"열역학적 베테 가설 풀이(thermodynamic Bethe ansatz)"의 두 판 사이의 차이

수학노트
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(새 문서: ==개요== * 산란행렬로부터 바닥상태의 에너지를 비섭동적으로 계산할 수 있는 방법 ==basic notions for particle scattering== * infinitely long cylinder...)
 
 
(같은 사용자의 중간 판 12개는 보이지 않습니다)
5번째 줄: 5번째 줄:
 
==basic notions for particle scattering==
 
==basic notions for particle scattering==
  
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*  infinitely long cylinder of radius <math>R</math><br>
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*  infinitely long cylinder of radius <math>R</math>
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*  N species of particles<br>
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*  N species of particles
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*  mass of particles <math>m_{a}, a=1,\cdots, N</math><br>
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*  mass of particles <math>m_{a}, a=1,\cdots, N</math>
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*  rapidity <math>\theta</math> (also called spectral parameter or wave number)<br>
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*  rapidity <math>\theta</math> (also called spectral parameter or wave number)
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**  a notion from relativity<br>
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**  a notion from relativity
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** http://en.wikipedia.org/wiki/Rapidity<br>
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** http://en.wikipedia.org/wiki/Rapidity
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*  energy <math>E=m_{a}R\cosh \theta</math><br>
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*  energy <math>E=m_{a}R\cosh \theta</math>
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*  momentum <math>p=m_{a}R\sinh \theta</math><br>
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*  momentum <math>p=m_{a}R\sinh \theta</math>
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*  energy-momentum vector <math>p^{\mu}=(E,P)</math><br>
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*  energy-momentum vector <math>p^{\mu}=(E,P)</math>
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*  S-matrix ([[factorizable scattering theory]])<br><math>S_{ab}(\theta)</math><br>
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*  산란행렬 S-matrix ([[factorizable scattering theory]]):<math>S_{ab}(\theta)</math>
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*  symmetric matrix kernel <br><math>\phi_{ab}(\theta)=-i\frac{d}{d\theta}\log S_{ab}(\theta)</math><br>
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*  symmetric matrix kernel :<math>\phi_{ab}(\theta)=-i\frac{d}{d\theta}\log S_{ab}(\theta)</math>
−
*  spectral density of particles <math>\epsilon_{a}(\theta)</math><br>
+
*  spectral density of particles <math>\epsilon_{a}(\theta)</math>
−
**  also called the pseudoenergy<br>
+
**  also called the pseudoenergy
−
*  Y-system <math>Y_{a}(\theta)=e^{-\epsilon_{a}(\theta)}</math> i.e. exponential of spectral density<br>
+
*  Y-system <math>Y_{a}(\theta)=e^{-\epsilon_{a}(\theta)}</math> i.e. exponential of spectral density
−
*  ground state energy <math>E(R)</math><br>
+
*  ground state energy <math>E(R)</math>
−
*  scaling function <math>c(R)</math> related to the central charge<br>
+
*  scaling function <math>c(R)</math> related to the central charge
−
*  TBA equation<br>
+
*  TBA equation
−
**  equation to find the spectral density functions <math>\epsilon_{a}(\theta)</math><br>
+
**  equation to find the spectral density functions <math>\epsilon_{a}(\theta)</math>
−
*  UV limit<br>
+
*  UV limit
−
**  plateau behaviour<br>
+
**  plateau behaviour
−
** <math>\epsilon_{a}(\theta)</math> becomes constant in a large region for <math>\theta</math> when r(inverse temperature) is small<br>
+
** <math>\epsilon_{a}(\theta)</math> becomes constant in a large region for <math>\theta</math> when r(inverse temperature) is small
−
*  IR limit<br>
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*  IR limit
  
 
   
 
   
33번째 줄: 33번째 줄:
 
==limit==
 
==limit==
  
−
*  energy <math>E=m_{a}R\cosh \theta</math><br>
+
*  energy <math>E=m_{a}R\cosh \theta</math>
−
*  momentum <math>p=m_{a}R\sinh \theta</math><br>
+
*  momentum <math>p=m_{a}R\sinh \theta</math>
−
*  in the CFT limit, we regard \theta \to \infty for right movers and -\infty for left movers<br>
+
*  in the CFT limit, we regard \theta \to \infty for right movers and -\infty for left movers
−
*  Thus we get, E=p and E=-p respectively in CFT limit<br>
+
*  Thus we get, E=p and E=-p respectively in CFT limit
  
 
   
 
   
44번째 줄: 44번째 줄:
 
==TBA equation==
 
==TBA equation==
  
−
*  a system which interacts dynamically via the scattering matrix and statistically via Fermi statistics<br><math>rm_{a}^{i}\cosh\theta = \epsilon_{a}^{i}(\theta)+\sum_{b=1}^{l}\sum_{j=1}^{\tilde{l}}\int_{-\infty}^{\infty} d\theta' \phi_{ab}^{ij}(\theta-\theta')\ln (1+e^{-\epsilon_{b}^{j}(\theta')})</math><br> where r is the inverse temperature and <math>m_{a}^{i}</math> the mass of particle (a,i)<br>
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*  a system which interacts dynamically via the scattering matrix and statistically via Fermi statistics:<math>Rm_{a}^{i}\cosh\theta = \epsilon_{a}^{i}(\theta)+\frac{1}{2\pi}\sum_{b=1}^{l}\sum_{j=1}^{\tilde{l}}\int_{-\infty}^{\infty} d\theta' \phi_{ab}^{ij}(\theta-\theta')\ln (1+e^{-\epsilon_{b}^{j}(\theta')})</math> where <math>R=T^{-1}</math> is the inverse temperature and <math>m_{a}^{i}</math> the mass of particle (a,i)
 +
 
