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Quench dynamics in the one-dimensional Sine-Gordon Model: Quantum kinetic equation approach Marco Tavora Aditi Mitra New York University Supported by: NSF DMR 1004589


Statement of the problem Ø  At  t=0,  system  of  1D  bosons  is  driven  out  of  equilibrium  via  an   interac7on  quench  and  addi7onally,  a  commensurate  periodic  poten7al   is  switched-­‐on  also  suddenly:   Hi =

⎡ u0 2 2⎤ 1 † dx K π Π(x) + ∂ φ (x) = u p η η (t < 0) ( ) ( ) ⎢ ⎥ ∑ 0 x 0 p p ∫ 2π K0 ⎣ ⎦ p

2 2⎤ ⎡ u 1 gu Hf = dx K π Π(x) + ∂ φ (x) − ( ) ( ) ⎥ α 2 ∫ dx cos[γφ (x)] (t > 0) 2π ∫ ⎢⎣ K x ⎦  

Vsg

K

K0

K0 t

Ø Kine7c equa7on  approach:     §  How  does  the  system  evolve  in  7me?   §  Does  it  thermalize?  If  so  what  are  the                7me  scales?   §  If  not,  what  is  the  long  7me  behaviour?  

K

Periodic potential switched-on (multi-particle scattering)


Mitra, Giamarchi  Phys.  Rev.  B  85,  075117  (2012)                                                              Iucci  ,  Cazalilla  2010  New  J.  Phys.  12  055019  (2010)   M.  A.  Cazalilla,  PRL  97,  156403  (2006)  

Quadratic Model

Ø  Consider an  interac7on  quench  from  K0  to  K  without  the  periodic   poten7al  first  (g=0).  In  this  case,  the  final  Hamiltonian  is:   Hf =

2 2⎤ ⎡ u 1 † dx K π Π(x) + ∂ φ (x) = u p γ γ ( ) ( ) ∑ ⎢ ⎥ x p p ∫ 2π K ⎣ ⎦ p

Ø  What is  the  number  occupa7on    n(          p)        =          γ      †p    γ      p        a>er  the  quench?                                 n( p)

(K

− K) 0

2

n( p) =

1 e

u p /T

− 1 Equilibrium (T>0)  

Quench

4KK 0

Equilibrium (T=0)   p

Ø  Even though  the  system  is  out  of  equilibrium,  each  p  mode  is  s7ll     infinitely  long  lived  and    n(          p)        is  conserved  in  the  dynamics  


Mitra, Giamarchi  Phys.  Rev.  B  85,  075117  (2012)                                                              Iucci  ,  Cazalilla  2010  New  J.  Phys.  12  055019  (2010)   M.  A.  Cazalilla,  PRL  97,  156403  (2006)  

Quadratic Model Ø  The  energy  a>er  the  quench  is:  

E 1 u ( K0 − K ) u ⎡ K neq ⎤ † = ∑ u | p | γ pγ p = = − 1⎥ 2 2 ⎢ L L p≠0 KK 0 4πα 2πα ⎢⎣ K eq ⎥⎦ 2

K neq

(  1/        α          is  the  UV  cut-­‐off      and  T    eff      α        /    u    finite)  

γ 2 ⎛ K2 ⎞ = K 1+ 2 ⎟ 8 0 ⎜⎝ K0 ⎠

γ 2K K eq = 8 It’s convenient  to  use   these  exponents  since   the  correlators  decay     according  to  them  

density correlator    

ρ (x) ρ (0) ∝ eiγφ ( x,t ) e− iγφ (0,t ) ⎯t→∞ ⎯⎯ → ρ (x) ρ (0) ∝ eiγφ ( x,t ) e− iγφ (0,t ) ⎯t→∞ ⎯⎯ →

K neq − K eq Quench amplitude  

1 x

2 K neq

1 x

2 K eq

Correla7ons decay   faster  a>er  the   quench  


Quantum Sine-Gordon (g≠0) Ø  For t>0,  the  q-­‐modes  are  no  longer  independent  but  due  to  Vsg    there  is   scagering  and  the  occupa7on  numbers  evolve  in  7me   q2 q

q3

Bosonic modes  acquire   finite  life7me  τ

Ø  This kind  of  term  is  irrelevant  in  equilibrium  i.e.  it  only  renormalizes  the   parameters  of  the  LL.  Out  of  equilibrium  the  periodic              poten7al  is  not  irrelevant  anymore,  genera7ng  inelas7c  scagering,              which  is  studied  here.  

