Quantum mechanical probability distributionsin and out of equilibrium

Fabian Essler (Oxford)

Osaka, September 2019

Characterizing Quantum Many-Body Systems
Complete description given by many-body state/wave-function

In practice: typically measure particular expectation values

Averages over many measurements.

A lot more info in the probability distribution of a given
observable O in a QM state |Ψ⟩PO (m) = ⟨Ψ|δ(O − m)|Ψ⟩ =!

2|⟨n|Ψ⟩|
δ(λn − m)n

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O|n⟩ = λn |n⟩0zqtp1zGiWGSjhe1EkFMRIbVkCZXjBrRswSp4vavhHZQITW2wKwtwZs+eZaUjwtnBe/GyxfPYYwM7MI+HIEHJ1CEKyhBGSg8whO8wKszcJ6dN+d9PDrnTDI78AfOxzd/y519</latexit>
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eigenstates of O

For this to be interesting Ο should have many distinct eigenvalues
Why should we care about these quantities?

- They can be interesting and universal- They are measured in cold atom experiments

Cf talks this morning
by BD & JMS

Cold Atom Experiments
Probability distribution of staggered magnetization for 2D Hubbard
model at finite temperature
Greiner group ‘17- Initialize system in some initial state- Measure m- RepeatzAs shown in F
q = qAFM. Dopin
Data
Heisenberg quantum Monte Carlo modelordering peak inbfind that mzc settoffset originates
T/t= 0.2 5
T/t= 0.35
still present at la
22butions of d < 2dependence mzc (Fig. 5). This fin00
Hubbard model
4 T/t= 0.6 4
4 T/t= 1.14
up to a critical hbe consistent wit
22
observed in highin our experime
0
0
–1.0
–0.5
0.0
0.5
1.0
–1.0
–0.5
0.0
0.5
1.0
might prevent um̂zm̂z
close to qAFM.
We have reali
Figure 3 | Full counting statistics of the staggered magnetization
operator. a, Selected images with one spin component removed
two-dimensiona(chequerboard overlaid to guide the eye) show a large variation in ordering possible to varystrength at the coldest temperature. This variation is a consequence of thethe Hubbard ph
Other examples:
SU(2) symmetry of the underlying Hamiltonian, which leads to different
Attainable paramorientations of the staggered spin-ordering vector mˆ relative to theconjectured pse
measurement
axis z, as shown
schematicallyby the spin
Probability
distributionof “relative
phase”
invectors
split(red
1DandBose temperatures
gases
z
T/t ≈
blue arrows) relative to the axis defined by Ŝ (black arrows). b, Measured
a transitiongroup to aˆ z ), are Schmiedmayer
plotted at
distributions of the staggered magnetization operator, p(mcould be achievedifferent temperatures T/t (histograms). We find excellent agreement with’10-‘17
Furthermore, enquantum Monte Carlo simulations of the Heisenberg model with no free
ˆ z)
p(m

ˆ z)
p(m

m̂z = 0.038m̂z = 0.189m̂z = 0.594

Few results available in the literature for such quantities:(homogeneous systems, no currents)•!ℓ
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0

dx e

iΦ(x)

in Luttinger liquid

Gritsev/Altman/Demler/Polkovnikov ’06
Kitagawa et al ’10• total magnetization in GS of critical Ising QFT• some numerics for GS of XXZ

Lamacraft/Fendley ‘08

Moreno-Cardoner et al ‘16• GS of Haldane-Shastry

Stephan/Pollmann ‘17• total transverse magnetisation in GS of
TFIM & related free fermion problems

Cherng & Demler ’07
Ivamov&Abanov’13
Klich ’14…

What to expect on “general grounds” ?
We consider
lattice models;observables O (quantized eigenvalues) that act on subsystems of linear size l, e.g. sub-system magnetization;-

Need ℓ to be large-ish s.t. Ο has many distinct eigenvalues
In states with finite correlation length ξ and ξ≪ℓ we expect

narrow, ≈ Gaussian(“thermodynamics”)
Cases with ξ→∞ or ξ≿ℓ will be most interesting.

in sizethe` =transverse
Ising
chain atInSubsystem
Fig. 1 we show magnetisation
Pw (m) for subsystem
20 and several fielddi↵erent
temperatures.
We T>0employ an order to make ℓthe deviations of the probability distributions from a Gaussian form (which would coparabolicWe can see from Fig. 1 (a) that the temperature dependence for h < 1,z form) more apparent.
z(ℓ) =σj ordered phase at zero temperature, is not very pronounced. In contrast w
o theSuferromagnetically∑stronger temperature
j=1 dependence in the paramagnetic phase, cf. Fig. 1 (b). At low temperatures t
distribution is as expected asymmetric as a result of the applied field and is seen to display an even/(u)
ThePlatter
quickly
whereas−theremains until th(m)disappears= Tr [ρ(β)δ(mas−temperatureSuz(ℓ)] = is2 increased,
Pw(u)(r)δ(m2r)asymmetry(ℓ even)∑exceeds the scale set by the magnetic field.
r∈ℤ(u)size ` for a range of temperatures. These are defined as the thermal e
"
#3
"
#4
D
E
D
E
X
X
p
p
,
3,
2
2
hX i
hX i

ange of temperatures. These are defined as the thermal expectation values"#3" Both skewness#4 and excess kurtosis are non-vanishing for finite andDE
D Gaussian.
AEvery peculiar feature is that at fixed ` skewness and exceXXzz at a fixed temperature th
temperature. Furthermore,
weSobserve
that
p
p
,
3, X=
(`)
hS
u
u (`)i .
22
hX i
hX i This signals that the corresponding probability distributio
increased.

for large subsystem sizes the laws of thermodynamics
p apply and theJ that scalesthat
Gaussianwith a standard
deviation
as the`. distribution
ss and excess kurtosis are non-vanishingfor finiteand `,h=0.5
whichestablishes

very peculiar feature is that at fixed ` skewness and excess kurtosis are non-monotonic functions(a)(b)both tend to zero as the subsystem s
Furthermore, we observe that at a fixed temperature theyV. FULL COUNTING STATISTICS AFTER
his signals that the corresponding probability distribution approaches a Gaussian. This is expec
FIG. 2: Skewness as a function of ` for several values of at (a) h = 0.5 and (b) h = 2.
system sizes the laws of thermodynamics
apply and the probability distribution is then approximpHighercumulants
behaveinway.
We nowthe
turn sameto the time
evolution of the characteristic function (h a standard deviation that scales as`.
two di↵erent classes of initial states:• We initialize the system in the ground state of H(h0 ) and time ev

Part I: FCS in equilibrium at a quantum
critical point

xxnfluctuations in Luttinger
liquids, =the( orderparameter
statistics in
hGS|S
S
|GSi
1)
1
. Theoretical results on full counting
statistics
in quantum criticalsystemsscar
j⌘4/⌘are4 relatively⌘+1/⌘
14 j+n
4n
n
4n
10–13
tizationin theIsingchain in andtheliquids
magnetization
the Haldane-She results includes
phasefluctuations
Luttinger, the orderinparameter
statistics
in!
Collura, FHLE
14
thetransverse
magnetization
in
the
Ising
chain
and
the
magnetization
inthe
Haldane-Sha
Groundstate of criticalspin-1/2XXZ
chained) subsystem
magnetizationintheanisotropic
one-dimensional
spin1
B̃
4
3⌘
z
&Groha‘17zz

arXiv:1706.07–9

consider the (staggered) subsystemin the anisotropic
spin-1hGS|Smagnetization
1 +one-dimensional+(
j+n Sj |GSi =
2
2
4/⌘
4
4⇡ ⌘n
2 2⌘
n
ain
L
L
20–23
X
X
where explicit
expressionsfortheamplitudes
in
(2)
are
known
and
y
y
x
x
z
z
-1<Δ≤1x xH = J y S ySzz
+ Sj SSj+1S+ Sj. Sj+1 .
j+1+
H=J
Sj Sj+1 + Sj Sjj+1
j j+1
j=1

j=1

=

cos(⇡⌘).

