Universal%20adiabatic%20dynamics%20across%20a%20quantum%20critical%20point

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Universal adiabatic dynamics across a quantum
critical point
Anatoli Polkovnikov, Boston University
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Consider slow tuning of a system through a
quantum critical point.
? ? ? t, ? ? 0
Gap vanishes at the transition. No true adiabatic
limit!
How does the number of excitations scale with ? ?
This question is valid for isolated systems with
stable excitations conserved quantities,
topological excitations, integrable models.
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Use a general many-body perturbation theory.
Expand the wave-function in many-body basis.
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Uniform system can characterize excitations by
momentum
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Caveats

1. Need to check convergence of integrals (no cutoff
   dependence)

Scaling fails in high dimensions.

1. Implicit assumption in derivation small density
   of excitations does not change much the matrix
   element to create other excitations.

1. The probabilities of isolated excitations

should be smaller than one. Otherwise need to
solve Landau-Zener problem. The scaling argument
gives that they are of the order of one. Thus the
scaling is not affected.
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Simple derivation of scaling (similar to
Kibble-Zurek mechanism)
In a non-uniform system we find in a similar
manner
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Example transverse field Ising model.
There is a phase transition at g1.
This problem can be exactly solved using
Jordan-Wigner transformation
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Spectrum
Critical exponents z?1 ? d?/(z? 1)1/2.
Correct result (J. Dziarmaga 2005)
Other possible applications quantum phase
transitions in cold atoms, adiabatic quantum
computations, etc.