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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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[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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nature physicshttps://doi.org/10.1038/s41567-025-02972-zArticleUniversality in quantum critical flow of charge and heat in ultraclean grapheneIn the format provided by the authors and uneditedSupplementary informationhttps://doi.org/10.1038/s41567-025-02972-zContents Page No.S1. The experimental strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2S2. Characterisation of cleanliness of our hBN-encapsulated graphene devices . . . . . . . . . 3S3. Calculation of the diffusive mean free path in our devices . . . . . . . . . . . . . . . . . . . . . . . . . 5S4. Thermal transport measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6S5. Equations for electron hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8S6. Electrical conductivity of hydrodynamic electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10S7. Violation of Wiedemann-Franz Law in lower mobility devices . . . . . . . . . . . . . . . . . . . . 12S8. Quantum critical conductivity for hydrodynamic electrons . . . . . . . . . . . . . . . . . . . . . . . 15S9. Relation between the shear viscosity and enthalpy density . . . . . . . . . . . . . . . . . . . . . . . .16S10. Thermal modelling of the hBN-encapsulated graphene devices . . . . . . . . . . . . . . . . . . . .171S1. The experimental strategy: Real graphene Ideal graphenengraphene incompressibleto external gate electric fieldnminabTF/Tgraphene incompressibleto external gate electric fieldaccessible only in very clean devicesGate voltage tunable(compressible) regime0 1-1Gate voltage tunable(compressible) regimeFigure S1. Carrier density dependence of electrical and thermal conductivity in practical graphenedevices: (a) The red-coloured curve represents an ideal graphene in which electrical conductivitynear the quantum critical Dirac point approaches universal conductivity σQ. The black-colouredcurve represents a realistic graphene device. The presence of a finite amount of charge inhomogene-ity nmin (depicted by the vertical black dashed lines) causes σ to saturate to σmin (depicted by ahorizontal black dashed line) for n ≤ nmin and hence, σQ falls inside an experimentally inaccessibleregion (denoted by yellow stripes). (b) Our experimental strategy relies on analysing the behaviourof the quantity κeσe/vFlmrsth as a function of TF/T , which is denoted by the black curve. Theyellow striped region, where TF/T = T 0F/T < 1 where T 0F = h̄vF√πnmin/kB remains experimentallyinaccessible, but the red striped region (T 0F ≤ TF ≤ T ) becomes accessible in ultra-clean devicesfor the evaluation of σQ.2S2. Characterisation of cleanliness of the hBN-encapsulated graphene devices:Experimental realisation of electron hydrodynamics in a solid-state system requires avery high degree of cleanliness because disorders and impurities tend to relax the electronmomentum, suppressing the fluidic nature of electron transport. Previous experimentalefforts to explore electron hydrodynamics have also suffered largely due to this reason. Overthe past decade, graphene has emerged as a viable candidate where one can achieve asignificantly high degree of cleanliness. Even then, recent experiments had to mainly resortto constriction-based and Corbino geometries, in the absence of disorder-free fabricationprocedures. In this scenario, in order to reach “ultra-clean”, we have adopted the followingstrategies:1. Surface topography scans of the hBN flakes: Hexagonal boron nitride is considered tobe an ideal substrate for graphene-based 2D transistors [1]. However, reports [2] haveconfirmed the presence of inhomogeneity and tape residue on the surface of thick hBNflakes (> 5 nm), originating during exfoliation, which can give rise to defect states,acting as traps during electrical transport. These surface contaminants are usuallydistributed non-uniformly over the flake and can be easily detected by scanning thetopography using atomic force microscopy (AFM). The hBN flakes chosen for devicefabrication (Fig. S2a, for example) were scanned for regions with minimal surfaceinhomogeneity as shown in Figs. S2b and S2c. The highlighted portion of the AFMline scans in Fig. S2c has a surface roughness of (0.2 ± 0.07) nm. These “flat” portionsof the hBN flakes are homogeneous and clean, and have been selectively used forplacing the