❝❛♠♣♦ ♠❛❣♥é✲ t✐❝♦✱ ❡t❝✳✮ ✜①❛❞❛s✱ ❡ ♦❜s❡r✈❛✲s❡ q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛❝r♦s❝ó♣✐❝❛s ♥ã♦ sã♦ ❛❧t❡r❛❞❛s✳ ❊♥tr❡✲ t❛♥t♦ ❡st❡ ♣r♦❝❡ss♦ ♥ã♦ ♣♦❞❡ s❡r r❡♣❡t✐❞♦ ✐♥✜♥✐t❛♠❡♥t❡✳ ❊✈❡♥t✉❛❧♠❡♥t❡✱ ❛♣ós ♠✉✐t❛s ✐t❡r❛çõ❡s✱ ❝♦♠♣♦rt❛♠❡♥t♦s ❡str❛♥❤♦s ✐rã♦ s✉r❣✐r✱ ✉♠❛ ✈❡③ q✉❡ q✉❛♥❞♦ ❛❧❝❛♥ç❛r♠♦s ❛ ❡s❝❛❧❛ ❛tô♠✐❝❛✱ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ✐♥❞✐✈✐❞✉❛❧ ❞❡ át♦♠♦s ❡ ♠♦❧é❝✉❧❛s ✭❞✐❢❡r❡♥t❡ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ♠❛t❡r✐❛❧ q✉❡ ❡❧❡s✴❡❧❛s ❝♦♥st✐t✉❡♠✮ r❡❣❡rá ❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞♦ s✐st❡♠❛✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❡s❝❛❧❛ ❡♠ q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❣❧♦❜❛✐s ❞♦s ♣❡❞❛ç♦s ❝♦♠❡ç❛♠ ❛ ❞✐❢❡r✐r ❛❝❡♥t✉❛❞❛♠❡♥t❡ ❞❛s ♦r✐❣✐♥❛✐s ❞á✲♥♦s ✉♠❛ ♠❡❞✐❞❛ ❞♦ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ ξ✳ ❖✉ s❡❥❛✱ ❛ ❞✐stâ♥❝✐❛ s♦❜r❡
✶❉❡s❞❡ q✉❡ ❛s ❤❛♠✐❧t♦♥✐❛♥❛s ♣♦ss✉❛♠ ❛s ♠❡s♠❛s s✐♠❡tr✐❛s ❡ ♠❡s♠❛ ❞✐♠❡♥sã♦✳
❈❆P❮❚❯▲❖ ✶✳ ❋❊◆Ô▼❊◆❖❙ ❈❘❮❚■❈❖❙ ❊ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ✼ ❛ q✉❛❧ ✢✉t✉❛çõ❡s ❛♣❛r❡❝❡♠ ❞❡✈✐❞♦ ❛♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ♠✐❝r♦s❝ó♣✐❝♦s s❡r❡♠ ❝♦rr❡❧❛❝✐♦♥❛❞♦s ✉♥s ❝♦♠ ♦s ♦✉tr♦s ✭✈❡❥❛ ♣♦r ❡①❡♠♣❧♦ ❛s ❘❡❢s✳ ❬✸✶✕✸✸❪✮✳
❯s✉❛❧♠❡♥t❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ é ❞❛ ♦r❞❡♠ ❞❡ ❛❧❣✉♥s ❡s♣❛ç❛♠❡♥t♦s ❛tô♠✐❝♦s✱ ❡♠❜♦r❛ ❡st❡ ❞❡♣❡♥❞❛ ❞❛s ❝♦♥❞✐çõ❡s ❡①t❡r♥❛s q✉❡ ❞❡t❡r♠✐♥❛♠ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛✳ ➱ ❜❡♠ ❝♦♥❤❡❝✐❞♦ q✉❡ ♦s s✐st❡♠❛s ♠✉❞❛♠ ❛❜r✉♣t❛♠❡♥t❡ s❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ♠❛❝r♦s❝ó♣✐❝♦ ♠❡s♠♦ ♣❛r❛ s✉❛✈❡s ♠✉❞❛♥ç❛s ❞❛s ✈❛r✐á✈❡✐s ❡①t❡r♥❛s✳ ◆❡st❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ ♦❝♦rr❡✉ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❛ ♣❛rt✐r ❞❡ ✉♠ ❡st❛❞♦ ♣❛r❛ ♦✉tr♦✳ ❊①✐st❡♠ ❜❛s✐❝❛♠❡♥t❡ ❞✉❛s ♠❛♥❡✐r❛s ♥❛s q✉❛✐s t❛✐s tr❛♥s✐çõ❡s ♦❝♦rr❡♠ ❬✸✷✱ ✸✸❪✿
