“Tensor renormalization group study of (1+1)-dimensionalO(3) nonlinear sigma model”

Center for Computational Sciences(CCS), Univ. of Tsukuba
Yoshinobu Kuramashi

The 1st TCHoU member meeting in 2024, July 2, 2024
1

Plan of talk

•
•
•
•
•

Introduction to Tensor Renormalization Group(TRG)
Application of TRG to Quantum Field Theories(QFTs)
Entanglement Entropy(EE) of (1+1)d O(3) NLSM at 𝜇 = 0
Quantum Phase Transition of (1+1)d O(3) NLSM at 𝜇 ≠ 0Summary and Outlook

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 }exp (−βH)
Z =
=

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
representation
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 half4

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=230〜50PHYSICAL, 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

TRG vs Monte Carlo
antithetical principles

Monte Carlostochastic

TRG
deterministic

Advantages of TRG
・ Free from sign problem/complex action problem in MC methodZ=!

Dφ exp(−SRe[φ] + iSIm[φ])D⎞systemsize ∝ D×log(L)・ Computational⎛costLx,µdϕ$ ! πfor$⎠Z = ⎝ −πT (ϕx,1, ϕx+1̂,2, ϕx+2̂,1, ϕx,2)・ Direct manipulationof2π
Grassmann
numbersx,µx
・ Direct evaluation of partition function⎛Z (density matrix⎞ ρ) itself

T (ϕx,1, ϕx+1̂,2, ϕx+2̂,1, ϕx,2) = exp ⎝β cos px + i
⎛

θ ⎠
qx
2π

⎞

dϕx,µ ⎠
Z = ⎝ −πexp (−S)x,µApplications in particle physics： 2π
(
(
' √
(
Finite density 'QCD, tQFTs
w/ θ-term,
Lattice
SUSY
etc.
)' √
Ti,j,k,l ⇒ T{j,k},{l,i} = U ΛV {j,k},{l,i} =U Λ {j,k},m V Λ {l,i},m$ ! πAlso, in condensed matter physicsm)= High(S1){j,k},m(S3){l,i},m
Hubbard model (Mott transition,
Tc superconductivity)
etc.
m
'

t

(

)'

√ (

'

√ (

TRG Approaches to QFTs (1)w/ sign problem
2d modelsCP(1) model w/ θ-term：Kawauchi-Takeda, EPJWoC175(2018)11015
O(3) NLSM：Luo-YK, JHEP03(2024)020
Real φ4 theory：
Shimizu, Mod.Phys.Lett.A27(2012)1250035
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP05(2019)184
Complex φ4 theory at finite density：
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP02(2020)161
U(1) gauge theory w/ θ-term：
YK-Yoshimura, JHEP04(2020)089
Schwinger(2d QED), Schwinger w/ θ-term：
Shimizu-YK, PRD90(2014)014508, 074503, PRD97(2018)034502
Gross-Neveu model at finite density：
Takeda-Yoshimura, PTEP2015(2015)043B01
N=1 Wess-Zumino model (SUSY)：
Kadoh-YK-Nakamura-Sakai-Takeda-Yoshimura, JHEP03(2018)141

Application to models w/ sign problem，
Development of calculational methods for scalar, fermion and gauge fields

7

TRG Approaches to QFTs (2)w/ sign problem

3d modelsZ2 gauge Higgs model at finite density：Akiyama-YK,JHEP05(2022)102
Real φ4 theory：Akiyama-YK-Yoshimura, PRD104(2021)034507
Z2gauge theory at finite temperature： YK-Yoshimura, JHEP08(2019)023
4d models
Ising model：Akiyama-YK-Yamashita-Yoshimura, PRD100(2019)054510
Complex φ4 theory at finite density：
Akiyama-Kadoh-YK-Yamashita-Yoshimura, JHEP09(2020)177
NJL model at finite density：
Akiyama-YK-Yamashita-Yoshimura, JHEP01(2021)121
Real φ4 theory：Akiyama-YK-Yoshimura, PRD104(2021)034507
Z2 gauge Higgs model at finite density：Akiyama-YK, JHEP05(2022)102
Z3 gauge Higgs model at finite density：Akiyama-YK, JHEP10(2023)077⇒ Research target is shifting from 2d models to 4d ones

8

supported in part by Grants-in-Aid for Scientific Research
2.68 ≤ μ ≤ 3.00. We obtain μc ðDÞ ¼ 2.698ð1Þ and ν ¼
from the Ministry of Education, Culture, Sports, Science and0.51ð2Þ at D ¼ 80. Our result for the critical exponent isTechnology (MEXT) (No. 20H00148) and JSPS KAKENHI
consistent with the theoretical prediction of ν ¼ 1=2. A
Grant No. JP21J11226 (S. A.).
previous quantum Monte Carlo simulation with small spatial
extension up to L ¼ 24 also yielded the same conclusion [22].
APPENDIX: GRASSMANN TENSOR FOR
In order to extrapolate the result of μc ðDÞ to the limit(d + 1)-DIMENSIONALw/
HUBBARD
MODEL
sign problem
D → ∞, we repeat the calculation changing D. The
numerical results
are summarized
In this Appendix, we consider the tensor network
Condensed
matterphysicsin Table III. In Fig. 7,
we plot μc ðDÞ as a function of 1=D, providing two types ofrepresentation for the path integral of the (d þ 1)-dimenSimilarity
btw
Hubbardmodelsand
NJL
ones
fittings. The solid line shows the fitting result with thesional Hubbard model, whose action is given by

