QCD in One Dimension at Nonzero Chemical Potential
L. Ravagli
Cyclotron Institute and Physics Department TEXAS A&M University, College Station, Texas 77843-3366, USA
J.J.M. Verbaarschot
Niels Bohr International Academy, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark,
Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark and
Department of Physics and Astronomy, SUNY, Stony Brook, New York 11794, USA
Using an integration formula recently derived by Conrey, Farmer and Zirnbauer, we calculate the
expectation value of the phase factor of the fermion determinant for the staggered lattice QCD action
in one dimension. We show that the chemical potential can be absorbed into the quark masses; the
theory is in the same chiral symmetry class as QCD in three dimensions at zero chemical potential.
In the limit of a large number of colors and fixed number of lattice points, chiral symmetry is broken
spontaneously, and our results are in agreement with expressions based on a chiral Lagrangian. In
this limit, the eigenvalues of the Dirac operator are correlated according to random matrix theory
for QCD in three dimensions. The discontinuity of the chiral condensate is due to an alternative
to the Banks-Casher formula recently discovered for QCD in four dimensions at nonzero chemical
potential. The effect of temperature on the average phase factor is discussed in a schematic random
matrix model.
I. INTRODUCTION
In spite of significant recent progress, QCD at nonzero chemical potential remains a notoriously hard problem
[1, 2, 3]. In particular, first principle nonperturbative results at low temperature are absent because the phase of the
fermion determinant invalidates probabilistic methods to evaluate the partition function. This problem is known as
the sign problem. The sign problem is particularly severe if the phase of the fermion determinant results in a different
free energy, i.e.
〈detNfD〉
〈|detNfD|〉
V (Fpq−FNf ) with FNf > Fpq. (1)
If the phase quenched free energy, Fpq differs from the free energy of QCD with Nf flavors, FNf , the number of
required gauge field configurations grows exponentially with the volume. The ratio defined in equation (1) can also
be interpreted as the average phase factor of the fermion determinant [4, 5]
〈eiNfθ〉pq =
〈eiNf θ|detNfD|〉
〈|detNfD|〉
. (2)
Alternatively, one can define the average phase factor with respect to the full QCD partition function [4, 5]
〈e2iθ〉Nf =
〈e2iθdetNfD〉
〈detNfD〉
, (3)
which has also been used in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12] as a measure for the severity of the sign problem.
When the chemical potential is sufficiently small, the average phase factor can be studied by means of chiral
perturbation theory [4, 5]. Recently, the average phase factor was analyzed in the microscopic domain of QCD [13, 14],
where only the constant fields in the chiral Lagrangian contribute to the mass and chemical potential dependence of
the partition function. It was found that the sign problem is not serious for µ < mπ/2. For µ > mπ/2 the chiral
condensate of the phase quenched theory rotates into a pion condensate resulting in a free energy that is different
from the full theory and a severe sign problem.
In this paper we study the sign problem for Euclidean QCD in one dimension. In one dimension, the only effect of
the gauge field is in the boundary conditions, and lattice QCD simplifies to a matrix integral. A general formula for
exactly this type of integrals was recently derived by Conrey, Farmer and Zirnbauer [15]. Using what we will call the
CFZ formula, exact analytical results for the one dimensional QCD partition function and the average phase factor
will be obtained.
The study of QCD in one dimension at finite chemical potential has had a long history. One reason to study this
model is that it is an effective model for the large Nc limit of strong coupling QCD [16, 17, 18, 19, 20, 21]. Among
http://arxiv.org/abs/0704.1111v2
others, it has been successfully used to explain [22, 23, 24] puzzling lattice results for quenched QCD at nonzero
chemical potential. More recently, one dimensional QCD was used to study the complex zeros of the partition
function [1, 25, 26].
The sign problem of QCD at nonzero chemical potential arises because of the nonhermiticity of the Dirac operator.
QCD in one dimension is not truly nonhermitean: Instead of being scattered in a two-dimensional domain of the
complex plane [27], the eigenvalues of the Dirac operator are localized on an ellipse in the complex plane [6, 22].
Another manifestation of the mild nonhermiticity is that the chemical potential can be absorbed into the quark
masses. For each flavor in one dimension with quark mass m, we can associate flavors with mass m+ µ and −m+ µ.
In the large Nc limit, this results in spontaneous chiral symmetry breaking according to U(2Nf) → U(Nf )× U(Nf ).
Therefore, QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. The microscopic
limit of QCD in one dimension is equivalent to the microscopic limit of QCD in three dimensions [28]. Another
peculiarity of QCD in one dimension is that, already in the free theory, the eigenvalues near zero are spaced inversely
proportional to the volume. Therefore, in the thermodynamic limit (i.e. at zero temperature) the chiral condensate
is discontinuous across the imaginary axis. However, this type of symmetry breaking is not a collective phenomenon,
and there are no associated Goldstone bosons. It is reminiscent to the alternative to Goldstone’s theorem proposed
by McKane and Stone [29]. Spontaneous symmetry breaking with Goldstone bosons takes place for a fixed number
of lattice points in the limit Nc → ∞.
A chemical potential excites color singlet excitations with baryon charge qk and mass Mk when qkµ > Mk. This is
also the case for one-dimensional QCD. For gauge group U(Nc), the only color singlet excitations are mesons that are
uncharged with respect to the chemical potential, whereas for SU(Nc) gauge group we have both neutral mesonic and
charged baryonic color singlet excitations. The complex conjugate of the fermion determinant can be interpreted in
terms of conjugate quarks [23, 30] which have a baryon charge that is opposite to that of regular quarks. Therefore,
both for U(Nc) and SU(Nc), the phase quenched partition function, where the fermion determinant has been replaced
by its absolute value, has charged mesonic excitations made out of quarks and conjugate anti-quarks. This results in
a phase transition at µ = µc = mπ/2. The SU(Nc) theory with Nf flavors has only charged baryonic excitations and
will have a phase transition at µ = µc = mB/Nc. In one dimension it turns out that the pion and the baryon have
the same mass per quark number so that the critical chemical potential of the phase quenched U(Nc) theory and the
SU(Nc) theory is the same.
For the U(Nc) partition function we expect the sign problem to be severe when µ > µc because the phase quenched
partition function has a phase transition at µ = µc whereas the normal theory remains in the same phase. For
SU(Nc), both the normal partition function and the phase quenched partition function do have a phase transition to
a phase of free quarks at µ = µc and have the same free energy not only for µ < µc but also for µ > µc. Therefore,
in one dimension there is no severe sign problem for SU(Nc) QCD.
Because one dimensional QCD with gauge group U(Nc) does not have a phase transition, the chiral condensate
in the thermodynamic limit or large Nc limit is discontinuous at m = 0, independent of the value of the chemical
potential. Because the Dirac eigenvalues are located on an ellipse for µ 6= 0, this discontinuity cannot be related to
the Dirac spectrum by means of the Banks-Casher formula. In [31] a different mechanism to explain the discontinuity
was discovered . It was found that an oscillating contribution to the spectral density with an amplitude that diverges
exponentially with the volume, is responsible for the discontinuity of the chiral condensate. We will show that a
similar mechanism is at work for U(Nc) gauge theory in one dimension.
Lattice QCD in one dimension will be introduced in section II, where we also discuss its continuum limit, mean field
results and the Conrey-Farmer-Zirnbauer formula. In section III we will evaluate the U(Nc) partition function and
its phase quenched version. The average phase factor for U(Nc) is calculated in section IV. In section V we study one
dimensional QCD with SU(Nc) as gauge group and evaluate the regular and phase quenched partition functions and
the average phase factor. The effect of temperature will be illustrated with results from a schematic random matrix
model in section VI. The connection with the microscopic domain of QCD in three dimensions is discussed in section
VII, and the Dirac spectrum and its relation with chiral symmetry breaking is analyzed in section VIII. Concluding
remarks are made in section IX.
II. QCD IN ONE DIMENSION
A. Lattice QCD
The staggered lattice QCD Dirac operator in one dimension is given by
mI eµU12/2 . . . e
−e−µU †12/2 mI · · · 0
0 · · · mI eµUn−1n/2
−eµUn1/2 · · · −e−µU †n−1n/2 mI
. (4)
We have used anti-periodic boundary conditions and the gauge fields on the links are taken to be in U(Nc) or SU(Nc).
