A robust and tunable Luttinger liquid in correlated edge of transition-metal second-order topological insulator Ta2Pd3Te5

The interplay between topology and interaction always plays an important role in condensed matter physics and induces many exotic quantum phases, while rare transition metal layered material (TMLM) has been proved to possess both. Here we report a TMLM Ta2Pd3Te5 has the two-dimensional second-order topology (also a quadrupole topological insulator) with correlated edge states - Luttinger liquid. It is ascribed to the unconventional nature of the mismatch between charge- and atomic- centers induced by a remarkable double-band inversion. This one-dimensional protected edge state preserves the Luttinger liquid behavior with robustness and universality in scale from micro- to macro- size, leading to a significant anisotropic electrical transport through two-dimensional sides of bulk materials. Moreover, the bulk gap can be modulated by the thickness, resulting in an extensive-range phase diagram for Luttinger liquid. These provide an attractive model to study the interaction and quantum phases in correlated topological systems.


Main Text: Introduction
Whereas topological systems without consideration of correlation have been studied widely in the last decades, the exotic quantum phases derived from the interaction between electrons and topological state have recently attracted great attention 1 , such as quantum spin liquids 2 and topological Mott insulators 3 which possess one-dimensional (1D) bosonic mode with spincharge separation, strongly correlated Chern insulators in flat bands of magic-angle twisted bilayer graphene 4,5 and topological superconductors for topological quantum computation 6,7 .
The aforementioned correlated 1D bosonic mode is described as Tomonoga-Luttinger liquid (simplified as Luttinger liquid, LL), and characterized by a power-law vanishing to anomaly zero density-of-state at the Fermi level.This power-law behavior in electrical transport measurements has been harnessed to experimentally test LL in the edge states of quantum spin Hall insulator candidates [8][9][10] and quantum Hall systems [11][12][13][14][15][16] .Thus, LL quantum phase provides a powerful tool to analyze the correlated 1D edge state emerging from topological order.However, in these systems, the LL relies on the microscopic details and is lack of experimental evidence for its universality and robustness, let alone extending to macro-scale, which is supposed to be a signature for topological order.Moreover, this vulnerability leads to the absence of the quantitative test of LL, e.g., its exponent factor as the Coulomb interaction strength, in the edge state of topological system, though a large amount of theories have discussed the possibility of taking into account different interactions [17][18][19][20][21] .
On the other hand, unlike a two-dimensional (2D) topological insulator with gapless edge states, a 2D second-order topological insulator (SOTI, or a quadrupole topological insulator) is supposed to have gapped edge states and in-gap corner states (in the presence of chiral symmetry) 22 .Although many 2D SOTI candidates have been proposed in literature [23][24][25][26][27] , rare transition-metal compound has been predicted to be a 2D SOTI.After considering many-body interactions, superconductivity, excitonic condensation and Luttinger liquid can be expected in bulk and edge states of the transition-metal SOTI.
Here, we have observed the LL inhabiting the edge states, which are as the remnants of the 2D second-order topology, of a TMLM Ta2Pd3Te5.Through the first-principles calculations, we demonstrate that this topological state originates from a remarkable double-band inversion and results in an unconventional nature with an essential band representation A"@4c 28 .Then we have performed the electrical transport measurements of tens of Ta2Pd3Te5 bulk and thinfilm devices, and detected the coexistence of the bulk insulating gap, the in-gap edge conducting state and the edge gap as the signature of the 2D SOTI.Interestingly, the edge state displays a universal scaling of power-law relation with a zero-energy anomaly as a function of temperature (T) and energy (signed as the voltage bias here), which is considered as the critical evidence of LL.The weak van der Waals (vdW) interaction between layers allows this LL to be reproducible in the edge channels of all the devices from atomic-to macro-scale, with a conductance proportional to the edge geometry.That is the 2D side facets of macroscopic bulk also show 1D LL anisotropic electrical transport.This reveals for the first time the evidence of a scale connection from atomic-to macro-LL, and supports this LL quantum state with universality and robustness irrespective of the microscopic details.In addition, the coexistence of the three states induces a tunability of the LL power-law exponent to represent the strength of Coulomb interaction, as well as a reversible LL-Fermi liquid (FL) transition and a full phase diagram.All these demonstrate a robust and universal LL with the many precise-controlled parameters, as an effective tool to characterize the interaction and the quantum phase transition in the correlated topological system.

