Table Of ContentEnsemble renormalization group for the random field hierarchical model
Aur´elien Decelle1, Giorgio Parisi1,2 and Jacopo Rocchi 1
1Dipartimento di Fisica, Universita` La Sapienza, Piazzale Aldo Moro 5, I-00185 Roma,
2INFN-Sezione di Roma 1, and CNR-IPCF, UOS di Roma. Italy.
The Renormalization Group (RG) methods are still far from being completely understood in
quenched disordered systems. In order to gain insight into the nature of the phase transition of
these systems, it is common to investigate simple models. In this work we study a real-space RG
transformationontheDysonhierarchicallatticewitharandomfield,whichledtoareconstruction
oftheRGflowandtoanevaluationofthecriticalexponentsofthemodelatT =0. Weshowthat
thismethodgivesveryaccurateestimationsofthecriticalexponents,bycomparingourresultswith
the ones obtained by some of us using an independent method.
4
1
0 The Renormalization Group (RG) is a fundamental Therefore,analyticalandnumericalstudieswerepursued
2
tool to study the changes of a system as observed at in this direction [10], [11], [12–15]. Among the numeri-
n different scales, which has been successfully employed calworks, anapproximaterealspacetransformationwas
a both in quantum field theory [1] and in the theory of suggested as a new approach to deal with the RG in dis-
J
second order phase transitions [2]. This process consists ordered systems. In the first numerical approach [11],
8
in integrating out small scale details of the physical sys- a real-space RG transformation for the Spin Glass (SG)
tems: the original interactions between the fundamental model was implemented using as a basis the transforma-
]
n degrees of freedom are replaced by renormalized inter- tion for the pure model (see also [12] for more details).
n
actions between effective degrees of freedom. Finally, However, on the SG problem the estimation of the criti-
-
s the critical exponents may be computed repeating this calexponentνgoverningthedivergenceofthecorrelation
i
d transformation over and over [2, 3]. The renormaliza- length did not always agree with the values obtained by
. tion group has two main flavours. On the one hand, the Monte-Carlo (MC) simulation [16]. Later on, a different
t
a process can be done in momentum space, by slowly inte- RG transformation was proposed in [13] for disordered
m
gratingouthighmomenta[4]. MomentumspaceRGwas systemsleadingtovaluesofν compatiblewithMCones.
- first developed in quantum field theory and then it also Still,inordertoconfirmthevalidityofthismethod,large
d
became a highly developed tool in statistical mechanics, systemsizeshavetobeconsidered. Thus,tothisaim,we
n
o where it is usually performed on a perturbation expan- adapt this RG transformation to the random field Ising
c sion to compute critical exponents. On the other hand, model on the hierarchical lattice (RFHM), for which we
[ a technically different approach is to integrate out small can numerically study large system sizes and compute
1 distancedegreesoffreedominrealspace[2]. Theadvan- the critical exponents with high accuracy. Our results
v tage of the latter is to provide a more physical picture canbethencomparedwiththeonesobtainedin[15], us-
1 of the process despite the difficulty to obtain accurate ing an independent method developed in [14], and they
5
results. In general, both methods need some approxima- are found to be in good agreement.
7
tions, but it is possible to find systems with a particular
1 The paper is organized as follows. In the first section
. topologyforwhichrealspacetransformationscanbeper- wedefinetheHM,anddiscusssomeofthemainfeatures
1
formed exactly, such as the two dimensional triangular
0 of the RF models. In the second section we briefly in-
lattice[5]andthediamondhierarchicallattice[6]. These
4 troducetheRGtransformationsbydefiningthemforthe
1 kinds of exactly soluble models could play a very impor- ferromagnetic model and then generalizing them for the
: tant rˆole to test new ideas for systems where the nature
v RF model. We directly illustrate our results by recon-
of the phase transition is difficult to understand.
i structing the complete phase diagram of the RFHM in
X
In this work, we focus on quenched disorder systems the T/J h/J plane, where J is a coupling constant be-
r −
a for which the renormalization group approach is still not tween spins and h the variance of the random field. We
deeply understood. Following the approach introduced show that we do correctly recover the positions of both
with the diamond hierarchical lattice [7], we concentrate the transition of the pure model (h = 0,T = Tc) and of
our effort on another hierarchical lattice which gave us the random field one (h = hc,T = 0), since they agree
theopportunitytodefineanapproximatetransformation very well with the results found in [10, 15]. In the third
for more realistic systems. section we discuss the computation of the critical expo-
nents, comparing them with the values found in [15].
