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The interface between different quantum phases of matter can give rise to novel physics, such as exotic topological phases or non-unitary conformal field theories. Here we investigate the interface between two spin chains in different chiral phases. Surprisingly, the mean-field theory description of this interacting composite system is given in terms of Dirac fermions in a curved space-time geometry. In particular, the boundary between the two phases represents a black hole horizon. We demonstrate that this representation is faithful both analytically, by employing bosonisation to obtain a Luttinger liquid model, and numerically, by employing Matrix Product State methods. A striking prediction from the black hole equivalence emerges when a quench, at one side of the interface between two opposite chiralities, causes the other side to thermalise with the Hawking temperature for a wide range of parameters and initial conditions., 16 pages, 10 figures

Break-in' Point, a 2012 arts and science performance and community engagement research initiative, was presented in the spring and fall semesters at the University of Leeds in the United Kingdom at Stage@Leeds. The outcome of a collaboration between dance artist A3 and theoretical physicist A2, under the direction of performance researcher A1, Break-in' Point is based on a series of real-life encounters at intersections of arts and science - exploring force, risk, exposure and resilience. The Break-in' Point performance offered an interrogation of the critical point at which physical, mental, and/or emotional strength give way under stress - causing structural degeneration and the experience of what lies beyond. This article is an examination of the performance, reviewing and analysing it as an imagined somatic zone - embodied encounters that transcend temporal bound-ness, compelling and igniting new possibilities - that engaged spiritual and epistemological transformation of performers and audiences. The article addresses three main periods in the life of Break-in' Point: (1) the development period - script building and rehearsals, (2) the performance - live encounters between and among performers and audiences and (3) beyond the theatre - digital engagements in the classroom and pedagogy. The article contributes new concepts and new ways of thinking about science education, the role of digital technology in pedagogy, dance/theatre public engagement and community arts practices as practices of healing, health and resilience., Comment: 20 pages, 11 figures

Unlike the fundamental forces of the Standard Model the quantum effects of gravity are still experimentally inaccessible. Rather surprisingly quantum aspects of gravity, such as massive gravitons, can emerge in experiments with fractional quantum Hall liquids. These liquids are analytically intractable and thus offer limited insight into the mechanism that gives rise to quantum gravity effects. To thoroughly understand this mechanism we employ a graphene-like system and we modify it appropriately in order to realise a simple (2+1)-dimensional massive gravity model. More concretely, we employ (2+1)-dimensional Dirac fermions, emerging in the continuous limit of a fermionic honeycomb lattice, coupled to massive gravitons, simulated by bosonic modes positioned at the links of the lattice. The quantum character of gravity can be determined directly by measuring the correlations on the bosonic atoms or by the interactions they effectively induce on the fermions. The similarity of our approach to current optical lattice configurations suggests that quantum signatures of gravity can be simulated in the laboratory in the near future, thus providing a platform to address question on the unification theories, cosmology or the physics of black holes., 13 pages, 3 figures, typos corrected, references added, matches the published version

The boundaries of quantum materials can host a variety of exotic effects such as topologically robust edge states or anyonic quasiparticles. Here, we show that fermionic systems such as graphene that admit a low energy Dirac description can exhibit counterintuitive relativistic effects at their boundaries. As an example, we consider carbon nanotubes and demonstrate that relativistic bulk spinor states can have non zero charge density on the boundaries, in contrast to the sinusoidal distribution of non-relativistic wave functions that are necessarily zero at the boundaries. This unusual property of relativistic spinors is complementary to the linear energy dispersion relation exhibited by Dirac materials and can influence their coupling to leads, transport properties or their response to external fields., Comment: 9 pages, 8 figures

The toric code is a simple and exactly solvable example of topological order realizing Abelian anyons. However, it was shown to support nonlocal lattice defects, namely twists, which exhibit non-Abelian anyonic behavior [Phys. Rev. Lett. 105, 030403 (2010)]. Motivated by this result, we investigated the potential of having non-Abelian statistics from puncture defects on the toric code. We demonstrate that an encoding with mixed-boundary punctures reproduces Ising fusion, and a logical Pauli- X upon their braiding. Our construction paves the way for local lattice defects to exhibit non-Abelian properties that can be employed for quantum information tasks.

