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Breakdown of the adiabatic approximation in low-dimensional gapless systems Anatoli Polkovnikov, Boston University Vladimir Gritsev Harvard University AFOSR

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Adiabatic theorem for integrable systems.

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Adiabatic theorem in quantum mechanics http://lab-neel.grenoble.cnrs.fr/themes/nano/fe8/15.gif

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Adiabatic theorem in QM suggests adiabatic theorem in thermodynamics: 1.Transitions are unavoidable in large gapless systems. 2.Phase space available for these transitions decreases with Hence expect Is there anything wrong with this picture? Hint: low dimensions. Similar to Landau expansion in the order parameter.

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More specific reason. Equilibrium: high density of low-energy states destruction of the long-range order, strong quantum or thermal fluctuations, breakdown of mean-field descriptions. Dynamics population of the low-energy states due to finite rate breakdown of the adiabatic approximation.

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This talk: three regimes of response to the slow ramp: A.Mean field (analytic): B.Non-analytic C.Non-adiabatic

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Example: crossing a second order phase transition. tuning parameter tuning parameter gap t, 0 t, 0 Gap vanishes at the transition. No true adiabatic limit! How does the number of excitations scale with ? Non-analytic regime B.

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Transverse field Ising model. There is a phase transition at g=1. 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): Linear response (Fermi Golden Rule): A. P., 2003 Interpretation as Kibble-Zurek mechanism: W. H. Zurek, U. Dorner, Peter Zoller, 2005

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Possible breakdown of the Fermi-Golden rule (linear response) scaling due to bunching of bosonic excitations. Zero temperature.

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Most divergent regime: Agrees with the linear response. Assuming the system thermalizes

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Same at a finite temperature. d=1,2 d=1; d=2; d=3 Artifact of the quadratic approximation or the real result?

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Numerical verification (bosons on a lattice). Expand dynamics in powers of U/Jn 0 (Truncated Wigner method + more, very accurate for these parameters.)

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Results (1d, L=128) Predictions : finite temperature zero temperature zero temperature

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T=0.02

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2D, T=0.2

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Conclusions. A.Mean field (analytic): B.Non-analytic C.Non-adiabatic Three generic regimes of a system response to a slow ramp: There are interesting and open problems beyond computing Z for various models.

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