Quantum criticality in condensed matter

Quantum criticality in condensed matter Inaugural Symposium of the Wolgang Pauli Center Hamburg, April 17, 2013 Subir Sachdev Scientific American 308, 44 (January 2013) HARVARD

Sommerfeld-Pauli-Bloch theory of metals, insulators, and superconductors: many-electron quantum states are adiabatically connected to independent electron states Metals E In Metal k

Boltzmann-Landau theory of dynamics of metals: Long-lived quasiparticles (and quasiholes) have weak interactions which can be described by a Boltzmann equation Metals E In Metal k

Modern phases of quantum matter Not adiabatically connected to independent electron states: many-particle quantum entanglement, and no quasiparticles

Outline 1. Superfluid-insulator transition of ultracold atoms in optical lattices: Conformal field theories and gauge-gravity duality 2. Metals with antiferromagnetism, and high temperature superconductivity The pnictides and the cuprates

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

T Quantum critical T KT h i6=0 h i =0 Superfluid Insulator 0 g c c g

T! a complex field representing the Bose-Einstein condensate of the superfluid T KT Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T T KT Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T V Particles and holes correspond to the 2 normal modes in the oscillation of about = 0. Quantum critical T KT Re( ) Ψ Im( ) Ψ h i6=0 h i =0 Superfluid Insulator 0 g c c g

Insulator (the vacuum) at large repulsion between bosons

Excitations of the insulator: Particles

Excitations of the insulator: Holes

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T 1 V Nambu Goldstone mode Re( Ψ) T KT Quantum Higgs mode Im( Ψ) j/j c 1 critical Nambu-Goldstone mode is the oscillation in the phase at a constant non-zero. h i6=0 h i =0 Superfluid Insulator 0 g c c g

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T A conformal field theory in 2+1 spacetime dimensions: T KT a CFT3 Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T Quantum state with complex, many-body, long-range quantum Quantum entanglement critical T KT h i6=0 h i =0 Superfluid Insulator 0 g c c g

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T No well-defined normal modes, or particle-like excitations Quantum critical T KT h i6=0 h i =0 Superfluid Insulator 0 g c c g

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T 1 V Nambu Goldstone mode Re( Ψ) T KT Quantum Higgs mode Im( Ψ) j/j c 1 critical Higgs mode is the oscillation in the amplitude. Thisdecays rapidly by emitting pairs of Nambu-Goldstone modes. h i6=0 h i =0 Superfluid Insulator 0 g c c g

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T 1 V Nambu Goldstone mode Re( Ψ) T KT Quantum Higgs mode Im( Ψ) j/j c 1 critical Despite rapid decay, there is a well-defined Higgs quasi-normal mode. This is associated with a pole in the lower-half of the complex frequency plane. h i6=0 h i =0 Superfluid Insulator 0 g c D. Podolsky and S. Sachdev, Phys. Rev. B 86, 054508 (2012). c g

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2! 1 V j/j c 1 Nambu Goldstone mode Re( Ψ) Higgs mode Im( Ψ) D. Podolsky and S. Sachdev, Phys. Rev. B 86, 054508 (2012). The Higgs quasi-normal mode is at the frequency! pole = i 4 + 1 N 16 4+ p 2 log 3 2 p 2 2 +2.46531203396 i! + O 1 N 2 where is the particle gap at the complementary point in the paramagnetic state with the same value of c, and N = 2 is the number of vector components of. The universal answer is a consequence of the strong interactions in the CFT3

