Topological Entanglement Research Articles

Topological pseudo entropy

We introduce a pseudo entropy extension of topological entanglement entropy called topological pseudo entropy. Various examples of the topological pseudo entropies are examined in three-dimensional Chern-Simons gauge theory with Wilson loop insertions. Partition functions with knotted Wilson loops are directly related to topological pseudo (Rényi) entropies. We also show that the pseudo entropy in a certain setup is equivalent to the interface entropy in two-dimensional conformal field theories (CFTs), and leverage the equivalence to calculate the pseudo entropies in particular examples. Furthermore, we define a pseudo entropy extension of the left-right entanglement entropy in two-dimensional boundary CFTs and derive a universal formula for a pair of arbitrary boundary states. As a byproduct, we find that the topological interface entropy for rational CFTs has a contribution identical to the topological entanglement entropy on a torus.

Boundary Topological Entanglement Entropy in Two and Three Dimensions

The topological entanglement entropy is used to measure long-range quantum correlations in the ground space of topological phases. Here we obtain closed form expressions for the topological entropy of (2+1)- and (3+1)-dimensional loop gas models, both in the bulk and at their boundaries, in terms of the data of their input fusion categories and algebra objects. Central to the formulation of our results are generalized {mathcal {S}}-matrices. We conjecture a general property of these {mathcal {S}}-matrices, with proofs provided in many special cases. This includes constructive proofs for categories up to rank 5.

Prediction of Toric Code Topological Order from Rydberg Blockade

The physical realization of Z2 topological order as encountered in the paradigmatic toric code has proven to be an elusive goal. We predict that this phase of matter can be realized in a two-dimensional array of Rydberg atoms placed on the ruby lattice, at specific values of the Rydberg blockade radius. First, we show that the blockade model—also known as a “PXP” model—realizes a monomer-dimer model on the kagome lattice with a single-site kinetic term. This model can be interpreted as a Z2 gauge theory whose dynamics is generated by monomer fluctuations. We obtain its phase diagram using the numerical density matrix renormalization group method and find a topological quantum liquid (TQL) as evidenced by multiple measures including (i) a continuous transition between two featureless phases, (ii) a topological entanglement entropy of ln2 as measured in various geometries, (iii) degenerate topological ground states, and (iv) the expected modular matrix from ground state overlap. Next, we show that the TQL persists upon including realistic, algebraically decaying van der Waals interactions V(r)∼1/r6 for a choice of lattice parameters. Moreover, we can directly access topological loop operators, including the Fredenhagen-Marcu order parameter. We show how these can be measured experimentally using a dynamic protocol, providing a “smoking gun” experimental signature of the TQL phase. Finally, we show how to trap an emergent anyon and realize different topological boundary conditions, and we discuss the implications for exploring fault-tolerant quantum memories.12 MoreReceived 24 November 2020Accepted 19 May 2021DOI:https://doi.org/10.1103/PhysRevX.11.031005Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTopological phases of matterTopological quantum computingPhysical SystemsRydberg atoms & moleculesCondensed Matter, Materials & Applied Physics

Elastic response of wire frame glasses. I. Two dimensional model.

We study the elastic response of concentrated suspensions of rigid wire frame particles to a step strain. These particles are constructed from infinitely thin, rigid rods of length L. We specifically compare straight rod-like particles to bent and branched wire frames. In dense suspensions, the wire frames are frozen in a disordered state by the topological entanglements between their arms. We present a simple, geometric method to find the scaling of the elastic stress with concentration in these glassy systems. We apply this method to a simple 2D model system where a test particle is placed on a plane and constrained by a random distribution of points with number density ν. Two striking differences between wire frame and rod suspensions are found: (1) The linear elasticity per particle for wire frames is very large, scaling like ν2L4, whereas for rods, it is much smaller and independent of concentration. (2) Rods always shear thin but wire frames shear harden for concentrations less than ∼K/kBTL4, where K is the bending modulus of the particles. The deformation of wire frames is found to be important even for small strains, with the proportion of deformed particles at a particular strain, γ, being given by (νL2)2γ2. Our results agree well with simple numerical calculations for the 2D system.

Chiral bosonic topological insulator on the honeycomb lattice with anisotropic interactions

We study hard-core bosons on the honeycomb lattice in the presence of anisotropic nearest-neighbor repulsive interactions. Using a quantum Monte Carlo (QMC) technique, we extract the phase diagram of the model in terms of the filling and the anisotropy. At half-filling we find a dimer insulator phase near maximum anisotropy that is characterized by a finite topological entanglement entropy $\ln(2)/2$, indicative of a fractional quantum Hall state for bosons. We identify the presence of edge states and derive a QMC-based method to extract and verify their chirality. Remarkably, this phase arises in the absence of magnetic flux and without explicit lattice frustration.