Crystal structure
Ta2Pd3Te5 is a vdW material with quasi-one-dimensional (Q1D) Ta2Te5 chains and crystallizes in needle-like single crystals along the b-direction (see the crystal structure and the optical image of the typical samples in Fig. 1a and ref. 29 ).The single crystal is synthesized with an orthorhombic structure Pnma (space group No.62) 29 .The Q1D characteristic and layered structure are confirmed by high-resolution scanning transmission electron microscope (STEM) images of Ta2Pd3Te5 single crystal along [010] and [001] projections, revealing a thickness of ~0.7 nm for monolayer (Fig. 1b and Supplementary Fig. 1, b and c).Because of the strong anisotropic bonding energy 29 , this material is easy to mechanically exfoliate into large and thin films with flat and uniform edges along the Q1D chain (Supplementary Fig. 1a).
Importantly, the Fermi level naturally locates in the bottom of conduction bands (Fig. 1c and Supplementary Fig. 1d), enabling a high-efficient modulation by the electrostatic gate.The coupling between vdW layers is weak, proved by the angle resolved photoemission spectroscopy (APRES) measurement (Fig. 1d and Supplementary Fig. 1, e and f).

Theoretical calculation
Recently, the Perdew-Burke-Ernzerhof (PBE) calculations with spin-orbit coupling (SOC) predict that its monolayer is a quantum spin Hall state with a tiny SOC gap and the bulk topological crystalline insulator without a global band gap (ref. 29).However, both of the scanning tunneling microscopy (STM) (ref. 30) and APRES (Fig. 1c) measurements detect a bulk gap of ~50 meV.As reported in ref. 29 , the nearly zero-energy band gap is sensitive to the lattice constant.On the other hand, the experimental energy gap can be induced by many-body interactions in the Q1D structure, like electron-phonon interaction or electron-hole attractive interaction, as the excitonic state is thermodynamically most stable in a zero-gap semiconductor.Although the exact reason of the bulk gap opening needs further investigations, the double-band inversion happens clearly between the {Y2;GM1-,GM3-} and {Y1;GM1+,GM3+} bands, which are 0.3 eV (or higher) away from the Fermi level in the monolayer structure of SG59 (imposing inversion; Supplementary Table. 1 and 2) 28 .The schematic of the double band inversion in the system is given in Fig. 1e.Hence, in this manuscript, we focus on the modified band structure of the monolayer Ta2Pd3Te5 (by adding small onsite energy to simulate the many-body gap opening) in Fig. 1f 31 .The band representation (BR) decomposition of the occupied bands shows that there is an essential BR at vacancies, suggesting the unconventional nature/obstructed atomic limit with second-order topology 31,32 .A 2D quadrupole topological insulator with second-order topology possesses 1D gapped edge states and isolated corner states (Detailed information can be found in Ref. 28).
In the computed (01)-edge spectrum of Fig 1g, the gapped edge states are clearly shown in the gap of the bulk continuum.The observed in-gap edge states as the remnants of the quadrupole topological insulator can exhibit 1D feature well, which is responsible for the observed LL behavior (will be analyzed later).In addition, since the layers of the bulk are vdW coupled (Fig. 1d and Supplementary Fig. 1, e and f), the 1D edge states can still survive on the side surfaces of the bulk.

