ËX ê s À W ¥ “ ¤X c l ü § Žö n ÚT ê s; c" e T Ç S Ë { ¢] k ù8 ý V ê s ¹ ÅT

V R

ËX ê s À W ¥ “ ¤X c l ü § Žö n ÚT ê s; c" e  T Ç S Ë { ¢] k ù8 ý V ê s ¹ ÅT 

 + 2­ £) כ

›

¸‚  @ /† < Ɠ §  ƒ  õ † < Æ@ /† < Æ Ó ü t o † < Æõ , F  g Å Ò 501-759

(2011¸   6 Z 4 1{ 9  ~ à Î6 £ §, 2011¸   6 Z 4 21{ 9  à º& ñ ‘ : r ~ à Î6 £ §, 2011¸   7 Z 4 25{ 9  > F  S X ‰& ñ )

$ í

 © œ   H Á º Œ •0 A Õ ªÓ ü t } © œ (growing random network) \ " f  s f ç — ¸+ þ A (Ising model) _  Á ºô  Ç 



© œ„  s  ‰ & ³ © œ (infinite order phase transition) `  ¦ M  g- ê ø Í Ä º Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O  (Wang-Landau sampling method) `  ¦ s 6   x # Œ à ºu > í ß –Z O  (numerical method) Ü ¼– Ð  Ž 7 £ x  9ô  Ç . s  " é ¶ ¨ î ~ ½ Ó  



(square lattice) ü < $ í  © œ   H Á º Œ •0 A Õ ªÓ ü t } © œ\ " f_   s f ç — ¸+ þ A_  M  g- ê ø Í Ä º Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ Ó Z O

Ü ¼– Ð  7 H _ …º ú ˜– Ð „  í ß – r Ð 3 x (Monte Carlo simulation) `  ¦ à º' Ÿ  # Œ  Ä »\  -t _  n>  • ¸† < Êà º\  ¦ ½ ¨ 

%

i  . s  " é ¶ ¨ î ~ ½ Ó   \ " f  s f ç — ¸+ þ A (Ising model) “ É r s    © œ„  s – Ð · ú ˜ 94 R e ”   H X < s \  ¦ „  í ß –r  Ð 3

x ~ ½ ÓZ O Ü ¼– Ð  Ä »\  -t _  2>  • ¸† < Êà º Ô  ¦ƒ  5 Å qe ” `  ¦  Ž 7 £ x % i  . ¢ ¸ô  Ç $ í  © œ   H Á º Œ •0 A Õ ªÓ ü t } © œ\ " f _

  s f ç — ¸+ þ A“    â Ä º\   H  Ä »\  -t _  n>  • ¸† < Êà º & h # Q• ¸ 3  t  ƒ  5 Å q“     õ \  ¦ % 3 # Q, s  כ s  Á

ºô  Ç   © œ„  s  ‰ & ³ © œ{ 9  0 p x$ í `  ¦ ˜ Ð% i  .

Ù þ

˜d ” # Q:  s f ç — ¸+ þ A, Á ºô  Ç   © œ„  s  ‰ & ³ © œ, Á º Œ •0 A Õ ªÓ ü t } © œ, M  g -ê ø Í Ä º Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O 

Phase Transition of the Ising Model on a Growing Random Network

Wooseop Kwak ^∗

Department of Physics, Chosun University, Gwangju 501-759 (Received 1 June 2011 : revised 21 June 2011 : accepted 25 July 2011)

We numerically investigate the infinite-order phase transition of the Ising model on a growing random network. We perform the simulations of the Ising models both on a 2-d (dimensional) square lattice and on a growing random network, and obtain their n-th derivatives of the free energies by using the Wang-Landau sampling method. The phase transition of the Ising model on the 2-d square lattice is a second-order phase transition and shows a discontinuity in the second derivative of free energy. However, the phase transition of the Ising model on the growing random network is not known well and does not show a discontinuity in any of the n-th derivatives of the free energies. These features can be understood in terms of an infinite-order phase transition.

