# Building temperature into the Hamiltonian

Given a quantum Hamiltonian $$H$$ (e.g. the quantum Ising Hamiltonian $$H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$$): we know that at temperature $$T$$, the system is in the state:

$$\rho(T) = e^{-H/T}.$$

It is clear $$\rho(0)=|0\rangle\langle0|$$, where $$|0\rangle$$ is the eigenvector of $$H$$ with smallest eigenvalue.

In other words, "$$\rho(0)$$ minimises $$\text{tr}(H\rho)$$".

Now, I'd like to make analogous statements to the above for $$\rho(T)$$ when $$T>0$$, i.e. I'd like to say that $$\rho(T)$$ minimises $$\text{tr}(\tilde{H}\rho)$$ for some $$\tilde{H}$$ which depends on $$T$$. My question is what is the $$\tilde{H}$$? (Or maybe first, does $$\tilde{H}$$ exist?) I think that the $$\tilde{H}$$ should be some "quantum version" of the free energy, but don't know what it should be exactly.

Thanks!

I don't think this is a well-defined question. If you put $$\tilde{H} = H e^{H T}$$ then $$\text{tr} \, \tilde{H} \rho(T) = \text{tr}\, H \rho(0).$$ The operator $$\tilde{H}$$ is not really well-defined, in the sense that for $$T>0$$ its eigenvalues grow very rapidly at infinity.
Okay, so I think this is correct, writing $$D$$ for the convex set of density operators and $$S$$ for the entropy: $$\rho(T) = \text{argmin}_{\rho \in \text{D}} [\text{tr}(\rho H)-TS(\rho)] = \text{argmin}_{\rho \in \text{D}} [\text{tr}(\rho (H+T\log \rho))].$$ The latter convex minimisation can be done via Lagrange multipliers giving $$\rho(T)$$ as desired. So if I were to invent some notation, I'd say something like: $$\tilde{H}=H+T\log ( \cdot ).$$