Real time confinement in a quantum quench
Pasquale Calabrese
SISSA-Trieste

Kyoto, June 15th 2016
Quantum matter, spacetime and information
Based on arXiv:1604.03571
with Mario Collura, Marton Kormos, Gabor Takacs

Quantum quench dynamics• A many-body quantum system is prepared in the groundstate of H0, i.e. |Ψ0⟩

• At t=0, H ➠H, i.e. a Hamiltonian parameter is quenched• For t>0, it evolves unitarily: |Ψ(t)⟩=e |Ψ ⟩• No contact with “external” world• What are the main features of the dynamics?
• What about a “stationary state”?
0

-iHt

0

The study of quench dynamics has been boosted by cold-atomexperiments in the last decade or soLight-cone spreading of entanglement entropy
PC, J Cardy 2005

• After a global quench, the initial state |ψ › has an extensive excess of energy0• It acts as a source of quasi-particles at t=0. A particle of momentum p hasenergy Ep and velocity vp=dEp/dp• For t > 0 the particles moves semiclassically with velocity v• particles emitted from regions of size of the initial correlation length arepentangled, particles from far points are incoherent• The point x ∈ A is entangled with a point x’ ∈ B if a left (right) moving particle
arriving at x is entangled with a right (left) moving particle arriving at x’. Thiscan happen only if x ± vp t ∼ x’∓ vpt

B

2t

2t
A

B
t

G. 7: Space-time picture illustrating how the entanglement between an interval A and the reLight-cone spreading of entanglement entropy

he system, due to oppositely moving coherent quasiparticles, grows linearly and then saturate

e case where the particles move only along the light cones is shown here forPC,clarity.
J Cardy 2005

The entanglement entropy of an interval A of length l	 is proportional to•x for which this can be satisfied. Thus the total entanglement entropy is

the total number of pairs of particles emitted from arbitrary points such
Z
Z
Z 1
Z
that at timet, x ∈00A and x’ ∈ B 0 00 0 00 0
0
0
00
00

A (t) ⇡

dx

dx

dx

f (p , p )dp dpxxv(p )txxv(p )t• Denoting with f(p) the rate of production of pairs of momenta ±p and their (4.1x0 2Ax00 2B1

contribution to the entanglement entropy, this implies

SA (t) ⇡

Z

/ t

dx0

x0 2A
Z 1
0

Z

dx00
x00 2Bdpf (p)2vp ✓(`Z 1

Z

f (p)dp x0 x vp t x00
1Z 1
2vp t) + `dpf (p)✓(2vp t `)dx

x + vp t(4.2

0

When v is bounded (e.g. Lieb-Robinson bounds) |v |<v , the second•
wice that between A and the real axis to the right of A, which corresponds to takin

Now specialize to the case where A is an interval of length `. The total entanglemen
p

p

max

term is vanishing for 2 vmax t<l and the entanglement entropy grows
withabove.
time upto integrations
a value linear
in l	 
< 0, p00linearly> 0 in theTheoverthe coordinates then give max (v( p0 )00))t, ` , so that!
nsequences of these analytic results are discussed and the e↵ectsY
f the Majorana operatorszx.60.Htaccount numerically.
ǎ2l 1 ⌘ml,!
!

.60.Htǎ2l 1 ⌘

Y

z
m

x
l ,

ǎ2l ⌘One example
Yyzml .the matrix
We introduce
m<l

m<l

(3.2)

ǎ2l ⌘Ym<lA` through the relation hǎm ǎn i =

z
m

!

y
l.

(3.

A
mn + i ` mn with 1  m, n 

d quantum
wherem<l j↵ are the Pauli matrices
at the site j. Periodic
It has the form of a block Toeplitz matrix
A
A
oduce
matrix boundary
relation hǎmareǎn i always= mn + iimposed.
 m, n of`. its 3
2Despite
ms oftheconconditions` through the` mn with 1
Transverse
field
Ising
chain
PC, J Cardy
2005⇧0 ⇧ 1 · · · ⇧1 `↵subject
simplicity,
the
model
shows
a
rich
phase
diagram
being
6
7
2
3
dhehequantum
where
are
the
Pauli
matrices
at
the
site
j.
Periodic
form of a block Toeplitz matrix
j
6
.. 7
6
7
⇧
⇧
fl g l2
30 with .
realization
for h conditions= 1 and anyareandfor imposed.
= 0A` =
and1,
6 h1 
7its
msof con- criticalboundaryalwaysDespite
of
4
5.
,
⇧
=
(3.
l
.
.
6
7
.
.
.
.
⇧
⇧
·
·
·
⇧
0 two 1critical
1 lines`. . 7
6 universality
g l fl.
of equilibthe
belongingtorich723di↵erent
hesubject 6
simplicity,
the
model
shows
a
phase
diagram
being
4
5
6
.. 7
6
⇧
⇧
. 7entanglement
fl entropygl⇧` 1 · · ·as· ·the
· ⇧0
th
high
declasses.
The
block
is
defined10realizationA` = 6
critical
for
h
=
1
any
and
for
=
0
and
h

