Benedikt Placke

Abstract:

We study the non-unitary dynamics of a class of quantum circuits based on stochastically measuring check operators of subsystem quantum error-correcting codes, such as the Bacon-Shor code and its various generalizations. Our focus is on how properties of the underlying code are imprinted onto the measurement-only dynamics. We find that in a large class of codes with nonlocal stabilizer generators, at late times there is generically a nonlocal contribution to the subsystem entanglement entropy which scales with the subsystem size. The nonlocal stabilizer generators can also induce slow dynamics, since depending on the rate of competing measurements the associated degrees of freedom can take exponentially long (in system size) to purify (disentangle from the environment when starting from a mixed state) and to scramble (become entangled with the rest of the system when starting from a product state). Concretely, we consider circuits for which the nonlocal stabilizer generators of the underlying subsystem code take the form of subsystem symmetries. We present a systematic study of the phase diagrams and relevant time scales in two and three spatial dimensions for both Calderbank-Shor-Steane (CSS) and non-CSS codes, focusing in particular on the link between slow measurement-only dynamics and the geometry of the subsystem symmetry. A key finding of our work is that slowly purifying or scrambling degrees of freedom appear to emerge only in codes whose subsystem symmetries are nonlocally generated, a strict subset of those whose symmetries are simply nonlocal. We comment on the link between our results on subsystem codes and the phenomenon of Hilbert-space fragmentation in light of their shared algebraic structure.

Abstract:

The insulating magnetic pyrochlore ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ has gained attention as a quantum spin-ice candidate with dipole-octupole character that arises from the crystal-electric-field ground-state doublet for the ${\mathrm{Ce}}^{3+}$ Kramers ion. This dipole-octupole character permits both spin-ice phases based on magnetic dipoles and those based on more-exotic octupoles. This work reports low-temperature neutron diffraction measurements on single-crystal ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ with $Q$ coverage both at low $Q$ , where the magnetic form factor for dipoles is near maximal, and at high $Q$ , covering the region where the magnetic form factor for ${\mathrm{Ce}}^{3+}$ octupoles is near maximal. This study was motivated by recent powder neutron diffraction studies of other Ce-based dipole-octupole pyrochlores, ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ , which each showed temperature-dependent diffuse diffraction at high $Q$ , interpreted as arising from octupolar correlations. Our measurements use an optimized single-crystal diffuse scattering instrument that allows us to screen against strong Bragg scattering from ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ . The temperature-difference neutron diffraction reveals a low- $Q$ peak consistent with dipolar spin-ice correlations reported in previous work, and an alternation between positive and negative net intensity at higher $Q$ . These features are consistent with our numerical-linked-cluster calculations using pseudospin interaction parameters previously reported for ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ , ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ , and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ . Importantly, neither the measured data nor any of the NLC calculations show evidence for increased scattering at high $Q$ resulting from octupolar correlations. We conclude that at the lowest attainable temperature for our measurements ( $T=0.05\mathrm{K}$ ), scattering from octupolar correlations in ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ is not present in the neutron diffraction signal on the level of our observation threshold of around 0.1% of the low- $Q$ dipole scattering. We compare these results to those obtained earlier on powder ${\mathrm{Ce}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ and ${\mathrm{Ce}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ , and to low-energy inelastic neutron scattering from single-crystal ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ .