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JHEP11(2015)079
Published for SISSA by Springer
Received:October 6, 2015
Accepted:October 20, 2015
Published:November 12, 2015
Lefschetz thimble structure in one-dimensional lattice
Thirring model at finite density
Hirotsugu Fujii,aSyo Kamataband Yoshio Kikukawaa
aInstitute of Physics, University of Tokyo,
Tokyo 153-8092, Japan
bDepartment of Physics, Rikkyo University,
Tokyo 171-8501, Japan
E-mail: hfujii@phys.c.u-tokyo.ac.jp,skamata@rikkyo.ac.jp,
kikukawa@hep1.c.u-tokyo.ac.jp
Abstract: We investigate Lefschetz thimble structure of the complexified path-integration
in the one-dimensional lattice massive Thirring model with finite chemical potential. The
lattice model is formulated with staggered fermions and a compact auxiliary vector boson
(a link field), and the whole set of the critical points (the complex saddle points) are sorted
out, where each critical point turns out to be in a one-to-one correspondence with a singular
point of the effective action (or a zero point of the fermion determinant). For a subset of
critical point solutions in the uniform-field subspace, we examine the upward and downward
cycles and the Stokes phenomenon with varying the chemical potential, and we identify
the intersection numbers to determine the thimbles contributing to the path-integration of
the partition function. We show that the original integration path becomes equivalent to a
single Lefschetz thimble at small and large chemical potentials, while in the crossover region
multiple thimbles must contribute to the path integration. Finally, reducing the model to
a uniform field space, we study the relative importance of multi-thimble contributions and
their behavior toward continuum and low-temperature limits quantitatively, and see how
the rapid crossover behavior is recovered by adding the multi-thimble contributions at low
temperatures. Those findings will be useful for performing Monte-Carlo simulations on the
Lefschetz thimbles.
Keywords: Lattice Quantum Field Theory, Phase Diagram of QCD, Lattice Integrable
Models
ArXiv ePrint: 1509.08176
Open Access,c
The Authors.
Article funded by SCOAP3.doi:10.1007/JHEP11(2015)079
JHEP11(2015)079
Contents
1 Introduction 1
2 One-dimensional massive Thirring model on the lattice 3
3 Lefschetz thimble approach 6
3.1 Preliminaries 6
3.2 Critical points and determinant zero points 7
4 Thimble structure at µ= 0 9
4.1 Bosonic theory 9
4.2 Thirring model 11
5 Stokes phenomenon and structure change at finite µ13
5.1 Stokes jumps with increasing µ14
5.2 Multi-thimble contributions and weight factor 16
6 Multi-thimble contributions in uniform-field model 19
6.1 Single-thimble approximation 20
6.2 Toward continuum limit 20
6.3 Toward low temperature limit 21
6.4 Multi-thimble contributions 21
7 Summary and discussions 23
A Exact expression and asymptotics of Z26
1 Introduction
The sign problem is the longstanding obstacle which prevents us from applying nonpertur-
bative lattice simulations directly to the physical systems with complex actions, including
quantum chromodynamics (QCD) at finite baryon chemical potential µ. The fermion deter-
minant at finite µbecomes complex, which invalidates the importance sampling algorithm.
In contrast, the determinant is real at finite temperature (T) with µ= 0, and lattice sim-
ulations of QCD have proved now to be a reliable nonperturbative method to evaluate
(e.g.) the equation of state of strongly interacting matter. Nonetheless, studies of QCD-
inspired models at finite Tand µhave suggested a variety of phase changes from nuclear
liquid-vapor transition, to chiral symmetry restoration, and to color-superconducting phase
transition, etc. With this situation, in order to unveil the QCD phase diagram from the
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JHEP11(2015)079
first principles, many attempts have been made to circumvent the sign problem in lattice
QCD simulations, although the complete resolution is still not available [1].
To study the physical systems with complex actions, two alternative approaches have
attracted much attention recently — complex Langevin equation [2–4] and Lefschetz thim-
ble integration [5–7], both of which involve complexification of the dynamical field variables.
Statistical sampling with the complex Langevin equation has been applied to various
models [8–47], including the massive Thirring model with chemical potential [21,22], as
testing grounds, and it is successful in some cases but not in other cases. A formal proof
for the correctness of the method has been elaborated under certain conditions [13,17],
but full justification of the complex Langevin approach is not established, where loga-
rithm terms such as the ferimon determinant in the action cause a subtlety [44]. Note-
worthily, complex Langevin simulations have been applied to full QCD at finite Tand
µ[26,30,34–36,38,40,45,47], showing consistent results with those obtained by the
reweighting method in the parameter region where both methods are stable [47].
Path integration on the Lefschetz thimbles was introduced in the study of analytic
property of gauge theories [5], and it was soon recognized as a mathematically sound way
to resolve the sign problem [48–50]. It can be regarded as a functional generalization of
the steepest descent method of complex analysis. In this approach the original integration
cycle is deformed to a sum of the curved manifolds, called Lefschetz thimbles, in the
complexified field space. On a thimble the imaginary part of the action ImSis constant,
and this property allows importance sampling with the weight e−ReS≥0. This advantage
was first applied to numerical simulations for 4-dimensional λφ4theory with chemical
potential with use of Langevin [49] and hybrid Monte Carlo (HMC) [50] algorithms on a
single thimble, and successfully reproduced the known results including the so-called Silver
Blaze behavior [51] — complete insensitivity of the system to µbelow a certain critical
value at T= 0. The residual phase problem from the Jacobian due to the curvature is mild
and can be efficiently taken into account by reweighting for this theory [50]. The Lefschetz
thimble integration has been examined in other models [52–55] and has been studied from
other aspects [56–62] which involve the sign problem. This approach also shed new light
on the complex Langevin sampling method [27,33,62], and vice versa [61].
In this paper, we study the path integration on the Lefschetz thimbles in the (0+1)
dimensional massive Thirring model at finite chemical potential µ[63], in order to clarify
the effects of the fermion determinant on the structure of the thimbles contributing to the
partition function [55]. The lattice model is formulated with the staggered fermions [64,65]
and a compact auxiliary vector boson (a link field). This model shows a crossover transition
from the low to the high density phase at finite Tas a function of µ, and the transition
becomes first order in T= 0 limit. Furthermore the exact solution of this model is available
on the finite lattice as well as in the continuum limit, and therefore one can assess the
validity of the approach precisely by comparing the results with the exact ones.
The fermion determinant has zero points on the complexified field space and actually
those zeros form continuous submanifolds on which the effective action Sbecomes singular.
At the same time, the determinant brings in many critical points, each of which a thimble
is associated to. We classify the critical points into subsets according to the subspaces they
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JHEP11(2015)079
belong to, and identify all the critical points and thimbles in each subspace by noting a
one-to-one correspondence between a critical point and a zero point of the determinent.
The thimbles whose critical points are located in the uniform-field subspace are shown to
dominate the integral toward the continuum limit. Hence we study within the uniform-
field subspace how the set of the contributing thimbles to the partition function changes
via the Stokes phenomenon as the chemical potential µvaries. We will see that in the
crossover region multi-thimble contributions are inevitable, and become more significant
for small inverse coupling and/or in low temperature limit. We study this interplay in more
detail by reducing the model degrees of freedom to the uniform-field subspace and show
how the crossover behavior is reproduced as adding the multi-thimble contributions to the
observables.
This paper is organized as follows. In section 2we introduce the (0+1) dimensional
massive Thirring model with chemical potential on the lattice in terms of the staggered
fermions and the compact auxiliary vector boson. In section 3, after a briefly review of the
Lefschetz thimble approach, we study the critical points and determinant zeros of the lattice
model in the complexified field space. The critical points are classified by the subspaces they
live, and all the critical points are identified. In section 4, we study the thimble structure of
the model at µ= 0, discuss the importance of each thimble by looking at the relative weight
at µ= 0. In section 5, we show within the uniform-field subspace the change of the thimble
structure with increasing the chemical potential µvia the Stokes phenomenon, and show
that the multiple thimbles contribute to the partition function in the crossover region. In
section 6, taking the uniform-field subspace, we examine the validity of the single thimble
approximation, and discuss the continuum and low temperature limits. Especially in the
low temperature limit, the importance of the multi-thimble contributions are clarified.
Section 7is devoted to summary and discussions. The exact solution of the model is
derived in appendix A.
2 One-dimensional massive Thirring model on the lattice
The (0+1)-dimensional lattice Thirring model we consider in this paper is defined by the
following action [21,22,63],
S0=β
L
X
n=1 1−cos An
−
L
X
n=1
Nf
X
f=1
¯χf
nneiAn+µa χf
n+1 −e−iAn−1−µa χf
n−1+ma χf
no,(2.1)
where β= (2g2a)−1,ma,µa are the inverse coupling, mass and chemical potential in
the lattice unit, and Lis the lattice size which defines the inverse temperature as 1/T ≡
La. The fermion field has Nfflavors and satisfies the anti-periodic boundary conditions:
χf
L+1 =−χf
1,χf
0=−χf
L, ¯χf
L+1 =−¯χf
1, and ¯χf
0=−¯χf
L. The partition function of this
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JHEP11(2015)079
lattice model is defined by the path-integration,
Z=ZDADχD¯χe−S0
=Zπ
−π
L
Y
n=1
dAne−βPL
n=1 1−cos An(det D[A])Nf,(2.2)
where Ddenotes the lattice Dirac operator,
(Dχ)n= eiAn+µa χf
n+1 −e−iAn−1−µa χf
n−1+ma χf
n.(2.3)
The functional determinant of Dcan be evaluated explicitly (see appendix Afor deriva-
tion) as
det D[A] = 1
2L−1hcosh(Lˆµ+iPL
n=1An) + cosh Lˆmi(2.4)
with ˆµ=µa, ˆm= sinh−1ma. It is not real-positive for µ6= 0 in general, but instead it has
the property (det D[A]|+µ)∗= det D[−A]|+µ= det D[A]|−µ. This fact can cause the sign
problem in Monte Carlo simulations.
