Abstract
Material line defects are onedimensional structures but the search and proof of electron behaviour consistent with the reduced dimension of such defects has been so far unsuccessful. Here we show using angle resolved photoemission spectroscopy that twingrain boundaries in the layered semiconductor MoSe_{2} exhibit parabolic metallic bands. The onedimensional nature is evident from a charge density wave transition, whose periodicity is given by k_{F}/π, consistent with scanning tunnelling microscopy and angle resolved photoemission measurements. Most importantly, we provide evidence for spin and chargeseparation, the hallmark of onedimensional quantum liquids. Our studies show that the spectral line splits into distinctive spinon and holon excitations whose dispersions exactly follow the energymomentum dependence calculated by a Hubbard model with suitable finiterange interactions. Our results also imply that quantum wires and junctions can be isolated in line defects of other transition metal dichalcogenides, which may enable quantum transport measurements and devices.
Introduction
1D electron systems (1DES) are sought for their potential applications in novel quantum devices, as well as for enabling fundamental scientific discoveries in materials with reduced dimensions. Certainly, 1D electron dynamics plays a central role in nanoscale materials physics, from nanostructured semiconductors to (fractional) quantum Hall edge states^{1,2}. Furthermore, it is an essential component in Majorana fermions^{3,4} and is discussed in relation to the highT_{c} superconductivity mechanism^{5}. However, truly 1D quantum systems that permit testing of theoretical models by probing the full momentumenergy (k, ω)space are sparse and consequently angleresolved photoelectron spectroscopy (ARPES) measurements have only been possible on quasi1D materials consisting of 2D or 3Dcrystals that exhibit strong 1D anisotropy^{6,7,8,9,10}.
Electrons confined in onedimension (1D) behave fundamentally different from the Fermiliquid in higher dimensions^{11,12,13}. While there exist various quasi1D materials that have strong 1D anisotropies and thus exhibit 1D properties, strictly 1D metals, that is, materials with only periodicity in 1D that may be isolated as a single wire, have not yet been described as 1D quantum liquids. Grain boundaries in 2D van der Waals materials are essentially 1D and recent DFT simulations on twin grain boundaries in MoS_{2} (ref. 14) and MoSe_{2} (ref. 15) have indicated that those defects should exhibit a single band intersecting the Fermi level. Therefore, such individual line defects are exceptional candidates for truly 1D metals.
In the case of quasi1D MottHubbard insulators (MHI)^{16,17,18,19}, there is strong evidence for the occurrence of the so called spincharge separation^{17,18}. Recently, strong evidence of another type of separation in these quasi1D compounds was found, specifically a spinorbiton separation with the orbiton carrying an orbital excitation^{16}.
The theoretical treatment of MHI is easier compared with that of the physics of 1D metals. The ground state of a MHI has no holons and no spinons and the dominant oneelectron excited states are populated by one holon and one spinon, as defined by the Tomonaga Luttinger liquid (TLL) formalism^{12}. For 1DES metals the scenario is however more complex, as the holons are present in both the ground and the excited states. Zero spindensity ground states have no spinons. Consequently, the experimental verification of key features of 1DES, especially the spincharge separation, remains still uncertain^{6,20,21,22}.
The theoretical description of 1DES lowenergy excitations in terms of spinons and holons, based on the TLL formalism, has been a corner stone of 1D electron lowenergy dynamics^{12}. The rather effective approximation of the relation of energy versus momentum in 1D fermions by a strictly linear dispersion relation, makes the problem accessible and solvable, by calculating analytically the valuable manybody lowenergy dynamics of the system. This drastic assumption has provided an effective tool to describe lowenergy properties of 1D quantum liquids in terms of quantized linear collective sound modes, named spinons (zerocharge spin excitations) and holons (spinless charge excitations), respectively. However, this dramatic simplification is only valid in the range of lowenergy excitations, very close to the Fermi level.
More recently, sophisticated theoretical tools have been developed that are capable to extend this description to highenergy excitations away from the Fermilevel^{13,23,24,25,26,27,28}. Particularly, the pseudofermion dynamical theory (PDT)^{24,25,26,27} allows to compute oneparticle spectral functions in terms of spinon and holon features, in the full energy versus momentum space ((k, ω)plane). The exponents controlling the low and highenergy spectralweight distribution are functions of momenta, differing significantly from the predictions of the TLL if applied to the highenergy regime^{23,24,25,26,27}. To the best of our knowledge, while other theoretical approaches, beyond the TLL limit, have also been recently developed^{13,28}, no direct photoemission measurements of spincharge separation in a pure metallic 1DES has been reported so far. Even more important, a theoretical 1D approach with electron finiterange interactions entirely consistent with the photoemission data in the full energy versus momentum space has never been reported before^{11,12,29}.
Here we present a description of the nonFermi liquid behaviour of a metallic 1DES with suitable finiterange interactions over the entire (k, ω)plane that matches the experimentally determined weights over spin and charge excitation branches. This has been accomplished by carrying out the first ARPES study of a 1DES hosted in an intrinsic line defect of a material and by developing a new theory taking electron finiterange interactions within an extended 1D Hubbard model into account. The mirror twin boundaries in a monolayer transition metal dichalcogenide^{30,31} are true 1D line defects. They are robust to high temperatures and atmospheric conditions, thus making them a promising material system, which is amendable beyond ultra high vacuum investigations and useful for potential device fabrication. Previously, the structural properties of these line defects have been studied by (scanning) transmission electron microscopy^{15,30,31,32} and by scanning tunnelling microscopy (STM) and tunnelling spectroscopy^{33,34,35}.
Results
Line defect characterization
Figure 1 shows STM results of the mono to bilayer MoSe_{2} grown on a MoS_{2} single crystal substrate. Three equivalent directions for the MTBs are observed in the hexagonal MoSe_{2} crystal. The high density of these aligned line defects in MoSe_{2} (ref. 30) provides a measurable ARPES signal for this 1DES and thus enables the ω(k) characterization of this line defect.
Peierls transition in MoSe_{2} grain boundary
For metallic 1D structures, an instability to charge density wave (CDW) is expected (see additional discussion in Supplementary Note 1), which has been previously reported for MoSe_{2} grain boundaries by low temperature STM studies^{35}. The CDW in MTBs gives rise to a tripling of the periodicity, as can be seen in the low temperatureSTM images shown in Fig. 2a,b. The CDW in 1D metals is a consequence of electronphonon coupling. The realspace periodicity of the CDW is directly related to a nesting of the Fermi wavevector, as schematically shown in Fig. 2c. ARPES measurements of the Fermisurface can thus directly provide justification for the periodicity measured in STM, which is shown below. In addition, the CDW transition is a metalinsulator transition and thus changes in the sample resistance occur at the CDW transition temperature. Figure 2d shows a fourpoint measurement with macroscopic contacts on a continuous mono to bilayer film (as shown in Fig. 1c). Clear jumps in the resistance are observed for three different samples at ∼235 K and ∼205 K, which are attributed to an incommensurate and commensurate CDW transitions, respectively. The drop in resistance at lower T is assigned to a depinning of the CDW from defects and socalled CDW sliding. CDW sliding is a consequence of the applied potential rather than a specific temperature.