PACS numbers: 05.10.-a, 05.10.Ln, 02.70.Lq

Keywords: Ising model, Infinite order, Random network, Wang-Landau sampling

∗

E-mail: [email protected]

-722-

I. " e  ] Ø



7 H _ …º ú ˜– Ð „  í ß –r Ð 3 x (Monte Carlo simulation) “ É r : Ÿ x >  Ó

ü

t o † < Æ\ " f ×  æ כ ¹ô  Ç % i ½ + É [1–26]`  ¦ “ ¦ e ” “ ¦ : £ ¤ y   © œ„   s

‰ & ³ © œõ  e ” > & h \ " f_  ( Ž É Ó' \ " f_  > í ß –r ç ß –`  ¦ ×  ¦ s 

“

¦ & ñ S X ‰ ô  Ç   õ \  ¦ 8 £ ¤& ñ l  0 A # Œ y Û ¼ž ÐÕ ªÏ þ › F ×  æ

~

½ ÓZ O  (histogram reweighting method) [1–3,12,13], { 9 ì ø Í



o  ) a € © œ © œ^  ¦ ~ ½ ÓZ O  (generalized ensemble method) [16,20–

22] ,  ' pà Ԗ Ðx  Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O  (entropy sampling method) [4,15,23–26] 1 p x s  > hµ 1 Ï÷ &“ ¦ e ”  . & ñ S X ‰ y   © œ„   s

 & h `  ¦ ¹ 1 Ô ? /“ ¦  © œ„  s  & h \ " f e ” > ‰ & ³ © œ`  ¦ ƒ  ½ ¨   H X

< y Û ¼ž ÐÕ ªÏ þ › F ×  æ ~ ½ ÓZ O õ   ' pà Ԗ Ðx  Á º Œ •0 A ³ ð‘ : r Æ Ò Ø

 ¦ ~ ½ ÓZ O  1 p x s   6   x ÷ &# Q M ® o  . Õ ª×  æ \ " f  ' pà Ԗ Ðx  Á º Œ •0 A

³

ð‘ : r Æ ÒØ  ¦“ É r “ : r • ¸\  _ ” > r t  · ú §l  M :ë  H \  ± ú “ É r “ : r • ¸\ 

"

f „  í ß –r Ð 3 x`  ¦ ½ + É Ã º e ” “ ¦, s  Qô  Ç  ' pà Ԗ Ðx  Á º Œ •0 A ³ ð

‘

: r Æ ÒØ  ¦ ~ ½ ÓZ O [ þ t ×  æ \  M  g- ê ø Í Ä º Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O  (Wang-Landau sampling method) [4] s  ™ è> h  ) a s Ê ê– Ð, s

 ~ ½ ÓZ O _  Z  t  î  r S X ‰  © œ$ í Ü ¼– Ð “   # Œ  Ö ¸ µ 1 Ïy   6   x ÷ &“ ¦ e ”

 . : £ ¤ y , s  ~ ½ ÓZ O “ É r ¢ - a  or ç ß – (relaxation time) s  š ¸ A

   o   H „  í ß –r Ð 3 x — ¸+ þ A\ " f > í ß –r ç ß –`  ¦ é ß –» ¡ ¤ “ ¦ f ”  ] X

  Ä »\  -t ,  ' pà Ԗ Ðx ,  © œI x 9 • ¸ 1 p x `  ¦ ½ ¨½ + É Ã º e ”   H

 '

pà Ԗ Ðx  Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O s  .

‰

&

³F  t  Ó ü t o — ¸+ þ A_  „  í ß –r Ð 3 x“ É r ŠҖ Ð     / B N ç ß –\ " f

 Ö

¸ µ 1 Ïy  s À Ò# Q4 R M ® o  H X <, 4 Ÿ ¤ ¸ ú š>  Õ ªÓ ü t } © œ (complex net- work) [27–31] s  ™ è> h  ) a s A – Ð 4 Ÿ ¤ ¸ ú š>  Õ ªÓ ü t } © œ 0 A\ " f _

 — ¸+ þ A_   © œ„  s  ‰ & ³ © œ`  ¦ V , o  ƒ  ½ ¨ “ ¦ e ” “ ¦, : £ ¤ y , Õ ª Ó

ü

t } © œ ” ¸× ¼ (node) _  ƒ    ‚  à º  © œ › ' a› ' a >  (degree-degree correlation)  4 Ÿ ¤ ¸ ú š>  Õ ªÓ ü t } © œ 0 A\  e ”   H — ¸+ þ A_   © œ„  s 

