The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension

In narrow-band conductors a weakly screened Coulomb interelectron repulsion can supress narrow-band electrons` hopping, resulting in formation of a "frozen" electron phase which differs principally from any known macroscopic self-localized electron state including the Wigner crystal. In a...

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Бібліографічні деталі
Опубліковано в: :Физика низких температур
Дата:1999
Автори: Slutskin, A.A., Slavin, V.V., Kovtun, H.A.
Формат: Стаття
Мова:Англійська
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 1999
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Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/139014
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension / A.A. Slutskin, V.V. Slavin, H.A. Kovtun // Физика низких температур. — 1999. — Т. 25, № 8. — С. 936-943. — Бібліогр.: 13 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
_version_ 1860209993759850496
author Slutskin, A.A.
Slavin, V.V.
Kovtun, H.A.
author_facet Slutskin, A.A.
Slavin, V.V.
Kovtun, H.A.
citation_txt The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension / A.A. Slutskin, V.V. Slavin, H.A. Kovtun // Физика низких температур. — 1999. — Т. 25, № 8. — С. 936-943. — Бібліогр.: 13 назв. — англ.
collection DSpace DC
container_title Физика низких температур
description In narrow-band conductors a weakly screened Coulomb interelectron repulsion can supress narrow-band electrons` hopping, resulting in formation of a "frozen" electron phase which differs principally from any known macroscopic self-localized electron state including the Wigner crystal. In a zero-bandwidth limit the "frozen" electron phase is a classical lattice system with a long-range interparticle repulsion. The ground state of such systems has been considered in the case of two dimensions for an isotropic pair potential of the mutual particle repulsion. It has been shown that particle ordering into stripes and effective lowering of dimension resides universally in the ground state for any physically reasonable pair potential and for any geometry of the conductor lattice. On the basis of this fact a rigorous general procedure to describe the ground state fully has been formulated. Arguments have been adduced that charge ordering into stripes in high-Tc superconductors testifies to the presence of a "frozen" electron phase in these systems.
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q�q x t h p � p�tQwAs�|4w i p l g w�p�l�~ iQg r i r x t � pAs r'r�nFo g p�v �Q� p � p�r'w�q)sut9l,oQq)s y9g vEp�v w i9x wÀw i p�l g w)pCl5r�sQtQl)w g w)n1w)p x oEq g3��g w gØy p � x w�w g r�p � g w g l r x �I� pCv i sul)w � x w�w g r�p � p � s ~ ��7 g�g � x t � | g ��� g t h | x r'w�sQq'� ρ = 8$«:9 � 8 → ∞ x tQv 9 → ∞ x q)pmw i pAw�sHw x � t�n ��j� p�q�lÂsQ|}w i p¶p � p'r�w�q)sut9l x tQvÃw i p i s�l�w j � x w�w g r'p¤l g w�p�l'� q p�l)o�p�r�w g6y p � � �67 g3g�g � x t � o i � l g r x ��� � q�p x l�sut x�� � p+� ( ¬ ) ; ¼ �¥�p�w x�� p�w i p@w�p�q � '<(+* w�s � p x tmw i1x w2w i p ·�¸ sQ|>w i p� �  N¡�¢ g l x l�p�w�sQ|zv g | |�p�q�p�t�w�pC| |�p'r�w g3y p § �  ?¡>¢ ~ i s�l)p � o x q�w g r � p�l�� x q p*oEp�q g s�v g r�l w�q g o�pCl�sutmw i p � x w�w g r�p s�|>w i p � � r'sQtQvQnEr�w)suqK�   sQq0p x r i § � l � l w�p � s�|�w i p*l�p�w w i pCq�p g l g w�l�su~�t ρ g t�w�p�q y�x � ~ i pCq)pÂw i9g l § �  ?¡>¢ q�p�oQq)p�l p�tQw�l:w i p �C� sutEpQ�Nw i p4~ i s � p$q x t h pQ� ¼ ≤ ρ ≤ § � r�s � oQq g l g t h�x �I� w i p g t�w�p�q y�x � lC� � i p '=(+* p�t x'� � p�l�n1l>w�s s�| |�p�q x q g�h s�q�sHn1l x t x � � w g r x � oQq s�r'pCv�nFq p¾|�suq w i p � �  ?¡0¢¾·*¸ v�pCl)r�q g o�w g sQtD�Nn1l g t h w i p,p'´ x r'w*q�pCl)n � w�l:s�|�w i ph pCt�pCq x � w i p'suq � su|{w i p § � � x w)w g r'p l � l w)p � lz~ g w i x � sQt h�j q x t h p g tEw�p�q�o x q w g r � p,q p�o�n � l g sut ¦�>H².?E¨ � @F\<ACB�D,eFE3_�a�^FeGB ^=\)HNeGD�I,EFJ$dK`�K,L�_2B%EFL � i p f*x���g � w�sut g3x tD�<M��Fs�|0w i p,l � l w�p � n1t1v�pCq�r's�tEl g3j vQp�q x w g sQt iQx l�w i p,|�suq � MON& ( P ) Q = § R ∑S ≠ S ′ � ( T ` − ` ′ T ) U (̀ ) ( ` ′) � ��§ � ~ i p�q p ` = �CV&W � + � � B � x q)p q x v g n1l y p�r'w�s�q�l su|�w i pi s�l�w j � x w�w g r'p;l g w�p�l'� � �� � x q�p g t�w)p h p�q�lK� B �� � x q p i sQl w j � x w j w g r'p*oQq g���g w g3y p�w�q x t9l � x w g s�t y p'r'w�suq�l ��¢?��X l ��7 w i p�sHr�r'n j o x w g sut4t�n �@� pCq�lNsQ|Dw i p i sul)w j � x w g r'p�l g w�p�l'�Y (̀ ) = ¼ suq § �x q p �*g rCq�s�l�r�sQo g r yQx q g�x'� � pCl2Z 7 w i p:l n �7g lNw x � p�t�s y p�q>w i p ~ i s � p i sQl w � x w)w g r'pE� � i p�o x�g q�o�s�w)pCt�w g�x � g l x l)l�n � p�vmw�s� p x tÃp y p�q � ~ i pCq�p r�sQt y p�´ |�n1tEr�w g s�tÃsQ|}w i p¤|�s�q �� ( [ ) = \] ( ¬ )̂:[ �:~ i p�q)p¿w i pµ|�nFt�r�w g sut �] ( ¬ ) vQp�o�p�tQv9lmsQtw i p*r i1x q x r'w�p�q2sQ|0w i p,l r�q�p'p�t g t h5� p�v g n �­x t1v g w�l�oQs�l g3j w g sQtz~ g w i q p�l�o�p�r'w�w�s{w i p � �  N¡0¢ �.