Estudo de interações competitivas e anisotropia em sistemas magnéticos

❊st✉❞♦ ❞❡ ■♥t❡r❛çõ❡s ❈♦♠♣❡t✐t✐✈❛s ❡ ❆♥✐s♦tr♦♣✐❛ ❡♠

❙✐st❡♠❛s ▼❛❣♥ét✐❝♦s

❊st✉❞♦ ❞❡ ■♥t❡r❛çõ❡s

❈♦♠♣❡t✐t✐✈❛s ❡ ❆♥✐s♦tr♦♣✐❛ ❡♠

❙✐st❡♠❛s ▼❛❣♥ét✐❝♦s

❘♦❞r✐❣♦ ❙❛♥t♦s ❞❛ ▲❛♣❛

❖r✐❡♥t❛❞♦r✿ ❆♥t♦♥✐♦ ❙ér❣✐♦ ❚❡✐①❡✐r❛ P✐r❡s

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s ✕ ❋ís✐❝❛✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛♦ ♣r♦❢❡ss♦r ❆♥tô♥✐♦ ❙ér❣✐♦ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❝♦♠♦ ♦r✐❡♥t❛❞♦r ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆❣r❛❞❡ç♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯❋▼●✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❡s♣♦s❛ P❛✉❧❛ ♣❡❧❛ ❛❥✉❞❛ ❡ ♣❛❝✐ê♥❝✐❛ ♥♦ ❞❡❝♦rr❡r ❞♦s ✹ ❛♥♦s ❞❡ ❞♦✉t♦r❛❞♦✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♠ã❡ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ♣❡❧♦ ❡s❢♦rç♦✳

❘❡s✉♠♦

❊st✉❞❛♠♦s ♦ ❡❢❡✐t♦ ❞❡ ✐♥t❡r❛çõ❡s ❝♦♠♣❡t✐t✐✈❛s ❡ ❛♥✐s♦tr♦♣✐❛s ♥♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❡♠ ❞✐❢❡r❡♥t❡s s✐t✉❛çõ❡s✳ ■♥✐❝✐❛❧♠❡♥t❡✱ tr❛t❛♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦ ❞❡ s♣✐♥ S = 1 ❝♦♠ ✐♥t❡r❛çõ❡s ❝♦♠♣❡t✐t✐✈❛s

❡♥tr❡ ♣r✐♠❡✐r♦s J1 ❡ s❡❣✉♥❞♦s J2 ✈✐③✐♥❤♦s ✭♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ J1−J2✮ ❡

❛♥✐s♦tr♦♣✐❛ ❞❡ ♣❧❛♥♦ ❢á❝✐❧ ♥✉♠❛ r❡❞❡ ❝ú❜✐❝❛ ❝♦♠ ✐♥t❡r❛çõ❡s J3 ❡♥tr❡ ♣❧❛♥♦s✳

❯t✐❧✐③❛♠♦s ❛ ❛♣r♦①✐♠❛çã♦ ❤❛r♠ô♥✐❝❛ ❛✉t♦✲❝♦♥s✐st❡♥t❡ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ t❡♠✲ ♣❡r❛t✉r❛ ❞❡ tr❛♥s✐çã♦ ❞❛ ❢❛s❡ ♣❛r❛♠❛❣♥ét✐❝❛ ♣❛r❛ ❛ ❢❛s❡ ♦r❞❡♥❛❞❛ ❛ ❜❛✐①❛ t❡♠♣❡r❛t✉r❛✳ ❚❛♠❜é♠ ♦❜t✐✈❡♠♦s✱ à t❡♠♣❡r❛t✉r❛ ♥✉❧❛✱ ♦ ✈❛❧♦r ❝rít✐❝♦ ❞❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ ♣❧❛♥♦ ❢á❝✐❧ q✉❡ s❡♣❛r❛ ❛ r❡❣✐ã♦ ♣❛r❛ ✈❛❧♦r❡s ❞❡D ♣❡q✉❡♥♦✱ ❞❛ ❢❛s❡ ♣❛r❛♠❛❣♥ét✐❝❛ q✉â♥t✐❝❛ ♣❛r❛ ✈❛❧♦r❡s ❞❡ D ❣r❛♥❞❡✳ ❊♥❝♦♥tr❛♠♦s ✉♠❛ ❢❛s❡ ❞❡s♦r❞❡♥❛❞❛ ❛ t❡♠♣❡r❛t✉r❛ ♥✉❧❛ q✉❡ ♣♦❞❡ s❡r ✉♠❛ ♣♦ssí✈❡❧ ❝❛♥❞✐❞❛t❛ ❛ ❢❛s❡ ❧íq✉✐❞♦ ❞❡ s♣✐♥✳ ❊♠ s❡❣✉✐❞❛✱ ✉t✐❧✐③❛♠♦s ❛ t❡♦r✐❛ ❞♦s ❜ós♦♥s ❞❡ ❙❝❤✇✐♥✲ ❣❡r ♣❛r❛ ❡st✉❞❛r ♦ ❡❢❡✐t♦ ❞❛s ✐♥t❡r❛çõ❡s ❝♦♠♣❡t✐t✐✈❛s J1 ❡ J2 ♥♦ ♠♦❞❡❧♦ ❞❡

❍❡✐s❡♥❜❡r❣ ❢❡rr✐♠❛❣♥ét✐❝♦ ✐s♦tró♣✐❝♦ ❞❡ s♣✐♥s ✶ ❡ ✶✴✷ ❡♠ ✉♠❛ ❡ ❞✉❛s ❞✐♠❡♥✲ sõ❡s✳ ❊♠ ✉♠❛ ❞✐♠❡♥sã♦✱ ❝♦♥s✐❞❡r❛♠♦s ❛✐♥❞❛ ♦ ❡❢❡✐t♦ ❞❡ ❞✐♠❡r✐③❛çã♦ ❛❧é♠ ❞❛s ✐♥t❡r❛çõ❡s ❝♦♠♣❡t✐t✐✈❛s✳ ❈❛❧❝✉❧❛♠♦s ❛ ♠❛❣♥❡t✐③❛çã♦ ❞❛s s✉❜r❡❞❡s✱ ♦ ❣❛♣ ❞♦ r❛♠♦ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦ ❡ ❛ ❡♥❡r❣✐❛ ❧✐✈r❡ à t❡♠♣❡r❛t✉r❛ ♥✉❧❛✳ ❆ ♦r❞❡♠ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ é ❡♥❝❛r❛❞❛ ❝♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❝♦♥❞❡♥s❛çã♦ ❞♦s ❜ós♦♥s ❙❝❤✇✐♥❣❡r✳ P♦r ú❧t✐♠♦✱ ❡st✉❞❛♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ s♣✐♥ S = 1✱ ❝♦♥s✐❞❡r❛♥❞♦ ✐♥t❡r❛çõ❡s ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s✳

❆❜str❛❝t

❲❡ st✉❞✐❡❞ t❤❡ ❡✛❡❝t ♦❢ ❝♦♠♣❡t✐t✐✈❡ ✐♥t❡r❛❝t✐♦♥s ❛♥❞ ❛♥✐s♦tr♦♣✐❡s ✐♥ t❤❡ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ✐♥ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s✳ ■♥✐t✐❛❧❧②✱ ✇❡ tr❡❛t t❤❡ ❛♥t✐❢❡rr♦♠❛❣✲ ♥❡t✐❝ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ♦❢ s♣✐♥ S = 1 ✇✐t❤ ❝♦♠♣❡t✐t✐✈❡ ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥

♥❡①t J1 ❛♥❞ ♥❡①t✲♥❡❛r❡st J2 ♥❡✐❣❤❜♦rs ✭❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ J1 −J2✮ ❛♥❞ ❡❛s②

♣❧❛♥❡ ❛♥✐s♦tr♦♣② ✐♥ ❛ ❝✉❜✐❝ ❧❛tt✐❝❡ ✇✐t❤ ✐♥t❡r❛❝t✐♦♥s J3 ❜❡t✇❡❡♥ ❛❞❥❛❝❡♥t

♣❧❛♥❡s✳ ❲❡ ✉s❡ t❤❡ s❡❧❢✲❝♦♥s✐st❡♥t ❤❛r♠♦♥✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♣❤❛s❡ tr❛♥s✐t✐♦♥ t❡♠♣❡r❛t✉r❡ ❢r♦♠ ♣❛r❛♠❛❣♥❡t✐❝ ♣❤❛s❡ t♦ t❤❡ ♦r❞❡r❡❞ ♣❤❛s❡ ❛t ❧♦✇ t❡♠♣❡r❛t✉r❡✳ ❲❡ ❛❧s♦ ♦❜t❛✐♥❡❞ ❛t ③❡r♦ t❡♠♣❡r❛t✉r❡✱ t❤❡ ❝r✐t✐❝❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ❡❛s② ♣❧❛♥ ❛♥✐s♦tr♦♣② t❤❛t s❡♣❛r❛t❡s t❤❡ r❡❣✐♦♥ ♦❢ s♠❛❧❧ D✈❛❧✉❡s✱ ❢r♦♠ t❤❡ q✉❛♥t✉♠ ♣❛r❛♠❛❣♥❡t✐❝ ♣❤❛s❡ ❢♦r ❧❛r❣❡ D ✈❛❧✉❡s✳ ❲❡ ❢♦✉♥❞ ❛ ❞✐s♦r❞❡r❡❞ ♣❤❛s❡ ❛t ③❡r♦ t❡♠♣❡r❛t✉r❡ ✇❤✐❝❤ ♠❛② ❜❡ ❛ ♣♦ss✐❜❧❡ ❝❛♥❞✐❞❛t❡ t♦ s♣✐♥ ❧✐q✉✐❞ ♣❤❛s❡✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇♦r❦✱ ✇❡ ✉s❡ t❤❡ ❙❝❤✇✐♥❣❡r ❜♦s♦♥s ♠❡❛♥ ✜❡❧❞ t❤❡♦r② t♦ st✉❞② t❤❡ ❡✛❡❝t ♦❢ ❝♦♠♣❡t✐t✐✈❡ ✐♥t❡r❛❝t✐♦♥s J1 ❛♥❞ J2 ✐♥ t❤❡ ❢❡rr✐♠❛❣♥❡t✐❝

❍❡✐s❡♥❜❡r❣ ✐s♦tr♦♣✐❝ ♠♦❞❡❧ ♦❢ s♣✐♥s ✶ ❛♥❞ ✶✴✷ ✐♥ ♦♥❡ ❛♥❞ t✇♦ ❞✐♠❡♥s✐♦♥s✳ ■♥ ♦♥❡ ❞✐♠❡♥s✐♦♥✱ ✇❡ ❤❛✈❡ ❛❧s♦ ❝♦♥s✐❞❡r❡❞ t❤❡ ❡✛❡❝t ❞✐♠❡r✐③❛t✐♦♥ ❜❡②♦♥❞ ❝♦♠✲ ♣❡t✐t✐✈❡ ✐♥t❡r❛❝t✐♦♥s✳ ❲❡ ❝♦♠♣✉t❡ s✉❜❧❛tt✐❝❡s ♠❛❣♥❡t✐③❛t✐♦♥s✱ t❤❡ ❛♥t✐❢❡rr♦✲ ♠❛❣♥❡t✐❝ ❜r❛♥❝❤ ❣❛♣ ❛♥❞ ❢r❡❡ ❡♥❡r❣② ❛t ③❡r♦ t❡♠♣❡r❛t✉r❡✳ ❚❤❡ ❧♦♥❣✲r❛♥❣❡ ♦r❞❡r ✐s s❡❡♥ ❛s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ ❙❝❤✇✐♥❣❡r ❜♦s♦♥ ❝♦♥❞❡♥s❛t✐♦♥✳ ❋✐♥❛❧❧② ✱ ✇❡ st✉❞② t❤❡ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ♦♥ t❤❡ sq✉❛r❡ ❧❛tt✐❝❡ ✇✐t❤ s♣✐♥S= 1✱ ❝♦♥s✐❞❡r✐♥❣

✈✐✐✐

❙✉♠ár✐♦

❆❣r❛❞❡❝✐♠❡♥t♦s ✈

❘❡s✉♠♦ ✈✐

❆❜str❛❝t ✈✐✐

✶ ■♥tr♦❞✉çã♦ ✶

✶✳✶ ❈♦♥s✐❞❡r❛çõ❡s ●❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❆♥✐s♦tr♦♣✐❛s ❡ ❚❡r♠♦s ❆❞✐❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✸ ▼❛❣♥❡t✐s♠♦ ❡♠ ❜❛✐①❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✹ ❙✐st❡♠❛s ❋r✉str❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✶✶

✷ ▼♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❆♥✐s♦tró♣✐❝♦ q✉❛s❡✲❜✐❞✐♠❡♥s✐♦♥❛❧ ✶✸ ✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ❆♣r♦①✐♠❛çã♦ ❍❛r♠ô♥✐❝❛ ❆✉t♦✲❈♦♥s✐st❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✸✸