To study a stable, gapless, 1DES, we determine the spectral weight together with the energy dispersion in momentum space, by performing ARPES measurements at room temperature, which is well above the CDW transition temperature. This is done on samples consisting predominantly of monolayer MoSe_{2} islands, as shown in the Supplementary Fig. 1. Figure 3; Supplementary Fig. 2 illustrate the Fermi surface of 1D metals, consisting of two parallel lines, separated by 2k_{F}, in the absence of interchain hopping. Because of the three equivalent real space directions of the MTBs in our sample, superpositioning of three rotated 1DES results in starshaped constant energy surface in reciprocal space, as shown in Fig. 3; Supplementary Note 1. In the three cases, a perfect nesting is noticeable, namely one complete Fermi sheet can be translated onto the other by a single wave vector ±2k_{F}.
Even more important, by using high energy and momentum resolution ARPES, the Fermiwave vector could be precisely determined, giving a value of k_{F}=0.30±0.02 Å^{−1}, which is about 1/3 of the BZboundary at . Hence a band filling of n=2/3 has been experimentally obtained. The Fermiwavevector also gives a direct prediction of the CDW periodicity of π/k_{F}=10.5±0.7 Å, which is in good agreement with measured in STM (Fig. 2).
Spin charge separation
While the perfect nesting conditions in 1D metals predicts a CDW transition, its occurrence is no proof for 1D electron dynamics. For obtaining evidence of 1D electron dynamics, a detailed analysis of the spectral function and its consistency with theoretically predicted dispersions need to be demonstrated. The photoemission spectral function of the 1D state is shown in Fig. 3e,f. Without any sophisticated analysis and considering only the raw ARPES data, it is evident that the experimental results are in complete disagreement with the single dispersing band predicted by ground state DFT simulations^{15,35}. Effectively, our data cannot be fit with a single dispersion branch (see also Supplementary Fig. 4 and Supplementary Note 2 for an analysis of the raw data in terms of energy distribution curves (EDC), momentum distribution curves (MDC) and lifetime.)
Using data analysis that applies a curvature procedure to raw data^{36}, as commonly used in ARPES, the experimental band dispersions in the full energy versus momentum space show two clear bands that exhibit quite different dispersions. We provisionally associate, which our theoretical results confirms below, the upper and lower dispersion with the spinon and the holon branch, respectively. Manifestly, the spin mode follows the lowenergy part of the 1D parabola, whereas the charge mode propagates faster than the spin mode. The extracted experimental velocity values are v_{h}=4.96 × 10^{5} ms^{−1} and v_{s}=4.37 × 10^{5} ms^{−1}, revealing a ratio v_{h}/v_{s} of the order of ≈0.88. Notice that these states lie entirely within the band gap of the MoSe_{2} monolayer, whose VBM is located at 1.0 eV below the Fermilevel, see Supplementary Fig. 3.
DFT simulations cannot predict the electron removal spectrum of the 1D electron dynamics. Thus the single dispersing band obtained in previous DFT simulations for this system is not expected to be consistent with the experiment. However, the singleband DFT results indicate that the electron dynamics behaviour can be suitably described by a single band Hubbard model and associated PDT. The PDT is a method that has been originally used to derive the spectral function of the 1D Hubbard model in the vicinity of highenergy branchline singularities^{24,25,26,27}. It converges with TLL for low energies^{37}. As reported below, here we use a renormalized PDT (RPDT) because the conventional 1D Hubbard does not include finiterange interactions.
Low energy properties and TLL electron interaction strength
Critical for calculating the spectral functions with RPDT is the knowledge of the electron interaction strength, which needs to be determined experimentally. Since very close to the Fermi level, in the lowenergy excitations limit, the RPDT converges to the TLL theory, we have evaluated the photoemission weight in the vicinity of the Fermilevel in accordance to TLL theory. A decisive lowenergy property of 1D metals is, according to that theory^{12,38}, the suppression of the DOS at the Fermilevel, whose power law exponent is dependent on the electron interaction range and strength. Figure 4 shows the angle integrated photoemission intensity, which is proportional to the occupied DOS, as a function of energy for the 1DES. It is compared with the photoemission from a gold sample under the same conditions. The suppression of the DOS for the 1D defects compared with Au is apparent in Fig. 4a.
According to the TLL scheme, the suppression of DOS follows a power law dependence whose exponent is determined by the electron interaction strength and range in the 1D system. An exponent of ∼0.8 is extracted from a logplot shown in Fig. 4b. A refined fitting for the exponent α that takes the temperature into account^{39} reveals that the data are best reproduced for α between 0.75 and 0.80 (Fig. 4c). The charge TLL parameter K_{c}, which provides information on the range of the electron interaction^{29}, is related to α by α=(1−K_{c})^{2}/4K_{c}. Hence K_{c} has values between 0.20 and 0.21.
Comparison of experiment to the theoretical model
Within the 1D Hubbard model with onsite repulsion U and hopping integral t, the charge TLL parameter K_{c} and related exponent α values should belong to the ranges K_{c}∈[1/2, 1] and α∈[0, 1/8], respectively. However, our experimental values are in the ranges K_{c}∈[0.20, 0.21] and α∈[0.75, 0.80], which is an unmistakable signature of electron finiterange interactions and therefore our system cannot be studied in the context of the conventional 1D Hubbard model^{29}. Consequently, we have developed a new theoretical scheme that successfully includes such interactions. As justified below in the Methods section, the corresponding RPDT specifically relies on the spectral function near the branch lines of the nonintegrable 1D Hubbard model with finiterange interactions being obtained from that of the integrable 1D Hubbard model PDT^{24,25,26,27} on suitably renormalising its spectra and phase shifts.
The renormalization using the PDT approach has two steps. The first refers to the U value, which loses its onsiteonly character and is obtained upon matching the experimental band spectra with those obtained within the 1D Hubbard model for n=2/3, leading to U=0.8t. Indeed, the ratio W_{h}/W_{s} of the observed c band (holon) and s band (spinon) energy bandwidths W_{h}=ɛ_{c}(2k_{F})−ɛ_{c}(0) and W_{s}=ɛ_{s}(k_{F})−ɛ_{s}(0), respectively, is achieved for that model at U/t=0.8. (The energy dispersions ɛ_{c}(q) for q∈[−π, π] and ɛ_{s}(q′) for q′∈[−k_{F}, k_{F}] and the related γ=c, c′, s exponents considered in the following are defined in more detail in the Methods section.) This renormalization fixes the effective U value yet does not affect t. The corresponding c and c′ (holon) and s (spinon) branch lines spectra ω_{c}(k)=ɛ_{c}(k+k_{F}) for k∈[−k_{F}, k_{F}], ω_{c′}(k)=ɛ_{c}(k−k_{F}) for ∈(−3k_{F}, 3k_{F}) and ω_{s}(k)=ɛ_{s}(k) for k∈[−k_{F}, k_{F}] are plotted in Fig. 5d–f; Supplementary Fig. 5. An important difference relative to the n=1 MottHubbard insulating phase is that for the present n=2/3 metallic phase the energy bandwidth W_{c}=ɛ_{c}(π)−ɛ_{c}(2k_{F}) does not vanish. That the renormalization does not affect t stems from a symmetry that implies that the full c band energy bandwidth is independent of both U and n and reads W_{h}+W_{c}=4t. Hence W_{h}=4t for the MottHubbard insulator whereas W_{h}<4t for the metal. Combining both the value of the ratio W_{h}/W_{c} for the 1D Hubbard model at U/t=0.8 and n=2/3 and the exact relation W_{h}+W_{c}=4t with analysis of Fig. 5d–f, one uniquely finds t≈0.58 eV. The parameter α is here denoted by α_{0} for the 1D Hubbard model. It reads with α_{0}=0 for U/t→0 and α_{0}=1/8 for U/t→∞ where is a superposition of pseudofermion phase shifts. (see Methods.)
