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Aniket Majumdar, Nisarg Chadha, Pritam Pal, Akash Gugnani, Bhaskar Ghawri, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Subroto Mukerjee, Arindam Ghosh

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This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1038/s41567-025-02972-z.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Universality in quantum critical flow of charge and heat in ultraclean graphene](https://mdr.nims.go.jp/datasets/6f293ba2-7f08-417f-a717-a8421b2624b7)

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Universality in quantum-critical flow of charge and heat in ultra-clean grapheneAniket Majumdar,1, ∗ Nisarg Chadha,1, 2 Pritam Pal,1 Akash Gugnani,1 Bhaskar Ghawri,1Kenji Watanabe,3 Takashi Taniguchi,4 Subroto Mukerjee,1, † and Arindam Ghosh1, 5, ‡1Department of Physics, Indian Institute of Science, Bangalore 560012, India2Department of Physics, Harvard University, Cambridge, MA 02138, USA3Research Center for Electronic and Optical Materials,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan5Center for Nano Science and Engineering, Indian Institute of Science, Bangalore 560012, IndiaClose to the Dirac point, graphene is expectedto exist in quantum critical Dirac fluid state,where the flow of both charge and heat can bedescribed with a dc electrical conductivity σQ,and thermodynamic variables such as the entropyand enthalpy densities [1, 2]. Although the fluid-like viscous flow of charge is frequently reportedin state-of-the-art graphene devices [3–15], thevalue of σQ, predicted to be quantized and deter-mined only by the universality class of the crit-ical point, has not been established experimen-tally so far [2, 16]. Here we have discerned thequantum critical universality in graphene trans-port by combining the electrical (σ) and ther-mal (κe) conductivities in very high-quality de-vices close to the Dirac point. We find that σand κe are inversely related, as expected fromrelativistic hydrodynamics, and σQ converges to≈ (4 ± 1) × e2/h for multiple devices, where e andh are the electronic charge and the Planck’s con-stant, respectively. We also observe, (1) a giantviolation of the Wiedemann-Franz law where theLorentz number exceeds the semiclassical valueby more than 200 times close to the Dirac point atlow temperatures, and (2) the effective dynamicviscosity (ηth) corresponds to ηth/sth → ℏ/4πkBwithin a factor of four, in the cleanest devices forTF/T → 0 , where TF, sth and kB are the Fermi tem-perature, thermal entropy density and the Boltz-mann constant, respectively. Our experiment ad-dresses the missing piece in the potential of high-quality graphene as a testing bed for some of theunifying concepts in physics.Transport in real graphene devices is determined bytwo competing length scales: The electron-electron scat-tering length lee and the momentum relaxation lengthlmr, both of which depend on the carrier density n (orthe Fermi Temperature TF = ℏvF√πn/kB, vF being theFermi velocity) and the temperature T (see schematic inFig. 1a). In the diffusive and inhomogeneous regimes,scattering from, for example, the Coulomb impurities orphonons, do not conserve the momentum, resulting inlmr ≪ lee. As the disorder decreases, the channel canbe ballistic when the device dimension is ≪ lee, lmr orviscous, when lee becomes the shortest length scale. Theviscous flow is described well by the electronic analogueof the Navier-Stokes equation [17] when TF/T ≫ 1, andmanifests in parabolic propagation profile [5, 13], neg-ative non-local resistance [4] or super-ballistic electricalconduction [7], among others.In the opposite (thermal) limit of TF/T ≪ 1, nearlyunscreened Coulomb interaction and linear dispersiontune to a Lorentz-invariant quantum critical point [2, 18],where the current relaxes by collisions among the ther-mally excited electrons and holes, while the energypropagates unimpeded due to conservation of momen-tum during such collisions. This naturally violates theWiedemann-Franz (WF) law. For relativistic hydrody-namics, quantum critical σ and κe are predicted to be[19–21],σ(n) = σQ +e2vFn2lmrH(1)andκe(n) =vFlmrHσQTσ(n)(2)where, H and vF are the enthalpy density and Fermi ve-locity, respectively. So far, several experimental reportson the breakdown of the WF law [3], enhanced thermo-electric power [10], giant quasi-linear