See also the related analysis of Ref. The Kitaev Model on the Honeycomb Lattice H = X K S i S j Superposition Majorana fermions in Section II and compute the disorder averaged partition function for the SYK model with q= 2 and q= 4 interactions. k = ( c k c k ) The energy spectrum for particle-hole symmetry is symmetric about zero. FIG. The results of this thesis indicate that these Majorana acquire a relevant physical meaning. The Sachdev-Ye-Kitaev (SYK) model describes a collection of randomly interacting Majorana fermions that exhibits profound connections to quantum chaos and black holes. Majorana edge modes in the Kitaev model Manisha Thakurathi, K. Sengupta, Diptiman Sen We study the Majorana modes, both equilibrium and Floquet, which can appear at the edges of the Kitaev model on the honeycomb lattice. 2009 , Kells et al. We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. However, the Majorana representation enlarges the physical Hilbert space of the half spin by twice. honeycomb lattice square-octagon lattice (Kitaev 2006) (Yang et al. The Kitaev model 3. The fact that the operator is localized means that these ground states differ only on the edge. 2020-11-26T15:00:00. In this chapter, we consider Kitaev's honeycomb lattice model (Kitaev, 2006). Two- and three-dimensional Kitaev magnets are prototypical frustrated quantum spin systems, in which the original spin degrees of freedom fractionalize into Majorana fermions and of \Kitaev" quantum spin liquids. Interactions and disorder intrinsic to the The prototypical toy model possessing Majorana zero modes is the Kitaev chain with open boundaries [12], a one-dimensional tight-binding model for spinless fermions in the presence of p-wave superconducting pairing. function and conductance of a Kitaev model. H = k k ( k k 1 / 2) (ii) We investigate the Majorana bound states. In the pure Kitaev model, when the magnetic field is rotated within the ac plane, C h changes its sign at +35; C h = +1 for = 60 and 45, and C h = 1 for = +60 and +45 (Fig. at the edges of the chain! Majorana bound states, Kitaev model 3. Clearly by setting = 0 you have cut the first and last Majorana mode from the rest of the chain. $\begingroup$ @RoderickLee My feeling is that your confusion is somehow related to what it means for "the Majorana mode to be localized".

of our model indicate that the bulk-edge correspondence can be extended to a single-band system with hidden topological feature. With the lattice, the topological number can be de ned naturally.

Review of Sachdev-Ye-Kitaev model Numerical studies of ETH in SYK model ETH in the Schwarzian sector of the SYK model ETH in the Conformal sector of the SYK model A few thoughts on the bulk duals Summary and conclusion!3. Abstract We investigate the topological properties of a ladder model of the Kitaev superconductor chains with a periodically modulated chemical potential. The pure Kitaev model is solved by representing the half spin on each site with four Majorana fermions , , , as [ 16] or a JordanWigner transformation of half spins. FIG. In this work, we demonstrate that a finite density of random vacancies in the Kitaev model gives rise to a striking pileup of low-energy Majorana eigenmodes and reproduces the apparent power-law upturn in the specific heat measurements of H 3 LiIr 2 O 6. The interest in studying Majorana end modes (MMs) was spurred on by a proposal by Kitaev and Preskill7 outlining a way to realise quantum com- Kitaev Model and Majoranas Som Phene(15D110001) Adway Gupta(175120034) Overview Majoranas as a transformation 1-D Kitaev Hamiltonian unpaired Majorana modes Imagine if we have a regular set of fermionic operators C and C t which we transform into a different set according to the rules Ct + 62) and

We study a system of interacting spinless fermions in one dimension which, in the absence of interactions, reduces to the Kitaev chain [A. Yu Kitaev, Phys.-Usp. the pure Kitaev model, how to solve it, and what types of Z2 QSLs can occur. 1) Kitaev honeycomb model as Majorana fermions in Z2 gauge field (0 or flux) characterized by integer Chern number, modulo 16 2) Triangular vortex lattices : Chern = 0,1,,6,8 but not 7 3) Effective models in the dilute vortex limit For hole, it is k / 2 and for electron it is k / 2. In Section 3, we briey explain the symmetry properties of materials and how Kitaev interactions arise. Kitaev Materials Laboratory Sighting of Majorana Fermions ?? Majorana fermion, one can readWilczek,2009. The model exhibits two characteristic temperatures, ${T}_{L}$ and ${T}_{H}$, which correspond to energy armadillo majorana nlohmann-json kitaev-model rashba-model Updated Mar 25, 2021; C++; Add a description, image, and links to the kitaev-model topic page so that developers can more easily learn about it.

