Theory of Quantum Matter: from Quantum Fields to Strings

Transcription

1 Theory of Quantum Matter: from Quantum Fields to Strings Salam Distinguished Lectures The Abdus Salam International Center for Theoretical Physics Trieste, Italy January 27-30, 2014 Subir Sachdev Talk online: sachdev.physics.harvard.edu HARVARD

2 Outline 1. The simplest models without quasiparticles A. Superfluid-insulator transition of ultracold bosons in an optical lattice B. Conformal field theories in 2+1 dimensions and the AdS/CFT correspondence 2. Metals without quasiparticles A. Review of Fermi liquid theory B. A non-fermi liquid: the Ising-nematic quantum critical point C. Holography, entanglement, and strange metals

3 Outline 1. The simplest models without quasiparticles A. Superfluid-insulator transition of ultracold bosons in an optical lattice B. Conformal field theories in 2+1 dimensions and the AdS/CFT correspondence 2. Metals without quasiparticles A. Review of Fermi liquid theory B. A non-fermi liquid: the Ising-nematic quantum critical point C. Holography, entanglement, and strange metals

4 Basic characteristics of CFTs Ordinary quantum field theories are characterized by their particle spectrum, and the S-matrices describing interactions between the particles. The analog of these concepts for CFTs are the primary operators O a (x) and their operator product expansions (OPEs). Each primary operator is associated with a scaling dimension a, definedbythe (T = 0) expectation value (for the simplest case of scalar operators): ho a (x)o b (0)i = ab x 2 a

5 Basic characteristics of CFTs The OPE describes what happens when two operators come together at a single spacetime point (considering scalar operators only) lim h a(x 0 ) b (x) c (0)i = x 0!x f abc x a+ b + c The values of { a,f abc } determine (in principle) all observable properties of the CFT, as constrained by a complex set of conformal Ward identities. For the Wilson-Fisher CFT3, systematic methods exist to compute (in principle) all the { a,f abc }, and we will assume this data is known. Thisknowledge will be taken as an input to the holographic analysis.

6 String theory Allows unification of the standard model of particle physics with gravity. Low-lying string modes correspond to gauge fields, gravitons, quarks...

7 A D-brane is a d-dimensional surface on which strings can end. The low-energy theory on a D-brane has no gravity, similar to theories of entangled electrons of interest to us. In d = 2, we obtain strongly-interacting CFT3s. These are dual to string theory on anti-de Sitter space: AdS4.

8 A D-brane is a d-dimensional surface on which strings can end. The low-energy theory on a D-brane has no gravity, similar to theories of entangled electrons of interest to us. In d = 2, we obtain strongly-interacting CFT3s. These are dual to string theory on anti-de Sitter space: AdS4.

9 A D-brane is a d-dimensional surface on which strings can end. The low-energy theory on a D-brane has no gravity, similar to theories of entangled electrons of interest to us. In d = 2, we obtain strongly-interacting CFT3s. These are dual to string theory on anti-de Sitter space: AdS4.

10 AdS/CFT correspondence at zero temperature AdS 4 R 2,1 The symmetry group of isometries of AdS 4 maps to the group of conformal symmetries of the CFT3 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

11 AdS/CFT correspondence at zero temperature AdS 4 R 2,1 The symmetry group of isometries of AdS 4 maps to the group of conformal symmetries of the CFT3 Minkowski CFT3 x i zr L 2 dr 2 ds 2 = r dt 2 + dx 2 + dy 2

12 AdS/CFT correspondence at zero temperature Consider a CFT in D space-time dimensions with primary operators O a (x) with scaling dimension a. This is presumed to be equivalent to a dual gravity theory on AdS D+1 with action S bulk. The bulk theory has fields a(x,r) corresponding to each primary operator. The CFT and the bulk theory are related by the GKPW ansatz Z Z D a exp ( S bulk ) = exp d D x a0 (x)o a (x) bdy where the boundary condition is lim r!0 a (x,r)=r D a0(x). CFT

13 AdS/CFT correspondence at zero temperature For every primary operator O(x) inthecft,thereis a corresponding field (x,r) in the bulk (gravitational) theory. For a scalar operator O(x) of dimension,the correlators of the boundary and bulk theories are related by ho(x 1 )...O(x n )i CFT = Z n lim r!0 r 1...r n h (x 1,r 1 )... (x n,r n )i bulk where the wave function renormalization factor Z = (2 D).