Years ago, Dyson [8] introduced the so-called Hierar-
chicalModel(HM)tostudytheproblemofphasetransi- The Hierarchical model: — The HM is a one-
tions in one dimensional long-range models. It was later dimensional model where the interaction between spins
understoodthatthetopologyofthismodelcouldbeused decreaseswiththedistancedefinedonabinarytree. The
to implement an exact real-space RG transformation [9]. simplest way to define the model is by an iterative con-
2
struction. First a pair of spins is coupled together with suggests that there is has a ferromagnetic phase transi-
a coupling 0 < J . Then a system of four spins is built tion when ρ [1;3/2). The situation for ρ = 3/2 is still
1
∈
by coupling two pairs of spins with a new ferromagnetic not clear. This transition is mean field-like for ρ < 4/3
coupling 0 < J < J . The operation is then repeated and non mean-field for ρ > 4/3, [14, 15, 17]. As for the
2 1
iteratively with these blocks of fours spins using another pure ferromagnet, similar results hold for the straight-
coupling 0<J <J <J : forward long range model with random fields, which has
3 2 1
been recently investigated in [18, 19]. In this work, we
show that our transformation is suitable to study the
1 = JJ1(s1+s2)2 (1) random field transition both at T = 0 and for T = 0.
H − (cid:54)
(cid:32) N (cid:33)2 Indeed thecomplete phase diagram ischaracterizedbya
= (L) + (R) J J (cid:88)s (2) critical line starting from the pure model and ending in
Hk Hk−1 Hk−1− k i
the true critical point at T = 0. By our method we ex-
i=1
plainhowtorecoverthefullphasediagramofthesystem
in the T/J h/J plane and how to compute the critical
k−1=2 Left k−1=2 Right exponent of−the true critical point.
The Ensemble RG transformation: —RealspaceRG
transformations can be defined as in [5]. These transfor-
mations connect a system A to a decimated system B
FIG. 1. Construction of a system of n = 3 levels from two
smaller ones. whose fundamental degrees of freedom can be obtained
from the ones of the system A via a coarse-graining pro-
whereLandRstandsfor“left”and“right”(seeFig. 1). cedure. They are the effective degrees of freedom we
The J and J are parameters of the model and N 2n mentioned in the introduction. It must be noted that
k n
≡
is the system size. In order to approach the behavior of the observables we compute in each system depend on
the straightforward one dimensional long range model, their respective parameters. For simplicity, consider a
where interaction between spins is J(i,j) = i j −ρ, it single parameter J. If we compute an observable in the
| − |
is common to define Jk = 2−ρk. The variable J sets the decimated system, we can use its dependance on JB and
type of interaction (ferromagnetic or anti-ferromagnetic) the relation between new spins and the old ones to write
anditsstrength. Inthefollowingwewillalwaysconsider an equation JB = f(JA). Suppose that this equation
J >0. The variable ρ controls the strength of the inter- has a fixed point J∗, then, this equation can be used to
action: forρ [1;2)themodelhasaferromagneticphase compute the critical exponent ν. In fact, as usual in RG
transition. W∈hen1<ρ<3/2thetransitionismeanfield theory, ∂f(J)/∂J J=J∗ = b1/ν, where b is the ratio be-
|
like whereas when 3/2 < ρ < 1 it is non-mean field [10]. tweenthesizeofthesystemAandthesizeofthesystem
Thus,HMcanbeusedtostudyphasetransitionsinnon- B. Thismappingdependsontheparametersofbothsys-
mean field systems, alike to D dimensional short range tems. At the critical point, if the chosen transformation
models,withDsmallerthanth−euppercriticaldimension iscorrect, JB =JA =J∗. Thisfixedpointwillbeunsta-
(i.e. Dc =4forthepureIsingmodel,Dc =6fortheRF ble respect to small perturbations from the fixed point
u u
Ising model and the SG model). Here the rˆole of the di- value J∗: if JA <J∗, then JB <JA, while for JA >J∗,
mension is played by the continuous parameter ρ, which JB >JA.
