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arXiv:1111.3050v2 [math-ph] 18 Nov 2011

A numerical approach to harmonic non-commutative spectral

field theory

Bernardino Spisso1and Raimar Wulkenhaar2

1,2Mathematisches Institut der Westf¨ alischen Wilhelms-Universit¨ at

Einsteinstraße 62, D-48149 M¨ unster, Germany

Abstract

We present a first numerical investigation of a non-commutative gauge theory defined via

the spectral action for Moyal space with harmonic propagation. This action is approximated

by finite matrices. Using Monte Carlo simulation we study various quantities such as the

energy density, the specific heat density and some order parameters, varying the matrix size

and the independent parameters of the model. We find a peak structure in the specific heat

which might indicate possible phase transitions. However, there are mathematical arguments

which show that the limit of infinite matrices is very different from the original spectral model.

1Introduction

Quantum field theory on noncommutative spaces [1, 2, 3] is an active subject of research. The

most-studied noncommutative spaces are the Moyal space [4] and fuzzy spaces [5]. Fuzzy spaces

are matrix approximations of manifolds and as such ideal for numerical investigations similar to

non-perturbative quantum field theory on the lattice. In this paper we focus on the Moyal space,

which is a continuous deformation of Rdfor which the usual Fourier techniques of perturbative

quantum field theory are available. It turned out that a renormalisable quantum field theory on

Rdis, in most cases, no longer renormalisable on d-dimensional Moyal space due to a phenomenon

called ultraviolet/infrared mixing [6].In [7] it was discovered that for the ϕ4-model on 4-

dimensional Moyal space the UV/IR-mixing generates an additional marginal coupling which

corresponds to a harmonic oscillator potential for the free scalar field. The resulting action

S[ϕ] =

?

d4x

?1

2ϕ ⋆ (−∆ + Ω2˜ x2+ µ2) ⋆ ϕ +λ

4ϕ ⋆ ϕ ⋆ ϕ ⋆ ϕ

?

(x)(1)

was then shown to be perturbatively renormalisable to all orders in λ. In (1), ˜ x = 2Θ−1·x, where

Θ is the deformation matrix defining the Moyal product. See also [8, 9, 10, 11, 12]. Moreover,

the frequency parameter can be restricted to Ω ∈ [0,1] by Langmann-Szabo duality [13].

1nispisso@tin.it

2raimar@math.uni-muenster.de

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The renormalisability of the action (1) raises the question whether a harmonic term can also

render Yang-Mills theory renormalisable on Moyal space (recall that the usual Yang-Mills action

on Moyal space has the same UV/IR-mixing problem [14]). Yang-Mills theories in noncommut-

ative geometry [15] are naturally obtained from the spectral action principle [16] relative to an

appropriate Dirac operator. In [17] it was shown that Moyal space (with usual Dirac operator)

is a (non-compact) spectral triple; its corresponding spectral action was computed in [18]. To

obtain a gauge theory with harmonic oscillator potential via the spectral action principle, a

differential square root of the harmonic oscillator hamiltonian is necessary as Dirac operator.

In absence of such a Dirac operator, in [19, 20] an effective gauge model was constructed as the

one-loop effective action of complex harmonic noncommutative quantum ϕ4-theory in a classical

external gauge field. As a result, the noncommutative Yang-Mills Lagrangian is extended by

two terms Xµ⋆ Xµand (Xµ⋆ Xµ)2, where Xµ=1

2˜ xµ+ Aµis the ‘covariant coordinate’.

A first outline of a candidate spectral triple for harmonic oscillator Moyal space was given

in [21]. Additionally, in [21] the linear and quadratic terms of the spectral action for a U(1)-

Yang-Mills-Higgs model were computed and then extended by gauge invariance. Thereby the

appearance of Xµ⋆ Xµwas traced back to a deep entanglement of gauge and Higgs fields in a

unified potential (αXµ⋆ Xµ+ β ¯ ϕ ⋆ ϕ − 1)2, with α,β ∈ R+.

It turned out that the candidate spectral triple proposed in [21] was the shaddow of a

new class of non-compact spectral triples with finite volume [22]. The spectral geometry of

Moyal space with harmonic propagation, which falls into this class, was fully worked out in

[23]. There are in fact two (even, real) spectral triples (A,H,D•,Γ,J), with • ∈ {1,2}, for

the d-dimensional Moyal algebra A and differential quare roots D•of the harmonic oscillator

hamiltonian. The spectral triples are of metric dimension d and KO-dimension 2d, have simple

dimension spectrum consisting of the integers ≤ d, and satisfy all regularity and compatibility

requirements of spectral triples. Additionally, the spectral action was rigourously computed in

[23], i.e. with H¨ older type estimates for the remainder of the asymptotic expansion and with

inclusion of the real structure J.

A completely new feature of the spectral action [23, 21] (and also of the effective action

[19, 20]) is that the expansion of Xµ⋆ Xµand its square produces a term which is linear in

the gauge field A. This means that the vacuum, i.e. the solution of the classical field equations,

is no longer taken at Aµ = 0 (or more generally at a flat connection Fµν = 0) but at some

non-constant value for the gauge field. A first discussion of the vacuum structure of this type of

gauge models was given in [24]. It turned out that generically there are infinitely many vacuum

solutions. Some of them were exposed, but it was not possible to give reasonable argument for

the right solution. In particular, it became completely impossible to study the gauge model as

a perturbative quantum field theory.

