Superfluid-insulator transition

Transcription

1 Superfluid-insulator transition Ultracold 87 Rb atoms - bosons M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

2 T Quantum critical T KT Superfluid Insulator 0 g c c g

3 T Boltzmann theory of Nambu- T KT Goldstone-Higgs excitations and vortices 0 Superfluid Quantum critical g c c Insulator Boltzmann theory of particles/holes g

4 CFT3 at T>0 T T KT Quantum No quasiparticles: Boltzmann critical theory does not apply Superfluid Insulator 0 g c c g

5 Title: Conformal field theories at non-zero temperature: operator product expansions, Monte Carlo, and holography Authors: Emanuel Katz, Subir Sachdev, Erik S. Sorensen, William Witczak-Krempa (Submitted on 12 Sep 2014 (v1), last revised 2 Dec 2014 (this version, v2)) Abstract: We compute the non-zero temperature conductivity of conserved flavor currents in conformal field theories (CFTs) in 2+1 spacetime dimensions. At frequencies much greater than the temperature, ħω >> k B T, the ω dependence can be computed from the operator product expansion (OPE) between the currents and operators which acquire a non-zero expectation value at T > 0. Such results are found to be in excellent agreement with quantum Monte Carlo studies of the O(2) Wilson-Fisher CFT. Results for the conductivity and other observables are also obtained in vector 1/N expansions. We match these large ω results to the corresponding correlators of holographic representations of the CFT: the holographic approach then allows us to extrapolate to small ħω/( k B T). Other holographic studies implicitly only used the OPE between the currents and the energy-momentum tensor, and this yields the correct leading large ω behavior for a large class of CFTs. However, for the Wilson-Fisher CFT a relevant "thermal" operator must also be considered, and then consistency with the Monte Carlo results is obtained without a previously needed ad hoc rescaling of the T value. We also establish sum rules obeyed by the conductivity of a wide class of CFTs pages; 5+2 figures. Single column. v2: Added new appendix about numerical Comments: methods, expanded discussion, typos corrected Strongly Correlated Electrons (cond-mat.str-el); General Relativity and Quantum Subjects: Cosmology (gr-qc); High Energy Physics - Lattice (hep-lat); High Energy Physics - Theory (hep-th) Journal reference: Phys. Rev. B 90, (2014) Recent extension away from the critical point

6 Quantum critical fluid in graphene ARO MURI meeting, Berkeley January 11, 2016 Subir Sachdev Talk online: sachdev.physics.harvard.edu HARVARD

7 Philip Kim Jesse Crossno Kin Chung Fong Andrew Lucas

8 Graphene k y k x

9 Graphene Electron Fermi surface

10 Graphene Hole Fermi surface Electron Fermi surface

11 Graphene T (K) Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 D. E. Sheehy and J. Schmalian, PRL 99, (2007) M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

12 Graphene Predicted T (K) strange metal Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

13 μ Superfluid Commensurate Mott insulator 0 For cuprates, this insulator is realized e.g. at hole density = 1/8. g c c Superfluid g

14 μ Superfluid Commensurate Mott insulator 0 For cuprates, this insulator is realized e.g. at hole density = 1/8. g c c Superfluid g

15 Ordinary metals: the Fermi liquid Fermi surface Fermi surface separates empty and occupied states in momentum space. Area enclosed by Fermi surface = Q. Momenta of low energy excitations fixed by density of all electrons. k y k x Long-lived electron-like quasiparticle excitations near the Fermi surface: lifetime of quasiparticles 1/T 2. (Thermal conductivity) T (Electrical conductivity) = 2 kb 2 3e 2 L 0

16 Fluid in Graphene iedemann-franz Law I Wiedemann-Franz law in a Fermi liquid: apple T 2 kb 2 3e W K ~'v 3"0 t ~ 2.5 O I ~ ' I Bi Carrier concentration (cm 3) ~ I 1 ~ t I t ; I SiGe 2 Cs AI Au Mg SiBe3 /. / Nb~b.,~o/Cu F i Sb As Ag Ga 9 i e c N 2.5 c- O " 2.0 SiGe z ~oe \ SiG% i \ SiGe 1 Se H~ Y W1 Cs2 Fe Ir Co z Li2 AUz Au3 AI2 All \." \\\!!,,L T, oytw, p, t Feq z,~/ Cr '' \ X~., K Pt,\ Ni2 u A ~" Yb ~o 1 Er / / 2 As Sb ~ Rh g t31~ Bi 2 Bi I Cq! I I r I I t f I [ I I [ I i I t I I I I I I e I 101o Electrical conductivity (~,~-1 cm-1) G. S. Kumar, G. Prasad, and R.O. Pohl, J. Mat. Sci. 28, 4261 (1993)

