(-%+*'),/#! "! 0. $-&/ 

$-&/

, 20191121

1

Plan of Talk
 Introduction
 Tensor Renormalization Group (TRG)­ Two Dimensional Ising Model­ Singular Value Decomposition Current Status for TRG Studies in Particle Physics
 Application to Finite Density System­ 2D Complex φ4 Theory at Finite Density
 SummaryIntroduction
Partition function of lattice gauge theory

Z!

−Sg ({U })
= DU
det
D({U
})
e
!
!
({U })−Sgg ({U })Z = DU det D({U }) e−S

U: gauge field
D: Dirac matrix−
ψDψ

DU quantity
det D({U }) e
! Zof=
Expectation value
physical⟨O⟩ = DU
O({U, D−1}) det D({U }) e−Sg ({U })!
−Sg ({U })⟨O⟩ = ! DU O({U, D−1−1}) det D({U }) e −Sg ({U })⟨O⟩ = DU O({U,D }) det D({U }) e""
−S(i,j,k,...)
ZMonte= Carloe=detDe"−S
T>0
TimnoimportanceTjpqr Tksuv Tlwxy···.
ijkl withforsampling" works−S(i,j,k,...)
i,j,k,...
Z i,j,k,...
=e=1
Tijkl TimnoTjpqr Tksuv Tlwxy · · · .
i,j,k,...
P = !i,j,k,...
det D({U }) e−Sg ({U })−Sg [U ]ZQCD(T, µ) = Z DU
det D(µ; U )! e−Sg [U ]ZQCD(T, µ) = " DU e
det D(µ; U )"In casedensity simulation
Z = of finite
e−S(i,j,k,...)
= ⟨Oe
Tijkl
iNf T
θ imno Tjpqr Tksuv Tlwxy · · · .
⟩||detD becomes a complex value,
i,j,k,...of chemical potential
i,j,k,... μ ⇒Introduction
iN
θ
⟨O⟩ =
⟨Oe
iNf θ f ⟩||
⟨e
⟩||
⟨O⟩
=
Importance sampling fails= statisticalbecomes uncontrollable!
iNerrorθf⟨ee−Sg⟩[U||] det D(µ; U )ZQCD(T, µ) = DU
⟨Oeiθ ⟩|| action problem
Sign problem / Complex
iθ
⟨O⟩ =
⟨Oe
iθ
⟩||iNf⟩θ||⟩||⟨O⟩ =⟨e⟨Oe⟨O⟩ = ⟨eiNiθf⟩θ||!
⟨e⟩||−Sg [U ]Z||(T, µ) = DU
| det D(µ; U )|! e
−Sg [U ]Z||(T, µ) = DU e
| det D(µ; U )|

Tensor Network Scheme
What is Tensor Network (TN) Scheme?
Theoretical and numerical methods for high precision analysesof many body problems with tensor network formalism
What is different from conventional methods?
Free from sign problem and complex action problem in Monte Carlo method
Computational cost for LD system size ∝ Dlog(L)Direct treatment of Grassmann numbers
Direct evaluation of partition function Z itself

Possible applications in particle physics
Light quark dynamics in QED/QCD, Finite density QCD,
Strong CP problem, Chiral gauge theories, Lattice SUSY etc.
Also many applications in condensed matter physics

χ(L) =

L2 ∂(1/2κ)2!
]ψ−Sg [U ]Tensor Renormalization
Group
(TRG)
Z = DψDψ̄DUe−ψ̄D[UD

cut"
Explain the algorithm with 2D
Ising
model
with N σsites
T≃UV

i,j,k,lHamiltonian H =

!

⟨i,j⟩

Z =

si sj

m=1

(i,j),m m m,(k,l)

si ± 1"
H =
si sj
⟨i,j⟩

!

Levin-Nave
PRL99(2007)120601

si ± 1exp (−βH)"
Partition Function
{Si }
Z =
exp (−βH)
=

2
!

