Critical Behavior of Systems with Complex Symmetries

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Corso di Perfezionamento in Fisica

Critical Behavior of Systems with

Complex Symmetries

Supervisor

Candidate

Prof. Ettore Vicari

Pasquale Calabrese

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Introduction

A phase transition is defined as a singularity in thermodynamic quantities of a system. Phase

transitions are classified, according to Ehrenfest, as first order, second order . . . , or first

order and continuous. The starting point of the classification is the rigorous result that

the free energy is always a continuous function (e.g. in the grancanonical ensemble) of the

temperature and chemical potential. But it may be nonanalytic. If one of its first derivatives

is discontinuous (internal energy, magnetization, volume, . . . ) the transition is first order; if

one of the second derivatives (specific heat, susceptibility, . . . ) is discontinuous the transition

is second order, and so on. Defining via singularities is the most general way of characterizing

a phase transition. For a large class of systems, singularities could occur due to ordering

after a phase transition (symmetry breaking), but this is not necessarily a requirement for

all transitions. In other words, the existence of an order parameter is not a prerequisite for

understanding phase transitions though it might be very useful in many contexts.

The behavior of a system close to a continuous phase transition is distinctly different from

the behavior far from it. This difference is so important that we refer to a continuous phase

transition as a critical phenomenon. Power laws are distinctive features of critical phenomena

and can be ascribed to diverging length-scale. This connection is now so well-characterized

that any phenomena, equilibrium or not equilibrium, thermal or non thermal, showing power

laws tend to be interpreted in the same manner as equilibrium critical phenomena through

the identification of relevant terms, scaling and diverging lengths. The exponents describing

the rate of divergence of physical quantities are named critical exponents and their proper

determination has been attracting generations of physicists in the last thirty years. Nowadays

there is a special dictionary for these exponents. For instance, in the case of a thermal

phase transition, whose driving parameter is the temperature, the exponent of specific heat

is α (C = A

±

|t|

α

, where t = T − T

c

), the one of magnetization (using the common magnetic

language) in the ordered phase β (M ∝ (−t)

β

), γ for the susceptibility χ = ∂M/∂h|

h=0

∝ t

−γ

.

The behavior of the diverging scaling-length at the critical point is fundamental, and it is

described by the exponent ν, i.e. ξ ∝ |t|

−ν

_{. At the critical point the correlation length}

diverges and the two point function has a power law behavior governed by the exponent η:

hM (x)M (0)i = |x|

2−d−η

_.

One of the most important contribution to the understanding of the critical phenomena

was the introduction of Renormalization Group (RG) ideas, which allowed the shift in point of

view to a classification of the terms of a Hamiltonian as relevant or irrelevant, rather than refer

to them as numerically weak or strong. It is not my aim here to give a complete overview of a

so long and complicated task, in this Introduction I only want to fix few general ideas behind

RG and I remand for an exhausting treatment to the textbooks quoted in the bibliography.

Close to the critical point the only important length scale is the correlation length, which

diverges at the critical point, leaving the system without a length scale, i.e. scale invariant.

Let us consider explicitly an Hamiltonian system. The “appropriate scale transformation”,

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called RG transformation, is a mapping in the abstract space of Hamiltonians H

−→ H

G 0

(on

a practical model this transformation may be done in several way, e.g. by a block

trans-formation in real space, by an integration on the fast modes in momentum space . . . ). In

particular the scale invariant Hamiltonians are those that do not change under this

transfor-mation GH

∗

_{= H}

∗

_{. Close to these fixed points the RG transformation may be linearized,}

leading to a linear operator L which possess an (infinite) number of eigenvalues y

i

and (left or

right) eigenoperators O

i

. A general Hamiltonian, close to the fixed point, may be expanded

as (l is a scale parameter)

H = H

∗

+

X

µ

i

O

i

,

where µ

i

satisfy

dµ

_i

dl

= y

i

µ

i

.

If y

i

has a positive real part the (even small) perturbation µ

i

grows up and the flow goes far

from the fixed point. Such terms are called relevant. If instead y

_i

has a negative real part,

the perturbation decays with the iteration of RG transformations and finally the flow reaches

the fixed point. These operator are called irrelevant. If y

i

= 0 the operator is called marginal

and the fate of the Hamiltonian will depend on the higher order terms in the RG equations.

From these abstract RG equations a fundamental feature of critical phenomena emerges:

Universality. Independently from the Hamiltonian (and, even more, independently from what

it describes), all the systems in the domain of attraction of a certain H

∗

_{have the same critical}

behavior. Usually at a thermal phase transition, one has to tune only two relevant

parame-ters to reach the critical point: the temperature to T

c

and the magnetic field h to 0 (more

generally for nonmagnetic systems h is the variable conjugated to the order parameter). Thus

we can divide all the critical phenomena in a relative small number of universality classes,

characterized by the dimensions of the space, the symmetry of the system (that reduces the

number of operators one may consider), and the range of interactions.

Finally predictions may be given writing the flow equation for the free energy. Close to a

fixed point the singular part of free-energy is

F

s

(h, t, . . . µ

i

. . . ) ∝ |t|

d/yt

F

µ

h

|t|

yh/yt

, . . . ,

µ

i

|t|

yi/yt

¶

,

from which we can derive the standard exponents

α = 2 − d/y

_t

,

β = (1 − y

_h

)d/y

t

,

γ = −(1 − 2y

h

)d/y

t

,

ν = 1/y

t

,

η = d + 2 − 2dy

h

.

