Quantum criticality, the AdS/CFT correspondence, and the cuprate superconductors

Transcription

1 Quantum criticality, the AdS/CFT correspondence, and the cuprate superconductors Talk online: sachdev.physics.harvard.edu HARVARD

2 Frederik Denef, Harvard Max Metlitski, Harvard Sean Hartnoll, Harvard Christopher Herzog, Princeton Pavel Kovtun, Victoria Dam Son, Washington Eun Gook Moon, Harvard HARVARD

3 Outline 1. The superfluid-insulator transition Quantum criticality and the AdS/CFT correspondence 2. Graphene `Topological Fermi surface transition 3. The cuprate superconductors Fluctuating spin density waves, and pairing by topological gauge fluctuations

4 Outline 1. The superfluid-insulator transition Quantum criticality and the AdS/CFT correspondence 2. Graphene `Topological Fermi surface transition 3. The cuprate superconductors Fluctuating spin density waves, and pairing by topological gauge fluctuations

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

6 Insulator (the vacuum) at large U

7 Excitations:

8 Excitations:

9 Excitations of the insulator: S = d 2 rdτ τ ψ 2 + v 2 ψ 2 +(g g c ) ψ 2 + u 2 ψ 4

10 T S = d 2 rdτ τ ψ 2 + v 2 ψ 2 +(g g c ) ψ 2 + u 2 ψ 4 T KT Quantum critical ψ =0 ψ =0 Superfluid Insulator 0 g c CFT3 g

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

12 Classical vortices and wave oscillations of the condensate Dilute Boltzmann/Landau gas of particle and holes T T KT Quantum critical Superfluid Insulator 0 g c g

13 CFT at T>0 T Quantum critical T KT Superfluid Insulator 0 g c g

14 Resistivity of Bi films Conductivity σ σ Superconductor (T 0) = σ Insulator (T 0) = 0 σ Quantum critical point (T 0) 4e2 h D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989) M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

15 Quantum critical transport Quantum perfect fluid with shortest possible relaxation time, τ R τ R k B T S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

16 Quantum critical transport Transport co-oefficients not determined by collision rate, but by universal constants of nature Electrical conductivity σ = 4e2 h [Universal constant O(1) ] K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

17 Quantum critical transport Transport co-oefficients not determined by collision rate, but by universal constants of nature Momentum transport η s viscosity entropy density = k B [Universal constant O(1) ] P. Kovtun, D. T. Son, and A. Starinets, Phys. Rev. Lett. 94, (2005)

18 T σ = 4e2 h Σ Quantum Σ, auniversalnumber. critical T KT Superfluid Insulator 0 g c g

19 σ = 4e2 h Σ ω k B T 1 ω k B T K. Damle and S. Sachdev, 1997

20 σ = 4e2 h Σ ω k B T Collisionless 1 ω k B T K. Damle and S. Sachdev, 1997

21 σ = 4e2 h Σ ω k B T Collision-dominated 1 ω k B T K. Damle and S. Sachdev, 1997

22

23

24 Now add Dirac fermions, generalize the gauge group to SU(N), and allow maximal supersymmetry in 2+1 dimensions. Yields a model whose transport properties can be computed exactly in the large N limit via the AdS/CFT correspondence. Most importantly, the large N limit exhibits hydrodynamic behavior, and the thermal equilibration time remains finite as N :this is a first for any solvable many body theory. Critical conductivity Σ = 2N 3/2 /3 ( selfdual value).

25 Now add Dirac fermions, generalize the gauge group to SU(N), and allow maximal supersymmetry in 2+1 dimensions. Yields a model whose transport properties can be computed exactly in the large N limit via the AdS/CFT correspondence. Most importantly, the large N limit exhibits hydrodynamic behavior, and the thermal equilibration time remains finite as N :this is a first for any solvable many body theory. Critical conductivity Σ = 2N 3/2 /3 ( selfdual value).

26 Now add Dirac fermions, generalize the gauge group to SU(N), and allow maximal supersymmetry in 2+1 dimensions. Yields a model whose transport properties can be computed exactly in the large N limit via the AdS/CFT correspondence. Most importantly, the large N limit exhibits hydrodynamic behavior, and the thermal equilibration time remains finite as N :this is a first for any solvable many body theory. Critical conductivity Σ = 2N 3/2 /3 ( selfdual value).

