"Talk on 'introduction to conformal field theory(CFT)'"의 두 판 사이의 차이

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(피타고라스님이 이 페이지의 이름을 talk on 'introduction to conformal field theory(CFT)'로 바꾸었습니다.)
 
(사용자 3명의 중간 판 23개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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==introduction==
  
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* scaling and power law
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* scale invariance and conformal invariance
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*  critical phenomena
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** http://scienceon.hani.co.kr/archives/13339
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** http://scienceon.hani.co.kr/archives/13941
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** http://scienceon.hani.co.kr/archives/14664
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==scale invariacne and power law==
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Scale Invariance of power law functions The function y=xp is "scale-invariant" in the following sense.  Consider an interval such as (x,2x), where y changes from xp to  2pxp.  Now scale x by a scale factor a.  We look at the interval (ax,2ax) where y changes from (ax)p to 2p(ax)p.  We see that y in this interval is the same (except for a scale change of ap) as y in the unscaled interval. Most functions do not behave this way.  Consider y=exp(x), which goes [from exp(x) to exp(2x)] over the interval (x,2x), and [from exp(ax) to exp(2ax)] in the scaled interval (ax,2ax).  There is no scale factor that can be removed from y to make the second interval of y appear the same as the first. The sum of two different powers is usually not scale invariant.  However, special cases may still be.  y=Ax2 + Bx can be written as A(x+B/2A)2 -B2/4A.  If the new variable X=x+B/2A is scaled to aX, then y(aX)=a2Y(X)+(a2-1)(B2/4A).  In other words, the function y(x) is scale invariant (with scaling exponent 2) after finding the right scaling variable X and allowing for a scale-dependent shift of y.
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==critical phenomena==
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* [[critical phenomena]]
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==correlation at large distance==
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* [[universality class and critical exponent]]
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* appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature.
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*  the critical exponent describes the behavior of physical quantities around the critical temperature e.g. magnetization
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:<math>M\sim (T_C-T)^{1/8}</math>
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* magnetization and susceptibility can be written as '''correlation functions'''
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*  large distance behavior of spin at criticality <math>\eta=1/4</math>
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:<math>\langle \sigma_{i}\sigma_{i+n}\rangle =\frac{1}{|n|^{\eta}}</math>
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*  correlation length critivel exponent <math>\nu=1</math>
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:<math>\langle \epsilon_{i}\epsilon_{i+n}\rangle=\frac{1}{|n|^{4-\frac{2}{\nu}}}</math>
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==conformal transformations==
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*  roughly, local dilations
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** this is also equivalent to local scale invariance
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* correlation functions do not change under conformal transformations
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==related items==
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* [[Expositions on conformal field theory]]
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* [[conformal field theory (CFT)]]
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==expositions==
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* [http://felix.physics.sunysb.edu/%7Eallen/540-05/scaling.html Scale Invariance of power law functions]
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[[분류:개인노트]]
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[[분류:talks and lecture notes]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q849798 Q849798]
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===Spacy 패턴 목록===
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* [{'LOWER': 'knot'}, {'LEMMA': 'theory'}]

2021년 2월 17일 (수) 03:00 기준 최신판

introduction


scale invariacne and power law

Scale Invariance of power law functions The function y=xp is "scale-invariant" in the following sense. Consider an interval such as (x,2x), where y changes from xp to 2pxp. Now scale x by a scale factor a. We look at the interval (ax,2ax) where y changes from (ax)p to 2p(ax)p. We see that y in this interval is the same (except for a scale change of ap) as y in the unscaled interval. Most functions do not behave this way. Consider y=exp(x), which goes [from exp(x) to exp(2x)] over the interval (x,2x), and [from exp(ax) to exp(2ax)] in the scaled interval (ax,2ax). There is no scale factor that can be removed from y to make the second interval of y appear the same as the first. The sum of two different powers is usually not scale invariant. However, special cases may still be. y=Ax2 + Bx can be written as A(x+B/2A)2 -B2/4A. If the new variable X=x+B/2A is scaled to aX, then y(aX)=a2Y(X)+(a2-1)(B2/4A). In other words, the function y(x) is scale invariant (with scaling exponent 2) after finding the right scaling variable X and allowing for a scale-dependent shift of y.



critical phenomena



correlation at large distance

  • universality class and critical exponent
  • appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature.
  • the critical exponent describes the behavior of physical quantities around the critical temperature e.g. magnetization

\[M\sim (T_C-T)^{1/8}\]

  • magnetization and susceptibility can be written as correlation functions
  • large distance behavior of spin at criticality \(\eta=1/4\)

\[\langle \sigma_{i}\sigma_{i+n}\rangle =\frac{1}{|n|^{\eta}}\]

  • correlation length critivel exponent \(\nu=1\)

\[\langle \epsilon_{i}\epsilon_{i+n}\rangle=\frac{1}{|n|^{4-\frac{2}{\nu}}}\]


conformal transformations

  • roughly, local dilations
    • this is also equivalent to local scale invariance
  • correlation functions do not change under conformal transformations



related items

expositions

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'knot'}, {'LEMMA': 'theory'}]
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