In this flux sector, the Majorana fermion system has linear dispersions and shows power-law behavior in Unpaired Majorana modes in the gapped phase of Kitaev's honeycomb model Here we construct fermionic all-to-all Floquet quantum circuits of random four-body gates designed to capture key features of SYK dynamics. The Kitaev honeycomb spin model provides a prominent example of exotic fractionalized quasiparticles, composed of itinerant Majorana fermions and gapped gauge fluxes. We propose a solid-state implementation based on a quantum dot coupled to an array of topological superconducting wires hosting Majorana zero modes. In this talk, we will discuss our approach using time-dependent Majorana mean-field theory to compute the results of inelastic neutron scattering experiments in models near, but beyond, the Kitaev model. We propose an approach to detect the peculiarity of Majorana fermions at the edges of Kitaev magnets. Majorana Fermions and Topological Quantum Computation 1. At large magnetic elds the system enters a spin-polarized paramagnetic phase. Special case: Topological regime: Majorana fermions (e= =0!!!) Introduction Kitaev model Non-abelian statistics Hosting and detecting Majorana particles Conclusion and future directions Pairs of Majorana fermions can be combined into ordinary fermions c = 1 2 (1 + i 2);cy= 1 2 (1 i 2); form a single 2 level system If the Majorana fermions are spatially separated, implies fermion state is delocalised, The magic stick rule still holds, but one Majorana species is free to hop in the presence of a static gauge eld. The red circles de-note the dierent, representative model parameter points that In the language of second quantization, this means that = y, i.e. 2.1 Hamiltonian. In the non-interacting case, a signal of topological order appears as zero-energy modes localized near the edges. 2. It is the first result of its kind in non-trivial quantum spin models. Kitaev chain model Kitaev provides a simple platform to study the Majorana zero modes, which has recently attracted a lot of attentions Wilczek ; Elliott . Majorana states on TI edges 5. The ground state is topo-logically This plays the same role as the NNN hopping inHaldane model [11] and opens a topological gap at theDirac cone of Majorana fermions. Topological superconductor 4. The numerics reveal an impurity entropy which can be explained by localized Majorana fermions. The Kitaev model exhibits a canonical quantum spin liquid as a ground state and hosts two fractional quasiparticles, itinerant Majorana fermion and localized flux excitation. Such modes are manifest in a toy model called Kitaev chain6 modelling a one-dimensional p-wave super-conducting wire. The S=3/2 Kitaev honeycomb model (KHM) is a quantum spin liquid (QSL) state coupled to a static Z 2 gauge field. [ 17] In this work, we adopt the Majorana representation. Magnetic fields can give rise to a plethora of phenomena in Kitaev spin systems, such as the formation of non-trivial spin liquids in two and three spatial dimensions. Codes for Majorana zero modes identification in non-interacting systems. Majorana in wires 6. We present certain exact analytical results for dynamical spin correlation functions in the Kitaev Model.

The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z 2 gauge fields. 2007 , Baskaran et al. A quantum spin liquid appears as the ground state of the Kitaev model in the flux-free sector, which has intensively been investigated so far. Koga, Akihisa We investigate the anisotropic S = 1/2 Kitaev model on the honeycomb lattice with the ordered-flux structure. The exactly solvable Kitaev model on the honeycomb lattice has recently received enormous attention linked to the hope of achieving novel spin-liquid states with fractionalized Majorana-like excitations. Intending to emulate Kitaev chain, we build a tight-binding model of a 3-site quantum dot chain.

