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Guo Yu, Pengjie Wang, Ayelet J. Uzan-Narovlansky, Yanyu Jia, Michael Onyszczak, Ratnadwip Singha, Xin Gui, Tiancheng Song, Yue Tang, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Robert J. Cava, Leslie M. Schoop, Sanfeng Wu

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[Evidence for two dimensional anisotropic Luttinger liquids at millikelvin temperatures](https://mdr.nims.go.jp/datasets/851ce5ae-582c-4a61-97cc-55c9e6358d8e)

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Evidence for two dimensional anisotropic Luttinger liquids at millikelvin temperaturesArticle https://doi.org/10.1038/s41467-023-42821-2Evidence for two dimensional anisotropicLuttinger liquids at millikelvin temperaturesGuo Yu 1,2,6, Pengjie Wang 1,6, Ayelet J. Uzan-Narovlansky 1, Yanyu Jia 1,Michael Onyszczak 1, Ratnadwip Singha 3, Xin Gui3, Tiancheng Song 1,Yue Tang1, Kenji Watanabe 4, Takashi Taniguchi 5, Robert J. Cava3,Leslie M. Schoop 3 & Sanfeng Wu 1Interacting electrons in one dimension (1D) are governed by the Luttingerliquid (LL) theory in which excitations are fractionalized. Can a LL-like stateemerge in a 2D system as a stable zero-temperature phase? This question iscrucial in the study of non-Fermi liquids. A recent experiment identifiedtwisted bilayer tungsten ditelluride (tWTe2) as a 2D host of LL-like physics at afew kelvins. Here we report evidence for a 2D anisotropic LL state down to50mK, spontaneously formed in tWTe2 with a twist angle of ~ 3o. While thesystem is metallic-like and nearly isotropic above 2 K, a dramatically enhancedelectronic anisotropy develops in the millikelvin regime. In the anisotropicphase, we observe characteristics of a 2D LL phase including a power-lawacross-wire conductance and a zero-bias dip in the along-wire differentialresistance. Our results represent a step forward in the search for stable LLphysics beyond 1D.The Luttinger liquid (LL) theory of 1D interacting conductors offers ademonstration of a gapless electronic phase beyond the standardFermi liquid (FL) paradigm1,2. Distinct features of a LL owing to strongcorrelations include the power-law suppression of the density of state(DOS) at the Fermi energy and the fractionalization of electronicexcitations into collective modes associated separately with spin andcharge degrees of freedom. These appealing properties of a 1D LL ledAnderson to explore the possibility of LL-like physics in dimensionshigher than one for explaining the unusual phenomena of cupratesuperconductors3–6. Theoretical searches for 2D or 3D LL-like non-Fermi liquids were put forward in the context of coupled-wireconstructions7–11, where identical parallel 1D nanowires, each beingdescribedby the LL theory, are placed together to form2Darrays or 3Dnetworks.One key question is whether the LL physics survives in thecoupled-wire systems at vanishing temperatures when interwire hop-ping is turned on. Similar problems have been considered in the studyof quasi-1D organic conductors12, which may be regarded as weaklycoupled wires. Typically, a crossover temperature T* from the LL stateat intermediate temperatures to a 2D FL or gapped state at low tem-peratures exists and may be estimated1,7,13 as T* ~ t⊥(t⊥/t‖)η/(1-η), where t⊥(t‖) is the interwire (intrawire) hopping term and η is the power lawexponent of the DOS at the 1D Fermi surface13. Here η reflects theintrawire interaction strength (see illustration in Fig. 1a, b) and t⊥ < t‖ ina coupled wire setting. Experimentally, T* in organic quasi-1D con-ductors is typically tens of kelvins12, below which the LL description isinvalid. Interestingly, in the above expression, one readily sees that theintrawire interaction suppresses T*. If η is large enough, i.e., η > 1, T* inprinciple vanishes, indicating a new regime where single-particlehopping is irrelevant1,7,13 even down to zero temperature. In realisticsystems, other competing phases, especially those arising from two-particle hopping processes, become important in this regime andprovide instabilities to the LL state. Evaluating competing phases, suchas charge density wave, FL