Quantum Glassiness and Topological Overprotection

1. Quantum Glassiness and Topological Overprotection Claudio Chamon Collaborators: Claudio Castelnovo (BU), Christopher Mudry (PSI), Pierre Pujol (ENS-Lyon) PRL 05, cond-mat/0404182 PRB 04, cond-mat/0310710 PRB 05, cond-mat/0410562 Annal of Phys. 05, /0502068 DMR 0305482

2. Classical glassiness viscosity Source: JOM, 52 (7) (2000)

3. Quantum glassy systems disordered systems eg. quantum spin glasses extensions of classical systems Bray & Moore, J. Phys. C (1980) Read, Sachdev, and Ye, PRB (1995) frustrated systems eg. 1 frustrated Josephson junctions with long-range interactions eg. II self-generated mean-field glasses Kagan, Feigel'man, and Ioffe, ZETF/JETP (1999) Westfahl, Schmalian, and Wolynes, PRB (2003) Does one need an order parameter? Does one need a thermodynamic or quantum phase transition? Why not simply remain in a mixed state and not reach the ground state instead!?

4. Does the classical glassy state need to be a phase? Does one need a phase transition? NO Kinetic constraints can lead to slow relaxation even in classical paramagnets! Ritort & Sollich - review of kinetic constrained classical models What about quantum systems? Not free to toy with the dynamics - it is given. Where to look: systems with hard constraints: ice models, dimer models, loop models,... Some clean strongly correlated systems with topological order and fractionalization Strong correlations that can lead to these exotic quantum spectral properties can in some instances also impose kinetic constraints, similar to those studied in the context of classical glass formers.

5. PART I A solvable toy model

6. Why solvable examples are useful? Classical glasses can be efficiently simulated in a computer; but real time simulation of a quantum system is doomed by oscillating phases (as bad as, if not worse, than the fermion sign problem)! Even a quantum computer does not help; quantum computers are good for unitary evolution. One needs a “quantum supercomputer”, with many qubits dedicated to simulate the bath. Solvable toy model can show unambiguously and without arbitrary or questionable approximations that there are quantum many body systems without disorder and with only local interactions that are incapable of reaching their quantum ground states.

7. 2D example (not glassy yet) Kitaev, Ann. Phys. (2003) - quant-phys/97 Wen, PRL (2003) topological order for quantum computing

8. Plaquettes with : defects Same spectrum as free spins in a magnetic field However,

9. Ground state degeneracy ground state: on a torus: NO two constraints: 4 ground states

10. Is the ground state reached? bath of quantum oscillators; acts on physical degrees of freedom Caldeira & Leggett, Ann. Phys. (1983) defects cannot simply be annihilated; plaquettes are flipped in multiplets

11. Is the ground state reached? defects must go away equilibrium concentration: defects cannot be annihilated; must be recombined simple defect diffusion (escapes glassiness) (Arrhenius law) activated diffusion Garrahan & Chandler, PNAS (2003) Buhot & Garrahan, PRL (2002) equivalent to classical glass model by

12. 3D strong glass model ground state degeneracy

13. 3D strong glass model always flip 4 octahedra: never simple defect diffusion (Arrhenius law)

14. What about quantum tunneling? defect separation: virtual process: topological quantum protection quantum OVER protection

15. PART II Beyond the toy model...

16. Josephson junction arrays of T-breaking superconductors Sr2RuO4 Constrained Ising model chirality Moore & Lee, cond-mat/0309717 Castelnovo, Pujol, and Chamon, PRB (2004)

17. What does the constraint do to thermodynamics?

18. Quantum model Loop updates ferro GS

19. Spin Ice Dy2Ti2O7 (Ho2Ti2O7) Snyder et al, Nature (2001) Source: Snyder et al, Nature (2001)

20. Conclusion Presented solvable examples of quantum many-body Hamiltonians of systems with exotic spectral properties (topological order and fractionalization) that are unable to reach their ground states as the environment temperature is lowered to absolute zero - systems remain in a mixed state down to T=0. New constraint for topological quantum computing: that the ground state degeneracy is protected while the system is still able to reach the ground states. Out-of-equilibrium strongly correlated quantum systems is an open frontier!