The Kitaev model is a quantum spin model with localized spin-1/2 magnetic moments with bond-dependent anisotropic interactions. domly interacting SU(M) spins [7], the model has seen a recent surge of interest due to the insight by Kitaev link-ing a Majorana fermion variant of the model to a gravi-tational dual [8]: a two-dimensional nearly anti-de Sitter space, which arises for near-extremal black holes [8{10]. tum computation. Kitaev chain is a theoretical model of a one-dimensional topological superconductor with Majorana zero modes at the two ends of the chain. 2010) In contrast to previous studiesbased on Hermitian chains in the thermodynamic limit, we focus on the Kitaev model on a nite lattice system.This is motivated by the desire to get a clear physicalpicture of the edge mode through the investigation of asmall system. In the year 1937, a new class of particles that are its own anti-particles were hypothesized by Ettore Majorana. Through the representation of the Kitaev model in terms of quasi-particles an elegant description of a complex, strongly correlated system is possible. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z 2 gauge fields. Kitaev model on a quantum computer using VQE with Majorana fermions Abstract We study the simulation of the Kitaev spin model on quantum computers. Topological quantum computation 6. We then apply our new formulas in chapter 6 to observe how we can manipulate the Majorana fermions in the Kitaev model. The key insight Kitaev provided is that this can be solved by \splitting" the fermionic site into two Majorana modes, which can then be spatially separated. This model can be rewritten as a free Majorana fermion system coupled with Z 2 variables. The (localized) Majorana operator transforms you between different (degenerate) ground states. In recent years, the Kitaev honeycomb model has been the focus of much attention, as it is a solvable example of a quantum spin liquid that hosts exotic Majorana excitations. (a) Kitaevs model describes a two-dimensional system of spins S = 1 / 2 on a honeycomb lattice interacting through a strongly anisotropic exchange interaction. 1. Gapped: topological.

We also discuss the nite temperature behavior. k = 2 J 1 + g 2 2 g cos ( k a) If we do Bogoliubov transformation of Fourier transformed Hamiltonian, we get. 2.

k. The model seems OK for a start, because it has some superconducting pairing and some normal dispersion given by terms proportional to Special case: What are Majorana fermions anyway? Honeycomb Lattice Model The honeycomb lattice is threefold coordinated. Periodic Table of topological insulators and superconductors 5. We call this model the twisted Kitaev chain, as these interactions are similar to those of the honeycomb Kitaev spin liquid. eld. which is not present in the ferromagnetically coupled model. Here we construct fermionic all-to-all Floquet quantum circuits of random four-body gates designed to capture key features of SYK dynamics. Consider 1- dimensional tight binding chain with spinless fermions and p-orbital hopping. 27) The model was originally introduced on a 2D honeycomb structure, while it can be extended to any tricoordinate structures in any spatial dimensions (some examples will be shown in Sect. A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. Bogoliubov de Gennes Theory 2. The topological class of the system is determined by the We discovered highly unusual

In this paper, we discuss the physics of the Kitaev model in a [001] magnetic eld on various two- and three-dimensional lattice geometries and compare and contrast the FM and AFM cases.

However, identification of the Majorana fermions in a three-dimensional honeycomb lattice remains elusive.

Reflecting both of this peculiarity of the Majorana In particular we focus on the models defined on the honeycomb, and square-octagon lattices. We rst study the present model from thedescription in terms of Majorana fermions. We show that the exact ground states can be obtained analytically even in the where. A. Kitaev, Physics Uspekhi (2001) Lutchyn, Sau, Das Sarma, PRL 2010 Oreg, Refael, von Oppen, PRL 2010 Solid Theoretical Foundation Toy model: First proposal based on existing materials topological insulators L. Fu and C. Kane PRL 2008 Semiconductor nanowire proposals: 1 1 2 2 A.Yu. The Kitaev model is one of the solvable quantum spin models, where the ground state is given by gapped and gapless spin liquids, depending on the anisotropy of the interactions. Hubbard model is provided. Depending on the relative strengths of the interactions, J x, J y, and J z, there are two phases of Eq. We also show that the flux fluctuations tend to open an energy gap in the Majorana spectrum near the gapless-gapped phase boundary. The aim will be to translate the Kitaev Chain Hamiltonian into a Matrix form to obtain energy spectrum and edge modes for an open chain. These results pave a new path to measurement of dynamical spinon or Majorana fermion spectroscopy of Kitaev and other spin-liquid materials. The Majorana bound states are located at domain walls between wire regions with a topological and normal superconducting phase - and this phase can be tuned This is an analytically tractable spin model that gives rise to quasiparticles with Abelian as well as non-Abelian statistics. We study the Kitaev model on a variety of 3D lattices with the goal of identifying characteristic signatures of its QSL phase.

