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Topological Defects in Floquet Circuits

by Mao Tian Tan, Yifan Wang, Aditi Mitra

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Submission summary

Authors (as registered SciPost users): Aditi Mitra
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-02-28
Date submitted: 2023-11-02 22:19
Submitted by: Mitra, Aditi
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We introduce a Floquet circuit describing the driven Ising chain with topological defects. The corresponding gates include a defect that flips spins as well as the duality defect that explicitly implements the Kramers-Wannier duality transformation. The Floquet unitary evolution operator commutes with such defects, but the duality defect is not unitary, as it projects out half the states. We give two applications of these defects. One is to analyze the return amplitudes in the presence of "space-like" defects stretching around the system. We verify explicitly that the return amplitudes are in agreement with the fusion rules of the defects. The second application is to study unitary evolution in the presence of "time-like" defects that implement anti-periodic and duality-twisted boundary conditions. We show that a single unpaired localized Majorana zero mode appears in the latter case. We explicitly construct this operator, which acts as a symmetry of this Floquet circuit. We also present analytic expressions for the entanglement entropy after a single time step for a system of a few sites, for all of the above defect configurations.

Author comments upon resubmission

Dear Referees and Editor,

Thank you for considering our manuscript and for the many positive
comments. Below, we respond to all the questions raised by the Referees, and
list the changes made to the manuscript.

Response to Referee 1

Q1: The duality transformation is not unitary. Maybe I missed it, but it is not entirely clear how to implement the transformation or to insert the twisted boundary in the Floquet circuit. Is measurement a necessary part of the protocol? Does one need to post-select based on the measurement result?

Our Response:
Inserting a duality defect in the space-like direction is not
straightforward, and there are
several suggestions that involve implementing the duality defect by unitaries and measurements \cite{tantivasadakarn2021long}. In the first part of the paper, where we simulate the duality defect in the space-like direction, and demonstrate fusion rules, we implement the duality defect by following the rules of category theory, which in particular produce a restricted set of linear transformations on the Hilbert space, not-necessarily unitary. Explicitly, the duality defect can be defined by its action on a basis as seen in (10) and produces states living on the dual lattice and does not in principle require measurements although measurements could be used to implement it as explained in \cite{tantivasadakarn2021long}. In their scheme, post-selection is not required to implement the Kramers-Wannier transformation as they can simply apply Pauli X matrices on the dual lattice depending on the measurement outcome to ensure that the Kramers-Wannier duality transformation is implemented correctly.

For the second part of the paper, when we consider the duality twist, the rules of category theory give us a modified time-evolution operator, which is perfectly unitary and can be simulated by two-site unitaries (as was done in a follow-up study on the quantum computer
(see response to Referee 2 below).

There is no local unitary transformation that can transform the untwisted unitary to the twisted unitary, and whether they can be transformed into one another through measurements, is an interesting open question.

Q2:While introducing topological defects into Floquet systems induces interesting features, most of the added features seem to reproduce what one expects in the equilibrium version of the system. Is this the general expectation? Could topological defect in Floquet system have new properties than their equilibrium counterpart?

Our Response:
This is an interesting open question that we plan to explore in the future.

Q3: It is not entirely clear whether the features describes in this paper are stable against errors in the implementation of the circuit.

Our Response:
If there are errors, then indeed, the exact commutation relations of the defects will not be obeyed, and in the long time and large system size limit, the localized zero mode will become unstable. However, if the system size is small, then the localized Majorana can still be stable even at infinite times. This was shown in a follow up study where perturbing terms were included ( Two kinds of perturbing terms were considered, those that commute with the $Z_2$ symmetry, and those that did not. It was then shown that for the zero mode to decay, the chain has to act like an effective reservoir, which requires going to large system sizes.

Response to Referee 2

Q1: A section or subsection could be added to discuss how to realize the proposed Floquet circuits in experiments and how to detect the related topological properties. Specially, are there any suggestions for the realization of the duality-twisted boundary conditions and the isolated Majorana zero modes in any real experimental setups?

Our Response:
The defects can be realized in current noisy intermediate scale quantum devices. The duality twisted unitary, for example, was recently implemented in the following follow-up study (

Q2: One unique feature of the Floquet Ising-Majorana chain is to hold Majorana edge modes with the quasienergy pi. Is there any possibility to find a single unpaired localized Majorana pi mode in the Floquet circuits considered by the authors? Under which conditions could we obtain Floquet pi modes in Floquet circuits with topological defects? The authors are suggested to comment on these issues.

Our Response:
We note that it is not possible to realize an isolated $\pi$ mode in the setup presented in the manuscript. This is explained on page 25-26 of the manuscript. The reason is that the determinant of the unitary here has to be unity. Since the unitary is odd-dimensional, this rules out an isolated eigenvalue of $-1$ which would correspond to an isolated $\pi$ mode.

List of changes

1. Typos corrected.
2. References updated and added.
3. A silly error, where site indices where mis-labelled in Eq. 23(b), has been corrected.
4. On page 8, and at the end of the second paragraph, we point out that a brief discussion of $\pi$ modes will be presented later, so that the reader anticipates it.
5. In Section 3.2, in response to the first question of Referee-1, we give some more details on how the system was numerically simulated.
6. A discussion has been added in the second paragraph on Page 16, on the non-local nature of the transformation between the defectless unitary and the duality twisted unitary.
7. In the conclusions, we have added discussion on both experimental realization as well as robustness to perturbations.

Published as SciPost Phys. 16, 075 (2024)

Reports on this Submission

Anonymous Report 1 on 2024-2-26 (Invited Report)


The authors have addressed my concerns and improved their manuscript accordingly. I think the present version of the manuscript is suitable for publication in Scipost Physics.

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