This is the point where the numerical treatment comes into play. The standard method

of numerical quantum field theory is to approximate the space by discrete points, for example

using a lattice approximation and then calculate the observables over that set of points [25]. For

Moyal space a position space approximation is not suitable due to the oscillator factor of the

Moyal product. Instead, we shall use the matrix Moyal base (which was already used in the first

renormalisation proof [7] of ϕ4-model), restrict it to finite matrices and perform a Monte Carlo

simulation of the resulting action. In this way we will study some statistical quantities such as

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energy density and specific heat, varying the parameters Ω,χ−1

some information on the various contributions of the fields to the action. The simulations are

quite cumbersome due the complexity of the action and the number of independent matrices

to handle. Nevertheless we are able to get an acceptable balance between the computation

precision and the computation time. For the simulations we apply a standard Metropolis-Monte

Carlo algorithm [26] with various estimators for the error and for the autocorrelation time of the

samples. The range of parameters is chosen to avoid problems with the thermalisation process,

thus permitting a relative small number of Monte Carlo steps to compute independent results

from the initial conditions.

χ0,α of the model and gathering

We are eventually interested in the continuum limit which corresponds to matrices of infinite

size. We thus compute our observables such as the energy density for various matrix sizes and

then look for a stabilisation of these observables as the matrix size increases. The specific

heat, which is a measure of the dispersion of the energy, will be used to identify possible phase

transitions in form of peaks of the specific heat at increasing matrix size.

2 Four-dimensional harmonic Yang-Mills model

The harmonic Yang-Mills model is defined as the spectral action resulting from the spectral

triples (A⋆,H,D•,Γ,J), with • ∈ {1,2}, analysed in [23]. The Moyal algebra A⋆is the space of

Schwartz class functions on R4equipped with the product

f ⋆ g(x) =

?

R4×R4

dy dk

(2π)4f(x+1

2Θ · k)g(x+y)ei?k,y?. (2)

The unbounded selfadjoint operators D• on the Hilbert space H are differential square roots

of the harmonic oscillator hamiltonian H = −∂µ∂µ+˜Ω2xµxµof frequency˜Ω, i.e. D2

(−1)•˜ΩΣ, for a certain spin matrix Σ. If L⋆(f) denotes left Moyal multiplication with a function

f ∈ A⋆, then one has

[D1,L⋆(f)] = L⋆(i∂µf) ⊗ Γµ,

•= H −

[D2,L⋆(f)] = L⋆(i∂µf) ⊗ Γµ+4, (3)

where the matrices Γ1,...,Γ8satisfy the anticommutation relations

{Γµ,Γν} = {Γµ+4,Γν+4} = 2(g−1)µν,{Γµ,Γν+4} = 0 ,(4)

relative to an induced metric g = (id −1

real structure satisfy JD•J−1= D•and JL⋆(f)J−1= R⋆(¯f), where R⋆denotes right Moyal

multiplication.

4˜Ω2Θ2)−1. The grading is Γ = Γ1···Γ8, and the

In order to implement the Higgs mechanism ` a la Connes-Lott [27] one considers the product

of the spectral triple (A⋆,H,D1,Γ,J) with the finite Higgs spectral triple (C⊕C,C2,Mσ1,Jf),

where σ1is a Pauli matrix, Jf any matricial real structure and M > 0. Then, a self-adjoint

fluctuation A =

?ai[D,bi] of the total Dirac operator D = (D1⊗ 1 + Γ ⊗ Mσ1) to give

DA= D + A + JAJ−1, for J = J ⊗ Jf, is of the form

A =

?

ΓµL⋆(Aµ)

ΓL⋆(¯φ)

ΓL⋆(φ)

ΓµL⋆(Bµ)

?

,(5)

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for two real one-forms Aµ,Bµ ∈ A⋆ and a complex scalar φ ∈ A⋆. Using D2 instead of D1

amounts to replace Γµby Γµ+4.

The spectral action principle [16] asserts that the bosonic action of a field theory with

fermionic Dirac operator DAhas the form

S(DA) = Tr(χ(D2

where χ is a smooth approximation of the characteristic function on [0,Λ2], for some scale

parameter Λ. For the fluctuation (5), the part of the spectral action which is relevant and

marginal for Λ → ∞ has been explicitly computed in [23], for general effective metric g. This

computation involved Laplace transformation, Duhamel expansion with H¨ older-type estimates

for the remainder and explicit use of the Mehler kernel for the harmonic oscillator hamiltonian.

For a special choice of the noncommutativity matrx Θ2= −θ2id, the result of [23] takes in

terms of Ω :=θ˜Ω

χ0:= χ(0) the form

SΛ(DA) =θ4Λ8

8Ω4

16Ω4

?52

+

π2(1 + Ω2)2

4Ω2

1+Ω2˜ XAµ⋆˜ Xµ

4Ω2

1+Ω2˜ XBµ⋆˜ Xµ

?(1 + Ω2)2

+ O(Λ−1) .

Here, Dµφ = ∂µφ − iAµ⋆ φ + iφ ⋆ Bµ− iM(Aµ− Bµ) is the covariant derivative of the scalar

field, FA

the field strength of B. Moreover,˜ XAµ:=˜ X0µ+ Aµand˜ XBµ:=˜ X0µ+ Bµare the covariant

derivatives of A and B, respectively, where˜ X0µ:=

of the spectral action (7) is that the Higgs field φ and the gauge fields A,B appear together

in a unified potential. In this way, also the gauge field shows a non-trivial vacuum structure.

Besides, the action is invariant under U(A⋆) × U(A⋆) transformations:

(φ + M) ?→ uA⋆ (φ + M) ⋆ uB,

A)) , (6)

2, the moments Λ2nχ−n:=?∞

8Ω4χ−4−M2θ4Λ6

45+M8θ4

?

