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Transient superconductivity in threedimensional Hubbard systems by combining matrix product states and selfconsistent meanfield theory
by Svenja Marten, Gunnar Bollmark, Thomas Köhler, Salvatore R. Manmana, Adrian Kantian
Submission summary
Authors (as Contributors):  Thomas Köhler 
Submission information  

Preprint link:  scipost_202208_00016v1 
Date submitted:  20220808 16:22 
Submitted by:  Köhler, Thomas 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We combine matrixproductstate (MPS) and meanfield (MF) methods to model the realtime evolution of a threedimensional (3D) extended Hubbard system formed from onedimensional (1D) chains arrayed in parallel with weak coupling inbetween them. This approach allows us to treat much larger 3D systems of correlated fermions outofequilibrium over a much more extended realtime domain than previous numerical approaches. We deploy this technique to study the evolution of the system as its parameters are tuned from a chargedensity wave phase into the superconducting regime, which allows us to investigate the formation of transient nonequilibrium superconductivity. In our ansatz, we use MPS solutions for chains as input for a selfconsistent timedependent MF scheme. In this way, the 3D problem is mapped onto an effective 1D Hamiltonian that allows us to use the MPS efficiently to perform the time evolution, and to measure the BCS order parameter as a function of time. Our results confirm previous findings for purely 1D systems that for such a scenario a transient superconducting state can occur.
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Submission & Refereeing History
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Reports on this Submission
Anonymous Report 2 on 20221223 (Invited Report)
Strengths
 Novel approach to nonequilibrium simulation embedded in a mean field
 Clearly written and presented
Weaknesses
 Lacks benchmarks to allow judgement of the trustworthiness
 Control of the results not convincing
Report
The authors present the extension of the approach of embedding 1D MPS simulations in a 3D mean field system to the nonequilibrium domain. The main advantage is that the approach allows to observe/simulate larger systems for longer realtime in an approximation best suited for highly anisotropic 3D systems. I think the manuscript/method presents a new approach with potential for more followup work in in and outofequilibrium systems and as such should be considered for publication, however only after convincingly addressing the aspects below:
1) The work aims to test and benchmark the method itself. Unfortunately there is no comparison against any other method. As such it is hard judge, whether and how far one should trust the approach. While I understand that the comparison is hard due to the limited capability of other methods due to the dimensionality, I ask the authors to elaborate on this, test limiting cases, or at the very least propose potential routes to do so.
2) I think the the notion of 'dynamically induced' is not appropriate in the context of a MF approximation, as the symmetry has to be broken explicitly to allow for finite order parameters (\alpha_{ini} must be finite).
3) The density is assumed to be stable and the chemical potential is not determined selfconsistently. However (1\rho), while initially on a small scale, diverges rapidly beyond t~40, in particular for larger system sizes. This raises the question if one can trust any results beyond t~40. Figures 5 and 6 go well beyond that time scale, without showing the deviation of \rho from 1.
4) The dependence of the time evolution on \alpha_{ini} appears to be more influential than just being a scale, i.e., the evolutions for different \alpha_{ini} cannot be rescaled to collapse onto each other (c.f. Fig. 5). Selfconsistent solutions in nonequilibrium MF do/should not depend on the initial MF parameters at the beginning of the selfconsistency cycles. Here the situation seems to very different, especially beyond (again) the times beyond t~40. How helpful is the method if the results (time evolution) so heavily depends on \alpha_{ini}?
5) What are the used thresholds \varepsilon in the selfconsistency cycle?
Anonymous Report 1 on 20221024 (Invited Report)
Strengths
 Novel approach to simulating the Hubbard model in nonequilibrium
 technically sound and explained in proper detail
Weaknesses
 not clear enough on possible applications of the method and relation to experiments
Report
The authors propose a combined meanfield and matrix product state ansatz to simulate the timeevolution of an extended threedimensional Hubbard model. The authors consider an attractive Hubbard interaction U at halffilling, where the equilibrium physics has already been intensely investigated. The manuscript offers a window into the dynamical properties of this system, and, hence, is highly relevant in the face of recent experiments on nonequilibrium superconductivity. The proposed simulation technique appears to be novel, and the technical skill of the authors allows them to perform the required simulations in a convincing manner. The authors convince especially with their accurate report of the convergence properties of their technique, something which is too often swept under the rug. A strong point of the manuscript is the selfcontained and comprehensive explanation of the method.
However, I think there are various aspects that the authors need to address in further detail, before the manuscript can be considered for publication.
1) It should be clarified, how these simulations are relevant to experiments. Is the considered quench something that could be implemented in a solidstate environment, or in a putative quantum optics experiment? What sorts of experiments could be simulated with this technique? This is an important point and should be discussed in thorough detail to convince the reader that the proposed method is useful.
2) A discussion whether different nonequilibrium processes, e.g. some Floquet dynamics, could similarly be simulated with their method is missing. Would the method also be applicable in this case?
3) The authors should discuss in detail the potential shortcomings of the meanfield approximation. Since meanfield theory typically overestimates order, the authors should comment how this could alter their results in the context of these nonequilibrium simulations. A comparison to meanfield approaches vs. morerigorous approaches in equilibrium might be a good guidance for this.
4) The figure captions are often a bit short require to go through the main text to understand the plot. As a reader, I would prefer to have more selfcontained figure captions, to quickly grasp the contents of the manuscript.
I also have some minor comments:
a) please explain in more detail what is meant by the term "transient" superconductivity
b) On page 5 leading up to Eq. 7, it is not clear what is meant by E_{i,\alpha}. Please clarify this.