New technique for replica symmetry breaking with application to the SK-model at and near T=0

0.69 MB·English

by  Manuel J. Schmidt

New technique for replica symmetry breaking with application to the SK-model at
and near T = 0
M.J. Schmidt and R. Oppermann
Institut f. Theoretische Physik, Universit¨at Wu¨rzburg, Am Hubland, 97074 Wu¨rzburg, FRG
(Dated: February 2, 2008)
We describe a novel method which allows the treatment of high orders of replica-symmetry-
breaking(RSB)atlowtemperaturesaswellasatT =0directly,withoutaneedforapproximations
or scaling assumptions. It yields the low temperature order function q(a,T) in the full range 0 ≤
8 a<∞ and is complete in thesense that all observables can be calculated from it. The behavior of
0 some observables and the finite RSB theory itself is analyzed as one approaches continuous RSB.
0 The validity and applicability of the traditional continuous formulation is then scrutinized and a
2 new continuousRSBformulation is proposed.

n
PACSnumbers: 75.10.Nr,75.10.Hk,75.40.Cx
a
J

1 I. INTRODUCTION RSB, high enough to obtain a confident extrapolation
1
κ . However,our approachallows for a numerically
→∞
exact description of the full SK-model with all its diffi-
] The ordered phase of spin-glass models [1] has gained
n culties and features at T =0. The numerical results can
much attention over the past three decades, but still
n be used to construct and test analytical approximations
- there are many open questions even in models which
or simpler alternative theories which capture the essen-
s
were designed to possess simple ’mean-field-type’ solu-
i tial physics while being generalizable to more complex
d tions. Paradigmatic for the difficulties encountered in
physical situations [8].
at. the descriptionofthe orderedphase is the appearanceof The low temperature formulation, which is developed
an ultrametric structure in the mean-field theory of spin
m in the present work, directly identifies and analyzes the
glasses, known as Sherrington-Kirkpatrick (SK) model
issues of the infinite RSB limit at zero temperature and
- [2]. ThecorrecttreatmentoftheSKmodelinvolvesahi-
d resolvesthembyappropriaterescalingofauxiliaryquan-
erarchicalscheme as introduced by Parisi[3], which only
n tities. It is based on the idea of rescaled Parisi block
o recentlyhasbeenproven[4]tobeexactinthelimitofan size parameters [3, 9] a = βm with β = T−1. By
c infinite number of hierarchicalsteps of replica symmetry i i
thistransformation,thenon-trivialstructureofthezero-
[ breaking (RSB).
temperature order function of the SK-model can be re-

1 Thoughattemperaturesrightbelowthe freezingtran- solved. Inthetraditionalformalism,thestructureiscon-
v sition TC, a small number κ of RSB steps is not a bad centratedandhiddenatthepointx=0,whereq˜(x)gets
6
5 approximation,attemperaturesT ≪TC theconvergence singular. Both, Pankov’s and our approach, investigate
oftheresultswithrespecttoκbecomesworse. Tradition- this point, but while the scaling approach is incomplete
7
1 ally, the κ= limit is formulated by a continuous the- in the sense that it e.g. only accounts for the short time
∞
. orywherethefunctionalfreeenergyf[q˜(x)]ismaximized observablesor that it cannotdescribe the situation atfi-
1
withrespecttotheParisiorderfunctionq˜(x), 0 x 1 nite externalfields correctly,our approachyields the full
0 ≤ ≤
8 [1, 3]. This has later been expressed by a closed self- zero temperature solution.