Z chain is a paradigmatic model for quantum critical behaviour in 1+1 dimensions. It featItexchange
follows from
(2) thatthroughout
thecriticalregime= thedominant coametrizedbytheanisotropy
1


1.
The
specialvalues±1
correspondtoatic Quasi-long-range
model forquantum
criticalbehaviourin
1+1
dimensions.
It
fea
order
the
xy-planefor|n|≫1antiferomagnet
tizations respectively.
in AFMthe xy-plane.
The
XXZ
exhibits
omagnet and ferromagnet
In theinregime1 chain< thus
1 thelow-energy
behaviour ofnge anisotropy
1theory Two-point 1. The
special= ±1
correspond
16–19
space.
functionssuchsvalues(2) area standard
for chara
bed by Luttingerspin
liquidor equivalently
a free,compact
boson. Themeans
long-distancerespectively.
Inthe<  1criticalthe low-energy
ground“phases”
in1 quantum
systems19e.g.
. Abehaviour
key objective
net
correlation
functionsis stateof
theregime
form
Lukyanov‘97 ocharacterizationa ofgroundstate◆properties
in thecritical! XXZ chain by
d theory or equivalently
free,boson16–19. The
long-distance✓ compact
B
Ãin the groundB̃ state. More preci
ationsofthesubsystem
magnetization
x
x
n A
1
1 + 2/⌘ 2 + . . . ,is of hGS|S
the form
j+n Sj |GSi = ( 1)
⌘
4/⌘
4
⌘+1/⌘
n
4n
n
of the following 4nobservables
!✓
!
◆
✓
◆
1
B̃z Ã 4 3⌘
Az
Bz
z
z
n
A
B
B̃
X̀
X̀
hGS|Sj+n
1
+
+
(
1)
1
+
...
n Sj |GSi =
↵
↵
↵ ,2
2
2
4/⌘
4
1/⌘
2/⌘
= ( 1)
1
1
+
+
.
.
.
4⇡
2 2⌘S (`) = 2/⌘4nSj2 ,n ⌘+1/⌘n (`) =
N(⌘4/⌘⌘n 4

4n

n

4n
n j=1
j=1
xplicit expressions for the amplitudes in (2) are!
known20–23 and ⌘ is related✓ to the anisotropy◆ pa
1
B̃z 4 3⌘
Bz
n Az
= +
cos(⇡⌘).
= N.B. 2Slowest
1+( 1) Δ≈-1
+ ..
decay
close
to
ferromagnet
! 1
2
4/⌘
4
1/⌘
2/⌘
2
4⇡ ⌘n
2 2⌘
n
4n
n

s from (2) that throughout the critical regime the dominant correlations are those of the stags in the xy-plane. The XXZ chain thus exhibits antiferomagnetic quasi-long range order in thwo-point
such s (2)are a standard means for characterizing physical properties antheyhavefunctions
the following
properties“phases” in quantum critical systems19 . A key objective of our work is to provide a com⇣ Subsystem⌘⇤critical Magnetization
on of ground state
properties
inthe
XXZ chain by determining
the quantum mecha
↵
↵
↵
` ↵
0)
= 1 , magnetizationX` ( ✓) = inXthe, state.
X` (✓More+ 2⇡)= ( 1)weXconsider
X probability= F, G.
subsystem
groundprecisely` (✓)` (✓) , thesmoothng observables

staggeredws us to restrict our attention to the interval 0  ✓ < 2⇡ and can be obtained↵

S (`) =

X̀

Sj↵ ,

↵

N (`) =j=1e↵i✓S (`)=

Ỳ ⇥Probability distribution:X̀( 1)j Sj↵ .

j=1cos(✓/2) + i sin(✓/2)

j=1

PNα (m, ℓ) = ⟨GS|δ<latexit sha1_base64="ymMhArBGp63sExghNUQV4+z/Oys=">AAAC0nicbZJdi9QwFIbT+rWOX6NeehMchA44Q7sI7oILC4J6tYxo3YXJTEnTdCZskpYkXRiyRcVbf513/gV/henHgDvrgZKX9zyHc3LStORMmzD87fk3bt66fWfv7uDe/QcPHw0fP/mii0oRGpOC
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F+osxZpyJmlsmOH0rFQUi5TT0/T8bZM/vaBKs0J+NpuSLgReSZYzgo2zkuGf2RJhXq5xchKIl4hyPj5CHMsVpxYpAd9/qi9RRrnBKGWr4KSng5aciMYcX25JpNrCI8SkSezEHbnZ1MvuRLnCxGbIrKnBtd1HJavp0k5Y50BRo0pmVKUOc713ZnDkFrw6Q73bvU7su+2d<latexit sha1_base64="ymMhArBGp63sExghNUQV4+z/Oys=">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</latexit>!

"
N (ℓ) − m |GS⟩ =e ↵ (r) =
X
`

α

Z ⇡

r)
m

⇤

.

dθ −iθmiθN α (ℓ)
e
⟨GS|e
|GS⟩
$
%&
'
−∞ 2π

d✓
e ir✓ X`↵ (✓) ,⇡ 2⇡butions of interest can be expressed as(Pe ↵ (r) (mF
PN↵ (m, `) = Pr2Z e`↵
1
F
r
+
r2Z `
2

n holds for PS↵ (m, `).

# ∞

↵
j

Focus on thisrFℓα (θ)X = F, G,

Characteristic function

1
2

if ` is even,if ` is odd.