and superconducting states, is indeed thekey focus of multiple theoretical works7–11 in the early 2000s, whichcarefully investigated the phase diagram of a 2D array of coupled LL.Received: 3 March 2023Accepted: 23 October 2023Check for updates1Department of Physics, Princeton University, Princeton, NJ 08544, USA. 2Department of Electrical and Computer Engineering, Princeton University, Prin-ceton, NJ 08544, USA. 3Department of Chemistry, Princeton University, Princeton, NJ 08544, USA. 4Research Center for Functional Materials, NationalInstitute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 5International Center for Materials Nanoarchitectonics, National Institute for MaterialsScience, 1-1 Namiki, Tsukuba 305-0044, Japan. 6These authors contributed equally: Guo Yu, Pengjie Wang. e-mail: sanfengw@princeton.eduNature Communications |         (2023) 14:7025 11234567890():,;1234567890():,;http://orcid.org/0000-0003-1812-9825http://orcid.org/0000-0003-1812-9825http://orcid.org/0000-0003-1812-9825http://orcid.org/0000-0003-1812-9825http://orcid.org/0000-0003-1812-9825http://orcid.org/0000-0002-1427-6599http://orcid.org/0000-0002-1427-6599http://orcid.org/0000-0002-1427-6599http://orcid.org/0000-0002-1427-6599http://orcid.org/0000-0002-1427-6599http://orcid.org/0000-0002-6508-4685http://orcid.org/0000-0002-6508-4685http://orcid.org/0000-0002-6508-4685http://orcid.org/0000-0002-6508-4685http://orcid.org/0000-0002-6508-4685http://orcid.org/0000-0001-6061-8441http://orcid.org/0000-0001-6061-8441http://orcid.org/0000-0001-6061-8441http://orcid.org/0000-0001-6061-8441http://orcid.org/0000-0001-6061-8441http://orcid.org/0000-0001-9411-2369http://orcid.org/0000-0001-9411-2369http://orcid.org/0000-0001-9411-2369http://orcid.org/0000-0001-9411-2369http://orcid.org/0000-0001-9411-2369http://orcid.org/0000-0002-3155-2137http://orcid.org/0000-0002-3155-2137http://orcid.org/0000-0002-3155-2137http://orcid.org/0000-0002-3155-2137http://orcid.org/0000-0002-3155-2137http://orcid.org/0000-0002-6845-6624http://orcid.org/0000-0002-6845-6624http://orcid.org/0000-0002-6845-6624http://orcid.org/0000-0002-6845-6624http://orcid.org/0000-0002-6845-6624http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3459-4241http://orcid.org/0000-0003-3459-4241http://orcid.org/0000-0003-3459-4241http://orcid.org/0000-0003-3459-4241http://orcid.org/0000-0003-3459-4241http://orcid.org/0000-0002-6227-6286http://orcid.org/0000-0002-6227-6286http://orcid.org/0000-0002-6227-6286http://orcid.org/0000-0002-6227-6286http://orcid.org/0000-0002-6227-6286http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-42821-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-42821-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-42821-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-42821-2&domain=pdfmailto:sanfengw@princeton.eduUnder some fine-tuned interactions, they predicted, remarkably, thepossible existence of a stable anisotropic phase in which both single-and two-particle interwire hopping processes are irrelevant, i.e., aphase akin to the LL surviving as a zero-temperature ground state in asmall parameter space. This anisotropic 2D phase, coined a “slidingLL”9–11 or “smectic metal”8, is surrounded by its competing orders andits realization requires careful control of parameters in the cou-pled wires.An experimental realization of coupled-wire constructions in acontrollable setting is however challenging. Very recently, new mate-rial systems have been identified for investigating LL physics beyond1D, notably including the moiré superlattice of twisted bilayer WTe2(tWTe2)14 and a quasi-2D material η-Mo4O1115. Evidence for LL physicshas been shown down to 1.8 K in tWTe2 with a twist angle near 5o and~10K in η-Mo4O11. An essential question is whether the LL state cansurvive in any realistic systems down to the lowest achievable tem-perature in experiments, particularly in the millikelvin regime. In thiswork, we address this fundamental question for establishing the con-cept of a 2D LL, based on tWTe2 moiré superlattices. The exceptionaltunability of electronic properties by moiré engineering places tWTe2as an outstanding system for such studies.ResultstWTe2 Moiré lattice and device designOwing to the rectangular unit cell of monolayer WTe2 (Fig. 1c), thesuper unit cell of themoiré lattice in tWTe2, when the twisted angle θ issmall, is also a rectangle but large. Figure 1d, e illustrate the atomicstructure of tWTe2, where only the W atoms are shown to bettervisualize it. The moiré pattern clearly resembles the coupled-wire lat-tice shown in Fig. 1a. By tuning the twist angle, one can arbitrarilyFig. 