27 Now add Dirac fermions, generalize the gauge group to SU(N), and allow maximal supersymmetry in 2+1 dimensions. Yields a model whose transport properties can be computed exactly in the large N limit via the AdS/CFT correspondence. Most importantly, the large N limit exhibits hydrodynamic behavior, and the thermal equilibration time remains finite as N :this is a first for any solvable many body theory. Critical conductivity Σ = 2N 3/2 /3 ( selfdual value).

28 For boson-vortex system, self-dual value is Σ = 1, closed to the observed values. Selfdual values are obtained for all models with simple gravity duals, analogous to η/s = /(4πk B ).

29 Collisionless to hydrodynamic crossover of SYM3 Imχ(k, ω)/k 2 Im K k2 ω 2 Collisionless P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

30 Collisionless to hydrodynamic crossover of SYM3 Imχ(k, ω)/k 2 Im Dχ c Dk 2 iω Collision-dominated P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

31 Outline 1. The superfluid-insulator transition Quantum criticality and the AdS/CFT correspondence 2. Graphene `Topological Fermi surface transition 3. The cuprate superconductors Fluctuating spin density waves, and pairing by topological gauge fluctuations

32 Outline 1. The superfluid-insulator transition Quantum criticality and the AdS/CFT correspondence 2. Graphene `Topological Fermi surface transition 3. The cuprate superconductors Fluctuating spin density waves, and pairing by topological gauge fluctuations

33 Graphene t

34 Graphene t Conical Dirac dispersion

35 Quantum phase transition in graphene tuned by a gate voltage Electron Fermi surface

36 Quantum phase transition in graphene tuned by a gate voltage Hole Fermi surface Electron Fermi surface

37 Quantum phase transition in graphene tuned by a gate voltage There must be an intermediate quantum critical point where the Fermi surfaces reduce to a Dirac point Hole Fermi surface Electron Fermi surface

38 Quantum critical graphene Low energy theory has 4 two-component Dirac fermions, ψ σ, σ = 1...4, interacting with a 1/r Coulomb interaction S = d 2 rdτψ σ τ iv F σ ψ σ + e2 d 2 rd 2 r dτψ 1 2 σψ σ (r) r r ψ σ ψ σ (r )

39 Quantum critical graphene Low energy theory has 4 two-component Dirac fermions, ψ σ, σ = 1...4, interacting with a 1/r Coulomb interaction S = d 2 rdτψ σ τ iv F σ ψ σ + e2 d 2 rd 2 r dτψ 1 2 σψ σ (r) r r ψ σ ψ σ (r ) Dimensionless fine-structure constant α = e 2 /(v F ). RG flow of α: dα d = α Behavior is similar to a conformal field theory (CFT) in 2+1 dimensions with α 1/ ln(scale)

40 Quantum phase transition in graphene T (K) Quantum critical Dirac liquid Hole Fermi liquid 1 n (1 + λ ln Λ n ) Electron Fermi liquid n /m 2

41 Quantum critical transport in graphene σ(ω) = e 2 h e 2 hα 2 (T ) π 2 + O O 1 ln(λ/ω) 1 ln(α(t )), ω k B T, ω k B T α 2 (T ) η s = k B α 2 (T ) where the fine structure constant is α(t )= α 1+(α/4) ln(λ/t ) T 0 4 ln(λ/t ) L. Fritz, J. Schmalian, M. Müller and S. Sachdev, Physical Review B 78, (2008) M. Müller, J. Schmalian, and L. Fritz, Physical Review Letters 103, (2009)

42 Previously unsolved: general quantum critical transport theory for arbitrary µ, applied magnetic field B, and small impurity density, and general ω/t. T (K) Quantum critical Dirac liquid n (1 + λ ln Λ n ) Hole Fermi liquid Electron Fermi liquid n /m 2 S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

43 Previously unsolved: general quantum critical transport theory for arbitrary µ, applied magnetic field B, and small impurity density, and general ω/t. maps onto quasinormal modes of a Reissner-Nordstorm black hole in AdS 4. T (K) Quantum critical Dirac liquid 1 n (1 + λ ln Λ n ) Hole Fermi liquid Electron Fermi liquid n /m 2 S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

