2404/2404.00965.md

Title: Inhomogeneous confinement and chiral symmetry breaking induced by imaginary angular velocity

URL Source: https://arxiv.org/html/2404.00965

Markdown Content:

Abstract

We investigate detailed properties of imaginary rotating matter with gluons and quarks at high temperature. Previously, we showed that imaginary rotation induces perturbative confinement of gluons at the rotation center. We perturbatively calculate the Polyakov loop potential and find inhomogeneous confinement above a certain threshold of imaginary angular velocity. We also evaluate the quark contribution to the Polyakov loop potential and confirm that spontaneous chiral symmetry breaking occurs in the perturbatively confined phase.

keywords:

Confinement, Rotation, Chiral Symmetry Breaking, Inhomogeneous States

††journal: Physics Letters B

\affiliation

[1]organization=School of Physics and Astronomy, University of Minnesota, addressline=Minneapolis, city=MN, postcode=55455, country=USA

\affiliation

[2]organization=Department of Physics, The University of Tokyo, addressline=7-3-1 Hongo, Bunkyo-ku, city=Tokyo, postcode=113-0033, country=Japan

1 Introduction

The physical mechanism and the interplay of confinement and chiral symmetry breaking are long-standing puzzles in Quantum Chromodynamics (QCD) that is a fundamental theory in terms of quarks and gluons. At finite temperature, if the quark mass is infinitely large to quench dynamical quarks, confinement and deconfinement of gluons are classified by center symmetry. Dynamical quarks explicitly break center symmetry[1], and the order parameter for center symmetry, i.e., the Polyakov loop is only an approximate measure of confinement. In the limit of zero quark mass, chiral symmetry is exact and the chiral condensate characterizes the state of matter. Various phases of QCD have been considered as functions of external environmental parameters such as the temperature T 𝑇 T italic_T, the quark chemical potential μ 𝜇\mu italic_μ, the isospin chemical potential[2, 3], the electric and magnetic fields[4, 5, 6], and the rotation angular velocity ω 𝜔\omega italic_ω[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], and their mixture[30, 31, 32]. In this way, the QCD phase diagram has been intensively studied in theoretical and experimental contexts[33].

Among various parameters, the angular velocity ω 𝜔\omega italic_ω has attracted much attention in recent years. An extraordinary value of ω∼10 22⁢s−1 similar-to 𝜔 superscript 10 22 superscript s 1\omega\sim 10^{22}\ \mathrm{s}^{-1}italic_ω ∼ 10 start_POSTSUPERSCRIPT 22 end_POSTSUPERSCRIPT roman_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT is estimated based on the data in the heavy-ion collision[34]. In nonrelativistic theories rotation effects are similar to those induced by magnetic fields, while relativistic rotation of quark matter has features analogous to finite density[30, 8, 9, 35]. In the same way as chiral symmetry restoration at high density, chiral effective models such as the (Polyakov–)Nambu–Jona-Lasino model exhibit the chiral phase transition induced by rotation[7, 8, 9, 10, 11]. Interestingly, rotation directly affects gluons, which makes a contrast to effects of finite density and magnetic fields, and the angular velocity turns out to be a useful probe to confinement and deconfinement. For the purpose to investigate a chance of (de)confinement caused by rotation, the hadron resonance gas model and the holographic QCD model have been adopted[11, 12, 13, 14, 15, 16], which predicted that real rotation favors deconfinement. Thus, it is expected that the deconfinement critical temperature should decrease for larger ω 𝜔\omega italic_ω.

The numerical results from lattice-QCD simulations[18, 19, 20] have invoked controversies reporting a trend opposite to the model predictions. That is, the Polyakov loop decreases as a result of real rotation, so that the deconfinement critical temperature should increase. A technical subtlety lies in the treatment of analytical continuation; the sign problem can be evaded with imaginary angular velocity, Ω I subscript Ω I\Omega_{\text{I}}roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT, and then the critical temperature is inferred from T c⁢(ω 2)=T c⁢(−Ω I 2)subscript 𝑇 c superscript 𝜔 2 subscript 𝑇 c superscript subscript Ω I 2 T_{\text{c}}(\omega^{2})=T_{\text{c}}(-\Omega_{\text{I}}^{2})italic_T start_POSTSUBSCRIPT c end_POSTSUBSCRIPT ( italic_ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_T start_POSTSUBSCRIPT c end_POSTSUBSCRIPT ( - roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) if the Wick rotation does not hit singularities.

The subtle connection between real and imaginary rotation has been studied recently[10, 20, 23, 24, 28, 29] but the validity condition for analytical continuation is only partially clarified for the moment. There are various scenarios; some support the lattice results[21, 22], some propose a modified model which can explain the lattice results[23], some suggest a non-monotonic function of T c⁢(ω)subscript 𝑇 c 𝜔 T_{\text{c}}(\omega)italic_T start_POSTSUBSCRIPT c end_POSTSUBSCRIPT ( italic_ω )[24, 25].

Previously in Ref.[36], we exploited the perturbative Polyakov loop potential to investigate rotation effects based on the first-principles approach. Thanks to the asymptotic freedom, QCD or the pure gluonic theory can be perturbatively analyzed if the temperature is sufficiently high. Without rotation, the perturbative Polyakov loop potential, known as the GPY-Weiss potential[37, 38, 39, 40, 41], spontaneously breaks center symmetry, which is consistent with deconfinement expected at high temperature; for a review, see Ref.[42]. We obtained the Polyakov loop potential in the rotating pure gluonic system and found that the Wick rotation, Ω I=−i⁢ω subscript Ω I i 𝜔\Omega_{\text{I}}=-\mathrm{i}\omega roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = - roman_i italic_ω, is hindered by singularity which corresponds to the causality violation. To cure this obstacle, the system must have a finite boundary, and then a closed analytical expression is no longer available.

Interestingly enough, the theory with imaginary angular velocity contains rich physics. In our previous work[36], we discovered exotic realization of confinement even at arbitrarily high temperature, i.e., the perturbatively confined phase. Thus, Ω I subscript Ω I\Omega_{\text{I}}roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT is a novel theoretical device to tackle the confinement mechanism. In the present work, we will discuss two nontrivial extensions from our previous study. One is exploring inhomogeneous structures away from the imaginary rotation center. In Ref.[36], we focused on r=0 𝑟 0 r=0 italic_r = 0 only (where r 𝑟 r italic_r is the radial distance) and argued that confinement is homogeneously realized for S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) at Ω I/T=π subscript Ω I 𝑇 𝜋\Omega_{\text{I}}/T=\pi roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT / italic_T = italic_π. There are some related works[27, 28, 29] in favor of inhomogeneous confinement/deconfinement with real and imaginary rotation. We will show that the perturbative Polyakov loop potential exhibits an inhomogeneous distribution of the Polyakov loop at r≠0 𝑟 0 r\neq 0 italic_r ≠ 0. Another extension is the inclusion of dynamical quarks, with which we can consider chiral symmetry breaking in the perturbatively confined phase. The relation between confinement and chiral symmetry breaking is not fully understood, but under reasonable assumptions, one can see that confinement leads to chiral symmetry breaking or spontaneous generation of quark mass. We will verify that quark mass is indeed nonzero in the perturbatively confined phase.