Thus the knowledge of RG eigenvalues leads to the critical exponents. From these relations

follow the well-known scaling laws

α + 2β + γ = 2 ,

γ = ν(2 − η) ,

dν = 2 − α .

The aim of this thesis is the characterization of some universality classes having rather

complicated symmetries and describing phase transitions in models with anisotropies,

ran-domness, frustration and competing order parameters. To obtain qualitative and quantitative

predictions we calculated and analyzed higher order perturbative expansions up to six or five

loops.

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Multicoupling Landau Ginzburg

Wilson Hamiltonian

In this introductory chapter we define the main subject of this thesis: The

Mul-ticoupling Landau-Ginzburg-Wilson Hamiltonians. In particular we will

empha-size on how perturbative field-theoretical methods, in conjunction with refined

resummation techniques, allow one to obtain very precise theoretical predictions

for universal quantities close to a continuous phase transition. We introduce the

perturbative approach in fixed dimension and the minimal subtraction

renormal-ization scheme with and without ² expansion. The application of these techniques

to systems with complex symmetries will be the main goal of this thesis. We end

the chapter with an overview of alternative theoretical methods to describe the

critical point, with which the perturbative methods must be confronted before of

the comparison with experiments.

A statistical model describing a second order phase transition is usually defined by a lattice

short-range Hamiltonian involving finite length spins. Typical examples are the O(N ) models

H = −

X

hiji

J

ij

~s

i

· ~s

j

,

(1.1)

where the sum is over next-neighbor and ~s

i

are N -component spin of unit length at the lattice

site i. We can replace fixed length spins with variable of unconstrained length ~

φ

i

by means of

a Hubbard-Stratonovich (HS) transformation

1

_{. Universality suggests that microscopic details}

are not essential, so we can take the formal limit of zero lattice spacing, arriving to an

Hamil-tonian depending on an N -component field φ

i

(x) (now i labels the components). Expanding

in power of φ

_i

one obtains a very complex object containing (at least in principle) all the

powers of the field, all the nth derivatives, and the products of all of them. Such Hamiltonian

is intractable. Fortunately not all these terms are relevant in the RG sense. Indeed by power

counting, one realizes that close to four dimensions the relevant terms in the Hamiltonian,

called Landau-Ginzburg-Wilson (LGW), are only the powers of the field up to the fourth

or-der (φ

4

_{terms) and the kinetic term (∂}

µ

φ

i

)(∂

ν

φ

j

). The φ

4

terms of the Hamiltonian may be

1_{The HS transformation for a general spin model on an arbitrary lattice is the Gaussian integral}

ePijPαβKαβij sαisβj _∝ Z dφαie 1 4 P ijPαβφαi(K−1)αβij φβj+ P i,αφαisαi _.

After this transformation, the spin variables decouple and it is straightforward to sum over them, ending in an effective Hamiltonian for unconstrained modes φα

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derived by the mapping into the lattice Hamiltonian. But this is not always needed. In fact

one can guess them requiring the most general form compatible with the symmetries of the

lattice Hamiltonian.

For an N -component order parameter φ

i

(x), the most general LGW Hamiltonian can be

written as

H =

Z

d

x

h 1

2 X

i

(∂

µ

φ

i

(x))

2

+

1 ₂

X

ij

r

ij

φ

i

(x)φ

j

(x) +

_4!

1 X

ijkl

u

ijkl

φ

i

(x)φ

j

(x)φ

k

(x)φ

l

(x)

i

, (1.2)

where the number of independent parameters r

_ij

and u

_ijkl

depends on the symmetry group

of the theory. Let us consider the O(N )-symmetric Hamiltonian, the introduced parameters

must have the form

u

_abcd

=

g

3 (δ

ab

δ

cd

+ δ

ac

δ

bd

+ δ

ad

δ

bc

) ,

r

ij

= δ

ij

r

0

.

(1.3)

Similarly, if we restrict the symmetry to the cubic one (given by the reflections and

permuta-tions of the field components) we have

u

abcd

=

g

₃

(δ

ab

δ

cd

+ δ

ac

δ

bd

+ δ

ad

δ

bc

) + vδ

ab

δ

cd

δ

ad

,

r

ij

= δ

ij

r

0

.

(1.4)

An interesting class of models is characterized by the fact that

P

_i

φ

2

i

is the only quadratic

polynomial that is invariant under the symmetry group of the theory. In this case, r

ij

is a

multiple of the identity, i.e. r

ij

= δ

ij

r

0

. Moreover, u

ijkl

must satisfy the trace condition

X

i

u

iikl

∝ δ

kl

,

(1.5)

in order to ensure that additional quadratic invariants are not generated by the RG

transfor-mations [61]. In these models, criticality is driven by tuning the single parameter r

0

, which in

a thermal phase transition corresponds to the temperature, i.e. r

0

= T − T

c

.

As clear from above, in the absence of a sufficiently large symmetry restricting the form

of the potential, many quartic couplings must be introduced and the study of the critical

behavior may become quite complicated.