magnetoresistance[8], near-Planckian scattering rates and efficient thermaldiffusion inferred from the terahertz [11] and pump-probe[9] spectroscopic techniques, provided valuable insight tothe formation of a Dirac fluid, but unambiguous exper-imental evaluation of σQ has not been possible. This isbecause the measured σ for n → 0 yields non-universalresults, limited by device-dependent spatial inhomogene-ity, or puddles, of charge close to the Dirac point [22] (seeSupplementary Section S1). In the absence of this, thequantum critical behaviour itself remains unsettled, espe-cially in the presence of alternative models that proposeviolation of the WF law in marginally gapped graphenewithin the Fermi liquid framework [23].In this work, we have combined dc charge and heattransport to explore the quantum critical conductivity210 1000.51310 D2S1 D4S4 D3S5 D1S5nmin (1010 cm-2)T (K)T20 2 4 6 8051015s (mS)W (mm)n = 8 x 1010 cm-2T = 160 K1 10110s (mS)W (mm)5 6 7 8 9 101.01.52.02.5 D2S1 D5S5 D1S6 D1S4n (m2 s-1)n (1010 cm-2)-1012 -1011 -1010 1010 1011 10120.11G (mS)n (cm-2)III II I II IIIDisordered Ön90 K260 KFL FLThermalThermal0.1 1 10 1000.40.60.81 D1S4 D4S4 D5S5 D4S2 D2S1G/Gballa b de g hcTF/TDisorder/phononsInhomogeneous1InteractionThermalFermi liquidViscous DiffusiveBallistic𝑙𝑒𝑒 ≪ 𝑊𝑙𝑒𝑒 ≫ 𝑊𝑙𝑒𝑒 ≫ 𝑙𝑚𝑟𝑙𝑒𝑒 ≪ 𝑙𝑚𝑟fSiO23 μmIV0.1 1 100.40.60.81 0.1 0.2 0.3 0.4 0.5G/Gballn (1012 cm-2)D4S4hBNhBNMLGFigure 1: Viscous electron flow in ultra-clean graphene devices: (a) Different regimes of electron transportin graphene, based on the interplay between momentum-conserving and momentum-relaxing collisions. (b) (Top)Optical micrograph of the device along with the circuit diagram used for conducting the electrical transportmeasurements. (Bottom) Schematic of the heterostructure. (c) Electrical conductance of device D4S4 as a functionof the carrier density n for 90 K ≤ T ≤ 260 K. The dashed line labelled Gball indicates the ballistic conductance forthe measured channel. (d) Charge inhomogeneity (nmin) as a function of T for five different devices. The colourgradient in the background indicates the transition from a disorder-driven regime to a thermally-driven transportregime. (e) Electrical conductance (normalised by the ballistic conductance, Gball) of D4S4 as a function of lee/Wfor different carrier densities, varying from n = 1× 1011 cm−2 to 5× 1011 cm−2. (f) Normalised conductance as afunction of the Knudsen number for five different devices. (g) Electrical conductivity as a function of width W atn = 8× 1010 cm−2 and T = 160 K. For the chosen values of n and T , we have observed that lee/W ≤ 0.5. The solidline scales as W 2 and serves as a guide to the eye. Inset shows the same dataset in double logarithmic scale wherethe straight line represents quadratic dependence. (h) Kinematic viscosity ν as a function of n for four differentdevices, measured at T = 180 K.in graphene. We measured both σ and κe in mul-tiple graphene devices with low charge inhomogeneity≃ 5 − 10 × 109 cm−2, where the momentum relaxation,particularly at low temperatures, occurs at the bound-ary in most of the devices [8]. Combining σ and κe as inEqn. 2, and assigning lmr = min(W, lmfp), where W andlmfp = hσ/2e2kF are the geometric channel width and themean free path of carriers in the channel, respectively, weobtain a device-insensitive estimate of σQ ≃ (4±1)×e2/h,where the error represents variation across multiple de-vices. Fundamentally, this all-experimental strategy todetermine σQ depends on the high quality of our deviceswhich makes the thermal limit (TF/T < 1) accessibleover a broad range of T for n ≳ nmin(0), i.e., when n iswell-defined.ELECTRON VISCOSITY IN THE ULTRA-CLEANGRAPHENE DEVICESAll devices are single layers of graphene encapsulatedwith hexagonal boron nitride (hBN), which were etchedinto small rectangles of length L (≈ 1 to 2 µm) and widthW , where W was varied between ≈ 1 to 8 µm (Figs. 1b,details in Methods, Table 1 and Supplementary SectionS2). The variation of conductance (G) in a typical device(D4S4, see Table 1) with n for T = 90− 260 K is shownin Fig. 1c. In the low-n limit, G becomes insensitive ton below a characteristic scale nmin(T ), which increaseswith increasing T . The T -dependence of nmin(T ), asshown for multiple devices in Fig. 1d, can be describedwith nmin(T ) = nmin(0) + βnth(T ), where nmin(0) andnth(T ) = (2π3/3)(kBT/hvF)2 are the intrinsic