 +
 
 +
==예 : Yang-Lee 모형==
 +
* 1 particle
 +
* 산란행렬
 +
:<math>
 +
S_{11}(\theta)=\tanh \left(\frac{1}{2} \left(\theta -\frac{2 i \pi }{3}\right)\right) \coth \left(\frac{1}{2} \left(\theta +\frac{2 i \pi }{3}\right)\right)
 +
</math>
 +
* 커널
 +
:<math>
 +
\phi_{11}(\theta)=-i\frac{d}{d\theta}\log S_{11}(\theta)=\sqrt{3} \left(\frac{1}{2 \cosh (\theta )+1}+\frac{1}{2 \cosh (\theta )-1}\right)
 +
</math>
 +
*
 +
:<math>
 +
N=\frac{1}{2\pi}\int_{-\infty}^{\infty}\phi_{11}(\theta)=1
 +
</math>
 +
 
 +
 
 +
 
 +
==계산 리소스==
 +
* http://msstp.org/?q=node/277
 +
** [http://msstp.org/sites/default/files/ex4_tba.pdf Romuald Janik, TBA Integral Equations exercise.pdf]
 +
** [http://msstp.org/sites/default/files/tba_answer.nb Romuald Janik, TBA Integral Equations solution.nb]
 +
 
 +
 
 +
==관련된 항목들==
 +
* [[대수적 베테 가설 풀이(algebraic Bethe ansatz)]]
 +
 
 +
 
 +
[[분류:통계물리]]
 +
[[분류:통계물리]]
 +
 
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q1366833 Q1366833]
 +
===Spacy 패턴 목록===
 +
* [{'LEMMA': 'rapidity'}]

2021년 2월 17일 (수) 03:30 기준 최신판

개요

  • 산란행렬로부터 바닥상태의 에너지를 비섭동적으로 계산할 수 있는 방법


basic notions for particle scattering

  • infinitely long cylinder of radius \(R\)
  • N species of particles
  • mass of particles \(m_{a}, a=1,\cdots, N\)
  • rapidity \(\theta\) (also called spectral parameter or wave number)
  • energy \(E=m_{a}R\cosh \theta\)
  • momentum \(p=m_{a}R\sinh \theta\)
  • energy-momentum vector \(p^{\mu}=(E,P)\)
  • 산란행렬 S-matrix (factorizable scattering theory)\[S_{ab}(\theta)\]
  • symmetric matrix kernel \[\phi_{ab}(\theta)=-i\frac{d}{d\theta}\log S_{ab}(\theta)\]
  • spectral density of particles \(\epsilon_{a}(\theta)\)
    • also called the pseudoenergy
  • Y-system \(Y_{a}(\theta)=e^{-\epsilon_{a}(\theta)}\) i.e. exponential of spectral density
  • ground state energy \(E(R)\)
  • scaling function \(c(R)\) related to the central charge
  • TBA equation
    • equation to find the spectral density functions \(\epsilon_{a}(\theta)\)
  • UV limit
    • plateau behaviour
    • \(\epsilon_{a}(\theta)\) becomes constant in a large region for \(\theta\) when r(inverse temperature) is small
  • IR limit



limit

  • energy \(E=m_{a}R\cosh \theta\)
  • momentum \(p=m_{a}R\sinh \theta\)
  • in the CFT limit, we regard \theta \to \infty for right movers and -\infty for left movers
  • Thus we get, E=p and E=-p respectively in CFT limit



TBA equation

  • a system which interacts dynamically via the scattering matrix and statistically via Fermi statistics\[Rm_{a}^{i}\cosh\theta = \epsilon_{a}^{i}(\theta)+\frac{1}{2\pi}\sum_{b=1}^{l}\sum_{j=1}^{\tilde{l}}\int_{-\infty}^{\infty} d\theta' \phi_{ab}^{ij}(\theta-\theta')\ln (1+e^{-\epsilon_{b}^{j}(\theta')})\] where \(R=T^{-1}\) is the inverse temperature and \(m_{a}^{i}\) the mass of particle (a,i)


예 : Yang-Lee 모형

  • 1 particle
  • 산란행렬

\[ S_{11}(\theta)=\tanh \left(\frac{1}{2} \left(\theta -\frac{2 i \pi }{3}\right)\right) \coth \left(\frac{1}{2} \left(\theta +\frac{2 i \pi }{3}\right)\right) \]

  • 커널

\[ \phi_{11}(\theta)=-i\frac{d}{d\theta}\log S_{11}(\theta)=\sqrt{3} \left(\frac{1}{2 \cosh (\theta )+1}+\frac{1}{2 \cosh (\theta )-1}\right) \]

\[ N=\frac{1}{2\pi}\int_{-\infty}^{\infty}\phi_{11}(\theta)=1 \]


계산 리소스


관련된 항목들

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LEMMA': 'rapidity'}]
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