{si }

Ti,m,n,l Ts,t,i,j
2 Tr,j,k,q Tk,l,o,p · · ·"i,j,k,l,···=1 =Tα,λ,ρ,δ Tσ,κ,α,β Tµ,β,γ,τ Tγ,δ,ν,χ · · ·Z ="α,β,γ,δ,···=1})
DU det D({U!}) Tensor
e−Sg ({UNetwork
formulation
Z = DU det D({U }) e−Sg ({U })Details of model
are specified in initial tensor"−Sg ({U })⟨O⟩= DUprocedureO({U, D−1
det D({U })
e
!is})
The
algorithmic
independentof−1 models⟨O⟩ = DU O({U, D }) det D({U }) e−Sg ({U })1
Of course, direct
is impossible
g ({U })for large N even with current
P =contraction
det D({U}) e−S
1Z
fastest supercomputerP =det D({U }) e−Sg ({U })Z⇒ How
to evaluate the!partition function?
!

Z=e−S(i,j,k,...) ="Tijkl TimnoTjpqr
Tksuv Tlwxy · · · .
"−S(i,j,k,...)
i,j,k,...
i,j,k,...
Z=e=
Tijkl TimnoTjpqr Tksuv Tlwxy · · · .

Schematic View of TRG Algorithm
1. Singular Value Decomposition of local tensor T
2. Contraction of old tensor indices (coarse-graining)
3. Repeat the iteration

Keep largest Dcut components⇒ Reduction of freedom

Tensors
w/ new indices#sites are reduced to half6

Quick Review of Singular Value Decomposition
Any m"n (m>n) real matrix A can be decomposed as A=UΣVT
U: m"m orthogonal matrix
V: n"n orthogonal matrix
Σ=diag(σ1, σ2, σ3, σ4, $, σn)(σ1≥σ2≥ σ3≥σ4≥$≥σn≥0)

σ1, σ2, σ3, σ4,%%%,σn: Singular Values (SV) of A (non-negative)!$
  $ 
#  #  $# 
σiuiviT=

"

"
σi

ui

viT
7

Approximation of Matrix
Truncation of sum of rank 1 matrices+%&+ % & ,+%&,+%&+%&+ % & ,+%&"!!" ($ '$"#$#"" " !!" ($ ))  )) + + ,+ '")) )) "   **   $  !"## #! !%"!!$ Image Compression with SVD
Image data with 200x320 pixel ⇒ 200x320 real matrix
Application of SVD to the matrix
A=σ1u1v1T+σ2u2v2T+…+σnunvnT (n=200)

Sample image (200x320 pixel)

Distribution of SVs

J. Demmel: Applied Numerical Linear Algebra, SIAM 19979

Approximation with SVD
Ak=σ1u1v1T+σ2u2v2T+…+σkukvkT (k≪200)

J. Demmel: Applied Numerical Linear Algebra, SIAM 1997

Numerical test for 2D Ising Model
The key element in the algorithm is low-rank approximation by SVD

Ti,j,k,l ≃

D!
cut

m=1

U(i,j),mσmVm,(k,l)!

H = by thesisj parametersi ± 1 Dcut
Truncation error is controlled⟨i,j⟩!
XIE, CHEN, QIN, ZHU, YANG, AND XIANGREVIEW B 86, 045139 (201
= offexp
Free energy Z
on andthe (−βH)transition point, lattice size=23050PHYSICAL, Dcut=24

{Si }

2
!

=

i,j,k,l,···=1(a)Ti,m,n,l Ts,t,i,j Tr,j,k,q TXie···
k,l,o,pet al.

(n)

z

(b)

(n)

T y'1
x1
x'1
y'2
y1 i
x2
x'2
(n)
T
y2
z'

T

PRB86(2012)045139"

DU det D({U }) e−Sg ({U })Comparison with analytic results−6"
Relative
error
of
free
energy≤10zy'⟨O⟩ = DU O({U, D−1}) det D({U }) e−Sg ({U })
V
Z =

(n)

(n)

(n)

1
P =
det D({U }) e−Sg ({U })
Z

(n)

M

U x'

x U
y

(n)

(n)

V

z'

M

importance of keeping a large D in the 3D TRG calculation.
The error in Tc , estimated from the difference between thevalues of Tc for D = 15 and D = 16, is also less than 10−6 . A
comparison of the values of Tc obtained by different methods
is shown in Table II. Our results agree with the Monte Carlodata.37–39
The above discussion indicates that the HOTRG works
very well in 3D. The accuracy of the results can be further
improved by applying the HOSRG. However, the HOSRG
calculation costs much more CPU time. A thorough study
with the HOSRG on the 3D Ising model is still in progress
and the results will be published separately.