The chemical potential µ is an imaginary vector field introduced according to the Hasenfratz-Karsch prescription [32].
In general, the lattice QCD partition function for Nf flavors with mass m is given by
ZNf (µc, µ) =
dUldet
NfD e−SYM/g
, (5)
where the product is over the links and SYM is the Yang-Mills action. In one dimension, the partition function
simplifies substantially. First, there is no Yang-Mills action, and second, by unitary transformations the integral over
the links can be reduced to a single integral. In the gauge where all gauge fields except Un 1 ≡ U are equal the unity,
the fermion determinant reduces to [24]
detD = 2−nNc det[enµc + e−nµc + enµU + e−nµU †], (6)
with µc given by
µc = sinh
−1m. (7)
This value will turn out to be the critical value of the chemical potential. From now on, the overall factor 2−nNc will
be absorbed into the normalization of the partition function. To avoid sign factors, n is taken to be even throughout
this paper.
The standard method to evaluate the QCD partition function in one dimension is to use the eigenvalues of U as
integration variables [6, 16, 17, 18, 24]. For example, one finds this way that the result for Nf = 1 with U(Nc) as
gauge group is given by [24]
ZNf=1(µc, µ) =
U(Nc)
dU detD =
sinh((Nc + 1)nµc)
sinh(nµc)
. (8)
For most partition functions that are considered in this paper it is not possible to obtain analytical results by means
of this method. Instead we will use powerful integration formulas for unitary integrals that were recently derived in
[15]. These integrals are based on the color-flavor transformation [33] which has also been applied to lattice QCD
with baryons in the canonical ensemble [34, 35].
B. Continuum Theory
The continuum limit of the staggered lattice action is given by
Dcont =
m ∂0 + iA0 + µ
∂0 + iA0 + µ m
, (9)
where A0 is a Hermitean Nc × Nc matrix and the off-diagonal blocks connect even and odd lattice sites. What is
special in one dimension is that the off-diagonal blocks are identical. The eigenvalue equation (∂0 + iA0)ψk = iukψk
is solved by
ψ = Pe−i
A0dtχ0 with ∂0χ0 = iEχ0, (10)
with Pe the path ordered exponent. Nontrivial eigenvalues are obtained by imposing boundary conditions on ψ. At
nonzero chemical potential the eigenvalues of the Dirac operator are given by
λk = m± (iuk + µ) with uk ∈ R. (11)
This is very different from QCD with d ≥ 2 where the eigenvalues of the QCD Dirac operator at µ 6= 0 are scattered in
the complex plane. Also the eigenvalues of the one dimensional staggered lattice Dirac at nonzero chemical potential
are localized on a curve in the complex plane.
Another consequence of the structure of the Dirac operator (9) is that the fermion determinant can be rewritten as
detDcont = det(∂0 + iA0 + µ+m) det(∂0 + iA0 + µ−m), (12)
which can be interpreted as a two-flavor partition function with masses m + µ and −m + µ. This Dirac operator
has the same structure as QCD in three dimensions: In the large Nc limit, chiral symmetry is broken spontaneously
according to U(2) → U(1) × U(1) with the squared Goldstone masses proportional to the difference of the positive
and negative quark masses, i.e. m2π ∼ m− (−m) = 2m. At low energy, these Goldstone modes interact according to a
chiral Lagrangian determined by the pattern of chiral symmetry breaking. In the large Nc limit and mπβ ≪ 1 (with
β the length of the one-dimensional box) the QCD partition function in one dimension is therefore equivalent to the
low-energy limit of QCD in three dimensions. In section VII this will be worked out explicitly for the microscopic
limit of the partition function. This is the limit
ρ(0) → ∞ with mπρ(0) = fixed and µπρ(0) = fixed, (13)
where ρ(0) is density of the eigenvalues, or their projections onto the imaginary axis, close to zero. Since ρ(0) = nNc/2π
(see section VIII), this is the limit nNc → ∞ with nNcm and nNcµ fixed. When we use the term microscopic limit,
we always mean the universal microscopic limit. This is the limit (13) associated with the formation of Goldstone
bosons, i.e. the limit Nc → ∞ at fixed n.
C. Mean Field Limit for Large Nc
In the large Nc limit, chiral symmetry is broken spontaneously even in one dimension, so that its low-energy limit
is a theory of Goldstone bosons. In this section we give general arguments that determine the chemical potential
dependence of the partition function in the microscopic domain where the Compton wavelength of the Goldstone
bosons is much larger than β.
As was argued in [4, 5], in the microscopic domain, the mean field limit of the partition function is given by
Z = J
mπ(µ)
e−V F , (14)
where J is the value of the integration measure at the saddle point, F is the free energy, V is the space time volume,
and mπ are the masses of the Goldstone bosons. Let us apply this result to the average phase factor in one dimension.
For µ < µc, in the limit T → 0, only the vacuum state contributes to the partition function, so that the free energy
is independent of µ. For quark mass m, the equivalent QCD3 mass matrix of the Dirac operator D in (12) is given
by diag(−m+µ,m+µ), whereas the hermitean conjugate Dirac operator D†, has diag(−m−µ,m−µ) as equivalent
QCD3 mass matrix. For Nf flavors, the average phase factor is the partition function with Nf + 1 fermionic quarks
and one conjugate bosonic quark. We thus have 2(Nf +1)
2 Goldstone bosons made out of two fermionic quarks with
squared mass m2π = 2mG (with G a constant), 4(Nf +1) fermionic Goldstone modes, half of them with squared mass
m2π = 2(m − µ)G, and the other half with squared mass m2π = 2(m + µ)G. Finally, we have 2 Goldstone bosons
composed out of two bosonic quarks with squared mass equal to m2π = 2mG. The fermionic partition function ZNf
has 2N2f Goldstone bosons all with squared mass m
π = 2mG. Using (14) we thus find
〈e2iθ〉Nf =
ZNf+1|1∗(µc, µ)
ZNf (µc, µ)
)Nf+1
for µ < µc. (15)
For phase quenched partition functions with Nf flavors and Nf conjugate flavors, the equivalent QCD3 symmetry
breaking pattern is U(4Nf )/(U(2Nf)× U(2Nf)), so that the total number of Goldstone bosons is equal to 8N2f . Of
these, 4N2f Goldstone bosons have squared massm
π = 2mG, half made out of two quarks and the other half out of two
conjugate quarks. The other 4N2f Goldstone bosons are composed out of a quark and a conjugate anti-quark quark,
half with squared mass m2π = 2(m−µ)G and the other half with squared mass m2π = 2(m+µ)G. All equivalent QCD3
Goldstone bosons of the full QCD partition function with 2Nf flavors have a squared mass equal to m
π = 2mG. For
µ < µc the free energy of the phase quenched partition function and the full QCD partition function is the same so
that, using (14), the phase quenched average phase factor is given by
〈e2iθ〉Nf+N∗f =
ZNf+1+(Nf−1)∗(µc, µ)
Z2Nf (µc, µ)
for µ < µc. (16)
Below we will show that the results derived in this section also follow from the zero temperature microscopic limit
of the exact evaluation of the partition function.
D. The Conrey-Farmer-Zirnbauer formula
Exactly the integrals that are required for the evaluation of the average phase factor of QCD at nonzero chemical
potential in one dimension were studied in a recent paper by Conrey, Farmer and Zirnbauer [15]. They considered
the partition function
Z({ψk, φk}) =
U(Nc)
det(1− eψjU)
det(1− eφjU)
l=p+1
det(1− e−ψlU †)
det(1− e−φlU †)
, (17)
with dU the Haar measure of U(Nc) and ψk, φk complex parameters with Re(φj) < 0 < Re(φl). Using the color
flavor transformation [33] and Howe’s theory of supersymmetric dual pairs, they derived the following formula
Z({ψk, φk}) =
π∈Sp+q/(Sp×Sq)
eNc(π(ψl)−ψl)
(1− eφj−π(ψl))(1 − eπ(ψj)−φl)
(1− eπ(ψj)−π(ψl))(1 − eφj−φl)
. (18)
The sum is over permutations in Sp+q/(Sp×Sq) that interchange any of the ψ1, · · · , ψp with any of the ψp+1, · · · , ψp+q.
Partition functions with an unequal number of bosonic and fermionic determinants can be obtained from special
limits of (18). In the case of only fermionic determinants, an equivalent expression was first obtained in [36, 37].
Expressions for degenerate parameters can be derived by carefully taking limits of the above formula.
