Dynamical phase transitionand prethermalization

Sarang Goplalakrishnan, Eugene Demler (Harvard)

Mobile magnetic impurity in Fermi superfluids

Pietro Smacchia, Alessandro Silva (SISSA, Trieste)Dima Abanin (Perimeter Institute, Waterloo)Michael Knap, Eugene Demler (Harvard)

Dynamical phase transitionand prethermalization

Prethermalization

We observe irreversibility and approximate thermalization. At large time the system approaches stationary solution in the vicinity of, but not identical to, thermal equilibrium. The ensemble therefore retains some memory beyond the conserved total energy…This holds for interacting systems and in the large volume limit.

Prethermalization in ultracold atoms, theory: Eckstein et al. (2009); Moeckel et al. (2010), L. Mathey et al. (2010), R. Barnett  et al.(2010)

Heavy ions collisionsQCD

Prethermalization in 1 dimensional systemProbing prethermolization in atomchip experiments

Initial T=120 nK (blue line). After  27.5 ms identical to thermal system at T= 15 nKAt all lengthscalesIn all correlation functions

Gring et al., Science (2012)

Quench and wait

Ultracold bosons in optical lattices

Quantum quench from Mott insulator to superfluid + wait:

We look at the long time limit of the system. What is the nature of the stationary state? Is it thermal?Does it become superfluid?Earlier work: Eckstein et al., PRL 103:056403 (2009)Schiro, Fabrizio, PRL 105: 076401 (2010)Sciola, Biroli, PRL 105:220401 (2010)Gambassi, Calabrese, EPL 95:66007 (2011)Sciola, Birolli, arXiv:1211.2572

Dynamical Phase transition

Experimental probes using quantum microscope Statistics of defects→ defect densi es and their higher moments (fluctuations, skewness,...)

local defects density d (lower case d): 

global defects D (upper case D):

average moments cumulants

average moments cumulants (same as before)

Large‐N model Dynamical phase transition from Mott to Superfluid

→ at quan zed densi es O(2) field theory

large‐N limit: interaction factorizes

decoupled harmonic oscillators with time dependent mass (to be determined self‐consistently)

Large‐N model

Stationary point at long times can be found (without crossing the dynamical phase transition)

For d>2 there is a transition when r*=0

Critical properties of dynamical phase transition

Experimental signatures of dynamical phase transition

Use field theory to calculate the number of excitations in the basisof the Hamiltonian before the quench. This corresponds to the number of defects in the system

Average number of defects saturates at long wait times withor without crossing the DPT. Nothing special at the dynamicalphase transition. (Equilibration?)

The variance in the number of defects shows very different behavior before and after crossing the dynamical phase transition.

withoutcrossing DPT

aftercrossing DPT

Defect counting in O(N) model model reduces to a set of harmonic oscillators with time dependent masses

initial state has no defects:        ground state of  quench the mass of the oscillators→ wave func on is squeezed state

from            we calculate

density: fluctuations:

Defect counting in O(N) model

Individual moments

Moment generating function

PrethermalizationDivergence in the variance of the number of defects demonstratesprethermalization (introduced by Wetterich et al. , see e.g. hep-th/0403234).The system has anomalous occupation of the low energy modes

Ratio of the cumulants 

C1/C2

C2/C3

Dynamical phase transition at r*=0 'Mott'

Prethermalization

Precursors ofhigher moments divergingat DPT

local defects: C1 C2

Is this prethermalization real?

exact dynamics in 1D with DMRG

Defect statistics

Steady state

cumulants of global defects D (as in field theory)

→ approach quickly the thermal value 

Steady state

cumulants of local defects

→ approach quickly the thermal value (~ 2‐3 1/J2times) 

(thermal value: solid line; temporal average: symbols)

Berges et al., Nucl. Phys. B 727:244 (2005)

Relaxation in 4 theory beyond large N

Open questions Does a prethermalized regime exist in higher dimension? this is hard (impossible?) to answer theoretically  experimental study would give insight in the validity of O(N) for dynamics of bosons

thermal values can be easily obtained by QMC in higher dimension

1D: O(N) and DMRG give even qualitative difference for cumulant ratio:

Mott Mott

Open questions

Does a prethermalized regime exist in higher dimension? this is difficult (impossible?) to answer theoretically  experimental study would give insight in the validity of O(N) 

for dynamics of bosons thermal values can be easily obtained by QMC in higher 

dimension

1D: agreement improves when going to larger filling (particle‐hole symmetry)

Mott

Mott

slight decrease?

Mobile magnetic impurities in Fermi superfluids

Impurities in solid state systems

Bound states on magnetic impurities in superconductors

Bound states on magnetic impurities in superconductorsYu, Acta Phys. Sin. 21, 75 (1965)Shiba, Prof. Theor. Phys. 40, 435 (1968).Rusinov, Sov. Phys. JETP Lett. 9, 85 (1969)

Salkola, Balatsky, Schrieffer, PRB 55:12648 (1997)

Science 275:1767 (1997)

Local density of states near 

Parity changing transitionSalkola, Balatsky, Schrieffer, PRB 55:12648 (1997)

Analogous to Kondo singlet formation

Possible realization: Cs impurities in Li fermionic condensate

e.g. Chen Chin’s group

Bound state of Bogoliubov quasiparticle and impurity atom

Sketch of the wavefunction

Bound state energy

Weak interaction: parity transition

Shiba‐Kondo molecule

Strong interaction: global minimum

Shiba‐Kondo molecule

“Exotic” molecule

Detection via RF

Exotic many-body phases?Reminiscent of electrons with spin-orbit coupling

in 2D and contact interaction, ground state:breaks rotational symmetry. Wigner crystal or nematic.Berg, Rudner, Kivelson (2012)

For contact interaction Wigner crystal with aspect ratiohas energy per particle parametrically better than uniform phase

Interaction between two Shiba states: Yao et al., 13092633

???

Dynamical phase transitionand prethermalization

Sarang Goplalakrishnan, Eugene Demler (Harvard)

Mobile magnetic impurity in Fermi superfluids

Pietro Smacchia, Alessandro Silva (SISSA, Trieste)Dima Abanin (Perimeter Institute, Waterloo)Michael Knap, Eugene Demler (Harvard)