controlthedecayoftheinteractionatlargedistance. The This method works very well in the ferromagnetic hi-
main advantage of these models is that it is possible to erarchical model, even for small system sizes [11, 12]. In
writeanexactrecursiverelationcomputingthepartition spin glasses, sample-to-sample real-space RG have been
function [11–13, 15] in polynomial time with the system tried as in [11, 12] but didn’t always get satisfying re-
size, which can be used to compute correlation functions sults. Indeed, after obtaining an estimate of the critical
and other observables. exponents, the comparison with Monte-Carlo measures
In this work we consider the random field model on clearly indicate that the method fails in the non-mean
the hierarchical lattice. The RFHM is naturally defined field regime. Recently, Angelini et al. [13] proposed a
by taking eqs. (1,2) and adding a random magnetic field new RG transformation called Ensemble Renormaliza-
of zero mean and variance h2 in the Hamiltonian. The tionGroup(ERG).Themaindifferencebetweenthisap-
random field is added only to the first level of the inter- proach and the previous ones came from the order in
actions (i.e. in eq. (1)) and therefore relation (2) still which the average over disorder and the mapping be-
holds. As a consequence, the computation of the par- tween the two systems are made. The general procedure
tition function and of the correlation functions is still can be described as follows. First we define observ-
N
tractable. We can therefore use the same algorithm of ables s for systems A and B, where s runs over dif-
OA,B
thepuremodel,theonlydifferencebeingthatwehaveto ferentobservablesand isthenumberofparametersto
N
average over the disorder which increases a bit the com- be determined. Second, we compute s s = 1,..., .
OA ∀ N
plexity. For this model, a simple domain-wall argument Third,wefindthenewparametersforthesystemB such
3
observables is found by fine tuning h and J. For the
HRFM we consider the two following observables:
11..55
11..22 PPAARRAAPPMMHHAAAAGGSSEEEENNTTIICC Ok1 = (cid:113)(cid:104)mL,k(cid:105)(cid:104)mR,k(cid:105) , (5)
m2 m2
(cid:104) L,k(cid:105)(cid:104) R,k(cid:105)
00..99
m m m m
h/Jh/J Ok2 = (cid:104) L,k(cid:113)R,k(cid:105)−(cid:104) L,k(cid:105)(cid:104) R,k(cid:105) . (6)
m2 m2
00..66 (cid:104) L,k(cid:105)(cid:104) R,k(cid:105)
corresponding to both disconnected and connected cor-
00..33 FFEERRRROOMMAAGGNNEETTIICC relation functions. Similarly to eq. (4), JB and hB can
PPHHAASSEE be inferred from
(cid:118)
2 (cid:117)n−1
00..33 00..66 00..99 11..22 11..55 (J ,h )=argmin(cid:88)(cid:117)(cid:116)(cid:88)(cid:104)ds(J,h;J ,h )(cid:105)2, (7)
B B k A A
J,h
TT//JJ s=1 k=2
where we defined
FIG. 2. Phase diagram of the RFHM for ρ = 1.4 (non-
mean field), obtained via the ERG method. The mapping ds(J ,h ;J ,h )=Os (J ,h ) Os (J ,h ).
