## Introduction

A one-dimensional (1D) electronic system is described as a Tomonaga-Luttinger liquid (TLL) rather than a Landau-Fermi liquid, since the strictly 1D nature of the Fermi surface with only two Fermi points (±kF) leads to a breakdown of screening1. As a consequence, electronic correlations are strong, even for weak electron–electron interactions, leading to collective excitations2. The elementary excitations of a TLL can be described as bosons, rather than fermions, and are given by charge and spin density waves with spin-charge separation1,3. For weak interactions, the correlation strength can be approximated by a single dimensionless parameter, the Luttinger parameter4,

$$K\approx {\left[1+\frac{W(q=0)}{\pi \hslash {v}_{{{{{{{{\rm{F}}}}}}}}}}\right]}^{-1/2}.$$
(1)

Here, πvF is the inverse density of states for 1D Dirac electrons, with a constant Fermi velocity vF, and W(q = 0) is the energy of electrostatic interactions in the long wavelength limit. The former depends on the detail of the electronic band dispersion directly, while the latter can be expected to be susceptible to the dielectric environment of the 1D charge distribution, screening the many-body interactions. The Luttinger parameter can thus be expected to be highly tunable, ranging within its theoretical bounds, 0 < K < 1, for repulsive (e.g. Coulombic) interactions, where K = 1 [W(q = 0) = 0] represents the limit of non-interacting electrons. A threshold K ≈ 1/2 is naturally given where the Coulomb and kinetic energy terms become equal, thus defining K < 1/2 as the limit of strong interactions.

Luttinger liquids have been detected in several realisations of 1D electronic structure, including linear chain organic conductors5, metallic single-wall carbon nanotubes (SWNTs)4,6, quantum Hall edge channels7, cold atoms8, atomic wires9, and even crystalline mirror-twin boundaries in semiconducting transition metal dichalcogenides10.

A key experimental signature of a TLL is a pseudogap suppression in the local density of states (LDOS), measured via tunnelling into the 1D electronic system, e.g., from the tip of a scanning tunnelling microscope (STM). Such pseudogap suppression is often referred to as zero-bias anomaly (ZBA) as it is strictly centred at the Fermi energy EF6,9,10,11. In a TLL, the tunnelling conductivity around such ZBA scales universally, that is, as a power law in both bias voltage and temperature, with the same scaling exponent α, related to the Luttinger parameter as $$\alpha=C\left(K+{K}^{-1}-2\right)$$12. Different from the spin-degenerate (spinful) parabolic dispersions of conventional 1D metallic systems (C = 1/4), quantum spin Hall (QSH) insulators host linearly dispersing 1D edge states, in which the spin polarity is locked to the crystal momentum (helicity)13,14. For a helical Luttinger Liquid, strong spin orbit coupling causes the spin and chirality indices to coincide, and the bosonized Hamiltonian can be expressed as an effectively "spinless" Luttinger liquid with C = 1/2. TLL formation in such 1D helical systems is not only of fundamental interest as a testbed to study the interplay of topology and strong correlations, but may further constitute a potential host platform to realise non-Abelian parafermions15,16,17,18,19.

Tuning of the many-body interaction strength has so far been demonstrated only for spinful Luttinger liquids, for instance, by applied gate voltages20,21, or magnetic fields22. In this work, we demonstrate control of the Luttinger parameter K in a helical TLL at the edges of the quantum spin Hall insulator 1T’-WTe2, in which K is found distinct for different crystal edges, and can be controlled by dielectric screening via the substrate. This becomes evident from a dependence of the Luttinger parameter on substrate dielectric constant and crystal edge, ranging between K = (0.21 ± 0.01) and K = (0.33 ± 0.01) in the strong coupling limit. Our conclusions are based on temperature-dependent scanning tunnelling spectroscopy, demonstrating universal scaling, combined with a detailed statistical analysis of tens of tunnelling points along WTe2 edge states, showing that values of K fall into normal distributions of distinct statistical mean for different substrates and crystal edges.