TRG Approaches to QFTs (3)Action consisting of hopping terms and 4-fermi interaction term

"
$
# X
d
X !
ψðn þ τ̂Þ − ψðnÞUϵ ψ̄ðnÞ
ðψ̄ðn þ σ̂ÞψðnÞ þ ψ̄ðnÞψðn þ σ̂ÞÞ þ ðψ̄ðnÞψðnÞÞ2 − μψ̄ðnÞψðnÞ ;
−t
S¼
ϵ
2
n∈Λ
σ¼1
dþ1

ðA1Þ

where n ¼ ððnσ Þσ¼1;…;d ; nτ Þ ∈ Λdþ1 , which denotes the (d þ 1)-dimensional anisotropic lattice. Since the hopping terms in
Eq. (A1) are all diagonal in the internal space, we can immediately have the following decompositions,First principle calculation at finite density(1+1)d Hubbard model：Akiyama-YK, PRD104(2021)014504
014504-5(2+1)d Hubbard model：Akiyama-YK-Yamashita,
PTEP2022(2022)023I01

In this talk we focus on (1+1)d O(3) NLSM
Entanglement Entropy (EE) at 𝜇 = 0
Direct evaluation of partition function Z (density matrix ρ) itself
Quantum phase transition at 𝜇 ≠ 0Free from sign problem/complex action problem
Determination of dynamical critical exponent z

9

as(n
S =
). s(n)(2.1)functio
n ,b·ns(nnd satisfiesD
the periodic
boundary
conditions
+
⌫ˆ+
L)⌫ˆ=
(⌫ = 1, 2).ofThelatticecut
the TRG
method.
The s(n)tensor
network
representation
partition
X
†
† (⌫)
†
n2⇤
'
U
(⌫)V
(2.11)
1+1 ,⌫ (⌫), (2)Ui0 (1)V0 (2),,ba,b,i(2.12)
⇥
V
(1)U
on
S
is
defined
as
Y
n
n
n
n
+ˆ
⌫ n+ˆ
⌫
,a
,b
i
a
,b
,x
an ,bn ,y
n
n
n
n+ˆ
⌫
n+ˆ
,an ,bn X
xn ,an ,bnO(3) nonlineary⌫nsigmann(1+1)-dimensionalmodel
Txn x0n yn yn0 = Tr [T · · · T ] .
i =1
X Z'
he partitionn function Z is EE
given
by
O(3)
NLSM
at1+1
𝜇 =precision
0 hypercubic
0 ···S =of T(1+1)d
s(n)
+
⌫
ˆ
).
(2.1)
s consider
the bondthe
dimension
tensorcontrols
numerical
inn2⇤
partitionoffunctionof, whichtheZ O(3)
NLSM
on
an
isotropic
latticex·0s(nx00 ythe0y0
network representation of the O(3)n2⇤
NLSM
on the site n 2 ⇤Luo-YK,
1+1,⌫
JHEP03(2024)020
1+1
thod.
The
tensor
network
representationofpartitionfunctionisgivenby ⇥ Nt . (2.2).,NvolumeisV= (2L)
The+1 = {(n1 , n2 ) |n1 = 1, . . . , 2L, n2Z==1, . .D[s]et } Swhose,qWe
employthehigherorder
tensorrenormalization
group(HOTRG)
algor
X
⇡
X
Y
tice
spacinga xis0 (1)set toa
10 unless
The O(3) matrix s(n) resides on the sites
partitionfunction
Zyn is
given
=
(1)
(2)=
(2)by 0 necessary.
w0an=
wbTr
xn(1+1)d
yNLSM
n [T
O(3)
free)
nlatticen Tx x(asymptotic
Z'···T] .
(2.13)uateZ.
n n yn y8nZ
nd satisfies
the O(3)periodic
conditionss(n + ⌫ˆL)= s(n)(⌫ = 1, 2). The latticehere
D[s] is the
Haarboundary
measure,whose expression
is givenlater.
an ,bnx0 x00 y0 y00 ··· n2⇤1+1
S
,
(2.2)
Z
=
D[s]e
†is defined as†ion
S
Calculationofentanglement
andRényientropies
0
0
(2)U
(2),
(2.12)⇥ Vxn ,an ,bn (1)Uan ,b2.3(1)V
an ,bn ,yn
n ,xn
yn ,an ,bn
2 higherTensor
networkof themodel algorithm [2] to evalXheorder
tensor representation
renormalization
group(HOTRG)S1 =s(n)· s(nprecision+).
(2.1)entropy
illustrates
thenumericalcalculationof the entanglement
re D[s] isofthe
O(3)TFigureHaarmeasure,whose
expressionis ⌫ˆprocedure
given
later.
mensiontensor,which
controlstheinhe O(3) matrix in the model can ben2⇤
expressed
as
1+1 ,⌫
systemtotwosubsystems
A andB, bothnsor network representation of partition function
is given
by of which have the same lattice s
1
Tensor
networkThe
representation
of
the
model
Tand
of
entanglementRényientropies
densitymatrixofsubsystemAisdefinedby⇢=· T ], where
s
(⌦)
=
(cos
✓,
sin
✓
cos
,
sin
✓
sin
)
eationpartitionfunction
Z
is
given
by
A
X
Y
Z TrB [T · ·(2.3)
0 y y⌦
0 =
Zthe
Tr
[Tof
·the
·, T
(2.13)
xntrace
xmodel
to
We
useWe
HOTRG
tothe
approximate the d
=can
(✓,be
)·expressed✓]entanglement
2. subsystem(0,as⇡], 2 B.
(0,2⇡].
n nrestricted
O(3) matrix
in T
then procedure
tratesthe calculation
entropy.
divideS 1x0 x00 y0 y00 ··· n2⇤1+1, TrB Txx0 y y0 Tx0 xy y0 = My ,y0 .(2.2)
Z
=
D[s]e
subsystem
A,
in
which
⇢
'
The entanAosubsystems
A
and
B,
both
of
which
have
the
same
size
withA LZ
(𝜃, function𝜙) is discretizedw/
B representation
A ⇥ Nt . A A
he partition
andmeasure
are
sTits
(⌦)
=Gauss-Legendre(cos ✓,
sin1written✓ cosquadrature,assin ✓latticesin →)BTN
is is
obtainedby(2.3)
matrixof subsystemAdefinedby⇢A = YTr
where
TrB denotes theZ
er tensor
renormalization
group(HOTRG)
algorithm[2]
toevalB [T · · · T ],Z⌦=(✓, ) whose, ✓ 2 (0,
⇡],
2(⌦
(0,
2⇡]. later.
ere D[s] is the O(3) Haar
measure,expression
is given
s
(⌦
)s
n
i
i
n+ˆ
⌫the
) , density matrix ofed to theEntanglement
subsystem B.Entropy(EE)
We use
HOTRG
e
(2.4)
Z=
D⌦ to approximate
S
=
Tr
⇢
log(⇢
).
A
A A
A
1 and its measure
writtenasn,⌫00, partition
inTensorwhich ⇢function
Txx0 yB yB0 Tx0 xyAare
=
M
.
The
entanglement
entropy
network
of
the
model
A ' Z TrBrepresentation
yis
yA ,y
A divided
Wholesystem(V=2L×Nt)
toAsubsystems A, B (VA,VB=L×Nt)
Z
tanglement
andRényientropies
V
Ycalculation procedure of the nth-order Rényi entropyyY
Figure2
depicts
the1(⌦n )si (⌦Rényi(n-th)n+ˆ⌫) ,e O(3) matrix inmodelD⌦
expressedvontheNeumanne spias(2.4)
Zcan==be D⌦
sin(✓
)d✓
d
.
(2.5)
p
p
lculation procedure of the entanglement
entropy.
We
dividethen4⇡n,⌫).
ln TrA ⇢A(2.14)
SA =
Tr
⇢
log(⇢
p=1
T
A
A
A ✓ cos , sin ✓ sin(n))
s
(⌦)
=
(cos
✓,
sin
,
A and B, both of which have theVsame lattice size withSLA ⇥=
Nt .
(2.3)
1
n
Y 1
1
⌦
=
(✓,
)
,
✓
2
(0,
⇡],
2
(0,
2⇡].
eystemdiscretize
the integration(2.4)
quadrature
is definedby⇢A =D⌦[T· Tsin(✓],Gauss-Legendre
where
the
cts
theAcalculation
procedureofBwiththe· ·thenth-orderRényidefined[11,by20] after changd Bpdenotes.entropy
(2.5)
=
Z Tr
p )d✓
pTr
4⇡
gsystem
integration
B. We
usevariables:
HOTRG
approximate
the density
matrix ofn written
e thepartitionfunctionand itsto
measureasln p=1Tr ⇢areCalculation of EE