✭✐✮ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞❡s❝♦♥tí♥✉❛ ♦✉ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ♥❛ q✉❛❧ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ é ✜♥✐t♦✳ ◆❡st❡ ❝❛s♦✱ ♣❛rt❡ ❞♦ s✐st❡♠❛ ❝♦♠♣❧❡t♦✉ ❛ tr❛♥s✐çã♦ ❡ ♦✉tr❛s ♣❛rt❡s ♥ã♦✱ ✐✳❡✳✱ ❡①✐st❡ ✉♠ r❡❣✐♠❡ ❞❡ ❡st❛❞♦ ♠✐st♦ ❡♥tr❡ ♦s ❞♦✐s ✭♦✉ ♠❛✐s✮ ❡st❛❞♦s✳ ❈❡rt❛♠❡♥t❡✱ ♦ ❡①❡♠♣❧♦ ♠❛✐s ❝♦♥❤❡❝✐❞♦ ❞❡st❛ ❝❧❛ss❡ ❞❡ tr❛♥s✐çã♦ é ❛ ❢✉sã♦ ❞❡ ✉♠ só❧✐❞♦ tr✐❞✐♠❡♥s✐♦♥❛❧✳ ❖ s✐st❡♠❛ ❛❜s♦r✈❡ ♦✉ ❧✐❜❡r❛ ❡♥❡r❣✐❛✱ ❛ ✉♠❛ t❛①❛ ✜①❛✱ ❡♥q✉❛♥t♦ q✉❡ ❛ t❡♠♣❡r❛t✉r❛ ♣❡r♠❛♥❡❝❡ ❝♦♥st❛♥t❡✳ ❆ ❡str✉t✉r❛ ❞♦ ♠❛t❡r✐❛❧ ♠✉❞❛ ❞r❛st✐❝❛♠❡♥t❡ ❛ ♣❛rt✐r ❞❡ ✉♠ ❡st❛❞♦ ❡stá✈❡❧ ♣❛r❛ ♦✉tr♦✱ ❡ ❛✐♥❞❛ ♦ só❧✐❞♦ ♥ã♦ ♣❛ss❛ ✐♥st❛♥t❛♥❡❛♠❡♥t❡ ♣❛r❛ ❧íq✉✐❞♦✱ ♠❛s s✐♠ ❢♦r♠❛ ✉♠❛ ♠✐st✉r❛ ❞❡ ❞♦♠í♥✐♦s só❧✐❞♦s ✭♠♦❧é❝✉❧❛s ❛rr❛♥❥❛❞❛s ❡♠ ✉♠❛ r❡❞❡ ❝r✐st❛❧✐♥❛✮ ❡ ❞♦♠í♥✐♦s ❧íq✉✐❞♦s ✭♥ã♦ ❡①✐st❡ r❡❞❡ ❡ ❛s ♠♦❧é❝✉❧❛s ♣❡r❝♦rr❡♠ ❝❛♠✐♥❤♦s ❞❡s♦r❞❡♥❛❞♦s✮ ♥♦ ♣♦♥t♦ ❞❡ ❢✉sã♦❀
✭✐✐✮ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❝♦♥tí♥✉❛ ♦✉ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ♥❛ q✉❛❧ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ t♦r♥❛✲s❡ ❡❢❡t✐✈❛♠❡♥t❡ ✐♥✜♥✐t♦✳ ◆❡st❡ ❝❛s♦✱ ❛s ✢✉t✉❛çõ❡s ❡stã♦ ❝♦rr❡❧❛❝✐♦♥❛❞❛s ❡♠ t♦❞❛s ❛s ❡s❝❛❧❛s ❞❡ ❞✐stâ♥❝✐❛ ♥♦ ♠❛t❡r✐❛❧✱ ❢♦rç❛♥❞♦ ❡♥tã♦ ♦ s✐st❡♠❛ ❛ ❡st❛r ❡♠ ✉♠❛ ú♥✐❝❛✱ ❡ ❝rít✐❝❛✱ ❢❛s❡✳ ➚ ♠❡❞✐❞❛ q✉❡ ❡st❡ ♣♦♥t♦ ❝rít✐❝♦ é ❛♣r♦①✐♠❛❞♦ ❛s ❞✉❛s ✭♦✉ ♠❛✐s✮ ❢❛s❡s t♦r♥❛♠✲s❡ ✐❞ê♥t✐❝❛s✳ ❘❡t♦r♥❛♥❞♦ ❛♦ ❡①❡♠♣❧♦ ❞♦ ✢✉í❞♦✱ ❛❣♦r❛ ♥❛ tr❛♥s✐çã♦ ❡♥tr❡ ❛ ❢❛s❡ ❧íq✉✐❞❛ ❡ ❛ ❣❛s♦s❛✱ q✉❡ é ✉♠❛ tr❛♥s✐çã♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ✈✐st♦ q✉❡✱ ❛ ❞❡♥s✐❞❛❞❡ ❞♦s ❞♦♠í♥✐♦s ❞♦ ♠❛t❡r✐❛❧ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❡♥tr❡ ❞♦✐s ✈❛❧♦r❡s ❜❡♠ ❞❡✜♥✐❞♦s✱ ✉♠ ❛❧t♦ ✭❧íq✉✐❞♦✮ ❡ ♦✉tr♦ ❜❛✐①♦ ✭✈❛♣♦r✮✳ ❊♥tr❡t❛♥t♦ ❡st❛ ❧✐♥❤❛ ❞❡ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ é ❜❡♠ ❝♦♥❤❡❝✐❞❛ ♣♦r t❡r♠✐♥❛r ❡♠ ✉♠ ♣♦♥t♦✱ ♦ ♣♦♥t♦ ❝rít✐❝♦ ❧íq✉✐❞♦✲❣ás✱ ♥♦ q✉❛❧ ❧íq✉✐❞♦ s❡ tr❛♥s❢♦r♠❛ ❡♠ ❣ás ❝♦♥t✐♥✉❛♠❡♥t❡ ❡ ❛ ❡s❝♦❧❤❛ ❡♥tr❡ ❞✉❛s ❞❡♥s✐❞❛❞❡s ❝❡ss❛✳ ❊♥tã♦✱ ❞❡✈✐❞♦ às ✢✉t✉❛çõ❡s ❞❡ ❞❡♥s✐❞❛❞❡ ❞✐③❡♠♦s q✉❡ ♥ã♦ ❡①✐st❡ ❞✐st✐♥çã♦ ❡♥tr❡ ❛s ❢❛s❡s ❧íq✉✐❞❛s ❡ ❣❛s♦s❛s✱ s❡ ♣ró①✐♠❛s ❞♦ ♣♦♥t♦ ❝rít✐❝♦✳ ❊st❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ é ♠❛r❝❛❞❛ ♣❡❧♦ ❝❤❛♠❛❞♦ ❢❡♥ô♠❡♥♦ ❞❛ ♦♣❛❧❡s❝ê♥❝✐❛ ❝rít✐❝❛✱ ♥♦ q✉❛❧ ♦ ❧íq✉✐❞♦ ✜❝❛ ❝♦♠ ✉♠❛ ❛♣❛rê♥❝✐❛ ❧❡✐t♦s❛✱ ❞❛❞♦ q✉❡ ❛s ✢✉t✉❛çõ❡s ❞❡ ❞❡♥s✐❞❛❞❡ ♣r♦❞✉③❡♠ ✢✉t✉❛çõ❡s ♥♦ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦ ❬✸✸❪✳
❈❆P❮❚❯▲❖ ✶✳ ❋❊◆Ô▼❊◆❖❙ ❈❘❮❚■❈❖❙ ❊ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ✽
T m
T=T
c
(h=0)
m
h
T=Tc T<T
c
T>T
c
(a)
(b)
❋✐❣✉r❛ ✶✳✶✳✶✿ ●rá✜❝♦s q✉❛❧✐t❛t✐✈♦s ♣❛r❛ ✉♠ ♠❛t❡r✐❛❧ ♠❛❣♥ét✐❝♦ ❣❡♥ér✐❝♦✳ ✭❛✮ ▼❛❣♥❡t✐③❛çã♦ ❡♠ ❢✉♥çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ♥❛ ❛✉sê♥❝✐❛ ❞❡ ❝❛♠♣♦ ❡①t❡r♥♦✳ ✭❜✮ ▼❛❣♥❡t✐③❛çã♦ ❡♠ ❢✉♥çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ ❛♣❧✐❝❛❞♦ ♣❛r❛ ❞✐✈❡rs❛s t❡♠♣❡r❛t✉r❛s✳ ◆♦t❡ q✉❡ ♣❛r❛ T < T❝ ❡ h→0± t❡♠♦s
♠❛❣♥❡t✐③❛çõ❡s ±m0✱ ✐✳❡✳✱ ❞✉❛s ♣♦ssí✈❡✐s✳
♦r❞❡♠ é ✉♠❛ ❞✐❢í❝✐❧ t❛r❡❢❛ r❡q✉❡r❡♥❞♦ ❝❡rt❛ ❤❛❜✐❧✐❞❛❞❡✳
❆♣❡s❛r ❞♦ ✐♥tr♦❞✉tór✐♦ ❡ ♣❡❞❛❣ó❣✐❝♦ ❡①❡♠♣❧♦ ❞❡ ✉♠ ✢✉í❞♦✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ ❛q✉✐ é ♦ tr❛✲ t❛♠❡♥t♦ ❞❡ s✐st❡♠❛s ♠❛❣♥ét✐❝♦s✳ ■♥✐❝✐❡♠♦s ♣♦rt❛♥t♦ ❝♦♠ ✉♠❛ ✈✐sã♦ q✉❛❧✐t❛t✐✈❛ ❞♦ q✉❡ é ♦ ❢❡rr♦♠❛❣♥❡t✐s♠♦✳ ❙❛❜❡♠♦s q✉❡ ♦s át♦♠♦s ❞❡ ❝❡rt♦s ♠❛t❡r✐❛✐s ♣♦ss✉❡♠ ✉♠ t✐♣♦ ❞❡ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r ✐♥trí♥s❡❝♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ s♣✐♥✱ ❡♥q✉❛♥t♦ q✉❡ ❛ ♥í✈❡❧ ♠❛❝r♦s❝ó♣✐❝♦ ❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛ ❛❝❡ssí✈❡❧ é ❛ ♠❛❣♥❡t✐③❛çã♦ m~✱ q✉❡ é s✐♠♣❧❡s♠❡♥t❡ ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ ✈❛❧♦r ♠é❞✐♦ ❞❛s ✈❛r✐á✈❡✐s ❞❡ s♣✐♥ ❞❡♥tr♦ ❞❡ ✉♠ ✈♦❧✉♠❡ ❞❡ ✐♥t❡r❡ss❡ ❞♦ ♠❛t❡r✐❛❧✳ P❛r❛ ♠✉✐t♦s ♠❛t❡r✐❛✐s ♦s s♣✐♥s ♦r✐❡♥t❛♠✲s❡ ❛❧❡❛t♦r✐❛♠❡♥t❡✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ t♦t❛❧ ♣r♦❞✉③✐❞❛ ♣♦r ❡❧❡s é ♥✉❧❛✱ ❡♥tr❡t❛♥t♦ ❡①✐st❡ ✉♠❛ ♦✉tr❛ ❝❧❛ss❡ ❞❡ ♠❛t❡r✐❛✐s ❡♠ q✉❡ s❡ ♦❜s❡r✈❛ ♠❛❝r♦s❝♦♣✐❝❛♠❡♥t❡ ✉♠❛ ♠❛❣♥❡t✐③❛çã♦ ♥ã♦ ♥✉❧❛✱ ❞❡✈✐❞♦ à ✐♥t❡r❛çã♦ ❞♦s s♣✐♥s✳ ❊♠ t❛✐s ♠❛t❡r✐❛✐s✱ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❢❡rr♦♠❛❣♥ét✐❝♦s✱ ✐st♦ é ♦❜s❡r✈❛❞♦ ♣❛r❛ ✈❛❧♦r❡s ❞❡ t❡♠♣❡r❛t✉r❛s ♠❡♥♦r❡s q✉❡ ✉♠ ✈❛❧♦r T❝✱ ❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ♦✉ ❞❡ ❈✉r✐❡✱ q✉❡ ❞❡♣❡♥❞❡ ❞♦ ♠❛t❡r✐❛❧ ❡♠ q✉❡stã♦ ❬✸✶✕✸✸❪✳
❈❆P❮❚❯▲❖ ✶✳ ❋❊◆Ô▼❊◆❖❙ ❈❘❮❚■❈❖❙ ❊ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ✾ ♠❛❣♥❡t✐③❛çã♦ t❡♠ ♦ s❡❣✉✐♥t❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♠ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❬✸✶✕✸✸❪
m=m0(T −T❝)β✱ ✭✶✳✶✳✶✮
♣❛r❛♠❡tr✐③❛❞♦ ♣❡❧♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ β✳
❖ ♦✉tr♦ ♣❛râ♠❡tr♦ ❡①t❡r♥♦ q✉❡ ♣♦❞❡♠♦s ✈❛r✐❛r é ♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ h✳ ❖❜s❡r✈❛✲s❡ q✉❡ ❛❜❛✐①♦ ❞❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ✭T < T❝✮ ♦ ✈❡t♦r ♠❛❣♥❡t✐③❛çã♦ −→m ♥ã♦ é ✜①♦ ❡ ❞❡♣❡♥❞❡ ❞❛
♦r✐❡♥t❛çã♦ ❞♦ ✈❡t♦r ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✱ ❡ ❛✐♥❞❛✱ ♣❛r❛h= 0 t❡♠♦s ✉♠❛ ❧✐✈r❡ ❡s❝♦❧❤❛ q✉❡ ❞❡♣❡♥❞❡ ❞♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❛♥t❡s ❞♦ ❝❛♠♣♦ s❡r ❞❡s❧✐❣❛❞♦✳ ❖✉ s❡❥❛✱ ❛ ♠❛❣♥❡t✐③❛çã♦ é ❞❡s❝♦♥tí♥✉❛✱ t❡♥❞♦ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ q✉❛❧✐t❛t✐✈♦ ❝♦♠♦ ♠♦str❛❞♦ ♥❛ ❋✐❣✉r❛ ✶✳✶✳✶✭❜✮✳ ❊st❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ é ❝❛r❛❝t❡ríst✐❝❛ ❞❡ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❛♣r♦①✐♠❛♥❞♦✲s❡ ❞❡ ③❡r♦ q✉❛♥❞♦ ❛ t❡♠♣❡r❛t✉r❛ t❡♥❞❡ à t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ❡ ❞❡✐①❛ ❞❡ ❡①✐st✐r ♣❛r❛ T > T❝✳ ❖ ♣♦♥t♦ h = 0 ❡ T =T❝ é ❡♥tã♦ ♦✉tr♦ ❡①❡♠♣❧♦ ❞❡ ♣♦♥t♦ ❝rít✐❝♦ ♥♦ q✉❛❧ ❛ tr❛♥s✐çã♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ t♦r♥❛✲s❡ ❝♦♥tí♥✉❛✳