This lattice model is exactly solvable in the following sense. The path-integration over
the field Ancan be done explicitly and the exact expression of the partition function is
obtained (Nf= 1) as
Z=e−βL
2L−1hI1(β)Lcosh Lˆµ+I0(β)Lcosh Lˆmi,(2.5)
where I0,1(β) are the modified Bessel functions of the first kind. The number density and
scalar condensate of the fermion field are then obtained as
hni ≡ 1
La
∂ln Z
∂µ
=I1(β)Lsinh Lˆµ
I1(β)Lcosh Lˆµ+I0(β)Lcosh Lˆm,(2.6)
h¯χχi ≡ 1
La
∂ln Z
∂m
=1
cosh ˆm
I0(β)Lsinh Lˆm
I1(β)Lcosh Lˆµ+I0(β)Lcosh Lˆm.(2.7)
The µ-dependence of these quantities are shown in figure 1for L= 8, ma = 1, and β= 1,3,
and 6. It shows a crossover behavior in the chemical potential ˆµ(in the lattice unit) around
ˆµ'ˆm+ ln(I0(β)/I1(β)).
The continuum limit (a→0) of this lattice model at finite Tmay be defined as
β=1
2g2a→ ∞, L =1
T a → ∞ with β/L =T /(2g2) fixed.(2.8)
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JHEP11(2015)079
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
<n>
µ/m
(a) L=8, ma=1
β=1
β=3
β=6
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
<χ
−χ>
µ/m
(b) L=8, ma=1 β=1
β=3
β=6
Figure 1. (a) Fermion number density and (b) scalar condensate with L= 8, ma = 1 for β= 1
(solid), 3 (dashed), and 6 (dotted).
Figure 2. Number density hnion the T-µplane in the continuum limit (g2/m = 1/2).
In this limit the partition function scales as
Z→1
2L−11
2πβ L/2
e3g2
4Tcosh µ
T+ eg2
Tcosh m
T,(2.9)
and the continuum limits of hniand h¯χχiare obtained as follows:
lim
a→0hni=sinh µ
T
cosh µ
T+ eg2
Tcosh m
T
,
lim
a→0h¯χχi=eg2
Tsinh m
T
cosh µ
T+ eg2
Tcosh m
T
.(2.10)
From these results, one sees that the model shows a crossover behavior in the chemical
potential µat non-zero temperatures T > 0, while in the zero temperature T= 0 limit it
shows a first-order transition at the critical chemical potential |µc|=m+g2. See figure 2.
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JHEP11(2015)079
3 Lefschetz thimble approach
3.1 Preliminaries
Now we consider the complexification of the Thirring model on the lattic and reformulate
the defining path-integration of eq. (2.2) by the integration over Lefschetz thimbles. In the
complexification, the field variables Anare extended to complex variables zn(∈CL) and
the action is extended to a holomorphic function given by S[z] = βPL
n=1(1 −cos zn)−
ln det D[z].1For each critical point z=σgiven by the stationary condition,
∂S[z]
∂znz=σ
= 0 (n= 1,·· · , L),(3.1)
the thimble Jσis defined as a union of all the (downward) gradient flow curves deter-
mined by
d
dtzn(t) = ∂¯
S[¯z]
∂¯zn
(t∈R) s.t. z(−∞) = σ. (3.2)
The thimble so defined is an L-dimensional real submanifold in CL. Then, according to
Picard-Lefschetz theory (complexified Morse theory), the original path-integration region
CR≡[−π, π]Lcan be replaced with a set of Lefschetz thimbles,2
CR=X
σ
nσJσ,(3.3)
where nσstands for the intersection number between CRand the dual submanifold Kσ,
which is another L-dimensional real submanifold associated to the same critical point σ
and is defined as a union of all the gradient flow curves s.t. z(+∞) = σ. With denoting the
set of the critical points as Σ ≡ {σ}, the partition function and the correlation functions
of the lattice model can be expressed by the formulas
Z=X
σ∈Σ
nσZσ, Zσ≡ZJσD[z] e−S[z],(3.4)
hO[z]i=1
ZX
σ∈Σ
nσZσhO[z]iσ,hO[z]iσ≡1
ZσZJσD[z] e−S[z]O[z].(3.5)
The functional measure D[z] along the thimble Jσis specified as dLzJσ=dL(δξ) det Uz
by the orthonormal basis of tangent vectors {Uα
z|(α= 1,··· , L)}which span the tangent
space as δz =Uα
zδξα(δz ∈CL, δξ ∈RL).
The integration on each Lefschetz thimble is convergent because the real part of the
action increases monotonically to ∞while the imaginary part stays constant along the
downward flow,
dReS
dt ≥0,dImS
dt = 0.(3.6)
1The logarithm has branch cuts, but it does not affect the gradient flows as discussed below.
2We will extend this original integration region to CR≡([−π+i∞,−π]⊕[−π, π]⊕[π , π +i∞])Las
the well-defined integration cycle. The value of the integral is unchanged by this extension thanks to 2π
periodicity of S.
– 6 –
JHEP11(2015)079
The sign problem remains in the Lefschetz thimble approach in two facts. First, it
seems that when we factor out the complex weight e−S[σ], the integrand of each thim-
ble, e−(S[z]−S[σ]) >0, is real positive. But a complex phase appears from the Jacobian
factor det Uzin the integration, which is called residual sign problem. For λφ4theory it
is demonstrated that the residual sign problem can be treated by the reweighting method
safely [50]. Second, the terms Zσand hO[z]iσare actually complex quantities although the
total averages Zand hO[z]ishould be real. If there is a certain symmetry in the thim-
ble structure of the system, one can show the cancellation of the phases in the sum [58].
The multi-thimble contributions to the partition function and observables will be more
elaborated in this paper.
3.2 Critical points and determinant zero points
Given the above mathematical results, however, it is not straightforward to work out for
general fermionic models all the critical points Σ = {σ}, the thimbles {Jσ}, and their
intersection numbers {nσ}. Fortunately in our lattice model, we can find all the critical
points determined by the stationary condition eq. (3.1).
The critical point condition for the Thirring model is written as
∂S
∂zn
=βsin zn−isinh(Lˆµ+is)
cosh(Lˆµ+is) + cosh Lˆm= 0 with s≡
L
X
`=1
z`.(3.7)
The key observation is that the second term depends on the field configuration only through
the sum s, so that all sin zn(n= 1,··· , L) of a critical point σmust have the same value
to cancel the common second term. Let us denote it as sin z, then the field components
can be either zn=zor π−zand the sum sis written as
s=n+z+n−(π−z)=(L−2n−)z+n−π, (3.8)
where n±are the numbers of zand π−zin the components {zn}with n++n−=L. The
critical point condition for zis now explicitly written as
βsin z−isinh [Lˆµ+i(L−2n−)z]
cosh [Lˆµ+i(L−2n−)z] + (−)n−cosh Lˆm= 0.(3.9)
This can be regarded as the critical point condition for a one-variable model;
Sn−= (L−2n−)β(1 −cos z)−log cosh [Lˆµ+i(L−2n−)z] + (−)n−cosh Lˆm.(3.10)
The case of n−= 0 corresponds to a uniform field configuration, where zn=z(n=
0,··· , L −1), and the case n−= 1 means that there is one flipped component π−z, ··· ,
etc. In the case of n−=L/2, the second term of (3.9) becomes independent of z. The case
of n−> L/2 gives the same critical points as in the case L−n−with z↔π−z. Hence
we need to consider n−= 0,·· · , L/2−1.
Thus we have classified the critical points with index n−. By solving the condition
eq. (3.9) of the one-variable model for each n−, we can locate all the critical points of
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JHEP11(2015)079
the model. Note that a critical point is L
n−-ply degenerated for n−6= 0 due to the
combination about which components to be flipped.
One of the distinctive features of fermionic theories is the fact that the fermion deter-
minant has many zero points within a compact domain in the complexified space. The real
part of the effective action ReSdiverges at these zeros, and therefore the downward cycle
Jσmay flow into one of these zeros, otherwise it must extend outward to the safe exterior
region where ReS= +∞. Hence, in addition to the critical points, we need to locate all
the zeros of the fermion determinant
det D[z]=0.(3.11)
Thanks to the concise expression of det D[z] in eq. (2.4), one can easily find all zero
points:
szero = iL(ˆµ±ˆm) + (2n+ 1)π(n∈Z).(3.12)
This only fixes s=PL−1
`=0 z`, and thus defines submanifolds with the complex dimension
L−1, embedded in the Ldimensional complexified configuration space. Note that these
zero points are independent of β, and that nonzero ˆµsimply shifts the zero points along the
imaginary axis. Restricting this submanifold of the zeros in the subspace n−= 0, where
s=Lz, we find 2Lisolated zeros of
zzero = i(ˆµ±ˆm) + 2n+ 1
Lπ(n∈Zmod L),(3.13)
while in the subspace (n−= 1) with a single link flipped to π−z(and thus s= (L−
2n−)z+π), we have 2(L−2) zeros of
zzero = i L
L−2(ˆµ±ˆm) + 2n
L−2π(n∈Zmod L−2).(3.14)
Figure 3shows two sections of the gradient flows in the uniform-field subspace (n−= 0;
left) and in the subspace with one link flipped (n−= 1; right) of the model with L= 4,
β= 3 −0.1i and ma = 1 at ˆµ= 0. (The reason for complex βwill be explained in the next
section.) Globally, the flows are streaming out of the remote points z=±i∞and flowing
away towards the safe remote points z=±π±i∞. We solve eq. (3.9) numerically, and
find ten (eight) critical points3for n−= 0 (1), as shown with green dots in figure 3. For
later convenience, we have numbered the critical points as shown here. We also put the
zero points of the determinant det D[z] with red dots. Each critical point apparently pairs
up with a zero point next to it, besides the two sitting at the origin and ±π.
Now that we have identified all the critical points and the zeros of the Thirring model,
we can study the structure of the Lefschetz thimbles of the model in detail.
3The two critical points located at z=±πare identical.
