Phases of fuzzy field theories

Phases of fuzzy ﬁeld theories
Juraj Tekel
Department of Theoretical Physics
Faculty of Mathematics, Physics and Informatics
Comenius University, Bratislava
Seminar aus Mathematischer Physik, University of Vienna
24.3.2015

Juraj Tekel Phases of fuzzy ﬁeld theories

Short outline

Short outline
what are the noncommutative/fuzzy spaces
properties of scalar ﬁeld theories on these spaces
basics of matrix models
matrix model corresponding to fuzzy scalar ﬁelds
results for the triple point and phase diagram
current research, challenges and outlook

Noncommutative spaces

NC spaces
The usual sphere S2
is given by coordinates
xixi = R2
, xixj − xjxi = 0
which generate the algebra of functions
f = ak1k2k3 xki
i xixi = R2
which is commutative by deﬁnition.
Using this algebra, we can reconstruct all the information about the
sphere.
It turns out that any commutative algebra corresponds to some
diﬀerentiable manifold in a similar way.

NC spaces
Noncommutative spaces are (by definition) spaces, which are in
correspondence with noncommutative algebras
[xi, xj] = iΘij
The choice of Θ determines what the kind of space we have.
xi are operators on a Hilbert space. If this space is finite dimensional, we
call the space "fuzzy". It is a finite mode approximation to compact
manifold.

NC spaces
For the fuzzy sphere S2
F
xixi = ρ2
, xixj − xjxi = iθεijkxk
This can be realized as a N = 2j + 1 dimensional representation of the
SU(2)
xi =
2r
√
N2 − 1
Li , θ =
2r
√
N2 − 1
, ρ2
=
4r2
N2 − 1
j(j + 1) = r2
The coordinates xi still carry an action of SU(2) and thus the space still
has the symmetry of the sphere.
xi are N × N matrices and functions on S2
F are combinations of products
→ a herminitian matrix M

NC spaces
Hermitian N × N matrix can be decomposed into
M =
N−1
l=0
m=l
m=−l
clmTlm
matrices Tl
m are called polarization tensors and
[Li, [Li, Tlm]] = l(l + 1)Tlm
Tr (TlmTl m ) = δll δmm

NC spaces
Hermitian N × N matrix can be decomposed into
M =
N−1
l=0
m=l
m=−l
clmTlm
there is an analogy with spherical harmonics on S2
f(φ, θ) =
∞
l=0
m=l
m=−l
al
mY l
m(φ, θ)
algebra of functions on S2
F is the algebra of functions on S2
with a basis
cut oﬀ, such that functions with l > L a L = N − 1
The multiplication rule has changed form a commutative pointwise
multiplication to a noncommutative matrix multiplication.
Limit of large N reproduces the original sphere S2
.

NC spaces
By introducing a cut oﬀ on l we introduce a maximal momentum ∼ N,
which in turn means a smallest possible distance ∼ 1/N ≈ θ.
Number of extrema of Y l
m is ∼ 2l2
, so one extremum corresponds to area
∼ 1/l2
and function Y l
m can give us information about distances ∼ 1/l.
Points on S2
are encoded in the algebra of functions as sequences of
gaussians converging to the corresponding δ-function. But after the cutoﬀ
we can not construct an arbitrarily narrow gaussian and we can no longer
construct and arbitrarily precise approximation of a point.
The points on S2
F no longer exist.

Fuzzy ﬁeld theory

Scalar field on S2
F
Scalar field is a function on S2
F , i.e. an N × N hermitian matrix.
Derivatives become commutators with generators Li, integrals become
traces and we can write and Euclidean field theory action
S = −
1
2
Tr ([Li, M][Li, M]) +
1
2
rTr M2
+ V (M) =
=
1
2
Tr (M[Li, [Li, M]]) +
1
2
rTr M2
+ V (M)
The theory is given by functional correlation functions
F =
dM F[M]e−S[M]
dM e−S[M]
These can be computed using a noncommutative version of the Feynmann
rules with a propagator
clmcl m =
δll δmm
l(l + 1) + r
Fuzzy scalar field theory is a matrix model!

