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Lea F. Santos, Yeshiva University Beyond Integrability 2015
Lea F. Santos
Department of Physics, Yeshiva University, New York, NY, USA
Dynamics of Interacting Quantum Systems: Effects of Symmetries, Perturbation Strength,
and Initial States
E. Jonathan Torres-Herrera (U. Puebla, Mexico) Huijie Guan (Rutgers University, USA) Natan Andrei (Rutgers University, USA) Marcos Rigol (Penn State University, USA) co
llabo
rato
rs F. Pérez-Bernal (U. Huelva, Spain)
Luca Celardo (U. Brescia, Italy) Fausto Borgonovi (U. Brescia, Italy)
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Lea F. Santos
Department of Physics, Yeshiva University, New York, NY, USA
Initial state +
Hamiltonian
Ø Isolated quantum systems with interacting particles.
Ø Analysis of quench dynamics. Experiments:
NMR, optical lattices, trapped ions
E. Jonathan Torres-Herrera (U. Puebla, Mexico) Huijie Guan (Rutgers University, USA) Natan Andrei (Rutgers University, USA) Marcos Rigol (Penn State University, USA) co
llabo
rato
rs F. Pérez-Bernal (U. Huelva, Spain)
Luca Celardo (U. Brescia, Italy) Fausto Borgonovi (U. Brescia, Italy)
Dynamics of Interacting Quantum Systems: Effects of Symmetries, Perturbation Strength,
and Initial States
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Same Hamiltonian Different Initial States
Hamiltonian with strong long-range interaction
(trapped ions)
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Trapped ions: long-range interactions
P. Richerme et al, Nature 511, 198 (2014) P. Jurcevi et al, Nature 511, 202 (2014)
H = B ! nz
n! +
J| n"m |"
! nx! m
x
n<m!
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Trapped ions: long-range interactions
H = B ! nz
n! +
J| n"m |"
! nx! m
x
n<m!
P. Richerme et al, Nature 511, 198 (2014) P. Jurcevi et al, Nature 511, 202 (2014)
P. Hauke and L. Tagliacozzo, PRL 111, 207202 (2013)
L=100, excitation on 50
!!..!!"!!..!!Z
! = 3Magnetization
in z of each site
Violation of the Lieb-Robinson
bound
! = 0.7
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Trapped ions: long-range interactions
H = B ! nz
n! +
J| n"m |"
! nx! m
x
n<m!
P. Richerme et al, Nature 511, 198 (2014)
P. Jurcevi et al, Nature 511, 202 (2014)
L=13, excitation on 7
!!..!!"!!..!!X
! = 3Magnetization in
x of each site
LFS, Celardo, Borgonovi arXiv:1507.xxxx
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Trapped ions: long-range interactions
H = B ! nz
n! +
J| n"m |"
! nx! m
x
n<m!
P. Richerme et al, Nature 511, 198 (2014)
P. Jurcevi et al, Nature 511, 202 (2014)
L=13, excitation on 7
! = 3 ! = 0Magnetization in
x of each site
Localization
!!..!!"!!..!!XLFS, Celardo, Borgonovi
arXiv:1507.xxxx
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Trapped ions: long-range interactions
H = B ! nz
n! +
J| n"m |"
! nx! m
x
n<m!
