Strong coupling problems in condensed matter and the AdS/CFT correspondence HARVARD arXiv: Reviews: Talk online: sachdev.physics.harvard.edu arXiv:

Frederik Denef, Harvard Sean Hartnoll, Harvard Christopher Herzog, Princeton Pavel Kovtun, Victoria Dam Son, Washington Frederik Denef, Harvard Sean Hartnoll, Harvard Christopher Herzog, Princeton Pavel Kovtun, Victoria Dam Son, Washington Yejin Huh, Harvard Max Metlitski, Harvard Yejin Huh, Harvard Max Metlitski, Harvard HARVARD

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

The Superfluid-Insulator transition Boson Hubbard model M.P. A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher, Phys. Rev. B 40, 546 (1989).

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Superfluid-insulator transition

Insulator (the vacuum) at large U

Excitations:

Excitations of the insulator:

CFT3

Classical vortices and wave oscillations of the condensate Dilute Boltzmann/Landau gas of particle and holes

CFT at T>0

D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989) Resistivity of Bi films M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

Quantum critical transport S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

Quantum critical transport K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

Quantum critical transport P. Kovtun, D. T. Son, and A. Starinets, Phys. Rev. Lett. 94, (2005), 8714 (1997).

Quantum critical transport

Density correlations in CFTs at T >0

Conformal mapping of plane to cylinder with circumference 1/T

Density correlations in CFTs at T >0 No hydrodynamics in CFT2s.

Density correlations in CFTs at T >0

K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

Density correlations in CFTs at T >0 K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007) Collisionless to hydrodynamic crossover of SYM3

P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007) Collisionless to hydrodynamic crossover of SYM3

Universal constants of SYM3 P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007) C. Herzog, JHEP 0212, 026 (2002)

Electromagnetic self-duality

K: a universal number analogous to the level number of the Kac-Moody algebra in 1+1 dimensions

C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/

The self-duality of the 4D abelian gauge fields leads to C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/

The self-duality of the 4D abelian gauge fields leads to C. Herzog, P. Kovtun, S. Sachdev, and D.T. Son, hep-th/

Electromagnetic self-duality

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

d-wave superconductivity in cuprates Electron states occupied

d-wave superconductivity in cuprates

4 two-component Dirac fermions

d-wave superconductivity in cuprates

Time-reversal symmetry breaking

Lattice rotation symmetry breaking

M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000) E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson,.-A. Kim,er, P. OretoE. Fradkivelson, Phys. Rev. BPhys. Rev. B 77, (2008).

M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000) Ising order and Dirac fermions couple via a “Yukawa” term. Nematic ordering Time reversal symmetry breaking

M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000) Ising order and Dirac fermions couple via a “Yukawa” term. Nematic ordering Time reversal symmetry breaking

Integrating out the fermions yields an effective action for the scalar order parameter Expansion in number of fermion spin components N f Y. Huh and S. Sachdev, Physical Review B 78, (2008).

Integrating out the fermions yields an effective action for the scalar order parameter Expansion in number of fermion spin components N f Y. Huh and S. Sachdev, Physical Review B 78, (2008).

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

1. Quantum-critical transport Collisionless-t0-hydrodynamic crossover of CFT3s 2. Exact solution from AdS/CFT Constraints from duality relations 3. Quantum criticality of Dirac fermions “Vector” 1/N expansion 4. Quantum criticality of Fermi surfaces The genus expansion

Quantum criticality of Pomeranchuk instability x y

Electron Green’s function in Fermi liquid (T=0)

Quantum criticality of Pomeranchuk instability x y

x y

x y

Quantum critical

Quantum criticality of Pomeranchuk instability Quantum critical Classical d=2 Ising criticality

Quantum criticality of Pomeranchuk instability Quantum critical D=2+1 Ising criticality ?

Quantum criticality of Pomeranchuk instability

Hertz theory

Quantum criticality of Pomeranchuk instability Hertz theory

Quantum criticality of Pomeranchuk instability Sung-Sik Lee, Physical Review B 80, (2009)

Quantum criticality of Pomeranchuk instability A string theory for the Fermi surface ?

AdS 4 -Reissner-Nordstrom black hole

T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: F. Denef, S. Hartnoll, and S. Sachdev, arXiv: Examine free energy and Green’s function of a probe particle

Short time behavior depends upon conformal AdS 4 geometry near boundary T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: F. Denef, S. Hartnoll, and S. Sachdev, arXiv:

Long time behavior depends upon near-horizon geometry of black hole T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: F. Denef, S. Hartnoll, and S. Sachdev, arXiv:

Radial direction of gravity theory is measure of energy scale in CFT T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: F. Denef, S. Hartnoll, and S. Sachdev, arXiv:

AdS 4 -Reissner-Nordstrom black hole

AdS 2 x R 2 near-horizon geometry

T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: Infrared physics of Fermi surface is linked to the near horizon AdS 2 geometry of Reissner-Nordstrom black hole

T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: AdS 4 Geometric interpretation of RG flow

T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: AdS 2 x R 2 Geometric interpretation of RG flow

Green’s function of a fermion T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: See also M. Cubrovic, J Zaanen, and K. Schalm, arXiv:

Green’s function of a fermion T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, arXiv: Similar to non-Fermi liquid theories of Fermi surfaces coupled to gauge fields, and at quantum critical points

Free energy from gravity theory F. Denef, S. Hartnoll, and S. Sachdev, arXiv:

General theory of finite temperature dynamics and transport near quantum critical points, with applications to antiferromagnets, graphene, and superconductors Conclusions

The AdS/CFT offers promise in providing a new understanding of strongly interacting quantum matter at non-zero density Conclusions