Emergent Phenomena in Quantum Hall Systems, Tata Institute for Fundamental Research, Mumbai January 9, 2016

Quantum phases of graphene

Subir Sachdev

Talk online: sachdev.physics.harvard.edu

HARVARD 1. Competing quantum orders in the zeroth Landau level of graphene

2. Dirac liquid in graphene: quantum matter without quasiparticles 1. Competing quantum orders in the zeroth Landau level of graphene

2. Dirac liquid in graphene: quantum matter without quasiparticles Junhyun Lee and Subir Sachdev, "Wess-Zumino-Witten Terms in Graphene Landau Levels," Physical Review Letters 114, 226801 (2015) Experimentally: single layer graphene at ν = 0 is an INSULATOR

Kharitonov (2012): Canted Anti-Ferromagnet Ferromagnet

c Ez = 2gxy

Ez cosψ = ψ 2gxy

A. F. Young et al., Nature 505 (2014): tilted magnetic field Figure: Efrat Shimshoni Conductance in bilayer graphene

• AB stacked bilayer graphene suspended in setup • Electric and magnetic ﬁeld perpendicular to the plane • ⌫ =0 ﬁlling

E B

R. T. Weitz et al., Science, 330, 812 (2010) P. Maher et al., Nature Physics 9, 154 (2013) Conductance in bilayer graphene

Phase I :

valence bond solid (VBS) or

Phase II :

R. T. Weitz et al., Science, 330, 812 (2010) P. Maher et al., Nature Physics 9, 154 (2013) Maxim Kharitonov, PRL, 109, 046803 (2012) Insulating ⌫ = 0 state in BLG at zero electric ﬁeld is ...

AF favored by anisotropy uz > u > 0: II. ) ? consistent with micro Partial layer polarized (PLP) = considerations - ok I. Interlayer-coherent (ILC) = Canted valence bond solid (VBS) 2 2 ✏V⇤ uz u Antiferromagnet Phase I : ⇡ ? ⇡ 20B q[T]K-ok (CAF) ? Maxim Kharitonov,linear B -dependence at ? PRL, 109, 046803B (20122T) - ok ? & ΕZ F 2!u!! Phase II : FLP"B# FLP"T# CAF PLP PLP

# # # 2 # 2 2 # 2 " Ε !u!! uz uz u! 0 uz u! !u!! uz V Conductance in bilayer graphene

Phase I :

Phase II :

R. T. Weitz et al., Science, 330, 812 (2010) P. Maher et al., Nature Physics 9, 154 (2013) Maxim Kharitonov, PRL, 109, 046803 (2012) Conductance in bilayer graphene

• What is the nature of this transition? • What is the origin of the conducting transition point?

E B

R. T. Weitz et al, Science, 330, 812 (2010) Main result

Neel Valence bond solid z x 1 = ⇢ x z y 4 = ⇢ 2 = ⇢ y z z 5 = ⇢ 3 = ⇢ Main result

qN @c` 2 = c† d r n (r, ⌧) †(r, ⌧) (r, ⌧) L ` @⌧ a a X`=1 Z

5 component unit vector na(r, ⌧) couples as naa

Neel Valence bond solid z x 1 = ⇢ x z y 4 = ⇢ 2 = ⇢ y z z 5 = ⇢ 3 = ⇢ Main result =2⇡iq W [n ] SWZW a 3 1 W [n ]= du d2rd⌧✏ n @ n @ n @ n @ n a 8⇡2 abcde a x b y c ⌧ d u e Z0 Z

5 component unit vector na(r, ⌧) couples as naa

Neel Valence bond solid z x 1 = ⇢ x z y 4 = ⇢ 2 = ⇢ y z z 5 = ⇢ 3 = ⇢ q =1, 2 for mono-,bi-layer graphene Main result =2⇡iq W [n ] SWZW a 3 1 W [n ]= du d2rd⌧✏ n @ n @ n @ n @ n a 8⇡2 abcde a x b y c ⌧ d u e Z0 Z