 


Quantum Sine-Gordon (g≠0) Ø  The exact  Green’s  func7on    G(xt,                      y    t  ′  )    obeys  the  Dyson  equa7on:  

(

G = g  (1+ Σ  G) Dyson equa0on    

)

1 ∂t2 + u 2 q 2 GR = δ (t − t ′ ) + (Σ R  GR )(t, t ′ ) π Ku GK = GR  F − F  G A −

(

)

1 ∂t2 − ∂t2′ F(t, t ′ ) = Σ K (t, t ′ ) − (Σ R  F )(t, t ′ ) + (F  Σ A )(t, t ′ ) π Ku Quantum Kine0c  Equa0on  

Ø  We simplify  the  Dyson  equa7ons  by  performing  a  gradient  expansion  to   lowest  order,  which  is  equivalent  to  a  quasi-­‐par7cle  approxima7on  (energy   levels  are  not  modified  but  only  the  occupa7on  numbers  change)   ⎛ iπ K ⎞ K ∂F R A (q,T ) = ⎜ ⎟ ⎡⎣ Σ (q,T ) − (Σ (q,T ) − Σ (q,T ))F(q,T ) ⎤⎦ ∂T ⎝ 2q ⎠ 1.0

AHk,wL

0.8

∂(GR − G A ) (q,T ) = 0 ∂T

0.6 0.4 0.2 0.0

0.0

Σ[G] =

+

0.2

0.4

0.6 w

0.8

1.0

A(k,ω )  δ (ω − | q |)

 


Quantum Sine-Gordon (g≠0) Ø  The conserved  energy  is:   u E(T ) = 4π

∫ dqe

−α q

q F(q,T )

−∞

T

(

)

iuK −α q ⎡ K R A ⎤ − d T dq e Σ (q, T ) − Σ − Σ (q, T )F(q, T ) ′ ′ ′ ′ ∫ ∫ ⎣ ⎦ 8 0 −∞

Ø  This poten7al  describes  mul7-­‐par7cle  scagering  and  even  to  leading  order  in                the  gradient  expansion  the  system  has  dynamics  and  has  the  capacity  to  thermalize. Vsg = −

2 4 ⎡ 1 ⎤ gu gu 1 dx cos γφ (x) = − dx 1− γφ (x) + γφ (x) + ... ( ) α 2 ∫ ⎢ 2 ( ) 4!( ) ⎥ α2 ∫ ⎣ ⎦

∂F (q,T ) = I[F(q,T )] ∂T (F(q,T ) = 1+ 2n(q,T ))

I[n(k)] 

∫ ⎡⎣(1+ n(k))(1+ n( p))n(q)n(r) − n(k)n( p)(1+ n(q))(1+ n(r)) ⎤⎦ 4 -­‐ theory   Φ ⎡ ⎤

p,q

I[n(q)]  exp ⎢ − ∫ n(q) ⎥ ⎢⎣ q ⎥⎦

Sine-­‐Gordon

,  


Numerical Results

Ø  For large  quench  amplitudes,  the  system  thermalizes  and  the  distribu7on   approaches  the  equilibrium  distribu7on:   ⎛ uq ⎞ Ψ(q,T ) = qF(q,T ) ⎯⎯⎯⎯→ qcoth ⎜ ⎟ ⎝ 2Teq ⎠ Equilibrium

Kneq =13.8, Keq =3

40

               

YHqL

30 20

T

10 0 0

2

4 q

6

8


Numerical Results

Ø  In  equilibrium,  the  temperature  is  given  by:   ⎛ uq ⎞ u Ψ(q) = qcoth ⎜ ⇒ T = Ψ(q = 0) ⎟ eq 2T 2 ⎝ eq ⎠

Ø  Defining an  effec7ve  temperature  analogously  we  see  that  it  converges  to   the  equilibrium  value  which  is  calculated  from   − H /T