‰

&

³ © œ\  ×  æ כ ¹ô  Ç % i ½ + É`  ¦ ô  Ç  [32–37].



s f ç — ¸+ þ A (Ising model) “ É r s  " é ¶,  Œ ™ " é ¶ Õ ªo “ ¦

œ

í{ 9 ~ ½ Ó    (hyper square lattice) \ " f — ¸¿ º s    © œ„  s 

‰

&

³ © œ ( second order phase transition) `  ¦ ˜ Ðs  9, ¨ î ç  H  © œ s

 : r (mean-field theory) `  ¦  Ø Ô  H  © œe ” >  " é ¶ (upper critical dimension) “ É r 4 " é ¶ Ü ¼– Ð · ú ˜ 94 R e ”  . Bauer et al. [35]   H $ í  © œ   H Õ ªÓ ü t } © œ\ " f y © œ $ í (ferromagnetic)



s f ç — ¸+ þ As  Á ºô  Ç   © œ„  s \  ¦ ˜ Г     H  כ `  ¦ s  : r& h  Ü

¼– Ð > í ß –`  ¦ % i  . t ë ß –, à ºu > í ß –Z O Ü ¼– Ð  Ž 7 £ x   H

 כ

“ É r ~ 1 t  · ú § .

s

  7 Hë  H \ " f Ä ºo   H $ í  © œ   H Õ ªÓ ü t } © œ 0 A_  y © œ $ í   s

f ç — ¸+ þ A_   Ä »\  -t [ þ t _  • ¸† < Êà º\  ¦ M  g- ê ø Í Ä º ³ ð‘ : r Æ

ÒØ  ¦ ~ ½ ÓZ O `  ¦  6   x # Œ ½ ¨ # Œ Á ºô  Ç   © œ„  s  e ” `  ¦ ˜ Ð s

 9 ô  Ç . Õ ªo “ ¦, s  " é ¶ ¨ î ~ ½ Ó    0 A\ " f y © œ $ í   s

f ç — ¸+ þ A_  „  í ß –r Ð 3 x`  ¦ à º' Ÿ  # Œ ¿ º — ¸+ þ As  Ó ü t o | ¾ Ó[ þ t (physical quantities) s  # Qb  G>    É r \  ¦ q “ § % i  .

II. Ž ì ŏ ŒU ê s0 n É õ m Í T  ] Ø

1. { ¢] k ù

y

© œ $ í  s f ç — ¸+ þ A_  K ‰x 9 ž Ðm î ß –“ É r H = −J X

<i,j>

S

i

S

j

(1)

–

Ð & ñ _   ) a  . # Œl " f  s f ç — ¸+ þ A_  Û ¼— 2 ;s  +z» ¡ ¤ Ü ¼– Ð

&

ñ § > =÷ &# Q e ” Ü ¼€   S

_i

= 1 s “ ¦, ì ø Í@ / ~ ½ ӆ ¾ ÓÜ ¼– Ð & ñ § > =÷ &# Q e ”

Ü ¼€   S

_i

= −1 s “ ¦, y © œ $ í  s f ç — ¸+ þ A_    ½ + Ë © œÃ º (coupling constant) “   J  H 0 ˜ Ð   H  © œÃ ºs  9, Û ¼— 2 ;_ 

½

+ ˓ É r þ j“  ] X s Ö  © (nearest neighbors) \  @ /K " fë ß – à º' Ÿ 

 ) a  .

