Anyons in Quantum Many-Body Systems

One of the major efforts in recent investigations of the fractional quantum Hall (FQH) effect is to understand the connections between topological order, quantum geometry and symmetry breaking. In this talk, I will discuss how bulk dynamics of FQH states can be studied using a "geometric quench" [Zhao Liu, Andrey Gromov, and Zlatko Papic, Phys. Rev. B 98, 155140 (2018)]. One of the physical consequences of geometric quenches in gapped FQH states is that they expose coherent oscillations of the emergent geometric degree of freedom, which can be viewed as the condensed matter analogue of the graviton particle. The geometric quench is a tool that allows analytical and numerical studies of new types of many-body quantum dynamics in strongly-correlated topological systems away from the zero temperature limit, which is governed by the topological quantum field theory.

One-dimensional systems with topological order are intimately related to the appearance of zero-energy modes localized on their boundaries. The most common example is the Kitaev chain, which displays Majorana zero-energy modes and it is characterized by a two-fold ground state degeneracy related to the global $\mathbb{Z}_2$ symmetry associated with fermionic parity. By extending the symmetry to the $\mathbb{Z}_N$ group, it is possible to engineer systems hosting topological parafermionic modes. In this talk, I will address one-dimensional systems with a generic discrete symmetry group $G$. I will define a ladder model of gauge fluxes that generalizes the Ising and Potts models and displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner transformation, I will map this flux ladder into a model of dyonic operators, defined by the group elements and irreducible representations of $G$. The so-obtained dyonic model has topological order, with zero-energy modes localized at its boundary. These dyonic zero-energy modes are in general weak topological modes, but strong dyonic zero-modes appear when suitable position-dependent couplings are included.

Kitaev magnets are prototypical examples of frustrated spin systems, in which the original spin degrees of freedom fractionalize into itinerant Majorana fermions and a static Z2 gauge field. For conventional Kitaev magnets in 3D, the gauge field orders at low temperatures and the system enters a spin liquid regime. Here we discuss an explicit example that goes beyond this paradigmatic scenario -- the Z2 gauge field is found to be subject to geometric frustration, the thermal ordering transition is suppressed, and an extensive residual entropy remains. Only very deep in the quantum regime, the degeneracy in the gauge sector is lifted by a subtle interplay with the Majorana fermions and the formation of a Majorana metal.

We propose a mean-field ansatz to study phase transitions from a topological phase to a trivial phase. We probe the efficiency of this approach by considering the string-net model in the presence of a string tension for any anyon theory. We also discuss how to go beyond this simple approach.

Symmetry and topology are two of the conceptual pillars that underlie our understanding of matter. While both ideas are old, over the past several years a new appreciation of their interplay has led to dramatic progress in our understanding of topological electronic phases. A paradigm that has emerged is that insulating electronic states with an energy gap fall into distinct topological classes. Interfaces between different topological phases exhibit gapless conducting states that are protected and are impossible to get rid of. In this talk we will discuss the application of this idea to the quantum Hall effect, topological insulators, topological semimetals and topological superconductors. The latter case has led to the quest for observing Majorana fermions in condensed matter, which opens the door to proposals for topological quantum computation. We will close by surveying the frontier of topological phases in the presence of strong interactions.

Ultracold atoms in optical lattices are powerful experimental platforms to study a variety of phenomena ranging from condensed-matter to statistical physics. Recently, a promising new direction was opened by the successful realization of two paradigmatic topological condensed matter models, i.e. the Hofstadter and the Haldane model. Topological states of matter exhibit unique conductivity properties, which are particularly robust against perturbations. One of the most prominent examples is the integer quantum Hall effect, where a two-dimensional electron gas under extreme conditions exhibits a quantized conductivity. Investigating related phenomena with charge-neutral atoms required the development of novel experimental techniques to mimic the behavior of charged particles in magnetic fields. I will introduce some of the most common methods using Floquet engineering, i.e. periodic driving of the system's parameters to emulate the properties of a system that is otherwise not available in static settings. These methods led to the successful generation of artificial magnetic fields in optical lattices and direct observations of the associated non-trivial topological properties in the topological condensed matter models mentioned above. The simulation of complete gauge theories requires additional key ingredients that allow for an interaction between matter and gauge fields. One of the main challenges consists in engineering synthetic gauge fields with internal degrees of freedom that interact with the matter fields in a suitable manner that respects the symmetry of the gauge theory. Here, I will present recent results, where we have used a two-component mixture of ultracold bosonic atoms and resonant periodic driving at the value of the inter-species Hubbard interactions. Choosing particular modulation parameters enables the implementation of a $Z_2$ symmetric model, which we study in an optical double-well potential -- the basic building block of the $Z_2$ lattice gauge theory.