P. Richerme et al, Nature 511, 198 (2014)
P. Jurcevi et al, Nature 511, 202 (2014)
! = 0
Magnetization in x of each site
Localization
!!..!!"!!..!!XLFS, Celardo, Borgonovi
arXiv:1507.xxxx
L=13, excitation on 7
Energies 10 20 30 40 50 60
Mx = ! nx / 2
n=1
L
!2JMx2
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Different Hamiltonians (integrable vs chaotic) Similar Initial States
Hamiltonian with short-range interaction
(optical lattices)
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Integrable vs chaos (1D)
XX model
Chaotic NNN model
H = Jn=1
L!1
" (#SnzSn+1
z + SnxSn+1
x + SnySn+1
y )+
H = J(SnxSn+1
x + SnySn+1
y )n=1
L!1
"
Integrable
H = J(!SnzSn+1
z + SnxSn+1
x + SnySn+1
y )n=1
L"1
#
XXZ model
H = J(!SnzSn+1
z + SnxSn+1
x + SnySn+1
y )n=1
L"1
#
XXZ model
Integrable
Integrable
+! Jn=1
L!2
" (#SnzSn+2
z + SnxSn+2
x + SnySn+2
y )
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Level Spacing Distribution
Integrable XXZ model
Chaotic NNN model
H = J(!SnzSn+1
z + SnxSn+1
x + SnySn+1
y )n=1
L"1
# H = Jn=1
L!1
" (#SnzSn+1
z + SnxSn+1
x + SnySn+1
y )+
+! Jn=1
L!2
" (#SnzSn+2
z + SnxSn+2
x + SnySn+2
y )
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Level Spacing Distribution: Integrable Models
! = 0.1 ! = 0cos(! / 2)
! = 0.5cos(! / 3)
! = cos(!" / N )Roots of units
Kudo and Deguchi J. Phys. Soc. Jpn (2005)
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Density of States
Chaotic NNN model
H = Jn=1
L!1
" (#SnzSn+1
z + SnxSn+1
x + SnySn+1
y )+
Integrable
H = J(!SnzSn+1
z + SnxSn+1
x + SnySn+1
y )n=1
L"1
#XXZ model
+! Jn=1
L!2
" (#SnzSn+2
z + SnxSn+2
x + SnySn+2
y )
-6 -4 -2 0 2 4 6E
0
0.1
0.2
(E)
-6 -4 -2 0 2 4 6E
-6 -4 -2 0 2 4 6E
(a) (b) (c)
-6 -4 -2 0 2 4 6E
0
0.1
0.2
(E)
-6 -4 -2 0 2 4 6E
-6 -4 -2 0 2 4 6E
(a) (b) (c)
! = 0.5
! = 0.5! =1
0.2
0.1
0
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Overlap between the initial state and the evolved state
F(t) = !(0) |!(t)"2= C!
ini
!
#2e$iE! t
2
Survival Probability (Fidelity)
|!(0)" = ini = C!ini
!
# |"! " |!(t)" = C!ini
!
# e$iE!t |"! "
Eigenvalues and eigenstates of the final Hamiltonian
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Overlap between the initial state and the evolved state
Survival Probability (Fidelity)
|!(0)" = ini = C!ini
!
# |"! " |!(t)" = C!ini
!
# e$iE!t |"! "
Eigenvalues and eigenstates of the final Hamiltonian
F(t) = !(0) |!(t)"2= C!
ini
!
#2e$iE!t
2
% Pini (E)2 e$iEt dE
$&
&
'2
Fourier transform of the weighted energy distribution of the initial state of the LDOS (local density of states), strength function
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Overlap between the initial state and the evolved state
F(t) = !(0) |!(t)"2= C!
ini
!
#2e$iE!t
2
% Pini (E)2 e$iEt dE
$&
&
'2
Fourier transform of the weighted energy distribution of the initial state of the LDOS (local density of states), strength function
Survival Probability (Fidelity)
|!(0)" = ini = C!ini
!
# |"! " |!(t)" = C!ini
!
# e$iE!t |"! "
Eigenvalues and eigenstates of the final Hamiltonian
-6 -4 -2 0 2 4E
0
0.1
0.2
0.3
0.4
|C |2
0
0.1
0.2
0.3
0.4
|C |2P
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Perturbation increases Fidelity decays faster
Hinitial = HXXZquench! "!! H final = HXXZ +!HNNN
-6 -4 -2 0 2 4
J-1
E!
0
1
2
3
4
5
P!
0 1 2 3 4
Jt
10-4
10-3
10-2
10-1
100
F
" = 0.2
delta function
Slow decay
L=16, 8 up spins, T=7.1 ! = 0.5
F(t) = Cini (E)2e!iEt dE
!"
"
#2
LDOS:
Integrable Chaotic
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Exponential decay
-6 -4 -2 0 2 4
J-1
E!
0
0.2
0.4
0.6
0.8
1
P!
0 1 2 3 4
Jt
10-4
10-3
10-2
10-1
100
F
" = 0.45 F(t) = exp(!"init)12!
!ini
(Eini "E)2 +!2ini / 4
Lorentzian
F(t) = 12!
!ini
(Eini "E)2 +!2ini / 4
e"iEt dE"#
#
$2
L=16, 8 up spins, T=7.1 ! = 0.5
C!ini 2
Hinitial = HXXZquench! "!! H final = HXXZ +!HNNN
LDOS:
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Exponential decay
-6 -4 -2 0 2 4
J-1
E!