❆ ♠❛❣♥❡t✐③❛çã♦✱ ❣r❛♥❞❡③❛ q✉❡ q✉❛♥t✐✜❝❛ ❛ ♦r❞❡♠ ♠❛❣♥ét✐❝❛✱ é ♦ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ ❞❡st❛ tr❛♥s✐çã♦✳ ❈♦♠♦ ♠❡♥❝✐♦♥❛❞♦ ♥♦ ♣r✐♠❡✐r♦ ♣❛rá❣r❛❢♦✱ ❛ ♠❛✐♦r✐❛ ❞❛s q✉❛♥t✐❞❛❞❡s ❢ís✐❝❛s ❞❡ ✐♥t❡✲ r❡ss❡ ❡①✐❜❡♠ ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❛♥ô♠❛❧♦ ❡♠ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ q✉❛♥❞♦ ♦ s✐st❡♠❛ ❡stá s✉✜❝✐❡♥t❡✲ ♠❡♥t❡ ♣ró①✐♠♦ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ✭♥♦ ❝❛s♦ h= 0 ❡ T =T❝✮✳ ❊♥tr❡t❛♥t♦✱ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ ❞❡♣❡♥❞❡ ❞❡ ❝♦♠♦ ♦s s♣✐♥s ❡stã♦ ❝♦rr❡❧❛❝✐♦♥❛❞♦s✳ ❈♦♥❢♦r♠❡ ❞✐t♦ ♥♦ s❡❣✉♥❞♦ ♣❛rá❣r❛❢♦✱ ✉♠❛ ✐♠♣♦rt❛♥t❡ ♠❡❞✐❞❛ ❞❡st❛ ❝♦rr❡❧❛çã♦ é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ ξ✱ q✉❡ é ✉♠❛ ♠❡❞✐❞❛ ❞❛ ❞✐stâ♥❝✐❛ ♥❛ q✉❛❧ ♦s s♣✐♥s ❡stã♦ ❝♦rr❡❧❛❝✐♦♥❛❞♦s✱ ♦✉ ♠❡❧❤♦r✱ ♦ t❛♠❛♥❤♦ tí♣✐❝♦ ❞❛s r❡❣✐õ❡s ❡♠ q✉❡ ♦s s♣✐♥s ❡stã♦ ❛❧✐♥❤❛❞♦s✳
■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ❢♦❝❛r ❛ ♥♦ss❛ ❛t❡♥çã♦ ♥♦s ❛s♣❡❝t♦s ❝❧áss✐❝♦s ❞♦ ♣r♦❜❧❡♠❛✱ ✐✳❡✳✱ ❝♦♥s✐❞❡✲ r❛♥❞♦ ♦s s♣✐♥s ❝♦♠♦ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❝❧áss✐❝♦s✳ ❙✉♣♦♠♦s q✉❡ ❛ ❝❛❞❛ s♣✐♥ ❡stá ❛ss♦❝✐❛❞♦ ❝♦♠ ✉♠ ✈❡t♦r −→S(−→r )q✉❡ ❡stá ❧♦❝❛❧✐③❛❞♦ ❡♠ ✉♠ sít✐♦−→r ❞❡ ✉♠❛ r❡❞❡d✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ ❤✐♣❡r✈♦❧✉♠❡ V✳ ▲♦❣♦ ❡st❡s s♣✐♥s ❢♦r♠❛♠ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ✭❝♦♠ ❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦ ❞❡ s♣✐♥s ❝♦♥s✐❞❡✲ r❛❞♦✮ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ d✳✷ ❙❡❥❛ H✐♥t ❛ ❡♥❡r❣✐❛ ❞❡✈✐❞♦ às ✐♥t❡r❛çõ❡s ❡♥tr❡ ♦s s♣✐♥s✱ ❡♥tã♦✱ ❛ ❤❛♠✐❧t♦♥✐❛♥❛ ❞♦ s✐st❡♠❛ éH =H✐♥t−−→h ·−S→T✱ s❡♥❞♦ ♦ ú❧t✐♠♦ t❡r♠♦ ❛ ✐♥t❡r❛çã♦ ❞♦ s♣✐♥
t♦t❛❧ −S→T = P
− →r
−
→S(−→r) ❝♦♠ ♦ ❝❛♠♣♦ ❡①t❡r♥♦ −→
h✱ ❞❡ ♠♦❞♦ q✉❡ ❛ ♠❛❣♥❡t✐③❛çã♦ é −→m = D−S→T
E
/V✳ ❱❛♠♦s ❡♥tã♦ ❞❡✜♥✐r ❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❞❡ ❞♦✐s ♣♦♥t♦s✱ q✉❡ ♠❡❞❡ ♦ ❣r❛✉ ❞❡ ❛❧✐♥❤❛♠❡♥t♦ ❡♥tr❡ ❞♦✐s s♣✐♥s ♥❛s ♣♦s✐çõ❡s −→r1 ❡ −→r2✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
