In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
We study topological order in a toric code in three spatial dimensions or a 3+1DZ2 gauge theory at finite temperature. We compute exactly the topological entropy of the system and show that it drops, for any infinitesimal temperature, to half its value at zero temperature. The remaining half of the entropy stays constant up to a critical temperature Tc, dropping to zero above Tc. These results show that topologically ordered phases exist at finite temperatures, and we give a simple interpretation of the order in terms of fluctuating strings and membranes and how thermally induced point defects affect these extended structures. Finally, we discuss the nature of the topological order at finite temperature and its quantum and classical aspects.
The entanglement properties of a class of topological stabilizer states, the so-called topological color codes defined on a two-dimensional lattice or 2-colex, are calculated. The topological entropy is used to measure the entanglement of different bipartitions of the 2-colex. The dependency of the ground-state degeneracy on the genus of the surface shows that the color code can support a topological order, and the contribution of the color in its structure makes it interesting to compare with Kitaev's toric code. While a qubit is maximally entangled with the rest of the system, two qubits are no longer entangled showing that the color code is genuinely multipartite entangled. For a convex region, it is found that entanglement entropy depends only on the degrees of freedom living on the boundary of two subsystems. The boundary scaling of entropy is supplemented with a topological subleading term, which for a color code defined on a compact surface, is twice over the toric code. From the entanglement entropy we construct a set of bipartitions in which the diverging term arising from the boundary term is washed out, and the remaining nonvanishing term will have a topological nature. Besides the color code on the compact surface, we also analyze the entanglement properties of a version of color code with a border, i.e., triangular color code.
We calculate exactly the von Neumann and topological entropies of the toric code as a function of system size and temperature. We do so for systems with infinite energy scale separation between magnetic and electric excitations, so that the magnetic closed loop structure is fully preserved while the electric loop structure is tampered with by thermally excited electric charges. We find that the entanglement entropy is a singular function of temperature and system size, and that the limit of zero temperature and the limit of infinite system size do not commute. The two orders of limit differ by a term that does not depend on the size of the boundary between the partitions of the system, but instead depends on the topology of the bipartition. From the entanglement entropy we obtain the topological entropy, which is shown to drop to half its zero-temperature value for any infinitesimal temperature in the thermodynamic limit, and remains constant as the temperature is further increased. Such discontinuous behavior is replaced by a smooth decreasing function in finite-size systems. If the separation of energy scales in the system is large but finite, we argue that our results hold at small enough temperature and finite system size, and a second drop in the topological entropy should occur as the temperature is raised so as to disrupt the magnetic loop structure by allowing the appearance of free magnetic charges. We discuss the scaling of these entropies as a function of system size, and how the quantum topological entropy is shaved off in this two-step process as a function of temperature and system size. We interpret our results as an indication that the underlying magnetic and electric closed loop structures contribute equally to the topological entropy (and therefore to the topological order) in the system. Since each loop structure per se is a classical object, we interpret the quantum topological order in our system as arising from the ability of the two structures to be superimposed and appear simultaneously.
The string-net model describes a vast family of topological orders in two spatial dimensions. Here, we consider the effect of thermal fluctuations on these topological phases. In the original string-net model, the description of charge (vertex) excitations can be problematic. Therefore, in order to describe all anyon excitations, we study an extended model [Y. Hu et al., Phys. Rev. B 97, 195154 (2018)]. Building on recent methods, we compute the spectral degeneracies of excited states and obtain the exact partition function. In the thermodynamic limit, the latter is dominated by the trivial (vacuum) anyon, so that topological order is destroyed at any nonzero temperature. In contrast, in a finite-size system, order survives up to a finite temperature, revealing a nontrivial scaling between temperature and size similar to that of the one-dimensional classical Ising model. We confirm this scaling by computing the thermal average of several observables such as Wegner-Wilson loops and topological mutual information.
Topologically ordered quantum phases are robust in the sense that perturbations in the Hamiltonian of the system will not change the topological nature of the ground-state wave function. However, in order to exploit topological order for applications such as self-correcting quantum memories and information processing, these states need to be also robust both dynamically and at finite temperature in the presence of an environment. It is well known that systems like the toric code in two spatial dimensions are fragile in temperature. In this paper, we show a completely analytic treatment of the toric code away from equilibrium, after a quantum quench of the system Hamiltonian. We show that, despite being subject to unitary evolution (and at zero temperature), the long-time behavior of the topological entropy is thermal, therefore vanishing. If the quench preserves a local gauge structure, there is a residual long-lived topological entropy. This also is the thermal behavior in presence of such gauge constraints. The result is obtained by studying the time evolution of the topological 2-R\'enyi entropy in a fully analytical, exact way.