W [na] is the Wess-Zumino-Witten term with a quantized coecient: it com- putes a Berry phase linking together spatial and temporal textures in the AF and VBS orders. It is a higher dimensional generalization of the Berry phase of a single spin S, which is equal to S times the area enclosed by the spin world-line on the unit sphere. Similarly, the WZW term measures the area on the surface of the sphere in the ﬁve-dimensional AF and VBS order parameter space.

q =1, 2 for mono-,bi-layer graphene Deconﬁned criticality • Second order transition between two ordered phases, different from Landau-Ginzburg-Wilson picture •Gappless ‘photon’ excitation of a emergent U(1) at critical point •Typical example is the Neel to VBS transition in square lattice

T. Senthil et al, Science, 303, 1490 (2004) Connection to deconﬁned criticality

O(5) non-linear sigma model + level 1 WZW term

CP1 model

O(5) non-linear sigma model + level 2 WZW term

CP2 model

T. Senthil and M. P. A. Fisher, Phys. Rev. B, 74, 064405 (2006) T. Grover and T. Senthil, Phys. Rev. Lett., 100, 156804 (2008) M. Levin and T. Senthil, Phys. Rev. B, 70, 220403(R) (2004) T. Grover and T. Senthil, Phys. Rev. Lett., 98, 247202 (2007) Experimental consequencces ~ • The AF Skyrmion N =( n 1 ,n 2 ,n 3 ) in bilayer graphene carries interlayer charge, and so there is a counterflow “critical superfluid”. • With a finite electric field, the interlayer symmetry is broken, and so there is enhanced conductivity, a vestige of the counterflow superfluidity.

R. T. Weitz et al, Science, 330, 812 (2010) 1. Competing quantum orders in the zeroth Landau level of graphene

2. Dirac liquid in graphene: quantum matter without quasiparticles 1. Competing quantum orders in the zeroth Landau level of graphene

2. Dirac liquid in graphene: quantum matter without quasiparticles Jesse Crossno Philip Kim

Kin Chung Fong Andrew Lucas Ordinary metals: the Fermi liquid Fermi surface separates empty • and occupied states in mo- Fermi mentum space. surface Area enclosed by Fermi sur- • face = . Momenta of low energyQ excitations ﬁxed by density of all electrons. ky Long-lived electron-like quasi- • particle excitations near the Fermi surface: lifetime of quasi- kx particles 1/T 2. ⇠ (Thermal conductivity) ⇡2k2 = B L • T (Electrical conductivity) 3e2 ⌘ 0 I in hydrodynamics one ﬁnds