graphene layer on top.2. Spectroscopic investigation of defects in graphene: In case of graphene, the layernumber and spatial distribution of defects can be easily identified using Raman spec-troscopy [3]. One such spectrum is depicted in Fig. S2d, for the graphene flake shownin Inset. The G and 2D peaks are clearly visible at wavenumbers 1581 cm−1 and2689 cm−1 respectively (Fig. S2e), which indicate a very low degree of intrinsic chargeinhomogeneity in graphene [4]. Further, the Full Width at Half-Maximum (FWHM)of the G and 2D peaks (15 cm−1 and 26 cm−1 respectively, (Fig. S2e)) indicates aweak electron-phonon coupling, as expected in pristine monolayer graphene [5]. Also,3the absence of the D Raman peak and a low signal-to-noise ratio are considered to becrucial evidence for a clean monolayer graphene [4].-0.6 -0.3 0.0 0.3 0.612 After annealing Before annealingr (kW)n (1012 cm-2)1500 2100 2700Intensity (arb. units)n (cm-1)_G 2D020400 5 10 15 20 25 3002040Z (nm)Z (nm)X (mm)a b c10 µmGrhBN10 µmde1540 1580 1620 2650 2690 2730Intensity (norm.)n (cm-1)G_ _n (cm-1)2DfFigure S2. Cleaning procedures adopted during fabrication: (a) Optical micrograph of an exfo-liated hBN flake. (b) 3D AFM imaging of the hBN flake. (c) Height profile of the hBN flake attwo different line cuts, indicated by red and black dashed lines in (b). The highlighted portion ofthe line scans has a surface roughness of ≃ 0.2 nm. (d) Raman spectrum of a pristine monolayergraphene flake shown in Inset. (e) Lorentzian fitting of the G and 2D Raman peaks to calculate theexact position of the two peaks, along with their respective linewidths. (f) Transfer characteristicsof device D1S2 before and after low-temperature current annealing.3. Viscoelastic dry transfer method using a van der Waals pick up layer: Once cleanflakes have been procured, and their cleanliness has been verified, the next key step isthe systematic transfer of the layers into a stack without introducing any additionalimpurities, organic residues or surface contaminants in the process. Dry transfer tech-niques are well known for introducing minimal contamination in the “pick-up anddrop-off” process. Further, using a hBN layer for picking up the stack [6] is currentlyone of the most effective ways for achieving a clean transfer of the exfoliated layers.Raman spectra of the graphene layer in the assembled van der Waals heterostructure4also showed low signal-to-noise ratio, sharp G and 2D peaks and an absence of D peak,confirming the cleanliness of the transferred graphene flake.4. Low-temperature current-induced annealing: Some of the fabricated devices exhib-ited low electron mobility, and hence, electron-electron scattering was suppressed byelectron-impurity scattering in those devices. To improve the carrier mobility, suchdevices were subjected to current annealing at low temperatures (T ∼ 20 K). Thedevices were annealed to a maximum current of 50 µA by increasing in steps of100 nA. Passing such a large current helps in removing any unwanted dopants from thegraphene-hBN interface and other adsorbates, if any [7]. Device D1S2 was subjectedto low-temperature current annealing - the transfer characteristics before and afterthe annealing process are shown in Fig. S2f. The mobility of the device at T = 10 Kimproved from 250, 000 cm2V−1s−1 to 650, 000 cm2V−1s−1, after the annealing process.S3. Calculation of the diffusive mean free path in our devices:Our ultra-clean devices exhibit a range of variation in their n dependence on electricalconductivity σ, i.e. σ ∼ nα with α ∈ [0.5, 2] (Fig. S3a). This variation owes its origin to the-100 -10 10 100110100 D1S1 D2S1 D1S5 D3S5s/sminn (109 cm-2) n2 nCNP-0.5 0 0.5110-0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5l mfp (mm)n (1012 cm-2)D2S1 D1S5 D1S4D3S5 D5S5a bFigure S3. Mean free path of electrons in hydrodynamic devices: (a) Normalised electricalconductivity of four devices as a function of n at T = 110 K. The red and blue dashed linesserve as guides to the eye. (b) Diffusive mean free path lmfp as a function of n for five devices atT = 110 K. The dashed lines indicate the widths of the respective electronic channels.5dominant nature of scattering affecting electrical transport, e.g. α = 0.5, 1, 2 respectivelyindicate ballistic, diffusive and viscous electronic conduction. From the σ − n data, wecalculate the differential field-effect mobility for the electronic channels in our devices which,as per Drude’s model, is given byµ =1e∣∣∣∣dσdn∣∣∣∣ (1)and the diffusive mean free path (lmfp) which is given