� t x t � r x l)p_�] ( ¬ )w pÍt�v�l>w s���p�q�s x l ¬ − � s q0| x l�w pÍq>~ i p�t ¬ → ∞ 7 � ] ( ¼ ) = � � « κ� � g l2w i p�p � p'r�w�q�s�tAr iQx q h pQ� κ g l�w i p�v g p � p'r�w�q g r,o�p�q ��g w j w g3y1g w � � � ( w i p�q ~ g l pE�`�] ( ¬ ) r x t � p�q)p'r � s�tQp�v x l x q ��g3j w�q x q � 5 x l�~ g �I� � pAl i s�~*t � p � sH~m� g w�l,l)o�p�r g | g rA|�sQq � g lg��*��x w�p�q gLx � w)s$s nFq x oEoQq)s x r i � � � sut h w i p ·*¸ r'sut9| g+h n1q x w g sQtQl N ( ` ) Q ~ g w i v g | |�p jq)pCt�w ρ w i pa��� � � �F� �'©*sutEp�l x q�p � � r�q � l�w x � l�~ g w i s�t�p p � p'r�w�q)sut¾o�p�qmr'p ���,� � ¸ j rCq � l)w x � l�� � Z�Z � � i p g q g t y p�q�l p ρy1x � nQp�l ��x�� p n9o x t g t1| g t g w�pÂl�p'wzsu| g tEw�p h pCq�lcb:d = = T vEp�w ( � κλ d ( B �:� B � ))T �1~ i pCq)p$e g t9v�p'´1p�l ¸ j r�q � l)w x � lK�1w i pg t�w)p h pCq)l � κλ d � κ � λ = § � �9� x q)p;r's � oEsutEpCt�w�l0su| ¸ j r�q � l j w x �,¢N�.X l g t w i p B λ �1x l g l 7 b d g l4w i pfe j w i ¸ j rCq � l)w x � p � p � p�t�w x q �Qj r�p ��� x q�p x@� p x l nFq p�v g t,n9t g w�l>su|Dw i1x w�sQ|Dw i pi sul)w � x w)w g r'pE� σ S = T B � ×W%g�h �( nFq*l�w�q x w�p hQ�¿g l@w�szvQp�q g3y p$w i p�|�n �I� v�p�l)rCq g o�w g s�t su| w i p ·*¸ |�suq x t � B �� � �=l w x q w g t h ~ g w i r'sut9l g v�pCq x w g sut5su| l ��x ��� yQg r g t g w g pCl�su| ρ = § « b:d�� ¸ g t�r�pÂl�o�p'r g | g r � κλ d y1x � nQp�l x q)p g q�q)p � p yQx tQw�w�smw iEg l�q)p x l�s�t g t h ��~�p,vQq)suo5w i pg tQvEp'´ie x w b x t9v x wzs w i p�q{r i9x q x r�w)pCq g l w g r�l{s�|5w i p¸ j rCq � l)w x � l�|�s�q x ~ i1g � pQ�* nQp{w�s�w i pÀv g l r�q)p�w)pCt�p�l�l,s�|�w i pzl � l)w�p � ~ g w i w i pf*x �@g � w)sQt g�x t � § � x �*x r�q)sul)r�s�o g r x �I� � l �*x ��� r iEx t h pE� δρ � g t ρ (δρ → ¼4jk8 �mlQ� Tδρ T → ∞ ~ i pCt 8 → ∞ � oQq)s j vEnQr�p�l}s�t � � ����� � ¯E© ��#n#,���o� K©F� g t x t ¸ j r�q � l�w x � ��w i p l�o x r'p$l w�q)nQr'w�n9q�p�sQ|0w i p$v�pC|�p'r�w�l�pCl)l�p�t�w g�x �I� � v�p�oEpCtQv g t h sQt�~ i p'w i p�qmw i p � q p�l)n � w�| q)s � x t g t�r�q�p x l pzsQq x v�p j r�q�p x l p g t ρ � � iQg l�| x r'w g l�p�´FoQq)pCl)l)pCv �E� w i p g vEp�t�w g w � p Û (8 ± Tδ 8 h j,9 ± Tδ 9 T ) − p Û ( 8 � 9 ) = = ± µ± Tδ 8 h +− q ± Tδ 9rT � ���Q� ~ i p�q)p p Û g l�w i p ·*¸ pCt�pCq hQ� � δ8 x t1v δ 9 x q�p:r i9x t h p�lg t 8 x t9v 9 oQq sQv�n9r g t h δρ � � i p¶oQq sQo�sQq�w g s�t x � g w � r'sup�| | g r g pCtEw�lC� µ− º µ+ �ks − ¹ s + � x q�pzw i p yQx � n1p�l4su| w i p�r i p ��g r x � o�s�w�p�t�w g�x � � µ � x t9v®w i p o9q p�l�l n1q)pE�fs�� ~ iQg r iAx q)p�w i p@pCt9vQoEs g t�w�l2sQ|>w i p µ x t9v s g t�w�p�q y�x � l�su|¸ j rCq � l)w x � p'´ g l)w�p�t�r'pQ� � i p �®x q)p v�p'w�p�q ��g tEpCv �Q� w i p p�tQp�q h1g p�l s�|¶|�s�q �*x w g sut su|7r's�q)q)pCl)oQs�t1v g t h vEp�|�p�r'w�lC�� i n9l'� g tÀl)s � p y1g r g t g w � su| ρ = § « b w i p ·�¸ g l � s�n1t9v w�s � p x l n1oEpCq)l)w�q nEr�w)n�q�p*s�|0w i p,v�p�|�p�r'w�lK� ( n1q�tQp'´Qw�l)w�p�og l�w�s�| g tQvmw i p � � tF\<u)JK`�a4v�bFewDCJ�^,L'eØa�^xByENbxJ�z{J dH_oL|B ^9bA_o}4J�e-` d�ayByEFJ�L'd�JK^�d�J ��v9v g t h sQt�p�p � p�r'w�q�s�t w�s s�q}q)p � s y1g t h | q)s � x t¸ j rCq � l)w x � q p�l)n � w�l g t�w i pÀ|�suq ��x w g sQt�s�| x ��p�q s j v g�� pCt j l g s�t x � v�p�|�p�r'w�� �y~ v�p�|�p�r'w�sQtF�{sQq � ² vEpC|�p'r'w�sutD�Z�>q�pCl�o�p'r j w gØy p � � � ( tQp:r x t � p g t�r � g tQp�v4w�s,w iEg t � w i9x w δ8 l i sHn � v� p g vQp�t�w g | g pCvÃp'´ x r'w � � ~ g w i w i p w�s�w x � t�n �@� pCq su| ± v�p'|�p�r'w sut�l�l�o x w g3x ��� � l p�o x q x w)p�vD� ± µ± � p g t h l g3� o � � w i p p�tQp�q hQ� sQ| ±vQp�|�p�r'w�s�t�|�suq ��x w g sQtD��� ± � f s ~�p y p�q'�Nw i9g l l)p�p ��g t h � � p yQg v�p�tQw¿l)w x w)p � p�t�w g l x r'w�n x ��� �¶g tQr'suq�q�p�r'w � 8�ù�ú ρ þ�î�ð¤ó�ú_ò)îÉü�ú î�òBòIó�í)ú ��ôBû���ú ö�ûÙô�ôIû�û���ð¤û�òIú¸ôBû ρ = � �òBû�ôBù�îÙô=��ú.�_îÙö�ö�ú "�ð�ú �Éô�ôBù�ú.�_ûÙö�#@ï "�ó�üIî�ô�ï ûÙö�ò?�)ï ôBù Æ ( S ) = g ï�öC���{�³îÉö���ôBîÉõ�ú ö�ûî�� �_ûÙó�ö�ô�ûÍ#�ôBù�ú)ò��0ï ö+þ�îÙü�ï î)�/ð¤ú�ò�ý �w� G�ò9��îÉò#òBù�û���öøï ö ÿ�ú�#Mý���� S�� 0ï-# ä ( à ) ��ú_üMú=��ûo��ìJðEï�õ�ú' 0ï�ô9��ûÙó/ð��2��ú)ö�ûÙô�ôBù�ú=�Éî�òBúN��ó�ú)ôBû³î���ðEó�òIôBú_üèï &Mî�ôMï û�ö+û�#�ô@ù�ú?��î�üIô�ï ��ðEú_ò�ý � Î ��Î Ì�Ò���Î ��Ì�Î Ìo�������������-ÒÍÊØÉ��IÐCÙ�Ú�Ú'ÚuÐ�ÓCÆ���� Ð��Q×���� Ú Ú���� vQnQp­w�s x �� ¯ �F� �� � k � sQ|ÂvQp�|�p'r�w)sut9lÂs�| w i p¤l x � p� l g6h tF�Z�F��t4s w i p�q�~@suq�vQlK� g |Dw i p:tQn �@� p�qK� Tν T �usQ| ¸ j rCq � l jw x � p � p�r'w�q)s�t1l,q)p � s y pCv � ν º ¼ � s�q x v9v�pCv � ν ;�¼ � g l� sQq pmw i1x t § � x�� �H�0 # �'©�¯E© � s�| Tν T ± vEpC|�p'r�w)sut9l x q g l)pCl~ i s�l)p�p�t�p�q h��¿g l � pCl�l@w iQx t Tν T � ± �=¥�p iQx'y pmq p y p x � p�v w i pzr�s x � p�l)r�p�t�r'p �Q� r's � o�nQw x w g sQtD��n9l g t h x � v g o�s � pu� vQp�l r�q g o�w g sut,s�|Nw i p ·,¸ ~ g w i ν = ± § � ± � ����������~ iQg r i ~:pi1x'y pÂl�o�p�r g�x ��� � ~:sQq � pCv¶sHnQwµ|�s�qzw i1g l�o�n1q�o�s�l�pE� � i p v g o�s � p x oQoQq)s x r i sQ| |�p�q�l x r � p x q yQg p'~£s�| i s ~ w i p vQp�|�p�r'w�s�t1l�� � sun9t1v�l�w x w�p x q g l�pCl4vQp�l�o g w�p{w i pz| x r�w4w iQx w w i p�vEp�|�p�r'w�s�t1l{s�|Aw i p l xH� p l g6h t q)p�o�p � p x r i s w i p�q'�� p g t h ~ g vEp � � l�o x r'pCv°� �0w ν ≠ ¼ x oEpCq�w�n1q � p�v ¸ j rCq � l w x � g l*|�s�q � p�vz~ i pCq�pQ�� p�l g v�pzp � p�r'w�q)s�t1lmo � x r�p�v x w i sQl w j � x w�w g r�p¿l g w�p�l g t w i pg tQw)pCq)l)w g r�p�l2s�|�w i p ¸ j rCq � l)w x ��� ν ; ¼ � sQq0p � o�w � ¸ j rCq � l jw x � l g w)pClC� � i s � p�l��Z� � ν º ¼ � ��w i p�q p x q)p h p�t�pCq x ��� �¶x r�p�q)w x�g t�tEn �@� p�q$s�| ¸ j r�q � l)w x � p � p'r�w�q sQtQlml i9g |�w�p�v | q�s � w i p g q2t x w g6y p ¸ j r�q � l w x � l g w p�lC� � i p � x w w�p'q0r x t � p:r�s�tQl g v�p j q�p�v x l � x tQw g o x q)w g r � p�l'�m~ i sul)p,r iEx q h p g l pC��n x � w�s5w i p p � p�r'w�q)s�tmsQt�p g t ��x'h t g w�n1vEp � n9w g l0sQoQo�sQl g w�p g t5l g6h t°� x o x�g q � x t,p � p�r'w�q)sutml iQg |�w)pCv �Q�$x@y p�r'w�sQq ξξ + g w�l x tEw g o x q j w g r � p � s r x w�p�v x w x t ¸ j rCq � l w x � l g w)p ` � � p g t h w i p � ` � ξξ j v g o�s � pu�Z� � i nFl'�1w i p*o�p�q)w�n1q �1x w g sutms�|�w i p ¸ j rCq � l�w x � r x t� pmp�t yQg l g sQt�pCv x l x t¿p�tQl)p �:� � pmr�sQtQl g l)w g t h su|�l p y p�q x � v g o�s � pCl x tQv Tν T g tQw)pCq)l)w g w g�x � o x q w g r � p�l � i s � p�l � � ¢ � f l � �� i p*v g o�s � p�l g tEw�p�q x r'w2~ g w i w i p,� ¢ � f l x t9v4~ g w i p x r i suw i p�q�� � i p,p�tEpCq h1� su| g tEw�p�q x r'w g sQt � p�w)~�p'pCt x t � ¢ � f� x w ` = ¼ � x tEv ` � ξξ j v g o�s � p g l � ξξ(̀ ) = l g�h t ν( � ( h P − ξξ T ) − − � ( T ` h)) ≡ ��� �� ν∆� ξξ � ( T ` T ) 7 w i p­pCt�p�q hQ� s�| g tQw)pCq x r'w g sQt� p'w�~:p'pCt ` � ξξ j x tQv ` ′ � ξξ′ j v g oQs � p g l � ξξ ξξ′( ` − ` ′) = = ∆ � ξξ∆ � ξξ′ � ( T ` − P ′ T ) � � i pÂ� ¢ � f l'� g t w�n1q�tD��n1t9v�p�q h s x � nQw)n x � q�pCoEn � l g sQt x t1v x q�p,p�´Fo�s�l�p�v5w�s x t � p�´Ew�p�q�t x � � | g p � v°�<� ( ` ) �<~ i9g r i g l�pC�En x � w�s − � � S |�sQq i s � p�l � i p�q px tQv¾|�n1q�w i pCq4sQtÂ� Â� s�|�w i p ¸ j r�q � l�w x � g lmvEpCt�s�w)pCv �Q� � S � � x tEvz|�s�q�� ¢ l g w g l:w i pm| g p � v¿oQq�s�v�n1r'pCv x w x oEs g t�w` �Q�$x ��� p � p�r'w�q)sut9l>su|Dw i p ¸ j rCq � l�w x � �9��t�w i pCl�p�w�p�q � l>w i p r i9x t h p g t4w i p ·*¸ p�t�pCq hQ�$x w x2hFg-y pCt ν �����6à (ν) ��w xH� pClw i p*|�sQq � � ��à (ν) = �*g t ( ����� � + � Ø + � ú2��� + � ) � ��±F� f p�q p � üBú � = Σ α ! β � ( T ` αβ T) g l4w i pÀp�tQp�q h9� sQ|�w i p � nEw�n x � q�p�o�n � l g sut{sQ|�w i pm� ¢ � f l 7 � Ø = Σ α Å � ξξ " ( ` α Å ) g l:w i p,pCt�p j q hQ� su| w i p g q g tQw)pCq x r�w g sQt ~ g w i w i p v g o�s � p�l 7 � ú&�'� = ΣÅ δ � ξξ # + ΣÅ ! Ä � ξξ " ξξ $ ( `&%(' ) ; ¼ g l w i p p'´1r g w x w g sQt pCtEpCq h1� sQ| x t ¸ j rCq � l w x � ~ g w i Ø v g o�s � p�l x w ν = ) 7 δ � ξξ ∼ � S hξξ T � «.¬ Â_Â� ;�¼ g l5w i p�pCtEpCq h1� s�|m|�s�q ��x w g sut¾s�| s�t�p�v g oQs � p 7 � = Σ α � (̀ α) g l�w i p�p�t�pCq hQ� su|0w i p$� ¢ � f l g t w i pÂp'´1w)pCq)t x � | g p � v � p�t�w g sutEpCv 7 w i p g t9v g r'pCl α = = § �*������� Tν h x tQv � = § �*�+�,�3� U.- pCt�n � p�q x w�p}w i pÂ� ¢ � f q x v�n g l jLy p�r'w�sQq�l x tQv¾v g oQs � p�lK�;q�p�l�o�p�r'w g3y p � � � Ø g lmw i p w�s�w x � tQn �@� p�qNsQ|Fw i p�v g oQs � p�l 7 ` Ó / ≡ ` Ó − ` / � � i p ��g t gLj� n � g l�w x�� p�tz~ g w i q)p�l�o�p'r�w�w�sO Ø �?w i p$v g o�s � p y1x q gLj x'� � p�l'� ` Å � ξξ Å � x t9v ` α � � i p�q p�|�sQq pE�*w i p�v g o�s � p x o j oQq)s x r i¶x �I� s ~*l w�s¾~�sQq � ~ g w i s�t � � x |�p'~ v g l)rCq�p�w)py1x q g�x'� � p�l'� � iEg l$| x r g � g w x w)pCl�r�s�t1l g vQp�q x'� � � w i p10 s�t�w�p j�<x q � s r�s � o�n9w�p�q l g�� n � x w g s�t sQ| w i p ± vEpC|�p'r�w)sut9l ( � �#à (± § ) = ± � ±) x tQvmw i p g q�r�s x � p�l)r�p�t�r'p x w Tν h ; § �� i p � p'r iQx t g l � sQ|:w i pmr�s x � pCl)r'pCt�r�pmr x t � p�p � nQr gLj v x w�p�v �Q� w i p�|�s ��� su~ g t h$i p'n1q g l�w g r x q h n � pCtEw�lC� � i p ·,¸ w�s�w x � v g oEs � p$p�tEpCq h�� � p Ø (ν) = � ú&�'� (ν) + �øØ (ν) � g l�t�p jh1x w g-y pQ�?l�sAw iQx w�|�suq x t � Tν T w i p ·*¸ l�o x r'p$l)w�q nEr�w)n�q�p g lvEp�w)pCq �*g t�pCv �1�¾x t g tEw�p�q�o � x'�¾� p�w)~�p'pCt7t�p h1x w g6y p2� Øx t9v®o�sQl g w g3y p3� ú&�'� �4� ü�ú � � � i p¾� ¢ � f v g oEs � pg t�w)pCq x r'w g s�t h1g3y p�l:w i p ��x ´ g���x � h�x�g t g t{p�tQp�q h1� ~ i p�t� ¯y / � ¢ � f g l0p �@� p�vQvEpCv g t x � l i p ��� �@s�|>|�s n1q2v g o�s � p�l w i9x w x q�p x w�w�q x r�w)pCvÂw)s g w��:w i p¿v g o�s � pCl5� x t�w g o x q�w g r � p�l |�sQq ��g t h x o x q x �I� p � s h q x�� sQ| x l g ��p ∼ ¬ Â_ ∼ b �76����  g6h � §C� � � i p l i p ��� l sQ|�t�p g6h1iQ� suq g t h � ¢ � f l x q�p� s�n1t9vAw)sÀl i9x q p$l)s � p,sQ|2w i p g q:v g o�s � pCl�|�s�q8� ú&�'� � x t9vi pCt�r'p Ø � w)s � p x l�l �*x ��� x l*o�sQl)l gØ� � pE� � i9g l�q p��Qn g q p j� pCt�w�r x t � p¾|�n � | g �I� p�v s�t � � ~ i p�t x ��� � ¢ � f l x q�p ¯ � ��! ��# �m µ¯ ¬ ���4�.w i pÀt�p x q j tEp g+hFi�� sQq4� ¢ � f l � p g t hl iQg |�w)pCv4q p � x w g6y p�w)s*sQt�p x t�s�w i pCq �Q� w i p;l x�� p ¸ j rCq � l)w x �¢?