properties. For example, the ground state can be studied by
the DMRG24 and the thermodynamics can be studied by thequantum transfer matrix renormalization group (TMRG).44,45
Here we will only discuss how to apply the HOTRG/HOSRG
to a 2D quantum lattice model.
As an example, we will take the 2D quantum Ising modelwith a transverse field to show
howet
these
Xie
al.methods work. The
Hamiltonian of this model is defined by!
! PRB86(2012)045139
σzi σzj − h
σxi .
(19)
H =−

Study of 3D Ising Model⟨ij ⟩iHigher-Order TRG (HOTRG): applicable to higher dimensionalmodelsIV. GROUND STATE AND THERMODYNAMICS
OF 2D
11log(V)Computational
cost
∝
(D
)
cut
QUANTUM LATTICE MODELS
XIE, CHEN, QIN, ZHU, YANG, AND XIANG

A d-dimensional quantum lattice model is equivalent
to a (d + 1)-dimensionalclassical model,oftheT HOTRG and
Dcut dependencec

We start by representing the partition function of this model
as a tensor-network model in the 2 + 1 dimensions. By using
the Trotter-Suzuki decomposition formula, we can express thepartition function as19 PHYSICAL REVIEW B 86, 045139 (2012)
Z = Tre−βH ≈ Tr[e−τ Hz e−τ Hx ]L + O(τ 2 ),Comparison with Monte Carlo data

Method1%

FIG. 10. (Color online) Temperature dependence of the magnetization for the 3D Ising model (D = 14). The Monte Carlo result isfrom Ref. 35. (Inset) Logarithmic plot of the magnetization around
FIG. 11.
(Color point.
online)The
Theslope
criticaltemperature
Tc asa function
the criticalof thefitting curve
gives
the criticalexponentof the D
magnetization,
γ = 0.3295.
of the bonddimensionfor the 3D Isingmodel obtained from the(20)

TABLE II. Comparison of the critical point Tc for the 3D Ising
model
where obtained by different methods.

Hz = −!

σzi σzj ,Tc (21)⟨ij ⟩HOTRG (D = 16, from U)
4.511544!
HOTRG (D = 16, from HM)x = −h4.511546
σxi .
(22)
4.511523
Monte Carlo37
i
38
4.511525
Monteβ=Lτ Carlois theinversetemperatureand τ is a small
Trotter
39
4.511516
Monte
Carlo
parameter. Thispartition function can be also expressed as a
4.511528
Monte Carlo35
productofevolution
matrix V ,4.511536
Series expansion40
4.5788CTMRG12
L(23)
Z = TrV ,4.5704
TPVA134.5393
CTMRG14
where
4.554
TPVA16
41
−τ
H
/2
−τ
H
−τ
H
/2
x
z
x
4.547 (24)
Algebraic variation
e
e
V =e

is the evolution operator between two neighboring Trotter
HOSRG
methods
introduced
canbetween
be also any
extendedlayers.
To insert
theabovecompletebasis set
two ofto study the ground-state and thermodynamic properties ofthe exponential terms on the right-hand side of Eq. (20), it isd-dimensional quantum lattice models. For one-dimensionalstraightforward
to show that V can be expressed as a product ofquantum lattice models, there are already many mature methlocal
Fromwestatecan asexpressfunctionodstensors.
for studying
thethis,groundwell asthe
thepartition
thermodynamic

Results show good agreement with Monte Carlo data at high precisioninternal energy (U ) and magnetization (M), respectively.
D. It becomes converged only when D ! 13, indicating theproperties. For example, the ground state can be studied byimportance of keeping a large D in the 3D TRG calculation.
045139-6
the DMRG24 and the thermodynamics can be studied by the
The error in Tc , estimated from the difference between thequantum transfer matrix renormalization group (TMRG).44,45
values of Tc for D = 15 and D = 16, is also less than 10−6 . A
Here we will only discuss how to apply the HOTRG/HOSRG
comparison of the values of T obtained by different methods

Collaborators
Y. Kuramashi, Y. Yoshimura
S. Akiyama

U. Tsukuba

Y. Nakamura, (Y. Shimizu)

R-CCS

S. Takeda, R. Sakai

Kanazawa U.