Universality

3

ed to be susceptible to short-distance physics, and this is indeed borne out
moments are not universal
n Appendix A.
3,24,25
usingbosonization)d on microscopic details, certain ratios can be(shown
universal. In particularxhibit universal behaviour
⌦ ↵
⌦ ↵
2n ↵
2n ↵
(S (`))(N (`))(14)⌦ But ratios↵n ,⌦↵n . are!
2
2(S ↵ (`))(N ↵ (`))

d generating functions“Rescaled” generating function is universal:|GSi ,✓✓Y = q,2↵
hGS| Y (`) |GSi

Y = S, N,(15)

eter ✓. This means in particular that they can be calculated by field theory
he moment generating functions calculated from field theory and computed
e up to an overall rescaling of the parameter ✓.

directly in the lattice model should agree up to an overall rescaling of the parameter✓.
23
lishedthatlattice
the long
distance
behaviourof local equaltimecorrelation
functionsin
a0 is thespacingandthe amplitudes
b
,
c
,
b
are
known
exactly.
01
120,22,23,26,27
described by (perturbed) Luttinger liquid theory. In absence of a magne
Havein studiedacombinationofnumerical(iTEBD)andbe castthe form Fℓ(θ) by
II. FIELD THEORY DESCRIPTION OF THE XXZ CHA
Z
1 2-loop RG) methods.
analytic (exact fieldv theory,S zfree
fermion,
(`)
(0)]
+
.
.
.
,
2 ⇡ 1 p [2 (`a0 )H( ) =
dx K(@ ✓) + (@x⇡ ) + . . . ,It is well established 2that the longx distance
K behaviour of local equal time correlat
Z ` liquid theory
20,22,23,26,27
Fieldchain
theorybasedonXXZ
is wellapproachdescribed by(perturbed)
Luttinger. In abpc1 y)] = (i/2)sgn(x y), the dots indicz [ (t, x), ✓(t,Bose
fields with
commutation
relations
Hamiltoniancanbe cast in theform
N (`) ⇡
dx sin 4⇡ (x) + . . . ,Za0 0
rrelevant in the renormalization group sense andv
1Z2`K(@x ✓) + p (@x )2 + . . . ,p
Low-energy field theory
H(
)
=
dx
b0
2
x
2
⇡ 1
⇡
1dx cos K⇡✓(x) + . . . .
N
(`)
⇡
v=
,
K= a
,
arccoswith commutation
2 ⇡0relations
arccos [ (t, x), ✓(t, y)] = (i/2)sgn(x0where and ✓ are Bose2 fields
bations
thatarespinirrelevantin thegroup sense
andLatticespin
operators
formulasfortheoperators
arerenormalizationing
the bosonizationprescription
to our generating
functionsand ignoring su
p
p
a
2 ✓
0
z
j
⌦
⇡
1
⇡i p✓ (0)1 ↵)Sj ' p @x (x) + ( 1) c1 sin(, (`azv = 4⇡ (x)) + .i,.p. ⇡0
K =e ⇡,G` (✓) 2⇡arccos
0|e
|0
,
⇡
2 ⇡ arccos
p
p
p⌦
p
x
j
c1 R `
↵
S
'
b
(
1)
cos
⇡
✓(x)
+
ib
sin
⇡
✓(x)
sin
4⇡
(x)
+
.
.
.
,
i✓
dx
sin
4⇡
(x)
z
0
1
The bosonization
formulas for the Fspin(✓)
operators
are a0 0
j
⇡
0|e
|0 ,
`
p 23
a
R
0
p
z
j
b0
ttice spacing and the amplitudes
exactlyFor
large
subsystemsi` 4⇡ . (x))↵
0 , cx
1 are
Sj ' bp@1 x, b(x)+ known(⌦ 1) ci✓
sin(
+
.
.
.
,
dx
cos
⇡
(x)
1 a
0 0
Characteristic functionF⇡` (✓) ⇡ 0|e
|0 ,
p
p
p
x
j
Sj ' b10 ( 1) cos ⇡ ✓(x) + ib1 sin ⇡ ✓(x) sin
4⇡ (x) +zz|0i is the Fock vacuum.
p representation
S (`) ⇡ The[ (`a0 )(0)] +(21). . . ,reveals that G` (✓) maps o
Relatedtopartition
fnofboundary-sine-Gordon
model
↵
⇡
23
function
boson
theory,whereasF(✓)
correspondto
expectation
where a0inisthe
the freelatticespacingandtheamplitudes
b
,
c
,
b
are
known
exactly. For lv
` 0 1 1Z `have will have noted
c1we did not p
eader
that
provideabosonized
expression
forGzBazhanov,
Lukyanov,
Zamolodchikov‘01Ndx sin 4⇡ (x) + . . . ,x (`) ⇡ acalculation of G` (✓) is easier
1di↵erent setup and we return t
0 0in a somewhatzS (`) ⇡ p [ (`a )(0)] + . . . ,xResults for Staggered Subsystem Magnetizations
Fℓx(θ) = ⟨GS | e

A. Transverse componentℓiθ ∑j=1 (−1) jSjx

| GS⟩

7
1.2

= 50
= 100
= 200

= 50
= 100
= 200

= 0.6

1
0.8

F x( )

0.6

very small
except at θ≈0,π

0.4
0.2
0
-0.2

1

2

3

-0.4

-3

-2

(✓), for representative values of-10and `.

1

2

3

Observe scaling collapse around θ=0,π:Scaling collapse around θ≈0:
𝓁 = 50𝓁 = 100𝓁 = 200

Δ = -0.6

1
0.8

1

𝓁 = 50𝓁 = 100𝓁 = 200

Δ=0

1

0.8

0.8

0.6

0.6

𝓁 = 50𝓁 = 100𝓁 = 200

Δ = 0.6

0.4

F𝓁x(θ)

F𝓁x(θ)

F𝓁x(θ)

0.6

0.4

0.4

0.2
0
-0.2
-0.4
-30

-20

-10

0

θ 𝓁1-𝜂/2

10

20

30

0.2

0.2

0

0

-0.2
-15

-10

-5

0

θ 𝓁1-𝜂/2

5

10

15

-0.2
-10

-5

0

θ𝓁

1-𝜂/2

5

Red lines: exact field theory results using BSG mapping.
Similar scaling collapse around θ≈π (subleading contribution)10

= 200
= 201

= 200
= 201

0.02

Δ = -0.6

m

50

100

0.01

0.01

PNx(m,𝓁)0- small- broad- bimodal

𝓁 = 200

0.02

PNx(m, )PNx(m, )0.02
Probability distribution:

=
=

0.01

0
-100

-50

0

m

50

100

0
-100

-50

0

50

m

-

x
x
y distributionfunctions
P
(m,
`)
for
=
0.6,
=
0
and
=
0.6.
As
P
(m, `) is a sum over -funNNeven/odd
effectinmponding weights at the appropriate values of m. The solid red lines show the field theory result,
he large-` limit.
even/odd effect in 𝓁

0-100

go away for large 𝓁

-50

0

m

50

100

behaviour of these contributions we conclude that⌘x
PN (m, `) ' ` 2

︸ ︸
1 ex
F0 (m/`1 ⌘/2 ) + (1)

b`/2c+bmc`⌘
2

5/4 exFe/o (m/`1 ⌘/2 ),

x
x
x
are
obtainedbyFouriertransforming
thefunctions
F
(z)
and
F
0
e/o
e/o (z) that describe the s

red lines,
generating function around✓ = 0fieldand ✓ = ⇡ respectively.
For large subsystem sizes ` the evensubleading⌘
1 expears and we are lefttheory
with ` 2 results
F0 (m/`1 ⌘/2 ), which can be calculated exactly using the bou
ping. The corresponding contribution is shown by a solid red line in Figs 5 and 6. We see thtudinal generating function F` (✓). As we will see, its behaviour
0.4 is rather di↵erent from
eld theory approach discussed in section III B we expect F`z (✓) to be described by thezℓj z we will see, its
We now turn to the longitudinal generating
function
F
(✓).
iθ ∑
(−1)As
Sj
` j=1
z
|
|
F
(θ)
=
⟨GS
e
GS⟩
B. Longitudinalcomponent
2
0.2
F`x (✓).
According
to
the
field
theoryapproachdiscussedin
section
III
B
we
exp
ℓ
z
z /4
1/2
F` (✓ ⇡ 0) = e, z = ✓`,
(44)

scaling function

onstant that encodes the fact that appropriate ratios of moments are universal, while thet, cf. eqn (15). Numerical results for F`z (✓) are shown in Fig.
z0 7 and are seen toz 2 be/4 inF` (✓⇡ 0) = e,
Scalingarounde scaling
form (44).collapse
The coefficientis foundθ≈0:
to be consistent
with
-6