2. V R ËX ê s À W ¥ “ ¤X c l ü § Žö n ÚT ê s

$ í

 © œ   H Á º Œ •0 A Õ ªÓ ü t } © œõ  s  " é ¶ ¨ î ~ ½ Ó   \ " f  s  f ç

— ¸+ þ A_   © œ„  s  ‰ & ³ © œ`  ¦ q “ § % i  . $ í  © œ   H Á º Œ •0 A Õ

ªÓ ü t } © œ“ É r Callaway et al. [32] s  ] jî ß –ô  Ç Á º Œ •0 A Õ ªÓ ü t } © œ

`

 ¦ €  •ç ß – à º& ñ # Œ ë ß –[ þ t% 3 Ü ¼ 9, Á º Œ •0 A Õ ªÓ ü t } © œ“ É r ë ß –× ¼  H

~

½ ÓZ O “ É r  6 £ § õ  ° ú    [13]. ” ¸× ¼_  Õ ü w   N“   $ í  © œ   H Á

º Œ •0 A Õ ªÓ ü t } © œ“ É r y Œ •y Œ •_  é ß –>    ô  Ç > hm ”  ” ¸× ¼\  ¦  8ô  Ç Ê

ê\  Á º Œ •0 A– Ð i ( v“ É r ô  Ç Š © œ_  ” ¸× ¼[ þ t`  ¦ ƒ     # Œ l> h_  a A ß

¼[ þ t`  ¦ ë ß –[ þ t% 3  . q “ § l  0 AK " f  6   x ô  Ç s  " é ¶ ¨ î ~ ½ Ó

 

   H Å Òl & h   â > › ¸|  (periodic boundary condition)

`

 ¦ & h 6   x % i  .

3. · Ï Ñ-Š ˜ m – ¤ “ ¤X c l ü ƒ »Ä ] Ø ˜ ¼û s Ú U ê s0 n É

Á

º Œ •0 A– Ð Û ¼— 2 ;`  ¦ ‚  × þ ˜ # Œ > _   © œI \  ¦    o  9 \ 



-t  / B N ç ß –\ " f Á º Œ •0 A   6 £ §`  ¦   H M  g- ê ø Í Ä º Á º Œ •0 A ³ ð

‘

: r Æ ÒØ  ¦ ~ ½ ÓZ O “ É r „  s  S X ‰Ò  ¦ (transition probability) s  é ß – t

  © œI x 9 • ¸ (density of state) \ ë ß – _ ” > r Ù ¼– Ð, \  -t  E

₁

`  ¦ ”    © œI \ " f \  -t  E

2

\  ¦ ”    © œI – Ð_  „  s  S X

‰Ò  ¦“ É r  6 £ § õ  ° ú  s  j þ t à º e ”  :

p (E

₁

→ E

2

) = min  g(E

₁

) g(E

2

) , 1



, (2)

g(E

₁

) ≥ g(E

₂

) s €   min 

_g(E

1) g(E2)

, 1 

= 1 s Ù ¼– Ð „  s  S X ‰Ò  ¦ p = 1 s  ÷ &# Q" f D h– Ðî  r \  -t  E

²

\  ¦ ° ú   H D h– Ðî  r  © œI 

 † ½ Ó © œ ‚  × þ ˜s  ÷ &“ ¦, ë ß –{ 9  g(E

₁

) < g(E

₂

) Ü ¼€   „  s  S X ‰Ò  ¦ p =

^g(E_g(E¹⁾

2)

\     D h– Ðî  r  © œI  ‚  × þ ˜÷ &# Q”   .

„

 í ß –r Ð 3 x`  ¦   H 1 l x î ß – \  -t  y Û ¼ž ÐÕ ªÏ þ › (histogram) H(E) ü < à º& ñ “    (modification factor) f\  ¦  6   x ô  Ç  © œI  x 9

• ¸ g(E)\  ¦ ½ ¨ # Œ > 5 Å q » ¡ ¤' ‘ r †   . # Œl " f » ¡ ¤' ‘ r v 



 H ~ ½ ÓZ O “ É r  6 £ § õ  ° ú   :

H(E

2

) → H(E

1

) + 1 (3) g(E

₂

) → g(E

₁

) × f (4)

#

Œl " f, œ íl  f = e

¹

`  ¦  6   x ô  Ç . \  -t  / B N ç ß –`  ¦ Á º Œ •0 A

 

6 £ §`  ¦ : Ÿ x # Œ \  -t  y Û ¼ž ÐÕ ªÏ þ ›s  ¨ î ¨ î K t €   y Û ¼ž Ð Õ

ªÏ þ › H(E)\  ¦ „  Â Ò 0Ü ¼– Ð F [ O & ñ “ ¦, D h– Ðî  r à º& ñ “    f

new

= f

^1/2

– Ð  Ë ¨# Q, þ j7 á x à º& ñ “    f

^final

= e

⁻⁹



| ¨

c M : t  „  í ß –r Ð 3 x`  ¦ à º' Ÿ  # Œ g(E)\  ¦ ½ ¨ô  Ç .

