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

Question

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?

Answer 1

• 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.