Scalar ﬁeld on CPn
F
Taking Li to be generators of SU(n + 1), we can obtain xi’s as coordinates
on fuzzy version of CPn
, with a corresponding ﬁeld theory action
S =
1
2
Tr (M[Li, [Li, M]]) +
1
2
rTr M2
+ V (M)
where i = 1, . . . , (n + 1)2
− 1, polarization tensors
[Li, [Li, Tlm]] = l(l + n)Tlm
Tr (TlmTl m ) = δll δmm
a more complicated dependence of N on l, but a theory which is still a
particular matrix model.

UV/IR mixing
The trademark property of the noncommutative field theories is the
UV/IR mixing, which arises as a consequence of the nonlocality of the
theory.
Quanta can not be compressed into arbitrarily small volume and if we try
to do so in one direction, they will stretch in a different direction.
Processes at high momenta contribute to processes at low momenta.
In theories on noncompact spaces, e.g. Moyal plane or R4
, divergence of
non-planar diagrams at small momentum
Due to no clear separation of scales the theory is not renormalizable.
The commutative limit of such noncommutative theory is (very) different
than the commutative theory we started with.

UV/IR mixing
To obtain a renormalizable theory with a well deﬁned commutative limit,
we have to modify the original action such that we remove the UV/IR
divergence
Highly nontrivial, several approaches
B.P. Dolan, D. O’Connor and P. Prešnajder
Matrix φ4
models on the fuzzy sphere and their continuum limits
[arXiv:/0109084]
H. Grosse and R. Wulkenhaar
Renormalization of φ4
theory on noncommutative R4
in the matrix base
[arXiv:0401128]
R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa
A translation-invariant renormalizable non-commutative scalar model
[arXiv:0802.0791]

UV/IR mixing
In our setting the UV/IR mixing will arise as an extra phase in the phase
diagram, not present in the commutative case.
The commutative field can oscillate around zero, in which case
Tr (M) = 0 or around some nonzero value Tr (M), which is a minimum of
the potential.
Noncommutative theories will exhibit a third phase, where the field does
not oscilate around any fixed value. This is a consequence of the non
locality of the theory.

Random Matrix Models

Some basic results of random matrix theory
The random variable is N × N hermitian matrix M, expressed as
M =
N−1
l=1
l
m=−l
cl
mTl
m
We will expect SU(N) symmetric probability distribution
1
Z
e−N2
S[M]
, S[M] =
1
N
1
2
rTr M2
+
N
n=3
gnTr (Mn
)
The mean value of matrix function f(M) is
f =
1
Z
dM e−N2
S[M]
f(M)
where
dM =
i≤j
dReMij
i<j
dImMij
What is the distribution of the eigenvalues of the random matrix M?

The key is diagonalization, we write M = UΛU−1
for some U ∈ SU(N)
and Λ = diag(λ1, . . . , λN ), the integration measure becomes
dM = dU
N
i=1
dλi ×
i<j
(λi − λj)2
The problem is now N in an eﬀective potential (eigenvalue repulsion)
Seff =
1
N
1
2
r
i
λ2
i + g
i
λ4
i −
2
N2
i<j
log(λi − λj)

The key is diagonalization, we write M = UΛU−1
for some U ∈ SU(N)
and Λ = diag(λ1, . . . , λN ), the integration measure becomes
dM = dU
N
i=1
dλi ×
i<j
(λi − λj)2
The problem is now N in an eﬀective potential (eigenvalue repulsion)
Seff =
1
N
1
2
r
i
λ2
i + g
i
λ4
i −
2
N2
i<j
log(λi − λj)
The eigenvalue distribution
ρ(λ) =
1
N
N
i=1
δ(λ − λi)
will become continuous in the large N limit