The second step of the renormalization corresponds to changing the ξ_{c} and phase shift values so that the parameter has values in the range α∈[α_{0}, α_{max}] where α_{0}≈1.4 × 10^{−3} for U/t=0.8 and n=2/3. As justified in the Methods section, α_{max}=49/32≈1.53. The effect of increasing α at fixed finite U/t and n from α_{0} to 1/8 is qualitatively different from that of further increasing it to α_{max}. As discussed in that section, the changes in the (k, ω) plane weight distribution resulting from increasing α within the latter interval α∈[1/8, α_{max}] are mainly controlled by the finite–range interactions.
For U/t=0.8, n=2/3 and T=0 the oneelectron spectral function of both the conventional 1D Hubbard model (α=α_{0}) and corresponding model with finite range interactions (α∈[α_{0},α_{max}]) consists of a (k, ω)plane continuum within which welldefined singular branch lines emerge. Most of the spectral weight is located at and near such singular lines. Near them, the spectral function has a powerlaw behaviour characterised by negative k dependent exponents. At T≈300 K such singular lines survive as features displaying cusps. Our general renormalization procedure leads to a oneelectron spectral function expression that for small deviations (ω_{γ}(k)−ω)>0 from the finiteenergy spectra ω_{γ}(k) of the γ=c, c′, s branch lines plotted in Fig. 5d–f reads, for α∈(α_{0}, α_{max}). The singular branch lines correspond to the γ=c, c′, s lines k ranges for which their exponents are negative. As confirmed and justified in the Methods section, for U/t=0.8, n=2/3 and t=0.58 eV there is quantitative agreement with the (k, ω)plane ranges of the experimentally observed spectral function cusps for α∈[0.75, 0.78]. This is fully consistent with the α experimental uncertainty range α∈[0.75,0.80]. The three γ=c, c′, s exponents momentum dependence for both the 1D Hubbard model with finiterange interactions corresponding to α=0.78 (full lines) and the conventional 1D Hubbard model for which α=α_{0}≈1.4 × 10^{−3} (dasheddotted lines) is plotted in (Fig. 5a–c).
Discussion
The agreement of the theoretical calculations with finite range interactions over the entire (k, ω)plane provides strong evidence for the assignment of the two spectral branches observed in the experiments to spin charge separation in a 1D metal. Despite this agreement, alternative explanations for the photoemission spectrum should be noted. Strongly asymmetric line shapes in photoemission spectra have been reported and thus an assignment of the cusps to yet unknown lineshape effects in 1D materials cannot be entirely excluded. However, the accurate prediction of the continuum between the cusp lines and the fit of the c and s branchline dispersions by the 1D Hubbard model with finite range interactions makes alternative effects unlikely to reproduce exactly such spectral features.
Concerning the DOS at the Fermi level, our measurements clearly show a suppression of the DOS that can be fit with a power law behaviour. DOS suppression has, however, also been observed due to finalstate pseudogap effects in nanostructures^{40,41}. While it is difficult to exclude such effects categorically, the expected 1D nature of the line defects and thus the breakdown of Fermiliquid theory requires application of TLL, as has been applied to other (quasi) 1D systems in the past^{6,38,42}, to interpret photoemission intensity at the Fermi level. Certainly, obtaining the same exponent α for the power law behaviour of TLL from the experimental fit of the DOS and the spectral features of the 1D Hubbard model with finite range interactions support the assignment of the DOS suppression at the Fermilevel to TLL effects.
We have presented a detailed experimental analysis of the electronic structure of a material line defect by angle resolved photoemission. High density of twin grain boundaries in epitaxial monolayer MoSe_{2} could be analysed by angle resolved photoemission spectroscopy. This enabled us to accurately determine the Fermi surface and demonstrate the CDW observed in this material is a consequence of Fermi wave vector nesting. Both the suppression of DOS at the Fermi level as well as broad spectral features with notable cusps are in agreement with 1D electron dynamics. While the lowenergy spectra are described by TLL, the dispersion of the cusps in the full energy versus momentum space in highenergy range could be only accurately reproduced by a 1D Hubbard model with suitable finite range interactions. Consequently, the cusps could be interpreted as spin and charge separation in these 1D metals. The accurate description of the experiment by RPDT calculations allows us to go beyond the low energy restriction of TLL, showing that the exotic 1D physics is valid for both low and highenergy, with nonlinear band dispersions and broad momentum values. Unlike other systems that only exhibit strong 1D anisotropy, the intrinsic line defects in TMDs have no specific repetition length and can thus be viewed as true 1D structures. Moreover, isolated twin grain boundaries of micrometre length have been recently reported in CVDgrown TMDs^{31}, which can be envisaged as remarkable candidates for quantum transport measurements on isolated 1D metals. Furthermore, 2D materials can be gated and this will exert control of transport properties of these quantum wires.
Methods
Sample preparation
Monolayer MoSe_{2} islands were grown by van der Waals epitaxy by codeposition of atomic Se from a hot wall Secracker source and Mo from a miniebeam evaporator. The MoS_{2} single crystal substrate was a synthetically grown and cleaved in air before introducing into the UHV chamber where it was outgassed at 300 °C for 4 h before MoSe_{2} growth. Mo has been deposited in a selenium rich environment at a substrate temperature of ∼300–350 °C. The MoSe_{2} monolayer was grown slowly with a growth rate of ∼0.16 monolayers per hour. While the detailed mechanism for the formation of MTBs during MBE growth is not completely understood, it has been noted that the structure shown in Fig. 1a is deficient in chalcogen atoms, i.e. the grain boundary has a stoichiometry of MoSe embedded in the MoSe_{2} matrix. Computational studies have shown that MTBs are thermodynamically favoured over the formation of high density of individual chalcogen vacancies^{15} and this may explain their presence in MBE grown samples. These samples were investigated by RT STM in a surface analysis chamber connected to the growth chamber. In Addition, characterization by VTSTM and ARPES were performed by transferring the grown samples in a vacuum suitcase to the appropriate characterization chambers. In addition, airexposed samples were characterised by ARPES. After vacuum annealing to ∼300 °C, the ARPES results were indistinguishable to the in vacuum transferred samples indicating the stability of the material in air against oxidation and other degradation. The stability of the sample also enables the fourpoint transport measurements described below.
ARPES measurements
MicroARPES measurements were performed at the ANTARES beamline at the SOLEIL synchrotron. The beam spot size was ∼120 μm. The angular and energy resolution of the beamline at a photon energy of 40 eV are ∼0.2° and ∼10 meV, respectively. Most of the data were collected around the Γpoint of the second Brillouin zone, corresponding to an emission angle of 42.5° with respect to the surface normal, for photon energy of 40 eV. Both left and right circular polarized light, as well as linear polarized light was used. The photonincident angle on the sample was normal incidence. For circular polarized light photoemission from all MTBs is obtained. Emission from a single MTB direction could be enhanced with linear polarized light and the Avector parallel to the surface. For azimuth rotation with the Avector aligned to the direction of one MTB enhanced emission from this direction was obtained as shown in Fig. 3c. All data shown here were obtained at 300 K.
