Breakdown of the adiabatic Breakdown of the adiabatic approximation in low-dimensional approximation in low-dimensional

gapless systemsgapless systems

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

Vladimir GritsevVladimir GritsevHarvard UniversityHarvard University

AFOSRAFOSR

Quantum Lunch Seminar, Quantum Lunch Seminar, LANL, 08//21/2007 LANL, 08//21/2007

Cold atoms:Cold atoms:

1. Equilibrium thermodynamics:1. Equilibrium thermodynamics:

Quantum simulations of equilibrium Quantum simulations of equilibrium condensed matter systemscondensed matter systems

2. Quantum dynamics:2. Quantum dynamics:

Coherent and incoherent dynamics, Coherent and incoherent dynamics, integrability, quantum chaos, …integrability, quantum chaos, …

3. = 1+2 Nonequilibrium thermodynamics?3. = 1+2 Nonequilibrium thermodynamics?

Adiabatic process.Adiabatic process.

Assume no first order phase transitions.Assume no first order phase transitions.

Adiabatic theorem:Adiabatic theorem:

““Proof”:Proof”: thenthen

Adiabatic theorem for integrable systems.Adiabatic theorem for integrable systems.

Density of excitationsDensity of excitations

Energy density (good both for integrable and nonintegrable Energy density (good both for integrable and nonintegrable systems:systems:

EEBB(0) is the energy of the state adiabatically connected to (0) is the energy of the state adiabatically connected to

the state A. the state A. For the cyclic process in isolated system this statement For the cyclic process in isolated system this statement implies no work done at small implies no work done at small ..

Adiabatic theorem in quantum mechanicsAdiabatic theorem in quantum mechanics

Landau Zener process:Landau Zener process:

In the limit In the limit 0 transitions between 0 transitions between different energy levels are suppressed.different energy levels are suppressed.

This, for example, implies reversibility (no work done) in a This, for example, implies reversibility (no work done) in a cyclic process.cyclic process.

Adiabatic theorem in QM Adiabatic theorem in QM suggestssuggests adiabatic theorem adiabatic theorem in thermodynamics:in thermodynamics:

Is there anything wrong with this picture?Is there anything wrong with this picture?

HHint: low dimensions. Similar to Landau expansion in the int: low dimensions. Similar to Landau expansion in the order parameter.order parameter.

1.1. Transitions are unavoidable in large gapless systems.Transitions are unavoidable in large gapless systems.

2.2. Phase space available for these transitions decreases with Phase space available for these transitions decreases with Hence expectHence expect

More specific reason.More specific reason.

Equilibrium: high density of low-energy states Equilibrium: high density of low-energy states

•destruction of the long-range order, destruction of the long-range order, •strong quantum or thermal fluctuations, strong quantum or thermal fluctuations, •breakdown of mean-field descriptions, e.g. Landau breakdown of mean-field descriptions, e.g. Landau theory of phase transitions.theory of phase transitions.

Dynamics Dynamics population of the low-energy states due to finite rate population of the low-energy states due to finite rate breakdown of the adiabatic approximation.breakdown of the adiabatic approximation.

This talk: three regimes of response to the slow ramp:This talk: three regimes of response to the slow ramp:

A.A. Mean field (analytic) – high dimensions: Mean field (analytic) – high dimensions:

B.B. Non-analytic – low dimensionsNon-analytic – low dimensions

C.C. Non-adiabatic – lower dimensionsNon-adiabatic – lower dimensions

We can view the response as parametric amplification of We can view the response as parametric amplification of quantum or thermal fluctuations. quantum or thermal fluctuations.

Some examples.Some examples.

1. Gapless critical phase (superfluid, magnet, crystal, …).1. Gapless critical phase (superfluid, magnet, crystal, …).

LZ condition:LZ condition:

Second example: crossing a QCP.Second example: crossing a QCP.

tuning parameter tuning parameter

gap

gap t, t, 0 0

Gap vanishes at the transition. Gap vanishes at the transition. No true adiabatic limit!No true adiabatic limit!

How does the number of excitations scale with How does the number of excitations scale with ? ?