bylmfp =hσ2e2kF(2)The diffusive mean free paths for a few of the measured devices, as a function of n, atT = 110 K are shown in Fig. S3b. This confirms that lmfp ≈ W for most of our devices andhence, we can easily consider the momentum relaxation length lmr to be equal to the widthW of the channel for most of our devices.S4. Thermal transport measurement circuit:The thermal transport measurements were carried out using a circuit, whose schematichas been detailed in Fig. S4a. For this measurement, the device was bonded onto a PCBmade of FR-4 dielectric, housing an impedance-matching LC tank circuit in coplanar waveg-uide geometry. The chosen inductor (L = 220 nH) and capacitor (C = 3.9 pF) result in aresonant frequency of fres ≃ 12π√LC≃ 100 MHz across the experimental range of tempera-tures (20−300 K). The resonant frequency of the tank circuit was experimentally confirmedusing a Keysight E5071C Vector Network Analyser (Fig. S4b and S4c). The impedance-matched RF signal, carrying the thermal noise from the device, was extracted from thecryostat via RG-316 cables and underwent a series of analog and digital processing steps atroom temperature. It started with 3 stages of low-noise amplification (ZFL-500LN+ andMITEQ-AU 1291), after which the amplified signal passed through an RF mixer (ZLW-3+),which was fed using a Keysight N5173B signal generator, followed by a low-pass filter (SLP-1.9+) with a cut-off frequency of 1.9 MHz. The resulting signal is then passed throughan RF-matched Schottky diode (DMS-104P) which integrates the signal strength over thefrequency range and converts the RF power into a DC voltage. The resulting DC voltage,as observed using the Digital Multimeter (Keithley 2002) can be written asVdc = c⟨V 2⟩R= c4GLkBTR∆f(1− Γ2)R= 4cGLkBT∆f(1− Γ2) (3)6Table I. List of all the measured samples: Details specifying the channel dimensions, the thicknessof the bottom hBN layer, intrinsic charge inhomogeneity, differential field-effect mobility (at n =1011 cm−2 and T = 300 K) and diffusive mean free path (at n = 1011 cm−2 and T = 300 K)Sample L (µm) W (µm) thBN (nm) nmin(0) (1010 cm−2) µ (105 cm2/V.s) lmfp (µm)D1S1 2.2 4.6 22 3.1 0.6 0.5D2S1 2.1 4.2 22 1.48 2.2 3D3S1 2.2 4.6 22 1.23 7.3 6D1S2 1.9 5.2 30 1 0.9 1.12D1S4 1 6.7 24 0.98 3.74 3.93D4S4 0.6 0.8 24 1.02 0.2 0.18D1S5 1.4 8.2 27 0.71 15.4 15.11D2S5 1 9.1 27 1.78 0.4 0.3D3S5 1.1 4.6 27 0.84 4.3 3.85D5S5 1.2 5.7 27 0.92 2.2 1.84D1S6 1.4 3.4 25 0.99 0.62 0.46D4S6 1.1 2.4 25 1.1 0.57 0.71D5S6 1 1.9 25 1.6 0.2 0.16D1S7 3 10 27 0.69 0.25 0.14D2S7 3 10 27 0.62 0.2 0.18D3S7 2.5 10 28 0.99 0.31 0.15where G is the amplification factor, L is the conversion loss through the mixer andfilter, c is the power-to-voltage conversion factor of the diode, ∆f is the bandwidth of themeasurement, R is the two-terminal resistance of the device under test, Γ2 = S11 is thereflectance coefficient and T is the electron temperature. Now, the device under test has atwo-terminal resistance (R) has 2 contributions: one from the channel resistance Rch andone from contact resistance Rc, and is given by R = Rch + 2Rc. Since Rch is the electricalresistance of interest, Eqn. 3 is modified toVdc = 4cGLkB∆f(1− Γ2)(RchTch + 2RcTcRch + 2Rc)(4)where Tch and Tc are respectively the electron temperatures of the channel and the con-7tacts. We further assume a cold phonon bath model for our calculations, which meansthat heating the channel using a DC bias does not change the temperature of the contacts,significantly [8].S5. Equations for electron hydrodynamics:The standard expression for the electrical conductivity in the Drude model isσ =ne2τm(5)where n is the carrier density, τ the relaxation time andm the mass of the carriers as obtainedfrom band structure. Note that the τ , which appears in this expression is the bulk momentumrelaxation time. In the hydrodynamic regime of electrical transport, the shear viscosity alsocontributes to the conductivity rendering the above expression incomplete. Further, for asystem like graphene with a linear dispersion, there is no effective mass coming from bandDMMSample LC tank circuitLow Noise Amplifier RF Mixer Low Pass FilterRF-matched Schottky DiodeLocal Oscillator-9-6-303660 80 100 120f (MHz)Vg (V)-20-15-10S11 (dB)60 80 100 120-30-20-100S11 (dB)f (MHz) Vg = 4 V 85.93 MHzab cFigure S4. Measurement of Johnson-Nyquist noise: (a) Schematic of the circuit for measuringJohnson-Nyquist noise. (b) Colour plot of the reflectance coefficient S11 of the device at T = 20 K,as a function of signal frequency (f) and applied gate