1 | The coupled-wire constructionand tWTe2moiré superlattices. aCartoonillustration of coupled-wire construction, where the interwire (intrawire) hoppingt⊥ (t‖) and the effective power lawexponentη in a singlewire is indicated.bAsketchof the 1D-2D crossover temperature T* v.s. η in coupled wires, only consideringsingle particle hopping. c Top, a top view of monolayer WTe2, with the unit cellmarked by the dashed rectangle. Bottom, a side view of a tWTe2 with the top layerrotated by 3°. d Moiré superlattice of tWTe2 (Only W atoms for a bettervisualization), showing alternating AB and AA stacking sites. The AA (or AB) sitesmimic the coupled-wire construction shown in (a). e A zoom-in view of the tWTe2moiré lattice. f A cartoon illustration of the device structure. See Methods andSupplementary Fig. 1 for details about device fabrication. g An optical image ofdevice 1. Yellow dashed line, white dotted line and red solid line indicate respec-tively the top WTe2, bottom WTe2, and the electric contact regions.Article https://doi.org/10.1038/s41467-023-42821-2Nature Communications |         (2023) 14:7025 2choose the size of the supercell in a wide range, which alters keyparameters that determine its ground phases. Our previous workrevealed that tWTe2 at θ ~ 5o indeed develops LL physics below ~30Kon the hole-rich side14, evidenced by the emergence of large transportanisotropy and power law scaling behaviors of its conductance. Therethe LL transport characteristics were confirmed down to 1.8 K, belowwhich the across-wire resistance becomes too large (>10MΩ) to beresolved quantitatively14, preventing transport access to the physics inthe sub-kelvin regime. The essential question of whether the LLdescription is valid at millikelvin temperatures in tWTe2, or any 2D/3Dexperimental system, remains unknown.In this work we focus on tWTe2 with a smaller θ, near 3o, where themoiré cell is larger (interwire distance d ~ 12 nm, as shown in Fig. 1d) andthe energy scale is in principle smaller. Similar to previous reports14,16,17,we fabricate devices with tWTe2 fully encapsulated by top and bottomhexagonal boronnitride (hBN) dielectrics andmetal (Pd) gate (see Fig. 1ffor device structure). A thin layer of selectively etched hBN is insertedbetween the tWTe2 and metal (Au or Pd) electrodes, to ensure directelectric contacts to the tWTe2 interior (indicated by the red squares inthe optical image of a typical device shown in Fig. 1g). With this devicegeometry, contributions to transport from the nearby monolayer WTe2regions, as well as its edge modes, are minimized. Details of devicefabrication are illustrated in Methods and Supplementary Fig. 1. Theapplication of gate voltages varies the carrier density in the sample. Wequantify the gate-induced doping as ng≡ εrε0(Vtg/dtg+Vbg/dbg)/e, whereVtg (Vbg) is the top (bottom) gate voltage; dtg (dbg) is the thickness of thetop (bottom) hBN dielectric; ε0, εr and e are respectively the vacuumpermittivity, relative dielectric constant of hBN, and elementary charge.The choice of near 3o twist angle is based on systematic studies of tWTe2with a range of small twist angles (see Supplementary Figs. 2–4). Themuch smaller resistivity, compared to ~5o devices or the monolayer,together with a large emergent anisotropy near this angle, provides akey condition for us to evaluate the LL transport characteristics quan-titatively down to temperatures as low as ~50mK.VIVIHardEasybc2 K320 mK50 mK104105106Rhard(Ω)103104Reasy(Ω)ed101102103104β-4 -2 0 2 4V (mV)104105106(dV/dI ) hard(Ω)-1 0 101234(dV/dI) easy(kΩ)V (mV)-4 -2 0 2 4101102103104V (mV)βi50 mK70901101301602002503204005207001.0 K1.42.0Rhard Reasy Rhard/Reasy(dV/dI)hard (dV/dI)easy β107105-6 -3 0 3 6Vbg (V)Device 1Vtg = 2 VDevice 1Vtg = 1.00 VVbg = -3.00 V103-6 -3 0 3 6Vbg (V)-6 -3 0 3 6Vbg (V)-5 0 5-5050 10 20Vtg(V)Vbg (V)a Rhard (kΩ) fβ10-1 100T (K)10010210410310110-14.6×1012 cm-2-2.6×1012 cm-2Device 14.7×1012 cm-2-1.0×1012 cm-2Device 2hgFig. 