inhomo-geneity and the thermally excited carrier density, re-spectively (the prefactor β is ≈ 0.7 − 0.8 for most de-vices, likely due to the residual charge inhomogeneity[24]). For the present experiments, we chose devices withnmin(0) ≲ 1010 cm−2, which usually exhibited lower Ra-3man linewidth and were occasionally subjected to post-fabrication current annealing (see Methods and Supple-mentary Section S2). The best devices exhibited carriermobility as high as 5×105 cm2/V.s and 6×106 cm2/V.sat room temperature and low T (≈ 10 K), respectively,for n ≈ 1011 cm−2 (see Supplementary Section S3 andTable I).Fig. 1c identifies three distinct ranges in n. I: n <nmin(0), where the system is spatially inhomogeneous;II: the quasi-thermal regime, where nmin(0) < n ≲8 × 1010 cm−2, for which TF is lower than the maxi-mum experimental T (300 K), and, III: n ≳ 1011 cm−2,where we expect the system to behave as Fermi liquidat all experimental T . In the large n (≳ 5× 1011 cm−2)limit of regime III, the (four-terminal) conductance Gapproaches ∼√n with a nearly T -independent magni-tude Gball (dashed line in Fig. 1c), that lies within afactor of about two of the Landauer-Sharvin conduc-tance (Gls = (4e2/πh)kFW ). This is the ballistic regimewhere the Landauer-Sharvin conductance is delocalisedinto the bulk as a result of hydrodynamic flow that me-diates the transfer of momentum among the propagatingmodes [12]. G/Gball reduces from unity as n is decreasedwhich may signify the onset of viscous flow (Fig. 1a). Toexamine this, we plot G/Gball as a function of lee/W fora broad range of n in regime III of device D4S4 for whichlmr ≈ W (Fig. 1e). We evaluated lee ≈ ℏvFTF/α2kBT2using the effective fine structure constant α ≈ 0.5 [25].The onset of decrease in G/Gball at lee/W ≈ 1 is evidentin Fig. 1e. Since lee/W effectively represents the Knud-sen number ζ(= lee/lmr) [6], this constitutes a strong ev-idence of the onset of Poiseuille flow. This behaviour wasfound to be generic in all high mobility devices, as shownfor five different devices at n = 1011 cm−2 in Fig. 1f,for which the momentum relaxation is expected to occurmainly at the boundaries (see Supplementary Section S3for further details).Strong Poiseuille-like flow of degenerate electrons re-quires the electronic Gurzhi length Dν =√νlmr/vFto exceed W , where the conductivity σ = GL/W =e2n2W 2/12η (Supplementary Sections S5 and S6) is thenexpected to vary quadratically with W [26] (here η,ν = η/ρm, and ρm are the dynamic viscosity, the kine-matic viscosity, and the areal density of effective mass,respectively). In Fig. 1g, σ in multiple devices at fixed n(≈ 8 × 1010 cm−2) and T (= 160 K), such that TF ≳ Tand lee/W < 1 are simultaneously satisfied, is consis-tent with a quadratic dependence on W (see Fig. 1ginset). The quantitative estimate of ν, obtained fromthe n-dependence of σ at 180 K, is shown in Fig. 1h,where its magnitude ∼ 1.5 − 2 m2/s is nearly device-independent and agrees with previous reports [4, 7], aswell as the theoretical calculations [27]. Since lmr is lim-ited by the boundaries in these devices, the observed νconfirms Dν ≳ W and hence the strong Poiseuille-likeflow. The differential resistance dV/dI also exhibits thecharacteristic non-monotonic dependence on the drivecurrent density (Extended data Fig. 1), where the ini-tial increase, followed by decrease, in dV/dI is attributedto crossover from the quasi-ballistic Knudsen flow to thePoiseuille regime.THERMAL CONDUCTIVITY AND THEWIEDEMANN-FRANZ LAWTo examine the current and energy relaxation path-ways, we have used Johnson noise thermometry [29] todetermine the effective Lorentz number L = κe/σT ,and thereby the electronic thermal conductivity (κe), atvarying n and T . The experimental strategy, schemat-ically shown in Fig. 2a (details in Supplementary Sec-tion S4), involves Joule heating the graphene layer withan external electric field E and evaluating the result-ing increase in the electron temperature (Te) from theJohnson-Nyquist noise magnitude [3]. The weak cou-pling of electrons and phonons in graphene [30, 31] makesthe Joule dissipation PJ = σE2 rather effective to heatthe electrons, leading to Te ≈ LE/√L at large E, asshown for sample D2S1 for different n (at T = 30 K)and different T (at n = 5 × 1011 cm−2) in ExtendedData Fig. 2a and 2b, respectively. The dashed lines arefits according to Eqn. 6 (Methods) obtained