Unitary matrix integrals can also be calculated by using an eigenvalue representation of the unitary matrices. The
integrals we are interested in are of the form
U(Nc)
F (eiθk), (19)
where exp(iθk) are the eigenvalues of U . Using an eigenvalue representation of the unitary matrices, the orthogonal
polynomial method can be used to express Z as [17]
Z = det(Bk−l)k,l=0,···,Nc−1, (20)
where
dθeikθF ({eiθ}). (21)
This is the method that was used in the literature on one-dimensional QCD prior to this paper [6, 16, 17, 18, 24].
In a few cases the determinant in (20) could be evaluated explicitly resulting in expressions that are similar to those
derived directly from the CFZ-formula. We have used (20) to numerically check the results obtained by means of the
CFZ-formula.
III. EXACT EVALUATION OF THE ONE-DIMENSIONAL U(Nc) PARTITION FUNCTION
A. Partition Function for arbitrary Nf
In this section we evaluate the one-dimensional U(Nc) QCD partition function for Nf flavors.
To apply the CFZ formula (18) we rewrite the determinant (6) as
detD = eµcnNc det(1 + en(µ−µc)U) det(1 + en(−µ−µc)U †). (22)
For arbitrary Nf we then find the remarkably simple answer
ZNf (µc, µ) ≡
U(Nc)
dUdetNfD =
σ∈S2Nf /SNf ×SNf
eNcmσ(+ k)
1− exp(mσ(− l) −mσ(+ k))
, (23)
m− k = −nµc k and m+ k = nµc k. (24)
The sum is over all permutations that interchange positive and negative masses. Notice that the µ-dependence has
canceled from this expression. This also follows from an expansion of the determinant in powers of U and U †. Only
terms with an equal number of factors U and factors U † are non-vanishing.
Although the result for degenerate positive and negative masses can be obtained by carefully taking limits of (23),
it is simpler to start from a different representation [36, 37] of partition function (23) given by
ZNf (µc, µ) =
1≤k<l≤2Nf
(eMl − eMk)
1 em− 1 · · · e(Nf−1)m− 1 e(Nc+Nf )m− 1 · · · e(Nc+2Nf−1)m− 1
m−Nf · · · e(Nf−1)m−Nf e(Nc+Nf )m−Nf · · · e(Nc+2Nf−1)m−Nf
1 em+1 · · · e(Nf−1)m+1 e(Nc+Nf )m+1 · · · e(Nc+2Nf−1)m+1
m+Nf · · · e(Nf−1)m+Nf e(Nc+Nf )m+Nf · · · e(Nc+2Nf−1)m+Nf
. (25)
For convenience we have introduced the mass matrix Mk = (m1 1, · · · ,m−Nf ,m+1, · · · ,m+Nf ). To obtain an expres-
sion for degenerate masses, we Taylor expand the exponential functions exp(m+ k) and exp(m− k) to order Nf − 1
about exp(−nµc) and exp(nµc), respectively, and write the resulting matrix as the product of two matrices, one
containing the Taylor coefficients, and the other containing powers of the expansion parameters. The determinant of
the second matrix can be written as a Vandermonde determinant which cancels against part of the prefactor in (25).
Our final expression for the partition function with degenerate masses is given by
ZNf (µc, µ) =
∏Nf−1
k=0 k!)
(enµc − e−nµc)N
1 · · · 0 1 · · · 0
e−nµc · · · e−nµc enµc · · · enµc
e−2nµc · · · δNf−1− e−2nµc e2nµc · · · δ
e−n(Nf−1)µc · · · δNf−1− e−n(Nf−1)µc en(Nf−1)µc · · · δ
n(Nf−1)µc
e−n(Nc+Nf )µc · · · δNf−1− e−n(Nc+Nf )µc en(Nc+Nf )µc · · · δ
n(Nc+Nf )µc
e−n(Nc+2Nf−1)µc · · · δNf−1− e−n(Nc+2Nf−1)µc en(Nc+2Nf−1)µc · · · δ
n(Nc+2Nf−1)µc
, (26)
d(−nµc)
, δ+ =
d(nµc)
. (27)
The k’th column (with k ≤ Nf) is given by the δ(k−1)− derivative of the first column, and the Nf + k’th column is
given by the δ
(k−1)
+ derivative of the Nf ’th column.
From this result one can easily derive explicit expressions for small values of Nf . The partition function for Nf = 2
reads
ZNf=2(µc, µ) =
(en(Nc+2)µc − e−n(Nc+2)µc)2
(enµc − e−nµc)4 −
(Nc + 2)
(enµc − e−nµc)2 . (28)
The microscopic limit of this partition function is given by
ZmicroNf=2(µc, µ) =
(eNcµc − e−Ncµc)2
16µ4c
. (29)
For Nf = 3 the partition function is given by:
ZNf=3(µc, µ) =
(en(Nc+3)µc − e−n(Nc+3)µc)3
(enµc − e−nµc)9
− (Nc + 3)3
(en(Nc+1)µc − e−n(Nc+1)µc)
(enµc − e−nµc)7
+Nc(Nc + 2)
n(Nc+3)µc − e−n(Nc+3)µc)
(enµc − e−nµc)7
(Nc + 2)
2(Nc + 3)
n(Nc+3)µc − e−n(Nc+3)µc)
(enµc − e−nµc)5
B. The Phase Quenched Partition Function
To calculate the average phase factor according to the definition (2) for Nf = 2, we also need the one-dimensional
phase quenched QCD partition function for two flavors which will be evaluated in this subsection. It is defined by
Z1+1∗(µc, µ) =
U∈U(Nc)
dU detD detD† (32)
= e2µnNc
U∈U(Nc)
dU det(1− en(µ−µc)U) det(1 − en(−µ−µc)U †) det(1− en(µ−µc)U †) det(1 − en(−µ−µc)U).
This partition function is easily evaluated using the integration formulae of [15]. We find
Z1+1∗(µc, µ) =
cosh(2n(Nc + 2)µc)
8 sinh(n(µc − µ)) sinh(n(µ+ µc)) sinh2(nµc)
cosh(2n(Nc + 2)µ))
8 sinh(n(µ− µc)) sinh(n(µ+ µc)) sinh2(nµ)
8 sinh2(nµ) sinh2(nµc)
. (33)
For µ = 0 this result agrees with the Nf = 2 partition function given in (28). Its microscopic limit simplifies to
Zmicro1+1∗ (µc, µ) =
e2nNcµc + e−2nNcµc
16n4(µ2c − µ2)µ2c
16n4µ2µ2c
2nNcµ + e−2nNcµ
16n4(µ2c − µ2)µ2
. (34)
In section VII we will show that this result is equal to the microscopic limit of the QCD3 partition function with
masses −(µc + µ), −(µc − µ), µc − µ, µc + µ.
The large Nc limit of the phase quenched partition function is given by
Z1+1∗(µc, µ) =
e2n(Nc+2)µc
16 sinh(n(µc − µ)) sinh(n(µ+ µc))(sinh(nµc))2
for µ < µc,
Z1+1∗(µc, µ) =
e2n(Nc+2)µ
16 sinh(n(µ− µc)) sinh(n(µ+ µc))(sinh(nµ))2
for µ > µc,
Z1+1∗(µc, µ) =
2n(Nc+2)µc
8 sinh2(nµc) sinh(2nµc)
1 +O(N−1c )
for µ = µc, (35)
which will be used to calculate the phase quenched average phase factor in this limit.
IV. AVERAGE PHASE FACTOR FOR U(Nc)
In this section we will evaluate the average phase factor, first from the ratio of the full QCD partition function and
the phase quenched partition function and in the next subsection starting from the definition in Eq. (3).
A. Phase Quenched Average Phase Factor
For Nf = 2, the average phase factor with the absolute value of the fermion determinant as weight is given by
〈e2iθ〉pq =
〈e2iθ| det(D)|2〉
〈| det(D)|2〉
ZNf=2(µc, µ)
Z1+1∗(µc, µ)
. (36)
The two-flavor partition function was evaluated in subsection III A, and the phase quenched partition function was
calculated in subsection III B. The large Nc limit of the phase quenched average phase factor is given by
〈e2iθ〉pq =
sinh(n(µc−µ)) sinh(n(µc+µ))
sinh2(nµc)
for µ < µc,
sinh2(nµ) sinh(n(µ−µc)) sinh(n(µ+µc))
sinh4(nµc)
e−2n(Nc+2)(µ−µc) for µ > µc,
2 cosh(nµc)
Nc sinh(nµc)
for µ = µc.
