# Triviality problem and high-temperature expansions of higher susceptibilities for the Ising and scalar-field models in four-, five-, and six-dimensional lattices.

**ABSTRACT** High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state. We have obtained a substantial extension, through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s≥1/2 and for the lattice scalar-field theory with quartic self-interaction on the simple-cubic and the body-centered-cubic lattices in four, five, and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality.

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arXiv:1112.5274v1 [hep-lat] 22 Dec 2011

Triviality problem and the high-temperature expansions of

the higher susceptibilities for the Ising and the scalar field

models on four-, five- and six-dimensional lattices

P. Butera∗

Dipartimento di Fisica Universita’ di Milano-Bicocca

and

Istituto Nazionale di Fisica Nucleare

Sezione di Milano-Bicocca

3 Piazza della Scienza,

20126 Milano, Italy

M. Pernici†

Istituto Nazionale di Fisica Nucleare

Sezione di Milano

16 Via Celoria, 20133 Milano, Italy

(Dated: December 23, 2011)

Abstract

High-temperature expansions are presently the only viable approach to the numerical calculation

of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices.

The critical amplitudes of these quantities enter into a sequence of universal amplitude-ratios which

determine the critical equation of state. We have obtained a substantial extension through order

24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the

Ising models with spin s ≥ 1/2 and for the lattice scalar field theory with quartic self-interaction,

on the simple-cubic and the body-centered-cubic lattices in four, five and six spatial dimensions.

A numerical analysis of the higher susceptibilities obtained from these expansions, yields results

consistent with the widely accepted ideas, based on the renormalization group and the constructive

approach to Euclidean quantum field theory, concerning the no-interaction (“triviality”) property

of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the

upper critical dimensionality.

PACS numbers: 03.70.+k, 05.50.+q, 64.60.De, 75.10.Hk, 64.70.F-, 64.10.+h

Keywords: Scalar field triviality, Ising model, magnetic field

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I.INTRODUCTION

The renormalization group theory (RG) theory1–3predicts the value dc= 4 for the upper

critical dimensionality of the N-component lattice scalar-field theory and of the short-range

classical Heisenberg N-component spin systems with O(N)-symmetric interaction. When

d ≥ 4, the critical fluctuations become too weak to drive the leading critical exponents

away from the “classical” values taken in the mean field (MF) approximation, and can only

induce minor corrections to scaling. In particular, in 4D the simple MF asymptotic forms

of the thermodynamical quantities at criticality should be corrected by logarithmic factors,

whose precise structure is also predicted by the RG. In higher dimensions, the dominant

singularities have purely MF forms and the fluctuations can only influence the critical am-

plitudes and the corrections to scaling. These RG predictions entail the “triviality”1,4–7of

the quantum N-component scalar-field theories in d ≥ dc, or, more precisely, the property

that the continuum (scaling) limit of the lattice approximation of the theories (or of the

spin models) describes fields whose connected fourth- and higher-order correlation-functions

vanish and therefore are free or generalized-free.

The main clues of this no-interaction property had been pointed out long ago8, but more

stringent arguments were produced only by the modern developments of the RG theory1–3.

In the same years, a rigorous constructive approach4,7,9–13based on the representation5of

the lattice scalar-field as a gas of polymers, made it possible to prove conclusively that the

continuum Euclidean quantum field theory, built as the scaling limit of a lattice theory

(with the simplest nearest-neighbor discretization of the Laplacian) in the symmetric phase,

is “non-trivial”14,15in d ≤ 3 and “trivial” in d ≥ 5 dimensions.

The rigorous results that exist in 4D (and, in general, for N > 4) are still incomplete,

although they strongly suggest that nevertheless the triviality property still holds. Therefore

some room is left not only to numerical studies, but also to a variety of efforts16–18(and the

related controversies19), aimed either to exploit possible gaps in the arguments, or to relax

some of the hypotheses underlying the constructive approach, in order to make the definition

of a “non-trivial” continuum theory feasible.

For d ≥ 4, the MonteCarlo (MC) simulation approach to the numerical verification of

the RG predictions is not yet completely satisfactory. The detailed exploration of the near-

critical behavior is hampered by the necessity of considering systems of very large sizes,

and in particular, at d = 4, by the difficulty of an accurate characterization of the slowly

varying logarithmic deviations from MF behavior. For d ≥ 4, also the finite-size-scaling

theory and the confluent corrections to scaling have been debated20–25. Thus relatively few

of the numerous available MC studies17,22,26–35are likely to be extensive enough to yield

a satisfactory overall description of these systems at criticality, in spite of the remarkable

progress in the simulation algorithms with reduced critical slowdown.

On the other hand, for these systems high-temperature (HT) expansions have been un-

til now derived only for a small number of observables and are too short36–38, or perhaps

barely adequate39–43to extract reliable information in the critical region. We believe how-

ever, that the HT series methods might bring further insight into this context, provided that

for a conveniently enlarged set of observables, the lengths of the expansions can be signif-

icantly extended. Recently, new stochastic algorithms44–47have shown promise of deriving

extremely long, although approximate, HT expansions valid for finite-size lattice systems.

The application47of these methods also to the triviality problem is particularly interesting.

The traditional graphical36–39,48–54or iterative55–58methods of calculation, although

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severely limited by the fast increase of their combinatorial complexity with the order of

expansion, remain necessary to derive the exact HT series coefficients, valid in the thermo-

dynamical limit, which are needed for a reliable use of the known analytic extrapolation

tools59–61, such as Pad´ e approximants(PA) or differential approximants (DA). It is finally

worth adding that, in the case of high-dimensional models, these exact HT expansions still

seem to be the only practicable method to compute the higher-order field-derivatives of the

free energy at zero field, usually called “higher susceptibilities”.

