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▼♦❞❡❧✐♥❣ ✜♥✐t❡ ❡♥tr♦♣② st❛t❡s ✇✐t❤ ❢r❡❡ ❢❡r♠✐♦♥s ❖❧❡❦s❛♥❞r ●❛♠❛②✉♥ ❯♥✐✈❡rs✐t② ♦❢ ❆♠st❡r❞❛♠ ✇♦r❦✲✐♥✲♣r♦❣r❡ss t♦❣❡t❤❡r ✇✐t❤✿ ◆✳ ■♦r❣♦✈✱ ❨✉✳ ❩❤✉r❛✈❧❡✈✱ ❉✳ ▼✳ ❈❤❡r♥♦✇✐t③ ❛♥❞ ❏✳❙✳ ❈❛✉① ❘❆◗■❙✱ ❆♥♥❡❝②✱ ✵✸✴✵✾✴✷✵✷✵ ✶✴✷✵

❇♦t❤ ❛♣♣r♦❛❝❤❡s ❢❛✐❧ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ✦✦✦ ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ✐♥ ✶❉ s②st❡♠s � |� q |O| ❦ �| ✷ e − itE ❦ + ixP ❦ � q |O ( x , t ) O ( ✵ , ✵ ) | q � = ❦ ◮ ◆✉♠❡r✐❝s ✭❆❇❆❈❯❙✮ ◮ ❋✐❡❧❞ t❤❡♦r② ✭ k F x ≫ ✶✱ k ✷ F t ≫ ✶✮ ◮ ▲✐♥❡❛r ❙♣❡❝tr✉♠ ◮ O = P ( ∂ϕ, ∂ ✷ ϕ, e i ϕ ) ◮ ❯♥✐✈❡rs❛❧✐t② ❢r♦♠ ♠✐❝r♦s❝♦♣✐❝ ▲❡❢t✿ ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ❆❇❆❈❯❙ ❛♥❞ ✐♥❡❧❛st✐❝ ♥❡✉tr♦♥ s❝❛tt❡r✐♥❣ ❢♦r ❑❈✉❋ ✸ ✳ ❬P❘▲ ✶✶✶ ✶✸✼✷✵✺❪✳ ❘✐❣❤t✿ ❚❤❡ t❤r❡s❤♦❧❞ s✐♥❣✉❧❛r✐t✐❡s ✐♥ t❤❡ ◆♦♥✲▲✐♥❡❛r ▲✉tt✐♥❣❡r ▲✐q✉✐❞✳ ✷✴✷✵

❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ✐♥ ✶❉ s②st❡♠s � |� q |O| ❦ �| ✷ e − itE ❦ + ixP ❦ � q |O ( x , t ) O ( ✵ , ✵ ) | q � = ❦ ◮ ◆✉♠❡r✐❝s ✭❆❇❆❈❯❙✮ ◮ ❋✐❡❧❞ t❤❡♦r② ✭ k F x ≫ ✶✱ k ✷ F t ≫ ✶✮ ◮ ▲✐♥❡❛r ❙♣❡❝tr✉♠ ◮ O = P ( ∂ϕ, ∂ ✷ ϕ, e i ϕ ) ◮ ❯♥✐✈❡rs❛❧✐t② ❢r♦♠ ♠✐❝r♦s❝♦♣✐❝ ▲❡❢t✿ ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ❆❇❆❈❯❙ ❛♥❞ ✐♥❡❧❛st✐❝ ♥❡✉tr♦♥ s❝❛tt❡r✐♥❣ ❢♦r ❑❈✉❋ ✸ ✳ ❬P❘▲ ✶✶✶ ✶✸✼✷✵✺❪✳ ❘✐❣❤t✿ ❚❤❡ t❤r❡s❤♦❧❞ s✐♥❣✉❧❛r✐t✐❡s ✐♥ t❤❡ ◆♦♥✲▲✐♥❡❛r ▲✉tt✐♥❣❡r ▲✐q✉✐❞✳ ❇♦t❤ ❛♣♣r♦❛❝❤❡s ❢❛✐❧ ❛t ✜♥✐t❡ t❡♠♣❡r❛t✉r❡ ✦✦✦ ✷✴✷✵

❖✉t❧✐♥❡ ◮ ❲❛r♠✲✉♣✿ s✐♥❡✲❦❡r♥❡❧ ◮ ▼✐❝r♦s❝♦♣✐❝ ❜♦s♦♥✐③❛t✐♦♥ T = ✵ ◮ P❤❛s❡ ❞r❡ss✐♥❣ ◮ ❙t❛t✐❝ ❝♦rr❡❧❛t✐♦♥s ◮ ❳❨✲♠♦❞❡❧ ◮ ❋❡rr♦♠❛❣♥❡t✐❝ ◮ P❛r❛♠❛❣♥❡t✐❝ ◮ ❉②♥❛♠✐❝s ✸✴✷✵