The large µcNc, µNc limit of the microscopic phase quenched partition function (34) for µ < µc is given by
Zpq(µc, µ) =
e2nNcµc
16n4(µ2c − µ2)µ2c
, (38)
whereas the two-flavor partition function in this limit reads
ZNf=2(µc, µ) =
e2nNcµc
16n4µ4c
, (39)
so that the average phase factor simplifies to
〈e2iθ〉pq = 1−
for µ < µc. (40)
This result is in agreement with the mean field result (16). Indeed, the denominator of (38) is the product of the
Goldstone masses µc−µ, µc+µ, µc and µc, whereas the denominator of (40) is the product of four Goldstone masses
µc. The free energy of both partition functions is equal to 2nNcµc.
For µ > µc, the large Nc limit of the two flavor partition function is still given by (39), but the large Ncµ limit of
the microscopic phase quenched partition function is now given by
Zpq =
e2nNcµ
16n4(µ2 − µ2c)µ2
, (41)
resulting in the average phase factor
〈e2iθ〉pq =
ZNf=2
− 1)e−2nNc(µ−µc) for µ > µc . (42)
This result can also be obtained from the small nµ, nµc expansion of the large Nc limit of the average phase factor
(see Eq. (37)) at fixed nNcµc. We conclude that the sign problem becomes exponentially hard for µ > µc.
In Fig. 1 we show the average phase factor for different values of n and Nc and a critical chemical potential of
µc = 0.1. In the thermodynamic limit at fixed Nc, the average phase factor is one for µ < µc and jumps to zero for
µ > µc. In the right figure we observe a rapid convergence to the mean field result discussed in section II C.
B. Average phase factor for full QCD
The average phase factor with the Nf -flavor quark determinant as weight is defined by
〈e2iθ〉Nf =
ZNf (µc, µ)
detD†
detNfD
enNfNcµc
ZNf (µc, µ)
det(1− Uenµ−nµc) det(1− U †e−nµ−nµc)
det(1− Ue−nµ−nµc) det(1− U †enµ−nµc)
detNf (1 − Uenµ−nµc)detNf (1− U †e−nµ−nµc).
FIG. 1: The average phase factor for one-dimensional QCD with gauge group U(Nc) calculated as the ratio of the two flavor
flavor partition function and the phase quenched partition function. In the left figure the number of lattice points varies as
indicated for fixed Nc = 3, and in the right figure the number of colors varies as indicated for fixed n = 4.
We evaluate this integral exactly for Nf = 0 and Nf = 1 but only give its large Nc limit for other values of Nf .
Using the CFZ-formula (18) [15] we obtain in the quenched case
〈e2iθ〉Nf=0 =
(1− e−2n(µ+µc))(1− e2n(µ−µc))
(1− e−2nµc)2
+ e−2nNcµc
(1 − e−2nµ)(1− e2nµ)
(1 − e2nµc)(1 − e−2nµc)
for µ < µc . (43)
The large Nc limit of this result coincides with the large Nc limit of the phase quenched average phase factor. In the
microscopic limit where µcnNc and µnNc remain fixed for Nc → ∞ we obtain
〈e2iθ〉Nf=0 = 1−
e−2nNcµc . (44)
In the large nNcµc limit the second term does not contribute resulting in
〈e2iθ〉Nf=0 = 1−
, (45)
in agreement with the results obtained in [4] (see subsection II C). In the thermodynamic limit at fixed Nc the average
phase factor converges to one for µ < µc.
After rewriting the determinants in (43) as
det(1− U †e−nµ−nµc)
det(1− U †enµ−nµc)
→ e−2nNcµ det(1− Ue
nµ+nµc)
det(1− Ue−nµ+nµc)
, (46)
the denominator of (43) can be expanded in powers of U for µ > µc. For Nf = 0 only the constant term in the
integrand yields a non-vanishing result. We thus find
〈e2iθ〉Nf=0 = e−2nNcµ for µ > µc, (47)
so that the average phase factor vanishes in the large nNcµ limit.
In Fig. 2 we show the quenched average phase factor for various values of n and Nc. Also in this case the average
phase factor jumps from 1 to 0 at µ = µc in the thermodynamic limit. The convergence to the mean field result for
increasing Nc (right figure) is very rapid. This can be understood from the expansion of the microscopic result given
by 1− (µ2/µ2c)(1− 13 (nµ)
2 + · · ·.
FIG. 2: The quenched average phase factor for one-dimensional QCD with gauge group U(Nc). In the left figure Nc = 3 and
the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.
Finally we calculate the average phase factor for an arbitrary number of flavors in the large Nc limit. The CFZ
formula can again be applied after inserting the factor
detNf (1− Ue−α)detNf (1− U †e−β)
in the denominator and taking the limit α → ∞ and β → ∞ at the end of the calculation.
For µ < µc, the leading order large Nc result is given by the identity permutation in the sum over permutations in
the CFZ formula. This results in
〈e2iθ〉Nf =
(1− e−2n(µ+µc))(1− e2n(µ−µc))
(1− e−2nµc)2
)Nf+1
for µ < µc. (48)
The microscopic limit of this result is given by
〈e2iθ〉Nf =
]Nf+1
for µ < µc, (49)
in agreement with the discussion given in section II C.
For µ > µc the CFZ formula can again be applied after rewriting the partition function according to (46) and the
insertion (48). Because of degenerate prefactors, all critical chemical potentials that occur in the combination µ+ µc
have to be taken different. After carefully taking limits we find
ZNf+1|1∗(µc, µ) =
e−2µnNc
eµcnNcNf
(1− e−2µcn)N
(1− e−2n(µ+µc))Nf (1− e−2nµ)Nf
(1− e−2nµc)Nf
. (50)
The average phase factor given by
〈e2iθ〉Nf =
ZNf+1|1∗
∼ e−2nNcµ for µ > µc (51)
vanishes in the thermodynamic limit. The exact result becomes rather cumbersome even for small values of Nf . As
an illustration we give the result for Z2|1∗(µc, µ) in Appendix A .
V. AVERAGE PHASE FACTOR FOR SU(Nc)
To calculate integrals over SU(Nc) we use the identity
SU(Nc)
dU · · · =
U(Nc)
dUdetp(U) · · · . (52)
For the partition functions discussed below, with a few exceptions the sum truncates to a small number of terms.
A. SU(Nc) QCD Partition Function
Let us first calculate the 1d SU(Nc) QCD partition function for Nf = 1. The partition function is defined by
SU(Nc)
(µc, µ) =
SU(Nc)
dUdetNfD =
U(Nc)
dUdetpUenNcµcdetNf (1 + Uenµ−nµc)detNf (1 + U †e−nµ−nµc). (53)
For p ≥ Nf the integrand can be rewritten in terms of determinant of matrices U only
ZNf ,p(µc, µ) =
U(Nc)
dUdetpUdetNfD
U(Nc)
dUenNcNfµcdetp−Nf (U)detNf (1 + Uenµ−nµc)detNf (U + e−nµ−nµc) . (54)
For p > Nf it follows immediately that the integral vanishes. For p = Nf only the constant term inside the
determinants contributes to integral resulting in
ZNf ,p=Nf (µc, µ) = e
−nNfNcµ,
ZNf ,p>Nf (µc, µ) = 0. (55)
For p < 0 we combine detpU = det−pU † with the factor detNf (1− Uenµ−nµc). We then find
ZNf ,p=−Nf (µc, µ) = e
nNfNcµ,
ZNf ,p<−Nf (µc, µ) = 0. (56)
Using that the hermitean conjugate of a unitary matrix is also unitary we obtain
ZNf ,−p(µc, µ) = ZNf ,p(µc,−µ). (57)
For |p| < Nf the integral can be calculated by means of the CFZ formula by choosing all µc different and introducing
the limit
detU = lim
det(U − e−α). (58)
The limit of degenerate µc and α → ∞ is taken at the end of the calculation. We will only give exact results for
Nf = 1 and Nf = 2.
Using (55,56) and the result for U(Nc) we obtain for Nf = 1
SU(Nc)
(µc, µ) = e
−nNfNfµ + enNfNfµ +
sinh(n(Nc + 1)µc)
sinh(nµc)
, (59)
in agreement with earlier work by Bilic and Demeterfi [24].