from thespatially averaged solution for Te from the heat diffu-sion equation with L as the (only) fit parameter. Evi-dently, L is strongly n and T -dependent, which under-lines the breakdown of the WF law. The magnitude ofL in the electron-doped part of D2S1 varies over six or-ders of magnitude as a function of n at low T (Fig. 2b,Extended Data Fig. 3), with a maximum value (L0) atthe Dirac point that can exceed the universal magnitudeLWF = (π2/3)2(kB/e)2 = 2.44 × 10−8 V2K−2 from theWF law by nearly 300 times. Correspondingly, κe = LσTclose to the Dirac point also exceeds the WF magnitudeby similar order at low T . At large n however, e.g. atn = 1011 cm−2, κe is much lower than that expected fromthe WF law at all T , even though the electrical conduc-tivity is significantly higher (Fig. 2b). The behaviourremains similar, albeit reduced, in lower mobility devices(D1S1, Extended Data Fig. 4) and indicates decouplingof the charge and heat flow expected in the hydrodynamicregime where L(n, T ) varies as [21, 26],L(n, T ) = 1e2[s(T )n0(T )n2 + n20(T )]2(3)Here, n0(T ) is an effective density scale that determinesthe intrinsic conductivity of the electron fluid [26] (Ex-tended data Fig. 5a). Semi-classical analysis consideringsmall band gap opening at the Dirac point and variousscattering mechanisms fail to explain the observed be-haviour (Supplementary Section S7). Eqn. 3, however,420 100 25010100(eV/mm2)T (K)e-dopedh-doped100 200110s ( 10-8 J m-2 K-1) T220 2000.111010020 200 20 200ke (nW K-1)T (K) T (K)n = 1010 cm-2 n = 1011 cm-2CNPT (K)-0.5 0.0 0.510-510-310-1101103 19 K 170 Kn (1012 cm-2)a dcbexTFT < Timp T > TimpT Timp TimpTElectron puddlesHole puddlesVg DMMLCVNA+-20 K300 KRBFigure 2: Violation of the WF Law for hydrodynamic electrons: (a) Circuit diagram for measuringJohnson-Nyquist noise of hot electrons in our graphene devices, when subjected to Joule heating by in-plane electricfields. The LC network acts as a tank circuit for impedance matching the graphene device to the 50 Ω noisemeasurement circuit. (b) Normalised Lorentz number for device D2S1 as a function of n for T = 19 K and 170 K.The solid lines are theoretical fits of the experimental data, as per Eqn. 3. The electron-doped and hole-doped datapoints have been fitted independently using different values for the fitting parameters. (c) Electronic component ofthermal conductivity for device D2S1 as a function of T for 3 different number densities, from the charge neutralitypoint to a highly electron-doped regime. The dashed lines indicate the variation of thermal conductivity with T ,assuming the normalised Lorentz number L/LWF = 1. (d) Enthalpy density (H) for D2S1 as a function of T . Thedashed lines (black - hole doped, red - electron doped) highlight T 3-like asymptotic behaviour. The colour gradientin the background signifies a transition from a disorder-driven regime to a thermally-driven transport regime. [Inset]Entropy density for D2S1 as a function of T for T > 80 K. The black dashed line is the T 2 fit of the experimentallyobtained data and is within a factor of 2 of what is expected from the theoretical expression in Ref. [28]. (e)Schematic showing the interplay between two different types of electrical transport - disorder-driven and thermalexcitation-driven. For T < Timp, the energy scale of the background potential fluctuations is greater than thethermal energy of electrons and hence the electrical transport is dominated by charged impurities and defects,whereas for T > Timp, the electronic thermal energy dominates and electron-electron interactions drive electricaltransport in this regime.fits very well (solid lines in Fig. 2c) to the experimen-tally observed dependence of L on n, especially in theelectron doped regime. These fits also provide experi-mental estimation of two key thermodynamic variablesassociated with the hydrodynamic flow, namely the en-tropy s(T ) = en0√L0 and, from Gibb’s equation, theenthalpy H(T ) = Ts(T ) densities. The dependence of Hand s on T are shown in Fig. 2d and its inset, respectively.An important temperature scale, identified by the non-monotonic behaviour of L0 (Extended data Fig. 5b), isT ∼ Timp = ℏvF√πnmin(0)/2kB ≈ 80 K, determinedby the fluctuations in local chemical potential from in-trinsic disorder. For T ≪ Timp, the incompressibility ofgraphene arrests the fluctuations in n to ∼ nmin(0), whichsaturates H to