1.1 Mean-Field Theory

A first simplified analysis that leads to a qualitative description of the phase diagram of

the models described by Eq. (1.2) is given by mean-field theory. In this approximation one

neglects fluctuations of the order parameter and considers it as uniform in the space and equal

to its mean value φ

i

(x) = hφ

i

i. The ground-state is the configuration that minimizes H. To

make solvable this minimum problem, the Hamiltonian must be bounded, which corresponds

to the fact that the “coupling matrix” u

_ijkl

is positive (in the sense that u

_ijkl

v

_i

v

_j

v

_k

v

_l

is always

positive for a generic choice of the v

i

’s). Assuming the positivity of u

ijkl

and r

ij

= r

0

δ

ij

, we

have that for r

0

≥ 0 the quartic form is minimized by hφ

i

i = 0. For r

0

< 0 several degenerate

minima appear. These satisfy the equations

r

₀

hφ

_i

i +

1 3!

u

ijkl

hφ

j

ihφ

k

ihφ

l

i = 0

with i = 1, . . . , N .

(1.6)

The solution is given by hφ

i

i = h|φ|iv

i

, where v

i

are unit vectors that minimize u

ijkl

v

i

v

j

v

k

v

l

.

In terms of these vectors the mean-field solution for r

₀

< 0 is

hφi

2

= −

6r

0

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The vanishing of the order parameter with r

0

→ 0, implies a second order phase transition with

β = 1/2. Calculating other thermodynamic quantities, one obtains the mean-field (sometimes

called classical) critical exponents ν = 1/2, η = 0, and γ = 1.

Note that when u

_ijkl

is not positive, Eq. (1.7) does not make sense. This is connected with

the fact that the Hamiltonian is not bounded. In such case we are not allowed to drop higher

order terms in φ in the LGW Hamiltonian, since they are responsible of the stability. Taking

into account these higher order contributions the transition is altered. The more relevant

term to be added is φ

6

. Let us consider for simplicity the one-component case, the mean-field

Hamiltonian is

H =

1

2 r

0

φ

2

₊

1 4!

uφ

4

₊

1 6!

gφ

6

_,

_(1.8)

where we assume g > 0 (elsewhere we continue in the expansion in powers of φ until we find a

positive term). When r

0

> 0, hφi = 0 is a minimum, but it is not always the lower one. The

result is a first order transition (a part from some special values of the couplings leading to a

tricritical transition).

1.2 RG approach

The mean-field theory gives exact values for the various critical exponents in a large enough

number of spatial dimensions, but it generally fails sufficiently close to the critical point in

physical dimensions, where the effect of fluctuations is important. The Ginzburg criterion leads

to the conclusion that for φ

4

_{models, fluctuations are important in less than four dimensions}

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US

G

H

S

v

u

Figure 1.1:

A possible RG flow for a theory with two couplings u and v .

1.3.1 The fixed-dimension expansion

In the fixed-dimension expansion one works directly in d = 3 or d = 2. In this case the theory

is super-renormalizable since the number of primitively divergent diagrams is finite (three in

three dimensions and only one in two dimensions). One may regularize the corresponding

integrals by keeping d arbitrary and performing an expansion in ² = 3 − d or ² = 2 − d. Poles

in ² appear in divergent diagrams. Such divergences are related to the necessity of performing

a renormalization of the parameter r

0

appearing in the bare Hamiltonian. This problem can

be avoided by replacing r with the mass m defined by

m

−2

=

1 Γ

(2)

₍₀₎

∂Γ

(2)

_(p

2

₎

∂p

2

¯

p2₌₀

,

(1.9)

where the function Γ

(2)

_(p

2

_{) is related to the one-particle irreducible two-point function by}

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where Γ

(1,2)

_{is the one-particle irreducible two-point function with an insertion of 1/2}

P

i

φ

2i

.

Since the renormalization is performed on zero-momentum correlation functions in the massive

theory, we will refer to this approach as Massive Zero-Momentum (MZM) scheme. From the

perturbative expansions of the correlation functions Γ

(2)

_{, Γ}

(4)

_{, and Γ}

(1,2)

_{, one derives the}

expansion of the RG functions we are going to define.

The FP’s of the theory are given by the common zeros g

∗_abcd

of the β-functions

β

ijkl

(g

abcd

) = m

∂g

_∂m

ijkl

¯

uabcd

.

(1.14)

In the case of a continuous transition, when m → 0, the couplings g

_ijkl

are driven toward an

infrared-stable zero g

∗

ijkl

of the β-functions. The stability properties of the FP’s are controlled

by the eigenvalues ω

i

of the matrix

Ω

ijkl,abcd

=

∂β

_ijkl

∂g

abcd

(1.15)

computed at the given FP: a FP is stable if all eigenvalues ω

i

are positive. The smallest

eigenvalue ω determines the leading scaling corrections, which vanish as m

ω

_{∼ |t|}

∆

_where

∆ = νω. Usually ω is associated with the leading irrelevant operator. If ω

i

is negative,

∆

i

= νω

i

is the crossover exponents which quantifies how the RG flow goes away from the

unstable FP (see section 3). If a stable FP has ω with a non-vanishing imaginary part, it is

called focus because the approach to the FP is spiral-like.

The critical exponents are obtained by evaluating the RG functions

η

φ

(g

ijkl

) =

∂ ln Z

_φ

∂ ln m

¯

u

= β

abcd

∂ ln Z

_φ

∂g

abcd

,

η

t

(g

ijkl

) =

∂ ln Z

t

∂ ln m

¯

u

= β

abcd

∂ ln Z

_∂g

t abcd

(1.16)

at the stable FP g

∗ ijkl

:

η = η

_φ

(g

_ijkl∗

),

ν = [2 − η

_φ

(g

_ijkl∗

) + η

_t

(g

_ijkl∗

)]

−1

.