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JHEP11(2015)079
Figure 3. Critical points (green dots) and determinant zeros (red dots) of the Thirring model
with L= 4 and ma = 1 at ˆµ= 0 within the subspaces of n−= 0 (left) and 1 (right). We set
β= 3 −0.1i. Gradient flows are drawn with arrows. We assign numbers to the critical points as σi,¯
i
here. The downward Jσand upward Kσcycles of a critical point σare shown with solid and dashed
lines, respectively. The brighter background indicates the larger value of ReS(in arbitrary unit).
Figure 4. Downward flow, critical points of free theory for β= 3 −0.1i in the complex zplane.
The horizontal (vertical) axis is for the real (imaginary) part.
4 Thimble structure at µ= 0
4.1 Bosonic theory
It would be instructive to start our discussion with the bosonic theory without fermions,
S[z]≡PL
n=1 Sn(zn) = PL
n=1 β(1 −cos zn), whose complexified configuration space is a
direct product of (S1×R)L. The downward flow is simply given by
dzn
dt =∂¯
S[¯z]
∂¯zn
=¯
βsin ¯zn,(4.1)
which is depicted in figure 4for a certain znwith β= 3 −0.1i.
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JHEP11(2015)079
Let us focus on this complex znplane for a moment. The action is periodic in the
direction of the real axis, so that the configuration space is equivalent to S1×R, a cylinder.
There are two critical points, zn= 0 and ±π(shown in green dots),4corresponding to the
Gaussian and doubler solutions, respectively. The downward cycle (thimble) J0associated
to zn= 0, extends to the “safe” exterior regions toward zn=±π±i∞depicted with
light-red color at the corners. There is another thimble J−πassociated to the doubler
solution zn=−π, which connects these two “safe” regions vertically along the imaginary
direction. In other words, the two safe regions are connected by two cycles with and
without winding around the cylinder. These cycles constitute the base of homology of this
restricted space S1×R.
We notice here that the original integration path from zn=−πto πis ill-defined as a
homology cycle. A well-defined downward cycle should extend to a “safe” region where the
Morse function (h=−ReS) approaches −∞ [5]. Actually, the thimble J0coincides with
this original path only for real β, which is the very parameter for the Stokes phenomenon
to occur between zn= 0 and ±π(the action is real at both points; Sn= 0 and 2β). Hence
in figure 4we have added nonzero imaginary part to the coupling β= 3 −0.1i,5to make
the thimble J0well-defined.
Thanks to the periodicity of the action Sn(z), we can exptend the original integration
path without changing the value of Zto a U-shaped integration cycle which starts at
zn=−π+i∞and comes down along the imaginary direction to zn=−πthen moves along
the real axis to zn=π, and goes up to zn=π+ i∞.6This U-shaped cycle (which we
simply denote with C) is equivalent to the sum of the two thimbles:
C ∼ J0+J−π.(4.2)
Here we set the orientation of the thimbles so that “+” sign is appropriate here. One can
confirm that both the upward cycles K0and K−πintersect this integration cycle C.
There are 2Lcritical points in the (0+1) dimensional bosonic theory with Llattice
sites from combinatorics, and its thimble structure is obtained as a direct product of the
thimbles J0and J−π. The integration cycle equivalent to the original integration path is
symbolically written as
CL∼(J0+J−π)L.(4.3)
The safe exterior region where the real part of the action ReSdiverges has the complex
dimension (L−1) because it is characterized by the condition PL
n=1(1 −cos zn) = ∞, i.e.,
at least one of {zn}is fixed to π±i∞.
We comment on the continuum limit (β→ ∞ with fixed β/L). In this limit the
contribution to the partition function from each variable becomes Gaussian:
Zπ
−π
dz
2πe−β(1−cos z)=I0(β)e−β→1
√2πβ .(4.4)
4Note that zn=±πare the same point on S1×R.
5If we take β= 3 + 0.1i, the flow structure is just reflected about the imaginary axis from figure 4.
6One may choose alternatively the cycle which connects zn=±π−i∞passing through zn= 0, which
does not change the discussions below.
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JHEP11(2015)079
The integration along the vertical path, which we have added to make the integration
cycle well-defined, becomes irrelevant giving only a contribution which is exponentially
suppressed. For example,
Zπ−i∞
π+i∞
dz
2πe−β(1−cos z)→ − i
√2πβ e−2β.(4.5)
Thus we see that the doubler contribution J−πis suppressed by e−2βand the free theory
with Ldegrees of freedom is correctly reproduced by the integration on the thimble JL
0.
4.2 Thirring model
We have already idintified the critical points and determinant zeros of the Thirring model in
the previous section and shown them in figure 3. There we also noticed a certain correlation
between a critical point and a determinant zero. Now let us look at the thimble structure
of the model at ˆµ= 0.
In the uniform-field (n−= 0) subspace shown in figure 3(left), the thimble Jσ0extends
from one safe remote z=−π−i∞to another safe remote z=π+ i∞passing through the
critical point σ0at the origin, and the U-shaped cycle is still equivalent to the sum of the
two thimbles, associated to the Gaussian and doubler critical points:
Cc
∼ Jσ0+Jσ¯
0,(4.6)
where “ c
∼” indicates the equivalence as the integration cycles under the constraint n−= 0.
In the subspace of n−= 1 (figure 3(right)), on the other hand, the critical point σ0
contains one doubler component π−z, and the thimble Jσ0ends at determinant zeros.7
The U-shaped cycle within n−= 1 space is covered by the sum of four thimbles:
Cc
∼ Jσ¯
2+Jσ0+Jσ¯
0− Jσ¯
2,(4.7)
with two Jσ¯
2contributions canceling out in the end.
The strong correlation between a critical point and a zero point may be expected by
noticing the fact that because a zero point zzero is a simple pole of the flow field, one can
always find in its vicinity the point on which the first term of eq. (3.9) can be counter-
balanced by the would-be pole contribution, especially when βis large.
One can also understand the paring between them by considering the thimble structure
of the one-variable model assigned by n−, where a thimble Jσbecomes a line segment
associated to a critical point σand connects between the zeros and/or safe remote points
z=±π±i∞. In n−= 0 case, for example, two safe remote points are connected by the
two thimbles, Jσ0and Jσ¯
0. Because the thimbles form the basis of independent cycles, no
trivial loops are allowed. That is, a set of thimbles must be a connected skeleton graph on
S1×Rsubspace assigned by n−. One can add a new critical point, which is accompanied
by a new thimble, only when one has a new zero point. Hence the number of thimbles
7Note that we use the same notation σ0for the critical points in n−= 0,1 subspaces without any
confusion.
– 11 –
JHEP11(2015)079
L n−0 1 2 3 4 ··· ¯
0 2β(L−2n−)
4 0 −0.8 1.6 25.5 — — — 23.2 24
1 5.3 −3.6 32.4 — — — 17.3 12
8 0 1.7 2.5 32.0 73.1 101.5 — 97.7 96
1 13.7 8.5 30.8 74.4 95.7 — 85.7 72
16 0 7.1 7.5 37.6 92.5 164 · ·· 391 384
1 31.2 26.7 44.6 93.5 164 · ·· 367 336
Table 1. ReSat the critical points σi(i= 0,1,2,3,4,¯
0) with L= 4k,β= 3k, and ma = 1/k
(k= 1,2,4) at µ= 0. The rightmost column shows the difference of ReSbetween σ0and σ¯
0.
coincides with the number of the critical points, and furthermore with the number of the
end-point zeros (including two safe remote points) in our model.
The formulas (3.13) and (3.14) with L= 4 give us eight and four zeros for n−= 0 and
1, respectively, as seen in figure 3. Adding two remote zeros, we have ten thimbles, ten
critical points, and ten zeros for n−= 0, and six of those for n−= 1. We have just two
thimbles in n−= 2 subspace because we have no determinent zeros there.
Note that a thimble Jσis not a simple curve but extends with real dimension L, and
its section with the subspace is seen as a curve in figure 3. For example, integration on the
thimble Jσ0associated to the Gaussian critical point z=σ0in n−= 0 subspace contains
the perturbative fluctuations in all the directions around z=σ0.
Finally in this subsection, let us look at the real part of the action ReS[σi] at these
critical points for real β= 3, which is listed in the first row (L= 4) of table 1. The
background brightness of figure 3actually indicates the value of ReS[z] (in arbitrary unit).
We only list the values at σ0,1,2,¯
0because the critical points which interchange with each
other by the reflection about the real and imaginary axes have the same ReS[σi] at ˆµ= 0
for real β.
The value ReS[σ¯
0] of the doubler solution is larger than ReS[σ0] by 2βL = 24 for
n−= 0 and 2β(L−2) = 12 for n−= 1. This difference comes from the bosonic part
β(1 −cos z) of the action. On the other hand, the action ReS[σ0] at σ0in n−= 1 sector
is larger than that in n−= 0 sector by a factor of order 2β= 6 because the former point
contains one doubler component zn=π. One may notice that ReS[σ1] in n−= 1 sector
takes a smaller value than ReS[σ0], indicating the larger weight for it. But Kσ1has no
intersection with Cat ˆµ= 0, and the thimble Jσ1is not a member of the integration
cycles for Z.
It is intriguing to check this behavior with changing the lattice size Ltowards the
continuum limit. By increasing Land βwith keeping β/L and Lma fixed, there appear
more zero points and accordingly the critical points aligned in two rows. We compute
ReS[σi] and list the results for L= 8 and 16 in the lower part of table 1. We observe that
the contributions from the n−= 1 sector to Zare more suppressed by the factor e−2βfor
the larger Land β. Within the n−= 0 sector, we can estimate the difference between
ReS[σ1] and ReS[σ2] as 4π2β/L for larger L, basing on the bosonic part Lβ(1 −cos z) and
expanding it with approximation σk∼zzero,k ≡(2k−1)π/L −im. This gives us a factor
– 12 –
JHEP11(2015)079
3π2∼30, which is consistent with the numerical result of table 1. For the smaller β/L we
have the smaller gap between ReS[σ1] and ReS[σ2]. The difference between ReS[σ0] and
ReS[σ1] is more sensitive to the choice of parameters β/L and Lma.