Title: Thermal Fractionalization of Quantum Spins in a Kitaev Model: Coherent Transport of Majorana Fermions and $T$-linear Specific Heat On the theoretical side, Kitaev introduced a celebrated model based upon the spinless p-wave superconductor in one dimension Kitaev , which is currently named as the Kitaev model, one of the simplest but highly nontrivial model to unveil MZM. One-dimensional (1D) dimerized Kitaev chain is a prototype model for MZMs, but its realization remains a challenge in material systems. Majorana Fermions. Kitaev used a simpli ed quantum wire model to show how Majorana modes might manifest as an emergent phenomena, which we will now discuss. \textbf{44}, 131 (2001)]. The Sachdev-Ye-Kitaev (SYK) model is an all-to-all interacting Majorana fermion model for many-body quantum chaos and the holographic correspondence. Map into a chain of Majorana modes using: Majorana states in the Kitaev model. The Kitaev model is ananisotropic spin- model with Ising interactions S xr S xr , S yr S yr and S zr S zr assigned to the three bonds in the hon-eycomb lattice. Authors: Petrova, Olga; Mellado, Paula; Tchernyshyov, Oleg Publication Date: 2014-10-01 NSF-PAR ID: 10009670 Journal Name: Physical Review B Volume: 90 Issue: 13 ISSN: 1098-0121 Publisher: American Physical Society 1) Kitaev honeycomb model as Majorana fermions in Z2 gauge field (0 or flux) characterized by integer Chern number, modulo 16 2) Triangular vortex lattices : Chern = 0,1,,6,8 but not 7 3) Effective models in the dilute vortex limit Using the time-dependent thermal pure quantum state method, we examine finite-temperature spin dynamics in the Kitaev model. Majorana zero modes (MZMs) have attracted tremendous attention in condensed matter and materials physics communities due to the implications in topological quantum computation.

First, there is a gapped Z 2 Abelian topologically ordered spin liquid phase. [59] and which focuses on the Majorana signatures of the model. We will then derive the Schwinger-Dyson equations on the closed time contour, and in turn use them to derive the Kadano -Baym equations on the real time axis. M1 and M2 are the spanning vectors of the lattice, and A Kitaev used a simplied quantum wire model to show how Majorana modes might manifest as an emergent phenomena, which we will now discuss. Consider 1- dimensional tight binding chain with spinless fermions and p-orbital hopping. In the following, there are three parts: (i) We present the Kitaev Hamiltonian on a square lattice and the phase diagram for the topological gapless phase. Majorana edge states in the Kitaev model 4. (Majorana, 1937) and become increasingly important in the analysis of solid state phenomena (Wilczek, 2009). Through the representation of the Kitaev model in terms of quasi-particles an elegant description of a complex, strongly correlated system is possible. k ) z + 2 y sin. Majorana MF results for the [001] eld dependence of the magnetization Mz, the Majorana gap , and the magnetic susceptibility zz for (a) the FM Kitaev model and (b) the AFM one. The numerics reveal an impurity entropy which can be explained by localized Majorana fermions. Majorana MF results for the [001] eld dependence of the magnetization Mz, the Majorana gap , and the magnetic susceptibility zz for (a) the FM Kitaev model and (b) the AFM one.