1, with
7 and
4
5
,
⇧
=
.
(3.3)
l
.
.
6
7
...
.. Neumann.. entropyAstronglyVonlogdi↵erent⇢` , whereuniversality⇢` =` = g l Tr⇢
67lines S
fl `towithof equilib- 4 the two criticalbelonging5
Z 2⇡ formed
these pheTr⇢
isthereduceddensitymatrix
of
the
block
n
`
1
⇧
· · · ⇧blocki'li✓as theth high declasses.
entanglement entropy
is defined` 1 · · · The
0
g
d'e
e
(cos ' i sin ' cos 2✏' t) ,l =methods toby ` contiguous spins. In the following we2⇡will
considerA strongly
Von Neumann entropy S` = Tr⇢` log
Z⇢0 `2⇡, where ⇢` =i
For meththe quench with parameters suddenlyf changed
at ti'l=sin0 sin 2✏ t ,
=
d'e
(3.
l
'
'
Z
these
pheTr
⇢
is
the
reduceddensitymatrixoftheblock
formed2⇡n`2⇡0malization
1from h0 , i'l
0 toi✓'h, .
gto
d'e
e
(cos spins.
i sinIn' thecos 2✏followingl ='' t) ,
methods`
contiguous
we willconsiderwhere
2⇡by[3] to a too
Our
result
is
that,
in
the
thermodynamic
limit
N
!
1
0
Z
q
For meththe2⇡subsequently
quench
withinparameterssuddenlychangedatt = 02
iand
ween
parts
the
limit
of
a
large
block
`
1,
the
i'l
✏' (3.4)= (h cos ') + sin2 ' ,fl =d'e
sin ' sin 2✏' t ,malizationfromh0 , 0 to h,. entanglement entropy isqa classical 2⇡time
0 dependenceof the02 + sin2 ' ,
✏
=
(h
cos
')
0
'
[3]
to a tooamountof
Our result
is that, in the thermodynamic
limit N ! 1
ZZ
cos ' h i sin '
i✓
q
d'
d'
ween parts
e
=
,
and
subsequently1,
the0in the limit of a large block `✏2S(t) =
2|✏
H(cos ' ) , '
✏' t=
(h 2⇡
cos
')2|H(cos+ sin ' , ' )+ `sta fromtheclassical
2⇡time dependenceof 1
thewithentanglement
entropy issin '(h0 h)Analyticallyfort/lconstant
q t, l ⨠00
sin ' =,
2|✏ |t>`0
dingof the0 2|✏ |t<`2
2✏✏amountof''✏' = Z(h0 cos ') + sin ' ,ZMFagotti, PC 2008
(2)
particular
1
cos
'(h
+
h
)
+
hh
0
0
d' i sin0 '
cosd'
.
(3.
0 cos ' h
' =
i✓
'
0,
where
✏
=
d✏/d'
is
the
derivativeof
thedispersion
ree
=
,
✏
✏
S(t)
=
t
2|✏
|H(cos)+`H(cos)
esults''' '✏st fromfrom
the
'
2
2⇡
lation ✏2 =0 (h 2⇡cos ')2Calling+ 2 thesineigenvalues' and0 representsA
merical onesof ` as ±i⌫mthe, m = 1 . . . `, the entanglement entropy is S
2|✏
|t<`
2|✏
|t>`
sin
'(h
h)
0
ing of the momentumP`
sin ' =, sounddependent
velocity(that
because of
ccepted [3]
0
H(⌫
)
where
H(x)
is
✏
✏
m
(2)
m=1
'
'
0
particular locality has a maximum
weindicate
as
v
⌘
max
|✏
|),
M
'
with time
0 1 cos '(h + h0 ) + hh0
1+x
1 + xre-1 x
1 x
where
✏
=
d✏/d'
is
the
derivativeof
thedispersion
2
2
sults
from
H(x)
=
log
log
.
(3.
cos '
= (hh0. + cos ' +
(3.5)
cos
cos
'(h
+
h
)
sin
')/✏✏
' =
0
0
0
nitial state
0
2
2
2
2
2
2
2
2
✏' ✏cos'lation✏=(h')information+ sin '
ericalones
contains
allthequench[8]andand represents
H(x) = thee excess of''Light-cone spreading of correlations
The same scenario is valid for correlations:

PC, J Cardy 2006/07• Horizon: points at separation r become correlated when left- and rightmoving particles originating from the same point first reach them