2 | Exceptional transport anisotropy at millikelvin temperatures. a A dual-gate map of resistance (Rhard, geometry shown in b) taken in device 1 at 4 K. bCartoon illustration of measurement geometries for Rhard and Reasy. c Rhard v.s. Vbgtaken at various T, ranging from 50mK (blue) to 2 K (red). See inset in (e–g) fortemperature legends. Vtg is kept at 2 V as indicated by the dashed line in (a).d Samemeasurements as in (c) but for Reasy. e The gate-dependent anisotropy ratio β ≡Rhard/Reasy under various T. Orange (blue) arrow indicates the gate where theorange (blue) curve in (f) is extracted. f β as a function of T, plotted for two typicalng in electron (hole) side. Data taken in both devices 1 and 2 are shown. g (dV/dI)hardv.s. d.c. bias V at various T at a selected gate parameter indicated by the cross in (a).h The same as (g) but for easy direction. i Bias-dependent anisotropy β at various T.Article https://doi.org/10.1038/s41467-023-42821-2Nature Communications |         (2023) 14:7025 3Emergent anisotropy at millikelvin temperaturesFigure 2a plots resistance measured in device 1 (θ ~ 3o) under varyinggate voltages at a sample temperature, T, of 4 K. The maximum isonly ~20 kΩ and no substantial transport anisotropy is found; bothaspects are distinct from the tWTe2 with θ ~ 5o at the sametemperature14, highlighting the key role of θ. Interestingly, anexceptionally large transport anisotropy develops at millikelvintemperatures, where the easy and hard transport directions can beclearly identified by measuring resistances between neighboringprobes in the ring contact geometry (see Supplementary Fig. 5). Toquantify the anisotropy, we carefully examine four-probe resistances(Reasy and Rhard) measured in both directions (as illustrated inFig. 2b). Figure 2c, d plot Rhard and Reasy, respectively, as a function ofVbg at a fixed Vtg = 2.00 V, taken at various T below 2 K. One clearlysees a dramatic increase of Rhard on the hole-rich side and near thecharge neutrality point (CNP) when T is lowered. At 50mK, Rhardreaches a value of >1 MΩ. In sharp contrast, this strong increase isabsent on the electron-rich side and, markedly, for Reasy at all doping.We define an anisotropy ratio β ≡ Rhard /Reasy (Fig. 2e). Near CNP andwith hole doping, β is large, reaching 10,000 at 50mK. Warming thesample β decreases dramatically (Fig. 2f). Anisotropy is absent in theelectron-rich regime at any T.The distinct transport along the two orthogonal directions (easyv.s. hard) manifests itself not only in the strong anisotropy (large β),but also in its bias-dependence. Figure 2g–i examine the effects of ad.c. bias (V) applied to the source contact indevice 1, atVtg = 1.00V andVbg = -3.00 V (indicated by the white cross in Fig. 2a), where a large β isseen.Near zerobias, a largepeak is clearly seenwhen transport is alongthe hard direction (Fig. 2g), exhibiting an insulating-like behavior.Warming up the device to ~2 K, the curveflattens. In sharp contrast, thesamemeasurement along the easy direction yields a clear zerobias dip(Fig. 2h). Similar behavior is found in device 2 (θ ~ 3.5o), as shown inSupplementary Fig. 6. A consistent zero bias dip is also seen in a ~ 5otWTe2 device (Supplementary Fig. 7). At first glance, this dip featureresembles that of a superconductor. However, it cannot be suppressedby magnetic fields (Supplementary Fig. 6) and only appears when it ismeasured along the easy direction. Instead of arising from super-conductivity, this remarkable feature can be well explained by thespontaneous formation of a new phase consisting of a 2D array of 1Delectronic channels, as illustrated by the gray lines in Fig. 2b. At smallbias, this new phase develops and the current flowing between sourceand drain contacts are restricted to the 1D channels connecting them.Consequently, current flow is minimized