Universal finite-­‐size scaling for a dynamical phase transi8on in a kine8cally constrained model and a quantum phase transi8on in a ferromagnet.
Takahiro Nemoto (Paris VII, LPMA) Collabora8on work with Vivien Lecomte (Paris VII, LPMA) Shin-­‐ichi Sasa (Kyoto) Frédéric van Wijland (Paris VII, MSC) T. N. V. Lecomte, S. Sasa, F. van Wijland, J. Stat. Mech. (2014) P10001 dynamical phase transi8ons in KCMs Dynamical heterogeneity
J. P. Garrahan, Proc. Natl. Acad. Sci. U. S. A. 2011 108 (12) 4701.
2
of statistical mechanics in the limit of a very large which is depicted schematically in Fig. 1. A firstsystem, what is usually called the “thermody- order phase transition is manifested by a disnamic” limit (11). For finite systems studied continuity at the pressure p = p*. At this value of
numerically, there are no such singularities. Evi- the pressure, two phases coexist with respective
dence of a phase transition in these cases is found volumes per particle v1 and v2.
At coexistence, the distribution function for
in the behaviors of crossovers from one phase to
another (12). Figure 1 illustrates the system-size the order parameter is bimodal. The two peaks in
coincide
with 1
the
two1equilibrium
behavior of a crossover.
the transition
we the distribution
M. MFor
erolle, J. P. Garrahan, D. Chandler, PNAS, 02, 0837 (2005)
consider, the partition function is a sum over dy- phases. There is a low probability to observe an
R.L. Jack, J.P.Garrahan, . Chandler, J. Cofhem. Phys. 84509 (2006)
value
V, between
Nv1125, and1Nv
namical histories (i.e.,
trajectories)
of the system, Dintermediate
2 in
This low(20007).
probability decreases exponenand the order parameter
the amount
ofRL, Fig.
J. P. measures
Garrahan et al, P
98, 1.195702 dynamical phase transi8ons in KCMs Dynamical phase transi2on by ac2vity bias
Equilibrium
Nonequilibrium
Fig. 1. Finite size effects
of equilibrium and nonequilibrium phase transitions. The mean volume
Vp manifests an equilibrium first-order phase transition at pressure p = p*,
whereas the mean dynamical activity Ks manifests a
dynamical first-order phase
transition at the dynamical
field s = s*. At conditions
of phase coexistence, the
volume distribution function,
Pp(V), and the dynamical
activity distribution, Ps(K),
are bimodal. Configurations
or trajectories with intermediate behaviors lie at much
L. O(or
. Hlower
edges, R. L. Jack, J. those
P. Garrahan, D. For
Chandler, science, 323, 2009
higher free energies
probabilities)
than
of the basins.
finite systems,
discontinuous
phase transitions become crossovers with widths that vanish as system size, N, and observation time, tobs,
Finite size scaling in first order transi8on Finite size scaling
Second order phase transi8on
-­‐  From small system size simula8ons, -­‐  Where is true cri8cal point? (with finite size simula8ons) -­‐  Precise order of the transi8on More on first order phase transi8on
-­‐  (i) Scaling speed ? -­‐  (ii) Scaling func8ons ? 4
magnetization at T T, in place of the
peracase of the same phenomenon.
and to
that do not have a degenerate
contour representation
models variation
pected
under critical-point
Now the
discontinuous
results of Mo(T) at
nor our vanishing
Pirogov-Sinai
which
neither
the
theory,
~
=B T T, Finite as T—
(3.7)
~T,size scaling T =fiT,
described
mayobe
by P=O in Eq. (3.7). The
i
n rst rder t
ransi8on symwith
continuous
are
Heisenberg-like
systems
apply,
scaling relation 2 —
a=P(1+
5) then implies 5= ao
a latent heat
Similarly,
may observe
(2) oneFinite size scaling: Classical (thermodynamic) transi2on
metries.