0 consistency form, involving a set of numerically solvable Thepaperisorganizedasfollows. InsectionII,wegive
: partial differential equations [5, 6]. The treatment of ashortreviewoftheissuesencounteredwhenperforming
v
RSB at finite temperatures is well established, mean- the zero temperaturelimit and showthe connectionsbe-
i
X while. NearTC,thereareevenanalyticalsolutionsavail- tween the two approaches mentioned above. In section
r able. At zero temperature, however, the proper form of III, we derive the non-trivial zero-temperature limit of
a the theory is still under discussion. The standard differ- the SK-model at finite RSB. In section IV, some results
entialequationsofthecontinuoustheoryandtheirinitial at low temperatures, obtained with the finite RSB for-
conditionsgetsingularinthezerotemperaturelimitand malism are shown. In section V, we discuss the κ
the traditional formalism breaks down. limit of our theory. → ∞
In the literature, one finds two main directions of ap-
proachingthezerotemperaturelimit. Pankovproposeda
beautifullysimplescalingAnsatzdirectlyatinfiniteRSB II. THE T =0 LIMIT OF q˜(x)
which becomes exact in a certain limit at zero tempera-
ture [7]. This limit, however, is far from being the only The Parisi order function q˜(x), x [0,1] which is
∈
importantregionofthe fullzero temperaturesolutionas the central quantity at finite temperatures becomes a
we shall show later. On the other hand, our group has constant function at zero temperature (lim q(x) =
T→0
developed an exact finite RSB approach, which involves 1, x > 0) with a singularity at x = 0. Since typical
some advanced numerics when trying to reach orders of observables as e.g. the internal energy u involve inte-
2
δa =a a ,i.e. thestepheightsandwidthsapproach
i i i+1
−
zero. ForT >0,those requirementsarefullfilled andare
the basis of the well known Parisi differential equations
[1, 3, 6]. At zero temperature and κ , we still find
→ ∞
δq 0 which is not surprising, since q is restricted
i i
→
to[0,1]independentoftemperature. However,somestep
widthsδa growindefinitelyinthislimit,sotheassertion
i
δa 0 which is needed to derive the Parisi differential
i
→
equations fails at zero temperature. The non-vanishing
δa correspond to those a which approach infinity for
i i
κ and so the continuous theory fails at a = ,
→ ∞ ∞
while for finite a, a differential equation formulation is
valid.
InsectionVofthispaper,weshowhowthefailureofa
FIG. 1: (color-online): The true T =0 order function q(a,0)
continuoustheoryata= canbeovercomebychanging
at 200 RSB (black) and the step approximation at 20 RSB. ∞
the initial condition of a partial differential equation.
Thelower(red)curveshowstheresult ofPankov-scaling,i.e.
q(a,0)=1−γa−2.
III. RECURSION TECHNIQUE FOR
grals of the form u=β 1dx(1 q˜(x))2, one is tempted ARBITRARILY HIGH RSB ORDERS κ
0 −
to absorb the divergent factor β by a transformation
R WeinvestigatethestandardSK-modelHamiltonian[1,
x a = βx. The non-trivial zero temperature struc-
→ 2] of N Ising spins s = 1 with external field H
ture which was concentratedatx=0 in the Parisiorder i
±
function and createdthe singularitythere is then ’blown
= J s s +H s . (1)
up’ and resolved by the new order function q(a). The HSK ij i j i
i<j i
transformation might remind the reader of PaT scaling X X
[13]. The idea here is different, though, as we don’t seek The quenched random coupling constants J are Gaus-
ij
forauniversalformoftheorderfunction(whichisknown sian distributed random variables with zero mean and
to be nonexistent) but for a non-singular zero tempera- varianceN−1[14]. The model yields afreezing transition
ture theory. at T =1. Since the coupling constants’ degrees of free-
C
The transformation x a is not limited to T = domarequenched,thefreeenergyperspinofthesystem
→
0, of course, so that we can formulate a general low- is given by f = T/N log exp( β ) where
− h h − HSK isiJ h·is
temperature theory which allows a smooth transition refers to an average in the space of Ising-spins s = 1
i
±
from a treatment directly at T = 0 to finite tempera- and refers to an average with respect to the cou-
h·iJ
tures. Theobjective ofourformalismis to findthe func- pling constants J . In order to bypass the average of
ij
tion q(a,T) with a [0,β] from which one can obtain the logarithm we avail ourselves of the standard replica-
∈
q˜(x,T) = q(βx,T) at finite temperatures. At κ < , trick. After transformation to a single site model [15],
∞
q(a,T)canbedisplayedasastepfunctionwiththenum- we introduce ParisiRSB for the replica coupling matrix,
ber of steps equal to κ (see Fig. 1). perform the replica limit n 0 and the thermodynamic
→
Inthisnotation,thePankov-scalingapproachisequiv- limit N and arrive at an expression for the free
→ ∞
alent to expanding q(a,0) 1 γa−2 near a = and energy per spin (see also [1])
calculating the coefficient γ≃. Th−e knowledge of th∞e a−2-
κ+1
termofthis expansionis sufficientfor short-timeobserv- f = β(1 2q )+ β q2(m m ) f˜ (2)
ablesastheentropyorthenon-equilibriumsusceptibility. −2 − 1 4 i i− i−1 −
i=1
The full solution, however, must respect the functional X
dependence of q(a,0) at all a [0, ] which is needed withκtheorderofRSBandqi thevaluesoftheelements
to calculate e.g. the ground sta∈te en∞ergy or the equilib- of the Parisi-blocks with size mi [16]. The non-trivial
rium susceptibility. Another point is, that the action of part f˜of the free energy is given by a κ+1-fold nested
a finite external field is reflected by the small a domain integral
of q(a,T) which is not resolved in [7]. In Fig. 1 the
difference between the two solutions at low a is shown. f˜= T G log GE GE G(2cosh(βh ))m1
1
In order to obtain a well defined zero temperature mκ Zκ+1 "Zκ ···Z2 Z1 #
limit, we start with a formulation of the theory at fi- (3)
nite orders of RSB (κ < ), where q(a,T) is a function where two Gaussian integral operators are defined as
∞
with κ plateaus. The height of the ith plateau is given
by the nunber qi ∈[0,1]and the step positions are given Gf(hi) ∞ dhi exp (hi−hi+1)2 f(hi)
by a [0,β]. If one aims at a continuous theory for ≡ √2π∆q − 2∆q
i ∈ Zi Z−∞ i (cid:18) i (cid:19)
κ , one must first check that δq = q q and (4)
i i i+1
→ ∞ −
3
with ∆q =q q , ∆q =q and while the deviation of the integrals from its asymptotic
i i i+1 κ+1 κ+1
−
formatsmallhisincorporatedinformofauxiliaryfunc-
GEf(hi) Gf(hi)ri−1, ri−1 = mi . (5) tions expCi(h) defined by
≡ m
Zi Zi i−1
GE G
ThelastGaussianintegralinthischainiscenteredatthe (2cosh(βh ))Ta1 =
1
external field h =H. ···
κ+2 Zi Z1
As explained above, it is convenient at low temper-
i
1
atures to transform to new variables a = βm . Since
i i exp a a ∆q + h +exp (h ) . (9)
r = mi+1 = ai+1 the definitions ofthe Gaussianintegral  i2 i i | i+1| Ci i+1 
i mi Gai GE Xj=1
operators and remain invariant. The explicit   
temperature dependence is located at the initial condi-
The functions exp are well defined and free of singu-
tion, whichRhas a wRell defined zero temperature limit in Ci
larities at each level of integration, for all temperatures
the a-formulation
includingT =0directlyandatallκ< . Itistherefore
∞
(2cosh(βh ))Ta1 T→0 ea1|h1|. (6) the object of our choice for the finite RSB numerics. At
1 −→ zeroth level of integration the ’initial condition’
The utility of our new formulation for low temperatures
lies in the non-singular limit of this inner integrand. It 2h
exp (h)=T log 1+exp | | (10)
has been shown before[9] that the first integral can be C0 − T
(cid:18) (cid:18) (cid:19)(cid:19)
performed analytically in the T 0 limit. For finite
→
temperatures, there is also an asymptotic regime h
1 has a smooth transition from finite temperatures to T =
→
of the integrand where
0, i.e. lim exp (h) = 0. Furthermore, the exp
±∞ T→0 C0 C
obeys a h h symmetry at each level of integration -
(2cosh(βh1))Ta1 h1→±∞ ea1|h1|. (7) irrespectiv→e of−temperature T or magnetic field H and is
−→
continuousath=0,thoughthereisakinkath=0with
In the asymptotic regime, the Gaussian integrals can
∂ exp (h) = 1, i. Therefore, it is sufficient to
be performed analytically at each level of integration h Ci |h=0+ − ∀
restrict ourselves to h > 0 and introduce a boundary
(h 1)
i+1 condition at h=0 in the continuous theory.
| |≫
GE 1 Atfiniteκ,therearesimplerecursionrelationsbetween
eai−1|hi| exp a h + a ∆q , (8) exp (h) and exp (h)
≃ i | i+1| 2 i i Ci Ci−1
Zi (cid:18) (cid:18) (cid:19)(cid:19)
1 G 1
exp (h)= log exp a h′ h +exp (h′) a ∆q (11)
Ci ai "Zi (cid:18) i(cid:18)| |−| | Ci−1 − 2 i i(cid:19)(cid:19)#
andintheend,thenon-trivialpartofthefreeenergycan β and the first line of (13) becomes proportional to an
be expressed in terms of an integral of expCκ integral 0βda(1−q(a))2.