Broadening of the ARPES spectral function and lifetime analysis
As it has already been reported in previous ARPES studies (see for instance Fig. 5 of ref. 17), the lifetime of a Fermiliquid quasiparticle, τ(k), can be directly determined from the width of the peak in the energy distribution curves (EDC), analysing the ARPES data defined by the spectral weight at fixed k as a function of ω, where ω is the energy. Specifically,
The consistency of a Fermiliquid picture can be also checked by studying the momentum distribution curves (MDC), that is, from the momentum width Δk of the spectral function peak at fixed binding energy, ω. As long as the Fermiliquid quasiparticle excitation is well defined, (that is, the decay rate is small compared with the binding energy), the energy bandwidth and momentum width are related as,
Here v_{F} is the renormalized Fermi velocity, which can be directly measured using high energy and momentum resolution ARPES. Because of the separation of charge and spin, one hole (or one electron) is always unstable to decay into two or more elementary excitations, of which one or more carries its spin and one or more carries its charge. Then elementary kinematics implies that, at T=0, the spectral function is nonzero only for negative frequencies such that,
where v_{c} and v_{s} are the charge and spin velocity, respectively. This analysis procedure is described in Fig. 6, where the spectral function particularly at ω values between 0.40 and 0.95 eV shows a continuum, which is valid for all momentum k values that fit Equation (3). MDC and EDC plots are sensitive to this detachment of the system with respect to a conventional Fermiliquid quasiparticle behaviour.
This type of analysis, based on the shape of EDC and MDC plots, is also well explained by Emery et al. (see Figs 2 and 3 of ref. 5). In Fig. 6 we present the results of a similar analysis. As it is shown in panels (d) and (e), the MDC and EDC cuts of the raw data at different binding energies and momentum, respectively, show a clear enlargement of the lifetime that can be extracted from the ARPES data. However, this experimental value is just proportional to various interaction strengths. This approximative methodology of the nature and magnitude of the present interactions can be improved by using more sophisticate theoretical approaches as the one reported in the present manuscript.
PDT as starting point of our theoretical method
The method used in our theoretical analysis of the spincharge separation observed in the 1D quantumline defects of MoSe_{2} was conceived for that specific goal. It combines the pseudofermion dynamical theory (PDT) for the 1D Hubbard model^{24,27,37} with a suitable renormalization procedure.
On the one hand, the 1D Hubbard model range α_{0}∈[0,1/8] corresponds to the intervals K_{c}∈[1/2,1] and of the TLL charge parameter^{29,12,43} and the related parameter . On the other hand, the range α∈[0.75, 0.78] for which the renormalized theory is found to agree with the experiments implies that and have values in the ranges and , respectively. Here and is our notation for the TTL charge parameter and related parameter, respectively, in the general case when they may have values within the extended intervals and thus . The minimum values and follow from corresponding phaseshift allowed ranges. (Below the relation of to phase shifts is reported.) The above experimental subinterval belongs to the interval for which the electron finiterange interactions must be accounted for ref. 29.
In the case of the conventional 1D Hubbard model, the PDT was the first approach to compute the spectral functions for finite values of U/t near singular lines at highenergy scales beyond the lowenergy TLL limit^{24}. (In the lowenergy limit the PDT recovers the TLL physics^{37}.) After the PDT was introduced for that integrable model, novel methods that rely on a mobile impurity model (MIM) approach have been developed to tackle the highenergy physics of both nonintegrable and integrable 1D correlated quantum problems, also beyond the lowenergy TLL limit^{13,28,44,45}. The relation between the PDT and MIM has been clarified for a simpler model^{46}, both schemes leading to exactly the same momentum dependent exponents in the spectral functions expressions. Such a relation applies as well to more complex models. For instance, studies of the 1D Hubbard model by means of the MIM^{44,45} lead to exactly the same momentum, interaction and density dependence as the PDT for the exponents that control the oneelectron removal spectral function near its branch lines.
For integrable models, in our case the 1D Hubbard model, there is a representation in terms of elementary objects called within the PDT c and s pseudofermions for which there is only zeromomentum forwardscattering at all energy scales. The c and s bands momentum values are associated with the 1D Hubbard model exact Betheansatz solution quantum numbers. The c pseudofermion and the s pseudofermion annihilated under transitions from the N electron ground state to the N−1 electron excited states refer to the usual holon and spinon, respectively^{12,13,43}.
That for the pseudofermions there is only zeromomentum forwardscattering at all energy scales, follows from the existence of an infinite number of conservation laws associated with the model integrability^{47,48}. This means that in contrast to the model underlying electron interactions, the pseudofermions, on scattering off each other only acquire phase shifts. Hence under their scattering events there is no energy and no momentum exchange, on the contrary of the more complex underlying physical particles interactions. In the vicinity of welldefined (k, ω)plane features called branch lines, the T=0 spectral functions of integrable 1D correlated models are of powerlaw form with negative momentum dependent exponents. Such properties apply to all integrable 1D correlated models.
Universality behind our method renormalization procedures
In the case of nonintegrable 1D correlated models, there is no pseudofermion representation for which there is only zeromomentum forwardscattering at all energy scales. This is because of the lack of an infinite number of conservation laws. The universality found in the framework of the MIM for the spectral functions of nonintegrable and integrable 1D models^{13,28} refers to specific energy scales corresponding to both the lowenergy TLL spectral features and energy windows near the highenergy nonTLL branch lines singularities. In the vicinity of these lines, the T=0 spectral functions of nonintegrable 1D correlated models are also of powerlaw form with negative momentum dependent exponents.
This universality means that at both these energy scales there is for such models a suitable representation in terms of pseudofermions that undergo only zeromomentum forwardscattering events and whose phase shifts control the spectral functions behaviours. Our renormalization scheme for adding electron finiterange interactions to the 1D Hubbard model and corresponding PDT relies on this universality. Indeed, the finiterange interactions render the model nonintegrable. However, in the vicinity of the branch lines singularities the spectral function remains having the same universal behaviour. Our normalization procedure can be used for any chosen α value in the range α∈[α_{0},α_{max}]. Here α_{0}∈(0,1/8) is the conventional 1D Hubbard model α value for given U/t and electronic density n values. For the U/t=0.8 and n=2/3 values found within our description of the 1D quantumline defects of MoSe_{2} it reads α_{0}≈1.4 × 10^{−3}. The maximum α value α_{max}=49/32=1.53125 refers through the relation , and thus to the above minimum values and .
The renormalization of the conventional 1D Hubbard model used in our studies refers to some 1D Hamiltonian with the same terms as that model plus finiterange interaction terms. The latter terms are neither a mere firstneighbouring V term nor a complete longrange Coulomb potential extending over all lattice sites. Interestingly, the specific form of the additional finiterange interaction Hamiltonian terms is not needed for our study. This follows from the above universality implying that both for the lowenergy TLL limit and energy windows near the highenergy branch lines singularities of the 1D Hubbard model with finite–range interactions under consideration the relation of α to the phase shifts remains exactly the same as for the conventional 1D Hubbard model.
