Discussions: Giuseppe Mussardo, Rosario Fazio (SISSA)

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Statistics of the Work done in a Quantum Quench
Alessandro Silva ICTP Trieste
Discussions Giuseppe Mussardo, Rosario Fazio
(SISSA) Natan Andrei
(Rutgers) Vadim Oganesyan
(Yale), Anatoli Polknovnikov (Boston)
arXiv0806.4301 to be published in Phys. Rev.
Lett
arXiv0806.4301 to be published in Phys. Rev.
Lett
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Nonequilibrium
Nonequilibrium Last unexplored frontier
Partition function Mean field theory Renormalizati
on group
Equilibrium tools
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Non equilibrium physics in many body systems
Prototype example Kondo effect in Quantum Dots
From L. Kouwenhoven and L. Glazman, Phys. World
14(1), 33 (2001) D.
Goldhaber-Gordon, et al., Nature 391, 156 (1998)
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Non equilibrium physics in many body systems
Nonequilibrium splitting of the Kondo resonance
From De Franceschi, et al, PRL 89, 156801 (2002)
Abrupt quench inside the Kondo valley
From Nordlander, et al PRL 83, 808 (1999) .
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Non equilibrium physics in many body systems
The nonequilibrium lab cold atomic gases
Superfluid
Mott
Superfluid
From Fisher et al, Phys Rev B 40, 546 (1989).
See also Jaksch et al, PRL 81, 3108 (1998).
From Greiner et al, Nature 419, 51 (2002)
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Non equilibrium physics in many body systems
From Kinoshita et al., Nature 440, 900 (2006)
40 periods without thermalization integrability
??
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A paradigm the quantum quench
Example
Can be quenched globally or locally
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Quantum quenches
Early works Baruch, McCoy, Dresden, Mazur,
Girardeau (70)
Universality ? Time dependence of
correlators Igloi, Riegel (01) Altman, Auerbach
(02) Sengupta, Powell, Sachdev (04) Calabrese
and Cardy (07) Generation of excitations
(defects) Zurek, Dorner, Zoller (05) Polkovnikov
(05) Dziarmaga (05) Cherng and Levitov
(06) Gritsev, Polkovnikov (07) D. Patane, A
Silva, et al. (08)
Thermalization and integrability ? Rigol et al,
(06) Kollath, et al. (07) Manmana et al.
(07) Cazalilla (07) Gangart and Pustilnik
(08) Cramer et al (08) Barthel and Schollwock
(08)
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A fundamental characterization
Think thermodynamics !!!!
A,B points in parameters space
A.Silva, arxiv0806.4301
g path
g1
B
Thermodynamic transformation
g
Work
Entropy Heat
g2
A
g3
Closed systems
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NonequilibriumStatistics
Quasistatic transformation
g1
B
g
g2
g
Out of equilibrium
A
g3
Statistics depends on path, time dependence, etc
Classical systems Jarzynski (97), Crooks (99)
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Outline
Statistics of the work done in a quantum quench
1- Work probability distribution P(W)
Loschmidt echo (dephasing !) 2-
In Quantum Critical Systems (Quantum Ising Model)
Criticality
Singularities in moments of P(W) Local
quenches Edge singularities

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Work statistics and Loschmidt echo
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Work and Loschmidt
Abrupt quench
Initial energy
To measure work
Final energy
Initial state probability
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Work and Loschmidt
Take a Fourier Transform
Characteristic function
Loschmidt echo, Core hole correlator,
etc appears in X-ray edge problems, quantum
chaos, DEPHASING
Z. P. Karkuszewski, C. Jarzynski, and W. Zurek,
Phys. Rev. Lett. 89, 170405 (2002) H. T. Quan,
et al. Phys. Rev. Lett. 96, 140604 (2006). D.
Rossini, et al. Phys. Rev. A 75, 032333 (2007).
Initial state
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Work and Loschmidt
At T0
Loschmidt echo Partition function (in real time)
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Jarzynski equalities
Arbitrary quench
Abrupt quench
Nonequilibrium
Equilibrium
Jarzynski equality
C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
P. Talkner, E. Lutz, and P. Haanggi, Phys. Rev. E
75, 050102 (2007)
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Homework
Given
Prove
Tasaki-Crooks fluctuation theorem G. E. Crooks,
Phys. Rev. E 60 2721 (1999) P. Talkner, P.
Haanggi, J. Phys. A 40, F569-F571 (2007)
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Using Jarzynsky-Loschmidt connection I Global
quench in the Quantum Ising Model
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Global quantum quench
Global Quench Small
Fluctuations
Work X unit volume
Fluctuations
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Ising model and Landau Zener dynamics
Jordan Wigner
Bogoliubov rotation
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Loschmidt echo for global quench
eigenmodes of
eigenmodes of
Determinant formula (full counting statistics)
Klich (02), Abanin and Levitov (03) Or
direct expansion re-exponentiation A. LeClair,
G. Mussardo, H. Saleur, S. Skorik, Nucl.Phys.
B453, 581 (1995)
Integrable boundary state
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Loschmidt echo for global quench
System size
Expand and get all cumulants
Difference in ground state energies
Excess work
Thermodynamics dixit
Its Ok !!!
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Loschmidt echo for global quench
Asymptotics for large t (low W)
Measurable by dephasing
Critical Casimir effect on a Cylinder
t it
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Using Jarzynsky-Loschmidt connection I LOCAL
quench in the Quantum Ising Model
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The setting
Expand in cumulants
Decay of Loschmidt echo Fluctuations, etc
Long timeasymptotics
Vanishing at criticality
Orthogonality Catastrophe !!!
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Edge Singularity
Start at Criticality
Edge Singularity
Let us get P(W) in the scaling limit !!
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Scaling Limit
Quenchlocal mass term
1- Double your Majoranas
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Scaling Limit
1-Form Dirac fermions
Quench Local Backscattering
2- Perform nonlocal rotation (at criticality m0)
d
Two chiral modes
Quench Phase shift
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Scaling Limit
Use bosonization
This is the characteristic function of the GAMMA
distribution
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Conclusions
Statistics of the work done in a quantum quench
1- Work probability distribution P(W)
Loschmidt echo (dephasing !) 2-
In Quantum Critical Systems (Quantum Ising Model)
Criticality
Singularities in moments of P(W) Local
quenches Edge singularities

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Outlook
Work, entropy, etc as fluctuating variables.
NONEQUILIBRIUM STATISTICS
1- Other exactly solvable models (zero
dimensions) with F. Paraan 2- General time
dependence (Ising) ?? 3- More complex integrable
models ?? 4- Impurity models ?? 5- Statistics of
entropy ??
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Non equilibrium physics in many body systems
From MacKay et al., Nature 453, 76 (2008)
Saturation of damping rate at low T quantum
phase slip !