At finite κ, the free energy must be maximized with
G 1 κ R
f˜=Zκ+1(expCκ(h)+|h|)+ 2k=1ak∆qk (12) breyspaecrotottose{aaric,hqi}o.f thTehigsraisdidenotneoffofrinTt=he02κ(fi(n2κite+T1))
X
dimensional parameter space [17].
so the free energy per particle can be written as
Therearetwomainadvantagesofperformingnumerics
intermsofexp . Thefirstisthatthenon-trivialdomain
κ C
T 1 of this function (i.e. where it is non-zero, see Fig. 2) is
f = χ2 + a (q 1)2 (q 1)2
−4 ne 4 i i− − i+1− strongly restricted to the region of small h and so the
Xi=1 (cid:2) (cid:3) numerical effort reduces considerably. This is illustrated
G
by Fig. 3 where the κ-dependence of typical calculation
(exp (h)+ h) (13)
− Cκ | | times is shown. A single calculation implies the calcu-
Zκ+1
lation of the free energy and all its derivatives. Since
with the non-equilibrium susceptibility χ = β(1 q ). the number of derivativesgrowsas 2κ+1,the full calcu-
ne 1
−
For the continuous RSB limit, one can fix the domain lation time grows stronger than linear. The calculation
boundaries of q(a,T) by defining a 0 and a time normalized to the number of derivatives to calcu-
κ+1 0
≡ ≡
4
50 initial condition given in equation (10).
40
i
30 IV. RESULTS
20
In this section, we discuss some results of our exten-
sivenumericalcomputationsaccordingtotheschemede-
10
scribed in section III. We have been able to perform cal-
culations at extremely high orders of RSB (up to 200
-6 -4 -2 0 2 4 6
at T = 0 and up to 53 at finite temperatures) with un-
h
precedentedaccuracywhichallowadeepinsightintothe
subtleties that arise as κ approaches its physical limit
FIG. 2: Non-trivial domain of expCi(h) of a typical 50-RSB
κ= .
calculation at T = 0 and H = 0 represented by the bars in
∞
hdirection. Thenon-trivialdomain is definedas theinterval In a previous publication we used 42 RSB data to ex-
where expCi(h) is numerically greater than zero. trapolate the zero temperature free energy to κ =
by fitting f to the function f = f + const. [18∞].
κ κ κ=∞ (κ+κ0)α
1.6 Theextrapolationisnowconfirmedaccordingto200RSB
9
high precision numerics with the new exp -formalism to
sec01..82 7 10 se-3 fesκt=im∞a=te−th0e.7e6r3ro1r66of72th6e56e6xt5r4a7p,oαlat=ed4.frIeCteiesndeirffigyc.ulTthtoe
c
0.4 fluctuations ofthe free energyofthe finiteκ calculations
5
are on an arbitrary small scale, due to the utilization of
30 50 70 90 20 40 60 80
κ κ arbitrary precision arithmetics, so that the accuracy of
f dependsonlyonthefitprocedure. Togivethereader
∞
FIG. 3: Calculation times at T = 0 in dependence of κ. an idea of the convergence level at 200 RSB, let us state
The calculation time of the free energy with all derivatives that f f 10−11. With standard fit methods, an
200 ∞
(left part) goes with κ1.6 and the calculation time of a single accuracyo−f10−≃13 is obtained, which representsthe min-
derivative(right part) goes with κ1.1 imum accuracy of f . With some technical tricks which
∞
are beyond the scope of this paper, however, it seems
that the accuracy might be up to 10−15. In any case,
late, however grows nearly linearly with κ and therefore this is by far the most precise ground state energy ever
highordersofRSBmaybe obtained. The secondadvan- obtainedfortheSK-modelanditprovidesagoodtestfor
tage is, that exp (h) (1) and no loss of numerical allcomingformalismswhichworkdirectlyatκ= and
C ∼ O ∞
precision due to large numbers happens. T = 0. In comparison with the literature, we find that
Figure 4 shows the functions exp at all integration the value is consistent with the estimate of Parisi [10],
Ci
levelsin caseofzero temperatureandfor finite tempera- but is slightly outside of the confidence interval given in
tures. The initial condition is represented by the lowest [5].
