Topological Quantum Glassiness

TL;DR

This work investigates how glassy dynamics can arise in disorder-free quantum systems with topological order. It constructs three exactly solvable lattice models (2D and 3D) with commuting stabilizers $O_I$ that host ground-state degeneracy and nonlocal topological operators, and demonstrates both strong and fragile quantum glass behavior: strong glassiness yields $\tau_q \sim \exp(L)$ at $T=0$, while fragile glassiness yields $\tau_q \sim \exp\big(L^{\log_2 3}\big)$ due to fractal defect membranes with dimension $d^* = \frac{\ln 3}{\ln 2}$. At finite temperature, relaxation can become polynomial in $L$ or remain activated depending on the bath coupling, highlighting a nonperturbative dynamical landscape in topological quantum glasses. The results have implications for understanding nonequilibrium dynamics in topologically ordered phases and for how environmental couplings interact with nonlocal order to produce dynamical arrest.

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 simple exactly solvable examples where the barriers each system encounters on its approach to lower and lower energy states become increasingly large and eventually scale with the system size. If the environment couples locally to the physical degrees of freedom in the system, tunnelling under large 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 thus 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 or regular open membranes, but fractal objects with dimension $d^* = ln 3/ ln 2$.

Figures (4)

• Figure 1: Square lattice, with spin operators defined on the sites. The square lattice is bipartite, and the two sets of points $A$,$B$ are shown in red solid dots and blue open circles. A diamond contains 4 vertices in an elementary plaquette, and the diamonds can also be divided into two sets (forming a red/blue checkerboard) according to which sublattice their topmost vertices belong to. Four-spin operators are defined on each plaquette using the $\sigma^{\rm x}$ and $\sigma^{\rm y}$ components of the spin, as described in the text. The green dots correspond to "defects" that are generated by applying a $\sigma^{\rm z}$ to the site encircled.
• Figure 2: Cubic cell of an fcc lattice. The centers of the six faces form an octahedron, with its sites labeled from 1 (topmost) to 6. In addition to the set of octahedra formed by the face centered sites, there are three more sets of octahedra that can be assembled from sites both on faces and on corners of the cubic cells, totalling 4 such sets. Six-spin operators are defined on these octahedra using the $\sigma^{\rm x,y,z}$ components of spin on each vertex as described in the text.
• Figure 3: Sites of an hcp (hexagonal close-packed) lattice. (a) The hcp lattice is comprised of two interpenetrating hexagonal lattices, which can be alternatively seen as stacked triangular lattices, shown in red and blue. Prisms are defined as sets of five sites, two of which belong to one sublattice (top and bottom of the prism), and three of which belong to the other and form a triangle that lies in the layer in between the top and bottom sites of the prism. Five-spin interactions are defined on each prism as explained in the text. (b) Top view of the hcp lattice, which shows that the blue sites stack on top of the red upward pointing triangles, and the red sites stack on top of the downward pointing blue triangles. (c) Two prisms with topmost sites belonging to different sublattices can share a common edge and the five-spin operators defined on the two prisms commute because minus signs from commuting the $\sigma^{\rm x}$ and $\sigma^{\rm z}$ components appear twice, once for each shared site, and cancel.
• Figure 4: To annihilate three defects (shown in green) at the corners of an equilateral triangle, one must flip the spins in a "fractal" membrane (containing sites shown in red) that stretches between the defects. For a triangle of size $2^\ell$, there are $3^\ell$ sites in the membrane. The annihilation of the three defects through quantum tunneling is a virtual process of order the number of sites that are involved (red sites). Hence, the amplitude for the quantum tunneling process vanishes exponentially with the "volume" of the membrane.