between the voltage probesplaced nearby, yielding a vanishing voltage, i.e., the zero-bias dip. Athigh V or high T, this strongly anisotropic phase is destroyed (Fig. 2i),leading to a finite voltage between the two probes. The shoulder nextto the zero-bias dip (Fig. 2h) signifies the transition between the ani-sotropic phase to an isotropic one at high V.LL Characteristics down to 50mKWe next examine transport characteristics expected for a LL state,namely the power low scaling behavior of the conductance.Device 1α = 1.16β ~ 1.3×104d Vtg = -1.40 VVbg = -0.29 VVtg = 1.00 VVbg = -3.00 Vα = 1.53aβ ~ 2.3×104cfeV (V)(dI/ dV)/Tα(dI/dV)/Tα10-1T (K)10-610-510-4G (S)100b10-610-510-410-5 10-3dI/dV(S)100 101 10210-410-3eV/kBT10-1 100T(K)10-510-4G (S)10-410-5dI/dV(S)V (V)10-410-5 10-310-1100 101 102eV/kBT10-110-410-3520400320250200160130110907050 mK700520400320250200160130110907050 mKFig. 3 | Luttinger Liquid behaviors down to 50mK. aGv.s. T in a log-logplot takenindevice 1 at thesamegate configurationas that in Fig. 2g–i. Thecontact configurationis the sameasRhard. The solid line is a power law fit to the lowTdata,with the resultingexponentα indicated. The corresponding anisotropy β is indicated aswell.bdI/dV v.s.V at various T, ranging from 50mK to 520mK (the same legends as (c)), taken at thesame gate voltages as in (a). c The scaled plot (dI/dV)/Tα v.s. eV/kBT for the same datashown in (b) using the same α extracted in (a). d–f The same plots for a selected gateconfiguration taken in device 2.Article https://doi.org/10.1038/s41467-023-42821-2Nature Communications |         (2023) 14:7025 4Conventionally, in a single 1D wire system one maymeasure tunnelingconductance from a Fermi liquid lead to the wire18–22. In our case of anarray of parallel 1D wires in themoiré system, electron transport in thehard direction involves tunneling between wires, providing an excel-lent opportunity for examining power law behaviors without the needof an external tunneling probe14. One consequence of the LL physics isthat the across-wire conductance G(T) ≡ 1/Rhard ∝ Tα where the expo-nent α reflects the power law suppression of the DOS near the Fermienergy7–11,13,23. This is indeed seen in the strongly anisotropic regime ofour devices, as shown in Fig. 3a for device 1. At this gate configuration,we find that a value of α ~ 1.53 captures well the low-T conductancefrom ~500mK down to 50mK. A transition to the high-T isotropicphase occurs near ~1 K, above which only a weak T-dependence is seenin G. In Supplementary Fig. 8, we further confirm that neither anexponential formexpected for anactivation gapnor the variable-rangehoping form for localization24 can describe the observed conductanceat millikelvin temperatures.Another essential LL feature lies in the bias-dependent differentialconductance, i.e., dI/dV ∝ Vα when eV » kBT, where kB is the Boltzmannconstant. The same exponent αmust be seen here as in the aboveG(T)since it reflects the same suppression of DOS. In other words, LLphysics1 dictates that the scaled conductance (dI/dV)/Tα is only afunction of eV/kBT. Figure 3b plots themeasured dI/dV as a function ofV varied from 10 μV to ~1mV, taken at various T. Remarkably, all datapoints, taken in the parameter space spanned over two decades in Vand one decade in T, collapse into a single curve in the scaled plot(Fig. 3c)! The only parameter used here is α, the same one extractedfromG(T) (Fig. 3a). Hence the exponentαprovides a key description ofthe transport behaviors of the system, well consistent with the emer-gence of LL physics in the moiré system. In a simplified argumentwhere only an effective intrawire Fermi surface exponent η is con-sidered, a calculation for acrosswire transport yields α = 2η − 113. If thisis valid, we estimate η ~ 1.26, a value that is larger than one. It istherefore consistentwith the condition for the single-particle interwirehopping to be irrelevant, as discussed in Fig. 1b. We note that in rea-listic system, two-particle hopping process are also important in thisregime and hence the interactions shall be described bymore complexparameters that involve both interwire and intrawire