Emergent gauge fields and the high temperature superconductors

Transcription

1 HARVARD Emergent gauge fields and the high temperature superconductors Nambu Memorial Symposium University of Chicago March 12, 2016 Subir Sachdev Talk online: sachdev.physics.harvard.edu

2 Nambu and superconductivity Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. P Y. NAMBU AND G. JONA-LASINIoj' The Enrico terms Institute for Nuclear StuCkes and the Department of Physics, The University of Chicago, Chicago, Illinois (Received October 27, 1960) Quasi-Particles and Gauge Invariance in the Theory of Superconductivity* YoicHmo NAMsU The Enrico Fermi Institute for Nuclear Studies and the Department of Physics, The University of Chicago, Chicago, Illinois (Received July 23, 1959) The problems of Nambu-Goldstone bosons linked to the spontaneous breaking of a global symmetry and the Higgs phase with a massive photon in a weakly-coupled gauge theory are closely connected

3 High temperature superconductors: Electrons in crystals provide a novel vacuum, and their interactions can lead to quantum ground states with long-range quantum entanglement. The dynamics of such states is described by emergent gauge fields, usually at strong coupling.

4 High temperature superconductors Cu O CuO 2 plane YBa 2 Cu 3 O 6+x

5 SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

6 SM Insulating Antiferromagnet FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

7 Undoped insulating antiferromagnet

8 Antiferromagnet with p mobile holes per square

9 Filled Band

10 Antiferromagnet with p mobile holes per square But relative to the band insulator, there are 1+ p holes per square

11 Antiferromagnet In a conventional metal (a Fermi liquid), with no broken symmetry, the area with p mobile holes per square enclosed by the Fermi surface must be 1+p But relative to the band insulator, there are 1+ p holes per square

12 SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

13 M. Platé, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, Ruixing Liang, D. A. Bonn, W. N. Hardy, S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli, Phys. Rev. Lett. 95, (2005) SM FL A conventional metal: the Fermi liquid

14 Ordinary metals: the Fermi liquid Fermi surface k y Fermi surface separates empty and occupied states in momentum space. Luttinger Theorem: volume (area) enclosed by Fermi surface = the electron density. Hall co-e cient R H = 1/((Fermi volume) e). k x

15 Hall effect measurements in YBCO b p* 1.5 SDW CDW FL n H = V / e R H p Fermi liquid (FL) with carrier density 1+p p p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

16 Hall effect measurements in YBCO b p* n H = V / e R H SDW CDW FL p 1 + p Spin density wave (SDW) breaks translational invariance, and the Fermi liquid then has carrier density p p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

17 Hall effect measurements in YBCO b p* 1.5 SDW CDW FL n H = V / e R H p 1 + p Charge density wave (CDW) leads to complex Fermi surface reconstruction and negative Hall resistance p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

18 Hall effect measurements in YBCO b p* n H = V / e R H SDW CDW FL p 1 + p p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016) Evidence for FL* metal with Fermi surface of size p and emergent gauge fields?!

19 FL* A metal with: A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p Emergent gauge fields and connections to topological field theories T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett 90, (2003)

20 FL* A metal with: A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p Emergent gauge fields and connections to topological field theories There is a general and fundamental relationship between these two characteristics. T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett 90, (2003)

21 1. The insulating spin liquid and topological field theory 2. Topology and the size of the Fermi surface 3. Transition between FL* and FL 4. Quantum matter with quasiparticles strange metals in superconductors, graphene, the quark-gluon plasma, the superfluid-insulator transition of ultra-cold atoms, and the dynamics of charged black holes horizons

22 1. The insulating spin liquid and topological field theory 2. Topology and the size of the Fermi surface 3. Transition between FL* and FL 4. Quantum matter with quasiparticles strange metals in superconductors, graphene, the quark-gluon plasma, the superfluid-insulator transition of ultra-cold atoms, and the dynamics of charged black holes horizons

