In quantum spin liquids, fractional spinon excitations carry half-integer spins and other fractional quantum numbers of lattice and time-reversal symmetries. Different patterns of symmetry fractionalization distinguish different spin liquid phases. In this work, we derive a general constraint on the symmetry fractionalization of spinons in a gapped spin liquid, realized in a system with an odd number of spin-$1/2$ per unit cell. In particular, when applied to kagome/triangular lattices, we obtain a complete classification of symmetric gapped $\mathbb{Z_2}$ spin liquids.
The angle resolved photoemission spectroscopy lineshapes of quasi one-dimensional (1d) Li0.9Mo6O17 display both agreement with and departures from the one-band Tomonaga-Luttinger model. We show that the departures can be understood by explicitly accounting for the four modes arising from the two quasi-1d bands known to cross the Fermi energy. The key assumption is that the antisymmetric charge mode is gapped with a magnitude near the temperature (T) of a mysterious 25K powerlaw resistivity upturn. The gap is consistent with the lack of a charge or spin density wave accompanying the upturn, is …
Bosonic model with unfrustrated hopping and short-range repulsive interaction is constructed that realizes $Z_3$ fractionalized insulator phase in two dimensions and in zero magnetic field. Such phase is characterized as having gapped charged excitations that carry fractional electrical charge 1/3 and also gapped $Z_3$ vortices above the topologically ordered ground state.
In this work we study symmetry fractionalization of vison excitations in topological $\mathbb{Z}_2$ spin liquids. We show that in the presence of the full $\mathrm{SO}(3)$ spin rotational symmetry and if there is an odd number of spin-$\frac12$ per unit cell, the symmetry fractionalization of visons is completely fixed. On the other hand, visons can have different classes of symmetry fractionalization if the spin rotational symmetry is reduced. As a concrete example, we show that visons in the Balents-Fisher-Girvin $\mathbb{Z}_2$ spin liquid have crystal symmetry fractionalization classes whic…
In quantum spin liquid states, the fractionalized spinon excitations can carry fractional crystal symmetry quantum numbers, and this symmetry fractionalization distinguishes different topologically ordered spin liquid states. In this work we propose a simple way to detect signatures of such crystal symmetry fractionalizations from the crystal symmetry representations of the ground state wave function. We demonstrate our method on projected $\mathbb Z_2$ spin liquid wave functions on the kagome lattice, and show that it can be used to classify generic wave functions. Particularly our method can…
We investigate charge fractionalizations in artificial Tomonaga-Luttinger liquids (TLLs) composed of two capacitively coupled quantum Hall edge channels (ECs) in graphene. The interaction strength of the artificial TLLs can be controlled through distance W between the ECs. We show that the fractionalization ratio r and the TLL mode velocity v vary with W. The experimentally obtained relation between v and r follows a unique function predicted by the TLL theory. We also show that charged wavepackets are reflected back and forth multiple times at both ends of the TLL region.
