In frustrated quantum magnetism, chiral spin liquids are a particularly intriguing subset of quantum spin liquids in which the fractionalized parton degrees of freedom form a Chern insulator. Here we study an exactly solvable spin-3/2 model which harbors not only chiral spin liquids but also spin liquids with higher-order parton band topology -- a trivial band insulator, a Chern insulator with gapless chiral edge modes, and a second-order topological insulator with gapless corner modes. With a focus on the thermodynamic precursors and thermal phase transitions associated with these distinct states, we employ numerically exact quantum Monte Carlo simulations to reveal a number of unconventional phenomena. This includes a heightened thermal stability of the ground state phases, the emergence of a partial flux ordering of the associated $\mathbb{Z}_2$ lattice gauge field, and the formation of a thermal Majorana metal regime extending over a broad temperature range.
Higher-order topological insulators are distinguished by the existence of topologically protected modes with codimension two or higher. Here, we report the manifestation of a second-order topological insulator in a two dimensional frustrated quantum magnet, which exhibits topological corner modes. Our exactly-solvable model is a generalization of the Kitaev honeycomb model to the Shastry-Sutherland lattice that, besides a chiral spin liquid phase, exhibits a gapped spin liquid with Majorana corner modes, which are protected by two mirror symmetries. This second-order Kitaev spin liquid remains…
Majorana fermions were originally proposed as elementary particles acting as their own antiparticles. In recent years, it has become clear that Majorana fermions can instead be realized in condensed-matter systems as emergent quasiparticles, a situation often accompanied by topological order. Here we propose a physical system which realizes Landau levels - highly degenerate single-particle states usually resulting from an orbital magnetic field acting on charged particles - for Majorana fermions. This is achieved in a variant of a quantum spin system due to Kitaev which is distorted by triaxia…
Motivated by recent experiments in the Kitaev honeycomb lattice, Kondo insulators, and the "Luttinger's theorem-violating" Fermi liquid phase of the underdoped cuprates, we extend the theoretical machinery of Landau-Fermi liquid theory to a system of itinerant, interacting Majorana-like particles. Building upon a previously introduced model of "nearly self-conjugate" fermionic polarons, a Landau-Majorana kinetic equation is introduced to describe the collective modes and Fermi surface instabilities in a fluid of particles whose fermionic degrees of freedom obey the Majorana reality condition. …
We study the temperature (T) dependence of the dynamical structure factor of the chiral spin liquid phase of Kitaev's honeycomb model. We find, using a recently developed analytical approach, direct signatures of the Majorana modes associated with the non-Abelian Z_2 vortices (visons). In particular, a characteristic multiplet of discrete peaks at low but nonzero-T and its field dependence reflect the existence of thermally activated visons whose varying separation yields different splittings of the Majorana `zero' modes. The resonant processes involved are specific to the Majorana modes, and …
Three-dimensional (3D) variants of the Kitaev model can harbor gapless spin liquids with a Majorana Fermi surface on certain tricoordinated lattice structures such as the recently introduced hyperoctagon lattice. Here we investigate Fermi surface instabilities arising from additional spin exchange terms (such as a Heisenberg coupling) which introduce interactions between the emergent Majorana fermion degrees of freedom. We show that independent of the sign and structure of the interactions, the Majorana surface is always unstable. Generically the system spontaneously doubles its unit cell at e…
The Raman scattering with local optical excitation from the Majorana edge modes of Kitaev spin liquids and topological superconductors is studied theoretically. Although the effective one-dimensional model is common between these two cases, the coupling to the electromagnetic field is different. It is found that the Raman spectrum at low energy scales with $ω^3$ in Kitaev spin liquids while it shows the gap in topological superconductors. This is in sharp contrast to the infrared absorption, where the spectrum shows the gap in Kitaev spin liquids, while it behaves as $\sim ω^2$ in topological …