0
0.2
0.4
0.6
0.8
1
P!
0 1 2 3 4
Jt
10-4
10-3
10-2
10-1
100
F
" = 0.45
F(t) = exp(!"init)12!
!ini
(Eini "E)2 +!2ini / 4
Lorentzian
L=16, 8 up spins, T=7.1 ! = 0.5
C!ini 2
Hinitial = HXXZquench! "!! H final = HXXZ +!HNNN
LDOS: F(t) = !(0) |!(t)"2
= C!ini
!
#2e$iE!t
2
~1$" ini2 t2
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Faster than exponential: Gaussian
-6 -4 -2 0 2 4
J-1
E!
0
0.1
0.2
0.3
0.4
0.5
P!
0 1 2 3 4
Jt
10-4
10-3
10-2
10-1
100
F
" = 1
Strong perturbation regime (global quench)
Gaussian
12!" ini
2e!(E!Eini )
2
2" ini2 F(t) = exp(!! 2
init2 )
L=16, 8 up spins, T=7.1 ! = 0.5
C!ini 2
Hinitial = HXXZquench! "!! H final = HXXZ +!HNNN
LDOS:
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Gaussian decay Gaussian DOS & LDOS
-6 -4 -2 0 2 4
J-1
E!
0
0.1
0.2
0.3
0.4
0.5
P!
0 1 2 3 4
Jt
10-4
10-3
10-2
10-1
100
F
" = 1
Strong perturbation regime (global quench)
12!" ini
2e!(E!Eini )
2
2" ini2 F(t) = exp(!! 2
init2 )
L=16, 8 up spins, T=7.1 ! = 0.5
C!ini 2
Hinitial = HXXZquench! "!! H final = HXXZ +!HNNN
E!
!
Density of States
Gaussian
LDOS:
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Decay slows down away from the middle of the spectrum
E!
!
French & Wong, PLB (1970)
DENSITY OF STATES of systems with 2-body interactions
-6 -4 -2 0 2E
0
0.2
0.4
0.6
Pini
0 1 2 3 4 5t
10-3
10-2
10-1
100
F
-3.5 -3 -2.5 -2 -1.5Eini
0
1
2
3
1
-3.5 -3 -2.5 -2 -1.5Eini
0
1
2
2
(a) (b)
(c) (d)
LDOS: skewed Gaussian
L=18, 6 up spins
HXXZ!! "! HXXZ +!HNNN
! =1
Torres & LFS PRA 90 (2014)
C!ini 2
Integrable Chaotic
Lea F. Santos, Yeshiva University Beyond Integrability 2015
0 2 4 6 8 10
0.2
0.40.60.81
F
0 2 4 6 8 10
0 3 6 9 1210-3
10-2
10-1
100
F
0 3 6 9 12
0 1 2 3Jt
10-4
10-2
100
F
0 1 2 3Jt
µ = 0.2 = 0.2
µ = 0.4 = 0.4
= 1µ = 1.5
Exponential and Gaussian F(t) Hfinal: Chaotic or Integrable
! = 0.4
! = 0.2
! =1.5
J(SnxSn+1
x + SnySn+1
y )n=1
L!1
"#$
J(SnxSn+1
x + SnySn+1
y +$SnzSn+1
z )n=1
L!1
"
HXXZ!! "! HXXZ +!HNNNHXX
!" #" HXXZ
Integrable to
integrable
Integrable to
chaotic
L=18, 6 up spins
Torres, Manan, LFS NJP 16 (2014)
Torres & LFS PRA 89 (2014)
Torres & LFS PRA 90 (2014)
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Exponential and Gaussian F(t) Hfinal: Chaotic or Integrable
J(SnxSn+1
x + SnySn+1
y )n=1
L!1
"#$
J(SnxSn+1
x + SnySn+1
y +$SnzSn+1
z )n=1
L!1
"
HXXZ!! "! HXXZ +!HNNNHXX
!" #" HXXZ
Integrable to
integrable
Integrable to
chaotic
L=18, 6 up spins
Torres, Manan, LFS NJP 16 (2014)
Torres & LFS PRA 89 (2014)
Torres & LFS PRA 90 (2014)
-1 -0.5 00246
8
P
-1 -0.5 0
-2 -1 0 10
1
2
P
-2 -1 0 1
-4 -2 0 2 4
J-1E
0
0.1
