Random Matrix Theory, Dirac Spectra and
the Dimensionality of Space-Time
Jacobus Verbaarschot
[email protected]
Stony Brook University
Krakow, July 2014
Dirac Spectra – p. 1/4
Acknowledgments
Collaborators: Gernot Akemann (Bielefeld)
Poul Damgaard (NBIA)
Mario Kieburg (Bielefeld)
Kim Splittorff (NBI)
Savvas Zafeiropoulos (Clermont-Ferrand)
Dirac Spectra – p. 2/4
Relevant Papers
M. Kieburg, J. J. M. Verbaarschot and s. Zafeiropoulos, Dirac Spectra of 2-dimensional QCD-like
theories, submitted to Phys. Rev. D (2014), [arXiv:1405.0433].
P.H. Damgaard, K. Splittorff and J. J. M. Verbaarschot, Microscopic Spectrum of the Wilson Dirac
Operator, Phys. Rev. Lett.105 (2010)
G. Akemann, P.H. Damgaard, K. Splittorff and J. J. M. Verbaarschot, Spectrum of the Wilson
Dirac Operator at Finite Lattice Spacing, Phys. Rev. D83 (2011).
K. Splittorff and J. J. M. Verbaarschot, The Wilson Dirac Spectrum for QCD with Dynamical
Quarks, Phys. Rev. D 84 (2011) 065031 [arXiv:1104.6229 [hep-lat]].
M. Kieburg, J. J. M. Verbaarschot, S. Zafeiropoulos Eigenvalue Density of the Non-Hermitian
Wilson Dirac Operator„ Phys. Rev. Lett. 108 (2012) 022001 arXiv:1109.0656 [hep-lat].
J.J.M. Verbaarschot and T. Wettig, The Spectrum of One-Flavor QCD at θ = 0 and the
Continuity of the Chiral Condensate, in preparation.
Dirac Spectra – p. 3/4
Contents
I. Dirac Spectra in QCD
II. Disordered Systems versus lattice QCD
III. Wilson Dirac Operator and Tail States
IV. Dirac Spectra in Two Dimensions
V. Conclusions
Dirac Spectra – p. 4/4
I. Dirac Spectrum in QCD
QCD Partition Function
Banks-Casher Formula
Picture of the Dirac Spectrum
Dirac Spectra – p. 5/4
QCD Partition Function
The QCD partition at temperature 1/β and chemical potential µ is
given by
X
ZQCD =
e−β(Ek −µ) ,
k
where the sum is over all states. At zero momentum the energy of the
states is equal to the mass of the particles.
This partition function can be rewritten as a Euclidean quantum field
theory
ZQCD (m)
= hdetNf (D + m)iYM .
Dirac
operator
quark mass
matrix
Dirac Spectra – p. 6/4
Dirac Operator in Four Space-Time Dimensions
Chiral Block Structure