Fractional chiral hinge insulator

We propose and study a wave function describing an interacting three-dimensional fractional chiral hinge insulator (FCHI) constructed by Gutzwiller projection of two non-interacting second order topological insulators with chiral hinge modes at half filling. We use large-scale variational Monte Carlo computations to characterize the model states via the entanglement entropy and charge-spin-fluctuations. We show that the FCHI possesses fractional chiral hinge modes characterized by a central charge $c=1$ and Luttinger parameter $K=1/2$, like the edge modes of a Laughlin $1/2$ state. By changing the boundary conditions for the underlying fermions, we investigate the topological degeneracy of the FCHI. Within the range of the numerically accessible system sizes, we observe a non-trivial topological degeneracy. A more numerically pristine characterization of the bulk topology is provided by the topological entanglement entropy (TEE) correction to the area law. While our computations indicate a vanishing bulk TEE, we show that the gapped surfaces host a two-dimensional topological order with a TEE per surface compatible with half that of a Laughlin $1/2$ state, a value that cannot be obtained from topological quantum field theory.

Topological entanglement entropy of interacting disordered zigzag graphene ribbons

Interacting disordered zigzag graphene nanoribbons have fractional charges, are quasi-one-dimensional, and display an exponentially small gap. Our numerical computations showed that the topological entanglement entropy of these systems has a small finite but universal value, independent of the strength of the interaction and the disorder. The result that was obtained for the topological entanglement entropy shows that the disorder-free phase is critical and becomes unstable in the presence of disorder.

Entanglement bootstrap approach for gapped domain walls

We develop a theory of gapped domain wall between topologically ordered systems in two spatial dimensions. We find a new type of superselection sector -- referred to as the parton sector -- that subdivides the known superselection sectors localized on gapped domain walls. Moreover, we introduce and study the properties of composite superselection sectors that are made out of the parton sectors. We explain a systematic method to define these sectors, their fusion spaces, and their fusion rules, by deriving nontrivial identities relating their quantum dimensions and fusion multiplicities. We propose a set of axioms regarding the ground state entanglement entropy of systems that can host gapped domain walls, generalizing the bulk axioms proposed in [B. Shi, K. Kato, and I. H. Kim, Ann. Phys. 418, 168164 (2020)]. Similar to our analysis in the bulk, we derive our main results by examining the self-consistency relations of an object called information convex set. As an application, we define an analog of topological entanglement entropy for gapped domain walls and derive its exact expression.

Extreme dwelling: Assembling domus horribilis

10 Rillington Place names the site of temporally extensive practices of murder (1943–1953), and offers an empirical entry point for critically advancing the conceptual innovations of relational approaches to the criminological study of ‘home’. In so doing, the paper, firstly, (re)conceptualises serial homicide as practice, more specifically as a mode of domestic labour which materialises in and is enacted through the relational dynamics of everyday residential life; and secondly, rejects the notion of ‘home’ and argues for the concept of dwelling to better capture the active, generative and fluid dynamics of domestic life. This subtle shift in conceptual approach acknowledges how domus horribilis is etched from, and woven through the topological entanglements of everyday and extreme practices, and moves us toward an alternative set of conceptual commitments in our research of domestic space. Drawing from a mixed portfolio of cultural media (including archival, epistolary, journalistic, photographic, filmic, architectural, museological and dramaturgical data), the paper takes forward Schatzki’s site ontology as an organising framework for practice-based analytics, and advances the critical insights of an embryonic criminology of the domestic.

The Entangled Conductive Structure of CB/PA6/PP MFCs and Their Electromechanical Properties.

In this work, carbon black (CB)/polyamide 6 (PA6)/polypropylene (PP) microfibrillar composites (MFCs) were fabricated through an extrusion (hot stretching) heat treatment process. The CB-coated conductive PA6 microfibrils with high aspect ratio were in situ generated as a result of the selective accumulation of CB at the interface. At the proper temperature, a 3D entangled conductive structure was constructed in the PP matrix, due to topological entanglement between these conductive microfibrils. This unique conductive structure provided the PP composites with a low electrical conductivity percolation threshold. Moreover, the electromechanical properties of conductive MFCs were investigated for the first time. A great stability, a high sensitivity and a nice reproducibility were achieved simultaneously for CB/PA6/PP MFCs. This work provides a universal and low-cost method for the conductive polymer composites’ (CPCs) fabrication as sensing materials.

Two-Dimensional Metal-Organic Framework Materials: Synthesis, Structures, Properties and Applications.