where ⇢nA can be calculated by just computing the nth matrix power of ⇢A .

Luo-YK, JHEP03(2024)020

where ⇢nA can be calculated by just computing the nth matrix power of ⇢A .
nwhere ⇢nA can be calculatedbyjust
thejustnthcomputing
matrix power⇢Amatrix.
where ⇢ can be computing
calculated by
the ofnth
power of ⇢A .

von Neumann EE

Awhere ⇢nA can be calculated by just computing the nth matrix power of ⇢A .

(a) Divide the system to two (b) Trace all legs for subsys- (c) Figure 1b is coarse-gra
subsystems A and B.
tem B.
to this with HOTRG.

Figure 1: Calculation of entanglement entropy.

(a) Divide the system to two (b) Trace all legs for subsys- (c) Figure 1b is coarse-grained(a) Divide the systemtwo (b)all tolegs
subsys1bB.
is coarse-grained(a)toDivide
theTracesystem
twofor(b)
Trace
all Figure
legs temforsubsys(c) Figure 1b is coarse-grained
subsystems
Aand
B. (c)
to this with HOTRG.
subsystems A and B.subsystemstem
B.
to
this
with
HOTRG.
A and B.
tem B.
to this with HOTRG.

Coarse-graining
Divided into
Trace out in terms of
1: Calculation of entanglementw/
entropy.
HOTRG(a)subsystems
Divide the system to A
twoand(b)
Trace
all legs
for subsys-of(c)Figure1bFigure
is coarse-grainedBsubsystem
B(Tr)
Figure1:
Calculation
entanglement
entropy.
B
Figure1:
Calculationofentanglement
entropy.
subsystems A and B.
tem B.
to this with HOTRG.

n-th Rényi EE Figure 1: Calculation of entanglement entropy.
(n)(a) Graphical representation of nth-order Rényi entropy ⇢A .