Three features in dI/dV curves
To investigate the electronic properties, we show the typical result for thin-film devices #A1,2 with thickness ~10 nm in Fig 2 , with a back-gate voltage Vbg tuning the chemical potential as illustrated in Fig. 2a. Figure 2b and c show the four-terminal measurement configurations and gate tunable dI/dV for devices #A1,2.Once applying the voltage bias (VDC), the differential conductance (dI/dV(VDC)) curve of #A1 at charge neutral point (CNP, Vbg =-28V) shows distinguishable three regions (Fig. 2d, indicated by the purple-green-orange background).The first region at small VDC indicates a gap structure (highlighted by the purple background, also see the conductance valley plotted by the symmetric VDC in the inset of Fig. 2d), which closes around T = 3.75 K; The second region at medium VDC reveals a power-law shape with an exponent of ~0.33 (highlighted by the green background); The third region at high V DC shows a new shape of enhanced conductance with increasing V DC .Those regions are also repeated in other devices (Fig. 2e for device #A2 and Supplementary Fig. 12, l and m).
Notably, for device #A2, the T-dependent dI/dV(T) curve at CNP (purple curve in Fig. 2f) shows similar three regions, which are semiconductor behavior with an energy gap of ~27 meV from 300 to ~50 K, a power-law dependence with the exponential factor α of ~0.79 from ~30 K to 6 K and another semiconductor behavior with a smaller energy gap of ~ 0.6 meV at lower T.This is clearer in the log(dI/dV) v.s.1/T plot in the inset where the yellow dot dash lines are fitting curves of Arrhenius equation dI/dV(T)  exp(−Δ/2kBT).Similar behavior is repeatable, e.g., device #A4 with a ~ 33 meV gap at high T and a ~ 0.9 meV gap at low T in Supplementary Fig. 2. Apparently, this three-region feature fits the theoretical result of band structure with the bulk gap Eb, edge conducting state and edge gap Ee very well.
It should be noted that these devices are measured with four-terminal configuration with standard lock-in measurement and the phases θ of the lock-in amplifier are also small enough.
So, the very-low-T behaviors do not come from the contact resistance which is not included in the measurement.In addition, the gap size we extracted from the Arrhenius equation is consistent with the thermal excitation energy of the temperature at which the gap behavior appeared.We also measured dI/dV(VDC) and dI/dV(T) from CNP to positive back-gate voltage (Vbg) (Fig. 2, e and f).As the Fermi level moves up via applying positive Vbg sketched in Fig. 2a, the three-region feature gradually transitions into that of two regions without Ee in both dI/dV(VDC) and dI/dV(T) curves, further consistent with the calculated band structure.

Validation of edge state
To better clarify where the three structures respectively come from, we perform the nonlocal transport measurement based on Hall bar configuration 33 .For a classical planar Hall bar, when a local voltage V0 is applied, the nonlocal response Vnl at remote contacts will decay exponentially with distance, irrespective of T, chemical potential μ and so on.Whereas, the nonlocal voltages through edge channels don't rely on the distance 34 .Fig. 3a is  Therefore, by comparing the local and nonlocal measurements at different T and Vbg, we deduce both of the power-law feature and the small gap Ee root in the edge channels.This also excludes Ee from any bulk insulating state, e.g., Mott insulator or Anderson insulator, in particular with the phenomena of the greatly tunable T range at samples with varying thickness (will be analyzed more later in section 'Luttinger liquid phase diagram').
More evidence to prove this edge state is that while both APRES (Fig. 1c) and STM (ref. 30) clearly detect a large gap up to ~ 50 meV on the top surface of Ta2Pd3Te5 single crystal, the transport measurements still show large conductance around CNP on thin films exfoliated from the same bulk sample.Some of the conductance even reach a value as high as 2G0 (G0 is the quantized conductance 2e 2 /h).STM measurement also reveals a distinct local conducting edge state in the bulk energy gap 30 .Additionally, the similar edge conductivity in power-law region for samples with various thickness further confirms the existence of robust edge states from atomic-to macroscopic-scale and will be analyzed in detail in section 'Robust and universal Luttinger liquid from micro-to macro-size'.Moreover, the finite conductance around CNP, compared with the close-to-zero conductance when grounding the edge, also supports the existence of edge states (Supplementary Fig. 7).In conclusion, all the local and nonlocal measurements prove the existence of bulk state, edge conducting state and correlated edge gap, namely the second-order topology in this material.

