Order parameters and color-flavor center symmetry in QCD

TL;DR

This paper reveals the existence of non-trivial order parameters in N-color QCD with multiple flavors, linked to a color-flavor center symmetry, affecting the understanding of phase structure at various conditions.

Contribution

It introduces the concept of a $ ext{Z}_d$ color-flavor center symmetry in QCD with multiple flavors and explores its implications for phase transitions and symmetry realization.

Findings

Existence of $ ext{Z}_d$ symmetry in certain QCD theories.

Dependence of symmetry realization on temperature and chemical potential.

Implications for conformal window and dense quark matter studies.

Abstract

Common lore suggests that $N$-color QCD with massive quarks has no useful order parameters which can be non-trivial at zero baryon density. However, such order parameters do exist when there are $n_f$ quark flavors with a common mass and $d\equiv\gcd(n_f,N) > 1$. These theories have a $\mathbb Z_d$ color-flavor center symmetry arising from intertwined color center transformations and cyclic flavor permutations. The symmetry realization depends on the temperature, baryon chemical potential and value of $n_f/N$, with implications for conformal window studies and dense quark matter.

Figures

Equations

$$⟨\text{tr}\Omega ⟩=⟨\text{tr}\mathcal{P}{e}^{i{\int }_0^{L}d{x}_1{A}_1}⟩,$$

$$⟨\text{tr}\Omega ⟩\to \omega ⟨\text{tr}\Omega ⟩,\omega \equiv e^{2\pi i/N}.$$

$$q_a(x_1+L)={\mathcal{U}}^{ab}{q}_b(x_1),$$

$$\mathcal{U}$$

$$d\equiv \gcd (n_{\mathrm{f}},N)>1,$$

$$\begin{array}{rl}\text{tr}{\Omega }^p& \to {\omega }^{Np/d}\text{tr}{\Omega }^p,{\mathcal{O}}_{\Gamma }^{(p)}\to {\nu }^{n_{\mathrm{f}}p/d}{\mathcal{O}}_{\Gamma }^{(p)}.\end{array}$$

$$⟨\text{tr}\Omega (\vec{x})\text{tr}{\Omega }^{†}(0){⟩}_{\mathrm{conn}}\equiv e^{-F(r)},r=|\vec{x}|.$$

$$V_{\mathrm{g}}(\Omega )$$

$$V_{\mathrm{f}}(\Omega )$$

$$=\frac{2}{{\pi }^2{L}^4{n_{\mathrm{f}}}^3}\sum _{n=1}^{\infty }\frac{(\pm 1{)}^n}{n^4}\left(\text{tr}{\Omega }^{n_{\mathrm{f}}n}+h.c.\right).$$

$$V_{\mathrm{g}}(\Omega )=-\frac{1}{L^4}\sum _{n=1}^{\infty }{f}_n(|\text{tr}{\Omega }^n{|}^2-1),$$

$$V_{\mathrm{f}}(\Omega )$$

$$\displaystyle\qquad\qquad{}\times\sum_{k\in\mathbb{Z}+\frac{1}{2}}(1+{n_{\rm f}}n\mathfrak{m}_{k}L)\>e^{-{n_{\rm f}}n\mathfrak{m}_{k}L}\Bigr{]}\,,$$

$$V_{\mathrm{f}}(\Omega )$$

$$\times \left(\text{tr}{\Omega }^{n_{\mathrm{f}}}+h.c.\right)+\text{(holonomy-independent)},$$

$${\mathcal{U}}_1\equiv \text{diag}(1,-1,1,-1),{\mathcal{U}}_2\equiv \text{diag}(1,1,-1,-1).$$

$$q(x_4=\beta )$$

$$\bar{q}(x_4=\beta )$$

$$q(x_1=L{)}_{aA}$$

$$\bar{q}(x_1=L{)}_{aA}$$

$$F$$

$$\displaystyle=-\frac{1}{\beta L}\int\frac{d^{2}p_{\perp}}{(2\pi)^{2}}\sum_{a,n,k}\ln\det\big{(}i\gamma^{\mu}p^{(a,n,k)}_{\mu}(\Omega)+m_{q}\big{)},$$

$$F$$

$$\Delta F(\Omega ,L)$$

$$=\frac{2}{\beta }\sum _k{\int }_0^{\infty }\frac{dz}{z}\int \frac{d^2{p}_{\perp }}{(2\pi {)}^2}$$

$$\times \left(\frac{1}{L}\sum _{a,n}-\int \frac{d{p}_1}{2\pi }\right)e^{-z(p_{\mu }^2+m_q^2)}.$$

$$\Delta F$$

$$\times \sum _{n\ne 0}{e}^{-n^2{L}^2/(4z)}\sum _{a,A}{e}^{i({\theta }_a+2\pi A/n_{\mathrm{f}})n}$$