23 Undoped Antiferromagnet

24 Insulating spin liquid =( "#i #"i) / p 2 The first proposal of a quantum state with long-range entanglement L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973)

25 Insulating spin liquid =( "#i #"i) / p 2 The first proposal of a quantum state with long-range entanglement L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973)

26 Insulating spin liquid =( "#i #"i) / p 2 The first proposal of a quantum state with long-range entanglement L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973)

27 Insulating spin liquid =( "#i #"i) / p 2 The first proposal of a quantum state with long-range entanglement L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973)

28 Insulating spin liquid =( "#i #"i) / p 2 The first proposal of a quantum state with long-range entanglement L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973)

29 Insulating spin liquid =( "#i #"i) / p 2 The first proposal of a quantum state with long-range entanglement L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973)

30 Modern description: Insulating spin liquid The Z 2 spin liquid: Described by the simplest, nontrivial, topological field theory with time-reversal symmetry: L = 1 4 K IJ Z d 3 xa I ^ da J where a I, I =1, 2 are U(1) gauge connections, and the K matrix is K = N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991) M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Annals of Physics 310, 428 (2004)

31 Modern description: Insulating spin liquid The Z 2 spin liquid: Described by the simplest, nontrivial, topological field theory with time-reversal symmetry: L = 1 Z 4 K IJ d 3 xa I ^ da J where a I, I =1, 2 are U(1) gauge connections, and the K matrix is See also E. Fradkin and S. H. Shenker, 0 2 Phase diagrams of lattice gauge theories with Higgs fields, Phys. Rev. D 19, 3682 K = 2 0 (1979); J. M. Maldacena, G. W. Moore and N. Seiberg, D-brane charges in five-brane N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) backgrounds, JHEP 0110, 005 (2001). X.-G. Wen, Phys. Rev. B 44, 2664 (1991) M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Annals of Physics 310, 428 (2004)

32 Ground state degeneracy Place insulator on a torus:

33 Ground state degeneracy Place insulator on a torus: Ground state becomes degenerate due to gauge fluxes enclosed by cycles of the torus

34 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus: obtain topological states nearly degenerate with the ground state: number of dimers crossing red line is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

35 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus: 4 obtain topological states nearly degenerate with the ground state: number of dimers crossing red line is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

36 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus: 4 obtain topological states nearly degenerate with the ground state: number of dimers crossing red line is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

37 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus: 4 obtain topological states nearly degenerate with the ground state: number of dimers crossing red line is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

38 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus: 2 obtain topological states nearly degenerate with the ground state: number of dimers crossing red line is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

39 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus: 2 obtain topological states nearly degenerate with the ground state: number of dimers crossing red line is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

40 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus: 0 obtain topological states nearly degenerate with the ground state: number of dimers crossing red line is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

41 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus: 2 obtain topological states nearly degenerate with the ground state: number of dimers crossing red line is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

42 1. The insulating spin liquid and topological field theory 2. Topology and the size of the Fermi surface 3. Transition between FL* and FL 4. Quantum matter with quasiparticles strange metals in superconductors, graphene, the quark-gluon plasma, the superfluid-insulator transition of ultra-cold atoms, and the dynamics of charged black holes horizons

43 Topology and the Fermi surface size M. Oshikawa, PRL 84, 3370 (2000) A. Paramekanti and A. Vishwanath, PRB 70, (2004) Φ L y L x We take N particles, each with charge Q, on a L x L y lattice on a torus. We pierce flux = hc/q through a hole of the torus. An exact computation shows that the change in crystal momentum of the many-body state due to flux piercing is P xf P xi = 2 N L x (mod 2 ) =2 L y (mod 2 ) where = N/(L x L y )isthedensity.