(1.17)

All the other standard exponents can be obtained using the scaling and hyperscaling relations

reported in the Introduction.

To obtain the critical exponents of the correlation functions of a generic operator O, one

introduces the new renormalization constant Z

O

of the operator O via the natural relation

Γ

(2)_O

(0) = Z

_O−1

(g

_abcd

)C

_ijO

,

(1.18)

where Γ

(2)_O

(0) is the zero-momentum two-point function with one insertion of the operator O,

and C

O

ij

is the tensorial structure of Γ

(2)

O

(0) at tree-level (e.g. for O = 1/2φ

2

, C

ijO

= δ

ij

). The

anomalous dimension of O is given by

η

O

(g

ijkl

) =

∂ ln Z

O

∂ ln m

¯

u

= β

_abcd

∂ ln Z

O

∂g

_abcd

,

(1.19)

calculated at the stable FP. The thermodynamics of the operator O defines a new set of critical

exponents given by

hOi = (−t)

−βO

_,

_{for t < 0 ,}

_(1.20)

χ

O

=

Z

d

xhO(x)O(0)i

c

∝ t

−γO

,

(1.21)

F

sing

(t, h

O

) = |t|

dν

F (h

O

|t|

−φO

) ,

(1.22)

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where h

O

is the field conjugated to O, φ

O

= y

O

ν, and y

O

is the RG dimension of O (i.e.,

y

_O

= 2 + η

_O

− η). These exponents satisfy the scaling relations

β

O

= dν − φ

O

,

γ

O

= −dν + 2φ

O

,

β

O

+ γ

O

= φ

O

.

(1.23)

We have computed the perturbative expansion of the correlation functions Eqs. (1.11),

(1.12), and (1.13) for several interesting LGW Hamiltonians up to six loops in three dimensions

and five loop in two dimensions. The diagrams contributing to the two-point and four-point

functions to six-loop order are reported in Ref. [44]: they are 789 for the four-point function and

88 for the two-point one (at five-loop they are 162 and 26 respectively). We have developed a

symbolic manipulation program which generates the diagrams using the algorithm described

in Ref. [43], and computes the symmetry and group factors of each of them. We did not

calculate the integrals associated to each diagram, but we used the numerical results compiled

in Ref. [44] for three dimensions and in Ref. [46] for two dimensions.

Technical remark

In fixed dimension expansion the β functions have the form

β

_g_i

4 − d

= −g

i

+ A

d

C

N

g

2

i

+ O(g

j2

, g

i3

, g

i

g

j

) ∀j 6= i,

(1.24)

where A

_d

is the value of the single one-loop Feynman integral for the four-point function

A

d

=

Γ(2 − d/2)

(4π)

d/2

,

and C

_N

the combinatorial factor of the same diagram, depending only on the symmetry of

the theory. In order to have a FP value of the order of unity for all N , the change of variable

¯

g

_i

= A

_d

C

_N

g

_i

⇒ β

_g_¯_i

= A

_d

C

_N

β

_g_i

(g

_i

(¯

g

_i

))

(1.25)

is performed. After this change the β functions are

β

¯gi

4 − d

= −¯

g

i

+ ¯

g

2

i

+ O(¯

g

j2

, ¯

g

i3

, ¯

g

i

g

¯

j

) ∀j 6= i .

1.3.2 The ² expansion

A useful tool that allows to find the FP’s of the RG flow and permits analytical manipulations

of the results is the so called ² expansion. It is based on the observation that d = 4 is a special

dimension for φ

4

_{LGW Hamiltonians (it is the upper critical dimension, i.e. the dimension}

above which the critical behavior is mean-field like). Indeed, using the definition of the β

functions we get

β

gi

({g}) = m

∂g

i

∂m

¯

0

= −(4 − d)

µ

d log g

i

Z

gi

dg

i

¶

₋₁

,

(1.26)

where the subscript 0 stands for derivative at fixed bare couplings. Being Z

gi

= 1 + O({g}),

the fixed point g

∗

i

= 0 is IR stable for d > 4 and unstable in the opposite case. Thus, for

d > 4 the critical behavior is governed by the FP g

∗

i

= 0 (Gaussian FP), corresponding to

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dimension from four, the new IR stable fixed point is expected to be close to the Gaussian

one, i.e., g

∗

_{= O(²), where ² ≡ 4 − d. As originally pointed out in a seminal paper by Wilson}

and Fisher [37], one can perform a double expansion of RG functions in terms of g and ² and

find the zeros of the β functions as series in ². The critical exponents and other universal

quantities are obtained expanding in ² the corresponding RG functions evaluated at the stable

fixed point.

This procedure allows to obtain the critical quantities as series in ². The analytical

contin-uation of such series at physical dimensions d = 3, 2 (i.e., ² = 1, 2) may not seem

straightfor-ward. However, the validity of this continuation to ² = 1, 2 is corroborated by the very good

agreement of the ²-expansion estimates with other theoretical and experimental values, as we

will discuss later on.

Practically, one first determines the expansion of the renormalization constants Z’s and

the renormalized couplings g

ijkl

in powers of the bare couplings u

ijkl

. In the first stages of RG

theory, they were obtained by requiring the normalization conditions (1.11), (1.12), and (1.13).