In this paper, we focus on the HT series approach to provide further numerical evidence

supporting the RG predictions in the case of the N = 1 lattice scalar-field models and

of the Ising spin-s systems. For this purpose, we have computed and analyzed exact HT

expansions of the higher susceptibilities, through order 24, to study their critical behavior

and an important class of universal combinations of critical amplitudes (UCCAs), whose

properties might also be of interest.

The paper is organized as follows.In Section II we define the spin-s Ising and the

lattice scalar-field systems for which we have substantially extended the HT expansions

of the specific free-energy in the presence of a uniform magnetic field.

due reference to the few HT data already in the literature. In Section III, we introduce

the higher susceptibilities and indicate how their expected critical behavior varies with the

lattice dimensionality. In Section IV, we review the definition of the dimensionless 2n-points

renormalized coupling constants in terms of the higher susceptibilities and indicate their role

in the discussion of the RG predictions. Then we introduce several classes of UCCAs related

to the latter quantities. The following Section V is devoted to a detailed numerical analysis of

our HT expansions including discussions of numerical estimates of the critical temperatures,

exponents and several UCCAs for the models under scrutiny. The final Section contains our

conclusions.

Then we make

II.ISING-TYPE MODELS. DEFINITIONS AND NOTATION

In what follows, we shall be concerned only with spin-s Ising or one-component lattice

scalar-fields, so that, unless explicitly needed, it will be convenient to drop the dependence

of the physical quantities on the number N of components of the spin or of the field.

In a bounded region Λ ⊂ Zdof the d-dimensional lattice Zd, the spin-s Ising model

interacting with an external uniform magnetic field H is described by the Hamiltonian48–51

HΛ{s} = −J

s2

?

<ij>

sisj−mH

s

?

i

si

(1)

where si = −s,−s + 1,...,s is the spin variable at the lattice site?i, m is the magnetic

moment of a spin, J is the exchange coupling. Within the region Λ, the first sum extends

over all distinct nearest-neighbor pairs of sites, the second sum over all lattice sites. Clearly,

the conventional Ising model is obtained simply by setting s = 1/2.

The self-interacting one-component scalar-field lattice model in a magnetic field is de-

scribed by the Hamiltonian38,39,51,54

HΛ{φ} = −

?

<ij>

φiφj+

?

i

(V (φi) + Hφi).(2)

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Here −∞ < φi< +∞ is a continuous variable associated to the site?i and V (φi) is an even

polynomial in the variable φi. In this study, for brevity we have only discussed the particular

model in which V (φi) = φ2

requires only simple changes in the computation.

The Gibbs specific free energy F(K,h) is defined as usual by

i+g(φ2

i−1)2, but considering interactions of a more general form

F(K,h) = − lim

|Λ|→∞

1

|Λ|kBTlnZΛ(K,h) = − lim

|Λ|→∞

1

|Λ|kBTln

?

conf

exp[−HΛ/kBT] (3)

Here |Λ| is the volume of the region Λ, K = J/kBT (or K = 1/kBT in the case of the

scalar-field models), called inverse temperature for short, is the HT expansion variable, with

kBthe Boltzmann constant and T the temperature, while h = mH/kBT denotes the reduced

magnetic field.

We have studied the models described by eqs.(1) and (2) on the hyper-simple-cubic (hsc)

and the hyper-body-centered (hbcc) lattices. Following Ref.[49], for d ≥ 4, the hbcc lattices

are defined as those in which the first neighbors qˆj of the siteˆi are such thatˆi −ˆj =

(±1,±1,...,±1). This choice has the technical advantage, decisive for the computations on

high dimensional lattices, that the “lattice free-embedding numbers”, that enter into the

contribution of each graph to the HT expansion, factorize so that they can be expressed as

powers of those referring to the 1D lattice. As a consequence of this drastic simplification,

the computing time of the expansions is independent of the lattice dimensionality, whereas,

in the case of the hsc lattices, it grows exponentially with the dimensionality. We should

also notice that, for d > 2, the coordination number q = 2dof the hbcc lattice is much

larger than the coordination number q = 2d of the hsc lattice and therefore the hbcc lattice

expansions share the advantage of being notably smoother and faster converging than the

hsc ones.

The expansions presented here are based on a calculation of the HT and low-field expan-

sion of the free energy of various models described by the Hamiltonians eqs.(1) and (2), in

presence of an external uniform magnetic field, that we have extended through the order

24. In the case of the conventional Ising model, i.e. the model with spin s = 1/2, such

an expansion, through order 17, was already in the literature62,63in the case of the four-

dimensional hsc lattice (h4sc). Our expansion agrees only up to order 16 with these data

and, as a consequence, with the series coefficients of the ordinary susceptibility χ2(K) and

of the fourth field-derivative of the free energy χ4(K), obtained from them and analyzed in

Ref.[37], as well as in some successive studies. For the conventional Ising model, in addi-

tion to the expansion in the case of the h4sc lattice, we have also computed the analogous

expansion in the case of the hbcc lattice in 4D (h4bcc). For both the h4sc and the h4bcc

lattices, we have moreover computed HT and low-field expansions in the case of the Ising

models with spin s = 1,3/2,...,3 and in the case of the Euclidean one-component scalar-

field lattice models with an even-polynomial self-interaction. We have finally repeated the

series derivation for the same set of models in 5D and 6D, but restricting ourselves to the

five-dimensional hbcc (h5bcc) and the six-dimensional hbcc (h6bcc) lattices, for the reasons

of computational simplification indicated above. All these expansions do not exist in the

literature.