The experimental extraction of K is further supported by both a numerical model to estimate the strength of screened Coulomb interactions for the edge state’s charge distribution and a material-specific real-space tight-binding model, which takes charge distribution and WTe2 edge dispersion self-consistently into account. Both models are able to reproduce the trend of the Luttinger parameter qualitatively in different screening environments. Interestingly, despite being in the strong coupling limit (K < 0.5), where deviations from Eq. (1) may be expected due to strong interactions and a renormalised Fermi velocity, we show that the perturbative approximation can allow for qualitative predictions of its fundamental dependencies in terms of the Coulomb interaction strength and edge state dispersion. In the limit of very strong interactions (K < 1/4) reached in some of our samples, two-particle back-scattering of helical electrons may become significant23,24, potentially leading to additional effects such as charge density wave formation, or even fractional Wigner crystallisation24. Eq. (1) can thus serve as an important indicator for the strength and tunability of Coulomb correlations in a Tomonaga–Luttinger liquid.

## Results

Helical TLLs have been reported for the 1D topological edge states of InAs/GaSb semiconductor heterostructures23 and, more recently, for the edge states of atomically flat bismuthene11. An alternative realisation of an atomically thin QSH insulator18,19 is 1T’-WTe215,25,26,27,28,29, with topological band gaps of a few tens of meV26,27, tunable by electric fields and strain. A typical 1T’-WTe2 monolayer crystal is shown in Fig. 1a, synthesised by van-der-Waals molecular-beam epitaxy (MBE)26,27 on highly oriented pyrolytic graphite (HOPG) (see Methods). We achieve typical island circumferences L = 100−400 nm with disordered edges. Locally, however, we find straight and well-defined sections, with typical lengths between 10−20 nm that are terminated either perpendicular (Y-edge) or parallel (X-edge) to the atomic rows.

Corresponding differential conductance (dI/dV) maps of the same area are shown in Fig. 1b, c, reflecting the expected QSH electronic signature of an enhanced local density of states (LDOS) at the crystal edge around an insulating 2D bulk26,30. A larger area of the same crystal can be viewed in Supplementary Fig. 1, showing that the electronic edge state is largely continuous along the island’s circumference, even in the presence of local edge roughness and disorder, and regardless of edge direction or termination, and at adjoining corners. The relevant length-scale determining the electrostatic energy of the screened Coulomb interactions within the edge state’s 1D charge distribution is therefore set by the islands’ circumference. Superimposed spatial modulations in the charge density can be shown to arise from edge roughness even in otherwise perfect monolayer crystals (Supplementary Fig. 7). However, as these modulations –more clearly pronounced on the X-edges –appear quasi-periodic, additional mechanisms may be at play, including spatial inhomogeneity of the Fermi velocity and/or interaction-mediated scattering of the edge state, leading to charge density wave order.

We note that we do observe semi-metallic behaviour and a variety of in-gap states in some monolayer crystals, especially in the vicinity of lattice defects and adsorbates (Supplementary Fig. 1). Some crystals also show a small but finite 2D LDOS within the bulk and slight variations in gap size. For this study, however, we focus on larger clean regions, such as shown in Fig. 1a–c, in which a well-defined suppression of the bulk LDOS is observed.

A spatial profile of the local density of states (LDOS), measured along the black arrow in Fig. 1a, is shown in Fig. 1e, alongside a corresponding STM height profile (Fig. 1d). The data confirms a well-defined energy gap over ~70 meV in the 2D bulk, similar in magnitude to earlier reports26,27, and in agreement with the tight-binding model adopted in this work31 (compare Fig. 3d, e). The gap is seen to narrow and eventually disappear at the edge, consistent with the presence of a metallic edge state. We identify the position of the edge, electronically, from a profile of the energy-integrated LDOS (−20 to −40 meV) in Fig. 1f, which shows a finite width Δx ≈ 2 nm with exponential decay into the 2D bulk. From an exponential fit, we extract a decay length of (2.1 ± 0.2) nm, significantly larger compared to the crystal lattice constant (a = 0.348 nm). Spatial modulations superimposed onto the overall edge state LDOS enhancement further indicate that the observed LDOS enhancement is not due to localised atomic orbitals, but rather are an electronic effect due to the metallic edge state.