Tunable Luttinger liquid in edge states
Now let's discuss the power-law relation of the edge state.Such behavior in 1D quantum system is a typical signature of LL, in particular with a universal scaling.We use device #A4 as a typical example whose behavior is similar to device #A2.Fig. 4a shows in dI/dV(VDC) curves for device #A4 at Vbg = 0 V, Ee disappears and power-law relation dominates at higher T (other systematic data of device #A4 are shown in Supplementary Fig. 2).For those dI/dV(VDC) curves without Ee behavior (From T = 12.9 K to 39.8 K), in Fig. 4b we scaled the conductance (dI/dV)/T 0.69 with exponent of ~0.69, which is extracted from dI/dV(T) curve in Supplementary Fig. 2b, as y-axis and eVDC/kBT (kB is the Boltzmann constant) as x-axis.By this, all the curves converge together with a zero-energy anomaly, confirming the universal power-law correlation as well as LL behavior (similar behavior of device #A2 is shown in Supplementary Fig. 4 and more examples are in Supplementary Fig. 2, 3, 12).We note that for dI/dV(VDC) curves at the lowest temperature (e.g.T < 12.9 K in Fig. 4a), the low-bias regions exhibit edge gap behavior due to the 2D SOTI nature which limits the power-law and scaling ranges, so these low-temperature dI/dV(VDC) curves shouldn't obey the scaling behavior physically.
Furthermore, we linearly tune the power-law exponent α from ~ 0.7 at CNP to below 0.1 at positive Vbg (Fig. 2e and 4c, here α = 0 means FL).This reveals a tunability of the Coulomb interaction strength of 1D edge state, as well as the transition from LL of correlated 1D edge state to FL of extended high-dimensional bulk state along with the rising of Fermi level.The exponent α is correlated with the strength of the Coulomb interaction and has been harnessed to test different types of interaction, e.g., Kondo effect in the helical edge state 19  interaction and provide a controllable approach to study the interaction strength and the quantum phase transition.Usually, one can use the exponent α to estimate the Luttinger liquid parameter K, which is an intrinsic parameter describing the electron-electron interactions in the system.But the numerical relationship between α and K intricately depends on the details of the system, such as the type/strength of tunnel barriers [35][36][37] , edge state of fractional quantum Hall effect 12,38 /quantum spin Hall effect 8,19 , spin-orbit coupling 37 , etc.The relation in our system is not clear yet, so we could not get the accurate calculation of K here and further theoretical modelling is needed.