✸ ▼♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❋❡rr✐♠❛❣♥ét✐❝♦ ■s♦tró♣✐❝♦ ❝♦♠ ■♥t❡✲

r❛çõ❡s ❈♦♠♣❡t✐t✐✈❛s ✸✺

❙❯▼➪❘■❖ ✐①

✸✳✺ ❋❛t♦r ❞❡ ❊str✉t✉r❛ ❉✐♥â♠✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✻ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✽

✹ ▼♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ J1−J2 ❜✐❞✐♠❡♥s✐♦♥❛❧ ❛♥t✐❢❡rr♦♠❛❣♥é✲

t✐❝♦ ❝♦♠ ❙❂✶ ♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ✻✵

✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✹✳✷ ❋♦r♠❛❧✐s♠♦ ❞♦s ❜ós♦♥s ❞❡ ❙❝❤✇✐♥❣❡r ❙❯✭✸✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✸ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✸

✺ ❈♦♥❝❧✉sõ❡s ✼✺

❆ ❈á❧❝✉❧♦ ❞❛s ♠❛❣♥❡t✐③❛çõ❡s ❞❛s s✉❜✲r❡❞❡s ❡ ❋❛t♦r ❞❡ ❡str✉✲

t✉r❛ ✼✼

❆✳✶ ❈á❧❝✉❧♦ ❞♦ ❋❛t♦r ❞❡ ❊str✉t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

❇ ❈á❧❝✉❧♦ ❞♦ ❖♣❡r❛❞♦r ❞❡ ◗✉❛❞r✉♣♦❧♦ ❡ ❋❛t♦r ❞❡ ❊str✉t✉r❛ ✽✶

✶

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

✶✳✶ ❈♦♥s✐❞❡r❛çõ❡s ●❡r❛✐s

❖ ❡st✉❞♦ t❡ór✐❝♦ s♦❜r❡ ♦ ♠❛❣♥❡t✐s♠♦ ❞♦s ♠❛t❡r✐❛✐s ❝♦♠❡ç♦✉ ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳ ❡ ❛t✐♥❣✐✉ ♦ ♣♦♥t♦ ♠❛✐s ❛❧t♦ ♥❛q✉❡❧❛ é♣♦❝❛ ❝♦♠ ❛ t❡♦r✐❛ ❢❡♥♦♠❡♥♦✲ ❧ó❣✐❝❛ ❞❡ P✐❡rr❡ ❲❡✐ss ❬✶✱✷❪✳ ❆ ✐❞é✐❛ ♣r♦♣♦st❛ ♣♦r ❲❡✐ss ❝♦♥s✐❞❡r❛✈❛ q✉❡✱ ♦ ♠❛❣♥❡t✐s♠♦ ❡r❛ r❡s✉❧t❛❞♦ ❞❛ ✐♥t❡r❛çã♦ ❞❡ ❝❛❞❛ ♠♦♠❡♥t♦ ♠❛❣♥ét✐❝♦ ❞❛ r❡❞❡ ❝r✐st❛❧✐♥❛ ❝♦♠ ✉♠ ❝❛♠♣♦ ❡❢❡t✐✈♦ ♣r♦❞✉③✐❞♦ ♣❡❧♦s ❞❡♠❛✐s ♠♦♠❡♥t♦s ❞♦s í♦♥s ❞❛ r❡❞❡✳ ◗✉❛❧✐t❛t✐✈❛♠❡♥t❡✱ ❛ t❡♦r✐❛ ❝♦♥s❡❣✉✐✉ ♣r❡✈❡r ❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ❞❡ ❝♦♠♣♦st♦s ❢❡rr♦♠❛❣♥ét✐❝♦s q✉❡ ❛♣r❡s❡♥t❛♠ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ Tc ❜❛✐①❛✱

❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ M nSb✱ CrT e✱ CrO2✱ CrM r3✱ EuO✱ EuS✳ P♦ré♠✱ ♣❛r❛

♠❛t❡r✐❛✐s ❝♦♠ ❛❧t❛ Tc✱ ❝♦♠♦ F e✱ N i ❡ Co✱ ❛ t❡♦r✐❛ ❢❛❧❤❛ ♣♦rq✉❡✱ ❞❡ ♠❛✲

♥❡✐r❛ ❣❡r❛❧✱ ✐♥t❡r❛çõ❡s ❞✐♣♦❧❛r❡s ♥ã♦ sã♦ ♦ ♠❡❝❛♥✐s♠♦ ♣r❡♣♦♥❞❡r❛♥t❡✱ ❛♣❡s❛r ❞❡st❛s ❡st❛r❡♠ s❡♠♣r❡ ♣r❡s❡♥t❡s✳ ❙❛❜❡✲s❡ ❛t✉❛❧♠❡♥t❡ q✉❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ♠❛❣♥ét✐❝❛s ❡stã♦ ❛ss♦❝✐❛❞❛s ❛♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❧♦❝❛❧✐③❛❞♦s ♥♦s í♦♥s ❡ t❡♠ ♦r✐❣❡♠ ✐✮ ❞❡✈✐❞♦ ❛♦s s♣✐♥s ❧♦❝❛❧✐③❛❞♦s ✭✐s♦❧❛♥t❡✮ ♦✉ ✐✐✮ ❞❡✈✐❞♦ ❛♦s s♣✐♥s ❞♦s ❡❧étr♦♥s ❡♠ ♠♦✈✐♠❡♥t♦ ♥❛ r❡❞❡ ✭♠❡t❛❧✮✳ ◆❡st❛ t❡s❡✱ ♦s tr❛❜❛❧❤♦s ❞❡s❡♥✲ ✈♦❧✈✐❞♦s sã♦ r❡str✐t♦s ❛♦ ❡st✉❞♦ ❞♦s ♠♦❞❡❧♦s ♠❛❣♥ét✐❝♦s ❧♦❝❛❧✐③❛❞♦s✱ ❝♦♠♦ ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❞❡s❝r✐t♦ ♣❡❧♦ ❤❛♠✐❧t♦♥✐❛♥♦✿

H =X

ij

JijS~i·S~j. ✭✶✳✶✮

❈❆P❮❚❯▲❖ ✶ ✷

t✐s♠♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠✐❝r♦s❝ó♣✐❝♦ ❞❡ ❛❧❣✉♥s ♠❛t❡r✐❛✐s✱ s✉♣♦♥❞♦ q✉❡ ♦ ❛❧✐♥❤❛♠❡♥t♦ ❞♦s s♣✐♥s ❞❡❝♦rr✐❛ ❞❛ ✐♥t❡r❛çã♦ ❝♦♠ s❡✉s ✈✐③✐♥❤♦s ♠❛✐s ♣ró①✐✲ ♠♦s✳ ❆ ✐♥t❡r❛çã♦ ❡❧❡tr♦stát✐❝❛ ❞♦s ❡❧étr♦♥s ❞❡ ❝❛♠❛❞❛s ♠❛✐s ❡①t❡r♥❛s ❞❡ í♦♥s ❛❞❥❛❝❡♥t❡s✱ tr❛t❛❞❛ ✈✐❛ t❡♦r✐❛ ❞❡ ♣❡rt✉r❜❛çã♦✱ ♣r♦❞✉③ ✉♠❛ s❡♣❛r❛çã♦ ❞♦s ♥í✲ ✈❡✐s ❞❡ ❡♥❡r❣✐❛ ❡❧❡trô♥✐❝♦s✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ❞♦✐s ❡❧étr♦♥s ✐♥t❡r❛❣❡♥t❡s ❬✷❪ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❡①❝❧✉sã♦ ❞❡ P❛✉❧✐ ❡①✐❣❡ q✉❡ ❛s ❛✉t♦❢✉♥çõ❡s ❞❡ ♦♥❞❛ t♦t❛❧ s❡❥❛♠ ❛♥t✐✲s✐♠étr✐❝❛s✳ ❆tr❛✈és ❞❡ t❡♦r✐❛ ❞❡ ♣❡rt✉r❜❛çã♦✱ ♦s ❛✉t♦✈❛❧♦r❡s ♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞♦s ❡ sã♦ ❞❛❞♦s ♣♦r✿

E± =E0±Jij, ✭✶✳✷✮

s❡♥❞♦

Jij =

Z Z

d~rid~rj φ∗i(~rj)φ∗j(~ri)

e2

|ri−rj|

φi(~rj)φj(~ri), ✭✶✳✸✮

♦♥❞❡ E0 é ❛ ❛✉t♦✲❡♥❡r❣✐❛ ♥❛ ❛✉sê♥❝✐❛ ❞❡ ♣❡rt✉r❜❛çã♦ ❝♦✉❧♦♠❜✐❛♥❛✱ φn(~ri)

é ❛ ❛✉t♦❢✉♥çã♦ ❞❛ ♣❛rtí❝✉❧❛ i = 1,2 ♥♦ ❡st❛❞♦ n ❞♦ s✐st❡♠❛ ♥ã♦ ♣❡rt✉r✲ ❜❛❞♦✳ ❆ ❡♥❡r❣✐❛ Jij✱ q✉❡ é ❝❤❛♠❛❞❛ ❞❡ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ ♦✉ ❡①❝❤❛♥❣❡✱ é ❛

❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ ❡♥❡r❣✐❛ ❞♦s ❡❧étr♦♥s ♥♦ ❡st❛❞♦ tr✐♣❧❡t♦ (S = 1) ❡ s✐♥❣❧❡t♦ (S = 0) ❬✺✕✼❪✳ ◗✉❛♥❞♦Jij <0✱ ♦ ❡st❛❞♦ ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ é ♦ ❡st❛❞♦ tr✐♣❧❡t♦

❝♦♠ ❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ s♣✐♥s ♣❛r❛❧❡❧♦s ✭❢❡rr♦♠❛❣♥❡t✐s♠♦ ✜❣✳ ✶✳✶✮✳ ◗✉❛♥❞♦ Jij > 0✱ ♦ ❡st❛❞♦ ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛ é ♦ s✐♥❣❧❡t♦ ❝♦♠ ❝♦♥✜❣✉r❛çã♦ ❞♦s s♣✐♥s

❛♥t✐♣❛r❛❧❡❧♦s ✭❛♥t✐❢❡rr♦♠❛❣♥❡t✐s♠♦ ✜❣✳ ✶✳✷✮✳ ❆ ✐♥t❡r❛çã♦ ❞❡ tr♦❝❛ Jij ❞❡✲

❝r❡s❝❡ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s í♦♥s✱ ❡♠ ❝♦♥tr❛st❡ ❝♦♠ ❛ ✐♥t❡r❛çã♦ ❝♦✉❧♦♠❜✐❛♥❛ q✉❡ ❞❡❝r❡s❝❡ ❝♦♠ ∼ 1

r✳ ❊st❛ ❡♥❡r❣✐❛ é ❛ ♣r✐♥❝✐♣❛❧

r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❢♦rt❡ ♠❛❣♥❡t✐s♠♦ ❞❛ ♠❛tér✐❛ ❡ ❝♦♠♦ ❝♦♠❡♥t❛♠♦s ❛❝✐♠❛✱ só ♣♦❞❡ s❡r ❡①♣❧✐❝❛❞❛ ❛tr❛✈és ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ✭s✉♣❡r♣♦s✐çã♦ ❞❛s ❢✉♥çõ❡s ❞❡ ♦♥❞❛✮ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❡①❝❧✉sã♦ ❞❡ P❛✉❧✐✳ P♦rt❛♥t♦✱ ❛♣❡✲ s❛r ❞❛ t❡♦r✐❛ ❞❡ ❲❡✐ss ❧❡✈❛r ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛s♣❡❝t♦s ♠✐❝r♦s❝ó♣✐❝♦s✱ ❡❧❛ ♥ã♦ ♣♦❞❡r✐❛ ❡①♣❧✐❝❛r ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ ♠❛❣♥❡t✐s♠♦ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛r❣✉♠❡♥t♦s ❞❡ ♥❛t✉r❡③❛ ❝❧áss✐❝❛✳

❈❆P❮❚❯▲❖ ✶ ✸

❋✐❣✉r❛ ✶✳✶✿ ❊st❛❞♦ ❞❡ ❞♦✐s ❡❧étr♦♥s q✉❡ ❛❝♦♣❧❛♠ ❢❡rr♦♠❛❣♥❡t✐❝❛♠❡♥t❡✳ ❋✐✲ ❣✉r❛ ♦❜t✐❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✼❪

❋✐❣✉r❛ ✶✳✷✿ ❊st❛❞♦ ❞❡ ❞♦✐s ❡❧étr♦♥s q✉❡ ❛❝♦♣❧❛♠ ❛♥t✐❢❡rr♦♠❛❣♥❡t✐❝❛♠❡♥t❡✳ ❋✐❣✉r❛ ♦❜t✐❞❛ ❞❛ r❡❢❡rê♥❝✐❛ ❬✼❪