Altogether, we have examined these Ising-type models in 28 cases distinct by spatial

dimensionality, value of the spin and structure of the lattice or of the interaction. In a given

dimension, all these models are expected to belong to the same critical universality class

and therefore to be characterized by the same set of critical exponents and UCCAs.

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Finally, let us also mention that our HT expansions for the Ising models in a magnetic

field can be readily transformed into low-temperature (LT) and high-field expansions, from

which the spontaneous magnetization and the LT higher susceptibilities can be derived.

In our calculation of the HT expansions, we have employed the linked-cluster graphical

method of Ref.[48]. We have used an algorithm of graph generation and series calculation

already described in Ref.[51]. The details of the computer implementation of this procedure,

its validation, and its performance are discussed in the same paper, that was devoted to

a study of the higher susceptibilities and the scaling equation of state for the 3D Ising

universality class. Our extensions of the HT and low field expansions are summarized in

Table I. The set of series coefficients cannot fit into this paper because of its large size and

will be tabulated elsewhere.

A.Available series expansions in zero field

It is appropriate to list here the few HT expansions of the higher susceptibilities for

high-dimensional models at zero field that can already be found in the literature. All of

them are restricted to the conventional spin-1/2 Ising model on the hsc lattices in zero

field. The ordinary susceptibility χ2(K) was derived36through order 11 in dimensions d =

2,..,6. More recently, these calculations were extended38to include, through the the same

order, also χ4(K) and the second moment of the correlation function µ2(K), in d = 2,3,4

dimensions and carried39up to order 14. The expansion of the susceptibility χ2(K) has been

recently pushed43to order 19 on the h4sc and h5sc lattices. An expansion of χ2(K) valid

for any dimension d was computed64through order 15. For χ2(K), χ4(K) and the sixth

field-derivative of the free energy χ6(K), strong coupling expansions through order 11, i.e.

expansions in powers of the second-moment correlation length ξ2(K) = µ2(K)/2dχ2(K),

instead of K, valid for any d, have also been obtained65. Of course, the usual HT expansions

in powers of K can be recovered simply by reverting the appropriate expansion of ξ2(K).

TABLE I: Maximal order in K of the HT and low-field expansions of the free energy for the Ising

and scalar-field models on four-, five- and six-dimensional lattices.

Existing data63This work

17

0

0

0

0

0

0

0

0

h4sc Ising s = 1/2

h4sc Ising s > 1/2

h4sc scalar field

h4bcc Ising s ≥ 1/2

h4bcc scalar field

h5bcc Ising s ≥ 1/2

h5bcc scalar field

h6bcc Ising s ≥ 1/2

h6bcc scalar field

24

24

24

24

24

24

24

24

24

III.THE HT EXPANSIONS OF THE HIGHER SUSCEPTIBILITIES

The assumption of asymptotic scaling66–70for the singular part Fs(τ,h) of the reduced

specific free energy, valid for dimension d ?= dc, when both h and τ approach zero, is usually

expressed in the form

Fs(τ,h) ≈ |τ|2−αY±(h/|τ|βδ). (4)

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where τ = (1 − K/Kc) is the reduced inverse temperature. The functions Y±(w) are

defined for 0 ≤ w ≤ ∞ and the + and − subscripts indicate that different functional

forms are expected to occur for τ < 0 and τ > 0. The exponent α specifies the divergence

of the specific heat, β describes the small τ asymptotic form of the spontaneous specific

magnetization M on the phase boundary (h → 0+,τ < 0)

M ≈ B(−τ)β

(5)

and B denotes the critical amplitude of M. The exponent δ characterizes the small h

asymptotic behavior of the magnetization on the critical isotherm (h ?= 0,τ = 0),

|M| ≈ Bc|h|1/δ

(6)

and Bcis the corresponding critical amplitude. For d ≥ dc, the MF values expected for

the exponents α, β and δ are α = αMF = 0, β = βMF = 1/2 and δ = δMF = 3, while for

the susceptibility exponent we have γ = γMF = 1 and for the correlation-length exponent

ν = νMF = 1/2. The usual scaling laws (but, of course, not the hyperscaling laws) follow

from eq.(4).

From our calculation of the magnetic-field-dependent free energy, we have gained exten-

sions of the existing HT expansions in zero field and, in addition, made available a large

body of data not yet existing in the literature, in particular for the n-spin connected corre-

lation functions at zero wave number and zero field (the “higher susceptibilities”), defined

by the successive field-derivatives of the specific free energy

χn(K) = (∂nF(K,h)/∂hn)h=0=

?

s2,s3,...,sn

< s1s2...sn>c.(7)

in the Ising model case, or by the analogous formula in the scalar field case. For odd values

of n, the quantities χn(K) vanish in the symmetric HT phase, while they are nonvanishing

for all n in the broken-symmetry LT phase.

For even values of n in the symmetric phase, the RG theory predicts that, for d > 4, we

have

χn(τ) ≈ C+

as τ → 0+along the critical isochore (h = 0,τ > 0). In eq.(8), b+

tively, the amplitude and the exponent of the leading confluent correction to the asymptotic

behavior. The explicit expressions obtained in the case of the spherical model71–73suggest

that in 5D one should expect θ = 1/2, whereas, in 6D, θ = 1, with possible multiplicative

logarithmic correction terms.

