Quantum Glassiness and Topological Overprotection Quantum Glassiness and Topological Overprotection Claudio Chamon DMR 0305482 PRL 05, cond-mat/0404182.

Quantum Glassiness and

Topological Overprotection

Quantum Glassiness and

Topological Overprotection

Claudio Chamon Claudio

Chamon

DMR 0305482

PRL 05, cond-mat/0404182PRB 04, cond-mat/0310710PRB 05, cond-mat/0410562Annal of Phys. 05, /0502068

PRL 05, cond-mat/0404182PRB 04, cond-mat/0310710PRB 05, cond-mat/0410562Annal of Phys. 05, /0502068

Collaborators: Claudio Castelnovo (BU), Christopher Mudry (PSI), Pierre Pujol (ENS-Lyon)

Classical glassinessClassical glassinessviscosity

Source: JOM, 52 (7) (2000)

Quantum glassy systemsQuantum glassy systems

disordered systemseg. quantum spin glasses

frustrated systemseg. 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)

Bray & Moore, J. Phys. C (1980)Read, Sachdev, and Ye, PRB (1995)

extensions ofclassical systems

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?

Does the classical glassy state need to be a phase?

Does one need a phase transition?

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.

NOKinetic constraints can lead to slow relaxation even in

classical paramagnets!Ritort & Sollich - review of kinetic constrained classical models

What about quantum systems?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 withtopological order and fractionalization

PART I

A solvable toy model

PART I

A solvable toy model

Why solvable examples are useful?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)

2D example(not glassy yet)

Kitaev, Ann. Phys. (2003) - quant-phys/97Wen, PRL (2003)

topological order for quantum computing

Same spectrum as free spins in a magnetic fieldHowever,

Plaquettes with :defectsdefects

Ground state degeneracyGround state degeneracy

on a torus:

ground state:

NONO

two constraints:

4 ground states4 ground states

Is the ground state reached?Is the ground state reached?bath of quantum oscillators;acts on physical degrees of freedomCaldeira & Leggett, Ann. Phys. (1983)

defects cannot simply be annihilated; plaquettes are flipped in multiplets

Is the ground state reached?Is the ground state reached?

defects must go awayequilibrium concentration:

defects cannot be annihilated; must be recombined

simple defect diffusion (escapes glassiness)

activated diffusionGarrahan & Chandler, PNAS (2003)Buhot & Garrahan, PRL (2002)

equivalent to classical glass model by

(Arrhenius law)

3D strong glass model3D strong glass model

ground statedegeneracy

3D strong glass model3D strong glass model

always flip 4 octahedra: never simple defect diffusion

(Arrhenius law)

What about quantum tunneling?What about quantum tunneling?

defect separation:

virtual process:

topological quantum protection quantum OVER protection

PART II

Beyond the toy model...

PART II

Beyond the toy model...

Josephson junction arrays

of T-breaking superconductors

Josephson junction arrays

of T-breaking superconductors

Moore & Lee, cond-mat/0309717Castelnovo, Pujol, and Chamon, PRB (2004)

Sr2RuO

4

Constrained Ising model

chirality

What does the constraint do to thermodynamics?

What does the constraint do to thermodynamics?

Quantum modelQuantum model

Loop updatesferro GS

Source: Snyder et al, Nature (2001)

Dy2Ti

2O

7 (Ho

2Ti

2O

7)Spin

IceSpin Ice Snyder et al, Nature

(2001)

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.

ConclusionConclusion

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!