��X ~ g w i w i p � s�vQn � n1l ∼ ¬ Â_ ��  g�h � § � ����t¾l nEr i¾x r x l�p T p Ø (ν) T g l � sQq p:w iQx t�w i p ��x!h t g w)n1v�p:su|Dw i p:v g oEs � pp�tQp�q hQ� su| Tν T g t9| g t g w)p � � l�p�o x q x w�p�v�v�pC|�p'r�w)sQtQlK� p Ø∞ = = Tν T p Ø (± § ) � � i p,r's x � p�l)r�p�t�r'p x q g l p�l:~ i p�tAw i p,p�t�p�q h��h1x g t°� ∆ = T 9 Ø (ν) T − T p Ø∞ T �°p'´Qr'p�p�v9l8� üBú�� su|:w i p{� ¢ � f lx � g6h tQp�v g t�w i p;q)s ~�� ¸ g t�r'p ∆ ∼ TνT � ( ¬ Â� ) w iQg l>r�sQtQv g w g sQtg l � p'w g |_� ( ¬ ) vQp'rCq�p x l)p�lÀt�suw5w�s�s l � s�~ � � �;sQq'� � sQq�p p'´ x r'w � � � g | γ = ∫Ã�:;: ∞ � ( ¬ ) # ¬�«<¬+<=< � ( [ Â� ) º > § � � ?1� � i p�r�s � o�nQw�p�q.l g�� n � x w g sQt�r x q�q g pCv�s nQw>~ g w i w i p � sQv�p � o�s�w�p�t�w g�x � � ( ¬ ) ∝ ¬ −β p'´Fo (− [ «A@ ) s y pCq x ~ g vQp�q x t h p�su|w i p*o x q x�� p�w)pCq)lK� β �CBm� iQx l0r�sQtQ| g q � pCvmw i1x w2w i p@r�s�t1v gLjw g sut � ?F� g l2q�p x ��� � w i p:r�q g w)pCq g s�t$s�|�w i p:r�s x � pCl)r'pCt�r�p�|�sQqx t � Tν T � x tQv x t � B �� � � �� q g w p�q g sut �&?9� g l�|�s q>w i p � sul�w.o x q�w.|Ln � | g ��� p�v?�Q�Iw i s � v�l |�sQq x t � �] ( ¬ ) ��¸ p�rE� �A� l nEr i w iQx w��] ( ¼ ) − � ] ( ¬ Â_ ) ∼ �] ( ¼ ) �� i1g lmr x l pz~ g �I� � pÀw i p¿|�sHr�n1l�su|,sHn1q x w)w�p�t�w g s�t | q)s �i pCq�p�sut°� � i p�o x q x�� p'w�p�q γ � p�r's � pCl ;:; § g | �] ( ¬ )vEp�r�q p x l)pCl4l n � l w x t�w g�x �I� � sut � � |�sQq ¬ ~ iQg r i�x q)pAp�´9oQs jtEpCt�w g�x ��� � � x q h p g t γ �N��tAw i1g l � g3�*g w:w i p � nEw�n x � q)p�o�n � j l g s�t{sQ|�w i pA� ¢ � f l�v g l�q�n�o�w�l:w i p g q*q�su~4� x t1v{w i p�q p g l tEs�r's x � pCl)r'pCt�r'pE� x w � p x l)w¿|�sQqzl)n1| | g r g p�t�w � � � x q h p Tν T � Ú���� � Î ��Î Ì�Ò���Î ��Ì�Î Ìo�������������-ÒÍÊØÉ��IÐCÙ�Ú'Ú�ÚuÐ�Ó�Æ���� Ð��u×$��� Ú f s ~@p y p�q�� g t ¸ p�rE��� g w g l{s nEw � g tQp�v¾w iQx w5w i p ')(+* h s y p�q�t9l2w i p ·*¸ g tmw iQg l�q x w i p�q�l�o�p'r g�x � r x l pE�Fw�s sD� �F\��.}4J+E�a���JK`�e3^��Àa0z�b=e D+J�^4L�e�a�^ k / �C���F����� �©�¯ ¬ 3 �'© ¬ � � � �|�m 5©�/ ��� � � ¬ ��� � �����w� K© ¬ �� +��/u¯%� � �2l2|�s �I� s ~*l2| q�s � w i p x |�s�q�p�l x�g vN�Qw i p � s nFtQvml w x w)p;s�| Tν T vQp�|�p�r'w�s�t1l g l4w�q x tQl�|�suq � p�v g t�w�s x o�p�q g suv g r ��© ¬ �G� � �� � � � l w�q)nQr�w)n1q)p ~ g w i£x t g tQ| g t g w�p g t�r�q)p x l)p g t Tν T��  g�h � § � �}� w�r'sut9l g l�w�l7su|Âp � p � pCt�w x q � § � vQp�|�p�r'w�l ~ i1g r i � x l4~ g �I� � pzl i s�~*t � p � s�~4��q�p�o�p � p x r i s�w i p�qK�� i p�q p�|�s�q�pQ� g w g l ©�/ � ��� � � �F� �'© § � #,���o� �©1� w i1x w x q p p�´Fo�p'r�w)pCv w�s�|�suq � w i p ·�¸ l n1oEpCq)l)w�q nEr�w)n�q�pQ�2��t x q ��g3j w�q x q � § � v�pC|�p'r�w2sQ|0l n9r imx w � o�p g l x l w�q g o�p�su|0q x q p�| x r j w g sQt suqmr's � oQq)pCl)l g sQt w i9x w x q g l)p�l$~ i pCt x t ¸ j r�q � l�w x � o x q�w x v�� x r'pCtEw2w�s x � g tQp@su|>p � p�r'w�q)s�t1l0~ g w i l)s � p ¢?��X �b � g l;l i1g |�w)pCv x l x ~ i s � p�q�p � x w g-y p�w�s4w i p�s�w i p�q2sQt�p �Q� xÂi s�l�w j � x w�w g r'pµw�q x t1l � x w g sQt y p'r�w)sQq'� ξξ �   sQq ��x w g sQt7su| sQt�p¿l)w�q g o�pzsQ| � pCt h w i�� È r i9x t h p�l 9 �E� δ 9 = ±σ ��� � σ = h b ×ξξ h j � È g l � p x l n1q)p�v g t4n1t g w�l>sQ| T b T � � � i p�r�s�q�q p jl�o�sut9v g t h r i9x t h p g t�p�tQp�q h9� � δ p � g l$oEq�s�o�sQq w g sQt x � w)s δ 9 5 δ p « Tδ 9 T = ε( b � ξξ) = σ− � ∑Æ = � ∞ ∑S ′ � ξξ( ` − z ) � ��³Q� f p�q�p ΣS ′ � p x t9lzl n ����x w g sQt7s y p�q{w i p ¸ j r�q � l)w x � l p j��g o � x tQp ` = � b + � z � −∞ º��$º ∞ � −∞ º � ≤ ¼ �67 z g lx t � ¸ j rCq � l)w x �0¢?��X s w i p�q2w iQx t b � � i p ·�¸ g l�q)p x � g ��p�v�Q� w i p l�w�q g oQp�l�~ g w i b = b ± x t9v ξξ = ξξ± ~ iQg r iµ��g t gLj��g ��p ε( b � ξξ) x w x�hFg-y pCt{l g6h tAsQ| −δ 9 � − s�q + l �1� � s j� g ��p�l�q x q�pC| x r'w g sQt suqmr's � oQq)p�l�l g s�tÂq)pCl)o�p'r�w g3y p � � � �2¥¾p ~ g ��� r x �I� w i pCl)p*l)w�q g o�p�l � ² l�w�q g oQp�l��4suq � + l�w�q g o�p�l��Z�� i p p�tQp�q hFg p�l ε± = Tε( b ± � ξξ±) T x q)p w i p¾��n x t�w g w g p�ls ± � l�p'p ���9��� x l�l s�r g�x w�p�v�~ g w i ± l w�q g oEp�l�|�sQq ��x w g s�tN� � i pr's�q)q)pCl)o�sQtQv g t h µ± � x l�|�s �I� s�~�l�| q s � h p�tEpCq x � w i p�q � s j v � t x ��g r�r'sut9l g v�p�q x w g s�t1lC� x q)p ε] ± = � S + b ε± � ��ÁQ� ' pCl)w:w i p�q p � p�t�sAr'sQt�w�q x v g r�w g sut{~ g w i w i p�| x r'w�su|�w i p r's x � p�l r'pCtEr�pE�FpCt�p�q h1g pCl ε] ± x t1v � ± x q p � s�n1tQv5w�sml x w g l�| �g t�p��En x � g w g p�l � − º ε� − º ε] + º � + � � >Q� ¥ i p�t b�� ;�� x tFv � ( [ ) h sEpCl�w�sm��p q)s�s y pCq:v g l)w x tEr�p�lB ¹ º [ Â_ ∼ b V l � �°w i p � |�s �I� s�~ | q�s � l g3� o � pAp�l w g���x�j w�p�lK� � x �Qg t h g tQw)s x r'r�s�n1tEw$w iQx w T ξξ± T∼ ¯uS � x t1vD��r'sQq p j l�o�sut9v g t h � � � T b ±×ξξ± T ∼ b �-l�� σ S ��| q)s � ¡ �D� ��³1� ~@p s � w x�g t�5 ε± ∼ ( ¯ S�b �1l�� « B ) » S � ( t�w i p�s�w i pCq i9x t1v?� T � − T ∼ � S ∼ � ( ¬ <,< ) � x tEv i p�t�r'pQ� ε� − ;:; h � − h �1��t@w i p;r x l!p n1tQvEpuqZr�s1tQl g v�p�q x w g s1t�� + ∼ � ( ¬ í³ï ö ) �?