interactions7–11,instead of a single η. In that case, dimensional crossover and com-peting orders provide new instabilities to the LL phase. Theoretically,eV/kBT100 101 102α = 0.98eVtg = 1.40 VVbg = -7.10 V10-310-410-1eV/kBTα = 1.15fVtg = -4.00 VVbg = 1.30 V100 101 10210-310-410-1100 101 10210-510-4eV/kBT(dI/dV/)TαVtg = -3.50 VVbg = -4.00 Vh10-1a-5 0 5-505Rhard/Reasycdefhgi10-1 101 103Vbg (V)Vtg(V)T = 150 mKbT (K)G (S)10-310-410-510-610-1 100hgi,x10crossoverdα = 1.75Vtg = -2.21 VVbg = -2.52 V10-310-4100 101 102eV/kBT10-12D anisotropic LLIsotropic electron liquidIsotropicmetalngTjh-rich e-rich~1 K50 mKCNPnon-LL520400320250200160130110907050 mK2.0 K1.4 K1.0 K700 mK×10-4i100 101 102eV/kBT10-1Vtg = -7.00 VVbg = -8.00 V(dI/dV/)Tα250 mK2.0 K2468(dI/dV/)Tαα = 1.34cVtg = -0.89 VVbg = -1.02 V10-310-4100 101 10210-1eV/kBT520 mK50 mK(i only)10-5 10-4 10-3V (V)012dI/dV(S)Vtg = 5.40 VVbg = 7.19 Vg×10-3Fig. 4 | Electronic phase diagram for the tWTe2 (θ ~ 3°). a A dual gate map ofanisotropy β taken at 150mK (device 2). b G v.s. T at selected gate voltages, cor-responding to the spotsmarked as (c–i) in (a). For better visualization, some curvesare multiplied by a factor, as indicated next to them. c–f Scaled differential con-ductance plot for the corresponding spots (c–f) in (a). Excellent power law scalingbehaviors are seen, indicating the LL physics. g dI/dV v.s. V for spot g (electron-richregion), showing a weak T or V dependence (an Ohmic behavior). h The scaleddifferential conductance plot for spot h, which develops a clear deviation from theuniversal scaling. An exponent α = 1.08 is used in the plot. i The same scaling plotfor spot i,whichalsodevelopsadeviation fromtheuniversal scaling.Data ispresentdown to 250mK, below which electric contacts become bad at this gate config-uration. An exponent α = 1.00 is used in the plot. j A preliminary phase diagram forthe ~3° tWTe2 system.Article https://doi.org/10.1038/s41467-023-42821-2Nature Communications |         (2023) 14:7025 5competing phases indeed reside in most regions in the calculatedphase diagram, expect that the predicted sliding LL under fine-tunedinteractions offers a possible realization of a 2D LL phase at vanishingtempeatures7–11. The experimental consequence is that although thewires are closely packed, the system, driven by interactions, behaves asan array of “independent” LL wires, and that transport across the wiresis fully suppressed unless a finite temperature or bias is applied. Fromthis perspective, the vanishing dV/dI (Fig. 2h) in the zero-bias dipmeasured along the wires, which does indicate that the wires areeffectively independent, provides an additional key characterizationofthe observed phase in tWTe2. We believe our observations of thedramatic zero-bias dip along the wire, together with the large aniso-tropy and the power law across-wire conductance, indicate the emer-gence of a highly intriguing new phase akin to the proposed “slidingLL”, although the understanding of the exact mechanism requiressubstantial future developments in both theory and experiments.The observations are reproduced in different contact geometries(Supplementary Fig. 9) and in device 2, as shown in Fig. 3d–f andSupplementary Fig. 6. We again highlight that here the LL descriptionis valid down to 50mK, an unprecedented regime. This temperature iswell below the energy scale (~meV) of the hopping and interactionterms in the system. Any dimensional crossover, if exists, must belower than this temperature.Electronic phase diagramWe discuss the gate-tuned phase diagram of tWTe2 at this small twistangle based on device 2. Figure 4a presents the gate-dependentanisotropy map taken at 150mK, where the red color indicates largeβ. G(T) and dI/dV were recorded at selected typical locations, asindicated in the map. Power law behaviors are found together withstrong anisotropy for locations labeled as c-f, while regions of g, hand i show clear deviations from a power law (Fig. 4b). Particularly, atc-f, electronic transport exhibits universal scaling characteristics(Fig. 4c–f) qualitatively like the observations in Fig. 3, hence implyingthe formation of a 2D