Random Matrix Theory at Nonzero $\mu$ and $T$
Random Matrix Theory at Nonzero µ and T
Kim Splittorff1,∗) and Jacobus Johannes Maria Verbaarschot1,2 ,3 ,∗∗)
1 The Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
2 Niels Bohr International Academy, Blegdamsvej 17, DK-2100, Copenhagen Ø
3 Department of Physics and Astronomy, SUNY, Stony Brook, New York 11794
We review applications of random matrix theory to QCD at nonzero temperature and
chemical potential. The chiral phase transition of QCD and QCD-like theories is discussed
in terms of eigenvalues of the Dirac operator. We show that for QCD at µ 6= 0, which
has a sign problem, the discontinuity in the chiral condensate is due to an alternative to
the Banks-Casher relation. The severity of the sign problem is analyzed in the microscopic
domain of QCD.
§1. Introduction
Starting from its introduction in nuclear physics by Wigner,1) random matrix
theories have been applied to a wide range of problems ranging from the physics of
proteins2) to quantum gravity (see3), 4) for a historical review). Three reasons for
the ubiquity of random matrix theory come to mind. First, eigenvalues of large ran-
dom matrices have universal properties determined by symmetries. Second, random
matrices are models for disorder present in many physical systems. Third, random
matrix theories have a topological expansion which is important for applications to
quantum field theory. One of the attractive features of random matrix theory is that
analytical information can be obtained for complex systems which otherwise only
can be studied experimentally or numerically.
In this review we discuss applications of random matrix theory to QCD at
nonzero temperature and chemical potential. Since the order parameter for the
chiral phase transition5), 6) and the deconfining phase transition7), 8) are determined
by the infrared behavior of the eigenvalues of the Dirac operator, these eigenvalues
are essential for the phase transitions in QCD. Remarkably, the distribution of the
smallest Dirac eigenvalues is given by universal functions9)–13) that depend only on
one or two parameters, the chiral condensate and the pion decay constant. This
offers an alternative way to measure these constants on the lattice.14)–22)
§2. Random Matrix Theory in QCD
Chiral Random Matrix Theory (chRMT) is a theory with the global symmetries
of QCD, but matrix elements of the Dirac operator replaced by random numbers9), 10)
iW † m
, P (W ) ∼ e−NTrW †W . (2.1)
∗) e-mail address: split@nbi.dk
∗∗) e-mail address: jacobus.verbaarschot@stonybrook.edu
http://arxiv.org/abs/0704.0330v1
2 K.Splittorff and J.J.M. Verbaarschot
This random matrix model has the global symmetries and topological properties of
QCD. It is confining in the sense that only color singlets have a nonzero expecta-
tion value. It is now well understood that fluctuations of low-lying eigenvalues of
the Dirac operator are described by chRMT (see23)–28) for lectures and reviews).
Philosphically, this is important because of the realization that chaotic motion dom-
inates the dynamics of quarks at low energy. Practically, this is important because
we can use powerful random matrix techniques to calculate physical observables.
The condition for the applicability of chRMT is that the Compton wavelength
of Goldstone bosons associated with the mass scale z of these eigenvalues is much
larger than the size of the box. With the squared mass of the associated Goldstone
boson given by 2zΣ/F 2π , this condition reads
≪ Λ2. (2.2)
The second condition is necessary to factorize the partition function into a contribu-
tion from the lightest degrees of freedom and all heavier degrees of freedom. These
two conditions determine the microscopic domain of QCD. We stress that z is a scale
in the Dirac spectrum so that, for sufficiently large volumes, we always have eigenval-
ues in the domain (2.2) where eigenvalues fluctuate according to chRMT. This can be
shown rigorously from the following two observations.30), 31) First, the infrared Dirac
spectrum follows from a (partially quenched) chiral Lagrangian determined by chiral
symmetry, and the inequality (2.2) is the condition for factorization of the partition
function into a factor containing the constant modes and another factor containing
the nonzero momentum modes. Second, the factor with the constant modes is equal
to the large N limit of chiral random matrix theory.
In32), 33) the condition (2.2) was imposed on the quark masses and was the bases
for a systematic expansion of the chiral Lagrangian known as the ǫ expansion.
One feature that underlies universal properties of eigenvalues is that they be-
have as repulsive confined charges. This follows from the joint probability distri-
bution ∼
k<l(λ
)2 exp(−N
). It can be shown that eigenvalues
correlations at the micrsocopic scale are universal.34) The reason is spontaneous
symmetry breaking and a mass gap so that they can be described in terms of a
chiral Lagrangian.
