Overview ofthe conventional Gross-Neveu phase diagram
and recent noteworthy reports (Review)

Sakura Itatani, M1 fromOK A ENDAI
Sep. 20th, SSI2023 @Hiroshima UniversitySelf-introduction and today’s talk
Name:

Sakura Itatani (板⾕さくら)

Mar., 2023 : bachelor’s degree (Hiroshima U.)• supervisor
Ken-Ichi Ishikawa san• graduate thesis ⊂ today’s talk“A review of chiral phase transitionin the 1+1 dimensional Gross-Neveu model”

Apr., 2023~ : M1 in SOKENDAI (KEK)Contents
I. The 1+1 dimensional Gross-Neveu model (~ 10 min.)- Introduction- Scheme for the GN phase diagram- Calculation results of the phase diagram
II. Phase structures in higher dimensional theories (~ 5 min.)- Probes for the phase structures

The theory of the strong interaction
Quantum Chromodynamics (QCD)

The early universe
Hadronic physics
Neutron strars
etc.

low density• Nonperturbative• Lattice QCD
finite density
Lattice QCD (sign problem)→ toy models
Ref. [1]••Nambu-Jona-Lasinio (1961)
Gross-Neveu(1974)QCD v.s. the Gross-Neveu model• QCD Lagrangian• 1+1 dim. Gross-Neveu model Ref.[2](Minkowski spacetime)(Euclidean spacetime)

1 & $%&$#ℒ!"# = 𝜓 𝒾𝛾 𝐷$ − 𝑀 𝜓 − 𝐺$% 𝐺4𝑔)# )ℒ'( = 𝜓# 𝒾𝛾$ 𝜕$ 𝜓 +𝜓𝜓2𝑁*

𝐷$ = 𝜕$ − 𝒾𝑔𝐴$& 𝑇 &• 𝑆𝑈(3) gauge symmetry• Asymptotic freedom• continuous chiral symmetry (𝑀 → 0)

• 𝑆𝑈(3) gauge symmetry
ü Asymptotic freedom# +)ü 𝑍) symmetry (𝑍): 𝜓 → 𝒾𝛾 +𝜓, 𝜓# → 𝒾𝜓𝛾

We can draw the phase diagram of 1+1 dim. Gross-Neveu modelby observing SSB of chiral 𝒁𝟐 symmetry.

Scheme for the GN phase diagram Ref.[3]1. order parameter=𝜓𝜓(𝑥)=1==' 𝒟𝜓𝒟𝜓𝑒 &(#$ 𝜓𝜓𝑍

2. introduce an auxiliary field 𝜎𝑆)

𝑆*' = ' d# 𝑥 ℒ*'

not integrable# ) ⟹ notℒ'( includes 𝜓𝜓integrable#over 𝒟𝜓𝒟𝜓

3. Ward-Takahashi identity𝑁"=𝜓𝜓(𝑥) = 𝒾 # 𝜎(𝑥)𝑔• translational invariance of PI measures
𝒟𝜎 = 𝒟(𝜎 + 𝑎)
new order parameter: 𝜎(𝑥)= ' d# 𝑥

𝑁" #ℒ*' + # 𝜎
2𝑔→ ' d# 𝑥= 𝛾$ 𝜕$ + 𝜎 𝜓 +𝜓𝒾𝑁" #𝜎2𝑔#physics does not change: (𝑐𝑜𝑛𝑠𝑡. )×𝑍

4. effective action
𝑆!"" =1' d# 𝑥 𝜎 # − log det (𝛾$ 𝜕$ + 𝜎 + 𝜇𝛾% )#2𝑔
1
𝜎 = ' 𝒟𝜎 𝑒 &'! ("!! 𝜎𝑍

search forsaddle point approx.
⟹ 𝐦𝐢𝐧 𝑺𝒆𝒇𝒇
𝑁* → ∞𝝈, 𝜷, 𝝁Result 1 : 𝝈(𝒙) = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕[Matplotlib]

Spatially varying order parameter Ref.[3,4]• solutions of the gap eq.