0.2
0.3
0.4
P
-4 -2 0 2 4
J-1E
µ = 0.2 = 0.2
µ = 0.4 = 0.4
= 1µ = 1.5
! = 0.4
! = 0.2
! =1.5
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Lower bound achieved
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
Pini
0 5 10 15 20 2510
-4
10-2
100
F
-6 -4 -2 0 2 4 6
E
0
0.1
0.2
0.3
0.4
0.5
Pini
0 1 2 3 4 5 6
t
10-4
10-2
100
F
(a) (b)
(c) (d)
L=16, 8 up spins ! = 0.48
F(t) = cos2 (dt / 2)exp(!! 2t2 )d=8
Torres & LFS PRA 90 (2014)
Hinitial = HXXZ = J(SnxSn+1
x + SnySn+1
y +!SnzSn+1
z )n=1
L"1
# $%$ H final = HXXZ + dJSL/2z
C!ini 2
impurity model
F(t) ! cos2 (! init)
Mandelstam-Tamm relation
! H! A ![H,A]2i
=12d Adt
LDOS
Lower bound
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Long-time decay
0 5 10 15 20Jt
10-6
10-4
10-2
100
FF(t) = C!ini
!
!4+ C!
ini
!""
!2C"
ini 2 ei(E!#E" )t
F(t) = !(0) |!(t)"2= C!
ini
!
#2e$iE!t
2
! F = C!ini
!
"4= IPRini
!"!"!"!"Néel state !"! HXXZ +!HNNN
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Long-time decay
!"!"!"!"Néel state !"! HXXZ +!HNNN
C(t) = 1t
F(! )0
t! d! " t#"
L=16, 8 up spins
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Many-body localization
DISORDER &
POWERLAW
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Disorder: Random Magnetic Fields
H final = hnSnz
n=1
L
! + J(SnxSn+1
x + SnySn+1
y + SnzSn+1
z )n=1
L
!
n-2 n-1 n n+1 n+2
Anderson localization Many-body localization
Ising model
Disordered XXZ model Hinitial = JSnzSn+1
z
n=1
L
!
Schreiber et al arXiv:1501.05661
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Survival Probability vs LDOS
L=16, 8 up spins
100 102 104t
10-4
10-3
10-2
10-1
100
<F(t)>
0
0.1
0.2
0.3
-4 -2 0 2 4E
0
1
2
0
0.4
0.8
(a)
(b)
(c)
(d)
J
h=0.5
Torres & LFS PRB(2015) - arXiv:1501.05662
h=1.5
h=2.7
100 102 104t
10-4
10-3
10-2
10-1
100<F(t)>
0
0.1
0.2
0.3
-4 -2 0 2 4E
0
1
2
0
0.4
0.8
(a)
(b)
(c)
(d)
h=1.5
h=2.7
C!ini 2
100 102 104t
10-4
10-3
10-2
10-1
100
<F(t)>
0
0.1
0.2
0.3
-4 -2 0 2 4E
0
1
2
0
0.4
0.8
(a)
(b)
(c)
(d)
C!ini 2
100 102 104t
10-4
10-3
10-2
10-1
100
<F(t)>
0
0.1
0.2
0.3
-4 -2 0 2 4E
0
1
2
0
0.4
0.8
(a)
(b)
(c)
(d)C!ini 2
h=0.5
h=2.7 C!
ini 4
!
! = IPRiniH final = hnSn
z
n=1
L
! + J(SnxSn+1
x + SnySn+1
y + SnzSn+1
z )n=1
L
! multifractal fluctuations
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Powerlaw Exponent
10-1 100 101 102 103t
10-4
10-3
10-2
10-1
100
<F(t)>
10-1 100 101 102 103t
10-1
100(a) (b)h=1.0 h=2.7
L=16 L=14 L=12 L=10
J J
Generalized dimension Multifractal dimension
!IPRini = C!ini 4
!
! "1
(Dim)"# " <1
F(t) = dE! dE" C!ini 2 C"
ini 2!! ei(E""E! )t # d#ei#t |# |$"1! #1t$
F(t) ! t"!