D=
0
id(A)
−id† (A)
0

.
Eigenvalues are zero or occur in pairs ±λk .
ZQCD (m) =
X
ν
νNf
m
Y
h (iλk + m)Nf i.
k
where ν is the topological charge and Nf is the number of flavors.
Dirac Spectra – p. 7/4
The Free Dirac Spectrum
The solution of the free Dirac equation are plane waves. Since
P 2
(γµ dµ ) =
dµ the eigenvalues are given by
λnk = ±(
d
X
π 2 n2
k 1/2
k=1
L2
)
.
The total number of eigenvalues < λ is equal to
N (λ) ∼
λL
π
d
.
Then the eigenvalue density is
dN (λ)
ρ(λ) =
∼ V λd−1 .
dλ
Dirac Spectra – p. 8/4
Free Dirac Spectrum
5
ΡHΛL
d=2
4
d=4
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Λ
The free Dirac spectrum in 2 and 4 dimensions for N = 1000 and 100,
respectively.
Dirac Spectra – p. 9/4
Euclidean Gauge Fields and the Dirac
Spectrum
The gauge fields fluctuate randomly with probability distribution
given the the Yang-Mills action.
In the presence of gauge fields the modes of the free Dirac
operator become coupled resulting in a repulsion of the
eigenvalues.
The eigenvalues move to the place where there are no
eigenvalues, i.e. to zero.
Since the free density is proportional to the volume, we expect that
the interacting density is also proportional to the volume.
Dirac Spectra – p. 10/4
Picture of the Dirac Spectrum
Zero Modes
3
ρ(λ )
∼V λ
VΣ
π
λ
Because of asymptotic freedom, the Dirac spectrum should
approximate the free one for λ ≫ ΛQCD
What is the origin of the small eigenvalues?
Dirac Spectra – p. 11/4
Facts About QCD
The QCD partition function is invariant under SU (Nf ) × SU (Nf )
which is broken spontaneously to SU (Nf ) .
The order parameter for the spontaneous breaking is the chiral
condensate hq̄qi ≈ −(240 M eV )3 .
According to Goldstones theorem, the spontaneous breaking is
associated with Goldstone Bosons, the pions. QCD is a confining
theory and does not have any other light excitations in the broken
phase so that the QCD partition function can be approximated by a
partition function of pions. In the chiral limit, this approximation
becomes exact.
QCD is a strongly coupled gauge theory and fist principle studies
require lattice QCD simulations.
Dirac Spectra – p. 12/4
Chiral Condensate
Order parameter for the chiral phase transition
Z
1
1 d
d4 xhq̄qi =
log ZQCD (m)
V
V dm
1 d ν Y
m h (iλk + m)i
=
V dm
k
+
*
Y
1 X
1
=
(iλk + m) .
V
m + iλk
k
k
Experimentally, hψ̄ψi = (−240M eV )3 .
Physically this indicates that QCD has a nontrivial vacuum structure.
The chiral condensate is the analogue of the magnetization of the
ground state.
Dirac Spectra – p. 13/4
Banks-Casher Formula
iλk
m
C
∆λ ∼
1
V
The chiral condensate, Σ(m) , is given
by the Banks-Casher formula
*
+
1 X
1
Σ(m) = lim
V →∞
V
iλk + m
k
I
dsΣ(s) = il(Σ(m) − Σ(−m))
Σ (m)
m
= 2πilρ(0)/V
Σ(m) =
πρ(0)
,
V
Banks − Casher
The chiral condensate is the discontinuity of the resolvent for V → ∞
when m crosses the imaginary axis. That is why we are interested in
the behavior of the small eigenvalues.
Eigenvalue spacing near zero: ∆λ =
1
ρ(0)
=
π
ΣV
.
Dirac Spectra – p. 14/4
II. Dirac Spectrum and Spontaneous Symmetry
Breaking
Generating Function for the Dirac Spectrum
Chiral Lagrangian
Chiral Random Matrix Theory
Dirac Spectra – p. 15/4
Partial Quenching and the Dirac Spectrum
The resolvent follows from the partial quenched generating
function
*
+
Nf
d det (D + m) det(D + z) .
G(z) =
′
dz
det(D + z )
′
z =z
This is a partition function of Nf + 1 fermionic quarks and 1
bosonic quark.
Because of spontaneous symmetry breaking, the low-energy limit
of the partial quenched generating function is given by a weakly
interacting systems of Goldstone bosons and fermions.
Dirac Spectra – p. 16/4
Spontaneous Symmetry Breaking
They interact according to a Lagrangian that is completely
determined by the symmetries of the microscopic theory.
Z
R
†
− 21 mΣTr(U +U † )]
−[ d4 x[ 14 Fπ Tr∂µ U ∂µ
Z=
dU e
.
U/inG/H
The masses of the Goldstone particles are given by
2
Mmm
=
2mΣ
,
2
Fπ
2
Mmz
=
(m + z)Σ
,
2
Fπ
2
Mzz
=
2zΣ
Fπ2
When the pion Compton wavelength is much larger than the size
of the box, the zero momentum part factorizes from the partition
function. This is the microscopic domain of QCD.
Dirac Spectra – p. 17/4
Random Matrix Theory
It is clear that in the microscopic domain all theories with a mass
gap and the same global symmetries and pattern of spontaneous
symmetry breaking have the same low energy effective theory and
therefore the same Dirac spectrum.
Nonuniversal contributions enter when fluctuations of nonzero
momentum modes complete with the fluctuations of the zero
momentum modes, i.e.
2zΣ
4π 2
∼ 2
2
Fπ
L
JV-1996,Janik-Nowak-Papp-Zahed-1998,Osborn-JV-1998,
Osborn-Toublan-JV-1999,Damgaard-Osborn-Toublan-JV-1999
Dirac Spectra – p. 18/4
Maximum Flavor Symmetry
In QCD the chiral condensate has maximum flavor symmetry.
Peskin-1980
The global symmetries of chiral random matrix theories are broken
spontaneously in the same way as is the case for QCD. Since
QCD is a strongly interacting dynamical theory, this is a highly
non-trivial result.
Dirac Spectra – p. 19/4
Chiral Random Matrix Theory
This is a theory with the global symmetries of QCD, but matrix
elements. of the Dirac operator replaced by random numbers
(JV-1994, Shuryak-JV-1992). In the sector of topological charge ν
the random matrix Dirac operator is given by


