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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.
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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
Classical glassiness
viscosity
Source: JOM, 52 (7) (2000)
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!?
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.
PART I
A solvable toy model
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.
2D example
(not glassy yet)
Kitaev, Ann. Phys. (2003) - quant-phys/97
Wen, PRL (2003)
topological order for quantum computing
Plaquettes with :
defects
Same spectrum as free spins in a magnetic field
However,
Ground state degeneracy
ground state:
on a torus:
NO
two constraints:
4 ground states
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
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
3D strong glass model
ground state
degeneracy
3D strong glass model
always flip 4 octahedra: never simple defect diffusion
(Arrhenius law)
What about quantum tunneling?
defect separation:
virtual process:
topological quantum protection
quantum OVER protection
PART II
Beyond the toy model...
Josephson junction arrays
of T-breaking superconductors
Sr2RuO4
Constrained Ising model
chirality
Moore & Lee, cond-mat/0309717
Castelnovo, Pujol, and Chamon, PRB (2004)
What does the constraint do to thermodynamics?
Quantum model
Loop updates
ferro GS
Spin Ice
Dy2Ti2O7 (Ho2Ti2O7)
Snyder et al, Nature (2001)
Source: Snyder et al, Nature (2001)
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!