~ i pCq)p ¬ í³ï ö g l@w i p� p x l)w�s�|=w i p:v g l)w x t�r�p�l � p�w)~�p�p�t�w i p�� ¢ x t1v�w i p ¸ j r�q � l j w x � l g w)pClK� � i9g l}p�t�pCq hQ�®g l � nQr i®� sQq p7w iQx t ε+ x l B º2º ¬ Â_Â��� s ��x�� pÀl n9q�pmw i1x w�w i p g t�p���n x � g w g p�l � >9� i s � v¿|�sQq s�w i p�qC� ( ¬ ) x t9v B «>¬ Â_Â;~@p i1x'y p,r's � oEnQw�p�v ε± � �1x l g t h s�n1q�l)p � y p�l2s�t ¡ �D� ��³9� x t9v ¡ �D� ��Á1��� g tmo x q x �I� p � w�s,w i p 0zs�tQw)p j��<x q � s r�s � o�nQw�p�qAl)w�n9v g p�lmsQ|,w i p¿r's x � pCl)r'pCt�r'pE�� i p �$i9x'y p@r�s�t1| g q � p�v,w i1x w0w i p g t�p��Qn x � g w g p�l x q p�q)p x ��� � w i p�r x l p$|�suq x ��� � ( [ ) n9tQvQp�q�r�s�t1l g vEpCq x w g sQt x tQvÀ|�sQq x ���b � ��x���� p�p�´Er�p�o�w b = � �� s h p�w i p�qN~ g w i w i p � nQw�n x � q)pCo�n � l g s�t,s�| ± l w�q g o�pCl>su|w i p�l x�� p�l g�h tzw i p g tQp���n x � g w g pCl � >9�$� p x vzw�s5w i p$r's�t j r � n1l g s�t�w i1x w + l)w�q g o�p�l$sQq ² l w�q g o�p�l�v�sµr'sut9l w g w�nEw�pzw i p·*¸ l n1oEp�q�l)w�q�nQr�w)n�q�p g tÃw i p y1g r g t g w � sQ| § « b � � i p o�sQl g w g s�t�s�|mp x r i ± l w�q g o�p x r'sQtQl)w g w�n9p�t�wÀs�|mw i p l)n1o�p�q�l w�q)nQr'w�n9q�p g l2vEp�w)pCq �*g t�pCv �Q� w i p�l�w�q g o�p � r's sQq j v g t x w�p��Z� � � ~ i9g r i�g lAw i p�w�s�w x � tQn ��� p�q5su|mo x q�w g r � p� g t�p�l � ~ g w i w i p ¢?��X b ± � � p�w)~�p�p�tÂw i9g lAl)w�q g o�p x t9v l)s � pÀ| g ´Qp�v}sQt�p � � = ) � �0� l)p'w,s�|@w i pCl)pAr's�suq�v g t x w�p�l �6ŤÛ/À�æ ý'8�ù�úN�_ûÙî�ðEú_ò��_ú_ö'�_ú�#Eû�ü�ò���ó�î�üBú�� Ó � = ���#îÙö��øô@ü_ï î�ö'"�ó�ðEî�ü)� / � = � P��#ù�û�òIô�ð�î�ôIô�ï �_ú_ò)� ν = − � �Ùý��)ú_üBú ❍❍ ��ú_ö�û�ôIú_ò#ù�ûÙòBôBì@ð�îÉôBô�ï �_ú)ò�ï ôBú_ò� ● ��îÙüBô�ï ��ð¤ú�ò"! 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pCl�w i p ')(+* � k / � ! ¬ � �0 # �'©�¯E© � ����� � ¬ ��© ¬ �0 K©�� ¬ � � � ��© ¬ � � � �� i p ·*¸ x q�q x t h p � p�tQw�s�|�w i p ± l w�q g o�pCl g l h s y p�q�tEp�v�1� w i p�o x�g q0oQsHw�p�tQw g�x � sQ|Dw i p;l)w�q g o�p j l)w�q g o�p g tEw�p�q x r'w g sQtD� � ÈBȱ ( � ) = ∑Æ = Þ + � ∞ ∑S ′ � ξξ ± −ξξ ± ( ` − z ± − ξξ±) � � � ~ i p�q p g tEw�p�q j l)w�q g o�p � v g l)w x t�r�p�� � = § � � �;����� 7 z ± g l x t¸ j r�q � l�w x ��¢N�.X s�w i p�q2w iQx t b ± � ΣS ′ � p x t9l0w i p*l x�� p x lg t ¡ �D� ��³9�@� b � z = b ± � z ± � �   s�q x �I� � ( ¬ ) n1tQvEp�q2r's�tEl g3jvQp�q x w g sQt ΣS ′( ) ;�¼ � x t1v ΣS ′( ) ; ΣS ′( + � ) � f p�t�r'pQ��+ÈBÈ ( � ) ; ) g l x r'sQt y p'´À|�n1t�r'w g sQtms�| � � � i9g l2p�t x'� � p�l2n1lw�sÀv�p�l)rCq g3� p,w i p ± l)w�q g o�p�l;l)n1o�pCq�l w�q)nQr�w)n1q)p x w θ − � ≠ )� θ = � « ρ � sut w i p �1x l g lms�|$w i pznFt g3y p�q�l x �m§ � x � h s j q g w iQ� ¦&>�²,?9¨ �.r�s�t1l g vEpCq g t h w i p{l w�q g oEpCl x l�w i p � o x q)w g3j r � pCl'�,sQ| x tmp�| |�p'r�w g-y p § �  ?¡0¢ 5 � Ñ = ¦ � ^ ± ��7�� ± = Tθ − � h ^ σ± j σ± = T b ±×ξξ± h � ?9� ~ i p�q p ¦ ����� ¨ g lzw i p g tQw)p h q x � o x q)w¿su| x t�n �@� p�q�� � pCtEn � pCq x w�p�l>w i p ± l)w�q g o�p�l 7 g tEw�p h pCq � Ñ g l>w i p:r�sHs�q)v g t x w�ps�| � j w i l�w�q g o�pQ�N~ i9g r iÀg l x o x�g q�sQ|�t�p g6h1iQ� s�q g t h � g tEpCl su|2p � p�r!w�q sutEl ` Ñ -� ( � ) = � b ± + � Ñ z ± + � ξξ± x t�v ` Ñ � ( � ) = = ` Ñ 3� ( � ) + z ± + ξξ± � � = ¼ � ± § �,�I��� � � � i p¿l n1oEpCq)l)w�q nEr j w�nFq p�vEpCl�rCq g6� p�v �Q� ¡ �D� � ?9� g l�w i n1l x���g ´Qw�nFq p¾s�|² l w�q g oEp�l � θ − b ; ¼ � s�q + l)w�q g o�pCl � θ − b º ¼ � x t9vn�tQo�pCq)w�n9q � pCv,l)w�q g o�p�lNsQ|Dw i p ¸ j rCq � l)w x � ~ i9g r i,x q)p�o x q x � j� p � w�s b ± �Zl)s4w i9x w ± = 8 È « 9 È g l�w i p�r's�t�r�p�tQw�q x w g sQt s�|0w i p ± l)w�q g o�pCl 7 8 È g l�w i p g q:t�n �:� p�q 7 9 È g l2w i p�w�sHw x � tQn �@� p�q:sQ|�w i p ± l w�q g oEpCl x t1v{w i p ¸ j r�q � l w x � sQt�pClK� � i ptQn �@� p�qAsQ|4n1t9o�p�q�w)n1q � p�v¾l�w�q g o�pCl � p'w�~�p'pCt � j w i�x t9v� + § j w i ± l w�q g oEp�l2p���n x � l � Ñ + � − ��� − � � � m¯ � ! � ¬ �)©&/ � � � ¬ ¯ ¬�¬ ¯0 6! ����� 9©$� � ���G� �© ¬ �� ���� � �& � �¸ g3� o � p�r�q � l�w x � l0~ g w i z ± = ±ξξ± �Q~ i p�q�p ± = b « σ±g l x t g tEw�p h p�q'� x q�pÀsQ|,| q�pC��n9p�t�w$sHr�r'n�q)q)pCt�r'pE� ¢ x q�w g r'n j� x q � � ��w i1g l�s r'r'n1q�l�s�|�t�p'r�p�l�l g w � |�suq x w�q g3x t h n � x q i sQl w� x w�w g r'p � ¸ p�rE� >1� � x t9v x � l)sA|�sQq σ± = σ � x l g l:w � o g r x �s�| ¸ j rCq � l)w x � l2sQt x,i sQl w � x w)w g r'p�su| x � su~:p�q:l �Q�*� p'w�q � � ��t4l)nQr imx r x l)p:w i p x'� s y p jI� pCtEw g s�tQp�v,p � p�r'w�q)sut � g tQp�l�s�|� sHw i w � o�pClK� ` Ñ -� ( � ) x tQv ` Ñ � ( � ) ��| x ��� g tEw�s,w i p�r � x l�l2s�|p � p�r'w�q)s�t � g t�p�l � b ± + � ξξ± � � = ¼ � ± § �2���I� 7 � g l x t g t�w)p jh pCq � ��~ iQg r i r x t � p}r�sQtQl g v�p�q)pCv x l §�� � o x q w g r � p�l�� ~ g w i � r's sQq�v g t x w)pCl'� � � � i p g q x q�q x t h1� p�t�wK� x l:|�s �I� s ~*l | q�s � ¡ �D� ��?9� �1s � p � l�w i p x � h suq g w iQ� 5 � Ñ = ¦ � � ¨ �f� = � ± +− ± � ~ i p�q p � Ñ g l�w i p � r's�suq�v g t x w�p��AsQ|�w i p � j w i � g t�pQ�)� g l w i p � p x t � g tEp*l)pCo x q x w g sut � p x l�n9q p�v g tmn1t g w�l2su| Tξξ± T � \��fJ��1eFE=L!_�BEe�`�d�ByL J � i p;vEpCo�p�tQvEpCt�r�p:su| ± � s�q ρ � s�t µ � � n1r i w i p�l x�� pw�s}w i p § �  N¡0¢¤¦&>u²,?9¨ � g l x ~�p ��� j vQp y p � sQo�p�v�| q x r�w x � l)w�q�nQr�w)n�q�pQ� x v�p y1g � l�w x�g q r x l�p,~ i sQl p$l�w�p�o1l�sHr�r'n�q x w x ��� ¬ ¯E©-�&�� E¯ � ± = �}« � ≤ § � � � � x q�p�r's�oEq g�� p g t�w�p jh pCq�l � �°�2w h1g3y p�t � � � w i p ·*¸ r�s�t1| g6h n1q x w g sut�su|@w i p�K�  N¡.