2 Inhomogeneous confinement and deconfinement

Real rotation is a real-time effect, while imaginary rotation is interpreted as a geometrical effect. The latter has a theoretical advantage as explicated below. In the presence of an angular velocity vector, 𝝎 𝝎\bm{\omega}bold_italic_ω, a thermal system is described by the following partition function:

𝒵 T,𝝎=tr⁡e−β⁢(H^−𝝎⋅𝑱^),subscript 𝒵 𝑇 𝝎 trace superscript e 𝛽^𝐻⋅𝝎^𝑱\mathcal{Z}_{T,\bm{\omega}}=\tr\mathrm{e}^{-\beta(\hat{H}-\bm{\omega}\cdot\hat% {\bm{J}})},,caligraphic_Z start_POSTSUBSCRIPT italic_T , bold_italic_ω end_POSTSUBSCRIPT = roman_tr roman_e start_POSTSUPERSCRIPT - italic_β ( over^ start_ARG italic_H end_ARG - bold_italic_ω ⋅ over^ start_ARG bold_italic_J end_ARG ) end_POSTSUPERSCRIPT ,(1)

where β=1/T 𝛽 1 𝑇\beta=1/T italic_β = 1 / italic_T is the inverse temperature and 𝑱^^𝑱\hat{\bm{J}}over^ start_ARG bold_italic_J end_ARG denotes the total angular momentum operator. We can regard the above expression as a thermal expectation value of the topological operator, e β⁢𝝎⋅𝑱^superscript e⋅𝛽 𝝎^𝑱\mathrm{e}^{\beta\bm{\omega}\cdot\hat{\bm{J}}}roman_e start_POSTSUPERSCRIPT italic_β bold_italic_ω ⋅ over^ start_ARG bold_italic_J end_ARG end_POSTSUPERSCRIPT. For pure imaginary 𝝎=i⁢𝛀 I 𝝎 i subscript 𝛀 I\bm{\omega}=\mathrm{i}{\bm{\Omega}}_{\text{I}}bold_italic_ω = roman_i bold_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT, this topological operator,

e i⁢β⁢𝛀 I⋅𝑱^,superscript e⋅i 𝛽 subscript 𝛀 I^𝑱\mathrm{e}^{\mathrm{i}\beta{\bm{\Omega}}_{\text{I}}\cdot\hat{\bm{J}}},,roman_e start_POSTSUPERSCRIPT roman_i italic_β bold_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ⋅ over^ start_ARG bold_italic_J end_ARG end_POSTSUPERSCRIPT ,(2)

is a unitary operator that generates a rotation by β⁢|𝛀 I|𝛽 subscript 𝛀 I\beta|{\bm{\Omega}}{\text{I}}|italic_β | bold_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT | angle along the rotation axis ∥𝛀 I\parallel!{\bm{\Omega}}{\text{I}}∥ bold_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT. In the Euclidean path integral, we can take account of 𝛀 I subscript 𝛀 I{\bm{\Omega}}_{\text{I}}bold_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT in the partition function(1) by imposing an aperiodic thermal boundary condition which in cylindrical coordinates takes the form,

(r,θ,z,τ)∼(r,θ+β⁢Ω I,z,τ+β).similar-to 𝑟 𝜃 𝑧 𝜏 𝑟 𝜃 𝛽 subscript Ω I 𝑧 𝜏 𝛽(r,\theta,z,\tau)\sim(r,\theta+\beta\Omega_{\text{I}},z,\tau+\beta),.( italic_r , italic_θ , italic_z , italic_τ ) ∼ ( italic_r , italic_θ + italic_β roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT , italic_z , italic_τ + italic_β ) .(3)

Alternatively, we can perform the coordinate transformation to change the condition(3) to the ordinary periodic one, but then the metric gains nontrivial components in a rotating frame, i.e.,

g μ⁢ν=(1 0 0 0 0 r 2 0 r 2⁢Ω I 0 0 1 0 0 r 2⁢Ω I 0 1+r 2⁢Ω I 2).subscript 𝑔 𝜇 𝜈 matrix 1 0 0 0 0 superscript 𝑟 2 0 superscript 𝑟 2 subscript Ω I 0 0 1 0 0 superscript 𝑟 2 subscript Ω I 0 1 superscript 𝑟 2 superscript subscript Ω I 2\displaystyle g_{\mu\nu}=\begin{pmatrix}1&0&0&0\ 0&r^{2}&0&r^{2}\Omega_{\text{I}}\ 0&0&1&0\ 0&r^{2}\Omega_{\text{I}}&0&1+r^{2}\Omega_{\text{I}}^{2}\end{pmatrix},.italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 1 + italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ) .(4)

Both approaches lead us to the same partition function.

We shall discuss inhomogeneous structures in the perturbatively confined phase in S⁢U⁢(N c)𝑆 𝑈 subscript 𝑁 c SU(N_{\text{c}})italic_S italic_U ( italic_N start_POSTSUBSCRIPT c end_POSTSUBSCRIPT ) pure gluonic matter with N c=2 subscript 𝑁 c 2 N_{\text{c}}=2 italic_N start_POSTSUBSCRIPT c end_POSTSUBSCRIPT = 2 and 3 3 3 3. Although we have already given a derivation in Ref.[36], we quickly review the formulation to make this paper self-contained. In our calculation, we take the cylindrical coordinates, (r,θ,z,τ)𝑟 𝜃 𝑧 𝜏(r,\theta,z,\tau)( italic_r , italic_θ , italic_z , italic_τ ), and assume rigidly rotating matter along the z 𝑧 z italic_z-axis.

We take a constant A B⁢4 subscript 𝐴 B 4 A_{\text{B}4}italic_A start_POSTSUBSCRIPT B 4 end_POSTSUBSCRIPT background and then ∂τ subscript 𝜏\partial_{\tau}∂ start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT is replaced by the covariant derivative D τ subscript 𝐷 𝜏 D_{\tau}italic_D start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT. The inhomogeneity will be studied within the local density approximation (in which the spatial derivatives on A B⁢4 subscript 𝐴 B 4 A_{\text{B}4}italic_A start_POSTSUBSCRIPT B 4 end_POSTSUBSCRIPT are neglected). Using the Cartan subalgebra 𝔥 𝔥\mathfrak{h}fraktur_h of Lie algebra 𝔤 𝔤\mathfrak{g}fraktur_g of gauge group G 𝐺 G italic_G, the covariant derivative is

D τ=∂τ+i⁢ϕ⋅𝑯 β,subscript 𝐷 𝜏 subscript 𝜏 i⋅bold-italic-ϕ 𝑯 𝛽 D_{\tau}=\partial_{\tau}+\mathrm{i}\frac{\bm{\phi}\cdot\bm{H}}{\beta},,italic_D start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT = ∂ start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT + roman_i divide start_ARG bold_italic_ϕ ⋅ bold_italic_H end_ARG start_ARG italic_β end_ARG ,(5)

where 𝑯 𝑯\bm{H}bold_italic_H is a vector of bases of 𝔥 𝔥\mathfrak{h}fraktur_h and ϕ⋅𝑯⋅bold-italic-ϕ 𝑯{\bm{\phi}}\cdot\bm{H}bold_italic_ϕ ⋅ bold_italic_H is normalized A B⁢4 subscript 𝐴 B 4 A_{\text{B}4}italic_A start_POSTSUBSCRIPT B 4 end_POSTSUBSCRIPT. Using this background field, we fix the gauge by setting D μ⁢A μ=0 subscript 𝐷 𝜇 subscript 𝐴 𝜇 0 D_{\mu}A_{\mu}=0 italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = 0.