Ultracold atoms in optical lattices are a versatile platform to study the fascinating phenomena of gauge fields and topological matter. Periodic driving can induce topological band structures with non-trivial Chern number of the effective Floquet Hamiltonian and paradigmatic models, such as the Haldane model on the honeycomb latticce, can be directly engineered. Here, I will present recent experiments, in which we realized three new approaches for measuring the Chern number in this system. First, we use a new topological effect - the quantization of the circular dichroism with the Chern number - which appears as the integrated difference of the depletion rates in chiral spectroscopy of both chiralities. Our experiments confirm this quantization and establish depletion-rate measurements as a versatile probe. Second, we study the dynamics of the system after a quench into the topological regime using time-resolved Bloch-state tomography and employ the mapping of the Chern number to the linking number manifesting in the emerging dynamical vortices. In these experiments, we obtain the Chern number directly from a topological object instead of infering it from a quantized response. Third, we apply state-of-the-art deep learning techniques to our momentum-space images and train a network to recognize the Chern number from single images. This allows mapping out the full two-dimensional Haldane phase diagram, which was unfeasible with conventional analysis. These new approaches to topology also define a promising starting point for probing topological order of interacting systems such as fractional Chern insulators.

In the quantum Hall effect, determining the statistic properties of quasi-electrons has remained more problematic in comparison to quasi-holes. In this talk, I will quickly review the properties of the various proposals for quasi-electrons, and their shortcomings. By using the MPS approach, we study the CFT formulation quasi-electrons. This reveals on the one hand that the shortcoming of this proposal is rooted in the screening properties, but also how to screen the quasi-electrons, so that they behave in the proper way.

We argue that a correlated fluid of electrons and holes can exhibit a fractional quantum Hall effect at zero magnetic field analogous to the Laughlin state at filling 1/m. We introduce a variant of the Laughlin wavefunction for electrons and holes and show that for m=1 it describes a Chern insulator that is the exact ground state of a free fermion model with p_x + i p_y excitonic pairing. For m>1 we develop a composite fermion mean field theory, and we will give several pieces of evidence that our wavefunction correctly describes this phase. We will present physical arguments that the m=3 state can be realized in a system with C_6 rotational symmetry in which energy bands with angular momentum that differ by 3 cross at the Fermi energy. This leads to a gapless state with (p_x + i p_y)^3 excitonic pairing, which we argue is conducive to forming the fractional excitonic insulator in the presence of interactions. Prospects for numerics on model systems and band structure engineering to realize this phase in real materials will be discussed.

Silicene, the silicon counterpart of graphene, possibly hosting a remarkable Quantum Spin Hall effect [1], has become a “Hot Research Front” [2], since its initial synthesis by epitaxy on Ag(111) in 2012 [3]. It has further given a quick to research on related elemental low-dimensional materials called Xenes. Typically, Kagome silicene [4], nearly planar two-dimensional (2D) germanene [5] and stanene [6], as well as 1D Si nanoribbons solely composed of pentagonal moieties [7] and 0D nanodots [7,8], with unique characteristics, have been artificially created on different substrates. The electronic properties of silicene have been rapidly exploited to fabricate ultra-thin planar devices, typically field-effect transistors with a monolayer silicene channel have been already fabricated in 2015 [9]. In my talk, I will present the realization and properties of these new exotic low-dimensional materials and draw perspectives for future research, e.g., the quest for Majoranas. References: [1] C.-C. Liu, W. Feng, Y. Yao, Phys. Rev. Lett. 107 076802 (2011). [2] C. Day, Physics Today, September 25, 2015 [3] P. Vogt et al., Phys. Rev. Lett. 108 155501 (2012). [4] Y. Sassa et al., to be submitted. [5] M. E. Dávila et al., New Journal of Physics 16 095002 (2014). [6] J. Yuhara et al., 2D Materials 5 025002 (2018). [7] J. Cerdá et al., Nature Comm. 7 13076 (2016). [8] S. Sheng et al., Nano Lett. 18 2937 (2018). [9] L. Tao et al., Nature Nanotechnol. 10 227 (2015). Guy Le Lay PIIM - CNRS, Aix-Marseille University, France E-mail: guy.lelay@univ-amu.fr