D. KadohN. Tsing-Hua U./
Keio U.

13

Application of TRGs to Particle Physics (1)2D models
Ising modelLevin-Nave, PRL99(2007)120601
X-Y modelMeurice+, PRE89(2014)013308
CP(1)Kawauchi-Takeda, PRD93(2016)114503
Real φ4 theory
Shimizu, Mod.Phys.Lett.A27(2012)1250035,
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1905(2019)184
Complex φ4 theory at finite density
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, in preparation
U(1) gauge theory+θ
YK-Yoshimura, arXiv:1911.06480
Schwinger, Schwinger+θ
Shimizu-YK, PRD90(2014)014508, PRD90(2014)074503,
PRD97(2018)034502
Gross-Neveu model at finite density
Takeda-Yoshimura, PTEP2015(2015)043B01
N=1 Wess-Zumino model
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP1803(2018)141

14

Application of TRGs to Particle Physics (2)3D models
IsingXie+, PRB86(2012)045139
Potts modelWan+, CPL31(2014)070503
Free Wilson fermion
Sakai-Takeda-Yoshimura, PTEP2017(2017)063B07,
Yoshimura-YK-Nakamura-Takeda-Sakai, PRD97(2018)054511
Z2 gauge theory at finite temperature
YK-Yoshimura, JHEP1908(2019)023
4D models
IsingAkiyama-YK-Yamashita-Yoshimura, PRD100(2019)054510
weak first-order phase transition (not second-order phase transition)

15

Application to Finite Density System
2D complex φ4 theory at finite density
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, in preparation
How to treat continuous dof?
Complex action with finite chemical potential μSign problem is really solved?

4 Theory at Finite Density
2D Complex
φ
2
2
2
2
2
∗
∗
4

Scont ="!

d x |∂ρφ| + (m − µ )|φ| + µ(φ ∂2φ − ∂2φ φ) + λ|φ|

!

2

"

Kadoh+, in preparation

2

2

2

2

∗

∗

4

Scont =actiond x of|∂2D+ (m −φµ4 theory)|φ| + atµ(φ∂2φ μ− ∂2φ φ) + λ|φ|
ρ φ|complex
Continuum
finiteScont =

!

2

"

#

2

Z =2

!

2 exp(−S)
2
∗
Dφ

#

∗

4

d x |∂ρφ| + (m − µ )|φ| + µ(φ ∂2φ − ∂2φ φ) + λ|φ|!

#Z =chemical
Dφ exp(−S)
Introduction of finite
potential ⇒ complex action

Lattice action!

!

Z(original)
2 exp(−S)
Z = =Dφ1DφDφ12Dφexp(−S)Z(original) =
⎡
⎡
$⎢
$⎣
⎢
⎣
n
⎡
n
$⎢
⎣

!

Dφ1Dφ2 exp(−S)

⎤

⎤
)
2 ( (
$
)
2
⎥
2
2
4
µδ
∗
−µδ
∗
$ ρ,2
ρ,2 φ
∗ +e
−µδ
∗φn ⎦ ⎥
S(4
| n+|2λ|φρ,2n | n−
nφ
n+ρ̂
S=
=
(4 +
+m
m2)|φ
) +n|φ+ λ|φ|4ρ=1− e eµδφρ,2
φn+ρ̂
φ
+
e
φ
n+ρ̂
n
n+ρ̂ φn ⎦
⎤
ρ=1
(
)
2
$
⎥
2
2
4
⎦
S=
(4 + m ) + |φn| + λ|φn| −eµδρ,2 φ∗nφn+ρ̂ + e−µδρ,2 φ∗n+ρ̂φn
4n
TN representationis constructed in theρ=1
same way as for the real φ case

φφn ==(φ, φn,2) → (rn cos θn, rn sin θn)(φn,1nn,1 , φn,2 ) → (rn cos θn , rn sin θn )Bose condensation is expected to occur at sufficiently large μφn = (φn,1, φn,2) → (rn cos θn, rn sin θn)exp(x cos z) =exp(x cos z) =

