A question on OQuPy spinboson symmetry-broken steady states

[yuchihe]

 I have a question regarding the "quickstart" code of spin boson model,  and will appreciate if you could have a look at my issue:

Basically, I just run the code https://oqupy.readthedocs.io/en/latest/pages/tutorials/quickstart.html , but adjust the parameter for the symmetry-broken phase alpha=1.5 [ref.Nat. Comm. 9, 3322 Fig2(a)]. After I increased the accuracy by adjusting epsrel=10*(-7), I was able to get the "steady" non-zero Sz as Fig2(a). Then I consider that the steady state (|sz|) should not depend on the initial state; to test this, I do the following modification, initial_state=0.9up_density_matrix+0.1*down_density_matrix. It turns out that this leads to a "steady" state with a smaller |sz|. My question is that is my expectation wrong or have I made some mistakes using the code.

 Edit:  clearly a steady state can be any mixtures of two other steady states, so an accurate algorithm should rely on the initial states. But from another point of view,  the  two maximally ordered states should be special because other states are cat states of them and could be absent because of environmental [in real life] or numerical noise [in Monte Carlo and closed-system DMRG]. It seems to me that in the current setting, the "collapse" to non-cat states does not happen. 

[gefux]

Hi @yuchihe, I would not expect that the steady state is independent of the initial state in the localised phase. Indeed, as I understand it: this is what makes the phase "localised". In the localised phase the effective tunneling between the two states vanishes, which means that the steady state occupation you see corresponds to the overlap of the initial state with the two polaron-type states. I attach a quote from [Vojita, page: 1828]. I am not an expert on the phase transitions of the spin boson model, but this (and its references) is where I'd start reading to learn more.
[Vojta] ... Vojta, Matthias. ‘Impurity Quantum Phase Transitions’. Philosophical Magazine 86, no. 13–14 (1 May 2006): 1807–46. https://doi.org/10.1080/14786430500070396.

[gefux]

Concerning your edit: The way I currently understand the physics of the localised phase (comments are welcome) is that there are two polaron type (hybridisation between the spin and bosons) eigen states, and the initial state may have a finite overlap with both of them. Because these polaron states are eigen states of the entire (system+environment) there is no reason why they should dephase with respect to each other, so the steady state may not just be a mixture, but indeed a superposition of those two polaron states. If one adds additional physical dephasing (by for example adding Lindbladian \sigma_z terms) this would create a mixture rather than a superposition, but the |\sigma_z| expectation would stay unchanged (and thus also dependent on the initial state). TEMPO is a nummerically exact method, so should thus ideally reflect all this. I would expect that the nummerical imperfections may lead to a slightly subnormalised state, or to some finite tunneling rate. Both should vanish with increasing precision. I would not expect that the nummerical imperfections would lead to something as drastic as a pinning to one of the possible steady states independently of the initial state as you seem to expect it in your comment above.

Does this help? @others: Feel free to weigh in!

[gefux]

I hope my reply above was helpful. I am closing this now. Feel free to reopen to continue the conversation!