In the actual calculations, the scalar Laplacian, −D s 2=−(D τ−Ω I⁢∂θ)2−r−1⁢∂r(r⁢∂r)−r−2⁢∂θ 2−∂z 2 subscript superscript 𝐷 2 s superscript subscript 𝐷 𝜏 subscript Ω I subscript 𝜃 2 superscript 𝑟 1 subscript 𝑟 𝑟 subscript 𝑟 superscript 𝑟 2 superscript subscript 𝜃 2 superscript subscript 𝑧 2-D^{2}{\mathrm{s}}=-\quantity(D{\tau}-\Omega_{\text{I}}\partial_{\theta})^{2% }-r^{-1}\partial_{r}\quantity(r\partial_{r})-r^{-2}\partial_{\theta}^{2}-% \partial_{z}^{2}- italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT = - ( start_ARG italic_D start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT - roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( start_ARG italic_r ∂ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_ARG ) - italic_r start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ∂ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, is the basic building block. We solve the equation of motion, −D s 2⁢Ψ=λ⁢Ψ subscript superscript 𝐷 2 s Ψ 𝜆 Ψ-D^{2}_{\mathrm{s}}\Psi=\lambda\Psi- italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT roman_Ψ = italic_λ roman_Ψ, to find the spectrum as

λ n,l,k,𝜶=(2⁢π⁢n+ϕ⋅𝜶 β−Ω I⁢l)2+k⟂2+k z 2,subscript 𝜆 𝑛 𝑙 𝑘 𝜶 superscript 2 𝜋 𝑛⋅bold-italic-ϕ 𝜶 𝛽 subscript Ω I 𝑙 2 superscript subscript 𝑘 perpendicular-to 2 superscript subscript 𝑘 𝑧 2\displaystyle\lambda_{n,l,k,\bm{\alpha}}=\quantity(\frac{2\pi{n}+\bm{\phi}!% \cdot!{\bm{\alpha}}}{\beta}-\Omega_{\text{I}}{l})^{2}+{k_{\perp}}^{2}+{k_{z}}% ^{2},,italic_λ start_POSTSUBSCRIPT italic_n , italic_l , italic_k , bold_italic_α end_POSTSUBSCRIPT = ( start_ARG divide start_ARG 2 italic_π italic_n + bold_italic_ϕ ⋅ bold_italic_α end_ARG start_ARG italic_β end_ARG - roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT italic_l end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,(6)

where quantum numbers are n,l∈ℤ 𝑛 𝑙 ℤ n,l\in\mathbb{Z}italic_n , italic_l ∈ blackboard_Z, k⟂∈ℝ+subscript 𝑘 perpendicular-to superscript ℝ k_{\perp}\in\mathbb{R}^{+}italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, k z∈ℝ subscript 𝑘 𝑧 ℝ k_{z}\in\mathbb{R}italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ∈ blackboard_R, and 𝜶∈Φ 𝜶 Φ\bm{\alpha}\in\Phi bold_italic_α ∈ roman_Φ with Φ Φ\Phi roman_Φ denoting the union of the 𝔰⁢𝔲⁢(N c)𝔰 𝔲 subscript 𝑁 c\mathfrak{su}(N_{\text{c}})fraktur_s fraktur_u ( italic_N start_POSTSUBSCRIPT c end_POSTSUBSCRIPT ) root system and the zero roots. The ghost contribution needs the determinant of −D s 2 subscript superscript 𝐷 2 s-D^{2}{\mathrm{s}}- italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_s end_POSTSUBSCRIPT, which is given in terms of λ n,l,k,𝜶 subscript 𝜆 𝑛 𝑙 𝑘 𝜶\lambda{n,l,k,\bm{\alpha}}italic_λ start_POSTSUBSCRIPT italic_n , italic_l , italic_k , bold_italic_α end_POSTSUBSCRIPT.

The contribution from the gauge field requires the vector Laplacian, −D v 2 subscript superscript 𝐷 2 v-D^{2}{\mathrm{v}}- italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_v end_POSTSUBSCRIPT. For the explicit form of −D v 2 subscript superscript 𝐷 2 v-D^{2}{\mathrm{v}}- italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_v end_POSTSUBSCRIPT, see Ref.[36]. The eigenvalue spectrum is the same as the scalar one. Each eigenmode has four polarization degrees of freedom, and two out of four are canceled by the ghost contribution. After some calculations, we arrive at the Polyakov loop potential resulting from the two physical (transverse) modes as

V g⁢(ϕ;ΩI)=T 4⁢π 2⁢∑𝜶∈Φ∑l∈ℤ subscript 𝑉 𝑔 bold-italic-ϕ subscriptΩ I 𝑇 4 superscript 𝜋 2 subscript 𝜶 Φ subscript 𝑙 ℤ\displaystyle V_{g}({\bm{\phi}};\tilde{\Omega}{\text{I}})=\frac{T}{4\pi^{2}}% \sum{{\bm{\alpha}}\in\Phi}\sum_{l\in\mathbb{Z}}italic_V start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ( bold_italic_ϕ ; over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ) = divide start_ARG italic_T end_ARG start_ARG 4 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT bold_italic_α ∈ roman_Φ end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_l ∈ blackboard_Z end_POSTSUBSCRIPT∫0∞k⟂⁢d k⟂⁢∫−∞+∞d k z superscript subscript 0 subscript 𝑘 perpendicular-to differential-d subscript 𝑘 perpendicular-to superscript subscript differential-d subscript 𝑘 𝑧\displaystyle\int_{0}^{\infty}!!k_{\perp}\mathrm{d}{k_{\perp}}\int_{-\infty}% ^{+\infty}!!!\mathrm{d}{k_{z}}∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT roman_d italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + ∞ end_POSTSUPERSCRIPT roman_d italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ×[J l−1 2⁢(k⟂⁢r)+J l+1 2⁢(k⟂⁢r)]⁢Re⁢ln⁡(1−e i⁢ϕ⋅𝜶−i⁢ΩI⁢l−β⁢|𝒌|).absent delimited-[]subscript superscript 𝐽 2 𝑙 1 subscript 𝑘 perpendicular-to 𝑟 subscript superscript 𝐽 2 𝑙 1 subscript 𝑘 perpendicular-to 𝑟 Re 1 superscript e⋅i bold-italic-ϕ 𝜶 i subscriptΩ I 𝑙 𝛽 𝒌\displaystyle\times\Bigl{[}J^{2}{l-1}(k{\perp}r)+J^{2}{l+1}(k{\perp}r)% \Bigr{]};\mathrm{Re}\ln\quantity(1-\mathrm{e}^{\mathrm{i}\bm{\phi}\cdot\bm{% \alpha}-\mathrm{i}\tilde{\Omega}_{\text{I}}l-\beta|\bm{k}|}),.× [ italic_J start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l - 1 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT italic_r ) + italic_J start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l + 1 end_POSTSUBSCRIPT ( italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT italic_r ) ] roman_Re roman_ln ( start_ARG 1 - roman_e start_POSTSUPERSCRIPT roman_i bold_italic_ϕ ⋅ bold_italic_α - roman_i over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT italic_l - italic_β | bold_italic_k | end_POSTSUPERSCRIPT end_ARG ) .(7)