44 Topology and the Fermi surface size Proof of P xf P xi = 2 N L x (mod 2 ) =2 L y (mod 2 ). The initial and final Hamiltonians are related by a gauge transformation! U G H f U 1 G = H i, U G =exp i 2 X x iˆn i. L x while the wavefunction evolves from ii to U T ii, whereu T is the time evolution operator. We want to work in a fixed gauge in which the initial and final Hamiltonians are the same: in this gauge, the final state is f i = U G U T ii. Let ˆT x be the lattice translation operator. Then we can establish the above result using the definitions ˆT x ii = e ip xi ii, ˆTx f i = e ip xf f i, i and the easily established properties ˆT x U T = U T ˆTx, ˆTx U G =exp i2 N L x U G ˆTx

45 Topology and the Fermi surface size P x =2 L y (mod 2 ), P y =2 L x (mod 2 ) Now we compute the momentum balance assuming that the only low energy excitations are quasiparticles near the Fermi surface, and these react like free particles to a su ciently slow flux insertion. So each quasiparticle picks up a momentum ~ p (2 /Lx, 0), and then we can write (with n p the quasiparticle density excited by the flux insertion) P x = X p n p p x. Now n p = ±1 on a shell of thickness ~ p d S ~ p on the Fermi surface (where S ~ p is an area element on the Fermi surface). So we can write the above as a surface integral I Lx L y P x = p x ~ p d Sp ~ FS 4 2 Z = ( ~ Lx L y p ˆx) 4 2 dv by the divergence theorem. So FV ~ p

46 Topology and the Fermi surface X size P x =2 L y (mod 2 ), P y =2 L x (mod 2 ) Now n p = ±1 on a shell of thickness ~ p d S ~ p on the Fermi surface (where S ~ p is an area element on the Fermi surface). So we can write the above as a surface integral I Lx L y P x = p x ~ p d Sp ~ FS 4 2 Z = ( ~ Lx L y p ˆx) 4 2 dv by the divergence theorem. So FV ~ p

47 Topology and the Fermi surface size P x =2 L y (mod 2 ), P y =2 L x (mod 2 ) P x = 2 L x Lx L y 4 2 V FS, P y = 2 L y Lx L y 4 2 V FS where V FS is the volume of the Fermi surface. So, although the quasiparticles are only defined near the Fermi surface, by using Gauss s Law on the momentum acquired by quasiparticles near the Fermi surface, we have converted the answer to an integral over the volume enclosed by the Fermi surface. Now we equate these values to those obtained above, and obtain N L x L y V FS 4 2 = L xm x, N L x L y V FS 4 2 = L ym y for some integers m x, m y. Now choose L x, L y mutually prime integers; then m x L x = m y L y implies that m x L x = m y L y = pl x L y for some integer p. Then we obtain = N = V FS L x L y p. This is Luttinger s theorem.

48 Topology and the Fermi surface size in FL* Φ L y L x To obtain a di erent Fermi surface size, we need low energy excitations on a torus which are not composites of quasiparticles around the Fermi surface. The degenerate ground states of a Z 2 spin liquid can provide the needed excitation, and lead to a Z 2 -FL* state with a Fermi surface size of p, rather than the Luttinger size of 1 + p. T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)

49 Topology and the Fermi surface size in FL* Φ L y L x The exact momentum transfers P x =2 (1 + p)l y (mod2 ) and P y =2 (1 + p)l x (mod2 ) due to flux piercing arise from A contribution 2 pl x,y from the small Fermi surface of quasiparticles of size p. The remainder is made up by the topological sector: flux insertion creates a vison in the hole of the torus. T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)

50 Start with a spin liquid and then remove electrons =( "#i #"i) / p 2

51 Start with a spin liquid and then remove electrons =( "#i #"i) / p 2

52 Start with a spin liquid and then remove electrons =( "#i #"i) / p 2

53 Start with a spin liquid and then remove electrons =( "#i #"i) / p 2

54 Start with a spin liquid and then remove electrons =( "#i #"i) / p 2

55 Start with a spin liquid and then remove electrons =( "#i #"i) / p 2

56 Start with a spin liquid and then remove electrons =( "#i #"i) / p 2

57 A mobile charge +e, but carrying no spin =( "#i #"i) / p 2

58 A mobile charge +e, but carrying no spin =( "#i #"i) / p 2

59 S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987) N. Read and B. Chakraborty, PRB 40, 7133 (1989) Spin liquid with density p of spinless, charge +e holons. These can form a Fermi surface of size p, but not of electrons =( "#i #"i) / p 2