Importantly, the only input parameters of our renormalization procedure are the effective U and transfer integral t values for which the theoretical branch lines energy bandwidths match the corresponding experimental bandwidths. Apart from the 1D quantumline defects bandfilling n=2/3, our approach has no additional ‘fitting parameters’.
The spectra in terms of pseudofermion energy dispersions
Within the PDT for the 1D Hubbard model^{24,25,26,27}, nearly the whole electron removal spectral weight is in the metallic phase originated by two ι=±1 excitations generated from the ground state by removal of one c pseudofermion of momentum q∈[−2k_{F}, 2k_{F}] and one s pseudofermion of momentum q′∈[−k_{F}, k_{F}]. The superposition in the (k, ω)plane of the spectral weights associated with the corresponding two ι=±1 spectra generates the multiparticle continuum. Such ι=±1 spectra are of the form,
They are twoparametric, as they depend on the two independent c and s bands momenta q and q′, respectively. Hence such spectra refer to twodimensional domains in the (k, ω)plane. They involve the energy dispersion ɛ_{c}(q) whose c momentum band interval is q∈[−π, π]and whose groundstate c pseudofermion occupancy is q∈[−2k_{F}, 2k_{F}] and the dispersion ɛ_{s}(q′) whose s momentum band range is q′∈[−k_{F}, k_{F}], which is full in the present zero spindensity ground state, are defined below.
The multiparticle continuum in the oneelectron removal spectral function that results from the superposition of the spectral weights associated with the two ι=±1 spectra contains three branch lines that display the cusps: two c,ι branch lines and a s branch line. The c,ι branch lines result from processes for which the removed c pseudofermion has momentum in the range q∈[−2k_{F}, 2k_{F}] and the removed s pseudofermion has momentum . Hence the excitation physical momentum is . The s branch line results from removal of one c pseudofermion of momentum . The removed s pseudofermion has momentum in the interval q′∈[−k_{F}, k_{F}]. The physical momentum is then given by k=−q′.
It is convenient to redefine the two c,ι branch lines in terms of related c and c′ branch lines. The spectra of the c, c′, and s branch lines are plotted in Fig. 5d–f for U/t=0.8, t=0.58 eV and electronic density n=2/3. On the one hand, the c branch line results from processes relative to the ground state that involve removal of one c pseudofermion with momentum belonging to the ranges q∈[−2k_{F}, −k_{F}] and q∈[k_{F}, 2k_{F}] and removal of one s pseudofermion with momentum q′=−ιk_{F} for ι=sgn{k}. The c branch line spectrum then reads,
On the other hand, the c′ branch line is generated by removal of one c pseudofermion with momentum belonging to the ranges q∈[−2k_{F}, k_{F}] and q∈[−k_{F}, 2k_{F}] and removal of one s pseudofermion with momentum q′=−ιk_{F} for ι=−sgn{k}. Its spectrum is given by,
The s branch line spectrum reads,
The dispersions ɛ_{c}(q) and ɛ_{s}(q′) appearing in these equations are uniquely defined by the following equations valid for U/t>0 and electronic densities n∈[0, 1],
Here the distributions 2tη_{c}(Λ) and 2tη_{s}(Λ) are the unique solutions of coupled integral equations given in supplementary Equations 1 and 2.
The q and q′ dependence of the dispersions ɛ_{c}(q) and ɛ_{s}(q′) occurs through that of the momentum rapidity function k=k(q) for q∈[−π, π] and spin rapidity function Λ=Λ(q′) for q′∈[−k_{F}, k_{F}], respectively. Those are defined in terms of their inverse functions q=q(k) for k∈[−π, π] and q′=q′(Λ) for Λ∈[−∞, ∞] in supplementary Equations 3 and 4. The distributions 2πρ(k) and 2πσ(Λ) in their expressions are the unique solutions of the coupled integral equations provided in Supplementary Equations 5 and 6.
Spectral function within the conventional 1D Hubbard model
Within the PDT for the 1D Hubbard model^{24,25,26,27}, the spectral weight distributions are controlled by the set of phase shifts ±2πΦ_{β,β′}(q, q′) acquired by the β=c and β=s pseudofermions with momentum q upon scattering off each β′=c and β′=s pseudofermion with momentum q′ created (+) or annihilated (−) under the transitions from the ground state to the excited energy eigenstates. (In contrast to otherwise in this section, here the momentum values q and q′ are not necessarily those of c and s pseudofermions, respectively.)
The expressions of the momentum dependent exponents that control the line shape in the vicinity of the γ=c, c′, s branch lines involve phase shifts whose β=c, s pseudofermions have momentum at the corresponding Fermi points, ±q_{Fc}=±2k_{F} and ±q_{Fs}=±k_{F}. This includes phase shifts 2πΦ_{β,β′}(ιq_{Fβ},ι′q_{Fβ′})=−2πΦ_{β,β′}(−ιq_{Fβ},−ι′q_{Fβ′}), where ι=±1, ι′=±1, acquired by such β=c, s pseudofermions on scattering off β′=c,s pseudofermions of momentum also at Fermi points annihilated under the transitions from the N electron ground state to the N−1 excited states. Furthermore, such exponents expressions also involve phase shifts−2πΦ_{β,c}(q_{Fβ}, q)=2πΦ_{β,c}(−q_{Fβ}, −q) and−2πΦ_{β,s}(q_{Fβ}, q′)=2πΦ_{β,s}(−q_{Fβ}, −q′) acquired by the same β=c,s pseudofermions upon scattering off β′=c and β′=s pseudofermions of momentum q∈[−2k_{F}, 2k_{F}] and q′∈[−k_{F}, k_{F}], respectively, annihilated under such transitions.
For energy windows corresponding to small energy deviations (ω_{γ}(k)−ω)>0 from the highenergy γ=c, c′, s branchline spectra ω_{c}(k)=ɛ_{c}(k+k_{F}) for k∈(−k_{F},k_{F}), ω_{c′}(k)=ɛ_{c}(k−k_{F}) for k∈(−3k_{F},3k_{F}) and ω_{s}(k)=ɛ_{s}(k) for k∈(−k_{F},k_{F}), equations 5, 6, 7, the electron removal spectral function has within the PDT the universal form^{25,26,27,37},
The exponents in this general expression are for U/t>0 and electronic densities n∈[0, 1] given in terms of pseudofermion phase shifts in units of 2π by,
At zero spin density, the entries of the conformalfield theory dressedcharge matrix Z and corresponding matrix (Z^{−1})^{T} can be alternatively expressed in terms of pseudofermion phase shifts in units of 2π and of the related parameters ξ_{c} and ξ_{s}, as given supplementary equations 7 and 8, respectively. (Here we use the dressedcharge matrix definition of ref. 37, which is the transposition of that of ref. 43.) Conversely, the pseudofermion phase shifts with both momenta at the Fermi points can be expressed in terms of only the charge TLL parameter and spin TLL parameter (ref. 43) and thus of the present related β=c,s parameters . Specifically,
Here β=c,s and β′=c,s.