44 Magnetohydrodynamics of quantum criticality We used the AdS/CFT connection to derive many new relations between thermoelectric transport co-efficients in the quantum critical regime. As a simple example, in zero magnetic field, we can write the electrical conductivity as σ = σ Q + e 2 ρ 2 v 2 ε + P πδ(ω) where σ Q is the universal conductivity of the CFT, ρ is the charge density, ε is the energy density and P is the pressure. The same quantities also determine the thermal conductivity, κ: k 2 2 κ = σ B T ε + P Q e 2 k B T ρ S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

45 Magnetohydrodynamics of quantum criticality We used the AdS/CFT connection to derive many new relations between thermoelectric transport co-efficients in the quantum critical regime. As a simple example, in zero magnetic field, we can write The same results were later obtained from the equations of the electrical conductivity as generalized relativistic magnetohydrodynamics, and from a solution of the quantum Boltzmann σ = σ Q + e 2 ρ 2 v 2 ε + P πδ(ω) equation. So where the σresults Q is the apply universal to experiments conductivity graphene, of the CFT, theρ cuprates, is the charge density, ε is the energy density and P is the pressure. and to the dynamics of black holes. The same quantities also determine the thermal conductivity, κ: k 2 2 κ = σ B T ε + P Q e 2 k B T ρ S.A. Hartnoll, P.K. Kovtun, M. Müller, and S. Sachdev, Phys. Rev. B (2007)

46 Magnetohydrodynamics of quantum criticality We used the AdS/CFT connection to derive many new relations between thermoelectric transport co-efficients in the quantum critical regime. As a simple example, in zero magnetic field, we can write the electrical conductivity as σ = σ Q + e 2 ρ 2 v 2 ε + P πδ(ω) where σ Q is the universal conductivity of the CFT, ρ is the charge density, ε is the energy density and P is the pressure. The same quantities also determine the thermal conductivity, κ: k 2 2 κ = σ B T ε + P Q e 2 k B T ρ

47 Magnetohydrodynamics of quantum criticality We used the AdS/CFT connection to derive many new relations between thermoelectric transport co-efficients in the quantum critical regime. As a simple example, in zero magnetic field, we can write The same quantities also determine a Wiedemann-Franz - the electrical conductivity as like relation for thermal conductivity, κ at B =0 σ = σ Q + e 2 ρ 2 v 2 ε + P πδ(ω) k 2 2 κ = σ B T ε + P Q e 2. k B T ρ where σ Q is the universal conductivity of the CFT, ρ is the charge At B density, = 0 andε ρ is= the 0 we energy havedensity a Wiedemann-Franz and P is the pressure. relation for vortices The same quantities also determine the thermal conductivity, κ: κ = 1 2 k 2 v(ε + P ) BT. σ k 2 2 Q κ = σ B T k Bε TB + P Q e 2 k B T ρ

48 Magnetohydrodynamics of quantum criticality We used the AdS/CFT connection to derive many new relations between thermoelectric transport co-efficients in the quantum critical regime. As A second a simple example: example, In in an zero applied magnetic magnetic field, field we B, can the write dynamic the electrical transport conductivity co-efficients as exhibit a hydrodynamic cyclotron resonance at a frequency ω c σ = σ Q + e 2 ρ 2 v 2 ω ε + P πδ(ω) c = e Bρv 2 c(ε + P ) where σ Q is the universal conductivity of the CFT, ρ is the charge and damping density, constant ε is the γenergy density and P is the pressure. B The same quantities also determine 2 v 2 γ = σ Q the thermal conductivity, κ: c 2 (ε + P ). k 2 2 The same constants κ = determine σ B T the ε quasinormal + P frequency Q of the Reissner-Nordstrom eblack 2 hole. k B T ρ

49 Outline 1. The superfluid-insulator transition Quantum criticality and the AdS/CFT correspondence 2. Graphene `Topological Fermi surface transition 3. The cuprate superconductors Fluctuating spin density waves, and pairing by topological gauge fluctuations

50 Outline 1. The superfluid-insulator transition Quantum criticality and the AdS/CFT correspondence 2. Graphene `Topological Fermi surface transition 3. The cuprate superconductors Fluctuating spin density waves, and pairing by topological gauge fluctuations

51 The cuprate superconductors

52 Central ingredients in cuprate phase diagram: antiferromagnetism, superconductivity, and change in Fermi surface Strange Metal Γ Γ

53 Fermi surface+antiferromagnetism Γ Hole states occupied Electron states occupied + Γ The electron spin polarization obeys S(r, τ) = ϕ (r, τ)e ik r where K is the ordering wavevector.