In this limit
i
g(xi) = N dx g(λ)ρ(λ)
So the eﬀective action
Seff =
1
N
1
2
r
i
λ2
i + g
i
λ4
i −
2
N2
i<j
log(λi − λj)
becomes
Seff = dλ
1
2
rλ2
+ gλ4
ρ(λ) − 2 dλ dλ log(λ − λ )ρ(λ)ρ(λ )

In this limit
i
g(xi) = N dx g(λ)ρ(λ)
So the eﬀective action becomes
Seff = dλ
1
2
rλ2
+ gλ4
ρ(λ) − 2 dλ dλ log(λ − λ )ρ(λ)ρ(λ )
Thanks to the factor N2
the large N average f will be given by the
conﬁguration of eigenvalues which minimizes this action
f =
i
f(xi) = dλ f(λ)ρ(λ)
Euler-Lagrange equation for ρ is
rλ + 4gλ3
= 2P dλ
ρ(λ )
λ − λ
which can be solved by complex analysis, with an additional assumption
on the support of the distribution

Assuming the support to be only a single interval (−a, a), we get the
distribution (normalized to 1, not N)
ρ(λ) =
1
π
1
2
r + ga2
+ 2gλ2
a2 − λ2
a2
=
1
6g
r2 + 48g − r

ρ(λ) =
1
π
1
2
r + ga2
+ 2gλ2
a2 − λ2
a2
=
1
6g
r2 + 48g − r
for g = 0 the Wigner semicircle

ρ(λ) =
1
π
1
2
r + ga2
+ 2gλ2
a2 − λ2
a2
=
1
6g
r2 + 48g − r
for nonzero g

ρ(λ) =
1
π
1
2
r + ga2
+ 2gλ2
a2 − λ2
a2
=
1
6g
r2 + 48g − r
for negative r the potential has two wells

ρ(λ) =
1
π
1
2
r + ga2
+ 2gλ2
a2 − λ2
a2
=
1
6g
r2 + 48g − r
for negative r less dense around origin

If we allow negative r, the dip in the potential can be so deep that it
splits the eigenvalues completely.

If we allow negative r, the dip in the potential can be so deep that it
splits the eigenvalues completely.
At that point, the formula
ρ(λ) =
1
π
1
2
r + ga2
+ 2gλ2
a2 − λ2
becomes negative and looses the interpretation of a probability
distribution
This happens, if
r ≤ −4
√
g
We need to change the one-cut assumption and look for a distribution
supported on two disjoint intervals (−a, −b) ∪ (b, a) with the solution
ρ(λ) =
2g|λ|
π
(a + λ)(a − λ)(λ + b)(λ − b)
a2
=
|r|
4g
+
1
√
g
, b2
=
|r|
4g
−
1
√
g
We thus obtain the phase diagram.

Asymmetric eigenvalue distributions
What if we start with probability distribution, which is not symmetric, eg.
S[M] =
1
N
1
2
rTr M2
+ hTr M3
+ gTr M4
we still get the one-cut and the two-cut distributions, but qualitatively
diﬀerent third possibility - an asymmetric one-cut distribution.
This happens if one of the wells of the potential is deep enough.

Fuzzy ﬁeld theory matrix model

As we have seen, action of the fuzzy CPn
is given by
S =
1
N
1
2
Tr (M[Li, [Li, M]]) +
1
2
rTr M2
+ gTr M4
= Skin[M] +
1
N
1
2
rTr M2
+ gTr M4
and the correlation functions are computed as
F =
dM F[M]e−N2
S[M]
dM e−N2S[M]
=→ dλF(λ)ρ(λ)
the same kind of integrals as for a matrix model before