Robust and universal Luttinger liquid from micro-to macro-size
Here, we find there are two types of devices, among which we name the devices with distinguishable three regions as type A and the devices without Ee as type B. Because the edge conducting state is dominating at low energy, device B owns straight power-law traces with the absence of edge gap for both dI/dV(T) and dI/dV(VDC) curves, as well as large values of conductance, at small T and VDC around CNP (Fig. 5a-c for device #B1 for example), similar to the curves at positive Vbg for device A (see the green curves in Fig. 2, e and f).These powerlaw traces could also be scaled (Fig. 5c).Prominently, different sections of the same device could be scaled with the similar exponent (Supplementary Fig. 8).All these further confirm the intrinsic LL property of the edge conducting state, no matter w/o Ee.Interestingly, the exponent α from dI/dV(T) curves is always larger than, sometimes nearly twice of, that from dI/dV(VDC) curve (Fig. 2 and 5, Supplementary Fig. 2-4 and 12-14), revealing both the inner LL-LL tunneling and FL-LL tunneling through electrodes are involved here 35,36,39,40 -the exponents of LL-LL junctions are twice of FL-LL junctions.So, the LL-LL tunneling becomes dominant and the exponent of dI/dV(T) is α(LL-LL) at low temperature due to the larger resistance of LL-LL junctions with larger exponents (with the assumption that both FL-LL and LL-LL junctions have similar resistance at room temperature).When we apply high bias, FL-LL junctions become dominant and the exponent of dI/dV(VDC) is α(FL-LL) (FL-LL junctions become more resistive because their resistance decreases more slowly) 39,40 .This edge LL is of great stability and reproducibility, because it shows up in almost all the devices and maintains as long as at least one year.Remarkably, bulk samples with macroscopic geometry of ~ 100 μm, which we name as type C, also exhibit the nice LL behavior with a universal power-law correlation down to 100 mK (Fig. 5d-f and Supplementary Fig. 14).
Interestingly, both of the thick devices and bulk samples behave like type B without Ee.Thus, it gives us a hint that the reason for the absence of Ee might be the larger fluctuation of chemical potential or slight band variation due to the coupling of increasing layers.It is important to emphasize that there is no substantial difference between type A and B devices (and type C bulk samples); the only difference is the edge gap size between them.Type B devices and type C bulk samples do not exhibit distinct edge gap behavior down to the lowest temperature (~ 1.5 K for type B devices and ~ 0.1 K for type C samples) even at CNP in the measurement.But we do not deny the possibility that they will exhibit edge gap behavior at a lower temperature (see Supplementary Fig. 8g for device #B4R as an example).
There is another point of interest, the kinks in the scaling curves of bulk samples with larger lead spacing are usually larger than that of thin films with smaller lead spacing (Fig. 5c and 5f as an example).In Luttinger liquid physics, the kink position in eV/kBT represents the number of the tunnel barriers in the 1D channel.We give a brief statistical result of this value for type B devices, type C bulk samples and Luttinger liquid region of type A devices in Supplementary Fig. 9a.It is clear that the number of tunnel barriers is positively correlated with the channel length.
Generally, Luttinger liquid behavior will be more credible if the power-law regions can extend to more than one decade.In our samples, at high temperature, thermal excitation will drive the sample exhibiting a bulk semiconducting behavior and at low temperature edge gap behavior will arise due to the 2D SOTI nature.The range of resistance change in power-law regions is smaller due to the small power exponents (usually < 1 for our devices/samples).Even so, the power-law range for almost all the type B devices/type C samples without edge gap and some of the type A devices can extend to more than one decade.
Apparent power-law dependence also shows up because of the environmental quantum fluctuation (EQF) or special region of variable range hopping (VRH) 10,[41][42][43][44] .For example, VRH from a disordered quasi-1D state due to inhomogeneity or folding lines in devices could show 'apparent' power-law behavior 44 .We can exclude the above-mentioned reasons as follows.Firstly, the high quality of our devices with PMMA protecting layers (see 'Devices fabrication' section in Methods for detail) can be verified by the optical images of the devices (inset of Fig. 2b,c, Fig. 3a, etc.).For a small number of devices with hBN as top capping layer (only 3 of the 22 devices -device #A6, Supplementary device #1 and #2), the optical images may exhibit contrast color (Supplementary Fig. 7) due to the unevenness of the top hBN on thick Ti/Au electrodes.These top capping layers are insulating and serve only as protecting layers, so they do not influence the transport behavior.This could also be confirmed by that we repeated robust power-law relation in various samples -with varying sizes from microscopic to macroscopic scale -at wide T and VDC range other than specific samples with certain parameter range.Secondly, for VRH transport, the power exponent should become larger with increasing channel length 44 .As a contrast, in our result, the power exponent is similar for microscopic devices and bulk samples with hundreds of micrometers (Supplementary Fig. 9b), inconsistent with VRH transport.Lastly, we have further proved the robust existence of clean edge states, instead of random 1D disordered channels by the sawtooth-shaped SQUID (superconducting quantum interference device) pattern in the Al-Ta2Pd3Te5-Al Josephson junction 45 .
The feature that semiconductor/Luttinger behavior at high/low T arising from bulk/edge will be more visual if we plot T-dependent curves of several devices B and bulk samples C together in bulk conductivity σ (Fig. 5g) or edge sheet conductivity σedge (Fig. 5h).Because the values in high T region converge into the same order of magnitude in the former (orange region in Fig. 5g) while that in low T region in the latter (green region in Fig. 5h).As a contrast, the curves in top (bottom) surface sheet conductivity diverge in the entire T region (Supplementary Fig. 10), which excludes the possibility of the existence of top/bottom surface states.Moreover, the starting resistance for LL (RLL, defined as 2% deviation of the extension resistance from powerlaw behavior in LL region) increases almost linearly with the number of squares in edge, □edge, for the devices with thickness of from ~ 4 nm to ~ 200 μm over five orders of magnitude (Fig. 5i).This is the first experimental observation of the scale connection between atomic-(equals to ~6 layers) and macro-LL, and confirms the LL quantum state with correlation on the length scale and a universal feature independent of the microscopic details 46,47 .It also supports the edge state calculated for the monolayer is generic for bulk sample due to the weak inter-layer coupling as the sketch in the inset of Fig. 5i shows.Such an in-gap 1D LL in the 2D sides of bulk Ta2Pd3Te5 might boost a potential application of this giant anisotropic electrical transport, as well as the anisotropic heating transport.
The temperature-dependent electrical resistance of the bulk Ta2Pd3Te5 has also been measured in previous works and the results are consistent with ours 30,48 .Interestingly, as reported in Ref. 48, the superconducting behavior can be observed in Ti-or W-doped Ta 2 Pd 3 Te 5 , indicating luxuriant physics phenomena hidden in Ta 2 Pd 3 Te 5 system.