54 Start from the spin-fermion model Z = Dc α Dϕ exp ( S) S = dτ k λ c kα dτ i τ ε k c kα c iα ϕ i σ αβ c iβ e ik r i + dτd 2 r 1 2 ( r ϕ ) 2 ζ + 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4

55 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

56 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

57 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron pockets Hot spots S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

58 Hole-doped cuprates Increasing SDW order Γ Hole pockets Γ Γ Γ Hole pockets Electron pockets Hot spots Fermi surface breaks up at hot spots into electron and hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

59 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron pockets Hot spots Fermi surface breaks up at hot spots into electron and hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

60 Evidence for small Fermi pockets Fermi liquid behaviour in an underdoped high Tc superconductor Suchitra E. Sebastian, N. Harrison, M. M. Altarawneh, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich arxiv:

61 Theory of underdoped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron pockets Hot spots Fermi surface breaks up at hot spots into electron and hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

62 Theory of underdoped cuprates Increasing SDW order Γ Γ Γ Γ Hot spots Begin with SDW ordered state, and rotate to a frame polarized along the local orientation of the SDW order ˆϕ c c = R ψ+ ψ ; R ˆϕ σ R = σ z ; R R =1 H. J. Schulz, Physical Review Letters 65, 2462 (1990) B. I. Shraiman and E. D. Siggia, Physical Review Letters 61, 467 (1988). J. R. Schrieffer, Journal of Superconductivity 17, 539 (2004)

63 Theory of underdoped cuprates With R = z z z z or ˆϕ = z α σ αβ z β the theory is invariant under z α e iθ z α ; ψ + e iθ ψ + ; ψ e iθ ψ. We obtain a U(1) gauge theory of bosonic neutral spinons z α ; spinless, charged fermions ψ ± with small pocket Fermi surfaces; an emergent U(1) gauge field A µ. S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Phys. Rev. B 80, (2009).

64 Begin with a CFT3: the CP 1 model. L z = 1 γ ( µ ia µ )z α 2 ; z α 2 =1 Higgs γ c Coulomb γ CFT3

65 Begin with a CFT3: the CP 1 model. L z = 1 γ ( µ ia µ )z α 2 ; z α 2 =1 Antiferromagnetic order z α =0 Higgs γ c Spin liquid/ Valence bond solid z α =0 Coulomb CFT3 γ

66 Begin with a CFT3: the CP 1 model. Add probe non-relativistic fermions, g + and g, with opposite gauge charges ε L f = g + τ ia τ 1 2 ia g + 2m + g τ + ia τ ia 2m g CFT3 k g +

67 Begin with a CFT3: the CP 1 model. Add probe non-relativistic fermions, g + and g, with opposite gauge charges Turn on fermion chemical potential: leads to a marginal Fermi liquid L f = g + τ ia τ µ 1 2 ia g + 2m + g τ + ia τ µ ia 2m ε g µ CFT3 k g +

68 Complete theory L = L z + L f L z = 1 γ ( µ ia µ )z α 2 ; z α 2 =1 L f = g + τ ia τ µ 1 2 ia g + 2m + g τ + ia τ µ ia 2m g V. Galitski and S. Sachdev, Phys. Rev. B 79, (2009).

69 Theory has many similarities to holographic superconductors (Gubser, Hartnoll, Herzog, Horowitz) solved via the AdS/CFT correspondence, which (presumably) describe SYM3 theories in which gluinos pair via exchange of gluons into color singlets, and then Bose condense: Fermi surfaces with non-fermi singularities in spectral functions Cooper pairs which are gauge neutral Are obtained after doping a CFT3 with finite density of a conserved global charge Fermion and current spectral functions in superconducting and normal states have many similarities to cuprates

70 Begin with a CFT3: the CP 1 model. Add probe non-relativistic fermions, g + and g, with opposite gauge charges Turn on fermion chemical potential: leads to a marginal Fermi liquid of g ± (not electrons) L f = g + τ ia τ µ 1 2 ia g + 2m + g τ + ia τ µ ia 2m ε g µ CFT3 k g +