¢ g l�w i n1l x �  N¡.¢ r�q � l�w x � �5~ g w i � p � p'r'w�q)sut9l o�p�q2r'p ��� x t1vm~ g w i ¢?��X l b ± � � z ± + � ξξ± � ��tmw i p*r's ��� sut � � sur'r'n1q�q g t h r x l)p�w iQx w z ± g l x,� n � j w g o � p2s | ξξ± ��¸ p�r�� ?4� w i p�l�w p�o�l � ~ g v�w i l!� ∆µ = ∆µ(� ^�� ) �r x t � p�|�s�n1tQv �1� v g q�p�r'w x oQo � g r x w g sQt4su|>w i p § � w i p�s�q �¦ �Z� ?9¨ �=r�s�t1l g v�p�q g t h w i p,pCt�pCq hQ� su|�w i p � g tQp j � g tEp�q)p�o�n � j l g s�t?� � ( � ) = ∑Ä =−∞ ∞ � ( T � b ± + � ξξ± h ) � � g l�w i p$v g l w x t�r'p � p�w)~:p�p�t g t�w�p�q x r�w g t h � g t�p�l � � x l�w i p§�� o x�g q�o�s�w�p�t�w g�x � � � iEg l�oQq)s�vQnQr'p�l ∆µ = = � ∑Ñ = � ∞ � ( � ( � � − § ) − ��� ( � � ) + � ( � � + § )) � ~ i p�q)p � = ± � +− � g l�w i p o�p�q g s�v sQ|}w i p � g tEpCl�� o x w�w�p�q�t°� � i p,p'´FoQq)pCl)l g sQt g t{w i p � q x r � p'w�l g l�o�sQl g w g3y pE� l g tQr'p � ( � ) g l x r�sQt y p'´�|�n�t�r�w g sut g t¿w i p5r x l�pAn�tEvQp�q r's�tEl g vEpCq x w g sQtD� · p�t�p�q x �I� � � ∆µ( ��« � ) x q p:p'´1oQq)p�l�l)pCv g t w�p�q � l,s�|8� ÈBȱ ( � ) �E�µx l � g�h�i w � suv g | g r x w g sQt�su|@w i p § � w i p'suq � � �F\��Yv��9`�B ^Fd�}4J�L|B ^Qb z!e-`�L!_ov a�`�bxJK`�_)`oBH^xLKe�_!eØa�^4L,e-^5_o}4J �9`�a�cE^1b LÍ_oBH_oJ4a0z2_2}xJC@ �! z `�a0L�J�^�" J�EFJHd�_�`�a�^I }kB%L�J � i p x � h sQq g w i9� � ?9� r x t � p�p'´Qw�p�t1v�pCv­s y pCq w i p ~ i s � p ± q x t h pQ� ¼ ¹ ± º � ��oQq�s yQg vEp�v�w i p rCq � l)w x � ~ g w i sQtEp*o x q)w g r � p�oQp q0r'p � �N� � ¸ ′ j r q � l�w x � �*~ g w i ¢°��X lb ± � z ± + ξξ± � ~ iQg r i}x q g l)p�l x w ± = §A� θ = b ± σ± � g l l)w x'� � p � g � pE� g w g l x tQsHw i pCq ¸ j r�q � l�w x ��� s�q � p'w x l)w x'� � pE�� i1g l{|�s �I� s ~*l{| q s � w i p}| x r'wAw iQx w g � s�~ g t h w�sÂw i p r's x � l r'p�tQr'p$s�|;vEp�|�p�r'w�s�t1l ��x r�q�s�l)r�s�o g r x ��� � l ��x �I� y1x q gLjx w g s�tEl g t θ h p�t�pCq x w)pQ� x w x t � ± � § � v�p�|�p'r�w�l0s�t � � 7 g�g � w i p�l p §�� vEpC|�p'r'w�l'� x r�r'suq�v g t h w�s5s n1q:r's � o�nQw)pCq r x � r�n j� x w g sQtQlK� i9x'y p*w i p�l x � p ¢?��X � b ± �=|�suq x ��� ± � 0{s�q p�s y p'q'�=v�n�p,w s � � p�w x � l�w x!�1g � g w � s |�w i p ¸ ′ j r�q � l w x � w i p x � h s q g w iE� ��?E� i s � vQlDs y p�q x θ q x t h p x v � x r�p�t w>w s�w i pg t�w p�q y�x � T b − σ− � b + σ+ T �,� t�l nEr i�x r x l p ¡ �Z� ��?E� vEp�w)pCq �*g t�pCl x��@g ´1w)n1q p}sQ|ml)w�q g o�pCl{s�|At�p'~ h p's � p�w�q � ~ iQg r izx q)p,r iQx q x r�w)pCq g ��p�v �Q� x t�p'~¤w�q g o � p,s�| y p'r'w�suq�lC�b ±′ � z ±′ � ξξ±′ �.w i p x t x � s h nQp�l�su| b ± � z ± � ξξ± � x t9vzw i p ± l�w�q g o�p�l r�sQt�r'p�tQw�q x w g s�t ±′ = Tθ − b ± σ± T «$T b ±′ ×ξξ±′ T � Ú$#&% � Î ��Î Ì�Ò���Î ��Ì�Î Ìo�������������-ÒÍÊØÉ��IÐCÙ�Ú'Ú�ÚuÐ�Ó�Æ���� Ð��u×$��� Ú � q x t9l g w g sutm| q�s � s�tQp h p�s � p'w�q � w)s x t�s�w i p�q g l0r�s�tQw g t�n j sun9l g t θ l g tEr'p ±′ h s�p�l2w)s$��pCq)s4~ i p�t θ → b + σ± �� sQt�w g tQnEs nFl � � p'´1w)pCtQv g t h w i p x � h sQq g w iQ� � ?Q� g t4w i p�*x t9t�p�q.l i s�~�t x'� s y pE�u~@p�s � w x�g t�w i p � e jL� q x t�r i � � ~:pg tQw�q sQv�nQr'p}w i p g t9v�p�´ e x!h�x�g t � ~ iQg r i r's � oQq g l)pCl x �I�� � p'w x � l�w x�� � p,l w�q�nEr�w)n1q)pCl ¡ �D� � ?Q� r'sut9t�p'r�w)pCv g tAr'sutEw g3j tQn g w � ~ g w i w i p:l)w x q w g t h ¸ j r�q � l�w x � � � i p�r'sQq�q)pCl)o�sQtQv g t h pCtEpCq h1� � p d (θ) �{r x t � p p x l g � � |�s nFtQv g tÃw�p�q � lÂs�| � ÈBȱ ( � ) ��n�l g t h ¡ �D� ��?9� ����l x q)n � pE�uw i p�q p:p�´ g l)w.v g | |�p�q)pCt�w¸ j r�q � l�w x � l � p � sut hEg t h w)szw i p�l x�� p�e jL� q x t�r i � ( t�w i p suw i p�q iQx tQv°� x l�~@p i9x'y pmr�s � o�nQw�p�v°� g tEw�p�q�l)p'r�w g sut9l�s�| v g | |�p�q p�tEw p d (θ) � x t9v i p�tEr�pE� ��� ¬ ��� © ��� � � ¬ ¯E©�� ¬ � � � ¬ �'©-�� ¬ #,� ¬ © ¬ ¯y ����)©&� �� �� �& �Q¯ ¬ � ¯ �k�F� � µ � ¬ s � x w � o�pµs�| oQs � �Q� s�q)o i9g l � � � x q)p{n1t g-y pCq)l x ��� � oQq p�l)pCt�w g t w i p � � ?¡0¢ � � ¸ p�pmp'´ x�� o � p g t ¸ p'rQ� >9� � � i p�v�pCo�p�t9v�pCtEr'p�s�|p Û�sQt θ g l@w i p�|�nFt�r�w g sut ~ i9g r i r's � oEq g l)p�l x ��� l)w x'� � p oQs�q�w g s�tEl2sQ| x �I� p d (θ) �� i n1lK�Qs�~ g t h w)s*w i p ')(+* v�p�l)rCq g3� p�v x'� s y p:w i p ·*¸ s�|°w i p � �  N¡�¢ g l2|�n �I� � v�p'w�p�q ��g tEpCv �1� w i p ¸ j r�q � l�w x � l¢?��X lK� � κλ d �=w i p � v g q)p�r'w�s�q�l��Z� b ± d � x t9vAw i p$v g l�o � x r'p j� p�tQw y p�r'w�sQq)l�� ξξ± d � w)s h p'w i pCq{~ g w i w i p�l)p�w{su| p d (θ)g tQw)pCq)l)p�r'w g sQt�o�s g tQw�l,~ iEg r i x q�p5w i pAsQt � � ·*¸ r iQx q x r j w�p�q g l w g rClmr iQx t h1g t h s�tÂl �*x ��� y1x q g�x w g sQtQl g t � ( [ ) ��� �I�w i pCl�p;�En x tQw g w g p�l�r x t � p�r's � o�nQw�p�v,s�t�w i p �1x l g l>su| ¡ �°���³1� x t1v ¡ �D� � ?9� �Q�¤x l�p � | j r'sQtQl g l w)pCtEw�oQq)s r'pCvEn1q�pQ� | g t1v g t h w i p ¸ j r�q � l)w x � l�w)s h p'w i pCq�~ g w i w i p�e jL� q x tQr i p�lK� ¥�p iQx'y p¿|�sHn1t9v�w i p ·�¸ |�s�qmw�q g�x t h n � x q x tEv¾l���n x q pi s�l�w � x w)w g r'pCl x l:~�p ��� x l�|�sQq x t�n �:� pCq:s�|�w i sul)p,~ g w i r�p�tQw�q x � l �1��� p�w�q � sut � � � � i p}r�s � o�nQw x w g sQt i9x lÀtEs w q�p y p x � p�v�o9q g t�r g o x � v g | |�p�q�p�t�r'pCl � p'w�~@p�p�t ·�¸ o9q sQo�p�q j w g p�lzs�| � �  ?