17 Key properties of a strange metal No quasiparticle excitations Shortest possible collision time, or more precisely, fastest possible local equilibration time ~ k B T Continuously variable density, Q (conformal field theories are usually at fixed density, Q = 0)

18 Graphene T (K) Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2 D. E. Sheehy and J. Schmalian, PRL 99, (2007) M. Müller, L. Fritz, and S. Sachdev, PRB 78, (2008) M. Müller and S. Sachdev, PRB 78, (2008)

19 Dirac Fluid in Graphene 28 J. Crossno et al. arxiv: ; Science, to appear Wiedemann-Franz Law Violations in Experiment Strange metal in graphene Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 L = [Crossno et al, submitted] (Thermal conductivity) T (Electrical conductivity) ; L 0 2 k 2 B 3e 2

20 Dirac Fluid in Graphene 28 J. Crossno et al. arxiv: ; Science, to appear L = Wiedemann-Franz Law Violations in Experiment Tbath (K) n (10 9 cm -2 ) [Crossno et al, submitted] Strange metal in graphene phonon-limited disorder-limited (Thermal conductivity) T (Electrical conductivity) ; L 0 2 k 2 B 3e L / L0 Wiedemann-Franz obeyed

21 Dirac Fluid in Graphene 28 J. Crossno et al. arxiv: ; Science, to appear Wiedemann-Franz Law Violations in Experiment Strange metal in graphene Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 Wiedemann-Franz violated! L = [Crossno et al, submitted] (Thermal conductivity) T (Electrical conductivity) ; L 0 2 k 2 B 3e 2

22 Quasiparticle transport in metals: Focus on infinite number of (near) conservation laws (momenta of quasiparticles on the Fermi surface) and compute how they are slowly violated by the lattice or impurities

23 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 (including contributions of the Fermi surface (if present) and all critical bosons) by the lattice or impurities

24 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 (including contributions of the Fermi surface (if present) and all critical bosons) by the lattice or impurities

25 Summary Transport in Strange Metals universal constraints on transport [Forster 70s] [Hartnoll, others] hydrodynamics [Lucas, Sachdev PRB] long time dynamics; renormalized IR fluid emerges few conserved quantities memory matrix appropriate microscopics for cuprates [Lucas 1506] [Donos, Gauntlett 1506] perturbative limit [Lucas JHEP] holography matrix large theory; Dynamics ofncharged non-perturbative computations black hole horizons figure from [Lucas, Sachdev, Physical Review B (2015)] S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007)

26 Prediction for transport in the graphene strange metal Recall that in a Fermi liquid, the Lorenz ratio L = apple/(t ), where apple is the thermal conductivity, and is the conductivity, is given by L = 2 kb 2 /(3e2 ). For a strange metal with a relativistic Hamiltonian, hydrodynamic, holographic, and memory function methods yield = Q 1+ e2 v 2 F Q2 imp H Q L = v2 F H imp T 2 Q, apple = v2 F H imp T 1+ e2 v 2 F Q2 imp H Q 1+ e2 v 2 F Q2 imp H Q 2, S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007) M. Müller and S. Sachdev, PRB 78, (2008) 1 where H is the enthalpy density, imp is the momentum relaxation time (from impurities), while = Q, an intrinsic, finite, quantum critical conductivity. Note that the limits Q! 0 and imp!1do not commute.