voltage (Vg). (c) S11 as a function of f atVg = 4 V.8structure. To incorporate the effects of the shear viscosity and also figure out what replacesthe band effective mass in general case, we consider a hydrodynamic description of transportbelow.We start with the expression for the stress tensor of a relativistic fluid [9]. The treatmentdeveloped below will also be applicable to a non-relativistic fluid but it is most convenientto think of it as the large rest mass limit of a relativistic fluid. The stress tensor isT µν = pηµν +(p+ ϵ)c2UµU ν (6)Here p is the pressure of the fluid, ϵ, its energy density including the rest mass energy andηµν is the metric tensor of special relativity. c is the speed of “light”, which we will later setequal to the dispersion velocity of graphene (vF). For the time being, we will call it c to beconsistent with the notation of most papers and books. The relativistic vector U has thecomponents (U0,U), where the space-like componentU =v√1− v2/c2(7)and the time-lime componentU0 =c√1− v2/c2(8)Here v is the velocity of the fluid. Further, the ith component of the momentum densityπi = T 0i/c. Here i takes on the values 1 and 2 (for a 2D system), which correspond to thex and y directions. We assume that v ≪ c, which is expected to be true for typical flowvelocities in graphene and in this limit, Eqns. 6, 7 and 8 give πi = (p + ϵ)vi or in morefamiliar notationπ =(p+ ϵ)c2v =Hc2v (9)The quantity H = p + ϵ is the enthalpy density (i.e. the enthalpy per unit volume). In themore familiar case of a non-relativistic fluidπ = ρv (10)where ρ is the density of the fluid. Thus, more generally H/c2 acts as the mass density ofa fluid. In particular for the case of the non-relativistic fluid (i.e. a fluid in which the the9rest mass energy of the particles is much higher than any other energy scale), H/c2 ≈ ρand we get Eqn. 10. We will see below that H/c2 replaces ρ in all the usual expressions weencounter in the context of transport.Having obtained the appropriate expression for π, let us now focus on the Navier-Stokesequation for the fluid. In particular, we are interested in steady-state laminar (low Reynoldsnumber) flow solutions due to an applied electric field E. The relevant equation for this is−η∇2v = neE (11)where η is the shear viscosity of the fluid and n its number density. The above equation isfor the flow of a fluid with strict momentum conservation in the bulk. In the presence ofmomentum relaxation in the bulk, the equation becomes−η∇2v = neE− πτ= neE− Hvv2Fτ(12)Here, τ is the bulk momentum relaxation time and vF is the Fermi velocity of graphene.S6. Electrical conductivity of hydrodynamic electrons:We now focus on a geometry in 2D in which the flow is along the x direction in a channelof width W . Eqn. 12 can be solved in this geometry with no-slip conditions at the channeledges. The solution isvx(y) = −env2FτH1− cosh(√Hv2Fητy)cosh(√Hv2FητW2)E (13)with y = 0 at the centre of the channel. The total currentI = −en∫ W/2−W/2vx(y)dy (14)=e2n2v2FτH[W − 2√v2FητHtanh(√Hv2FητW2)]E (15)The conductivity is thusσ =e2n2v2FτH1− tanh(√Hv2FητW2)√Hv2FητW2 (16)10In the limit in which there is perfect momentum relaxation in the bulk (i.e. τ → ∞), Eqn. 16reduces toσ =e2n2W 212η(17)In the other limit, in which there is no effect of the shear viscosity (i.e. η → 0),σ =e2n2v2FτH(18)which is just the usual expression obtained from the Drude model with the quantity H/(v2Fn)playing the role of the mass of the charge carrier. From the forms of Eqns. 16 and 17, wecan define an effective shear viscosity ηeff when both the effects of viscosity and momentumrelaxation are present, which isηeff =W 2H12v2Fτ1− tanh(√Hv2FητW2)√Hv2FητW2−1(19)so that the conductivityσ =e2n2W 212ηeff(20)In the limit τ → ∞, ηeff = η as expected whereas in the opposite limit η → 0,ηeff =W 2H12v2Fτ(21)to yield Eqn. 18. Note that the conductivity obtained from all four Eqns. 16, 17, 18 and 20goes to zero as n goes to zero. While this seems intuitively obvious (there should be no trans-port if there are no carriers), it is not quite correct in the case of a system like graphene withparticle-hole symmetry and a charge neutrality point (CNP). When the chemical potentialis at the CNP, the net carrier density is indeed zero. However, electrons and holes (which areof equal density and hence combine to give n = 0) can individually have non-zero densities.Each species can thus carry a current which can be degraded giving rise to an electricalconductivity σmin at the CNP even when n = 0. The total conductivity is thus