Energy density for chiral lattice fermions with chemical potential
Energy density for chiral lattice fermions with chemical potential
Christof Gattringera and Ludovit Liptakb
Institut für Physik, FB Theoretische Physik, Universität Graz,
Universitätsplatz 5, 8010 Graz, Austria
Institute of Physics, Slovak Academy of Sciences,
Dúbravská cesta 9, 845 11 Bratislava 45, Slovak Republic
We study a recently proposed formulation of overlap fermions at finite density. In particular we
compute the energy density as a function of the chemical potential and the temperature. It is shown
that overlap fermions with chemical potential approach the correct continuum behavior.
PACS numbers: 11.15.Ha, 12.38.Gc
I. INTRODUCTION
Over the last two decades lattice gauge theory was
turned into a powerful qualitative tool for analyzing
QCD. This progress is in part due to the advances in
algorithms and computer technology, but also on the con-
ceptual side important breakthroughs were made. Most
prominent among these is the correct implementation of
chiral symmetry on the lattice based on the Ginsparg-
Wilson equation for the Dirac operator [1].
An application of lattice techniques which has seen a
lot of attention in recent years, is the study of QCD at
finite temperature. The lattice implementation of the
chemical potential µ, necessary for such an analysis, is
not straightforward, however. It is well known [2], that
a naive introduction leads to µ2/a2 contributions which
diverge in the continuum limit when the lattice spacing
a is sent to zero. For more traditional formulations, such
as the Wilson or staggered Dirac operators, the problem
has been solved by introducing the chemical potential in
the same way as the 4-component of the gauge field.
A satisfactory implementation of the chemical poten-
tial should be compatible with chiral symmetry on the
lattice based on the Ginsparg-Wilson equation. When
attempting to introduce the chemical potential into the
only solution of the Ginsparg-Wilson equation know in
closed form, the overlap operator [3], a potential prob-
lem quickly surfaces: defining the sign function of a non-
hermitian matrix. In [4] Bloch and Wettig proposed a
solution based on an analytic continuation of the sign
function into the complex plane. It was shown, that the
eigenvalue spectra of this construction match the expec-
tations from random matrix theory.
In this letter we analyze the proposal [4] further
and study the energy density of free, massless overlap
fermions with chemical potential. The dependence of the
energy density on µ and the temperature T allows for a
detailed analysis of the lattice formulation at finite den-
sity. Of particular interest will be the question whether
the analytic continuation of the sign function produces
divergent µ2/a2 terms. Our study indicates the absence
of such contributions and we find that the µ and T de-
pendence of the energy density is approached correctly.
II. SETUP OF THE CALCULATION
The overlap Dirac operator D(µ) for fermions with a
chemical potential µ is given as
D(µ) =
[1− γ5 signH(µ)] ,
H(µ) = γ5 [1− aDW (µ)] . (1)
The sign function may be defined through the spectral
theorem for matrices. DW (µ) denotes the usual Wilson
Dirac operator,
DW(µ)x,y = 1
δx,y − (2)
Uj(x)δx+ĵ,y +
Uj(x− ĵ)†δx−ĵ,y
U4(x)e
µa4δx+4̂,y −
U4(x−4̂)†e−µa4δx−4̂,y .
For later use we distinguish between the lattice spacing
a in spatial direction and the temporal lattice constant
a4. Periodic boundary conditions are used in the spatial
directions, while in time direction we apply anti-periodic
boundary conditions. The chemical potential µ is cou-
pled in the usual exponential form [2].
For vanishing µ the Wilson Dirac operator is γ5-
hermitian, i.e., γ5DW (0)γ5 = DW (0)
†. This implies that
H(0) is a hermitian matrix. As soon as the chemical po-
tential µ is turned on, γ5-hermiticity no longer holds, and
H(µ) is a non-hermitian, general matrix. This fact has
two important consequences: Firstly, the eigenvalues of
H(µ) are no longer real and the sign function for a com-
plex number has to be defined in the spectral representa-
tion of signH(µ). Secondly, the spectral representation
has to be formulated using left and right eigenvectors.