♠é❞✐♦ ❡ ♥✉♠ér✐❝❛s ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ♣❛r❛ ❡st✉❞❛r ♦ ♠♦❞❡❧♦ s♦❜ ❞✐✈❡rs❛s s✐t✉❛çõ❡s✳ ❆s té❝♥✐❝❛s ❛♥❛❧ít✐❝❛s ❞❡ ❝❛♠♣♦ ♠é❞✐♦ ✈✐s❛♠ ❜❛s✐❝❛♠❡♥t❡ ❞❡s❝r❡✲ ✈❡r ♦ ♣r♦❜❧❡♠❛ ❛tr❛✈és ❞❡ ♦♥❞❛s ❞❡ s♣✐♥ ✭❝♦♥❝❡✐t♦ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❇❧♦❝❤ ❬✽❪✮✱ ❡♥q✉❛♥t♦ q✉❡ ❛s té❝♥✐❝❛s ♥✉♠ér✐❝❛s ✈✐s❛♠ ♦❜t❡r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♠♦❞❡❧♦ ❞❡N s♣✐♥s ✐♥t❡r❛❣❡♥t❡s✱ ❛tr❛✈és ❞❡ ✉♠❛ ♣❡q✉❡♥❛ ♣♦rçã♦ ❞♦ s✐st❡♠❛✱ s❡❥❛ ♣♦r s✐♠✉❧❛çõ❡s ♦✉ ❞✐❛❣♦♥❛❧✐③❛çõ❡s ♥✉♠ér✐❝❛s✱ ♣♦r ❡①❡♠♣❧♦✳

✶✳✷ ❆♥✐s♦tr♦♣✐❛s ❡ ❚❡r♠♦s ❆❞✐❝✐♦♥❛✐s

❈❆P❮❚❯▲❖ ✶ ✹

♥❛♠❡♥t♦ ♠❛❣♥ét✐❝♦ ♥❛ ♠❛tér✐❛✱ s❡♥❞♦ ❞❡ ♥❛t✉r❡③❛ ✐s♦tró♣✐❝❛ ❡ ✐♥❝❛♣❛③ ❞❡ ❞❡✜♥✐r ❛❧❣✉♠❛ ♦r✐❡♥t❛çã♦ ❞♦s ♠♦♠❡♥t♦s ♠❛❣♥ét✐❝♦s ❝♦♠ r❡s♣❡✐t♦ ❛♦s ❡✐①♦s ❝r✐st❛❧♦❣rá✜❝♦s✱ ♠❛s ❡❧❛ ♣r♦❞✉③ ✉♠ ♦r❞❡♥❛♠❡♥t♦ ♠út✉♦ ❞♦s s♣✐♥s ❡♠ ✈ár✐♦s sít✐♦s ❞❛ r❡❞❡✳ P♦ré♠✱ ❡♠ ❛❧❣✉♥s ❝❛s♦s ❡st❛s ✐♥t❡r❛çõ❡s ♣♦❞❡♠ ♥ã♦ s❡r ❡s♣❛✲ ❝✐❛❧♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡s✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❡♠ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧ ❝♦♠♦✿

H =X

ij

J_ijxS_ixS_jx+J_ijyS_iyS_jy +J_ijyS_izS_jz. ✭✶✳✹✮

◗✉❛♥❞♦ Jx ij = J

y

ij = Jijz t❡♠♦s ♦ ♠♦❞❡❧♦ ✐s♦tró♣✐❝♦✳ P♦❞❡♠♦s ❛✐♥❞❛ t❡r

Jx ij =J

y

ij 6=Jijz q✉❡ é ♠❛✐s ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ❳❳❩✳ ◆❡st❡ ❝❛s♦✱ q✉❛♥❞♦

J_ijx,y _≫ Jz

ij✱ ♣♦❞❡♠♦s ♦♠✐t✐r ♦ ú❧t✐♠♦ t❡r♠♦ ❞♦ ❤❛♠✐❧t♦♥✐❛♥♦ ✭✶✳✹✮✱ q✉❡ é

♠❛✐s ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠♦❞❡❧♦ ❳❨ ♦✉ ♣❧❛♥❛r ❬✾❪✳ ◆♦ ❝❛♣ít✉❧♦ ✷ ❞❡st❛ t❡s❡✱ ❡st✉❞❛r❡♠♦s ♦ ♠♦❞❡❧♦ ❳❳❩ ❡♠ ✉♠❛ r❡❞❡ ❝ú❜✐❝❛ ❝♦♠ ✐♥t❡r❛çõ❡s ❡♥tr❡ ♣❧❛♥♦s ❛❞❥❛❝❡♥t❡s✳ ❚♦♠❛♠♦s t❛♥t♦ ♦ ❧✐♠✐t❡ t♦t❛❧♠❡♥t❡ ✐s♦tró♣✐❝♦✱ q✉❛♥t♦ ♦ ❧✐♠✐t❡ ❳❨✱ ♣❛r❛ ❞❡s❝r❡✈❡r ❛s ❣r❛♥❞❡③❛s t❡r♠♦❞✐♥â♠✐❝❛s✳

❊①✐st❡ ❛✐♥❞❛ ♦ ❧✐♠✐t❡ J_ijx,y _≪ Jz

ij✱ ♦♥❞❡ ♣♦❞❡♠♦s ♦♠✐t✐r ♦s ❞♦✐s ♣r✐♠❡✐✲

r♦s t❡r♠♦s ❞♦ ❤❛♠✐❧t♦♥✐❛♥♦ ✭✶✳✹✮✳ ❊st❡ ❧✐♠✐t❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❡ r❡♣r❡s❡♥t❛ ♦ ♠♦❞❡❧♦ ♠❛✐s s✐♠♣❧❡s ❝♦♠ s♦❧✉çã♦ ❡①❛t❛ ❡♠ ✉♠❛ ❡ ❞✉❛s ❞✐♠❡♥sõ❡s s❡♠ ❝❛♠♣♦ ❡①t❡r♥♦ ❬✶✵❪✳

❖✉tr❛ ❝❛r❛❝t❡ríst✐❝❛ ✐♥t❡r❡ss❛♥t❡ ♥♦s s✐st❡♠❛s ♠❛❣♥ét✐❝♦s é ❛ ♣r❡s❡♥ç❛ ❞❡ ❛♥✐s♦tr♦♣✐❛s q✉❡ sã♦ ❝❛♣❛③❡s ❞❡ ♦r❞❡♥❛r ♦ s✐st❡♠❛ ❡♠ ❛❧❣✉♠❛ ❞✐r❡çã♦ ♣r✐✈✐❧❡✲ ❣✐❛❞❛✱ ♦✉ ❞✐r❡çã♦ ❞❡ ❢á❝✐❧ ♠❛❣♥❡t✐③❛çã♦✳ P♦❞❡♠♦s ❞✐③❡r q✉❡ ♥♦ ❝r✐st❛❧ ❡①✐st❡♠ ❝❛♠♣♦s ♠❛❣♥ét✐❝♦s ❡❢❡t✐✈♦s s♦❜r❡ ♦s í♦♥s✱ q✉❡ t❡♥❞❡♠ ❛ ♦r✐❡♥tá✲❧♦s ♥✉♠❛ ❞✐✲ r❡çã♦ ♣r✐✈✐❧❡❣✐❛❞❛✳ P♦rt❛♥t♦✱ ❞❡✈❡♠♦s t❡r ❛❧❣✉♠ t✐♣♦ ❞❡ ✐♥t❡r❛çã♦ q✉❡ t♦r♥❡ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥✐s♦tró♣✐❝♦✳ ❯♠ t✐♣♦ ❞❡ ❛♥✐s♦tr♦♣✐❛ ♠✉✐t♦ ❝♦♥❤❡❝✐❞❛ ❡ q✉❡ q✉❡❜r❛ ❛ s✐♠❡tr✐❛ r♦t❛❝✐♦♥❛❧ ❞♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❞❡ ❍❡✐s❡♥❜❡r❣ é ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ í♦♥ ú♥✐❝♦✳ ❙✉❛ ♦r✐❣❡♠ ♠✐❝r♦s❝ó♣✐❝❛ ❡stá ♥❛ ✐♥✢✉ê♥❝✐❛ ❞♦ ♣♦t❡♥❝✐❛❧ ❡❧étr✐❝♦ ❞❛ r❡❞❡ ❝r✐st❛❧✐♥❛ s♦❜r❡ ♦ ❛❝♦♣❧❛♠❡♥t♦ s♣✐♥✲ór❜✐t❛✱ ❝✉❥♦ t❡r♠♦ é ❞❛❞♦ ♣♦r✿

∆E = 1

2m2_c2_r

dV(r)

❈❆P❮❚❯▲❖ ✶ ✺

❡st❡ ❛❝♦♣❧❛♠❡♥t♦ ♦r✐❣✐♥❛ ✉♠ t❡r♠♦ ❛❞✐❝✐♦♥❛❧ ♥♦ ❤❛♠✐❧t♦♥✐❛♥♦ ✭✶✳✹✮✱ ❞♦ t✐♣♦✿

H′ =

N

X

i

D(Siz)2, ✭✶✳✻✮

♦♥❞❡ ❉ é ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ í♦♥ ú♥✐❝♦ ♦r✐✉♥❞♦ ❞❛ ✐♥t❛r❛çã♦ s♣✐♥✲ó❜✐t❛✳ ◗✉❛♥❞♦ ♦ ♣❛râ♠❡tr♦ D ❢♦r ♥❡❣❛t✐✈♦✱ ♦s s♣✐♥s t❡♥❞❡♠ ❛ ❛❧✐♥❤❛r✲s❡ ♥❛ ❞✐r❡çã♦ z✱ ♣♦✐s ❡st❛ ❛♥✐s♦tr♦♣✐❛ ✐♥❞✉③ t❛❧ ♦r✐❡♥t❛çã♦✳ ❊st❡ t✐♣♦ ❞❡ ❛♥✐s♦tr♦♣✐❛ ♣♦❞❡ ❛✐♥❞❛ ❢❛✈♦r❡❝❡r ❛❧✐♥❤❛♠❡♥t♦s ♥❛s ❞✐r❡çõ❡s x ♦✉ y✱ ♥❡st❡ ❝❛s♦ ❝♦♥s✐❞❡r❛♠♦s ❡♠ z ♣♦r q✉❡stõ❡s ❞❡ s✐♠♣❧✐✜❝❛çã♦✳ ❙❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❢♦r ♣♦s✐t✐✈♦✱ ♦s s♣✐♥s t❡♥❞❡♠ ❛ ❛❧✐♥❤❛r✲s❡ ♥♦ ♣❧❛♥♦ ❳❨✳ ❊st❛ ❛♥✐s♦tr♦♣✐❛ só é r❡❧❡✈❛♥t❡ ♣❛r❛ ❝❛s♦s ♦♥❞❡ S > 1/2✳

❆❧❣✉♥s ♦✉tr♦s t❡r♠♦s ✭♦r✐❣✐♥ár✐♦s ❞❡ ✐♥t❡r❛çõ❡s ❝♦✉❧♦♠❜✐❛♥❛s✮ ♣♦❞❡♠ s❡r ❞❡❞✉③✐❞♦s ✈✐❛ t❡♦r✐❛ ❞❡ ♣❡rt✉r❜❛çã♦ ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦ t❡r♠♦ ❞❡ ✐♥t❡r❛çã♦ ❜✐q✉❛❞rát✐❝❛ ❞❛❞❛ ♣♦r✿

❍2 =−

X

<i,j>

KS~i·S~j

2

, ✭✶✳✼✮

♦✉ ❛✐♥❞❛ ❛ ✐♥t❡r❛çã♦ ❞❡ q✉❛tr♦ s♣✐♥s

❍4 =−

X

<i,j,k,l>

Jijkl

~

Si·S~j S~k·S~l

, ✭✶✳✽✮

q✉❡ ♣♦❞❡♠ t❡r ❛❧❣✉♠❛ r❡❧❡✈â♥❝✐❛ ❢ís✐❝❛✱ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ♥❛t✉r❡③❛ ❞♦ ♣r♦✲ ❜❧❡♠❛✳

❈❆P❮❚❯▲❖ ✶ ✻

❞❡ ❖♥s❛❣❡r ❞♦ ♠♦❞❡❧♦ ❞❡ ■s✐♥❣ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❬✶✷❪✳ ❈♦♠ ♦ ❝r❡s❝❡♥t❡ ❞❡s❡♥✲ ✈♦❧✈✐♠❡♥t♦ ❞❡ té❝♥✐❝❛s ❡①♣❡r✐♠❡♥t❛✐s ❡ sí♥t❡s❡ ❞❡ ♠❛t❡r✐❛✐s✱ ❡st❡s s✐st❡♠❛s ❞❡✐①❛r❛♠ ❞❡ s❡r ❞❡ ✐♥t❡r❡ss❡ ❛♣❡♥❛s t❡ór✐❝♦ ❡ ♣❛ss❛r❛♠ ❛ s❡r ❛❧✈♦ ❞❡ ✐♥t❡♥s❛s ✐♥✈❡st✐❣❛çõ❡s ❡①♣❡r✐♠❡♥t❛✐s ✭✈❡❥❛ r❡❢❡rê♥❝✐❛ ❬✶✸❪✱ ♦♥❞❡ ❡①✐st❡♠ ❝♦♠♣❛r❛çõ❡s ✐♥t❡r❡ss❛♥t❡s ❡♥tr❡ t❡♦r✐❛ ❡ ❡①♣❡r✐♠❡♥t♦✮✳

❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ❛✉sê♥❝✐❛ ❞❡ ♦r❞❡♠ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ♣❛r❛ s✐st❡✲ ♠❛s ❝♦♠ s✐♠❡tr✐❛ ❝♦♥tí♥✉❛ ❛ t❡♠♣❡r❛t✉r❛s ✜♥✐t❛s ❡♠ d _≤2 é ✉♠ r❡s✉❧t❛❞♦

❡①❛t♦ q✉❡ ❞❡✈❡ s❡r r❡♣r♦❞✉③✐❞♦ ♣♦r té❝♥✐❝❛s ❛♣r♦①✐♠❛t✐✈❛s ❬✶✹❪✳ ❆ t❡♠♣❡✲ r❛t✉r❛ ♥✉❧❛✱ ❡s♣❡r❛✲s❡ q✉❡ ♦ s✐st❡♠❛ t❡♥❤❛ ❛❧❣✉♠ t✐♣♦ ❞❡ ♦r❞❡♠✱ ♣♦ré♠ ❡♠ ❛❧❣✉♥s ❝❛s♦s é ♣♦ssí✈❡❧ ❛✐♥❞❛ q✉❡ ♠❡s♠♦ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ s❡❥❛ ♠❛❣✲ ♥❡t✐❝❛♠❡♥t❡ ❞❡s♦r❞❡♥❛❞♦ ❬✶✺❪✳ ❊♠ ❢ís✐❝❛ ❡st❛tíst✐❝❛✱ ✉♠ ❡st❛❞♦ ♦r❞❡♥❛❞♦ é ❞❡s❝r✐t♦ ♣♦r ❛❧❣✉♠ ♣❛râ♠❡tr♦ ❞❡ ♦r❞❡♠ q✉❡ é ✉♠❛ q✉❛♥t✐❞❛❞❡ ❝✉❥❛ ♠é❞✐❛ é ♥ã♦ ♥✉❧❛ ❡ ❝❛r❛❝t❡r✐③❛ ❛ ♦r❞❡♠✳ ◆♦s s✐st❡♠❛s ❞❡ ❜❛✐①❛ ❞✐♠❡♥sã♦✱ ❛s ✢✉t✉❛✲ çõ❡s q✉â♥t✐❝❛s sã♦ ♠✉✐t♦ ♠❛✐s ❡①♣r❡ss✐✈❛s ❞♦ q✉❡ ♥♦s s✐st❡♠❛s tr✐❞✐♠❡♥s✐♦♥❛✐s ♣♦r ❡①❡♠♣❧♦✳ P♦rt❛♥t♦✱ ❡st❛s ✢✉t✉❛çõ❡s t❡♠ ❝❛rát❡r ❞♦♠✐♥❛♥t❡ ♥❛ ❞❡t❡r♠✐✲ ♥❛çã♦ ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧✳ ❇❛✐①❛ ❞✐♠❡♥s✐♦♥❛❧✐❞❛❞❡ ❬✶✻❪✭❞❂✶✱✷✮ ❛❧✐❛❞❛ ❛ ❛♥✐s♦tr♦♣✐❛s ❡ ✐♥t❡r❛çõ❡s ❝♦♠♣❡t✐t✐✈❛s✱ sã♦ ♠❡❝❛♥✐s♠♦s ♣❡❧♦s q✉❛✐s ❡st❛s ✢✉✲ t✉❛çõ❡s ♣♦❞❡♠ s❡r ❝♦♥tr♦❧❛❞❛s✱ t♦r♥❛♥❞♦ ♦ ❡st✉❞♦ ❜❛st❛♥t❡ ♠♦t✐✈❛♥t❡✳ ◆♦s ❝❛s♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s✱ ❛s ✢✉t✉❛çõ❡s ❡ ✐♥t❡r❛çõ❡s ♣♦❞❡♠ ❡st❛r ♠❡❧❤♦r ❜❛❧❛♥✲ ❝❡❛❞❛s✱ ♦ q✉❡ ♣♦❞❡ ❧❡✈❛r ❛ tr❛♥s✐çõ❡s ❞❡ ❢❛s❡ q✉â♥t✐❝❛s ❬✶✼❪ ❝♦♠ ❛ ✈❛r✐❛çã♦ ❞❡st❡s ♣❛râ♠❡tr♦s✳

❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡ ♥♦s s✐st❡♠❛s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s é ❛ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ❣❛♣ ✭❞✐❢❡r❡♥ç❛ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥✲ t❛❧✱ ❡ ♦ ♣r✐♠❡✐r♦ ❡st❛❞♦ ❡①❝✐t❛❞♦✮ ♥♦ ❡s♣❡❝tr♦ ❞❡ ❡①❝✐t❛çõ❡s✳ ❖ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ✐s♦tró♣✐❝♦ ❝♦♠ ✐♥t❡r❛çõ❡s s♦✲ ♠❡♥t❡ ❡♥tr❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ♥♦ ❧✐♠✐t❡ ❝❧áss✐❝♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❡①✐❜❡ ♦r❞❡♠ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ q✉❡ ✐♥❞❡♣❡♥❞❡ ❞♦ ✈❛❧♦r ❞♦ s♣✐♥✱ ❡ ♦ s❡✉ ❡st❛❞♦ ❢✉♥❞❛♠❡♥✲ t❛❧ ♥ã♦ ♣♦ss✉✐ ❣❛♣✳ P❛r❛ ♦ ❝❛s♦ q✉â♥t✐❝♦ ❡①✐st❡ ✉♠❛ ❞✐❢❡r❡♥ç❛ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ✈❛❧♦r ❞♦ s♣✐♥✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❝❛❞❡✐❛ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝❛ ❞❡ s♣✐♥1/2q✉❡ ❢♦✐ ❡st✉❞❛❞❛ ♣♦r ❇❡t❤❡ ❬✶✽❪ ❡ ❍✉❧t❤❡♥ ❬✶✾❪ ❡♠♣r❡❣❛♥❞♦ ♦ ♠ét♦❞♦

❈❆P❮❚❯▲❖ ✶ ✼

❣❛♣✱ ♣♦ré♠ ❛s ✢✉t✉❛çõ❡s q✉â♥t✐❝❛s ❞❡str♦❡♠ ❛ ♦r❞❡♠ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡✳ ❊st❡ r❡s✉❧t❛❞♦ ❢♦✐ ♣r♦✈❛❞♦ ❛tr❛✈és ❞❡ ✉♠ r✐❣♦r♦s♦ t❡♦r❡♠❛ ❡st❛❜❡❧❡❝✐❞♦ ♣♦r ▲✐❡❜✱ ❙❝❤✉t③ ❡ ▼❛tt✐s ❬✷✵❪✱ q✉❡ ❝❛❞❡✐❛s ❞❡ s♣✐♥1/2♥ã♦ ♣♦ss✉❡♠ ❣❛♣✳ ❖ t❡♦r❡♠❛ ❢♦✐

♠❛✐s t❛r❞❡ ❣❡♥❡r❛❧✐③❛❞♦✱ ❡ ❛❜r❛♥❣❡ t♦❞❛s ❛s ❝❛❞❡✐❛s ❞❡ s♣✐♥ s❡♠✐✲✐♥t❡✐r♦ ❬✷✶❪✳ ◗✉❛♥❞♦ s❡ tr❛t❛ ❞❡ ❝❛❞❡✐❛s ❞❡ s♣✐♥ ✐♥t❡✐r♦✱ ❛ ❝♦♥❥❡❝t✉r❛ ❞❡ ❍❛❧❞❛♥❡ ❬✾❪ ❡s✲ t❛❜❡❧❡❝❡ q✉❡ ❛s ❝♦rr❡❧❛çõ❡s ❞❡ ❞♦✐s ♣♦♥t♦s t❡♠ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ✭❢❛s❡ ❍❛❧❞❛♥❡✮✱ q✉❡ ✐♠♣❧✐❝❛ ❡♠ ✉♠ ❣❛♣ ♥❛s ❡①❝✐t❛çõ❡s✳ ❆ ❡①✐stê♥❝✐❛ ❞❡st❡ ❣❛♣ é ✉♠ ❢❡♥ô♠❡♥♦ ♣✉r❛♠❡♥t❡ q✉â♥t✐❝♦✱ s❡♠ ❛♥❛❧♦❣✐❛ ❝❧áss✐❝❛✳ ❊♠ ❝♦♥tr❛st❡ ❝♦♠ ♦ ❝❛s♦ ❝❧áss✐❝♦ ♥♦✈❛♠❡♥t❡✱ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❢ís✐❝♦ ❞❛s ❝❛❞❡✐❛s ❞❡ s♣✐♥ ✈❛r✐❛♠ ❝♦♠ ♦ ✈❛❧♦r ❞♦ s♣✐♥✳ ❖ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝♦ ✐s♦tró♣✐❝♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ s♣✐♥ 1 é ♦ ❡①❡♠♣❧♦ ♠❛✐s s✐♠♣❧❡s ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠ ❢❛s❡

❞❡ ❍❛❧❞❛♥❡✳ ❊♥tr❡ ♦s ✈ár✐♦s ❝♦♠♣♦st♦s ♠❛❣♥ét✐❝♦s ✉s❛❞♦s ♣❛r❛ ❝♦♠♣r♦✈❛r ♦ ❣❛♣ ❞❡ ❍❛❧❞❛♥❡✱ ♣♦❞❡♠♦s ❝✐t❛r ♦ CsN iCl3 ❬✷✸✱✷✹❪ ❡ ♦ Y2BaN iO5 ❬✷✺❪✳

❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡st✉❞❛r ♠♦❞❡❧♦s ❡♠ ❜❛✐①❛ ❞✐♠❡♥sã♦ sã♦ ❜❡♠ ❛♠♣❧❛s✳ ❆t✉❛❧♠❡♥t❡ ❡①✐st❡♠ ❞✐✈❡rs❛s ♠❡t♦❞♦❧♦❣✐❛s ❛♥❛✲ ❧ít✐❝❛s ❡ ♥✉♠ér✐❝❛s q✉❡ ♣♦❞❡♠ s❡r ❡♠♣r❡❣❛❞❛s ❝♦♠ s✉❝❡ss♦✱ ❝♦♠♦ ♦ ❝❛s♦ ❞❛ ❞✐❛❣♦♥❛❧✐③❛çã♦ ❡①❛t❛ ❡ ▼♦♥t❡ ❈❛r❧♦ q✉â♥t✐❝♦✱ ♦✉ ❛✐♥❞❛ ♦♥❞❛s ❞❡ s♣✐♥ ♣❛❞rã♦ ❡ ❛ t❡♦r✐❛ ❞❡ ❝❛♠♣♦ ♠é❞✐♦ ❞♦s ❜ós♦♥s ❞❡ ❙❝❤✇✐♥❣❡r✳ ❊st❛ ú❧t✐♠❛ ❢♦✐ ❡♠✲ ♣r❡❣❛❞❛ ♥❛ ♠❛✐♦r✐❛ ❞♦s ♥♦ss♦s tr❛❜❛❧❤♦s q✉❡ ❝♦♠♣õ❡♠ ❡st❛ t❡s❡✱ ❝♦♠♦ s❡rá ✈✐st♦ ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳

✶✳✹ ❙✐st❡♠❛s ❋r✉str❛❞♦s

❈❆P❮❚❯▲❖ ✶ ✽

❋✐❣✉r❛ ✶✳✸✿ ❙♣✐♥s ✐♥t❡r❛❣✐♥❞♦ ❛♥t✐❢❡rr♦♠❛❣♥❡t✐❝❛♠❡♥t❡ ❡♥tr❡ ♣r✐♠❡✐r♦s(J1>

0)❡ s❡❣✉♥❞♦s (J2 >0)✈✐③✐♥❤♦s✳

◆♦ ❝❛s♦ ❞❛ r❡❞❡ q✉❛❞r❛❞❛✱ q✉❛♥❞♦ ❛s ✐♥t❡r❛çõ❡s ❡♥tr❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s J2 ♥ã♦ sã♦ ❞❡s♣r❡③í✈❡✐s ❡♠ r❡❧❛çã♦ ❛ J1✱ ♥♦t❛♠♦s q✉❡ ❤á ✉♠ ❝♦♥✢✐t♦ ♥❛s

✐♥t❡r❛çõ❡s ❡♥tr❡ ♣r✐♠❡✐r♦s ❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s✱ ♦ q✉❡ ♣♦❞❡ ❧❡✈❛r à ❢r✉str❛çã♦✳ ❖ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣J1−J2 ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ❢♦✐ ❡①t❡♥s❛♠❡♥t❡ ❡st✉❞❛❞♦