the sum ofσmin and the expression obtained from Eqn. 16 (or equivalently Eqn. 20),σ = σmin +e2n2W 212ηeff(22)11and in the limit η → 0, this becomesσ = σmin +e2n2vFlmH(23)with the momentum relaxation length lm = vFτ , which is the expression that has been usedto obtain H assuming that lm = W (which is the condition for Knudsen flow). Note thatthe above expression is obtained only in the limit η → 0 or more correctly,η ≪ W 2H4vFlm(24)S7. Violation of Wiedemann-Franz Law in lower mobility devices:The violation of the WF law in disordered monolayer graphene may not necessarily implythe existence of a hydrodynamic Dirac fluid near the charge neutrality point. Violationsoriginating due to the presence of disorder [10, 11], band gap opening at the Dirac point[12], bipolar diffusion [13], etc. have similar orders of magnitude as those observed in thelow-mobility graphene devices considered here. As a result, the origin of the WF violationin these devices needs to be understood properly. Among the devices listed in Table 1, D1S1is the most disordered, with nmin(0) ≈ 3 × 1010 cm−2 and a detailed study of this deviceis expected to provide valuable information to resolve the origin of WF violation in thesedisordered graphene systems.To understand the role of electron-impurity scattering in this scenario, we construct aphenomenological model based on a combination of short- and long-range impurity scatteringprocesses. Short-range scattering is independent of the electronic energy (ϵ) and mainlyincludes neutral defects with a delta function potential. In contrast, long-ranged impurityscattering involves the Coulombic interaction between electrons and charged impurities.The resulting relaxation time for such a combination of scattering processes follows theMatthiessen rule and is given by (up to some overall constant):τ−1(ϵ) = γ|ϵ|+ 1− γ|ϵ|(25)Here, γ = 1 corresponds to purely short-range (and acoustic phonon) scattering and γ =0 corresponds to purely long-range scattering. Although γ usually exhibits a non-trivialtemperature dependence, this can be ignored for studying the qualitative behaviour of L.12-0.3 0.0 0.3036 Data at 100 K  Phenomenological model Fit to hydrodynamic modeln (1012 cm-2)-0.4 -0.2 0.0 0.2 0.41357  Long range (D = 0 meV) Long range (D = 20 meV) Short range (D = 0 meV) Short range (D = 20 meV)n (1012 cm-2)-50 0 50EF (meV)030303-0.3 0.0 0.3010n (1012 cm-2)40 K57 K90 K100 KabcΤℒ ℒ𝑊𝐹 = 1Figure S5. Fitting the experimental data to a Dirac fluid model and a phenomenological diffusivescattering-based model: (a) Theoretically calculated L/LWF vs n for long-range and short-rangeinteractions, both in the presence (∆ = 20 meV) and absence (∆ = 0) of a small band gapat the Dirac point. (b) Comparison of the experimental L/LWF vs n data for D1S1 at 100 Kwith theoretical predictions of the phenomenological model Eqn. 25 and the hydrodynamic modelEqn. 30. (c) L/LWF vs n for D1S1 at different temperatures. The solid circles represent theexperimental data points, the solid line represents the fit as per the hydrodynamic model Eqn. 29,and the dashed line represents the fit as per the linear-density corrected model Eqn. 30. The R2values for the fits of Eqn. 29 to the data, respectively, vary between 90− 95 %.Within the relaxation-time approximation, the electrical and thermal conductivities aregiven by:σ = e2∫dϵ(− ∂f 0∂ϵ)g(ϵ)v2(ϵ)τ(ϵ) (26)κ =1T∫dϵ(ϵ− µ)2(− ∂f 0∂ϵ)g(ϵ)v2(ϵ)τ(ϵ) (27)where g(ϵ) is the electronic density of states, µ = EF is the chemical potential, v(ϵ) is thedispersion velocity, and f 0(ϵ) is the equilibrium distribution function. By using the graphenedispersion relation ϵ = h̄vk, we get the respective transport coefficients, which, in turn, yieldthe Lorentz number L for different carrier densities. Fig. S5a shows the variation of L/LWF as13a function of n for different combinations of scattering mechanisms. In the presence of purelyshort-ranged scattering processes, the Wiedemann-Franz law remains valid for all values ofn since the scattering process is independent of the electronic energy. On the other hand,long-ranged Coulomb scattering results in an enhancement of L/LWF above 1 near the DiracPoint. However, the saturation of L/LWF to 1 at large n owes its origin to the dominantcontribution from short-ranged scattering for ϵ ≫ ϵc =√1− γγin Eqn. (25). Thus, forµ > ϵc, the integrals in Eqn. (26) and Eqn. (27) are approximately the same as the coefficientsin the presence of short-ranged scattering, restoring the Lorentz ratio to the value dictatedby the Wiedemann-Franz