In this work we study the crystal symmetry fractionalization in chiral spin liquids with the chiral-semion topological order. We show that if such a chiral spin liquid is realized in a two-dimensional lattice model with odd number of spin-$\frac12$ per unit cell and the state preserves spin rotation symmetry and translation symmetries, the semion excitation must carry both half-integer spin and fractional crystal quantum numbers. As a result, only a unique symmetry enriched topological phase can be realized in chiral spin liquids in a spin-$\frac12$ kagome lattice model. These fractional symme…
In this paper, we consider analytically the density evolution of a spinless Fermi liquid with a nonlinear dispersion relation into which one particle is injected. The interaction is point-like and the temperature is zero. We obtain a formula for the evolution of the density and discuss the picture it gives as well as the physics behind it. Compared to the case of a linear spectrum, we find further and more complex fractionalization of the initial density hump: it splits into three humps instead of two, moreover, all three change their shapes in a complicated manner. We analyze the mechanisms o…
Based on large-scale quantum Monte Carlo simulations, we examine the dynamical spin structure factor of the Balents-Fisher-Girvin kagome lattice quantum spin-$1/2$ model, which is known to harbor an extended $Z_2$ quantum spin liquid phase. We use a correlation-matrix sampling scheme combined with a stochastic analytic continuation method to resolve the spectral functions of this anisotropic quantum spin model with a three-site unit-cell. Based on this approach, we monitor the spin dynamics throughout the phase diagram of this model, from the XY-ferromagnetic region to the $Z_2$ quantum spin l…
We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with Z2 topological order, that is, on gapped Z2 spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group H^2(G,Z2). This result leads us to a symmetry classification of gapped Z2 spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a p…
We present the theory of dynamical spin-response for the Kitaev honeycomb model, obtaining exact results for the structure factor (SF) in gapped and gapless, Abelian and non-Abelian quantum spin-liquid (QSL) phases. We also describe the advances in methodology necessary to compute these results. The structure factor shows signatures of spin-fractionalization into emergent quasiparticles -- Majorana fermions and fluxes of $Z_2$ gauge field. In addition to a broad continuum from spin-fractionalization, we find sharp ($δ$-function) features in the response. These arise in two distinct ways: from …
We describe ground states of correlated electron systems in which the electron fractionalizes into separate quasiparticles which carry its spin and its charge, and into real Majorana fermions which carry its Fermi statistics. Such parent states provide a unified theory of previously studied fractionalized states: their descendants include insulating and conducting states with neutral spin S=1/2 fermionic spinons, and states with spinless fermionic charge carriers. We illustrate these ideas on the honeycomb lattice, with field theories of such states and their phase transitions.
We consider gapped Z2 spin liquids, where spinon quasiparticles may carry fractional quantum numbers of space group symmetry. In particular, spinons can carry fractional crystal momentum. We show that such quantum number fractionalization has dramatic, spectroscopically accessible consequences, namely enhanced periodicity of the two-spinon density of states in the Brillouin zone, which can be detected via inelastic neutron scattering. This effect is a sharp signature of certain topologically ordered spin liquids and other symmetry enriched topological phases. Considering square lattice space g…
Quantum spin liquids (QSLs) are intriguing phases of matter possessing fractionalized excitations. Several quasi-two dimensional materials have been proposed as candidate QSLs, but direct evidence for fractionalization in these systems is still lacking. In this paper, we show that the inter-plane thermal conductivity in layered QSLs carries a unique signature of fractionalization. We examine several types of gapless QSL phases - a $Z_2$ QSL with either a Dirac spectrum or a spinon Fermi surface, and a $U(1)$ QSL with a Fermi surface. In all cases, the in-plane and $c-$axis thermal conductiviti…
The anyonic excitations of a spin-liquid can feature fractional quantum numbers under space-group symmetries. Detecting these fractional quantum numbers, which are analogs of the fractional charge of Laughlin quasiparticles, may prove easier than the direct observation of anyonic braiding and statistics. Motivated by the recent numerical discovery of spin-liquid phases in the kagome Heisenberg antiferromagnet, we theoretically predict the pattern of space group symmetry fractionalization in the kagome lattice chiral spin liquid. We provide a method to detect these fractional quantum numbers in…