However, in this framework it is simpler to use the minimal-subtraction (MS) scheme [38]. In

this renormalization scheme, the renormalized one-particle irreducible correlation functions

are obtained by subtracting to the bare ones the poles in ² in a dimensional regularization (in

this sense it is the minimal subtraction that render the correlation functions finite). Once the

renormalization constants are determined, one computes the RG functions β

ijkl

, η

φ

, and η

t

as

in Sec. 1.3.1.

Also in this expansion, we calculated the RG functions for several LGW Hamiltonians, by

means of the same manipulation algorithm of the MZM scheme. We did not calculate the

integrals, but we used the analytical results reported up to five loops in Ref. [33].

2

1.3.3 The MS scheme without ² expansion

The ² expansion is surely the most natural way to analyze MS RG functions, because it allows

analytical manipulations of the results and so it can be used to show some particular features

in an analytical manner. However, to get predictions in physical dimensions d = 2, 3, one

usually assumes a smooth behavior decreasing the dimensions and continues the series in ² to

² = 1, 2. Although this procedure works almost perfectly for O(n) models, there is no general

reason to believe blindly to this analytical continuation. In fact nowadays there are several

examples were ² expansion fails.

3

Thus it is desirable to have a fixed dimension way of analyzing the MS RG functions

which does not assume smooth behavior on ²: the minimal-subtraction scheme without ²

expansion [41]. In this approach, the RG functions are those of the MS scheme, but ² is no

longer considered as a small quantity but it is set to its physical value, i.e. in three dimensions

it is ² = 1. The β-functions have a simple dependence on d, indeed

β

gi

= (d − 4)g

i

+ B

i

({g

i

}) ,

(1.27)

where the functions B

i

are independent of d (they are essentially four dimensional).

The biggest difference between the MS scheme in fixed dimensions (we will refer to it

with 3d-MS scheme) and the

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massless (critical) theory. Indeed in d = 4 no IR divergences occur in the massless Feynman

diagrams, contrarily to lower dimensions. This property obviously simplify the evaluations

of Feynman diagrams, and in what follows, when referring to the 3d-MS scheme, it will be

always understood that we are working directly in the massless theory.

In the 3d-MS scheme the couplings are normalized so that g

_i

= g

_i,0

µ

−²

_/A

d

with A

d

=

2

d−1

π

d/2

Γ(d/2), and µ is the renormalization scale. The fixed points of the theory and the

critical exponents are determined as in the fixed dimension approach. Notice that the FP

values g

∗

i

are different from the FP values of the renormalized quartic couplings of the MZM

renormalization scheme, only the values of RG functions at the FP’s (i.e. the exponents) must

be the same in the two schemes.

1.3.4 The pseudo-² expansion

An alternative method to analyze MZM RG functions is the so called pseudo-² expansion. It

was introduced by B. Nickel (see footnote 19 in [71]). It starts from the observation that when

calculating the critical exponents, one has to solve first the equations β

gi

({g

j

}) = 0 and then

to calculate the others series γ({g

_j

}), η({g

_j

}) . . . at the FP {g

_j

} = {g

∗

j

} (here {g

j

} stands for

the set of all the couplings of the theory). As a result the final errors on the exponents is the

sum of the error due to the direct uncertainty of the series of the exponent, and of the error

coming from g

∗

j

. To avoid this cumulation of errors, one can define

β

gi

({g

j

}, τ ) = β({g

j

}) + g

i

(1 − τ ).

(1.28)

Since β

gi

= −g

i

+ O({g

j

}

2

), it is possible to calculate g

∗j

as power series in τ and to substitute

these series in the exponents. Cumulation of errors is therefore avoided, at the price of having

series with more complicated structures.

1.4 The Analysis of perturbative series

It is a well-known fact, that increasing the order in the ² expansion the estimates of physical

quantities at ² = 1 or 2 get worse. For example the ² expansion of the exponent ν of the

one-component φ

4

_{theory (i.e. Ising universality class) is}

ν =

1

2 +

1

12 ² + 0.043²

2

_{− 0.019²}

3

_{+ 0.071²}

4

_{− 0.217²}

5

_{+ O(²}

6

_{) .}

_(1.29)

From this expression, it is evident (being the fifth order term three times bigger than the first

one!) that considering more and more terms in this sum the result becomes unreliable. As first

realized in Ref. [66] from heuristic arguments, this strange feature reflects the divergent nature

of the series both in ² and in fixed dimension expansion. A first way to attack this problem is

to account only the first few terms in the expansions that seem to give a convergent sequence

of results. However, it will be preferable to have some (almost) rigorous manipulations

al-lowing us to use all the expansions (that costed us the evaluations of thousand integrals and

combinatorial factors). This can be done by exploiting the Borel summability of RG functions,

that has been proved for the fixed-dimension expansion of φ

4

_{theories in d < 4 [47, 49, 48, 50]}

and has been conjectured for the ² expansion. Once the Borel summability is assumed (or

proved), we can give sense to expansions like Eq. (1.29) using resummation techniques.

In the next subsection 1.4.1 we consider the large order behavior of perturbative

expan-sions, from which the origin of the strange behavior of series like Eq. (1.29) will become clear.