At the marginal dimension dc, the homogeneity property described by eq.(4) is not strictly

true, because of the expected logarithmic corrections. In this case, for even values of n, in

the symmetric phase, the RG theory predicts for the higher susceptibilities the following

asymptotic behavior

n|τ|−γn(1 + b+

n|τ|θ+ ...).(8)

nand θ denote, respec-

χn(τ) ≈ C+

nτ−γn|ln(τ)|Gn(N)?

1 + O

?

ln(|ln(τ)|)/ln(τ)

??

(9)

in the τ → 0+limit. In both eqs. (8) and (9), one has γn= γMF+ (n − 2)∆MF, with the

gap exponent ∆MF= βMFδMF= 3/2. The general expression for Gn(N) is

Gn(N) = (3

2n − 2)N + 2

N + 8− n/2 + 1(10)

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so that in the N = 1 case, Gn(1) = G = 1/3, independently of n.

Clearly, the usual hyperscaling relation 2∆ = dν +γ, which is valid for d < dc, fails by a

power when d > 4, while it is only logarithmically violated in d = 4.

The simplest consequence of the usual scaling hypothesis eq.(4), which will be tested

using our HT expansions, is that the critical exponents of the successive derivatives of

F(τ,h) with respect to h at zero field, are evenly spaced by the gap exponent ∆MF. Also

in 4D, this property can be simply and accurately checked by a HT analysis of the higher

susceptibilities.

IV.RENORMALIZED COUPLINGS AND RELATED QUANTITIES

It is useful here to recall the definitions of the universal quantities g+

momentum n−spin dimensionless renormalized coupling constants (RCC’s) in the symmetric

phase. They enter into the approximate representations of the scaling equation of state and

moreover play a key role in the discussion of the triviality properties of the d ≥ 4 systems.

They are defined as the critical limit when K → K−

2n, called zero-

cof the expressions

g4(K) = −

v

ξd(K)

χ4(K)

χ2

2(K)

(11)

g6(K) =

v2

ξ2d(K)

?

−χ6(K)

χ3

2(K)+ 10

?χ4(K)

χ2

2(K)

?2?

?χ4(K)

χ2

(12)

g8(K) =

v3

ξ3d(K)

?

−χ8(K)

χ4

2(K)+ 56χ6(K)χ4(K)

χ5

2(K)

− 280

2(K)

?3?

(13)

g10(K) =

v4

ξ4d(K)

?

−χ10(K)

χ5

−4620χ6(K)χ2

2(K)+ 120χ8(K)χ4(K)

χ6

2(K)

+ 126χ2

6(K)

χ6

?4?

2(K)

(14)

4(K)

χ7

2(K)

+ 15400

?χ4(K)

χ2

2(K)

g12(K) =

v5

ξ5d(K)

?

−χ12(K)

χ6

4(K)

2(K)

2(K)+ 220χ10(K)χ4(K)

− 36036χ2

χ8

χ7

2(K)

+ 792χ8(K)χ6(K)

χ7

+ 560560χ6(K)χ3

2(K)

−17160χ8(K)χ2

χ8

6(K)χ4(K)

2(K)

4(K)

χ9

2(K)

−1401400

?χ4(K)

χ2

2(K)

?5?

(15)

and so on. The constant v is a lattice-dependent geometrical factor called the volume

per lattice site74. A longer list of the RCC’s appears in Ref.[51], where the equation of state

is discussed only for the 3D case. For technical reasons, we have not yet extended the HT

expansions of µ2(K) and, correspondingly, of the second-moment correlation length ξ(K),

so that in this paper we shall study only the ratios of RCC’s, for n > 2,

r2n(K) =

g2n(K)

g4(K)n−1

(16)

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which share the computational advantage of being independent of ξ(K). The critical limits

of these ratios are universal quantities that will be denoted by r+

At the upper critical dimension dc, the quantities g2n(K) are expected to vanish like

powers of 1/ln(τ), when τ → 0+. Therefore the continuum limit theory is consistent only

for vanishing renormalized coupling, i.e. it is trivial. We can check numerically that, in the

same limit, the lowest ratios r2n(K) remain finite in 4D. For d ≥ 5, both the g2n(K) and

the r2n(K) vanish in the critical limit like powers of τ, so that the mentioned property of

triviality is also true for d > dc.

Briefly recalling more detailed discussions3,51,75, we can also observe that, for d > 4, in

the small magnetization region, where the reduced magnetic field h(M,τ) has a convergent

expansion in odd powers of the magnetization M, the critical equation of state can be written

in terms of an appropriate variable z ∝ Mτ−βas

2n.

h(M,τ) =¯h|τ|βδF(z)(17)

where¯h is a constant and F(z) is normalized by the equation F′(0) = 1. In general, the

small z expansion of F(z) can be written as

F(z) = z +1

3!z3+r+

6

5!z5+r+

8

7!z7+ ...(18)

In the MF approximation, all r+

At the upper critical dimension, the following form of the critical universal equation of

state for an N-component system is obtained2,3from the RG

2nvanish, and F(z) reduces to FMF(z) = z +1

3!z3.

H ∝

?

Mτ|lnM|(N+2)/(N+8)+

1

(N + 8)

M3

|lnM|

?

(1 +const.

|lnM|)(19)

valid for small M and H. From eq.(19) the general formula eq.(9) and the expression (10)

for Gn(N) can be deduced.