• If |v |<v , connected correlations are then frozen for t < r/2v
p

max

max

Example: Ising model within ferromagnetic phase
PC, F Essler, M Fagotti 2011/12
Quantum Quench in the Transverse Field Ising chain I

9Figure 4. Numerical data for a quench within the ferromagnetic phase from h0 = 1/3 toh = 2/3. Left: The two-point function against the asymptotic prediction Eq. (19) for ` = 30 (upLight cone in interacting models
5

0.1

+

4

3

3

2
1

0.001

0
r=2
r=3
r=4
r=5
r=6

<n0nr>(t)rescaled+

0.01

time t<b0br>(t)low pass

0.01

0.1

t

0

Carleo et al, ’14
Bose-Hubbard

10
8

-1
0

(a)

0.2

2
0.4

0.6

0.8
-1

time t [J ]

1

1.2

0.3
1.4
0.25

vinst

2
1

(a) V = 0

4

3

3

2

00

2
1

1
0

(c) V = 5

0

4

0

10
|i−j|

(b) V = 2
20

0

0

10
|i−j|

(d) V = 20
20 25

−0.005−5Bonnes, Essler, Lauchli ’14
XXZ spin chain

FIG. 1: (color online) Time evolution of the equal-time density correlation function Ci,j (t) of spinless fermions after a quench
from the CDW ground state of H(V0 ) with V0 = 10, evolved by the Hamiltonian H(V ), with (a) V = 0, (b) V = 2, (c) V = 5,(d) and V = 20.

to u(V = 0) = 2vF = 4th , as expected, where vF denotes
the Fermi velocity for V = 0. In addition to the light
15
cone, additional
propagation fronts at later times can be
identified in Fig. 1(a), which, however, possess a lower velocity. This signals that slower quasiparticles stemming
10 without linear dispersion also participate
from regionsin spreading information. Figure 1(c) shows the evolution of the correlation function for a quench within thei 5 i.e., a case which should not be describable
CDW phase,†by conformal field theory. Interestingly, we neverthelessr
0 light-conefind a pronounced
behavior in the correlation
function. 0
Although the conformal field theory underlying
r of Calabrese and Cardy is not valid in thisthe treatment
region, the physical picture that ballistically propagating
5 are generated by the quench seems to hold.
quasiparticles
However, in contrast to the case of the quench to the LL
displayed in Figs. 1(a) and (b), we see that a strong alternating pattern
10 forms in the density correlation function
and remains present and qualitatively unchanged after
the onset of the light
0 cone.
-0.0150.2 from U = 2J toFigureafter a quench
6 3. Time-evolution of correlation functions⇥
R
Uf = 40J. The upper panel shows the single particle correlation
0.15 functions b b
0
8 16 24for different
distances r. The correlations
show partial revivals up to a time t when
4
5.2
0.1
they start to reach a quasi-steady state. This time tr grows approximately
linearly
5
with2the distance r as marked by the vertical lines. The central panel shows the
0.05
4.8
same correlations functions after filtering out the high frequencies, see text
for details.
⇥
4.6density correlations function n n after15shifting
The 0lowest panel shows the density
0 r
0
15
30
45
and rescaling their amplitude for
0 lines1
Rbetter visibility. The common vertical dashed1.4
denote the arrival of the minima as determined from the density-density
correlations.

t−3

x0.00510
5

5

4
time ttime t<b0br>(t)1time tKollath-Lauechli ’08
and entanglement after a quench in the Bose-Hubbard model
Bose-Hubbard

Manmana et al ’08
Fermi-Hubbard
8

3

2

3

A more detailed view of the temporal evolution of thecorrelation functions is shown in Fig. 2, in which we plot
the values of Ci,j (t) as a function of time for increasing
distance | i−j | for V = 0 and V = 2, the two extremes of
the Luttinger-liquid phase. After the arrival of the firstsignal, oscillatory behavior as a function of time can beobserved at each distance. However, as V is increased,
the observed oscillations both decrease in magnitude andare damped out more rapidly. Comparing the results forthe free case to the ones obtained for V = 2 in Fig. 2, it
can be seen that the incoming front travels with a higher
velocity when V is larger, as can also be seen in Fig. 1.
In contrast to the oscillatory behavior in the Luttingerliquid phase, a steady increase of the correlations is observed when the quench occurs within the CDW phase, as
can be seen in Fig. 3. The alternating pattern imprinted
at the onset of the light cone is preserved. Presumably,
the correlation functions saturate at some time that issignificantly longer than the maximum time reached here.
While results for both V < V0 and V > V0 show the same

4

5

6

7

Light cone in experiment
M. Cheneau et al, Nature 481, 484 (2012)aquenchposition
b

time

~
~

d=vt

FIG. 1. Spreading of correlations in a quenched atomic
Mott insulator. a, A 1d ultracold gas of bosonic atoms(black balls) in an optical lattice is initially prepared deepin the Mott-insulating phase with unity filling. The lattice
depth is then abruptly lowered, bringing the system out of
equilibrium. b, Following the quench, entangled quasiparticle
pairs emerge at all sites. Each of these pairs consists of adoublon (red ball) and a holon (blue ball) on top of the unityfilling background, which propagate ballistically in opposite
directions. It follows that a correlation in the parity of thesite occupancy builds up at time t between any pair of sites
separated by a distance d = vt, where v is the relative velocityof the doublons and holons.