anisotropic LL phase robust down to 50mK.This region occurs near CNP, with ng roughly within ± 4 × 1012cm-2.We find that while LL behaviors are sensitive to carrier density, thedisplacement field (D) effect is less dramatic especially if D is not toolarge. At high D, we do find a drop in the power law exponent(Supplementary Fig. 10), potentially indicating a transition to a dif-ferent phase if D is further increased. On the electron dominant side(e.g., location g), one sees almost no T- or V- dependence of theconductance, indicating a transition to a metallic-like state with anOhmic behavior (Fig. 4g). On the hole dominant side, althoughstrong transport anisotropy starts to develop below ~1 K, the con-ductance deviates from the power law typically when T is furtherlowered to, e.g., ~400mK for location i and ~100mK for h, as shownin Fig. 4b, h & i. The data suggests a crossover to a non-LL phase atlower T. In Fig. 4j we summarize the observation by presenting apreliminary phase diagram describing tWTe2 at this twist angle,under varying T and ng (here we limit the electric displacement fieldto small values for simplicity). The presence of a finite parameterregion (blue) that hosts an anisotropic 2D phase mimicking the LL isthe key finding.DiscussionIn this work, we experimentally address the long-standing questionof whether a 2D non-Fermi liquid phase resembling a LL can exist as astable ground state. We conclude that such a phase does develop inthe tWTe2 moiré system down to at least 50mK. We note that at thisearly stage a concrete theoretical modeling of this strongly corre-lated phase is still lacking. It is a challenging task to compute theelectronic structure of tWTe2moiré system even at the single particlelevel due to the large number of atoms and orbits involved in eachmoiré cell, together with the presence of spin-orbital coupling andpossible superlattice reconstructions. Future theoretical considera-tion would also need to consider strong electron interactions, whichseems to be essential in the system. The phenomenology of a 2Dsliding LL, independently proposed two decades ago based on the-oretical analysis of coupled-wire models7–11, is well consistent withour observations here, including the large transport anisotropy,power law conductance across the wires and the vanishing differ-ential resistance along the wires. However, establishing exact con-nections between the experiments and the models requires futureefforts from both theory and experimental sides. Our experimentsopen new possibilities to further study topics related to 2D LLground states and phase transitions, including spin-chargeseparation1,20,25, novel quantum oscillations and quantum Halleffects in non-Fermi liquids26–28.MethodsSample fabricationWe used Pdmetal bottom and top gates in both devices. To create themetal bottom gate, a ~ 2 nm Ti/~6 nmPd film was deposited onto aninsulating Si/SiO2 substrate using the standard e-beam lithography(EBL) and metal deposition tools. The bottom hBN was then trans-ferred onto the metal bottom gate in a dry-transfer setup, followed byEBL andmetal deposition of the metal contacts (~2 nm Ti/~6 nmPd fordevice 1 and ~2 nm Ti/~6 nm Au for device 2). After a tip-cleaningprocess using an atomic force microscope (AFM), a thin hBN wastransferred to cover the metal electrodes and selective areas of thethin hBN were etched using EBL and reactive ion etching (RIE) tech-niques. The top metal gate consists of either a ~15 nmPd layer (device1) or a ~ 3 nm Ti/~12 nmPd layer (device 2), which was deposited ontoexfoliated hBN flakes with the help of EBL for defining the location andshape. Monolayer WTe2 was exfoliated on Si/SiO2 substrates in an Ar-filled glovebox. The top metal/hBN stack was picked up as a wholefollowed by the ‘tear-and-stack’29,30 procedures that create the tWTe2stack. All processes that involveWTe2wereconducted in the glovebox.A cartoon illustration of the fabrication process can be found in Sup-plementary Fig. 1.Transport measurement at ultralow temperaturesElectronic transport measurements were performed in a Blueforsdilution refrigerator with bottom loading probe system. The