열역학적 베테 가설 풀이(thermodynamic Bethe ansatz)

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)
  
  • a notion from relativity
    
  • http://en.wikipedia.org/wiki/Rapidity
    
• 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\[rm_{a}^{i}\cosh\theta = \epsilon_{a}^{i}(\theta)+\sum_{b=1}^{l}\sum_{j=1}^{\tilde{l}}\int_{-\infty}^{\infty} d\theta' \phi_{ab}^{ij}(\theta-\theta')\ln (1+e^{-\epsilon_{b}^{j}(\theta')})\]
  where r 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

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

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

관련된 항목들

• 대수적 베테 가설 풀이(algebraic Bethe ansatz)