In quantum mechanics, string-net condensed states---a family of prototypical states exhibiting nontrivial topological order---can be classified via their long-range entanglement properties, in particular, topological corrections to the prevalent area law of the entanglement entropy. Here we consider classical analogs of such string-net models whose partition function is given by an equal-weight superposition of classical string-net configurations. Our analysis of the Shannon and Renyi entropies for a bipartition of a given system reveals that the prevalent volume law for these classical entropies is augmented by subleading topological corrections that are intimately linked to the anyonic theories underlying the construction of the classical models. We determine the universal values of these topological corrections for a number of underlying anyonic theories including SU(2)${}_{k}$, SU${(N)}_{1}$, and SU${(N)}_{2}$ theories.
In quantum mechanics, string-net condensed states---a family of prototypical states exhibiting nontrivial topological order---can be classified via their long-range entanglement properties, in particular, topological corrections to the prevalent area law of the entanglement entropy. Here we consider classical analogs of such string-net models whose partition function is given by an equal-weight superposition of classical string-net configurations. Our analysis of the Shannon and Renyi entropies for a bipartition of a given system reveals that the prevalent volume law for these classical entropies is augmented by subleading topological corrections that are intimately linked to the anyonic theories underlying the construction of the classical models. We determine the universal values of these topological corrections for a number of underlying anyonic theories including SU(2)k, SU(N)1, and SU(N)2 theories.
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.
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.
Recent lattice studies have revealed that the color flux tube between static quark-antiquark pairs exhibits an excess entanglement entropy (flux tube entanglement entropy, FTE2) that scales linearly with the quark separation L. In this paper, we demonstrate similar behavior in a string-net model based on SU(2)k fusion categories, where the nontrivial object j=1/2 (analogous to color charge) cannot exist in isolation due to the fusion rules, naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two j=1/2objects using the topological definition S(R)=ln(dim(Hom(R))) and find an entropy density σk=ln d1/2=ln(2cos(π/(k+2))). For k=3, the category reduces to the Fibonacci case, yielding an entropy density σ3 = ln φ ≈ 0.481 (φ is the golden ratio), which is qualitatively comparable in magnitude to the scale inferred from lattice studies and the entropy-surface mechanism. Minimizing the free energy F=E-TS yields a confinement–deconfinement transition at Tc=J/σk.The parameter k offers a tunable knob, making the SU(2)k family a computable laboratory for entropic confinement. The predicted entropy-density jump and critical scaling can be directly tested in quantum simulator platforms (e.g., Rydberg arrays or superconducting circuits) that realize Fibonacci anyonic models.
At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multisheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2+1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling.
Abstract The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S3 complements of a two-component link which is a connected sum of a knot K K and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z( MKm M K m ), which is the partition function of MKm M K m . Here MKm M K m is a closed 3-manifold associated with the knot K K m, where K K m is a connected sum of m-copies of K K (i.e., K K # K K . . . # K K ) which mimics the well-known replica method. We analayse the partition functions Z( MKm M K m ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z( MKm M K m ) can grow at most polynomially in k. On the contrary, we conjecture that Z( MKm M K m ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z( MKm M K m ) is the hyperbolic volume of the knot complement S3\ K K m. We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.
Abstract We study the multi-boundary entanglement structure of the state associated with the torus link complement S3\Tp,q in the set-up of three-dimensional SU(2)k Chern-Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of k → ∞. We present a detailed analysis of several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large k limiting value of the Rényi entropy of torus links of type Tp,pn is the sum of two parts: (i) the universal part which is independent of n, and (ii) the non-universal or the linking part which explicitly depends on the linking number n. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang-Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological 2d Yang-Mills theory with SU(2) gauge group. Further, the universal parts appearing in the large k limits of the entanglement entropy and the minimum Rényi entropy for torus links Tp,pn can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of Tp,pn link in the double scaling limit of k → ∞ and n → ∞ and propose that the entropies converge in the double limit as well.