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

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Presentation transcript:

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

Adiabatic theorem for integrable systems.

Adiabatic theorem in quantum mechanics

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.

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.

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

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.

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

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

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

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

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

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

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

T=0.02

2D, T=0.2

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.