14 AdS/CFT correspondence at zero temperature For a U(1) conserved current J µ of the CFT, the corresponding bulk operator is a U(1) gauge field A µ. With a Maxwell action for the gauge field S M = 1 4g 2 M Z d D+1 x p gf ab F ab we have the bulk-boundary correspondence hj µ (x 1 )...J (x n )i CFT = (Zg 2 M )n lim r 2 D...r 1 n 2 D r!0 ha µ (x 1,r 1 )...A (x n,r n )i bulk with Z = D 2.

15 AdS/CFT correspondence at zero temperature A similar analysis can be applied to the stress-energy tensor of the CFT, T µ. Its conjugate field must be a spin- 2 field which is invariant under gauge transformations: it is natural to identify this with the change in metric of the bulk theory. We write g µ =(L 2 /r 2 ) µ, and then the bulk-boundary correspondence is now given by ht µ (x 1 )...T (x n )i = CFT ZL 2 n lim r D...r 1 n D h µ (x 1,r 1 )... (x n,r n )i, r!0 bulk apple 2 with Z = D.

16 AdS/CFT correspondence at zero temperature So the minimal bulk theory for a CFT with a conserved U(1) current is the Einstein-Maxwell theory with a cosmological constant S = 1 4gM 2 Z + Z d 4 x p g d 4 x p gf ab F ab apple 1 2apple 2 R + 6L 2 This action is characterized by two dimensionless parameters: g M and L 2 /apple 2, which are related to the conductivity (!) =K and the central charge of the CFT..

17 AdS/CFT correspondence at zero temperature This minimal action also fixes multi-point correlators of the CFT: however these do not have the most general form allowed for a CFT. To fix these, we have to allow for highergradient terms in the bulk action. For the conductivity, it turns out that only a single 4 gradient term contributes S bulk = 1 + Z g 2 M d 4 x p g Z apple d 4 x p apple 1 g 4 F abf ab + L 2 C abcd F ab F cd R 1 2apple 2 + 6L 2, where C abcd is the Weyl tensor. The parameter can be related to 3-point correlators of J µ and T µ. Both boundary and bulk methods show that apple 1/12, and the bound is saturated by free fields. R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011) D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Physical Review B 87, (2013).

18 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R There is a family of solutions of Einstein gravity which describe non-zero S = Z d 4 x p g apple 1 2apple 2 R + 6L 2 temperatures

19 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R There is a family of solutions of Einstein gravity which describe non-zero temperatures A 2+1 dimensional system at its quantum critical point: k B T = 3~ 4 R. 2 apple L dr ds 2 2 = f(r)dt 2 + dx 2 + dy 2 r f(r) with f(r) =1 (r/r) 3

20 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R Black-brane at temperature of 2+1 dimensional quantum critical system A 2+1 dimensional system at its quantum critical point: k B T = 3~ 4 R. 2 apple L dr ds 2 2 = f(r)dt 2 + dx 2 + dy 2 r f(r) with f(r) =1 (r/r) 3

21 Black Holes Objects so massive that light is gravitationally bound to them.

22 Black Holes Objects so massive that light is gravitationally bound to them. In Einstein s theory, the region inside the black hole horizon is disconnected from the rest of the universe. Horizon radius R = 2GM c 2

23 Black Holes + Quantum theory Around 1974, Bekenstein and Hawking showed that the application of the quantum theory across a black hole horizon led to many astonishing conclusions

24 Quantum Entanglement across a black hole horizon _

25 Quantum Entanglement across a black hole horizon _

26 Quantum Entanglement across a black hole horizon _ Black hole horizon

27 Quantum Entanglement across a black hole horizon _ Black hole horizon

28 Quantum Entanglement across a black hole horizon There is a non-local quantum entanglement between the inside and outside of a black hole Black hole horizon

29 Quantum Entanglement across a black hole horizon There is a non-local quantum entanglement between the inside and outside of a black hole Black hole horizon