G(2)(−→r1,−→r2) =
D−→
S(−→r1)·−→S(−→r2)
E
✳
➱ ✐♠♣♦rt❛♥t❡✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ✐♠❛❣✐♥❛r♠♦s ❛❣♦r❛ q✉❡ ♦ s✐st❡♠❛ é ✐♥✈❛r✐❛♥t❡ à tr❛♥s❧❛çã♦ ❡ r♦t❛çã♦✱ ❞❡ ♠♦❞♦ q✉❡ G(2)(→−r
1,−→r2) =G(2)(|−→r2 − −→r1|) ❡
D−→
S(−→r1)
E
=D−→S(−→r2)
E
=−→S0✳
✷P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ s✉♣♦r❡♠♦s q✉❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❡ s♣✐♥s é ✉♥✐tár✐❛✱ ✐✳❡✳✱N =V✱ ♦♥❞❡N é ♦ ♥ú♠❡r♦ ❞❡ s♣✐♥s
❈❆P❮❚❯▲❖ ✶✳ ❋❊◆Ô▼❊◆❖❙ ❈❘❮❚■❈❖❙ ❊ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ✶✵ ❯♠❛ ♠❡❧❤♦r ♠❡❞✐❞❛ é ❛ ❢✉♥çã♦ ❞❡ ❝♦rr❡❧❛çã♦ ❝♦♥❡❝t❛❞❛
G(2)
❝ (|−→r2 − −→r1|) =
Dh−→
S(−→r1)−−→S0
i
·h−→S(−→r2)−−→S0
iE
✱ ✭✶✳✶✳✷✮
q✉❡ q✉❛♥t✐✜❝❛ ❛ ❝♦rr❡❧❛çã♦ ❞♦ ❞❡s✈✐♦ ❞❛ ♠é❞✐❛ −→S0 ❞❡ ❝❛❞❛ s♣✐♥✳ ❙❡♥❞♦ ❡♥tã♦ ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛
♠❡❞✐r ❛s ✢✉t✉❛çõ❡s ♣❛r❛ ♦ ❝❛s♦ T < T❝✱ ✉♠❛ ✈❡③ q✉❡ ♥❡st❡ ❝❛s♦ ♦s s♣✐♥s ❡stã♦ ♣r❡❞♦♠✐♥❛♥t❡✲ ♠❡♥t❡ ❛❧✐♥❤❛❞♦s✳ ◆♦t❡ q✉❡✱
G(2)❝ (|−→r2 − −→r1|) =
D−→
S(−→r1)·−→S(−→r2)
E
−D−→S(−→r1)
E
·D−→S(−→r2)
E
é ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❡st❛tíst✐❝❛ ❡♥tr❡ ♦s s♣✐♥s ❞✐st❛♥t❡s ❞❡|−→r2−−→r1|✳ ❙♦♠❛♥❞♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ s♦❜r❡
t♦❞♦s ♦s ✈❛❧♦r❡s ❞❡ −→r1 ❡−→r2 t❡♠♦s P − →r1,−→r2G
(2)
❝ (|−→r2− −→r1|) =
−→
ST −
D−→
ST
E2
=k❇T ∂hhSTi✳ ❙❡♥❞♦
k❇ ❛ ❝♦♥st❛♥t❡ ❞❡ ❇♦❧t③♠❛♥♥✱ ST ❛ ❝♦♠♣♦♥❡♥t❡ ❞♦ s♣✐♥ t♦t❛❧ ❛♦ ❧♦♥❣♦ ❞❛ ❞✐r❡çã♦ ❞♦ ❝❛♠♣♦
♠❛❣♥ét✐❝♦ ~h ST = ˆh·−S→T
❡ ❛ ❞❡r✐✈❛❞❛ t❡r♠♦❞✐♥â♠✐❝❛ é ♣r♦♣♦r❝✐♦♥❛❧ à s✉s❝❡♣t✐❜✐❧✐❞❛❞❡ χ✳ ❊st❛ ú❧t✐♠❛ é ❞❛❞❛ ♣♦r
χ= ∂m
∂h =
∂ ∂h
hSTi
V
✱
q✉❡ ♣♦rt❛♥t♦ ✐♥❞✐❝❛ ❛ r❡s♣♦st❛ ❞❛ ♠❛❣♥❡t✐③❛çã♦ ❛ ✉♠❛ ♣❡q✉❡♥❛ ✈❛r✐❛çã♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦✳ ❋✐♥❛❧♠❡♥t❡✱ ❞❡✈✐❞♦ às s✐♠❡tr✐❛s ❞❡ r♦t❛çã♦ ❡ tr❛♥s❧❛çã♦✱ P
− →
r1
G(2)❝ (|−→r2 − −→r1|) é ✐♥❞❡♣❡♥❞❡♥t❡