probebase temperature is ~24mK. Thermocoax wires and a series of heatsinks are used to keep the electron temperature low.We calibrated theelectron temperatures using a high-quality GaAs quantumwell samplebased on the activation behavior of a fractional quantumHall state andfound that the electron temperature in our setup starts to deviate fromthe probe thermometer temperature only below 45mK. We thereforeperform measurements in this work at temperatures of 50mKor above.In the measurements, an a.c. excitation of typically 10 μV with afrequency of 7 ~ 17Hz, together with a d.c. bias, was applied to thesource electrode via a Keysight 33511B function generator. Current andvoltage signals were collected using lock-in amplifiers after a current(DL Instruments model 1211, with an internal impedance of 20Ω) andvoltage pre-amplifier (DL Instruments model 1201, with an internalimpedance of 100MΩ) to improve the signal. Gate voltages wereapplied via Keithley 2400 or 2450. The setup can reliably measurefour-probe resistance up to a few MΩ and all data presented in thiswork are in the reliable regime.Data availabilityThedata that support the findings of this study are available at HarvardDataverse (https://doi.org/10.7910/DVN/FVYOPF) or from the corre-sponding author upon request.Article https://doi.org/10.1038/s41467-023-42821-2Nature Communications |         (2023) 14:7025 6https://doi.org/10.7910/DVN/FVYOPFReferences1. Giamarchi, T. Quantum Physics in One Dimension. (Oxford Uni-versity Press, 2003).2. Haldane, F. D. M. ‘Luttinger liquid theory’ of one-dimensionalquantum fluids. I. 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A. Parameswaran, and S.L. Sondhi, S. A. Kivelson, and E. H. Fradkin. This work was supported byONR through a Young Investigator Award (N00014-21-1-2804) to S.W.Measurement systems and data collection were supported by NSFthrough a CAREER award (DMR-1942942) to S.W. Materials synthesis anddevice fabrication were partially supported by the Materials ResearchScience and Engineering Center (MRSEC) program of the NSF (DMR-2011750) through support to R.J.C., L.M.S., and S.W. S.W. and L.M.S.acknowledge support from the Eric and Wendy Schmidt TransformativeTechnology Fund at Princeton. A.J.U. acknowledges support from theRothschild Foundation and the Zuckerman Foundation. K.W. and T.T.acknowledge support from the JSPS KAKENHI (Grant Numbers19H05790, 20H00354, and 21H05233). L.M.S. acknowledges supportfrom theGordon andBettyMoore Foundation throughGrantsGBMF9064,the David and Lucile Packard Foundation and the Sloan Foundation.Author contributionsG.Y. and P.W. fabricated the devices, performed measurements, andanalyzed the data, assisted by A.J.U., Y.J., M.O., T.S., and Y.T., andsupervised by S.W. R.S., L.M.S., X.G., and R.J.C. grew and characterizedbulk WTe2 crystals. K.W. and T.T. provided hBN crystals. S.W., G.Y., andP.W. wrote the paper with input from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-42821-2.Correspondence and requests for materials should be addressed toSanfeng Wu.Peer review information Nature Communications thanks RaymondAshoori and the anonymous reviewer(s) for their contribution to the peerreview of this work. A peer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Article https://doi.org/10.1038/s41467-023-42821-2Nature Communications |         (2023) 14:7025 7https://doi.org/10.1038/s41467-023-42821-2http://www.nature.com/reprintsOpen Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons license and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023Article https://doi.org/10.1038/s41467-023-42821-2Nature Communications |         (2023) 14:7025 8http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Evidence for two dimensional anisotropic Luttinger liquids at millikelvin temperatures Results tWTe2 Moiré lattice and device�design Emergent anisotropy at millikelvin temperatures LL Characteristics down to�50 mK Electronic phase diagram Discussion Methods Sample fabrication Transport measurement at ultralow temperatures Data availability References Acknowledgements Author contributions Competing interests Additional information