Systems of interacting fermions can give rise to ground states whose correlations become effectively free-fermion-like in the thermodynamic limit, as shown by Baxter for a class of integrable models that include the one-dimensional XYZ spin-$\frac{1}{2}$ chain. Here, we quantitatively analyse this behaviour by establishing the relation between system size and correlation length required for the fermionic Gaussianity to emerge. Importantly, we demonstrate that this behaviour can be observed through the applicability of Wick's theorem and thus it is experimentally accessible. To establish the relevance of our results to possible experimental realisations of XYZ-related models, we demonstrate that the emergent Gaussianity is insensitive to weak variations in the range of interactions, coupling inhomogeneities and local random potentials., Comment: 8 pages, 4 figures

Here, we analyse two Dirac fermion species in two spatial dimensions in the presence of general quartic contact interactions. By employing functional bosonisation techniques, we demonstrate that depending on the couplings of the fermion interactions the system can be effectively described by a rich variety of topologically massive gauge theories. Among these effective theories, we obtain an extended Chern–Simons theory with higher order derivatives as well as two coupled Chern–Simons theories. Our formalism allows for a general description of interacting fermions emerging, for example, at the gapped boundary of three-dimensional topological crystalline insulators., SCOPUS: ar.j, info:eu-repo/semantics/published

Quasiparticle poisoning, expected to arise during the measurement of Majorana zero mode state, poses a fundamental problem towards the realization of Majorana-based quantum computation. Parafermions, a natural generalization of Majorana fermions, can encode topological qudits immune to quasiparticle poisoning. While parafermions are expected to emerge in superconducting fractional quantum Hall systems, they are not yet attainable with current technology. To bypass this problem, we employ a photonic quantum simulator to experimentally demonstrate the key components of parafermion-based universal quantum computation. Our contributions in this article are twofold. First, by manipulating the photonic states, we realize Clifford operator Berry phases that correspond to braiding statistics of parafermions. Second, we investigate the quantum contextuality in a topological system for the first time by demonstrating the contextuality of parafermion encoded qudit states. Importantly, we find that the topologically-encoded contextuality opens the way to magic state distillation, while both the contextuality and the braiding-induced Clifford gates are resilient against local noise. By introducing contextuality, our photonic quantum simulation provides the first step towards a physically robust methodology for realizing topological quantum computation., 11+6 pages, 5+2 figures, 1+2 tables, presentation extended and improved, analysis and results the same, to appear in PRX Quantum

We demonstrate that $\mathbb{Z}_2$ gauge transformations and lattice deformations in Kitaev's honeycomb lattice model can have the same description in the continuum limit of the model in terms of chiral gauge fields. The chiral gauge fields are coupled to the Majorana fermions that satisfy the Dirac dispersion relation in the non-Abelian sector of the model. For particular values, the effective chiral gauge field becomes equivalent to the $\mathbb{Z}_2$ gauge field, enabling us to associate effective fluxes to lattice deformations. Motivated by this equivalence, we consider Majorana-bounding $\pi$ vortices and Majorana-bounding lattice twists and demonstrate that they are adiabatically connected to each other. This equivalence opens the possibility for novel encoding of Majorana-bounding defects that might be easier to realise in experiments., Comment: 17 pages, 11 figures

It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents. Nevertheless, it is not known how accurately curved geometry can describe actual microscopic models. Here, we demonstrate that the low-energy properties of the Kitaev honeycomb lattice model, a topological superconductor that supports localised Majorana zero modes at its vortex excitations, are faithfully described in terms of Riemann-Cartan geometry. In particular, we show analytically that the continuum limit of the model is given in terms of the Majorana version of the Dirac Hamiltonian coupled to both curvature and torsion. We numerically establish the accuracy of the geometric description for a wide variety of couplings of the microscopic model. Our work opens up the opportunity to accurately predict dynamical properties of the Kitaev model from its effective geometric description., 16 pages, 10 figures

5 pags., 5 figs., Here we provide a general methodology to directly measure the topological currents emerging in the optical lattice implementation of the Haldane model. Alongside the edge currents supported by gapless edge states, transverse currents can emerge in the bulk of the system whenever the local potential is varied in space, even if it does not cause a phase transition. In optical lattice implementations the overall harmonic potential that traps the atoms provides the boundaries of the topological phase that supports the edge currents, as well as providing the potential gradient across the topological phase that gives rise to the bulk current. Both the edge and bulk currents are resilient to several experimental parameters such as trapping potential, temperature, and disorder. We propose to investigate the properties of these currents directly from time-of-flight images with both short-time and long-time expansions., This work was supported by the EPSRC Grant No. EP/R020612/1; Spanish Projects PGC2018-094792-B-I00 (MCIU/AEI/FEDER, EU), PGC2018-094180-B-I00 (MCIU /AEI/FEDER, EU), and FIS2015-63770-P (MINECO /FEDER, EU); CAM/FEDER Project No. S2018/TCS-4342 (QUITEMAD-CM), and CSIC Research Platform PTI-001.