For notational brevity, we introduced a dimensionless imaginary angular velocity; ΩI=Ω I/T subscriptΩ I subscript Ω I 𝑇\tilde{\Omega}{\text{I}}=\Omega{\text{I}}/T over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = roman_Ω start_POSTSUBSCRIPT I end_POSTSUBSCRIPT / italic_T. By expanding the logarithm, we can perform the momentum integration to simplify the above form into

V g⁢(ϕ;ΩI)=−2⁢T 4 π 2⁢∑𝜶∈Φ∑n=1∞cos⁡(n⁢ϕ⋅𝜶)⁢cos⁡(n⁢ΩI){n 2+2⁢r2⁢[1−cos⁡(n⁢ΩI)]}2.subscript 𝑉 𝑔 bold-italic-ϕ subscriptΩ I 2 superscript 𝑇 4 superscript 𝜋 2 subscript 𝜶 Φ superscript subscript 𝑛 1⋅𝑛 bold-italic-ϕ 𝜶 𝑛 subscriptΩ I superscript superscript 𝑛 2 2 superscript𝑟 2 delimited-[]1 𝑛 subscriptΩ I 2\displaystyle V_{g}({\bm{\phi}};\tilde{\Omega}{\text{I}})=-\frac{2T^{4}}{\pi^% {2}}\sum{{\bm{\alpha}}\in\Phi}\sum_{n=1}^{\infty}\frac{\cos(n{\bm{\phi}}\cdot% {\bm{\alpha}})\cos(n\tilde{\Omega}{\text{I}})}{\Bigl{{}n^{2}+2\tilde{r}^{2}% \bigl{[}1-\cos(n\tilde{\Omega}{\text{I}})\bigr{]}\Bigr{}}^{2}},.italic_V start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT ( bold_italic_ϕ ; over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ) = - divide start_ARG 2 italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG start_ARG italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT bold_italic_α ∈ roman_Φ end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG roman_cos ( start_ARG italic_n bold_italic_ϕ ⋅ bold_italic_α end_ARG ) roman_cos ( start_ARG italic_n over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT end_ARG ) end_ARG start_ARG { italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 over~ start_ARG italic_r end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ 1 - roman_cos ( start_ARG italic_n over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT end_ARG ) ] } start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .(8)

The potential is dependent on the dimensionless radial distance; r=r⁢T𝑟 𝑟 𝑇\tilde{r}=rT over~ start_ARG italic_r end_ARG = italic_r italic_T. The potential is minimized at the optimal value of ϕ bold-italic-ϕ{\bm{\phi}}bold_italic_ϕ, and the Polyakov loop expectation value, L⁢(ϕ)𝐿 bold-italic-ϕ L({\bm{\phi}})italic_L ( bold_italic_ϕ ), is evaluated accordingly. The denominator has singularities at r≠0𝑟 0\tilde{r}\neq 0 over~ start_ARG italic_r end_ARG ≠ 0 even for ϕ=0 bold-italic-ϕ 0{\bm{\phi}}=0 bold_italic_ϕ = 0 (i.e., free theory), which is consistent with Ref.[43].

Specifically, for the S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) Yang-Mills theory, the background gauge field is A B⁢4=(ϕ 1⁢T 3+ϕ 2⁢T 8)/g⁢β subscript 𝐴 B 4 subscript italic-ϕ 1 superscript 𝑇 3 subscript italic-ϕ 2 superscript 𝑇 8 𝑔 𝛽 A_{\text{B}4}=(\phi_{1}T^{3}+\phi_{2}T^{8})/g\beta italic_A start_POSTSUBSCRIPT B 4 end_POSTSUBSCRIPT = ( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_T start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT ) / italic_g italic_β, where T 3 superscript 𝑇 3 T^{3}italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT and T 8 superscript 𝑇 8 T^{8}italic_T start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT constitute the Cartan subalgebra of 𝔰⁢𝔲⁢(3)𝔰 𝔲 3\mathfrak{su}(3)fraktur_s fraktur_u ( 3 ). The traced Polyakov loop in the fundamental representation of S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) is

|L|=1 3⁢|tr⁡exp(i⁢g⁢∫0 β A B⁢4⁢𝑑 τ⁢missing)|=1 3⁢4⁢cos 2⁡(ϕ 1 2)+4⁢cos(ϕ 1 2⁢missing)⁢cos(3⁢ϕ 2 2⁢missing)+1.𝐿 1 3 trace i 𝑔 superscript subscript 0 𝛽 subscript 𝐴 B 4 differential-d 𝜏 missing 1 3 4 superscript 2 subscript italic-ϕ 1 2 4 subscript italic-ϕ 1 2 missing 3 subscript italic-ϕ 2 2 missing 1|L|=\frac{1}{3}\biggl{|}\tr\exp\biggl(\mathrm{i}g\int_{0}^{\beta}A_{\text{B}4}% ,d\tau\biggr{missing})\biggr{|}=\frac{1}{3}\sqrt{4\cos^{2}\Bigl{(}\frac{\phi_% {1}}{2}\Bigr{)}+4\cos\Bigl(\frac{\phi_{1}}{2}\Bigr{missing})\cos\Bigl(\frac{% \sqrt{3}\phi_{2}}{2}\Bigr{missing})+1},.| italic_L | = divide start_ARG 1 end_ARG start_ARG 3 end_ARG | roman_tr roman_exp ( start_ARG roman_i italic_g ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT B 4 end_POSTSUBSCRIPT italic_d italic_τ roman_missing end_ARG ) | = divide start_ARG 1 end_ARG start_ARG 3 end_ARG square-root start_ARG 4 roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ) + 4 roman_cos ( start_ARG divide start_ARG italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG roman_missing end_ARG ) roman_cos ( start_ARG divide start_ARG square-root start_ARG 3 end_ARG italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG roman_missing end_ARG ) + 1 end_ARG .(9)

In our previous work[36], we showed that confinement, |L|=0 𝐿 0|L|=0| italic_L | = 0, is realized for ΩI≥π/2 subscriptΩ I 𝜋 2\tilde{\Omega}_{\text{I}}\geq\pi/2 over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ≥ italic_π / 2 at r=0𝑟 0\tilde{r}=0 over~ start_ARG italic_r end_ARG = 0.