The large thermal Hall conductivity recently detected in Mott insulating cuprates has been attributed to chiral neutral spin excitations. A quantum spin liquid with Majorana excitations, Chern number +/-4 and large thermal Hall conductivity is found to be an excited state of a frustrated Heisenberg model on the square lattice. Using a Majorana mean-field theory and exact diagonalizations, we explore two possible routes to achieve this chiral quantum spin liquid, an orbital effect of an applied magnetic field and spin orbit couplings as present in cuprates. In particular, we show how only the o…
In this letter, we propose a local detection scheme for the Majorana zero mode (MZM) carried by a vison in Kitaev's chiral spin liquid (CSL) using scanning tunneling microscopy (STM). The STM introduces a single Majorana into the system through hole/charge injection and the Majorana interacts with the MZM to form a stable composite object. We derive the exact analytical expression of single-hole Green's function in the Mott insulating limit of Kitaev's model and show that the differential conductance has split peaks, as a consequence of resonant tunneling through the vison-hole composite. The …
Quantum dot devices allow one to access the critical point of the two-channel Kondo model. The effective critical theory involves a free Majorana fermion quasiparticle localized on the dot. As a consequence, this critical point shows both the phenomenon of non-Fermi liquid physics and fractionalization. Although a violation of the Wiedemann-Franz law is considered a defining feature of non-Fermi liquid systems, we show by exact calculations that it holds at the critical point, providing a counterexample to this lore. Furthermore, we show that the fractionalized Majorana character of the critic…
A three-dimensional Kitaev model on a hyperhoneycomb lattice is investigated numerically at finite temperature. 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. This model can be rewritten as a free Majorana fermion system coupled with $Z_2$ variables. The density of states of Majorana fermions shows an excitation gap in the gapped region, while it is semimetallic in the gapless region reflecting the Dirac node. Performing the Monte Carlo simulation, we calculate the…
Magnetoelastic interaction couples spin and lattice degrees of freedom and plays a key role in thermal transport properties of magnetic insulators. In the Kitaev quantum spin liquid, the low energy excitations are charge neutral Majorana fermions, which transform the magnetoelasctic interaction into Majorana-phonon scattering. Motivated by anomalous thermal properties of the Kitaev quantum spin liquid candidate RuCl$_3$, in this letter, we combine meV resolution inelastic x-ray scattering and theoretical calculation to examine the Majorana-phonon scattering. We analytically derive the velocity…
Majorana-assisted nonlocal spin correlation in quasi-one-dimensional Kitaev spin liquids
RIKEN Cluster for Pioneering Research（理化学研究所先锋研究集群）；Department of Applied Physics, The University of Tokyo（东京大学应用物理系）；RIKEN Center for Emergent Matter Science（理化学研究所 emergent matter 科学中心）
Fractionalization is a hallmark of spin-liquid behavior; it typically leads to response functions consisting of continua instead of sharp modes. However, microscopic processes can enable the formation of short-distance bound states of fractionalized excitations, despite asymptotic deconfinement. Here we study such bound-state formation for the $Z_2$ spin liquid realized in Kitaev's honeycomb compass model, supplemented by a kekule distortion of the lattice. Bound states between flux pairs and Majorana fermions form in the Majorana band gaps. We calculate the dynamic spin susceptibility and sho…
We consider a helical spin liquid system which shows majorana fermion modes at the edge. The interaction between the quasiparticles in this system induces phase transition, Majorana-Ising transition. We comply the density matrix renormalization group method to study this phase transition for the entire regime of the parameter space. We observe the presence of topological quantum phase transition for repulsive interaction, however this phase is more stable for the attractive interaction. The length scale dependent study shows many new and important results and we show explicitly that the major …