For Nf = 2, the only integrals that have not yet been calculated are those for p = ±1 which are related by (57).
For the p = 1 contribution we find
ZNf=2,p=1(µc, µ) = e
nNc(µc−µ)
e−2n(Nc+1)µc + e2nµc
(enµc − e−nµc)2
−2nNcµc(e−3nµc − 3e−nµc)− e3nµc + 3enµc
(enµc − e−nµc)3
. (60)
For µ < µc the large nNcµ limit of the Nf = 2 partition function is dominated by the p = 0 term, whereas for
µ > µc the large nNcµc limit is given by the p = −2 term. For µ = µc the large Nc limit is dominated by the p = −1
with 1/Nc corrections from the p = 0 and p = −2 terms. We thus find as leading large Nc result
SU(Nc)
(µc, µ) =
e2n(Nc+2)µc
16 sinh4(nµc)
for µ < µc,
SU(Nc)
(µc, µ) = e
2nNcµ for µ > µc,
SU(Nc)
(µc, µ) =
2n(Nc+1)µc
4 sinh2(nµc)
1 +O(N−1c )
for µ = µc. (61)
Both for µ < µc and µ > µc the terms that are canceled by the integration over the unitary group are subleading in
the thermodynamic limit. In other words there is no serious sign problem.
By inspection one can easily show that the dominance of the p = 0 terms for µ < µc and the p = −Nf term for
µ > µc is a feature of the large Nc limit that is valid for any number of flavors. Therefore, for µ < µc, the chiral
condensate is the same as for the U(Nc) theory. For µ > µc though, the dominant p = −Nf term is mass-independent.
This results in a vanishing chiral condensate so that the large Nc limit of the U(Nc) theory and the SU(Nc) theory
is different.
B. The Phase Quenched Partition Function for SU(Nc)
A second ingredient for the average phase factor is the phase quenched SU(Nc) partition function. Using the same
arguments as for the full QCD partition function, one easily obtains
Z1+1∗,|p|>2(µc, µ) = 0,
Z1+1∗,|p|=2(µc, µ) = 1,
Z1+1∗,−p(µc, µ) = Z1+1∗,p(µc,−µ). (62)
The partition function for p = 0 is the U(Nc) partition function which was already given in (33). What remains to
be calculated is the partition function for p = +1. Using the CFZ formula one easily arrives at
Z1+1∗,p=1(µc, µ) =
4 sinh(nµ) sinh(nµc)
sinh(n(Nc + 2)(µ+ µc))
sinh(n(µ+ µc))
− sinh(n(Nc + 2)(µ− µc))
sinh(n(µ− µc)))
. (63)
Both for µ < µc and µ > µc, the leading large Nc result of the phase quenched partition function resides in the
p = 0 term given in Eq. (35).
C. Average Phase Factor for SU(Nc)
The average phase factor can be either obtained from the ratio of the full QCD partition function and the phase
quenched partition function or can be calculated with the full QCD partition function as weight (see Eqs. (2) and
(3)). In the first case we find for large Nc,
〈e2iθ〉SU(Nc)pq =
sinh(n(µc−µ)) sinh(n(µc+µ))
sinh2(nµc)
for µ < µc,
16 sinh(n(µ− µc)) sinh(n(µc + µ)) sinh2(nµ) for µ > µc,
1− e−4nµc for µ = µc.
The microscopic limit of the average phase factor is obtained by expanding this result for small µc, µ. We again find
the usual mean field result
〈e2iθ〉SU(Nc)pq =
θ(µc − µ). (65)
In the thermodynamic limit, the average phase factor converges to one in both for µ < µc and µ > µc.
FIG. 3: The phase quenched average phase factor for one-dimensional QCD with gauge group SU(Nc). In the left figure
Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.
Exact results for the average phase factor are displayed in Fig. 3. At finite Nc, we observe a rapid convergence
to the asymptotic value of 1. At fixed n, the approach to the large Nc limit is slow. Also clearly visible is that for
µ ≈ µc the corrections to the microscopic limit are large. As was already discussed before, this is due to the p = −1
contribution.
In the quenched case the average phase factor for SU(Nc) defined according to Eq. (3) can be written as
〈e2iθ〉SU(Nc)Nf=0 =
U∈U(Nc)
dUdetp(U)
det(1 + Uenµ−nµc) det(1 + U †e−nµ−nµc)
det(1 + Ue−nµ−nµc) det(1 + U †enµ−nµc)
. (66)
The integrals can again be calculated by means of the CFZ formula. For Nc ≥ 3 one can easily show that the limit
introduced in (58) gives vanishing results for |p| > 2. However, for Nc = 1 and Nc = 2 additional terms may contribute
to this limit. For Nc = 2, where the average phase factor is equal to one, contributions for |p| > 2 vanish for µ < µc,
but terms with p < −2 are nonzero for µ > µc. The formulae for |p| ≤ 2 given below are valid for Nc = 2. For Nc = 1
the integrals in (66) only vanish for p ≥ 2 and µ > µc. Also the general answer for p = −2 is not correct in this case.
For p = −1 we obtain
〈e2iθ〉Nf=0 =
e−Ncn(µc−µ)
(1−e−2n(µc+µ))(1−e−2nµ)
1−e−2nµc
for µ < µc,
enNc(µc−µ)(1−e−2nµ)
1−e−2nµc
[(1− e−2nµc−2nµ)− e−2nNcµc(e−2nµc − e−2nµ)] for µ > µc.
For p = 1 we find
〈e2iθ〉Nf=0 =
e−Ncn(µc+µ)
(1−e−2n(µc−µ))(1−e2nµ)
1−e−2nµc
for µ < µc,
0 for µ > µc.
Finally, for p = −2 the result is
〈e2iθ〉Nf=0 =
0 for µ < µc,
(1− e−2nµ)(1 − e−2n(µ+µc))(1 − e2n(µc−µ))(1 − e−2nµ) for µ > µc.
For p = 2 the average phase factor vanishes. Notice that the relation
〈e2iθ〉Nf=0,−p(µc, µ) = 〈e2iθ〉Nf=0,p(µc,−µ) (70)
FIG. 4: Average phase factor for one-dimensional QCD with gauge group SU(Nc). In the left figure Nc = 3 and the number
of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.
is also valid in this case. It can be used to derive results for negative values of µ.
In the large Nc limit, the average phase factor is dominated by the p = 0 contribution for µ < µc and by the p = −2
contribution for µ > µc. For µ = µc, the p = 0, p = −1 and p = −2 contributions are of equal order in Nc, but only
the p = −1 term is nonvanishing. As large Nc limit of the average phase factor we thus find
〈e2iθ〉SU(Nc)Nf=0 =
sinh(n(µc−µ)) sinh(n(µc+µ))
sinh2(nµc)
for µ < µc,
16 sinh(n(µ− µc)) sinh(n(µc + µ)) sinh2(nµ) for µ > µc,
1− e−4nµc for µ = µc,
which is the same expression as obtained in (64) for the phase quenched average phase factor. The microscopic limit
of the average phase factor is again given by the small µ-µc expansion of this result which is equal to the usual mean
field result
〈e2iθ〉SU(Nc)Nf=0 =
θ(µc − µ). (72)
In Fig. 4 we show the exact result for the SU(Nc) average phase factor. We observe a rapid approach to the
thermodynamic limit at fixed Nc just like in Fig. 3.
In the large Nc limit the result for the quenched average phase calculated according to (66) is also given by (64). As
already was argued in the introduction, we conclude that QCD in one dimension with SU(Nc) as gauge group does
not have a serious sign problem, not only for µ < µc, but also for µ > µc. Indeed one-dimensional QCD at nonzero
chemical potential could be simulated reliably by the Glasgow method [26].
VI. THE SIGN PROBLEM AT NONZERO TEMPERATURE
The results of the previous two sections show that the sign problem is severe for QCD in one dimension with
gauge group U(Nc) and µ > µc. Although this theory does not have a critical temperature, it is clear from Fig. 1
that the average phase factor for µ > µc increases significantly for higher temperatures (i.e. for lower values of n).