m iW
†

 , P (W ) ∼ e−N TrW W
D=
iW † m
where W a n × (n + ν) matrix so that D has exactly ν zero
modes.
Example.
The matrix
D3 has exactly one zero
mode

0

D3 = 
 0
a∗

0
a
0

b 
.
0
b∗
Dirac Spectra – p. 20/4
Remarks
The chRMT partition function is given by
Z
ν
ZchRMT
= dW detNf (D + m)P (W ).
It came as a surprise that the behavior of the smallest Dirac
eigenvalues of a complicated nonlinear field theory can be
obtained analytically.
Powerful RMT techniques can be used to evaluate observables.
chiral Random Matrix Theory has become a standard tool in lattice
QCD.
The random matrix theory is determined by the anti-unitary
symmetries and by the dimensionality of spacetime.
Dirac Spectra – p. 21/4
III. Dirac Spectrum and the Dimensionality of
Space- Time
Anti-Unitary Symmetries
Chiral Blocks
Classification
Topology and Random Matrix Theory
Schnyder-Ryu-Fursaki-Ludwig-2008, DeJonge-Frey-Imbo-2012,
Kieburg-JV-Zafeiropoulos-2014
Dirac Spectra – p. 22/4
Dimensionality and Symmetries
Dirac operator:
D=
Pd
µ=1
γµ Dµ (Aµ ).
d=4
{γ5 , D} = 0,
Anti-unitary symmetry: [γµ , γ2 γ4 K] = 0
[γ2 γ4 K, 21 (1 ± γ5 )]
d=2
γ1 = σ1 , γ2 = σ2 , γ5 = σ3
{γ5 , D} = 0,
Anti-unitary symmetry: [iσµ , σ2 K] = 0
[σ2 K, 21 (1 ± σ3 )] 6= 0.
d=3
γ1 = σ1 , γ2 = σ2 , γ3 = σ3
There is no γ5 that anti-commutes with D .
Anti-unitary symmetry: [iσµ , σ2 K] = 0
Dirac Spectra – p. 23/4
Chiral Blocks
d=4
d=2