❙✐♥❡ ❑❡r♥❡❧ ✶ + e ✷ π i ν − ✶ � sin x ( q − p ) / ✷ � τ ( x , t = ✵ ) = det π q − p [ − k F , k F ] ❋♦r♠✲❢❛❝t♦r ♣r❡s❡♥t❛t✐♦♥ � |� q |O| ❦ �| ✷ e − itE ❦ + iP ❦ x τ ( x , t ) = �O ( x , t ) O ( ✵ , ✵ ) � = k ✶ < k ✷ ··· < k N | q � ✿ ❢r❡❡ ❢❡r♠✐♦♥s✿ q i = ✷ π n i | ❦ � ✿ s❤✐❢t❡❞ ❢r❡❡ ❢❡r♠✐♦♥s✿ k i = ✷ π ( n i − ν ) L ❀ L � � k ✷ P ❦ = k i , E ❦ = i / ✷ i i ❚❤❡ ❢♦r♠✲❢❛❝t♦r ✭♦✈❡r❧❛♣✮✿ � ✷ � ✷ N � � ✷ ✶ |� q |O| ❦ �| ✷ = L sin πν det . k i − q j N × N ✹✴✷✵

❋✐❡❧❞ t❤❡♦r② tr❡❛t♠❡♥t ❂ ▼✐❝r♦s❝♦♣✐❝ ❜♦s♦♥✐③❛t✐♦♥ ✭ T = ✵ ✮ ◮ ❋♦r♠✲❋❛❝t♦r s✉♠♠❛t✐♦♥ � |� q |O| ❦ �| ✷ e − ixP ❦ + itE ❦ = det( ✶ + ˆ τ ( x , t ) = V ) ❦ ◮ ❖rt❤♦❣♦♥❛❧✐t② ❈❛t❛str♦♣❤❡✿ |� q |O| ❦ ✈❛❝ �| ✷ = A / N ✷ α ◮ ❙♦❢t✲♠♦❞❡ s✉♠♠❛t✐♦♥ √ αϕ ( x , t ) e −√ αϕ ( ✵ , ✵ ) � = A � |� q |O| ❦ �| ✷ e − ixP ❦ + itE ❦ = � e τ ( x , t ) ∼ ( x − k F t ) α ( x + k F t ) α ■❘ ◮ ◆♦♥❧✐♥❡❛r ❝♦♥tr✐❜✉t✐♦♥s B F ) / ✷ = |� q |O| ❦ �| ✷ e − ixP ❦ + itE ❦ + ix ( Q − k F )+ it ( Q ✷ − k ✷ � τ ( x , t ) ∼ √ t ( x − k F t ) ˜ α ( x + k F t ) α Q + ■❘ ❙❧❛✈♥♦✈ ✭✶✾✽✾✮❀ ❙❧❛✈♥♦✈ ❛♥❞ ❑♦r❡♣✐♥ ✭✶✾✾✶✮❀ ❆✳ ❙❤❛s❤✐✱ ▲✳ ■✳ ●❧❛③♠❛♥✱ ❏✳✲❙✳ ❈❛✉①✱ ❛♥❞ ❆✳ ■♠❛♠❜❡❦♦✈ ✭✷✵✶✶✮❀ ◆✳ ❑✐t❛♥✐♥❡✱ ❑✳❑✳ ❑♦③❧♦✇s❦✐✱ ❏✳✲▼✳ ▼❛✐❧❧❡t✱ ◆✳❆✳ ❙❧❛✈♥♦✈✱ ❛♥❞ ❱✳ ❚❡rr❛s ✭✷✵✵✾✲✷✵✶✷✮❀ ❑✳❑✳ ❑♦③❧♦✇s❦✐✱ ❏✳✲▼✳ ▼❛✐❧❧❡t ✭✷✵✶✺✮❀ ✺✴✷✵

❈♦♠❜✐♥❛t♦r✐❝s ♦❢ ♦rt❤♦❣♦♥❛❧✐t② ❝❛t❛str♦♣❤❡ ●❡♥❡r✐❝ ♦✈❡r❧❛♣ ( k i − k j ) ✷ � ( q i − q j ) ✷ � � ✷ � ✷ N |� ❦ ✈❛❝ | q �| ✷ = i > j i > j L sin πν . � ( k i − q j ) ✷ i , j ❋❡r♠✐ s❡❛ ✐♥t❡❣❡rs k j = ✷ π q j = ✷ π n j = − N − ✶ L ( n j − ν ) , L n j , + j − ✶ , j = ✶ , ✷ . . . N ✷ � sin πν � ✷ N � � − ✷ = G ✷ ( ✶ − ν ) G ✷ ( ✶ + ν ) G ✹ ( N + ✶ ) � ν |� ❦ ✈❛❝ | q �| ✷ = ✶ − G ✷ ( N − ν + ✶ ) G ✷ ( N + ν + ✶ ) . πν i − j i � = j |� ❦ ✈❛❝ | q �| ✷ = G ✷ ( ✶ − ν ) G ✷ ( ✶ + ν ) N ✷ ν ✷ ❋♦r ν = ν ( k ) ✱ ✭ ν ± = ν ( ± k F ) , k F = π L / N ✮ �� � |� ✈❛❝ | ❋❙ �| ✷ = G ✷ ( ✶ − ν − ) G ✷ ( ✶ + ν + )( ✷ π ) ν − − ν + ν ( λ ) ν ( µ ) exp ( λ − µ + i ✵ ) ✷ d λ d µ N ν ✷ − + ν ✷ + [ − k F , k F ] ✷ ✻✴✷✵