We propose a standard time-of-flight experiment as a method for observing the anyonic statistics of quasiholes in a fractional quantum Hall state of ultracold atoms. The quasihole states can be stably prepared by pinning the quasiholes with localized potentials and a measurement of the mean square radius of the freely expanding cloud, which is related to the average total angular momentum of the initial state, offers direct signatures of the statistical phase. Our proposed method is validated by Monte Carlo calculations for $\nu = 1/2$ and 1/3 fractional quantum Hall liquids containing a realistic number of particles.

We discuss the stabilization of chiral phases in realistic models of ultra cold fermions with SU(N) symmetry in optical lattices. Concentrating on the triangular lattice with one particle per site, we show that a chiral phase is stabilized by a complex ring exchange term for N from 3 to 9. Such a term, which breaks time-reversal symmetry, is expected to be the leading correction to the Heisenberg model in a strong coupling expansion of the Hubbard model with an artificial gauge field. In addition, we provide numerical evidence suggesting that, even in the absence of an artificial gauge field, a chiral phase that spontaneously breaks time-reversal symmetry is stabilized for the SU(3) Hubbard model between the three-sublattice ordered phase at very strong coupling and the weak-coupling metallic phase.

We consider the behaviour of quantum Hall edges away from the Luttinger liquid fixed point that occurs in the low energy, large system limit. Using the close links between quantum Hall wavefunctions and conformal field theories we construct effective Hamiltonians from general principles and then constrain their forms by considering the effect of bulk symmetries on the properties of the edge. In examining the effect of bulk interactions on this edge we find remarkable simplifications to these effective theories which allow for a very accurate description of the low-energy physics of quantum Hall edges relatively far away from the Luttinger liquid fixed point, and which apply to small systems and higher energies Richard Fern, Roberto Bondesan, Steven H. Simon, arXiv:1805.04108

Superfluidity is one of the most intriguing quantum phenomena that emerges at temperatures close to absolute zero. Vortices with quantized circulation are the hallmark of superfluidity and are created under external rotation. If the number of vortices is much smaller than the number of particles, vortices arrange themselves in a triangular lattice that exhibits intriguing collective dynamics. Quantum vortex crystals in superfluids and superconductors have been one of the exciting avenues of research for many decades. In this talk I will introduce a low-energy effective field theory of a two-dimensional bosonic superfluid on the lowest Landau level at zero temperature and identify a Berry term that governs the dynamics of coarse-grained superfluid degrees of freedom. For an infinite vortex crystal I will show how the Berry term affects the low-energy spectrum of Tkachenko collective oscillations and will present non-dissipative Hall responses of the particle number current and stress tensor.

Inhomegenous quantum many body systems exhibit novel phenomena such as the breaking of the area law of entanglement entropy. A notable example is the so called rainbow model which consists of a XX spin chain where the exchange coupling constant decreases exponentially from the center towards the edges. This system has been analyzed in great detail using the strong disorder renormalization group method and conformal field theory. These methods explain the appearance of long range valence bonds across the spin chain. In this talk, we shall explain yet another feature which is the appearance of topological order in the rainbow chain for an odd number of sites. The latter order seems to be related to the Haldane topological order of a folded chain. This results suggest that the short range order that characterizes the SPT phases may be related to the long range critical phases characterized by global anomalies.