A clear problem is that the diagonalization trick no longer works
dM e−N2
S[M]
= dΛ ∆2
e−NTr(1
2 rM2
+gM4
)dU e−N2
Skin[U†
ΛU]
As long as the function F is invariant, we can write
dU e−N2
Skin[U†
ΛU]
= e−N2
Seff [Λ]
and the model becomes a quartic model with new terms
S[Λ] = Seff [Λ] +
1
N
1
2
r
i
λ2
i + g
i
λ4
i −
2
N2
i<j
log(λi − λj)
the new terms change the EOM for the eigenvalue distribution

Numerical results
dM integral can be treated numerically and computed by Monte-Carlo.
Most recently by B. Ydri [arXiv:1401.1529]
The following picture from F. Garcia Flores, X. Martin and D. O’Connor
[arXiv:0903.1986]

Numerical results

Perturbative calculation
One possible analytic treatment due to D. O’Connor and Ch. Sämann
[arXiv:0706.2493,arXiv:1003.4683]
Expand the exponential in powers of kinetic term
e−aTr(M[Li,[Li,M]])
= 1−aTr (M[Li, [Li, M]])+
1
2
a2
[Tr (M[Li, [Li, M]])]
2
+. . .
and perform the dU integral explicitly.
For the fuzzy sphere, the result is (up to third order in a)
Skin = a
1
2
c2 − c2
1 − a2 1
24
c2 − c2
1
2
+ a3 1
432
c3 − 3c1c2 + 2c3
1
2
where
cn = Tr (Mn
)
Note, that we obtain a multi-trace model, since the eﬀective action
contains powers of traces of M.
We will see how odd multi-traces generate asymmetric solutions.
The terms become very nasty and diﬃcult to handle beyond the third
order.

Bootstrap
Most recently, Ch. Sämann [arXiv:1412.6255] introduced a method using
a version of Schwinger-Dyson equations to compute the eﬀective action up
to fourth order in a, with and extra term (in the symmetric c1 = 0 case)
a4 1
2880
m4
2 − a4 1
3456
m4 − 2m2
2
2
From here on, we rescale the matrix M and couplings r, g with N in such
a way, that all the terms in the eﬀective action contribute in the large N
limit, i.e. are of order N2
.

Different Bootstrap
Finaly, A. Polychronakos [arXiv:1306.6645] used the following method to
compute a part of the effective action non-perturbatively.
It starts from previous result, that the free theory g = 0 eigenvalue
distribution is still a semicircle, with a rescaled radius. The effective
action can thus be split into two parts
Seff =
1
2
F(m2) + R[mn − tn]
where mn are the symmetrized moments of the distribution
mn = Tr M −
1
N
Tr (M)
n
and tn are the (symmetrized) moments of the semicircle distribution
t2n = Cnmn
2
and odd vanish.

Computation of F(t)
We will concentrate on F part, i.e. we work with the action
S[Λ] =
1
2
F
i
λ2
i +
1
N
1
2
r
i
λ2
i + g
i
λ4
i −
2
N2
i<j
log(λi − λj)
=
1
2
F (c2) +
1
N
1
2
rc2 + gc4 −
2
N2
i<j
log(λi − λj) (1)
we have assumed a symmetric distribution c1 = 0, keeping this in mind
later.
The corresponding EOM for the minimizing eigenvalue distribution is
1
2
F (c2)
dc2
dλi
+
1
2
r
dc2
dλi
+ g
dc4
dλi
= [F (c2) + r] λi + 4gλ3
i
= 2
j
1
λi − λj

Computation of F(t)
[F (c2) + r] λi + 4gλ3
i = 2
j
1
λi − λj
We see, that the multi-trace term containing c2 acts as an eﬀective mass,
which depends on the second moment of the distribution. In the same
way, terms containing c1 and c3 will introduce terms into the eﬀective
potential that break the symmetry.
One can use this equation to derive the equation for the radius of the
g = 0 semicircle. Then, we need to make sure that the resulting
distribution yields a correct second moment.
We get two equations that determine the F(t) function completely.