In summary, we have clarified the thimble structure of the Thirring model in this
section. The determinant zeros form submanifolds with complex dimension L−1, and
their sections in the subspace assigned with n−appear as isolated zero points. The critical
points of the model are classified with n−, and each of them pairs up with a zero point in
the subspace assigned with n−(except for the Gaussian critical point σ0and its doubler
counterpart σ¯
0). Thus all the thimbles are identified in the (0+1) dimensional Thirring
model. Towards the continuum limit (β→ ∞), ReS[σ] with nonzero n−, which contains
n−“doubler” components, acquire the large values of order 2n−βcompared to ReS[σ0] in
the n−= 0 subspace. This implies that the relative weights of their contributions to Z
are strongly suppressed toward the continuum limit, even when they join the set of the
integration cycles as µincreases.
5 Stokes phenomenon and structure change at finite µ
In this section, with increasing µ, we study the change of the intersection numbers and
thimbles which contribute to the partition function Zfrom the viewpoint of the Stokes
phenomenon and jumps. We restrict our discussion in the uniform configuration space
n−= 0.
Figure 5shows the downward gradient flows of the model with L= 4, β = 3 and
ma = 1 for ˆµ= 0.6, 1.2 and 1.8. The zero points zzero’s and their associated critical points
σ’s move upward as ˆµincreases. Then the critical points which align on the lower side, cross
the real axis at certain values of ˆµ∼ˆm(see eq. (3.13)), and accordingly the intersection
numbers of Kσ’s with Cchange on the way. Now one encounters the situation where certain
thimbles join and/or leave the set of integration cycles for the partition function Z. For
a large enough ˆµ, as can be inferred from figure 5(c), the single thimble Jσ0comes to
connect the two safe remote points z=±π+i∞, to become an equivalent cycle to the
original U-shaped cycle: C ∼ Jσ0.
The downward and upward cycles Jσand Kσof a critical point σgenerally extend to
“safe” and “unsafe” remote regions, respectively, without crossing other cycles Jσ0and Kσ0
which have different values of ImS. When multiple critical points share the same value of
ImS, the cycle associated to one of those critical points may meet another critical point.
This is the so-called Stokes phenomenon. Change of the intersection number is achieved
only by a jump of one endpoint of a upward cycle Kσfrom (e.g.) z=−i∞to z= i∞, and
in between the critical point σmust undergo the Stokes phenomenon with another critical
point σ0in which Kσand Jσ0just overlap.
As has been discussed in the previous section, zeros of the fermion determinant be-
come endpoints of the thimbles. Because the determinant appears as −Nflog det Din
the action S, the imaginary part ImSchanges by −2πNfwhen we encircle a zero point
counterclockwise from one side to the other side of a thimble which terminates at this
zero. However this difference is not reflected in the gradient flow. Therefore the necessary
– 13 –
JHEP11(2015)079
(a) ˆµ= 0.6 (b) ˆµ= 1.2 (c) ˆµ= 1.8
Figure 5. Downward gradient flows, critical points (green) and zero points (red) in complex z
plane of the Thirring model with ma = 1, β= 3 and L= 4 for (a) µ= 0.6, (b) 1.2, and (c) 1.8.
The downward (upward) cycles, Jσ(Kσ), are depicted with solid (dashed) lines.
condition for the Stokes phenomenon to occur between critical points σand σ0is
ImSσ= ImSσ0+ 2πk k ∈Z.(5.1)
Incidentally, the imaginary part ImSon the upward cycle (e.g.,) Kσmay differ by a multiple
of 2πdepending on which side of the thimble Jσthe cycle starts. Moreover, since the value
of ImSchanges around a zero point, two thimbles can meet at the zero point making an
angle determined by the difference of their ImS(σi). Thus, one can read the relative phase
of the two thimbles from their relative angle when they meet at the zero point.
5.1 Stokes jumps with increasing µ
Let us study the Stokes phenomenon with increasing µin more details. We set Nf= 1.
Because the configuration subspace for real βis symmetric under reflection about the
imaginary axis as seen in figure 5, we discuss the thimble structure on the right-half plane
hereafter. Even at finite chemical potential µ6= 0, this reflection symmetry z→ −¯zguar-
antees the realness of Z; the thimbles which interchange under this transformation give the
contributions which are complex conjugate to each other and whose sum becomes real [58].
In figure 6, we compare the values of the action at the critical points σi. We first
note that ImS= 0 at σ0and σ¯
0independently of the chemical potential µ. Indeed, in
figure 5(a), we see the Stokes phenomenon between σ0and σ¯
0, where the cycles Jσ0and
Kσ¯
0overlap, and
Cc
∼ Jσ0+Jσ¯
0.(5.2)
At ˆµ= 0 the critical points σ¯
ion the upper side have positive values of ImS(σ¯
i) and
their associated upward cycles Kσ¯
iextend to the unsafe region toward z= +i∞. With
increasing ˆµthe values of ImS(σ¯
i) increase monotonically and Kσ¯
icontinue to have no
intersection with C. On the other hand, the critical points σion the lower side move
– 14 –
JHEP11(2015)079
-10
0
10
20
30
0 0.5 1 1.5 2
µ
Im Sσ0
Im Sσ1
Im Sσ2
Im Sσ1
-
Im Sσ2
-
(a)
-3
-2
-1
0
1
2
3
0.6 0.7 0.8 0.9 1
µ
Im Sσ0
Im Sσ1
Im Sσ2
(b)
-20
0
20
40
60
0 0.5 1 1.5 2
µ
Re Sσ0
Re Sσ1
Re Sσ2
Re Sσ1
-
Re Sσ2
-
Re Sσ0
-
(c)
Figure 6. (a) ImS(σi) on the right half plane as a function of ˆµ. (b) Enlarged plot of (a).
(c) ReS(σi). The dashed line indicates min.x∈RReS(x). Parameters are set to L= 4, β= 3 and
ma = 1.
upward and the imaginary parts ImS(σi) at these points increase from negative to positive
values with increasing ˆµ, as seen in figure 6(a). In the enlarged plot in the panel (b), the
lines of ImSσishow three crossings at ˆµ= ˆµ∗
1= 0.7, ˆµ∗
2= 0.735 and ˆµ∗
3= 0.86. We now
discuss the Stokes phenomenon and the change of the intersection numbers at each ˆµ∗
i.
In figure 7we show typical thimble structures at several values of ˆµ. At ˆµ < ˆµ∗
1, the
cycles Kσ1,2starting from σ1,2extend to the lower unsafe region toward z=−i∞, while
Kσ¯
1,¯
2extend to the upper unsafe region toward z= +i∞. None of them has nonzero
intersection with C, and Cc
∼ Jσ0+Jσ¯
0as was discussed previously. At ˆµ= ˆµ∗
1, ImSσ0=
ImSσ2is achieved, and the two cycles Jσ0and Kσ2overlap. Across ˆµ∗
1, one end of the
upward cycle Kσ2jumps from −i∞to +i∞, to give the intersection number n2= 1 with C
(see panel (b)). (And one end of the cycle Jσ0jumps from σ¯
0to zzero,2.)
At the same value of ˆµ= ˆµ∗
1, the point σ2shows the Stokes phenomenon with another
critical point σ¯
0because ImSσ0= ImSσ¯
0= 0. (This coincidence could be avoided by
adding a small imaginary part to β, again.) In this case, the two cycles, Jσ2and Kσ¯
0
overlap, and one end of the cycle Jσ2jumps from π−i∞to π+ i∞across ˆµ= ˆµ∗
1. Hence,
we have the equivalence of the cycles8
Cc
∼ Jσ−2+Jσ0+Jσ2for ˆµ∗
1<ˆµ < ˆµ∗
2.(5.3)
At ˆµ= ˆµ∗
2(panel (c)), the Stokes phenomenon happens between the critical points,
σ0and σ1. The two cycles Jσ0and Kσ1overlap there. When ˆµpasses ˆµ∗
2, one end of the
cycle Kσ1jumps from −i∞to +i∞and one end of the cycle Jσ0from zzero,2to zzero,1, and
therefore the critical point σ1now acquires the intersection number nσ1= 1. Hence,
Cc
∼ Jσ−2+Jσ−1+Jσ0+Jσ1+Jσ2for ˆµ∗
2<ˆµ < ˆµ∗
3.(5.4)
At ˆµ= ˆµ∗
3(panel (e)), ImSof σ1and σ2coincide, which allows the Stokes phenomenon
between them. Across ˆµ= ˆµ∗
3one end of the cycle Kσ2flips down from +i∞to −i∞, while
one end of the cycle Jσ1jumps from zzero,2to π+ i∞, so that the intersection number n2
changes from 1 to 0. Thus, we have (ˆµ∗
4introduced below)
Cc
∼ Jσ−1+Jσ0+Jσ1for ˆµ∗
3<ˆµ < ˆµ∗
4.(5.5)
8We define the orientation of a thimble as the direction where Rezincreases, and define it for thimble
Jσ¯
0on the left as the direction of decreasing Imz.
– 15 –
JHEP11(2015)079
So far we have discussed only the cases where the Stokes phenomenon occurs between
the critical points having the same value of ImS. For ˆµlarger than ˆµ∗
3we need to take
into account the multivaluedness of the logarithm because the edge of Jσ0is now going
around the zero points. The condition for the Stokes phenomenon to occur is the equality
of ImSmodulo 2πbetween the two critical points as announced in eq. (5.1). For our
model parameters, there are three more critical values ˆµ∗
4,5,6. At ˆµ= ˆµ∗
4(figure 8(g)), the
condition ImSσ0+2π= ImSσ1is fulfilled, and for ˆµ∗
4<ˆµ < ˆµ∗
5(figure 8(h)) the equivalent
integration cycle becomes
Cc
∼ Jσ0for ˆµ∗
4<ˆµ < ˆµ∗
5.(5.6)
At ˆµ= ˆµ∗
5(figure 8(i)) the condition ImSσ0+ 2π= ImSσ2is fulfilled, and the equivalent
integration cycle changes to
Cc
∼ Jσ−2+Jσ0+Jσ2for ˆµ∗
5<ˆµ < ˆµ∗
6.(5.7)
At ˆµ= ˆµ∗
6(figure 8(k)), the condition ImSσ0+ 4π= ImSσ2holds and the equivalent
integration cycle now consists of a single thimble
Cc
∼ Jσ0for ˆµ∗
6<ˆµ . (5.8)
5.2 Multi-thimble contributions and weight factor
We have seen how the original integration cycle Cis decomposed equivalently into a set
of thimbles with increasing ˆµ. The partition function Zis correctly reproduced only if we
evaluate the contributions from all the thimbles in the set, in principle. Especially in the
crossover region of ˆµ, multiple thimbles take part in the set of the integration cycles.