φ⋆¯φ + M(φ+¯φ) +

0ds sn−1χ(s) of the “characteristic function” and

χ−3+

?M4θ4Λ4

χ0

+8θ2Λ4

12Ω2

?

χ−2−

?M6θ4Λ2

48Ω4

+2M2θ2Λ2

3Ω2

?

χ−1

+

192Ω4+M4θ2

?

3Ω2

?

χ0

d4x2(1 + Ω2)Dµφ ⋆ Dµφ

+

?

?¯φ⋆φ + M(φ+¯φ) +

A+ M2−χ−1

B+ M2−χ−1

χ0

Λ2?2

Λ2?2

−

?4Ω2

?4Ω2

1+Ω2˜ X0µ⋆˜ Xµ

0+ M2−χ−1

0+ M2−χ−1

χ0

Λ2?2

Λ2?2

+

χ0

−

1+Ω2˜ X0µ⋆˜ Xµ

χ0

+

2

−(1 − Ω2)4

6(1 + Ω2)2

??FA

µν⋆ FAµν+ FB

µν⋆ FBµν??

(7)

µν:= ∂µAν− ∂νAµ− i(Aµ⋆ Aν− Aν⋆ Aµ) the field strength of A and similarly FB

µν

˜ xµ

2= (Θ−1)µνxν. The remarkable outcome

˜ XA

µ?→ uA⋆˜ Xµ

A⋆ uA,

˜ Xµ

B?→ uB⋆˜ Xµ

B⋆ uB.(8)

3Discretisation by Moyal base

The 2-dimensional Moyal algebra with deformation parameter θ > 0 has a natural basis of

eigenfunctions fmn of the harmonic oscillator, where m,n ∈ N.

coordinates by

These are given in radial

fmn(ρcosϕ,ρsinϕ) = 2(−1)m

?

m!

n!eiϕ(n−m)

??

2

θρ

?n−m

e−ρ2

θ Ln−m

m

?2

θρ2

?

(9)

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and satisfy

(fmn⋆ fkl)(x) = δnkfml(x)

?

(10)

d2xfmn(x) = 2πθδmn, (11)

see [4, 17] for details. The expansion of Schwartz functions on R4in the Moyal base,

A⋆∋ a = a(x0,...,x3) =

?

m1,m2,n1,n2∈N

am1n1

m2n2fm1n1(x0,x1)fm2n2(x2,x3) , (12)

then provides an isomorphism of Fr´ echet spaces between A⋆and the space of rapidly decreasing

double sequences (amn)m,n∈N2 equipped with the family of seminorms

pk((amn)m,n∈N2) :=

∞

?

m,n∈N2

?

(2|m| + 1)2k(2|n| + 1)2k|amn|2?1

2,|m| := m1+ m2.(13)

According to (10), Moyal product and integral reduce in the (fmn)-basis to product and trace of

infinite N2-labelled matrices, with convergent index sums due to (13). By duality, the covariant

derivatives XA

µcan also be expanded in the (fmn)-basis, but the expansion coefficients

XA

µm1n1

µm1n1

m2n2

To any a ∈ A⋆we can associate a sequence (aN)N∈Nof cut-off matrices

?

0else .

µand XB

diverge for mi,ni→ ∞.

m2n2, XB

aN

m1n1

m2n2

=

am1n1

m2n2

if max(m1,m2,n1,n2) ≤ N ,

Then, (aN) is a Cauchy sequence in any of the semi-norms pkand converges to a in the Fr´ echet

topology of A⋆.

In quantum field theory we are confronted with the converse problem. To deal with diver-

gences, a regularisation has to be introduced which restricts the system to a finite number of

degrees of freedom. After re-normalisation from bare to physical quantities one has to show that

the limit to an infinite number of degrees of freedom is well-defined. In our case, the natural

regularisation is to restrict the matrix indices to mi≤ N, which corresponds to a cut-off in the

energy. Even if we could solve the renormalisation problem, the removal of the cut-off, i.e. the

limit N → ∞ to infinite matrices, will fail: A sequence of (N × N)-matrix algebras does not

converge in the Fr´ echet topology.

The question of convergence of matrix algebras has been studied in a sequence of articles

[28, 29, 30] by Marc Rieffel, David Kerr and Hanfeng Li [31, 32, 33]. Rieffel defined in [28]

the quantum Gromov-Hausdorff distance for compact quantum metric spaces, which are order-

unit spaces equipped with a generalised Lipschitz seminorm, the Lip-norm. In [29] he proved

that the sequence of matrix algebras which defines the fuzzy sphere converges in the quantum

Gromov-Hausdorff distance to the continuous functions on the sphere. While it seems possible

to give along these lines a reasonable notion of convergence of (N × N)-matrix algebras to a

limiting C∗-algebra (which describes continuity), we see at the moment no possibility to specify

convergence to the Fr´ echet algebra A⋆which describes smoothness.

This lack of convergence in the Fr´ echet topology implies that the results we obtain for finite

approximations of the action (7) are probably of little relevance for the smooth model. Thus,

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this work should be regarded as a first and limited expedition into the world of noncommutative

gauge theories. We show what is possible to do, knowing that the picture we obtain might be

misleading.

Using the identites Dµφ = i(φ+M)⋆˜ XBµ−i˜ XAµ⋆(φ+M) and FA

(and similarly for FB

finite number, we can recast the restriction of the action (7) to finite matrices in the following

form:

???1 − Ω2?2

2

µν= −i[XA

µ,XA

ν]+i[X0

µ,X0

ν]

µν), and ignoring all contributions of X0

µwhich for finite matrices yields some

S(φ,˜ XA,˜ XB) =

1

(1 + Ω2)2Tr

−

?1 + Ω2?4

?

