Emergent light and the high temperature superconductors
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1 HARVARD Emergent light and the high temperature superconductors Pennsylvania State University State College, January 21, 2016 Subir Sachdev Talk online: sachdev.physics.harvard.edu
2 Maxwell's equations: 150 years of light A century and a half ago, James Clerk Maxwell submitted a long paper to the Royal Society containing his famous equations. Inspired by Michael Faraday s experiments and insights, the equations unified electricity, magnetism and optics. Their far-reaching consequences for our civilisation, and our universe, are still being explored Jon Butterworth Sunday 22 November GMT, The Guardian
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4 Maxwell's equations: 150 years of light A century and a half ago, James Clerk Maxwell submitted a long paper to the Royal Society containing his famous equations. Inspired by Michael Faraday s experiments and insights, the equations unified electricity, magnetism and optics. Their far-reaching consequences for our civilisation, and our universe, are still being explored Jon Butterworth Sunday 22 November GMT, The Guardian
5 Modern point-of-view: Electromagnetic gauge fields are needed to describe the long-range quantum entanglement of the vacuum. Electrons in crystals provide a new vacuum, and their interactions can naturally lead to quantum states which have long-range quantum entanglement, and require emergent gauge fields.
6 Modern point-of-view: Electromagnetic gauge fields are needed to describe the long-range quantum entanglement of the vacuum. Electrons in crystals provide a new vacuum, and their interactions can naturally lead to quantum states which have long-range quantum entanglement, and require emergent gauge fields.
7 High temperature superconductors Cu O CuO 2 plane YBa 2 Cu 3 O 6+x
8 SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis
9 SM Insulating Antiferromagnet FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis
10 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
11 1. Emergent gauge fields and long-range entanglement in insulators 2. Theory of ordinary metals: Fermi liquids (FL) (a) Quasiparticles (b) Luttinger theorem for volume enclosed by Fermi surface 3. Fractionalized Fermi liquids (FL*) Quasiparticles with a non-luttinger volume, and emergent gauge fields 4. The pseudogap metal of the cuprate superconductors
12 Undoped Antiferromagnet
13 Insulating spin liquid =( "#i #"i) / p 2 An insulator with emergent gauge fields: 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)
14 Insulating spin liquid =( "#i #"i) / p 2 An insulator with emergent gauge fields: 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)
15 Insulating spin liquid =( "#i #"i) / p 2 An insulator with emergent gauge fields: 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)
16 Insulating spin liquid =( "#i #"i) / p 2 An insulator with emergent gauge fields: 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)
17 Insulating spin liquid =( "#i #"i) / p 2 An insulator with emergent gauge fields: 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)
18 Insulating spin liquid =( "#i #"i) / p 2 An insulator with emergent gauge fields: 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)
19 Ground state degeneracy Place insulator on a torus;
20 Ground state degeneracy 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)
21 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)
22 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)
23 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)
24 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)
25 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)
26 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)
27 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)
28 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)
29 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus; 1 to change dimer number parity across red line, it is necessary to create a pair of unpaired spins and move them around the sample. D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)
30 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus; 1 to change dimer number parity across red line, it is necessary to create a pair of unpaired spins and move them around the sample. D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)
31 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus; 1 to change dimer number parity across red line, it is necessary to create a pair of unpaired spins and move them around the sample. D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)
32 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus; 1 to change dimer number parity across red line, it is necessary to create a pair of unpaired spins and move them around the sample. D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)
33 Ground state degeneracy =( "#i #"i) / p 2 Place insulator on a torus; 1 to change dimer number parity across red line, it is necessary to create a pair of unpaired spins and move them around the sample. D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)
34 Ground state degeneracy Place insulator on a torus; The sensitivity of the degeneracy to the global topology indicates long-range quantum entanglement 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 Place insulator on a torus; The degenerate states are conjugate to the flux of an emergent gauge field piercing the cycles of the torus D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)
36 Emergent gauge fields ˆn 1 ˆn 2 ˆn 3 Local constraint on dimer number operators: ˆn 1 +ˆn 2 +ˆn 3 +ˆn 4 =1. ˆn 4 Identify dimer number with an electric field, Êi =( 1) i x+iy ˆn i, ( = x, y); the constraint becomes Gauss s Law : Êi =( 1) i x+i y. The theory of the dimers is compact U(1) quantum electrodynamics in the presence of static background charges. The compact theory allows the analog of Dirac s magnetic monopoles as tunneling events/excitations. G. Baskaran and P. W. Anderson, Phys. Rev. B 37, 580(R) (1988) E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B 4, 225 (1990)
37 Emergent gauge fields ˆn 1 ˆn 2 ˆn 3 ˆn 4 Including dimers connecting the same sublattice leads to a Z 2 gauge theory in the presence of Berry phases of static background charges. This has a stable deconfined phase in 2+1 dimensions. By varying parameters it can undergoes a confinement transition to a valence bond solid, described by a frustrated Ising model. N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); R. A. Jalabert and S. Sachdev, Phys. Rev. B 44, 686 (1991); X.-G. Wen, Phys. Rev. B 44, 2664 (1991); S. Sachdev and M. Vojta, J. Phys. Soc. Jpn 69, Supp. B, 1 (1999)