the T -independent Fermi liquid contribu-tion (H0). The thermal component Hth ∼ T 3 dominatesonly for T ≳ Timp (dashed lines in Fig. 2d). The scenariois schematically explained in Fig. 2e. For D2S1, we findH0 ∼ ρm0v2F = 3.5 eV/µm2 within a factor of ∼ 3 ofthe observed enthalpy at low temperatures (here, ρm0 isthe mass density at nmin(0)). Moreover, an estimate ofL0 ≈ vFWH0/σminT2 at T ≪ Timp from Eqn. 2 suggestsL0/LWF ≈ 500 at T = 20 K, which is within a factor oftwo of that observed experimentally.QUANTUM CRITICAL TRANSPORT IN THETHERMAL REGIMEHaving established the dominance of the electron-electron scattering rate over that of momentum relax-ation in our devices, we now focus on the thermal regime510 1 0.10.1110sQ (e2/h) 110 K, D2S1      170 K, D2S1   260 K, D2S1     140 K, D1S1 296 K, D1S7     296 K, D2S7   296 K, D3S7         Theoretical Model1 10TF/Tde - dopedh - doped0 100 200100101102103  0.5 0.2 0.05 0sQ (e2/h)T (K)n (1012 cm -2) -0.05 -0.15 -0.34 8110100s (mS)ke (nW K-1) 110 K 170 K 260 Kµ 1/s-0.5 0.0 0.50.110 110 K 170 K 260 Kke (nW K-1)n (1012 cm-2)cba0.01 0.02 0.03110100s (e2/h) sQ sminnmin (0) (1012 cm-2)eFigure 3: Universality of the quantum critical conductivity: (a) Electronic component of thermalconductivity (κe) for device D2S1 as a function of n for T = 110 K, 170 K and 260 K. (b) κe as a function ofelectrical conductivity σ for device D2S1 at three different temperatures from the region indicated by the boundingbox in panel (a). The dashed lines indicate a 1/σ dependence and serve as a guide to the eye. (c) σQ for D2S1 as afunction of T for a range of densities from n = −3× 1011 cm−2 (hole-doped) to n = 5× 1011 cm−2 (electron-doped).The shaded region corresponds to temperatures greater than the scale of background chemical potential fluctuations.(d) σQ as a function of the ratio of Fermi temperature to absolute temperature (TF/T ), calculated at differenttemperatures for four different devices. The dashed line is based on theoretical calculations performed in Ref. [25].(e) Comparison of the quantum critical conductivity σQ (obtained from thermal transport measurements) and theminimum electrical conductivity σmin (obtained from electrical transport measurements at T < 60 K), as a functionof the intrinsic charge inhomogeneity nmin(0). The dashed line scales as n2min(0) and serves as a guide to the eye.where we expect the system to behave as a quantum-critical Dirac fluid. The rapid increase in H(T ) at T ≳Timp (Fig. 2d) suggests the onset of the thermal regime,where the corresponding entropy density sth(T ) = s(T ≳100 K) = H/T is ∝ T 2 (inset of Fig. 2d). The observedmagnitude of sth(T ) = cstheory(T ) agrees closely with thetheoretically predicted entropy density stheory(T ) [28],where the numerical constant c lies between 0.5 and 2for all devices measured. Subsequently, multiplying σ tothe fitted expression of L(n) yields n-dependence of κeas shown for three values of T > Timp in Fig. 3a. Ob-taining κe and σ for the same n (≲ 1011 cm−2), we findκe to decrease with increasing σ (Fig. 3b), which can-not be explained within a Fermi liquid framework, andpoints towards the relativistic hydrodynamic descriptiondescribed by Eqn. 2.Combining κe, σ and sth, each of which is obtainedfrom either directly or through analysis of the trans-port and noise measurements, we then compute σQ =κeσ/vFlmrsth (Details in Supplementary Section S8),and first show it as a function of T for different n inFig. 3c. Beyond the inhomogeneous puddle-dominatedregime (i.e. T ≲ 80 K) (shaded region in Fig. 3c),it is evident that close to the charge neutrality point(|n| ≲ 5 × 1010 cm−2), σQ approaches ≈ 4e2/h, andis nearly independent of temperature. For larger n(≳ 5 × 1010 cm−2), σQ is lower than this value in boththe electron and the hole-doped regimes. This is furtheremphasized when σQ is plotted against TF/T for differentdevices and temperatures (≳ Timp), where it follows verysimilar trajectories that converge to (4 ± 1) × e2/h forTF/T ≪ 1, and drops sharply for TF/T ≫ 1 (Fig. 3d).6To distinguish between σmin and σQ, we have comparedthese two parameters for all devices in Fig. 3e. σmin isclearly non-universal and increases rapidly with disorder(nmin(0)) by more than an order with the dashed linesuggesting ∼ n2min(0) dependence, whereas σQ varies by≲ 25% (shaded region).The device independence of σQ can be readily at-tributed to the