6(1 + Ω2)2

???

4Ω2

1 + Ω2˜ Xµ

?2

˜ XAµ,˜ XAν

?

⋆

?

˜ Xµ

A,˜ Xν

A

?

⋆

+

?

?

˜ XBµ,˜ XBν

?

4Ω2

1 + Ω2˜ Xµ

⋆

?

˜ Xµ

B,˜ Xν

B

?

⋆

?

+ φ ⋆¯φ +

A⋆˜ XAµ− Λ2χ−1

χ0

?2

+

¯φ ⋆ φ +

B⋆˜ XBµ− Λ2χ−1

χ0

+ 2(1 + Ω2)

?

φ ⋆˜ XBµ−˜ XAµ⋆ φ

??¯φ ⋆˜ Xµ

A−˜ Xµ

B⋆¯φ

??

.(14)

The restriction to finite matrices shows crucial differences to the smooth model. Only these

differences make the numerical simulation possible, but their use drives our results away from

the original smooth model.

1. The action (14) has an obvious family of minima given appropriate multiples of the identity

matrices. We thus define

?χ−1

˜ XAµ= YAµ+1

2Λ

φ + M = ψ + Λ

χ0

cosαI

(15)

?χ−1

?χ−1

χ0

?

?

2Ω2

(1 + Ω2)Iµsinα (16)

˜ XBµ= YBµ+1

2Λχ0

2Ω2

(1 + Ω2)Iµsinα . (17)

Note that the corresponding minimum configurations for Aµ,Bµbadly violate, in the limit

N → ∞, the Fr´ echet condition.

2. For finite matrices, the N2-indexed double sequences can be written as tensor products of

ordinary matrices,

K

?

Since the matrix product and trace also factor into these independent components, the

action factors into S =?K

partition function factors, too:

Xm1n1

m2n2=

i=1

Xi

m1n1⊗ Xi

m2n2. (18)

i=1S(ψ1i,Y1i

A,Y1i

B)S(ψ2i,Y2i

A,Y2i

B). Then, regarding all ψ1i,Y1i

A,

Y1i

B,ψ2i,Y2i

A,Y2i

Bas random variables over which to integrate in the partition function, the

?

=

D(ψ11,Y11

??

A,Y11

B,ψ21,Y21

A,Y21

B)···D(ψ1K,Y1K

A ,Y1K

B ,ψ2K,Y2K

A ,Y2K

B ) e−S

D(ψ1i,Y1i

A,Y1i

B,ψ2i,Y2i

A,Y2i

B) e−S(ψ1i,Y1i

A,Y1i

B)·S(ψ2i,Y2i

A,Y2i

B)

?K

. (19)

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We may therefore restrict ourselves to K = 1. Again, for the limit N → ∞ we would have

K → ∞ and no chance of convergence.

3. Instead of integrating in the partition function over all gauge-equivalence classes of ψ,YA,YB

as required we may integrate over all ψ,YA,YB. The difference is the volume of the gauge

group, a well-defined factor for finite matrices but ill-defined in the limit N → ∞.

It is convenient to pass, for each factor in the tensor product (18), to complex matrices [24]:

Z0= YA

Z1= YB

Z2= YA

Z3= YB

0+ iYA

0+ iYB

2+ iYA

2+ iYB

1,

¯Z0= YA

¯Z1= YB

¯Z2= YA

¯Z3= YB

0− iYA

0− iYB

2− iYA

2− iYB

1

1,

1

2,

3

3,

3

(20)

The convention that the bar denotes the hermitian conjugate will also be used for the complex

matrix ψ. In the end the discretised action is:

S4=

1

(1 + Ω2)Tr?LF+ LV0+ LV1+ LD0¯LD0+ LD1¯LD1+ LD2¯LD2+ LD3¯LD3

?

,(21)

with

LF=D

+?Z0−¯Z0,Z2+¯Z2

+?Z1+¯Z1,Z3−¯Z3

LV0=?ψ¯ψ + µcosα(ψ +¯ψ) +1

+µsinα

2√C

LV1=?¯ψψ + µcosα(ψ +¯ψ) +1

+µsinα

2√C

LD0=

LD1=

LD2=

LD3=

2

??¯Z0,Z0

?2+?¯Z1,Z1

?2+1

4

??Z0+¯Z0,Z2−¯Z2

?2−?Z0+¯Z0,Z2+¯Z2

?2−?Z1+¯Z1,Z3+¯Z3

?2−?Z1−¯Z1,Z3−¯Z3

?+?¯Z2,Z2

?2

?2−?Z0−¯Z0,Z2−¯Z2

?2+?Z1−¯Z1,Z3+¯Z3

2

((−1 + i)(Z0+ Z2) + (1 + i)(¯Z0+¯Z2))?2

2

((−1 + i)(Z1+ Z3) + (1 + i)(¯Z1+¯Z3))?2

2(1 + Ω2)?µcosα(Z1+¯Z1− Z0−¯Z0) + ψ(Z1+¯Z1) − (Z0+¯Z0)ψ?

2(1 + Ω2)?µcosα(Z1−¯Z1− Z0+¯Z0) + ψ(Z1−¯Z1) − (Z0−¯Z0)ψ?

2(1 + Ω2)?µcosα(Z3+¯Z3− Z2−¯Z2) + ψ(Z3+¯Z3) − (Z2+¯Z2)ψ?