∞
$
∞
$

Ik (x) exp(ikz) x ∈ R, z ∈ C
Ik (x) exp(ikz) x ∈ R, z ∈ C#(fields if& first' function&point' expectation
networktorepresentation
partition
and
the
value
of
the
%the
Refs. [4,important
to
note
is
that
the
latticeactioncontains the1 $ 5]. 1 of2The
λ2
4= exp − 4 + m|z|2 + |z ′ | −|z|4 + |z ′ | + eµδν,2 z ∗ z ′ + e−µδν,2 zz ′∗ . (5)nessesary.
In the4 following
we shall derive
it4 especiallyforweight
the partition
nearest-neighborinteraction.
Thus theBoltzmann
can be function
expressedaccording
as a product ofTensor Network Representation1
to Refs.
[4, 5].
first important
to noteis thatlattice which
actioniscontains
thefactors:
Thelocalnext
step isTheto derivethe discretepoint
structurelabeledby thean integera candidate

nearest-neighborThus theBoltzmannbe Kadoh+,
expressedrulea productof
inaspreparation
2weight can quadratureof a tensor index.interaction.
For that purpose
we usethe"Gauss–Hermiteasin refs. [4,"e−S =fν (φn , φn+ν̂ ) ,local
factors:
5].
For
one-variable
integration,thequadratureprovidesa discretization as follows. For a
Boltzmann weight is expressed as(4)n∈Γ ν=1

2
proper function g(x),""−Swhere the local factor is explicitly
given
e =
fν by
(φn , φn+ν̂ ) ,
(4)
) ∞
K
n∈Γ ν=1*−x2
dxe g (x) ≈wα g (yα )
(6)
′
f
(z,
z
)
ν
−∞
where the local factor# is explicitly &given by ' α=1&
(
'
%
1$
λ
2
4
2
′ 2
′ 4
µδν,2 ∗ ′−µδν,2
′∗
exp
−
4
+
m
|z|
+
|z
|
−
|z|
+
|z
|
+
e
z
z
+
e
zz
. (5)
where yα and=w
are
the
α-th
root
of
the
K-th
Hermitepolynomial
andthecorresponding′ α
44fν (z, z )#weight,HeretheK
theand for large K (the pre& dictates' order& o.off.approximation'⇒
Needrespectively.
todiscretize
continuous
d.
$
%
1
λ
2
4
The−next4step
is2 to |z|derivetheis a (5)candidate′ 2 discrete structure′ 4 labeledµδν,2by∗ an′ integer−µδν,2which
′∗
=
exp
+
m
+
|z
|
−
|z|
+
|z
|
+
e
z
z
+
e
zz
.
1
√
cision of integration
is expected to be better.
4
4 Even in two-variable case (z = (z1 + iz2 ),of a tensor index. For that purpose we use the Gauss–Hermite quadrature2 rule as in refs. [4,z1 , z2 ∈ R), one can simply apply the quadrature rule
The
next
step
is to derivequadrature
the discrete thestructurelabeledby an integer
which is a as
candidate
Useof5].
Gauss-HermiteFor
one-variable
integration,
quadratureprovidesa discretization
follows. For a+,) ∞) ∞
K*of a tensor
index.
For thatpurpose −z
we2 −zuse
the Gauss–Hermite
quadraturerule as in refs. [4,y+iy2properfunction
g(x),
α
√ β ,
dz1 dz2 e 1 2 h (z) ≈wα wβ h(7)2−∞−∞5]. For one-variable integration, the )quadrature α,β=1
providesKa discretization as follows. For a∞*
as long as the integrand h is a proper −xfunction.
After applying the replacement for each
2
dxe
g
(x)
≈
wα g (yα )(6)properfunctiong(x),Discretizedversionofpartitionfunction−∞ function is now
complex field on a site, the partition
α=1expressed in a discrete form
1
The expectation value of the fields
canbetreated in a similar
way.
) ∞
K
*
%
&
2
2
−x
"
where yα and wα!
are"the α-thHermitepolynomial
andthecorresponding# 2the
$
y
+
iy
y
+
iy
dxerootg of(x)≈ K-th
w
g
(y
)
α
β
α
β
n+ν̂
n+ν̂ (6)
n
n
α
α
2
√
√
Z ≈ Z (K) =
w
w
exp
y
+
y
f
,
, (8)
α
β
ν
n
n
α
β
−∞
n
n
α=1
2
weight, respectively.
Here K dictates3 the orderν=1
of approximationand for 2large K the pre{α,β} n∈Γwhere ycision
w integration
are the α-throot of theHermitepolynomial
and thecorresponding
α and ofexpectedto K-thbeEven
in two-variablecase(z = √12 (z1 + iz2 ),where
we αhaveused anis abbreviation
for better.
the summation
weight,zrespectively.
Here
dictatesthetheorder of approximationand for large K the precan K
simplyapply
Krule
1 , z2 ∈ R), one! quadrature
" !