The main result of this section is that a given RG function

S({xg

i

}) ≡

X

(17)

has coefficients that for large k behave as

s

_k

({g

i

}) = c k!(−a)

k

b0

[1 + O(k

−1

)],

(1.31)

where the value of the constant a > 0 is independent of the particular quantity considered,

unlike the constants b

0

and c, which depend on the RG function. The function

B

_S

(t) =

X

k

s

_k

({g

_i

})

k!

t

k

_,

_(1.32)

is expected to be convergent in the circle in the complex plane |t| < 1/a (i.e. a is the inverse

of the singularity of the Borel transform closest to the origin a = −1/g

b

). If we are able to

analytically continue the function B

_S

(t) to all the real axis, a finite function with the same

perturbative expansion of S is

S(g) =

Z

_∞

0

e

−t

B(xt)dt.

(1.33)

The subsection 1.4.2 is devoted to review several methods to analytically continue B(t) to the

real axis, so to make sense to asymptotic series like (1.29).

1.4.1 Large order behavior

The main purpose of this subsection is to show Eq. (1.31) for φ

4

_{theories and to explicitly}

calculate the value of a for some LGW Hamiltonians. Unfortunately as this derivation is quite

technical, the uninterested reader may skip this section and blindly trust Eq. (1.31).

The large order behavior of a field theoretical perturbative series can be determined by

means of a steepest-descent in which the relevant saddle point is a finite-energy solution

(instanton) of the classical field equations with negative coupling [51, 52].

Let us start our discussion from the case of one scalar field φ with one coupling g in fixed

d < 4. We closely follow Ref. [52]. A generic correlation function may be written as

A(g) =

Z

[dφ]A(φ)e

−S(φ)

,

(1.34)

where S(φ) is the Euclidean action, A(φ) stands for the product of a generic number of fields

φ (composite operators) at different points. The kth order of the perturbative series of A(g)

may be obtained by the contour integral

A

k

=

Z

[dφ]

1 2πi

I

dge

−S(φ)

g

(k+1)

A(φ) .

(1.35)

For large enough k this integral is given by a steepest-descent approximation in the variables

g and φ(x), that leads to the coupled integral-differential equations

(−∂

2

+ m

2

)φ

c

(x) +

1 ₆

g

c

m

4−d

φ

3c

(x) = 0 ,

(1.36)

k + 1

g

c

= −

m

4−d

4!

Z

d

xφ

4_c

(x) .

(1.37)

Eq. (1.36) is the classical equation of motion [for this reason the solutions are named g

c

and

φ

c

(x)]. In terms of the solutions of these equations A

k

is

(18)

The change of variable φ

c

(x) = (−6/g)

1/2

m

d/2−1

f (mx) leads to dimensionless equations

kg

c

= −

3

2 Z

d

xf

_c4

(x),

and

(−∂

2

+ 1)f

c

(x) − f

c3

(x) = 0 ,

(1.39)

and to a renormalized action

S(f ) = −

6 g

c

Z

d

x

·

1

2 (∂

µ

f (x))

2

₊

1

2 f (x)

2

₋

1

4 f (x)

4

¸

.

(1.40)

The equation (1.39) for f

c

(x) has not unique solution, also requiring for a finite action (that

translate in the boundary condition f (x) → 0 for |x| → ∞). All these solutions contribute to

the asymptotic behavior of (1.31), but the leading term is given the solution with minimum

action. Notice that S(f

c

) = −(g

c

a)

−1

after simple algebraic manipulations may be written as

1 a

=

6 4 − d

Z

d

xf

_c2

(x) =

3

2 Z

d

xf

_c4

(x) =

6 d

Z

d

x[∂

_µ

f

_c

(x)]

2

.

(1.41)

These relations can only be true for d < 4. Thus dimension four is singular.

The solution of Eq. (1.39) for f

c

(x) may be simply worked out numerically (but not

analytically). The results are [26]

1 a

=

(

35.1026 for d = 2 ,

113.3835 for d = 3 ,

and so

1/g

c

= −ak,

S(φ

c

) = −(ag

c

)

−1

= k .

(1.42)

Inserting the last result in (1.38) we have

A

k

∝ (−)

k

e

−[S(φc)+k log(−gc)]

' e

−k

k

(−a)

k

' k!(−a)

k

,

(1.43)

and a introduced so far is exactly the one appearing in Eq. (1.31). The calculation of b

0

and

c in Eq. (1.31) is much more cumbersome and it requires the study of the fluctuations around

φ

_c

. We remand the interested reader to Ref. [52].

Generalization to N -component models

Let us now discuss how these results extend to the general N -component LGW Hamiltonian

(1.2) with only one quadratic invariant. The system of coupled equations is

(−∂

2

+ m

2

)φ

i

(x) +

1 ₆

m

4−d

u

ijkl

φ

j

(x)φ

k

(x)φ

l

(x) = 0 ,

(1.44)

k + 1 = −

m

4−d

4!

u

ijkl

Z

d

x φ

i

(x)φ

j

(x)φ

k

(x)φ

l

(x) .

(1.45)

It is convenient to extract the O(N )-symmetric term from the coupling-matrix

u

_ijkl

=

u

3 (δ

ab

δ

cd

+ δ

ac

δ

bd

+ δ

ad

δ

bc

) + v

ijkl

,

(1.46)

so that the previous equations are

(−∂

2

+ m

2

)φ

i

(x) +

u

₆

m

4−d

h

φ

i

(x)φ

2

(x) +

v

ijkl

_u

φ

j

(x)φ

k

(x)φ

l

(x)

i

= 0 ,

(1.47)

k + 1

u

= −

m

4−d

4!