2(1 + Ω2)?µcosα(Z3−¯Z1− Z2+¯Z2) + ψ(Z3−¯Z3) − (Z2−¯Z2)ψ?

?1 − Ω2?2

2

In this case, (21) becomes an action for 5 complex matrices. For the partition function we need

the independent product of two copies of (21), i.e. we are dealing with a complex 10-matrix

model. This is already cumbersome and shows that there is little hope to treat the original

model (7) where the simplifying consequences of finite matrices are not available.

?2

?2??

??¯Z0,Z0

??

??¯Z1,Z1

?+?¯Z3,Z3

??

?

?

?

?

and

C =1 + Ω2

4Ω2

,D =

−

?1 + Ω2?4

6(1 + Ω2)2,Λ2χ−1

χ0

= µ2.(22)

The next step is to define the estimator for the average values of interest and to specify some

numerical parameters in order to analyse the numerical results.

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4 Definition of the observables

Following Monte Carlo methods, we will produce a sequence of configurations {(ψ,Zi)j}j=1,2,···,TMC

and evaluate the average of the observables over that set of configurations. These sequences of

configurations, called Monte Carlo chain, are representatives of the configuration space at given

parameters. In this framework the expectation value is approximated as

?O? ≈

1

TMC

TMC

?

j=1

Oj, (23)

where Oj is the value of the observable O evaluated in the j-sampled configuration, (ψ,Zi)j,

Oj= O[(ψ,Zi)j]. The internal energy is defined as

E(Ω,µ,α) = ?S? , (24)

and the specific heat takes the form

C(Ω,µ,α) = ?S2? − ?S?2. (25)

These quantities correspond to the usual definitions for energy

E(Ω,µ,α) = −1

Z

∂Z

∂β

(26)

and specific heat

C(Ω,µ,α) =∂E

∂β,

(27)

where Z is the partition function. It is very useful to compute separately the average values of

the four contributions:

F(Ω,µ,α) = ?TrLF? ,

V0(Ω,µ,α) = ?TrLV0? ,

V1(Ω,µ,α) = ?TrLV1? ,

D(Ω,µ,α) = ?Tr?LD0¯LD0+ ··· + LD3¯LD3

Order parameters

(28)

(29)

(30)

?? . (31)

4.1

The previous quantities are not enough if we want to measure the various contributions of

different modes of the fields to the configuration (ψ,Zi).

parameters usually called order parameters. As a first idea we can think about a quantity

related to the norms of the fields, for example the sums?

trace of the square:

Therefore, we need some control

nm|ψnm|2,

?

nm|Zinm|2. These

quantities are called the full-power-of-the-field [34, 35, 36, 37]; they can be computed as the

ϕ2

Z2

a= Tr(|ψ|2)

ia= Tr(|Zi|2)

(32)

(33)

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In contrast, ?ϕa? alone is not a good order parameter because it does not distinguish contribu-

tions from the different modes. But we can use it as a reference to define the quantities

ϕ2

0=

N

?

N

?

n=0

|ann|2,

Z2

i0=

n=0

|zinn|2,(34)

where amnand zimnare the expansion coefficents of ψ and Zi, respectively, in the matrix base

(9) . Referring to (9) it is easy to see that these parameters (34) are connected with the purely

spherical contribution. These quantities will be used to analyse the spherical contribution to

the full-power-of-the-field. We can generalise the previous quantity and define parameters ϕlin

such a way that they form a decomposition of the full-power-of-the-fields:

ϕ2

a= ϕ2

0+

?

l>0

ϕ2

l,Z2

ia= Z2

i0+

?

l>0

Z2

il. (35)

Following this prescription, the other quantities for l > 0 can be defined as:

ϕ2

l=

l?

n,m=0

|anm(1 − δnm)|2,Z2

il=

l?

n,m=0

|zlnm(1 − δnm)|2.(36)

If the contribution is dominated by the spherically symmetric parameter we expect to have

?ϕ2

In the next simulations we will evaluate, apart from l = 0, the quantity with l = 1 as

representatives of those contributions where the rotational symmetry is broken. According to

(36) we have

ϕ2

a? ∼ ?ϕ2

0?, ?Z2

ia? ∼ ?Z2

i0?.

1= |a10|2+ |a01|2, Z2

i1= |zi10|2+ |zi01|2. (37)

Using higher l in (36) we could analyse the contributions of the remaining modes, but it turns

out that the measurements of the first two modes are enough to characterise the behaviour of

the system.

5 Numerical results

Now we discuss the results of the Monte Carlo simulation of the approximated spectral model. As

a first approach we use some restrictions on the parameters. Starting point is the approximation

(21) of the spectral action. Since (21) is symmetric under the transformation µ ?→ −µ we can

assume µ ≥ 0 and µ2≥ 0. In this first treatment we explore the range µ ∈ [0,3.1], which is

enough to show a particular behaviour of the system for fixed Ω. The parameter Ω appears only

with its square and is defined as a real parameter, therefore also for Ω we require Ω ≥ 0. For the

scalar model [7] it was possible to restrict to Ω ∈ [0,1], because Langmann-Szabo duality maps

Ω to1

Ω. In the gauge model under consideration, Langmann-Szabo duality is not realised. Due

to the prefactor in front of the integral (21), the action vanishes for Ω → ∞. Studying the plots

for the energy and the specific heat, we have chosen the range Ω ∈ [0,2π] in which the action

is significantly different from zero. The last parameter to consider is α, which is connected to

9

Page 10

the choice of the vacuum state, with range α ∈ [0,2π]. The study of the system varying α is

quite important from a theoretical point of view because it is related to the vacuum invariance.