The difference between bulk and edge state LDOS is further highlighted by the direct comparison of individual STS point spectra shown in the insert to Fig. 1f. For a linear (Dirac) dispersion in 1D, one would expect a constant (energy-independent) edge state LDOS throughout the gap. Instead, we observe a clear V-shaped suppression with a minimum at the Fermi energy E = EF (ZBA). This ZBA – a first indication of the presence of 1D correlations11,26,30– is strictly confined to the crystal edges as confirmed in Fig. 1g, where we plot individual dI/dV point spectra away from the Y-edge into the monolayer bulk. Black solid lines are fits to the theoretically predicted tunnelling density of states into a TLL at 4.5 K, following the universal scaling relationship4,

$${\rho }_{{{\mbox{TLL}}}}(E,\, T)\propto {T}^{\alpha }\cosh \left(\frac{E}{2{k}_{{{{{{{{\rm{B}}}}}}}}}T}\right){\left | {{\Gamma }}\left(\frac{1+\alpha }{2}+i\frac{E}{2\pi {k}_{{{{{{{{\rm{B}}}}}}}}}T}\right)\right | }^{2}$$
(2)

Here, α is the scaling exponent, E − EF ≡ eV is the STM bias voltage, T is temperature, and $${{\Gamma }}\left(x\right)$$ is the gamma function. For our fits, we further take broadening of the LDOS into account, consistent with the procedure in Ref. 11 (as shown in Supplementary Fig. 2).

Particular to 1D helical systems, the linear (Dirac) dispersion supports the bosonisation picture of TLL theory over a much larger energy range. Contrasting the spinful parabolic dispersions of conventional metals, which deviate from a linear approximation already at moderately low energy, the 1T’-WTe2 edge state dispersion remains linear over a wide energy range within the QSH band gap31 (compare Fig. 3d, e), consistent with the range of our TLL fits (typically −20 meV to +15 meV). From 33 edge spectra (two of which are shown in Fig. 1g), we extract α = 0.59 ± 0.02 and K = 0.35 ± 0.02, assuming helicity of the QSH edge state (C = 1/2). These numbers are in good agreement with values recently reported for the helical edges of bismuthene (K ≈ 0.42)11, which features similarly tightly confined 1D edge states, in contrast to the larger edge state decay lengths of semiconductor-based QSH systems23.

We note that, in principle, a narrow suppression in the LDOS of 1D electronic systems1,4,11,26,30 may have its origin in a number of different mechanisms, most notably the formation of a Coulomb gap (CG) due to long-range interactions32. However, as we show in Supplementary Fig. 4 and 5, a Coulomb gap (CG) model cannot explain the detailed functional form of the ZBA observed. To further rule out the presence of a CG, we turn to temperature-dependent scanning probe spectroscopy, to confirm universal scaling of the TLL tunnelling conductance. In Fig. 2a, d, we compare dI/dV spectra measured on WTe2/HOPG with those taken on WTe2/BLG. We limit the temperature range explored to T < 25 K, as higher temperatures are found to give rise to a significant bulk contribution to the overall LDOS (Supplementary Fig. 5e). Black solid lines in Fig. 2a, b are fits to the universal scaling relationship (Eq. (2)), yielding comparable power-law exponents α ≈ 0.45 (HOPG) and α ≈ 0.98 (BLG) at all temperatures. The corresponding bare power law is indicated by dotted lines. Universal scaling is confirmed by normalising the measured dI/dV with Tα and plotting the result as a function of the dimensionless energy parameter eV/kBT. Following this procedure, all spectra collapse onto a single universal curve (Fig. 2c, d) with power-law exponents α = 0.45 ± 0.03 (K = 0.40 ± 0.01) for WTe2/HOPG and α = 0.98 ± 0.03 (K = 0.27 ± 0.01) for WTe2/BLG. The uncertainty in the fitted values of α, indicated by the shaded confidence bands in the figure, arises from the mean absolute error of TLL fits across the different measurement temperatures. As a further confirmation of universal scaling, we plot the zero-bias (E = EF) tunnelling conductance as a function of temperature in Fig. 2e, f, comparing the expected temperature dependencies of TLL and CG theories, respectively. The temperature dependence of the TLL is in excellent agreement with the universal scaling exponents extracted from Fig. 2a–d, while the CG model shows substantial deviations from the measured data on both substrates, especially at temperatures <10 K.