Luttinger liquid phase diagram
We have mentioned the weak inter-layer coupling in Fig. 1d, but interestingly the bulk gap still varies with thickness.A plausible reason for the increasing bulk gap with decreasing thickness (summarized in Fig. 6a, calculated from the Arrhenius equation Δ = -2kB(∂ln(dI/dV)/∂T -1 ) by using the dI/dV(T) curve) is that the insulating state is more robust in thinner devices due to the reduced screening effect to the Coulomb interaction.As the appearance of LL relies on both of the bulk gap and edge gap, it is easy to perform a quantitative study of LL phase diagram.Fig. 6b illustrates the summary of starting and ending temperature, T1 and T2, of LL from dI/dV(T) curves of devices with different thicknesses and results in a clear thickness-dependence phase diagram of bulk semiconducting FL, edge LL and edge gap.
The temperature range of LL is adjustable over three orders of magnitude from ~20 -100 K in thin film devices to 0.1 -7 K in bulk samples, exhibiting a large parameter space.We also find T1, which indicates the energy size of Eb, varies for a large temperature range with thickness but is still correlated with T2 (indicating the size of Ee, Supplementary Fig. 11).This reveals that Ee originates from the intrinsic band structure instead of other insulating state.

Summary
As such, this LL has not only extended the LL families to the more general boundary, beyond helical and chiral edges, of high-dimensional systems, but also remarkably promotes the reproducibility and enlarges parameter space including geometry, temperature (T) and chemical potential (μ).We conclude this LL possesses universality and robustness, as well as a full phase diagram with extensive range, complete data set and strong tunability.Therefore, this model system would have wide-reaching implications to understand quantum phases towards more complex systems, with a combination of strong correlation and topology, which is lack of useful and practical method to analyze before.

Density functional theory calculations
The first-principles calculations were performed within the framework of the density functional theory (DFT) using the projector augmented wave (PAW) method 49,50 , as implemented in Vienna ab-initio simulation package (VASP) 51,52 .The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation exchange-correlations functional 53 was used.In the selfconsistent process, 16 × 4 × 4 k-point sampling grids were used, and the cut-off energy for plane wave expansion was 500 eV.The monolayer structure is extracted from the experimental bulk crystal, where inversion symmetry (IS) is slightly broken.With relaxation the symmetry of the relaxed monolayer becomes Pmmn (No. 59).The PBE band structure shows that there is a small band overlap at Γ.The band representation analysis of the monolayer band structure shows that there is a remarkable double-band inversion on the Γ-Y line.In order to match the experimental energy gap, we artificially upshift the conduction bands (Ta-dz 2 in the Wannierbased Hamiltonian) by 0.1 eV to remove the small band overlap.

Growth of Ta2Pd3Te5 crystals
Single crystals of Ta2Pd3Te5 were synthesized by self-flux method.Details can be found in Ref 29 .