However, importance of their contributions depends on the weight factor
exp[−ReS(σ)]. For example, the integration cycle consists of Jσ0and Jσ¯
0for 0 ≤ˆµ < ˆµ∗
1.
But the contribution from Jσ¯
0is numerically negligible because ReS(σ¯
0) is larger than
ReS(σ0) by a large amount ∼2Lβ as seen in figure 6(c) (see also table 1for ˆµ= 0 value).
For ˆµ∗
1<ˆµ < ˆµ∗
4and ˆµ∗
5<ˆµ < ˆµ∗
6, the thimbles Jσ±1and/or Jσ±2are in the set of
the integration cycles in addition to Jσ0. According to the weight factor exp(−ReS(σ)) in
figure 6(c), the thimble Jσ0will give the largest contribution and Jσ±1will contribute as
the second largest. The contributions from Jσ±2will be strongly suppressed. This behavior
is mainly controlled by the bosonic part Lβ(1 −cos z) of the action. (The thimble Jσ¯
1is
not a member of the integration cycle, although ReS(σ¯
1) becomes smallest as ˆµincreases.)
In figure 9we plot the βdependence of the critical chemical potential ˆµ∗
ifor L= 4
and ma = 1. Outside of the interval ˆµ∗
1<ˆµ < ˆµ∗
6the single thimble Jσ0becomes (almost)
equivalent to the original integration cycle C, but within this interval multiple thimbles need
to be considered. Especially, the second-dominant thimbles Jσ±1contribute in the interval
ˆµ∗
2<ˆµ < ˆµ∗
4. We notice that the crossover region ˆµ∼ˆmis indeed covered by this interval
ˆµ∗
2<ˆµ < ˆµ∗
4, which indicates that the multi-thimble contribution is requited to reproduce
the crossover behavior correctly. The interval becomes wider (narrower) for smaller (larger)
β. From this β-dependence there may be a possibility that the approximate evaluation of
– 16 –
JHEP11(2015)079
(a) ˆµ= ˆµ∗
1(b) ˆµ∗
1<ˆµ < ˆµ∗
2
(c) ˆµ= ˆµ∗
2(d) ˆµ∗
2<ˆµ < ˆµ∗
3
(e) ˆµ= ˆµ∗
3(f) ˆµ∗
3<ˆµ < ˆµ∗
4
Figure 7. Stokes phenomena at ˆµ= ˆµ∗
i(ˆµ < ˆµ∗
4).
– 17 –
JHEP11(2015)079
(g) ˆµ= ˆµ∗
4(h) ˆµ∗
4<ˆµ < ˆµ∗
5
(i) ˆµ= ˆµ∗
5(j) ˆµ∗
5<ˆµ < ˆµ∗
6
(k) ˆµ= ˆµ∗
6(l) ˆµ∗
6<ˆµ
Figure 8. Stokes phenomena at ˆµ= ˆµ∗
i(ˆµ∗
4≤ˆµ).
– 18 –
JHEP11(2015)079
0
0.5
1
1.5
2
2.5
3
3.5
1 2 3 4 5 6
µ
β
ImSσ1= 0
2π
ImSσ2= 0
4π
Figure 9. Critical values of ˆµ∗
1,6at which ImSσ2= 0,4π, and ˆµ∗
2,4at which ImSσ1= 0,2π, as a
function of the coupling β(L= 4 and ma = 1).
Zwith the single thimble Jσ0becomes better for larger β. Note that for larger βthe
difference in the relative weights among the critical points also becomes more significant
and the thimbles whose critical point locates away from σ0in the real axis direction is
expected to less contribute to Z.
In summary, for L= 4 case, we have clarified the change of the Lefschetz thimble
structure and the set of the thimbles contributing to Zas ˆµincreases. At small and large
chemical potentials outside of the interval ˆµ∗
2<ˆµ < ˆµ∗
4, the evaluation of Zwith the single
thimble Jσ0is legitimate, provided that Jσ±2contributions are negligibly small. But in the
crossover region Jσ±1contributions must be taken into account in addition to that of Jσ0.
The approximate evaluation by taking only one thimble Jσ0is performed in numerical
simulations for several models so far [49,50,52,53]. Hence it would be worthwhile to
examine the validity of the single thimble approximation across the crossover region with
varying β. Furthermore it would be intriguing to study how the crossover behavior is
reproduced by the multi-thimble contributions with increasing the lattice size Ltoward
the continuum and/or low temperature limits.
6 Multi-thimble contributions in uniform-field model
In order to examine the single thimble approximation and to investigate how the crossover
behavior is reproduced by contributions from multiple thimbles, we study the Thirring
model in the uniform-field subspace. The limitation to uniform-field configurations cor-
responds to the classical approximation with neglecting the quantum fluctuations. The
partition function of this restricted model is analytically evaluated to be
Z0=Zπ
−π
dx
2π
1
2L−1hcosh(L(ˆµ+ ix)) + cosh Lˆmie−Lβ(1−cos x)
=e−βL
2L−1IL(βL) cosh Lˆµ+I0(βL) cosh Lˆm,(6.1)
– 19 –
JHEP11(2015)079
and the fermion number density and chiral condensate are obtained by
hni0=1
La
∂
∂µ log Z0,h¯χχi0=1
La
∂
∂m log Z0.(6.2)
Interestingly, in the T= 0 limit this classical model shows a first order transition at the
same value of |µc|=m2+g2as the original model.
6.1 Single-thimble approximation
We compare the values evaluated on the single thimble Jσ0to the exact ones by taking
their ratios in figure 10. We show the results with L= 4 and ma = 1 for β= 1 (left) and
3 (right). The critical values of the chemical potential ˆµ∗
i=1,...,6for the Stokes phenomenon
are found to be {0.40, 0.56, 0.73, 2.10, 2.31, 3.0}for β= 1, and {0.70, 0.735, 0.86, 1.39,
1.48, 2.01}for β= 3. We see that the single thimble integration gives us practically the
exact results outside the region of ˆµ∗
2<ˆµ < ˆµ∗
4in both cases. This is because, compared
to the thimbles Jσ0and Jσ±1, the thimbles Jσ±2and Jσ¯
0have so small weight factor
exp(−ReS) that their participation in the integration cycle are numerically negligible.
On the other hand, the results deviate from unity in the range of ˆµ∗
2<ˆµ < ˆµ∗
4,
indicating that the contributions from Jσ±1need to be included to reproduce the original
integral quantitatively. The much smaller deviation for β= 3 case can be understood if
one recalls the rough estimate for the weight factor exp(−ReS(σ±1)) ∼exp(−βπ2/(2L))
as discussed in subsection 4.2. Furthermore, we notice that the missing Jσ±1contribution
to Zchanges the sign from positive to negative, and back to positive again, as ˆµincreases.
This is the reflection of the fact that ImS(σ1) increases from 0 at ˆµ= ˆµ∗
2to 2πat ˆµ= ˆµ∗
4.
Because ImS(σ0) = 0 for any ˆµ, the two thimbles Jσ1and Jσ0contribute additively just
above ˆµ= ˆµ∗
2. But when ImS(σ1) = π, they contribute with opposite signs. At this point
they are connected at z=zzero,1with an angle πbetween their edges as seen in figure 7
(f). The Jσ±1contributions return to be positive as ˆµapproaches the critical value ˆµ∗
4
for the Stokes phenomenon. Regarding hniand h¯χχi, their integrands have non-constant
imaginary parts on Jσ±1, and the contributions of Jσ±1to these densities alternate in
different ways in the interval ˆµ∗
2<ˆµ < ˆµ∗
4.
6.2 Toward continuum limit
In figure 11 we examine the behavior of the fermion number density hniJ0evaluated only
on the single thimble Jσ0as a function of µ/m for L= 4,8,16 toward the continuum
limit. The parameters are set to (a) (β/L, Lm) = (1/4,4) and (b) (β/L, Lm) = (3/4,4).
In figure 11 (a), some discrepancy from the exact value (dashed line) is seen between
ˆµ∗
2<ˆµ < ˆµ∗
4for L= 4, where the thimbles Jσ±1have the nonzero intersection number
and need to be included in the integration. This behavior persists when we increase the
lattice size to L= 8,16 toward the continuum limit (thin black dashed curve). The critical
values µ∗
i/m for the Stokes phenomenon with the thimbles Jσ±1only slightly shift to larger
ˆµtoward the continuum limit. In figure 11 (b), The discrepancy from the exact values is
practically invisible and again the results are relatively insensitive to the size of the lattice
with our parameters. This implies that at finite temperatures Monte Carlo simulations on
a single thimble may work well for a certain parameters.
– 20 –
JHEP11(2015)079
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3
µ/m
(a) β=1
ZJ0 / Z0
<n>J0 / <n>0
<χ
−χ>J0/<χ
−χ>0
0.9
0.95
1
1.05
1.1
0 0.5 1 1.5 2
µ/m
(b) β=3
ZJ0 / Z0
<n>J0 / <n>0
<χ
−χ>J0/<χ
−χ>0
Figure 10.Z0(solid), hni0(dashed) and h¯χχi0(dotted) evaluated on the single thimble Jσ0
normalized by the exact values of the uniform-field model for β= 1 (a) and 3 (b) with L= 4 and
ma = 1. Arrows indicate the values of ˆµ∗
i(i= 1,·· · ,5).