Z

d

x

h

(φ

2

(x))

2

+

v

ijkl

u

φ

i

(x)φ

j

(x)φ

k

(x)φ

l

(x)

i

,

(1.48)

(19)

where φ

2

_{(x) =}

P

i

φ

2i

.

The solution of these equations proceed as for mean-field equations, with hφi replaced by

φ

c

(x). Indeed, searching for solutions of the form φ

ci

= v

i

φ

c

, one arrives to an equation for φ

c

equal to that of the one-component case. At fixed v

_ijkl

/u, one has also to minimize the term

1 +

v

ijkl

u

v

i

v

j

v

k

v

l

,

by an appropriate choice of v

i

. As for mean-field theory, the set of v

i

may be guessed from

symmetries.

Note that the positivity condition in mean-field theory, i.e. the condition for the stability

of the quartic form, is also the condition that determines the Borel summability of the theory,

in fact when u

_ijkl

is no more positive, a singularity of the Borel transform falls on the real

positive axis.

We now report the large-order behavior of all the models studied in this thesis.

O(N ) models

For the O(N ) model there is only one coupling and the choice of the vector ~v is arbitrary,

due to rotational symmetry. So 1/g

b

of the O(N ) model is the same of Ising model. The RG

functions are usually expressed in terms of the rescaled coupling (1.25), so that

−

1 ¯

g

_b

= ¯aR

N

,

(1.49)

where R

K

= 9/(8 + K) and

¯a = 0.147 744 220 . . .

in d = 3 ,

¯a = 0.238 659 217 . . .

in d = 2 .

(1.50)

The Cubic Model

For the cubic model the vector ~v is along the diagonal of an N -dimensional hypercube for

v > 0 (i.e., ~v ∝ (1, 1 . . . , 1)) and along the axis for v < 0 (i.e., ~v = (1, 0, 0 . . . , 0)). Thus, in

Eq. (1.31), the singularities of the Borel transform (we use the non rescaled variable) are

1 x

_b

(u, v)

= −a(u + v)

for

v

u

> 0,

1 x

b

(u, v)

= −a

³

u +

v

N

´

for −

2N

N + 1

u < v < 0.

(1.51)

For v < −

_{N +1}2N

u the RG functions are not Borel summable.

The M N Model

The calculation of the singularities of the Borel transform proceeds along the same line of the

Cubic model. The result is

1 x(u, v)

= −a (u + v)

for

0 < v and v < −

2N

N + 1

u,

(1.52)

1 x(u, v)

= −a

³

u +

v

N

´

for

0 > v > −

2N

N + 1

u,

(20)

Note that the condition of Borel summability (that coincides with the mean-field boundness

condition) is v > −u for v < 0. So in this region, even if the second of Eq. (1.52) takes into

account the singularity of the Borel transform closest to the origin, there is another singularity

on the real positive axis that makes the series not Borel summable.

O(n) ⊗ O(m) models

In this case we can argue that the expansion is Borel summable when

u ≥ 0,

u −

m − 1

m

v ≥ 0 .

(1.53)

In this region we have

1 x

b

(u, v)

= a Max

· u, u −

µ

1 −

1 m

¶

v

¸

.

(1.54)

We notice that (even outside the region (1.53)), if the condition

u −

m − 1

2m

v > 0

(1.55)

holds, then the Borel-transform singularity closest to the origin is still in the negative axis,

and therefore the large-order behavior is still oscillating with x

_b

(u, v) given by (1.54).

Borel summability in MS scheme and ² expansion

As we already mentioned the Borel summability in ² is only conjectured. The source of the

problem is that in the MS scheme the perturbative series are essentially four-dimensional. In

d = 4, there are singularities of the Borel transform that are not detected by a semiclassical

analysis, see e.g. [55] (they are connected with the renormalization procedure, and in fact the

quasi-particle responsible of them are called renormalons). Such singularities, that should be

on the real positive axis, may make the expansion non Borel summable for any value of the

coupling(s). In any case, it is commonly believed that the large-order behavior of the series

is still given by the istantons. This fact is corroborated by the good agreement between the

results obtained from the analyzes of MS series [41] and those obtained by other methods,

indicating that the renormalon effects are either very small or absent (note that, as shown in

Ref. [54], this may occur in some renormalization schemes).

From the analysis of the classical equations of motion, we can find the “conjectured”

large-order behavior of the perturbative series. It is given exactly by the same formulas in the MZM

scheme, with the modification that the constant a, in this scheme, is equal to 1/2.

From the order behavior of the MS perturbative series, one easily obtains the

large-order behavior of the ² expansion. We have only to further assume that in the inversion of

the series (needed to find the zeros of the β functions), no new singularities are generated.

For instance, for O(N ) models, the large order behavior of all RG functions is (1/2)

k

_{k!, the}

one-loop FP is g

∗

= (N + 8)/6, so the large-order behavior of ² expansion series is governed

by the singularity of the Borel transform

²

b

= −

N + 8

₃

.

(1.56)

In a similar fashion, one can deduce the large-order behavior of ² expansion for all models.