❞❡ −→r2 ❡ ♣♦rt❛♥t♦
χ= (k❇T)−1
∞
X
r=0 G(2)
❝ (r)✱ ✭✶✳✶✳✸✮
♦ q✉❡ ♥♦s ❧❡✈❛ ❛ ❝♦♥❝❧✉✐r q✉❡ ❛ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡ é ✉♠❛ ♠❡❞✐❞❛ ❞❛ ❝♦❡rê♥❝✐❛ ❡st❛tíst✐❝❛ ❞❡ ✉♠ s✐st❡♠❛✱ ✐✳❡✳✱ ❞❛ ❝♦rr❡❧❛çã♦ ❡♥tr❡ ♦s s♣✐♥s ❞♦ s✐st❡♠❛✳ ❉❡st❡ ♠♦❞♦✱ ♣ró①✐♠♦ ❛♦ ♣♦♥t♦ ❝rít✐❝♦ ✭h= 0 ❡ T →T❝✮✱ ♥♦ q✉❛❧ ♦s s♣✐♥s ❡stã♦ t♦❞♦s ❝♦rr❡❧❛❝✐♦♥❛❞♦s✱ ❞❡✈❡ ❤❛✈❡r ✉♠❛ ❞✐✈❡r❣ê♥❝✐❛ ♥❛ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡✳ ■st♦ é ❞❡ ❢❛t♦ ♦❜s❡r✈❛❞♦✱ ❡ t❛♠❜é♠ ❡♠ ❢♦r♠❛ ❞❡ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❬✸✶✕✸✸❪
χ∝|T −T❝|−γ✳ ✭✶✳✶✳✹✮
❆ ❝♦♥✜❣✉r❛çã♦ ❞❡ s♣✐♥ é ♦❜t✐❞❛ ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ ❛tr❛✈és ❞❡ ❡s♣❛❧❤❛♠❡♥t♦ ❞❡ ♥ê✉tr♦♥s✱ ♣♦r ❡①❡♠♣❧♦✳ ◆❡ss❡ ❡①♣❡r✐♠❡♥t♦ ♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❞♦s ♥ê✉tr♦♥s ✐♥t❡r❛❣❡♠ ❝♦♠ ♦s s♣✐♥s ❞♦s ❡❧étr♦♥s✳ ❖ q✉❡ é ♠❡❞✐❞♦ é ❛ s❡çã♦ ❞❡ ❝❤♦q✉❡✱ ♦✉ t❛①❛ ❞❡ ❡s♣❛❧❤❛♠❡♥t♦✱ q✉❡ é ♣r♦♣♦r❝✐♦♥❛❧ à tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ G(2)❝ (r) ✭✈❡❥❛ ♣♦r ❡①❡♠♣❧♦ ❛ ❘❡❢✳ ❬✸✶❪✮✱ ❞❡ ♠♦❞♦ q✉❡ ♣♦❞❡♠♦s t❡r
✉♠❛ ✐♠❛❣❡♠ ❞❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ s♣✐♥ ✈✐❛ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛✳
▲♦♥❣❡ ❞♦ ♣♦♥t♦ ❝rít✐❝♦ T 6= T❝ é ♦❜s❡r✈❛❞♦ q✉❡ ❛ ❢✉♥çã♦ ❝♦rr❡❧❛çã♦ t❡♠ ✉♠ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❛ss✐♥tót✐❝♦ ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ❬✸✷❪✱
G(2)❝ (r)∝ e−
r/ξ
❈❆P❮❚❯▲❖ ✶✳ ❋❊◆Ô▼❊◆❖❙ ❈❘❮❚■❈❖❙ ❊ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ✶✶
❡①♣♦❡♥t❡ ❞❡✜♥✐çã♦ ❝♦♥❞✐çã♦
β m∝(T❝−T)β T < T❝ ❡ h= 0
δ m∝h1/δ T =T
❝
γ χ=∂hm∝ |T −T❝|−γ h= 0
α C ∝ |T −T❝|−α h= 0
ν ξ∝ |T −T❝|−ν h= 0
η G(2)❝ (r)∝r−(d−2+η) T =T❝
❚❛❜❡❧❛ ✶✳✶✿ ❉❡✜♥✐çã♦ ❞♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ♠❛✐s ❝♦♠✉♥s ❬✸✶✕✸✹❪✳ ❙❡♥❞♦ C ♦ ❝❛❧♦r ❡s♣❡❝í✜❝♦✳