Torres & LFS PRB(2015) - arXiv:1501.05662
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Time-Averaged Survival Probability
C(t) = 1t
F(! )0
t! d! " t#"
L=16 L=14 L=12 L=10
10-1 100 101 102 103t
10-3
10-2
10-1
100
C(t)
10-1 100 101 102 103t
10-1
100(a) (b)h=1.0 h=2.7
J J
PRtypini ! exp lnPRini( )" (Dim)! typ
Torres & LFS PRB(2015) - arXiv:1501.05662
IPRini = C!ini 4
!
! "1
(Dim)"# " <1
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Summary
H = B ! nz
n! +
J| n"m |"
! nx! m
x
n<m!
!!..!!"!!..!!z !!..!!"!!..!!x! <1
0
100
200 Same Hamiltonian,
different initial states
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Summary
H = B ! nz
n! +
J| n"m |"
! nx! m
x
n<m!
!!..!!"!!..!!z !!..!!"!!..!!x! <1
0
100
200 Same Hamiltonian,
different initial states
Similar initial states, different Hamiltonians
0 2 4 6 8 10
0.2
0.40.60.81
F
0 2 4 6 8 10
0 3 6 9 1210-3
10-2
10-1
100
F
0 3 6 9 12
0 1 2 3Jt
10-4
10-2
100
F
0 1 2 3Jt
µ = 0.2 = 0.2
µ = 0.4 = 0.4
= 1µ = 1.5
HXXZ + !HNNNHXXZintegrable chaotic
-8 -4 0 4 80
0.1
0.2
0.3
0.4
0 2 40
0.6
1.2
2 4 6
J-1
E!
0
0.3
0.6
0.9
1.2
0 2 4 6
J-1
E!
0
0.2
0.4
0.6
0.8-8 -4 0 4 8
0
0.2
0.4
0.6
PN
S
-8 -4 0 4 80
0.1
0.2
0.3
0 2 4 6
J-1
E!
0
0.5
1
1.5
2
2.5
PD
W
(a) (c)
(d) (e) (f)
(b)
C!ini 2
E!
! = 0.5
Gaussian LDOS
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Dynamics at an ESQPT
Same Hamiltonian, initial states with similar energy,
but different structures
LFS & Pérez-Bernal arXiv: 1506.06765
Lipkin
H = (1!! ) N2+ " n
z
n=1
N
"#
$%
&
'(!
4!N" n
x" mx
n,m=1
N
"
L=2000
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Dynamics at an ESQPT
Same Hamiltonian, initial states with similar energy,
but different structures
LFS & Pérez-Bernal arXiv: 1506.06765
Lipkin
H = (1!! ) N2+ " n
z
n=1
N
"#
$%
&
'(!
4!N" n
x" mx
n,m=1
N
"! = 0.6
L=2000
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Dynamics at an ESQPT
Same Hamiltonian, initial states with similar energy,
but different structures
Lipkin
H = (1!! ) N2+ " n
z
n=1
N
"#
$%
&
'(!
4!N" n
x" mx
n,m=1
N
"LDOS
LFS & Pérez-Bernal arXiv: 1506.06765
! = 0.6
L=2000
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Dynamics at an ESQPT
Same Hamiltonian, initial states with similar energy,
but different structures
Lipkin
H = (1!! ) N2+ " n
z
n=1
N
"#
$%
&
'(!
4!N" n
x" mx
n,m=1
N
"LDOS
LFS & Pérez-Bernal arXiv: 1506.06765
U ! = 0.6
L=2000
U(n+1) QPT
U(n) SO(n+1)
Lea F. Santos, Yeshiva University Beyond Integrability 2015
Conclusions
Ø Equilibration process depends on the structure of the initial state with respect to the final Hamiltonian that evolves it.
Ø The shape and width of the LDOS: short-time decay of the survival probability
PRA 89 (2014) PRA 90 (2014) NJP 16 (2014) PRE 89 (2014)
• Lorentzian LDOS >>> exponential decay. • Gaussian LDOS >>> Gaussian decay. • Bimodal LDOS >>> uncertainty relation bound • Fragmented LDOS >>> powerlaw behavior at long times. PRB (2015)
arXiv:1501.05662 arXiv:1506.08904
THANK YOU
Ø The filling of the LDOS: long-time decay of F(t)
Ø Presence of a critical point affects the LDOS arXiv:1506.06765