1. Equilibrium dynamics of entangled states near quantum critical points Talk online at http://sachdev.physics.harvard.edu Physical Review Letters 78, 843 (1997); 78, 2220 (1997); 95, 187201 (2005). Kedar Damle (TIFR, Mumbai) Subir Sachdev (Harvard) Peter Young (UCSC)

2. T Quantum-critical Why study quantum phase transitions ? g gcgc Theory for a quantum system with strong correlations: describe phases on either side of g c by expanding in deviation from the quantum critical point. Important property of ground state at g=g c : temporal and spatial scale invariance; characteristic energy scale at other values of g: Critical point is (often) a novel (entangled) state of matter without quasiparticle excitations Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.

3. I. Quantum Ising Chain 2Jg

4. Full Hamiltonian leads to entangled states at g of order unity

5. LiHoF 4 Experimental realization

6. Weakly-coupled qubits Ground state: Lowest excited states: Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p Entire spectrum can be constructed out of multi-quasiparticle states p

7. Three quasiparticle continuum Quasiparticle pole Structure holds to all orders in 1/g S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997) ~3  Weakly-coupled qubits

8. Quantum mechanical S-matrix has a universal form at low momenta (in one dimension) (-1)

12. Ground states: Lowest excited states: domain walls Coupling between qubits creates new “domain- wall” quasiparticle states at momentum p p Strongly-coupled qubits

13. Two domain-wall continuum Structure holds to all orders in g S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997) ~2  Strongly-coupled qubits

17. Entangled states at g of order unity g gcgc “Flipped-spin” Quasiparticle weight Z A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994) g gcgc Ferromagnetic moment N 0 P. Pfeuty Annals of Physics, 57, 79 (1970) g gcgc Excitation energy gap 

18. Critical coupling No quasiparticles --- dissipative critical continuum

19. Quasiclassical dynamics P. Pfeuty Annals of Physics, 57, 79 (1970) S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).

20. II. Quantum O(3) non-linear  model

21. Quantum mechanical S-matrix has a universal form at low momenta (in one dimension) (-1)

22. x t

23. t (0,0) (x,t) x

25. III. Quantum sine-Gordon model

26. Quantum mechanical S-matrix has a universal form at low momenta (in one dimension) (-1)

29. III. Quantum sine-Gordon model

30. Conclusions 1.Large, entangled quantum systems close to equilibrium have an “intrinsic” relaxation rate independent of the strength of the coupling to a heat bath. 2.This relaxation rate has a universal form near interacting quantum critical points 3.In one-dimensional systems