However, this is a geometric effect due to the Boltzmann factor. Recent lattice simulations [10, 11, 12, 26, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47] suggest that the sign problem becomes much milder for T around Tc. Because of the
absence of a critical temperature, this temperature dependence of the average phase factor cannot be investigated in
FIG. 5: Scatter plot of the eigenvalues of the random matrix Dirac operator for µ =0.3 and T = 0, 0.7, 0.9, 1.0, 1.1 . The
quark mass at m = 0.1 is indicated by the black dot.
one dimensional QCD. Instead we study this problem in a schematic random matrix model at nonzero temperature
and chemical potential [30, 48, 49]. This model has a severe sign problem as well as a critical temperature and its
eigenvalues are scattered in a finite domain of the complex plane, just like in QCD. The quenched average phase factor
in this model is given by the matrix integral
〈e2iθ〉(m,µ, T ) =
dCe−NTrCC
† det(D +m)
det(D† +m)
, (73)
with Dirac operator given by
0 iC + µ+ it
iC† + µ+ it 0
, (74)
and t is the traceless diagonal matrix t = diag(−T, · · · ,−T, T, · · · , T ). The integral is over the real and imaginary
parts of the matrix elements of the complex N ×N matrices C. The phase quenched average phase factor is defined
〈e2iθ〉pq =
dCdet2(D +m)e−NTrCC
dCdet(D +m) det(D† +m)e−NTrCC
If at all possible, it certainly takes a significant effort to evaluate these integrals analytically. Therefore we have
studied the quenched (73) and phase quenched (75) average phase factor numerically. In Fig. 5, we give scatter plots
of the eigenvalues of the random matrix Dirac operator. For each of the temperatures T = 0, 0.7, 0.9, 1, 1.1, the
figures show results of 40 matrices with N = 100 and µ = 0.3. The black dot represents the quark mass for which the
average phase factor is calculated. In Fig. 6 we show the µ-dependence of the phase quenched (left) and quenched
(right) average phase factor obtained by averaging over 10,000 to 80,000 matrices C with N = 100. The results for
the phase quenched average phase factor for T = 0 and T = 0.7 and µ > µc are not displayed because they do not
converge because of the severity of the sign problem. The average phase factor depends only weakly on the size of
FIG. 6: Phase quenched (left) and quenched (right) average phase factor versus chemical potential. Results have been obtained
from the chiral random matrix model (74).
the matrices. The conclusion of Figs. 5 and 6 is that the sign problem is not serious when the quark mass is outside
the domain of the eigenvalues. This confirms the conjecture that was made in [4, 5, 50]. It also agrees with the
observation in [30] that the free energy of the full theory and the phase quenched theory are the same for T = 0 if
the quark mass is outside the domain of the eigenvalues (this is the case for µ < µc). The µ-dependence is due to the
curvatures at the saddle point.
VII. COMPARISON WITH THE QCD3 PARTITION FUNCTION
According to the general arguments given in section II B, the microscopic limit of the one dimensional QCD
partition function at nonzero chemical potential is equal to the microscopic limit of the QCD3 partition function at
zero chemical potential, but with shifted quark masses. In this section we show this explicitly for some of the results
derived before. We first discuss the microscopic limit of the QCD3 partition function which can be derived from either
a chiral Lagrangian or a random matrix model with the same global symmetries [28]. For a discussion of QCD3 in
terms of chiral Lagrangians we refer to [51].
A. Random Matrix Model
From the continuum Dirac operator (9) it follows that the random matrix model that describes the fluctuations of
the low-lying Dirac eigenvalues is given by
dHP (H) det(iH + µ+m) det(iH −m+ µ), (76)
where the probability distribution can conveniently be taken to be the Gaussian distribution
P (H) = e−2NΣ
2TrH2 . (77)
Such random matrix partition functions have been studied elaborately in the literature to analyze the microscopic
limit of QCD in three dimensions [28, 52, 53, 54, 55]. The microscopic limit of the partition function (76) with quark
masses given by M = diag(−m1, · · · ,−mNf ,m1, · · · ,mNf ) is equal to [53, 54]
ZQCD3 =
A(m) A(−m)
A(−m) A(m)
, (78)
where the matrix elements of the Nf ×Nf matrix A(m) are given by
A(m)kl = m
−mk , (79)
and ∆ is the Vandermonde determinant
∆(M) =
(Mkk −Mll). (80)
Although general expressions for QCD3 partition functions with an arbitrary number of bosonic and fermionic
determinants are also known [56], for our purposes we only need the partition function
ZNf+2|2 =
det(D +mk) det(D −mk)
det(D + z1) det(D − z2)
det(D + z̄1) det(D − z̄2)
. (81)
This partition function was evaluated in [57] by means of the supersymmetric method and is given by
ZNf+2|2 =
(z1 − z̄1)(z2 − z̄2)
∆2Nf (m)
expNc(z̄1 + z̄2)
Nc(z̄1 + z̄2)
(mk − z̄1)(mk − z̄2)
(mk − z1)(mk − z2)
sinhNc(mk+ml)
Nc(mk+ml)
sinhNc(mk+z1)
Nc(mk+z1)
sinhNc(z2+ml)
Nc(z2+ml)
sinhNc(z1+z2)
Nc(z1+z2)
e−Nc(z1−z̄1)+Nc(z2−z̄2)
∆2Nf (m)
sinhNc(mk +ml)
Nc(mk +ml)
. (82)
The last term is a so-called Efetov-Wegner term, and the Vandermonde determinant is over positive masses only
∆(m) =
(mk −ml). (83)
B. Microscopic Limit
We will now show that the microscopic limit of the partition function (25) is equal to the microscopic limit of the
QCD3 partition function. To this end we multiply row k of the first Nf rows of the determinant by exp(−Ncm− k/2)
and row k of the second Nf rows of the determinant in (25) by exp(−Ncm+ k/2). In the microscopic limit we keep
Ncm±,k fixed so that we can expand the masses that do not occur in this combination. By subtracting successive
columns starting with the first one we obtain
ZNf =
1≤k<l≤2Nf
(Ml −Mk)
e−Ncm− 1/2 m− 1e
−Ncm− 1/2 · · · mNf−1− 1 e−Ncm− 1/2 m
+Ncm− 1/2 · · · m2Nf−1− 1 eNcm− 1/2
−Ncm−Nf /2 m−Nf e
−Ncm−Nf /2 · · · mNf−1−Nf e
−Ncm−Nf /2 m
+Ncm−Nf /2 · · · m2Nf−1−Nf e
Ncm−Nf /2
e−Ncm+1/2 m+1e
−Ncm+1/2 · · · mNf−1+ 1 e−Ncm+1/2 m
+Ncm+1/2 · · · m2Nf−1+1 eNcm+1/2
−Ncm+Nf /2 m+Nf e
−Ncm+Nf /2 · · · mNf−1+Nf e
−Ncm+Nf /2 m
+Ncm+Nf /2 · · · m2Nf−1+Nf e
Ncm+Nf /2
where the masses MK are defined below Eq. (25). By multiplying the columns l = Nf + 1, · · · , 2Nf by (−1)l−1 and
introducing microscopic masses, we obtain exactly the expression for the QCD3 partition function given in (78).
For masses −(µc + µ), −(µc − µ), µc − µ, µc + µ corresponding to the microscopic limit of the two-flavor phase
quenched partition function given in (34) the determinant in (84) is given by
4nµ2(e2nNcµc + e−2nNcµc)− 4nµ2c(e2nNcµ + e−2nNcµ) + 8n(µ2c − µ2). (85)
and the prefactor is equal to
1≤k<l≤2Nf
(Ml −Mk) = 64n6(µ2c − µ2)µ2cµ2. (86)
FIG. 7: The number variance for n = 4 and Nc = 100.
Their ratio coincides with the microscopic phase quenched partition function given in (34).
As second example, we consider the quenched average phase factor for U(Nc) given in (43). For Nf = 0 the partition
function (82) simplifies to
Z2|2 = (z1 − z̄1)(z2 − z̄2)
eNc(z̄1+z̄2)
Nc(z̄1 + z̄2)
sinhNc(z1 + z2)
Nc(z1 + z2)
+ e−Nc(z1−z̄1−z2+z̄2). (87)
The masses in this partition function corresponding to (43) are given by
z1 = nµc + nµ, z2 = nµc − nµ,
z̄2 = nµ− nµc, z̄1 = −nµ− nµc. (88)
Substituting them into (87) we obtain
Z2|2 = 1−
e−4Ncnµ (89)
which is exactly the microscopic limit of the quenched average phase factor (43) given in Eq. (44).