60 S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987) N. Read and B. Chakraborty, PRB 40, 7133 (1989) Spin liquid with density p of spinless, charge +e holons. These can form a Fermi surface of size p, but not of electrons =( "#i #"i) / p 2

61 S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987) N. Read and B. Chakraborty, PRB 40, 7133 (1989) Spin liquid with density p of spinless, charge +e holons. These can form a Fermi surface of size p, but not of electrons =( "#i #"i) / p 2

62 S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, PRB 35, 8865 (1987) N. Read and B. Chakraborty, PRB 40, 7133 (1989) Spin liquid with density p of spinless, charge +e holons. These can form a Fermi surface of size p, but not of electrons =( "#i #"i) / p 2

63 =( "#i #"i) / p 2

64 =( "#i #"i) / p 2

65 =( "#i #"i) / p 2

66 =( "#i #"i) / p 2

67 =( "#i #"i) / p 2

68 =( "#i #"i) / p 2

69 =( "#i #"i) / p 2

70 =( "#i #"i) / p 2

71 FL* S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996) R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, (2007) Mobile S=1/2, charge +e fermionic dimers: form a Fermi surface of size p of electrons =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

72 FL* S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996) R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, (2007) Mobile S=1/2, charge +e fermionic dimers: form a Fermi surface of size p of electrons =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

73 FL* S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996) R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, (2007) Mobile S=1/2, charge +e fermionic dimers: form a Fermi surface of size p of electrons =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

74 FL* S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996) R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, (2007) Mobile S=1/2, charge +e fermionic dimers: form a Fermi surface of size p of electrons =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

75 FL* S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996) R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, (2007) Mobile S=1/2, charge +e fermionic dimers: form a Fermi surface of size p of electrons =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

76 FL* S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996) R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, (2007) Mobile S=1/2, charge +e fermionic dimers: form a Fermi surface of size p of electrons =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

77 FL* S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996) R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, (2007) Mobile S=1/2, charge +e fermionic dimers: form a Fermi surface of size p of electrons =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

78 FL* S. Sachdev PRB 49, 6770 (1994); X.-G. Wen and P. A. Lee PRL 76, 503 (1996) R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, PRB 75, (2007) Mobile S=1/2, charge +e fermionic dimers: form a Fermi surface of size p of electrons =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

79 FL* Place FL* 2 on a torus: obtain topological states nearly degenerate with quasiparticle states: number of dimers crossing red line is conserved modulo 2 =( "#i #"i) / p 2 =( " i + "i) / p 2 T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)

80 FL* Place FL* 0 on a torus: obtain topological states nearly degenerate with quasiparticle states: number of dimers crossing red line is conserved modulo 2 =( "#i #"i) / p 2 =( " i + "i) / p 2 T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)

81 FL* Place FL* 0 on a torus: obtain topological states nearly degenerate with quasiparticle states: number of dimers crossing red line is conserved modulo 2 =( "#i #"i) / p 2 =( " i + "i) / p 2 T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)

82 FL* Place FL* 2 on a torus: obtain topological states nearly degenerate with quasiparticle states: number of dimers crossing red line is conserved modulo 2 =( "#i #"i) / p 2 =( " i + "i) / p 2 T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)

83 1. The insulating spin liquid and topological field theory 2. Topology and the size of the Fermi surface 3. Transition between FL* and FL 4. Quantum matter with quasiparticles strange metals in superconductors, graphene, the quark-gluon plasma, the superfluid-insulator transition of ultra-cold atoms, and the dynamics of charged black holes horizons