❬✷✽❪✳

H =J1

X

<i,j>

~

Si·S~j+J2

X

<<i,j>>

~

Si·S~j. ✭✶✳✾✮

❈❧❛ss✐❝❛♠❡♥t❡✱ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ é ♦r❞❡♥❛❞♦ q✉❛♥❞♦ J2/J1 < 0.5✱

q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❡st❛❞♦ ❞❡ ◆é❡❧ ❝♦♠ ♦s s♣✐♥s ❛❧✐♥❤❛❞♦s ❛♥t✐♣❛r❛❧❡❧❛♠❡♥t❡ ♥♦s ✈ért✐❝❡s ❞♦ q✉❛❞r❛❞♦ ❬✷✾❪✳ ◗✉❛♥❞♦ J2/J1 > 0.5✱ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧

é ✉♠ ❡st❛❞♦ ❝♦❧✐♥❡❛r ❞❡ s♣✐♥s q✉❡ ♣♦❞❡ ❛♣r❡s❡♥t❛r ❞✉❛s ❝♦♥✜❣✉r❛çõ❡s ❝♦♠ ✈❡t♦r❡s ~q = (π,0) ♦✉ ~q = (0, π)✱ q✉❡ ❝♦rr❡s♣♦♥❞❡♠ ❛ ❧✐♥❤❛s ♦✉ ❝♦❧✉♥❛s

❢❡rr♦♠❛❣♥ét✐❝❛s ✐♥t❡r❛❣✐♥❞♦ ❛♥t✐❢❡rr♦♠❛❣♥❡t✐❝❛♠❡♥t❡ ✉♠❛s ❝♦♠ ❛s ♦✉tr❛s✳ ◗✉❛♥❞♦ J2/J1 = 0.5✱ ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❝❧áss✐❝♦ é ❛❧t❛♠❡♥t❡ ❞❡❣❡♥❡r❛❞♦✳

P❛r❛ ♦ ❝❛s♦ q✉â♥t✐❝♦ ♥❛ r❡❞❡ q✉❛❞r❛❞❛ ❞♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣✱ té❝♥✐✲ ❝❛s ❝♦♠♦ ▼♦♥t❡ ❈❛r❧♦ q✉â♥t✐❝♦✱ ❞✐❛❣♦♥❛❧✐③❛çã♦ ❡①❛t❛ ❡ ❜ós♦♥s ❞❡ ❙❝❤✇✐♥❣❡r ✐♥❞✐❝❛♠ ❛ ❛✉sê♥❝✐❛ ❞❡ ♦r❞❡♠ ❞♦ t✐♣♦ ◆é❡❧ ♥❛ r❡❣✐ã♦ 0.4 _≤ J2/J1 ≤ 0.6✳

❈❆P❮❚❯▲❖ ✶ ✾

❋✐❣✉r❛ ✶✳✹✿ ❙♣✐♥s ❞♦ t✐♣♦ ■s✐♥❣ ✐♥t❡r❛❣✐♥❞♦ ❛♥t✐❢❡rr♦♠❛❣♥❡t✐❝❛♠❡♥t❡J >0✳

tr❛❜❛❧❤♦s t❡ór✐❝♦s✱ ♥♦ ❡♥t❛♥t♦✱ ❛❧❣✉♠❛s ❞✐✜❝✉❧❞❛❞❡s ♣❛r❛ r❡❛❧✐③❛r ❡①♣❡r✐♠❡♥✲ t♦s ❝♦♠ ❡st❛s ✐♥t❡r❛çõ❡s ❢♦rt❡ ♦ s✉✜❝✐❡♥t❡s ♣♦❞❡♠ s✉r❣✐r✳ ➱ ♥❡st❡ ♣♦♥t♦ q✉❡ ♦s s✐st❡♠❛s ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❢r✉str❛❞♦s ❣❛♥❤❛♠ ❜❛st❛♥t❡ ✐♥t❡r❡ss❡✱ ♣♦✐s sã♦ s✐st❡♠❛s ❡♠ q✉❡ ❛ ❡str✉t✉r❛ ♣♦❞❡ ❞❡s❡st❛❜✐❧✐③❛r ❛ ♦r❞❡♠ s♦♠❡♥t❡ ❝♦♠ ✐♥t❡✲ r❛çõ❡s ❞❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s✳

❊♠ s✐st❡♠❛s ❞♦ t✐♣♦ ❍❡✐s❡♥❜❡r❣✱ ❛ ❝♦♠♣❡t✐çã♦ ❞❡ ✐♥t❡r❛çõ❡s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s ✢✉t✉❛çõ❡s q✉â♥t✐❝❛s ♣♦❞❡♠ s✉♣r✐♠✐r ♦r❞❡♠ ❞❡ ❧♦♥❣♦ ❛❧❝❛♥❝❡ ♠❡s♠♦ ❡♠ T = 0✳ ❊st❡s ❢❛t♦r❡s ♣♦❞❡♠ ❧❡✈❛r ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❢❛s❡s q✉â♥t✐❝❛s✱

❛❧❣✉♠❛s ✈❡③❡s ❝❤❛♠❛❞❛s ❞❡ ❧íq✉✐❞♦s ❞❡ s♣✐♥✳ ◆❛ r❡❢❡rê♥❝✐❛ ❬✷✾❪ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❞❡✜♥✐çã♦ ❜ás✐❝❛ ❞❡ ✉♠ ❧íq✉✐❞♦ ❞❡ s♣✐♥✳

❉❡✜♥✐çã♦✿ ❯♠ ❧íq✉✐❞♦ ❞❡ s♣✐♥ q✉â♥t✐❝♦ é ✉♠ ❡st❛❞♦ ❡♠ q✉❡ ❛s ❝♦rr❡❧❛✲ çõ❡s s♣✐♥✲s♣✐♥ hSα

i S β

ji✱ ❞❡❝❛❡♠ ❛ ③❡r♦ ❛ ❣r❛♥❞❡s ❞✐stâ♥❝✐❛s |~ri−~rj| → ∞✳ ➱

✉♠ ❡st❛❞♦ q✉â♥t✐❝♦ ❝♦♠ ❡①❝✐t❛çõ❡s ❢r❛❝✐♦♥ár✐❛s s❡♠ q✉❛❧q✉❡r q✉❡❜r❛ ❡s♣♦♥✲ tâ♥❡❛ ❞❡ s✐♠❡tr✐❛✳

❈❆P❮❚❯▲❖ ✶ ✶✵

♠❛❣♥❡t✐s♠♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛ s✉s❝❡♣t✐❜✐❧✐❞❛❞❡ q✉❡ t❡♠ ✉♠ ✈❛❧♦r ♥✉❧♦ ♥♦ ❧✐♠✐t❡T _→0❬✸✶❪✳ ❉✐❢❡r❡♥t❡ ❞♦s ❝❛s♦s ❝♦♥❤❡❝✐❞♦s✱ ❡st❡ é ✉♠ ❡❢❡✐t♦ ♣✉r❛♠❡♥t❡

q✉â♥t✐❝♦✳ ◆♦ ❝❛♣ít✉❧♦ ✷ ♥ós ✐♥✈❡st✐❣❛♠♦s ♦ ♣❛♣❡❧ ❞❛s ✐♥t❡r❛çõ❡s ❝♦♠♣❡t✐t✐✈❛s ♥♦ s✉r❣✐♠❡♥t♦ ❞❡ ✉♠ ♣♦ssí✈❡❧ ❡st❛❞♦ ❧íq✉✐❞♦ ❞❡ s♣✐♥ q✉❡ s❡♣❛r❛ ❞✉❛s ❢❛s❡s ♠❛❣♥❡t✐❝❛♠❡♥t❡ ♦r❞❡♥❛❞❛s✳

❊st❡ tr❛❜❛❧❤♦ é ❞✐✈✐❞✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ◆♦ ❝❛♣ít✉❧♦ ✷ ❡st✉❞❛r❡♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ q✉❛s❡✲❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦♠ ✐♥t❡r❛çõ❡s ❞❡ ♣r✐♠❡✐r♦s ❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s✳ ❈♦♠♦ ♦ ♥♦ss♦ ❤❛♠✐❧t♦♥✐❛♥♦ ❛♣r❡s❡♥t❛ ✉♠❛ ❛♥✐s♦tr♦♣✐❛ ❞♦ t✐♣♦ ❳❨ ❡ í♦♥ ú♥✐❝♦✱ ✉t✐❧✐③❛r❡♠♦s ❛ ❛♣r♦①✐♠❛çã♦ ❤❛r♠ô♥✐❝❛ ❛✉t♦✲❝♦♥s✐st❡♥t❡✱ q✉❡ é ✉♠❛ ❜♦❛ ♠❡t♦❞♦❧♦❣✐❛ ♣❛r❛ ❡ss❡ t✐♣♦ ❞❡ ♠♦❞❡❧♦✱ ❡ é ❝❛♣❛③ ❞❡ ♣r❡✈❡r ❛ tr❛♥s✐çã♦ ❞❡ ❑♦st❡r❧✐t③✲❚❤♦✉❧❡ss ✭❑❚✮✳ ◆♦ ❝❛♣ít✉❧♦ ✸ ❡st✉❞❛r❡♠♦s ♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❢❡rr✐♠❛❣♥ét✐❝♦ ❛♥✐s♦tró♣✐❝♦ ❡♠ ✉♠❛ ❡ ❞✉❛s ❞✐♠❡♥sõ❡s✳ ■♥✈❡st✐✲ ❣❛r❡♠♦s ❛ ✐♥✢✉ê♥❝✐❛ ❞❛s ✐♥t❡r❛çõ❡s ❝♦♠♣❡t✐t✐✈❛s ♥♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧✳ ◆♦ ❝❛♣ít✉❧♦ ✹ ❡st✉❞❛r❡♠♦s ❛ ❢❛s❡ ❞❡s♦r❞❡♥❛❞❛ ❞♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥✐s♦✲ tró♣✐❝♦ ❝♦♠ ✐♥t❡r❛çõ❡s ❝♦♠♣❡t✐t✐✈❛s ♥❛ r❡❞❡ q✉❛❞r❛❞❛✱ ♦♥❞❡ ✐♥✈❡st✐❣❛r❡♠♦s ❛ ❢❛s❡ ♥❡♠át✐❝❛ ❞❡ s♣✐♥✳ ◆♦ ❝❛♣ít✉❧♦ ✺ ❛♣r❡s❡♥t❛r❡♠♦s ❛s ♥♦ss❛s ❝♦♥❝❧✉sõ❡s✳ ❈♦♠♦ ❢♦r♠❛ ❞❡ s✐♠♣❧✐✜❝❛r ♦s ❝á❧❝✉❧♦s✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠ s✐st❡♠❛ ❞❡ ✉♥✐❞❛❞❡s ♦♥❞❡ ~ _{= 1} ❡ _{kB = 1}✱ ♦♥❞❡ ~ é ❛ ❝♦♥st❛♥t❡ ❞❡ P❧❛♥❝❦ ❡ _kB ❡

✶✶

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶❪ ❏✳ ❲❡✐ss ❏✳ ❞❡ P❤②s✐q✉❡ ✻✱ ✻✻✼ ✭✶✾✵✼✮✳

❬✷❪ ❉✳ ❈✳ ▼❛tt✐s ❚❤❡ ❚❤❡♦r② ♦❢ ▼❛❣♥❡t✐s♠ ■ ❙♣r✐♥❣✲❱❡r❧❛❝❤ ✭✶✾✽✶✮✳

❬✸❪ ❲✳ ❑✳ ❍❡✐s❡♥❜❡r❣ ❩✳ P❤②s✳ ✹✾✱ ✻✶✾ ✭✶✾✷✽✮✳

❬✹❪ P✳ ❆✳ ▼✳ ❉✐r❛❝ ❚❤❡ Pr✐♥❝✐♣❧❡s ♦❢ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝s ❈❧❛r❡♥❞♦♥ Pr❡ss✱ ❖①❢♦r❞ ✭✶✾✺✽✮✳

❬✺❪ ❨✳ ❋r❡♥❦❡❧ ❩✳ P❤②s✳ ✹✾✱ ✻✶✾ ✭✶✾✷✽✮✳

❬✻❪ ❨✳ ❉♦r❢♠❛♥ ◆❛t✉r❡ ✶✶✾✱ ✸✺✸ ✭✶✾✷✽✮✳

❬✼❪ ❆✳ ❆✉❡r❜❛❝❤ ■♥t❡r❛❝t✐♥❣ ❊❧❡❝tr♦♥s ❛♥❞ ◗✉❛♥t✉♠ ▼❛❣♥❡t✐s♠ ❙♣r✐♥❣✲ ❱❡r❧❛❝❤✱ ◆❡✇ ❨♦r❦ ■♥❝✳ ✭✶✾✾✹✮✳