¡0¢ ~ g w i v g | |�pCq)p�t�w h p�s � p�w�q � s�| i sQl w� x w�w g r'pClK� x w � p x l)wz|�suq{w i sul)p}~ iQg r i¶x q pÂt�s�w l g6h t g | g3j r x t�w � �Ax t g l)s w�q�s�o g rE� �9\����0B�D I4EFJ f p�q p�~@p g �I� n1l)w�q x w�p�w i p x'� s y p h pCt�pCq x � q)pCl�n � w�l>~ g w ix w�q g�x t h n � x q i s�l)w � x w�w g r'p ��� f '0� �=� ��� w�q g�x t h n � x q � x w j w g r'pCl5sQt¾w i p � f ' x q p�tEp�r'pCl)l x q g � � ¸ j r�q � l)w x � l'� � iQg l |�s ��� s ~*l2| q�s � w i p*| x r'w.w iQx w g w g l0w i p:w�q g3x t h n � x q � x w�w g r�p w i9x w�q)p x � g ��pCl4w i p¿¯ � � � � �9© � p�t�p�q hQ�¾��g t g3� n � sQ|�w i p l � l)w�p � ~ i s�l�p¶p � p'r'w�q)sut9l x q p | q)p�p¶w)s � s y pE� ¸ nQr i¸ j r�q � l�w x � l x q�p � �0� j r�q � l)w x � l!�m~ g w i ¢?��X l�� B � + W �4�� B � + W�� x tQv θ = � � + � − � � �0� x q�p x q �1g w�q x q �g tQw)p h p�q�lK� B �� �� � g l x w�q g o � pmsu| � f '¾¢N�.X l@~ i9g r i}x q p pC�En x � g t�w i p � s�vQn � n1l x tQv,|�sQq �¾x t x t h � p2s�| §�� ¼�� ~ g w ip x r i s�w i pCq � �.�*l g t h w i p�oQq�s�r�p�v�nFq p�v g l)r�n1l)l)pCv g t ¸ p'rE�Á �9~�p iQxKy p�|�sun9tQvmw iEx w x ��� ¼ � j r�q � l�w x � l � p � sQt h w�s4w i pl x�� p:e j�� q x tEr i � w i p ��xHg t � q x t�r i � ��~ i9g r i r�s y pCq�l�w i p q x t h p ? ≤ θ º ∞ � � i p ¸ ′ j r�q � l)w x � l}s�|�w i p ¼ � j sutEpClx q p ¸ j rCq � l)w x � lÃw�sHsD� � i p � sur'r'n1q x w θ = ( � + § ) ( � ≤ º ∞) x tEv i1x'y p ¢N�.X l B κ � ( + § ) B λ � κ � λ = § �� � ±�7 κ ≠ λ � � � i p:l�w�q g o�p l�w�q)nQr'w�n1q)p�l ��?9� i1x'y p�w i p;l x � p¢��4X � B κ � |3s q x ��� θ s |9w i p g tHw�pÍq yux ��¦ ( − § ) � ( � + § ) ¨ � w i p g q ξξ± � p g t h ± W λ (κ ≠ λ) � ¥ i pCt_�0� ≠ ¼ �xe jL� q x t�r i p�l@su|�v g | |�pCq)p�t�w �<� � j r�q � l jw x � l x q p,v g l)w g t�r�wK� � i p � vQsmt�s�w i1x'y p � nEw�n x � g t�w)pCq�l p'r j w g sut9l'� � n9w x ��� g tEw�p�q�l)p'r�w@w i p ��x�g t � q x t�r i �?w i p g tQw)pCq j l)p�r'w g sQtQl�s�r�r'n1q�q g t h x w�q x w i p�q:l �@x ��� r'sutEr�p�t�w�q x w g sut9l�su| w i p:�0� � j rCq � l�w x � l5� ± l)w�q g 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r'w g w l i s�n � v � p p'´1oQp'r'w�p�v w iQx w w i p ·�¸ |�sQq ρ − § ^ b º2º § ^ b g l x l�n9o�pCq�l�w�q)nQr'w�n1q)p¿s�|ml p�o x q x w�p�v¸ j rCq � l)w x � ��� ¬ ��� # � ��� ����m�� Q¯ � v�p�|�p�r'w�lK� � n9w�t�suw�sutEp j v gLj� pCtQl g s�t x � s�t�pCl � l w�q g oEpCl � �*¥¾p i9x'y p q p y p x � p�v7w i9x w w i p�l p¾��p�q)s j v g3� pCt9l g s�t x � vEp�|�p�r'w�l x q�p � �9g vQp�|�p'r�w)sut9l��Z� ~ iQg r imx q p:r's � o � p'´QpCl0r'sut9l g l�w g t h sQ|Nw)~;s � sun9tQvmv�pC|�p'r j w�sQtQlK�=¥¾p �I� j l�p�o x q x w�p�v �1g v�pC|�p'r�w)sQtQl:r x t � p,r'sQtQl g v�p�q�p�vx l�t�p�~ o x q)w g r � p�l2s�t4w i p ¸ j r�q � l�w x � x l0w i p i sul)w � x w)w g r'pE� w i p � p x t �9g vQp�|�p'r�w)sut{l�p�o x q x w g sut°� ¬ Øz� � p g t h p���n x � w)s ∼ T � (ρ − § « b ) T− �mlQ� � � i p*p�| |�p�r'w g6y p�o x�g q�o�s w)pCtEw gLx � sQ| x� nQw�n x � �Fg vQp�|�p'r�w)sutzq p�o�n � l g sut g l�r i1x q x r'w�p�q g ��p�v �Q� w i p l x�� p¿l)o x r'p¿o x q x�� p�w)pCqK� Bm� x l � ( ¬ ) ����| ¬ Ø ; > Bm��w i p�1g vEp�|�p�r'w�s�t1lC� x r�r'sQq�v g t h w�s w i p h p�tQp�q x � q)p�l n � w�l su|¸ p'rClC� ± � ? � x q p � sun9tQv¶w�s � p�suq�v�p�q)pCv g t�w)s l�w�q g oQp�lx q�q x t h pCv �Q� w i p x � h s�q g w i1� � ?9� � ¡ ´Qw�p�tQw g s�tÂsQ|�w i9g l q)p x l�sQt g t h w)s¤w i p¶r x l�p s�| B � ;�¬ Ø � p x v9l}w)s t�p'~ l)w�q g o�p j � g�� p,l nFo�p�q�l)w�q�n1r'w�n1q�pCl�r'sut9l g l�w g t h su|0��p�q s j v g�� pCt j l g s�t x � vEp�|�p�r'w�l>sQ| � tEp�~$� ¸ j r�q � l w x � lK� x tQvml s�s�t?� ¡ y pCt j w�n x ��� ��x ~�p �I� j vQp y p � sQo�pCv | q x r'w x � x q g l p�lK� � i s�n h1i v�p j w x�g � l�su| g w�l�l)w�q n9r�w)n1q p x q�p�l�w g ��� w)s � p,vEp�w)pCq �*g t�pCv°� g w g l l x |�pmw�szl x'� tEs ~Ãw i9x w*w i p '=(+* vQsHpCl�w x�� pAo � x r�pA|�sQq γ ;:; � ��w�sEs�� �F\)H9c0D D�Bu`{K � i p x'� s y p,r'sut9l g vQp�q x w g sQtÀl i sH~*l�w i9x w;w i p,p � p'r�w�q sQt sQq�vEpCq g t h g tQw)s¿l)w�q g o�pCl x tQv�w i pAp�| |�p�r'w g6y p � s ~@pCq g t h su| v g�� p�t1l g sut­q)pCl g v�p g t w i p � �  N¡0¢ n1t g6y pCq)l x �I� � ����t p�l�l�pCtEr'pQ� x r's �@�1g t x w g s�t{s�|�v g l)r�q p'w�p�tQp�l�l:sQ|�p � p�r'w�q)s�t1l�� o�sQl g w g s�t1l0~ g w i�x � sut hEj q x t h p�v g tEw�p�q p � p�r'w�q)sutmq�pCoEn � l g sQtg l�w i p�sQt � � | x r�w)suq2~ i1g r imh1g3y p�l�q g l p�w�s4w i9g l;o i pCtEs � p j tEsut°�   suq>w iQg l.q�p x l sQt g w g l x � l s � s�n1tQv,w)s x q g l p�~ g w i,x t p'´1w)pCq)t x � v g l)suq�v�p�q}oQq p�l)pCt�w��,w i p l�w�q g oQp�l � p g t h | q x r j w�n9q�p�v x tQv o g tQtEp�v �9� w i p v g l sQq�vEp�q'� � i n�lC�Àl�w�q g o�p |�sQq ��x w g s�t g t � � x tQv � x'� p�q�p�v¿t x q�q)sH~ j��9x tQv r�sQtQvEnQr j � Î ��Î Ì�Ò���Î ��Ì�Î Ìo�������������-ÒÍÊØÉ��IÐCÙ�Ú�Ú'ÚuÐ�ÓCÆ���� Ð��Q×���� Ú Ú #9Ù w�sQq�l�r x t � p�r's�tEl g v�pCq)p�vmw�s � p�w i p�o9q g t�r g o � p,l g6h t x w)n1q p s�| x � �  N¡�¢ � k / � /�¯ ¬ ! � � ¬ #�� ¬ �m �! �m ��6� ¬ ¯E© � ��¯���¯� ¯% k� �o� ��©�¯�©-�m�� � � ¯ � � � ¬ ��� � � �6��� K© ¬ �� ��/Q¯6� �  q s � w i p xK� s y p{l�w x tQv9o�s g 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t gL� l x � w�p�q�t x w�pE� q)pCl�n � w g t hÀg tÀoEq�s�tQsHn�t�r�p�v � �  ?