2lattice depths to 20.0(5) Er , where Er = (2⇤~)2 /(8ma2lat )is the recoil energy of the lattice and m the atomic mass of87
Rb. The e⇢ective interaction strength along the chains
was tuned via a third optical lattice along the x-axis. The
number of atoms per chain ranged between 10 and 18, resulting in a lattice filling n̄ = 1 in the Mott-insulating domain. The inital state was prepared by adiabatically increasing the x-lattice depth until the interaction strength
reached a value of (U/J)0 = 40(2). We then brought thesystem out of equilibrium by lowering the lattice depth
typically within 100 µs, which is fast compared to theinverse tunnel coupling ~/J, but still adiabatic with respect to transitions to higher Bloch bands. The final
lattice depths were in the Mott-insulating regime, close
to the critical point. After a variable evolution time, we“froze” the density distribution of the many-body stateby rapidly raising the lattice depth in all directions to⌅ 80 Er . Finally, the atoms were detected by fluorescenceimaging using a microscope objective with a resolution
on the order of the lattice spacing and a reconstruction
algorithm extracted the occupation number at each lattice site [17]. Because inelastic light-assisted collisions
during the imaging lead to a rapid loss of atom pairs, we
directly detected the parity of the occupation number. FIG
Our experimental results for the time evolution of the pro

9.0

Some no light-cone spreadings
MBL, logarithmic growth ofentanglement:

Jz /J
0.0
0.01
0.1

0.3

Jurcevic et al Nature 511, 202 (2014)

0.2
0.1, L = 20abĮa

cĮa

Įa

20

12
4

4

Time (1/J)

0.4

2Time (ms)Bardason, Pollmann, Moore, ’12

Long-range interaction:

dï7

4

1 ï9

30

ï8

10

ï
15

3

3

8

10

6
2

0.14

S

2

2

0

0.5

1

e 25

10
1

0.07

1

ï8
ï

4

S

0.4

ï6
0

0.1

0.2

ï4

3

20

0.2

0

5

20

1
ï

2

15
10
5

0

0
0
0.1

1

0.1

10
J t

1

10 100
Jz t

100

1000

see also: De Chiara et al ’05
0.6 J /J = 0.2
Burrell & Osborne ’07
Vosk and Altman ’13
0.3

0.8

S

z

L = 4, 6, 8, 10(bottom to top)

S1

0.6

0.0

100

103 106
J t

109

Jz /J
0.0

5

ion 1015

0 0

5

ion 1015

0 0

5

ion 1015

0When the range of interaction is longenough there is no light cone

0
0.5

1

FIG. 4: Break-down of the light-cone picture as the underlying interaction range is increased. a-c. In a chain of 15
spin is flipped and the subsequent evolution of the magnetisation h zi (t)i is measured. From a to c, the interaction range
power laws with ↵ ⇠ 1.41, 1.07, 0.75. Red lines: fits to the experimentally observed magnon arrival times (see methods;
in panel d). With increasing interaction range, the signal clearly propagates faster than what is allowed by the nearest-nei(white lines, derived from the maximal group velocity of the nearest-neighbour model). d. Magnetisation evolution of spinfrom panel a (bottom) and c (top). Solid lines: Gaussian fits to measured magnon arrival. Top: In the longest-range case,clearly before what is permitted by the nearest-neighbour Lieb–Robinson bound (shaded region). Bottom: In the shortest-ran
the slight violation of the nearest-neighbour Lieb–Robinson bound at short times is almost not distinguishable. e. Maximu
With increasing ↵, the measured arrival velocities (red circles) of the magnon excitation approach the group velocity of thelight-cone (grey dash-dotted line). For smaller ↵, the measured values are consistent with the divergent behaviour predicted
the full power-law interactions (black line).

see also: Hauke & Tagliacozzo, ’13
Schachenmayer et al ’13
Richerme et al ’14

hx = 0.025, 0.05, 0.1, 0.2, 0.4.