• allow 𝜎 to depend on 𝑥:⃗𝜎 → 𝜎 𝑥⃗

at high densities:

only 𝜎S±: ≠ 0𝑆2** 𝜎, 𝛽, 𝜇 → 𝑆2** 𝜎(𝑥),⃗ 𝛽, 𝜇at low densities:

𝜎S±: dominant• self-consistency eq. (gap eq.)
𝛿𝑆2**=0𝛿𝜎(𝑥)⃗assume𝝈 𝒙

𝜎 𝑥 =𝜎 𝑥+𝑎 ;𝑎 = 𝐿/𝑁𝜇Fourier expansion
𝜎 𝑥 = R 𝜎S33∈ℤ𝑒 𝒾)7! 8 ;low𝑞3 = 𝜋𝑚/𝑎high𝒙[Matplotlib]

ignore higher-order Fourier coefficients⟹ 𝜎 𝑥 ∝ sin 2𝑞:𝑥

Result 2 : 𝝈(𝒙) ≠ 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕symmetric phase (SP)homogeneous
phase (HBP)

inhomogeneous
phase (IP)[Matplotlib]

Result 2 : 𝝈(𝒙) ≠ 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕symmetric phase (SP)homogeneous
phase (HBP)

What about higher
dimensional theories?

inhomogeneousphase (IP)[Matplotlib]

Indirect detection of IP Ref.[5]Stability analysis of HBP against inhomogeneous perturbationin 𝑑 + 1-dim. GN modelinstable𝛤 (#) < 0• effective action=:
𝑆2** = " ∫< d𝜏 ∫ d> 𝑥 𝜎 ) − log det(𝛾$ 𝜕$ + 𝜎 + 𝜇𝛾<));• add an inhomogeneous perturbation
𝜎(𝑐𝑜𝑛𝑠𝑡. ) → 𝜎 𝑥⃗ = 𝜎(𝑐𝑜𝑛𝑠𝑡.
#) + 𝛿𝜎 𝑥⃗ ,• (Taylor) expansion of 𝑆@AA(B)(C)(D)
𝑆@AA = 𝑆@AA + 𝑆@AA + 𝑆@AA + ⋯ ,𝑆2**stable𝛤 (#) > 0

j + 𝛿𝜎𝐷→𝐷𝐷 = 𝛾$ 𝜕$ + 𝜎 + 𝜇𝛾<
E G HI (D)(D)
𝑆@AA = ∫FG 𝛤 𝛿 𝜎(𝑞)- DDDJ“curvature” of 𝑆2**

Results of stability analysis (𝒅 = 𝟏) Ref.[5]curvature 𝛤 ()) (𝜎,# 𝜇, 𝑇, 𝑞)< 0 ⇔ instable HBP : imply IP> 0 ⇔ stable HBP or SP𝜇 = 𝜇"B ≈ 0.75

𝜞(𝟐)𝒒IP appears at very low 𝑇

Ref.[6]

𝑇 = 0.15

𝜞(𝟐)𝒒
IP appears when 𝜇 ~ 0.8

Ref.[6]

IP in higher dimensions (𝑻 = 𝟎) Ref.[6]curvature 𝛤 ()) (𝜎,# 𝜇, 𝑇, 𝑞)< 0 ⇔ instable HBP : imply IP> 0 ⇔ stable HBP or SP𝝁 = 𝝁𝑪 𝒅 B

𝜞(𝟐)𝒒Ref.[6]𝑑 < 2.0 : IP exists at higher densities

𝝁 = 𝝁𝑪 𝒅 B

𝜞(𝟐)𝒒𝑑 ≥ 2.0 : IP does not exist

Ref.[6]

Summary and Prospect𝑑 + 1-dim. Gross-Neveu model(toy model of QCD)• SSB of chiral 𝑍) symmetry• HBP for all spatial dimensions 𝑑• IP at high densities (𝑑 ≤ 2)
“Moat regime”?
• “quantum pion liquid (Q𝜋L) phase”
in QCD?

𝒅𝝁Ref.[6]phase structure of the GN model at 𝑇 = 0References[1] W. Weise, Nuclear Chiral Dynamics and Phases of QCD, Progress in Particle and Nuclear
Physics 67, 299 (2012).
[2] D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field
Theories, Physical Review D 10, 3235 (1974).
[3] J. Lenz et al. Inhomogeneous Phases in the Gross-Neveu Model in 1+1 Dimensions at
Finite Number of Flavors, Physical Review D 101, (2020).
[4] M. Thies and K. Urlichs, Revised Phase Diagram of the Gross-Neveu Model, PhysicalReview D 67, (2003).
[5] A. Koenigstein et al. Detecting Inhomogeneous Chiral Condensation from the Bosonic
Two-Point Function in the (1 + 1)-Dimensional Gross–Neveu Model in the Mean-Field
Approximation*, Journal of Physics A: Mathematical and Theoretical 55, 375402 (2022).
[6] L. Pannullo, Inhomogeneous Condensation in the Gross-Neveu Model in Noninteger
Spatial Dimensions 1 ≤ D < 3, Physical Review D 108, (2023).
[7] B. Cowan, Topics in Statistical Mechanics (Second Edition) (World Scientific, 2021).