Computation of F(t)
This approach is applicable for a more general theory, where
Skin[M] =
1
2
Tr (MKM)
for which the kinetic action is diagonal, i.e.
KTlm = K(l)Tlm

Computation of F(t)
The ﬁnal result is
F (t) + t =
N2
f(t)
where
f(z) =
l
dim(l)
K(l) + z
For the fuzzy sphere, we can compute F(t) in closed form
F(t) = log
t
1 − e−t
For a general CPn
, f(z) is given in terms of hypergeometric function and
the equation is transcendental. We are left only with a perturbative
calculation
F(t) = A1t + A2t2
+ A3t3
+ . . .

Computation of F(t)
We can compare the coeﬃcients An with the perturbative result
Skin = a
1
2
c2 − c2
1 − a2 1
24
c2 − c2
1
2
+ a4 1
2880
c4 − 2c2
2
2
without too much surprise they do agree.
Also for CP2
and CP3
, which involve also a a3
term.

Phase transition
We can now compute the phase transition line. The two conditions we
have are
vanishing of the distribution → F (t) + r = −4
√
g
consistency on the second moment → t =
1
√
g
This gives
r(g) = −4
√
g − F (1/
√
g)
For the fuzzy sphere this becomes
r(g) = −5
√
g −
1
1 − e1/
√
g
For higher CPn
only a perturbative expansion in powers of 1/
√
g.

Phase transition
The series from
r(g) = −4
√
g − F (1/
√
g)
is asymptotic for small g. We could use Borel summation to resum the
series, but not enough terms are known for practical purposes.
To obtain numerical results, we use Pade approximation by a rational
function.
But this teaches us a lesson. The phase transition lines and other
expressions we are going to encounter are not analytic around g = 0 and
we need to be very careful when interpreting the results of perturbative
computations. For example for the above formula
r(g) = −4
√
g −
1
2
+
1
12
√
g
−
1
720g3/2
+
1
30240g5/2
+ . . .

Asymmetric distributions
To introduce asymmetry, we recall that we should have
F(t) → F(c2 − c2
1)
yielding an equation for the distribution
F (c2 − c2
1)(λi − c1) + rλi + 4gλ3
i = 2
j
1
λi − λj
We can think of this as an effective potential
V (λ) = − F (c2 − c2
1)c1 λ +
1
2
F (c2 − c2
1) + r λ2
+ gλ4
i.e. the advertised asymmetry. Note that for symmetric distributions the
asymmetric term vanishes, so the one-cut/two-cut phase transition is not
modified.
We now look for the solution in the form of an asymmetric one cut,
supported on one interval (b, a). We again need to impose self consistency
equations on the first and second moments of the distribution.

Asymmetric distributions
We write the interval as (D −
√
δ, D +
√
δ). The solution then is
ρ(λ) = 4D2
g + 2δg + r + F
δ
4
+
δ3
g
16
−
9D2
δ4
g2
16
+ 4Dgλ + 4gλ2
×
×
1
2π
δ − (D − λ)2
The conditions on D, δ can be solved only perturbatively.
To ﬁnd the domain of existence of the asymmetric one cut solution, we
seek the values of parameters when this distribution becomes negative.

Asymmetric phase transition
We can do this perturbatively
r = −2
√
15
√
g +
4A1
5
+
135A2
1 + 848A2
1500
√
15
1
√
g
+
+
2025A3
1 + 10170A1A2 + 11236A3
562500
1
g
+ . . .
where Ai’s from
F(t) = A1t + A2t2
+ A3t3
+ . . .
Note, that we again obtain a series in 1/
√
g, but this time the series is
convergent.