71 Begin with a CFT3: the CP 1 model. Add probe non-relativistic fermions, g + and g, with opposite gauge charges Turn on fermion chemical potential: leads to a marginal Fermi liquid of g ± (not electrons) G( k, ω) = 1 ω v F ( k k F )+cω[ln( ω )+iπsgn(ω)] ε g ± µ CFT3 k g +

72 Begin with a CFT3: the CP 1 model. Add probe non-relativistic fermions, g + and g, with opposite gauge charges Turn on fermion chemical potential: leads to a marginal Fermi liquid of g ± (not electrons) Low T state is a superconductor with g + g = = 0 ε g CFT3 k g +

73 Why is the pairing d-wave? ncreasing SDW Electron c 2α, spinless fermion g ± Γ Electron c 1α, spinless fermion g ± Focus on pairing near (π, 0), (0, π), where ψ ± g ±, and the electron operators are g+ c2 g+ = R c z ; = R 1 g c z 2 g z z R z z z. c1

74 Why is the pairing d-wave? ncreasing SDW Electron c 2α, spinless fermion g ± Γ Electron c 1α, spinless fermion g ± Focus on pairing near (π, 0), (0, π), where ψ ± g ±, and the electron operators are g+ c2 g+ = R c z ; = R 1 g c z 2 g z z R z z z. c1

75 Why is the pairing d-wave? Fluctuating pocket theory for electrons near (0, π) and (π, 0) Attractive gauge forces lead to simple s-wave pairing of the g ± g + g = For the physical electron operators, this pairing implies c 1 c 1 = z α 2 c 2 c 2 = z α 2 i.e. d-wave pairing! R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Phys. Rev. B 78, (2008).

76 T=0 Phase diagram L z = 1 γ ( µ ia µ )z α 2 ; z α 2 =1 Antiferromagnetic order z α =0 Higgs Spin liquid/ Valence bond solid z α =0 Coulomb γ c CFT3 γ

77 T=0 Phase diagram L z = 1 γ ( µ ia µ )z α 2 ; z α 2 =1 L f = g + τ ia τ µ 1 2 ia g + 2m + g τ + ia τ µ ia 2m g d-wave superconductivity Antiferromagnetic order z α =0 Spin liquid/ Valence bond solid z α =0 γ c E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) CFT3 γ

78 T=0 Phase diagram L z = 1 γ ( µ ia µ )z α 2 ; z α 2 =1 L f = g + τ ia τ µ 1 2 ia g + 2m + g τ + ia τ µ ia 2m g d-wave superconductivity Antiferromagnetic order z α =0 Spin liquid/ Valence bond solid z α =0 γ c E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) CFT3 γ

79 T=0 Phase diagram Competition between antiferromagnetism and superconductivity shrinks region of antiferromagnetic order: feedback of probe fermions on CFT is important d-wave superconductivity Antiferromagnetic order z α =0 Spin liquid/ Valence bond solid z α =0 γ c E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) CFT3 γ

80 Theory of quantum criticality in the cuprates T * Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface V. Galitski and S. Sachdev, Phys. Rev. B 79, (2009). Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave superconductor E. G. Moon and S. Sachdev, Phys. Rev. B 80, (2009) Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

81 Theory of quantum criticality in the cuprates T * Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface V. Galitski and S. Sachdev, Phys. Rev. B 79, (2009). Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave superconductor E. G. Moon and S. Sachdev, Phys. Rev. B 80, (2009) Spin density wave (SDW) Physics of competition: d-wave SC and SDW eat up same pieces of the large Fermi surface.

82 T T * Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface T sdw d-wave SC E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) SDW (Small Fermi pockets) H SC+ SDW M SC "Normal" (Large Fermi surface)

83 Similar phase diagram for CeRhIn5 G. Knebel, D. Aoki, and J. Flouquet, arxiv:

84 Similar phase diagram for the pnictides Ort Tet Ishida, Nakai, and Hosono arxiv: v1 50 AFM / Ort 0 SC S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman, arxiv:

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

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

87 Conclusions Gauge theory for pairing of Fermi pockets in a metal with fluctuating spin density wave order: Many qualitative similarities to holographic strange metals and superconductors