In terms of the higher susceptibilities, the simple sequence of quantities was defined76

long ago

I2r+4(K) =χr

2(K)χ2r+4(K)

χr+1

4

(K)

(20)

with r ≥ 1. The finite and universal critical values

I+

2r+4=(C+

2)rC+

(C+

2r+4

4)r+1

(21)

of the functions I2r+4(K) in the limit K → K−

in the literature.

Together with the sequence I+

the critical limits of the functions

c, include some of the UCCAs first described

2r+4of UCCAs, the sequences A+

2r+4and B+

2r+8, obtained as

A2r+4(K) =χ2r(K)χ2r+4(K)

(χ2r+2(K))2

(22)

B2r+8(K) =χ2r(K)χ2r+8(K)

(χ2r+4(K))2

(23)

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with r ≥ 1, were also defined in Ref.[76].

In 4D the general formula eq.(10) for Gn(N) implies that the powers of the logarithms that

enter into the leading critical singularities eq.(9) cancel in the quantities I2r+4(K), A2r+4(K),

and B2r+8(K) at the critical limit. Conversely, eq.(10) can also be obtained recursively from

the knowledge of only G2(N) and G4(N) by requesting that such a cancellation occurs.

The ratios r2n(K) can be simply expressed in terms of the functions I2r+4(K). For

example

r6(K) =g6(K)

g4(K)2= −I6(K) + 10

r8(K) =g8(K)

g4(K)3= I8(K) − 56I6(K) + 280

and so on. Taking the K → K−

the quantities r2n(K) vanish as K → K−

critical valuesˆI+

ˆI+

values of the first few terms of the sequences A+

ˆ A+

In the next Section, we shall study numerically the first few terms of the sequences I+

A+

(24)

(25)

c limit in the eqs.(24), (25), etc.

c in the MF approximation2, the corresponding

2r+4of the quantities in eq.(20) can be simply evaluated, obtainingˆI+

10= 15400,ˆI+

2r+4and B+

10= 55/28ˆB+

and observing that

6= 10,

8= 280,ˆI+

12= 1401400, etc. It is also not difficult71to compute the MF

2r+8. For example:

ˆ A+

8= 14/5,

10= 154, andˆB+

12= 143/8.

2r+4,

2r+4and B+

2r+8and observe that they share similar properties.

V.RESULTS OF THE SERIES ANALYSIS

We address the reader to Refs.[50,51], for a more detailed description of the numeri-

cal approximation techniques necessary to estimate the critical parameters in the models

under study, i.e. the locations of the critical points, their exponents of divergence and

the critical amplitudes for the various susceptibilities. We shall employ the DA method, a

generalization59–61of the elementary PA method to resum the HT expansions nearby the

border of their convergence disks. This technique consists in estimating the values of the

finite quantities or the parameters of the singularities of the expansions of the singular quan-

tities from the solution, called differential approximant, of an initial value problem for an

ordinary linear inhomogeneous differential equation of the first- or of a higher-order. The

equation has polynomial coefficients defined in such a way that the series expansion of its

solution equals, up to an appropriate order, the series under study. In addition to this tech-

nique, we shall also use a smooth and faster converging modification61,77of the traditional

method of extrapolation of the coefficient-ratio sequence, sometimes called modified-ratio

approximant (MRA) method, to determine the location and the exponents of critical points.

Using PA or DA approximants, one can achieve, in some cases, evaluations of the param-

eters which are unbiased, i.e. obtained without using independent estimates of the critical

temperature in the construction of the approximants. In some cases however, accurate es-

timates of the critical inverse temperatures are necessary to bias the determination of the

critical exponents and amplitudes. As a general comment on the uncertainties of the es-

timates obtained by these methods, we have to observe that, inevitably, they are rather

subjective. Therefore, we should be very cautious, compare the estimates obtained from

the different approximation methods and also check how effectively our tools perform when

applied to artificial model functions having the expected singularity structure. In our DA cal-

culations, the uncertainties are taken as a small multiple of the spread among the estimates

9

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TABLE II: Our estimates of the critical inverse temperatures of the spin-s Ising models for various

lattices and several values of the spin in 4D, in 5D and in 6D.

lattice

h4sc

h4bcc 0.0690114(8) 0.101165(2)

h5bcc 0.0326478(3) 0.0484554(3) 0.0579714(3) 0.0643290(3) 0.0688769(3) 0.0722915(3)

h6bcc 0.0159390(2) 0.0237914(2) 0.0285102(2) 0.0316592(2) 0.0339099(2) 0.0355989(2)

s = 1/2

0.149693(3)

s = 1s = 3/2

0.255641(3)

0.120592(2)

s = 2s = 5/2

0.301919(3)

0.142930(2)

s = 3

0.215597(3)0.282568(3)

0.133605(2)

0.316497(3)

0.149942(2)

TABLE III: Our estimates of the critical inverse temperatures of the scalar-field models for various

lattices and several values of the quartic self-coupling in 4D, in 5D and in 6D.

lattice

h4sc

h4bcc 0.136451(2)

h5bcc 0.0665777(2) 0.0657078(2) 0.0625961(2) 0.0604114(2) 0.0582806(2) 0.0562586(2)

h6bcc 0.0329566(1) 0.0324976(1) 0.0308921(1) 0.0297796(1) 0.0287008(1) 0.0276811(1)

g = 0.5

0.283025(3)

g = 0.6

0.280704(3)

0.134940(2)

g = 0.9

0.270597(3)

0.129177(2)

g = 1.1

0.262806(3)

0.124991(2)

g = 1.3

0.254915(3)

0.120852(2)

g = 1.5

0.247233(3)

0.116885(2)

obtained from an appropriate sample of the highest-order approximants i.e. those using

most or all available expansion coefficients. Similarly, in the case of the MRAs, the error

bars will be defined as a small multiple of the uncertainty of an appropriate extrapolation50

of the highest-order approximants.