We compute the intersection of the two curves
rC = −0.73 , gC = 0.022
The numerical result is

rC = −0.73 , gC = 0.022
[arXiv:0601012] , rC = −0.8 ± 0.08 , gC = 0.036 ± 0.013

rC = −0.73 , gC = 0.022
[arXiv:0601012] , rC = −0.8 ± 0.08 , gC = 0.036 ± 0.013
[arXiv:0903.1986] , rC = −2.3 ± 0.2 , gC = 0.13 ± 0.005

rC = −0.73 , gC = 0.022
[arXiv:0601012] , rC = −0.8 ± 0.08 , gC = 0.036 ± 0.013
[arXiv:0903.1986] , rC = −2.3 ± 0.2 , gC = 0.13 ± 0.005
[arXiv:1401.1529] , rC = −2.49 ± 0.07 , gC = 0.145 ± 0.025
What is the source of discrepancy?

Some topics of current research

Fourth order contribution
We can use Sämann’s result for the effective action for the computation.
Skin =a
1
2
c2 − c2
1 − a2 1
24
c2 − c2
1
2
+ a3 1
216
c3 − 3c1c2 + 2c3
1
2
+
+ a4 1
2880
(c2 − c2
1)4
− a4 1
3456
(c4 − 4c − 1c3 + c2c3
1 − 3c − 14
) − 2(c2
The main distinction is the presence of c2
4 term, which acts as an effective
coupling.
The properties of the asymmetric transition line are not altered
dramatically. But the symmetric transition line gets shifted quite
drastically, moreover loosing Borel summability! Also Pade approximation
is quite involved. The overall effect is to shift the triple point to even
lower values of r and g, further away from the numerical result.

Fourth order contribution
The fourth order effective coupling has another interesting feature. It can
be expressed as
geff = g + 2F1 c40 − 2c2
20
with F1 negative. So for distributions that give negative second term, one
can have geff > 0 even for g < 0.
Such distributions are "spread out" more than semicircle, so for a
negative reff , we might have a region of parameter space where we obtain
a stable solution even for negative coupling.
Two caveats
seems that this is not the case for fuzzy sphere,
might be only a perturbative effect and the full effective action might
remove this effect.

Free energies
The sole existence of the asymmetric one-cut solution does not guarantee,
that the solution is going to be realized. Even with no kinetic term,
asymmetric solution exists as long as
r ≤ −2
√
15
√
g
but is never realized since it has higher free energy as the double-cut.

Free energies
The sole existence of the asymmetric one-cut solution does not guarantee,
that the solution is going to be realized. Even with no kinetic term,
asymmetric solution exists as long as
r ≤ −2
√
15
√
g
but is never realized since it has higher free energy as the double-cut.
The result is still interesting, because it gives a top bound on the triple
point, but for a complete result we should compute the free energy of the
distributions.
Or do something else.

Outlook

What is next?
Complete treatment of the eﬀective action.
What is the non-perturbatve formula for the kinetic term eﬀective action?
Can it be computed?

What is next?
Diﬀerent sources of asymmetry
Contributions of nontrivial conﬁgurations to the saddle point, instantons.

What is next?
Limit CPn
→ R2n
.
Phase diagram in the non-compact case, phase structure of the theory on
non-commutative plane, etc.

What is next?
Limit CPn
→ R2n
.
Phase diagram of the theories with no UV/IR mixing.
Does the non-uniform order phase disapear from the diagram? What is
the mechanism? What are the lessons to be learned

What is next?
Limit CPn
→ R2n
.
Phase diagram nonenormalizable theories.
Is there anything we can learn from the phase structure of theories, which
are not renormalizable even at the commutative level, eg. φ6
on R4
or φ4
on R6
.

What is next?
Limit CPn
→ R2n
.
Theory on R × S2
F .
Theory with a commutative time. There are similar perturbative and
numerical results available to compare with.

What is next?
Limit CPn
→ R2n
.
Theory on R × S2
F .
Theory on S4
.
There is a realization of the theory on fuzzy S4
that is suitable for our
approach.

What is next?
Limit CPn
→ R2n
.
Theory on R × S2
F .
Theory on S4
.

Thanks for your attention!

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