Universal scaling as demonstrated is usually considered sufficient proof for the presence of a TLL11. Further to this, however, below we show that the observed differences in the Luttinger parameter across substrates can be explained by substrate-induced screening of the many-body interactions. To this end, we have performed a detailed statistical analysis of K values extracted from a total of 69 locations, measured along the edge states of crystals on both substrates, plotted in the histograms of Fig. 3a, b. We find that K follows normal distributions with distinct statistical mean but comparable standard deviation for both edge terminations and substrates. This is further confirmed by fits to Gaussian normal distributions (solid lines), from which we extract K = 0.27 ± 0.01 (Y-edge) and K = 0.21 ± 0.01 (X-edge) for WTe2/BLG and K = 0.33 ± 0.01 (Y-edge) and K = 0.24 ± 0.01 (X-edge) for WTe2/HOPG. The uncertainty is given by the standard error of the mean $$\sigma /\sqrt{n}$$ which quantifies the accuracy of the extracted mean, reflecting that K was obtained from a population of n sample points with standard deviation σ. Further detail on the statistical analysis, including normality tests, and statistical significance of the differences in the mean is given in Supplementary Table 1. Finally, we include K values extracted from edge state tunnelling spectra in literature26,33,34,35, and find that their values fall within the same distributions as shown in Fig. 3a, b.

The mean and standard error of the mean for both edges are plotted as a function of the substrate dielectric constant ϵr in Fig. 3c. Across all data, the Luttinger parameter K increases at low ϵr and saturates for ϵr → , where the latter corresponds to HOPG36. We note that although there have also been reports of a finite dielectric constant for HOPG (ϵr ≈ 53)37, K starts to saturate at (ϵr ≈ 50), implying that ϵr →  provides a reasonable approximation in this context.

To explain this trend, we devise a qualitative model for K38, based on the perturbative approximation (Eq. 1) and the assumption of a substrate-dependent mirror charge, screening many-body Coulomb interactions within the helical edge. In our model, edge state and mirror charge are approximated by space-charge distributions ρ(x, y, z) and $$\rho ^{\prime} (x,\,y,\,z)=-\frac{{\epsilon }_{r}-1}{{\epsilon }_{r}+1}\rho (x,\,y,-z)$$, respectively, allowing us to calculate the electrostatic energy W(q = 0) of the total charge numerically (see Section S6 of the Supplementary Information). In order to estimate K for the respective Y- (X-) edges, we extract the Fermi velocities vF = 1.16 × 105 m/s (2.69 × 105 m/s) from the tight-binding dispersions in Fig. 3b–e. The red and blue shaded bands in Fig. 3c have been calculated for an average edge length L = 200 nm, and space-charge densities ρ0,y = 0.0039 ± 0.0007 e/nm3 and ρ0,x = 0.0082 ± 0.0009 e/nm3 to reflect the standard deviations consistent with Fig. 3a, b. The observed trend is further confirmed by our material-specific tight-binding model, taking edge state charge density and band dispersion self-consistently into account (Supplementary Fig. 7). The solid red line in Fig. 3a represents the calculated the Luttinger parameter K along the Y-edge.