In the action there appear some contributions proportional to (sinα)/Ω which seem to diverge

for Ω = 0. Numerically we have verified that this is an eliminable divergence and the curves

of the observables can be extended to Ω = 0 by continuity. Studying the dependence on α we

can conclude that in the limit N → ∞ the observables are independent from α, therefore for

our purposes α will be fixed equal to zero avoiding the annoying terms. In general, for each

observable we compute the plots for matrix size approximations N =5, 10, 15, 20.

5.1Varying α

We start looking at the variation of the energy density and of the full-power-of-the-fields density

for fixed µ and Ω, varying α ∈ [0,2π]. As representatives we present the plots for µ = 1,

Ω ∈ {1, 0.5}, but we obtain the same behaviour for any other choice of the parameters allowed

in the considered range. All three plots show an oscillating behaviour of the values, and this

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123456

Α

0.005

0.010

0.015

?Z0a

2??N2

Figure 1: Total energy density and full-power-of-the-fields density for ?ϕ2

a?, ?Z2

0a? (from the left to the right)

fixing µ = 1, Ω = 1, varying α and N. N = 5 (circle), N = 10 (square), N = 15 (triangle), N = 20 (cross).

oscillation is present in all other quantities measured. The amplitude of this oscillation becomes

smaller and smaller increasing the size of the matrix and this is true for all the quantities. The

same trend is described in fig.2 which shows different positions of the maxima, but again smaller

amplitudes for increasing N. These results allow us to consider α = 0 for all next plots, since we

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0.02

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0.04

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0.06

0.07

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123456

Α

0.02

0.04

0.06

0.08

?Z0a

2??N2

Figure 2: Total energy density and full-power-of-the-fields density for ?ϕ2

a?, ?Z2

0a? (from the left to the right)

fixing µ = 1, Ω = 0.5, varying α and N.

are interested in the behaviour of the system for N → ∞. This occurrence simplify all the next

simulations thanks to the vanishing of terms ∼ (sinα)/Ω appearing in the discretised action.

10

Page 11

5.2Varying Ω

As already mentioned we chose [0,3] as range for Ω. In fact, if we look at the plots in fig.3 of

the total energy density ?S?/N2for µ ∈ {0, 1}, we notice that the action tends to zero for Ω

outside the selected interval. This behaviour of the action is the same for all possible choices of

parameters and for the specific heat, too.

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2

3

4

E? N2

Figure 3: Total energy density and the various contributions for µ = 0 (left), µ = 0 (right), α = 0 varying Ω

and N. With N = 5 (circle), N = 10 (square), N = 15 (triangle), N = 20 (cross).

In the rest of this section we ignore for the computations of ?E?,?D?,?V ?,?F? the global

prefactor (1+Ω2)−1. In this way we focus our attention to the integral as the source of possible

phase transitions. Now we will analyse three cases in which µ is fixed to 0,1,3. In all cases α

is zero and we vary Ω ∈ [0,3]. The plots in fig.4 show the total energy density and the various

contributions: the potential V/N2, the Yang-Mills part F/N2and the covariant derivative part

D/N2, for µ = 1. There is no evident discontinuity or peak, and increasing the size of the

matrices the curves remain smooth.

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0.51.01.5 2.02.53.0

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2

3

4

E?N2

Figure 4: Total energy density and the various contributions for µ = 1, α = 0 varying Ω and N. From the left

to the right E, V , D, F with N = 5 (circle), N = 10 (square), N = 15 (triangle), N = 20 (cross).

Comparing the energy density and the various contributions in fig.5 we notice that the

contributions between F and V balance each other and the total energy follows the slope of D,

and this behaviour continues increasing the size of the matrices.

11

Page 12

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Eng?N2

Figure 5: Comparison of the total energy density and the various contributions for µ = 1, α = 0. E (circle), F

(triangle), D (cross), V (square). With N = 5 (left) and N = 20 (right).

The specific heat density in fig.6 shows a small peak in Ω = 0. This peak does not increases

as N increases, therefore is not related to a phase transition.

Figure 6: Specific heat for µ = 1.

In order to gain some information on the composition of the fields we look at the order

parameters defined in the previous section. Starting from the scalar field ψ, fig.7 shows the

plots for ?ϕ2

constant, where the spherical contribution ?ϕ2

a?, ?ϕ2

0? and ?ϕ2

1? for N = 5. The three values ?ϕ2

a?, ?ϕ2

0? and ?ϕ2

1? seem essentially

0? to the full-power-of-the-field is dominant. The

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0.51.02.02.5

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0.5

1.0

1.5

2.0

2.5

?Z0i

2??N2

Figure 7: On the left comparison of ?ϕ2

comparison of ?Z2

a? (circle), ?ϕ2

0? (square) and ?ϕ2

1? (triangle) density. On the right

0a? (circle), ?Z2

00? (square) and ?Z2

01? (triangle) density.

behaviour of the Z0 fields (which describe the covariuant coordinates) is different. Here the

spherical contribution becomes dominant only for Ω approaching 0, starting from a zone in

12

Page 13

which the contribution of ?Z2

for ?Z2

plots are compatible to the Z0-case. The dependence of the previous quantities on N is shown

in the following plots fig.8. All previous parameters decrease with N, but the dominance of ϕ0

00? and ?Z2

01?, but taking into account the statistical errors, the other Zi-related

01? are comparable. For brevity we only show the plots

0a?, ?Z2

00? and ?Z2

Figure 8: Starting from the up left corner and from the left to the right the densities for ?ϕ2

?Z2

01? for µ = 1 varying Ω and N.

a?, ?ϕ2

0?, ?ϕ2

1?, Z2

0a,

00? and ?Z2

on the total-power-of-the-field is independent by N. The peak related to Z0decreases with N,

but if we look at the single plot for the spherical contribution at N = 20, the peak persists as Ω

approaches Ω = 0.