Figure 1: (Left) Polyakov loop |L|𝐿|L|| italic_L | for S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) as a function of dimensionless radial distance r~~𝑟\tilde{r}over~ start_ARG italic_r end_ARG and dimensionless imaginary angular velocity ΩI subscriptΩ I\tilde{\Omega}_{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT. (Right) The same plot of |L|𝐿|L|| italic_L | for S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ). The confined phase with |L|=0 𝐿 0|L|=0| italic_L | = 0 is bounded by the first-order phase transitions.

It is a straightforward exercise to find the global minima of the potential(8) for r≠0𝑟 0\tilde{r}\neq 0 over~ start_ARG italic_r end_ARG ≠ 0. Figure1 shows the results from such extensive analyses of Eq.(8) for S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) (left) and S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) (right). It is notable that both cases generally develop r 𝑟 r italic_r-dependent spatial structures. For the S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) case as shown in the left of Fig.1, the Polyakov loop changes to zero, indicating confinement, with second-order phase transition as ΩI subscriptΩ I\tilde{\Omega}{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT grows up. In the previous paper[36], we focused on two edges of r=0𝑟 0\tilde{r}=0 over~ start_ARG italic_r end_ARG = 0 and ΩI=π subscriptΩ I 𝜋\tilde{\Omega}{\text{I}}=\pi over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π only. Our present results imply that, for ΩI≃3⁢π/4 similar-to-or-equals subscriptΩ I 3 𝜋 4\tilde{\Omega}{\text{I}}\simeq 3\pi/4 over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ≃ 3 italic_π / 4 for example, there should appear a spatial interface separating the confined phase for r≲0.5 less-than-or-similar-to𝑟 0.5\tilde{r}\lesssim 0.5 over~ start_ARG italic_r end_ARG ≲ 0.5 and the deconfined phase for r≳0.5 greater-than-or-equivalent-to𝑟 0.5\tilde{r}\gtrsim 0.5 over~ start_ARG italic_r end_ARG ≳ 0.5. We can confirm a qualitatively similar trend for the S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) case, but the detailed shape looks different as shown in the right of Fig.1. In this case of S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ), the homogeneously confined phase is realized around ΩI≲3⁢π/4 less-than-or-similar-to subscriptΩ I 3 𝜋 4\tilde{\Omega}{\text{I}}\lesssim 3\pi/4 over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ≲ 3 italic_π / 4, and the interface between confinement and deconfinement emerges only when ΩI subscriptΩ I\tilde{\Omega}_{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT becomes larger.

We should emphasize that this r𝑟\tilde{r}over~ start_ARG italic_r end_ARG dependence of inhomogeneity is qualitatively consistent with the lattice-QCD results[29]. As closely discussed in our previous work[36], our ΩI subscriptΩ I\tilde{\Omega}_{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT dependence of the Polyakov loop is opposite to the lattice-QCD case; |L|𝐿|L|| italic_L | goes smaller with larger imaginary rotation in our perturbative calculation, while |L|𝐿|L|| italic_L | goes smaller with larger real rotation in the lattice-QCD simulation. However, in the both cases of ours and the lattice-QCD calculations, |L|𝐿|L|| italic_L | becomes larger (smaller) with larger r𝑟\tilde{r}over~ start_ARG italic_r end_ARG for a given imaginary (real) angular velocity. This makes a sharp contrast to preceding works[30, 14, 15, 27, 28] and might be a key observation to resolve the controversy of rotating QCD matter.

Figure 2: S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) effective potential as a function of ϕ 1 subscript italic-ϕ 1\phi_{1}italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ϕ 2 subscript italic-ϕ 2\phi_{2}italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for ΩI=π/2 subscriptΩ I 𝜋 2\tilde{\Omega}{\text{I}}=\pi/2 over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π / 2 (left) and ΩI=π subscriptΩ I 𝜋\tilde{\Omega}{\text{I}}=\pi over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π (right) with r=0𝑟 0\tilde{r}=0 over~ start_ARG italic_r end_ARG = 0 fixed. The dark (light) colored region has smaller (larger) potential values. The triangular domain indicated by the red line is sufficient for the minimum search.

Now, let us turn our attention to the Polyakov loop potential and the symmetry properties. Since we sum up all the roots in Φ Φ\Phi roman_Φ, different (ϕ 1,ϕ 2)subscript italic-ϕ 1 subscript italic-ϕ 2(\phi_{1},\phi_{2})( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) pairs can have the same potential and the Polyakov loop value according to Weyl symmetry of the S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) root lattice. We can see the characteristic patterns of the potential minima in Fig.2. The repetition of the minima reflects Weyl symmetry. The red triangle region with three edges, (ϕ 1,ϕ 2)=(0,0)subscript italic-ϕ 1 subscript italic-ϕ 2 0 0(\phi_{1},\phi_{2})=(0,0)( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = ( 0 , 0 ), (2⁢π,2⁢π/3)2 𝜋 2 𝜋 3(2\pi,2\pi/\sqrt{3})( 2 italic_π , 2 italic_π / square-root start_ARG 3 end_ARG ), (2⁢π,−2⁢π/3)2 𝜋 2 𝜋 3(2\pi,-2\pi/\sqrt{3})( 2 italic_π , - 2 italic_π / square-root start_ARG 3 end_ARG ), is the fundamental domain and we can identify the state of matter from the minimum inside this triangular domain. The triangle has S 6 subscript 𝑆 6 S_{6}italic_S start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT geometric symmetry, including three-fold rotational symmetry, which manifests center symmetry, and two-fold reflective symmetry, which manifests charge conjugation. The potential shape visualized by the pattern in Fig.2 does not change by 2⁢π/3 2 𝜋 3 2\pi/3 2 italic_π / 3 rotation but the phase of the Polyakov loop, L 𝐿 L italic_L, does. The center of the triangle at (4⁢π/3,0)4 𝜋 3 0(4\pi/3,0)( 4 italic_π / 3 , 0 ) corresponds to L=0 𝐿 0 L=0 italic_L = 0, that is, a center symmetric vacuum. Although the potential minima may break center symmetry, the charge conjugation symmetry is never broken.

We point out a nontrivial observation in Fig.2; an emergent symmetry is realized at ΩI=π/2 subscriptΩ I 𝜋 2\tilde{\Omega}{\text{I}}=\pi/2 over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π / 2. We observe a reflective mirror on the line of ϕ 1=π subscript italic-ϕ 1 𝜋\phi{1}=\pi italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_π. This emergent symmetry comes from the vanishing of odd-n 𝑛 n italic_n terms in the one-loop potential(8) at ΩI=π/2 subscriptΩ I 𝜋 2\tilde{\Omega}_{\text{I}}=\pi/2 over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π / 2 and exists for not only r=0𝑟 0\tilde{r}=0 over~ start_ARG italic_r end_ARG = 0 but any radius. It could be either a one-loop artifact or a genuine symmetry. In the latter case, it has to be a non-invertible symmetry like that in the 2D critical Ising model since it exchanges the unbroken and broken vacua.