The two sets of two coupled integral equations, Supplementary equations 1, 2, 5 and 6, respectively, that one must solve to reach the momentum dependence of the exponents, equation 10, have no simple analytical solution. Within our study, these equations are solved by exact numerical methods. The exponents found from such a numerical solution are plotted as a function of the momentum k in Fig. 5a–c (dasheddotted lines) for U/t=0.8, t=0.58 eV and electronic density n=2/3. The c and s exponent expressions in Equation 10 are not valid at the lowenergy limiting values k=±k_{F}.
In the present zero spindensity case, the spin SU(2) symmetry implies that the parameter ξ_{s} appearing in Equation 11 is u independent and reads . The parameter ξ_{c} in Equations 10 and 11 is in turn given by ξ_{c}=f(sinQ/u) where the function f(r) is the unique solution of the integral equation given the Supplementary Equation 9 whose kernel D(r) is defined in Supplementary Equation 10. The parameter has limiting values for u→0 and ξ_{c}=1 for u→∞. This is why for the 1D Hubbard model the exponent in the lowω power law dependence of the electronic density of states suppression ,
has corresponding limiting values α_{0}=0 for u→0 and α_{0}=1/8 for u→∞.
The c pseudofermion phase shifts 2πΦ_{c, c}(ι2k_{F}, q) for q∈[−2k_{F}, 2k_{F}] and 2πΦ_{c,s}(ι2k_{F}, q′) for q′∈[−k_{F}, k_{F}] that determine the momentum dependence of the exponents in equation (10) are beyond the reach of the TTL. Such exponents also involve the s pseudofermion phase shifts 2πΦ_{s,c}(ιk_{F}, q) and 2πΦ_{s,s}(ιk_{F}, q′). Because of the spin SU(2) symmetry, at zero spin density the latter phase shifts are u independent. They are given in the supplementary Equations 14 and 15. Their values provided in these equations have been accounted for in the derivation of the exponents expressions in Equation (10) and contribute to them.
The c pseudofermion phase shifts explicitly appearing in the exponents expressions, Equation (10), can be written as and where the parameters ±Q=k(±2k_{F}) define the c pseudofermion Fermi points in rapidity space. The corresponding general c pseudofermion phase shifts are given by and where the related rapidity phase shifts and are the unique solutions of the integral equations given in the Supplementary Equations 11 and 12. The free term D_{0}(r) of the former integral equation is provided in Supplementary Equation 13.
One finds from manipulations of integral equations that the energy dispersions ɛ_{c}(q) and ɛ_{s}(q), equation (8), can be expressed exactly in terms of the c pseudofermion rapidity phase shifts as follows,
and
respectively. Here k=k(q) and Λ=Λ(q′) are the momentum rapidity function and spin rapidity function, respectively, considered above.
Description of the finiterange interactions within our method
Below it is confirmed that except for the effective U value the energy dispersions, equations (13) and (14), are not affected by the renormalization that accounts for the short–range interactions. As reported above, the effective value U=0.8t is determined by the ratio W_{h}/W_{s} of the experimentally observed c band (holon) and s band (spinon) energy bandwidths W_{h}=ɛ_{c}(2k_{F})−ɛ_{c}(0) and W_{s}=ɛ_{s}(k_{F})−ɛ_{c}(0), respectively. Indeed, within the 1D Hubbard model the W_{h}/W_{s} value only depends on U/t and the electronic density n. For n=2/3 the agreement with the observed energy bandwidths is then found to be reached for U/t=0.8.
However, the renormalization fixes the effective U value yet does not affect t. This is because of symmetry implying that within the 1D Hubbard model the full c band energy bandwidth ɛ_{c}(π)−ɛ_{c}(0) is independent of U and n and exactly reads 4t. That energy bandwidth can be written as W_{h}+W_{c}=4t, where for the present metallic phase the energy bandwidth W_{c}=ɛ_{c}(π)−ɛ_{c}(2k_{F}) is finite. Within our pseudofermion representation, W_{h} and W_{c} are the c band filled and unfilled, respectively, groundstate Fermi sea energy bandwidths. Again, the value of the ratio W_{h}/W_{c} only depends on U/t and the electronic density n. Accounting for the W_{h}/W_{c} value at U/t=0.8 and n=2/3 together with the exact relation W_{h}+W_{c}=4t one finds from analysis of Fig. 5d–f that t≈0.58 eV for the MoSe_{2} 1D quantumline defects.
Such defects experimental uncertainty interval α∈[0.75, 0.80] of the exponent that controls the lowω electronic density of states suppression ω^{α} is outside the corresponding 1D Hubbard model range, Equation (12). Hence the U=0.8t value obtained from matching the corresponding ARPES cusps lines spectra with those of the 1D Hubbard model for electronic density n=2/3 refers to an effective interaction having contributions both from electron onsite and finiterange interactions. In addition to the interaction U renormalization, both the parameter ξ_{c} and the corresponding c pseudofermion phase shifts 2πΦ_{c,β′}(ι2k_{F}, q_{Fβ′}) in equation (11), where β′=c,s whose expressions involve ξ_{c} undergo a second renormalization. It is such that ξ_{c} is replaced by a parameter associated with α values in the range α∈(α_{0},α_{max}).
The universality referring to lowenergy values in the vicinity of the c and s bands Fermi points implies that for the nonintegrable model with finiterange interactions the relation given in Equation (12) remains having the same form for α∈[α_{0},α_{max}] and , so that,
(The first equation other mathematical solution, , is not physically acceptable.)
On the one hand, the spin SU(2) symmetry imposes that the values of the U/tindependent parameter and s pseudofermion phase shifts 2πΦ_{s,β′}(ιk_{F}, q_{Fβ′}) in equation (11) where β′=c, s remain unchanged for the model with finiterange interactions. On the other hand, the general relations, equation (11), are universal so that for that model corresponding to any α value in the range α∈[α_{0}, α_{max}] the c pseudofermion phase shifts 2πΦ_{c,β′}(ι2k_{F}, q_{Fβ′}) are for β′=c, s given by,
The universality on which our scheme relies refers both to the lowenergy TLL limit and to energy windows near the highenergy c, c′ and s branchlines singularities. The expression of the exponents that control the spectral function behaviour at low energy and in the vicinity of such singularities only involves the phase shifts of c and s pseudofermions with momenta at their Fermi points q=±2k_{F} and q′=±k_{F}, respectively. On the one hand, as result in part of the spin SU(2) symmetry, at zero spin density the general s pseudofermion phase shifts and remain unchanged for their whole momentum intervals. On the other hand, the general phase shifts and of c pseudofermions whose momenta have absolute values q<2k_{F} inside the c band Fermi sea contribute neither to the TLL lowenergy spectral function expression nor to the highenergy branchlines exponents. Consistently, similarly to the s pseudofermion phase shifts and , they remain unchanged upon increasing α from α=α_{0}.
Hence the main issue here is the renormalization of phase shifts of c pseudofermions with momenta at the Fermi points, and for ι=±1. Multiplying and by the phase factor−1 gives the phase shifts acquired by the c pseudofermions of momenta q=ι2k_{F}=±2k_{F} on scattering off one c band hole (holon) created under a transition to an excited state at any momentum q in the interval q∈[−2k_{F}, 2k_{F}] and one s band hole (spinon) created at any momentum q′ in the domain q′∈(−k_{F},k_{F}), respectively. The overall phaseshift renormalization must preserve the c pseudofermion phaseshifts values given in Equation (16) for (i) q=ι2k_{F}=±2k_{F} and (ii) q′=ιk_{F}=±k_{F}. Hence it introduces suitable factors multiplying 2πΦ_{c, c}(ι2k_{F}, q) and 2πΦ_{c,s}(ι2k_{F}, q′). In the case of , this brings about a singular behaviour at q =−ι2k_{F} for α>α_{0} similar to that in the s pseudofermion phase shift 2πΦ_{s,s}(ιk_{F}, q′) at q′=ιk_{F}, Supplementary equation 15, for the conventional 1D Hubbard model, which remains having the same values for the renormalized model.
