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Real-Time Evolution in the Hubbard Model with Infinite Repulsion
by Elena Tartaglia, Pasquale Calabrese, and Bruno Bertini
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Submission summary
| Authors (as registered SciPost users): | Bruno Bertini |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202110_00006v1 (pdf) |
| Date submitted: | 2021-10-06 21:38 |
| Submitted by: | Bertini, Bruno |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider the real-time evolution of the Hubbard model in the limit of infinite coupling. In this limit the Hamiltonian of the system is mapped into a number-conserving quadratic form of spinless fermions, i.e. the tight binding model. The relevant local observables, however, do not transform well under this mapping and take very complicated expressions in terms of the spinless fermions. Here we show that for two classes of interesting observables the quench dynamics from product states in the occupation basis can be determined exactly in terms of correlations in the tight-binding model. In particular, we show that the time evolution of any function of the total density of particles is mapped directly into that of the same function of the density of spinless fermions in the tight-binding model. Moreover, we express the two-point functions of the spin-full fermions at any time after the quench in terms of correlations of the tight binding model. This sum is generically very complicated but we show that it leads to simple explicit expressions for the time evolution of the densities of the two separate species and the correlations between a point at the boundary and one in the bulk when evolving from the so called generalised nested Néel states.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-11-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202110_00006v1, delivered 2021-11-25, doi: 10.21468/SciPost.Report.3901
Strengths
- exact calculations
- all steps are well explained
Weaknesses
- lack of test for the results
- physical arguments lacking
Report
This is an interesting paper, where all calculations are clearly conducted on a professional level, and they are correct, but it would be nice to see some comparison with another approach (a numerical method for example). The main problem I see is the lack of a more global picture. We understand that solving the dynamics in this model is non-trivial due to the mapping which is non-local, but what do we learn from these calculations? For example, it would be intriguing to understand how to obtain the large time correlations after a quench: is there a GGE only expressed in terms of the free fermion correlations, or it is more complicated than that? And what about the thermodynamic limit of the overlap functions? And anyway, how the dynamic of infinite U-Hubbard then differ from a free fermion dynamics? It would be physically-relevant to compare the plots in the paper with a free fermion calculation to see this. The manuscript would greatly benefit from such discussions.
Requested changes
In order of priority
-- add some numerical check, if possible
-- discuss GGE and steady states after quenches
--compare the result with the one of a free fermion theory
Report #1 by Anonymous (Referee 1) on 2021-11-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202110_00006v1, delivered 2021-11-23, doi: 10.21468/SciPost.Report.3883
Report
In this work authors study the time evolution in the Hubbard model following the quantum quench. Authors focus on the infinite repulsion limit in which the model can be mapped into a non-interacting theory, through a Kumar's transformation. Whereas the resulting Hamiltonian is simple the relation between the operators of initial and final theories is more intricate and therefore the time evolution of expectation values is certainly worth exploring.
The results presented by the authors concern two classes of operators: (i) arbitrary analytic function of arbitrary number of local number operators and (ii) quadratic monomials of creation, annihilation operators. As initial states they consider kinds of Neel states. The discussion of the results from the physical perspective is minimal.
The results are interesting and the paper is well written. Before suggesting the publication I would like the authors to fix some small issues that I list below.
Requested changes
- In section 4.1 concerning the expectation values of total number of spinfull fermions the authors consider $C_{\Psi}$, eq. (4.15), which contains operators not of this form. Please comment.
- Could the the authors be more explicit with explaining the step leading from (4.48) to (4.50). Especially, please expand the phrase between (4.49) and (4.50).
- Does the $\in$ symbol (4.69) and (4.70) refer to the values of $K_i$? If so, it's better to split the definition of $K_i$ from their properties. Otherwise, please rewrite because the meaning is not clear.
Typos:
- in the text between (3.17) and (3.18): t should be J.
- is $N$ missing in $C_{\Psi_N}$ and $D_{\Psi_N}$ of eqs. (4.15)?
- no $N$ on the l.h.s of eqs. (4.16).
- constrains -> constraints, below (4.74).
- "decays very rapidly to zero", below eq. (4.87)
- double 'of the' in the conclusions.