Disappearance of the light cone
the Hamiltonian

with the disStarting from the ferromagnetic state (all spins up) and evolving
p with2J 1 2hz cos
L
X
⇥ x x
⇤
z
x
H= J
(1)
j j+1 + hz j + hx j , FIG.
1: Death of the lj=1x
At quenchinghz = 1 t
ic afterwith hz= 0.25. Connected longitudinal correlation h 1x m+1

where
hx =0

↵ are the Pauli matrices and we imposejhx =0.025
hx =0.05

cal point separat

periodic boundary conditions. The parameters magnetic phases
hz and hx are dimensionless, while J setsen- gapped ferroma
thetheHamiltonian

ergy scale. For hx = 0 we recover the integrable fermionsL
can be
X
⇥
Why??
x x
H
=
J
j+1 + htransverse field Ising model (TFIM) which can domainj wallssehx =0.1

hx =0.2

hx =0.4

j=1

be diagonalized by a Jordan-Wigner mapping to tion ¯ = (1 hwhere j↵ are the Pauli mat
free spinless Majorana fermions ak [20, 21]:
field hx induces
periodic boundary conditionX
tween pairs dom†
hx are dimensionless,
HTFIM ="(k)ak ak + const.h,z and(2)of length d and okmapping it to free spinless fermions:

Confinementin the IsingXmodel with
the Hamiltonian
HTI =

L
X

k

⇥ x x

z
x
H
=
J
+
h
+
h
x j
j+1
j
where the dispersionj relation
isz given
by
j=1

•

⇤

the
†
"(k)ak ak + const.
p ,McCoy & Wu ’78
2J
1 2hz cos,p
For hx= 0, free fermions
with dispersion "(k) = 2J 1
↵

where

dis

Bhaseen, Tsvelik ’04(1) + many more, sorry

At hz = 1 t

2hz cos k + hz 2 .

are the Pauli matrices and we impose cal point separathz= 1 separates
twocaremassive
periodic
boundary
The parametersSome
has tophases
beconditions.
taken with respect
to the boundary
conditions
magnetic
phasesfj•themassive
momentumk, but Jwe
dobe
notseengothese details herhquantizationandh areofthe
dimensionless,
while
sets
the
en-intophase),
fermions
can
asgapped
domain• For h < 1 (ferro
ferroma
⅛
z

z

x

2
hcriticalz
1 thea quantum
point separating theergy scale.= Forhx system= 0 wehas
recoverthe integrable
fermions can bewalls separating domains of magnetization σ=(1- )
At hz

•

z < 1 the system is in the gapped ferromagnetic
magnetic
phases.
Forhh induces an attractive interaction between DW that can be approximated as atransverse field Ising model (TFIM) which can domain walls se
fermions
be thoughtof asxfreely
propagating domain walls separating dlinear potential
V(x)=canχ |x|,
with χ=2Jhσbe diagonalized by a Jordan-Wigner mapping to tion ¯ = (1 hxhz 2 )1/8 . This picture becomes more and more accurate as hz ap

freelyand theyget confinedinto• DW do not propagate
free spinless
Majorana
fermionsa [20,
21]:mesons field h induces
A small (?) non-zero field h induces a linear attractive potential betw
¯ = (1xkx

pairs domwalls which borderXa domain† having magnetization intween
the direction
op
HTFIM ="(k)ak ak + const. ,(2)

distance between thek domain walls, the potential is V (d)· d with
of=lengthd and=oandstate
they=getconfined into bound
Bound
meson
Freewalls
DW do not propagate freely anymore

Ω!Θ,

Ω!Θ;

X=
,⇥ = ✓ x+=✓ x,2 x1✓ ,= ✓2 ✓1 .
1
2
2
r semiclassicalparticles are
fermionic
so
only
odd
valuesof⌫areallowed,⌫=2n
1. The integral
energy levels are given by the Bohr–Sommerfeld
quantization condition,
2
4.2
✓2 ✓1
3.5
I
⇥ =takes✓1 +heuristic✓meson=spectrum.
simpleapproximationforthe
1 . Consider
2 ,form ✓picture
nsformedAascan
Then
the
Hamiltonian
thespectrumbe understoodinthefollowingtw
dp q = 2⇡(⌫4.0+ 1/2) ,
(19)
2
3.0
0
q
q
q
qa
a
a
a
Z
Z
Z
Z
Rutkevich
’08
!2
!1
0
1
2
3
!3
!2
!1
0
1
2
3
dp(q)
ṗ(t)
2
@!(q;
⇥)
2
greinthe!31Dasaclassical
systemwith
theHamiltonian
integration pathtaken along the takes
periodicthe
path
in the phase
use
that
the
form
H
=we
!(✓;
⇥)
+ |x| ,
Θ is HamiltonianΘ space. Now@
dpConsider
q = 2 Then
dq
q
=
2
dq
q
=
dq
q
=
2q
a !(qa ; ⇥)two fermions
in
1D
with
Hamiltonian@q(a) Boundare
statesfor h =dq
0.25,
= 0.1,⇥ =values0.
(b) Boundstates⌫for=h 2n= 0.5,
h =0.1, ⇥= 3.
particles
fermionic
so honlyoddof ⌫ areallowed,1.
The
integralcan be
4.0