universality of dc conductivity of hydro-dynamic transport in the presence of incoherent electron-hole collisions, for example, that at superfluid-insulatortransition [16, 32]. This quantum critical conductivityis predicted to be quantized to 4e2Φ/h, where Φ ∼ O[1]is a dimensionless number dependent only on the uni-versality class of the critical point [2, 16]. The observedconvergence of σQ to ∼ 4e2/h in Fig. 3d for TF/T ≪ 1 isconsistent with this scenario. As |n| increases away fromthe Dirac point, σQ scales as ∼ (4e2/h)f(T/TF), where fis a function of the dimensionless parameter TF/T . ForTF/T ≳ 1, f decreases rapidly, which is because of thesuppression of the relativistic physics and quantum crit-icality as the system crosses over from the zero to thefinite-momentum mode [25].DISCUSSIONWith the temperature as the only energy scale left, wewill now estimate the effective viscosity ηth = HthτP/2 inthe thermal regime, where we assume the current relax-ation to occur by electron-hole collisions at the Planckianrate τ−1P ≈ α2kBT/ℏ [11]. Notably, no such assumptionwas required in the evaluation of the quantum-criticalconductivity σQ. The hydrodynamicity of charge flow isthen expected to result in a quadratic n-dependence of σ(Eqn. 1), which we can indeed identify in our high-qualitydevices between |n| ≳ nmin(0) ∼ 1010 cm−2, and TF ≲ T ,as shown for D2S1 in the inset of Fig. 4a (see ExtendedData Fig. 6 for other devices). This allows us to estimateηth = e2vFWτP/2Ath(T ), where Ath(T ) is the coefficientof n2 in the n-dependence of σ.ηth varies nonmonotonically with a minimum aroundT ∼ Timp. For T ≳ Timp, ηth increases and approaches∼ T 2 dependence at high T [28] (Fig. 4a, details in Sup-plementary Section S9). Finally, with ηth estimated fromthe σ as above and sth obtained from thermal conduc-tivity (dashed line in Fig. 2d, inset), we have computedthe ratio ηth/sth normalized by its universal limit ℏ/4πkB[33]. Within the holographic description, the normalisedηth/sth is expected to approach unity for minimally dissi-pative flow of strongly interacting quantum liquids, lim-ited only by the Heisenberg’s uncertainty principle [34].We have plotted the normalised ηth/sth as a function ofT for both electron and hole-doped regimes for severaldevices in Fig. 4b. At T < Timp ∼ 100 K, the variationin ηth/sth ∼ 1/T 3, represented by the open symbols andsolid lines in the left inset of Fig. 4b, is consistent with D1S5  D3S5 D5S5  D4S4  D2S140 100 30015202530hth (10-19 J m-2 s)T (K)a250 150 5010010110210310450 150 250hth/sth (ħ/4pkB)T (K)e-dopedh-dopedb1010 1011101103hth/sth nmin(0) (cm-2)300 K100 10 10 100T (K) 1/T3  1/T310 10010-510-310-1s - smin (S)n (109 cm-2) n2D2S1 26 K260 KFigure 4: Thermal viscosity in ultra-cleangraphene: (a) Thermal shear viscosity (ηth) for D2S1as a function of T . The solid line indicates anasymptotic T 2 dependence. [Inset] The quadraticdensity dependence of σ − σmin where theinhomogeneity regime (n < nmin(0) ≈ 1010 cm−2) isindicated through shading. The upper limit of thisrange is marked by a dashed line, which representsnmin(0).(b) ηth/sth as a function of T for both theelectron- and hole-doped regimes, in five differentdevices. The dashed line indicates the holographic lowerbound. The data for T < Timp have been indicatedusing open symbols. [Left inset] ηth/sth in doublelogarithmic scale to indicate the power law dependenceon temperature (solid lines). [Right inset] Dependenceof ηth/sth on nmin(0) at T = 300 K.that expected from a degenerate Fermi liquid [33], wherethe chemical potential close to the Dirac point is pinnedto the inhomogeneity scale (Fig. 2e). For T ≫ Timp, how-ever, ηth/sth tends to saturate, but the saturation occurs7to lower values when the channel disorder is decreased,and for the cleanest channel D1S5 (see Table 1), ηth/sthapproaches ℏ/4πkB within a factor of 4. The sensitivityof the limiting ηth/sth to disorder, quantified by nmin(0),in Fig. 4b (right inset), is quite striking, and could bedue to increase in the shear viscosity in the presence ofdisorder scattering in Dirac fluids [35].In conclusion, we report signatures of viscous electronflow and quantum critical transport in extremely cleanmonolayer graphene devices with charge inhomogeneityscale as low as ∼ 5 × 109 cm−2. At the charge neu-trality point, we find the electronic thermal conductiv-ity to exceed that expected from the