(b) Figure 2a is coarse-grained to this with HOTRG.
(a) Graphical representationof nth-order(n)Rényi
entropy ⇢A .
(n)(a) Graphical representationofnth-orderRényientropy⇢.
A(a) Graphical representation of nth-orderRényi entropy ⇢A .
(n)(a) Graphical representation of nth-order Rényi entropy ⇢A .
Figure 2: Calculation of nth-order Rényi entropy.
(n)

TrB for subsystem B on each sheet ofn- times copied system

Coarse-graining
w/ HOTRG
3 Numerical results

The density matrix ⇢A is evaluated using HOTRG with the bond dimension Dcut(b) Figure 2a is coarse-grained to this with HOTRG.
(b) Figure 2a is coarse-grainedtothiswith HOTRG.to this with HOTRG.
Note that the correlation length ⇠ in this model was precisely measured over(b)toFigure
2aHOTRG.
is coarse-grained(b) Figure 2a is coarse-grained
this with
1.4   1.9 with the interval of= 0.1 in Ref. [29]. We list the values of

1.0

1

2

4

8

32
Nt

16

64

128

256

512

0.0

1024

1

2

4

8

32
Nt

16

64

128

256

512

1024

Spatial sizeofRényi
EE entropy at = 1.5. The bond dimen= Figure
1.5.(L)The
bonddimension ofis 2nd-order
8:dependence
N dependence

ure 3: Nt dependence of entanglement entropy at= 130.

t

Dcut = 130.

Luo-YK, JHEP03(2024)020

2nd Rényi EE

von Neumann EE
3.0

3.0
β=1.4
β=1.5
β=1.6
β=1.7

β=1.4
β=1.5
β=1.6
β=1.7

2.0

SA

SA

(2)

2.0

1.0

1.0

Nt=1024, Dcut=130

Nt=1024, Dcut=130
0.0

2

4

8

16

32

64

128

256

512

1024

L

0.0

2

4

8

32

16

64

128

256

512

1024

L

"

(%)

"

'

#

%

𝑆! ~ ofln(𝜉)(𝑐: central charge, 1024
𝜉: correlation
length)𝑆 ~ (1 + )ln(𝜉)ure 4: L dependence
entanglement entropy with Nt =Figure
at 9: =L 1.4,
1.5, 1.6 anddependenceof 2nd-order!Rényi entropy with N = 1024 at#t= 1.4, 1

and 1.7. The bond dimension is Dcut = 130.
Convergenceat 𝐿 ≫ 𝜉 is confirmed

The bond dimension is Dcut = 130.

β

1.4

1.5

1.6

1.7

ξ

6.90(1)11.09(2)19.07(6)34.57(7)Wolff,
NPB334(1990)581

directly measured entanglement entropy. It seems difficult to obtain the correct value of theentanglement entropy by an extrapolation of the nth order Rényi entropy at n 2. On theother hand, the interpolations of the Rényi entropy at n = 1/2, 2, ..., 4 and n = 1/2, 2, ..., 11with the third and sixth order polynomial functions, respectively, which are denoted by thepink and purple curves in Fig. 12, give consistent results with the entanglement entropy at
Luo-YK, JHEP03(2024)020
n = 1.

von Neumann vs Rényi

Comparison btw von Neumann EE(n=1) and Rényi EE(n≠1)
3.0extrapolate with n=2~5, 3rd order polynomial
extrapolate with n=2~11, 6th order polynominalinterpolate with n=1/2~4, 3rd order polynomialinterpolate with n=1/2~11, 6th order polynomial

2.5

2.0
SA

(n)

β=1.5, L=128, Nt=1024
Dcut→∞

1.5

1.0

0.5

0

1

2

3

4

5

6
n

7

8

9

10

11

12

SA(2) is not a good approximation of von Neumann EE(n=1)
Figure 12: n dependence of nth-order Rényi entropy with Nt = 1024 at = 1.5. Solid(n≥2) toat high
precision values
Difficultto extrapolate
SA(n) entropy.
symbolat n = 1 denotes
the entanglement
All n=1
the results
are extrapolated
at Dcut! 1. interpolation to n=1 using S (1/2) and S (n) (n≥2)
Reliable
A
A
4

Summary and outlook

(2)Figure10:1.5,
1/D
dependenceof SA (L = 128) with Nt = 1024 at
cutandFigure 5: 1/Dcut dependence of SA (L = 128) with Nt = 1024 at= 1.4,
1.6
1.7.
1.7. Solid lines denote linear extrapolation.
Solid lines denote linear extrapolation.