-4

-2

z = ✓`1/2 ,

0

2

4

6

1/2
(
)
where is a=100-dependent constantthat
encodes
the
fact
that
appropriate
ratios of= -0.6
simple Gaussian
1=-0.4
second moment itself is not, cf. eqn (15). Numerical results for F`z (✓) are show
=0z
excellentwiththe scaling
form (44).
TheFcoefficientis found to be co.
FIG. agreement
7: Staggered
longitudinalfunction= 0.4generating` (✓ ⇡ 0) for several values of

0.8

= 0.6
collapse that is in excellent agreementwith the universal scaling function exp(

z 2 /4) (rfind that rescaling parameter is
=100

0
-6

-4

-2

0

(

)

1/2

2

4

F z( )F z( )= -0.6
1= -0.4
1
0.4
=
.
=0
2 2⌘
= 0.4
0.8
0.2
= 0.6
z
We conclude that the fluctuations of N (`) have a very simple form: the second m
0.6

6

0.6

z

2

`

hGS|
N (`)= exact|GSi = results+ o(`) ,red line
8 8⌘

0.4 now turn to the behaviour of F z (✓ ⇡ ⇡). G
while all higher
cumulants
vanish.
We`
nal generating function F`z (✓ ⇡ 0) for several values of . The numerical data exhibit scaling= 0 we have attempted to describe our numerical data by the ansatz
reement with the universal scaling function exp( z 2 /4) (red line).
Similar scaling collapse around(θ≈π(subleading
contribution)2
0.2 `/2
↵
z /4
(
1)
A
`
e
` evenz
1

Probability distribution:
12
0.1

= 100
= 101
= 200
= 201

= -0.6

0.08

PNz(m, )PNz(m, )

0.06

0.04

0.02

0.04

0.02

-20

0

m

20

0.02

0
-40

40

= 100
= 101
= 200
= 201

= 0.6

0.08

0.06

0.04

0
-40

=0

0.08

0.06

0.1

= 100
= 101
= 200
= 201

PNz(m, )

0.1

-20

0

m

20

40

0
-40

-20

0

m

20

40

z
FIG. 10: Probability distribution functions PN(m, `) for= 0.6,= 0 and= 0.6. In the noninteracting case thefull lines are the exact results obtained using the determinant formula. In the interacting cases, the red dashed lines are theleading smooth contribution coming from the scaling behaviour of the generating function in the vicinity of ✓ = 0. Notice inparticular that, in the ferromagnetic regime, the sub-leading staggered corrections are almost invisible for the sizes consideredhere. Otherwise, as the antiferromagnetic regime is approached, the sub-leading parity e↵ects become more significant.

PD is Gaussian and much narrower than transverse analog.
1.

The XX point=0

As it is straightforward to take into account a magnetic field along the z-direction in this case, we present results
for the subsystem magnetization in the ground state of the Hamiltonian
L
X

L
X

Part II: FCS in isolated out-of-equilibrium
systemsms and has been explored in a variety of areas including condensed matter4,5
cal results on full counting statistics in quantum critical systems are relatively sc
M. Collura&FHLE’19 iudes phasefluctuations
in Luttinger
liquids10–13 , Quench”
the order parameterstatistics
Meltingof LRO
after a “Quantum
erse magnetization in the Ising chain14 and the magnetization in the Haldane-Se (staggered) subsystem magnetization in the anisotropic one-dimensional spin

• Consider the spin-1/2 XXZ chain
H=J

L
X

y y
x x
Sj Sj+1 + Sj Sj+1 +z
Sjz Sj+1.

j=1

a paradigmatic model for quantum critical behaviour in 1+1 dimensions. It fe
Prepare anisotropy
the system 1at time
t=0Thein aspecialclassical
Néel=statey the• exchange
 1.
values±1 correspondferromagnet respectively. In the regime 1 <  1 the low-energy behaviour o
16–19
inger liquid theoryorequivalentlyafree,compact
boson. The long-distance|Ψ(0)⟩ =|↑↓↑↓↑↓↑↓↑↓…⟩
functions is of the form
!
✓
◆
j z
|
⟨Ψ(0)
(−1)
Sj | Ψ(0)⟩≠0Ã
B̃orderBxx - AFM Long-rangen A∑
1
1 + 2/⌘ 2 + . . . ,j+n Sj |GSi = ( 1)
⌘
4/⌘
4
⌘+1/⌘
4n
n
4nj
n
- not an eigenstate of H!
✓◆1
B̃z 4 3⌘
Bz
z
z
n Az
Sj+n Sj |GSi =1 + 4/⌘ 4+ ( 1)
1
+.
2
2
1/⌘
2/⌘
2
4⇡ ⌘n
2 2⌘−iHtn4nn• time-evolve with H| Ψ(t)⟩ = e| Ψ(0)⟩

ssions for the amplitudes in (2) are known20–23 and ⌘ is related to the anisotropyWe are interested in the prob. dist. of

Nℓz =

ℓ
!

(−1)j Sjz

j=1
<latexit sha1_base64="lxfPwV86m3pTBN0k2iOEh2SUm8s=">AAACDXicbVDLSsNAFJ3UV62vqEs3g6VQF5ZEBBUsFNy4korGFpo0TKaTdtrJg5mJUEO/wI2/4saFilv37vwbJ20X2nrgwplz7mXuPV7MqJCG8a3lFhaXllfyq4W19Y3NLX17505ECcfEwhGLeNND/ue1EumfOikN40SSEE8+8hMGZQSzbGCHcoIlGyqCMKdqV4h7iCMsVYIFFYI5e/I8sY4qZxXz+rhYO5+mkQd7YB+UgQlOQA1cgjqwAAaP4Bm8gjftSXvR3rWPSWtOm87sgj/QPn8AslOayA==</latexit>
gjAaEktSyUgz5gQFHiMNb3CR+Y17wgWNwls5jIkToG5IfYqRVJKrl67aD65NGKvaIgnctF81R+3sDcuH5kG7D2+U33f1olExxoDzxJySIpii7upfdifCSUBCiRkSomUasXRSxCXFjIwKdiJIjPAAdUlL0RAFRDjp+JwRLCmlA/2IqwolHKu/J1IUCDEMPNUZINkTs14m<latexit sha1_base64="lxfPwV86m3pTBN0k2iOEh2SUm8s=">AAACDXicbVDLSsNAFJ3UV62vqEs3g6VQF5ZEBBUsFNy4korGFpo0TKaTdtrJg5mJUEO/wI2/4saFilv37vwbJ20X2nrgwplz7mXuPV7MqJCG8a3lFhaXllfyq4W19Y3NLX17505ECcfEwhGLeNNDgjAaEktSyUgz5gQFHiMNb3CR+Y17wgWNwls5jIkToG5IfYqRVJKrl67aD65NGKvaIgnctF81R+3sDcuH5kG7D2+U33f1olExxoDzxJySIpii7upfdifCSUBCiRkSomUasXRSxCXFjIwKdiJIjPAAdUlL0RAFRDjp+JwRLCmlA/2IqwolHKu/J1IUCDEMPNUZINkTs14m/ue1EumfOikN40SSEE8+8hMGZQSzbGCHcoIlGyqCMKdqV4h7iCMsVYIFFYI5e/I8sY4qZxXz+rhYO5+mkQd7YB+UgQlOQA1cgjqwAAaP4Bm8gjftSXvR3rWPSWtOm87sgj/QPn8AslOayA==</latexit>

require⟨Ψ(t) | eiθNℓz

| Ψ(t)⟩

This will initially depend on time, but eventually relax to astationary value (“local relaxation”).