In the framework of Hofstadter`s approach we provide a detailed analysis of a realization of exotic topological states such as the Chern insulator with large Chern numbers. In a transverse homogeneous magnetic field a one-particle spectrum of fermions transforms to an intricate spectrum with a fine topological structure of the subbands. In a weak magnetic field $H$ for a rational magnetic flux, a topological phase with a large Chern number is realized near the half filling. There is an abnormal behavior of the Hall conductance ${σ_{xy}\simeq \frac{e}{2 H}}$. At half-filling, the number of chir…
We describe scanning tunneling spectroscopic signatures of Majorana zero modes (MZMs) in Kitaev spin liquids. The tunnel conductance is determined by the dynamical spin correlations of the spin liquid, which we compute exactly, and by spin-anisotropic cotunneling form factors. Near a $\mathbb{Z}_2$ vortex, the tunnel conductance has a staircase voltage dependence, where conductance steps arise from MZMs and (at higher voltages) from additional vortex configurations. By scanning the probe tip position, one can detect the vortex locations. Our analysis suggests that topological magnon bound stat…
Quantum spin liquids hosting Majorana excitations have recently experienced renewed interest for potential applications to topological quantum computation. Performing logical operations with reduced poisoning requires to localize such quasiparticles at specific point of the system, with energies that are well defined and inside the bulk energy gap. These are two defining features of second order topological insulators (SOTIs). Here, we show two spin models that support quantum spin liquid phases characterised by Majorana excitations and that behave as SOTIs, one of which is analytically solvab…
We study the spin transport through the quantum spin liquid (QSL) by investigating the real-time and real-space dynamics of the Kitaev spin system with a zigzag structure in terms of the time-dependent Majorana mean-field theory. After the magnetic field pulse is introduced to one of the edges, the spin moments are excited in the opposite edge region although no spin moments are induced in the Kitaev QSL region. This unusual spin transport originates from the fact that the $S=1/2$ spins are fractionalized into the itinerant and localized Majorana fermions in the Kitaev system. Although both Ma…
We analyse the dynamical response of a range of 3D Kitaev quantum spin-liquids, using lattice models chosen to explore the different possible low-energy spectra for gapless Majorana fermions, with either Fermi surfaces, nodal lines or Weyl points. We find that the behaviour of the dynamical structure factor is distinct in all three cases, reflecting the quasiparticle density of states in two fundamentally different ways. First, the low-energy response is either straightforwardly related to the power with which the low-energy density of states vanishes; or for a non-vanishing density of states,…
The honeycomb Kitaev spin model provides a quantum spin liquid in the ground state, where the spin excitations are fractionalized into itinerant and localized Majorana fermions; the former spectrum has a broad continuum ranging up to a high energy, while the latter has a sharp peak at a low energy. Despite tremendous efforts, it remains elusive to clearly identify these distinct Majorana excitations in experiments. Here we show their manifestation in the time evolution after quenching the magnetic field, by using the time-dependent Majorana mean-field theory for both the ferromagnetic and anti…
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. For the original honeycomb Kitaev model, it has recently been observed that the sign of the bond-directional exchange is of crucial relevance for the field-induced physics, with antiferromagnetic couplings giving rise to an intermediate spin liquid regime between the low-field gapped Kitaev spin liquid and the high-field polarized state, which is not present in the ferromagnetically coupled model. Here, by employing a Majorana m…
We map an interacting helical liquid system, coupled to an external magnetic field and s-wave superconductor, to an XYZ spin system, and it undergoes Majorana-Ising transition by tuning of parameters. In the Majorana state, lowest excitation gap decays exponentially with system size, and the system has degenerate ground state in the thermodynamic limit. On the contrary, the gap opens in the Ising phase even in the thermodynamic limit. We also study other criteria to characterize the transition, such as edge spin correlation with its neighbor $C(r=1)$, local susceptibility $χ_i$, superconductin…