4. ö n ÚP S ë s



© œI x 9 • ¸ g(E)\  ¦ · ú ˜€  , “ : r • ¸ T \  _ ” > r   H ì  r C † < Êà º (partition function) Z(T ) = P

E

g(E)e

^−E/kBT

\  ¦ ½ ¨½ + É Ã

º e ” Ü ¼ 9  Ä »\  -t  (free energy) F ü <  Ä »\  -t _  beta \  @ /ô  Ç n>  • ¸† < Êà º F

⁽ⁿ⁾

  H  6 £ § õ  ° ú  s  j þ t à º e ”  :

F

⁽⁰⁾

= − 1 β logZ, F

⁽¹⁾

= + 1

β

²

logZ + 1

β < E >, F

⁽²⁾

= − 2!

β

³

logZ − 2!

β

²

< E >

+ 1

β (< E >

²

− < E

²

>), F

⁽³⁾

= + 3!

β

⁴

logZ + 3!

β

³

< E >

− 3

β

²

(< E >

²

− < E

²

>) + 1

β (2 < E >

³

−3 < E >< E

²

>

− < E

³

>), F

⁽⁴⁾

= − 4!

β

⁵

logZ − 4!

β

⁴

< E >

+ 12

β

³

(< E >

²

− < E

²

>)

− 4

β (2 < E >

³

−3 < E >< E

²

>

− < E

³

>) + 1

β (6 < E >

⁴

−12 < E >

²

< E

²

> +3 < E

³

>

−4 < E >< E

³

> − < E

⁴

>),

. . . (5)

Fig. 1. (Color online) log(g(E)) as a function of energy E.

#

Œl " f, β = 1/k

^B

T   H % i “ : r • ¸ (inverse temperature) s

 .

III. + s ÇÊ Ý õ m Í w Š²  o

s

 : r& h Ü ¼– Ð r Û ¼% 7 ›ß ¼l  N“    s f ç — ¸+ þ A_   © œI x 9 

•

¸  H 2

^N

Ü ¼– Ð B Ä º  H à ºs  . & ñ S X ‰ • ¸\  ¦ Z  } s l  0 AK  „   í

ß –r Ð 3 x“ É r Û ¼— 2 ;_  Ì  à º N = 100“   s  " é ¶ ¨ î ~ ½ Ó   ü < $ í



© œ   H Á º Œ •0 A Õ ªÓ ü t } © œ 0 A\ " f à º' Ÿ ÷ &% 3 “ ¦, # Œl " f s  

"

é

¶ ¨ î ~ ½ Ó   \   H Å Òl & h   â > › ¸| `  ¦ & h 6   x ÷ &“ ¦ $ í  © œ   H Á

º Œ •0 A Õ ªÓ ü t } © œ\ " f  H ¨ î ç  H ƒ    ‚  à º < k >  H 4 s  .



© œI x 9 • ¸_  ß ¼l  B Ä º ß ¼Ù ¼– Ð, Õ ªa Ë > 1“ É r \  -t  E\ 

@

/6 £ x   H  © œI x 9 • ¸\  – ÐÕ ª (log) \  ¦ 2 [ # Œ • ¸³ ð– Ð Õ ª§ 4 



. s  • ¸³ ð\ " f W 1— ¸  Ҡ ñ  H s  " é ¶ ¨ î ~ ½ Ó    0 A_    s

f ç — ¸+ þ A`  ¦ ³ ðr  “ ¦, 1 l x Õ ª p   Ҡ ñ  H $ í  © œ   H Á º Œ • 0

A Õ ªÓ ü t } © œ 0 A_   s f ç — ¸+ þ A`  ¦ ³ ðr ô  Ç .

Õ

ªa Ë > 2\ " f r Û ¼% 7 ›_  ¨ î ç  H \  -t  < E > “ : r • ¸\   



 7 £ x    H  כ `  ¦ ˜ Ð# ŒÅ ғ ¦ e ”  .