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
<n>
µ/m
(a) β/L=1/4, Lm=4
L=4
L=8
L=16
L→ ∞
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
<n>
µ/m
(b) β/L=3/4, Lm=4
L=4
L=8
L=16
L→ ∞
Figure 11. (a) Fermion number density as a function of ˆµ, evaluated on the single thimble Jσ0for
L= 4 (red), 8 (green), 16 (blue) with fixed (β/L, Lm) = (1/4,4). (b) The same as (a) but with
(β/L, Lm) = (3/4,4). The uniform-field exact ones are shown in dashed lines for comparison.
6.3 Toward low temperature limit
Next we change Las 4, 8, and 16 with fixed β= 1 and ma = 1, toward the zero temperature
limit in figure 12. We find that the agreement between hniJ0and hni0is getting worse as
Lincreases. Even in β= 3 case (figure 12 (b)) we see a significant discrepancy from the
exact result (dashed line) for larger L. As Lincreases, the slope of the exact curve becomes
steeper in the crossover region and eventually converges to a step function, while the single
thimble result hniJ0behaves almost as a linear function between two kink points. The
singular points indicate the Stokes jump occurring there, through which the thimbles Jσ±1
join or leave the set of the integration cycles for the partition function Z.
6.4 Multi-thimble contributions
We draw the thimble structure on the right-half plane for L= 16 at ˆµ= 0.8,1.0,1.35,1.7
in the crossover region with β= 1, ma = 1 in figure 13. At ˆµ= 0.8 the three thimbles Jσ0
and Jσ±1have the nonzero intersection numbers with the original integration cycle, while
– 21 –
JHEP11(2015)079
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
<n>
µ
(a) β=1, m=1
L=4
L=8
L=16
L→ ∞
σ0 + σ± 1
σ0 + σ± 1 + σ± 2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
<n>
µ
(b) β=3, m=1
L=4
L=8
L=16
L→ ∞
σ0 + σ± 1
Figure 12. (a) Fermion number density as a function of ˆµ, evaluated on the single thimble Jσ0
for L= 4 (red), 8 (green), 16 (blue) with fixed (β, m) = (1,1). (b) The same as (a) but with
(β, m) = (3,1). The uniform-field exact ones are shown in dashed lines. For L= 16, improved
evaluations by including thimbles Jσ±1(±2) are shown with dots (crosses). L→ ∞ limit is shown in
a thin solid line.
Figure 13. Thimble structure on the right-half plane of zwith L= 16, β = 1, ma = 1 for
ˆµ= 0.8,1.0,1.35,1.7 (from bottom to top). The critical (zero) points are indicated with green (red)
dots.
at larger ˆµthe thimbles Jσ0,±1,±2,±3,±4(according to our numbering) intersect, and they
need to be included as the integration cycles to reproduce the partition function Z0.
Based on this observation, we extend the evaluation by including the contributions
from Jσ±1for β= 1,3 and those from Jσ±2further for β= 1, as shown with dots and
crosses in figure 12. Indeed, the agreement between the exact and multi-thimble evaluations
becomes systematically improved by taking into acount the multi-thimble contributions.
In table 2we listed the contributions to the partition function Z0and the fermion
density hni0from each thimble with L= 16, β= 1 and ma = 1. The thimbles Jσ±igive the
contributions which are complex conjugate to each other so that their sum becomes always
real. Regarding partition function Z0, the thimble Jσ±0gives the largest contribution, but
the thimbles Jσ±1also provide a substantial contribution in this crossover region. Those
from Jσ±i(i≥2) decrease rather quickly as i= 2,3,4 increases, which will be very
favorable for a systematic expansion. But we notice that a cancellation occurs between the
Jσ0and Jσ±1contributions at ˆµ= 1.35 owing to the negative sign of the Jσ±1contributions.
– 22 –
JHEP11(2015)079
ˆµ Z0,hni0σ0σ1σ2σ3σ4
0.8 2.04 1.19 (0.43, 0.04) — — —
1.3E-4 -7.33E-3 (3.73E-3, -7.351E-2) — — —
1.0 2.05 1.50 (0.28, -0.42) (-0.005, -0.021) (-1E-4, -1E-4) (-3E-7, -2E-7)
3.2E-3 0.1186 (-0.0508, -0.0774) (-6.9E-3, 0.6E-3) (-5E-5, 5E-5) (-9E-8, 2E-7)
1.35 3.80 9.09 (-2.72, -0.39) (0.07, 0.05) (1E-3, -4E-4) (-3E-7, -3E-7)
0.46 1.17 (-0.37, 0.23) (0.016, -0.008) (-9E-5, -4E-5) (-1E-7, 8E-8)
1.7 474.2 374.7 (51.0, 80.7) (-1.3, 0.9) (1E-3, -2E-3) (-7E-7, -2E-7)
1.00 0.67 (0.16, 0.09) (-1E-4, 4E-3) (-4E-6, -5E-6) (-7E-10, 2E-9)
1.35 54.91 569.97 (-298.63, -30.39) (42.60, 13.20) (-1.51,-1.27) (5E-3, 2.8E-2)
0.47 5.05 (-2.72, 0.84) (0.45, -0.20) (-0.025, 6.6E-3) (4E-4, 1E-4)
Table 2. Contributions of thimbles Zσiand Zσihniσi/Z0(i= 0,1,2,3,4) on the right-half plane
to Z0(upper) and to hni0(lower) with L= 16, β= 1 and ma = 1 for ˆµ= 0.8,1.0,1.35,1.7.
Thimbles on the left-half plane give the values complex conjugate to those in the list. Below the
double line, those values with L= 32 are listed.
For the fermion density hni0the cancellation between the Jσ0and Jσ±1contributions
becomes more delicate at ˆµ= 0.8 and 1.0, while those come to contribute additively at
ˆµ= 1.7. Insensitivity of the observables in small chemical region at low temperatures,
especially at zero temperature, is sometimes called Silver Blaze phenomenon. We find
here that when multiple thimbles contribute to the partition function they show a delicate
cancellation between them.
The alternating sign exp(−iImS) of the thimbles at ˆµ= 1.35 manifests in figure 13 as
the fact that the critical points and zero points are aligned and the thimbles are connected
at each zero point with the angle about π. In order to check this alternating pattern, we
extend our calculation to L= 32 as listed in the bottom row in table 2. We find that the
thimble-by-thimble alternating sign and cancellation become more striking not only for Z0
but also for hni0. In this case, we need to include the thimbles up to Jσ±3to evaluate the
observables with a few % accuracy. At larger Lmore zero points appear near the imaginary
axis (eq. (3.13)), and in between the critical points and associated thimbles are aligned at
ˆµin the crossover region. The weight factor from the bosonic part of the action does
not suppress these thimble contributions as far as Re(Lβ(1 −cos σi)) <1. Therefore we
need to treat the neat cancellation in multiple thimble contributions in order to reproduce
the sharp rise of the fermion density at low temperature (large L). Implication of this
observation to the feasibility of the numerical simulations with large lattice size is left for
future study.
7 Summary and discussions
We have studied the Lefschetz thimble structure of the (0+1) dimensional Thirring model
at finite chemical potential, which is formulated on the lattice of size Lwith the staggered
fermions and a compact auxiliary vector field. This model suffers from the sign problem
by the complex fermion determinant.
– 23 –
JHEP11(2015)079
The fermion determinant brings in two important features in the complexified field
space: many isolated critical points of the gradient flow and submanifolds of the zero
points with complex dimension (L−1). Those critical points accompany the Lefschetz
thimbles and the submanifolds of the zeros serve the ending points for the thimbles. We
have identified all the critical points of this model, and furthermore we have pointed out
a one-to-one correspondence between a critical point and a zero point within a projected
configuration subspace assigned with n−.
We argued that the thimbles associated with the critical points in n−= 0 subspace
become more important toward the continuum limit because the relative weights of the
other critical points located in n−6= 0 subspaces are suppressed by powers of e−2β. The
critical points with nonzero n−actually involve the doubler components and they are
expected naturally to decouple from the system in the continuum limit.
Hence, restricting our analysis to the critical points in the n−= 0 subspace, we have
shown how the thimble structure changes via the Stokes jumps as the chemical potential µ
increases. We found that at small and large chemical potentials the single thimble Jσ0is
sufficient as the integration cycle to reproduce the partition function of the model. However
in the crossover region we must include multiple thimbles in the set of the integration cycles
for the partition function Z. Their relative weights depend on the lattice size Land the
coupling strength β.
Taking the uniform-field model as a concrete example, we have examined the impor-
tance of the multi-thimble contributions and how the crossover behavior is generated by
them. The single-thimble approximation is justified for large β/L ∼T /g2, even in the
continuum limit. But as we increase the lattice size L, i.e., lower the temperature Twith
βand ma fixed, we have seen the breakdown of the single-thimble approximation, which
indicates the necessity of the multi-thimble contributions. The sign of those contributions
is alternating, which yields a neat cancellation to reproduce the correct values of Zand
observables at large L. We notice that the contributions from the thimbles away from the
origin diminish rather quickly. The Silver Blaze behavior and the following abrupt rise of
the density hniwith increasing µare achieved by the interplay among the multi-thimble
contributions in the crossover region.
We have performed HMC simulations for the (0+1) dimensional Thirring model with
finite chemical potential on the single thimble Jσ0in ref. [66]. We observed scaling behavior
of the results to the continuum limit at finite temperature and to the low-temperature limit.
The single thimble evaluation in the crossover region is getting worse for smaller βand/or
larger L, which is consistent with the results obtained in the uniform-field model. We show
one example of the simulation results for L= 16, β= 3 and ma = 1 in figure 14.
For comparison, we also tried the complex Langevin simulation as yet another approach
with complexification and as a possible way to include the “multi-thimble” contributions,
which is shown in figure 14. We find that the Langevin result also deviates from the exact
one in the crossover region, but in a different manner. We observed that the sampling
points in the Langevin simulation are distributed around the thimbles Jσ0and Jσ±1. The
details of the Langevin simulation will be reported elsewhere.
