"Talk on 'introduction to conformal field theory(CFT)'"의 두 판 사이의 차이
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2020년 11월 13일 (금) 03:52 판
introduction
- scaling and power law
- scale invariance and conformal invariance
- critical phenomena
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
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