The ability of our model to reproduce the trends in K may seem surprising at first, given that Eq. (1) should strictly only apply in the perturbative limit of weak interactions (K > 0.5). However, we note that several factors can extend the range of validity of Eq. (1), especially in helical systems. For instance, Coulomb interactions do not allow for backscattering with large momentum transfer q = 2kF in a helical TLL. This supports a perturbative formalism which primarily includes the long wavelength forward scattering modes (q ≈ 0). More so, our observation of a finite width TLL in which the LDOS falls exponentially from the edge into the QSH bulk (compare Fig. 1f) leads to a Coulomb interaction extending up to the fifth nearest neighbour ion of the lattice (Supplementary Fig. 8). This justifies the approximation g2 = g4 implied in our perturbative analysis, where g2 and g4 correspond to interaction terms in a Hamiltonian contribution, $${g}_{4}{({\psi }_{R\uparrow }^{{{{\dagger}}} }{\psi }_{R\uparrow })}^{2}+{g}_{2}({\psi }_{R\uparrow }^{{{{\dagger}}} }{\psi }_{R\uparrow }{\psi }_{L\downarrow }^{{{{\dagger}}} }{\psi }_{L\downarrow })$$, where ψR annihilates a right mover electron with spin up, and ψL annihilates a left mover electron with spin down. We conclude that while the precise quantitative match of K may be fortuitous, the perturbative treatment is able to correctly predict the trend in K, for different edge dispersions and substrate mediated screening. This work therefore presents a precision experimental determination of the effect of kinetic and potential energy terms on the Luttinger parameter, and may form a benchmark for future theoretical studies.

To summarise, we have demonstrated tunability of the many-body interactions in a Tomonaga-Luttinger Liquid (TLL) at the helical edges of the quantum spin Hall insulator 1T’-WTe2. Synthesising WTe2 on metallic van-der-Waals substrates, we have been able to demonstrate effective screening of the many-body Coulomb interactions, reflected in a significant enhancement of the Luttinger parameter. The cleanest signatures of the TLL are observed on the well-screened Y-edges of WTe2/HOPG, where K = (0.33 ± 0.01). Here, the higher degree of tunability observed would suggest that systems of lower Fermi velocity and low carrier density may allow to explore a wider range of K, something which may be confirmed in other QSH material systems. In turn, the strongest many-body interactions are found on the unscreened X-edges for WTe2/BLG where we measure K = (0.21 ± 0.01) – the lowest reported for any helical TLL system to date. Indeed, this value represents the limit of very strong interactions (K < 0.25), where additional mechanisms, such as charge density wave formation due to two-particle scattering or even Wigner crystallisation24 may be expected. Indeed, the quasi-periodic charge density modulations observed on X-edges are not inconsistent with such phenomena and will be a focus for future investigation into the role of edge disorder and scattering. We thus expect that our results will stimulate further experimental and theoretical investigations of the interplay of topology and strong electronic interactions in 1D, also in the superconducting state, in which non-Abelian parafermions are predicted18,39.

## Methods

### MBE growth

Monolayer 1T’-WTe2 was synthesised by molecular-beam epitaxy (MBE) on highly oriented pyrolytic graphite (HOPG) and bilayer graphene (BLG) grown on 6H-SiC(0001) substrates in an Omicron Lab10 ultrahigh vacuum (UHV) MBE chamber with a base pressure of 2 × 10−10 mbar. Prior to MBE, the HOPG substrates were mechanically cleaved in ambient conditions and then degassed at 300 °C overnight. The 6H-SiC(0001) substrates were degassed at 600 °C overnight in UHV and then flash annealed at 2000 °C for a few cycles to graphitise producing BLG. The WTe2 islands were grown by evaporating high-purity W(99.998%) and Te(99.999%) with a flux ratio of 1:280 and substrate temperature of 165 °C for 2 h to cover approximately 50% of the monolayer area. In total, we measured three 1T’-WTe2/HOPG samples and another three of 1T’-WTe2/BLG.

### Scanning tunnelling spectroscopy

Low-temperature scanning tunnelling microscopy and spectroscopy (STM/STS) measurements were performed in an Omicron low-temperature STM (4.5 K) under UHV conditions ( 5 × 10−11 mbar). For temperature-dependent STM/STS, a heater element fitted to the sample stage was used for counter-heating from 4.5 K. For all spectroscopy measurements, we used a platinum/iridium tip calibrated against the Au(111) Shockley surface state. The spectroscopy measurements were carried out using standard lock-in techniques with a modulation amplitude of Vac = 2 mV and a modulation frequency of 1.331 kHz. The value of Vac was optimised to maximize the signal-to-noise ratio while eliminating instrumental broadening (Supplementary Fig. 3). Differential conductance maps were taken in constant height mode.