curve, which is exactly zero for T = 0 and remains fi- At zero temperature the free energy f equals the in-
nite forfinite temperaturecorrespondingtothe non-zero ternal energy u. At finite temperatures, they are still
closely related. We have checked numerically, that the
free energy can be expanded in a Taylor series with reg-
ular exponents at T =0
f(T)=f s T +f T2+f T3+ (T4) (14)
0 0 2 3
− O
where s = χ2 /4 is the zero temperature entropy,
0 − ne
which can also be expressed in terms of the non-
equilibrium, or single-valley susceptibility χ . The in-
ne
ternal energy can be expressed by the same coefficients
u(T) = f T df = f f T2 2f T3 + (T4), and
− dT 0 − 2 − 3 O
also the temperature dependence of the entropy can be
written as s(T)=s 2f T 3f T2+ (T4). Fromour
0 2 3
− − O
results we find f 0 and f 0.24, in agreement
FIG.4: (color-online): expC (h)atevenlevelsofintegrations 2 3
i → → −
at T = 0 and κ = 50. The lower lines (red) correspond to with the literature [5, 7].
small iandtheupperlines(blue)tolarge i. Theinsetshows Thezerotemperatureentropyisdirectlyrelatedtothe
a T =0.1 calculation with κ=30. non-equilibrium susceptibility which can be written at
5
finite RSB as [19]
κ
1
χ = a (q2 q2 ) f. (15)
ne −2 i i − i+1 −
i=1
X
One finds, that χ (κ+κ )−γ with γ = 1.666664
ne 0
∝ ±
5 10−6. For an exponent, this numerical accuracy is
·
sufficient to claim γ = 5/3. As a result, both, the non-
equilibrium susceptibility and the entropy vanish with
irregularexponents5/3and10/3forκ atzerotem-
→∞
perature. The implications of those irregular exponents
will be discussed elsewhere [12].
As we have explained above, the natural formulation
of the order function at low temperatures is in terms of
a function q(a,T) since it resolves the structure at the
singular point x=0 of the original Parisiorder function
q˜(x,T) at T =0. In order to better understand the crit-
ical properties of the zero temperature order function,
we have to discuss the finite RSB step approximation to
q(a,T). InFig. 5the logarithmofthe Parametersa are
i
plotted as a function of logκ. One can clearly see that
for largea the spacing between successive a-points does
i
not vanish, while for moderate a , a continuum emerges.
i
Alsoatsmalla ,a spacingappearsonaloga-scale. This
i FIG.5: logai asafunctionoflogκforκ=1,..,200at T =0
small-loga spacing, however, does not imply a spacing
and for κ=1,..,50 for T =0.03.
on an a scale, where the differential equations of a con-
tinuous theory appear. At finite temperatures, the dis-
creteness at large a disappears due to the restrictedness
equationisgivenatm [21]. Thereis,however,theregion
1
of a to the interval [0,β], while the small a discreteness
i ofdiscrete m where a continuous theoryis invalid, soat
i
remains. A finite external field would destroy also the
T = 0 the initial condition of the continuous theory is
small-a discreteness. The notion of non-zero plateau-
disconnected from the validity domain of the differential
widths on a log scale is directly related to the discrete
equation. This is why the traditional theory fails for
spectraofther -levels,whichwillbediscussedmorethor-
i T =0.
oughly elsewhere [12].
Two other important and closely related parameters
At T > 0, the identification of a break point x¯ is im- aretheT2-coefficientoftheT-expansionoftheEdwards-
portant. One finds, that Anderson order parameter q (T) = 1 αT2 + (T3)
EA
− O
and the a−2-coefficient of the 1/a-expansionof q(a,0)=
x¯= lim Ta . (16)
1 1 γa−2 + (a−3). At finite orders of RSB, there is
κ→∞
− O
also a linear term in the T-expansion of q with the
EA
From our computations at finite temperatures, we can coefficient equal to χ , but this coefficient vanishes for
ne
extract a confident value for x¯ at temperatures down to κ . From our numerical data, we can extract α =
T = 0.015. Below this temperature, calculations at κ > 1.5→94∞11 0.00002 and γ =0.4108 0.0001.
