# How to calculate the ground state of Ising model at non-zero temperature

I'm studying the quantum Ising model, i.e. with Hamiltonian $$H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$$.

I know conceptually how to compute the ground state of the Ising model at zero temperature - just find the lowest eigenvalue $$E$$ and eigenstate $$|\psi\rangle$$ of $$H$$. This is equivalent to minimising $$\langle \psi|H|\psi\rangle$$ over states $$|\psi\rangle$$.

But I don't know how to do the same computation when we have non-zero temperature, I think it has something to do with minimising the free energy $$F=\langle E \rangle - TS$$, where think $$\langle E \rangle$$ is just $$\langle \psi|H|\psi\rangle$$, however looking at the answers on here, it would seem that the entropy $$S$$ of any pure state $$| \psi \rangle$$ is zero so if I were to minimise $$F$$, it's just the same as minimising the energy. But I thought it should be different, e.g. at very high termperatures shouldn't the "ground state" be a more uniform superposition over the spin configurations.

I feel like I'm missing something here and may be using wrong terminology for things. I would really appreciate a practical method of finding the ground state (and energy) at non-zero temperature.

A system is not usually in its ground state at finite temperature. Indeed the notion of state'' is not really applicable unless you use a microcanonical distribution. In that case the system us in whatever excited energy state that you chose to give it. A microcanonical system only has a "temperature" if the system/state satisfies the "eigenstate thermalization" condition.
For the canonocal distribution, on the other hand, you have a temperature but must describe the system by a density matrix $$\rho=e^{-\beta \hat H}$$ rather than a pure state.
• @Daochen The "eigenstate thermalization hypothesis" is topic of current research. If true for a particular system/state, it means that the finite, isolated, quantum system in an excited state of energy $E$ has the same local properties as a the same system coupled to a heat bath at the temperature that gives the system the same average energy $<E>$. It's a the quantum analogue of the ergodicity that makes a microcanonical ensemble equivalent to a canonical one. For more details Google "eigenstate thermalization hypothesis." Commented Nov 25, 2018 at 13:19