VIII. DIRAC SPECTRUM
We consider the Dirac operator (4) in a gauge where all gauge fields except Un1 ≡ U and U1,n = U † are equal to
unity. If the eigenvalues of U are equal to exp(iθk) the eigenvalues of the Dirac operator are given by
λk,l =
2πi(k+1/2)+iθl
+µ − e−
2πi(k+1/2)−iθl
−µ), k = 1, · · · , n, l = 1, · · · , Nc. (90)
Contrary to QCD at µ 6= 0 in more than one dimension, where the eigenvalues are scattered in the complex plane
[27, 30, 58, 59, 60, 61], the eigenvalues are located on an ellipse in the complex plane with real and imaginary parts
related by
Re(λk,l)
eµ − e−µ
Im(λk,l)
eµ + e−µ
= 1. (91)
A. Universal Fluctuations
In section II B we have shown that the continuum limit of the staggered Dirac operator in one dimension is in the
same universality class as QCD in three dimensions. The continuum limit of the staggered three dimensional Dirac
operator, though, is in the chiral symmetry universality class of QCD in four dimensions. For example, in [62] this
was found for the distribution of the small Dirac eigenvalues. The reason for the chiral structure is that the staggered
Dirac operator only couples even and odd lattice sites. This is also the case in one dimension, but the off-diagonal
blocks, containing the gauge fields, are the same in the continuum limit (or occur in the combination Un + U †n),
resulting in a two flavor theory with opposite masses (see (12)). The secular equation is given by
det[(∂0 + iA0 + µ+ λ)(∂0 + iA0 + µ− λ)] = 0. (92)
This corresponds to the superposition of the spectrum of ∂0 +A0 + µ and −∂0 − iA0 − µ. In the domain of the Dirac
spectrum where ∂0 can be neglected, the eigenvalues on each of the lines ±µ are given by the Hermitean random
matrix ensemble A0. For µ = 0 we have the superposition of the Hermitean random matrix ensembles of A0 and −A0.
From the eigenvalues (90) it is clear that the staggered lattice Dirac eigenvalues show a similar superposition. One
ensemble is localized on the left half of the ellipse and the other ensemble on its right half. Superpostions of more
ensembles arise for correlations on scales larger than Nc level spacings.
Eigenvalues of U(Nc) matrices are correlated according to the circular unitary ensemble. Therefore, the eigenvalues
λk,l are correlated according the Gaussian Unitary Ensemble (GUE) on a scale where the variations in the average
spectral density can be neglected, i.e. on a scale much less than Nc level spacings. By correcting for the variation of
the average level spacing, the scale on which universal random matrix correlations are found can be extended, but this
scale should always remain well below Nc level spacings. Then we are necessarily within one of the n successive copies
of the eigenvalues along the ellipse. At a scale of a finite number of average level spacings, we expect convergence to
universal random matrix correlations in the large Nc limit.
As measure of the correlations between eigenvalues, we use the number variance Σ2(n̄) defined as the variance on
the number of eigenvalues in an interval that contains n̄ eigenvalues on average. We calculate the number variance for
intervals starting from zero. Because of the superposition of an enemble and its negative, this is equal to the number
variance of a single ensemble for an interval that is symmetric about zero which is given by the GUE result. In Fig.
7 we show the number variance obtained from an average over 105 gauge field configurations for Nc = 100 and µ = 0.
The number variance is calculated for an interval starting at zero. The discrepancy between the analytical result and
the GUE (solid curve) is only barely visible.
The increase of the domain of validity of chiral random matrix theory with increasing Nc was also observed for
Dirac operator of QCD in four dimensions [63]. This can be explained as follows: The number of eigenvalues that is
described by random matrix theory is equal to ≈ F 2π
V (with V the space-time volume), but F 2π scales with Nc in
the large Nc limit.
B. Chiral Symmetry Breaking at µ 6= 0
We first discuss the case of µ = 0. Then the average spacing of the eigenvalues scales as 1/(nNc). This means
that the chiral condensate develops a discontinuity at m = 0 when nNc → ∞. Two cases of interest where a nonzero
chiral condensate can be obtained are the limit n → ∞ for fixed Nc, i.e. at zero temperature, or at fixed n in the
limit Nc → ∞. In both cases the chiral condensate is discontinuous for nNc → ∞ when the quark mass crosses the
imaginary axis where the eigenvalues are located. However, Goldstone bosons and universal behavior is only found
in the large Nc limit. At fixed Nc, chiral symmetry breaking is not associated with universal behavior.
For µ 6= 0 the Dirac eigenvalues are located on the ellipse (91). When the quark mass is inside the ellipse of
eigenvalues the chiral condensate is zero for nNc → ∞ for the SU(Nc) partition function and the phase quenched
U(Nc) partition function [24]. For gauge group U(Nc) with Nf ≥ 1 flavors, though, the partition function is µ-
independent so that the chiral condensate is also nonzero if the mass is inside the eigenvalue ellipse. The Banks-Casher
formula fails in this case which is known as the “Silver Blaze Problem” [64]. The resolution is the same as for QCD
in four dimensions [31]: the re-weighted eigenvalue distribution defined as
ρfull(z) =
U(Nc)
dUdetNfD
δ2(z − λk) (93)
shows oscillations with an amplitude that diverges exponentially with n and a period that is proportional to 1/n.
Below we will illustrate this for U(1).
For Nc = 1 one easily derives that the spectral density is given by
ρfull(z) = 4
enµc + e−nµc − en(iα+µ) − e−n(iα+µ)
(enµc + e−nµc)(e2r + e−2r + e2iα + e−2iα)
δ(r − µ), (94)
where z is parameterized as
(er+iα − e−r−iα), r > 0, α ∈ [0, 2π], (95)
and we have used that ZNf=1 = 2 cosh(nµc) (see eq. (8)). For µ > µc this spectral density has oscillations with
an amplitude that diverges exponentially with n and a period of order 1/n which are the essential properties of the
spectral density of the Dirac operator of QCD in four dimensions [31, 65, 66]. It can be decomposed as
ρfull(z) = ρq(z) + ρosc(z). (96)
The average quenched spectral density is defined by
ρq(z) =
δ2(z − λk) (97)
with the eigenvalues λk given by Eq. (90). After changing variables according to (95) and integrating over θ we find
ρq(z) =
(e2r + e−2r + e2iα + e−2iα)
δ(r − µ). (98)
The oscillatory part of the spectral density, ρosc(z) is equal to the difference ρfull(z)− ρq(z).
The chiral condensate is given by
Σfull(m) =
ρfull(z)
, (99)
and can also be decomposed as
Σfull(m) = Σq(m) + Σosc(m). (100)
The Jacobian for the transformation of d2z to drdα is given by
d2z =
(e2r + e−2r + e2iα + e−2iα)drdα, (101)
so that the chiral condensate after integration over r can be simplified to
Σfull(m) =
enµc + e−nµc − en(iα+µ) − e−n(iα+µ)
(enµc + e−nµc)(eµ+iα − e−µ−iα − 2m)
tanh(nµc)
coshµc
. (102)
This result remains finite for n → ∞. We remind the reader that the mass is parameterized as m = sinhµc. This
expression represents the resolvent at the quark mass which was evaluated in [49] and is in agreement with earlier
work [6, 24]. Decomposing the spectral density and the chiral condensate according to (96) we obtain
Σosc(m) = θ(sinh(µ)−m)Σfull(m),
Σq(m) = θ(m− sinh(µ))Σfull(m), (103)
so that when the quark mass is inside the ellipse of eigenvalues, the entire chiral condensate is due to the oscillatory
part of the spectral density. The discontinuity in the chiral condensate is reminiscent to a Stokes phenomenon. The
alternative to the Banks-Casher relation proposed in [31] is also at work for QCD in one dimension. This solves the
“Silver Blaze Problem” [64].
IX. CONCLUSIONS
We have studied QCD in one dimension at nonzero chemical potential. Both the full theory, its quenched and
phase quenched versions and gauge groups U(Nc) and SU(Nc) have been considered. In one dimension, the QCD or
QCD-like partition functions can be reduced to a single matrix integral, which because of recent advances by Conrey,
Farmer and Zirnbauer, could be evaluated analytically. In this paper we have analyzed the small mass behavior of the
partition function, the nature of the sign problem, and the relation between the Dirac spectrum and chiral symmetry
breaking.