This latter problem will be dealt with later when we dis-
cuss the evaluation of signH(µ). For the sign function
of a complex number we use the analytic continuation
proposed in [4] and define the sign function through the
sign of the real part
sign (x+ iy) = sign (x) . (3)
http://arxiv.org/abs/0704.0092v2
The observable we study here is the energy density
defined as
ǫ(µ) =
〈H〉 = 1
H e−β (H−µN )
= (4)
e−β(H−µN )
= − 1
∂ lnZ
Here H is the Hamiltonian of the system, N denotes the
number operator and β = 1/T is the inverse temperature
(in our units the Boltzmann constant k is set to k = 1).
The derivatives in the second line are taken such that
βµ = c = const.
The continuum result for the subtracted energy density
of free massless fermions reads (see, e.g., [5])
ǫ(µ)− ǫ(0) = µ
µ2T 2 . (5)
When working on the lattice, the inverse tempera-
ture β is given by the lattice extent in 4-direction, i.e.,
β = N4a4. Thus the derivative ∂/∂β in (4) turns into
N−14 ∂/∂a4. The partition function Z is given by the
fermion determinant detD which we write as the prod-
uct over all eigenvalues λn. We thus find
ǫ(µ) = − 1
∂ ln detD
a4µ=c
= − 1
a4µ=c
= − 1
a4µ=c
. (6)
III. EVALUATION OF THE EIGENVALUES
According to (6), for the evaluation of ǫ(µ) the eigen-
values λn of the Dirac operator D have to be computed.
This is done in three steps: First we bring the Dirac oper-
ator for free fermions to 4× 4 block-diagonal form, using
Fourier transformation. Subsequently the spectral repre-
sentation is applied to the 4× 4 blocks of H to evaluate
sign H . Finally the eigenvalues of the blocks of D are
computed and by summing over the discrete momenta
all eigenvalues are obtained.
Following this strategy, one finds for the Fourier trans-
form Ĥ of H ,
Ĥ = γ5h5 + iγ5
γνhν , (7)
h5 = 1−
[1− cos(apj)]−
[1− cos(a4(p4 − iµ))] ,
hj = − sin(apj) for j = 1, 2, 3 ,
h4 = −
sin(a4(p4 − iµ)) . (8)
The spatial momenta are given by pj = 2πkj/aN ,
where N is the number of lattice points in the spatial
directions and kj = 0, 1 ... N − 1. The momenta in time-
direction are p4 = π(2k4 + 1)/a4N4, k4 = 0, 1 ... N4 − 1.
The remaining diagonalization of Ĥ is similar to the
construction of the left- and right-eigenfunctions for the
free Dirac operator. One finds that Ĥ has two different,
doubly degenerate eigenvalues
α1 = α2 = + s , α3 = α4 = − s , s =
h2 + h25 , (9)
where h2 =
ν . The corresponding left- and right-
eigenvectors, lj and rj are given by
l1 = l
1 [Ĥ + s1] , l2 = l
2 [Ĥ + s1] ,
l3 = l
3 [Ĥ − s1] , l4 = l
4 [Ĥ − s1] ,
r1 = [Ĥ + s1]r
1 , r2 = [Ĥ + s1]r
r3 = [Ĥ − s1]r(0)3 , r4 = [Ĥ − s1]r
4 . (10)
The constant spinors l
j , r
j are (T is transposition)
1 = r
(0) T
1 = c (1, 0, 0, 0) , l
2 = r
(0) T
2 = c (0, 1, 0, 0) ,
3 = r
(0) T
3 = c (0, 0, 1, 0) , l
4 = r
(0) T
4 = c (0, 0, 0, 1) .
The constant c = (2s(s + h5))
−1/2 ensures the correct
normalization, such that the eigenvectors obey lirj = δij .