50 are needed. There is, however, no reason to expect a Inthe±followingparagraph,wean±alyzeourresultsfrom
large deviation of x¯(T = 0.015) from its true extrapola- the viewpoint of PaT scaling [13]. In the originalformu-
tiontoT =0. Wefindx¯(T =0.015)=0.54684 2 10−5, lation, PaT scaling has been used to evaluate a (approx-
± ·
in consistence with the literature [5, 7]. At zero temper- imately) universal scaling function q˜(x,T) f(x/T) for
ature, x¯ is ill defined because q˜(x,0) = 1, x > 0. As a x < x¯. This scaling is known to not hol≃d exactly. If
∀
consequence, one does not find x¯ directly in the T = 0- it held exactly, then f(x/T)= q(x/T,0) at all tempera-
theory. In fact, there is a subtle non-commutativity of tures. In Fig. 6 we compare q(a,0) to the result of the
the T 0 and κ limits for quantities like the PaThypothesis,whichwasobtainedaccordingtothede-
→ → ∞
break-point [12]. scription in [13].
At finite κ and T = 0, no mi remains finite, all go to In order to discuss the correction to PaT scaling near
zero. Fromscalingarguments,however,one cansee that T =0, we write
there are finite m even at T = 0 in the κ = limit.
i
∞
These are exactly the mi = Tai corresponding to the q(a,0)+q˜(a,T) for a<a¯
q(a,T)= (17)
discretea-points(seeFig. 5)andsothefinitemi arealso q(a¯,0)+q˜(a¯,T) for a a¯
(cid:26) ≥
discrete[20]. Inthecontinuousformulationoftheinfinite
RSB limit, an initial condition of a partial differential with a¯=βx¯ and q˜(a,T) is the correctionto PaT scaling
6
1 V. ∞-RSB LIMIT
0.8
As κ , the number of parameters (q ,a ) goes to
i i
→ ∞
infinity and a smooth order function q(a,T), defined on
q(a)0.6 0.02 [0,β] rises for T > 0. In this limit, the set of functions
exp become one continuous function of the variables a
0.4 0.01 andChi defined as
0.2 0 exp (a ,h)= lim exp (h) (25)
C i κ→∞ Ci
0 1 2 3 4 5 6
and the free energy for H =0 can be written as
1 2 3 4 5 6
a
1 β
f = da(q(a) 1)2 exp (a=0,h=0) (26)
FIG. 6: (color-online): Comparison of the universal scaling −4 − − C
functionobtainedbythePaT-hypothesis(black,dashed)and Z0
the zero temperature 200 RSB order function q(a,0) (red, where exp (a,h) is obtained by solving the partial dif-
solid). The inset shows thedifference of thetwo curves. ferential eqCuation
q˙(a)
nearT =0. Fromtheidentity1= 0βda(1−q(a,T))one ∂aexpC =− 2 ∂h2expC+2a∂hexpC+a(∂hexpC)2
can derive the correctionto the PaT-scalingbreak point h (2i7)
R
at zero temperature with boundary and initial conditions
1 1 a¯ d ∂ exp (a,h) = 1; exp (a, )=0
x¯(T =0)= lim da q˜(a,T) (18) h C |h=0+ − C ∞
2 − 2αT→0Z0 dT exp (β,h)=T log(1+exp( 2βh)) (28)
C −
with α the quadratic temperature coefficient of the
for h 0.
EαTdw2.ardIfs-wAendaesrssuomne(EthAa)toq˜r(dae,rTp)acraamnebteereqxEpAa(nTd)ed≃i1n−a Atfi≥nitetemperatures,wherethespacingbetweensuc-
cessive a and q go to zero as κ , the exp is
Taylor-seriesnear (a,T)=( ,0), i.e. i i → ∞ C
∞ well-behaved and the differential equation can be solved
q˜(a,T)= Ti q˜(a), for T 1 (19) numerically. In this case equations (27,28) are merely
i
≪ a reformulation of the Parisi theory [1, 3], convenient
i
X for investigations at low temperatures. At exactly zero
and temperature, however, two problems arise which force
us again to reformulate. Firstly the discreteness in the
q˜(a)= bja−j, for a 1 (20)
i i ≫ large a regime of q(a) formally invalidate the differential
j
X equation at a = , where the initial condition is given.
∞
one can derive relations between the lowest coefficients Secondlythesecondderivativeoftheinitialconditionhas
of the q˜(a,T) expansion, the quadratic temperature co- a divergence at h=0 as T 0.