 hydro = L , hydro 1. T (1 + ( / )2)2 L Q Q0

hence the Lorenz ratio, L, departs from the Sommer- at low temperatures the deviations are due to the feld value, L o inelastic nature of electron-phonon interactions. In some cases, a higher Lorenz number is due to the L- efT (4) presence of impurities. The phonon contribution to thermal conductivity sometimes increases the Lorenz The important scattering processes in thermal and number, and this contribution, when phonon electrical conduction are: (i) elastic scattering by solute Umklapp scattering is present, is inversely propor- atoms, impurities and lattice defects, (ii) scattering of tional to the temperature. The deviations in Lorenz the electrons by phonons, and (iii) electron-electron number can also be due to the changes in band interactions. In the elastic scattering region, i.e. at very structure. In magnetic materials, the presence of mag- low temperature, IE = IT and hence L = L 0. At higher nons also can change the Lorenz number at low temperatures, electron-electron scattering and elec- temperatures. In the presence of a magnetic field, the tron-phonon scattering dominate and the collisions Lorenz number varies directly with magnetic field. are inelastic. Then IE#l T and hence L deviates Changes in Lorenz number are sometimes due to from L o. structural phase transitions. In recent years, the Deviations from the Sommerfeld value of the Lorenz number has also been investigated at higher Lorenz number are due to various reasons. In metals, temperatures and has been found to deviate from the Sommerfeld value [14-20] and it is sometimes attributed to the incomplete degeneracy (Fermi 1 0 2 Ag /e smearing) [21] of electron gas. The Lorenz number Cu / o~ ,", has also been found to vary with pressure [-22, 23]. In alloys, the thermal conductivity and hence the Lorenz number have contributions from the electronic and lattice parts at low temperatures. The apparent Brass OV," Lorenz ratio (L/Lo) for many alloys has a peak at low "73 t- temperatures. At higher temperatures the apparent O C~ Sn o z~/v Fe Lorenz ratio is constant for each sample and ap- "~I01 proaches Lo as the percentage of alloying, x, increases. GerP~nPt Q~ z~/~z~ In certain alloys at high temperatures, the ordering 0) causes a peak in L/L o. > silvU The Lorenz number of degenerate semiconductors rr" also shows a similar deviation to that observed in Wiedemann-Franz metals and alloys. Up to a certain temperature, in- Dirac Fluid in Graphene elastic scattering determines the Lorenz number value, 26 ;< and below this the scattering is elastic which is due to 10 ~ I I I .I impurities. Supression of the electronic contribution Wiedemann-Franz0 o Law101 10 2 Relative electrical conductivity to thermal conductivity and hence the separation of the lattice and electronic parts of conductivity can be FigureI Wiedemann-Franzl Relative thermal conductivities, A, law measured in aby Fermidone by liquid:application of a transverse magnetic field and Wiedemann and Franz (AAg assumed to be = 100) and relative hence the Lorenz number can be evaluated. The devi- electrical conductivities, ~, measured by ( 9 Riess, (A) Becquerel, 2 2 ation of the Lorenz number in some degenerate semi- and (V) Lorenz. C~Agassumed to be - 100. After ⇡WiedemannkB and 8 W ⌦ Franz [1]. 2.45conductors10 is attributed· to .phonon drag. In some T ⇡ 3e2 ⇡ ⇥ K2

Carrier concentration (cm 3) 10 16 1017 1018 1 019 1 0 2~ 1 0 21 1 0 22 1 0 23 1 024 I ~ ' I I 1 ~ t I t ; I SiGe2 Cs AI Au Mg Bi ~'v 3"0 t SiBe3 /. / Nb~b.,~o/Cu Fi 9 i e ~ 2.5 O 2.0 Sb As Ag Ga

SiGez ~oe Se H~ Y W1 Cs2 Fe Ir Coz Li2 AUz Au3 AI2 All 30 \ c SiG% \." \\\!! c-N 2.5 i \ O SiGe1 , ,L T, oytw, p, t Feq z,~/ Cr '' \ X~., K Pt,\ Ni2 u A~" ! " 2.0 Yb ~o1 Er / / 2 As Sb ~ Rh g t31~ Bi2 BiI Cq

I I r I I t f I [ I I [ I i I t I I I I I I I 1 0 2 1 0 3 1 0 4 1 0 6 1 0 e 1 0 7 1 0 8 1 0 9 101o Electrical conductivity (~,~-1 cm-1)

Figure 2 Experimental Lorenz number of elemental metals in the low-temperature residual resistance regime, see Table I. Also shown are our own data points on a doped, degenerate semiconductor (Table III). Data are plotted versus electrical conductivity and also versus carrier concentration, taken from Ashcroft and Mermin [24] G.except S. Kumar,for the semiconductors. G. Prasad, and R.O. Pohl, J. Mat. Sci. 28, 4261 (1993) 4262 Graphene

kx Graphene

Electron Fermi surface Graphene

Hole Electron Fermi surface Fermi surface Graphene T (K) 600 Quantum critical 500 Dirac liquid 400 1 (1 + λ ln Λ ) 300 ∼ √n √n 200 Hole Electron 100 Fermi liquid Fermi liquid n -1 -0.5 0 0.5 1 1012/m2

D. E. Sheehy and J. Schmalian, PRL 99, 226803 (2007) M. Müller, L. Fritz, and S. Sachdev, PRB 78, 115406 (2008) M. Müller and S. Sachdev, PRB 78, 115419 (2008) Graphene Predicted T (K) strange metal 600 Quantum critical 500 Dirac liquid 400 1 (1 + λ ln Λ ) 300 ∼ √n √n 200 Hole Electron 100 Fermi liquid Fermi liquid n -1 -0.5 0 0.5 1 1012/m2