To put our results in perspective, we emphasize that QCD in one dimension, is quite different from QCD in
more dimensions. In particular, we wish to mention the following three points. First, instead of being scattered in
the complex plane, the eigenvalues of the staggered Dirac operator are located on an ellipse in the complex plane.
Second, phase transitions can only take place for zero temperature or for Nc → ∞. At the critical chemical potential,
a transition from the vacuum state to a state of free quarks takes place. Third, color singlets are made out of
noninteracting quarks so that the critical chemical potential is given by the quark mass. In particular, the critical
chemical potential for the meson state and the baryon state is the same.
Because the Dirac eigenvalues are located on a curve, the large Nc limit of staggered lattice QCD in one dimension
is in the same chiral symmetry class as QCD in three dimensions. We have shown this both by an explicit evaluation
of the partition function, and by analyzing the fluctuations of the Dirac eigenvalues for large Nc. We have used this
equivalence to explain the behavior of the partition functions and the average phase factor in the microscopic limit.
Contrary to QCD in four dimensions, the sign problem for QCD in one dimension is not severe in the case of gauge
group SU(Nc). One reason is that the critical chemical potential for full QCD and phase quenched QCD is the same,
so that the parameter domain mπ/2 < µ < mN/3, where the sign problem becomes severe in four dimensions, is
absent in one dimension. A second reason is that for µ > µc both the full theory and the phase quenched theory
become a theory of free quarks with the same free energy in the thermodynamic limit.
For gauge group U(Nc), on the other hand, the sign problem is severe when µ > mπ/2 both in one dimension and in
four dimensions. The reason is that the U(Nc) theory does not have charged excitations, whereas the phase quenched
theory has charged mesons resulting in a phase transition at µ = mπ/2. We have evaluated the average phase factor
both by averaging with respect to the phase quenched partition function and the full partition function, and similar
conclusions have been reached.
The condititon µ > mπ/2 for having a severe sign problem can be rephrased as the quark mass being inside the
ellipse of eigenvalues. In more dimensions this condition is that the quark mass is inside the support of the Dirac
spectrum. It also applies to nonzero temperature as we have demonstrated explicitly in the framework of a chiral
random matrix model.
Also for U(Nc) QCD in one dimension, the chiral condensate is discontinuous across the imaginary axis in spite
of the fact that there are no Dirac eigenvalues. This can only mean that the phase of the fermion determinant
is responsible for the discontinuity. This is what happens in four dimensions where the discontinuity is due to a
contribution to the spectral density that oscillates with a period of the inverse volume and amplitude that diverges
exponentially with the volume. Exactly the same mechanism, where the discontinuity in the chiral condensate arises
due to a Stokes like phenomenon, is at work in one dimension. This suggests that this is a universal mechanism for
theories with a sign problem.
We end by repeating that QCD in one dimension with SU(Nc) as gauge group has no serious sign problem. Our
hope is that part of this conclusion translates to four dimensions ameliorating the sign problem for µ > MN/3.
Acknowledgments. We wish to thank K. Splittorff for numerous comments and criticism and P.H. Damgaard, H.
Neuberger and R.Pisarski are thanked for valuable discussions. K. Splittorff is also thanked for a careful reading of
the manuscript. This work was supported by U.S. DOE Grant No. DE-FG-88ER40388 (JV), the Angelo Della Riccia
Foundation (LR), the Villum Kann Rassmussen Foundation (JV) and the Danish National Bank (JV).
APPENDIX A: AVERAGE U(Nc) PHASE FACTOR FOR Nf = 1
In this appendix we give explicit results for the average phase factor for Nf = 1 defined by
〈e2iθ〉Nf=1 =
Z2|1∗(µc, µ)
ZNf=1(µc, µ)
. (A1)
The partition function ZNf=1(µc, µ) was already given in Eq. (8). The numerator can be obtained from the CFZ
formula. After taking the limit of degenerate critical chemical potentials at the end of the calculation, we obtain for
µ < µc
Z2|1∗(µc, µ) = e
n(Nc+1)µce−4nµ
(1− e2n(µ+µc))2(1− e2n(µ−µc))2
(enµc − e−nµc)5
+ e−3n(Nc+1)µc
(enµ − e−nµ)4
(enµc − e−nµc)5
+e−n(Nc+1)µc [f0 +Ncf1 +N
c f2]. (A2)
2e2n(µ+µc) + 2e2n(µ−µc) − e4nµ − e−4nµ + 4e2nµ + 4e−2nµ − e4n(µ−µc) − e4n(µ+µc) + 2e6nµc−2nµ + 2e6nµc+2nµ
(enµc − e−nµc)5
−10e2nµc + 6e4nµc − 2− 5e6nµc − e−2nµc
(enµc − e−nµc)5
(1− e−2nµ)2(3e2nµc + e−2nµc − 2e2nµ − 2e−2nµ)
(enµc − e−nµc)3
f2 = −
(1− e−2nµ)2(1 − en(µ+µc))(1 − en(µ−µc))
(enµc − e−nµc)3
. (A3)
For µ > µc a similar calculation results in
Z2|1∗ = Nce
nNc(µc−2µ)
(1− e−2n(µ+µc))(1− e−2nµ)
(1− e−2nµc)2
) +Nce
−nNc(2µ+µc)
(1− e2n(µc−µ))(1− e−2nµ)
(1− e2nµc)2
g1 = 1 +
e2nµ − 1
e2n(µ+µc) − 1
e2nµc − 1
g2 = 1 +
e2n(µ−µc) − 1
e2nµ − 1 −
e−2nµc − 1 . (A5)
We have checked that the results in this appendix agree with the large Nc limit of the average phase factor given in
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	Introduction
	QCD in One Dimension
	Lattice QCD
	Continuum Theory
	Mean Field Limit for Large Nc
	The Conrey-Farmer-Zirnbauer formula
	Exact Evaluation of the One-Dimensional U(Nc) Partition Function
	Partition Function for arbitrary Nf
	The Phase Quenched Partition Function
	Average Phase Factor for U(Nc)
	Phase Quenched Average Phase Factor
	Average phase factor for full QCD
	Average Phase Factor for SU(Nc)
	SU(Nc) QCD Partition Function 
	The Phase Quenched Partition Function for SU(Nc)
	Average Phase Factor for SU(Nc) 
	The Sign Problem at Nonzero Temperature
	Comparison with the QCD3 Partition Function
	Random Matrix Model
	Microscopic Limit
	Dirac Spectrum
	Universal Fluctuations
	Chiral Symmetry Breaking at =0
	Conclusions
	Average U(Nc) Phase Factor for Nf = 1
	References

  Using an integration formula recently derived by Conrey, Farmer and
Zirnbauer, we calculate the expectation value of the phase factor of the
fermion determinant for the staggered lattice QCD action in one dimension. We
show that the chemical potential can be absorbed into the quark masses; the
theory is in the same chiral symmetry class as QCD in three dimensions at zero
chemical potential. In the limit of a large number of colors and fixed number
of lattice points, chiral symmetry is broken spontaneously, and our results are
in agreement with expressions based on a chiral Lagrangian. In this limit, the
eigenvalues of the Dirac operator are correlated according to random matrix
theory for QCD in three dimensions. The discontinuity of the chiral condensate
is due to an alternative to the Banks-Casher formula recently discovered for
QCD in four dimensions at nonzero chemical potential. The effect of temperature
on the average phase factor is discussed in a schematic random matrix model.

Introduction
	QCD in One Dimension
	Lattice QCD
	Continuum Theory
	Mean Field Limit for Large Nc
	The Conrey-Farmer-Zirnbauer formula
	Exact Evaluation of the One-Dimensional U(Nc) Partition Function
	Partition Function for arbitrary Nf
	The Phase Quenched Partition Function
	Average Phase Factor for U(Nc)
	Phase Quenched Average Phase Factor
	Average phase factor for full QCD
	Average Phase Factor for SU(Nc)
	SU(Nc) QCD Partition Function 
	The Phase Quenched Partition Function for SU(Nc)
	Average Phase Factor for SU(Nc) 
	The Sign Problem at Nonzero Temperature
	Comparison with the QCD3 Partition Function
	Random Matrix Model
	Microscopic Limit
	Dirac Spectrum
	Universal Fluctuations
	Chiral Symmetry Breaking at =0
	Conclusions
	Average U(Nc) Phase Factor for Nf = 1
	References