Observation of Higgs quasi-normal mode across the superfluid-insulator transition of ultracold atoms in a 2-dimensional optical lattice: Response to modulation of lattice depth scales as expected from the LHP pole a h 0 /U j 0 0.03 0.06 0.09 0.12 0.15 1.2 Mott Superfluid 1 1 Insulator 0.8 0.6 0.4 0.2 3 0 0 0.5 1 1.5 2 2.5 j/j c 2 b Lattice depth (E r ) 20 15 10 5 Lattice loading Modulation Hold time Ramp to atomic limit 0 T mod = 20 0 100 200 300 400 k B T/U V 0 0.12 0.14 T tot = 200 ms Time (ms) 0.12 2012 0 400 800 mod (Hz) Temperature measurement A = 0.03V 0 Figure 1 Illustration of the Higgs mode and experimental sequence. a, Classical energy b density 0.17V as a function of the order parameter Y. Within the ordered (superfluid) phase, Nambu Goldstone 1 and Higgs modes arise from phase and amplitude0.15 modulations (blue and red arrows in panel 1). As the coupling j 5 J/U (see main text) approaches 0 the critical value j c, the energy density transforms into a function with a minimum at Y 5 0 (panels 2 and 3). 0.13 Simultaneously, the curvature in the radial direction decreases, leading to a characteristic reduction of the excitation frequency for the Higgs mode. In the 0.11 disordered (Mott insulating) phase, two gapped modes exist, respectively corresponding to particle and hole excitations in our case (red and blue arrow in 0.18 panel 3). b, The Higgs mode can 2 be excited with a periodic modulation of the coupling j, which amounts to a shaking of the classical energy density 0.16 potential. In the experimental sequence, 0 this is realized by a modulation of the optical lattice potential 0.14 (see main text for details). t 5 1/n mod ; E r, lattice recoil energy. V 0 = 8E r j/j c = 2.2 V 0 = 9E r j/j c = 1.6 system with a recently developed scheme based on single-atomresolved detection0.18 24. It is 3the high sensitivity of this method that allowed us to reduce the modulation amplitude by almost an order of magnitude compared 0.16 with earlier experiments 20,21 and to stay well within the linear response regime 0 (Supplementary Information). V 0 = 10E r j/j c = 1.2 Macmillan Publishers Lim e Higgs Manuel mode. Endres, a, The Takeshi fittedfukuhara, gap valuesdavid hn 0 /UPekker, Marc region. Cheneau, b, Temperature Peter Schaub, response Christian to lattice Gross, modulati stic softening Eugene close Demler, to the Stefan critical Kuhr, point and inimmanuel quantitative Bloch, blue Nature line) 487, and454 fit with (2012). an error function (solid black

[Δ(-g)/J]2/ν 3SSF(ω)/J SSF(ω)/J Hence, the scalar susceptibility defined3 in Eq. (2) Physics criticality in terms of the order parameter fields and 45, 660 120, (198 inates the experimental responsemode at low frequencies [7] J. P. Pouget Observation of Higgs quasi-normal in quantum Monte Carlo [4] P. Littlewood derivatives, Sato, Phys. R wave 3vectors. (1982). g =0.2424 2 2 2 2 Scaling of spectral response [8] R. Yusupov ~have ~ ~ g =0.0924 summary, we calculated the scalar susceptibil (x, ) = + @ + ( ) +... (10) [5] C. Ru egg et µ g =0.0462 2.5 2 functions predicted ini. A eck an [9] or relativistic O(2) and O(3) models in 2+1D.dimen(2008). 1.5 Podolsky and S. Sachdev, 8934 (1992). ong as2 6= 0, the 1first term is more relevant than the [6] Y. Ren, Z. X Phys. Rev. B 86, 054508 (2012). [10] Physics K. G. Wilson Hence, the scalar susceptibility defined in Eq. (2) 0.5 120, 1.5 (1972). inates the experimental response [7] J. P. Pouget 2 4 6 8 at low frequencies ω/j [11] Sato, N. Lindner 1 Phys. R wave vectors. 054512 (2010 Kun Chen, Longxiang Liu, [8] R. Yusupov summary, we have calculated the scalar susceptibil0.5 Youjin Deng, Lode Pollet,[12] D. Podolsky A eck an and Nikolay Prokof ev, [9] I. or relativistic O(2) and O(3) models in 2+1 dimenreview B 84 0 0 5 10 15 20 ω/δ(-g) 25 30 35 40 IG. 3. (Color online) Collapse of spectral functions for difrent values of U along trajectory i in the SF phase, labeled y the detuning g = (U Uc )/J. Inset: original data for SF (!). 2.5 ν 3 1.5 SSF(ω)/J SSF(ω)/J 2 1.5 1 0.5 gµ =0.40 gµ =0.30 gµ =0.20 8934 (1992). [13] K. S. Sachdev, [10] G. Wilson [14] (1972). W. Zwerger, [15] N. N. Dupuis, [11] Lindner P [16] 054512 M. Endres e (2010 [17] D. D. Podolsky [12] Podolsky (2012). B 84 Review [18] S. L. Sachdev, Pollet and [13] Snir Gazit, Daniel Podolsky, 10401 (2012) W. Zwerger, and Assa [14] Auerbach, [19] N. S. Sachdev, arxiv:1212.3759 [15] Dupuis, P ed. Endres (Cambrid [16] M. e [20] M. P. A. Fish arxiv:1301.3139