27 Tbath (K) n (10 9 cm -2 ) L / L0 Temperature (K) Temperature (K) D T dis T el-ph L/L C H +V e h h e C -V H (ev/µm 2 ) T (K) n (10 10 cm 2 ) κ e E 10 mm σtl 0 0 V -0.5 V T bath (K) FIG. 3. Disorder in the Dirac fluid. (A) Minimum carrier density as a function of temperature for all three samples. At low temperature each sample is limited by disorder. At high temperature all samples become limited by thermal excitations. Dashed lines are a guide to the eye. (B) The Lorentz = v2 F H imp 1 ratio of all three samples as a function of bath temperature. TThe 2 Qlargest (1 WF + eviolation 2 vf 2 Q2 is imp seen/(h in the Q cleanest )) 2 sample. (C) The gate dependence of the Lorentz ratio is well fit to hydrodynamic theory of Ref. [5, 6]. Fits of all three samples are shown at 60 K. All samples return to the Fermi liquid value (black dashed line) at high density. Inset shows the fitted enthalpy density as a function of temperature and the theoretical value in clean graphene (black dashed line). Schematic inset illustrates the di erence between heat and J. Crossno et al. arxiv: ; Science, to appear charge current in the neutral Dirac plasma. Lorentz ratio L = apple/(t )

28 Relativistic hydrodynamics I hydrodynamics when l l ee, t t ee I long time dynamics governed by conservation T µ = J F ext J µ =0. dynamics of relaxation to equilibrium I expand T µ, J µ in perturbative parameter l µ : T µ = P µ +( + P )u µ u 2P µ ( u ) P µ J µ = Qu µ qp µ P µ µ + u µ u, Q i = T ti Q i = J i µj i µt ti µ T u F ext +, 2 u +, New (and only) transport co-e cient, Q: I quantum physics! values of P, q, etc... quantum critical conductivity at Q = 0. S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007)

29 Translational symmetry breaking Momentum relaxation by an external source h coupling to the operator O Z H = H 0 d d xh(x) O(x). Z M Leads to an additional term in equations of µ T µi =... T it imp +... S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007) A. Lucas and S. Sachdev, PRB 91, (2015)

30 Translational symmetry breaking Momentum relaxation by an external source h coupling to the operator O Z H = H 0 d d xh(x) O(x). Z M Leads to an additional term in equations of µ T µi =... T it imp +... Memory function methods yield an explicit expression for imp : Z M = lim imp!!0 d d q h(q) 2 q 2 x Im G R OO (q,!) H 0! +... S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007) A. Lucas and S. Sachdev, PRB 91, (2015)

31 AdS/CFT correspondence at zero temperature Einstein gravity S E = Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) L 2 AdS d+2 R d,1 Minkowski CFTd+1 x i r ds 2 = L r 2 dr 2 dt 2 + d~x 2

32 AdS/CFT correspondence at non-zero temperature Einstein gravity S E = Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) L 2 AdS-Schwarzschild R d,1 Minkowski CFT d+1 at Hawking temperature T of horizon x i r R Entropy density of CFT d+1, S T d Bekenstein-Hawking entropy density, S BH T d

33 Einstein-Maxwell theory S EM = Charged black branes Z d d+2 x p g apple 1 2apple 2 R + d(d + 1) R 2 L 2 gf 2 F 2 AdS-Reissner-Nordstrom r Electric flux hqi =0 Quantum matter on the boundary with a variable charge density Q of a global U(1) symmetry. Realizes a strange metal: a state with an unbroken global U(1) symmetry with a continuously variable charge density, Q, at T = 0 which does not have any quasiparticle excitations. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, 99

34 Charged black branes r Electric flux hqi =0 Quantum matter on the boundary with a variable charge density Q of a global U(1) symmetry. More general theories have hyperscaling violating metric ds 2 = 1 r 2 dt 2 r 2d(z 1)/(d ) + r2 /(d ) dr 2 + dx 2 i at T=0 C. Charmousis, B. Gouteraux, B. S. Kim, E. Kiritsis and R. Meyer, JHEP 1011, 151 (2010). N. Iizuka, N. Kundu, P. Narayan and S. P. Trivedi, JHEP 1201, 94 (2012). L. Huijse, S. Sachdev, B. Swingle, Phys. Rev. B 85, (2012)