WF law by nearly300 times at low temperatures. Combining electrical andthermal transport measurements, we could evaluate theintrinsic dc quantum critical conductivity σQ ≈ 4e2/h,with device-to-device variation ≲ ±25%. Considering aPlanckian scattering-limited electronic viscosity, we alsodemonstrate that the ratio of viscosity to thermal en-tropy density → ℏ/4πkB, the holographic limit, withinabout a factor of four in the cleanest devices at roomtemperature. Apart from unifying some of the deep con-ceptual frameworks in physics, our results will impactthe analysis and interpretation of both dc charge andheat transport in monolayer graphene, as the quality ofsuch devices continues to improve.METHODSFabrication of hBN-encapsulated grapheneheterostructuresAll the devices presented in this work have been as-sembled using a manually operated transfer setup. Thegraphene (Gr) and hexagonal boron nitride (hBN) flakeswere mechanically exfoliated from bulk graphite (ob-tained from KishGraphite) and bulk hBN crystals (ob-tained from NIMS, Japan) respectively, using a combi-nation of both scotch-tape and hot exfoliation. Subse-quently, the thickness of the flakes was confirmed us-ing atomic force microscopy (AFM) and Raman spec-troscopy. Further, the cleanliness of the individual flakeswas qualitatively characterised using the surface rough-ness of flakes obtained from AFM line scans and the peakline widths obtained from Raman spectroscopy (Sup-plementary Section S2). These flakes were thereafterstacked into a van der Waals (vdW) heterostructure ona Si/SiO2 substrate using a dry-transfer technique thatinvolves coating a PPC (Polypropylene Carbonate) filmon a hemispherical PDMS (Polydimethylsiloxane) drop.The transferred vdW heterostructure was etched into arectangular shape using e-beam lithography, followed byreactive ion etching using a mixture of O2 and CHF3gases. Finally, electrical contacts were deposited on theheterostructure via thermal evaporation of Cr (5 nm) andAu (60 nm).Transport measurementsThe electrical and thermal transport characterisationof the devices was carried out in a home-made He-4 Cryo-stat and in the s200 4K Cryostation (Montana Instru-ments), respectively. The temperature-dependent trans-fer characteristics were measured using a SR830 Lock-inAmplifier and a Keithley 2400 Source Meter. The dif-ferential conductance measurements were carried out us-ing an AC-DC mixer circuit, where the AC signal wassourced and measured using a Lock-in Amplifier, whilethe DC signal was sourced from a Keithley 2400 SourceMeter and measured using a Keithley 2182A Nanovolt-meter. The thermal transport measurements were car-ried out using a high-frequency compatible Johnson noisethermometry setup, whose schematic has been detailedin Supplementary Section S4.Solution of the heat diffusion equationThe DC electric field applied across the contacts ofthe device leads to the formation of a thermal gradientacross the device. The spatial distribution of the thermalgradient is ideally governed by the heat diffusion equa-tion. However, under experimental conditions, it can bereduced to the 1D Fourier’s equation, which is given byq = −κe∇Te(x) (4)where q is the rate of heat flow into the channel andTe(x) is the electron temperature at any point x alongthe direction of current flow.For our graphene device, the rate of power influx isequal to the power dissipated at the contacts.q = JWEL (5)where W and L are, respectively, the width and lengthof the channel.If we assume the thermal gradient to primarily be inthe direction of the applied electric field, the above equa-tion can be solved to get the following solution:Te(x) =√T 2c +2LLE2x for 0 ≤ x ≤ L/2√T 2c +2LLE2(L− x) for L/2 ≤ x ≤ Lwhere Tc is the temperature of the metallic electrode.The measured thermal noise gives us an average elec-tron temperature (Te), integrated over the entire surfacearea of the device and it is given byTe =2L3L2E2[(T 2c +L2LE2)3/2−(T 2c)3/2] (6)8Detailed derivation of this calculation has been pre-sented in Supplementary Section S10.DATA AVAILABILITYAll data files are available from the corresponding au-thor upon request.CODE AVAILABILITYThe code used for theoretical simulations present inthe Supplementary Information is available in the linkedGithub repository.ACKNOWLEDGMENTThe authors gratefully acknowledge the usage of theMNCF and NNFC facilities at CeNSE, IISc. The