Devices fabrication
We have two kinds of fabrication for devices: one is coating devices with PMMA to prevent from surface oxidization and contamination, and using SiO2/Si+ as bottom gate (devices #A1-A5, #A7-A14, and #B1-B6); the other is using a graphite/hexagonal boron nitride (hBN) bottom gate and covering devices with hBN on top to protect devices (device #A6, Supplementary device #1 and #2).
For the first kind of devices: The Ta2Pd3Te5 thin films were mechanically exfoliated onto SiO2/Si + substrates and coated with PMMA in a glove box.The two-terminal/fourterminal/Hall bar devices were patterned using electron-beam lithography and moved into the glove box to develop.The Ti/Au electrodes were deposited using thermal evaporation instrument installed in the glove box.Then lifted-off the metal film and coated with PMMA once again without being moved out of the glove box.After these, we performed electron-beam lithography for the second time to pattern the Ti/Au pads in order that we could connect the pads and transport measurements sample holder using wire-bonding out of the glove box.The Ta2Pd3Te5 thin films were in the nitrogen atmosphere glove box or coated with PMMA in the air during the whole fabrication process to prevent oxidation.
For the second kind of devices: We used polydimethylsiloxane (PDMS) to exfoliate graphite/hBN/Ta2Pd3Te5 flake and drop them on SiO2/Si + substrates respectively on a micropositioning stage in the glove box to build a graphite/hBN/Ta2Pd3Te5 stack.Then coat the substrate with PMMA without moving out of the glove box.The two-terminal/fourterminal/Hall bar devices were patterned using electron-beam lithography and moved into the glove box to develop.The Ti/Au electrodes were deposited using thermal evaporation instrument installed in the glove box.Then lifted-off the metal film and coated the device with hBN once again without being moved out of the glove box.

Electrical transport measurements
The electrical transport measurements were performed in cryostats (Oxford instruments dilution refrigerator with temperature range ~10 mK -30 K, Oxford instruments 3 He cryostats with temperature range ~ 400 mK -30K and Oxford instruments TeslatronPT with temperature range ~ 1.5 K -300 K).The bottom gate voltage Vbg and d.c.bias were applied using Keithley 2400 or 2612.Standard lock-in measurements were taken with a frequency of 3 -30 Hz with a small a.c.excitation below the thermal excitation range without specially marked.Note that due to the disorders and defects which could break the conducting channels in the edges of Ta2Pd3Te5, the tunneling behaviors appear both in the electrodes -edge junctions (FL -LL) and edge -edge junctions (LL -LL) 8,36,37 .So, we can observe the power law behavior of Luttinger liquid both in two-terminal and four-terminal/Hall bar configurations.
Note that due to the thickness of SiO2 dielectric layers used to apply Vbg are different in device #A1 (200 nm) and #A2 (100 nm), the adjusting abilities are different.The thickness of SiO2 is 100 nm if not specially marked in other devices.
Fig. 5 for nonlocal measurement of device #A2).For the nonlocal conductance in Fig 3, c and d, both of nonlocal ratios Vnl1/V0 and Vnl2/V0 at CNP increase with decreasing T gradually between 50-120 K (see the orange region in insets of Fig. 3, c and d), which are due to the reduced contact resistance of local electrodes and still behave as the classical ohmic planar fluid (the dash lines in the inset are classical planar values obtained by the simulation in Supplementary Fig. 6).Afterwards, they grow drastically on cooling from 50 K to 9 K (the green region), reaching values far above the classical values and indicating the edge transport case.Particularly, the sum of these two nonlocal ratios is close to 1 around 6 K.In the same T region (Fig. 3b), a power-law region appears in the local conductance fitted by the dash line (inset) and revealing it is the edge state.Interestingly below 4 K when the local conductance enters the Ee (purple region), Vnl1/V0 comes up to 1 and Vnl2/V0 goes to zero, suggesting that the inhomogeneity in Ee between different edge channels is enhanced.Vnl/V0(Vbg) curves at low T in Fig 3, c and d show nonlocal values become classic ohmic planar-like when Fermi level goes towards the conduction band at positive Vbg, excluding these unusual nonlocal values are due to planar fluid with extreme-disordered cases, e.g., with cracks or being broken into pieces.
, filling in the fractional quantum Hall 17 .The similar slope of tunability by gate voltage (also indicating the electron carrier density) in Fig 4c from three devices might indicate a constant intrinsic