The c and s pseudofermion phase shifts of the 1D Hubbard model with electron finiterange interactions are for the whole range α∈[α_{0}, α_{max}] thus of the general form,
Our theoretical results refer to the thermodynamic limit at T=0. In that case the phaseshifts renormalization, Equation (17), only affects those of the c pseudofermion scatterers with momentum values ±2k_{F} corresponding to the zeroenergy Fermi level. Note however that the corresponding c and s pseudofermion scattering centres have momenta q∈[−2k_{F}, 2k_{F}] and q′∈[−k_{F}, k_{F}], respectively, that correspond to a large range of highenergy values. At finite temperature T≈300 K one has that k_{B}T≈0.045t where t≈0.58 eV is within the present theoretical description the transfer integral value suitable for the MoSe_{2} 1D quantumline defects. The derivation of some of the theoretical expressions involves a T=0 c band momentum distribution that reads one for q<2k_{F} and zero for 2k_{F}<q<π. At finite temperature T≈300 K, such a distribution is replaced by a c pseudofermion FermiDirac distribution. This implies for instance that the q=±2k_{F} c pseudofermion phaseshift renormalization in Equation (17) is extended from the zeroenergy Fermi level to a small region of energy bandwidth 0.045t≈0.026 eV near the c band Fermi points q=±2k_{F}. This refers to a corresponding small region with the same energy bandwidth near the physical Fermi points k=±k_{F} in Fig. 5d–f. Interestingly, finitesize effects have at T=0 the similar effect of slightly enhancing the energy bandwidth of the c pseudofermion phase shifts renormalization, Equation (17), in the very vicinity of the zeroenergy Fermi level. Hence any small finite temperature and/or the system finite size remove/s the singular behaviour of the phaseshifts renormalization being restricted to the zeroenergy Fermi level.
Fortunately, both the finite size of the MoSe_{2} 1D quantumline defects and the experimental temperature ≈300 K lead though to very small effects, as confirmed by the quantitative agreement reached between the T=0 theoretical results associated with the 1D Hubbard model with electron finiterange interactions and the experimental data. Hence for simplicity in the following we remain using our T=0 theoretical analysis in terms of that model in the thermodynamic limit.
Spectral function accounting for finite–range interactions
For energy windows corresponding to small γ=c, c′,s energy deviations (ω_{γ}(k)−ω)>0 from the highenergy branchline spectra ω_{γ}(k) given in equations 5, 6, 7, which as confirmed below remain unchanged upon increasing α from α_{0}, the general form of the electron removal spectral function, equation 9 and corresponding exponent, equation 10, prevails for the model with finiterange interactions corresponding to α∈[α_{0}, α_{max}]. Hence for these energy windows that spectral function has the same universal form as in equation 9,
Both within the PDT (α=α_{0}) and RPDT (α>α_{0}), most of the oneelectron spectral weight is located in the (k, ω)plane at and near the singular branch lines. Those refer to the k ranges of the γ=c, c′, s branch lines for which the corresponding exponent in equation 18 is negative. For further information on the validity of the spectral functions expressions, equations (9) and (18), and the definition of some quantities used in our theoretical analysis, see Supplementary note 3.
We start by confirming that the c and s pseudofermion energy dispersions in the expressions of the γ=c, c′, s branchlines spectra ω_{γ}(k), equations 5, 6, 7, remain unchanged. This follows from the behaviour of the phase shifts appearing in these pseudofermion energy dispersions expressions, equations (13) and (14). In the case of the conventional 1D Hubbard model, the integral over the rapidity momentum k in the integrand rapidity phase shifts and of equations (13) and (14) can be transformed into a momentum integral over the whole c band Fermi sea with the integration momentum q′′∈[−2k_{F}, 2k_{F}] appearing in corresponding integrand c pseudofermion phase shifts 2πΦ_{c, c}(q′′, q) and 2πΦ_{c,s}(q′′, q′), respectively.
Under the electron finiterange interactions renormalization, the latter phase shifts become and , respectively, as defined in Equation (17). As given in that equation, the latter c pseudofermion phase shifts are only renormalised at the Fermi points, q′′=±2k_{F}. Hence such phase shifts renormalized values refer only to the limiting values of the integration . The phaseshift contributions associated with such limiting momentum values−2k_{F} and+2k_{F} have in the thermodynamic limit vanishing measure relative to the phaseshift contributions from the range−2k_{F}<q′′<2k_{F} in . For q′′<2k_{F} the phase shifts and remain unchanged, see equation (17). Hence the energy dispersions , equation (13), and , equation (14), remain as well unchanged. The same thus applies to the γ=c, c′, s spectra ω_{γ}(k), equations 5, 6, 7, in the spectral function expression, equation (18).
In contrast, one finds from the combined use of equations (10) and (17) that for the model with finite–range interactions the momentum dependent exponents in that expression are renormalised. For U/t>0, electronic densities n∈[0, 1] and α∈[α_{0}, α_{max}] they are given by,
Plotting the momentum dependence of these exponents requires again the use of exact numerical methods to solve the corresponding sets of coupled integral equations. The momentum dependences found from that exact numerical solution are plotted in Fig. 7 as a function of the momentum k for U/t=0.8, t=0.58 eV, n=2/3 and representative α values α=α_{0}≈1.4 × 10^{−3}, α=0.70, α=0.7835≈0.78 and α=0.85. Their choice is confirmed below to be suitable for the discussion of the relation between the theoretical results and the observed spectral features.
The physics associated with the α range α∈[α_{0}, 1/8] is qualitatively different from that corresponding to α∈[1/8, α_{max}]. Note that at α=1/8 and thus the c pseudofermion phase shift in equation (17) exactly vanishes. This vanishing marks the transition between the two physical regimes. The c pseudofermion phase shift 2πΦ_{c, c}(ι2k_{F}, q) of the conventional 1D Hubbard model also vanishes in the limit of infinity onsite repulsion in which α_{0}=1/8. Increasing α from α=α_{0} within the interval α∈[α_{0}, 1/8] indeed increases the actual onsite repulsion, which for α>α_{0} is not associated anymore with the renormalised model constant effective U value. In addition, it introduces electron finiterange interactions. On the one hand, in that α interval the effects on the γ=c, c′, s exponents, equation (19), of increasing α are controlled by the increase of the actual onsite repulsion. On the other hand, as α changes within the interval α∈[α_{0}, 1/8] the fixed effective U value accounts for both effects from the actual onsite interaction and emerging finiterange interactions. It imposes that the c and s pseudofermion energy dispersions in equations (13) and (14) remain as for that U value. This means that the effects of increasing the actual onsite repulsion due to increasing α are on the matrix elements of the electron annihilation operator between energy eigenstates that control the branchlines exponents, equation (19), and thus the spectral weights.