AB• Entire System: A∪B• Take A infinite, B finite• Ask questions only about B:
lim lim ⟨Ψ(t) | 𝒪B | Ψ(t)⟩ = Tr [ρSS𝒪B]

t→∞ L→∞

ρSS = steady state density matrix
Physical Picture: A acts like a bath for B.

Probability distribution in initial state (t=0):

Pℓ (m)

1.0→ delta-function at m=ℓ/2

0.8
0.6
0.4
0.2
0.0
-20

-10

0

10

20

m

What do we expect in the stationary state?

Stationary State:

Finite correlation length ξ
0.15

SRO has melted

Pℓ (m)

ξ<ℓ

0.10
0.05
0.00
-20

-10

0

10

20

m
part. relevant for“large quenches”

Approximately Gaussian

ξ>ℓ
SRO remains, but
spin-flip symmetry
should be restored.

Finite correlation length ξPℓ (m)

Stationary State:

0.30
0.25
0.20
0.15
0.10
0.05
0.00
-20

short-ranged
order

-10

0

10

20

m
part. relevant for“small quenches”

Analytic understanding
for large Δ.

cf finite temperature equilibrium states

ξ(T)<ℓξ(T)>ℓTime evolution for a “large quench”(obtained from iTEBD)

Prob. dist. =narrow Gaussian
average

Here the prob. dist. does not give a lot of extra info (except at
short times)…

Time evolution for a “small quench”(obtained from iTEBD)
peak atm=ℓ/2
remains
averagepeak atm=-ℓ/2develops
Prob. dist. reveals a lot of physics beyond the average!

Time evolution for an “intermediate quench”

Broad prob. dist.

averageFor XXZ we understand the “small-Δ” and short time regimes by aself-consistent time-dependent mean-field approximation in thefermionic representation; no analytic results.

To make analytic progress consider TFIM.

A.

Transverse Field Ising chain

S. Groha, FHLE
Analytic
results:
Transverse-Field
Ising
Chain
g we consider the spin-1/2 transverse field Ising model on an infinite chain& P. Calabrese ‘18

H(h) =

1
X
⇥

x x
z
+
h
j j+1
j

j= 1

⇤

.

e phase diagram features ferromagnetic (h < 1) and paramagnetic (h > 1) phases that are sexx [26]. The order pa
ritical point
intheuniversality
class
of
the
two-dimensional
Ising
model
σ
→
−σ
Z2 symmetry: rotations around z-axis by πjjxes the transition is the longitudinal magnetisation hGS| j |GSi. At finite temperature spon
2 symmetry of H(h) is forbidden and hence the order present in the ground state at h < 1 m
per to be self-contained we now briefly summarize the relevant steps for diagonalizing the Ham
iled discussion
can be found in e.g. theTAppendix in [40]. The TFIC is mapped to a model ofPhase Diagram:
Quantumrdan-Wigner transformation<latexit sha1_base64="5Y3MqIv6Mgn/n4IN7mMgn0xUan4=">AAACJXicbVDLSgMxFM34tr6qLt0Ei+DGMiOCDxQENy4rWFvslJJJb9tgJhmSO+I49Gvc+CtuXKgIrvwV08fC14HA4Zx7yT0nSqSw6Psf3sTk1PTM7Nx8YWFxaXmluLp2ZXVqOFS5ltrUI2ZBCgVV
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aEI8TTUjzVQSxENGGmHvbOg3bolUVCSXup+SNkdxQiOKkTZSYENf0Zij6/vgxpc07mokpbjb/1EDu+JUnRHgNHELUgEF6oH96XcEzjhJNGZIqZbrpLqdI6kpZmRQ9jNFUoR7KCYtQxPEiWrno08GcNcoHRgJaSrRcKT+nsgRV6rPQ9PJke6qSW8o/ue1Mh0dt3OapJkm<latexit sha1_base64="kkbjNelz99tiBAYCQGFzkJNHLsI=">AAACCHicbVBNS8NAEN34WetX1KOXxSJ4sSQiqOCh4MVjBWMLTQ2b7SZdu5sNuxu1hF69+Fe8eFDx6k/w5r9x2wbU1gcDj/dmmJkXpowq7Thf1szs3PzCYmmpvLyyurZub2xeKZFJTDwsmJDNECnCaEI8TTUjzVQSxENGGmHvbOg3bolUVCSXup+SNkdxQiOKkTZSYENf0Zij6/vgxpc07mokpbjb/1EDu+JUnRHgNHELUgEF6oH96XcEzjhJNGZIqZbrpLqdI6kpZmRQ9jNFUoR7KCYtQxPEiWrno08GcNcoHRgJaSrRcKT+nsgRV6rPQ9PJke6qSW8o/ue1Mh0dt3OapJkmCR4vijIGtYDDWGCHSoI16xuCsKTmVoi7SCKsTXhlE4I7+fI08Q6qJ1X34rBSOy3SKIFtsAP2gAuOQA2cgzrwAAYP4Am8gFfr0Xq23qz3ceuMVcxsgT+wPr4B4h6aow==</latexit>z
j =12c†j cj ,

jY1

x
j =

Critical Point(12c†l cl )(cj + c†j ) ,l= 10
mion operators obeying canonical anticommutation
relations {c†j , ck } =ditions the Hamiltonian takes the form 01h0
H(h) =

J

1
X

j= 1

(c†j

†
x+ cj+1 )cj )(cj+1⟨σ ⟩ =
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Jhcj c†jc†j cj .

j,k .

Setting aside t

T>0: order melts

perto be self-contained
we now briefly summarize the relevant steps for diagonalizing the H
Jordan-Wignertransformation
iled discussion can be found in e.g. the Appendix in [42]. The TFIC is mapped to a mode
jY1
TFIM canbe mappedtofree
fermionsbyJW
trafo:
rdan-Wignertransformation
†
†
†
z
x
j =1

2cj cj ,j =(1

2cl cl )(cj + cj ) ,l= j1 1Y††x
2c†j cj ,
=
(1
2c
c
)(c
+
c
),
†j
j
j
l
l
fermion operators obeying canonical anticommutation relations {cj , ck } = j,k .
l= 1
z
j =1

conditions the Hamiltonian takes the form

Setting aside the i†1Xmion operators obeying canonicalanticommutation
relations{c, ck } =†††† jH(h) = Jtakes (ccj )(cj+1 + cj+1 ) Jh(cj cj cj cj ).
j formditions the Hamiltonian
the

j,k .

