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

4. Adiabatic theorem for integrable systems.

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

6. 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.

7. 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.

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

9. 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.

10. Transverse field Ising model. There is a phase transition at g=1. This problem can be exactly solved using Jordan-Wigner transformation:

11. 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

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

13. Most divergent regime:    Agrees with the linear response. Assuming the system thermalizes

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

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

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

17. T=0.02

18. 2D, T=0.2

19. 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.