Central Charge= 1.4, 1.5, 1.6 an

Luo-YK, JHEP03(2024)020
3.0

3.0

c

terms of Nt so that Nt = 1024 is large enough to be regarded as the zero temperature limit.
c w/ entanglement entropySA
c w/ Renyi entropy
In Fig. 4 we plot SA (L)= 1.4, 1.5, 1.6 and 1.7. The entanglement(2) with Nt = 1024 atSA
c=2
entropy shows plateau behavior once the interval L goes beyond the correlation
length.
2.5
of Nbehavior
is large enoughregardedas the zero
limit.
This is anterms
expectedunder
the conditionof ⇠ ⌧toLbe[27].
As ⇠ increases
for temperaturelarger ,t so that N
t = 1024
2.0
In of
Fig.
4 westarts
plot atSAlarger(L) with
Nt its= value
1024 isatincreased= 1.4, according
1.5, 1.6 and1.7.theoretical
The entanglement
the plateau
SA (L)
L andto thecentropyshows
behavior
goesatbeyond
theexpectationof SA (L)⇠ 3plateauln ⇠ [27].
In Fig. once
5 we theplotinterval
SA (L =L128)= 1.4,
1.5,correlation
1.6 and length.
2.0
1.7 as a functionof expected
1/Dcut . We
observean increasing
trend
SAL(L[27].
= 128)
vanishingThis is an
behaviorunder
the conditionof of⇠⌧As ⇠forincreases
for larger ,1/Dcut . The
solid lines
the linear
extrapolationof Sis= 128)according
at 1/Dcuttothe0.02
the 1.0
plateauof Sexpressat larger
L and its value
increasedtheoretical
A (L
A (L) starts
to obtain expectation
the valueDS! 1,
are listed
1. SA (L = 128) 1.5⇠ 3cwhichln ⇠ [27].
In Fig.in5Table
we plot
at = 1.4, 1.5, 1.6 and
cut
A (L)
(2) atof
β=1.5,
behavior of SA (L) is observedin the L=128,
large Nt Nt=1024
region so that Nt = 1024 is essentially
The mass
m in theof(1+1)dis expressed as [30]1.7 asgap
a function
1/DcutO(3). We NLSM
observe(L = 128) for vanishing(2) an increasing trend of SA(2)regarded as the zero temperature limit Dcut→∞of SA (L). Figure 9 compares SA (L) at = 1.4,
1/Dcut . The solid lines express the linear extrapolation of SA (L = 128) at 1/Dcut  0.02
8 1024 fixed. Our128⇡
1.5, 1.6 and1.7 withm
N=observationconsistent
with
t = ⇤
1.0 the theoretical
( is
2⇡
),
(3.1) 3
=!
64⇤
0.0
L =
to obtain
the
value
Dcut
1,
which
areexplistedin Table
1.
7
8
9
10
11
12
0
1 (2)2456(2) 1.4 eat MS
1.3
1.7
1.8
1.5
1.6 a
n
expectation that SA (L) should stay constantin the range of L
⇠ according to SA (L) ⇠β
The mass gap m in the (1+1)d O(3) NLSM is expressedas [30](2)c1/n)
ln ⇠ [27].
In Fig. 10
of S
(L =
at equation.
= 1.4, 1.5,
where
the
two-loop
expression
forwe
theshowbetaD
function
at ! 1
isAusedin128)
the last
cut dependence
6 (1 +(2)
8 proportional
128⇡
Since
the correlation
length is inversely
to
the
mass
gap
entanglement
1.6 and1.7. The extrapolated
SA (L = 128)
at D
by the (3.1)
cut( !
exp
2⇡1the) is, obtained
mvalue= ⇤ofMS = 64⇤L =massgap(twoloop):
c=1.97(9)(von charge
NeumannEE)from nth-ordr Rényi (open) an(2)
Figure11:ndependencec obtainedeaFigure6: isfitdependence
entanglement
andRényientropies= 128 withofofScentral
entropy
rewritten
asof inlinearof the data
terms of 1/D
with 1/Dcut
 0.02.
Theat Ldependencecut 2nd-orderA
entanglement
entropies.
Solid different
line denotes cmethods= 2 to guide your eyes.
consistentbtw
Nt = 1024.
at L = where
128 with
is plottedin theFig.beta
6 together
withWeis(closed)
extract
the N
two-loopfunctionat S!
usedin thethe central
last equation.
t = 1024expression
A. 1
c for
SA= (2⇡ln ) + const.
(3.2)
vonSince
Neumannchargec fromthecorrelation
dataEE:in 1.4
1.7inversely
employing
the following
with= 2:by Matrix Product State (MPS)thelengthproportional
to fitthefunction
mass c=2.04(4)gapthen entanglement
3 is
In the Monte Carlo approach it is difficult to calculate the entanglement entrop◆
entropy
is rewritten
as c .✓In Fig.
PRD99(2019)074501
in terms ofthe coupling
constant
1 6 we plot the dependence of SBruckmann+,(n)
A at L = 128
S
1
+
(2⇡
ln
)
+
const.
(3.3)=
A
Rényi EE:
6 centraln ccharge c by fitting the data inc〜2
with Nn-th
the rangeoft = 1024. We determine theby finite
size spectrum w/ TNR
SA = (2⇡ln ) + const.
(3.2)
3
1.4   1.7 with the function of Eq. (3.2), where
the condition of ⇠ ⌧ L is well satisfied.
The value of c = 2.27(16) is slightly larger than that determined from SA . Ueda+, PRE106(2022)014104
centralchargeis1.97(9),determinedfromdependence
We obtainthevalueof c coupling=
which is .consistentwithc =the2.04(14)
obtainedofby
termsof theconstant
Fig. β6 weplot
dependence
SAthe
at L = 128
Weinrepeatthesame calculationfor otherInnth-orderRényientropy.
The n dependence– 11 –MPS method
in
Ref.
[31].
For
an
instructive
purpose
Fig.
7
shows
an
alternative
plot
of
–
7
–
with Ntcharge= 1024.
determine
the11.
centralcharge cthat
by fitting
the ofdata
in the torange ofof the central
c isWeplottedin Fig.
We observe
the valuec seems
SA (L = 128)
withN1.71024theas afunction
functionofofEq.
ln ⇠(3.2),
measuredin Ref.
[29]. Thist =with
1.4to
whereconditionof ⇠isRényi⌧motivated
L isentropy
well satisfied.
convergec = 2 asn increases.
Here we considertheerrortheof thenth-order
by a concern
that
thethe(1+1)dO(3)c NLSM
doeswhich
not haveconsistenta good asymptotic(n)property
Wefromobtainvalue= 1.97(9),withSupposec = scaling
2.04(14)
stemming
the errorsof ofthe eigenvalues
in the isdensityc matrix.
S̄A isobtained
the true by the
below ⇠ 2.0 [29, 32]. The use of the fit function S = ln ⇠ + const. gives the central