In this talk I will describe latest effort in visualizing anyons in condensed matter systems. I will describe our recent efforts in creating Majorana using a higher order topological insulator where combination of magnetism and superconductivity can be used to realize the Fu-Kane proposal using one-dimensional topological hinge states. In addition, I will describe proposal on how to probe anyons in fractional quantum Hall states using STM spectroscopic imaging near individual defects.

Geometric phases, generated by cyclic evolutions of quantum systems, offer an inspiring playground for advancing fundamental physics and technologies alike. The exotic statistics of anyons realized 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. I will present an all-optical quantum system that simulates the statistical evolution of Majorana fermions. This experiment realizes non-Abelian Berry phases with the topological characteristic that they are invariant under continuous deformations of their control parameters. A universal set of Majorana-inspired gates is implemented by performing topological and nontopological evolutions and their resilience against perturbative errors is investigated.

The standard implementation of topological quantum computation (TQC) relies on non-Abelian anyons and their exotic braiding statistics. However, there are also other routes to achieve and even facilitate the realization of TQC. In particular, two attractive possibilities were recently proposed. One of them is to introduce and braid twist defects which carry non-Abelian properties even when hosted by Abelian topological states. The other strategy is to utilize gapped boundaries in topological states, which allow universal TQC even though the underlying intrinsic anyons alone do not support it. The first step to implement these beautiful ideas is to investigate the realizations of topological phases compatible with twist defects or gapped boundaries. In this talk, instead of using the effective field theory, I will visit this problem by numerical approaches in microscopic lattice models. I will show compelling results that explicitly demonstrate the existence of lattice fractional quantum Hall states on effective high-genus surfaces induced by twist defects or gapped boundaries.

Abelian and non-Abelian topological phases exhibiting protected chiral edge modes are ubiquitous in the realm of the Fractional Quantum Hall (FQH) effect. Here, we investigate a spin-1 Hamiltonian on the square lattice which could, potentially, host the spin liquid analog of the (bosonic) non-Abelian Moore-Read FQH state, as suggested by Exact Diagonalisation of small clusters. Using families of fully SU(2)-spin symmetric and translationally invariant chiral Projected Entangled Pair States (PEPS), variational energy optimization is performed using infinite-PEPS methods, providing good agreement with Density Matrix Renormalisation Group (DMRG) results. A careful analysis of the bulk spin-spin and dimer-dimer correlation functions in the optimized spin liquid suggests that they exhibit long-range "gossamer tails". We argue these tails are finite-D artifacts of the chiral PEPS, which become irrelevant when the PEPS bond dimension D is increased. From the investigation of the entanglement spectrum, we observe sharply defined chiral edge modes following the prediction of the SU(2)2 Wess-Zumino-Witten theory and exhibiting a conformal field theory (CFT) central charge c=3/2, as expected for a Moore-Read chiral spin liquid. We conclude that the PEPS formalism offers an unbiased and efficient method to investigate non-Abelian chiral spin liquids in quantum antiferromagnets.

Dynamical response functions encode characteristic features of the emergent excitations in quantum many body systems. We introduce a matrix-product state based method to efficiently obtain these dynamical response functions for general two-dimensional lattice Hamiltonians. First, we apply this method to different phases of the Kitaev-Heisenberg model. Here we find significant broad high energy features beyond spin-wave theory even in the ordered phases proximate to spin liquids. This includes the phase with zig-zag order of the type observed in α-RuCl3, where we find high energy features like those seen in inelastic neutron scattering experiments. Second, we consider gapped spin liquid systems and study the dynamics of anyonic excitations.