M. Müller, L. Fritz, and S. Sachdev, PRB 78, 115406 (2008) M. Müller and S. Sachdev, PRB 78, 115419 (2008) Key properties of a strange metal

No quasiparticle excitations • Shortest possible “collision time”, or • more precisely, fastest possible local ~ equilibration time ⇠ kBT Continuously variable density, • (conformal ﬁeld theories are usuallyQ at ﬁxed density, = 0) Q Graphene T (K) 600 Quantum critical 500 Dirac liquid 400 1 (1 + λ ln Λ ) 300 ∼ √n √n 200 Hole Electron 100 Fermi liquid Fermi liquid n -1 -0.5 0 0.5 1 1012/m2

D. E. Sheehy and J. Schmalian, PRL 99, 226803 (2007) M. Müller, L. Fritz, and S. Sachdev, PRB 78, 115406 (2008) M. Müller and S. Sachdev, PRB 78, 115419 (2008) Dirac Fluid in Graphene 28 J. Crossno et al. arXiv:1509.04713; Science, to appear Wiedemann-Franz Law Violations in Experiment Strange metal in graphene 100 20 90 phonon-limited 80 16 70

12 L / 60 (K) 50 0

bath 8 T 40 30 disorder-limited 4 20 10 0 −15−10 −5 0 5 10 15 n (109 cm-2)

[Crossno et al, submitted] (Thermal conductivity) ⇡2k2 L = ; L B T (Electrical conductivity) 0 ⌘ 3e2 Dirac Fluid in Graphene 28 J. Crossno et al. arXiv:1509.04713; Science, to appear Wiedemann-Franz Law Violations in Experiment Strange metal in graphene 100 20 90 phonon-limited 80 16 70

12 L / 60 (K) 50 0

bath 8 T 40 30 disorder-limited 4 20 10 0 −15−10 −5 0 5 10 15Wiedemann-Franz 9 -2 n (10 cm ) obeyed [Crossno et al, submitted] (Thermal conductivity) ⇡2k2 L = ; L B T (Electrical conductivity) 0 ⌘ 3e2 Dirac Fluid in Graphene 28 J. Crossno et al. arXiv:1509.04713; Science, to appear Wiedemann-Franz Law Violations in Experiment Strange metal in graphene 100 20 90 phonon-limited 80 16 70

12 L / 60 (K) 50 0

bath 8 T 40 Wiedemann-Franz 30 disorder-limited 4violated ! 20 10 0 −15−10 −5 0 5 10 15 n (109 cm-2)

[Crossno et al, submitted] (Thermal conductivity) ⇡2k2 L = ; L B T (Electrical conductivity) 0 ⌘ 3e2 Quasiparticle transport in metals:

Focus on inﬁnite number of (near) conserva- • tion laws (momenta of quasiparticles on the Fermi surface) and compute how they are slowly violated by the lattice or impurities Transport in strange metals

There are no quasiparticles, and so the Fermi • surface is not a central actor in transport (although a Fermi surface can be precisely deﬁned in some cases). Focus on relaxation of total momentum (in- • cluding contributions of the Fermi surface (if present) and all critical bosons) by the lattice or impurities Transport in strange metals

universal constraints on transport [Forster ’70s] hydrodynamics [Lucas 1506] [Hartnoll, others] [Donos, Gauntlett 1506] [Lucas, Sachdev PRB] long time dynamics; few conserved quantities “renormalized IR ﬂuid” emerges

perturbative memory matrix limit holography [Lucas JHEP] appropriate microscopics Dynamicsmatrix large of Ncharged theory; for cuprates non-perturbativeblack hole horizons computations

ﬁgure from [Lucas, Sachdev, Physical Review B91 195122 (2015)]

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007) Prediction for transport in the graphene strange metal Recall that in a Fermi liquid, the Lorenz ratio L = /(T ), where  is the thermal conductivity, and is the conductivity, is given by 2 2 2 L = ⇡ kB/(3e ). For a strange metal with a “relativistic” Hamiltonian, hydrodynamic, holographic, and memory function methods yield