→
efficient α of the EA order parameter, the break point Thesecondissuecanberesolvedbyafurtherrescaling
and the a−2 coefficient γ of the expansion of q(a,0) at y = (a+1)h and introduction of the function g(a,y) =
a= . (a+1)exp (a,y/(a+1)). The initial condition at β for
∞ exp transClates to g(β,y) = log(1+e−2y) = g (y). The
0
b01 =b11 = 0 (21) leftCpartofFig. 7showstheaandy-dependenceofg(a,y)
γ
α = b0 (22) (for visualization purposes, the a-axis has been rescaled
x¯2 − 2 to the interval [0,1] by the introduction of the variable
∞ γ ζ = a/(1+a)). It shows that for finite a (i.e. ζ < 1),
daq˜ (a) = (1 2x¯) b0. (23)
1 x¯2 − − 2 g(a,y)variessmoothlyasafunctionofaandthedescrip-
Z0
tionintermsofapartialdifferentialequationisvalid. In
The firstrelationstates thatq˜ (a) mustgoto zerofaster
1 the large a limit, however,a singularity appears. To un-
thana−1 asa . Theparametersγ,α,x¯areverywell
derstandthissingularityandthesolutionofthisissue,we
→∞
known from the literature and have been obtained from
go back again to the finite RSB formulation. The inves-
our numerical data, too. From the above relation, one
tigation of the 200 RSB calculations shows that at large
can thus extract b0 = 0.22035 0.00012 and write the
2 − ± a, the difference between g(ai,y) and g(ai+1,y) does not
correction to PaT-scaling near (a,T)=( ,0) as
vanish with κ as it does for moderate a a [22]
∞ 1
→ ∞ ≪
q(a,T)=q(a,0) 0.22T2+ (T,a−1)3. (24) (see rightside ofFig. 7). This means, that the recursion
− O relation(11)doesnotapproachadifferentialequationat
Relation (23) can be used as a check for q˜ (a) extracted a = . Instead, one can see that the recursion drives
1
∞
from numerics. the functionto alimiting function g (y)fromwhichthe
∞
7
FIG.7: (color-online): Left: g(ζ,y)withζ =a/(1+a). Right: Theinitialconditiong0(y)(topblackline)atx=1,a=∞and
thefirst 20 integrations (red to blue). The bottom black line shows the initial condition g∞(y) at x=0,a=∞.
find g˙ = 1F[g] with
a
F[g]=g y g′ γ(g′′+2g′+(g′)2). (30)
− −
Obviously, an initial condition g( ,y) of (29) with
∞
F[g( ,y)] = 0 would lead to a logarithmic singularity
∞ 6
of g(a,y) at a = . The only non-singular initial con-
∞
dition is therefore the solution to the ordinary differen-
tial equation F[g˜ (y)] = 0. Indeed, the solution g˜(y) of
∞
this differential equation seems to be the limiting func-
tiong (y)ofthe recursionstarting fromg (y) discussed
∞ 0
above. In Fig. 7, the function g (y) is the numerical
FIG. 8: (color-online): g(ζ,0) with ζ = a/(1+a) for up to ∞
200ordersofRSB.Thedotswiththedifferentcolors referto solution of F[g∞] = 0. In some sense, the partial dif-
low (red) and high (blue) RSBorders. The solid line (red) is ferential equation governing the function g(a,y) yields
obtained bythe ∞ RSBformalism proposed in thetext. its own initial condition - it is the only initial condition
which makes sense.
To further illustrate and confirm this line of reasoning
continuous part of g(a,y) starts. In the right part of fig-
and to better understand the transition κ we shall
ure 7 the first 10 results of the recursion relation for a →∞
restrict our discussion of g(a,y) to y = 0 as representa-
κ = 200 calculation are shown together with g (y) and
0 tive for the a-dependence of g. In Fig. 8 we plot g(a ,0)
g (y), whichis obtainedby numericallysolvinganordi- i
∞ at T = 0 for κ < varying from 10 to 200. Again,
nary differential equation, as explained below. ∞
one can see the discreteness at a = . For demonstra-
Tocompleteourdiscussion,wenowapproachthepoint tion purposes, a numerical solution o∞f (29) is plotted as
a = from below. In the finite a regime, the behavior
the drawn through line. To obtain this solution at large
∞
of g(a,y) is governed by the partial differential equation a, equation (29) has been expanded up to order a−2 at
(thedotreferstoaderivativewithrespecttoa,whilethe
a = . The solution of this expansion taken at a = 8
prime means a y-differentiation) (ζ ∞0.89) is then used as an initial condition for the
≃
full partial differential equation (29) at a = 8. By thor-
q˙ g y g′ oughlylookingatthe line, one cansee asmallerrornear
g˙ = (a+1) (a+1)g′′+2ag′+a(g′)2 + − .