Figure 3: S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) effective potential for r=0𝑟 0\tilde{r}=0 over~ start_ARG italic_r end_ARG = 0 (left) and r=0.5𝑟 0.5\tilde{r}=0.5 over~ start_ARG italic_r end_ARG = 0.5 (right) with ΩI=π subscriptΩ I 𝜋\tilde{\Omega}_{\text{I}}=\pi over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π fixed.

In this work, we also quantify the spatial inhomogeneity. In Fig.3, we plot the Polyakov loop potential for r=0𝑟 0\tilde{r}=0 over~ start_ARG italic_r end_ARG = 0 (confined phase) and r=0.5𝑟 0.5\tilde{r}=0.5 over~ start_ARG italic_r end_ARG = 0.5 (deconfined phase) at ΩI=π subscriptΩ I 𝜋\tilde{\Omega}_{\text{I}}=\pi over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π in the S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) case. It is intriguing that the potential minima are located in a different way from the ordinary perturbative vacuum where three vertices of the triangle minimize the potential. In this sense, this deconfined phase discovered in the upper right region in the right panel of Fig.1 may have exotic properties different from the ordinary one.

We mention the difference from the previous arguments[27] based on the Tolman-Ehrenfest (TE) effect. In our calculation, we treat T 𝑇 T italic_T as a Lagrange multiplier to conserve the total energy. By construction, T 𝑇 T italic_T is a global variable without r 𝑟 r italic_r dependence. Therefore, in our study, we do not need to introduce the apparent temperature as a result of the TE effect. Nevertheless, the calculations naturally lead to r 𝑟 r italic_r dependent structures.

3 Chiral symmetry breaking in the perturbatively confined phase

We can repeat similar calculations including dynamical quark contributions that break center symmetry explicitly. We can also address a relation between confinement and chiral symmetry breaking from two (approximate) order parameters, namely, the Polyakov loop and the dynamical quark mass, m 𝑚 m italic_m, which is rooted in the chiral condensate. In the perturbative treatment, the pressure (the free energy) is maximized (minimized) for m=0 𝑚 0 m=0 italic_m = 0, and the dynamical mass generation is energetically disfavored. It is quite interesting what would happen in the perturbatively confined phase with imaginary rotation.

The theoretical treatments of fermions in the rotating frame are found in Refs.[30, 44, 7]. We should be careful about the fact that the Polyakov loop coupling with quarks is given by the fundamental representation. We can calculate the fermionic partition function by imposing an aperiodic thermal boundary condition or equivalently considering the ordinary anti-periodic boundary condition in the rotating frame. In this paper, for fermions, we choose the latter. It should be noted that the fermion interactions appear from gauge fluctuations which are beyond the one-loop perturbative order. In this way, we can locate the onset of instability toward chiral symmetry breaking, but finding the physical value of m≠0 𝑚 0 m\neq 0 italic_m ≠ 0 needs non-perturbative interactions that we do not include in the present study. The fermionic partition function is the determinant of the Dirac operator γ μ⁢G B⁢μ+m superscript 𝛾 𝜇 subscript 𝐺 B 𝜇 𝑚\gamma^{\mu}G_{{\rm B},\mu}+m italic_γ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT roman_B italic_μ end_POSTSUBSCRIPT + italic_m in the rotating frame, i.e.,

𝒵 f⁢T,ω=Det⁢(γ μ⁢G B⁢μ+m).subscript 𝒵 f 𝑇 𝜔 Det superscript 𝛾 𝜇 subscript 𝐺 B 𝜇 𝑚\mathcal{Z}{{\rm f}T,\omega}=\mathrm{Det}(\gamma^{\mu}G{{\rm B},\mu}+m),.caligraphic_Z start_POSTSUBSCRIPT roman_f italic_T , italic_ω end_POSTSUBSCRIPT = roman_Det ( italic_γ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_G start_POSTSUBSCRIPT roman_B italic_μ end_POSTSUBSCRIPT + italic_m ) .(10)

Here, G B⁢μ=D μ−Γ μ subscript 𝐺 B 𝜇 subscript 𝐷 𝜇 subscript Γ 𝜇 G_{{\rm B},\mu}=D_{\mu}-\Gamma_{\mu}italic_G start_POSTSUBSCRIPT roman_B italic_μ end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT - roman_Γ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT is the covariant derivative including the A B4 subscript 𝐴 B4 A_{{\rm B}4}italic_A start_POSTSUBSCRIPT B4 end_POSTSUBSCRIPT background field with Γ μ=−i 4⁢σ i⁢j⁢ω μ⁢i⁢j subscript Γ 𝜇 𝑖 4 superscript 𝜎 𝑖 𝑗 subscript 𝜔 𝜇 𝑖 𝑗\Gamma_{\mu}=-\frac{i}{4}\sigma^{ij},\omega_{\mu ij}roman_Γ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = - divide start_ARG italic_i end_ARG start_ARG 4 end_ARG italic_σ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT italic_μ italic_i italic_j end_POSTSUBSCRIPT, where σ i⁢j=i 2⁢[γ^i,γ^j]superscript 𝜎 𝑖 𝑗 𝑖 2 superscript^𝛾 𝑖 superscript^𝛾 𝑗\sigma^{ij}=\frac{i}{2}[\hat{\gamma}^{i},\hat{\gamma}^{j}]italic_σ start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT = divide start_ARG italic_i end_ARG start_ARG 2 end_ARG [ over^ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT , over^ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ] and ω μ⁢i⁢j=g ρ⁢σ⁢e i ρ⁢(∂μ e j σ+Γ μ⁢ν σ⁢e j ν)subscript 𝜔 𝜇 𝑖 𝑗 subscript 𝑔 𝜌 𝜎 superscript subscript 𝑒 𝑖 𝜌 subscript 𝜇 subscript superscript 𝑒 𝜎 𝑗 subscript superscript Γ 𝜎 𝜇 𝜈 subscript superscript 𝑒 𝜈 𝑗\omega_{\mu ij}=g_{\rho\sigma},e_{i}^{\ \rho},\quantity(\partial_{\mu}e^{\ % \sigma}{j}+\Gamma^{\sigma}{\mu\nu},e^{\ \nu}{j})italic_ω start_POSTSUBSCRIPT italic_μ italic_i italic_j end_POSTSUBSCRIPT = italic_g start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ( start_ARG ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + roman_Γ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ). We denote the gamma matrices of the flat space-time by γ^i superscript^𝛾 𝑖\hat{\gamma}^{i}over^ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT and γ μ=e i μ⁢γ^i superscript 𝛾 𝜇 superscript subscript 𝑒 𝑖 𝜇 superscript^𝛾 𝑖\gamma^{\mu}=e{i}^{\ \mu}\hat{\gamma}^{i}italic_γ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT over^ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT. After all, we arrive at the expression for the Polyakov loop potential per one fermion (particle or anti-particle but without flavor degrees of freedom) as