au-thors would also like to acknowledge fruitful discussionswith D. Sen, A. Lucas, S. Sachdev, A. Hui, B. Skin-ner, N. Trivedi, M. Randeria, B. Dora, R. Moessner, A.Green and S. Sondhi. A.G. acknowledges financial sup-port from a project under NanoMission, Department ofScience and Technology, India and J. C. Bose Fellow-ship. P.P. and Ak.G. thank the Ministry of Education,Govt. of India for the Prime Minister’s Research Fellow-ship (PMRF). K.W. and T.T. acknowledge support fromthe JSPS KAKENHI (Grant Numbers 21H05233 and23H02052) and World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.ETHICS DECLARATIONThe authors declare no competing interests.∗ aniketm@iisc.ac.in† smukerjee@iisc.ac.in‡ arindam@iisc.ac.in[1] D. Son, Quantum critical point in graphene approachedin the limit of infinitely strong coulomb interaction, Phys.Rev. B - Cond. Mat. and Mat. Phys. 75, 235423 (2007).[2] S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. 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B 106, 014205 (2022).10-0.40.00.4-10 60dV/dI (W)12 K200 K110 K50 K-0.40.00.4-0.40.00.4n (1012 cm-2)-10 -5 0 5 10-0.40.00.4J (A/m)020002000200-10 0 100200  12 K50 K110 K200 K   J (A/m)Extended Data Figure 1: Differential resistance measurements showing crossover from Knudsen toPoiseuille regime: (Left side) Colour plot showing the variation of dV/dI for D3S5 as a function of n and appliedelectrical current density (J) at four different T . (Right side) Line plots, obtained from the respective colour plotson the left, depicting the variation of dV/dI as a function of applied electrical current density (J) for two distinctnumber densities: n = 1010 cm−2 (marked in black) and n = 1011 cm−2 (marked in red).110 1000 200020120220Te (K)E (V m-1)19 K110 K60 K170 K260 Kab0 1000 2000 3000205080110140Te (K)E (V m-1)n ( 1011 cm-2)53210-0.4 0.0 0.40500R (W)n (1012 cm-2)Extended Data Figure 2: Electronic temperature across the graphene channel at low electric fields: (a)Te in D2S1 as a function of the applied electric field (E) at different electron densities, recorded at T = 19 K. Thedashed lines are theoretically fitted curves for the experimental data, following Eqn. 6. [Inset] Transfercharacteristics of the device D2S1 at 19 K, with colored dots indicating the resistance at the different n at which Tevs E data has been recorded. (b) Te as a function of the applied electric field (E) at different temperatures, forn = 5× 1011 cm−2.12-0.4 0.410-510-310-1101103-0.4 0.4 -0.4 0.4n (1012 cm-2)-0.4 0.4 -0.4 0.4 -0.4 0.4D2S119 K 30 K 60 K 110 K 170 K 260 KExtended Data Figure 3: Violation of WF Law in D2S1 at six different temperatures: Normalised Lorentzratio (L/LWF ) for D2S1, as a function of n for T = 19 K, 30 K, 60 K, 110 K, 170 K and 260 K. The solid lines aretheoretical fits for the experimental data, as per Eqn. 3.13-0.5 0.0 0.510-310-1101  19 K 110 K 160 K n (1012 cm-2)-3 0 3200400600Extended Data Figure 4: Carrier density- and temperature-dependence of the Lorentz number in D1S1:Normalised Lorentz ratio (L/LWF ) for D1S1, as a function of n for three different T . The solid lines are theoreticalfits for the experimental data, as per Eqn. 3. Inset shows the transfer characteristics of D1S1 at T = 40 K. They-axis shows the electrical resistance in ohms while the x-axis shows the applied gate voltage Vg in volts. Furtherelectrical characterisation of D1S1 indicating its intrinsic charge inhomogeneity and mobility are mentioned inTable I in Supplementary Information.1420 50 100 2500.1 h-doped e-dopedn0 (1012 cm-2)T (K) T3/220 50 100 250110100T (K)TimpabExtended Data Figure 5: Characteristic scales of Lorentz number variation with density andtemperature in D2S1: (a) The characteristic density scale n0 as a function of T, for the electron- and hole-dopedranges of D2S1. The solid lines show a T 3/2 fit of the experimental data points. (b) Normalised Lorentz ratio at theDirac point (L0/LWF ) of D2S1, as a function of T. The dashed line indicates the Timp for D2S1.1510 1003301010 100 10 100 10 100hth (10-19 J m-2 s)T (K)D3S5 D5S5 D1S6 D4S6Extended Data Figure 6: Non-monotonic temperature dependence of electron viscosity for additionaldevices: ηth vs T for four devices D3S5, D5S5, D1S6 and D4S6. The non-monotonic trend is consistent across allthe devices and approaches ∼ T 2 for T > Timp. The black solid lines in each panel represent T 2 dependence.