latticen spacing
a is alreadyset
a given= 1 unlessTheO(3)matrixs(n)resides
on
the
sites
representationareinthenecessary.
appendixof Ref.
give
relevantsites
and satisfies
thetoperiodic
boundary
conditionss(n[28],+ ⌫ˆweL) briefly=s(n)(⌫ =the1, 2).
TheonS
isdefinedasnatticeand satisfies
periodic
conditionss(n + ⌫ˆL) = s(n) (⌫ = 1, 2). The lattice
expressions
forSthethiswork tomake thispaper
self-contained.
actionis definedasboundaryXactionS
isdefinedas
Quantumphase
0 lattice (2.1)
S = function· s(n +
⌫ˆ). atWe consider the partition
thetransition
O(3)
NLSM
on𝜇an≠(1+1)d⇤1+1 =
3 of s(n)X X
X,⌫ s (⌦volume{(n1 , n2 ) |n1 = 1, . . . ,SL,=n2 = S1, =. . .n2⇤, Nt1+1} whoseis+⌫ˆV)s=(⌦Ln+ˆ
⇥arXiv:2406.08865(µ,),t . The lattice spacing(2.1)
n )D· s(n⌫Ns(n)⌫ˆLuo-YK,).
(2.1)
n2⇤1+1A,⌫ real
, =1three-component unit vector s(n) resides on thea partitionis set to afunction= 1 unlessnecessary.
Z
is
given
by
n2⇤
(1+1)d lattice O(3) NLSM at 𝜇≠01+1 ,⌫
Z
sites
nthe
andspinsatisfies
theperiodic
boundary
conditionss(n + ⌫ˆL) = s(n) (⌫ = 1, 2). The
where
s(⌦)
and
matrix
D(µ,
⌫
ˆ
)
are
expressed
as
The partition function Z is givenZby
S
,
(2.2)
attice action S is defined as 0 = D[s]eZ1cos ✓ 3
X
(2.2)
ZX
= C
D[s]e S ,B
re D[s] is the O(3) Haar
measure,whose
expression
isgiven
later.
, n )D (µ, ⌫ˆ)s (⌦n+ˆ⌫ ),(2.2)
S s(⌦)= = @ sin ✓ cos sA(⌦
(2.1)
n2⇤1+1, =1sin,⌫✓ sin
where
D[s]
is
the
O(3)
Haar
measure,later.
Tensor network representationof theexpression
model is given 1
0 whose1 ⌫ˆ) are expressed aswhere the spin s(⌦) and matrix D(µ,O(3) Tensor
matrix in
the model
can0beBexpressed
as modelC
2.2
network
representation
of
the
12,⌫
complex action
D(µ,⌫ˆ) = @ cosh((2.3)µ) i sinh( 2,⌫ µ) Acos ✓T (⌦) =sinh(
The O(3) matrix in thesmodel
can
as
(cos
sin ✓Ccos, sin ✓cosh(sin )2,⌫ µ)
2,⌫ µ)
B be✓,iexpressed(2.3)s(⌦) = @ sin ✓ cos A ,(2.2)⌦ = T(✓, ) , ✓ 2 (0, ⇡], 2 (0, 2⇡].
withs (⌦)sin= (cos✓ sin✓, sin ✓ cos , sin ✓ sin )
(2.3)
1
(2.4)
partition function and its ⌦measureas= (✓,0 )are,, written
✓✓ 2
(0,
⇡],
2
(0,
2⇡].
2 (0, ⇡], 2 (0, 2⇡].
1
ZB
C the second and third
Y
Note
that
we
introducethechemicalpotential
totherotation
between
D(µ,
⌫
ˆ
)
=
(2.3)
cosh(
µ)
i
sinh(
µ)
s
(⌦
)s
(⌦
@
A
The partition
and Z
its
measuren i asn+ˆ
i quadrature2,⌫⌫ ) , →2,⌫
(𝜃, 𝜙)function
is discretized
w/=Gauss-Legendre
TN representation(2.4)
D⌦ aree written
Z i sinh(components.
cosh(2,⌫ µ)2,⌫ µ)n,⌫ Y
Symmetry btw spatial and temporal
directionsisbrokenby 𝜇si (⌦n )si (⌦n+ˆ
⌫)
e written as ,(2.4)
D⌦ areThe partition function andZitsV=measure
Y⇒ Spatial correlationZlength (𝜉)correlation length (𝜉( )with1 ≠ Temporaln,⌫
P3 p )d✓p d p .
(2.5)
D⌦ = ) Y sin(✓
s
(⌦
)D
(µ,ˆ
⌫
)s
(⌦
)
=
𝜉
with
z
dynamical
criticalexponent
𝜉
n
n+ˆ
⌫
(2.4)
(✓, Y
)4⇡ , ✓ 2(0, ⇡], 2 (0, 2⇡].
, =1
Z =⌦( =D⌦,
(2.5)
p=1 V e 1

f=

LNt

ln Z.