1 e2v2 2⌧ v2 ⌧ e2v2 2⌧ = 1+ F Q imp ,  = F H imp 1+ F Q imp Q T ✓ H Q ◆ ✓ H Q ◆ 2 v2 ⌧ e2v2 2⌧ L = F H imp 1+ F Q imp , T 2 Q ✓ H Q ◆ where is the enthalpy density, ⌧ is the momentum relaxation time H imp (from impurities), while = Q, an intrinsic, ﬁnite, “quantum critical” conductivity. Note that the limits 0 and ⌧imp do not commute. Q ! !1

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007) M. Müller and S. Sachdev, PRB 78, 115419 (2008) 4

11 25 impurities. Two parameters in Eqn. 2 are undetermined 10 A B S1 for any given sample: lm and . For simplicity, we as- 20 ∆Vg = 0 H S2 sume we are well within the DF limit where lm and S3 are approximately independent of n.Weﬁttheexperi-H )

-2 15 mentally measured (n)toEqn.(2) for all temperatures 10 0 10 and densities in theL Dirac ﬂuid regime to obtain l and (cm m

L / 10

min for each sample. Fig 3C shows three representative ﬁts n H 5 to Eqn. (2) taken at 60 K. lm is estimated to be 1.5, 0.6, Disorder Thermally and 0.034 µm for samples S1, S2, and S3, respectively. Limited Limited 109 0 For the system to be well described by hydrodynamics, 1 10 100 1000 0 100 200 lm should be long compared to the electron-electron scat- Temperature (K) Temperature (K) tering length of 0.1 µm expected for the Dirac ﬂuid at

100 20 ⇠ 20 C 10 60 K [18]. This is consistent with the pronounced sig- 90 natures of hydrodynamics in S1 and S2, but not in S3,

) 8 80 16 e 2

16 H C m where only a glimpse of the DF appears in this more

µ 6 70 h /

12 L / disordered sample. Our analysis also allows us to es-

60 (eV 4

12 (K) 0

0 timate the thermodynamic quantity (T ) for the DF.

50 e H bath 8 2 H T L/L +V -V The Fig. 3C inset shows the ﬁtted enthalpy density as 40 8 h 0 a function of temperature compared to that expected in 30 4 40 60 80 100 T (K) 20 4 clean graphene (dashed line) [18], excluding renormal- 10 0 ization of the Fermi velocity. In the cleanest sample −15−10 −5 0 5 10 2 15 0 2 H 9 -2 varies from 1.1-2.3 eV/µm for Tdis

R (k e ⌘ to entropy density s is an indicator of the strength of 4 ples. At low temperature each sample is limited by disorder. 0 Lorentz ratio L = /(T ) the interactions between constituent particles. It is sug- h+ e- 4 At high temperature all samples become limited by thermal /h) 2 2