Õ

ªa Ë > 3  H r Û ¼% 7 › ß ¼l  N = 100“   s  " é ¶ ¨ î ~ ½ Ó     0

A_   s f ç — ¸+ þ A_   © œ„  s  “ : r • ¸ T

c

(N ) ' 2.324 s “ ¦, $ í



© œ   H Á º Œ •0 A Õ ªÓ ü t } © œ 0 A_   s f ç — ¸+ þ A_   © œ„  s  “ : r • ¸ T

_c

(N ) ' 2.873e ” `  ¦ ˜ Ð# Œ Šғ ¦ e ”  .

Õ

ªa Ë > 4  H s  " é ¶ ¨ î ~ ½ Ó     0 A_   s f ç — ¸+ þ A_   © œ„   s

 “ : r • ¸   H % ƒ\ " f n>  • ¸† < Êà º F

⁽ⁿ⁾

/N \  ¦ ˜ Ð# ŒÅ ғ ¦ e ”  .

#

Œl " f, n“ É r 1 õ  2s  .

Õ

ªa Ë > 5  H $ í  © œ   H Á º Œ •0 A Õ ªÓ ü t } © œ 0 A_   s f ç — ¸+ þ A _

  © œ„  s  “ : r • ¸   H % ƒ\ " f n>  • ¸† < Êà º F

⁽ⁿ⁾

/N \  ¦ ˜ Ð# ŒÅ Ò

“

¦ e ”  . # Œl " f, n“ É r 1, 2, Õ ªo “ ¦ 3 s  .

Fig. 2. (Color online) Average energy < E > as a func- tion of temperature T .

Fig. 3. (Color online) Specific heat C(T ) as a function of T .

Fig. 4. (Color online) Normalized nth derivatives of free energy F (T ) on the square lattice.

IV. À X Ø8 ý õ m Í + s Ç Â ] Ø

Fig. 5. (Color online) Normalized nth derivatives of free energy F (T ) on the growing random network.

M 

g- ê ø Í Ä º Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O `  ¦ : Ÿ x # Œ f ” ] X   Ä »

\

 -t \  ¦ > í ß – # Œ  Ä »\  -t  • ¸† < Êà º_  Ô  ¦ƒ  5 Å q$ í `  ¦ s  6

 

x # Œ  © œ„  s ‰ & ³ © œ`  ¦ ƒ  ½ ¨ % i  . $ í  © œ   H Á º Œ •0 A Õ ª Ó

ü

t } © œõ  s  " é ¶ ¨ î ~ ½ Ó    0 A\  e ”   H  s f ç — ¸+ þ A_   © œI  x 9

• ¸ g(E)\  ¦ M  g- ê ø Í Ä º Á º Œ •0 A ³ ð‘ : r Æ ÒØ  ¦ ~ ½ ÓZ O `  ¦ s 6   x

# Œ, r Û ¼% 7 › ß ¼l  N = 100\ " f ½ ¨ % i  . Õ ªo “ ¦, $ í



© œ   H Á º Œ •0 A Õ ªÓ ü t } © œ_  ¨ î ç  Hƒ    ‚  à º 7 £ ¤, ¨ î ç  H þ j“  ] X  s

Ö  © _  Õ ü w    H s  " é ¶ ¨ î ~ ½ Ó      s f ç — ¸+ þ A_  þ j“   ] X

s Ö  © _  Õ ü w   4ü < ° ú  • ¸2 Ÿ ¤ # Œ, $ í  © œ   H Á º Œ •0 A Õ ªÓ ü t }

© œ`  ¦ ½ ¨$ í # Œ  © œ„  s  ‰ & ³ © œ`  ¦ " f– Ð q “ § % i  .  © œI x 9 

•

¸ g(E)– Ð “ : r • ¸\    É r q \ P `  ¦ C(T )\  ¦ ½ ¨ # Œ Ä »ô  Çß ¼ l

(finite size) e ” >  “ : r • ¸ T

^c

(N ) s , s  " é ¶ ¨ î ~ ½ Ó    0 A_ 



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