84 SM FL Figure: K. Fujita and J. C. Seamus Davis

85 SM FL FL* FL Figure: K. Fujita and J. C. Seamus Davis

86 Strange Metal SM FL FL* FL Figure: K. Fujita and J. C. Seamus Davis

87 S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Phys. Rev. B 80, (2009) D. Chowdhury and S. Sachdev, PRB 91, (2015) SU(2) gauge theory for transition between Z 2 -FL* and FL A SU(2) gauge boson. Fermion, transforming as a gauge SU(2) fundamental, with dispersion " k from the band structure, at a nonzero chemical potential: has a large Fermi surface, and carries electromagnetic charge A complex Higgs field, H, transforming as a gauge SU(2) adjoint, carrying non-zero lattice momentum. A SU(2) fundamental scalar z, carrying electron-spin and electromagnetically neutral.

88 SU(2) gauge theory for transition between Z 2 -FL* and FL The Higgs phase with hhi has a deconfined Z 2 gauge field, and realizes a Z 2 -FL* with Ising-nematic order. The confining phase is a FL or a superconductor, and no Ising-nematic order. When the Higgs potential is critical, we obtain a non- Fermi liquid of a Fermi surface coupled to Landaudamped gauge bosons, and critical Landau-damped Higgs field. This is a candidate for describing the strange metal. S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Phys. Rev. B 80, (2009) D. Chowdhury and S. Sachdev, PRB 91, (2015)

89 1. The insulating spin liquid and topological field theory 2. Topology and the size of the Fermi surface 3. Transition between FL* and FL 4. Quantum matter with quasiparticles strange metals in superconductors, graphene, the quark-gluon plasma, the superfluid-insulator transition of ultra-cold atoms, and the dynamics of charged black holes horizons

90 Fermi liquids: quasiparticles moving ballistically between impurity (red circles) scattering events Strange metals: electrons scatter frequently off each other, so there is no regime of ballistic quasiparticle motion. The electron liquid then flows around impurities

91 Graphene k y k x

92 Graphene Electron Fermi surface

93 Graphene Hole Fermi surface Electron Fermi surface

94 Graphene T (K) 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)

95 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)

96 Wiedemann-Franz ;< metals and alloys. Up to a certain temperature, inelastic scattering determines the Lorenz number value, and below this the scattering is elastic which is due to impurities. Supression of the electronic contribution to thermal conductivity and hence the separation of the lattice and electronic parts of conductivity can be done by application of a transverse magnetic field and hence the Lorenz number can be evaluated. The deviation of the Lorenz number in some degenerate semi8 conductors is attributed to phonon drag. In some c Fluid in Graphene Thermal and electrical with quasiparticles 10 ~ I I 101 I 0o Wiedemann-Franz Law.I conductivity 10 2 Relative electrical conductivity Figure I l Relative thermal conductivities, A, measured by Fermi liquid: Wiedemann-Franz law in a Wiedemann and Franz (AAg assumed to be = 100) and relative electrical conductivities, ~, measured by ( 9 Riess, (A) Becquerel, and (V) Lorenz. C~Agassumed to be After Wiedemann and Franz [1]. L0 = T I ~ ' I I 1 ~ 30 c Nc- 2.5 SiBe3 Se i " 2.0 I I r 10 3 \ SiGe1 I t I ; H~ Y,,L T, Yb ~o1 I t 10 4 f I [ 10 6 AI /. / Au W1 Cs2 Mg Nb~b.,~o/Cu Fi 9 i e As \." SiG% O I Cs Sb SiGez ~oe \ t SiGe2 Bi ~ W. K2 Carrier c o n c e n t r a t i o n (cm 3) ~ ~'v3"0t 2.5 O kb 3e2 Ag Fe Ir Coz Li2 AUz Au3 Ga AI2 All \\\!! o y t w, p, t F e q z, ~ / Cr '' Er / / 2 As t31~ Bi2 BiI \ X~., I t I [ I 10 e Electrical c o n d u c t i v i t y i I 10 7 Sb ~ K Pt,\ Rh Cq I Ni2 I 10 8 I I ua ~" I 10 9 I! g I 101o (~,~-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 G.(Table S. Kumar, G. Prasad, R.O. Pohl,conductivity J. Mat. Sci.and 28,also 4261 (1993) own data points on a doped, degenerate semiconductor III). Data are plottedand versus electrical versus carrier