law. In addition to impurity scattering events contributing to WFviolation, we also note that in graphene-based vdW heterostructures, the presence of defectsand interfacial disorder [14] as well as strain caused by the encapsulating hBN substrate [15]can open up a gap at the Dirac point. This band gap [12] can be responsible for a non-trivialvariation of L/LWF with n, usually characterised by an enhancement at low densities. Theexperimentally obtained band gap for device D1S1 is ≃ 23 meV, which has also been usedfor the theoretical calculation of L/LWF in Fig. S5a, assuming the simple model proposedin [12], where the density of states g(ϵ) for the conduction (+) and valence (-) band is givenby the same linear spectrum now displaced by the gap ∆.g±(ϵ) =2πh̄v2F|ϵ∓∆/2| (28)As a function of carrier density, L/LWF for D1S1 exhibits a range of values similar towhat is expected from a combination of short- and long-range scattering mechanisms. How-ever, certain features in the data cannot be explained by the above-mentioned mechanisms,as can be observed in Fig. S5b. Firstly, any combination of short- and long-range scatteringprocesses leads to a simultaneous enhancement and suppression of L/LWF as a function ofn with the dip reaching a minimum value of ∼ 0.8 LWF, but the experimental data for D1S1shows that L can be as low as 0.01 LWF at high densities. Secondly, the experimental L/LWFfor D1S1 does not saturate to 1 for higher values of n (>∼ 1015 m−2). Such saturation wasobserved in the behaviour of L in Ref. [16] and the authors attributed it to the absence ofhydrodynamic electrons at high densities. Both these factors point towards the incompati-bility of our phenomenological model in explaining the experimental data for D1S1, whichis further exemplified by the quality of fit in Fig. S5b. Lastly, opening up a band gap atthe Dirac point can partially account for the enhancement of L/LWF near n = 0 but this14mechanism does not allow for any suppression of L/LWF below 1. Hence, this mechanismalso is not the right choice for the origin behind the WF violation in D1S1.We also believe that bipolar diffusion (BD) of electrons and holes does not play a signifi-cant role in the observed non-trivial variation of L/LWF for D1S1. Previous reports [12, 16]show that in the presence of BD, there is an additional contribution to the electronic thermalconductivity, thereby resulting in an enhancement of L/LWF. However, this mechanism doesnot provide any avenue for a possible suppression of L/LWF below 1 at high densities.Finally, we fit the observed variation of L/LWF as a function of n at different tempera-tures, for device D1S1, to Eqn. 3 from the main text, given by:L/LWF =a[1 +(nb)2]2 (29)where a and b are fitting parameters. The fit is shown in Figs. S5b and S5c. We obtaina significantly high coefficient of determination (R2), indicating the presence of electronhydrodynamics rather than diffusive scattering even in D1S1. We further observe a consistentunderfitting of the observed data for higher densities, indicating the presence of finite densitycorrections to the terms evaluated at the critical point. Motivated by the conductivitybecoming linear for large n, we introduce a linear finite density correction to the enthalpydensity, giving the new fit as:L/LWF =a (1 + c|n|)1 +(nb)21 + c|n|2 (30)where c is an additional fitting parameter. The inclusion of this additional parameter im-proves the fit at large densities, as shown by the dashed curves in Fig. S5c for differenttemperatures. Thus, we expect D1S1 to remain in the hydrodynamic regime, albeit deviateaway from quantum criticality as we stray further away from the charge neutrality point.S8. Quantum critical conductivity for hydrodynamic electrons:The expression for the electronic thermal conductivity κe from the general theory of crit-ical hydrodynamic transport can be obtained as follows. The expression for the electricalconductivity at zero frequency and zero magnetic field is [17]σ = σQτmr (ψ + 1/τmr) (31)15whereψ =4e2ρ2v2FσQHand τmr is the momentum relaxation time, ρ, the density of carriers, vF, the Fermi velocityand H is the enthalpy density. The quantum of conductanceσQ = Φσ4e2h(32)where Φσ is a universal dimensionless number that depends only on the universality class ofthe critical point. The electronic thermal conductivity at zero frequency and zero magneticfield is given by the expression [17]κe =1Φσh4e2HσQv2Fψ + 1/τmr(33)Combining Eqns. 31, 32 and 33, we getκe =σQHvFlmrTσ(34)which is Eqn. 2 of the main text. Here the momentum