Experimental identification of quantum spin liquids remains a challenge, as the pristine nature is to be seen in asymptotically low temperatures. We here theoretically show that the precursor of quantum spin liquids appears in the spin dynamics in the paramagnetic state over a wide temperature range. Using the cluster dynamical mean-field theory and the continuous-time quantum Monte Carlo method, which are newly developed in the Majorana fermion representation, we calculate the dynamical spin structure factor, relaxation rate in nuclear magnetic resonance, and magnetic susceptibility for the h…
Visons in Kitaev Spin Liquids with Majorana Fermi Surfaces
Universidade Federal do Rio Grande do Norte（北里奥格兰德联邦大学）；International Institute of Physics, Universidade Federal do Rio Grande do Norte（北里奥格兰德联邦大学国际物理研究所）
Temperature evolution of the spin correlation and excitation spectrum is investigated for a Kitaev model defined on a decorated honeycomb lattice by using the quantum Monte Carlo simulation in the Majorana fermion representation. The ground state of this quantum spin model is given by two kinds of chiral spin liquids: one is topologically trivial with Abelian anyon excitations, and the other is topologically nontrivial with non-Abelian anyon excitations. While lowering temperature, the model exhibits several crossovers in the paramagnetic state, which originate from the fractionalization of qu…
The point contact tunnel junctions between a one-dimensional topological superconductor and single-channel quantum Hall (QH) liquids are investigated theoretically with bosonization technology and renormalization group methods. For the $ν=1$ integer QH liquid, the universal low-energy tunneling transport is governed by the perfect Andreev reflection fixed point with quantized zero-bias conductance $G(0)=2e^{2}/h$, which can serve as a definitive fingerprint of the existence of a Majorana fermion. For the $ν=1/m$ Laughlin fractional QH liquids, its transport is governed by the perfect normal re…
We present the theoretical results for the Majorana band structure and thermal transport properties in the Kitaev honeycomb model under a staggered magnetic field in addition to a uniform one. The Majorana band structure is asymmetrically modulated in momentum space, and the valley degree of freedom is activated and controlled by the magnitudes and directions of the uniform and staggered magnetic fields; as a result, the Majorana excitation is either gapped or gapless, the latter of which in general has Majorana Fermi surfaces, in contrast to the point nodes in the absence of the fields. We sh…
We discuss the signatures of a Kramers pair of Majorana modes formed in a Josephson junction on top of a quantum spin Hall system. We show that, while ignoring interactions on the quantum spin Hall edge allows arbitrary Andreev process in the system, moderate repulsive interactions stabilize Andreev transmission - the hole goes into the opposite lead from where the electron has arrived. We analyze the renormalization group equations and deduce the existence of a non-trivial critical point for sufficiently strong interactions.
We propose and study the charge transport through single and double quantum point contacts setup between helical Majorana modes and an interacting helical Luttinger liquid. We show that the differential conductance decreases for stronger repulsive interactions and that the point contacts become insulating above a critical interaction strength. For a single point contact, the differential conductance as a function of bias voltage shows a series of peaks due to Andreev reflection of electrons in the Majorana modes. In the case of two point contacts, interference phenomena make the structure of t…
Time-reversal invariant topological superconductors are characterized by the presence of Majorana Kramers pairs localized at defects. One of the transport signatures of Majorana Kramers pairs is the quantized differential conductance of $4e^2/h$ when such a one-dimensional superconductor is coupled to a normal-metal lead. The resonant Andreev reflection, responsible for this phenomenon, can be understood as the boundary condition change for lead electrons at low energies. In this paper, we study the stability of the Andreev reflection fixed point with respect to electron-electron interactions …