A.The critical inverse temperatures of the models

If in 4D, as predicted by the RG theory, logarithmic factors modify the structure of the

leading critical singularities and also appear in the corrections to scaling, as described by

eq.(9), we should expect that the numerical procedures mentioned above might suffer from

a slower convergence than in the case of pure power-law scaling. For the determination of

the critical temperatures, different approximation methods such as PAs, DAs and extrapo-

lated MRAs have been used to study the expansions of the ordinary susceptibility χ2(K),

the quantity which generally shows the fastest convergence. Independently of the lattice

type and dimensionality, our best estimates of the critical inverse temperatures for the sys-

tems under study are obtained extrapolating to large order r of expansion, a few (from

four to seven) highest order terms of the MRA sequences of estimates (Kc)rof the critical

inverse temperatures. To perform the extrapolation, we rely on the validity50of the simple

asymptotic form

Γ(γ)

Γ(γ − θ)

(Kc)r= Kc−

θ2(1 − θ)b2

r1+θ

+ o(

1

r1+θ). (26)

In general, θ and b2 indicate respectively the exponent and the amplitude of the leading

confluent correction to scaling of χ2(K).

In the 4D case, in which the asymptotic critical behavior of χ2(K) is described by eq.(9),

we can take θ = 0. Therefore the second term on the right-hand side of eq.(26) vanishes

and it must be replaced by a higher-order term depending on the exponent of the next-to-

leading correction to scaling in eq.(9). A similar argument applies in the 6D case in which we

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expect θ = 1. In the 5D case, in which we expect θ = 1/2, the coefficient of 1/r1+θin eq.(26)

also appears to be negligible, so that the situation is similar to that of the 4D and the 6D

cases. Since we do not know the values of the exponents of the next-to-leading correction

to scaling, the simplest procedure of extrapolation might consist in assuming an asymptotic

form (Kc)r= Kc+ w/r1+ǫand in determining Kc, w, and ǫ by a best fit to our data. We

obtain the values ǫ = 0.6(2) in 4D, ǫ = 1.1(2) in 5D and ǫ = 1.5(2) in 6D. These estimates

are compatible with our previous remarks, indicating that the asymptotic behavior of eq.(26)

is determined by the next-to-leading rather than the leading correction to scaling. At the

same time, as suggested by M.E. Fisher78, the expectations71–73concerning the exponent

of the leading corrections to scaling, whose amplitudes are probably not negligible in spite

of the fact that they are not seen by the MRAs, can be essentially confirmed studying by

DAs the critical behavior of quantities like I2r+4(K), A2r+4(K), B2r+8(K) etc. and of their

derivatives. As above remarked, in these quantities the dominant critical singularities cancel,

while the leading corrections to scaling should survive and could be detected by DAs. In

particular, a study of the derivatives of I6(K) and I8(K), for the spin-s Ising models, leads

to the values θ = 0.25(10) in 4D, θ = 0.45(10) in 5D and θ = 0.95(10), in very reasonable

agreement with the predictions71–73.

Our final results for the critical inverse temperatures of some spin-s Ising and scalar-field

models are collected in the Tables II and III. In the 4D case, we have attached particu-

larly generous error bars to our estimates. In d > 4 dimensions, no logarithmic factors are

expected to modify the leading MF behavior of the physical quantities, so that our approx-

imation tools are likely to yield estimates of a higher accuracy, which moreover appear to

improve with increasing lattice dimensionality, both because of the decreasing size of the

corrections to scaling and of the increasing lattice coordination number. All these results

are confirmed also by the analyses employing DAs.

Only in the case of the Ising model with spin s = 1/2 on the h4sc lattice, we can compare

our estimates with those obtained in other studies by extrapolation of shorter HT series.

In Ref.[42] the estimate Kc = 0.149696(4) was obtained from a series of order 17, while

in Ref.[43] the result Kc = 0.149691(3) was derived from a series of order 19. As far as

the most recent large-scale MC simulations are concerned, the estimate Kc= 0.149697(2)

was obtained in Ref.[42], the value Kc = 0.149697(2) in Ref.[34], while the value Kc =

0.1496947(5) is reported in Ref.[35]. Our result in Table II is fully consistent with the older

estimates. No comparison is possible either for higher values of the spin on the h4sc lattice,

or for any value of the spin on the h4bcc lattice, since no studies are available for these

systems. In the case of the higher dimensional lattices our analysis includes only the h5bcc

and h6bcc lattices, which have not been studied elsewhere until now.

B.The logarithmic corrections in 4D

Also in the computation of the critical exponents, it is convenient to distinguish the 4D

case from the higher dimensional ones.

In 4D, when computing the exponent γ of χ2(K) by PAs or DAs, we obtain estimates

very near to, but slightly larger than unity. These estimates should then be regarded as

the values of “effective exponents” which reflect the presence of a small correction to the

leading classical behavior (and of subleading corrections). If we assume that the leading

correction to MF behavior has the multiplicative logarithmic structure predicted by the

RG, we can resort to a variety of procedures proposed40,61,73,79in the literature to isolate

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Page 12

the logarithmic factor from the main power behavior and to measure its exponent. These

prescriptions generally amount to cancel out the main power-singularity in favor of the weak

logarithmic one and therefore they need to be biased with an estimate of the inverse critical

temperature, to which, in turn, the values obtained for the exponent of the logarithm are

very sensitive.