The infinite Projected Entangled-Pair State (iPEPS) algorithm is one of the most efficient techniques for studying the ground-state properties of two-dimensional quantum lattice Hamiltonians in the thermodynamic limit. Here, we show how the algorithm can be adapted to explore nearest-neighbor local Hamiltonians on the ruby and triangle-honeycomb lattices, using the Corner Transfer Matrix (CTM) renormalization group for 2D tensor network contraction. Additionally, we show how the CTM method can be used to calculate the ground state fidelity per lattice site and the boundary density operator and entanglement entropy (EE) on an infinite cylinder. As a benchmark, we apply the iPEPS method to the ruby (two-body color-code) model with anisotropic interactions and explore the ground-state properties of the system. We further extract the phase diagram of the model in different regimes of the couplings by measuring two-point correlators, ground state fidelity and EE on an infinite cylinder. Our phase diagram is in agreement with previous studies of the model by exact diagonalization.

Strongly interacting many-body systems consisting of fermions or bosons can host exotic quasi- particles with anyonic statistics. Here, we demonstrate that many-body systems of anyons can also form anyonic quasi-particles. The charge and statistics of the emergent anyons can be different from those of the original anyons.

Kitaev's honeycomb model is a remarkable exactly solvable spin-1/2 model, consisting of bond dependent Ising interactions, and with a gapless quantum spin liquid ground state. Even more remarkably such interactions are actually realized in a number of spin-orbit entangled Mott insulators, alongside other more conventional interactions. Here, motivated by the recent surge in interest in the behaviour of these materials in a magnetic field, we map out the complete phase diagram of the pure Kitaev model in tilted magnetic fields. Besides the expected gapped quantum spin liquid at low fields and the trivial polarized state at high fields we report the emergence of a distinct gapless quantum spin liquid at intermediate field strengths. Analyzing a number of static, dynamical, and finite temperature quantities using numerical exact diagonalization techniques, we find strong evidence that this phase exhibits gapless fermions coupled to a massless gauge field resulting in a dense continuum of low-energy states. We discuss its stability in the presence of perturbations, Heisenberg and off-diagonal symmetric exchange interactions, that naturally arise in spin-orbit entangled Mott insulators alongside Kitaev interactions.

Compressible fractional quantum Hall state provides a unique arena to explore non-fermi liquid behaviors, described by a gapless fermi surface coupled to emergent gauge fields. Based on exact diagonalization (ED), we observe a direct transition from composite fermi liquid (CFL) state to stripe ordered phase as a function of the mixing between wavefunction of zeroth and first Landau levels. Distinct from the pairing instability of CFL, which leads to Moore-Read state, we find by both ED and RG analysis that the transition to stripe phase shows strong discontinuity due to the fluctuations of gauge field. Such finding may shed light to the possible instability of a non-fermi liquid state.

In this talk, I will present theoretical studies on two topics about fusions of Majorana fermions. In the first topic [1], we propose a single Josephson junction to detect a non-Abelian statistics effect of Majorana fermions, formed by two finite-size s-wave superconductors on a topological insulator under a magnetic field. At certain field strengths, a train of three localized Majorana fermions appears along the junction, while an extended chiral Majorana fermion encircles the train and the superconductors. A DC voltage bias across the junction causes the train to move and collide with the extended Majorana fermion. This involves interchange of fusion partners among the four Majorana fermions, leading to non-Abelian state evolution. The evolution gives rise to a $2n \pi$ fractional AC Josephson effect with an arbitrary integer $n \ge 2$ tunable by the voltage. In the second topic [2], we study topological entanglement in a one-dimensional topological superconductor. We find that the entanglement appears in the bulk of the 1D system, its properties depend on the topological classification of the superconductors (the number of end Majorana fermions), and the entanglement exhibits nontrivial temperature dependence. In a system having one Majorana fermion at the each end, the topological entanglement has a Bell-state form at zero temperature and decays as the temperature increases, vanishing suddenly at a certain finite temperature. In a system having two Majorana fermions at each end, it is in a cluster-state form and its nonlocality is more noticeable at a finite temperature. [1] Sang-Jun Choi and H.-S. Sim, Non-Abelian Evolution of a Majorana Train in a Single Josephson Junction, arXiv:1808.08714 [2] YeJe Park, Jeongmin Shim, S.-S. B. Lee, and H.-S. Sim, Nonlocal Entanglement of 1D Thermal States Induced by Fermion Exchange Statistics, Phys. Rev. Lett. 119, 210501 (2017).