# 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

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

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

## 메타데이터

### Spacy 패턴 목록

• [{'LEMMA': 'rapidity'}]