Fig. 1
Fig. 1 Crystal structure and band structure for Ta2Pd3Te5.a, Crystal structure viewed from the chain direction along the b-axis.Inset: Optical image of single crystals.Side length of the square is 1 mm.b, Cross-sectional STEM images of Ta2Pd3Te5 single crystal along [010] (up) and [001] (down) direction.c, ARPES intensity plot of the band structure along theΓ -Y direction at T = 100 K.For clarity, the data is divided by the Fermi-Dirac distribution function to visualize the conduction band bottom above EF.d, Intensity plot of ARPES data at E = EF -0.05 eV (valence band maximum) collected in a range of photon energies from 25 to 60 eV which shows weak coupling between Ta2Pd3Te5 vdW layers.e, The diagram of double-band inversion along Y -Г line (adapted from Ref. 28).f, The modified band structure by adding 0.1 eV onsite energy on Ta-dz 2 states to simulate the gapped band structure with interactions.g,The 1D gapped edge state of the 2D quadrupole topological insulator.The onsite energy is slightly modified in order to match the experimental energy gap.

Fig. 2
Fig. 2 Electrical transport measurement for Ta2Pd3Te5.a, Sketch of Fermi level rising with Vbg.b, dI/dV versus Vbg measured in device #A1 at T = 0.42 K. Inset: Optical image and schematic measurement configuration.Scale bar, 1 μm.c, dI/dV versus Vbg measured in device #A2 at T = 6.01 K. Inset: Optical image and schematic measurement configuration.Scale bar, 2 μm.d, Log-log plot of dI/dV versus VDC measured in device #A1 at Vbg = -28 V and different T. Inset: The same data as main figure but plotted in linear scale and symmetric VDC.e, Loglog plot of dI/dV versus VDC measured in device #A2 at T = 6.01K and different Vbg.f, Loglog plot of temperature dependence dI/dV of device #A2 at different Vbg.Inset: log (dV/dI) v.s.1/T plot of the temperature dependence resistance of device #A2 at V bg = -12.5 V.The gray dot

Fig. 3
Fig. 3 Nonlocal transport and edge states in Ta2Pd3Te5.a, Schematic nonlocal measurement configuration with Hall bar shape contacts (blue) on a Ta2Pd3Te5 thin film (gray).The red lines are edges of the thin film and red arrows are current flow along edges.Inset: Optical image of device #A3.Scale bar, 2 μm.b, Local differential conductance dI0/dV0 measured in device #A3 as a function of Vbg at differential T. c,d, Nonlocal voltage ratio Vnl1/V0 (c) and Vnl2/V0 (d) versus Vbg at different T. Inset in b, c and d: dI0/dV0 (b), Vnl1/V0 (c) and Vnl2/V0 (d) versus T taken as the mean value at Vbg = -22 V to -18 V.The error bars show the full range of values spanned by the fluctuations.Dotted-lines are labels with simulation values of homogeneous 2D ohmic resistivity situation.

Fig. 4 Fig. 5 Universal
Fig. 4 Tunable Luttinger liquid behaviors.a, dI/dV versus VDC measured in device #A4 at differential T with Vbg = 0 V. b, The dI/dV(VDC) curves without edge gap behavior in (a) (From T = 12.9 K to 39.8 K) are plotted as scaled conductance (dI/dV)/T 0.69 versus scaled temperature eVDC/kBT.All data collapses to a single curve indicating Luttinger liquid behaviors.(dI/dV(VDC) curves below T = 12.9 K exhibit edge gap behavior due to the 2D SOTI nature, so they shouldn't obey the scaling behavior physically.)c, Gate tunable α in Luttinger liquid behaviors in dI/dV v.s.VDC curves.The error bars reflect the uncertainty due to measurement noise.(Raw data are included in Supplementary Fig. 12.)

Fig. 6
Fig. 6 Luttinger liquid phase diagrams.a, Thickness-dependent bulk gap extracted from dI/dV(T) curves.The error bars in x-axis reflect the uncertainty from thickness measurement and error bars in y-axis show the ranges of bulk gap variation.b, The Luttinger liquid phase transition temperature versus sample thickness.T1/T1*/T1**: the transition temperatures from bulk Fermi-liquid semiconductor behaviors to edge Luttinger liquid behaviors for type A/B/C samples; T2: the transition temperatures from edge Luttinger liquid behaviors to edge gap behaviors for type A samples.The error bars reflect the uncertainty from thickness measurement.Uncertainties in y-axis are smaller than point size.