❬✽❪ ❋✳ ❇❧♦❝❤ ❩✳ P❤②s✳ ✻✶✱ ✷✵✻ ✭✶✾✸✵✮✳

❬✾❪ ❚✳ ▼❛ts✉❜❛❞❛✱ ❍✳ ▼❛ts✉❞❛ Pr♦❣✳ ❚❤❡♦r✳ P❤②s✳ ✶✻✱ ✹✶✻ ✭✶✾✺✻✮✳

❬✶✵❪ ❘✳ ❏✳ ❇❛①t❡r ❊①❛❝t❧② s♦❧✈❡❞ ♠♦❞❡❧s ✐♥ st❛t✐t✐❝❛❧ ♠❡❝❤❛♥✐❝s ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦ ✭✶✾✽✷✮✳

❬✶✶❪ ❯✳ ❙❝❤♦❧❧✇ö❝❦✱ ❏✳ ❘✐❝❤t❡r✱ ❉✳ ❏✳ ❏✳ ❋❛r♥❡❧❧✱ ❘✳ ❋✳ ❇✐s❤♦♣ ◗✉❛♥✲ t✉♠ ▼❛❣♥❡t✐s♠ ✲ ▲❡❝t✉r❡s ◆♦t❡s ✐♥ P❤②s✐❝s ❙♣r✐♥❣❡r✲❱❡r❧❛❣ ✭✷✵✵✹✮✳

❬✶✷❪ ▲✳ ❖♥s❛❣❡r P❤②s✳ ❘❡✈✳ ✻✺✱ ✶✶✼ ✭✶✾✹✹✮✳

❬✶✸❪ ❲✳ P✳ ❲♦❧❢ ❇r❛③✳ ❏✳ P❤②s✳ ✸✵✱ ✼✾✹ ✭✷✵✵✵✮✳

❘❊❋❊❘✃◆❈■❆❙ ❇■❇▲■❖●❘➪❋■❈❆❙ ✶✷

❬✶✺❪ P✳ ❋❛③❡❦❛s ❛♥❞ P✳ ❲✳ ❆♥❞❡rs♦♥ P❤✐❧♦s✳ ▼❛❣✳ ✸✵✱ ✷✸ ✭✶✾✼✹✮✳

❬✶✻❪ ❆✳ ❆✳ ❑❛t❛♥✐♥✱ ❱✳ ❨✉ ■r❦❤✐♥ P❤②s✐❝s✲❯s♣❡❦❤✐ ✺✵ ✭✻✮✱ ✻✶✸ ✭✷✵✵✼✮✳

❬✶✼❪ ❙✉❜✐r ❙❛❝❤❞❡✈ ◗✉❛♥t✉♠ P❤❛s❡ ❚r❛♥s✐t✐♦♥s ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss ✭✷✵✶✶✮✳

❬✶✽❪ ❍✳ ❇❡t❤❡ ❩✳ P❤②s✳ ✼✶✱ ✷✵✺ ✭✶✾✸✶✮✳

❬✶✾❪ ▲✳ ❍✉❧t❤é♥ ❆r❦✳ ▼❛t✳ ❆str♦♥♦♠✳ ❋②s✐❦ ✷✻✱ ✶✶ ✭✶✾✸✽✮✳

❬✷✵❪ ❊✳ ▲✐❡❜✱ ❚✳ ❙❝❤✉❧t③✱ ❉✳ ❈✳ ▼❛tt✐s ❆♥♥✳ P❤②s✳ ✶✻✱ ✹✵✼ ✭✶✾✻✶✮✳

❬✷✶❪ ■✳ ❆✤❡❝❦ ❛♥❞ ❊✳ ▲✐❡❜ ❍✳ ▲❡tt✳ ▼❛t❤✳ P❤②s✳ ✶✷✱ ✺✼ ✭✶✾✽✻✮✳

❬✷✷❪ ❋✳ ❉✳ ▼✳ ❍❛❧❞❛♥❡ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✺✵✱ ✶✶✺✸ ✭✶✾✽✸✮✳

❬✷✸❪ ❲✳ ❏✳ ▲ ❇✉②❡rs✱ ❘✳ ▼✳ ▼♦rr❛✱ ❘✳ ▲✳ ❆r♠str♦♥❣✱ ▼✳ ❏✳ ❍♦❣❛♥✱ P✳ ●❡r❧❛❝❤ ❛♥❞ ❑✳ ❍✐r❛❦❛✇❛ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✺✻✱ ✸✼✶ ✭✶✾✽✻✮✳

❬✷✹❪ ▼✳ ❙t❡✐♥❡r✱ ❑✳ ❑❛❦✉r❛✐✱ ❏✳ ❑✳ ❑❥❡♠s✱ ❉✳ P❡t✐❣r❛♥❞ ❛♥❞ ❘✳ P②♥♥ ❏✳ ❆♣♣❧✐❡❞ P❤②s✳ ✻✶✱ ✸✾✺✸ ✭✶✾✽✼✮✳

❬✷✺❪ ❚✳ ❙❛❦❛❣✉❝❤✐✱ ❑✳ ❑❛❦✉r❛✐✱ ❚✳ ❨♦❦♦♦ ❛♥❞ ❏✳ ❆❦✐♠✐ts✉ ❏✳ P❤②s✳ ❙♦❝✳ ❏♣♥ ✻✺✱ ✸✵✷✺ ✭✶✾✾✻✮✳

❬✷✻❪ ●✳ ❚♦✉❧♦✉s❡ ❈♦♠♠✉♥✳ P❤②s✳ ✷✱ ✶✶✺ ✭✶✾✼✼✮✳

❬✷✼❪ ❏✳ ❱✐❧❧❛✐♥ ❏✳ P❤②s✳ ❈ ✶✵✱ ✶✼✶✼ ✭✶✾✼✼✮✳

❬✷✽❪ P✳ ❈❤❛♥❞r❛✱ ❇✳ ❉♦✉❝♦✉t P❤②s✳ ❘❡✈✳ ❇ ✸✽✱ ✾✸✸✺ ✭✶✾✽✽✮✳

❬✷✾❪ ❈✳ ▲❛❝r♦✐①✱ P✳ ▼❡♥❞❡❧s✱ ❋✳ ▼✐❧❛ ■♥tr♦❞✉❝t✐♦♥ t♦ ❋r✉str❛t❡❞ ▼❛❣♥❡✲ t✐s♠ ❙♣r✐♥❣❡r ❙❡r✐❡s ✐♥ ❙♦❧✐❞ ❙t❛t❡ ❙❝✐❡♥❝❡s✱ ◆❡✇ ❨♦r❦ ✭✷✵✶✶✮✳

❬✸✵❪ ▲❡♦♥ ❇❛❧❡♥ts ◆❛t✉r❡ ✹✻✹✱ ✶✾✾ ✭✷✵✶✵✮✳

✶✸

❈❛♣ít✉❧♦ ✷

▼♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣

❆♥✐s♦tró♣✐❝♦ q✉❛s❡✲❜✐❞✐♠❡♥s✐♦♥❛❧

✷✳✶ ■♥tr♦❞✉çã♦

◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥♦ ❡st✉❞♦ ❞♦ ♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ❛♥✐s♦tró♣✐❝♦ q✉❛s❡✲❜✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ s♣✐♥ S = 1✱

q✉❡ é ❞❡s❝r✐t♦ ♣❡❧♦ s❡❣✉✐♥t❡ ❤❛♠✐❧t♦♥✐❛♥♦✿

H = J1 2

X

r,a

S_rxS_rx_+a+S_rySry+a+λSrzSrz+a

+DX

r

(S_rz)2 ✭✷✳✶✮

+ J2 2

X

r,d

S_rxS_rx_+d+S_ryS_ry_+d+λS_rzS_rz_+d

+ J3 2

X

r,δ

S_rxS_rx_+δ+S_ryS_ry_+δ+λS_rzS_rz_+δ,

♦♥❞❡ J1 (J2) é ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ♣r✐♠❡✐r♦s ✭s❡❣✉♥❞♦s✮ ✈✐③✐♥❤♦s ♥✉♠❛ r❡❞❡

q✉❛❞r❛❞❛ q✉❡ é ❛❝♦♣❧❛❞❛ ♣❡❧❛ ✐♥t❡r❛çã♦ ✐♥t❡r✲♣❧❛♥♦ J3✱ λ é ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡

❡①❝❤❛♥❣❡ ❡Dé ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ ♣❧❛♥♦ ❢á❝✐❧✳ ◆❡st❡ tr❛❜❛❧❤♦ ♥ós ❝♦♥s✐❞❡r❛♠♦s ❛♣❡♥❛s ❞♦✐s ✈❛❧♦r❡s ♣❛r❛ ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ ❡①❝❤❛♥❣❡ λ = 0 ❡ λ = 1✳ ❖

❈❆P❮❚❯▲❖ ✷ ✶✹

◗✉❛♥❞♦ J3 =J2 = 0✱ λ = 0 ❡ D = 0 t❡♠♦s ♦ ♠♦❞❡❧♦ ❳❨ ❜✐❞✐♠❡♥s✐♦♥❛❧

♥✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ s♣✐♥ S = 1✳ ❊st❡ ♠♦❞❡❧♦ ❢♦✐ ❡①t❡♥s❛♠❡♥t❡ ❡①♣❧♦✲

r❛❞♦ ❡ ❤♦❥❡ ❡①✐st❡ ✉♠ ❝♦♥s❡♥s♦ ❛ r❡s♣❡✐t♦ ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❈♦♠♦ s❡ s❛❜❡✱ ♦ ♠♦❞❡❧♦ ❡①✐❜❡ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❛ ✉♠❛ t❡♠♣❡r❛t✉r❛ ✜♥✐t❛ TBKT

❝❤❛♠❛❞❛ ❞❡ t❡♠♣❡r❛t✉r❛ ❞❡ ❇❡r❡③✐♥s❦✐✐✲❑♦st❡r❧✐t③✲❚❤♦✉❧❡ss ❬✶✱✷❪✳ ❊st❛ tr❛♥✲ s✐çã♦ ❡stá ❛ss♦❝✐❛❞❛ ❝♦♠ ♦ s✉r❣✐♠❡♥t♦ ❞❡ ✉♠ ♦r❞❡♠ t♦♣♦❧ó❣✐❝❛✱ r❡s✉❧t❛♥t❡ ❞❛ ❢♦r♠❛çã♦ ❞❡ ♣❛r❡s ❞❡ ✈órt✐❝❡s ❡ ❛♥t✐✲✈órt✐❝❡s ❬✸❪✳

◗✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦sJ3 6= 0 ❡J2 = 0♦ ♠♦❞❡❧♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ q✉❛s❡✲

❜✐❞✐♠❡♥s✐♦♥❛❧✱ ♣♦✐s ❛❣♦r❛ ♦s s♣✐♥s ❞♦s sít✐♦s ♥♦s ♣❧❛♥♦s ❛❞❥❛❝❡♥t❡s ✐♥t❡r❛❣❡♠✳ ❖ ❝♦♠♣♦st♦BaN i2V2O3 é ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❡❧❤♦r ♣r♦tót✐♣♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❳❨

q✉❛s❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❞❡ s♣✐♥ ✶✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❛ ✐♥t❡r❛çã♦ ✐♥t❡r♣❧❛✲ ♥❛r ♦ ❝♦♠♣♦st♦ s♦❢r❡ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ ❞♦ t✐♣♦ ♦r❞❡♠✲❞❡s♦r❞❡♠ ❡♠ TN = 47.4K✱ ❝♦♠ ♦s s♣✐♥s ❛❧✐♥❤❛❞♦s ♥♦ ♣❧❛♥♦ ❳❨ ❬✹❪✳ ▼❡s♠♦ ♣❛r❛ ✉♠

♣❡q✉❡♥♦ ✈❛❧♦r ❞❡ J3✱ ♦ ❝❛rát❡r ❞❛ tr❛♥s✐çã♦ ❇❑❚ é ♠♦❞✐✜❝❛❞♦ ♣❛r❛ ✉♠❛

tr❛♥s✐çã♦ ❞♦ t✐♣♦ ♦r❞❡♠✲❞❡s♦r❞❡♠✳

❖ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❝❧áss✐❝♦ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ✐♥t❡r❛çõ❡s ❞❡ s❡❣✉♥✲ ❞♦s ✈✐③✐♥❤♦s J2/J1 < 0.5✱ J3 = 0✱ λ = 1 ❡ D = 0 t❡♠ ♦r❞❡♠ ◆é❡❧✳ P♦r

♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦ J2/J1 > 0.5 ♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ t❡♠ ♦r❞❡♠ ❝♦❧✐♥❡❛r✳