S = Z d 2 rdt @ t 2 c 2 r r 2 V ( ) V ( ) = ( c) 2 + u 2 2 T A conformal field theory in 2+1 spacetime dimensions: T KT a CFT3 Quantum critical h i6=0 h i =0 Superfluid Insulator 0 g c c g

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

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

CFT3 at T>0 T Quantum critical T KT Superfluid Insulator 0 g c c g

CFT3 at T>0 T T KT Quantum Boltzmann theory of particles/holes/vortices critical does not apply Superfluid Insulator 0 g c c g

CFT3 at T>0 T T KT Quantum Needed: Accurate critical theory of quantum critical dynamics Superfluid Insulator 0 g c c g

Quantum critical dynamics Quantum nearly perfect fluid with shortest possible local equilibration time, eq eq = C ~ k B T where C is a universal constant. Response functions are characterized by poles in LHP with! k B T/~. These poles (quasi-normal modes) appear naturally in the holographic theory. (Analogs of Higgs quasi-normal mode.) S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

Quantum critical dynamics Transport co-oe cients not determined by collision rate of quasiparticles, but by fundamental constants of nature Conductivity = Q2 h [Universal constant O(1) ] (Q is the charge of one boson) M.P.A. Fisher, G. Grinstein, and S.M. Girvin, Phys. Rev. Lett. 64, 587 (1990) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

Renormalization group: ) Follow coupling constants of quantum many body theory as a function of length scale r r x i J. McGreevy, arxiv0909.0518

Renormalization group: ) Follow coupling constants of quantum many body theory as a function of length scale r r x i Key idea: ) Implement r as an extra dimension, and map to a local theory in d + 2 spacetime dimensions. J. McGreevy, arxiv0909.0518

Holography x i r For a relativistic CFT in d spatial dimensions, the metric in the holographic space is fixed by demanding the scale transformation (i =1...d) x i! x i, t! t, ds! ds

Holography x i r This gives the unique metric ds 2 = 1 r 2 dt 2 + dr 2 + dx 2 i This is the metric of anti-de Sitter space AdS d+2.

AdS/CFT correspondence AdS 4 R 2,1 Minkowski CFT3 x i zr This emergent spacetime is a solution of Einstein gravity with a negative cosmological constant Z S E = d 4 x p apple R 1 g 2apple 2 + 6L 2

Gauge-gravity duality at non-zero temperatures r There is a family of solutions of Einstein gravity which describe non-zero temperatures S E = Z d 4 x p g apple 1 2apple 2 R + 6L 2

Gauge-gravity duality at non-zero temperatures r A 2+1 dimensional system at T > 0 with couplings at its quantum critical point S E = Z d 4 x p g apple 1 2apple 2 R + 6L 2

Gauge-gravity duality at non-zero temperatures r A 2+1 dimensional system at T > 0 with couplings at its quantum critical point A horizon, similar to the surface of a black hole! Z S E = d 4 x p g apple 1 2apple 2 R + 6L 2

Gauge-gravity duality at non-zero temperatures r A 2+1 dimensional system at T > 0 with couplings at its quantum critical point The temperature and entropy of the horizon equal those of the quantum critical point

AdS4 theory of quantum criticality Most general e ective holographic theory for linear charge transport with 4 spatial derivatives: S bulk = 1 + Z g 2 M Z d 4 x p g d 4 x p g apple 1 2apple 2 apple 1 4 F abf ab + L 2 C abcd F ab F cd R + 6L 2 This action is characterized by 3 dimensionless parameters, which can be linked to data of the CFT (OPE coe cients): 2-point correlators of the conserved current J µ and the stress energy tensor T µ, and a 3-point T, J, J correlator. R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, 066017 (2011) D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Phys. Rev. B 87, 085138 (2013),

AdS4 theory of quantum criticality h(!) Q 2 1.5 (1) 1.0 = 1 12 =0 0.5 = 1 12 0.0 0.0 0.5 1.0 1.5 4 T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)

AdS4 theory of quantum criticality h(!) Q 2 1.5 (1) 1.0 = 1 12 =0 0.5 0.0 = 1 12 Stability constraints on the e ective theory ( < 1/12) allow only a limited!-dependence in the conductivity. This contrasts with the Boltzmann theory in which (!)/ 1 becomes very large in the regime of its validity. 0.0 0.5 1.0 1.5 4 T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, 066017 (2011)