We employ the interaction distance to characterise the physics of a one-dimensional extended XXZ spin model, whose phase diagram consists of both integrable and non-integrable regimes, with various types of ordering, e.g., a gapless Luttinger liquid and gapped crystalline phases. We numerically demonstrate that the interaction distance successfully reveals the known behaviour of the model in its integrable regime. As an additional diagnostic tool, we introduce the notion of "integrability distance" and particularise it to the XXZ model in order to quantity how far the ground state of the extended XXZ model is from being integrable. This distance provides insight into the properties of the gapless Luttinger liquid phase in the presence of next-nearest neighbour spin interactions which break integrability., 12 pages, 9 figures

Majorana fermions are a fascinating medium for discovering new phases of matter. However, the standard analytical tools are very limited in probing the non-perturbative aspects of interacting Majoranas in more than one dimensions. Here, we employ the holographic correspondence to determine the specific heat of a two-dimensional interacting gapless Majorana system. To perform our analysis we first describe the interactions in terms of a pseudo-scalar torsion field. We then allow fluctuations in the background curvature thus identifying our model with a (2 + 1)-dimensional Anti-de Sitter (AdS) geometry with torsion. By employing the AdS/CFT correspondence, we show that the interacting model is dual to a (1 + 1)-dimensional conformal field theory (CFT) with central charge that depends on the interaction coupling. This non-perturbative result enables us to determine the effect interactions have in the specific heat of the system at the zero temperature limit., SCOPUS: ar.j, info:eu-repo/semantics/published

10 pags., 8 figs., 1 tab., 1 app., We provide evidence that, alongside topologically protected edge states, two-dimensional Chern insulators also support localized bulk states deep in their valence and conduction bands. These states manifest when local potential gradients are applied to the bulk, while all parts of the system remain adiabatically connected to the same phase. In turn, the bulk states produce bulk current transverse to the potential difference. This occurs even when the potential is always below the energy gap, where one expects only edge currents to appear. Bulk currents are topologically protected and behave as edge currents under an external influence, such as temperature or local disorder. Detecting topologically resilient bulk currents offers a direct means to probe the localized bulk states., This work was supported by the EPSRC Grant No. EP/R020612/1, Spanish Projects No. PGC2018-094792-B-I00 (MCIU/AEI/FEDER, EU), No. PGC2018-094180-B-I00 (MCIU/AEI/FEDER, EU), No. FIS2015-63770-P (MINECO/FEDER, EU), CAM/FEDER Project No. S2018/TCS-4342 (QUITEMAD-CM), and CSIC Research Platform PTI-001.

Free fermion systems enjoy a privileged place in physics. With their simple structure they can explain a variety of effects, ranging from insulating and metallic behaviours to superconductivity and the integer quantum Hall effect. Interactions, e.g. in the form of Coulomb repulsion, can dramatically alter this picture by giving rise to emerging physics that may not resemble free fermions. Examples of such phenomena include high-temperature superconductivity, fractional quantum Hall effect, Kondo effect and quantum spin liquids. The non-perturbative behaviour of such systems remains a major obstacle to their theoretical understanding that could unlock further technological applications. Here, we present a pedagogical review of "interaction distance" [Nat. Commun. 8, 14926 (2017)] -- a systematic method that quantifies the effect interactions can have on the energy spectrum and on the quantum correlations of generic many-body systems. In particular, the interaction distance is a diagnostic tool that identifies the emergent physics of interacting systems. We illustrate this method on the simple example of a one-dimensional Fermi-Hubbard dimer., Comment: 27 pages, 8 figures; submission to SciPost

Thermalisation is a probabilistic process. As such, it is generally expected that when we increase the temperature of a system its classical behaviour dominates its quantum coherences. By employing the Gibbs state of a translationally invariant quantum spin liquid - Kitaev's honeycomb lattice model - we demonstrate that an insulating phase at $T=0$ becomes metallic purely by increasing temperature. In particular, we compute the finite temperature distribution of energies and show that it diverges logarithmically, as we move to small energies. The corresponding wave functions become critical alike as at Anderson transitions. These characteristics are obtained within a new exact Monte Carlo method that simulates the finite temperature behaviour of the Kitaev model. In particular, we take into account the projection onto the physical parity sectors, required for identifying the topological degeneracy of the model. Our work opens the possibility to detect thermal metal behaviour in spin liquid experiments., Comment: some content rearranged, small text changes