For example, in the case of the ordinary susceptibility χ2(K), one might study the aux-

iliary function l(K;˜Kc) defined by

l(K;˜Kc) = −(˜Kc− K)ln(˜Kc− K)d

dKln[(˜Kc− K)χ2(K)](27)

where˜Kc is some accurate approximation of the true Kc. By eq.(9), l(K;Kc) = G +

O(ln|lnτ|), i.e. it yields the value of the exponent G when K → K−

˜Kc enters as a parameter into the definition of this biased indicator, we should consider

how the estimate G(˜Kc) of the exponent depends on the choice of˜Kcin a small vicinity of

our best estimate of the critical inverse temperature reported in Tabs.II or III. As a typical

example, we show in Fig.1 the plots of G(˜Kc) vs˜Kc(normalized to our MRA estimate of

Kc), computed by PAs of various orders, in the case of the Ising model with spin s = 1 on

the h4bcc lattice. It is reasonable to expect that the value of G(˜Kc) should depend slowly

on˜Kcnear the exact value of the critical inverse temperature so that its best value might

perhaps correspond to a stationary point. We observe that, for most PAs of G(˜Kc), such

a point does indeed exist and also that the curves obtained by various PAs touch nearby

this point, which is generally not much different from our best estimate of Kcas reported

in our Tables. In the literature40,61,73,79, the value of G(˜Kc) at the point where the various

curves touch, is generally taken as the most accurate estimate of the exponent G. However,

this choice may be questioned, since the result appears to be insensitive to the order of

approximation. As shown in Fig.1, if we take the value of G(˜Kc) at the stationary point

as the best approximation, the estimates are also close to the expected value G = 1/3.

Unfortunately, also the choice of the stationary value as the best approximation is open to

doubt, since in this case the successive approximations seem to worsen as the order of the

series increases. We must moreover mention that, in the h4bcc Ising system, the values of

G computed in this way, range between ≈ 0.4 and ≈ 0.3, as the spin varies from s = 1/2

to s = 3. Finally, it is also unclear how to estimate the uncertainties involved in these

procedures and thus how to interpret the spread of exponent estimates, which might be

related to a strong spin-dependence of the slowly decaying corrections appearing in eq.(27).

Other prescriptions to study the exponent of the logarithmic corrections, do not lead to

better results.

cand˜Kc= Kc. Since

C. The critical exponents of the higher susceptibilities

For each model under study, we have computed the exponent γ of the susceptibility by

second- or third-order DAs biased with our estimate of the inverse critical temperature,

namely by resorting to the standard prescription of imposing that the approximants are

singular at the values of Kc reported in our tables II and III and then computing the

exponents. For d > 4, it is rigorously proved9,10that the systems must exhibit a MF critical

behavior (with non trivial subleading corrections). Let us discuss first how our numerical

tools perform in the 5D and 6D cases. We shall then argue that the differences between the

features of this computation and those of the 4D case can be simply ascribed to the expected

12

Page 13

presence of a multiplicative logarithmic correction to the dominant MF power behavior. In

Fig.2, we have plotted our estimates of the exponent of the ordinary susceptibility vs the spin

in the case of various spin-s Ising models for d = 4,5,6. For d > dc, our estimates reproduce

to a very good accuracy the expected value γMF= 1, so that the small deviations from this

value can be safely viewed as only the residual effects of the confluent corrections to scaling.

These deviations also show the expected decreasing size as the dimensionality of the system

increases. Moreover the critical universality, i.e. the independence of the exponents on the

interaction structure, is well verified. On the contrary, at the upper critical dimension our

calculations yield “effective” exponents larger than unity by ≈ 3%. We can interpret this

result as an indication that the leading critical singularity of the susceptibility is slightly

stronger than a pure MF singularity so that it might indeed contain the logarithmic factor

predicted by the RG, which is detected by the DAs as a power-like factor with a very small

exponent. This is confirmed by observing that, if the expected logarithmic singularity is

canceled by dividing out from the susceptibility the ln(1 − K/Kc)1/3correction factor, the

resulting estimate of the exponent γ generally gets within ≈ 0.5% of the MF value. Thus

the deviations are reduced to a smaller size and become compatible with the effects of the

corrections to scaling.

Very accurate estimates can be obtained also for the differences Dnbetween the exponents

of χ2n(K) and χ2n−2(K)

Dn= γ2n− γ2n−2= 2∆MF= 3 (28)

They can be computed from the ratios χ2n(K)/χ2n−2(K) by second- or third-order DAs

biased with the critical inverse temperature. In the 4D case, we should not expect any effects

from the logarithmic factors appearing in the leading singular behavior eq.(9) of the higher

susceptibilities, since these factors cancel in the above indicated ratios. Instead of the results

of the biased prescription, we prefer to show here the estimates from a computation by the

unbiased “critical point renormalization” method59. This procedure consists in determining

the difference Dnof eq.(28) from the exponent of the singularity in x = 1 of the series?arxr

with coefficients ar= c2n

r

, where cs

The biased DA calculation of the Dn, mentioned above, gives quite comparable results, so

that it is not necessary to report the corresponding figures.