Using these eigenvectors and the spectral theorem we
find for sign Ĥ the simple result
sign Ĥ =
sign (λj) rj lj =
sign(s)
Ĥ . (12)
Plugging this back into the overlap formula (1) and di-
agonalizing the remaining 4 × 4 problem one finds two
different eigenvalues for the overlap operator at a given
momentum,
1− sign (
h2 + h25 )h5 ± i
h2 + h25
, (13)
where each of the two eigenvalues is twofold degenerate.
The momentum dependence enters through the compo-
nents hν , h5 defined in (8). In the spectral sum (6) the la-
bel n runs over all momenta and the eigenvalues at fixed
momentum as given in (13). The necessary derivative
with respect to a4 is straightforward to compute in closed
form, and the spectral sum (6) can then be summed nu-
merically. The argument of the sign function cannot be-
come purely imaginary on a finite lattice, and no δ-like
terms occur. We remark, that after taking the derivative
with respect to a4, we set a = a4 = 1, i.e., all the results
we present are in lattice units.
0.00 0.02 0.04 0.06 0.08 0.10
0.000
0.001
0.002
0.003
0.004
 / 4π
FIG. 1: The energy density ǫ(µ)−ǫ(0) as a function of µ4 (all
in lattice units). The symbols (connected to guide the eye)
are for various lattice sizes, the dashed line is the continuum
result.
IV. RESULTS
We begin the discussion of our results with Fig. 1,
where we show the subtracted energy density ǫ(µ)− ǫ(0)
as a function of µ4 for three different lattice volumes. For
those lattices all 4 sides have equal length, i.e., in the
thermodynamic limit they correspond to zero tempera-
ture. Thus, according to (5), we expect the data (sym-
bols in Fig. 1) to approach the continuum form µ4/4π2
(dashed line) as the 4-d volume is sent to infinity.
The figure clearly shows that the lattice data are pre-
dominantly linear when plotted versus µ4 and that for
small µ they approach the continuum curve when the
volume is increased. It is, however, obvious that also on
our largest lattice still a discrepancy remains for larger
µ. In particular one finds a slight curvature upwards, a
discretization effect which here, since the lattice spacing
is just the inverse lattice extension, is also a finite size
effect. Furthermore, for small µ one expects to see finite
temperature corrections according to (5).
In order to study these finite temperature corrections
systematically, we analyzed lattices with short tempo-
ral extent, i.e., lattices with non-vanishing temperature.
Fig. 2 shows the corresponding results, where we again
plot the subtracted energy density as a function of µ4.
The lattice with the shortest temporal extent, 1283×8,
which corresponds to the largest temperature, shows a
clear curvature. This curvature is due to the T 2µ2/2
term in (5), which appears as a square root when plotted
as function of µ4. The effect is visible also for the other
lattices, but becomes less pronounced as the temporal
extent is increased, i.e., the temperature T is lowered.
In order to study this effect quantitatively, we fit the
finite temperature results to the continuum form (5) plus
two terms even in µ which parameterize the cutoff effects
0.00 0.02 0.04 0.06 0.08 0.10
0.000
0.001
0.002
0.003
0.004
 x 12
 x 16
 x 24
FIG. 2: The energy density ǫ(µ) − ǫ(0) as a function of µ4,
now for finite temperature lattices (all in lattice units).
observed in Fig. 1. The fit function is given by
2 + c4 µ
4 + c6 µ
6 + c8 µ
8 . (14)
Due to (5) the coefficient of the quadratic term should
scale with the temperature such that one expects
c2 ∼ T 2/2 = N−24 /2 . (15)
The coefficient for the quartic term should be constant,
c4 ∼ 1/4π2 = 0.02533 . (16)
The results of the fit for the data used in Fig. 2, and for
the largest lattice of Fig. 1 are given in Table 1.