AdS4 theory of quantum criticality PRL 95, 180603 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending 28 OCTOBER 2005 Universal Scaling of the Conductivity at the Superfluid-Insulator Phase Transition Jurij Šmakov and Erik Sørensen Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada (Received 30 May 2005; published 27 October 2005) The scaling of the conductivity at the superfluid-insulator quantum phase transition in two dimensions is studied by numerical simulations of the Bose-Hubbard model. In contrast to previous studies, we focus on properties of this model in the experimentally relevant thermodynamic limit at finite temperature T.We find clear evidence for deviations from! k scaling of the conductivity towards! k =T scaling at low Matsubara frequencies! k. By careful analytic continuation using Padé approximants we show that this behavior carries over to the real frequency axis where the conductivity scales with!=t at small frequencies and low temperatures. We estimate the universal dc conductivity to be 0:45 5 Q 2 =h, distinct from previous estimates in the T 0,!=T 1 limit. σ ( ω /T ) 0.8 0.6 0.4 0.2 (b) L τ =32 L τ =40 L τ =48 L τ =64 SSE σ(iω k ) 0.4 0.3 0.2 (c) β=2 β=4 β=6 β=8 β=10 ω k /(2π 0 0 25 50 75 ω/t 0.1 0 10 20 30 ω k 0

AdS4 theory of quantum criticality PRL 95, 180603 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending 28 OCTOBER 2005 Universal Scaling of the Conductivity at the Superfluid-Insulator Phase Transition Jurij Šmakov and Erik Sørensen Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada (Received 30 May 2005; published 27 October 2005) The scaling of the conductivity at the superfluid-insulator quantum phase transition in two dimensions is studied by numerical simulations of the Bose-Hubbard model. In contrast to previous studies, we focus on properties of this model in the experimentally relevant thermodynamic limit at finite temperature T.We find clear evidence for deviations from! k scaling of the conductivity towards! k =T scaling at low Matsubara frequencies! k. By careful analytic continuation using Padé approximants we show that this behavior carries over to the real frequency axis where the conductivity scales with!=t at small frequencies and low temperatures. We estimate the universal dc conductivity to be 0:45 5 Q 2 =h, distinct from previous estimates in the T 0,!=T 1 limit. QMC yields (0)/ 1 1.36 Holography yields (0)/ 1 =1+4 with apple 1/12. Maximum possible holographic value (0)/ 1 =1.33 W. Witzack-Krempa and S. Sachdev, arxiv:1302.0847

Traditional CMT Identify quasiparticles and their dispersions Compute scattering matrix elements of quasiparticles (or of collective modes) These parameters are input into a quantum Boltzmann equation Deduce dissipative and dynamic properties at nonzero temperatures Holography and black-branes Start with strongly interacting CFT without particle- or wave-like excitations Compute OPE co-efficients of operators of the CFT Relate OPE co-efficients to couplings of an effective graviational theory on AdS Solve Einsten-Maxwell equations. Dynamics of quasinormal modes of black branes.

Iron pnictides: a new class of high temperature superconductors Ishida, Nakai, and Hosono arxiv:0906.2045v1

Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, 184519 (2010)

Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " Short-range entanglement in state with Neel (AF) order 2.0 AF SDW 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, 184519 (2010)

Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity Superconductor Bose condensate of pairs of electrons S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, 184519 (2010) Short-range entanglement 0

Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity 0 Ordinary metal S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, 184519 (2010) (Fermi liquid)

Resistivity 0 + AT Strange Metal no quasiparticles, Landau-Boltzmann theory does not apply BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, 184519 (2010)

High temperature superconductors YBa 2 Cu 3 O 6+x

Strange Metal

Strange Metal Pseudogap

Fermi surface+antiferromagnetism Metal with large Fermi surface + The electron spin polarization obeys S(r, ) = (r, )e ik r where K is the ordering wavevector.

Fermi surface+antiferromagnetism Metal with large Fermi surface

Fermi surface+antiferromagnetism Fermi surfaces translated by K =(, ).

Fermi surface+antiferromagnetism Hot spots

Fermi surface+antiferromagnetism Electron and hole pockets in antiferromagnetic phase with antiferromagnetic order parameter h~' i6=0

Fermi surface+antiferromagnetism ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface r

V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt- Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010) Unconventional pairing at and near hot spots

D c k c k E = " S (cos k x cos k y ) V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt- Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010) S S Unconventional pairing at and near hot spots

Sign-problem-free Quantum Monte Carlo for antiferromagnetism in metals P ± (x max ) 10 x 10 4 8 6 4 2 _ P _ + P _ r c L = 14 L = 10 L = 12 0 2 2 1 0 1 2 3 r s/d pairing amplitudes P + /P as a function of the tuning parameter r E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