s❡♥❞♦ n ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡ ξ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♥♦ ♣♦♥t♦ ❝rít✐❝♦ ♦ s✐st❡♠❛ ❝♦♠♣♦rt❛✲s❡ ❝♦♠♣❧❡t❛♠❡♥t❡ ❞✐❢❡r❡♥t❡✱ ♥❡st❡ ❧✐♠✐t❡ ξ → ∞✱ ❡ ❛ ❝♦rr❡❧❛çã♦ ❞❡❝❛✐ ❝♦♠ ♦✉tr❛ ❧❡✐ ❞❡ ♣♦tê♥❝✐❛ ❛♥ô♠❛❧❛ ❬✸✶✱ ✸✷❪
G(2)❝ (r)∝ 1
rd−2+η✱ ✭✶✳✶✳✻✮
s❡♥❞♦ ηs❡✉ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ❡d ❛ ❞✐♠❡♥sã♦ ❞♦ s✐st❡♠❛✳ ❊♥tã♦✱ ❡♠ ❣❡r❛❧ ❞❡✈❡♠♦s t❡rG(2)❝ (r)∝
g(r/ξ)
rd−2+η ✱ ♦ q✉❡ ❡♥❢❛t✐③❛ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❝♦rr❡❧❛çã♦ ❞❡✈❡ ❞✐✈❡r❣✐r ♥♦ ♣♦♥t♦ ❝rít✐❝♦✳ ❆ ✜♠ ❞❡ ❝❛r❛❝t❡r✐③❛r ❡st❛ ❞✐✈❡r❣ê♥❝✐❛ ❞❡✜♥✐♠♦s ♦✉tr♦ ❡①♣♦❡♥t❡ ❝rít✐❝♦ ν✱ t❛❧ q✉❡ ♣❛r❛h= 0
ξ ∝ |T −T❝|−ν✳ ✭✶✳✶✳✼✮
◆❛ ❚❛❜❡❧❛ ✶✳✶ ❛♣r❡s❡♥t❛♠♦s ❛s ❞❡✜♥✐çõ❡s ❞♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s ❛ss♦❝✐❛❞♦s ❛ ✈ár✐❛s q✉❛♥t✐✲ ❞❛❞❡s ❢ís✐❝❛s✳ ➱ ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ♦❜s❡r✈❛❞❛s ♥❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ sã♦ ❞✐t❛s ✉♥✐✈❡rs❛✐s ♥♦ s❡♥t✐❞♦ q✉❡✿ s✐st❡♠❛s ♠✐❝r♦s❝♦♣✐❝❛♠❡♥t❡ ❞✐st✐♥t♦s t❡rã♦ ♦ ♠❡s♠♦ ❝♦♥❥✉♥t♦ ❞❡ ❡①♣♦❡♥t❡s ❝rít✐❝♦s✱ ❞❡s❞❡ q✉❡ ❛s ❤❛♠✐❧t♦♥✐❛♥❛s ❝♦♠♣❛rt✐❧❤❡♠ ❛s ♠❡s♠❛s s✐♠❡tr✐❛s ❡ t❡♥❤❛♠ ❛ ♠❡s♠❛ ❞✐♠❡♥sã♦✳ ➱ ❞❡✈✐❞♦ ❛ ❡st❡ ❢❛t♦ q✉❡ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s é ❡♥❝♦♥tr❛❞♦ ❡♠ s✐st❡♠❛s ❝♦♠♣❧❡t❛♠❡♥t❡ ❞✐❢❡r❡♥t❡s✱ ♣♦r ❡①❡♠♣❧♦ ♥♦ ✢✉í❞♦✱ ✐✳❡✳✱ ♦s ❡①♣♦❡♥t❡s ❝rít✐❝♦s s❡❣✉❡♠ ✉♠❛ r❡❣r❛ ✐❞ê♥t✐❝❛ à ❞♦ ❢❡rr♦♠❛❣♥❡t♦ ✭❢❛③❡♥❞♦ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❞❡ ❝❛❞❛ ♣r♦❜❧❡♠❛✮✳
➱ ✐♠♣♦rt❛♥t❡ s❛❧✐❡♥t❛r q✉❡ ❡st❡s ❡①♣♦❡♥t❡s ♥ã♦ sã♦ t♦❞♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♣♦r ❡①❡♠♣❧♦✱ ✉s❛♥❞♦ ❛s ❊qs✳ ✭✶✳✶✳✸✮ ❡ ✭✶✳✶✳✼✮✱ t❡♠♦s χ ∼ ´∞
0 dr
g(r/ξ)
rd−2+η
r=ξz
= ξ2−η´∞
0 dzz
1−ηg(z) ∼ ξ2−η ∼
|T −T❝|−ν(2−η) ❡ ❡♥tã♦✱ ❛tr❛✈és ❞❡ ✉♠❛ s✐♠♣❧❡s ❛♥á❧✐s❡ ❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ❛ ❊q✳ ✭✶✳✶✳✹✮✱ é ♣♦ssí✲ ✈❡❧ ♦❜t❡r ❛ ❧❡✐ ❞❡ ❡s❝❛❧❛ γ =ν(2−η)✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❧❡✐ ❞❡ ❋✐s❤❡r ❬✸✹✱ ✸✺❪✳