30 Quantum Entanglement across a black hole horizon There is a non-local quantum entanglement between the inside and outside of a black hole This entanglement leads to a black hole temperature (the Hawking temperature) and a black hole entropy (the Bekenstein entropy)

31 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R A 2+1 dimensional system at its quantum critical point: k B T = 3~ 4 R. Black-brane at temperature of 2+1 dimensional quantum critical system Beckenstein-Hawking entropy of black brane = entropy of CFT3

32 AdS/CFT correspondence at non-zero temperatures AdS4-Schwarzschild black-brane r R A 2+1 dimensional system at its quantum critical point: k B T = 3~ 4 R. Black-brane at temperature of 2+1 dimensional quantum critical system Friction of quantum criticality = waves falling into black brane

33 AdS/CFT correspondence at non-zero temperatures r 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 Characteristic damping time of quasi-normal modes: (k B /~) Hawking temperature

34 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

35 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.

36 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.

37 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.

38 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.

39 AdS4 theory of quantum criticality Most general e ective holographic theory for linear charge transport with 4 spatial derivatives: S bulk = 1 Z gm 2 d 4 x p apple 1 g 4 F abf ab + L 2 C abcd F ab F cd Z + d 4 x p apple R 1 g 2apple 2 + 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. Constraints from both the CFT and the gravitational theory bound apple 1/12 = R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, (2011) D. Chowdhury, S. Raju, S. Sachdev, A. Singh, and P. Strack, Phys. Rev. B 87, (2013)

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

41 AdS4 theory of quantum criticality h(!) Q (1) 1.0 = 1 12 =0 0.5 = 1 12 The > 0 result has similarities to the quantum-boltzmann result for transport of particle-like excitations T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

42 AdS4 theory of quantum criticality h(!) Q (1) 1.0 = 1 12 =0 0.5 = 1 12 The < 0 result can be interpreted as the transport of vortex-like excitations T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

43 AdS4 theory of quantum criticality h(!) Q (1) 1.0 = 1 12 = = 1 12 The = 0 case is the exact result for the large N limit of SU(N) gauge theory with N =8supersymmetry(the ABJM model). The -independence is a consequence of self-duality under particle-vortex duality (S-duality) T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

44 AdS4 theory of quantum criticality h(!) Q (1) 1.0 = 1 12 =0 0.5 = 1 12 Stability constraints on the e ective theory ( < 1/12) allow only a limited -dependence in the conductivity T R. C. Myers, S. Sachdev, and A. Singh, Physical Review D 83, (2011)

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46 Quantum Monte Carlo for lattice bosons σ(iω n )/σ Q T -> 0 βu=110 βu=100 βu=80 βu=70 βu=60 βu=50 βu=40 βu=30 βu= ω n /(2πT) (a) σ(iω n )/σ Q σ ω n /(2πT)= L τ T->0 L τ =160 L τ =128 L τ =96 L τ =80 L τ =64 L τ =56 L τ =48 L τ =40 L τ =32 L τ =24 L τ = ω n /(2πT) (b) FIG. 2. Quantum Monte Carlo data (a) Finite-temperature conductivity for a range of U in the L!1limit for the quantum rotor model at (t/u) c. The solid blue squares indicate the final T! 0 extrapolated data. (b) Finite-temperature conductivity in the L!1limit for a range of L for the Villain model at the QCP. The solid red circles indicate the final T! 0 extrapolated data. The inset illustrates the extrapolation to T = 0 for! n /(2 T ) = 7. The error bars are statistical for both a) and b). 4 W. Witczak-Krempa, E. Sorensen, and S. Sachdev, arxiv: See also K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof ev, arxiv:

47 AdS4 theory of quantum criticality shiwlêsq g = 0.08 g = wê2pt Good agreement between high precision Monte Carlo for imaginary frequencies, and holographic theory after rescaling e ective T and taking Q =1/g 2 M. W. Witczak-Krempa, E. Sorensen, and S. Sachdev, arxiv: See also K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof ev, arxiv:

48 AdS4 theory of quantum criticality 0.50 (!)/(e 2 /h) shwlêsq wê2pt Predictions of holographic theory, after analytic continuation to real frequencies W. Witczak-Krempa, E. Sorensen, and S. Sachdev, arxiv: See also K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof ev, arxiv: