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We study a one-dimensional spin-$1/2$ model with three-spin interactions anda transverse magnetic field $h$. The model has a $Z_2 \times Z_2$ symmetry, anda duality between $h$ and $1/h$. The self-dual point at $h=1$ is a quantumcritical point with a continuous phase transition. We compute the criticalexponents $z$, $\beta$, $\gamma$ and $\nu$, and the central charge $c$numerically using exact diagonalization (ED) for systems with periodic boundaryconditions. We find that both $z$ and $c$ are equal to $1$, implying that thecritical point is governed by a conformal field theory. The values obtained for$\beta/\nu$, $\gamma/\nu$, and $\nu$ from ED suggest that the model exhibitsAshkin-Teller criticality with an effective coupling that is intermediatebetween the four-state Potts model and two decoupled transverse field Isingmodels. An analysis on larger systems but with open boundaries usingdensity-matrix renormalization group calculations, however, shows that theself-dual point may be in the same universality class as the four-state Pottsmodel. An energy level spacing analysis shows that the model is not integrable.For a system with periodic boundary conditions, there are an exponentiallylarge number of exact mid-spectrum zero-energy eigenstates. A subset of theseeigenstates have wave functions which are independent of $h$ and have unusualentanglement structure, suggesting that they are quantum many-body scars. Thenumber of such states scales at least linearly with system size. Finally, westudy the infinite-temperature autocorrelation functions close to one end of anopen system. We find that some of the autocorrelators relax anomalously intime, with pronounced oscillations and very small decay rates if $h \gg 1$ or$h \ll 1$. If $h$ is close to the critical point, the autocorrelators decayquickly to zero except for an autocorrelator at the end site.

Weak universality, quantum many-body scars and anomalous infinite-temperature autocorrelations in a one-dimensional spin model with duality