Now we will analyse the model for µ = 0. Fig.9 shows the plots for total energy density and

the contributions V , D, F. The slope of the total energy density seems to be constant. The

D-contribution and the F-contribution do not balance each other like in the previous case, but

all three contributions balance themselves to produce a constant sum.

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2

3

Eng?N2

Figure 9: Total energy density, various contributions and the comparison among them for µ = 0 varying Ω and

N. From the left to the right E, V , D, F and comparison.

13

Page 14

The specific heat density fig.10 shows again the small peak in Ω = 0 without N-dependence.

Figure 10: Specific heat density for µ = 0 varying Ω and N.

For the other quantities ?ϕ2

the same behaviour as in the case µ = 1.

a?, ?ϕ2

0?, ?ϕ2

1? and ?Z2

0a?, ?Z2

00?, ?Z2

01? we have according to fig.11

Figure 11: Starting from the up left corner and from the left to the right the densities for ?ϕ2

?Z2

01? for µ = 0 varying Ω and N.

a?, ?ϕ2

0?, ?ϕ2

1?,

0a?, ?Z2

00? and ?Z2

A completely different response of the system is obtained in the plots for µ = 3, as we can

see from fig.12. The slope of total energy density is very similar to the F-component instead of

D. However, there appears a sharp minumum around Ω = 0.1 and two maxima at Ω ≈ 0.6 and

Ω ≈ 1.8 for large N. This dramatic change in the plots might be interpreted as consequence of

a phase transition in the parameter µ. Actually, in the next section we will find a peak in the

specific heat density for some fixed Ω and varying µ ∈ [0,3].

The specific heat density fig.13 displays a strong change, too. In fact, instead of the peak

at Ω = 0, the peak appears close to the origin arround Ω = 0.15. This peak, in contrast to the

previous ones, grows as N increases and therefore could indicate a phase transition.

The fig.14 describes the behaviour of the order parameters densities ?ϕ2

0a?, ?Z2

µ = 0. For the ψ field the spherical contribution remains dominant. However, in the ?ϕ2

there appears a deviation from the constant slope. This deviation is evident for N = 5 but still

a?, ?ϕ2

0?, ?ϕ2

1? and

?Z2

00?, ?Z2

01?. They show a similar behaviour as the corresponding plots for µ = 1 and

1? plot

14

Page 15

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3

4

5

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Figure 12: Total energy density, various contributions and the comparison among them for µ = 3 varying Ω

and N. From the left to the right E, V , D, F and comparison.

Figure 13: Specific heat density for µ = 3 varying Ω and N.

Figure 14: Starting from the up left corner and from the left to the right the densities for ?ϕ2

?Z2

01? for µ = 3 varying Ω and N.

a?, ?ϕ2

0?, ?ϕ2

1?,

0a?, ?Z2

00? and ?Z2

15

Page 16

present for higher N. The order parameters for Z0display a peak close to the origin without

oscillations even for N = 5. This maximum for higher N does not move closer to the origin, in

other words, this shift is not caused by finite volume effects. Even for Z2

at Ω = 0 which becomes shifted and smoother for higher N.

01there appears a peak

5.3Varying µ

In this section we analyse the response of the system varying µ ∈ [0,3] while Ω is fixed at 0, 1 or

3, and α is always zero. We start displaying the plots fig.15 of the total energy density and of

various contributions for Ω = 0. There is no evident discontinuity but there appears a peak in

the total energy density around µ ≈ 2.5 for N = 20. Comparing all the contributions it is easy

to notice that the slope of the total energy is dictated by the curve V of the potential part.

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0.5

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Figure 15: The total energy density and the various contributions for Ω = 0 varying µ and N. From the left to

the right E, V , D, F an comparison with N = 5 (circle), N = 10 (square), N = 15 (triangle), N = 20 (cross).

For the comparison: E (circle), V (square), D (triangle), F (cross).

Figure 16: Specific heat density for Ω = 0 varying µ and N.

As mentioned before, the specific heat density fig.16 features a peak around µ ≈ 2.5 for

N = 20. Again, since the peak increases with N, we could relate this to a phase transition. The

plots for the quantities ?ϕ2

a? and ?ϕ2

0? show a strong dependence on µ, in particular the slope of

16

Page 17

?ϕ2

first three plots of fig.17 we deduce that close to the origin the non-spherical contribution ?ϕ2

is bigger than the spherical one ?ϕ2

dominant over ?ϕ2

0? seems mostly linear. The plot for ?ϕ2

1? also increases with µ, but not linearly. From the

1?

0?. Increasing µ, this situation capsises and ?ϕ2

0? becomes

1?.

Figure 17: Starting from the up left corner and from the left to the right the densities for ?ϕ2

?Z2

01? for Ω = 0 varying µ and N.

a?, ?ϕ2

0?, ?ϕ2

1?,

0a?, ?Z2

00? and ?Z2

The behaviour of the Z0fields as shown in the last three plots of fig.17 is quite different.

The spherical contribution is always dominant for the whole interval µ ∈ [0,3]. The curves for

?Z2

in particular there is a smooth descending step which becomes smoother for bigger N. However,

we admit that due to some cancellation effects, the statistical errors are quite big so that this

interpretation is not fully conclusive. Anyway, this result demonstrates the dependence of the

order parameter for Zi, and in general of the system, on the two choices Ω = 0 or Ω ?= 0.