The parameter space of the three-component unit vector is discretized with K = 10
NumberDensitywhich gives much smaller
uncertainties
than Dcut . We evaluate the number density wit
the numerical di↵erentiation of f :Luo-YK, arXiv:2406.08865

number density: hni =@1 ln Z(µ +
f⇡
@µ
LNt

µ) ln Z(µ
2 µ

µ)

.

0.010

In order to discuss the phase transition we introduce ⌘ |µ µc | to measure th
25×225, K=100
β=1.4,
V=2
distance fromthetransition
point
µ
.
The
correlation length in the space direction, whicc0.008
D =125⌫ with the critical exponent ⌫ at the criticality
D =130is denoted by ⇠, should
divergeas
D =1350.006
the second order phase transition. If the system has the symmetry between the spati
and temporal directions, the spatial correlation length should be the same as the temporwith2current[125, 135].
The
temperatureand spatial
extensionby thecut0.004
one⇠t . DThemodel
definedin Eq. (2.1),
however,breaksnormalized
the symmetry
duemassto th
7
6
are given byofTthe/m chemical= 2.1 ⇥ 10potential
and Lm
= 4.9⇠t⇥should
10 , whichis large fromenoughbe relatio
regar
introduction
so thatbe deviated
⇠. to
The
0.002
z ⇠observez⌫ withas the thermodynamic
at zerotemperature.
the zSilver
Blaze phenome
betweentwo correlation limit
lengths
are givenby ⇠t ⇠ ⇠We
the dynamical
criticin Fig. 2: the vanishing number density hni up to µ ⇡ 0.1455 and sudden increase of
exponent.
0.000
beyondAlthoughresults withofDthe= 125, 130
and hni
135 inarethealmostdegenercut number
Figureµc2.0.1440
shows
the theµdependence
densityvicinityof th
0.14700.14450.14500.14550.14600.14650.1475µ= L
25 ⇥ 225
we find slight
apply
fit =
to 33554432
the data ⇥at33554432
Dcut 2 [125,transition
point D
atcut dependence.
= 1.4 on a V We
⇥ Ntthe= 2global
latti1
employing
the functionof hni(µ, Dcut ) = An · {µ (µc + Bn /Dcut )}⌫ with An , µc , Bn
Simultaneous
fit with⌫ the fit parameters. The fit range is chosen to be 0.14575  µ  0.14700. The s⇒ 𝜈 = 0.512 15 , 𝜇* = 0.14512(11)curves show the fit results with An = 0.20(2), µc = 0.14512(11), Bn = 0.068(14)Wolff,⌫=
0.512(15)
at
D
=
125,
130
and
135.
Note
that
the
value
of µc is consistent with
with mass gap m=0.1449(2)𝜇* is consistent cut– 4 – at 𝜇 = 0 NPB334(1990)581
mass gap m = 1/⇠0 = 1/6.90(1) = 0.1449(2) at µ = 0 obtained by a high precision Mo
cut
cut

<n>

cut

sha1_base64="g0joCDWo/zVvqkMnIDowwu2WVaA=">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</latexit><latexit

sha1_base64="ziakfbMbXsGJA3sAaxJatY3EPP8=">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</latexit><latexit

are theandlargestsecondlargesteigenvaluesof the density matrix
where 0 and 1𝜆+isandthe𝜆'largest
theandsecondlargesteigenvalues⇤⇤ the reduced single tensor obtained by HOTRG.
⇢yy0 = Trx Txxyy
0 with T
ex. density matrix on 4×4 lattice
In order to evaluate Z we employ the higher order tensor renormalization group (HOTRG)
algorithm [2].
Temporal Correlation Length (1)The density matrix is defined by ⇢ = Trx [T · · · T ], where Trx denotes the trace in terms
Luo-YK, arXiv:2406.08865
of the spatial direction of the legs of tensors. Figure 1 illustrates
the density matrix in thecase of a 4 ⇥ 4 lattice. The temporal correlation length is obtained by𝜉( is obtained from the eigenvalues of density matrix
Nt
⇣
⌘,
⇠t =
(2.15)
0
ln 1

t

–3–

x

Eigenvalues are calculated on reduced single tensor obtained by HOTRG

Temporal Correlation Length (2)Luo-YK, arXiv:2406.08865

𝜉( as a function of coarse-graining steps near 𝜇*8.0

β=1.4, Dcut=135, K=100

7.0
6.0

ln(ξt)

5.0µ=0.14200µ=0.14225µ=0.14250µ=0.14275µ=0.14300µ=0.14325µ=0.14350µ=0.14375µ=0.14400µ=0.14425µ=0.14450µ=0.14475µ=0.14500