Fermi surface+antiferromagnetism T * Fluctuating Fermi pockets Quantum Strange Critical Metal Large Fermi surface ncreasing SDW Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = x m

Fermi surface+antiferromagnetism Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Quantum Strange Critical Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) QCP for the onset of SDW order is actually within a superconductor

What about the pseudogap? Strange Metal Pseudogap

There is an approximate pseudospin symmetry in metals with antiferromagnetic spin correlations. The pseudospin partner of d-wave superconductivity is an incommensurate d-wave bond order These orders form a pseudospin doublet, which is responsible for the pseudogap phase. M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) T. Holder and W. Metzner, Phys. Rev. B 85, 165130 (2012) C. Husemann and W. Metzner, Phys. Rev. B 86, 085113 (2012) K. B. Efetov, H. Meier, and C. Pépin, arxiv:1210.3276. S. Sachdev and R. La Placa, arxiv:1303.2114

Pseudospin symmetry of the exchange interaction H J = X i<j J ij ~ Si ~S j with ~ S i = 1 2 c i ~ c i is the antiferromagnetic exchange interaction. Introduce the Nambu spinor i" = ci" c i#, i# = ci# c i" Then we can write X H J = 1 8 J ij i ~ i j ~ i i<j which is invariant under independent SU(2) pseudospin transformations on each site i! U i i This pseudospin (gauge) symmetry is important in classifying spin liquid ground states of H J. It is fully broken by the electron hopping t ij but does have remnant consequences in doped spin liquid states. I. A eck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B 38, 745 (1988) E. Dagotto, E. Fradkin, and A. Moreo, Phys. Rev. B 38, 2926 (1988) P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

Pseudospin symmetry of the exchange interaction H tj = X i,j t H J = ij c i c j + X J ij Si ~ ~S j i<j with ~ S i = 1 2 c i ~ c i is the antiferromagnetic exchange interaction. Introduce the Nambu spinor i" = ci" c i#, i# = ci# c i" Then we can write X H J = 1 8 J ij i ~ i j ~ i i<j which is invariant under independent SU(2) pseudospin transformations on each site i! U i i This pseudospin (gauge) symmetry is important in classifying spin liquid ground states of H J. It is fully broken by the electron hopping t ij but does have remnant consequences in doped spin liquid states. I. A eck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B 38, 745 (1988) E. Dagotto, E. Fradkin, and A. Moreo, Phys. Rev. B 38, 2926 (1988) P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

Pseudospin symmetry of the exchange interaction H tj = X i,j t H J = ij c i c j + X J ij Si ~ ~S j i<j with ~ S i = 1 2 c i ~ c i is the antiferromagnetic exchange interaction. Introduce the Nambu spinor i" = ci" c i#, i# = ci# c i" Then we can write X H J = 1 8 J ij i ~ i j ~ i i<j which is invariant under independent SU(2) pseudospin transformations on each site i! U i i This pseudospin (gauge) symmetry is important in classifying spin liquid ground states of H J. It is fully broken by the electron hopping t ij but does have remnant consequences in doped spin liquid states. We will start with the Néel state, and find important consequences of the pseudospin symmetry in metals with antiferromagnetic correlations. M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010)

Fermi surface+antiferromagnetism ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface r

Fermi surface+antiferromagnetism ncreasing SDW Focus on this region h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface r

Fermi surface+antiferromagnetism Hot spots

Fermi surface+antiferromagnetism Low energy theory for critical point near hot spots

Theory has fermions 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ~', interacting with coupling v 1 v 2 k y k x 1 fermions occupied Ar. Abanov and A. V. Chubukov, Phys. Rev. Lett. 84, 5608 (2000). 2 fermions occupied

S = Z apple d 2 rd 1 (@ iv 1 r r ) 1 + 2 (@ iv 2 r r ) 2 + 1 2 (r r ~' ) 2 + s 2 ~' 2 + u 4 ~' 4 ~' 1 ~ 2 + 2 ~ 1 v 1 v 2 k y k x 1 fermions occupied Ar. Abanov and A. V. Chubukov, Phys. Rev. Lett. 84, 5608 (2000). 2 fermions occupied

S = Z apple d 2 rd 1 (@ iv 1 r r ) 1 + 2 (@ iv 2 r r ) 2 + 1 2 (r r ~' ) 2 + s 2 ~' 2 + u 4 ~' 4 ~' 1 ~ 2 + 2 ~ 1 v 1 v 2 This low-energy theory is invariant under independent SU(2) pseudospin rotations on each pair of hot-spots: there is a global SU(2) SU(2) SU(2) SU(2) pseudospin symmetry. k y k x 1 fermions occupied M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) 2 fermions occupied

