Topological quantum glassiness

Abstract

Quantum tunneling often allows pathways to relaxation past energy barriers which are otherwise hard to overcome classically at low temperatures. However, this is not always the case. In this paper we provide exactly solvable examples where the barriers each system encounters on its approach to lower and lower energy states become increasingly wide and eventually scale with the system size. If the environment couples locally to the physical degrees of freedom in the system, tunneling under these barriers requires processes whose order in perturbation theory is proportional to the width of the barrier. This results in quantum relaxation rates that are exponentially suppressed in system size: For these quantum systems, no physical bath can provide a mechanism for relaxation that is not dynamically arrested at low temperatures. The examples discussed here are drawn from three-dimensional generalizations of Kitaev's toric code, originally devised in the context of topological quantum computing. They are devoid of any local order parameters or symmetry breaking and are examples of topological quantum glasses. We construct systems that have slow dynamics similar to either strong or fragile glasses. The example with fragile-like relaxation is interesting in that the topological defects are neither open strings nor regular open membranes, but fractal objects with dimension d* = ln3/ln2.

Keywords:

• glassiness
• stochastic processes
• quantum tunneling

Acknowledgements

This work was supported in part by EPSRC Postdoctoral Research Fellowship EP/G049394/1, and by DOE Grant DEFG02-06ER46316.

Notes

Notes

1. Note however that these time-scales may already be in practice longer than any experimentally accessible times.

2. In any realistic (i.e., extensive) system, time-scales cannot grow faster than exponential in the volume of the system.

3. These time scales are dimensionless for convenience. They are intended as factors multiplying the characteristic microscopic time of the system in the absence of barriers. For instance, in the quantum mechanical tunneling case, the characteristic time would be dictated by the inverse hopping amplitude, 1/t.

4. Systems where the glass transition occurs in the zero temperature limit ought to be treated with care, as the order of limits (thermodynamic vs. T → 0) matters. For instance, time-scales that are exponential in system size with a trivial Arrhenius activated prefactor exp(Δ/T) diverge in the T → 0 limit irrespective of system size.