Inhomogeneous phases
in 1+1 dimensional chiral Gross-Neveu
model on the lattice
ChudenCTI Co.,Ltd.A,
Hiroshima UniversityB
Kobayashi Maskawa Institute, Nagoya UniversityC,
Department of Physics, Nagoya UniversityD

Keita HorieA and Chiho NonakaB,C,D

Contents✔ Motivation ＆ Goal
✔ Model：(1+1) d chiral Gross-Neveu (GN) model✔ Results：1. analysis on GN model: phase diagram
2. analysis on chiral GN model: inhomogeneous phase✔ New data analysis：K-shape clustering✔ SummaryMotivation: QCD Phase diagram atnite µ1/20✣ Possible interesting phases at high μ
E ective

eld theory

inhomogeneous chiral condensation
Quark-Gluon Plasma

Order parameter of chiral symmetry

spatial dependence of chiral condensate σ※ Space-dependent condensates

space

Speci c Ansatz
ex. constant, chiral density wave

fi

fi

fi

ff

C.NonakaModel: (1+1)d chiral Gross-Neveu model✣ Lagrangian<latexit sha1_base64="d9QzeYXiDut47fYTtx230pMbNFs=">AAADmHichVFNaxNBGH6360etH031InoZDJWKEiaxVRGEgIgKIm1q2kI2CbPjZDNkv5idjcZh/4B/wIMnBQ/iz/Di1YOHnjyLxwpePPjuJn60oe0sM+/HvM+zz7yvG/sy0ZRuWzP2kaPHjs+emDt56vSZ+dLC2Y0kShUXTR75kdpyWSJ8GYqmltoXW7ESLHB9sekO7ub3m0OhEhmFT/QoFu2AeaHsSc40prqlr07AdJ8z3zzKyB3iuEwZJ05kRiRxPBYErGOcMM2IEzOlJfO7f8JEkqvE6SnGjdepZab2GLO+6OnW2CwRR4vnupBo+IiFmfmfviBAo6TX11c6tZxsL6ylPLdtqhV6jVZWcNPdFH8Vrkyzjb026ZbKiCsWmXaqE6cMk7UalT6DA08hAg4pBCAgBI2+DwwS/FpQBQox5tpgMKfQk8W9gAzmEJtilcAKhtkBnh5GrUk2xDjnTAo0x7/4uBUiCSzSL/Q93aGf6Af6jf7al8sUHLmWEVp3jBVxd/7l+fWfh6ICtBr6/1AHatbQg1uFVona4yKTv4KP8cMXr3bWbzcWzWX6ln5H/W/oNv2ILwiHP/i7NdF4fYAeF9/NMBbIuX/nDCpOsUt5lxX26llei0Ot7h3htLNRq1RvVK6vLZfr9cl4Z+EiXIIlnOFNqMMDWIUmcOueNbC0ldoX7Lp93344Lp2xJphzsGvZjd/NMOr2</latexit>

L = ¯i ⌫ @⌫g2 h ¯+2N⇠ h¯ i2+ ¯i 5

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i
2

5
¯
⇡⇠h i
i

<latexit sha1_base64="vUTv2k8SmJbsiO4ciyngmEW6Dwc=">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</latexit><latexit sha1_base64="sVG4SBIL5mYVhiyb1Je4oTRfsAg=">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</latexit>➤ Important features from comparison with QCD
Asymptotic freedom

Spontaneous symmetry breaking of discreet chiral symmetry
No sign problem : Monte Carlo simulation
Imhomogeneous chiral condensate in Nf → ∞@ continuous theory→ 1st step: (1+1)d Gross-Neveu model<latexit sha1_base64="SM5XQDc3DtW8XiO9jvt39sx5rDM=">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</latexit>

L = ¯i ⌫ @⌫

g2
+
2N

¯

2

D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974)