Spin-orbital liquids are quantum disordered states in systems with entangled spin and orbital degrees of freedom. We study exactly solvable spin-orbital models in two dimensions with selected Heisenberg-, Kitaev-, and $Γ$-type interactions, as well as external magnetic fields. These models realize a variety of spin-orbital-liquid phases featuring dispersing Majorana fermions with Fermi surfaces, nodal Dirac or quadratic band touching points, or full gaps. In particular, we show that Zeeman magnetic fields can stabilize nontrivial flux patterns and induce metamagnetic transitions between states…
We design a driven superconducting box with four spins S=1/2 (qubits) such that coupled devices can give insight on the occurrence of quantum spin liquids and many-body Majorana states. Within one box or island, we introduce a generalized nuclear magnetic resonance algorithm to realize our models and study numerically the spin observables in time as well as the emergent gauge fields. We discuss the stability of the box towards various detuning effects and we include dissipation effects through a Lindblad master equation. Coupling boxes allows us to realize quantum spin liquid phases of Kitaev …
We exhibit an exactly solvable example of a SU(2) symmetric Majorana spin liquid phase, in which quenched disorder leads to random-singlet phenomenology. More precisely, we argue that a strong-disorder fixed point controls the low temperature susceptibility $χ(T)$ of an exactly solvable $S=1/2$ model on the decorated honeycomb lattice with quenched bond disorder and/or vacancies, leading to $χ(T) = {\mathcal C}/T+ {\mathcal D} T^{α(T) - 1}$ where $α(T) \rightarrow 0$ as $T \rightarrow 0$. The first term is a Curie tail that represents the emergent response of vacancy-induced spin textures spre…
We study the effect of site dilution in Kitaev's model. We derive an analytical solution of the dynamical spin correlation functions for arbitrary configurations of $Z_2$ fluxes. By incorporating this solution into classical Monte Carlo scheme, we address how a site vacancy affects the experimental observables, such as the static spin susceptibility and the spin lattice relaxation rate, $1/T_1$. As a result, we found an enhancement of dynamical magnetic response in the vicinity of vacancy, which leads to Friedel-like oscillation in local $1/T_1$, in contrast to limited influences on the static…
Quantum matter provides an effective vacuum out of which arise emergent particles not corresponding to any experimentally detected elementary particle. Topological quantum materials in particular have become a focus of intense research in part because of the remarkable possibility to realize Majorana fermions, with their potential for new, decoherence-free quantum computing architectures. In this paper we undertake a study on high-quality single crystal of $α-RuCl_3$ which has been identified as a material realizing a proximate Kitaev state, a topological quantum state with magnetic Majorana f…
The unambiguous experimental detection of quantum spin liquids and, in particular, of the long-sought Kitaev quantum spin liquid (KQSL) with its Majorana fermion excitations remains an outstanding challenge. One of the major obstacles is the absence of signatures that definitively characterize this phase. Here we propose the Landau levels known to form in the Majorana excitation spectrum of the KQSL when certain strain fields are applied as a direct signature of Majorana fermions with Dirac-like dispersion. In particular, we show that the Majorana Landau level quantization of strained films of…
The 5d-electron honeycomb compound H$_3$LiIr$_2$O$_3$ [K. Kitagawa $et$ $al.$, Nature 554, 341-345 (2018)] exhibits an apparent quantum spin liquid (QSL) state. In this intercalated spin-orbital compound, a remarkable pile up of low-energy states was experimentally observed in specific heat and nuclear magnetic (NMR) spin relaxation. We show that a bond disordered Kitaev model can naturally account for this phenomenon, suggesting that disorder plays an essential role in its theoretical description. In the exactly soluble Kitaev model, we obtain, via spin fractionalization, a random bipartite h…
In an exciting development, SmB6, a Kondo insulator has been shown to exhibit bulk quantum oscillations. We propose that SmB6 is a bulk scalar Majorana Fermi Liquid} (MFL) with a finite gap for charge and spin excitations. In their study of Kondo insulators in 1993, Coleman, Miranda and Tsvelik (CMT) envisaged such a remarkable possibility, using a mean field ansatz. We generalize CMT theory to non-zero magnetic fields and show a counter intuitive result that the scalar Majorana fermi liquid, while remaining electrically insulating, responds to external magnetic field and exhibits Landau diama…