D c k Q/2, c k+q/2, E = Q (cos k x cos k y ) After pseudospin rotation on Q is 2k F wavevector half the hot-spots Q M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) Q Unconventional particle-hole pairing at and near hot spots

Incommensurate d-wave bond order Q M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) Q Q Q D c k Q/2, c k+q/2, E = Q (cos k x cos k y )

Incommensurate d-wave bond order +1 Bond density measures amplitude for electrons to be in spin-singlet valence bond. -1 M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010) X X c r c s = Q k e iq (r+s)/2 e ik (r s) D c k Q/2, c k+q/2, E where Q extends over Q =(±Q 0, ±Q 0 )withq 0 =2 /(7.3) and D E c k Q/2, c k+q/2, = Q (cos k x cos k y ) Note c r c s is non-zero only when r, s are nearest neighbors.

A Four Unit Cell Periodic Pattern of Quasi-Particle States Surrounding Vortex Cores in Bi 2 Sr 2 CaCu 2 O 8 J. E. Hoffman, 1 E. W. Hudson, 1,2 * K. M. Lang, 1 V. Madhavan, 1 H. Eisaki, 3 S. Uchida, 3 J. C. Davis 1,2 SCIENCE VOL 295 18 JANUARY 2002

Direct observation of competition between superconductivity and charge density wave order in YBa 2 Cu 3 O 6.67 NATURE PHYSICS VOL 8 DECEMBER 2012 J. Chang 1,2 *, E. Blackburn 3, A. T. Holmes 3, N. B. Christensen 4, J. Larsen 4,5, J. Mesot 1,2, Ruixing Liang 6,7, D. A. Bonn 6,7, W. N. Hardy 6,7, A. Watenphul 8, M. v. Zimmermann 8, E. M. Forgan 3 and S. M. Hayden 9 a 200 300 T C 250 T N T YBCO Intensity (cnts s ) 150 100 50 17 T 15 T 7.5 T 0 T Q = (1.695, 0, 0.5) Q = (0, 3.691, 0.5) (x4) T CDW Temperature (K) 200 150 100 50 T SDW T H T NMR T CDW a 0 0 50 100 150 T (K) b I (cnts s 1 ) 450 400 350 300 250 200 150 17 T c 0 T T = 2 K AF 0 0.00 0.05 0.10 Doping, p (holes/cu) Cut 1 17 T c Cut 1 17 T c Cut 1 400 400 0 T 0 T I (cnts s 1 ) 450 350 300 250 200 150 T = 66 K T = 150 K 1.64 1.66 1.68 1.70 1.72 1.74 1.64 1.66 1.68 1.70 1.72 1.74 1.64 1.66 1.68 1.70 1.72 1.74 h in (h, 0, 0.5) (r.l.u.) h in (h, 0, 0.5) (r.l.u.) h in (h, 0, 0.5) (r.l.u.) c I (cnts s 1 ) 450 350 300 250 200 150 0.15 SC 0.20

Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa 2 Cu 3 O y Tao Wu 1, Hadrien Mayaffre 1, Steffen Krämer 1, Mladen Horvatić 1, Claude Berthier 1, W. N. Hardy 2,3, Ruixing Liang 2,3, D. A. Bonn 2,3 & Marc-Henri Julien 1 8 S E P T E M B E R 2 0 1 1 V O L 4 7 7 N AT U R E 1 9 1 erved 120 80 Superconducting T (K) 40 Field-induced charge order Spin order 0 0.04 0.08 0.12 0.16 p (hole/cu)

Summary Conformal quantum matter New insights and solvable models for diffusion and transport of strongly interacting systems near quantum critical points using the methods of gauge-gravity duality. The description is far removed from, and complementary to, that of the quantum Boltzmann equation which builds on the quasiparticle/vortex picture. Good prospects for experimental tests of frequencydependent, non-linear, and non-equilibrium transport

Summary Antiferromagnetism in metals and the high temperature superconductors Antiferromagnetic quantum criticality leads to d-wave superconductivity (supported by sign-problemfree Monte Carlo simulations) Metals with antiferromagnetic spin correlations have nearly degenerate instabilities: to d-wave superconductivity, and to a charge density wave with a d- wave form factor. This is a promising explanation of the pseudogap regime.