2.1. Chiral Random Matrix Theory at µ 6= 0 and T 6= 0
A nonzero temperature does not change the fluctuating behavior of the Dirac
eigenvalues provided that chiral symmetry remains broken. However, a transition to
a different universality class takes place at the critical temperature. A random matrix
model that reproduces this universal behavior of QCD is obtained by replacing the
off-diagonal elements in (2.1) by35)
iW → iW + t, iW † → iW † − t with t = diag(−πT, πT ). (2.3)
This model has been studied elaborately in the literature (see e.g.35)–40)).
A nonzero chemical potential can be introduced analogously to the quark mass.
The requirement is that the small µ behaviour of the QCD partition function should
Random Matrix Theory 3
0.0 1.0 2.0 3.0 4.0
2µ/mπ
m=0.10
m=0.05
m=0.01
Fig. 1. Lattice results for Nc = 2 (taken from
55)) and phase quenched QCD with Nc = 3 (taken
from56))
be reproduced by the random matrix partition function. This achieved by modifying
(2.1) by41)
iW → iW + µ, iW † → iW † + µ, (2.4)
resulting in a nonhermitean Dirac operator with eigenvalues scattered in the complex
plane. The prescription (2.4) is not unique. A random matrix model that has had a
strong impact on recent developments is defined by42)
iW → iW + µH, iW † → iW † + µH with H† = H, (2.5)
where H is drawn from a Gaussian ensemble of random matrices. This model is in
the same universality class as (2.4) but is technically simpler since it can be worked
out by means of the complex orthogonal polynomial method.42)–46)
There are other types of random matrix models that have been applied to QCD.
For example models with random gauge fields such as the Eguchi-Kawai model47) or
its 2-dimensional version.48) QCD in 1 dimension49), 50) is a random matrix model
as well, with universally fluctuating Dirac eigenvalues. Also models with random
Wilson loops51), 52) have attracted significant interest.
§3. Phases of QCD and RMT
QCD-like theories with charged Goldstone bosons have a critical chemical poten-
tial equal to mπ/2. The phase transition to the Bose condensed phase can therefore
be described completely in terms of a chiral Lagragian. At the mean field level,53)
the kinetic terms of this chiral Lagrangian do not contribute, so that these results
can also be obtained from chiral random matrix theory. Indeed, the static part of
the chiral Lagrangian53), 54)
F 2πµ
2Tr[U,B][U †, B]−
ΣTr(MU +MU †). (3.1)
can also be obtained from the large N limit of the models (2.4) or (2.5).
4 K.Splittorff and J.J.M. Verbaarschot
Tricritial point
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Fig. 2. QCD phase diagram in the µTm-space (taken from58))
In Fig. 1 we display lattice results for QCD with Nc = 2
55) and phase quenched
QCD.56) They show an impressive agreement with the results from (3.1) given by
the solid curves in both figures.
3.1. Schematic RMT Phase Diagram
The phase transition in QCD with Nc = 3 at µc = mN/3 cannot be analyzed
by means of chiral Lagrangians. Because of the sign problem lattice studies are not
possible either. In such situation there is long tradition to analyze the same problem
in a much simpler theory in the hope of obtaining at least a qualitative understanding
of the problem. For example, one dimensional QCD,49), 50) or more recently, super
Yang-Mills theory and AdS-CFT duality,57) been explored as toy models for QCD.
We will use random matrix theory at T 6= 0 and µ 6= 0, introduced in (2.3) and
(2.4) to obtain a qualitive understanding of the QCD phase diagram. Lattice QCD
simulations show that the chiral phase transition at µ = 0 is of second order or a
steep cross-over. At T = 0 we expect a first order phase transition at µc = mN/3.
It is natural that the first order line ends in a critical end point or joins the second
order critical line at the tricritical point (see Fig. 3.1, left). This is indeed what
is observed in random matrix theory58), 59) (see Fig. 3.1, right). A similar phase
diagram has also been obtained from the NJL model.60)–62)
Another scenario that was discovered in RMT is the splitting of the first order
line into two at nonzero isospin chemical potential.63) This behavior was also found
in a NJL model64), 65) but might not be stable against flavor mixing interactions.66)
§4. Dirac Spectrum in Theories Without a Sign Problem
Since the spectrum of the Dirac operator determines the chiral condensate, phase
transitions in QCD can be understood in terms of its spectral flow. In this section we
discuss theories with a positive fermion determinant such as QCD with two colors and
phase quenched QCD, where a probabilistic interpretation of the eigenvalue density
is possible. The relation between chiral symmetry breaking and Dirac spectra is
much more complicated when the fermion determinant is complex and its discussion
will be postponed to the next section.