(since
a
and similar arguments,
(1)
as before, indiion corresponding to a discontinuity in
—b".
the
from
proven
here
fact, eigenvalue
The theory
presented
catestarts
aPrenormalization-group
AH
M. E
. F
isher a
nd A
. N
. B
erker, RB, 2
6, 2
507 (
1982)
energy, in place of the normal critical
Thermodynamically,
a discontinuous
describ- spontaneous
function
a model
in Ref. [8],
theR. partition
thatand C. Borgs Kotecky, PRL, 68, 1734 of
(1992)
metric
magnetization
does is,notatnecessarily
imply a latent
tem=h,
low
of
N
at
h
coexistence
the
phases
ing
'
In general — ~ as T +T, + .
— but, via scaling, we would have y=2 —
to
pt
heat
T —T,
a. A
U, =+A+
Number of p[10]
hases
peratures, very well approximated
by
latent heat can evidently be described in Eq. (3.8) by
was
Par88on func8on
(3.8)
a=1. Parallel arguments then yield a correlation
disv= 1/d and(5) a thermal
length
exponent —
exP[ fv(h)PL
Zt. , (h, L)
g-like
l
—,
l
i
l
=g
j,
—b,
ver, that the absence of a latent heat on
renormalization-group
eigenvalue Az
as beattice
field
fermagnetic
across a first-order
fore.
System volume
transition below
acis
an
energy"
merely
"metastable
free
T,
sort
of
Unforsome
is
where
fv(h)
Note, however, that in
principle ofone may also
Index f
or p
hases
energy”
o the
phase boundary
hap-“metastable free is equal to the free enerTwofact thatthethisphase
q. The quantity
arallel to the temperature axis; by conslope g „(Lj
whenever
rela- Ex.) Ising model
of theH, (T)
q is stable, and
pins (boundary
d-­‐dimension) gy sphase
g the first-order
& point if
anetwaybelow its
fv(h)
q dis unstable. d While it is not expected
tricritical
is associhmL
it
an analytic function [I
tran-heat]. that
atent
canβ hmL
be chosen− βas
per
The most
in such a way that it is differit is now be
stilldistinguished:
],must
may be introduced
uation in practice is illustrated in Fig.
d
it is seen thatperthe first-order transition
zero field, at the point now labeled F, is
cribed as a trE'pIe point at which three
5
noncritical, may coexist. This is the
FIG. 2. Schematic magnetization vs field curves fo
.
f(h)
f(h)
Z (h,f„(h)
L) ≈ e
f„(h)
+e
m (h, L) ≈ m tanh(β hmL )
I],
no problems
except near
the critical
divergesbyas Itl
whichwhere
can 400
be treated
all those
modelspoint
more generally,
More
serious is thetheory
fact that
asserting
the predominance
of the
class of
the Pirogov-Sinai
important
[9].inOne
temperastatesmodels
of "up"
total magnetization
Vmo(T),
we have
to
representation +_and
thatordo"down"
not havewith
a contour
(under
overlooked
all configurations
in whichtheory,
some nor
regions
the system are
our of
results
which neither
the Pirogov-Sinai
magnetized
"up"sHeisenberg-like
while others
aresystems
magnetized
in Fig.
Finite caling: Classical (thermodynamic) transi2on
symwith "down,"
continuousas illustrated
are
apply, size configurations are, of course, suppressed by a Boltzmann factor
(2)1. Such
metries.
M. E. Fisher and A. N. Berker, PRB, 26, 2507 (1982)
representing
the excess
free energy
associated
the the
from with
proven (or domain
here starts
fact,interface
The theory
presented
C. Borgs oppositely
and R. Kotecky, PRL, 68, 1regions:
734 (1992)describ- them does,
wall) inbetween
magnetized
function of
a modelIncluding
Ref. [8],the
that the partition
symmetric
however,
increase
the entropy.
For block
of this
In general =h, is, at configurations
low temof N phases
at h geometry,
coexistence
ing the
attempt tosort may thus yield corrections Number of phases
which, relative
[10]
peratures, very well approximated
by to the bulk, will be of order
odels was
APar88on / V ~ 1/Lfunc8on
o. This already suggests corrections to (2.18) [or (2.6)] of order
disbility
1 / V 1/d which
(See, however,—
Sections 3(5)
and 4 below.)
