  Passive error correction protects logical information forever (in the
thermodynamic limit) by updating the system based only on local information and
few-body interactions. A paradigmatic example is the classical two-dimensional
Ising model: a Metropolis-style Gibbs sampler retains the sign of the initial
magnetization (a logical bit) for thermodynamically long times in the
low-temperature phase. Known models of passive quantum error correction
similarly exhibit thermodynamic phase transitions to a low-temperature phase
wherein logical qubits are protected by thermally stable topological order.
Here, in contrast, we show that constant-rate classical and quantum low-density
parity check codes have no $\textit{thermodynamic}$ phase transitions at
nonzero temperature, but nonetheless exhibit $\textit{ergodicity-breaking}$
dynamical transitions: below a critical nonzero temperature, the mixing time of
local Gibbs sampling diverges in the thermodynamic limit. We conjecture that
the circuit complexity of preparing extensive-energy states may diverge without
crossing any thermodynamic transition. Fault-tolerant passive decoders,
inspired by Gibbs samplers, may be amenable to measurement-free quantum error
correction and may present a desirable experimental alternative to conventional
quantum error correction based on syndrome measurements and active feedback.
