In this paper we study the behavior of the entanglement measure dubbed negativity in the context of the toric code model. Using a replica method introduced recently by Calabrese, Cardy, and Tonni [Phys. Rev. Lett. 109, 130502 (2012)], we obtain an exact expression which illustrates how the nonlocal correlations present in a topologically ordered state reflect in the behavior of the negativity of the system. We find that the negativity has a leading area-law contribution if the subsystems are in direct contact with one another (as expected in a zero-range correlated model). We also find a topological contribution directly related to the topological entropy, provided that the partitions are topologically nontrivial in both directions on a torus. We further confirm by explicit calculation that the negativity captures only quantum contributions to the entanglement. Indeed, we show that the negativity vanishes identically for the classical topologically ordered eight-vertex model, which on the contrary exhibits a finite von Neumann entropy, inclusive of topological correction.
In this paper we study the behavior of the entanglement measure dubbed negativity in the context of the toric code model. Using a replica method introduced recently by Calabrese, Cardy, and Tonni [Phys. Rev. Lett. 109, 130502 (2012)], we obtain an exact expression which illustrates how the nonlocal correlations present in a topologically ordered state reflect in the behavior of the negativity of the system. We find that the negativity has a leading area-law contribution if the subsystems are in direct contact with one another (as expected in a zero-range correlated model). We also find a topological contribution directly related to the topological entropy, provided that the partitions are topologically nontrivial in both directions on a torus. We further confirm by explicit calculation that the negativity captures only quantum contributions to the entanglement. Indeed, we show that the negativity vanishes identically for the classical topologically ordered eight-vertex model, which on the contrary exhibits a finite von Neumann entropy, inclusive of topological correction.
We use the entanglement negativity, a measure of entanglement for mixed states, to probe the structure of entanglement in the ground state of a topologically ordered system. Through analytical calculations of the negativity in the ground state(s) of the toric code model, we explicitly show that the pure-state entanglement of a region A and its complement B is the sum of two types of contributions: boundary entanglement and long-range entanglement. Boundary entanglement is seen to be insensitive to tracing out the degrees of freedom in the interior of regions A and B, and therefore it entangles only degrees of freedom in A and B that are close to their common boundary. We recover the well-known result that boundary entanglement is proportional to the size of each boundary separating A and B and it includes an additive, universal correction. The second, long-range, contribution to pure-state entanglement appears only when A and B are noncontractible regions (e.g., on a torus) and it is seen to be destroyed when tracing out a noncontractible region in the interior of A or B. In the toric code, only the long-range contribution to the entanglement depends on the specific ground state under consideration.
We propose a diagnostic for finite temperature topological order using "topological entanglement negativity," the long-range component of a mixed-state entanglement measure. As a demonstration, we study the toric code model in d spatial dimensions for d=2,3,4, and find that when topological order survives thermal fluctuations, it possesses a nonzero topological entanglement negativity, whose value is equal to the topological entanglement entropy at zero temperature. Furthermore, we show that the Gibbs state of 2D and 3D toric code at any nonzero temperature, and that of 4D toric code above a certain critical temperature, can be expressed as a convex combination of short-range entangled pure states, consistent with the absence of topological order.
Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.
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.
Concepts of information theory are increasingly used to characterize collective phenomena in condensed matter systems, such as the use of entanglement entropies to identify emergent topological order in interacting quantum many-body systems. Here, we employ classical variants of these concepts, in particular R\'enyi entropies and their associated mutual information, to identify topological order in classical systems. Like for their quantum counterparts, the presence of topological order can be identified in such classical systems via a universal, subleading contribution to the prevalent volume and boundary laws of the classical R\'enyi entropies. We demonstrate that an additional subleading O(1) contribution generically arises for all R\'enyi entropies S(n) with $n\ensuremath{\ge}2$ when driving the system towards a phase transition, e.g., into a conventionally ordered phase. This additional subleading term, which we dub connectivity contribution, tracks back to partial subsystem ordering and is proportional to the number of connected parts in a given bipartition. Notably, the Levin-Wen summation scheme, typically used to extract the topological contribution to the R\'enyi entropies, does not fully eliminate this additional connectivity contribution in this classical context. This indicates that the distillation of topological order from R\'enyi entropies requires an additional level of scrutiny to distinguish topological from nontopological O(1) contributions. This is also the case for quantum systems, for which we discuss which entropies are sensitive to these connectivity contributions. We showcase these findings by extensive numerical simulations of a classical variant of the toric code model, for which we study the stability of topological order in the presence of a magnetic field and at finite temperatures from a R\'enyi entropy perspective.
Entanglement measures find frequent application in the study of topologically ordered systems, where the presence of topological order is reflected in an additional contribution to the entanglement of the system. Obtaining this topological entropy from analytical calculations or numerical simulations is generally difficult due to the fact that it is an order-one correction to leading terms that scale with the size of the system. In order to distill the topological entropy, one resorts to extrapolation as a function of system size or to clever subtraction schemes that allow one to cancel out the leading terms. Both approaches have the disadvantage of requiring multiple (accurate) calculations of the entanglement of the system. Here we propose a modification of conventional entanglement calculations that allows one to obtain the topological entropy of a system from a single measurement of entanglement. In our approach, we replace the conventional trace over the degrees of freedom of a partition of the system with a projection onto a given state (which need not be known). We show that a proper choice of partition and projective measurement allows one to rid the entanglement measures of the typical boundary terms, thus exposing the topological contribution alone. We consider specifically the measures known as von Neumann entropy and entanglement negativity and we discuss their application to both models that exhibit quantum as well as classical topological order.