z

x

z

qa

qa

x

qa

qa

where
H
= !(✓; ⇥) + |x| ,
H
=
"(✓
)
+
"(✓
)
+
|x
x
|
.
(4
1
2
2
1
nsformedas
FIG. 1: Semiclassical bound state energy levels in the “relative potential”0!(✓, ⇥) from the
1
Zqa
Zqa
Zqa
Zqa
solutionsofEqs. (21). Thethe
dashedverticallines show
✓of
dp(q)
ṗ(t)
2
@!(q;the⇥)turning2 points
a,b .=
tting
everything,
semiclassical
bound
the
two fer⇥)"(✓ +states
⇥/2)
+of
"(✓
.
@!(✓;
A . ⇥/2)
p q = 2together
dq qwhere=2dq q=
dq q energy=spectrum
2qa !(qdq !(q;⇥)a ; ⇥)
are taken todq be real numbers (continuum@q system), and ✓1 , ✓2 are the canonica

qa
qa
qa
qa
determinedbytheequation
This
can
be
quantized
semiclassicallya’laBohr-Sommerfeld
Now we have to cover the case whenEnergy!(✓; ⇥) has
twominima (see conservation
Fig. 1b). This happens
for
(20)
and
momentumgive
bles. Let us now make the ✓canonical transformation!(✓;depend⇥) = "(✓⇥/2)
The number and the energies
of mesonson+h⇥/2)
Θ "(✓Zax, hz, +hx =0.1,hz=. 0.25, Θ=0ting together everything, the semiclassical energy spectrumof bound states of the two fermionsz⇥ > ⇥m = 2 arccos h(23)Ω!Θ;0"

2Ena(⇥)✓⇥) =2⇡ (n 1/4) ,n5.0 = ⇥(t)1, 2, .=. .⇥ = const.
a x1minimum+ xd✓
2 !(✓;oneeterminedby theWhenω equationhas
singleobtains
X
=
,
x
=
x
x
,
(5
2
1
Energyandmomentumconservation
give
4.5
when the second derivative of !(✓;
becomes positive and we get a double well potential.
✓zeroaZ✓a ⇥) at2! (✓(t); ⇥) +|x(t)| = E = const.
4.0
For E > !(0; ⇥)2E
thenaboveis correct,
if (nE < !(0;✓⇥),(⇥)✓isd✓ !(✓;
⇥) of
=but
2⇡
1/4)
2, . . . takes 3.5place (21a)2, the✓classical
1 n = 1,motion
atreatment
ere ✓a = ✓a (n;⇥)
thesolutiontheequation
⇥
=
✓
+
✓
,
✓
=
.
(6
1
2
⇥(t)
=
⇥
=
const.
4
mesons✓ in the interval [✓ , ✓ ] where the turning points are given by then one of the two separated wells,b aand the canonicalequationsof motion are
2

a

3.0

!3

!2

!1

two
solutionsof Eq.
(16).
This
is notinvariant
under
✓! =
✓, so thePauli principle doesre ✓distinct✓a (n;⇥)θisthesolutionof theequationΘ
a =where!(✓a (n;⇥);
thesolution
of
a is
! ⇥)
(✓(t); E⇥)n (⇥)+ .|x(t)| = E =const.
@!(✓;⇥)
tonian
takes
the
form
h
=0.1,
hz =
x
Ẋ(t)levels=, 0.5, Θ=3
not restrict the allowed values of ⌫ = 0, 1, 2, . . . in Eq. (19). Thus the semiclassical energy
0

1

2

3

@⇥
!(✓a (n; ⇥); ⇥) = En (⇥) .
(21b)
For
two
minimaarete nowthatgiven
thebyperiod of the “breathing” motion is T (⇥) = 4✓ (n; ⇥)/ , and the sizes of the@!(✓; ⇥)
ẋ(t) =,
=motion!(✓;is⇥)+ =|x|
e that the period of the “breathing”
Tn (⇥)
4✓a,(n; ⇥)/ , and the sizes of the bound@✓aZ✓H
tes areEn (⇥)(✓a ✓b )d✓ !(✓; ⇥) = ⇡ (n 1/2) ,n = 1, 2, . .@!(✓;.
˙⇥)(24a)es are✓(t)=sgn(x(t)) .

andnand the canonical equations of motion are

a

4.6

(7

Ω!Θ,$"

4.4

Ẋ(t) =
,
@⇥ .
|x
x
|
=
(E
(⇥)
!(0;
⇥))/
2 = (E
1 max
3 mesons
x1 |max!(0; n⇥))/ .
n (⇥)@!(✓; ⇥)⇥ these(22)For a given value of the total momentum
equations descri
4.2

✓b

|x2

4.0

!3

!2

!1

0

1

2

3

approximations. Continuous lines: 1-meson states ofthe first meson. Dashed lines: 1-meson states of the
A simple approximation for themeson spectrum
second meson.