¡>¢ q)pCl g l)w g6y1g w � sQl�r g ��� x�j w g sut9l�~ i9g r i q�pC| � p�r'w�w i p h q�sun9t1v j l)w x w)p�v�p y1g � j l)w x�g q r x l�p vEpCo�p�tQvEpCt�r�p�su| ρ sQt µ 5�w i pzsQl r g ��� x w g sQtQl5�;o�p x�� l x q�p� s�n1t9v{w�s{s�r�r'n1q:r � sQl�p$w�s{w i p ¬ ¯E©-�&�� �¯ � | g ��� g t h | x r�w)suq�l sQ|Dw i p  N¡�¢ r�q � l)w x � l>~ i9g r i l)n1q y�gØy p x w x�h1g3y p�t k � � i9g l o i pCt�s � p�t�sQt g l � p�w x tEs w i p�q�v g l w g t�r'w g3y p ��x q � s�|:w i p� �  N¡.¢ ��¥¾p iQxKy p |�sHn�tEv g w0w�s � p y p�q � l g3�*g � x q>w�s,w i p q)pCl g w gØy�g w � sQl r g ��� x w g sut9l�sQ| x r's�tEvQnQr'w gØy p�l i p'p'w g t x l � l w�p � � p'w x ��² j w � o�p®l�p ��g r'sQtQvEnQr�w)suq ² � j w � oEp¤l)p ��gLj r's�tEvQnQr'w�s�q ¦�±�¨ ��~ i9g r i l w g ��� q)p ��x g t�w�s � p�p'´1o � x g tEpCv?� ¥¾p x q�p h s g t h w�smo�n � � g l i w i p*q)p�l n � w�l�r�sQt�r'pCq�t g t h w i9g lg l)l�nEp g tmw i p*tEp x q�|�nQw)n�q�pQ� � w g l0q p ��x q �Ex'� � p�w iQx w x t x q)w g | g r g�x �I� � r�q p x w�p�v,p'´1w)pCq j t x � oEp�q w�n9q �9x w g sQt � sHr x � g ��p�v4~ g w i9g t x l ��x ��� q p h1g sQt,r x t� � s�r � n9o}r�s�t1v�nQr�w g sQt}s y pCq x �I�� N¡0¢¶� g ��n g vD��o g tQt g t h sQt � � sQt�p}l w�q g o�pQ� � i p � sQl w x o1oQq�sQoQq g�x w�p¿l � l)w�p � lAw)s w�p�l)w$w iQg l,p'´Fo�pCq g3� pCtEw x ��� ��x q)pzo�pCq iQx o9l h q x tEn � x q�w i9g t | g � � l � g�� p�w i s�l�p;v�p�l)rCq g3� p�v g t ¦�?F¨ � ��l g��*g � x q�o i pCtEs � p j tEsutm~ x l�q)pCo�sQq�w�p�v g t ¦ ³9¨ ���;p'w h q x t�n � x q�| g � � l2n1l)p�v g t w i p$p'´1o�p�q g�� p�tQw�l ¦�³E¨ ~�p�q)p i1g�hEi � � v g l sQq�vEpCq�pCv°� x tEv g wg l@n1tEr � p x q�t�s�~®~ i p'w i p�q�w i p x'� s y pmw i p�sQq � ~�suq � l g t l)nQr iAx l g w)n x w g s�t?� �;d��1^1a �|EmJ�b �0D`JK^Q_oL ¥7p h q x w)pC|�n �I� � x r � t�s�~ � p�v h p v g l�r�nFl�l g sQtQl®~ g w i¢ q)su|!� 0 � ¢ pCoQo�pCqK� ¢ q)su|!� :� ¸ i9� � s y l �1g3g � x t9v ¢ q)s�|!�¢ 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id nasplib_isofts_kiev_ua-123456789-139014
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 0132-6414
language English
last_indexed 2025-12-07T18:13:53Z
publishDate 1999
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
record_format dspace
spelling Slutskin, A.A.
Slavin, V.V.
Kovtun, H.A.
2018-06-19T18:28:06Z
2018-06-19T18:28:06Z
1999
The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension / A.A. Slutskin, V.V. Slavin, H.A. Kovtun // Физика низких температур. — 1999. — Т. 25, № 8. — С. 936-943. — Бібліогр.: 13 назв. — англ.
0132-6414
https://nasplib.isofts.kiev.ua/handle/123456789/139014
In narrow-band conductors a weakly screened Coulomb interelectron repulsion can supress narrow-band electrons` hopping, resulting in formation of a "frozen" electron phase which differs principally from any known macroscopic self-localized electron state including the Wigner crystal. In a zero-bandwidth limit the "frozen" electron phase is a classical lattice system with a long-range interparticle repulsion. The ground state of such systems has been considered in the case of two dimensions for an isotropic pair potential of the mutual particle repulsion. It has been shown that particle ordering into stripes and effective lowering of dimension resides universally in the ground state for any physically reasonable pair potential and for any geometry of the conductor lattice. On the basis of this fact a rigorous general procedure to describe the ground state fully has been formulated. Arguments have been adduced that charge ordering into stripes in high-Tc superconductors testifies to the presence of a "frozen" electron phase in these systems.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
Физика низких температур
Низкоразмерные и неупорядоченные системы
The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension
Article
published earlier
spellingShingle The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension
Slutskin, A.A.
Slavin, V.V.
Kovtun, H.A.
Низкоразмерные и неупорядоченные системы
title The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension
title_full The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension
title_fullStr The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension
title_full_unstemmed The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension
title_short The ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. Stripe formation and effective lowering of dimension
title_sort ground state of the "frozen" electron phase in two-dimensional narrow-band conductors with a long-range interelectron repulsion. stripe formation and effective lowering of dimension
topic Низкоразмерные и неупорядоченные системы
topic_facet Низкоразмерные и неупорядоченные системы
url https://nasplib.isofts.kiev.ua/handle/123456789/139014
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