It has been known that by encoding the Boltzmann weights of a classical Ising model in the amplitudes of the wave function of the ground state of the toric code model, the classical phase transition in Ising model is mapped to a topological phase transition in a perturbed toric code model. Since such topological phase transitions cannot be characterized by any local order parameter, it will be an important challenge to find an order parameter which describes the above topological phase transition. In this paper, using a simple technic based on mapping between classical Ising model and the ground state of the toric code model, we find a non-local order parameter which well reveals the topological nature of the above phase transition. We show that such an order parameter is, in fact, a kind of string order parameter which has been recently introduced for some topological phase transitions.
A quantum Ising model in a transverse field is of the simplest quantum many-body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well known that such phase transitions can be mapped to topological phase transitions in toric code models. Therefore, one can expect that well-known properties of the transverse Ising model are used for characterizing topological phase transitions in toric code models. In this paper, we consider the magnetization of the Ising model and show, while it is a local order parameter, it is mapped to a nonlocal order parameter on the topological side. Consequently, we introduce a string order parameter for topological phase transitions in toric code models defined on different lattices with different dimensions in the presence of a uniform magnetic field. Since such a string order parameter is dual of the Ising order parameter with well-known properties, it leads to a topological (nonlocal) characterization of phase transition in the toric code model in uniform field. Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might still be possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.
Although the topological order is known as a quantum order in quantum many-body systems, it seems that there is not a one-to-one correspondence between topological phases and quantum phases. As a well-known example, it has been shown that all one-dimensional (1D) quantum phases are topologically trivial [X. Chen et al., Phys. Rev. B 83, 035107 (2011)]. By such facts, it seems a challenging task to understand when a quantum phase transition between different topological models necessarily reveals different topological classes of them. In this paper, we make an attempt to consider this problem by studying a phase transition between two different quantum phases which have a universal topological phase. We define a Hamiltonian as interpolation of the toric code model with
Z2 topological order and the color code model with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ topological order on a hexagonal lattice. We show such a model is exactly mapped to many copies of 1D quantum Ising model in transverse field by rewriting the Hamiltonian in a new complete basis. Consequently, we show that the universal topological phase of the color code model and the toric code model reflects in the 1D nature of the phase transition. We also consider the expectation value of Wilson loops by a perturbative calculation and show that behavior of the Wilson loop captures the nontopological nature of the quantum phase transition. The result on the point of phase transition also shows that the color code model is strongly robust against the toric code model.
It has been known that encoding Boltzmann weights of a classical spin model in amplitudes of a many-body wave function can provide quantum models whose phase structure is characterized by using classical phase transitions. In particular, such correspondence can lead to finding new quantum phases corresponding to well-known classical phases. Here, we investigate this problem for the Kosterlitz-Thouless (KT) phase in the
d-state clock model, where we find a corresponding quantum model constructed by applying a local invertible transformation on a
d-level version of Kitaev's toric code. In particular, we show the ground-state fidelity in such a quantum model is mapped to the heat capacity of the clock model. Accordingly, we identify an extended topological phase transition in our model in the sense that, for $d\ensuremath{\ge}5$, a KT-like quantum phase emerges between a
Zd topological phase and a trivial phase. Then, using a mapping to the correlation function in the clock model, we introduce a nonlocal (string) observable for the quantum model which exponentially decays in terms of distance between two end points of the corresponding string in the
Zd topological phase, while it shows a power law behavior in the KT-like phase. Finally, using well-known transition temperatures for the
d-state clock model, we give evidence to show that while the stability of both the
Zd topological phase and the KT-like phase increases by increasing
d, the KT-like phase is even more stable than the
Zd topological phase for large
d.
We study the phase transition from two different topological phases to the ferromagnetic phase by focusing on points of the phase transition. To this end, we present a detailed mapping from such models to the Ising model in a transverse field. Such a mapping is derived by rewriting the initial Hamiltonian in a new basis so that the final model in such a basis has a well-known approximated phase transition point. Specifically, we consider the toric codes and the color codes on various lattices with Ising perturbation. Our results provide a useful table to compare the robustness of the topological codes and to explicitly show that the robustness of the topological codes depends on triangulation of their underlying lattices.
We investigate the quantum phase diagram of the
K-layer Ising toric code corresponding to
K layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays
Z2K topological order originating from the toric codes in each layer, the system shows
Z2 topological order in the high-Ising limit. The latter is demonstrated for general
K by deriving an effective low-energy model in
Kth-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases
K=3 and
K=4 we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both
K separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.
Order parameters are key to our understanding of phases of matter. Not only do they allow to classify phases, but they also enable the study of phase transitions through their critical exponents which identify the universal long-range physics underlying the transition. Topological phases are exotic quantum phases which are lacking the characterization in terms of order parameters. While probes have been developed to identify such phases, those probes are only qualitative in that they take discrete values, and thus provide no means to study the scaling behavior in the vicinity of phase transitions. In this paper, we develop a framework based on variational tensor networks (infinite Projected Entangled Pair States, or iPEPS) for the quantitative study of topological phase transitions through topological order parameters. These order parameters allow to quantitatively probe the behavior through a phase transition and thus to identify universal signatures of topological phase transitions. We apply our framework to the study of the Toric Code model in different magnetic fields, which along some special lines maps to the (2+1)D Ising model. Our method identifies 3D Ising critical exponents for the entire transition, consistent with those special cases and general belief. However, we in addition also find a novel critical exponent
β∗≈0.02 for one of our topological order parameters, which we relate to disorder parameters in the 3D Ising model. This shows that our topological order parameters can provide additional means to characterize the universal data at topological phase transitions, and altogether demonstrates the power of this framework to microscopically study topological phase transitions and identify the universal data underlying the transition.