∞
$

p(x cos z) =

k=−∞

0

Ik (x) exp(ikz) x ∈ R, z ∈ C

L=6
L=8

0
0

0.5

1

1.5

2

0

0.5

1

µ

1.5
µ

Simple(st) Test Bed for Sign Problem
0.3

4
!
⎟ 0 ∞
⎠

0

2
0

0.2
Mori-Kashiwa-Onishi,
PTEP(2018)023B04
42 )− λ (r 4 +r
− 14 (4+m2 )(rn2 +rn+ρ̂
4 n n+ρ̂ )2πr
e 023B04
PTEP
n 2018,Previous study with path optimization method (Monte Carlo)
n 0

n

ρ=1

1 −k
·Ikn,ρ (2rnrn+ρ̂)ekn,ρµδρ,2 δ(kn,1+kn,2−kn−1̂,1n−2̂,2 ),0Average Phase Factor

Sq {U, ψ̄, ψ} =

#

m,n

1

0.6

iθ

Dφ1Dφ2 exp(−Re(S))"

Re <eiθ>pq

Zpq =

!

L=4
L=6
L=8

0.8

Re <e >pq⟨eiθ ⟩ = Z/Zpq!

0.1ψ̄(m)D({U })ψ(n)

0.4

0

-0.1
1
-0.2
0.8
-0.3
0.6 0

Y. Mori et al.

L=4
L=6
L=8
0.5

1

1.5

2

µ

0.4

0.2 Fig. 2. The top left and top right panels0.2
show the real part ofL=4
the average phase factor
L=6part of the aver0 optimization as a function of µ. The bottom
0 panel shows the imaginary
L=8

thesquares,and crosses
results1for L 1.5= 4, 6,2and 8, respective
0 optimization.
0.5
1 Circles,
1.5
2
0 are0.5µStill, it seems difficult to perform
1#
a MC simulation
L≳10= Fµν Fµν +q̄ [γµ(∂on
igAµ) + mq ] qµ−
4
q=u,d,s,c,b,t

1#− F µν Fµν +q̄ [iγ µ(∂µ − igAµ) − mq ] q
4Theauthors claim
q=u,d,s,c,b,tiθ>pq
ImRe<e<n><eiθ> becomes close to 1

µ

16
0.3
L=4
14
L=6
0.2
12
L=8
10
0.1 Mean Field approx.
8
0
6
-0.1
4
2
-0.2
0
-0.3
-2
0.5
00
0.5
11
µµ

0.2

L=4
L=6
L=8

0.1

m2=1, λ=1

Im <n>

n,1 ,kn,2 =−∞drn

5

Im <eiθ>pq

∞
$

⎞

-0.1

L=4
L=6
L=8
1.5
1.5

0

2

-0.2

0

0.5

1

1.5
µ

Fig. 2. The top left andtoppanels showvalue
the real
partof the density
averageinphasefactorCarlo
without
and with (solitheFig.
3. right
The expectationof thenumberthe Monte
calculations”We show thattheaveragephase
factorissignificantly
enhancedafterthe
optimization as a function
oftheµ. Thepanel shows theimaginary
the averagephase thefactorwithand inmeanbottom
field approximation(dashedline).part
Theofshaded
area shows
expectat
optimization theandthen
weCircles,
canperformCarlo
method.”optimization.
squares,are= 4, 6, and8, respectively.
pathsafely
optimization
methodthe
at results
L hybrid= 8. for LMonte&'and crosses