Thank you for your attention.
Questions in English or Japanese
are welcome!

Introducing the auxiliary 𝝈
𝑍 { 1 = 𝑍 { | 𝒟𝜎 exp− | 𝑑)𝑥𝑁* )𝜎)2𝑔

Gauss integration
with a normalization factor)
𝑁* )𝑔)<)## 𝛾$ 𝜕$ + 𝜇𝛾 𝜓 +#= | 𝒟𝜓𝒟𝜓𝒟𝜎exp − | 𝑑 𝑥 𝜓𝒾𝜓𝜓+ )𝜎2𝑁*2𝑔## 𝛾$ 𝜕$ + 𝜇𝛾 < + 𝜎 𝜓 += | 𝒟𝜓𝒟𝜓𝒟𝜎
exp − | 𝑑)𝑥 𝜓𝒾𝜎 → 𝜎- = 𝜎 −𝒾𝑔#𝑁"=# 𝜓𝜓𝑁* )𝜎)2𝑔# 𝜓bilinear form w.r.t. 𝜓,#→ 𝒟𝜓𝒟𝜓can be performed

Result 1 : effective potential 𝑼𝒆𝒇𝒇 = 𝑺𝒆𝒇𝒇 /𝜷𝑳at very low 𝑇𝑺𝒆𝒇𝒇𝜷𝑳𝑵𝒇

𝑺𝒆𝒇𝒇𝜷𝑳𝑵𝒇at very low 𝜇

𝑻 = 𝑻𝑪𝝁 = 𝝁𝑪𝝈[Python]

𝑻 = 𝑻𝑪𝝈[Python]minimum jumps: 𝜎 = ±1 → 0

minimum gradually shifts: 𝜎 = ±1 → 0

1st order phase transition

2nd order phase transition

Mermin-Wagner theorem Ref.[5]For theories in 𝑑 dimensions,• 𝑑≥3Both continuous and discrete symmetries can be broken spontaneously.

• 𝑑=2
Only discrete symmetries can be broken spontaneously.
ex.) the 2d Ising model• 𝑑=1
Any symmetries cannot be broken spontaneously.

Result 1 : 𝝈(𝒙) = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕Mermin-Wagner theoremRef. [7]in 2D theories:
SSB of discrete sym.
SSB of continuous sym.
𝑍) symmetry: discrete# +𝜓 → 𝒾𝛾 +𝜓,𝜓# → 𝒾𝜓𝛾⇓phase transition is allowed[Python]Result 2 : 𝝈(𝒙) ≠ 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕Mermin-Wagner theorem

symmetric phase (SP)

Ref. [7]in 2D theories:
SSB of discrete sym.
SSB of continuous sym.

homogeneous
phase (HBP)SSB of translational sym.?
(continuous)
Large 𝑁* → SSB is allowed
so far: my graduate thesis

What about higher
dimensional theories?

inhomogeneousphase (IP)[Python]𝒅 + 𝟏-dim. Gross-Neveu model Ref.[6]𝑑 + 1-dim. Gross-Neveu effective action (Euclidean spacetime)=1𝑆2** = ) | d𝜏 | d> 𝑥 𝜎 ) − log det(𝛾$ 𝜕$ + 𝜎 + 𝜇𝛾<)
2𝑔 <j 𝒆𝒇𝒇𝑼HBP – SP transition
𝑇, 𝜇 = (0, 𝜇" (𝑑))dimensional
renormalization+assume𝜎: 𝑥-independent

renormalized effective action (𝑇 = 0)̅
8@AA [𝜎, 𝜇, 𝑑]
𝑆@AA∝𝑈

2ndorder1storder𝒅𝝈Ref.[6]

Existence of the moat regime (𝑻 = 𝟎) Ref.[6]1 𝑑)𝛤 ())
𝑍=2 𝑑𝑞)< 0 ⇔ moat regime> 0 ⇔ not moat regime
𝒁𝒁𝝁𝑑 ≤ 2.0

Ref.[6]𝝁𝑑 ≥ 2.0

Ref.[6]