$$=\frac{1}{\pi L^3\beta }\sum _{n=1}^{\infty }\frac{1}{n^3}\left(\text{tr}{\mathcal{U}}^n\text{tr}{\Omega }^n+h.c.\right)$$

$$\times \sum _{k\in \mathbb{Z}+\frac{1}{2}}(1+nL{\mathfrak{m}}_k)e^{-nL{\mathfrak{m}}_k},$$

$${\mathfrak{m}}_k^2\equiv (2\pi kT+i\mu {)}^2+m_q^2.$$

$$\Delta F$$

$$\displaystyle\quad{}\times\Bigl{[}\pi{n_{\rm f}}n\,TL\cos({n_{\rm f}}n\,\mu L)\,\coth(\pi{n_{\rm f}}n\,TL)$$

$$\displaystyle\qquad{}+{n_{\rm f}}n\,\mu L\,\sin({n_{\rm f}}n\,\mu L)+\cos({n_{\rm f}}n\,\mu L)\Bigr{]}.$$

$$\Delta F$$

$$\displaystyle\qquad{}\times\big{[}{n_{\rm f}}n\,\mu L\sin({n_{\rm f}}n\mu L)+2\cos({n_{\rm f}}n\mu L)\big{]}\,.$$

Full text

††preprint: INT-PUB-17-021

Introduction. Defining order parameters in QCD is notoriously subtle. In pure $SU(N)$ Yang-Mills (YM) theory, the simplest non-trivial order parameter is the expectation value of a line operator:

[TABLE]

when the $x_{1}$ dimension is compactified with circumference $L$. If $x_{1}$ is regarded as Euclidean time, then the gauge theory functional integral with periodic boundary conditions calculates the thermal partition function with temperature $T=1/L$. The thermal expectation value (1) is the Polyakov loop confinement order parameter for the $\mathbb{Z}_{N}$ center symmetry of pure YM, whose realization changes with temperature.

Adding fundamental representation quarks $\{q_{a}\}$, $a=1,2,{\cdots},{n_{\rm f}}$ explicitly breaks the $\mathbb{Z}_{N}$ center symmetry and complicates the story. The Polyakov loop ceases to be an order parameter. With massless quarks, the flavor symmetry is $G=SU({n_{\rm f}})_{V}\times SU({n_{\rm f}})_{A}\times U(1)_{Q}$ and the chiral condensate $\langle\sum_{a}\bar{q}_{a}q_{a}\rangle$ is an order parameter for the $SU({n_{\rm f}})_{A}$ chiral symmetry, whose realization depends on temperature. But if the quarks are massive, as in real-world QCD, then chiral symmetry is explicitly broken and $\langle\bar{q}_{a}q_{a}\rangle$ ceases to be an order parameter. The remaining vector-like $U({n_{\rm f}})$ symmetry cannot break spontaneously at vanishing baryon number density Vafa and Witten (1984), leading to the common understanding that QCD with dynamical massive quarks lacks non-trivial order parameters at zero baryon density.

This standard lore overlooks the possibility of symmetries which intertwine center and flavor transformations. Our discussion generalizes earlier work Kouno et al. (2012); Sakai et al. (2012); Kouno et al. (2013a, b); Iritani et al. (2015); Kouno et al. (2016); Hirakida et al. (2016, 2017), showing that special boundary conditions (BCs) for quarks can lead to an unbroken $\mathbb{Z}_{3}$ symmetry. These works interpreted this choice as defining a “QCD-like” theory that they termed “$Z_{3}$-QCD”. Other works have considered different applications of the same BCs Briceno et al. (2014); Cherman et al. (2016); Liu et al. (2016); Cherman et al. (2017). Here, we generalize and reinterpret the constructions of these BCs and use them to define order parameters for both quantum and thermal phase transitions in QCD.

Color-flavor center symmetry. Center symmetry Weiss (1981) acts only on topologically non-trivial observables. It can be viewed as a topologically non-trivial gauge transformation, with the action

[TABLE]

By itself, this is not a symmetry when fundamental representation fields are present in the theory.

