열역학적 베테 가설 풀이(thermodynamic Bethe ansatz)

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개요

  • 산란행렬로부터 바닥상태의 에너지를 비섭동적으로 계산할 수 있는 방법


basic notions for particle scattering

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



limit

  • energy <math>E=m_{a}R\cosh \theta</math>
  • 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
  • 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:<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


관련된 항목들

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Spacy 패턴 목록

  • [{'LEMMA': 'rapidity'}]
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