V f⁢(ϕ;ΩI)subscript 𝑉 𝑓 bold-italic-ϕ subscriptΩ I\displaystyle V_{f}({\bm{\phi}};\tilde{\Omega}{\text{I}})italic_V start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( bold_italic_ϕ ; over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT )=−T 4⁢π 2⁢∑𝝁∈Φ f∑l∈ℤ∫0∞k⟂⁢d k⟂⁢∫−∞∞d k z absent 𝑇 4 superscript 𝜋 2 subscript 𝝁 subscript Φ 𝑓 subscript 𝑙 ℤ superscript subscript 0 subscript 𝑘 perpendicular-to differential-d subscript 𝑘 perpendicular-to superscript subscript differential-d subscript 𝑘 𝑧\displaystyle=-\frac{T}{4\pi^{2}}\sum{\bm{\mu}\in\Phi_{f}}\sum_{l\in\mathbb{Z% }}\int_{0}^{\infty}k_{\perp},\mathrm{d}k_{\perp}\int_{-\infty}^{\infty}% \mathrm{d}k_{z}= - divide start_ARG italic_T end_ARG start_ARG 4 italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT bold_italic_μ ∈ roman_Φ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_l ∈ blackboard_Z end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT roman_d italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT roman_d italic_k start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ×[J l 2⁢(k⟂⁢r)+J l+1 2⁢(k⟂⁢r)]⁢Re ln(1+e i⁢ϕ⋅𝝁−i⁢ΩI⁢(l+1/2)−β⁢𝒌 2+m 2⁢missing),absent delimited-[]superscript subscript 𝐽 𝑙 2 subscript 𝑘 perpendicular-to 𝑟 superscript subscript 𝐽 𝑙 1 2 subscript 𝑘 perpendicular-to 𝑟 1 superscript e⋅i bold-italic-ϕ 𝝁 i subscriptΩ I 𝑙 1 2 𝛽 superscript 𝒌 2 superscript 𝑚 2 missing\displaystyle\qquad\times\Bigl{[}J_{l}^{2}(k_{\perp}r)+J_{l+1}^{2}(k_{\perp}r)% \Bigr{]}\real\ln\Bigl(1+\mathrm{e}^{\mathrm{i}{\bm{\phi}}\cdot\bm{\mu}-\mathrm% {i}\tilde{\Omega}_{\text{I}}(l+1/2)-\beta\sqrt{\bm{k}^{2}+m^{2}}}\Bigr{missing% }),,× [ italic_J start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT italic_r ) + italic_J start_POSTSUBSCRIPT italic_l + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_k start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT italic_r ) ] start_OPERATOR roman_Re end_OPERATOR roman_ln ( start_ARG 1 + roman_e start_POSTSUPERSCRIPT roman_i bold_italic_ϕ ⋅ bold_italic_μ - roman_i over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ( italic_l + 1 / 2 ) - italic_β square-root start_ARG bold_italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_POSTSUPERSCRIPT roman_missing end_ARG ) ,(11)

where Φ f subscript Φ 𝑓\Phi_{f}roman_Φ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT denotes the set of the fundamental weights of 𝔰⁢𝔲⁢(N c)𝔰 𝔲 subscript 𝑁 c\mathfrak{su}(N_{\text{c}})fraktur_s fraktur_u ( italic_N start_POSTSUBSCRIPT c end_POSTSUBSCRIPT ). The limit of ϕ=0 bold-italic-ϕ 0{\bm{\phi}}=0 bold_italic_ϕ = 0 renders the above expression to the free quark and anti-quark gas energy with imaginary rotation derived from statistical mechanics.

First, we shall consider the interplay of |L|𝐿|L|| italic_L | and m 𝑚 m italic_m at r=0𝑟 0\tilde{r}=0 over~ start_ARG italic_r end_ARG = 0 by setting N c=N f=2 subscript 𝑁 c subscript 𝑁 f 2 N_{\text{c}}=N_{\text{f}}=2 italic_N start_POSTSUBSCRIPT c end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT f end_POSTSUBSCRIPT = 2. Then, the total effective potential is given by V g+2⁢N f⁢V f subscript 𝑉 𝑔 2 subscript 𝑁 f subscript 𝑉 𝑓 V_{g}+2N_{\text{f}}V_{f}italic_V start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT + 2 italic_N start_POSTSUBSCRIPT f end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT (where 2 2 2 2 comes from the sum over particle and anti-particle). In our numerical calculation, we consider ϕ∈[0,2⁢π]italic-ϕ 0 2 𝜋\phi\in[0,2\pi]italic_ϕ ∈ [ 0 , 2 italic_π ] and m=m/T∈[0,10]𝑚 𝑚 𝑇 0 10\tilde{m}=m/T\in[0,10]over~ start_ARG italic_m end_ARG = italic_m / italic_T ∈ [ 0 , 10 ]. We note that the former region for ϕ italic-ϕ\phi italic_ϕ is sufficient thanks to S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) Weyl symmetry. Because we do not model quark interactions to keep our analysis as model independent as possible, m 𝑚 m italic_m blows up in the phase where chiral symmetry is fully broken. We thus set a numerical cutoff on m~~𝑚\tilde{m}over~ start_ARG italic_m end_ARG in the practical calculation.

Figure 4: Potential contour as functions of ϕ italic-ϕ\phi italic_ϕ (Polyakov loop) and m~~𝑚\tilde{m}over~ start_ARG italic_m end_ARG (dimensionless dynamical mass) for N c=N f=2 subscript 𝑁 c subscript 𝑁 f 2 N_{\text{c}}=N_{\text{f}}=2 italic_N start_POSTSUBSCRIPT c end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT f end_POSTSUBSCRIPT = 2, ΩI=π/2 subscriptΩ I 𝜋 2\tilde{\Omega}{\text{I}}=\pi/2 over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π / 2 (left) and ΩI=π subscriptΩ I 𝜋\tilde{\Omega}{\text{I}}=\pi over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π (right). The darker (lighter) color indicates the smaller (larger) value and the potential minimum has the darkest color.

Figure4 shows the potential contour as functions of two order parameters in the S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) case. The minimum of the potential is given by the darkest spot in the figures. In the left of Fig.4 for ΩI=π/2 subscriptΩ I 𝜋 2\tilde{\Omega}{\text{I}}=\pi/2 over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π / 2, the minimum is still located at m=0𝑚 0\tilde{m}=0 over~ start_ARG italic_m end_ARG = 0 and ϕ=0 italic-ϕ 0\phi=0 italic_ϕ = 0 (i.e., L=1 𝐿 1 L=1 italic_L = 1). This is different from the pure gluonic case in the left of Fig.1; fermions generally favor deconfinement. Interestingly in the right of Fig.4 for ΩI=π subscriptΩ I 𝜋\tilde{\Omega}{\text{I}}=\pi over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π, however, the minimum moves to ϕ≃π similar-to-or-equals italic-ϕ 𝜋\phi\simeq\pi italic_ϕ ≃ italic_π and m~~𝑚\tilde{m}over~ start_ARG italic_m end_ARG blows up to the cutoff value, which signifies confinement and spontaneous chiral symmetry breaking.

