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arXiv:2304.05488 · Physical review. B./Physical review. B

Nonintegrable Floquet Ising model with duality twisted boundary conditions

Aditi Mitra, Hsiu-Chung Yeh, Fei Yan, Achim Rosch

Published Jun 14, 2023• 8 citations
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Abstract

Results are presented for a Floquet Ising chain with duality twisted boundary conditions, taking into account the role of weak integrability breaking in the form of four-fermion interactions. In the integrable case, a single isolated Majorana zero mode exists which is a symmetry in the sense that it commutes both with the Floquet unitary and the ${Z}_{2}$ symmetry of the Floquet unitary. When integrability is weakly broken, both in a manner so as to preserve or break the ${Z}_{2}$ symmetry, the Majorana zero mode is still found to be conserved for small system sizes. This is reflected in the dynamics of an infinite-temperature autocorrelation function which, after an initial transient that is controlled by the strength of the integrability breaking term, approaches a plateau that does not decay with time. The height of the plateau agrees with a numerically constructed conserved quantity and is found to decrease with increasing system sizes. It is argued that the existence of the plateau and its vanishing for larger system sizes is closely related to a localization-delocalization transition in Fock space triggered by the integrability-breaking interactions.

DOI
10.1103/physrevb.107.245416

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Results are presented for a Floquet Ising chain with duality twisted boundary conditions, taking into account the role of weak integrability breaking in the form of four-fermion interactions. In the integrable case, a single isolated Majorana zero mode exists which is a symmetry in the sense that it commutes both with the Floquet unitary and the Z2{Z}_{2}Z2​ symmetry of the Floquet unitary. When integrability is weakly broken, both in a manner so as to preserve or break the Z2{Z}_{2}Z2​ symmetry, the Majorana zero mode is still found to be conserved for small system sizes. This is reflected in the dynamics of an infinite-temperature autocorrelation function which, after an initial transient that is controlled by the strength of the integrability breaking term, approaches a plateau that does not decay with time. The height of the plateau agrees with a numerically constructed conserved quantity and is found to decrease with increasing system sizes. It is argued that the existence of the plateau and its vanishing for larger system sizes is closely related to a localization-delocalization transition in Fock space triggered by the integrability-breaking interactions.
The stability and dynamics of almost strong zero and $\ensuremath{\pi}$ modes in weakly nonintegrable Floquet spin chains are investigated. Such modes can also be viewed as localized Majorana modes at the edge of a topological superconductor. Perturbation theory in the strength of integrability breaking interaction Jz{J}_{z}Jz​ is employed to estimate the decay rates of these modes, and compared to decay rates obtained from exact diagonalization. The structure of the perturbation theory and thus the lifetime of the modes is governed by the conservation of quasienergy modulo $2\ensuremath{\pi}/T$, where TTT is the period of the Floquet system. If there are $4n\ensuremath{-}1$ bulk excitations whose quasienergies add up to zero (or $\ensuremath{\pi}/T$ for a $\ensuremath{\pi}$ mode), one obtains a decay channel of the Majorana mode with a decay rate proportional to Jz2n{J}_{z}^{2n}Jz2n​. Thus the lifetime is sensitively controlled by the width of the single-particle Floquet bands. For regimes where the decay rates are quadratic in Jz{J}_{z}Jz​, an analytic expression for the decay rate in terms of an infinite temperature autocorrelation function of the integrable model is derived, and shown to agree well with exact diagonalization.
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.
Physical quantities with long lifetimes have both theoretical significance in the study of quantum many-body systems and practical implications for quantum technologies. In this manuscript, we investigate the roles played by topological defects in the construction of quasiconserved quantities, using as a prototypical example the Kramers-Wannier duality defect in a deformed one-dimensional quantum transverse-field Ising model. We construct the duality defect Hamiltonian in three different ways: half-chain Kramers-Wannier transformation, utilization of techniques in the Ising fusion category, and defect-modified weak integrability breaking deformation. The third method is also applicable for the study of generic integrable defects under weak integrability breaking deformations. We also work out the deformation of defect-modified higher charges in the model and study their slower decay behavior. Furthermore, we consider the corresponding duality defect twisted deformed Floquet transverse-field Ising model and investigate the stability of the isolated zero mode associated with the duality defect in the integrable Floquet Ising model, under such weak integrability breaking deformation.
We study the robustness of the Floquet quantum Ising model against integrability-breaking perturbations, focusing on the phase hosting both Majorana zero and πππ modes. A recent work [Phys. Rev. B 110, 075117, (2024)] observed that the Floquet product mode, a composite edge mode constructed from both Majorana operators, is considerably more robust than the individual Majorana edge modes. We analyze these strong modes from the point of view of the eigenphase order present in finite chains with open boundary conditions. As a result of the Majorana modes, all Floquet eigenstates come in quadruplets in the integrable limit. We show that the robustness of the various modes as well as the behavior of the boundary spin correlation functions can be understood in terms of the spectral statistics of these quadruplets in the presence of integrability-breaking perturbations.
Investigating the interplay of dualities, generalized symmetries, and topological defects beyond theoretical models is an important challenge in condensed matter physics and quantum materials. A simple model exhibiting this physics is the transverse-field Ising model, which can host a topological defect that performs the Kramers-Wannier duality transformation. When acting on one point in space, this duality defect imposes the duality twisted boundary condition and binds a single zero mode. This zero mode is unusual as it lacks a localized partner in the same Z 2 sector and has an infinite lifetime, even in finite systems. Using Floquet driving of a closed Ising chain with a duality defect, we generate this zero mode in a digital quantum computer. We detect the mode by measuring its associated persistent autocorrelation function using an efficient sampling protocol and a compound strategy for error mitigation. We also show that the zero mode resides at the domain wall between two regions related by a Kramers-Wannier duality transformation. Finally, we highlight the robustness of the isolated zero mode to integrability- and symmetry-breaking perturbations. Our findings provide a method for exploring exotic topological defects, associated with noninvertible generalized symmetries, in digitized quantum devices.
Integrable trotterization} provides a method to evolve a continuous time integrable many-body system in discrete time, such that it retains its conserved quantities. Here we explicitly show that the first order trotterization of the critical {\it transverse field Ising model} is integrable. The discrete time conserved quantities are obtained from an inhomogeneous transfer matrix constructed using the {\it quantum inverse scattering method}. The inhomogeneity parameter determines the discrete time step. We then focus on the non-invertible {\it Kramers-Wannier} duality-symmetry for the trotterized evolution. We find that the discretization of both space and time leads to a doubling of these duality operators. They account for discrete translations in both space and time. As an interesting application, we find that these operators also provide maps between trotterizations of different orders. This helps us extend our results beyond the trotterization scheme and investigate the Kramers-Wannier duality-symmetry for finite time Floquet evolution of the critical transverse field Ising chain. {Finally, we investigate how these non-invertible operators shape the phase diagram of the discrete-time evolution. This question is particularly interesting in the Floquet setting, which is known to host a richer phase structure than its undriven counterpart. We systematically construct the necessary operators which relate different phases away from criticality for both trotterized and Floquet evolutions.