FIG.
Semiclassicalbound state dispersion relation and the low energy spectrum.
hx =0.1,hz=2:0.25,
Θ=0

The four masses are
m2=4.127m3=4.48m4=4.77
1=3.662
momenta ⇥, at least semiclassically, andm
higherlying mesons
have flatter
dispersion
relations. Sinc

5.0

En(θ) is the dispersion relation of the mesons

their velocities are given by

4.0

dEn (⇥)
vn (⇥) =,d⇥

3.5

4 mesons
3.0

4.8

(25

4.6
4.4

!2

!1

0

1

2

En

vmax = 0.274, 0.166,0.094, 0.004
Θ
this means that the heavier bound states move very slowly. The maximal velocites of the four meson
vmax of DW = 0.5
!3

3

4.2
4.0
3.8

are v = 0.274, 0.166, 0.094, 0.04. For hx = 0.2 the velocities are v = 0.188, 0.052. For comparison
!3

!2

!1

0
"

1

2

3

at this transverse field with zero longitudinal field the maximal velocity of the unbound domai◉ Comparison with exact diagonalization:walls is v = 0.5
4.6

The finite size spectrum can be computed via exact diagonalization for small lattices. In Fig. 2

hx =0.1, hz= 0.5

4.4

we plot the low energy4.2part of the spectrum together with one-particle dispersions E1,2 (k ⇤ 2⇡/L

2nd meson 1pt statesEΩ!Θ;0"

4.5

with k = 0, 1, . . . . It is 4.0obvious that the spectrum can be interpreted as energy levels of travelin

particles, and also that3.8the semiclassical approximation is quite accurate already for the lightes

1st meson 1pt states

3.6

meson. Note that the lightest
two-body
states678910lie beyond
11
12 the plotted energy range having energie
LBack to quenches
What happens if there are mesons in the spectrum of the postquench
Hamiltonian in the quasi-particle picture?

|ψ0› acts as a source of quasi-particles at t=0
pairs of quasi-particles move in opposite directions with velocity vp
moving away the quasi-particle feel the attractive interaction
Interaction will eventually turn the particle and start oscillations
ls get confi
ain wal
ned
dom

all

in
w

ll
wa

time

domaininto mesons

a
do m

space

x
1-pt function ⟨σ ⟩Quenches ferro to ferro
1

1

L=8
L = 10
L = 12
L = 14iTEBD

hz = 0.5 → hz = 0.25

0.99

L=8
L = 10
L = 12iTEBD

hz = 0.5 → hz = 0.25

0.99

⟨σx⟩

0.98

⟨σx⟩

2

0.97

0.98

0.97
0.96

0.95

0.96

hX = 0.1
0

5

10

15

20

25

30

hX = 0.2
0

5

10

15

t ⟨σx⟩
Power spectrum of
1

0.1

1

m1 m2 m3 m4

30

m1

m2

hz = 0.5 → hz = 0.25, hx = 0.2
m12

0.1

|σx(ω)|2

1x10-2|σx(ω)|2

1x10-2

1x10-3

1x10-3

1x10-4

1x10-525thz = 0.5 → hz = 0.25, hx = 0.1
m12 m13 m14

20

1x10-4

0

1

2

3

ω

4

5

1x10-5

0

1

2

3

ω

4

5

m2-m1 =0.46, m1 =3.7,
m2-m1 =0.68, m1 =4.0,
m2 =4.1, m3 =4.5
m2 =4.7
FIG. 2: Upper panels: Time evolution of the longitudinal magneti-

Half-chain entanglement entropy
Ferro-to-Ferro
1

Para-to-Ferro
5hz = 0/0.5 → hz = 0.25
hx = 0hx = 0.025
hx = 0.05
hx = 0.1
hx = 0.2
hx = 0.4

0.8

0.6

S

S

0.2

1

10

20

hx = 0hx = 0.1
hx = 0.2
hx = 0.4
hx = 0.8
hx = 1.6

320hz = 2 → hz = 0.25

4

0.4

0

4

30

t

40

50

60

0

0

10

20

30

40

50

t

entanglemententropy
entropy grows
FIG.
Time evolutionof theentanglement
after aThe6:entanglement
entropy
doeshalf-chainThebut muchslowerin thenot seemto grow
indefinitely
quenchto theconfiningphase.butLeft: startingfromthethanferromagnetic
integrable
caseoscillates
arounda finite value
phase
(hz =
0). Right:
starting from theparamagnetic
phase hz =Connected correlation functions
h0z=0, h0x =0, hz =0.25

FIG. 3:Connected longitudinal spin-spin correlation function

FIG. 4:

tion
0.25
0.4.