35 Inhomogeneous charged black branes r Electric flux hqi =0 Add a source term, Z d d xh(x) O(x) Weakly disordered charged black branes yield results identical to those obtained from memory functions and holography G.T. Horowitz, J.E. Santos, and D. Tong, JHEP 1207, 168 (2012), JHEP 1211, 102 (2012). D. Vegh, arxiv: M. Blake, D. Tong, and D. Vegh, PRL 112, (2013). M. Blake and D. Tong, PRD 88, (2013). A. Lucas, S. Sachdev, and K. Schalm, PRD 89, (2014). A. Lucas, JHEP 1503, 071 (2015). R. A. Davison and B. Goutéraux, arxiv: ; arxiv: M. Blake, arxiv: ; arxiv:

36 Prediction for transport in the graphene strange metal Recall that in a Fermi liquid, the Lorenz ratio L = apple/(t ), where apple is the thermal conductivity, and is the conductivity, is given by L = 2 kb 2 /(3e2 ). For a strange metal with a relativistic Hamiltonian, hydrodynamic, holographic, and memory function methods yield = Q 1+ e2 v 2 F Q2 imp H Q L = v2 F H imp T 2 Q, apple = v2 F H imp T 1+ e2 v 2 F Q2 imp H Q 1+ e2 v 2 F Q2 imp H Q 2, S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007) M. Müller and S. Sachdev, PRB 78, (2008) 1 where H is the enthalpy density, imp is the momentum relaxation time (from impurities), while = Q, an intrinsic, finite, quantum critical conductivity. Note that the limits Q! 0 and imp!1do not commute.

37 A. Lucas, J. Crossno, K.C. Fong, P. Kim, and S. Sachdev, arxiv: , PRB to appear 2 (k 1 ) hole FL Dirac fluid elec. FL apple (nw/k) hole FL Dirac fluid elec. FL Q n (µm 2 ) Q n (µm 2 ) Figure 1: A comparison of our hydrodynamic 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 as circular red data markers, and numerical results of our theory, averaged over 30 disorder realizations, are shown as the solid blue line. Our theory assumes the equations of state described in (27) with parameters C 0 11, C 2 9, C 4 200, 0 110, 0 1.7, and (28) with the density dependence of electrical conductivity u The yellow shaded region shows where Fermi liquid behavior is observed and the 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 apple(n). Comparison to theory with a single momentum relaxation time imp. Best fit of density dependence to thermal conductivity does not capture

38 Non-perturbative treatment of disorder l ee s(x) > 0 Note n Q n(x) > 0 n(x) < 0 n x 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 B T, and so the entropy density s/k B is much larger than the charge density n ; both electrons and holes are 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 l ee, the electron-electron interaction length. Numerically solve the hydrodynamic equations in the presence of a x-dependent chemical potential. The thermoelectric transport properties will then depend upon the value of the shear viscosity,. A. Lucas, J. Crossno, K.C. Fong, P. Kim, and S. Sachdev, arxiv: , PRB to appear

39 (k 1 ) hole FL Dirac fluid elec. FL apple (nw/k) hole FL Dirac fluid elec. FL Q n (µm 2 ) Q n (µm 2 ) Figure 1: A comparison of our hydrodynamic 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 Solution densities of the n near hydrodynamic the charge neutrality equations point. the Experimental presence data is shown as circular red data markers, and numerical results of our theory, averaged over 30 disorder of a space-dependent chemical potential. realizations, are shown as the solid blue line. Our theory assumes the equations of state described in (27) with the parameters C 0 11, C 2 9, C 4 200, 0 110, 0 1.7, and (28) with uthe 0 density The dependence yellow shaded of region the electrical shows where conductivity Fermi liquid(for behavior /s is observed 10). The andt the Wiedemann-Franz dependencies lawofis other restored, parameters and our hydrodynamic also agree well theorywith 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 apple(n). Best fit of density dependence to thermal conductivity now gives a better fit to A. Lucas, J. Crossno, K.C. Fong, P. Kim, and S. Sachdev, arxiv: , PRB to appear 2

40 Quantum matter without quasiparticles No quasiparticle excitations Shortest possible collision time, or more precisely, fastest possible local equilibration time ~ k B T Continuously variable density, Q (conformal field theories are usually at fixed density, Q = 0) Theory built from hydrodynamics/holography /memory-functions/strong-coupled-field-theory Exciting experimental realization in graphene.