❙t❛t✐❝ ✰ ③❡r♦ t❡♠♣❡r❛t✉r❡ τ ( x , t = ✵ ) = G ✷ ( ✶ − ν ) G ✷ ( ✶ + ν ) e − ✷ i ν x + ( ν → ν + Z ) ( − ✷ ix ) ν ✷ ( ✷ ix ) ν ✷ ✼✴✷✵

❋✐♥✐t❡ t❡♠♣❡r❛t✉r❡ � ✶ + n F ( q ) ( e ✷ π i ν − ✶ )sin( x ( p − q )) � τ ( x ) = det π p − q ◮ ■t ✐s ❝❤❛❧❧❡♥❣✐♥❣ t♦ ❞♦ ♠✐❝r♦s❝♦♣✐❝ ◮ ❖✈❡r❧❛♣s ❛r❡ t♦ s♠❛❧❧ ∼ e − cN ◮ ❚♦♦ ♠❛♥② s♦❢t ♠♦❞❡s ∼ e cN ◮ ❍❡✉r✐st✐❝ ❛♣♣r♦❛❝❤ ✐♥st❡❛❞✦ ❉r❡ss✐♥❣✿ ✐♥❤♦♠♦❣❡♥❡♦✉s ❛♥❞ ❝♦♠♣❧❡① ✈❛❧✉❡❞✦✦✦ ( e ✷ π i ν − ✶ ) = e ✷ π i ν T ( q ) − ✶ n F ( q ) ✶ ✷ π i log( ✶ +( e ✷ π i ν − ✶ ) n F ( q )) , ν → ν T ( q ) = π π ◮ � ∞ ν T ( q ) ν T ( q ′ ) � � � ( q ′ − q + i ✵ ) ✷ dqdq ′ τ ( x , t ) ≈ exp − i ( x − qt ) ν T ( q ) dq + −∞ R ✷ ✽✴✷✵

T = 0.1 T = 1.0 Re [ τ [ x ]] Re [ τ [ x ]] 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 x 5 10 15 20 25 - 0.2 x 5 10 15 20 25 z → z ′ = e ✷ π z /β ❚r ( e − β H O ( x , t ) . . . ) / ❚r❡ − β ❍ = �O ( z = x + it ) . . . � S ✶ × R ✶ ∼ �O ( z ′ ) . . . � R ✷ = ❈❋❚ ♣r❡❞✐❝t✐♦♥ ❢♦r ❝♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤✿ T = ✵ = A A (sinh( xT ) / T ) ν ✷ = e − x /ξ = � τ ( x ) x ν ✷ = ⇒ τ ( x ) = ⇒ ✶ /ξ ∼ T ??? � � ✾✴✷✵

❊①❛♠♣❧❡s ❛♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥s ◮ ▼♦❜✐❧❡ ✐♠♣✉r✐t② ❬❙❝✐P♦st P❤②s✳ ✽✱ ✵✺✸ ✭✷✵✷✵✮✱ ◆❡✇ ❏✳ P❤②s✳ ✶✽ ✭✷✵✶✻✮✱ ✵✹✺✵✵✺❪ [ − ✶ , ✶ ] ( ✶ + ˆ K + δ ˆ [ − ✶ , ✶ ] ( ✶ + ˆ ρ ( y ) = det K ) − det K ) K ( p , q ) = sin[ π F ( p )] e i ( p − q ) y / ✷ (cot[ π F ( p )] + i ) − e i ( q − p ) y / ✷ (cot[ π F ( q )] + i ) sin[ π F ( q )] π ( p − q ) δ K ( p , q ) = ✶ π sin[ π F ( p )] e − i ( p + q ) y / ✷ sin[ π F ( q )] ◮ ❘❡t✉r♥ ♣r♦❜❛❜✐❧✐t② ❢r♦♠ t❤❡ ❞♦♠❛✐♥ ✇❛❧❧ ✐♥✐t✐❛❧ st❛t❡ | ❉❲ � = | ↑↑ . . . ↑↓↓ . . . ↓� ❬❏✳▼✳ ❙t❡♣❤❛♥✱ ❏✳ ❙t❛t ✭✷✵✶✼✮❪ ✶ − e − p ✷ / ✹ sin √ τ ( p − q ) � � e − q ✷ / ✹ � ❉❲ | e τ H XXX | ❉❲ � = det R + π ( p − q ) ◮ P❡rs✐st❡♥❝❡ ♦❢ s♣✐♥ ❝♦♥✜❣✉r❛t✐♦♥s ❬■✳ ❉♦r♥✐❝ ✭✷✵✶✽✮❪ n F ( q ) = ✶ / cosh( q ) ◮ ❈❧❛ss✐❝❛❧ ✐♥t❡❣r❛❜❧❡ s②st❡♠s n F ( q ) = r ( q ) ✶✵✴✷✵