❖ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❝❧áss✐❝♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ♦r✐❡♥t❛çã♦ r❡❧❛t✐✈❛ ❞❡ ❛♠❜❛s ❛s s✉❜r❡❞❡s✳ ◆♦ ❡♥t❛♥t♦✱ ✢✉t✉❛çõ❡s q✉â♥t✐❝❛s r❡♠♦✈❡♠ ❛ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ❡ ✉♠ ❡st❛❞♦ ❝♦❧✐♥❡❛r ♦♥❞❡ ♦s s♣✐♥s ❛❧✐♥❤❛♠ ❢❡rr♦♠❛❣♥❡t✐❝❛♠❡♥t❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠ ❡✐①♦ ❞❛ r❡❞❡ q✉❛❞r❛❞❛ ❡ ❛♥t✐❢❡rr♦♠❛❣♥❡t✐❝❛♠❡♥t❡ ❛♦ ❧♦♥❣♦ ❞❡ ♦✉tr♦ é s❡❧❡❝✐♦♥❛❞♦ ❬✺✱✻❪✳

❖s t❡r♠♦s ❛❞✐❝✐♦♥❛✐s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❛♥✐s♦tr♦♣✐❛ ❞❡ í♦♥ ú♥✐❝♦✱ sã♦ ♣♦ssí✈❡✐s q✉❛♥❞♦ S > 1/2 ❡ ♣♦❞❡ ❧❡✈❛r ❛ ♥♦✈❛s ❝❛r❛❝t❡ríst✐❝❛s ❢ís✐❝❛s✱

t❛✐s ❝♦♠♦ ✉♠❛ tr❛♥s✐çã♦ ❞❡ ❢❛s❡ q✉â♥t✐❝❛ ♣❛r❛ ✉♠❛ ❢❛s❡ ❝♦♠ ✈❛❧♦r❡s ❞❡ D ❣r❛♥❞❡✳ ❖ ❊st✉❞♦ ❞❡st❡s ♠♦❞❡❧♦s ♥ã♦ sã♦ ❛♣❡♥❛s ❞❡ ✐♥t❡r❡ss❡ ❛❝❛❞ê♠✐❝♦ ❥á q✉❡ ♠❛t❡r✐❛✐s ❝♦♠ S = 1 ❡ ❛♥✐s♦tr♦♣✐❛ ❞❡ í♦♥ ú♥✐❝♦ ❢♦r❛♠ s✐♥t❡t✐③❛❞♦s

r❡❝❡♥t❡♠❡♥t❡ ❬✼❪✳

❈❆P❮❚❯▲❖ ✷ ✶✺

❋✐❣✉r❛ ✷✳✶✿ ❛✮ ❊st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❆♥t✐❢❡rr♦♠❛❣♥ét✐❝♦ ❜✮ ❊st❛❞♦ ❢✉♥❞❛♠❡♥✲ t❛❧ ❈♦❧✐♥❡❛r✭❝♦❧✉♥❛s✮✳

❛✉t♦✲❝♦♥s✐st❡♥t❡✳ ❈♦♠♦ é ❝♦♥❤❡❝✐❞♦✱ ✉♠❛ tr❛♥s✐çã♦ ❞♦ t✐♣♦ ❑♦st❡r❧✐t③✲❚❤♦✉❧❡ss ✭❑❚✮ ♦❝♦rr❡ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❞♦ t✐♣♦ ❳❨ q✉❛♥❞♦ J2 = J3 = 0✳ P♦rt❛♥t♦ é

✐♥t❡r❡ss❛♥t❡ ❡st✉❞❛r ❛ tr❛♥s✐çã♦ ❑❚ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❳❨ ❝♦♠ ✐♥t❡r❛çõ❡s ❝♦♠✲ ♣❡t✐t✐✈❛s✳ ❆s ✢✉t✉❛çõ❡s q✉â♥t✐❝❛s ♠✉❞❛♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ q✉❛♥t✐t❛t✐✈♦ ❞♦ ♠♦❞❡❧♦✱ ♠❛s ♦ ♣❡r✜❧ q✉❛❧✐t❛t✐✈♦ ❞♦ s✐st❡♠❛ ❝❧áss✐❝♦ ♣❡r♠❛♥❡❝❡✳

P❛r❛ ✉♠ ❍❛♠✐❧t♦♥✐❛♥♦ ❞♦ t✐♣♦ ❳❨ ✉♠❛ t❡♦r✐❛ ♠✉✐t♦ ❝♦♥✈❡♥✐❡♥t❡ é ❛ ❛♣r♦①✐♠❛çã♦ ❤❛r♠ô♥✐❝❛ ❛✉t♦✲❝♦♥s✐st❡♥t❡✱ q✉❡ s✉❜st✐t✉✐ ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ♣♦r ♦✉tr♦ ❡❢❡t✐✈♦ ❝♦♠ ♣❛râ♠❡tr♦s ❞❡♣❡♥❞❡♥t❡s ❞❛ t❡♠♣❡r❛t✉r❛ r❡♥♦r♠❛❧✐③❛❞♦s ❬✽✕✶✷❪✳ ❆♣❡s❛r ❞❡ s❡r ✉♠❛ t❡♦r✐❛ s❡♠✐❝❧áss✐❝❛✱ ❡❧❛ t❡♠ ❛ ✈❛♥t❛❣❡♠ ❞❡ s❡r ✉♠❛ t❡♦r✐❛ ❞❡ ♦♥❞❛ ❞❡ s♣✐♥ q✉❡ ❢♦r♥❡❝❡ ❛ tr❛♥s✐çã♦ ❑❚✳ ❊s❝r❡✈❡♠♦s ♣r✐♠❡✐r♦ ❛s ❝♦♠♣♦♥❡♥t❡s ❞♦ s♣✐♥ ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❡♠ t❡r♠♦s ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❱✐❧❧❛✐♥ ❬✶✸❪✱ ❞❛❞❛ ♣♦r

S_r+ = eiϕrp_{(S+ 1/2)2}

−(Sz

r + 1/2)2, ✭✷✳✷✮

S_r− = p(S+ 1/2)2_−(Sz

r + 1/2)2e−iϕr,

♦♥❞❡ ϕr é ♦ ♦♣❡r❛❞♦r ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ â♥❣✉❧♦ ❛③✐♠✉t❛❧ ❞♦ s♣✐♥ ❡♠ t♦r♥♦

❞♦ ❡✐①♦ z✳ ❙✉❜st✐t✉✐♥❞♦ ❛ r❡♣r❡s❡♥t❛çã♦ ✭✷✳✷✮ ❡♠ ✭✷✳✶✮✱ ❝♦♠ ❛ ❝♦♥❞✐çã♦ ϕ =φ+π ♣❛r❛ ♦s t❡r♠♦s ❞❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s ✭♦r❞❡♠ ❛♥t✐❢❡rr♦♠❛❣♥ét✐❝❛✮✱ ❡ ❡①♣❛♥❞✐♥❞♦ ❡♠ t❡r♠♦s ❞❡(Sz

r)2 ❡(φr−φr+a)2✱ ❛ ♣❛rt❡ ❞❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s

❡ ❛♥✐s♦tr♦♣✐❛ D ✜❝❛

H1 =

J1 2

X

r,a

h

ρ1Se2(φ2r−φrφr+a) + (Srz)2+λSrzSrz+a

i

+DX

r

❈❆P❮❚❯▲❖ ✷ ✶✻

♦♥❞❡ Se2 _{=S(S+ 1) ❡ ρ}

1 ♣❛r❛ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s é ❞❛❞♦ ♣♦r ❬✶❪

ρ1 =

*" 1₋ Sz r e S 2#+

hcos(φr−φr+a)i. ✭✷✳✹✮

❆ q✉❛♥t✐❞❛❞❡ ❛❝✐♠❛ é ❝❤❛♠❛❞❛ ❞❡ r✐❣✐❞❡③ ✭st✐✛♥❡ss✮ ❡ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ ♦r❞❡♠ ❞♦ s✐st❡♠❛✳ ❆ r✐❣✐❞❡③ t❡♠ ✈❛❧♦r ✜♥✐t♦ ♥ã♦✲♥✉❧♦ ♥❛ r❡❣✐ã♦ ♦r❞❡♥❛❞❛✱ ❡ ✈❛✐ ❝❛✐♥❞♦ ❛ ③❡r♦ ❛ ♠❡❞✐❞❛ q✉❡ ♦ s✐st❡♠❛ ❞❡ ❛♣r♦①✐♠❛ ❞❛ r❡❣✐ã♦ ❞❡s♦r❞❡♥❛❞❛✳ ❚♦♠❛♥❞♦ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ♥ós ♦❜t❡♠♦s

H1 = 2J1

X

q

h

ρ1Se2(1−γq)φqφ−q+ (1 +d+λγq)SqzS−zq

i

✭✷✳✺✮

♦♥❞❡ γq = 1₂(cosqx+ cosqy) ❡D˜ =D/2J1✳

❖ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦ é ✉s❛❞♦ ♣❛r❛ ❛ ♣❛rt❡ ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s ❡ ❛❝♦✲ ♣❧❛♠❡♥t♦ ❡♥tr❡ ♣❧❛♥♦s✱ ❧❡♠❜r❛♥❞♦ q✉❡ ♣❛r❛ ♦s t❡r♠♦s ❞❡ s❡❣✉♥❞♦s ✈✐③✐♥❤♦s✱ ♥❛ ❢❛s❡ ◆é❡❧✱ ♦s s♣✐♥s ❡stã♦ ♥❛ ♠❡s♠❛ ❞✐r❡çã♦ ❡ ♣♦rt❛♥t♦ ϕ = φ✳ ❆♣ós ✉♠ ♣r♦❝❡ss♦ ❛❧❣é❜r✐❝♦✱ ♦❜t❡♠♦s ✉♠ ❤❛♠✐❧t♦♥✐❛♥♦ ❡❢❡t✐✈♦ ♣❛r❛ ❛ ❢❛s❡ ◆é❡❧ ❡s❝r✐t♦ ❝♦♠♦✿

H =X

q

a(q)φqφ−q+b(q)SqzS−zq

, ✭✷✳✻✮

s❡♥❞♦

a(q) = 2hρ1(1−γq)−ηρ2(1−ζq) +

αρ3

2 (1−cosqz)

i e

S2 _{✭✷✳✼✮}

b(q) = 2h1 + ˜D₋η+ α

2 +λ(γq+ηζq+

α

2 cosqz)

i

. ✭✷✳✽✮

❖sρ✬s r❡♥♦r♠❛❧✐③❛❞♦s sã♦ ❞❛❞❛s ♣♦r

ρi =

*" 1₋ Sz r e S 2#+ exp " −X q

gi(q)hφqφ−qi

#

✭✷✳✾✮

s❡♥❞♦

❈❆P❮❚❯▲❖ ✷ ✶✼

ζq = cosqxcosqy , η=J2/J1 , α=J3/J1. ✭✷✳✶✶✮

■♥tr♦❞✉③✐♥❞♦ ❛ tr❛♥s❢♦r♠❛çã♦ ❝❛♥ô♥✐❝❛

φq =

b(q)

a(q)

1/4

a†_q+a−q

, S_qz =i

a(q)

b(q)

1/4

a†_q−a−q

✭✷✳✶✷✮

♦♥❞❡ a†

q ❡ aq sã♦ ♦♣❡r❛❞♦r❡s ❜♦sô♥✐❝♦s ❞❡ ❝r✐❛çã♦ ❡ ❛♥✐q✉✐❧❛çã♦✱ r❡s♣❡❝t✐✈❛✲

♠❡♥t❡✳ ❈♦♠ ✐ss♦ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ♦ ❤❛♠✐❧t♦♥✐❛♥♦ ♥❛ ❢♦r♠❛

