Quasi-local Edge Mode in XXX Spin Chain/Circuit with Interaction Boundary Defect

Abstract

We study the Heisenberg spin-1/2 model on a semi-infinite chain - or, equivalently, a trotterized unitary SU(2) symmetric six-vertex quantum circuit - with a boundary defect where the interaction between the two spins nearest the edge differs from that in the bulk. For sufficiently strong boundary interaction we explicitly construct a conserved operator quasi-localized near the boundary using a matrix-product ansatz. This quasi-local edge mode leads to non-decaying boundary correlation functions, corresponding to a nonzero boundary Drude weight. The correlation length of the edge mode diverges at a finite critical value of the boundary interaction, signaling a transition to ergodic boundary dynamics for subcritical interactions.

Figures (6)

• Figure 1: Schema of the six vertex staricase circuit with boundary interaction defect. On the last qubit ($L$) we apply depolarizing channel obtained by placing a virtual identity on site $L+1$ and then tracing out that site.
• Figure 2: Cancelation algebra diagrams (a): Eq. (\ref{['bulk']}), (b): Eq. (\ref{['boundary']}), and their implementation to demonstrate fixed point (conservation law) condition (c) for the boundary matrix product ansatz (\ref{['MPA']}).
• Figure 3: Partial norm profiles $\mathcal{P}_{\ell}$ of the subleading eigenoperator $Q^{(1)}$ of the staricase channel (Fig. \ref{['fig:scheme']}), obtained with quasi-exact numerics for $\omega=\tan 0.9$, and for $\tau = \tan 0.4$: orange (red), $\tau = \tan 0.5$: green (blue), $\tau = \tan 1.3$ brown (gray) dots, for $L=12$ ($L=14$). Straight lines indicate analytical results: exponentials $\propto\xi^\ell$ where $\xi=0.18667$ and $\xi= 0.35529$, respectively, are the spectral gaps $\xi=\Lambda_1/\Lambda_0$ of the transfer matrix (\ref{['HST']}). The inset shows the spectral gap $\Delta$ of the channel $\mathcal{M}_{[L]}$ vs. $L$ (dots) compared to $\xi^L$ (line).
• Figure 4: QLEM vector components $q_{\underline{\nu}}$ vs. integer ordered basis $\nu=\sum_{n=1}^L \nu_n 4^{n-1}$ for $\omega=\tan 0.9$, $\tau=\tan 0.5$ computed from analytical solution (\ref{['MPA']}), evaluated up to $L=9$. Black line indicates partial norm scaling $\nu^{\log_4\!\xi}$ from the spectral gap $\xi= 0.35529$. Inset: 256 eigenvalues of $Q$ for $L=8$, compared to analytical predictions $\|Q\|_{\rm HS}\sqrt{3}$,$-\|Q\|_{\rm HS}/\sqrt{3}$ (dashed).
• Figure 5: (a) The relative leading and subleading eigenvalues of the HS transfer matrix $\mathbb T$ (\ref{['HST']}), $\Lambda_0/\lambda_0^2$ (red), $\Lambda_1/\lambda_0^2$ (black), vs. $\tau$ for $\omega=1$. The inset gives a $(\omega,\tau)$phase-diagram with the region of existence of QLEM (where $\Lambda_0=\lambda_0^2$) indicated in light-blue. (b) Boundary Drude weights vs $\tau/\omega$ for $\omega=4$ (blue), $2$ (magenta), $1$ (red), $1/2$ (orange), $1/4$ (brown), and countinuous time case $\omega \to 0$ (black). Dashed vertical line denotes the critical point when the gap $1-\xi$ closes for $\omega=1$.