# Real-Time Evolution in the Hubbard Model with Infinite Repulsion

### Submission summary

 As Contributors: Bruno Bertini Preprint link: scipost_202110_00006v2 Date accepted: 2021-12-17 Date submitted: 2021-11-29 12:59 Submitted by: Bertini, Bruno Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics 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:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### Author comments upon resubmission

We thank both referees for their careful reading of our manuscript, for their relevant comments, and for their overall positive assessment. We have made a number of modifications to the manuscript to accomodate their comments. To help identifying the changes we highlighted them in red in the new version.

Response to Referee 1

We thank the referee for finding our paper interesting and clear and for providing us with constructive feedback. In the following we proceed to address the main criticisms of the referee, i.e.,

1) lack of testing for the results" 2)lack of physical arguments".

in a separate fashion.

Concerning the first point, even though we understand the reasoning the referee, we disagree with their conclusions and accordingly we decided not to add numerical tests. This for two main reasons. First, a philosophical one: our results are exact, there is no approximation that one should check. A numerical check could merely prove that we did not make mistakes in our algebraic manipulations (which the referee confirmed are sound). Second, obtaining our results numerically is very hard and can be done only for short times. Indeed, even the most advanced numerical techniques available to stimulate non-equilibrium dynamics for quantum many-body systems in 1d, i.e. those based on tensor-network techniques, are severely hampered by the growth of entanglement and are then limited to short times. In particular, for the problem at hand we will not be able to reach times larger than $20J$ (possibly less). This is actually one of the main points of interest of our paper and to highlight it we decided to slightly modify the discussion in the introduction.

Concerning the second point we partially agree with the spirit of the comment and we accordingly expanded some of the discussions making more obvious the comparison with the free-fermion results (which, as a matter of fact, was already present). In particular we expanded the discussions after Eqs. (4.12), (4.88), (4.110). Let us point out, however, that this paper is only focussed in the finite-time dynamics, not on the infinite time limit. A thorough description of the stationary state and its properties is presented in our previous paper on the topic, i.e., Ref. [77].

Response to Referee 2

We sincerely thank the referee for their very thorough reading of our manuscript, for their positive assessment, and for their useful comments and suggestions. Below we address them point-by-point

"- 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."

The referee is right. However $C_{\Psi_N}(x,x,t)$, which is the one used in Eq.~(4.14), is indeed of the right form. We added a brief comment in the new version.

" - 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).} "

In that step we used (4.49) and the fact that $[f^{(\dag)}_x,U^{(\dag)}_y]=0$ for $x\neq y$ to replace $U^\dag_x$ and $U^\dag_y$ on the left of $f_x^\dag f_y$ with ${\cal X}^\dag_x$ and $I$ respectively. We also replaced $U_x$ and $U_y$ on the right with $I$ and ${\cal X}_y$. This gives the first line of (4.50). Analogous reasonings give the second line. In the new version we expanded the explanation as requested.

"- 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."

Yes, it does. We agree with the referee: in the new version we performed the requested split.

"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 $\rightarrow$ constraints, below (4.74). - decays very rapidly to zero", below eq. (4.87) - double 'of the' in the conclusions."

We thank the referee for signalling these typos. They are all fixed in the new version.

### Submission & Refereeing History

Resubmission scipost_202110_00006v2 on 29 November 2021
Submission scipost_202110_00006v1 on 6 October 2021

## Reports on this Submission

### Report

The authors addressed my concerns.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -