形状ベースクラスタリング手法を用いた 有限密度格
子 Gross-Neveu 模型の相分類
ChudenCTI Co.,Ltd.A,
Graduate School of Advanced Science and Engineering, Hiroshima UniversityB,
International Institute for Sustainability with Knotted
Chiral Metar Matter/SKCM2,
CHiroshima University ,Kobayashi Maskawa Institute, Nagoya UniversityD

Keita HORIEA and Chiho NONAKAB,C,DCONTENTS❖ Motivation & Goal
❖ Method : Shape-based Clustering Method❖ Results : 1. Configuration Centroid
2. Analysis on GN model: inhomogeneous phase❖ Summary1

Motivation: Understanding of QCD Phase Diagram● QCD Phase Diagram
TPhases
Phase boundary•

At low densityLattice QCD
High-energy heavy-ion collisionsµNew method for classifying phases
Shape-based classifying method
Configurations of 1+1 d GN model on the lattice
Inhomogeneous phase,
homogeneously broken phase, chiral symmetric phase•

At high density
Interesting phases:
Inhomogeneous phase
X Lattice QCD
experiment ?

2

Motivation: (1+1)-dimensional Gross-Neveu Model❖ Lagrangian density
D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974)

Important features from comparison with QCD❏ Asymptotic freedom❏ Spontaneous symmetry breaking of discrete chiral symmetry❏ No sign problem : Monte Carlo simulation❏ Inhomogeneous chiral condensate in large Nf limit
3Motivation: (1+1)-dimensional GN Model @ Continuous Theory❖ Lagrangian density in the continuous theory
in the large Nf limit❖ Specific ansatz & phase diagram

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

To calculation without ansatz, we use lattice field theory
J. Lenz et. al, Phys. Rev. D 101, no.9, 094512 (2020)4

Motivation: (1+1)-dimensional GN Model @ Lattice Theory❖ Spatial correlators in three phases
Chiral Symmetric Phase

HomogeneouslyBroken Phase<latexit sha1_base64="SCOk7y4gr7BXexeE8ZRvFGFEwrw=">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</latexit>

C(x) =1 X
h (t, y + x) (t, y)i
Nt Nx t,yInhomogeneous Phase❖ Classification by minimum of the spatial correlator

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

Knowledge of Ansatz is used in the interpretation of the results.
<latexit sha1_base64="u6sxAvyg/O00w7pprhKCaNW4Jdo=">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</latexit>

Can we directly extract the spatial dependence of the configurations

(x) ?
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<latexit sha1_base64="u6sxAvyg/O00w7pprhKCaNW4Jdo=">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</latexit>(x)

Method: Shape-based Clustering Method❖ Difficulty in direct extraction of the spatial dependence
The configurations shifted at each Monte Carlo step

When we generate enough number of configurations,the expectation value at each point becomes zero.
spatial correlators

Is there a more general and direct method?

Ns

We need shift-invariant clustering method that focuses on the shape of configurations.
Time Series Clustering Method
6Method: Shape-based Clustering Method❖ Clustering Method❏ Unsupervised Learning
A method for grouping data without labeled training data
❏ It is important to choose a “similarity” that represents how similar two
data points are.

The choice of similarity can lead todifferences in the data that can be grouped.
Example:Moon-shaped data are well-suited fordensity-based similarity calculations such as
DBSCAN.
https://scikit-learn.org/stable/modules/clustering.html

7解析手法：形状ベースクラスタリング手法の応用❖ K-Shape法➢ 教師なし学習手法の一種➢ 時系列データのクラスタリング手法➢ シフト不変性とスケール不変性の特徴を持ったアルゴリズムを使用
J. Paparrizos, L. Gravano, PROC. ACM SIGMOD Int. Conf. Manage. Data, pp. 1855-1870, 2015

❖ K-Shape法の特徴➢ DTW法よりも高速※DTW法もシフト不変性とスケール不変性を持つ➢ インプット ：時系列データ, クラスタ分割数, etc…
アウトプット：クラスタリング結果, クラスタ重心
右図の赤い実線がクラスタ重心（クラスタ代表値）
R. Tavenard et. al., JMLR, 21, no.118 (2020)
http://jmlr.org/papers/v21/20-091.html

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Method: Shape-based Clustering Method❖ Time Series Clustering Method❏ One of the Clustering methods❏ Clustering methods for time series (1-dimensional data):❏ Major methods:
Dynamic Time Wraping, K-Shape, k-means, etc…
❖ Simple Example: k-means method with Euclidean distance“Similarity”:
Note：k-means method is not well suited for time series data because thesimilarity decreases when the phase is shifted.

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Method: Shape-based Clustering Method❖ K-Shape Method
This method is characterized by shift-invariance and scale-invariance❖ Shape-based Distance (SBD)：the similarity with shift-invariance
CCF：cross-correlation function
ACF：auto-correlation function
J. Paparrizos, L. Gravano, PROC. ACM SIGMOD Int. Conf. Manage. Data, pp. 1855-1870, 2015

SBD evaluates similarity
ignoring the phase shift.

10Method: Shape-based Clustering Method❖ Algorithm - Refinement step

1.

INPUT
X is an n-by-m matrix containing n time series oflength m that are initially z-normalized.
k is the number of clusters to produce.

Shift each data to overlap with the centroidwith minimized SBD
cluster-1
shift

while cluster labels don’t change or iter < max|| Refinement step|
| Assignment step
LOUTPUTIDX is as n-by-1 vector containing the assignment ofn time series to k clusters (initialized randomly).
C is a k-by-m matrix containing k centroids oflength m (initialized as vectors with all zeros).

2.

zeropadding

Calculate the optimal centroid for each data inthe cluster (Time Series Shape Extraction)cluster-1

Time Series
Shape Extraction

11Method: Shape-based Clustering Method❖ Algorithm - Assignment step

1.

INPUT
X is an n-by-m matrix containing n time series oflength m that are initially z-normalized.
k is the number of clusters to produce.

Calculate the similarity (SBD) between eachdata and the centroid
datacalculation SBD
while cluster labels don’t change or iter < max|| Refinement step|
| Assignment step
LOUTPUTIDX is as n-by-1 vector containing the assignment of
n time series to k clusters (initialized randomly).
C is a k-by-m matrix containing k centroids oflength m (initialized as vectors with all zeros).

centroid2.

Assign the centroid with the maximum similarity
to each datacluster-1

cluster-2

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Method: Shape-based Clustering Method❖ Modify K-Shape for lattice simulation
The Original K-Shape：we peform zero padding in the refinement step.

The Modified K-Shape：we impose periodic boundary condition in the refinement step.

13Results❖ Lattice Simulation Setup

We use a standard hybrid Monte Carlo algorithm.
Lattice discretization of fermions is a naive fermion.
To set the scale, we use the expectation value at zero temperature and zero chemical potential.
We use the same coupling constant as the previous study [J. Lenz et. al, PRD 101, no.9, 094512(2020)].
➢ The other simulation parameters are described in the table below:➢➢➢➢❖ Preprocessing

In the equilibrium, configurations do not have time dependence.
Therefore, we calculate the average of configurations along the time axis.
14Results: Shape-based Clustering Method❖ Extraction the spatial dependence of the configurations

inhomogeneous phase➢ Set the cluster number k = 1

homogeneously
broken phase

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

We can extract the spatial dependence without the ansatz!

15Summary:❖ Summary➢ We applied a shape-based clustering method, a type of unsupervised learning, to the analysis of latticeconfigurations.
➢ We modified the method to make it suitable for lattice calculations, including periodic boundary condition.

➢ We succeeded in extracting the spatial
dependence without the knowledge of ansatz.

❖ Future Work➢ We will apply this method to the configurations to classify the phases of GN model.
➢ We will use this method to other models with spatial dependent phases.

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