Setting asid

j= 1

1
X

nalized by a Bogoliubov
H(h) = transformationJ(c†j cj )(cj+1 + c†j+1 ) Jh(cj c†j c†j cj ).
Consider QQs e.g.
Z ⇡ from hground states of H(hi0) and determine
j=dk 1 ikj
†
c
=
e
cos(✓
/2)↵
+
i
sin(✓
/2)↵
j
kmagnetisation:kk
PD of transverse
subsystemk ,⇡ 2⇡

zed by a Bogoliubov transformation
ℓ
ℓ
!
!
†
Zz ⇡ angle is h †iz the Bogoliuobovp } = p,k andSu (ℓ) =σj = dk 1ikj
− 2cj cj
local in fermions†
cj =ecos(✓k /2)↵
ik k + i sin(✓k /2)↵ k ,
j=1
j=1
h e
⇡ 2⇡ i✓ke

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1 + h22h cos k.

∞

PSuz(ℓ),|Ψ(t)⟩(μ) = ⟨Ψ(t) | δ(Su(ℓ) − μ) | Ψ(t)⟩ =dλ enian takes
the form
∫−∞
 ik
Z ⇡
dk h e†
1
i✓k
H(h)
,.
e = = p "(k) ↵k ↵k1 + h2 2h cos2k⇡ 2⇡=p,k and the Bogoliuobov anglez isrelation
npersiontakes theformis given by∑r∈ℤ=2

Pw(u)(r, t) δ(m − 2r)Zp⇡"(k) =2J

H(h) =

1 + h2dk

(ℓ even)

2h cos(k).
1†"(k) ↵ ↵k,−iμλ⟨Ψ(t) | eiλSuz(ℓ)χu(λ,ℓ)| Ψ(t)⟩

✓ ◆n Z ⇡ RESULTSZ FORn1
1X
ANALYTICTHE
PROBABILITY
DISTRIBUTION
⇥ VI.
⇤
`
dk
.
.
.
dk
⇠j
1
n
0 n
~
~
@
Tr ( ) g = = Tr a ad⇠. . d⇠n C
k F k, exp i`(kj+1 kj )A ,
1 .a
=
Tr
a
n
l
2n
2n+2l
1
2n 1 2n 2l
2
(2⇡)
2
⇡
1
j=0
How
to
obtainanalytic
results
after
QQ:
now restrict our discussion to the particular case of transverse field quenches. As we have seen ab
fl = Tr a2n a2n+2l
l0 .
(u)
se
the
characteristic
functions(,`,t)
a scaling collapse at DISTRIBUTION
late times, cf. (58). This s
VI.
ANALYTIC
FOR exhibit
THE PROBABILITY
we havedefined k0 ⌘ knRESULTS
andmight be possible to obtain analytic results for the late time asymptotics by a!suitable generalization
1 correlation matrix of the auxiliary
n
1
he
block
diagonal
form
of
the
density thematrixin (27) ofweth
Y
Y
Step
1:
exact
determinant
representationforgenerating
function
imensional stationary state
approximation
methodpreviouslyusedtodetermine
asymptotics
k
k
j
j 1
0
~
~
t̂
(ki )such
. As
C(
k)
=
,
F
(
k)
=
Tr
nction
in
the
form
our
discussion
to [40]the andparticular
caseoftransverse
field
quenches.
we have seister
two-pointfunctiontheentanglement
entropy
[45].
As
we
will
see,
a generalization
2
sin
[(k
k
)/2]
j
j
1
j=0
i=0
acteristic
functions( , is`, significantlyt) exhibitascalingcollapseatlate times, cf. (58). Te, even though
the case at(u)
hand
more
complicated.
s✓◆
0startingis thefollowing
expression
1
tan(
)
w
changepoint
ossibletovariables
obtainanalytic
results
for
the
late
time
asymptotics
by 2ℓx2ℓ
a suitable
generaliknownmatrix(u)`( , `) = (2 cos )det,stationary state approximation
methodpreviously
usedto determine
the asymptotics
2
1, i = 1,
(u)
0
⇣
=
⇠
,
⇣
=
⇠
⇠
.
.
.
,
n
1
.
0( , `,1 t) = `i ln (cosi+1 ) +i Tr (ln(1lntan,nt function [40] and the entanglement entropy2[45]. As wewill))see,such a generalizat
tegration
rangesinhandthe ⇣ isvariables
is determinedbythe constraints
eplitzmatrix
ugh
the
case
at
significantly
morecomplicated.
s derived from (32) by using the identity ln (det (A)) = Tr (ln (A)). The second term in (71) can be ex
2: multiple
integral representation
k1
nt isStep
the following
werseries0 0expression
1
X
0
0
⇧0

⇧ 1 ... ⇧1 1 ` ⇣j  1 , 1 k = 1, . . . ,nn.
1X ✓tan( ) ◆ ⇥ 00 n ⇤
B (u)
C
1
1
.
j=0
0
.
0 t) = `B ⇧0 ( Tr
C
(ln(1
tan
)) =) + Tr
Tr ( ))) .,
ln
,
`,
ln
(cos
(ln(1
tan
⇧
.
g
f
l
l
0
0
1
0
B
C
2
2
n
2⇧n=1
= Bnow. be carried out as.theCintegrand,= not depend. on it. This gives
tegral over ⇣0 candoeslfl g l
..
.. A
@ ..
.
0
1
0 n (A)) = Tr (ln (A)). The secondom(32)usby
identity
ln
(det
term
in
(71)
can
en leads
to using
examineinteger
powers()
ofthecorrelation
matrix.
Unlike
in
the
case
of
the
order
pa
✓ ◆nthe
Z
Z
n
1
⇡
1
0
0
X
⇥ 0 n⇤` ⇧` in1 [41]dk
. . . dk⇣j. . .1odd.powers. n ⇧0do not vanish because~~ 0 is~ not a@real anti-symmetricnt function
analyzedmatrA

Tr ( ) =d⇣1 . . . d⇣n 1 µ(⇣) C k F k expi`(kj k0 ) ,n
0
2 to the(2⇡)
2t (k) corresponding
matrixis defined by⇡ correlation
1
j=1
n
1
X
Z 1⇡
⇥ 0 n⇤
tan(
)
1
0
dk
0 the
n)k
))
=
.
µ({⇣}) is the sizeof (ln(1the rangetanof ⇣0forunder
theconstraints(78)t̂0 (k) . Tr ( )B.TrExpressions
first
fewei(lcumulants()=
2⇡ n=1n
2⇡ !stationary
Step 3: 2asymptotics" fromlnmulti-dim
phase
approx!#jj
XX~ result
andsummingover
n1correlation
licit
expression
for aµ(magnetic
field
fromh0provides
tonminall⇣)= max0,0 )quench⇣kh is+ matrix.
minefficient
1 + way⇣in. determining
presentation
(29)
of
the
generating
function
an
the
k for
to
examineinteger
powers(oftheUnlike
the
case
of
the
ord
0jn 10jn 1✓◆ are obtai
k=1
ility
distribution,
which
is
the
main
purposeofthefunctionThe
cumulants
0 itself.k=1
i✓k
(cos
ido
sin notsin(2"t) anti-symmetricanalyzed
odd
powersvanishak(u,s)
k cos(2"k t)) because sin is knot
kreal
0 in [41] iet̂ (k)=,
theseriesexpansions
of
ln
(
,
`)
i✓k
difficult.
0sin
sin(2"
t)
ie
(cos
+
i
sin
cos(2"
t))
k
k
k
k
k
ponding to the correlation matrix
is defined by0Here Cdiscussed
is a constantthatis beyondthe accuracyof thestationary
phase approximation and the functions f The( ,kfunctionsabove
providesa Gaussian.
very
usefulThis
wayorganizing
fu0
probability
distribution
approaches
isof expected
l= as
1 the complicated summation required.n,m
given
in
terms
of
series.
Basedonthenthe approximately
first(u,s)
15 terms inseriesconjecture
expl
Inare
order
obtainthelogarithmof thecharacteristic
function( ,these`, t) we
nowweneedto sumthe
overfollowing
all contribu
resultcanbetoandexpressed
in infinite
the form
namicsapply
theprobability
distributionispResult:o(93)theexpressions
time
evolution of
the incharacteristic
function( , t) after
quantum
quenches.
We consider†
withcoefficients
given
(71).
This
is
a
formidable
task.
It
turns
out
that
the
structureof Setting
Heaviside
as
`.
e
c
are
fermionoperators
obeying
canonical
anticommutation
relations
{c
,
c
}
=
.
a
j
k
j,k