The Sachdev-Ye-Kitaev (SYK) model is an all-to-all interacting Majorana fermion model for many-body quantum chaos and the holographic correspondence. Event Date: Thursday, November 26, 2020 2:00 pm - 3:00 pm 2020-11-26T14:00:00. They occur in a laboratory 4B, solid and dashed orange arrows). Our circuits can be built using local ingredients in Majorana devices, namely, B. Kitaev model We now study the Kitaev model of a 1D p-wave SC (Kitaev,2001). Kitaev model can be exactly solved by decomposing each spin into 4 Majorana fermions. Kitaevs toy-models key ingredient is spinless nearest neighbour p-wave superconductivity which has not been realised in real materials. In 2010, however, two seminal papers show how to map the Kitaev p-wave quantum wire to an s-wave quantum wire in the presence of strong spin orbit coupling and a magnetic eld. the fermionic operator squares to 1. We used neutron scattering on single crystals of -RuCl 3 to reconstruct dynamical correlations in energy-momentum space. a model of N Majorana fermions Kitaev model (2) on the square-octagon lattice as a function /J z = 1/ 2 and a phase with non-Abelian (nA) Majorana excitations that emerges in eld above the gapless line. Where. GENERAL INTRO In 1937 Ettore Majorana speculated that there could be a particle that is its own antiparticle. Due to only small non-Kitaev terms a magnetic continuum consistent with Majorana fermions and the existence of a Kitaev QSL can be induced by a small out-of-plane-magnetic field. 8.Here, the superconductor is a ladder of density-modulated Kitaev superconductor chains described by the tight-binding Hamiltonian .According to the famous work of Andreev, the local Andreev H Katsura, Majorana excitations in Kitaev spin liquids, Proceedings of the Samahang Pisika ng Pilipinas 36, SPP-2018-INV-3A-02 (2018). The Kitaev quantum spin liquid (KQSL) is an exotic emergent state of matter exhibiting Majorana fermion and gauge flux excitations. Majorana Fermions ?? The result is also novel: in spite of the presence of gapless propagating Majorana fermion excitations, dynamical two spin c At this point you might however object: Unpaired Majoranas appear because you chose one particular, and perhaps even unreachable, set of parameters! We study how stable the Majorana-mediated spin transport in a quantum spin Kitaev model is against thermal fluctuations. The result is also novel: in spite of presence of gapless propagating Unpaired Majorana modes on dislocations and string defects in Kitaev's honeycomb model. Here, we focus on spin transport in the presence of In recent years, the Kitaev honeycomb model has been the focus of much attention, as it is a solvable example of a quantum spin liquid that hosts exotic Majorana excitations. CM Seminar - Magnetoelectric generation of a Majorana-Fermi surface in Kitaev's honeycomb model CM Seminar - Magnetoelectric generation of a Majorana-Fermi surface in Kitaev's honeycomb model. 7,14 To this end, Kitaev, in a seminal paper, introduced a model of interacting spins on a honeycomb lat-tice which reduces to the problem of Majorana fermions coupled to a static Z 2 gauge eld.

I. 1Kitaev model Majorana Operators Exact Solution Spectrum and phase diagram 2Gapped Abelian Phase 3Magnetic Field: NonAbelian Phase Spectral Gap Edge Modes Non-Abelian Anyons Michele Burrello Kitaev Model Anyons and topological quantum computation Quantum phenomena do not occur in a Hilbert space. H Katsura, Majorana excitations in Kitaev spin liquids, Proceedings of the Samahang Pisika ng Pilipinas 36, SPP-2018-INV-3A-02 (2018).

Employing an SO(6) Majorana representation of

We propose an approach to detect the peculiarity of Majorana fermions at the edges of Kitaev magnets. Floquet topological insulators (?) It is the lattice version of the continuum model introduced in previous section. It is found that the system has two different topological classes, the class BDI characterized by the Z index and the class D characterized by the Z 2 index. Majroana and non-abelian statistics 7. Our results indicate that the fractionalization is experimentally observable in the specific heat, spin correlations, and transport properties. In condensed matter physics and black hole physics, the SachdevYeKitaev ( SYK) model is an exactly solvable model initially proposed by Subir Sachdev and Jinwu Ye, [1] and later modified by Alexei Kitaev to the present commonly used form. . (Color online) Kitaev model on the honeycomb lattice with xx coupling J1, yy coupling J2 and zz coupling J3. The magnetic insulator -RuCl 3 is thought to realize a proximate KQSL. Now, we will calculate the differential conductance of the NS junction by means of the recursive Green's function method , .The model of NS junction is shown in Fig. 20. So once again, here is our skeleton, the Kitaev model Hamiltonian written in momentum space: H K i t a e v = ( 2 t cos. . Majorana states in the Kitaev model. Technically, we adopt a Majorana mean-eld approach that was rst applied by Nasu et al. To order 1/N, moments are given by those of the weight function of the Q-Hermite polynomials.Representing Wick contractions by rooted chord 2.2 Quantum circuit constructions To illustrate how to construct the quantum circuit that prepares the ground states of Kitaev-inspired models, we focus on the original Kitaev model con-structed on the honeycomb lattice with 8 sites (result-ing in 2 2 unit cells) as shown in Fig.1(a). The results of this thesis indicate that these Majorana acquire a relevant physical meaning. Unpaired Majorana modes in the gapped phase of Kitaev's honeycomb model A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. As is well known, a pair of Majorana edge modes is realized when a single complex fermion splits into real and imaginary parts which are, respectively, localized at the left and right edges of a sample magnet.