4.0
3.0
2.0
1.0
0.0

0

2

4

6

8

10

12

14

16

18

coarse-graining steps

Plateau behaviors are observed for 𝐿 > 𝜉 at sufficiently low temperature

with Dcut 2 [125, 135]. The temperature and spatial extension normalized by the mass gap
are given by T /m = 2.1 ⇥ 10 7 and Lm = 4.9 ⇥ 106 , which is large enough to be regarded
as the thermodynamic limit at zero temperature. We observe the Silver Blaze phenomenon
in Fig. 2: the vanishing number density hni up to µ ⇡ 0.1455 and sudden increase of hni
beyond µc . Although the results with Dcut = 125, 130 andLuo-YK,
135 arearXiv:2406.08865
almost degenerate,
we find slight Dcut dependence. We apply the global fit to the data at Dcut 2 [125, 135]
𝜇 dependence of 𝜉 near 𝜇*⌫
employing the function 8.0
of hni(µ, Dcut ) = An · {µ (µ(c + Bn /Dcut )} with An , µc , Bn and⌫ the fit parameters. The fit range is chosen to be 0.14575  µ  0.14700. The solid
curves show the fit results β=1.4,
with ADcut=135,µc = 0.14512(11),Bn = 0.068(14) andK=100
Dcut=125n = 0.20(2),⌫ = 0.512(15) at Dcut = 125, 130 and 135. Note that the valueDDcutof=130µ is consistent with the=135 ccut
7.0
mass gap m = 1/⇠0 = 1/6.90(1)= 0.1449(2) at µ = 0 obtained by a high precision Monte
Carlo result with a collective algorithm [31].
The temporal correlation length ⇠t is calculated from the largest and second largest
eigenvalues 0 and 1 , respectively, of the density matrix as already explained at the end
6.0
of Sec. 2. Figure 3 plots ln(⇠t ) as a function of the number of the coarse-graining steps at= 1.4 with Dcut = 135. The temporal correlation length shows plateau behavior once thespatial lattice size becomes larger than ⇠ at the sufficiently low temperature. The plateau
5.0
values increases as µ approaches
the criticalvalue ofµ = 0.14512.
We -5.5
make a fit of the
-7.0
-7.5
-6.5 c-6.0ln|µ−(µcwith+Bξ/Dthe)| density matrix of 16-th coarsetemporal correlation length, which is measured
cutgraining step, andSimultaneous
employ the fitfit
form
of ln ⇠t (µ, Dcut ) = A⇠ + ↵ ln |µ (µc + B⇠ /Dcut )| with
with
µc = 0.14512. The fit parameters A⇠ , B⇠ and ↵ are determined to be A⇠ = 0.030(29),⇒ 𝛼 = 𝑧𝜈 = 1.003 5 , 𝑧 = 1.96(6)B⇠ = 0.0599(9) and ↵ = 1.003(5). In Fig. 4 we plot ln ⇠t (µ, Dcut ) as a function of ln |µ(µc + B⇠ /DcutThe)| with135]. The solid
curve represents
the fit result
withthecut 2 [125,calculation
first D
successfulof dynamical
critical exponent
withTRG
choice of Dcut = 1, which shows fairly linear behavior. Using the relation of ↵ = z⌫ withln(ξt)

Temporal Correlation Length (3)Summary and Outlook
Advantages of TRG
・ Free from sign problem/complex action problem in MC methodZ=!

Dφ exp(−SRe[φ] + iSIm[φ])D⎞systemsize ∝ D×log(L)・ Computational⎛costLx,µdϕ$ ! πfor$⎠Z = ⎝ −πT (ϕx,1, ϕx+1̂,2, ϕx+2̂,1, ϕx,2)・ Direct manipulationof2π
Grassmann
numbersx,µx
・ Direct evaluation of partition function⎛Z itself⎞

T (ϕx,1, ϕx+1̂,2, ϕx+2̂,1, ϕx,2) = exp ⎝β cos px + i
⎛

θ ⎠
qx
2π

⎞

dϕx,µ ⎠
Z = ⎝ −πexp (−S)x,µApplications in particle physics： 2π
(
(
' √
(
Finite density 'QCD, tQFTs
w/ θ-term,
Lattice
SUSY
etc.
)' √
Ti,j,k,l ⇒ T{j,k},{l,i} = U ΛV {j,k},{l,i} =U Λ {j,k},m V Λ {l,i},m$ ! πAlso, in condensed matter physicsm)= High(S1){j,k},m(S3){l,i},m
Hubbard model (Mott transition,
Tc superconductivity)
etc.
m
√ (

√ (

)
Current
status：Calculationsoft 4d theories(scalar，fermion，gauge)
are possible
Ti,j,k,l⇒ T{k,l},{i,j} = U ΛV
=
U
Λ
V
Λ
{k,l},{i,j}{k,l},m{i,j},mm)⇒ Extension to Abelian (U(1)) and non-Abeliangroups(SU(2),
SU(3))=(S2){k,l},m (S4){i,j},mm
Non-perturbative study of Entanglement Entropy'(new)
To,n,m,p=)i,j,k,l(''(S4){l,k},o (S3){k,j},n (S2){j,i},m (S1){i,l},p