2 gested that the DF can behave as a nearly perfect fluid σTL0 excitations.vF Dashed⌧imp lines are a guide1 to the eye. (B) The 0 V Lorentz= ratioH of all three samples as a function of bath tem- [18]: ⌘/s approaches a “universal” lower bound conjec- 0 2 2 100 2 E T Q 2 2 2 T perature. The largest(1 WF + e violationvF is⌧imp seen/ in( theQ cleanest)) ture by Kovtun-Son-Starinets, (⌘/s)/(~/kB) 1/4⇡ for bath 6 sample. (C) The gate dependence ofQ the LorentzH ratio is well 0 a strongly interacting system [40]. Though we cannot (K) fit to hydrodynamic theory of Ref. [5, 6]. Fits of all three 1 40 K 10 mm 4 directly measure ⌘, we comment on the implications of samples are shown at 60 K. All samples return to the Fermi our measurement for its value. Within relativistic hy- 10 0 liquid value (black dashed line) at high density. Inset shows 1 20 K 2 Thermal Conductivity (nW/K) drodynamics, we can estimate the shear viscosity of the B -0.5 V the fitted enthalpy density as a function of temperature and Elec. ConductivityElec. (4 e 0 0 the theoretical value in clean graphene (black dashed line). electron-hole plasma in graphene from the enthalpy den- -1012 -1011 -1010 1010 1011 1012 -0.50 0.5 0 50 100 150 -2 ∆ Schematic inset illustrates the di↵erence between heat and sity as ⌘ ⌧ee [40], where ⌧ee is the electron-electron n (cm ) Vg (V) Tbath (K) J. Crossno et al. arXiv:1509.04713; Science, to appear charge current in the neutral Dirac plasma. scattering⇠ time.H Increasing the strength of interactions FIG. 1. Temperature and density dependent electrical and thermal conductivity. (A) Resistance versus gate voltage decreases ⌧ee, which in turn decreases ⌘ and ⌘/s.Employ- at various temperatures. (B) Electrical conductivity (blue) as a function of the charge density set by the back gate for di↵erent bath temperatures. The residual carrier density at the neutrality point (green) is estimated by the intersection of the minimum ing the expected Heisenberg limited inter-particle scat- conductivity with a linear fit to log() away from neutrality (dashed grey lines). Curves have been o↵set vertically such that more pronounced peak but also a narrower density de- the minimum density (green) aligns with the temperature axis to the right. Solid black lines correspond to 4e2/h.Atlow tering time, ⌧ee ~/kBT [5, 6], we find a shear viscosity 20 ⇠ temperature, the minimum density is limited by disorder (charge puddles). However, above Tdis 40 K, a crossover marked pendence, as predicted [5, 6]. ⇠ of 10 kg/s in two-dimensional units, corresponding in the half-tone background, thermal excitations begin to dominate and the sample enters the non-degenerate regime near More quantitative analysis of (n) in our experiment ⇠ 10 the neutrality point. (C-D) Thermal conductivity (red points) as a function of (C) gate voltage and (D) bath temperature L to 10 Pa s. The value of ⌧ee used here is consistent compared to the Wiedemann-Franz law, T 0 (blue lines). At low temperature and/or high doping ( µ kBT ), we find the ⇠ · L | | can be done by employing a quasi-relativistic hydrody- WF law to hold. This is a non-trivial check on the quality of our measurement. In the non-degenerate regime ( µ

I hydrodynamics when l l , t t ee ee I long time dynamics governed by conservation laws:

µ⌫ ext µ⌫ µ @⌫T = J⌫ F , @µJ =0. dynamics of relaxation to equilibrium µ⌫ µ I expand T , J in perturbative parameter lee@µ :

µ⌫ µ⌫ µ ⌫ µ⇢ ⌫ µ⌫ 2⌘ ⇢ T = P ⌘ +(✏ + P )u u 2 ⌘@ u ⇣ @⇢u + , P P (⇢ ) P d ··· ✓ ◆ µ µ µ⇢ µ ⌫ ext J = u q @⇢µ @⇢T u F + , Q P T ⇢⌫ ··· µ⌫ ⌘µ⌫ + uµu⌫ , ⇣ ⌘ P i ⌘ ti i Qi = TJi µTµJti New (and only) transport co-ecient, Q: I quantum physics values of P , ,etc... “quantum critical”! conductivity atq = 0. [Hartnoll, Kovtun, M¨uller,Sachdev, Physical Review B76Q 144502 (2007)] S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007) Translational symmetry breaking

Momentum relaxation by an external source h coupling to the operator O H = H ddxh(x) (x). 0 O Z R Im G (q, !) H d 2 2 OO 0 LeadsM to an=lim additionald termq h(q in) equationsqx of motion: + higher orders in h ⌧ ! 0 | | ! ! Z it µi T @µT = ... + ... ⌧imp

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007) A. Lucas and S. Sachdev, PRB 91, 195122 (2015) Translational symmetry breaking

“Memory function” methods yield an explicit expression for ⌧imp:

