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 \(R\)
  • 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

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

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


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Spacy 패턴 목록

  • [{'LEMMA': 'rapidity'}]
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