L) undisplayed.
exP[ fv(h)PL
Zt. , (h,remain
Ising-like
per lattice
System volume
Unforwhere fv(h) is some sort of "metastable free energy" of
for phases
to the free enerequal
3-5]. Two Index the phase
f„(h) isfree q. The quantity
“metastable energy”
], the relaq is stable, and
gy f(h) of the model whenever
oundary , cylinder-­‐shaped)** it is not expected
While
&bf(h)
is unstable. (2 d
if q condi8ons ch a**Depending way
fv(h)on -­‐ Exponen2ally scaling can be chosen as an analytic function [I I], it
that f„(h)
t the tranLII
~
.LII--~
in such a way that it is differintroduced
be
is
still
[5],V. itPrivman may
(b)
and M. E. Fisher, J. Stat. Phys. 33, (1983).
bulk mag-
Finite size scaling in first order transi8on =g
L•
(C)
I
j,
--
+
-
LII >> ~II(L~'T)
+
6
ofSect.
the active
region
is penalised
asfrom
the activity
n. Whenthe
λ>
λc , the growth
However,
numerical
results of
3 support
the scaling
derived
to the area of the active droplet.
elproportional
(11).
(t)
and
x− (t) of to
the
active
region
perform
non-crossing
ely,
thethat
boundaries
Finite ize scaling in the
first order transi8on cture
the
finitex+ssize
corrections
large
deviation
function
mirror rates for
of
rate pto (resp.
q)size to the
left (resp.
right) forpxhase + (t) and
λc jump
are related
Finite scaling: Dynamical transi2on
" from x = 0 at time 0, and to come
plicity the walks are constrained !to start
λ Math. Phys. 311 (2012)
odineau C. T1oninelli, omm. al time T. t B(this
assumption
does
the large time asymptotics).
In
ϕ̂L (λ)
log
ZeffCnot
,change
t
(9)
=and lim
t→∞
t
T. Bodineau, Lecomte ain
nd the
C. TL
oninelli, Jis
. Stat. Phys. 147 (2012)
escription,
the totalV. activity
system
proportional
to the area of the
"t
2
dy of
interfaces
in
the
static
Ising
model
[21,
22],
and
using
results
dτ
[x
(τ
)
−
x
(τ
)]
with
K
=
4c
(1 −from
c) the mean
nd
approximated
by
K
+
−
1d-­‐FA ! 2 dimensional s
pin p
roblem? 0
t ⟩ reads
heory
[23],the
we counterpart
show in Appendix
A that
this leads to the following scalingT. Bodineau et al.
ity.4 Thus
of ⟨e−sK
"
Fig. 1 Model for the space-time
−sK 0t dτ [x+ (τ )−x− (τ )]
⟨e in!the
δ(x± (t) = 0)⟩p,q
2
"
configuration
of
the
system
(8)
Zeff (s, t) ≡
3
λK
1
interfacial regime λ > λ√
.
An
−0)⟩
⟨δ√
c
± (x(t) =
3 α p,q
ϕ̂
(λ)
=
−4
pq
2
(10)
L
1
island of activity density
Brownian mo8on
4L pq
2
K =the
4c (1
− c) is delimited
by
notes
average
over trajectories
x± (τ )0≤τ ≤t without constraint at final time.
biasedofrandom
1 . .two
. isnon-crossing
the first zero
the Airy function
on the mnegative
Brownian o8on real axis. As a
walks x+ (τ ) and x− (τ ),
xpect
that thetofinite
of the microscopic model should be given
constrained
start at size
0 andscaling
end
0Dynamical at −Σ
time t forfree tra atcost
creating
the interfaces
energy
√
!
λK
ϕL (λ) = −Σ − 4 pq
√
4L pq
" 23
2
− 13
α1
(11)
oice of the effective parameters p, q (see Sect. 4.1 for a discussion on the
− 23
s). In other words the interface model we have considered leads to L
7
ge
ng
om
ng
lain
eck).
finite
hile
line
or
Brownian bridge theory [23], we show in Appendix A that this leads to the followin
T. Bodineau et al.
at large6 L
2
Finite s
ize caling i
n fi
rst t
ransi8on Fig. 2 Evaluation
of thes
large
!order "
3
λK
deviation function ϕL (λ) using
√
− 13
ϕ̂L (λ) D
=ynamical −4 pq phase 2
α1
Finite size scaling: t
ransi2on
the cloning
algorithm
(blue
√
4L pq
circles, increasing sizes
L ∈ {8, 16,
bottom Comm. Math. Phys. 311 (2012)
T. Bodineau a32,
nd 64}
C. from
Toninelli, to top
at positive λ), and using
where α
1 ≈ 2.3381 . . . is the first zero of the Airy function on the negative real a
direct diagonalisation
of theand C. Toninelli, J. Stat. Phys. 147 (2012)
T. Bodineau, V. Lecomte consequence,
expect
that the finite size scaling of the microscopic model should
operator ofwe
evolution
(54) (plain
green line, L = 8 run as a check).