1$ %
#
=
dUµdqdq̄ O[Uµ, q, q̄] exp − Llatt[Uµ, q, q̄]n
Z n,µResults for Z(original) with TRG
Kadoh+, in preparation

the phase quenched one exhibits the continuous behavior. To see the diﬀerence in mo

2=0.01, λ=1, K=64, D =64
Parameters: mdetail,cut in fig. 4. In the infinite volume limit the full-versi
we show its volume dependent
convergesfrom
to zero 44while thetophase
quenched one takes a finite value.
V=LL is changed256256

Average phase factor
1

Number density
3

3.5

3

V=24×24
V=25×25
V=26×26
V=27×27
V=28×28
V=2 ×2

0.9
0.8
0.7

3
2.5
2

0.6
0.5

n

Z/Zpq

V=224×224
V=26×26
V=28×28
V=2 ×2

1.5

0.4

1

0.3

0.5

0.2
0

0.1
0

-0.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

1.2

µ

µ

Sign problem: <eiθ> ≪ 1

Bose condensation@μ≳0.95

1.4

2 = 0.01,
FIG. 2. The
number density (n) as a function of µ. The lattice volume is varied fro
FIG. 1. The average of phase factor (Re[Z/Zpq ]) as a function of µ. The parameters
are mparticle
2

2

8

8

to 2sign
× problem
2 . The other parameters (m, λ, K and D) are the same as those of fig. 1.
= 1, K = 64 and D = 64. The lattice volume is varied from 23 × 23 to 28 2× 2×8 .2 Thes harder for larger µ and larger volume.

Z=

⎛

⎞

∞
$

⎜0
⎝

⎟
⎠

4
0! ∞

n 0

drn

5

0

2πrn

2
0

e

2 )− λ (r 4 +r 4 )− 14 (4+m2 )(rn2 +rn+ρ̂
4 n n+ρ̂

Effectsof Phase·Ik (2rnrn+ρ̂)ek µδ δ(k +k −kn kn,1 ,kn,2 =−∞

n

n,ρ

n,ρ

ρ=1

ρ,2

n,1

),0
n−1̂,1 −kn−
Kadoh+,
in2̂,2preparationn,2

Parameters: m2=0.01, λ=1, K=64, Dcut=64
Zpq =

!

Dφ1Dφ2 exp(−Re(S))<n> w/ and w/o phase
3.5

0.45

Full
Phase quenched3

μ=0.904

Full
Phase quenched

0.4

V=LL=10241024

2.5

1/L dependence of <n> @ μ=0.904

1

0.35
0.3
0.25

1.5

n

n

2

0.2
0.15

1

0.1

0.5

4 ≤ L ≤ 1024

0.05

0
-0.5

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.05
0

0.05

0.1

µ

Clear phase effect

0.15

0.2

0.25

1/L

No Silver Blaze phenomena
in phase quenched caseuenched version for m2 = 0.01, λ = 1, K = 64, D = 64 and V = 1024×K
1024.
full= 64. The density for the phase quenched case takes a finite value while
λ = 1,= 64 Theand D

IG. 3. A comparison of the particle number density between the full FIG.
theoryphase
4. andThe thethermodynamic
limit of the particle number density at µ = 0.904 with m2 = 0.

heory result clearly shows the Silver Blaze phenomenon while the phase quenchedcase showsthethe full theory
it converges
to zero in the limit.

$⎢
$⎢
⎣
⎣
n
⎡
n
$⎢
⎣

2

2

4

2 (2 ($$µδρ,2 µδ∗
4∗nρ,2 n+ρ̂

S =S = (4 +(4m+)m
+2|φ
| −
e
) +n||φ+n|2λ|φ+ nλ|φn| −S=n

ρ=1 ρ=12 ($))−µδρ,2
∗
−µδ
∗n ⎥
⎦
φ
φ
+
e
φ
φ
ρ,2
e
φnφn+ρ̂ + e n+ρ̂φn+ρ̂φn ⎥⎦

⎤
)
⎥
n ⎦

Sign-Problem-Free
Representation(4+ m2) + |φn|2 + λ|φn|4 −eµδρ,2 φ∗nφn+ρ̂ + e−µδρ,2 φ∗n+ρ̂φ
ρ=1Endres, PRD75(2007)065012