38 1. Emergent gauge fields and long-range entanglement in insulators 2. Theory of ordinary metals: Fermi liquids (FL) (a) Quasiparticles (b) Luttinger theorem for volume enclosed by Fermi surface 3. Fractionalized Fermi liquids (FL*) Quasiparticles with a non-luttinger volume, and emergent gauge fields 4. The pseudogap metal of the cuprate superconductors
39 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
40 1. Emergent gauge fields and long-range entanglement in insulators 2. Theory of ordinary metals: Fermi liquids (FL) (a) Quasiparticles (b) Luttinger theorem for volume enclosed by Fermi surface 3. Fractionalized Fermi liquids (FL*) Quasiparticles with a non-luttinger volume, and emergent gauge fields 4. The pseudogap metal of the cuprate superconductors
41 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
42 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
43 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
44 Undoped Antiferromagnet
45 Antiferromagnet with p holes per square
46 Filled Band
47 Antiferromagnet with p holes per square But relative to the band insulator, there are 1+ p holes per square, and so a Fermi liquid has a Fermi surface of size 1+ p
48 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 Fermi liquid Area enclosed by Fermi surface =1+p
49 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 Fermi liquid Hall co-efficient R H =+1/((1 + p)e).
50 Kyle M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, Z.-X. Shen, Science 307, 901 (2005) Pseudogap SM FL 2. Pseudogap metal at low p
51 The PG regime behaves in many SM respects like a Fermi liquid, but with a Fermi FL surface size of p and not 1+p
52 The PG regime behaves in many SM respects like a Fermi liquid, but with a Fermi FL surface size of p and not 1+p.e.g. Hall co-e cient R H =+1/(pe).
53 1. Emergent gauge fields and long-range entanglement in insulators 2. Theory of ordinary metals: Fermi liquids (FL) (a) Quasiparticles (b) Luttinger theorem for volume enclosed by Fermi surface 3. The FL* phase: Quasiparticles with a non-luttinger volume, and emergent gauge fields 4. The pseudogap metal of the cuprate superconductors
54 Antiferromagnet with p holes per square
55 Antiferromagnet with p holes per square Can we get a Fermi surface of size p? (and full square lattice symmetry)
56 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 this is not visible in electron photo-emission =( "#i #"i) / p 2
57 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 this is not visible in electron photo-emission =( "#i #"i) / p 2
58 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 this is not visible in electron photo-emission =( "#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 this is not visible in electron photo-emission =( "#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 this is not visible in electron photo-emission =( "#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 this is not visible in electron photo-emission =( "#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 this is not visible in electron photo-emission =( "#i #"i) / p 2
63 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 this is not visible in electron photo-emission =( "#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 =( "#i #"i) / p 2
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 visible in electron photoemission =( "#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 visible in electron photoemission =( "#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 visible in electron photoemission =( "#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 visible in electron photoemission =( "#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 visible in electron photoemission =( "#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 visible in electron photoemission =( "#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 visible in electron photoemission =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
79 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 visible in electron photoemission =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)
80 Ground state degeneracy Place FL* on a torus:
81 Ground state degeneracy Place Place FL* insulator on a torus: on torus; obtain topological states nearly degenerate with quasiparticle states: number of dimers crossing red line is conserved modulo 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 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)
84 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)
85 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)
86 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
87 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. M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000) T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)
88 FL* Following the evolution of the quantum state under adiabatic insertion of a flux quantum leads to a non-perturbative argument for the volume enclosed by the Fermi surface M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000) T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)
89 1. Emergent gauge fields and long-range entanglement in insulators 2. Theory of ordinary metals: Fermi liquids (FL) (a) Quasiparticles (b) Luttinger theorem for volume enclosed by Fermi surface 3. The FL* phase: Quasiparticles with a non-luttinger volume, and emergent gauge fields 4. The pseudogap metal of the cuprate superconductors
90 1. Emergent gauge fields and long-range entanglement in insulators 2. Theory of ordinary metals: Fermi liquids (FL) (a) Quasiparticles (b) Luttinger theorem for volume enclosed by Fermi surface 3. The FL* phase: Quasiparticles with a non-luttinger volume, and emergent gauge fields 4. The pseudogap metal of the cuprate superconductors
91 Kyle M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, Z.-X. Shen, Science 307, 901 (2005) Pseudogap SM FL 2. Pseudogap metal at low p
92 Y. Qi and S. Sachdev, Phys. Rev. B 81, (2010) M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015) SM FL*? Pseudogap FL A new metal - FL*: with electron-like quasiparticles on a Fermi surface of size p and emergent gauge fields
93 Recent evidence for pseudogap metal as FL* Density wave instabilities of FL* have wave vector and form-factors which agree with STM/X-ray observations in DW region (D. Chowdhury and S. Sachdev, PRB 90, (2014)). T -independent positive Hall co-e cient, R H, corresponding to carrier density p in the higher temperature pseudogap (Ando et al., PRL92, (2004)) and in recent measurements at high fields, low T, and around p 0.16 in YBCO (Proust-Taillefer-UBC collaboration, Badoux et al., arxiv: ).