Z
j
1
n+1
ses
of
initialstates:2⇡
X
X
functions discussed above provides
a veryof organizing`dk
2n|vk |tthe complicated summation required. The
0 usefuli✓k0 theway
oundary
conditions
theHamiltonian
takes
form
f
(
,
k
)
=
2
ln
1
+
i
cos
tan
e
,
ln (can, `,bet) ⇡⇥(` 2n|vk |t) 1
cos 2m"(k0 )t fn,m ( , k0 ) + C . (980,00 ` log(cosin) the form k0
resultexpressed22⇡`e the system in the groundstate
of 0H(hand time evolve with2 H(h).
2 0) 1m=0Such transverse field quenches
sin2n=0k0 tan
(cos
✓
+
i
cos
tan
)
TATISTICS AFTER A QUANTUMQUENCHk0
X k0 †
 2 2 †,
tudied in[39–46,53–63].
f1,0detail( , k0 )in=theln literature
1
Z
1
n+1
†
†
2⇡
2X
X
`
dk
2n|v
|t
(sin
✓
+
(cos
✓
+
i
cos
tan
)
)
H(h)
=
J
(c
c
)(c
+
c
)
Jhc
c
c
.
0
k
k
k
k
j
j+1
0of the0
HerelnC is( a ,constantthat "is beyond
the0 accuracy ⇥(`stationary
phase j+1
approximationand the( , k0 ,j 2n|v
j j2m"(kj c)tjfunctions`, t) ⇡ ` log(cos
)
+
|t)
1
cos
fn,m ( , fkn,mk00 ) + C .#(u,s)
racteristic
function( ,statet)series.
after
quantum
We4terms
consider4firstbreakingearethesystemin the. . . j="#i,thus
by2 onesite. This2|"#"#
2⇡quenches.
` ✓kseries
1
given
in
terms
of Néelinfinite
Based
on
the
15
in
we conjecture
the following
explicsin
sin2translational✓k0these(cos+ isymmetrycos
k0 tank0 tan )
n=0 0
0 m=0
f2,0at( ,latek0 ) =
ln 1after+ the.
sexpressions
restoredtimes
question
how
this
is
reflected
inthe
2 quench and it is an interesting
2
2
2
2
2
2
((sin ✓k0transformation+ (cos ✓k0 + i cos k0 tan ) )sin k0 tan (cos ✓k0 + i cos k0 tan ) )isdiagonalizedbyaBogoliubovHere
C 0is) aandconstantthat isbeyondaccuracyof thefieldstationary
phase approximation and the functions fn,m ( ,of observables.
edistributionsof H(htime
evolvewith
H(h).theSuch
quenchesi✓ktransverse(
f0,0( , kin= 2 lnof 1infinite+ i cos series.
e⇡ 0 on, the firstZ0 ) terms
k0 tanBased
are
given
15
terms
in
these
seriesweconjecture
thefollowingexhie [39–46, 53–63].

dk ikj†222
expressions
sin
tan
(cos
✓
+
i
cos
tan
)
c
=
e
cos(✓
/2)↵
+
i
sin(✓
/2)↵
, negligible for all cases
k0 j further terms
k0
k0 theirk contribution
k
k
k be
In
principle
one
could
determine
f
but
turns
out
to
n,0
f
(
,
k
)
=
ln
1
,
2⇡
"#"# 1,0. . . "#i,0 thus breaking
translational
symmetryby one hsite.
This
2⇡field2!
2 hA.
Transverse
quench
0
(sin
✓
+
(cos
✓
+
i
cos
tan
)
)
have considered. The contributions
fn,m>0to simplify. While the term f0,1 can stillk0
ki✓
k0 are more difficult
0 k0 ( , k0 , t)e quench
is anf0,0 ( and, k0 )it =2"lninteresting
1 + i cos questione this, is reflected in the#
k0 tan how
obtained
without further approximations, in
4 order to
2 closed form expressions for 2m > 1 we have resorted
4 obtain†sin
tan sin ✓k0 (cose {↵an} = p,kin and
the
angle
2Bogoliuobov
k0 is2
2 ✓k0 + i cos k0 tan )
k, ↵
p
sin
tan
(cos
✓
+
i
cos
tan
)
expansion
powers
of
sin(
).
This
is
expected
to
give
very
accurate
results forThesmallquenches, which
f
(
,
k
)
=
ln
1
+
.
k
k
k
k0 the initial
0and0
0 translationally0
protocolboththeHamiltonian
state
are
invariant.
characteristic
2
2
2
2
2
2
2
f2,0
(
,
k
)
=
ln
1
,
1,0
0 producing ((sin✓kdensity+ (cosof✓k0elementary+ i cos k0excitationstan ) ) 2 2throughsin k0thetanquench(cos ✓[39,
i cosThek0leadingtan ) terms
)
k0 +
2
0
definedasa small
40].
(sin
✓
+
(cos
✓
+
i
cos
tan
)
)
k0 (33) withk0 [44]k0
determinant representation
(32),
h eik
#
i✓k
then conjectured "to be of the form
(99e.
nsverse field quench h0 Z! h
4 = p 42
2
sin k0 tan
sin✓kk0 + i cos k0 tan )⇡ 2 ✓k2h0 (cos1+hcos
dk
f2,0 ( , k0 ) = ln 1 +i✓k0i✓kikl
2+
2
2 to be negligible for all1
iecos✓(cos
tan
2 )2
2 )2w
k+
g
=
i
e
e termssin(54)
0
In principle
one
could
furtherfn,0buti theircontribution
out
cases
((sin
✓
+
(cos
i
cos
tan
)
sin
tan
(cos
✓
+
i
cos
tan
)
l i tandeterminekk cos(2"k t))turns
k
k
k
k
k
k
0
0
0
0
0
0
f
=
ln
,
0,1
k0
and theinitialstatetranslationallyThecharacteristici✓kinvariant.
⇡ 2⇡
0 cos
Hamiltonian
takes
the
form
1
+
ie
tan
have
considered.
Thearecontributions
f
(
,
k
,
t)
are
more difficult to simplify. While the term f0,1 can still b
k
n,m>0
0
0
Z
),
(33) with[44] further⇡ approximations,
 form expressions for m > 1!we have resorted t
Z obtainobtained
without
closeddk  1ikl
i✓k0 in order to
⇡
+ sin
ie
cos
4 cos
sin ✓k0
k0 tank0 tandk
1results
3(55)
f
=
e
sin(2"
t)
,
†
l
k
k
f
=
tan
i
log
+
O
sin
(.ar
1
an
expansioninpowers
of
sin(
).
This
is
expected
to
give
very
accuratefor
small
quenches,
1,1kk0 )case
k
0
0
H(h)
=
"(k)
↵
↵
,
2negligible for(whichi✓k0 terms fn,0 but their
2kIniklprinciple
one could determine
contribution
turns
out
to
be
all
i✓k
k
⇡ 2⇡ 1 +furtherie
cos k0 tan
sin ✓through✓k0 quench+
i cos [39,) The leading terms ar
k0 + (costhe
k0 tanedefinede as(cos
i sina small
(54)
2⇡
2
producingdensityexcitations
40].
kk cos(2"k t)) of elementary⇡
have considered. The contributions fn,m>0 ( , k0 , t) are more difficult to simplify. While the term f0,1 can stthen conjectured to be of the form
obtained
without
further
approximations,in order toobtainclosedform
expressionsform >an1 extremely
we have resort
As
we
will
see
below,
thecontributions
described
by
(99)
and
(100)
are
sufficienttoobtainaccurethedispersion
relation
is
given
by
kl
of iksinexpansiont)of,powers
ansin(
is expected
to comparing
give(55)
very accurate
resultssmallwhich
k sin(2"k ini✓kk00 ). This
description( ,h`, t).
Theconstant
Ck canbe fixed
bythekt +
!11
limit forof (98)to quenches,
the result obtain
1
+
ie
cos
tan
e
hh
(h
+
h
)
cos
p0
0
0
dϵ(k)
i✓kproducing
fpreviously=tan,definedasalnsmall densityofelementary
excitations
throughthequench[39,
40].
The leading
p thek0behaviour
ini✓thestationary
state.
For
later
convenience
wedefinetwo
approximations
as term=i for
,
cos
=
4
.
(56)
0,1 e
2
k =2J
v
=
k
"(k)
1
+
h
2h
cos(k).
0k+ ie"h (k)"h0 (k)2hcos k cos k0 tan
then conjectured 1
to+beh2of 1the
form
! dk