✔︎

✔︎

✔︎

✔︎

C.Nonaka

Phase Diagram of (1+1)d Gross-Neveu model

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O. Schnetz, M. Thies and K. Urlichs, Annals Phys. 314, 425-447 (2004)

Spatial dependent chiral condensate

Without ansatz ?
Lattice Gauge simulation
J. Lenz et. al, Phys. Rev. D 101, no.9, 094512 (2020)

AnsatzC.Nonaka

V. Schon and M. Thies, Phys. Rev. D 62, 096002 (2000)

(1+1)d Gross-Neveu model on the Lattice

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J. Lenz et. al, PRD 101, no.9, 094512 (2020)chiral symmetric phasechiral broken phase

imhomogeneous phase➤ They con rmed the following:

J. Lenz et. al, PRD 101, no.9, 094512 (2020)

Existence of inhomogeneous phase at low T and high μ
Naive fermions and SLAC fermions: same results
lattice spacing dependence
volume dependence:
avor number dependence:
fi

fl

✔︎

✔︎

✔︎

✔︎

✔︎

C.Nonaka

GN modelcalculation

Simulation setup for GN model

5/20➤ Preparation for analysis of chiral GN model on the lattice• First we carry out GN model calculation based on J. Lenz et. al, PRD 101, no.9, 094512 (2020)• We choose naive fermion.
• Hybrid Monte Carlo• Parameters<latexit sha1_base64="dcN3/1bZDY8Zz9Q8lks4ZaQRU8M=">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</latexit>

fermion
naive

Nf
8

Ns = L/a32

Nt = 1/T a
2, 4, 6, . . ., 64

g2
1.9132a 00.4190(1)➤ Investigation of phase diagram of GN model→ inhomogeneous phase• Spatial dependent chiral condensate
X
1• Correlator: C(x) =h (t, y + x) (t, y)i<latexit sha1_base64="SCOk7y4gr7BXexeE8ZRvFGFEwrw=">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</latexit>

Nt Nx t,yAnalysis on chiral GN model
C.Nonakaµ/ 00.0, 0.1, . . .

Phase Diagram of GN model

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1 X
Correlator: C(x) =h (t, y + x) (t, y)i
Nt Nx t,y<latexit sha1_base64="SCOk7y4gr7BXexeE8ZRvFGFEwrw=">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</latexit>

ChiralInhomogeneous

symmetric phase

phase

Homegeneously
chiral broken phase

Nt=24, mu=0.25

C.Nonaka

Nt=04, mu=0.40
Nt=24, mu=0.65

Phase Diagram of GN model

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1 X
C(x) =h (t, y + x) (t, y)i
Nt Nx t,y<latexit sha1_base64="SCOk7y4gr7BXexeE8ZRvFGFEwrw=">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</latexit>

ChiralInhomogeneous

symmetric phase

phase

Homegeneously
chiral broken phase

Nt=04, mu=0.40

Nt=24, mu=0.25

Nt=24, mu=0.65

Classi cation of three phases
Homegeneously chiral broken phase
Chiral symmetric phasefi

C.Nonaka

Inhomogeneous
phase J. Lenz et. al, PRD 101, no.9, 094512 (2020)8

Phase Diagram of GN Model

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Homegeneously chiral broken phase
Chiral symmetric phase
Inhomogeneous phase

Existence of inhomogeneous phase, consistent with previous study
J. Lenz et. al, PRD 101, no.9, 094512 (2020)

C.NonakaChiral Gross-Neveu model<latexit sha1_base64="d9QzeYXiDut47fYTtx230pMbNFs=">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</latexit>⌫¯
L = i @⌫➤ Parameters<latexit sha1_base64="IVtRvQVnCXLdR9UBgB2B8LbHhvU=">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</latexit>

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ig2 h ¯ 2
2
5
¯
+
+ i
2N
⇡ ⇠ h ¯i 5 i⇠ h¯ i<latexit sha1_base64="sVG4SBIL5mYVhiyb1Je4oTRfsAg=">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</latexit><latexit sha1_base64="vUTv2k8SmJbsiO4ciyngmEW6Dwc=">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</latexit>

fermionNf

Ns = L/a

Nt = 1/T a

naive

8

32

4, 6, . . ., 32

g21.9332
1.8132
1.7132a⇤00.4153(3)0.3791(2)0.3436(2)µ/⇤00.0, 0.1, . . ., 0.9
0.0, 0.1, . . ., 0.9
0.0, 0.1, . . ., 0.9
GN➤ Numerical results
1. Analysis on homogeneous phase
The order parameter of discrete chiral symmetry Λ2@ T/Λ0 ≠ 0, μ/Λ0 = 0

2. Analysis on inhomogeneous phase
Using the minimum value of spatial correlation function Cσ(x), Cπ(x)C.Nonaka@ T/Λ0 ≠ 0, μ/Λ0 ≠ 0

1. Analysis on Homogeneous Phase➤T ≠ 0, μ = 0

9/20(T/Λ0, μ/Λ0) = (0.2408, 0.0000)
2
h¯
i
2
⌃ = 2
⇤0
<latexit sha1_base64="eATpu32amTp0mVZrRzbb5d0bBBA=">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</latexit>

2
h¯
⇡
i
2
⇧ = 2
⇤0
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2
2
Π
Σ
and
are uctuating as a function of time trajectory.
•
2
2
Π
Σ
First,
thevaluesofandare almost the same.
•Λ2 = Π2 + Σ2 is• Σ2 increases and Π2 decreases.
• Around 12500 Π2 becomes larger than Σ2.

constant.

fl

C.Nonaka1. Analysis on Homogeneous Phase➤T ≠ 0, μ = 0

11/20(T/Λ0, μ/Λ0) = (0.2408, 0.0000)
2
h¯
i
2
⌃ = 2
⇤0
<latexit sha1_base64="eATpu32amTp0mVZrRzbb5d0bBBA=">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</latexit>

2
h¯
⇡
i
2
⇧ = 2
⇤0
<latexit sha1_base64="xiJP66JrNgIUrZTm69DyqzMYlFg=">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</latexit><latexit sha1_base64="FFOhieEQHxk4P9BCwqH7yh6TH8E=">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</latexit>

22⇤ =⌃ +⇧

2
2
Π
Σ
and
are uctuating as a function of time trajectory.
•
2
2
Π
Σ
First,
thevaluesofandare almost the same.
•Λ2 = Π2 + Σ2 is• Σ2 increases and Π2 decreases.
• Around 12500 Π2 becomes larger than Σ2.

constant.

fl

C.NonakaΛ2 is an order parameter for chiral symmetry.

21. Analysis on homogeneous phase

12/20

2
Λ
Temperaturedependenceofat μ = 0.
➤Λ2 ≠ 0Chiralbroken phase
Chiralsymmetric phase

Chiral GN

Λ2 = 02Λ• is an order parameter for chiral symmetry. ex. GN model

C.Nonakacoupling

lattice1.7132

0.34361.8132
1.93320.37910.4153

1. Analysis on homogeneous phase

13/20

2
Λ
Temperaturedependenceofat μ = 0.
➤Λ2 ≠ 0Chiralbroken phase
Chiralsymmetric phase

Chiral GN

Λ2 = 02Σ2Λ
• is an order parameter for chiral symmetry. ex. GN model

GN

C.Nonakacoupling

lattice1.7132

0.34361.8132
1.93320.37910.4153

1. Analysis on homogeneous phase

14/20

2
Λ
Temperaturedependenceofat μ = 0.
➤Λ2 ≠ 0Chiralbroken phase

coupling

lattice1.7132

0.34361.8132
1.93320.37910.4153Chiralsymmetric phaseΛ2 = 0Chiral GN

2
2
is
an
order
parameterforchiralsymmetry.
ex.
in GN model.
ΛΣ•• No lattice spacing dependence.
• Consistent with mean eld theory analysis.
Restoration of chiral symmetry at high T. No lattice spacing dependence.
fi