Now we will analyse the model for Ω = 1. As fig.18 shows, the plots have a different slope

compared to the previous case. The maximum of total energy density follows the one of the

V -component. If we focus only on the total energy plot and compare it with the one for Ω = 0,

we notice a shift of the maximum for each N. In particular, in fig.18 some maxima are moved

outside the considered interval. We can find this shift very clearly looking at specific heat density

plotted in fig.19. Here again the peak both increases with N and is shifted to µ ≈ 3.3.

Fig.20 shows for ?ϕ2

displays an almost constant curve. However, close to the origin, the spherical contribution and

the first non-spherical one are comparable. The introduction of Ω ?= 0 creates, in the Z0-order

parameters shown in fig.20, a dependence similar to the plots for ψ. The full-power-of-the-field

density and the spherical contribution are no longer constant, they grow as µ increases. Even

in this case the spherical contribution is always dominant excluding the region around µ = 0.

0a?, ?Z2

00? are compatible to the constant slope. For ?Z2

01? we have the same dependence on µ,

a?, ?ϕ2

0? the same behaviour as in the case Ω = 0. The plot for ?ϕ2

1?

The last set of plots treats the case Ω = 3. The following diagrams for the energy and its

contributions show the absence of the previous peak. They show a sort of dilatation of the

former plots of fig.18.

17

Page 18

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3

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4

5

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Figure 18: Total energy density and contributions for Ω = 1 varying µ and N. From the left to the right E, V ,

D, F.

Figure 19: Specific heat density for Ω = 1 varying µ and N.

Figure 20: Starting from the up left corner and from the left to the right the densities for ?ϕ2

?Z2

01? for Ω = 1 varying µ and N.

a?, ?ϕ2

0?, ?ϕ2

1?,

0a?, ?Z2

00? and ?Z2

18

Page 19

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0.02

0.03

0.04

E?N2

Figure 21: Total energy density and the various contributions for Ω = 3 varying µ and N. From the left to the

right E, V , D,F .

The specific heat density fig.22 does not show the peak in zero anymore, and the curves do

not show any particular point as N increases. Actually, the peak can be found for higher µ.

?????? ?? ? ??

0.04

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0.02

0.03

0.05

0.06

C?N2

Figure 22: Specific heat density for Ω = 3 varying µ and N.

At last in fig.23 we find a behaviour of the density of the order parameters for the Z0and

ψ fields similar to the former plots for Ω = 1, and they are compatible with a dilatation of the

previous plots.

Conclusions and prospectives

We have studied a noncommutative gauge theory which arises by restriction of the spectral

action for harmonic Moyal space to finite matrices. For this quantum field theoretical model

we have performed Monte Carlo simulations to obtain, as function of the parameters (µ,Ω),

non-perturbative information for the energy density, for various contributions to the energy

density and for the specific heat density, as well as for a set of order parameters related to

sphericity. Despite the complexity of the approximated spectral action considered here, we were

able to obtain some reliable numerical results, showing that a numerical treatment of this kind

19

Page 20

Figure 23: Starting from the up left corner and from the left to the right the densities for ?ϕ2

?Z2

01? for Ω = 3 varying µ and N.

a?, ?ϕ2

0?, ?ϕ2

1?,

0a?, ?Z2

00? and ?Z2

of noncommutative gauge models seems feasible. However, as the restriction to finite matrices

shows severe differences to the original smooth action, the relevance of our results to the smooth

case is not clear.

The specific heat density shows various peaks which could indicate phase transitions. In

particular, studying the behaviour for some fixed mass parameter µ we found a relevant peak

close to Ω = 0 for µ = 3, and we noticed a big change in the energy density and in its contri-

butions between the cases µ ∈ {0,1} and µ = 3. Other peaks in the specific heat density can

be found varying µ and fixing Ω. The plots show that increasing Ω, the peak in the specific

heat which starts at µ ≈ 2.4 for Ω = 0 is moved towards higher µ. The order parameters we

introduced show a strong dependence on the cases Ω = 0 versus Ω ?= 0. Referring to the fixed-µ

plots we found a peak in the spherical contribution for the gauge fields Zi. Its behaviour can be

interpreted as a sort of symmetry breaking introduced by Ω ?= 0. Additionally, varying µ and

fixing Ω, the other parameters display a slope increasing with µ for all fields and all situations

but one: the plots of the order parameters ?Z0a?, ?Z00? for Ω = 0 show a constant behaviour.

The natural next steps in the numerical study of this model could be the computation of

the transition curves in order to separate the phase regions and to classify them using possibly

additional order parameters. Our treatment, forced by limited resource, was conducted varying

Ω in the range [0,3.1], since the Langmann-Szabo duality does not hold anymore in our case.

Actually, the computed plots do not show any periodicity in Ω ∈ [0,1] so that we can infer that

in contrast to the scalar case the range [0,1] is not enough to describe the system.

It will be very interesting to relax the condition µ2> 0. Implementing µ2< 0 amounts

to conduct the calculation no longer around the minimum of the action. The extension of the

space of parameters, together with the classification of the different phase regions, would allow

us to compare our model with the results of the simulation performed for the fuzzy spaces. In

particular, one could have a look at the occurrence of the so-called non-uniformly ordered phase

which is connected to UV/IR mixing. Since we have constructed our model starting from a

renormalisable one, this study is very desirable.

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Acknowledgements

This work has been supported by the Marie Curie Research Training Network MRTN-CT-2006-

031962 in Noncommutative Geometry, EU-NCG. Of particular importance was the interaction

with the Dublin Institute for Advanced Studies.

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