Geometric phases, generated by cyclic evolutions of quantum systems, offer an inspiring playground for advancing fundamental physics and technologies, alike. Intriguingly, the exotic statistics of anyons realised in physical systems can be interpreted as a topological version of geometric phases. However, non-Abelian statistics has not yet been demonstrated in the laboratory. Here we employ an all optical quantum system that simulates the statistical evolution of Majorana fermions. As a result we experimentally realise non-Abelian Berry phases with the topological characteristic that they are invariant under continues deformations of their control parameters. We implement a universal set of Majorana inspired gates by performing topological and non-topological evolutions and investigate their resilience against perturbative errors. Our photonic experiment, while it is not scalable, it suggests the intriguing possibility of experimentally simulating Majorana statistics with scalable technologies., Comment: 8 pages, 4 figures

Density functional theory maps an interacting Hamiltonian onto the Kohn-Sham Hamiltonian, an explicitly free model with identical local fermion densities. Using the interaction distance, the minimum distance between the ground state of the interacting system and a generic free fermion state, we quantify the applicability and limitations of the exact Kohn-Sham model in capturing the various properties of the interacting system. As a byproduct, this distance determines the optimal free state that reproduces the entanglement properties of the interacting system as faithfully as possible. The parent Hamiltonian of the optimal free state identifies a system that can determine the expectation value of any observable with controlled accuracy. This optimal entanglement model opens up the possibility of extending the systematic applicability of auxiliary free models into the non-perturbative, strongly-correlated regimes., Comment: 7+2 pages, 3+1 figures, presentation improved

Sir Peter Knight is a pioneer in quantum optics which has now grown to an important branch of modern physics to study the foundations and applications of quantum physics. He is leading an effort to develop new technologies from quantum mechanics. In this collection of essays, we recall the time we were working with him as a postdoc or a PhD student and look at how the time with him has influenced our research.

The boundary of a 2D topological superconductor can be modeled by a conformal field theory. Here we demonstrate the behaviors of this high level description emerging from a microscopic model at finite temperatures. To achieve that, we analyze the low energy sector of Kitaev's honeycomb lattice model and probe its energy current. We observe that the scaling of the energy current with temperature reveals the central charge of the conformal field theory, which is in agreement with the Chern number of the bulk. Importantly, these currents can discriminate between distinct topological phases at finite temperatures. We assess the resilience of this measurement of the central charge under coupling disorder, bulk dimerisation and defects at the boundary, thus establishing it as a favorable means of experimentally probing topological superconductors., Comment: updated -- minor tweaks to text and figures

We experimentally simulate the exchange of Majorana zero modes with a photonic quantum simulator. Our work opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures.

In this review we present an introduction to topological quantum computation – quantum computing with anyons. This approach is inherently resilient against errors, thus promising to overcome one of the main obstacles for the realisation of quantum computers. We first provide an introduction to anyon models and discuss the general steps how to use them to encode and process quantum information. Then we present a toy microscopic model, Kitaev’s p-wave wire, that supports localised Majorana modes. These are the simplest non-Abelian anyons, which are the subject of intense theoretical and experimental research. We outline how quantum computation could be carried out in this microscopic system in a topologically protected manner, discuss the nature of the topological fault tolerance and review the recent experimental developments in realizing Majorana modes.

Very few topological systems with long-range couplings have been considered so far due to our lack of analytic approaches. Here we extend the Kitaev chain, a 1D quantum liquid, to infinite-range couplings and study its topological properties. We demonstrate that, even though topological phases are intimately linked to the notion of locality, the infinite-range couplings give rise to topological zero and nonzero energy Majorana end modes depending on the boundary conditions of the system. We show that the analytically derived properties are to a large degree stable against modifications to decaying long-range couplings. Our work opens new frontiers for topological states of matter that are relevant to current experiments, where systems with interactions of variable range can be designed.

