"Talk on 'introduction to conformal field theory(CFT)'"의 두 판 사이의 차이
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In this sense, '''the thermodynamic functions seem to display scale invariance around the critical point (for systems that have a critical point!), with the scaling variable t=(1-T/Tc).''' Note that the logarithm y=log x obeys y(ax)=y(x) + log a. It is scale invariant with exponent 0 (and a scale-dependent shift.) This is related to the famous formula<br> limp-->0 (x^p-1)/p = log x<br> which shows that logs are a special case of power law functions with power 0. | In this sense, '''the thermodynamic functions seem to display scale invariance around the critical point (for systems that have a critical point!), with the scaling variable t=(1-T/Tc).''' Note that the logarithm y=log x obeys y(ax)=y(x) + log a. It is scale invariant with exponent 0 (and a scale-dependent shift.) This is related to the famous formula<br> limp-->0 (x^p-1)/p = log x<br> which shows that logs are a special case of power law functions with power 0. | ||
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| + | * [[basics of magnetism]] | ||
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<h5>correlation at large distance</h5> | <h5>correlation at large distance</h5> | ||
| − | appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. | + | * [[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. | |
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* the critical exponent describes the behavior of physical quantities around the critical temperature<br> e.g. magnetization <math>M\sim (T-T_C)^{1/8}</math><br> | * the critical exponent describes the behavior of physical quantities around the critical temperature<br> e.g. magnetization <math>M\sim (T-T_C)^{1/8}</math><br> | ||
* magnetization and susceptibility can be written as '''correlation functions''' | * 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><br><math><\sigma_{i}\sigma_{i+n}>=\frac{1}{|n|^{\eta}}</math><br> | * large distance behavior of spin at criticality <math>\eta=1/4</math><br><math><\sigma_{i}\sigma_{i+n}>=\frac{1}{|n|^{\eta}}</math><br> | ||
* correlation length critivel exponent <math>\nu=1</math><br><math><\epsilon_{i}\epsilon_{i+n}>=\frac{1}{|n|^{4-\frac{2}{\nu}}}</math><br> | * correlation length critivel exponent <math>\nu=1</math><br><math><\epsilon_{i}\epsilon_{i+n}>=\frac{1}{|n|^{4-\frac{2}{\nu}}}</math><br> | ||
2011년 1월 26일 (수) 12:51 판
introduction
- scaling and power law
- scale invariance and conformal invariance
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
In this sense, the thermodynamic functions seem to display scale invariance around the critical point (for systems that have a critical point!), with the scaling variable t=(1-T/Tc). Note that the logarithm y=log x obeys y(ax)=y(x) + log a. It is scale invariant with exponent 0 (and a scale-dependent shift.) This is related to the famous formula
limp-->0 (x^p-1)/p = log x
which shows that logs are a special case of power law functions with power 0.
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 \(M\sim (T-T_C)^{1/8}\) - magnetization and susceptibility can be written as correlation functions
- large distance behavior of spin at criticality \(\eta=1/4\)
\(<\sigma_{i}\sigma_{i+n}>=\frac{1}{|n|^{\eta}}\) - correlation length critivel exponent \(\nu=1\)
\(<\epsilon_{i}\epsilon_{i+n}>=\frac{1}{|n|^{4-\frac{2}{\nu}}}\)
conformal transformations
- roughly, local dilations
- this is also equivalent to local scale invariance
- correlation functions do not change under conformal transformations
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
expositions
articles
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
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experts on the field