– 24 –
JHEP11(2015)079
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
<n>
µ/m
ThimbleHMC
CLE
exact
Figure 14. Result of HMC simulation on the thimble Jσ0with L= 16, β= 3 and ma = 1. The
curve indicates the exact value. Result of complex Langevin simulation (step size 10−4, and 104
samples taken every 100 steps) is also shown for comparison.
We have seen that an interplay among multi-thimble contributions are necessary and
important to describe the rapid crossover behavior of the fermion system. However it is a
difficult task to identify all the critical points in generic models. Our analysis suggests that
the thimbles whose critical points locate in the uniform-field subspace will give dominant
contributions, while those with critical points in non-uniform-field subspace will decouple
by the suppressed weight factor toward the continuum limit because they have doubler
components. Assuming that we can identify all the relevant thimbles to be integrated
over, we will face another challenge — how to add up the multi-thimble contributions
in the Monte Carlo simulation. In our model analysis we can sum up them by knowing
the partition function values hZiσprecisely, but in Monte Carlo simulations we compute
only the average of the observables not the partition function. It is, therefore, extremely
important to devise an efficient way to perform the multi-thimble integration by extending
the Monte Carlo algorithm for practical applications of the Lefschetz thimble integration
to fermionic systems with the sign problem.
The multi-thimble structure, which stems from the fermion determinant, will be rela-
vant also in QCD at finite chemical potential. One surely needs to elaborate the study
on the complex saddle points, intersection numbers and the Stokes jumps in QCD on the
lattice, especially in the cross-over and first-order transition regions. We have seen that in
the Thirring model the sum of multi-thimble contributions with alternating signs is rapidly
converging, but it is quite intriguing to clarify the scaling properties of the situation in lat-
tice QCD as a funcition of the lattice parameters and the choice of the lattice fermion
formulations, as well as the spacetime dimensionality. We remark here that the thimble
analysis of SU(3) Yang-Mills thoery without fermions is already worthwhile to be pursued,
which will provide more insights on the Lefschetz thimbles in gauge theories. We leave
these points for future study.
– 25 –
JHEP11(2015)079
Acknowledgments
H.F. was partially supported by JSPS KAKENHI (# 24540255). S.K. was supported
by the Advanced Science Measurement Research Center at Rikkyo University. Y.K. was
supported in part by JSPS KAKENHI (# 24540253).
A Exact expression and asymptotics of Z
In this appendix, we give the exact expression for the partition function of the Thirring
model with the compact action. We assume Nf= 1 and Lis even.
A useful formula for a matrix determinant is known in [67]:
det
a1b10c0
c1......0
0......bL−1
bL0cL−1aL
=−(bL··· b1+cL−1···c0)
+ tr " aL−bL−1cL−1
1 0 !··· a2−b1c1
1 0 ! a1−bLc0
1 0 !#.(A.1)
In application of this formula to the Dirac operator D, the components an, bnand cnread
a1=··· =aL=m,
bn=(1
2e+ˆµUn−1for n<L
−1
2e+ˆµUL−1for n=L,
cn=(−1
2e−ˆµU−1
n−1for n > 0
1
2e−ˆµU−1
L−1for n= 0 (A.2)
with Un= eiAnand U−1
n= e−iAn. Then the 2-by-2 matrix under the trace turns out to
be an L-th power of a constant matrix m1
4
1 0 . Now it is straightforward to reach the
expression
det D[A] = 1
2L2 cosh Lˆµ+i
L−1
X
n=0
An+mL
++mL
−,(A.3)
where m±=m±√m2+ 1. With ˆm≡sinh−1mand with even L, this can be written as
in eq. (2.4).
Because the An-odd terms in the determinant vanish after the integration over An
with weight e−β(1−cos An), we can write the partition function as
Z=1
2L−1Zπ
−π
L−1
Y
n=0
dAn
2πhcosh Lˆµ
L−1
Y
n=0
cos An+ cosh Lˆmiexp −β
L−1
X
n=0
(1 −cos An).
(A.4)
– 26 –
JHEP11(2015)079
This integration is easily performed to yield
Z=e−Lβ
2L−1I1(β)Lcosh Lˆµ+I0(β)Lcosh Lˆm,(A.5)
where I0(x) and I1(x), respectively, are the zeroth and first order modified Bessel functions
of the first kind. The fermion number density and the scalar density can be derived by
differentiating ln Zwith respect to µand m, respectively.
Using the asymptotic expression of the modified Bessel function I0,1(β) for a large β,
we find that in the continuum limit at finite T, the partition function eq. (2.5) scales as
Z→1
2L−11
2πβ L/2
e−3g2
4Tcosh µ
T+ eg2
Tcosh m
T,(A.6)
where we have used L/β = 2g2/T and Lˆµ=µ/T . For the uniform-field model (6.1),
applying the asymptotic form for large L,
IL(Lβ)→eLη
√2πL (1 + β2)1/4(A.7)
with η= (1 + β2)1/2+ log β
1+(1+β2)1/2, we find
Z0→1
2L−11
2πLβ 1/2
e−g2
Tcosh µ
T+ eg2
Tcosh m
T.(A.8)
It is interesting to observe that in the T→0 limit both models show a first order transition
at the same point |µc|=m+g2.
If we take Llarge with βfixed, we find
Z→1
2L1
2πβ L/2hI1(β)LeL|ˆµ|+I0(β)LeLˆmi,(A.9)
and for the uniform-field model
Z0→1
2L1
2πLβ 1/2"√βeL(η−β)
(1 + β2)1/4eL|ˆµ|+ eLˆm#.(A.10)
In the infinite-Llimit these models show a first-order transition at |ˆµc|= ˆm+
ln(I0(β)/I1(β)) and |ˆµc|= ˆm+β−η, respectively.
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
– 27 –
JHEP11(2015)079
References
[1] P. de Forcrand, Simulating QCD at finite density,PoS(LAT2009)010 [arXiv:1005.0539]
[INSPIRE].
[2] G. Parisi, On complex probabilities,Phys. Lett. B 131 (1983) 393 [INSPIRE].
[3] J.R. Klauder, A Langevin Approach to Fermion and Quantum Spin Correlation Functions,J.
Phys. A 16 (1983) L317 [INSPIRE].
[4] J.R. Klauder, Coherent State Langevin Equations for Canonical Quantum Systems With
Applications to the Quantized Hall Effect,Phys. Rev. A 29 (1984) 2036 [INSPIRE].
[5] E. Witten, Analytic Continuation Of Chern-Simons Theory,AMS/IP Stud. Adv. Math. 50
(2011) 347 [arXiv:1001.2933] [INSPIRE].
[6] E. Witten, A New Look At The Path Integral Of Quantum Mechanics,arXiv:1009.6032
[INSPIRE].
[7] F. Pham, Vanishing homologies and the n-variable saddlepoint method, in Proceedings of
Symposia in Pure Mathematics. Vol. 40, Part 2: Singularities, AMS Press, Providence
U.S.A. (1983).
[8] G. Aarts and I.-O. Stamatescu, Stochastic quantization at finite chemical potential,JHEP 09
(2008) 018 [arXiv:0807.1597] [INSPIRE].
[9] G. Aarts, Can stochastic quantization evade the sign problem? The relativistic Bose gas at
finite chemical potential,Phys. Rev. Lett. 102 (2009) 131601 [arXiv:0810.2089] [INSPIRE].
[10] G. Aarts, Complex Langevin dynamics at finite chemical potential: Mean field analysis in the
relativistic Bose gas,JHEP 05 (2009) 052 [arXiv:0902.4686] [INSPIRE].
[11] G. Aarts, Can complex Langevin dynamics evade the sign problem?,PoS(LAT2009)024
[arXiv:0910.3772] [INSPIRE].
[12] G. Aarts, F.A. James, E. Seiler and I.-O. Stamatescu, Adaptive stepsize and instabilities in
complex Langevin dynamics,Phys. Lett. B 687 (2010) 154 [arXiv:0912.0617] [INSPIRE].
[13] G. Aarts, E. Seiler and I.-O. Stamatescu, The Complex Langevin method: When can it be
trusted?,Phys. Rev. D 81 (2010) 054508 [arXiv:0912.3360] [INSPIRE].
[14] G. Aarts and F.A. James, On the convergence of complex Langevin dynamics: The
Three-dimensional XY model at finite chemical potential,JHEP 08 (2010) 020
[arXiv:1005.3468] [INSPIRE].
[15] G. Aarts and K. Splittorff, Degenerate distributions in complex Langevin dynamics:
one-dimensional QCD at finite chemical potential,JHEP 08 (2010) 017 [arXiv:1006.0332]
[INSPIRE].
[16] G. Aarts, F.A. James, E. Seiler and I.-O. Stamatescu, Complex Langevin: Etiology and
Diagnostics of its Main Problem,Eur. Phys. J. C 71 (2011) 1756 [arXiv:1101.3270]
[INSPIRE].
[17] G. Aarts, F.A. James, E. Seiler and I.-O. Stamatescu, Complex Langevin dynamics: criteria
for correctness,PoS(LATTICE 2011)197 [arXiv:1110.5749] [INSPIRE].
[18] G. Aarts and F.A. James, Complex Langevin dynamics in the SU(3) spin model at nonzero
chemical potential revisited,JHEP 01 (2012) 118 [arXiv:1112.4655] [INSPIRE].
– 28 –
JHEP11(2015)079
[19] E. Seiler, D. Sexty and I.-O. Stamatescu, Gauge cooling in complex Langevin for QCD with
heavy quarks,Phys. Lett. B 723 (2013) 213 [arXiv:1211.3709] [INSPIRE].
[20] G. Aarts, F.A. James, J.M. Pawlowski, E. Seiler, D. Sexty and I.-O. Stamatescu, Stability of
complex Langevin dynamics in effective models,JHEP 03 (2013) 073 [arXiv:1212.5231]
[INSPIRE].
[21] J.M. Pawlowski and C. Zielinski, Thirring model at finite density in 0+1 dimensions with
stochastic quantization: Crosscheck with an exact solution,Phys. Rev. D 87 (2013) 094503
[arXiv:1302.1622] [INSPIRE].