Use a general many-body perturbation theory. Use a general many-body perturbation theory. (A.P. 2003)(A.P. 2003)

Expand the wave-function in many-body basis.Expand the wave-function in many-body basis.

i Ht

Substitute into SchrSubstitute into Schrödinger equation.ödinger equation.

Uniform system: can characterize excitations by momentum:Uniform system: can characterize excitations by momentum:

Use scaling relations:Use scaling relations:

Find:Find:

Transverse field Ising model.Transverse field Ising model.

0 1z zi jg

1xig

There is a phase transition at There is a phase transition at g=g=11..

This problem can be exactly solved using Jordan-Wigner This problem can be exactly solved using Jordan-Wigner transformation:transformation:

† † †

1

2 1, ( 1) ( )x zi i i i j j j j

j i

c c c c c c

SpectrumSpectrum::

Critical exponents: Critical exponents: z=z===1 1 dd/(z/(z +1) +1)=1/2.=1/2.

Correct result (J. Dziarmaga 2005):Correct result (J. Dziarmaga 2005): 0.11exn

0.18exn

Linear response (Fermi Golden Rule):Linear response (Fermi Golden Rule):

A. P., 2003A. P., 2003

Interpretation as the Kibble-Zurek mechanism: Interpretation as the Kibble-Zurek mechanism: W. HW. H. . Zurek, U. Dorner, Peter Zoller, 2005Zurek, U. Dorner, Peter Zoller, 2005

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

)( 2qq sq

Zero temperature.Zero temperature.

Density of excitations:Density of excitations:

Energy density:Energy density:

Agrees with the linear Agrees with the linear response.response.

Most divergent regime: Most divergent regime:

Agrees with the linear response.Agrees with the linear response.

Assuming the system thermalizes Assuming the system thermalizes

Finite temperatures.Finite temperatures.

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

ResultsResults

d=1,2d=1,2

d=1;d=1; d=2;d=2;

d=3d=3

Artifact of the quadratic approximation or the real result?Artifact of the quadratic approximation or the real result?

Non-adiabatic Non-adiabatic regime!regime!

Numerical verification (bosons on a lattice).Numerical verification (bosons on a lattice).

How do we simulate such system?How do we simulate such system?

High temperatures – use Gross-Pitaevskii equations High temperatures – use Gross-Pitaevskii equations with initial conditions distributed according to the with initial conditions distributed according to the thermal density matrix.thermal density matrix.

Note implies we are dealing with strongly interacting Note implies we are dealing with strongly interacting theory.theory.

What shall we do at low or zero temperatures?What shall we do at low or zero temperatures?We want to go beyond quadratic Bogliubov theory!We want to go beyond quadratic Bogliubov theory!

Quantum fluctuations drive the system to the Mott transition atQuantum fluctuations drive the system to the Mott transition at

In our caseIn our case

We have two fields propagating in time forward and backward .We have two fields propagating in time forward and backward .

Idea: expand quantum evolution in powers of Idea: expand quantum evolution in powers of ..

Take an arbitrary observableTake an arbitrary observable

Treat Treat exactly, while expand in powers of exactly, while expand in powers of ..

Results:Results:

Leading order in Leading order in : start from random initial conditions distributed : start from random initial conditions distributed according to the Wigner transform of the density matrix and according to the Wigner transform of the density matrix and propagate them classically (propagate them classically (truncated Wigner approximationtruncated Wigner approximation):):

•Expectation value is substituted by the average Expectation value is substituted by the average over the initial conditions. 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)Subsequent orders: quantum scattering events (quantum jumps)

Numerical verification (bosons on a lattice).Numerical verification (bosons on a lattice).

Results (1d, L=128)Results (1d, L=128)

3/13/4 LTE PredictionsPredictions::

finite temperaturefinite temperature

2/1E

zero temperaturezero temperature

0.1 1

0.1

1

Ene

rgy

dens

ity

TWA Quantum correction

T=0.02T=0.02

3/13/4 LTE

2D, T=0.22D, T=0.2

3/13/1 LTE

Conclusions.Conclusions.

A.A. Mean field (analytic): Mean field (analytic):

B.B. Non-analyticNon-analytic

C.C. Non-adiabaticNon-adiabatic

Three generic regimes of a system response to a slow ramp:Three generic regimes of a system response to a slow ramp:

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