Quantum critical dynamics: CFT, Monte Carlo & holography. William Witczak-Krempa Perimeter Institute

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1 Quantum critical dynamics: CFT, Monte Carlo & holography William Witczak-Krempa Perimeter Institute Toronto, HEP seminar, Jan

2 S. / PI E. E. U. E. Katz, S. Sachdev, E. Sørensen, WWK, PRB 90, 2014 [Ed. Sugg.] WWK, E. Sørensen, S. Sachdev, Nat. Phys. 10, 2014 WWK, S. Sachdev, PRB 87, 2013 [Ed. Sugg.] WWK, S. Sachdev, PRB 86, 2012

3 Real time finite T response of strongly correlated quantum fluids (quasi)particles: lifetime (poles) If have sharp (quasi)particles, get dynamics from Boltzmann Eq. Systems without quasiparticles: Quantum critical phase transition Strongly coupled gauge theories, etc.

4 Quantum critical dynamics Field Theory AdS/ CFT Simulations Experiments

5 Conformal Field Theory Scale + Lorentz invariant QFT Best characterized quantum fluid w/out (quasi)particles Many open questions: dynamics, etc Why care? Experiments! Realistic models: numerics Gravity = CFT? AdS/CFT [Maldacena, etc]

6 Bosons in 2+1D H = t X hi,ji b i b j + U X i n i (n i 1) µ X i n i T Quantum critical t U Insulator CFT Superfluid têu O(2) universality class: Bose-Hubbard, quantum rotors, XY spins, etc

7 SSB of O(2) order parameter L =(@ t ~ ) 2 +(r ~ ) 2 + m u 4 Strongly coupled fixed point in 2d (Wilson-Fisher)

8 Superfluid [Bloch] Insulator 87 Rb [Endres et al]

9 Break conformal symmetry Characteristic time scale: 1 / T Short times ω >> T: probing near vacuum Long times ω << T: excitations interact strongly with thermal background

10 Operator Product Expansion A scalar primary op O(x), w/ scaling dim Δ: O(x) O(0) = 1 / x 2Δ OPE: [Wilson; Polyakov; Ferrara et al; etc]

11 Current µ J µ =0 (!) = 1 i! hj x(!)j x (!)i T Quantum critical Universal scaling function:! T Insulator CFT Superfluid têu (!) =

12 Current OPE J µ (x)j (0) = I µ 1 x C JJO x µ x O(0) T µ (0) x 6 + C JJT x + O(N) CFT: Relevant Stress tensor scalar ~ φ 2 Get OPE coefficients from JJ

13 Conductivity via OPE lim J O g (p) x(q)j x ( q + p) = q 1 (p) C JJO q p q 1 + C JJT q 2 [T xx T yy 12 (T xx + T yy )] p + Thermal average hoi T = BT (i! n )! n T T = 1 + b 1! n + b 2 T! n 3 +

14 Dominant op in OPE? For O(N) Wilson-Fisher, it s SCALAR: N = : Δ = 2 N = 2: Δ = O(x) =φ 2 (x) since Δ = 3 1/ν All CFTs have 1/ω 3 term because there s always a stress tensor Sometimes it dominates (Dirac CFT, QED3, N=4 super-y-m, etc)

15 Quantum Monte Carlo [WWK, Sorensen, Sachdev, Katz] Simulate lattice model for O(2) QCP imaginary frequencies 0.40 Δ = Fit: : /n /n /n 3 QMC Villain Model 2 (i!n)/ Q (i! n )! n T T = 1 + b 1! n + b 2 T! n n =! n /(2 T )

16 Long times & analytic continuation

17 Use AdS/CFT to generate a family of physically motivated (constrained) variational functions σ(ω/t)

18 σ via AdS/CFT [Maldacena etc] spacetime: B-Hole in AdS4 BH A m CFT J m u u = r 0 êr 1 0 A µ (t, x, y; u) $ J CFT µ (t, x, y) Solve classical EoM for A get J-correlator in CFT (!/T )= u A y (!,~0; u) A y (!,~0; u) u=0

19 1st try S bulk [A µ ]= Z d 4 x p g 1 g 2 4 F 2 + C abcd F ab F cd Derivative expansion C = traceless part of Riemann Asymptotics [WWK]: σ ~ σ + (T/ωn) CFT Tµν AdS gµν [Herzog, Kovtun, Sachdev, Son; Ritz, Ward; Myers, Sachdev, Singh]

20 Holographic fit: take 1 [WWK, Sorensen, Sachdev] shiwlêsq g = 0.08 [Not rescaled] [Rescaled] wê2pt apple! T ;

21 2nd try S bulk [A µ ]= Z d 4 x 1 g 2 4 [1 + '(x)] F ab F ab Add scalar field (dilaton) CFT O AdS φ Simplest Ansatz: fix profile using OPE of O(2) Wilson-Fisher CFT '(u) =u [Katz, Sachdev, Sorensen, WWK]

22 Holographic fit: take 2 [Katz, Sachdev, Sorensen, WWK] shiwl 2pêsQ Ê Ê =1.5; =0.6 Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê w n ê2pt

23 Real frequencies 2pêsQ wê2pt

24 Why dabble with black holes? Physical properties: Tailored holographic description to match asymptotics Sum rules Z 1 0 d! Re[ (!/T ) (1)] = 0 Analytic properties in common with field theory results Practical: real-time, small number of params, fast numerics

25 Sum rules [WWK, Sachdev; Gulotta, Herzog, Kaminski] Re8s< Z 1 0 d! [Re (!/T ) (1)] = S-dual version: Z 1 0 d! apple Re 1 (!/T ) 1 (1) =0 1 wê4pt N=8 supersymmetric ABJM O(N) N= Dirac CFT

26 Sum rule proof via OPE (i! n )! n T T = 1 + b 1! n + b 2 T! n 3 + σ analytic in upper half-plane Δ > 1 Integrable Close contour in UHP

27 Conclusions Quantum critical dynamics (CFTs) in 2+1D OPE to constrain short time physics Large ω conductivity (i! n )! n T T = 1 + b 1! n + b 2 T! n 3 + Input OPE data of CFT into simple holographic ansatz Can match Monte Carlo data of O(2) CFT w/out unphysical tweaks

28 Outlook Apply this program (OPE, sum rules) to other correlators & CFTs [WWK, arxiv:1501.xxxxx] Go beyond simplest holographic Ansatz [in progress with T. Sierens & R. Myers] Unitarity in proof of sum rules?

29 Thanks! E. Katz, S. Sachdev, E. Sørensen, W.Witczak-Krempa, PRB 90, 2014 [Ed. Sugg.] W.Witczak-Krempa, E. Sørensen, S. Sachdev, Nat. Phys. 10, 2014 W.Witczak-Krempa, S. Sachdev, PRB 87, 2013 [Ed. Sugg.] W.Witczak-Krempa, S. Sachdev, PRB 86, 2012