The surface of a three-dimensional topological electron system often hosts symmetry-protected gapless surface states. With the effect of electron interactions, these surface states can be gapped out without symmetry breaking by a surface topological order, in which the anyon excitations carry anomalous symmetry fractionalization that cannot be realized in a genuine two-dimensional system. We show that for a mirror-symmetry-protected topological crystalline insulator with mirror Chern number $n=4$, its surface can be gapped out by an anomalous $\mathbb Z_2$ topological order, where all anyons c…
It is well known that the matrix product state (MPS) description of a gapped ground state with a global on-site symmetry can exhibit "symmetry fractionalization". Namely, even though the symmetry acts as a linear representation on the physical degrees of freedom, the MPS matrices---which act on some virtual degrees of freedom---can transform under a projective representation. This was instrumental in classifying gapped symmetry protected phases that manifest in one dimensional quantum many-body systems. Here we consider the multi-scale entanglement renormalization ansatz (MERA) description of …
Quantum spin liquids have fascinated condensed matter physicists for decades because of their unusual properties such as spin fractionalization and long-range entanglement. Unlike conventional symmetry breaking the topological order underlying quantum spin liquids is hard to detect experimentally. Even theoretical models are scarce for which the ground state is established to be a quantum spin liquid. The Kitaev honeycomb model and its generalizations to other tri-coordinated lattices are chief counterexamples --- they are exactly solvable, harbor a variety of quantum spin liquid phases, and a…
Helical Luttinger liquids, appearing at the edge of two-dimensional topological insulators, represent a new paradigm of one-dimensional systems, where peculiar quantum phenomena can be investigated. Motivated by recent experiments on charge fractionalization, we propose a setup based on helical Luttinger liquids that allows to time-resolve, in addition to charge fractionalization, also spin-charge separation and pure spin fractionalization. This is due to the combined presence of spin-momentum locking and interactions. We show that electric time-resolved measurements can reveal both charge and…
The interplay of symmetry and topological order leads to a variety of distinct phases of matter, the Symmetry Enriched Topological (SET) phases. Here we discuss physical observables that distinguish different SETs in the context of Z$_2$ quantum spin liquids with SU(2) spin rotation invariance. We focus on the cylinder geometry, and show that ground state quantum numbers for different topological sectors are robust invariants which can be used to identify the SET phase. More generally these invariants are related to 1D symmetry protected topological phases when viewing the cylinder geometry as…
Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or on-site unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B \textbf{83}, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phase…
We report a new kagome quantum spin liquid candidate Cu$_3$Zn(OH)$_6$FBr, which does not experience any phase transition down to 50 mK, more than three orders lower than the antiferromagnetic Curie-Weiss temperature ($\sim$ 200 K). A clear gap opening at low temperature is observed in the uniform spin susceptibility obtained from $^{19}$F nuclear magnetic resonance measurements. We observe the characteristic magnetic field dependence of the gap as expected for fractionalized spin-1/2 spinon excitations. Our experimental results provide firm evidence for spin fractionalization in a topologicall…
It is well established that at low energies one-dimensional (1D) fermionic systems are described by the Luttinger liquid (LL) theory, that predicts phenomena like spin-charge separation, and charge fractionalization into chiral modes. Here we show through the time evolution of an electron injected into a 1D t-J model, obtained with time-dependent density matrix renormalization group, that a further fractionalization of both charge and spin takes place beyond the hydrodynamic limit. Its dynamics can be understood at the supersymmetric point (J=2t) in terms of the excitations of the Bethe-Ansatz…
Motivated by the ongoing effort to search for high-resolution signatures of quantum spin liquids, we investigate the temperature dependence of the indirect resonant inelastic x-ray scattering (RIXS) response for the Kitaev honeycomb model. We find that, as a result of spin fractionalization, the RIXS response changes qualitatively at two well-separated temperature scales, $T_L$ and $T_H$, which correspond to the characteristic energies of the two kinds of fractionalized excitations, $\mathbb{Z}_2$ gauge fluxes and Majorana fermions, respectively. While thermally excited $\mathbb{Z}_2$ gauge fl…
Charge fractionalization is the phenomenon where quasi-particle excitations in a many-particle system appear with non-integer values relative to the fundamental charge unit. Examples of such systems are known from field theoretical models and from physical realizations. Recently there has been an interest in charge fractionalization in one-dimensional systems described by Luttinger liquid theory. These are gapless systems and that gives rise to the question whether non-integer charges should be regarded as sharp in the same meaning as in a gapped system. In this talk I first give an introducti…
A Falicov-Kimball model which thermodynamically reduces the local Coulomb interaction of particles to attraction or repulsion is studied within the dynamical mean-field theory. In the strong interaction regime a fractionalization of particles into charge and spin objects, the physical properties of which are different from the whole particles, is observed in both high- and low-temperature phases. At high temperature and strong interaction the single-particle density of states opens an excitation gap, but the charge compressibility and the spin susceptibility exhibit the features of gapless exc…
The fractionalization of quantum numbers in interacting quantum-many body systems is a central motif in condensed matter physics with prominent examples including the fractionalization of the electron in quantum Hall liquids or the emergence of magnetic monopoles in spin-ice materials. Here we discuss the fractionalization of magnetic moments in three-dimensional Kitaev models into Majorana fermions (and a $\mathbb Z_2$ gauge field) and their emergent collective behavior. We analytically demonstrate that the Majorana fermions form a Weyl superconductor for the Kitaev model on the recently synt…
Long-range entanglement in quantum spin liquids (QSLs) lead to novel low energy excitations with fractionalised quantum numbers and (in 2D) statistics. Experimental detection and manipulation of these excitations present a challenge particularly in view of diverse candidate magnets. A promising probe of fractionalisation is their coupling to phonons. Here we present Raman scattering results for the S = 1/2 honeycomb iridate Cu2IrO3, a candidate Kitaev QSL with fractionalised Majorana fermions and Ising flux excitations. We observe anomalous low temperature frequency shift and linewidth broaden…
We consider Dirac electrons on the honeycomb lattice Kondo coupled to spin-1/2 degrees of freedom on the kagome lattice. The interactions between the spins are chosen along the lines of the Balents-Fisher-Girvin model that is known to host a $\mathbb{Z}_2$ spin liquid and a ferromagnetic phase. The model is amenable to sign free auxiliary field quantum Monte Carlo simulations. While in the ferromagnetic phase the Dirac electrons acquire a gap, they remain massless in the $\mathbb{Z}_2$ spin liquid phase due to the breakdown of Kondo screening. Since our model has an odd number of spins per uni…
We study fractionalization in a spin-liquid Mott insulator defined by a Gutzwiller projected BCS state |0> at half-filling. We construct a trial vison (Z2 vortex) state |V> by projecting an hc/2e vortex and determine when it is orthogonal to |0>. Using a combination of analytical arguments and Monte Carlo calculations we show that generically the spin-liquid is Z2 fractionalized. For microscopic parameters appropriate for high Tc cuprates, we estimate that the vison gap Ev << J, consistent with recent experimental bounds, due to proximity to the bipartite symmetric point where Ev = 0.
The emergence of fractionalized quasiparticles in quantum spin liquids has served a wealth of unconventional phenomena in frustrated magnets. In our work, we explore the various domain patterns of such fractionalized quasiparticles, especially focusing on charge defects in U(1) quantum spin liquids. We claim that emergent long range interaction between charge defects leads to characteristic structures with distinct length scales, where they can be controlled via the ratio of interaction strengths. In this context, the spin ice phase is the dilute gas of weakly interacting charges, whereas, the…
We argue, based on general principles, that topological order is essential to realize fractionalization in gapped insulating phases in dimensions $d \geq 2$. In $d=2$ with genus $g$, we derive the existence of the minimum topological degeneracy $q^g$ if the charge is fractionalized in unit of $1/q$, irrespective of microscopic model or of effective theory. Furthermore, if the quasiparticle is either boson or fermion, it must be at least $q^{2g}$.
We propose a route toward realizing fractionalized topological phases of matter (i.e. with intrinsic topological order) by literally building on un-fractionalized phases. Our approach employs a Kondo lattice model in which a gapped electronic system of non-interacting fermions is coupled to non-interacting local moments via the exchange interaction. Using general entanglement-based arguments and explicit lattice models, we show that in this way gapped spin liquids can be induced in the spin system. We demonstrate the power of this topological bootstrap concept with two examples: (1) a chiral s…
Finite-temperature ($T$) properties of a Kitaev model defined on a honeycomb lattice are investigated by a quantum Monte Carlo simulation, from the viewpoint of fractionalization of quantum $S=1/2$ spins into two types of Majorana fermions, itinerant and localized. In this system, the entropy is released successively at two well-separated $T$ scales, as a clear indication of the thermal fractionalization. We show that the high-$T$ crossover, which is driven by itinerant Majorana fermions, is closely related with the development of nearest-neighbor spin correlations. On the other hand, the low-…
We employ a variational Monte Carlo approach to efficiently obtain the dynamical structure factor for the spin-1/2 $J_1-J_2$ Heisenberg model on the square lattice. Upon increasing the frustrating ratio $J_2/J_1$, the ground state undergoes a continuous transition from a Néel antiferromagnet to a $\mathbb{Z}_{2}$ gapless spin liquid. We identify the characteristic spectral features in both phases and highlight the existence of a broad continuum of excitations in the proximity of the spin-liquid phase. The magnon branch, which dominates the spectrum of the unfrustrated Heisenberg model, gradual…
The Fermi sign structure plays a crucial role for a Landau's Fermi liquid. In this work, we identify the exact sign structure for the bipartite Hubbard model at an arbitrary strength of the on-site Coulomb repulsion $U$. This general sign structure naturally reproduces the conventional fermion signs in the small $U$ limit, and the phase string signs of the $t$-$J$ model in the large $U$ limit. We focus on the half-filling case as an example to illustrate why such a generic sign structure is important to understand the transition from the weakly correlated Fermi liquid regime to the strongly co…
We construct an effective topological Landau-Ginzburg theory that describes general SU(2) incompressible quantum liquids of strongly correlated particles in two spatial dimensions. This theory characterizes the fractionalization of quasiparticle quantum numbers and statistics in relation to the topological ground-state symmetries, and generalizes the Chern-Simons, BF and hierarchical effective gauge theories to an arbitrary representation of the SU(2) symmetry group. Our main focus are fractional topological insulators with time-reversal symmetry, which are treated as generalizations of the SU…
Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties of the spin-liquid phase of the spin-$\frac{1}{2}$ $J_1$-$J_2$ Heisenberg model on the triangular lattice. We find four distinct ground-states characteristic of a non-chiral, $Z_2$ topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to coexistence of symmetry protected a…
We report Luttinger liquid line shapes better revealed by new angle resolved photoemission data taken with a much improved angle resolution on a quasi-1-dimensional metal Li$_{0.9}$Mo$_6$O$_{17}$. The new data indicate a larger spinon velocity than our previous lower resolution data indicated.
We show that the gapless boundary signatures - namely, chiral/helical hinge modes or localized zero modes - of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on surface is `anomalous' in the sense that it can be made consistent with symmetry only on the surface of a three dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-or…
We consider a strongly correlated "t,t'-J" spin model with a positive uniaxial anisotropy. We show that in 2+1 dimensions, this model is equivalent to spin-charge excitations coupled to Z_2 Ising gauge fields and not to a U(1) gauge fields (as in the "t-J" model). As a result a phase of frozen Z_2 gauge fluctuations is possible, giving rise to a spin-charge-separated liquid.
We study dynamical spin structure factor (DSSF) of S = 1/2 Heisenberg model on the kagome lattice (KAFM) by means of density-matrix renormalization group. By comparison with the well-defined magnetic ordered state and chiral spin liquid sitting nearby in the phase diagram, the KAFM with the nearest-neighbor interaction shows distinct dynamical response behaviors. First of all, the DSSF displays important spectral intensity predominantly at low frequency region around Q = M point in momentum space, and shows a broad spectral distribution at high frequency region for momenta along the boundary o…
Gapped non-liquid state (also known as fracton state) is a very special gapped quantum state of matter that is characterized by a microscopic cellular structure. Such microscopic cellular structure has a macroscopic effect at arbitrary long distances and cannot be removed by renormalization group flow, which makes gapped non-liquid state beyond the description of topological quantum field theory with a finite number of fields. Using Abelian and non-Abelian topological orders in 2-dimensional (2d) space and the different ways to glue them together via their gapped boundaries, we propose a syste…
While phase transitions between magnetic analogs of three states of matter --- a long-range ordered state, paramagnet, and spin liquid --- have been extensively studied, the possibility of "liquid-liquid" transitions, namely, between different spin liquids, remains elusive. By introducing the additional Ising coupling into the honeycomb Kitaev model with bond asymmetry, we discover that the Kitaev spin liquid turns into a spin-nematic quantum paramagnet before a magnetic order is established by the Ising coupling. The quantum phase transition between the two liquid states accompanies a topolog…
Transient localization from fractionalization: vanishingly small heat conductivity in gapless quantum magnets
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与工程中心）；The Ohio State University（俄亥俄州立大学）；Imperial College London（帝国理工学院）
We measure by inelastic neutron scattering the spin excitation spectra as a function of applied magnetic field in the quantum spin-ladder material (C5H12N)2CuBr4. Discrete magnon modes at low fields in the quantum disordered phase and at high fields in the saturated phase contrast sharply with a spinon continuum at intermediate fields characteristic of the Luttinger-liquid phase. By tuning the magnetic field, we drive the fractionalization of magnons into spinons and, in this deconfined regime, observe both commensurate and incommensurate continua.
Symmetry fractionalization describes the fascinating phenomena that excitations in a 2D topological system can transform under symmetry in a fractional way. For example in fractional quantum Hall systems, excitations can carry fractional charges while the electrons making up the system have charge one. An important question is to understand what symmetry fractionalization (SF) patterns are possible given different types of topological order and different symmetries. A lot of progress has been made recently in classifying the SF patterns, providing deep insight into the strongly correlated expe…
We study a three-components fermionic gas loaded in a one-dimensional optical trap at half-filling. We find that the system is fully gapped and may order into 8 possible phases: four 2$k_F$ atomic density wave and spin-Peierls phases with all possible relative $π$ phases shifts between the three species. We find that trionic excitations are unstable toward the decay into pairs of kinks carrying a {\it fractional} number, $Q=3/2$, of atoms. These sesquions eventually condense upon small doping and are described by a Luttinger liquid. We finally discuss the phase diagram of a three component mix…
Recent numerical studies have provided strong evidence for a gapped $Z_2$ quantum spin liquid in the kagome lattice spin-1/2 Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of symmetry fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons obeying either bose (Schwinger bosons) or fermi (Abrikosov fermions) statstics, which are then treated within the mean-fie…
The nontrivialness of quantum spin liquid (QSL) typically manifests in the non-local observables that signifies their existence, however, this fact actually casts shadow on detecting QSL with experimentally accessible probes. Here, we provide a solution by unbiasedly demonstrating dynamical signature of anyonic excitations and symmetry fractionalization in QSL. Employing large-scale quantum Monte Carlo simulation and stochastic analytic continuation, we investigate the extended XXZ model on the kagome lattice, and find out that across the phase transitions from Z2 QSLs to different symmetry br…
Gapped $Z_{2}$ spin liquid as the simplest spin liquid has been proved to be the most difficult to realize in realistic models. Here we show that the frustration from a third-neighbor exchange $J_{3}$ on the spin-$\frac{1}{2}$ $J_{1}-J_{2}$ model on the square lattice may serve to stabilize such a long-sought state. We argue that a Bosonic RVB description is more appropriate than a Fermonic RVB description for such a gapped spin liquid phase. We show that while the mean field ansatz of the proposed state has a $U(1)$ gauge symmetry, the gauge fluctuation spectrum on it is actually gapped and t…
We describe a mechanism for order fractionalization in a two-dimensional Kondo lattice model, in which electrons interact with a gapless spin liquid of Majorana fermions described by the Yao-Lee (YL) model. When the Kondo coupling to the conduction electrons exceeds a critical value, the model develops a superconducting instability into a state where the spinor order parameter carries charge $e$ and spin $S=1/2$. The broken symmetry state develops a gapless Majorana Dirac cone in the bulk. By including an appropriate gauge string, we can show that the charge $e$, spinorial order develops off-d…
The type-II terminated $1T$-TaS$_2$ surface of a three-dimensional $1T$-TaS$_2$ bulk material realizes the effective spin-$1/2$ degree of freedom on each David star cluster with ${\mathcal{T}^2=-1}$ such that the time-reversal symmetry is realized anomalously, despite the fact that bulk three-dimensional $1T$-TaS$_2$ material has an even number of electrons per unit cell with ${\mathcal{T}^2=+1}$. This surface is effectively viewed as a spin-$1/2$ triangular lattice magnet, except with a fully gapped topological bulk. We further propose this surface termination realizes a spinon Fermi surface …