relaxation length lmr = vFτmr.S9. Relation between the shear viscosity and enthalpy density:Within standard kinetic theory, the shear viscosity of a non-relativistic 2D fluid isη =12nmv̄l (35)where n is the density, m, the mass of the fluid particles, v̄, their mean speed and l, themean free path for collision among the fluid particles. For a relativistic fluid, this becomesη =Hl2vF(36)where we have substituted H/v2F for the mass density nm as per the argument after Eqn. 10and also set v̄ = vF. Thus, H obtained from the experimental measurements using Eqn. 23can be substituted in Eqn. 36 to obtain η provided l is known and the consistency condition 24is satisfied. In fact the consistency condition can be checked after η has been obtained thisway. Using Eqn. 36, the consistency condition can also be written asW ≫√2llm (37)16which only involves the three relevant length scales W , lm and l. Further if lm = W , thecondition simply becomes W ≫ 2l. The important quantity here is thus l, which, for theelectron fluid, becomes the momentum-conserving electron-electron scattering length. Fora Dirac fluid of electrons (i.e. when the Fermi energy is “very close” to the CNP), this hasbeen argued to be of the form,l ∼ h̄vFα2kBT(38)where α = e2/(h̄vF) is the “effective” fine structure constant. Substituting, Eqn. 38 inEqn. 36, we getη ∼ Hh̄α2kBT(39)Let us now consider the enthalpy densityH = ϵ+ p = µn+ Ts (40)from the Gibbs-Duhem relation, where µ is the chemical potential and s, the entropy density.At the CNP, µ = 0, n = 0 and thus, H = Ts at all temperatures. Further s ∼ kB(kBTh̄vF)2,which implies thatH =(kBT )3h̄2v2F(41)substituting which into Eqn. 39, we find thatη ∼ (kBT )2 h̄e4(42)at the CNP. Note that vF does not appear in this expression, which is fortuitous since vFgets renormalised to infinity at the CNP.S10. Thermal modelling of the hBN-encapsulated graphene devices:The flow of Joule heat within the device can be understood using a simple toy model. Toachieve thermal equilibrium, the hot electrons in graphene can undergo thermal relaxationvia four different processes -• out-diffusion through the metallic electrodes,• interaction with the graphene phonons,17• conduction through the adjacent hBN layers, and• radiative infrared emission into the environmentUsing the heat diffusion equation, this can be expressed asQ̇(r⃗) = −∇⃗ ·[κe∇⃗Te(r⃗)]+ AΣ[T 4e (r⃗)− T 4ph]+ Psubstrate + Prad (43)where Q is the thermal charge injected into the channel due to Joule heating, κe is thethermal conductivity, Te(r⃗) is the electron temperature at any point denoted by r⃗, Tph isthe lattice temperature, A is the sample area, Σ is the electron-phonon coupling constant,Psubstrate is the thermal power lost to the substrate and Prad is the radiated thermal powerdensity.Now, the thermal energy leaking into the electromagnetic environment [18] formed bythe electrical measurement system via the emission and absorption of blackbody photonsis given by the thermal radiative conductance, Grad =π2kBB3where B is the bandwidthof measurement. Usually for a measurement bandwidth of 4 MHz, Grad ∼ 10−15 W/K andhence, Grad and subsequently, Prad can be neglected in Eqn. 43.Further, due to the weak electron-phonon coupling of graphene [3], the heat dissipated bythe electronic subsystem to the lattice is quite small and, for |Te − Tph| ≪ Tph, the thermalconductance of the electron-phonon channel is reduced to Gep = 4AΣT 3ph, which can also beneglected in comparison to the heat reaching the metallic electrodes.In addition to this, the hBN substrate used in our experiment has a much larger surfaceoptical phonon energy (102 meV) and hence can influence the electron cooling process athigh Te but can be neglected in this case since experimentally Te never exceeds 700 K[19]. Also the hyperbolic phonon-polariton modes of hBN, responsible for dissipating heataway from the graphene electrons, have two Restrahlen bands around 100 meV [20], whichindicates that these modes also do not take away heat from the graphene channel under theexperimental conditions described in our paper. Hence, we can conclude that heat leakagethrough the hBN substrate (Psubstrate) can also be neglected.Further, assuming heat flow predominantly only along the direction of current flow, wecan rewrite Eqn. 43 as the 1D Fourier’s equationq = −κe∇Te(x) (44)18where q is the rate of heat flow into the channel and Te(x) is the electron temperature atany point x along the direction of current flow.Under thermal equilibrium, the total heat injected into the