Transcript Document

Level statistics and self-induced
decoherence in disordered spin-1/2 systems
Mikhail Feigel’man
L.D.Landau Institute, Moscow
In collaboration with:
Lev Ioffe
Rutgers University
Marc Mezard
Orsay University
Emilio Cuevas
University of Murcia
Previous publications on the related subjects:
Phys Rev Lett. 98, 027001(2007) (M.F.,L. Ioffe,V. Kravtsov, E.Yuzbashyan)
Annals of Physics 325, 1368 (2010) (M.F., L.Ioffe, V.Kravtsov, E.Cuevas)
Phys.Rev. B 82, 184534 (2010) (M.F.,L. Ioffe, M. Mezard)
Nature Physics, 7, 239 (2011) (B.Sacepe,T.Doubochet,C.Chapelier,M.Sanquer,
M.Ovadia,D.Shahar, M. F., L..Ioffe)
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Plan of the talk
1.
Superconductivity with pseudogap and effective
spin-1/2 model
2. Bethe lattice model of quantum phase transition.
Critical lines from the analitical solution
3. Level statistics on small random graph: exact
numerical diagonalization.
4. Summary of results
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D.Shahar & Z.Ovadyahu
amorphous InO 1992
On insulating side
(far enough):
Kowal-Ovadyahu 1994
T0 = 15 K
R0 = 20 kW
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Amorphous InOx films
Nature Physics, 7, 239 (2011)
SC side: local tunneling conductance
Superconductive state with a pseudogap: Fermi-level in the localized band
Superconductive state near SIT is very unusual:
The spectral gap appears much before (with T decrease) than
superconductive coherence does
Coherence peaks in the DoS appear together with resistance
vanishing
Distribution of coherence peaks heights is very broad near SIT
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Single-electron states suppressed by
pseudogap ΔP >> Tc
“Pseudospin”
approximation
2eV1 = 2Δ
eV2 = Δ+ ΔP
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Andreev point-contact spectroscopy
arXiv:1011.3275
Nature Physics 2011
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S-I-T: Third Scenario
• Bosonic mechanism: preformed Cooper pairs +
competition Josephson v/s Coulomb – S I T in arrays
• Fermionic mechanism: suppressed Cooper attraction, no
pairing – S M T
• Pseudospin mechanism: individually localized pairs
- S I T in amorphous media
SIT occurs at small Z and lead to paired insulator
How to describe this quantum phase transition ?
Bethe lattice model is solved
Phys.Rev. B 82, 184534 (2010)
M. Feigelman, L.Ioffe, M. Mezard
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Distribution function for the order parameter
General recursion:
Linear recursion (T=Tc)
Diverging 1st moment
Solution in the RSB phase:
T=0
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Vicinity of the
Quantum Critical Point
<< 1
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Order parameter:
scaling near transition
Typical value near the critical point:
(at T=0)
Near KRSB :
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K=4
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Insulating phase: continuous
v/s discrete spectrum ?
Consider perturbation expansion over Mij in H below:
Within convergence region the many-body spectrum is
qualitatively similar to the spectrum of independent spins
No thermal distribution, no energy transport,
distant regions “do not talk to each other”
What will happen when Mij are increasing ?
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Recursion relations for level widths
Spectral function of external noise
We look for the distribution function of the form
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Threshold energy at T=0
Low-energy limit
ω << 1
Full band localization
ω =1
Now set T>0. What happens to level width
at low excitation energies ?
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Threshold for activated transport
Nonzero line-width appears above
threshold frequency only:
This is T = 0 result !
Nonzero activation energy for transport of pairs
is due to the absence of thermal bath at low ω
Nonzero but low temperatures:
Activation
law
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Phase diagram
What else could
one expect?
Major feature: green and red line
Temperature
meet at zero energy
Energy
Hopping insulator
Superconductor
Full localization:
Insulator with
Discrete levels
MFA line
RSB state
gc
g
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Phase diagram-version 2
Here green and red line do not meet at zero energy
Temperature
Do gapless delocalized excitations exist
WITHOUT Long-range order ?
Full localization:
Insulator with
Discrete levels
Hopping insulator
with Mott (or ES) law
Energy
gc1
Superconductor
gc2
g
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Phase diagram-version 3
Temperature
Energy
Here green and red line
cross at non-zero energy:
first-order transition??
Full localization:
Insulator with
Discrete levels
Superconductor
gc
g
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Major results from Bethe lattice study
- Full localization of eigenstates with
E ~ W at weakest coupling between
spins, g < g* (or K < K*(g))
- No intermediate phase without both
order parameter and localization of lowenergy modes
Questions:
1) what about highly excited states with E >> W
2) how universal is the absence of intermediate phase ?
3) How to avoid the use of Bethe lattice ?
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Different definitions for the
fully many-body localized state
• 1.
• 2.
• 3.
• 4.
• 5.
No level repulsion (Poisson statistics of the full
system spectrum)
Local excitations do not decay completely
Global time inversion symmetry is not broken
(no dephasing, no irreversibility)
No energy transport (zero thermal conductivity)
Invariance of the action w.r.t. local time
transformations t → t + φ(t,r):
d φ(t,r)/dt = ξ (t,r) – Luttinger’s gravitational potential
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Level statistics: Poisson v/s WD
• Discrete many-body spectrum with zero
level width: Poisson statistics
• Continuous spectrum (extended states) :
Wigner-Dyson ensemble with level
repulsion
V.Oganesyan & D.Huse
Phys. Rev. B 75, 155111 (2007)
Model of interacting fermions
(no-conclusive concerning
sharp phase transition)
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Numerical results for random Z=3 graph
E.Cuevas (Univ. of Murcia, Spain)
Low-lying excitations,
Jc=0.10
Middle of the band, Jc = 0.07
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Density of States
T = (dS/dE)-1 = # (E/Ns)1/2
E ~ Ns T2
(at T << 1)
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Phase diagram
Red line: analytical theory
Orange ovals: numerical data
Green line: correction of analytical result for finite-size effect
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Role of Jz Siz Sjz interaction
J = 0.1
0.07 → 0.02 (midband)
0.10 → 0.075 (low E)
What is the reason
for such a strong effect?
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Original model:
XY exchange + transverse field
Full model with Sz-Sz coupling
Summation over large number of configurations with different
makes it easier to meet resonant conditions
Conclusion: critical coupling g depends on temperature, i.e. on E/Ns
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SzSz interaction results in
temperature-controlled transition to
the state with zero level widths and
zero conductivity
Basko et al 2006, Mirlin et al 2006
T ~ (E/Ns)1/2
T
Г>0
Г=0
g1
g2
g
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Conclusions
New type of S-I phase transition is described
On insulating side activation of pair transport is
due to ManyBodyLocalization threshold
Results from level statistics studies support general
shape of the phase diagram, but the possibility of
intermediate phase cannot be excluded this way
Interaction in the “density channel” is crucially
important for the shape of the phase diagram
at extensive energies E ~ Ns
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Open problems
- Analitical study of energy localization in
Euclidean space or RGM: order parameter ?
anything to do with compactification of space and black holes ?
-
Is it possible to modify the model in a way
to find an intermediate phase or 1st order?
- How to calculate electric and thermal
conductivities directly within recursion
relations approach?
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The End
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