Breakdown of the adiabatic approximation in low-dimensional gapless systems Anatoli Polkovnikov, Boston University Vladimir Gritsev Harvard University AFOSR Quantum Lunch Seminar, LANL, 08//21/2007

Cold atoms: 1. Equilibrium thermodynamics: Quantum simulations of equilibrium condensed matter systems 2. Quantum dynamics: Coherent and incoherent dynamics, integrability, quantum chaos, … 3. = 1+2 Nonequilibrium thermodynamics?

Adiabatic process. Assume no first order phase transitions. Adiabatic theorem: “Proof”: then

Adiabatic theorem for integrable systems. Density of excitations Energy density (good both for integrable and nonintegrable systems: E B (0) is the energy of the state adiabatically connected to the state A. For the cyclic process in isolated system this statement implies no work done at small .

Adiabatic theorem in quantum mechanics Landau Zener process: In the limit  0 transitions between different energy levels are suppressed. This, for example, implies reversibility (no work done) in a cyclic process.

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

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, e.g. Landau theory of phase transitions. 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) – high dimensions: B.Non-analytic – low dimensions C.Non-adiabatic – lower dimensions We can view the response as parametric amplification of quantum or thermal fluctuations.

Some examples. 1. Gapless critical phase (superfluid, magnet, crystal, …). LZ condition:

Second example: crossing a QCP. 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  ?

Use a general many-body perturbation theory. (A.P. 2003) Expand the wave-function in many-body basis. Substitute into Schrödinger equation.

Uniform system: can characterize excitations by momentum: Use scaling relations: Find:

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 the 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.

Density of excitations: Energy density: Agrees with the linear response.

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

Finite temperatures. Instead of wave function use density matrix (Wigner form).

Results d=1,2 d=1; d=2; d=3 Artifact of the quadratic approximation or the real result? Non-adiabatic regime!

Numerical verification (bosons on a lattice). How do we simulate such system?

High temperatures – use Gross-Pitaevskii equations with initial conditions distributed according to the thermal density matrix. Note implies we are dealing with strongly interacting theory.

What shall we do at low or zero temperatures? We want to go beyond quadratic Bogliubov theory! Quantum fluctuations drive the system to the Mott transition at In our case We have two fields propagating in time forward and backward. Idea: expand quantum evolution in powers of . Take an arbitrary observable Treat  exactly, while expand in powers of .

Results: Leading order in  : start from random initial conditions distributed according to the Wigner transform of the density matrix and propagate them classically (truncated Wigner approximation): Expectation value is substituted by the average over the initial conditions.Expectation value is substituted by the average over the initial conditions. Exact for harmonic theories!Exact for harmonic theories! Not limited by low temperatures!Not limited by low temperatures! Asymptotically exact at short times.Asymptotically exact at short times. Subsequent orders: quantum scattering events (quantum jumps)

Numerical verification (bosons on a lattice).

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: Open questions: general fate of linear response at low dimensions, non-uniform perturbations,…