97 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)

98 J. Crossno, Jing K. Shi, Ke Wang, Xiaomeng Liu, A. Harzheim, A. Lucas, S. Sachdev, Philip Kim, Takashi Taniguchi, Kenji Watanabe, T. A. Ohki, and Kin Chung Fong, Science 351, 1058 (2016) Dirac Fluid in Graphene 2 Strange metal in graphene Wiedemann-Franz Law Violations in Experiment L = T apple L 0 = 2 k 2 B 3e 2 Tbath (K) phonon-limited L / L disorder-limited n (10 9 cm -2 ) 4 0 [Crossno et al, submitted]

99 J. Crossno, Jing K. Shi, Ke Wang, Xiaomeng Liu, A. Harzheim, A. Lucas, S. Sachdev, Philip Kim, Takashi Taniguchi, Kenji Watanabe, T. A. Ohki, and Kin Chung Fong, Science 351, 1058 (2016) Dirac Fluid in Graphene 2 Strange metal in graphene Wiedemann-Franz Law Violations in Experiment L = T apple L 0 = 2 k 2 B 3e 2 Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 Wiedemann-Franz [Crossno et al, submitted] obeyed

100 J. Crossno, Jing K. Shi, Ke Wang, Xiaomeng Liu, A. Harzheim, A. Lucas, S. Sachdev, Philip Kim, Takashi Taniguchi, Kenji Watanabe, T. A. Ohki, and Kin Chung Fong, Science 351, 1058 (2016) Dirac Fluid in Graphene 2 Strange metal in graphene Wiedemann-Franz Law Violations in Experiment L = T apple L 0 = 2 k 2 B 3e 2 Tbath (K) phonon-limited disorder-limited n (10 9 cm -2 ) L / L0 Wiedemann-Franz [Crossno et al, submitted] violated!

101 V) n (109 cm-2) L/L L / L0 Tbath (K) 70 Temperature (K) C 10 8 H e H (ev/µm2) 100 Temperature (K) C h V e h -V T (K) 4 6 n (1010 cm 2) Tdis Tel-ph 8 D 6 κe 4 2 σtl 0 0V 0 E 6 10 mm V Tbath (K) FIG. 3.LDisorder Lorentz ratio = /(Tin )the Dirac fluid. (A) Minimum carrier 2 density as a function of temperature for all three samvples. imp At low temperature each 1sample is limited by disorder. F H = At high temperature2all2samples become limited 2by thermal 2 2 T Q (1 + e lines vf Q /(H )) (B) The Q eye. excitations. Dashed are aimp guide to the Lorentz ratio of all three samples as a function of bath temq! electron density; H! enthalpy density perature. The largest WF violation is seen in the cleanest sample. (C) The gateconductivity dependence of the Lorentz ratio is well critical Q! quantum fit to hydrodynamic theory of Ref. [5, 6]. Fits of all three imp! momentum relaxation from impurities samples are shown at 60 K. Alltime samples return to the Fermi liquid value (black dashed line) at high density. Inset shows S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, (2007) 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., plasma. Science 351, 1058 (2016) charge current in the neutral Dirac