Helical Majorana edge states at the 2D boundaries of 3D topological superconductors can be gapped by a surface Zeeman field. Here we study the effect nested defects imprinted on the Zeeman field can have on the edge states. We demonstrate that depending on the configuration of the field we can induce dimensional reduction of gapless Majorana modes from 2D to 1D or quasi-0D at magnetic domain walls. We determine the nature of the Majorana localisation on these defects as a function of the magnitude and configuration of the Zeeman field. Finally, we observe a generalisation of the index theorem governing the number of gapless modes at the interface between topologically non-trivial systems with partial Chern numbers., 11 pages, 9 figures

We evaluate the entanglement entropy of exactly solvable Hamiltonians corresponding to general families of three-dimensional topological models. We show that the modification to the entropic area law due to three-dimensional topological properties is richer than the two-dimensional case. In addition to the reduction of the entropy caused by a nonzero vacuum expectation value of contractible loop operators, a topological invariant emerges that increases the entropy if the model consists of nontrivially braiding anyons. As a result the three-dimensional topological entanglement entropy provides only partial information about the two entropic topological invariants.

The non-Abelian Kaluza–Klein reduction of conformally flat spaces is considered for arbitrary dimensions and signatures. The corresponding equations are particularly elegant when the internal space supports a global Killing parallelization. Assuming this imposes the generalized ‘spacetime’ to be maximally symmetric with holonomy in the unitary quaternionic group S p ( d / 4 ) . Recalling an analogous result for the complex case, we conclude that all special manifolds with constant properly ‘holonomy-related’ sectional curvature, are in natural correspondence with conformally flat, possibly non-Abelian, Kaluza–Klein spaces.

We consider topological quantum memories for a general class of abelian anyon models defined on spin lattices. These are non-universal for quantum computation when restricting to topological operations alone, such as braiding and fusion. The effects of additional non-topological operations, such as spin measurements, are studied. These are shown to allow universal quantum computation, while still utilizing topological protection. Our work gives an insight into the relation between abelian models and their non-abelian counterparts., 10 pages, 3 figures, presented at 6th QPL workshop

A study of the thermal properties of two-dimensional topological lattice models is presented. This work is relevant to assess the usefulness of these systems as a quantum memory. For our purposes, we use the topological mutual information $I_{\mathrm{topo}}$ as a "topological order parameter". For Abelian models, we show how $I_{\mathrm{topo}}$ depends on the thermal topological charge probability distribution. More generally, we present a conjecture that $I_{\mathrm{topo}}$ can (asymptotically) be written as a Kullback-Leitner distance between this probability distribution and that induced by the quantum dimensions of the model at hand. We also explain why $I_{\mathrm{topo}}$ is more suitable for our purposes than the more familiar entanglement entropy $S_{\mathrm{topo}}$. A scaling law, encoding the interplay of volume and temperature effects, as well as different limit procedures, are derived in detail. A non-Abelian model is next analysed and similar results are found. Finally, we also consider, in the case of a one-plaquette toric code, an environment model giving rise to a simulation of thermal effects in time.

A doublet of three-dimensional Dirac fermions can effectively describe the low energy spectrum of a fermionic cubic lattice. We employ this fermion doubling to encode a non-Abelian SU(2) charge in the fundamental representation. We explicitly demonstrate that suitable distortion of the tunnelling couplings can introduce a scalar and a Yang-Mills field in the effective low energy description, both coupled to the Dirac fermions. The simplicity of the model suggests its physical implementation with ultra-cold atoms or molecules.

Fox’s H-function provide a unified and elegant framework to tackle several physical phenomena. We solve the space fractional diffusion equation on the real line equipped with a delta distribution initial condition and identify the corresponding H-function by studying the small x expansion of the solution. The asymptotic expansions near zero and infinity are expressed, for rational values of the index α , in terms of a finite series of generalized hypergeometric functions. In x-space, the α = 1 stable law is also derived by solving the anomalous diffusion equation with an appropriately chosen infinitesimal generator for time translations. We propose a new classification scheme of stable laws according to which a stable law is now characterized by a generating probability density function. Knowing this elementary probability density function and bearing in mind the infinitely divisible property we can reconstruct the corresponding stable law. Finally, using the asymptotic behavior of H-function in terms of hypergeometric functions we can compute closed expressions for the probability density functions depending on their parameters α , β , c , τ . Known cases are then reproduced and new probability density functions are presented.