R Im G (q, !) H d 2 2 OO 0 M =lim d q h(q) qx + ... ⌧ ! 0 | | ! imp ! Z

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007) A. Lucas and S. Sachdev, PRB 91, 195122 (2015) 2 10 hole FL Dirac fluid elec. FL hole FL Dirac fluid elec. FL 1.5 8 ) 1 K) / 6 ⌦ 1 (k (nW 4  0.5 2 0 0 400 200 0 200 400 400 200 0 200 400 2 2 n (µm ) n (µm ) Q Q Figure 1: A comparison of our hydrodynamicFigure 1:testing theory of transport with the experimental results of [33] in clean samples of graphene at T = 75 K. We study the electrical and thermal conductances at various charge densities n near the charge neutrality point. Experimental data is shown Comparisonas circular red to data theory markers, with and anumerical single results momentum of our theory, relaxation averaged time over 30⌧imp disorder. Bestrealizations, fit of density are shown dependence as the solid blue to line. thermal Our theory conductivity assumes the equations does not of state capture described in (27) with the parameters C 11, C 9, C 200, ⌘ 110, 1.7, and (28)with the density dependence0 ⇡ 2 ⇡ of electrical4 ⇡ conductivity0 ⇡ 0 ⇡ u 0.13. The yellow shaded region shows where Fermi liquid behavior is observed and the 0 ⇡ Wiedemann-Franz law is restored, and our hydrodynamic theory is not valid in or near this regime. We also show the predictions of (2) as dashed purple lines, and have chosen the 3 parameter fit to be optimized for (n).

A. Lucas, J. Crossno, K.C. Fong, P. Kim, and S. Sachdev, arXiv:1510.01738, PRB to appear 2 where e is the electron charge, s is the entropy density, n is the charge density (in units of length ), is the enthalpy density, ⌧ is a momentum relaxation time, and is a quantum critical e↵ect, whose H q existence is a new e↵ect in the hydrodynamic gradient expansion of a relativistic ﬂuid. Note that up to q, (n) is simply described by Drude physics. The Lorenz ratio then takes the general form

DF (n)= L 2 2 , (3) L (1 + (n/n0) ) where 2 vF ⌧ DF = 2H , (4a) L T q 2 q n0 = H2 2 . (4b) e vF⌧ (n) can be parametrically larger than (as ⌧ and n n ), and much smaller (n n ). L LWF !1 ⌧ 0 0 Both of these predictions were observed in the recent experiment, and fits of the measured to (3)were L quantitatively consistent, until large enough n where Fermi liquid behavior was restored. However, the experiment also found that the conductivity did not grow rapidly away from n = 0 as predicted in (2), despite a large peak in (n) near n = 0, as we show in Figure 1. Furthermore, the theory of [25] does not make clear predictions for the temperature dependence of ⌧,whichdetermines(T ). In this paper, we argue that there are two related reasons for the breakdown of (2). One is that the dominant source of disorder in graphene – fluctuations in the local charge density, commonly referred to as charge puddles [43, 44, 45, 46] – are not perturbatively weak, and therefore a non-perturbative treatment of their e↵ects is necessary.3 The second is that the parameter ⌧, even when it is sharply defined, is

3See [47, 48] for a theory of electrical conductivity in charge puddle dominated graphene at low temperatures.

4 Non-perturbative treatment of disorder

✏

⇠ Note s(x) > 0 n lee ⌘ Q

n(x) > 0 x n(x) < 0 n

Figure 3: A cartoon of a nearly quantum critical fluid where our hydrodynamic description of transport is sensible. The local chemical potential µ(x) always obeys µ k T , and so the | | ⌧ B entropy density s/k is much larger than the charge density n ; both electrons and holes are B | | everywhere excited, and the energy density ✏ does not fluctuate as much relative to the mean. Near charge neutrality the local charge density flips sign repeatedly. The correlation length of disorder ⇠ is much larger than lee, the electron-electron interaction length.