The table shows that with increasing N4 the two phys-
ically significant parameters c2 and c4 approach the val-
ues expected from the continuum formula (5): c2 gets
closer to N−24 /2 as listed in the second column, and c4
approaches 1/4π2 = 0.02533. For the largest finite tem-
perature lattice 1283×24 the discrepancy is down to 9 %
/2 c2 c4 c6 c8
8 0.007812 0.010125 0.03519 0.010 -0.021
12 0.003472 0.004125 0.03178 0.023 -0.013
16 0.001953 0.002192 0.02803 0.029 -0.015
24 0.000868 0.000947 0.02587 0.025 -0.030
128 0.000030 0.000032 0.02543 0.015 0.016
TABLE I: Results of the fits to the form (14). The spatial
volume is always 1283. The temporal extension N4 is given
in the first column. In the second column we list the corre-
sponding value of N−2
/2 which is what one expects for the
fitting coefficient c2 in the third column. The coefficient c4 in
the fourth column is expected to approach the constant value
1/4π2 = 0.02533.
0.0 0.5 1.0 1.5 2.0
-0.05
Overlap ,  128
Wilson ,  128
FIG. 3: The ratio (ǫ(µ) − ǫ(0))/µ4 as a function of µ (in
lattice units). We compare the results for overlap to those
from Wilson fermions.
for c2, and 2 % for c4. The larger discrepancy for small
N4 can be understood as a discretization effect, since the
temporal lattice spacing a4 is related to the temporal ex-
tension through a4 = 1/N4 and thus larger N4 implies a
smaller a4. For comparison we also display the fit results
for the 1284 lattice, which corresponds to zero temper-
ature. There we find excellent agreement (less than 1%
discrepancy) for the parameter c4, governing the leading
term at T = 0. The overall picture obtained from the fit
results is that overlap fermions with chemical potential
reproduce very well both, the µ4 term, as well as the fi-
nite temperature contribution T 2µ2/2. We conclude that
the analytic continuation of the sign function does not in-
troduce lattice artifacts, such as the µ2/a2 term known
to be present in a naive implementation of the chemical
potential.
In the final step of our analysis we study the discretiza-
tion effect for larger values of µ and compare the re-
sults to the data from the standard Wilson operator. In
Fig. 3 we plot the ratio (ǫ(µ)− ǫ(0))/µ4 as a function of
µ. In the continuum at T = 0 this ratio has the value
1/4π2 = 0.02533 indicated by the horizontal line. For
small µ, up to about µ ∼ 0.7, the Wilson and overlap data
fall on top of each other. For very small µ both opera-
tors show a prominent increase which is a left-over finite
temperature effect, which for the ratio (ǫ(µ) − ǫ(0))/µ4
shows up as a 1/µ2 term. In the range between µ = 0.1
and 0.5 the data are close to the continuum value. Be-
yond 0.5 the discretization effects kick in and the overlap
and Wilson results start to differ. A comparison with
the equivalent plot in [6], where the results from various
other lattice Dirac operators were presented, shows that
the discretization effects of the overlap operator at large
µ are comparable to other formulations.
V. SUMMARY
In this article we have analyzed the energy density of
the overlap operator at finite chemical potential. Follow-
ing [4], the sign function in the overlap was implemented
through the spectral theorem using the analytic continu-
ation of the sign into the complex plane. The subtracted
energy density ǫ(µ) − ǫ(0) was analyzed for finite and
zero temperature lattices. Fits of the data show that the
expected continuum behavior is approached. No trace
of unphysical µ2/a2 terms was found. We conclude that
overlap fermions with chemical potential [4] provide both,
chiral symmetry and the correct description of fermions
at finite density.
Acknowledgments: We thank Leonard Fister,
Gabriele Jaritz, Christian Lang, Stefan Olejnik, Tilo
Wettig, and Florian Wodlei for discussions and check-
ing some of our calculations. This work is supported by
the Slovak Science and Technology Assistance Agency
under Contract No. APVT–51–005704, and the Austrian
Exchange Service ÖAD.
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http://arxiv.org/abs/hep-lat/0609020