[22] J.M. Pawlowski and C. Zielinski, Thirring model at finite density in 2+1 dimensions with
stochastic quantization,Phys. Rev. D 87 (2013) 094509 [arXiv:1302.2249] [INSPIRE].
[23] G. Aarts, Complex Langevin dynamics and other approaches at finite chemical potential,
PoS(Lattice 2012)017 [arXiv:1302.3028] [INSPIRE].
[24] G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty and I.-O. Stamatescu, Controlling complex
Langevin dynamics at finite density,Eur. Phys. J. A 49 (2013) 89 [arXiv:1303.6425]
[INSPIRE].
[25] G. Aarts, P. Giudice and E. Seiler, Localised distributions and criteria for correctness in
complex Langevin dynamics,Annals Phys. 337 (2013) 238 [arXiv:1306.3075] [INSPIRE].
[26] D. Sexty, Simulating full QCD at nonzero density using the complex Langevin equation,
Phys. Lett. B 729 (2014) 108 [arXiv:1307.7748] [INSPIRE].
[27] G. Aarts, Lefschetz thimbles and stochastic quantization: Complex actions in the complex
plane,Phys. Rev. D 88 (2013) 094501 [arXiv:1308.4811] [INSPIRE].
[28] P. Giudice, G. Aarts and E. Seiler, Localised distributions in complex Langevin dynamics,
PoS(LATTICE 2013)200 [arXiv:1309.3191] [INSPIRE].
[29] A. Mollgaard and K. Splittorff, Complex Langevin Dynamics for chiral Random Matrix
Theory,Phys. Rev. D 88 (2013) 116007 [arXiv:1309.4335] [INSPIRE].
[30] D. Sexty, Extending complex Langevin simulations to full QCD at nonzero density,
PoS(LATTICE 2013)199 [arXiv:1310.6186] [INSPIRE].
[31] G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty and I.-O. Stamatescu, Complex Langevin
simulation for QCD-like models,PoS(LATTICE 2013)451 [arXiv:1310.7412] [INSPIRE].
[32] L. Bongiovanni, G. Aarts, E. Seiler, D. Sexty and I.-O. Stamatescu, Adaptive gauge cooling
for complex Langevin dynamics,PoS(LATTICE 2013)449 [arXiv:1311.1056] [INSPIRE].
[33] G. Aarts, L. Bongiovanni, E. Seiler and D. Sexty, Some remarks on Lefschetz thimbles and
complex Langevin dynamics,JHEP 10 (2014) 159 [arXiv:1407.2090] [INSPIRE].
[34] G. Aarts, E. Seiler, D. Sexty and I.-O. Stamatescu, Simulating QCD at nonzero baryon
density to all orders in the hopping parameter expansion,Phys. Rev. D 90 (2014) 114505
[arXiv:1408.3770] [INSPIRE].
[35] D. Sexty, Progress in complex Langevin simulations of full QCD at non-zero density,Nucl.
Phys. A 931 (2014) 856 [arXiv:1408.6767] [INSPIRE].
[36] D. Sexty, New algorithms for finite density QCD,PoS(LATTICE2014)016 [arXiv:1410.8813]
[INSPIRE].
– 29 –
JHEP11(2015)079
[37] L. Bongiovanni, G. Aarts, E. Seiler and D. Sexty, Complex Langevin dynamics for SU(3)
gauge theory in the presence of a theta term,PoS(LATTICE2014)199 [arXiv:1411.0949]
[INSPIRE].
[38] G. Aarts, F. Attanasio, B. J¨ager, E. Seiler, D. Sexty and I.-O. Stamatescu, Exploring the
phase diagram of QCD with complex Langevin simulations,PoS(LATTICE2014)200
[arXiv:1411.2632] [INSPIRE].
[39] G. Aarts, B. J¨ager, E. Seiler, D. Sexty and I.-O. Stamatescu, Systematic approximation for
QCD at non-zero density,PoS(LATTICE2014)207 [arXiv:1412.5775] [INSPIRE].
[40] G. Aarts, F. Attanasio, B. J¨ager, E. Seiler, D. Sexty and I.-O. Stamatescu, QCD at nonzero
chemical potential: recent progress on the lattice,arXiv:1412.0847 [INSPIRE].
[41] A. Mollgaard and K. Splittorff, Full simulation of chiral random matrix theory at nonzero
chemical potential by complex Langevin,Phys. Rev. D 91 (2015) 036007 [arXiv:1412.2729]
[INSPIRE].
[42] H. Makino, H. Suzuki and D. Takeda, Complex Langevin method applied to the 2DSU(2)
Yang-Mills theory,Phys. Rev. D 92 (2015) 085020 [arXiv:1503.00417] [INSPIRE].
[43] G. Aarts, E. Seiler, D. Sexty and I.O. Stamatescu, Hopping parameter expansion to all
orders using the complex Langevin equation,arXiv:1503.08813 [INSPIRE].
[44] J. Nishimura and S. Shimasaki, New insights into the problem with a singular drift term in
the complex Langevin method,Phys. Rev. D 92 (2015) 011501 [arXiv:1504.08359]
[INSPIRE].
[45] G. Aarts, F. Attanasio, B. J¨ager, E. Seiler, D. Sexty and I.-O. Stamatescu, The phase
diagram of heavy dense QCD with complex Langevin simulations,Acta Phys. Polon. Supp. 8
(2015) 405 [arXiv:1506.02547] [INSPIRE].
[46] K. Nagata, J. Nishimura and S. Shimasaki, Justification of the complex Langevin method
with the gauge cooling procedure,arXiv:1508.02377 [INSPIRE].
[47] Z. Fodor, S.D. Katz, D. Sexty and C. T¨or¨ok, Complex Langevin dynamics for dynamical
QCD at nonzero chemical potential: a comparison with multi-parameter reweighting,
arXiv:1508.05260 [INSPIRE].
[48] AuroraScience collaboration, M. Cristoforetti, F. Di Renzo and L. Scorzato, New
approach to the sign problem in quantum field theories: High density QCD on a Lefschetz
thimble,Phys. Rev. D 86 (2012) 074506 [arXiv:1205.3996] [INSPIRE].
[49] M. Cristoforetti, F. Di Renzo, A. Mukherjee and L. Scorzato, Monte Carlo simulations on
the Lefschetz thimble: Taming the sign problem,Phys. Rev. D 88 (2013) 051501
[arXiv:1303.7204] [INSPIRE].
[50] H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu and T. Sano, Hybrid Monte Carlo
on Lefschetz thimbles — A study of the residual sign problem,JHEP 10 (2013) 147
[arXiv:1309.4371] [INSPIRE].
[51] T.D. . Cohen, Functional integrals for QCD at nonzero chemical potential and zero density,
Phys. Rev. Lett. 91 (2003) 222001 [hep-ph/0307089] [INSPIRE].
[52] A. Mukherjee and M. Cristoforetti, Lefschetz thimble Monte Carlo for many-body theories: A
Hubbard model study,Phys. Rev. B 90 (2014) 035134 [arXiv:1403.5680] [INSPIRE].
– 30 –
JHEP11(2015)079
[53] F. Di Renzo and G. Eruzzi, Thimble regularization at work: from toy models to chiral
random matrix theories,Phys. Rev. D 92 (2015) 085030 [arXiv:1507.03858] [INSPIRE].
[54] Y. Tanizaki, Lefschetz-thimble techniques for path integral of zero-dimensional O(n)
σ-models,Phys. Rev. D 91 (2015) 036002 [arXiv:1412.1891] [INSPIRE].
[55] T. Kanazawa and Y. Tanizaki, Structure of Lefschetz thimbles in simple fermionic systems,
JHEP 03 (2015) 044 [arXiv:1412.2802] [INSPIRE].
[56] M. Cristoforetti et al., An efficient method to compute the residual phase on a Lefschetz
thimble,Phys. Rev. D 89 (2014) 114505 [arXiv:1403.5637] [INSPIRE].
[57] Y. Tanizaki and T. Koike, Real-time Feynman path integral with PicardLefschetz theory and
its applications to quantum tunneling,Annals Phys. 351 (2014) 250 [arXiv:1406.2386]
[INSPIRE].
[58] Y. Tanizaki, H. Nishimura and K. Kashiwa, Evading the sign problem in the mean-field
approximation through Lefschetz-thimble path integral,Phys. Rev. D 91 (2015) 101701
[arXiv:1504.02979] [INSPIRE].
[59] A. Cherman, D. Dorigoni and M. ¨
Unsal, Decoding perturbation theory using resurgence:
Stokes phenomena, new saddle points and Lefschetz thimbles,JHEP 10 (2015) 056
[arXiv:1403.1277] [INSPIRE].
[60] A. Behtash, T. Sulejmanpasic, T. Sch¨afer and M. ¨
Unsal, Hidden topological angles and
Lefschetz thimbles,Phys. Rev. Lett. 115 (2015) 041601 [arXiv:1502.06624] [INSPIRE].
[61] K. Fukushima and Y. Tanizaki, Hamilton dynamics for the Lefschetz thimble integration akin
to the complex Langevin method,arXiv:1507.07351 [INSPIRE].
[62] S. Tsutsui and T.M. Doi, An improvement in complex Langevin dynamics from a view point
of Lefschetz thimbles,arXiv:1508.04231 [INSPIRE].
[63] J.M. Pawlowski, I.-O. Stamatescu and C. Zielinski, Simple QED- and QCD-like Models at
Finite Density,Phys. Rev. D 92 (2015) 014508 [arXiv:1402.6042] [INSPIRE].
[64] J.B. Kogut and L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories,
Phys. Rev. D 11 (1975) 395 [INSPIRE].
[65] P. Hasenfratz and F. Karsch, Chemical Potential on the Lattice,Phys. Lett. B 125 (1983)
308 [INSPIRE].
[66] H. Fujii, S. Kamata and Y. Kikukawa, Monte Carlo study of Lefschetz thimble structure in
one-dimensional Thirring model at finite density,arXiv:1509.09141 [INSPIRE].
[67] L.G. Molinari, Determinants of block tridiagonal matrices,Lin. Algebra Appl. 429 (2008)
2221.
– 31 –