by (10)!
plus
the
extra
cost
−Σ
forpcreating
theinterfaces
1d-­‐FA red
2 dashed
d
imensional spin roblem? The
line is the
infinite
L result −Kλ for λ < λc , while
!
" 23
the purple dotted horizontal line
λK
√
− 13
is the
infinite
L result −Σ for
Dynamical free energy
ϕ
(λ)
=
−Σ
−
4
pq
2
α1
√
L
λ > λc . We took c = 12
4L pq
T. Bodineau et al.
for appropriate choice of the effective parameters p, q (see Sect. 4.1 for a discussio
effective jump rates). In other words the interface model we have considered lead
corrections to the constant −Σ .
3 Numerical Results
3.1 Results from the Cloning Algorithm (i): The Free Energy
To
Fig. 3 (Left) Log-log plot of ϕL (λ0 ) + Σ , for Σ = 0.077, at fixed λ0 = 4.6 as a function of L: the numerical8
evaluation (blue
dots) fitsthe
withfinite
a power size
law corresponding
to the(11)
exponent
α = 23 (red
line). the
(Right)
Plot
investigate
whether
corrections
inferred
from
interface
m
Finite size scaling in first order transi8on Purpose of this talk
On the contrary, -­‐ With the simplest model showing dynamical phase transi8on -­‐ Directly checking if classical approach can work Answer: -­‐ It can work. (But another procedure is needed) -­‐ It is more close to quantum phase transi8on 9
Construc2on of this talk
1.  Preliminary: Introduc2on of mean-­‐field FA 2.  Problems and how to overcome it 3.  Discussions 10
1. Preliminary: Intro of mean-­‐field FA Model
What is KCMs?
1. Preliminary: Intro of mean-­‐field FA Model
(e.g. J. P Garrahan et al, J. Phys. A, Math. Theor. 42 (2009) 075007)
1. Preliminary: Intro of mean-­‐field FA Model
(e.g. J. P Garrahan et al, J. Phys. A, Math. Theor. 42 (2009) 075007)
1. Preliminary: Intro of mean-­‐field FA Model
(e.g. J. P Garrahan et al, J. Phys. A, Math. Theor. 42 (2009) 075007)
Detailed balance condi2on
1. Preliminary: Intro of mean-­‐field FA Dynamical phase transi2on
s -­‐ ensemble
An ensemble biased by a fic88ous field s
n(t) = ∑ ni = 1, 2,..., L
i
: state of the system (total spin)
1. Preliminary: Intro of mean-­‐field FA Dynamical phase transi2on
Ac2vity ?
Ex. K(t0 ≤ t ≤ t2 ) = 2
K(t0 ≤ t < t1 ) = 0
...
1. Preliminary: Intro of mean-­‐field FA Dynamical phase transi2on
s -­‐ ensemble
An ensemble biased by a fic88ous field s
1. Preliminary: Intro of mean-­‐field FA Dynamical phase transi2on
s -­‐ ensemble
An ensemble biased by a fic88ous field s
Large devia8on func8on of x
= K /τ
I(x) = max s [ xs − f (s)]
1. Preliminary: Intro of mean-­‐field FA Dynamical phase transi2on
How to calculate dynamical free energy?
-­‐ Popula8on dynamics method (In general, numerical method) C. Giardin`a, J. Kurchan, and L. Peli8, Phys. Rev. Lej. 96, 120603 (2006). -­‐ Transfer matrix (largest eigenvalue problem, solvable model) 1. Preliminary: Intro of mean-­‐field FA Dynamical phase transi2on
How to calculate dynamical free energy?
-­‐ Transfer matrix (largest eigenvalue problem, solvable model) 1. Preliminary: Intro of mean-­‐field FA Dynamical phase transi2on
How to calculate dynamical free energy?