1.2 NumericallyOutline solve the hydrodynamic equations in the presence of a Thex outline-dependent of this paper chemical is as follows. potential. We briefly review The thethermoelectric definitions of transport transport coecients properties in Section 2. In Section 3willwe develop then a depend theory of upon hydrodynamic the value transport of the in the shear electron viscosity, fluid, assuming⌘. that it is Lorentz invariant. We discuss the peculiar case of the Dirac fluid in graphene in Section 4, and argue that deviations from Lorentz invariance are small. We describe the results of our numerical simulations of this theory in Section 5, andA. Lucas, directly J. Crossno, compare K.C. our Fong, simulations P. Kim, withand S. recent Sachdev, experimental arXiv:1510.01738, data from PRB graphene to appear [33]. The experimentally relevant e↵ects of phonons are qualitatively described in Section 6. We conclude the paper with a discussion of future experimental directions. Appendices contain technical details of our theory. In this paper we use index notation for vectors and tensors. Latin indices ij run over spatial ··· coordinates x and y;Greekindicesµ⌫ run over time t as well. We will denote the time-like coordinate ··· of Aµ as At. Indices are raised and lowered with the Minkowski metric ⌘µ⌫ diag( 1, 1, 1). The Einstein ⌘ summation convention is always employed.

2 Transport Coecients

Let us begin by defining the thermoelectric response coecients of interest in this paper. Suppose that we drive our fluid by a spatially uniform, externally applied, electric field Ei (formally, an electrochemical potential gradient), and a temperature gradient @ T .Wewillreferto @ T as T ⇣ ,with⇣ = T 1@ T , i j j j j for technical reasons later. As is standard in linear response theory, we decompose these perturbations into various frequencies, and focus on the response at a single frequency !. Time translation invariance implies that the (uniformly) spatially averaged charge current J and the spatially averaged heat current h ii Q are also periodic in time of frequency !, and are related to E and ⇣ by the thermoelectric transport h ii i i coecients: Ji i!t ij(!) T ↵ij(!) Ej i!t h i e = e . (5) Q T ↵¯ (!) T ¯ (!) ⇣ ✓ h ii ◆ ✓ ij ij ◆✓ j ◆

6 2 10 hole FL Dirac fluid elec. FL hole FL Dirac fluid elec. FL 1.5 8 ) 1 K) / 6 ⌦ 1 (k (nW 4  0.5 2 0 0 400 200 0 200 400 400 200 0 200 400 2 2 n (µm ) n (µm ) Q Q Figure 1: A comparison of our hydrodynamicFigure 1:testing theory of transport with the experimental results of [33] in clean samples of graphene at T = 75 K. We study the electrical and thermal conductances at various chargeSolution densities of then near hydrodynamic the charge neutrality equations point. in the Experimental presence data is shown as circular red data markers,of a space-dependent and numerical results chemical of our potential. theory, averaged over 30 disorder realizations, are shown as the solid blue line. Our theory assumes the equations of state described Bestin (27 fit) of with density the parameters dependenceC to11, thermalC 9, C conductivity200, ⌘ now110, gives1 a.7, better and (28 fit)with to 0 ⇡ 2 ⇡ 4 ⇡ 0 ⇡ 0 ⇡ uthe density0.13. The dependence yellow shaded of region the electrical shows where conductivity Fermi liquid (for behavior⌘/s is observed10). The andT the 0 ⇡ ⇡ Wiedemann-Franzdependencies law of is other restored, parameters and our hydrodynamic also agree well theory with is not expectation. valid in or near this regime. We also show the predictions of (2) as dashed purple lines, and have chosen the 3 parameter fit to be optimized for (n).

3See [47, 48] for a theory of electrical conductivity in charge puddle dominated graphene at low temperatures.

4 Quantum matter without quasiparticles

No quasiparticle excitations • Shortest possible “collision time”, or more precisely, fastest • ~ possible local equilibration time ⇠ kBT Continuously variable density, • (conformal field theories are usuallyQ at fixed density, = 0) Q Theory built from hydrodynamics/holography • /memory-functions/strong-coupled-field-theory Exciting experimental realization in graphene. • 1. Competing quantum orders in the zeroth Landau level of graphene

2. Dirac liquid in graphene: quantum matter without quasiparticles