(JA,hA) → (JB,hB) shown in this figure has been obtained k B B A A B,k−1 B B − A,k A A(8)
with a transformation from systems with n=5 levels to sys-
This method has been proven to work better than the
tems with n−1 = 4 levels of interactions. For each couple
sample-to-sample RG transformations in the field of SG
(JA,B,hA,B) an arrow is drawn. The blue line is a sketch of
[13]. In fact, sample-to-sample RG transformations tend
the critical line inferred by the directions of the vector field.
to reduce the frustration introduced by the disorder, re-
spect to the ERG method, where they are automatically
that s = s s=1,..., . taken into account. Here we exploit this transformation
OA OB ∀ N
Wecanbetterdescribethismethodinthehierarchical on the RFHM where large system sizes are accessible.
model. Firstletusconsidertheferromagneticcasewhere Our method is able to capture the whole phase dia-
foreachlevelk =2,...,n 1ofthesystemA(containing gram, as is illustrated on Fig. 2. In order to compute
−
2n spins) we compute the RG flow, we first we pick a point in the (T/J, σ/J)
plane, corresponding to the parameters of a system A.
m m We then implement a one-step transformation and get a
L,k R,k
Ok = (cid:113)(cid:104) (cid:105) , (3) new point (T/J(cid:48), σ(cid:48)/J(cid:48)) for the system B. The temper-
m2 m2
(cid:104) L,k(cid:105)(cid:104) R,k(cid:105) ature acts just as a multiplicative factor. The RG flow
is then defined by a vector whose application point cor-
where m is the magnetization of the first (second)
L(R),k responds to the initial condition, and whose direction is
block made by 2k spins at the level k. Then, for each
givenbythearrivalpoint(magnitudeshavebeenrescaled
level k = 1,...,n 2 of the system B (containing 2n−1
to obtain a nicer plot). This procedure may be repeated
−
spins)wecomputethesameobservables. Fromtheequa-
all over the plane (T/J, σ/J) and allows us to charac-
tionsO (J )=O (J )fork =2,...,n 1weob-
B,k−1 B A,k A terize the entire phase space as illustrated for ρ=1.4 on
−
tain the mapping J =f(J ). In principle we just need
B A Fig. 2. Even for modest sizes (i.e. n = 5) the phase di-
oneobservabletodetermineJ . Inpracticeweobserved
B agram that we obtain is quite good, and the critical line
that the results are much more robust if we enforce that
givestheexpectedvaluesforh andT , apartfromfinite
c c
corresponding observables match at each level. In gen-
size corrections [10, 15] (see [20]).
eral,thisisanimpossiblerequirementsincewehavemore
Estimation of the critical exponent ν: — We now de-
equations than unknowns. Thus, this condition can be
scribe how to compute critical exponents. In the region
replaced by the weaker one
1 < ρ < 4/3 the critical exponent ν takes its mean-field
value ν = 1/(ρ 1) [17]. In another work [15], numer-
n(cid:88)−1(cid:104) (cid:105)2 ical estimates fo−r the region 4/3 < ρ < 3/2 has been
J =argmin O (J) O (J ) . (4)
B B,k−1 A,k A
J − computed using an algorithm developed in [14]. This
k=2
algorithm provides the ground state configuration of a
For disordered systems the observables have to be aver- RFHM sample and can be used to compute the critical
aged over the many realization of the disorder. We have exponents by mean of the Finite Size Scaling method.
considered here Gaussian distributed random fields for Our goal here is to both give a quantitative validation
both systems,A and B, parametrized by the variance h2 of the ERG method by applying it in a disordered sys-
(the mean being zero). Equality between corresponding temwhereitispossibletostudynumericallylargesystem
4
sizes and to estimate the critical exponents of this sys- 1.16
tem in the non-mean field regime. Therefore we finally 1.14
compare the estimations of the critical exponents given 1.12
by the ERG method with the values obtained in [15], 1.1
see Fig. 4. We studied the T 0 limit of the ERG 1.08 21/νn
Exponential Fit
→ 1.06
transformations [21] and computed the critical exponent Asymptotic Value (with error)
1.04 True Value
ν and the critical point (h/J) for different values of ρ.