Among the recent developments in metal-organic frameworks (MOFs), porous layered coordination polymers (CPs) have garnered attention due to their modular nature and tunable structures. These factors enable a number of properties and applications, including gas and guest sorption, storage and separation of gases and small molecules, catalysis, luminescence, sensing, magnetism, and energy storage and conversion. Among MOFs, two-dimensional (2D) compounds are also known as 2D CPs or 2D MOFs. Since the discovery of graphene in 2004, 2D materials have also been widely studied. Several 2D MOFs are suitable for exfoliation as ultrathin nanosheets similar to graphene and other 2D materials, making these layered structures useful and unique for various technological applications. Furthermore, these layered structures have fascinating topological networks and entanglements. This review provides an overview of different aspects of 2D MOF layered architectures such as topology, interpenetration, structural transformations, properties, and applications.

Abelian topological order of ν=2/5 and 3/7 fractional quantum Hall states in lattice models

Determining the statistics of elementary excitations supported by fractional quantum Hall states is crucial to understanding their properties and potential applications. In this paper, we use the topological entanglement entropy as an indicator of Abelian statistics to investigate the single-component $\nu=2/5$ and $3/7$ states for the Hofstadter model in the band mixing regime. We perform many-body simulations using the infinite cylinder density matrix renormalization group and present an efficient algorithm to construct the area law of entanglement, which accounts for both numerical and statistical errors. Using this algorithm, we show that the $\nu=2/5$ and $3/7$ states exhibit Abelian topological order in the case of two-body nearest-neighbor interactions. Moreover, we discuss the sensitivity of the proposed method and fractional quantum Hall states with respect to interaction range and strength.

Selective entanglement coupling of nanoparticles in polymer nanocomposite with high shape recovery stress

Nanoparticles have been widely used to enhance mechanical properties of polymer matrix, however, the complex mechanisms of interfacial entanglements, coupled confinements and coordinative motions among the polymer macromolecules and nanoparticles have not been well understood. In this study, we develop a coupling model to describe the thermodynamic entanglements and confinement effects within nanoparticle reinforced shape memory polymer (SMP) nanocomposites. We found that thermodynamic relaxation of the SMP nanocomposite is determined by cooperative effects of sub-entanglement and topological entanglement of macromolecule chains, which selectively enhance the relaxation behaviors and enable a high recovery stress of SMP nanocomposites. Finally, the effectiveness of this model is demonstrated by applying it to predict the constitutive relationships and shape memory behaviors, and this fundamental approach can reveal toughening mechanisms of shape recovery strength in the SMP nanocomposites by nanoparticles.

The Molecular Origins of the Influence of Translation-Elongation Kinetics on the Specific Activity of Enzymes

Synonymous codon substitutions, which change the mRNA sequence but not the protein sequence, have been found to affect co-translational folding and post-translational structure of some proteins due to their impact on translation elongation speed. For example, the specific activities of chloramphenicol acetyltransferase III (CAT-III) and Firefly luciferase (LUC) have been found to change due to synonymous codon substitutions. In this study, we use multi-scale modeling to investigate how synonymous mutations lead to changes in the kinetics and structural ensemble of enzymes that alter their specific activities for relatively long time scales. We simulated the synthesis of the enzymes CAT-III, DDLB and DHFR on the ribosome under fast and slow translation schedules, arising from synonymous mRNA variants, followed by post-translational simulations off the ribosome. The specific activity of each protein mutant was then estimated in silico. Our simulation model recapitulates the experimental trends of CAT-III specific activity changes and identifies a reduced activity of the slow-synthesizing DDLB mutant. For the misfolding-prone proteins CAT-III and DDLB, we identified diverse near-native-like, soluble misfolded kinetic traps with topological entanglements formed near the active site, which could establish a memory of the co-translational structural divergence arising from the different translation speeds that persists in the post-translational folding process and causes the altered enzymatic activity. The fast-folding protein DHFR without kinetic traps has no memory of co-translational structural divergence persisting in the post-translation and hence no altered function could be observed in synonymous mutants. We conclude that synonymous mutations lead to populations shifts that can promote the formation of entangled, near-native like structures that are kinetic traps with reduced functionality, but are native enough to bypass the proteostasis quality control systems of the cell.