C.Nonaka2. Analysis on inhomogeneous phase

15/20

1 X
C (x) =h (t, y + x) (t, y)i
Nt Nx t,y
1 X
C⇡ (x) =h⇡(t, y + x)⇡(t, y)i
Nt Nx t,y

➤ Correlators of σ and π

GN

Nt=24, mu=0.65

NT=20chiral GN(T/Λ0, μ/Λ0) = (0.1204, 0.8886)
Existence of inhomogeneous phase• Cσ and Cπ are uctuating• Amplitude of correlates of chiral GN is smaller than that of GN.

→ The uctuation is decomposed into Cσ and Cπ.

fl

C.NonakaflGN2. Analysis on inhomogeneous phase

NT=20chiral GN(T/Λ0, μ/Λ0) = (0.1204, 0.8886)

NT=24

16/20

chiral GN(T/Λ0, μ/Λ0) = (0.1003, 0.8886)• Existence of inhomogeneous phase at lower T• Amplitude at lower T becomes larger. The tendency is also found in GN model.
• More detailed analysis: correlator of Λ, cross term of σ and π,phase di erence between σ and π

Existence of inhomogeneous phase, no lattice spacing dependence
ff

C.NonakaAnalysis Method for Phases

17/20✣ Characteristic structure in con gurations• Correlators:

Homegeneously chiral broken phase
Chiral symmetric phase
Inhomogeneous phase✣ Is it possible to extract a pattern without any assumption?

• Con gurations of σin Hybrid Monte Carlo algorithmt1t2
Nt<latexit sha1_base64="uGv4xdB9dRaPdqvp5xHqXxK/QSU=">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</latexit><latexit sha1_base64="UW/NclgjJFTcXIJmHRqxnQ7v2SY=">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</latexit><latexit sha1_base64="ZQl0UW6IaDtqeRy0lINibiQzau4=">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</latexit>

Nt<latexit sha1_base64="ZQl0UW6IaDtqeRy0lINibiQzau4=">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</latexit>

Nt

t3<latexit sha1_base64="ZQl0UW6IaDtqeRy0lINibiQzau4=">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</latexit>

Nx

Nx<latexit sha1_base64="GNVznCqsLXrZm8rfCOJe2XbzQP4=">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</latexit><latexit sha1_base64="GNVznCqsLXrZm8rfCOJe2XbzQP4=">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</latexit>fi

C.Nonakafi<latexit sha1_base64="j08aUGUQtAqXzuwBGpTkczmDnM4=">AAACZnichVFNLwNBGH66vuujRYTEpdEQp+ZdGsRJuDiiqk2QZneN2nS/sjttQuMPSFw5OJGIiJ/h4g84+AfEkcTFwdvtJoLgnczMM8+8zzvPzOieZQaS6CGmtLS2tXd0dsW7e3r7Esn+gY3ArfqGyBuu5fpFXQuEZToiL01piaLnC83WLVHQK0uN/UJN+IHpOuty3xPbtlZ2zF3T0CRTOVmaLiXTlKEwUj+BGoE0olhxk1fYwg5cGKjChoADydiChoDbJlQQPOa2UWfOZ2SG+wKHiLO2ylmCMzRmKzyWebUZsQ6vGzWDUG3wKRZ3n5UpjNM9XdML3dENPdH7r7XqYY2Gl32e9aZWeKXE0Uju7V+VzbPE3qfqT88Su5gLvZrs3QuZxi2Mpr52cPqSm18br0/QBT2z/3N6oFu+gVN7NS5XxdoZ4vwB6vfn/gk2pjLqTCa7mk0vLEZf0YlRjGGS33sWC1jGCvJ8bhnHOMFp7FHpU4aU4WaqEos0g/gSSuoDYWGKow==</latexit>