D=
0
C
†

,
−C
0
Chiral ensembles


0
C

,
D=
−C † 0
βD = 1, 2, 4.
JV-1993
Because [σ2 K, ( 21 (1 ± γ5 )] 6= 0 we cannot use the Dyson
index. to classify the Dirac operator. Instead, because of
the anti-unitary symmetry, C , is either complex
symmetric or complex anti-symmetric.
Class CI or DIII.
Kieburg-JV-Zafeiropoulos-2013
d=3
No decomposition into chiral blocks, βD = 1, 2, 4 .
JV-Zahed-1994
Wigner-Dyson Ensembles
Dirac Spectra – p. 24/4
Spontaneous Symmetry Breaking and
Dimensionality
Dimension
βD
Symmetry Breaking Pattern
RMT
2
1
U Sp(2Nf ) × U Sp(2Nf ) → U Sp(2Nf )
(CI)
2
2
2
4
O(2Nf ) × O(2Nf ) → O(2Nf )
(DIII)
3
1
U Sp(4Nf ) → U Sp(2Nf ) × U Sp(2Nf )
GOE (AI)
3
2
3
4
O(2Nf ) → O(Nf ) × O(Nf )
GSE (AII)
4
1
U (2Nf )/Sp(2Nf )
chGOE (BDI)
4
2
U (Nf ) × U (Nf ) → U (Nf )
chGUE (AIII)
U (Nf ) × U (Nf ) → U (Nf )
U (2Nf ) → U (Nf ) × U (Nf )
chGUE (AIII)
GUE (A)
4
4
U (2Nf )/O(2Nf )
chGSE (CII)
Chiral symmetry breaking in two, three and four dimensions for
different values of the Dyson index βD . Also indicated is the random
matrix theory with the same breaking pattern and the corresponding
symmetric space. JV-1994, Altland-Zirnbauer-1996, Kieburg-JV-SZ-2013
Dirac Spectra – p. 25/4
Topology and Random Matrix Theory
As we have seen for the 4-dimensional chiral ensembles, topology
is implemented by means of the index of the Dirac operator.
If the random matrix theory cannot be deformed to have a nonzero
index, the gauge fields of the underlying microscopic theory cannot
have topology.
Dirac Spectra – p. 26/4
Topology and Random Matrix Theory
d=4
Π3 (SU (Nc )) = Z .
There are instantons with the number of zero modes
given by the Atiyah-Singer index theorem. The
off-diagonal block can be deformed trivially to
have a nonzero index.
d=2
Π1 (SU (Nc )) = 0,
Π1 (SU (Nc )/ZNc ) = ZNc .
There no zero modes for fundamental fermions, and for
adjoin fermions there are zero modes in the complexified
space. A symmetric or anti-symmetric off-diagonal block
cannot be deformed to have a nonzero index
d=3
Π2 (SU (Nc )) = 0.
There are no topological zero modes.
The Wigner-Dyson ensembles cannot be deformed
to have a nonzero index.
Dirac Spectra – p. 27/4
Sum over Topology
The QCD partition function includes a sum over topology and so
do all observables such as for example the chiral condensate.
In random matrix theory it is natural to work at fixed topology.
In the QCD the primary objects are averages that include the sum
over topology.
What is the spectral density of Dirac operator at θ = 0?
ρ(m, x) =
P
Zν (m)ρν (x)
νP
ν Zν (m)
.
This sum can be evaluated in the microscopic domain of one-flavor
QCD.
Note that Zν (−|m|) ∼ (−|m|)ν for Nf = 1
Dirac Spectra – p. 28/4
Dirac Spectrum of One-Flavor QCD at θ = 0
The quenched part of the spectral
density at θ = 0 , ρq (x, m) .
ρq (x, m)
=
1
π
ρZM (x, m)
Z
1
0
=
2
e−2mV t dt
J1 (2xV t).
√
t 1 − t2
|m|V
e
X
ν
The dynamical part of the spectral
density at θ = 0 , ρd (x, m) .
2
e−2mV t dt
ρd (x, m) =
√
1 − t2
0
ˆ
˜
2
× xtJ1 (2xV t) + m(1 − 2t )J0 (2xV t) .
x
2
−
π x2 + m2
Z
1
JV-Wettig-2014
−mV
|ν|Iν (mV )δ(x) = e
(I0 (mV ) + I1 (mV ))δ(x)
Dirac Spectra – p. 29/4
Mathematical Detail
To evaluate the microscopic spectral density at fixed θ-angle we need
sums of the form
Sa,b,c (x, m, θ) =
∞
X
eiνθ Iν+a (m)Jν+b (x)Jν+c (x)
ν=−∞
They can be reduced to one-dimensional integrals. Examples are
Z
1
2
Iν (m)Jν+1 (x)Jν−1 (x) = −
π
Z
X
ν
X
ν
Iν (m)Jν2 (x)
=
2
π
0
0
dt
m−2mt2
√
e
J0 (2xt) ,
2
1−t
1
dt
m−2mt2
√
e
J2 (2xt) .
2
1−t
JV-Wettig-2014, Kieburg-JV-Wettig-2014
Dirac Spectra – p. 30/4
IV. Dirac Spectra in Two Dimensions
Dirac Spectra of Lattice QCD in Two Dimensions
Dirac Spectra of the Schwinger Model
The Mermin-Wagner-Coleman Theorem
Possible Solutions
Dirac Spectra – p. 31/4
Lattice QCD
L
hopping matrix elements
on links
Gauge Fields
on links
Even lattice sites are only coupled to odd lattice sites.