The quantities Dnwith n = 2,3,...11, obtained by the unbiased method in the case of

the the scalar-field model on the h5bcc and h6bcc lattices, for several values of the coupling

g, are plotted vs n in Fig.3. The same computations for the spin-s Ising models with various

values of the spin on the h5bcc and h6bcc lattices yield completely similar results and

therefore we do not report the corresponding figure. Our estimates of Dnagree, generally

within 0.1%, with the expected value 2∆MF = 3. Thus the small size of these deviations

from the MF value suggests that they can safely be related with the confluent corrections to

scaling. The critical universality is also well verified. On the other hand, our results in the

4D case reported in Fig.4 in the case of the scalar-field model on the h4bcc lattice, those

reported in Fig.5 for the Ising model on the h4bcc lattice and those of Fig.6 for the same

system in the case of the h4sc lattice, show relative deviations from the value of 2∆MF, five

times larger than those in d > 4 dimensions (i.e. of the order of 0.5%), but still sufficiently

small to reflect only the residual influence of the expected subleading logarithmic corrections

to the critical behavior.

r/c2n−2

ris the r-th coefficient of the expansion of χs(K).

13

Page 14

D.Universal combinations of critical amplitudes

For d > 4, using second- or third-order DAs, the first few terms of the sequence of the

UCCAs I+

K = K−

we have introduced the ratios of these quantities to their MF values, and denoted them by

Q2r+4, R2r+4 and S2r+8, respectively. In Fig.7, we have reported our estimates for the

ratios Q6, Q8, Q10and Q12vs the value s of the spin for Ising models on the h5bcc and

h6bcc lattices. In complete agreement with the proven MF nature of the critical behavior,

these ratios generally equal unity, within the accuracy expected from our approximations

that, in this case, allows not only for the influence of the confluent corrections to scaling,

but also for the uncertainties in the estimates of the critical temperatures needed to bias

the calculations. Correspondingly, the critical limits of the ratios r2n(K) vanish and the

equation of state takes the MF form. Quite similar results are shown in Fig.8 for the other

normalized UCCAs R+

with various values of the spin in the case of the h5bcc and h6bcc lattices.

Also in the 4D case, as shown in Fig.9 for the first few UCCAs defined by eq.(21) in the

case of the scalar-field model, and in Fig.10 for a few UCCAs defined by eqs.(22) and (23)

in the case of the spin-s Ising model, the various quantities probably have been evaluated

with reasonable accuracy, because the logarithmic factors, expected to appear in the leading

critical behavior of the higher susceptibilities, cancel in the ratios defining the UCCAs.

As a consequence, the uncertainties in the critical temperatures and the influence of the

corrections to scaling should still be considered as the main sources of error. However, we

observe that generally the first few ratios Q2r+4, R2r+4and S2r+8are slightly, but definitely

smaller than unity. We can imagine two possible explanations of this result: either the

deviations from unity have to be related only to (unlikely) residual effects of the logarithms

in the leading and subleading behavior of the higher susceptibilities, or the UCCAs are

accurately estimated and they really do not take their MF values. Whatever the case, it is

clear that also these data on the UCCAs confirm that, consistently with the RG predictions,

the critical behavior in 4D is not MF-like.

2r+4, A+

cthe estimates of the functions I2r+4(K), A2r+4(K) and B2r+8(K). For convenience,

2r+4and B+

2r+8can be evaluated to a good accuracy, by extrapolating to

8, R+

10, S+

10and S+

12which are plotted vs the spin s for Ising models

VI.CONCLUSIONS

By analyzing our HT expansions of the zero-field higher-susceptibilities, extended through

order 24, in the case of the N = 1 lattice scalar-field models and of the spin-s Ising systems,

we have provided further numerical evidence consistent with the critical behavior predicted

by the RG in this class of models.

We have estimated the critical exponents of the ordinary and the higher susceptibilities

and the values of a class of universal combinations of their critical amplitudes, which de-

termine the form of the critical equation of state and are presently inaccessible by other

computational methods. In 4D, the results of our analysis suggest that, within a good

approximation, the critical exponents and this class of UCCAs, show small, but definitely

nonvanishing deviations from their values in the MF approximation. For the UCCAs, this

fact had been already predicted long ago also within the RG formalism, by showing80that,

at the upper critical dimension, at least one of the quantities in the above mentioned class

does not take the MF value. More generally, in 4D the deviations from the MF critical

behavior are compatible with the small effects associated to the logarithmic corrections pre-

14

Page 15

dicted by the RG. Our direct numerical checks concerning in particular the exponents of the

logarithmic corrections to the dominant power behavior of the higher susceptibilities have

only a rather limited accuracy, due to the modest sensitivity of the DAs to the logarithmic

singularities, either in the leading behaviors and in the confluent corrections.

Quite on the contrary, the same kind of analysis performed on five- and six-dimensional

lattices, shows no numerical evidence of deviations from the leading classical behavior by an

extent larger than the expected numerical uncertainties: both the exponents and the UCCAs

appear to take the MF values within a high approximation, so that the RG predictions

concerning the triviality property are rather convincingly confirmed.

VII.ACKNOWLEDGEMENTS

We thank Ian Campbell, Michael E. Fisher, Ralph Kenna and Ulli Wolff for their patience

in reading and commenting a preliminary draft of this paper. The hospitality and support

of the Physics Depts. of Milano-Bicocca University and of Milano University are gratefully

acknowledged. Partial support by the MIUR is also acknowledged.

∗Electronic address: paolo.butera@mib.infn.it

†Electronic address: mario.pernici@mi.infn.it

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