c R*(n)
y
Wedescribehereafterthedetailsofourmethod. Givena 1.65 Exponential Fit
Asymptotic Value (with error)
value of ρ, we considered different transformation sizes. True Value
1.45
Let’s consider a size n. We first took M samples of such
a system and for each, we compute the observables of 1.25
eq. (6). This is the main non–trivial observable in this 1.05
4 5 6 7 8 9 10
limit, since the averaged connected correlation function,
n
eq. (5), goestozeroasT 0. Wethenaverageoverthe
→
M samples (typically, M = 107 for n = 4 and M = 105 FIG.3. Extrapolationofthevalueof21/ν andRc forρ=1.2
y
for n=11) and we computed O2A,k for k =2,...,n 1. (mean-field). Thesevaluesareobtainedusingthemethodde-
−
Starting from the initial values (J ,h ), we found the scribedinthetext,fittingthedatapointsbyanexponential:
A A
values(J ,h )usingeq. (7). Thishasbeendonethanks f(n) = a+b2−cn. The extrapolated estimations are very
B B
to the C++ open source library Dlib [22]. These trans- close to the exact parameters of the system, as can be seen
from the error bars given in Fig. 4.
formations have been done for size up to n=11.
In order to measure the dispersion of (J ,h ) around
B B
their mean values, at a given (J ,h ) and n, we ran
A A 1.25 Data from [15]
M(cid:48) times the algorithm (typically M(cid:48) =102), each time ERG method
getting independent estimations of (J ,h ). Since we Exact values
B B 1.2
are making the transformation at T = 0, we only care
about the ratio R = h/J; thus we have to compare
y 1.15
hA/JA and hB/JB. If RyB > RyA the flow is directed ()νρ
toward the paramagnetic region, while if RB <RA, it is 1/2
y y 1.1
directed toward the totally ordered fixed point. At the 0.95
critical point, linearization of the transformation RB = 0.55
y 1.05 (h/J)
f(RyA)allowstomeasurethecriticalexponentν byusing 0.15 c
tthhee rseylsatteimonB∂Risy(cid:48)/tw∂Ricye|Rsmy∗ a=llesr1/tνhawnhetrheehseyrsetesm=A2. sAintcae 1 0.05 0.1 0.15 0 0.1.25 0 .02.525 0.3 0. 305.35 0 0.4.45 0.45 0.5
given n, we obtained curves as the one shown in Fig. 3. ρ−1
The error bars on ν and (h/J)c have been obtained by FIG.4. Estimationof21/ν usingtheERGfordifferentvalues
a bootstrap resampling method: the M(cid:48) pairs of RA,B
y ofρ. Wecomparethesevalueswiththeestimationsobtained
are divided into a finite number of groups and for each in [15]. We can observe that the two methods agree very
group, an estimate of ν and (h/J)c is computed. Their well. Intheinset,theresultsfor(h/J)c,witherrorbars(very
dispersions characterize the error bars on ν and (h/J) . small). Inthatcasebothmethodsareincompletagreement.
c
We studied the ERG transformation n (n 1) for
→ −
n = 4,...,11 and studied finite size corrections like in
Fig. 3, in order to extrapolate the infinite size limit of perspective it would be interesting to find a SG model
the critical ratio Rc and the critical exponent ν, see Fig. for which the ERG can be implemented for large sys-
y
4. tem sizes. This would also be useful in order to confirm
Conclusion and Acklnowledgements: — We devel- the choice of observables taken in [13]. Another devel-
oped an RG treatment of the RF problem on the hi- opment would be to find a possible implementation for
erarchicaltopology. TheRGtechniqueweusedhasbeen other topologies like the Euclidean one.
first proposed in [13] where they applied it to study spin The research leading to these results has received
glasses and dilute magnets. Here, we finally confirm the funding from the European Research Council under
validity of the method thanks to the possibility to study the European Unions Seventh Framework Programme
big systems. In addition, we show that it was possible (FP7/2007-2013) / ERC grant agreement n. [247328].
to recover the whole phase diagram of the model. This A. Decelle has been supported by the FIRB project n.
study also demonstrates how well the method works, as RBFR086NN1. Wewouldalsoliketoacknowledgestimu-
can be seen on the excellent agreement for the value of latingdiscussionswithF.Ricci-Tersenghi,DavidYllanes
the critical point and for the critical exponent. As a and Beatriz Seoane.
5
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