graphene channel is given byq = JWEL (45)Further, as per Wiedemann Franz Law, we also haveκe = LσTe(x)dTe(x)dxW (46)This leads to the following equality:12ddx[T 2e (x)]= E2LL(47)Since the metallic electrodes have a much higher thermal conductivity, they can be con-sidered to have the same temperature Tc despite the Joule heating in the channel. Hence,Te(x = 0) = Te(x = L) = Tc and the temperature profile is symmetric about x =L2.Thus, Eqn. 47 gives∫ Te(x)Tcd(T 2e (x))=2LLE2∫ x0dx⇒ T 2e (x)− T 2c =2LLE2f(x) (48)where f(x) = x when x ∈ [0, L/2] and f(x) = L− x when x ∈ [L/2, L].This leads to a temperature profile which can be written as:Te(x) =√T 2c +2LLE2x for 0 ≤ x ≤ L/2√T 2c +2LLE2(L− x) for L/2 ≤ x ≤ LThe thermal profile peaks at x = L/2, where the maximum temperature is given byTmaxe (x) = Te(x = L/2) =√T 2c +L2E2L(49)The measured thermal noise gives a non-weighted spatial average of the electronic tem-perature (Te) along the channel length.Te = ⟨Te(x)⟩ =1L∫ L0Te(x)dx (50)19Solving the integral, we getTe =2L3L2E2[(T 2c +L2LE2)3/2−(T 2c)3/2](51)[1] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi,P. Kim, K. L. Shepard, et al., Boron nitride substrates for high-quality graphene electronics,Nat. Nanotechnol. 5, 722 (2010).[2] W. Wang, X. Ma, Z. Dai, S. Zhang, Y. Hou, G. Wang, Q. Li, Z. Zhang, Y. Wei, and L. Liu,Mechanical behavior of blisters spontaneously formed by multilayer 2D materials, Adv. Mater.Int. 9, 2101939 (2022).[3] A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec,D. Jiang, K. S. Novoselov, S. Roth, et al., Raman spectrum of graphene and graphene layers,Phys. Rev. Lett. 97, 187401 (2006).[4] Y.-C. Lin, C. Jin, J.-C. Lee, S.-F. Jen, K. Suenaga, and P.-W. Chiu, Clean transfer of graphenefor isolation and suspension, ACS Nano 5, 2362 (2011).[5] J. Yan, Y. Zhang, P. Kim, and A. Pinczuk, Electric field effect tuning of electron-phononcoupling in graphene, Phys. Rev. Lett. 98, 166802 (2007).[6] L. Wang, I. Meric, P. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. Campos,D. Muller, et al., One-dimensional electrical contact to a two-dimensional material, Science342, 614 (2013).[7] J. Moser, A. Barreiro, and A. Bachtold, Current-induced cleaning of graphene, Appl. Phys.Lett. 91 (2007).[8] M. L. Roukes, M. Freeman, R. Germain, R. Richardson, and M. Ketchen, Hot electrons andenergy transport in metals at millikelvin temperatures, Phys. Rev. Lett. 55, 422 (1985).[9] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theoryof Relativity (Wiley, 1972).[10] R. Raimondi, G. Savona, P. Schwab, and T. Lück, Electronic thermal conductivity of disor-dered metals, Phys. Rev. B - Cond. Matt. and Mat. Phys. 70, 155109 (2004).[11] K. Michaeli and A. M. Finkel’stein, Quantum kinetic approach for studying thermal transport20in the presence of electron-electron interactions and disorder, Phys. Rev. B - Cond. Matt. andMat. Phys. 80, 115111 (2009).[12] Y.-T. Tu and S. D. Sarma, Wiedemann-Franz law in graphene, Phys. Rev. B 107, 085401(2023).[13] H. Yoshino and K. Murata, Significant enhancement of electronic thermal conductivity of two-dimensional zero-gap systems by bipolar-diffusion effect, Journ. of the Phys. Soc. of Jap. 84,024601 (2015).[14] M. Evaldsson, I. V. Zozoulenko, H. Xu, and T. Heinzel, Edge-disorder-induced Andersonlocalization and conduction gap in graphene nanoribbons, Phys. Rev. B - Cond. Matt. andMat. Phys. 78, 161407 (2008).[15] J. Jung, A. M. Da Silva, A. H. MacDonald, and S. Adam, Origin of band gaps in grapheneon hexagonal boron nitride, Nat. Commun. 6, 6308 (2015).[16] J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim, A. Lucas, S. Sachdev, P. Kim,T. Taniguchi, K. Watanabe, et al., Observation of the Dirac fluid and the breakdown ofthe Wiedemann-Franz law in graphene, Science 351, 1058 (2016).[17] S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, Theory of the Nernst effect nearquantum phase transitions in condensed matter and in dyonic black holes, Phys. Rev. B 76,144502 (2007).[18] J. B. Pendry, Quantum limits to the flow of information and entropy, Jour. of Phys. A: Math.and Gen. 16, 2161 (1983).[19] A. Betz, S. H. Jhang, E. Pallecchi, R. Ferreira, G. Fève, J.-M. Berroir, and B. Plaçais, Super-collision cooling in undoped graphene, Nat. Phys. 9, 109 (2013).[20] K.-J. Tielrooij, N. C. Hesp, A. Principi, M. B. Lundeberg, E. A. Pogna, L. Banszerus, Z. Mics,M. Massicotte, P. Schmidt, D. Davydovskaya, et al., Out-of-plane heat transfer in van der waalsstacks through electron–hyperbolic phonon coupling, Nat. Nanotechnol. 13, 41 (2018).21