Z
2
a
2⇡
`k0X
2n|v
|t X
1 + iei✓cos Bogoliubov
tan0
4 cos
tan
sin ✓byk
k0 dk
k0 definedk0

3
ground
state
of
H(h)
is
equal
to
the
vacuumstatei✓
mmaf1,1
itlnistoobtainthelarge-`
asymptotics
intheinitial(t=0)
andstationary( , `, hht)=`log(cos
)
+
⇥(`
2n|v
|t)
1
cos
2m"(k
)t
f
(
,
k(0 ) +
(1
(h
+
h
)
cos
k
+
1
= astraightforward
tan
i
log
+
O
sin
) (., 100
k
k
0
n,m
0
k0 0
k0 C
0
1
+
ie
cos
tan
2
i✓
k
2
k
0
0
2⇡
, f0,1
cos =k =i4tan k ln 1 + ie 2 n=0(56)
cos .0 kstate
tan at field
sin h✓k0and+(cos✓k`0been+ i m=0cos),
0k0 tan earlier.
The
t
=
0
resultcorrespondsagroundhas
discussed0" (k)" to
i✓
0
(k)
k0
sk
h
h
0
1 + iecos k0 tan↵k |0i = 0.
!
 we set i✓
where
a
=
1,
2
and
where
f
=
0.
Shouldprobably
give
a
few
more
details.
As
we
will
see
below,
thecontributions
described
by
(99)
and
(100)
are
sufficienttoobtainanextremely accurat
2,1
obtain the large-` asymptotics 1in+the
0 cos (t =ie kinitial
tan0) and stationary
4 cos
tan sin ✓<latexit sha1_base64="IqR3Vmezblb3KOmRBHs3NOSKTx0=">AAACA3icbVBNS8NAEN3Ur1q/oh57CRahXkoiggoKBS8eKxhbaELYbDbtks1u2N0USujBi3/FiwcVr/4Jb/4bt20O2vpg4PHeDDPzwowSqWz726isrK6tb1Q3a1vbO7t75v7Bg+S5QNhFnHLRC6HE
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lDDsKqIo7mUCwzSkuBsmN1O/O8JCEs7u1TjDfgoHjMQEQaWlwKyPguTaiwVEReThTBLKWTM5mRRRMgnMht2yZ7CWiVOSBijRCcwvL+IoTzFTiEIp+46dKb+AQhFE8aTm5RJnECVwgPuaMphi6RezJybWsVYiK+ZCF1PWTP09UcBUynEa6s4UqqFc9Kbif14/V/GFXxCW<latexit sha1_base64="IqR3Vmezblb3KOmRBHs3NOSKTx0=">AAACA3icbVBNS8NAEN3Ur1q/oh57CRahXkoiggoKBS8eKxhbaELYbDbtks1u2N0USujBi3/FiwcVr/4Jb/4bt20O2vpg4PHeDDPzwowSqWz726isrK6tb1Q3a1vbO7t75v7Bg+S5QNhFnHLRC6HElDDsKqIo7mUCwzSkuBsmN1O/O8JCEs7u1TjDfgoHjMQEQaWlwKyPguTaiwVEReThTBLKWTM5mRRRMgnMht2yZ7CWiVOSBijRCcwvL+IoTzFTiEIp+46dKb+AQhFE8aTm5RJnECVwgPuaMphi6RezJybWsVYiK+ZCF1PWTP09UcBUynEa6s4UqqFc9Kbif14/V/GFXxCW5QozNF8U59RS3JomYkVEYKToWBOIBNG3WmgIdSJK51bTITiLLy8T97R12XLuzhrtqzKNKqiDI9AEDjgHbXALOsAFCDyCZ/AK3own48V4Nz7mrRWjnDkEf2B8/gCBzJgx</latexit>

How well does this work?
→ Compare to numerically exact results.

λ=0.1, ℓ=200, h0=0, h=0.2

Not bad.

Slight caveat: when χ (u)(λ, ℓ, t) becomes very small as a fn of λ ourapproximation becomes poor. Not a problem for getting the PD.

“Transverse field quench”: prepare system in GS of H(h0),time evolve with H(h)h0=3, h=0.2

0

1

h0

even/odd structure thatwashes out over timeSummary
1. Full counting statistics for subsystems can be interesting;can be universal at critical points.
2. FCS is not easy to calculate analytically.
3. FCS in ground state of quantum critical XXZ chain
4. FCS of transverse magnetisation in TFIM in equilibrium &after quantum quenches
5. FCS after Neel quench in XXZ: melting of LRO
Many open questions:• Subleading scaling functions in ground state of XXZ?
• Order-parameter PDF for TFIM in equilibrium/after QQs?
• How to get PDF’s from Bethe Ansatz?
•…