We notice that the large order behavior in pseudo-² expansion is similarly obtained

replac-ing 1/2 with the right a, but even in this case we assume that no new sreplac-ingularity is generated

in inverting the β functions.

(21)

1.4.2 Resummation of the perturbative series

Let S(x) be an asymptotic series we want to resum

S({xg

i

}) =

X

s

k

({g

i

})g

k

,

(1.57)

with large-order behavior of the coefficients given by Eq. (1.31). We do not know all the

coefficients s

k

, but only up to the order p. We introduce the Borel-Leroy transform B(t) of

S(x) (from now on we understood the dependence upon {g

i

}) as

S(x) =

Z

_∞

0

t

b

e

−t

B(xt)dt,

(1.58)

where b is arbitrary. Its series expansion is given by

B

exp

(t) =

X

k

s

k

Γ(k + b + 1)

t

k

_.

_(1.59)

The constant a that characterizes the large-order behavior of the original series is related to

the singularity t

s

of the Borel transform B(t) closest to the origin: t

s

= −1/a. The series

B

exp

(t) is convergent in the disk |t| < |t

s

| = 1/a in the complex plane. In this domain, one

can compute B(t) using B

exp

(t). However, in order to compute the integral (1.58), one needs

B(t) for all positive values of t. It is thus necessary to perform an analytic continuation of

B

exp

(t).

The most common employed analytic continuation are the Pad´e approximants and the

conformal mapping. The former is defined as the ratio of two polynomials of order L and

M (with L + M ≤ p) whose series expansion is B

exp

(t). Explicitly the Pad´e approximants of

order [L/M ] is

P

p

(S)(b, L, M )(t) =

a

0

+ a

1

t + a

2

t

2

_{+ · · · + a}

L

t

L

1 + b

₁

t + · · · + b

_M

t

M

,

(1.60)

where the coefficients a

i

and b

i

are fixed by the condition that the expansion of P

p

(S)(b, L, M )(t)

in powers of t reproduces B

exp

(t).

At the order p, estimates of S(x = 1) are given by

E

p

(S)(b, L, M ) =

Z

_∞

0

dt t

b

e

−t

P

p

(S)(b, L, M )(t) ,

(1.61)

with varying the considered Pad´e (i.e. L and M ) and the value of the free parameter b. Since

the integral (1.61) should be defined, the approximant (1.60) must not have poles on the real

positive axis. The Pad´e with real positive poles are usually called defective and they must be

discarded in the average procedure.

A fruitful way to find a final estimate with a proper, well-weighted error bar is to search

for the value of b, named b

opt

, minimizing the difference between different approximants.

Then the final estimate is the mean value of the approximants at b

opt

(eventually discarding

too far estimates) and the uncertainty is the variance of the approximants in the range b ∈

[b

_opt

− ∆b, b

_opt

+ ∆b] (a standard choice for ∆b may be 1, but it depends on the case).

A more refined resummation procedure exploits the knowledge of the large-order behavior

of the expansion, and in particular of the constant a. One performs a conformal transformation

[71]

y(t) =

√

1 + at − 1

√

1 + at + 1

,

that allows to rewrite B(t) as B(t) =

X

k

(22)

If all the singularities belong to the real interval [−∞, −t

c

], the previous expansion converges

everywhere in the complex t-plane except on the negative axis for t < −t

_c

. After the

trans-formation, a new expansion for S(x) is obtained:

S(x) =

Z

_∞ 0

dt e

−t

t

b

X

k

f

k

[y(tx)]

k

.

(1.63)

This sequence of operations has transformed the original divergent series into an integral of a

convergent one, which can then be easily calculated.

In this context, another free parameter is usually introduced to optimize estimates of

critical quantities. The trick is to expand in powers of y the function B

b

(g(y))(1 − y)

α

and

then applies the conformal-mapping resummation. With this parameter more, we obtain the

approximants of the series S(x = 1)

E

p

(S)(α, b) =

p

X

k=0

B

k

(α, b)

Z

_∞ 0

dt t

b

e

−t

y(t)

k

[1 − y(t)]

α

,

(1.64)

in terms of the two free parameters b and α. The best choices for these parameters are usually

considered those minimizing the difference between the last perturbative orders. Explicitly,

one considers sets of approximants such that α ∈ [α−∆α, α+∆α] and b ∈ [¯b−∆b, bb+∆b]. In

order to obtain a final estimate, one should devise a procedure to determine optimal values for

the parameters α, ¯b, ∆α, and ∆b. Reasonable values of α and ¯b can be obtained by requiring

that the estimates are approximately independent of the number of terms one is considering.

It is less clear how to determine the width of the intervals ∆α and ∆b. Reasonable choices

are ∆α = 2 and ∆b = 3, or ∆α = 1 and ∆b = 6.

Finally, we mention that it is possible to improve the ²-expansion results for a quantity

R when its value is known for some values of the dimensions. The method works as follows

[67, 90]. One considers a quantity R such that for ² = ²

1

the exact value R

ex

(²

1

) is known.

Then, one defines

R(²) =

· R(²) − R

ex

(²

1

)

(² − ²

1

)

¸

,

and

R

imp

(²) = R

ex

(²

1

) + (² − ²

1

)R(²).

(1.65)

New estimates of R at ² = 1 can then be obtained by resumming the ² expansion of R(²) and

then computing R

imp

Critical Behavior of Systems with Complex Symmetries

Corso di Perfezionamento in Fisica