Nx<latexit sha1_base64="GNVznCqsLXrZm8rfCOJe2XbzQP4=">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</latexit>

K-shape: Analysis Method for Phases

18/20✣Clustering: partitioning n observables into k clusters<latexit sha1_base64="MHhbKerGY5i49WtlrthEp+mQnz8=">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</latexit><latexit sha1_base64="Rm4ZWpuMrK1TyYREK946HiT41RM=">AAACZHichVFNSwJBGH7cvswsLQmCICQxOskYUtFJ6tLRj/wAE9ndRltcd5fdVTDpD9S16NCpICL6GV36Ax38A0F0NOjSodd1IUqqd5iZZ555n3eemZEMVbFsxroeYWR0bHzCO+mb8k/PBIKzc3lLb5oyz8m6qptFSbS4qmg8Zyu2youGycWGpPKCVN/p7xda3LQUXduz2wYvN8SaplQVWbSJSmuVYITFmBPhYRB3QQRupPTgLfZxAB0ymmiAQ4NNWIUIi1oJcTAYxJXRIc4kpDj7HMfwkbZJWZwyRGLrNNZoVXJZjdb9mpajlukUlbpJyjCi7IndsR57ZPfshX38Wqvj1Oh7adMsDbTcqAROFrLv/6oaNNs4/FL96dlGFZuOV4W8Gw7Tv4U80LeOLnrZrUy0s8Ku2Sv5v2Jd9kA30Fpv8k2aZy7how+I/3zuYZBfi8XXY4l0IpLcdr/Ci0UsY5XeewNJ7CKFHJ3LcYoznHueBb8QEuYHqYLH1YTwLYSlT+U6ifc=</latexit>• Model-based: a nite combination of component models• Feature-based: picking up characteristic structure with reduced dimensional vectors• Shape-based: similarity search in clusters✣ K-shape clustering algorithm: shaped-based clustering

Paparrizos and Gravano, SIGMODʼ15• Shape-based distance (SBD)

K-shapeσ con gurationst
1
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Nx<latexit sha1_base64="GNVznCqsLXrZm8rfCOJe2XbzQP4=">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</latexit>

1 clusterfi

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C.Nonakat2<latexit sha1_base64="j08aUGUQtAqXzuwBGpTkczmDnM4=">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</latexit>t3

Nt<latexit sha1_base64="ZQl0UW6IaDtqeRy0lINibiQzau4=">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</latexit>

Nx<latexit sha1_base64="GNVznCqsLXrZm8rfCOJe2XbzQP4=">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</latexit>

Nx<latexit sha1_base64="GNVznCqsLXrZm8rfCOJe2XbzQP4=">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</latexit>

K-shape: Analysis Method for Phases

19/20✣ Interesting structureσπPreliminary• Interesting structure is observed.
• More statistics• Larger lattice
Existence of inhomogeneous phase ?
C.NonakaSummary

20/20

Existence of inhomogeneous phasein 1+1 dimensional chiral Gross-Neveu model on the lattice.
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fermionNf

Ns = L/a

Nt = 1/T a

naive

8

32

4, 6, . . ., 32

g21.9332
1.8132
1.7132a⇤00.4153(3)0.3791(2)0.3436(2)µ/⇤00.0, 0.1, . . ., 0.9
0.0, 0.1, . . ., 0.9
0.0, 0.1, . . ., 0.9 Inhomogeneous phase• T ≠ 0, μ = 0: restoration of chiral symmetry at high T .
• T ≠ 0, μ ≠ 0: existence of inhomogeneous phase.
• Correlator of Λ: cross term of Σ and Λ, phase di erence• Investigation of phase diagram of chiral GN• long range correlator

chiral GN• Application of K-shape to nding structures in σ and πcon gurations➤ Future task

fi

ff

fi

fl

• Baryon and thermodynamic quantities• Number of avor, color degrees of freedom• Superconducting term
C.Nonaka