102 Strange metal in graphene Negative local resistance due to viscous electron backflow in graphene D. A. Bandurin1, I. Torre2,3, R. Krishna Kumar1,4, M. Ben Shalom1,5, A. Tomadin6, A. Principi7, G. H. Auton5, E. Khestanova1,5, K. S. NovoseIov5, I. V. Grigorieva1, L. A. Ponomarenko1,4, A. K. Geim1, M. Polini3,6 1 School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom 2 National Enterprise for nanoscience and nanotechnology, Scuola Normale Superiore, I Pisa, Italy 3 Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I Genova (Italy) 4 Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom 5 National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom 6 National Enterprise for nanoscience and nanotechnology, Istituto Nanoscienze Consiglio Nazionale delle Ricerche and Scuola Normale Superiore, I Pisa, Italy 7 Radboud University, Institute for Molecules and Materials, NL 6525 AJ Nijmegen, The Netherlands Graphene hosts a unique electron system that due to weak electron phonon scattering allows micrometer scale ballistic transport even at room temperature whereas the local equilibrium is provided by frequent electron electron collisions. Under these conditions, electrons can behave as a viscous liquid and exhibit hydrodynamic phenomena similar to classical liquids. Here we report unambiguous evidence for this long sought transport regime. In particular, doped graphene exhibits L. Levitov and G. Falkovich, Nature Physics online FIG. 1: Current streamlines and injection potential mapwhich for visfig.to2:theno an anomalous (negative) voltage drop arxiv: , near current contacts, is attributed cous and ohmic whirlpools flows. White lines show current streamviscosity formation of submicrometer size in the electron flow. The viscosity of graphene s electron r

103 Strange metal in graphene Science 351, 1055 (2016) Negative local resistance due to viscous electron backflow in graphene D. A. Bandurin1, I. Torre2,3, R. Krishna Kumar1,4, M. Ben Shalom1,5, A. Tomadin6, A. Principi7, G. H. Auton5, E. Khestanova1,5, K. S. NovoseIov5, I. V. Grigorieva1, L. A. Ponomarenko1,4, A. K. Geim1, M. Polini3,6 1 School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom 2 National Enterprise for nanoscience and nanotechnology, Scuola Normale Superiore, I Pisa, Italy 3 Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I Genova (Italy) 4 Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom 5 National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom 6 National Enterprise for nanoscience and nanotechnology, Istituto Nanoscienze Consiglio Nazionale delle Ricerche and Scuola Normale Superiore, I Pisa, Italy 7 Radboud University, Institute for Molecules and Materials, NL 6525 AJ Nijmegen, The Netherlands Graphene hosts a unique electron system that due to weak electron phonon scattering allows micrometer scale ballistic transport even at room temperature whereas the local equilibrium is provided by frequent electron electron collisions. Under these conditions, electrons can behave as a Figure 1. Viscous backflow in doped graphene. (a,b) Steady state distribution of current injected through liquids. we report viscous exhibit hydrodynamic phenomena to classical a narrowliquid slit for and a classical conducting medium with zero (a)similar and a viscous Fermi liquid (b). Here (c) Optical unambiguous evidence long sought transport regime. In particular,geometry doped graphene micrograph of one of ourfor SLGthis devices. The schematic explains the measurement for vicinityexhibits resistance. (d,e) (negative) Longitudinalvoltage conductivity forinjection this device as a function induced by to the an anomalous drop nearandcurrent contacts, whichof is attributed applying gate 0.3 A; whirlpools 1 m. Forinmore detail, seeflow. Supplementary Information. formation of voltage. submicrometer size the electron The viscosity of graphene s electron

104 Graphene: a metal that behaves like water

105 Strange Metal SM FL Similar memoryfunction/ hydrodynamic/ holographic analyses for strange metal in cuprates.. Figure: K. Fujita and J. C. Seamus Davis

106 Pseudogap SM metal matches properties of Z2-FL* phase FL Figure: K. Fujita and J. C. Seamus Davis

107 FL* We have described a metal with: A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p Additional low energy quantum states on a torus not associated with quasiparticle excitations i.e. emergent gauge fields

108 FL* We have described a metal with: A Fermi surface of electrons enclosing volume p, and not the Luttinger volume of 1+p Additional low energy quantum states on a torus not associated with quasiparticle excitations i.e. emergent gauge fields There is a general and fundamental relationship between these two characteristics. Promising indications that such a metal describes the pseudogap of the cuprate supercondutors