Figure 5: Order parameters, |L|𝐿|L|| italic_L | and m𝑚\tilde{m}over~ start_ARG italic_m end_ARG, as functions of ΩI subscriptΩ I\tilde{\Omega}{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT for N f=2 subscript 𝑁 f 2 N{\text{f}}=2 italic_N start_POSTSUBSCRIPT f end_POSTSUBSCRIPT = 2, N c=2 subscript 𝑁 c 2 N_{\text{c}}=2 italic_N start_POSTSUBSCRIPT c end_POSTSUBSCRIPT = 2 (left) and N c=3 subscript 𝑁 c 3 N_{\text{c}}=3 italic_N start_POSTSUBSCRIPT c end_POSTSUBSCRIPT = 3 (right). The blue solid line represents the Polyakov loop |L|𝐿|L|| italic_L |, and the green dashed line represents the dynamical mass m𝑚\tilde{m}over~ start_ARG italic_m end_ARG. The horizontal axis is discretized by 200 200 200 200 points with equal spacing to capture spiky structures.

We make the plots for the behavior of the Polyakov loop |L|𝐿|L|| italic_L | and the dynamical mass m𝑚\tilde{m}over~ start_ARG italic_m end_ARG in Fig.5. The blue solid line represents |L|𝐿|L|| italic_L |, while the green dashed line represents m𝑚\tilde{m}over~ start_ARG italic_m end_ARG. With increasing ΩI subscriptΩ I\tilde{\Omega}{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT, the Polyakov loop goes smaller and the dynamical mass grows up. Then, a confined and chiral symmetry broken state is favored near ΩI≃π similar-to-or-equals subscriptΩ I 𝜋\tilde{\Omega}{\text{I}}\simeq\pi over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT ≃ italic_π. For the S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) case as shown in the left of Fig.5, the behavior is 2⁢π 2 𝜋 2\pi 2 italic_π periodic because the S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) gauge group covers the fermionic parity (−)F superscript 𝐹(-)^{F}( - ) start_POSTSUPERSCRIPT italic_F end_POSTSUPERSCRIPT. Also we see that the behavior is symmetric around ΩI=π subscriptΩ I 𝜋\tilde{\Omega}{\text{I}}=\pi over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT = italic_π. For the S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) case in the right of Fig.5, in contrast, the order parameters depend on ΩI subscriptΩ I\tilde{\Omega}{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT in a complicated way and the confined phase seems to be less favored. Also the periodicity becomes 4⁢π 4 𝜋 4\pi 4 italic_π because the S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) gauge group does not contain (−)F superscript 𝐹(-)^{F}( - ) start_POSTSUPERSCRIPT italic_F end_POSTSUPERSCRIPT.

Figure 6: Order parameters in the S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) case; the Polyakov loop (left) and the dynamical mass (right) as functions of dimensionless radial distance r~~𝑟\tilde{r}over~ start_ARG italic_r end_ARG and dimensionless imaginary angular velocity ΩI subscriptΩ I\tilde{\Omega}_{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT. To reduce the numerical cost, the mesh resolution is chosen to be 100×100 100 100 100\times 100 100 × 100.

In the same way as the pure gluonic theory, we have quantified the inhomogeneous structures in Fig.6. The left and the right panels show the Polyakov loop and the dynamical mass, respectively, as functions of r~~𝑟\tilde{r}over~ start_ARG italic_r end_ARG and ΩI subscriptΩ I\tilde{\Omega}{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT. We have performed this calculation for the S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) case only. As compared to the left of Fig.1 in the pure gluonic case, the perturbatively confined region is shrunk to the upper left corner. Interestingly, in view of the right of Fig.6, confinement and chiral symmetry breaking are locked together and the phase transitions occur simultaneously even at large imaginary rotation. It would be an interesting future problem to probe a possibility of fractal structures for ΩI/(2⁢π)subscriptΩ I 2 𝜋\tilde{\Omega}{\text{I}}/(2\pi)over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT / ( 2 italic_π ) given by a rational number, which is not visible in the present analysis.

4 Conclusions

We considered the inhomogeneous structures and the chiral symmetry breaking in the perturbatively confined phase which is realized by rotation with imaginary angular velocity, ΩI subscriptΩ I\tilde{\Omega}{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT. In both color S⁢U⁢(2)𝑆 𝑈 2 SU(2)italic_S italic_U ( 2 ) and S⁢U⁢(3)𝑆 𝑈 3 SU(3)italic_S italic_U ( 3 ) cases, the perturbatively confined phase is induced at the rotation center, r=0𝑟 0\tilde{r}=0 over~ start_ARG italic_r end_ARG = 0, for ΩI subscriptΩ I\tilde{\Omega}{\text{I}}over~ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT I end_POSTSUBSCRIPT above a certain threshold. We found that the perturbative confinement may break down in the r≠0𝑟 0\tilde{r}\neq 0 over~ start_ARG italic_r end_ARG ≠ 0 region away from the rotation center. This indicates the existence of phase interface between confinement and deconfinement and the critical temperature is also r𝑟\tilde{r}over~ start_ARG italic_r end_ARG dependent accordingly. Our results predict that the critical temperature should decrease with increasing r𝑟\tilde{r}over~ start_ARG italic_r end_ARG in the presence of imaginary rotation. This trend agrees with the latest results from the lattice simulation. Although the analytical continuation to real rotation has some subtle points, the inhomogeneity we have confirmed from the perturbative effective potential can presumably persist in real rotating systems.

We then added the quark contribution to the Polyakov loop effective potential, which is also a function of the dynamical quark mass, m𝑚\tilde{m}over~ start_ARG italic_m end_ARG. Our analysis of the potential minimum search showed that confinement and chiral symmetry breaking are tightly correlated in such unconventional systems with imaginary rotation and even without fermionic interaction. In the presence of quarks that explicitly break center symmetry, the Polyakov loop is only an approximate order parameter. Still, in the region where the Polyakov loop is vanishingly small, m𝑚\tilde{m}over~ start_ARG italic_m end_ARG departs from zero signifying spontaneous breaking of chiral symmetry. In our treatment, in fact, m𝑚\tilde{m}over~ start_ARG italic_m end_ARG blows up when chiral symmetry is broken, so that large m𝑚\tilde{m}over~ start_ARG italic_m end_ARG suppresses center symmetry breaking. In this way, we established simultaneous realization of confinement and chiral symmetry breaking. We also found highly complicated substructures in the deconfined regions. The imaginary-rotating inhomogeneous states with dynamical quarks should deserve further investigations in the future. The full understanding of imaginary-rotating matter should be the indispensable foundation for a more important and challenging problem with real rotation which requires a boundary to satisfy the causality condition. We are making progresses along these lines.

Acknowledgement

The authors thank Maxim Chernodub and Xu-Guang Huang for useful discussions. This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Nos.22H01216 (K.F.) and 22H05118 (K.F.) and 21J20877 (S.C.) and JST SPRING, Grant Number JPMJSP2108 (Y.S.).

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