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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.

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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.

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  • $\begingroup$ could you say a bit more about what the "eigenstate thermalization" condition is please? $\endgroup$ – dcw Nov 25 '18 at 0:29
  • $\begingroup$ @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." $\endgroup$ – mike stone Nov 25 '18 at 13:19
  • $\begingroup$ Hi Mike, I wonder if you could please comment on my new related question physics.stackexchange.com/questions/444366/…? Thanks a lot :) $\endgroup$ – dcw Dec 1 '18 at 0:19

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