For U/t=0.8, t=0.58 eV and n=2/3 the c, c′ and s branchlines exponents, equation (19), corresponding to α=1/8 are represented in Fig. 7a–c, respectively, by the dotted lines. The changes in these exponents caused by increasing the α value from α_{0} to 1/8 relative to the exponents curves given for the α_{0}≈1.4 × 10^{−3} conventional 1D Hubbard model in that figure are qualitatively similar to those originated by increasing U/t from 0.8 to infinity within the latter model. Such an increase also enhances α_{0} from α_{0}≈1.4 × 10^{−3} to 1/8. The main difference relative to the conventional 1D Hubbard model is that the c and s pseudofermion energy dispersions remain unchanged on increasing α. Comparison of the momentum intervals of the γ=c, c′, s branch lines for which the exponents, Equation (19), are negative for α∈(α_{0},1/8) with those in which there are cusps in the experimental dispersions of Fig. 5e,f reveals that there is no agreement between theory and experiments for that α range.
Further increasing α within the interval α∈[1/8, α_{max}] corresponds to a different physics. The changes in the branchlines exponents, equation (19), are then mainly due to the increasing effect of the electron finite–range interactions on increasing α. It leads in general to a corresponding increase of the three γ=c, c′, s exponents , equation (19). For U/t=0.8, t=0.58 eV, n=2/3 and both α∈[1/8, 0.75] and α∈[0.78, α_{max}] the momentum intervals of the γ=c, c′, s branch lines for which these exponents are negative do not agree to those for which there are cusps in the MoSe_{2} 1D quantumline defects measured spectral function. To illustrate the α dependence of the γ=c, c′, s branch lines exponents, Equation (19), their k dependence has been plotted in Fig. 7 for the set of representative α values α=α_{0}≈1.4 × 10^{−3}, α=0.70, α=0.7835≈0.78 and α=0.85.
The following analysis refers again to the values U/t=0.8, t=0.58 eV and n=2/3 associated with the MoSe_{2} 1D quantumline defects. For α<0.75 the momentum width of the γ=c′ branch line k range for which its exponent is negative is larger than that of the experimental dispersion shown in Fig. 5(e) near the corresponding excitation energy≈0.95 meV. On increasing α from α=0.75, the γ=c′ branch line momentum width for which is negative continuously decreases, vanishing at α=0.7835≈0.78. Comparison of the momentum ranges for which the exponents plotted in Fig. 7 are negative with those in which there are cusps in the experimental dispersions of Fig. 5 (e) e (f) reveals that there is quantitative agreement for α∈[0.75, 0.78]. Further increasing α from α=0.78 leads to a c branch line momentum width around k=0 in which the exponent becomes positive. This disagrees with the observation of experimental cusps near the excitation energy≈0.85 meV around k=0 and for decreasing energy along the c branch line upon further increasing α.
That there is quantitative agreement between theory and the experiments for α∈(0.75,0.78) is fully consistent with the corresponding α uncertainty range α∈[0.75, 0.80] found independently from the DOS suppression experiments. The momentum dependence of the γ=c, c′, s branch lines exponents corresponding to α=0.78 is represented by full lines in Fig. 5a,c and d for U=0.8t, t=0.58 eV and electronic density n=2/3.
As for the exponents expressions, Equation (10), those of the c and s branchline exponents given in Equation (19) are not valid at the lowenergy limiting values k=±k_{F}. While in the thermodynamic limit this refers only to k=±k_{F}, for the finitesize MoSe_{2} 1D quantumline defects it may refer to two small lowenergy regions in the vicinity of k=±k_{F}. Both this property and the positivity of the s branch exponent for α∈(0.75,0.78) in these momentum regions are consistent with the lack of lowenergy cusps in the ARPES data shown in Fig. 5e,f.
We have calculated the k and ω dependence of the spectral function expression of the 1D Hubbard model with finiterange interactions near the c and s branch lines in the momentum ranges for which they display cusps, Equation (18). If one goes away from the (k, ω)plane vicinity of these lines, one confirms that both such a model spectral function and that of the conventional 1D Hubbard model have the broadening discussed in the Supplementary Note 2.
For a short discussion on whether the RPDT is useful to extract information beyond that given by the conventional 1D Hubbard model and corresponding PDT about the physics of quasi1D metals and a comparison of the PDT and RPDT theoretical descriptions of the line defects in MoSe_{2}, see Supplementary Note 4.
Data availability
The data sets generated during and/or analysed during the current study are available from the corresponding authors on reasonable request.
Additional information
How to cite this article: Ma, Y. et al. Angle resolved photoemission spectroscopy reveals spin charge separation in metallic MoSe_{2} grain boundary. Nat. Commun. 8, 14231 doi: 10.1038/ncomms14231 (2017).
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Acknowledgements
The USF group acknowledges support from the National Science Foundation (DMR1204924). V.K., R.D. and M.H. P. acknowledges support from the Army Research Office (W911NF1510626) and thank Prof. Hari Srikanth for resistance measurements in his laboratory. M.C.A., J.A. and C.C. thank enlightening exchanges with Gabriel Kotliar and ZhiXun Shen. The Synchrotron SOLEIL is supported by the Centre National de la Recherche Scientifique (CNRS) and the Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA), France. T.Č. and J.M.P.C. thank Eduardo Castro, HaiQing Lin and Pedro D. Sacramento for illuminating discussions. The theory group acknowledges the support from NSAF U1530401 and computational resources from CSRC (Beijing), the Portuguese FCT through the Grant UID/FIS/04650/2013 and the NSFC Grant 11650110443.
Author information
Affiliations
Department of Physics, University of South Florida, Tampa, Florida 33620, USA
 Yujing Ma
 , Horacio Coy Diaz
 , Vijaysankar Kalappattil
 , Raja Das
 , ManhHuong Phan
 & Matthias Batzill
Synchrotron SOLEIL, L'Orme des Merisiers, Saint AubinBP 48, Gif sur Yvette Cedex 91192, France
 José Avila
 , Chaoyu Chen
 & Maria C. Asensio
Université ParisSaclay, L'Orme des Merisiers, Saint AubinBP 48, Gif sur Yvette Cedex 91192, France
 José Avila
 , Chaoyu Chen
 & Maria C. Asensio
Beijing Computational Science Research Center, Beijing 100193, China
 Tilen Čadež
 & José M. P. Carmelo
Center of Physics of University of Minho and University of Porto, Oporto P4169007, Portugal
 Tilen Čadež
 & José M. P. Carmelo
Department of Physics, University of Minho, Campus Gualtar, Braga P4710057, Portugal
 José M. P. Carmelo
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Contributions
Y.M. and H.C.D. contributed equally to this work. They both grew samples by MBE and characterized them by STM. The ARPES data have been obtained and analysed by J.A., H.C.D., C.C. and M.C.A.. The fourpoint transport measurements have been conducted and discussed by R.D., V.K. and M.H.P. The project has been conceived by M.B. and M.C.A. who directed its experimental part. The theoretical description has been conceived by J.M.P.C. and the corresponding theoretical analysis was carried out by T.Č. and J.M.P.C.. The manuscript has been written by M.B., M.C.A. and J.M.P.C.. All authors contributed to the scientific discussion, contributed to and agreed on the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to José M. P. Carmelo or Maria C. Asensio or Matthias Batzill.
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