-­‐ Transfer matrix (largest eigenvalue problem, solvable model) Escape rate
s
L
Largest eigenvalue of = dynamical free energy
1. Preliminary: Intro of mean-­‐field FA Dynamical phase transi2on
Numerical example (c=0.3)
f (s) : Dynamical free energy
1. Preliminary: Intro of mean-­‐field FA Finite size scaling
Biased total spin, biased suscep2bility
Biased total spin (per site) ρ (s) =
∑
Ps (hist)
history
Biased suscep8bility
1
χ (s) = ∑ Ps (hist)
τL
history
Numerical example (c=0.3)
χ (s)
ρ (s)
1
τL
∫
τ
0
∫
τ
0
dt n(t)
dt [ n(t) − L ρ (s)]
2
1. Preliminary: Intro of mean-­‐field FA Finite size scaling
Exponen2al scaling (numerical check)
1. Preliminary: Intro of mean-­‐field FA Finite size scaling
Scaling func2on (numerical check)
1. Preliminary: Intro of mean-­‐field FA Finite size scaling
Scaling func2on (numerical check)
Q1. Analy8cal expression of these func8ons? Q2. How to derive the exponen8al scaling?
Construc2on of this talk
1.  Preliminary: Introduc2on of mean-­‐field FA 2.  Problems and how to overcome it 3.  Discussions 27
2. Problems and how to overcome it Problem…?
Metastable free energy?? (No canonical distribu8on)
2. Problems and how to overcome it Idea to solve
-­‐ Auxiliary dynamics
R. L. Jack and P. Sollich, Prog. Theor. Phys. Supp. 184, 304 (2010). -­‐ Donsker-­‐Varadhan type varia2onal formula
e.g. J. P Garrahan et al, J. Phys. A, Math. Theor. 42 (2009) 075007
T. N. and S. Sasa, Phys. Rev. E 84, 061113 (2011)
aux
s
Ps (hist) ≈ P (hist)
2. Problems and how to overcome it Idea to solve
-­‐ Modifying free energy
2. Problems and how to overcome it Idea to solve
Numerical example of h* (L=100) c = 0.3
DYNAMICAL free energy
2. Problems and how to overcome it Idea to solve
-­‐ Non analy2c point of h* at s = sc
2. Problems and how to overcome it Idea to solve
-­‐ Non analy2c point of h* at s = sc
2. Problems and how to overcome it Idea to solve
-­‐ Non analy2c point of h* at s = sc
2. Problems and how to overcome it Idea to solve
-­‐ Non analy2c point of h* at s = sc
2. Problems and how to overcome it Idea to solve
-­‐ Our ansatz
For finite L, we assume that we know, for Distribu8on: 2. Problems and how to overcome it Idea to solve
-­‐ Our ansatz
2. Problems and how to overcome it Idea to solve
-­‐ Our ansatz
2. Problems and how to overcome it Idea to solve
-­‐ Scaling func2on
T. N. V. Lecomte, S. Sasa, F. van Wijland, J. Stat. Mech. (2014) P10001 2. Problems and how to overcome it Idea to solve
-­‐ Scaling factor
κ ≅e
aL
1
a = − log(1− c)
2
A is the poten2al height separa2ng two phases
T. N. V. Lecomte, S. Sasa, F. van Wijland, J. Stat. Mech. (2014) P10001 2. Problems and how to overcome it Idea to solve
-­‐ Scaling factor
1
a = − log(1− c)
2
Construc2on of this talk
1.  Preliminary: Introduc2on of mean-­‐field FA 2.  Problems and how to overcome it 3.  Discussions 42
3. Discussions Quantum phase transi2ons
3. Discussions Quantum phase transi2ons
3. Discussions Quantum phase transi2ons
3. Discussions Quantum phase transi2ons
-­‐ Scaling func2on
3. Discussions Quantum phase transi2ons
-­‐ Scaling factor
Conclusion
•  For mean field FA, we performed finite size scaling around 1st order phase transi8on. •  Idea is similar to classical one, but we used auxiliary poten8al and varia8onal principle •  The dynamical phase transi8on is close to quantum phase transi8on: mean field quantum ferromagnet
48