The spectrum of an anti-Hermitean Dirac operator is purely imaginary with an
eigenvalue density that is proportional to the volume. If chiral symmetry is broken
spontaneously, the chiral condensate becomes discontinuous across the imaginary
axis in the thermodynamic limit. Chiral symmetry is restored if such discontinuity
Random Matrix Theory 5
mm m m m m
T < Tc
µ = 0
T > Tc
µ = 0 T < Tc
µ < µc
T < Tc
µ = µc
T < Tc
µ > µc
T > Tc
µ > µc
Fig. 3. Critical behavior of the Dirac spectrum. µc = mπ/2 for T = 0 and increases with T .
is absent for example by the formation of a gap in the Dirac spectrum, see eg.71) .
For µ 6= 0, the Dirac spectrum broadens into a strip of width 4µ2F 2π/Σ.49), 67)
The chemical potential becomes critical when the quark mass hits the edge of this
strip. At this point the chiral condensate starts rotating into a pion condensate.
Chiral symmetry restoration takes place when a gap forms at zero. A schematic
picture of the critical behavior of Dirac eigenvalues is shown in Fig. 3 and the spectral
flow of the Dirac eigenvalues with respect to increasing µ and T is summarized in
Fig. 4. One conclusion from this behavior is that Tc(µ) is a concave function
of µ, and that µc(T ) is a convex function of T . The spectral flow discussed in this
section is supported by lattice simulations at T 6= 0 and µ 6= 0 (See Fig. 5)
4.1. Dirac spectrum in the µ-plane
We could equally well have diagonalized the Dirac operator in a representation
where µγ0 is proportional to the identity,
det(D +m+ µγ0) = det(γ0(D +m) + µ). (4.1)
These eigenvalues are relevant to the baryon number density. A gap in the spectrum
develops at m 6= 0 (see Fig. 6), and the chemical potential becomes critical, µ =
mπ/2 when it hits the inner edge of the domain of eigenvalues.
Increasing µ
Increasing T
Fig. 4. Spectral flow of the Dirac spectrum (left) and phase diagram (right) with respect to µ and
T in phase quenched QCD and QCD with two colors.
6 K.Splittorff and J.J.M. Verbaarschot
1 1.5 2 2.5
b=0.35
b=0.3525
b=0.355
b=0.3575
b=0.36
1.76(t-0.93)
0.0 0.1 0.2 0.3
β=5.5
β=5.66
β=5.71
β=5.75 β=5.9
Fig. 5. Temperature and chemical potential dependence of Dirac eigenvalues. From left to right
taken from.70), 72)–74)
4.2. Quenched Lattice QCD Dirac Spectra at µ 6= 0
Small Dirac eigenvalues at µ 6= 0 have been computed in quenched QCD. The
analytical formulas for the average density of the small Dirac eigenvalues are avail-
able.68), 69) They were first derived68) by exploiting the Toda lattice hierarchy in the
flavor index. Comparisons of random matrix predictions68) for the radial spectral
density and lattice QCD results75), 76) are shown in the left panel of Fig. 7. In other
cases, such as the overlap Dirac operator77) and QCD with Nc = 2,
78) a similar
degree of agreement was found. Both the spectral density and two-point correlations
can be derived from the Lagrangian (3.1), i.e. they are determined by two param-
eters, Fπ and Σ. This can be exploited to extract these low-energy constants. For
example, Fπ and Σ were determined
19), 21) (see also20)) from the correlators shown
in the two right panels of Fig. 7.
§5. Chiral Symmetry Breaking at µ 6= 0
The full QCD partition function at µ 6= 0 which is the average of
det(D +m+ µγ0) = |det(D +m+ µγ0)|eiθ, θ 6= 0, (5.1)
has properties which are drastically different from the phase quenched partition
function where the phase factor is absent. In particular, µc = mN/3 instead of mπ/2,
so that the free energy remains µ-independent until µ = mN/3. For µ < mN/3 the
Fig. 6. Eigenvalues of γ0(D + m) for a random matrix Dirac operator at m = 0 (left), m 6= 0
(middle) (both taken from79)), and lattice QCD at m 6= 0 (right, taken from49)).