Topological phases of matter possess intricate correlation patterns typically probed by entanglement entropies or entanglement spectra. In this work, we propose an alternative approach to assessing topologically induced edge states in free and interacting fermionic systems. We do so by focussing on the fermionic covariance matrix. This matrix is often tractable either analytically or numerically and it precisely captures the relevant correlations of the system. By invoking the concept of monogamy of entanglement we show that highly entangled states supported across a system bi-partition are largely disentangled from the rest of the system, thus appearing usually as gapless edge states. We then define an entanglement qualifier that identifies the presence of topological edge states based purely on correlations present in the ground states. We demonstrate the versatility of this qualifier by applying it to various free and interacting fermionic topological systems., 4+3 pages, 3+3 figures

Summarization: This editorial explains the scope of the special issue and provides a thematic introduction to the contributed papers. Presented on

We consider a graphene sheet folded in an arbitrary geometry, compact or with nanotube-like open boundaries. In the continuous limit, the Hamiltonian takes the form of the Dirac operator, which provides a good description of the low energy spectrum of the lattice system. We derive an index theorem that relates the zero energy modes of the graphene sheet with the topology of the lattice. The result coincides with analytical and numerical studies for the known cases of fullerene molecules and carbon nanotubes and it extend to more complicated molecules. Potential applications to topological quantum computation are discussed., Comment: 4 pages, 1 figure, new version to appear in IJMPB

We analytically derive a compatible family of effective field theories that uniquely describe topological superconductors in 3D, their 2D boundary and their 1D defect lines. We start by deriving the topological field theory of a 3D topological superconductor in class DIII, which is consistent with its symmetries. Then we identify the effective theory of a 2D topological superconductor in class D living on the gapped boundary of the 3D system. By employing the holographic correspondence we derive the effective chiral conformal field theory that describes the gapless modes living on the defect lines or effective boundary of the class D topological superconductor. We demonstrate that the chiral central charge is given in terms of the 3D winding number of the bulk which by its turn is equal to the Chern number of its gapped boundary., 4 pages

The entanglement properties of some novel quantum systems are studied that are inspired by recent developments in cold-atom technology. A triangular optical lattice of two atomic species can be employed to generate a variety of spin-1/2 Hamiltonians including effective three-spin interactions. A variety of one or two dimensional systems can thus be realized that possess multi-degenerate ground states or non-vanishing chirality. The properties of these ground states and their phase transitions are probed with appropriate measures such as the entropic entanglement and the spin chirality., 9 pages, 6 figures, talk presented at the De Giorgi Workshop on Entanglement in Physical and Information Science, Pisa 2004

We present an idealized model involving interacting quantum dots that can support both the dynamical and geometrical forms of quantum computation. We show that by employing a structure similar to the one used in the Aharonov-Bohm effect we can construct a topological two-qubit phase-gate that is to a large degree independent of the exact values of the control parameters and therefore resilient to control errors. The main components of the setup are realizable with present technology., 8 pages, 3 figures, submitted to Jour. of Opt. B (special issue on Quantum Computing)

We discuss an experimental setup where two laser-driven atoms spontaneously emit photons and every photon causes a ``click'' at a point on a screen. By deriving the probability density for an emission into a certain direction from basic quantum mechanical principles we predict a spatial interference pattern. Similarities and differences with the classical double-slit experiment are discussed., Comment: 6 pages, 2 figures, contribution to the symposium "100 Years of Werner Heisenberg - Works and Impact" in Bamberg, to appear in Fortschritte der Physik

The realization of Majorana zero modes is in the centre of intense theoretical and experimental investigations. Unfortunately, their exchange that can reveal their exotic statistics needs manipulations that are still beyond our experimental capabilities. Here we take an alternative approach. Through the Jordan-Wigner transformation, the Kitaev's chain supporting two Majorana zero modes is mapped to the spin-1/2 chain. We experimentally simulated the spin system and its evolution with a photonic quantum simulator. This allows us to probe the geometric phase, which corresponds to the exchange of two Majorana zero modes positioned at the ends of a three-site chain. Finally, we demonstrate the immunity of quantum information encoded in the Majorana zero modes against local errors through the simulator. Our photonic simulator opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures., main text (10 pages, 4 figures) and detailed supplementary information

The quasi one-dimensional transport of Abelian and non-Abelian anyons is studied in the presence of a random topological background. In particular, we consider the quantum walk of an anyon that braids around islands of randomly filled static anyons of the same type. Two distinct behaviours are identified. We analytically demonstrate that all types of Abelian anyons localise purely due to the statistical phases induced by their random anyonic environment. In contrast, we numerically show that non-Abelian Ising anyons do not localise. This is due to their entanglement with the anyonic environment that effectively induces dephasing. Our study demonstrates that localisation properties strongly depend on non-local topological interactions and it provides a clear distinction in the transport properties of Abelian and non-Abelian statistics., Comment: 9 pages, 5 figures