−2 a+1 this junction point. With more effort like higher order
(cid:2) (cid:3) (29) expansionsata= oradvancednumericalmethods for
∞
In orderto investigatethis equationatzero temperature partial differential equation (e.g. pseudo-spectral meth-
in the limit a , where the initial condition is given, ods), the quality of the full continuous RSB solution at
we expand th→e o∞rder function q(a,0) = 1 γa−2 and zero temperature can be strongly improved, but this is
equation(29) itself neara= . To first ord−erin a−1 we beyond the scope of this work.
∞
8
VI. CONCLUSION near zero temperature and a = x/T = has also been
∞
discussed. Furthermore,wehaveproposedanAnsatz for
WehavedevelopedanRSBtechniquewhichallowscal- a full treatmentof the zero temperature limit of the SK-
culations at extremely high orders of RSB near and at model directly at infinite RSB - in analogy to the con-
T =0. We have indeed performed calculations at T =0 tinuous RSB formalism at finite temperatures. It would
for up to 200 RSB and for finite temperatures up to 53 be usefull to derive a closed set of self-consistency equa-
RSB.Weextractedthedependenceofvariousobservables tions for q(a) in the sense of Sommers and Dupont [6]
on the order of RSB and on temperature and obtained, and solve them numerically in order to obtain the zero
to ourbestknowledge,the byfarmostprecisenumerical temperature order function directly in its physical limit.
value for the ground state energy of the SK model. The
connection to PaT-scaling has been discussed and a cor- We gratefully acknowledge useful discussions with
rection to the PaT-scaling assumption q(x,T)=f(x/T) David Sherrington, Kay Wiese and Markus Mu¨ller.
[1] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 [14] The variance of the disorder distribution must scale as
(1986). N−1 in order to obtain a non-trivial thermodynamic
[2] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, limit. This choice also defines the energy and temper-
1972 (1975). ature scale of thesystem.
[3] G. Parisi, Phys. Rev. Lett. 43, 1754 (1979); G. Parisi, [15] Duetotheinfiniterangecouplingbetweenthespins,the
Phys.Rev.Lett. 50, 1946 (1983). effectivedimensionoftheproblemis∞andsothemean-
[4] M. Talagrand, Annals of Mathematics 163, 221 (2006). field theory is exact
[5] A.CrisantiandT.Rizzo,Phys.Rev.E65,046137(2002). [16] We usethe convention qi <qi−1 and mi<mi−1.
[6] H. J. Sommers and W. Dupont, J. Phys. C 17, 5785 [17] At zero temperature, q1 =1, because q1 is theEdwards-
(1984). Anderson orderparameter.
[7] S.Pankov, Phys.Rev.Lett. 96, 197204 (2006). [18] κ0 ≃ 1.27 is an offset which is used to improve the fit
[8] R.Oppermann,M.J.SchmidtandD.SherringtonPhys. quality.
Rev. Lett. 98, 127201 (2007); R. Oppermann and D. [19] Alternatively, it is also obtained by varying the free en-
Sherrington,Phys. Rev.Lett. 95, 197203 (2005). ergywithrespecttoq1.Thishasbeencheckedandyields
[9] H. Feldmann and R. Oppermann, Phys. Rev. B 62 9030 the same result.
(2000). [20] This can also be seen from thediscrete spectra, because
[10] G. Parisi, J. Phys.A 13, L115 (1980). the definition of ri=ai/ai−1 =mi/mi−1
[11] J. R. L. de Almeida and D. J. Thouless, J. Phys. A 11, [21] Sometimesintheliteratureonefindstheinitialcondition
983 (1978). at x=1, which is equivalent due to the triviality of the
[12] R.Oppermann and M. J. Schmidt (tobe published). differential equations beyondthe break point.
[13] J. Vannimenus, G. Toulouse and G. Parisi, J. Phys. 47, [22] a1 is the largest a-parameter in the finite RSB-
565 (1981). formulation