This introduces a lattice “chiral” symmetry if the total number of
lattice points in both directions is even.
New symmetry classes arise for even-odd lattices. In fact we find
to two remaining random matrix ensembles from the Cartan
Kieburg-JV-Zafeiropoulos-2014.
classification.
For three dimensions, see also
Bialas-Burda-Peterson-2012
Dirac Spectra – p. 32/4
Two and Four Dimensional Dirac Spectra for
βD = 1
0.7
d)
analytical
numerical Lx=Ly=6
0.6
numerical Lx=Ly=8
numerical Lx=6 Ly=8
0.5
0.4
ρ
0.3
0.2
0.1
SU(3) adjoint
0
0
2
4
6
8
10
λ
12
14
16
18
20
of
Microscopic spectral density of the
opera-
quenched QCD Dirac operator in 2
tor in 4 dimensions for QCD with
dimensions for QCD with three col-
two colors and quarks in the adjoint
ors and adjoint quarks ( β = ∞ ).
Microscopic
quenched
spectral
staggered
representation.
density
Dirac
Edwards-Heller-
Kieburg-JV-Zafeiropoulos-2013
Narayanan-1999
Dirac Spectra – p. 33/4
Two and Four Dimensional Dirac Spectra for
βD = 2
0.7
a)
analytical
numerical Lx=Ly=5
0.6
numerical Lx=Ly=6
numerical Lx=Ly=7
0.5
numerical Lx=Ly=8
numerical Lx=5 Ly=7
0.4
ρ
numerical Lx=6 Ly=8
0.3
0.2
0.1
SU(3) fundamental
0
0
Microscopic
spectral
density
of
quenched staggered Dirac operator
in 4 dimensions for QCD with three
colors.
Wettig-etal-1999
5
10
λ
15
20
Microscopic spectral density of the
quenched QCD Dirac operator in 2 dimensions for QCD with three colors
and fundamental quarks ( β = ∞ ).
Kieburg-JV-Zafeiropoulos-2013
Dirac Spectra – p. 34/4
Two and Four Dimensional Dirac Spectra for
βD = 4
0.7
c)
analytical
numerical Lx=Ly=6
0.6
numerical Lx=Ly=8
numerical Lx=6 Ly=8
0.5
0.4
ρ
0.3
0.2
0.1
SU(2) fundamental
0
0
Microscopic
λ
15
20
Microscopic spectral density of the
opera-
quenched QCD Dirac operator in 2
tor in 4 dimensions for QCD with
dimensions for QCD with two colors
two colors and fundamental quarks.
and fundamental quarks ( β = ∞ ).
staggered
density
10
of
quenched
spectral
5
Dirac
Wettig-JV-etal-1999
Kieburg-JV-Zafeiropoulos-2013
Dirac Spectra – p. 35/4
Schwinger Model
Cumulative eigenvalue density of the two flavor Schwinger model
Bietenholz-Hip-Scheredin-Volkholz-2011
Eigenvalues are rescaled by λ → λV 5/8 because the eigenvalue
density of the two flavor Schwinger model ρ(λ) ∼ V λ3/5 .
See also Damgaard-Heller-Narayanan-Svetitsky-2005
Dirac Spectra – p. 36/4
Dirac Spectra in Two and Four Dimensions
Universal spectral correlations arise as a consequence of
spontaneous symmetry breaking.
Dirac spectra in two dimensions and four dimensions show the
same degree of agreement with random matrix predictions.
Yet according to the Mermin-Wagner-Coleman theorem a
continuous symmetry cannot be broken spontaneously in two
dimensions.
Dirac Spectra – p. 37/4
Possible Solutions
The generating function for the resolvent is given by
det(D + z)
d
Nf
det (D + m)
.
G(z) =
dz z ′ =z
det(D + z ′ )
This partition function has both fermionic and bosonic
“ghost”-quarks.
The flavor group is a supergroup and the boson-boson part to be
noncompact. Otherwise the integrals in the partition diverge.
A trivial ground state cannot exist for a noncompact Goldstone
manifold because the integration over the noncompact group will
be divergent.
Zirnbauer
Dirac Spectra – p. 38/4
Mermin-Wagner-Coleman Theorem
We conclude that the Mermin-Wagner-Coleman theorem does not
apply to noncompact continuous symmetries and the flavor
symmetry remains spontaneously broken in two dimensions.
Zirnbauer, Niedermaier-Seiler
Since the compact part of the symmetry group is not broken
spontaneously, one would expect different universal eigenvalue
correlations. At this moment it is not yet clear what is going on.
Dirac Spectra – p. 39/4
Bosonization
Two dimensional Yang-Mills theory can be bosonized. The
effective Lagrangian has all possible terms allowed by symmetries
1
1
†
L = F Tr∂µ U ∂µ U − mΣTr[U + U † ] + · · ·