(

Tr e

3.0

)

H f = Ψ in H f Ψ in

Teq

2.0 Teff

eq

Kneq =13.8, Keq =3

2.5

1.5

u Teff (T ) = Ψ(q = 0,T ) → Teq 2

1.0 0.5 0.0

f

0

500

1000 1500 2000 2500 3000 T


Numerical Results

Ø  For smaller  quenches  however  the  system  reaches  a  steady  state  but  the   effec7ve  temperature  does  not  converge  to  the  equilibrium  value  anymore:  

2.5

Keq =3

Kneq=14

-­‐-­‐-­‐-­‐-­‐ Teq  

Teff

2.0 1.5

Kneq=11

Kneq=8.4

1.0

Kneq=6.5

Decreasing quench   amplitude    

Kneq=4

0.5

Kneq=5.10

0.0

Kneq=3.56

0

200 400 600 800 1000 1200 1400

Kneq=3.25

lack of   thermaliza7on  

T Ø  Decay rates  vary  with  the  quench  amplitude  in  a  non-­‐monotonic  way  


Numerical Results Ø  How far  is  the  asympto7c  temperature  from  the  Teq  for  small  quenches?  

0.30 0.25 Kneq =3.25, Keq =3

Teff

0.20

Teq  

0.15 0.10 0.05 0.00

0

200

400 T

600

800


Numerical Results Ø  How far  is  the  asympto7c  temperature  from  the  Teq  for  small  quenches?  

0.6 Keq =3

0.5 Tasympt &Teq

-­‐-­‐-­‐-­‐-­‐-­‐ Teq Tasympt

0.4 0.3 0.2 0.1 0.0

3.2

3.4

3.6 Kneq

3.8

4.0

4.2


Numerical Results Ø  For very  large  quenches,  the  relaxa7on  rate  decreases  with  the  quench   amplitude.  For  very  small  quenches,  the  relaxa7on  rate  increases.   2.5

Keq=3

Decreasing quench   amplitude    

Teff

2.0 1.5 1.0 0.5

0.012 0

200 400 600 800 1000 1200 1400

Decay rate

0.0

T

0.30

Keq=3

Teff

0.25 0.20

--- Analytical expression

0.010 0.008 0.006 0.004 0.002

0.15 0.10

0.000

0.05 0.00

0

10

20

30

40

Kneq 0

100

200

300 T

400

500

50

60

70


Conclusions Ø  We have  studied  dynamics  in  the  gapless  phase  of  the  sine-­‐Gordon  model  using  a   quantum  kine7c  equa7on  approach.  The  physical  realiza7on  we  have  in  mind  is  that   of  one  dimensional  bosons  whose  interac7on  parameter  is  quenched  at  the  same   7me  as  a  commensurate  periodic  poten7al  is  switched    on.   Ø  Even  when  the  periodic  poten7al  is  irrelevant  in  equilibrium,  for  the  quench  it  gives              rise  to  inelas7c  mul7-­‐par7cle  scagering.     Ø  The  relaxa7on  7mes  agree  well  with  analy7c  es7mates  and  show  an  interes7ng              non-­‐monotonic  dependence  on  the  quench  amplitude  of  the  interac7on  parameter              (    K      neq        −    K      eq      )   Ø  For  large  quench  amplitudes        K        −        K      0                1        the  system  thermalizes     Ø  For  decreasing  quench  amplitudes      K        −        K      0                1      we  find  an  intriguing  transi7on  from  a   thermalized  steady  state  to  a  non-­‐thermal  steady  state.     Ø  The  non-­‐thermal  steady  state  for  small  quench  amplitudes  is  characterized  by  an   effec7ve  temperature,  but  its  value  is  different  from  the  equilibrium  temperature.  The   smaller  the  quench  amplitude,  the  greater  is  the  devia7on  from  the  equilibrium   temperature.  

Mmv6  

Quench dynamics in the one-dimensional Sine-Gordon Model: Quantum kinetic equation approach (2013)

Mmv6  

Quench dynamics in the one-dimensional Sine-Gordon Model: Quantum kinetic equation approach (2013)