φ = (φ , φ ) → (r cos θn, rn sin θn)
θn, rn sin θn)Mathematical tools n φn =n,1(φn,1n,2, φn,2) →n (rn cos• Polar coordinate
φn = (φn,1, φn,2) → (rn cos θn, rn sin θn)•∞Character expansion$∞
exp(x cos z) =Ik (x) exp(ikz) x ∈ R, z ∈ C$
exp(x cos z)k=−∞=Ik (x) exp(ikz) x ∈ R, z ∈ C
k=−∞
∞
$

exp(x cos z) =Ik (x) exp(ikz) x ∈ R, z ∈ CPartition function can be expressedink=−∞ a sign-problem-free form⎛⎞
Z(positive)
4
5
! ∞
∞⎞
2
⎛
2 )− λ (r 4 +r 4 )
0
$
0
0
0
− 14 (4+m2 )(rn2 +rn+ρ̂⎜⎟!
45
4 4 n n+ρ̂∞
2 n 1 e 2 2 2
⎠
Z⎜0
=⎝ $
dr
2πr
0n
0
− 4 (4+m )(rn +rn+ρ̂ )− λ4 (rn4 +rn+ρ̂)⎟ 0 ∞ 0nn
⎠ ⎞ 4 n drn= ⎝ ⎛ n kn,1,kn,2=−∞
2πr
e
ρ=1
5
∞
2
1 (4+m2 )(r 2 +r 2 )− λ (r 4 +r 4 )n⎜n 0 !
n 0 ρ=1 0
0 ,kn,2 =−∞
$
kn,1
−
⎟ 0 ∞
n n+ρ̂
kµδ
44 n n+ρ̂n,ρ ρ,2
⎠
Z=⎝
dr
2πr
e
n
n
·I
(2r
r
)e
δ
k
n
n+ρ̂
(k
+k
−k
−k
n,ρ
n,1
n,2
kn,ρnµδρ,2n kn,1 ,kn,2 =−∞n r0n−1̂,1n−2̂,2 ),0ρ=1·Ikn,ρ (2r)eδ
n n+ρ̂
(kn,1 +kn,2 −kn−1̂,1 −kn−2̂,2 ),0kn,ρ µδρ,2
·I
(2r
r
)e
δ(kn,1+kn,2−kn−1̂,1−kn−2̂,2),0
k
n
n+ρ̂
n,ρ
TRG should work for Z(positive)1!

Zpq = forDφ1 Dφ2 exp(−Re(S))
Consistency check btw the resultsZ(original)and Z(positive)

Zpq =

!

Dφ1Dφ2 1exp(−Re(S))

Comparison btw Z(original) and Z(positive)
Kadoh+, in preparation

Parameters: m2=0.01, V=1024x1024, λ=1, K=256, Dcut=64
1.1
1

1

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.5

0.6

naive TRG
TRG w/ sign problem free form

0.9<|φ|2><|φ|2>

0.9

1.1

Dce=5
Dce=9
Dce=17
Dce=33
Dce=129

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0.6

µ

0.7

0.8

0.9

1
µ

1.1

1.2

1.3

1.4

FIG. 10. ⟨|φ| ⟩ as a function of µ at D = 64 on the lattice V = 1024 × 1024. For the tensor netwo
Good convergenceGood agreement in large μ & V(TN) using the Gauss-Hermite quadrature, the truncation order is K = 64. For the case of t
in character expansion character expansion, the truncation⇒ Freeorderfromsign problemof the expansion is set to 128 and that for the rad2

integration is K = 256.

complex field: (φn,1 , φn,2 ) → (rn cos θn , rn sin θn ) and the character expansion:

Summary"+."-"#-*' )• (($#+#'&' +' *$) )%#'&&!,!+"')#*
2 '&*+),+#'&' )( ')*$) #$+"')#*
2 -$'(%&+' )**%&&  ') )%#'&*0*+%*2,** ,$&$0**'  %'$*.'%($/+#'&*• &$0**' "#!")#%&*#'&$%'$*
2 "*+)&*#+#'&'   !,!+"')02 "*+)&*#+#'&'  *#&! %'$/+*+(• &$0*#*'   %'$*
2 %,2'&*#&#' %'$+ #&#+&*#+0
2 '& $#&!,!+"')#*2 )$$$#1+#'&'   #&#*"