3
2
Because of strong fluctuations in two-dimensions, the chiral
condensate averages to zero.
Condition for decoupling of zero modes and nonzero modes
4π 2
mG ≪ 2 .
L
Eigenvalue spacing: ∆λ = 1/ρ(0) = π/ΣV . Number of
eigenvalues in the microscopic domain
4πLd−2 .
Dirac Spectra – p. 40/4
IV. Universal Deformations
Universality Beyond the Classical Random Matrix
Ensembles
The Wilson Dirac Operator
QCD at Nonzero Chemical Potential
Dirac Spectra – p. 41/4
Universal Deformations
If a random matrix theory is deformed by a interaction that breaks
the symmetry, the resulting ensemble also has universal
eigenvalue correlations.
This follows immediately from the existence of a low-energy
effective partition function. The symmetry breaking interactions
give rise to additional terms that can be identified by means of the
spurion formalism.
In particular we can deform each random matrix ensemble by
another random matrix ensemble.
Dirac Spectra – p. 42/4
chGUE + iα GUE
This ensemble applies to the Wilson Dirac operator, with α equal
to the lattice spacing.
It results in a pseudo-Hermitian Dirac operator with a complex
spectrum. This random matrix theory has been solved analytically.
Kieburg-JV-Zafeiropoulos-2012,2013
The hermitian random matrix theory related to this
pseudo-hermitian random matrix theory is solvable as well.
Splittorff-JV-2011,Akemann-Nagao-2011
Generally, the solution can be written in terms of a diffusion kernel
Guhr-2000, Splittorff-JV-2011
in the parameter α .
Dirac Spectra – p. 43/4
Eigenvalue Density of γ5 (DW + m)
3
Ev_279_20_HYPclov_Q1andQm1_m027
m=2.3 a6=i0.0 a8=0.06 ν=1
m=2.3 a6=i0.06 a8=0.0 ν=1
ρ5
ν=1
5
(λ ,m=2.3,a)
2.5
2
1.5
1
0.5
0
-10
-8
-6
-4
-2
0
λ
5
The microscopic spectrum of γ5 (DW + m) for ν = 1
2
4
6
8
10
Damgaard-Heller-Splittorff-2012.
The red and blue curves represents the analytical result for the resolvent Splittorff-JV-2011,
Guhr-1999
p
dsdt −[(s+iz)2 +(t−z ′ )2 ]/16a2 (m − is)ν ν p 2
2
e
Z̃1|1 ( m + s , m2 − t2 ; a = 0
ν
t − is
(m − t)
yν
ν
where Z̃1|1 (x, y; a = 0) = ν [yKν+1 (y)Iν (x) + xKν(y)Iν+1 (x
x
√
The width of the topological peak behaves as ∼ a/ V .
1
Gν (m, z; a) =
16a2 π
Z
Dirac Spectra – p. 44/4
chGUE +iα chGUE
This ensemble is describes the microscopic Dirac spectrum of
QCD at nonzero chemical potential. Stephanov-1996,Janik-NowakPapp-Zahed-1996,Halasz-Jackson-JV-1996,Osborn-2000,
Since the sign problem prohibits lattice simulations at nonzero
chemical potential, this random matrix theory has contributed
substantially to our understanding of QCD at nonzero chemical
potential.
This random matrix theory is analytically solvable.
Osborn-2000, Akemann-Osborn-Splittorf-JV-2001
Among other we now understand that the Banks-Casher formula
has to be replaced by the OSV mechanism.
Osborn-Splittorff-JV-2006.
Dirac Spectra – p. 45/4
V. Conclusions
The low-lying Dirac eigenvalues of fundamental Quantum Field
Theories are described by a Random Matrix Theory with the same
global symmetries
Dirac Spectra – p. 46/4
V. Conclusions
The low-lying Dirac eigenvalues of fundamental Quantum Field
Theories are described by a Random Matrix Theory with the same
global symmetries
Despite the Mermin-Wagner-Coleman theorem, there is no
difference in the agreement between two and four dimensions of
spectral fluctuations and Random Matrix Theory.
Dirac Spectra – p. 46/4
V. Conclusions
The low-lying Dirac eigenvalues of fundamental Quantum Field
Theories are described by a Random Matrix Theory with the same
global symmetries
Despite the Mermin-Wagner-Coleman theorem, there is no
difference in the agreement between two and four dimensions of
spectral fluctuations and Random Matrix Theory.
Weak breaking of the global symmetries of QCD is also described
by random matrix theory. It gives rise to universal distributions of
the low-lying Dirac eigenvalues.
Dirac Spectra – p. 46/4