Random Matrix Theory 7
—– Splittorff-Verbaarschot-2004
—– Wettig-2004
0 2 4 6 8
−0.15
−0.05
V = 8
  10000 configs
µisoFπV
 = 0.159
1.27 1.37 1.47 π/2 1.67 1.77 1.87
angle (θ)
lattice: 6
, µa = 0.006
fit: µFV
 = 0.14
Fig. 7. The radial spectral density for (left, taken from75), 76)) and two-point correlations (middle
taken from19) and right taken from21)).
chiral condensate remains discontinuous at m = 0, whereas the chiral condensate
of the phase quenched theory approaches zero for m → 0 (see Fig. 5). The only
difference between the phase quenched partition function and the full QCD partition
function is the phase of the fermion determinant. We conclude that the phase factor
is responsible for the discontinuity of the chiral condensate. How can this happen if
for each configuration the support of the spectrum is approximately the same? This
problem known as the “Silver Blaze Problem”80) was solved in.6)
5.1. Unquenched Spectral Density
The spectral density for QCD with dynamical fermions is given by
ρNf (λ) = 〈
δ2(λ− λk)detNf (D +m+ µγ0)〉. (5.2)
Because of the phase of the fermion determinant, this density is in general complex
and can be decomposed as ρNf (λ) = ρNf=0(λ) + ρU (λ). The chiral condensate can
then be decomposed as ΣNf (m) = ΣNf=0(m) +ΣU (m), so that the discontinuity in
Σ(m) is due to ρU . Asymptotically it behaves as
ρU ∼ e
µ2F 2V e
iIm(λ)ΣV
and vanishes outside an ellips starting at Re(λ) = m (see Fig. 9).6) In the right part
of this figure we show the real part of the spectral density for QCD with one flavor
at nonzero chemical potential.
Scatter plot of Dirac eigenvalues
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quark mass m
Support of spectrum
Chiral condensate
condensate
Quenched chiral
in full QCD
µ2F 2
Σ(m) = 1
Fig. 8. Chiral condensate of quenched and full QCD.
8 K.Splittorff and J.J.M. Verbaarschot
Dirac spectrum for Full QCD.
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Oscillating Region
quark mass m
-1000100
0.001
0.002
2F 2µ2
µ2F 2
Fig. 9. Support (left) and real part (right, taken from27)) of Dirac spectral density for QCD with
Nf = 1 and µ 6= 0.
This result explains the mechanism of chiral symmetry breaking at nonzero
chemical potential. The phase of the fermion determinant rotates the pion conden-
sate back into a chiral condensate, but it does so in an unexpected way.6) The same
mechanism is at play for 1d QCD at µ 6= 0.82)
§6. Phase of the Fermion Determinant
The magnitude of the sign problem can be measured by means of the expectation
value of the phase factor of the fermion determiant which can be defined in two ways
〈e2iθ〉Nf =
det(D + µγ0 +m)
det∗(D + µγ0 +m)
detNf (D + µγ0 +m)
, 〈e2iθ〉1+1∗ =
ZNf=2
Z1+1∗
The average 〈· · · 〉 is with respect to the Yang-Mills action. The sign problem is
managable when the average phase factor remains finite in the thermodynamic limit.
In the microscopic domain it is possible to obtain exact analytical expressions for
the average phase factor by exploiting the equivalence between QCD and RMT in
this domain. For µ < mπ/2 the free energy of both QCD and phase quenched
QCD are independent of µ. This does not imply that the average phase factor is
µ-independent. The µ-dependence originates from the charged Goldstone bosons
with mass mπ ± 2µ, and for Nf flavors the mean field result83), 84) for 〈exp(2iθ)〉
reads (1 − 4µ2/m2π)Nf+1. The exact result for the average phase factor for Nf = 2
is shown in Fig. 10 (right), where lattice results85) are also shown (left). The exact
result has an essential singularity at µ = 0, but its thermodyanmic limit agrees with
the mean result.
0 0.5 1 1.5
2µ/mπ
mΣV  =  4
mΣV >> 1
Fig. 10. Average phase factor. Lattice QCD results are shown left (taken from85)) and the exact
microscopic result83) is shown right.
Random Matrix Theory 9
§7. Conclusions
The equivalence of chiral random matrix theory and QCD has been exploited
succesfully to derive a host of analytical results. Among others, eigenvalue fluctua-
tions predicted by chRMT have been observed in lattice simulations, the phases of
QCD can be understood in terms of spectral flow, observables can be extracted from
the fluctuations of the smallest eigenvalues, the sign problem is not serious when the
quark mass is outside the domain of the eigenvalues, and mean field results can be
obtained from random matrix theory. Summarizing, chiral random matrix theory is
a powerful tool for analyzing the infrared domain of QCD.
Acknowledgements
The YITP is thanked for its hospitality. G. Akemann, J. Osborn and P.H.
Damgaard are acknowledged for valuable discussions. This work was supported by
US DOE Grant No. DE-FG-88ER40388 (JV), the Villum Kann Rasmussen Foun-
dation (JV), the Danish National Bank (JV) and the Carslberg Foundation (KS).
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