Transparent and mechanically strong hydrogen-bonded polymer complex elastomers with improved self-healability under ambient conditions

A major challenge in designing self-healing polymeric materials is to combine robust mechanical properties and high healing efficiency. Herein, we report a kind of transparent, hydrogen-bonded polymer complex elastomers with both good mechanical strength and rapid self-healing capability under ambient conditions. Such elastomers were simply fabricated by in situ photoinitiated random copolymerization of acrylic acid (AA) and N-acryloyl-6-aminocaproic acid (A6ACA), in the presence of a linear polyethylene oxide (PEO). By evaporating water molecules from the samples via free drying, the dynamical hydrogen bonding between PEO and poly(A6ACA-co-AA) were substantially formed, which endowed the polymer chains to form physically crosslinked three-dimensional polymer network, as well as the topological chain entanglements. The resulting optically transparent elastomers exhibited both good mechanical strengths (at break) as high as 1.73 MPa, and excellent stretchability that could be stretched up to 24 times their initial length, respectively. Comparing with the elastomers based on PEO and PAA homopolymer without the A6ACA comonomer, the introduction of A6ACA as a comonomer afforded complex elastomers with comparable mechanical strength and, more importantly, significantly improved self-healing efficiency even under ambient conditions, probably due to the significantly decreased glass transition temperature that facilitated chain mobility and reformation of hydrogen bonds. It is therefore believed that the copolymerization strategy offers a practical method to tune the compositions and to tailor the properties of hydrogen-bonded polymer complexes for engineering elastomers with balanced mechanical and self-healing performance under ambient environment for various applications.

Fatigue-resistant adhesion II: Swell tolerance

In a previous paper (Zhang et al., 2020), we have demonstrated an adhesive of high fatigue threshold ∼300 J/m2 using an as-prepared hydrogel of long-chain polymer network. In most applications, however, hydrogel adhesives are in contact with water. The long-chain hydrogel swells and reduces the fatigue threshold. Here we demonstrate a type of swell-tolerant and fatigue-resistant hydrogel adhesive: a long-chain polymer network in topological entanglement with short-chain microgels. The short-chain microgels restrict swell, and the long-chain network retains the high fatigue threshold. After the hydrogel adhesives swell in water to equilibrium, the fatigue threshold is 61 J/m2 for an adhesive of a long-chain network without microgels, and is 262 J/m2 for an adhesive of a long-chain network with microgels.

Quantum critical phase transition between two topologically ordered phases in the Ising toric code bilayer

We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly-coupled $\mathbb{Z}_2\times\mathbb{Z}_2$ and the strongly-coupled $\mathbb{Z}_2$ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the 3d Ising$^*$ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local $\mathbb{Z}_2$ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a non-local order parameter.

A tough and self-fusing elastomer tape

A multifunctional polymer self-fusing elastomer tape strengthened by the topological entanglement and H-bonds demonstrates an antibacterial activity, and appealing ability to seal liquid leakage (water, oil) and stop bleeding. • H-bonding and topological entanglements contribute to a self-fusing elastomer. • The tape seals the liquid leakage and bleeding, independent of surface nature. • This self-fusing tape kills harmful bacteria by intrinsic weak acidity. Pressure-sensitive and adhesive tapes have been widely used for many purposes. However, their intrinsically strong sticky nature and weak strength severely limits their applications to vulnerable surfaces and harsh conditions. Herein, a multifunctional tough and self-fusing elastomer tape is fabricated for the first time by simple copolymerization of a model hydrophobic monomer 2-methoxyethyl acrylate (MEA) and a model weak acid-containing monomer acrylic acid (AAc). The PMEA-PAAc copolymer elastomers demonstrate high strengths and toughness with a maximum Young’s modulus, tensile stress, breaking strain, toughness, and recovery stress up to 74.19 MPa, 10.77 MPa, 2260%, 48.95 MJ/m 3 , and 0.60 MPa, respectively, due to hydrogen bonds and hydrophobic interactions. Furthermore, these reversible physical interactions contribute to the room temperature self-fusing ability of the elastomer. Studies on rheological behavior, dynamic mechanical properties and time–temperature equivalence reveal that topological entanglement and hydrogen bonds collectively contribute to the toughness and self-fusing capability. Intriguingly, the weak acidity of PAAc imparts antibacterial activity to the elastomer. Importantly, this elastomer can seal liquid (water and oil) leakage and stop bleeding, which is independent of the adherend surface properties, outperforming the conventional surface-dependent pressure sensitive adhesive and adhesive tape, due to elastic retraction force and self-fusing capability.

Generating W states with braiding operators

Braiding operators can be used to create entangled states out of product states, thus establishing a correspondence between topological and quantum entanglement. This is well-known for maximally entangled Bell and GHZ states and their equivalent states under Stochastic Local Operations and Classical Communication, but so far a similar result for W states was missing. Here we use generators of extraspecial 2-groups to obtain the W state in a four-qubit space and partition algebras to generate the W state in a three-qubit space. We also present a unitary generalized Yang-Baxter operator that embeds the W_n state in a (2n-1)-qubit space.

Kitaev's quantum double model as an error correcting code

Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes --- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the ``log dim R'' term is included in the definition of entanglement entropy.