H =X

q

ωq(a†qaq+ 1/2), ✭✷✳✶✸✮

s❡♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❞✐s♣❡rsã♦ ωq ❞❛❞❛ ♣♦r

ωq = 2

p

a(q)b(q) ✭✷✳✶✹✮

❯s❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✷✮✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦s t❡r♠♦s ❞❛❞♦s ❛❜❛✐①♦✳

h(S_rz)2_i= 1 2π3

Z π 0 Z π 0 Z π 0

d3q s

a(q)

b(q) coth

βωq

2

✭✷✳✶✺✮

hφqφ−qi=

1 2

s b(q)

a(q)coth

βωq

2

✭✷✳✶✻✮

♦♥❞❡ β = 1/kBT ❡ s✐♠♣❧✐✜❝❛♠♦s ❛❞♦t❛♥❞♦ kB = 1✳

P❛r❛ ❛ ❢❛s❡ ❝♦❧✐♥❡❛r✱ s❡❣✉✐♠♦s ♦s ♠❡s♠♦s ♣r♦❝❡❞✐♠❡♥t♦s q✉❡ ✜③❡♠♦s ♣❛r❛ ♦ ❝❛s♦ ◆é❡❧✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛❣♦r❛ q✉❡ ♦s ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦s tê♠ ❛ ♠❡s♠❛ ♦r✐❡♥t❛çã♦ ♥❛ ❞✐r❡çã♦ y ❡♥q✉❛♥t♦ q✉❡ ♥❛ ❞✐r❡çã♦ x ❛❝r❡s❝❡♥t❛♠♦s π ❡♠ ϕ ✭✈❡❥❛ ✜❣✉r❛ ✷✳✶ ✭❜✮ ✮✳ P♦rt❛♥t♦ ❛s ❢♦r♠❛s ❞❡a(q)❡b(q)t❛♠❜é♠ sã♦ ❛❧t❡r❛❞❛s

s❡♥❞♦ ❡s❝r✐t♦s ❝♦♠♦✿

a(~q) =J1ρe1(cosqy−cosqx) + 2ηρe2(1−cosqxcosqy) +αρez(1−cosqz) ✭✷✳✶✼✮

❈❆P❮❚❯▲❖ ✷ ✶✽

❝♦♠ ♦s ρe′

ist❡♥❞♦ ❛ ♠❡s♠❛ ❢♦r♠❛ ❛♥t❡r✐♦r♠❡♥t❡ ♣♦✐s só ❞❡♣❡♥❞❡♠ ❞♦ t✐♣♦ ❞❡

r❡❞❡✳ ❆ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ Tc ❡ ♦ ✈❛❧♦r ❝rít✐❝♦ ❞❡ D ♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞♦s

q✉❛♥❞♦ ρ ✈❛✐ ❛ ③❡r♦✳ ❘❡❢♦rç❛♠♦s ❛ ✐❞é✐❛ ❞❡ q✉❡ ♣❛r❛ α= 0✱ Tc ❝♦rr❡s♣♦♥❞❡

❛ t❡♠♣❡r❛t✉r❛ ❞❡ tr❛♥s✐çã♦ ❞❡ ❑♦st❡r❧✐t③✲❚❤♦✉❧❡ss✳

❆ ♣r✐♥❝✐♣❛❧ ♠♦t✐✈❛çã♦ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦ é ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❝rít✐✲ ❝❛s ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ ✭✷✳✶✮✱ ♠❛s t❛♠❜é♠ ❝❛❧❝✉❧❛r♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s t❡r♠♦❞✐♥â♠✐❝❛s ❝♦♠♦ ❡♥❡r❣✐❛ ✐♥t❡r♥❛✱ ❝❛❧♦r ❡s♣❡❝í✜❝♦✱ ❡♥tr♦♣✐❛ ❡ ❡♥❡r❣✐❛ ❧✐✈r❡✳ ◆❛ ❛♣r♦①✐♠❛çã♦ ❤❛r♠ô♥✐❝❛ ❛✉t♦✲❝♦♥s✐st❡♥t❡ ♣♦❞❡♠♦s r❡❛❧✐③❛r ♦s ❝á❧✲ ❝✉❧♦s s♦♠❡♥t❡ ❛❜❛✐①♦ ❞❛s t❡♠♣❡r❛t✉r❛s ❝rít✐❝❛s✳ ❖s ❝á❧❝✉❧♦s ❢♦r❛♠ ❢❡✐t♦s ✉s❛♥❞♦ ❛s ❡①♣r❡ssõ❡s ❛❜❛✐①♦✳

U = 1 (2π)d

Z

nkωk ddk Energia Interna ✭✷✳✶✾✮

C = ∂U

∂T Calor especif ico ✭✷✳✷✵✮

S =₋ 1 (2π)d

Z

ddk [nk lnnk−(1 +nk) ln (1 +nk)] Entropia ✭✷✳✷✶✮

F =T

Z _dd_k

(2π)dln

1₋e−ωk_{T Energia Livre ✭✷✳✷✷✮}

♦♥❞❡ d é ❛ ❞✐♠❡♥sã♦ ❡ nk = 1/[exp(ωk/T) −1]✳ ▲❡♠❜r❛♥❞♦ q✉❡ ♥❡ss❛s

❡q✉❛çõ❡s✱ ωk é ❞❡♣❡♥❞❡♥t❡ ❞❛ t❡♠♣❡r❛t✉r❛✳

✷✳✸ ❘❡s✉❧t❛❞♦s

◆❛ ✜❣✉r❛ ✷✳✷ ♥ós ♠♦str❛♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ♣❛râ♠❡tr♦ ❝rít✐❝♦Dc/J1✱

q✉❡ s❡♣❛r❛ ❛s ❢❛s❡s ❆❋ ❡ ❈❆❋ ❞❛ ♣❛r❛♠❛❣♥ét✐❝❛ q✉â♥t✐❝❛✱ ❝♦♠♦ ❢✉♥çã♦ ❞❡ η ♣❛r❛ T = 0✱ λ = 0 ❡ α = 0 ✭♠♦❞❡❧♦ ❳❨ ✷❞✮✳ ❖ ♣❛râ♠❡tr♦ Dc s❡ ❛♥✉❧❛

♥♦s ♣♦♥t♦sη1c = 0.49❡η2c = 0.52✱ ❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❝✉r✈❛ é ✉♠ ❛rt❡❢❛t♦

❈❆P❮❚❯▲❖ ✷ ✶✾

✜❣✉r❛ ✷✳✸ ❛♣r❡s❡♥t❛♠♦s ♦s ♠❡s♠♦s r❡s✉❧t❛❞♦s✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛❣♦r❛ λ = 1

✭♠♦❞❡❧♦ ❞❡ ❍❡✐s❡♥❜❡r❣ ✷❞✮ ♦❜t❡♥❞♦ η1c = 0.49 ❡ η2c = 0.51✱ ❝♦♠ ❛ ❢❛s❡

♣❛r❛♠❛❣♥ét✐❝❛ ✉♠ ♣♦✉❝♦ ♠❡♥♦r ❡♠ r❡❧❛çã♦ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳ ❊♥❝♦♥tr❛♠♦s Dc/J1 = 7.32 ♥♦ ❧✐♠✐t❡ η = 0 q✉❡ ❡stá ❡♠ ❜♦♠ ❛❝♦r❞♦ ❝♦♠ ♦ r❡s✉❧t❛❞♦ 6.38 ♦❜t✐❞♦ ♣♦r ❲♦♥❣ ❡t✳ ❛❧ ❬✶✹❪ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ❡①♣❛♥sã♦ ❞♦ ❝❧✉st❡r

❛❝♦♣❧❛❞♦✳ ◗✉❛♥❞♦ ❛✉♠❡♥t❛♠♦s α✱ ❛ r❡❣✐ã♦ ❞❛ ❢❛s❡ ❞❡s♦r❞❡♥❛❞❛ ❞✐♠✐♥✉✐ ❡ ❞❡s❛♣❛r❡❝❡ q✉❛♥❞♦ α = 1✳ ❊st❡ r❡s✉❧t❛❞♦ é ❡s♣❡r❛❞♦ ✉♠❛ ✈❡③ q✉❡ ♦s

❡❢❡✐t♦s ❞❛s ✢✉t✉❛çõ❡s q✉â♥t✐❝❛s sã♦ ♠❡♥♦r❡s ❡♠ três ❞✐♠❡♥sõ❡s✳ ◆❛ ✜❣✉r❛ ✷✳✹ ❛♣r❡s❡♥t❛♠♦s✱ ❛ ✜♠ ❞❡ ❝❡rt✐✜❝❛çã♦ ❞❛ ❛✉sê♥❝✐❛ ❞♦ ❡st❛❞♦ ❞❡s♦r❞❡♥❛❞♦ ✐♥t❡r♠❡❞✐ár✐♦✱ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ Dc/J1 ✈❡rs✉s η ♣❛r❛ ♦ ♠♦❞❡❧♦ ❳❨ ✸❞

✭α= 1 ❡λ= 0✮✱ ♦♥❞❡ é ✈❡r✐✜❝❛❞♦ q✉❡Dc/J1 é ♠❛✐♦r ❞♦ q✉❡ ♦ ✈❛❧♦r ❡♠ ❞✉❛s

❞✐♠❡♥sõ❡s ✭α= 0✮✳

◆❛ ✜❣✉r❛ ✷✳✺ ♠♦str❛♠♦s ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ Tc

❝♦♠♦ ❢✉♥çã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ ❢r✉str❛çã♦ η ♣❛r❛ ♦ ♠♦❞❡❧♦ ❳❨ ❛♥t✐❢❡rr♦♠❛❣✲ ♥ét✐❝♦ ✷❞ ✭α=λ= 0✮ ♣❛r❛ ❞♦✐s ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ❞❛ ❛♥✐str♦♣✐❛ D/J1 = 5❡ 6✳ ❖ ✈❛❧♦r ❞❡Tc ♥ã♦ ❡stá ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ tr❛♥s✐çã♦ ❡♥tr❡ ♦r❞❡♠ ❡ ❞❡s♦r❞❡♠

❛❣♥ét✐❝❛✱ ❡ s✐♠ ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈❛❧♦r ❞❛ t❡♠♣❡r❛t✉r❛ ❞❡ ❑♦st❡r❧✐t③✲❚❤♦✉❧❡ss ✭TKT✮ ♥❛ q✉❡❜r❛ ❞❡ ♦r❞❡♠ t♦♣♦❧ó❣✐❝❛ ❞❛ ❡str✉t✉r❛ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❞♦s ✈ór✲

t✐❝❡s ♥❛ r❡❞❡ q✉❛❞r❛❞❛✳ P❛r❛ ✉♠ ❞❛❞♦ ✈❛❧♦r ❞❡ D/J1 ✜①♦✱ t❡♠♦s q✉❡ Tc

❞❡❝r❡s❝❡ ♠♦♥♦t♦♥✐❝❛♠❡♥t❡✱ ♥❛ ❢❛s❡ ❆❋✱ ❝♦♠ ❛✉♠❡♥t♦ ❞❛ ❢r✉str❛çã♦ ✭♠❡✲ ❞✐❞❛ ♣❡❧♦ ♣❛râ♠❡tr♦η✮ ❞❡ ♠♦❞♦ q✉❡ ❡①✐st❡ ✉♠❛ ❢❛s❡ ❞❡s♦r❞❡♥❛❞❛ ♣❛r❛ ❝❡rt❛ r❡❣✐ã♦ ❞♦ ♣❛râ♠❡tr♦ η✳ ❆❝✐♠❛ ❞❡ ✉♠ ✈❛❧♦r ❝rít✐❝♦ ηc(D) ❡ ❡♠ ❜❛✐①❛s t❡♠✲

♣❡r❛t✉r❛s T < Tc t❡♠♦s ✉♠ ♦r❞❡♥❛♠❡♥t♦ ❝♦❧✐♥❡❛r ✭❈❆❋✮✱ ♥♦ q✉❛❧ ❛ ♠❡❞✐❞❛

q✉❡η❝r❡s❝❡ ❛ t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ❝r❡s❝❡ ♠♦♥♦t♦♥✐❝❛♠❡♥t❡ ❛♣r❡s❡♥t❛♥❞♦ ✉♠ ❛✉♠❡♥t♦ ❞♦ ♦r❞❡♥❛♠❡♥t♦ ❈❆❋ ✭✧✈órt✐❝❡s✧✮✳ ◆♦ ❧✐♠✐t❡η _≫1♣♦❞❡♠♦s ✐♥t❡r✲

♣r❡t❛r ❝♦♠♦ s❡♥❞♦ ✉♠❛ r❡❞❡ q✉❛❞r❛❞❛ ❝♦♠ ✐♥t❡r❛çã♦ J2 r♦t❛❝✐♦♥❛❞❛ ❞❡ ✉♠

â♥❣✉❧♦ ❞❡π/4✱ ✉♠❛ ✈❡③ q✉❡ ❛ ✐♥t❡r❛çã♦ ❞❡ ♣r✐♠❡✐r♦s ✈✐③✐♥❤♦sJ1 é ❞❡s♣r❡③í✈❡❧

❡♠ r❡❧❛çã♦ ❛ J2✳ ❆ss✐♠ s❡♥❞♦✱ ♦ ✈❛❧♦r ❡♥❝♦♥tr❛❞♦ ♣❛r❛ Tc q✉❛♥❞♦ η = 0✱

Tc(0, D)✱ ❝♦rr❡s♣♦♥❞❡rá ♥♦ ❧✐♠✐t❡ η ≫ 1 ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❧✐♥❡❛r ♣❛r❛ ❛

t❡♠♣❡r❛t✉r❛ ❝rít✐❝❛ ❞♦ t✐♣♦ Tc(η, D) = Tc(0, D)η✳ ❆ ❛çã♦ ❞❛ ❛♥✐s♦tr♦♣✐❛ D

é ❞✐♠✐♥✉✐r ♦s ♦r❞❡♥❛♠❡♥t♦s ❆❋ ❡ ❈❆❋✱ ❞❡st❛ ♠❛♥❡✐r❛ Tc(η, D) ❞✐♠✐♥✉✐✱ ❡

❛ r❡❣✐ã♦ ❞♦ ❡st❛❞♦ ❞❡s♦r❞❡♥❛❞♦ ❡♠ T = 0 ❛✉♠❡♥t❛ ❣r❛❞✉❛❧♠❡♥t❡✱ ♦♥❞❡ ♥❛

✜❣✉r❛ ✷✳✷ t❡♠♦s ❛ ❛♥á❧✐s❡ ❞♦ ❡st❛❞♦ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ♣❧❛♥♦ D₋η✳