# What's the relation between zero temperature and ground state of interacting many body system?

In the famous monograph "Many body physics" by Mahan there is a statement about the corresponding relation between the zero temperature and the ground state:

"Furthermore,the zero temperature property of a system is an important conceptual quantity--the ground state of an interacting system."

As far as I know the ground state represents the lowest eigenenergy for our system obtained by solving Schrodinger equation and hence is a description based on the microscopic viewpoint. However the temperature is a macroscopic parameter for describing our system. How can I understand this statement? What's the relation between both?

• I would understand this as the limit of boltzmann factors, giving that at vanishing temperature only the ground state is occupied. Thus, if you know the zero temperature behavior, you know something about the ground state
– Bort
Commented Nov 24, 2016 at 15:13

## 1 Answer

• The eigenequation of Hamitonian $H$: $$\hat{H}|\phi_n\rangle = E_n|\phi_n\rangle \qquad (n=0,1,2,\cdots) \tag{1}$$
• The spectrum decomposition of Hamiltonian: $$\hat{H}=\sum_nE_n|\phi_n\rangle\langle\phi_n| \tag{2}$$
• Equilibrium density matrix: $$\hat{\rho} \equiv \dfrac{e^{-\beta \hat{H}}}{Z}=\sum_n \dfrac{e^{-\beta E_n}}{Z}|\phi_n\rangle\langle \phi_n| = \dfrac{\sum_{n}e^{-\beta E_n}|\phi_n\rangle\langle\phi_n|}{\sum_{n}e^{-\beta E_n}} \tag{3}$$ where $\beta=\dfrac{1}{k_B T}$ is the inverse temperature and we have used the fact $\hat{H}$ commutes with $\hat{\rho}$ in equilibrium.
• The zero tmeperature limit $T \rightarrow 0$ or $\beta\rightarrow\infty$: $$\lim_{\beta\rightarrow\infty}\hat{\rho} = \lim_{\beta\rightarrow\infty} \dfrac{\sum_{n}e^{-\beta E_n}|\phi_n\rangle\langle\phi_n|}{\sum_{n}e^{-\beta E_n}} = \lim_{\beta\rightarrow\infty} \dfrac{\sum_{n}e^{-\beta (E_n-E_0)}|\phi_n\rangle\langle\phi_n|}{\sum_{n}e^{-\beta (E_n-E_0)}} = |\phi_0\rangle\langle\phi_0| \tag{4}$$ For Hamiltoian $\hat{H}$ with degenerate ground states $(4)$ reduces instead to an equally weighted ensemble of degenerate ground states.