Topological Defects in Floquet Circuits

TL;DR

This paper develops a Floquet Ising circuit incorporating two topological defects: a spin-flip defect $D_{\\psi}$ and a non-invertible Kramers-Wannier duality defect $D_{\\sigma}$. The defects commute with the Floquet unitary, enabling space-like and time-like configurations that reveal fusion rules and twisted-boundary physics. A key result is the emergence of a single localized Majorana zero mode under duality-twisted boundaries, with explicit lattice constructions and auto-correlation signals, plus analytic entanglement expressions for small systems. The work connects lattice defects to Ising CFT defect structures and fusion categories, suggesting avenues for Floquet SPT classification, robustness to interactions, and experimental probes on NISQ devices.

Abstract

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.

Figures (22)

• Figure 1: Diagram depicting the matrix elements of the Floquet Ising unitary $U_{\mathds{1}, \text{even}}$ for a single time step acting on $\mathcal{H}_{\rm even}$ with $L=6$. The indices $h_i$ ($h_i'$) label the states $0,1$ of the incoming (outgoing) states of the Floquet unitary. The unlabelled vertices correspond to the label $\sigma$.
• Figure 2: Comparison of amplitudes for a $L=3$ circuit without defects and with two spin-flip defects inserted at times $n=50,100$. The physical spins are on even sites, and the couplings correspond to the paramagnetic phase with transverse magnetic fields $u_{2r} = \frac{\pi}{4}$ and Ising couplings $u_{2r-1} = \frac{\pi}{8}$ for $r=1,2,3$. (Left): Plots of the matrix elements $\text{Re}\langle u|U_{\mathds{1},\text{even}}|u\rangle$ and the corresponding quantity with spin-flip defects inserted at the $50\textsuperscript{th}$ and $100\textsuperscript{th}$ step, $\text{Re}\langle u|U_{\mathds{1},\text{even}}D_{\psi,\text{even}}(100)U_{\mathds{1},\text{even}}D_{\psi,\text{even}}(50)U_{\mathds{1},\text{even}}|u\rangle$, versus the number of Floquet steps $n$. More precisely, spin-flip defects are applied after $49$ and $99$ defect-less unitaries $U_{\mathds{1},\text{even}}$ have been applied. (Right): The difference between these two matrix elements.
• Figure 3: As with figure \ref{['HorizontalSpinFlipPMDefect']}, only in the ferromagnetic phase with transverse magnetic field $u_{2r} = \frac{\pi}{8}$ and Ising couplings $u_{2r-1} = \frac{\pi}{4}$.
• Figure 4: (Left): Plots of the real part of the amplitudes with a single duality defect inserted at $n=50$, $\text{Re}\langle u'|U_{\mathds{1},\text{odd}} D_\sigma(50)U_{\mathds{1},\text{even}}|u\rangle$ (blue), a single spin-flip inserted at $n=50$ followed by a single duality defect inserted at $n=100$, $\text{Re}\langle u'|U_{\mathds{1},\text{odd}}D_\sigma(100)U_{\mathds{1},\text{even}}D_\psi(50)U_{\mathds{1},\text{even}}|u\rangle$ (orange) as well as a duality defect at $n=50$ followed by a spin-flip defect at $n=100$, $\text{Re}\langle u'|U_{\mathds{1},\text{odd}}D_\psi(100)U_{\mathds{1},\text{odd}}D_\sigma(50)U_{\mathds{1},\text{even}}|u\rangle$ (green), where $U_{\mathds{1},\text{even}}$ and $U_{\mathds{1},\text{odd}}$ are the defectless Floquet unitary circuits acting on the even and odd sites respectively. Just as in the previous figures, inserting the defects at the Floquet steps $n=50$ and $n=100$ means that the defects are inserted after $49$ and $99$ defectless unitaries $U_\mathbb{I}$ are inserted. Note that these values of $n$ are chosen arbitrarily. The couplings are set to be $u_{2r}= \frac{\pi}{4}$ and $u_{2r+1}= \frac{\pi}{8}$. (Right): The difference between applying only a single duality defect and applying a single spin-flip followed by a single duality defect, $\text{Re}\langle u'|U_{\mathds{1},\text{odd}} D_\sigma(50)U_{\mathds{1},\text{even}}|u\rangle-\text{Re}\langle u'|U_{\mathds{1},\text{odd}}D_\sigma(100)U_{\mathds{1},\text{even}}D_\psi(50)U_{\mathds{1},\text{even}}|u\rangle$ (blue), as well as the difference between applying only a single duality defect and applying a single duality defect followed by a single spin-flip defect, $\text{Re}\langle u'|U_{\mathds{1},\text{odd}} D_\sigma(50)U_{\mathds{1},\text{even}}|u\rangle-\text{Re}\langle u'|U_{\mathds{1},\text{odd}}D_\psi(100)U_{\mathds{1},\text{odd}}D_\sigma(50)U_{\mathds{1},\text{even}}|u\rangle$ (orange).
• Figure 5: (Left): The real part of the matrix element with two duality defects inserted at $n=50,100$ i.e, $\text{Re}\langle {\rm u} |U_{\mathds{1}, \text{even}} D_\sigma(100) U_{\mathds{1},\text{odd}}D_\sigma(50) U_{\mathds{1}, \text{even}}|{\rm u}\rangle$, where $U_{\mathds{1}, \text{even}}$ and $U_{\mathds{1},\text{odd}}$ are the defectless Floquet unitary circuits acting on the even and odd sites respectively. Also shown is the sum of the matrix elements for the defectless circuit and a circuit with a single spin-flip defect inserted at $n=75$, i.e., $\text{Re}\langle {\rm u} |U_{\mathds{1}, \text{even}}|{\rm u}\rangle+\text{Re}\langle {\rm u} |U_{\mathds{1}, \text{even}} D_\psi(75) U_{\mathds{1}, \text{even}}|{\rm u}\rangle$. (Right): Difference between the two curves of the left plot. A total of $L=3$ physical sites are taken and the couplings are chosen to be $u_{2r}=\frac{\pi}{4},u_{2r+1}=\frac{\pi}{8}$. Both the initial and final states $|{\rm u}\rangle$ live on the even sites.