Connected correlation functions3Let’s zoom:-3)
Conclusions:
there
is
a
feeblelight-cone(afactor10
FIG. 4: The same correlation function in Fig. 3 but on a different
havingis themesonsscale: the plotted signal
around
10 3velocity!
times the one in Fig. 3 andthe orange regions represent out of range values of the correlation

Quench para→ferro
h0z=2, h0x =0, hz =0.25

Conclusions: The light-cone is visible, but it has the mesons
velocity!

Fqpson mcons
Physicalinterpretation
duce
almost invisible portion of the space-time. This effect can beof thexplained by considering that for a quench with hx = 0 the
For hxstate=0, theinitialstate canasbe[17]written in the postquench basis as
fract
initialcanbe written
duce
Y
† †
ever,
| 0i =(1 + iK(k)ak a k )|0i
(2)
a que
k>0
ment
hx confines
walls intomesons.wall excitation (K(k) isinterms ofthe
thedomain
post-quench
domain
tion
given
in
SI).
Addingafield
hconfinesthedomain
walls
into
the
d
x
When K(k) is small, the state is dominated by the linear terms which only
mesons.
K(k)is small,theintostate
is dominatedby thement
contain (k,When-k) pairsthat getconfinedmesonsat rest.
linear terms which only contain (k, k) domain walls pairs
Di
Quadratic
terms (andinto
higher)lead tomesons,but canbe seenmeri
that
get confinedmesonsat propagating
rest. Quadratic
terms
(and
only when
K(k)ones)is large
enough.
higherorder
lead
to propagating mesons, but they are
the n
only substantial when K(k) is large enough. These have veThe
Mesons have velocities that are very different from the domain walls,
locities that are significantly different from those of the doture
main walls, see SI.
the s
For the quench in Fig. 3 (a), K(k) ⌧ 1 for all momenta andfrequpractically only zero-momentum mesons are formed. Zoombe ea

Quench para→para

F
q
p
2
1.2

hz = 2 → hz = 1.75

x x
=0FIG. 5:hh Absolute
value
of
the
correlation
function
h
i
for:
c
1
m+1
= 0.1
h = 0.2
No dramatic
qualitative
difference(upper panel)
quenchfromtheparamagnetic
phase
(h
=
2, hx = 0)zh = 0.4
with increasing
toS the ferromagnetic one (hz = 0.25,
varying hhxx,);reflecting(lower thepanel)
absencequench within the paramagnetic casefrom ofhzmesons= 2 and hx = 0 tohz = 1.75 and varying hx . While in the confining phase the lightcone experiences a drastic non-perturbative change, in the paramagnetic phase it is only perturbatively modified.
1

x
x
x

0.8

x

0.6

s
t

0.4

0.2

0

0

5

10

15

t

20

25

p
fi

No transverse (mass) quench
0.0015hz = 0.25 → hz = 0.25

hx = 0.1
hx = 0.2
hx = 0.4

0.001

1

1

mij

0.1

m1 m2 m3 m4|S(ω)|2|S(ω)|2

5x10-40

1x10-35x10-40

1x10-4

1x10-4

1x10-5

1x10-5

1x10-6

0

1

2

3

4

5

1x10-6

ω

0

1

2

3

4

ω
hz = 0.25 → hz = 0.25, hx = 0.4

1
0.1

0

10

20

30

40

50

m2

60

t

5

m1

1x10-2
1x10-3|S(ω)|2Sm1

1x10-2

1x10-3

0

m12

0.1

1x10-2

5x10-4hz = 0.25 → hz = 0.25, hx = 0.2

hz = 0.25 → hz = 0.25, hx = 0.1

1x10-4The masses of the mesons are seen even more
neatly due to the absence of transient and drift⇒ entanglement quench spectroscopy

1x10-5
1x10-6
1x10-7

0

1

2

3

ω

4

5

Conclusions
In the Ising chain, mesons freeze the light cone spreadingof correlations and entanglement

Questions:
Is it a general property of the many models displaying
confinement? Presumably yes, possible to check numerically
Can it prevent thermalization? as in Banuls et al 2011?
What about prethermalization?
Is it true in higher dimensions? e.g. in QCD?
maybe holographically one can have some hints