94 Y. Kohsaka et al., Science 315, 1380 (2007) Pseudogap Density wave (DW) order at low T and p SM FL
95 M. A. Metlitski and S. Sachdev, PRB 82, (2010).S.SachdevR.LaPlaca,PRL111, (2013). K. Fujita, M. H Hamidian, S. D. Edkins, Chung Koo Kim, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, H. Eisaki, S. Uchida, A. Allais, M. J. Lawler, E.-A. Kim, S. Sachdev, and J. C. Davis, PNAS 111, E3026(2014) Identified as a predicted d-form SM factor density wave Pseudogap Q =( /2, 0) FL
96 SM FL Q =( /2, 0)
97 SM X FL Q =( /2, 0)
98 The high T FL* can help explain the d-form factor density wave observed at low T SM FL* FL Q =( /2, 0) d-form factor density wave D. Chowdhury and S.S., Phys. Rev. B 90, (2014).
99 Recent evidence for pseudogap metal as FL* Density wave instabilities of FL* have wave vector and form-factors which agree with STM/X-ray observations in DW region (D. Chowdhury and S. Sachdev, PRB 90, (2014)). T -independent positive Hall co-e cient, R H, corresponding to carrier density p in the higher temperature pseudogap (Ando et al., PRL92, (2004)) and in recent measurements at high fields, low T, and around p 0.16 in YBCO (Proust-Taillefer-UBC collaboration, Badoux et al., arxiv: ).
100 Recent evidence for pseudogap metal as FL* Density wave instabilities of FL* have wave vector and form-factors which agree with STM/X-ray observations in DW region (D. Chowdhury and S. Sachdev, PRB 90, (2014)). T -independent positive Hall co-e cient, R H, corresponding to carrier density p in the higher temperature pseudogap (Ando et al., PRL92, (2004)) and in recent measurements at high fields, low T, and around p 0.16 in YBCO (Proust-Taillefer-UBC collaboration, Badoux et al., arxiv: ).
101 SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis
102 High field, low T measurements show a positive Hall co-e cient corresponding to SMcarriers of density 1 + p Proust-Taillefer-UBC collaboration, Badoux et al. arxiv: FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis
103 High field, low T measurements show a positive Hall co-e cient corresponding to SMcarriers of density p. This is likely due to the presence of antiferromagnetic order. FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis
104 High field, low T measurements show a positive Hall co-e cient corresponding to SMcarriers of density p Proust-Taillefer-UBC collaboration, Badoux et al. arxiv: FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis
105 Recent evidence for pseudogap metal as FL* Proust-Taillefer-UBC collaboration, Badoux et al. arxiv:
106 Recent evidence for pseudogap metal as FL* a Proust-Taillefer-UBC collaboration, Badoux et al. arxiv:
107 Recent evidence for pseudogap metal as FL* b p* 1.5 SDW CDW FL n H = V / e R H p p p Proust-Taillefer-UBC collaboration, Badoux et al. arxiv:
108 Recent evidence for pseudogap metal as FL* b p* 1.5 SDW CDW FL n H = V / e R H p p FL*?! p Proust-Taillefer-UBC collaboration, Badoux et al. arxiv:
109 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
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