We assume ${n_{\rm f}}$ quark flavors have a common mass $m_{q}$, so the theory (on $\mathbb{R}^{4}$) has a $U({n_{\rm f}})_{V}$ flavor symmetry. We consider flavor-twisted quark boundary conditions,

[TABLE]

where $\mathcal{U}$ is a $U({n_{\rm f}})$ matrix, and regard $x_{1}$ as a spatial direction. The flavor twist $\mathcal{U}$ may be assumed diagonal without loss of generality. The $SU({n_{\rm f}})_{V}$ flavor subgroup has center $\mathbb{Z}_{{n_{\rm f}}}$, which motivates a $\mathbb{Z}_{{n_{\rm f}}}$ symmetric choice of BCs Kouno et al. (2012); Sakai et al. (2012); Kouno et al. (2013a, b); Iritani et al. (2015); Kouno et al. (2016); Hirakida et al. (2016, 2017) for which the set of eigenvalues of $\mathcal{U}$ is invariant under multiplication by elements of $\mathbb{Z}_{{n_{\rm f}}}$. If the theory is to retain charge conjugation and $x_{1}$-reflection symmetries (suitably redefined), then the set of eigenvalues must also be invariant under complex conjugation. Two possibilities result, namely ${n_{\rm f}}$’th roots of $+1$ or $-1$,

[TABLE]

With the BCs in (4), the finite $L$ flavor symmetry is reduced to $G_{\rm L}=U(1)^{{n_{\rm f}}-1}_{V}\times U(1)^{{n_{\rm f}}-1}_{A}\times U(1)_{Q}\subset G$.

The key observation is that if

[TABLE]

then the circle-compactified theory, with either boundary condition (4), also remains invariant under an intertwined $\mathbb{Z}_{d}\subset\mathbb{Z}_{N}\times\mathbb{Z}^{\rm perm}_{{n_{\rm f}}}$ color-flavor center (CFC) symmetry, generated by the combination of a center transformation with phase $\omega^{N/d}=e^{2\pi i/d}$ and a $\mathbb{Z}_{d}$ cyclic flavor permutation. To see this note that, given either choice (4), a $\mathbb{Z}_{d}$ center transformation effectively permutes the eigenvalues of $\mathcal{U}$. Combining the center transformation with the opposite cyclic flavor permutation (which is part of the $U({n_{\rm f}})_{V}$ flavor symmetry) leaves the boundary condition invariant.

CFC symmetry intertwines center and flavor transformations and so has both local and extended order parameters. Examples of CFC order parameters include Polyakov loops such as (1) with winding numbers which are non-zero mod $d$, as well as $\mathbb{Z}_{n_{\rm f}}$ Fourier transforms of fermion bilinears, $\mathcal{O}^{(p)}_{\Gamma}\equiv\sum_{a=1}^{{n_{\rm f}}}\nu^{-ap}\,\bar{q}_{a}\Gamma q_{a}$, where $\Gamma$ is an arbitrary Dirac matrix and $p\bmod d\neq 0$. The action of the $\mathbb{Z}_{d}$ CFC symmetry is

[TABLE]

Other related choices of boundary conditions, and generalizations to multiple compactified directions are discussed in our Supplemental Materials.

Center symmetry and confinement. Consider the Polyakov loop connected correlator in QCD compactified on $x_{1}$ with circumference $L$,

[TABLE]

Suppose there is a non-zero lower bound $E$ on the energy of states that can contribute to the correlator, so $F(r)\sim E\,r$ as $r\to\infty$. When ${n_{\rm f}}=0$, the theory has a $\mathbb{Z}_{N}$ center symmetry. If the ground state is $\mathbb{Z}_{N}$ invariant, then no intermediate state created by a local operator acting on the vacuum can contribute to the correlator. All contributions to the correlator (7) must involve flux tubes which wrap the compactified dimension, so that $E=L\sigma$ with $\sigma$ the string tension.

On the other hand, if center symmetry is broken, explicitly or spontaneously, then intermediate states created by local operators can also contribute to the correlator (7). The minimal energy $E$ need not grow with $L$. This is interpreted as a signal of string breaking. It is tempting to conclude that there is a tight link between unbroken center symmetry and confinement of static test quarks by unbreakable flux tubes.

Now suppose that $d=\mathrm{gcd}({n_{\rm f}},N)>1$, all quarks have a common mass $m_{q}$, and we engineer the existence of $\mathbb{Z}_{d}$ CFC symmetry by using the BCs (4). As seen above, CFC symmetry acts on both Polyakov loops and appropriate local operators. Intermediate states created by local operators transforming the same as $\text{tr}\,\Omega$ under all unbroken symmetries can contribute to the correlator (7). For example, states created by $\mathcal{A}\equiv\sum_{a=1}^{{n_{\rm f}}}\nu^{-ap}\,\bar{q}_{a}\gamma_{1}D_{1}q_{a}$ and $\mathcal{B}\equiv\sum_{a=1}^{{n_{\rm f}}}\nu^{-ap}\,\bar{q}_{a}\gamma_{1}q_{a}$ can contribute to correlators of $\mathrm{Re}\,\text{tr}\,\Omega^{p}$ and $\mathrm{Im}\,\text{tr}\,\Omega^{p}$, respectively, even when CFC symmetry is not spontaneously broken. Consequently, the string tension as defined by the asymptotic behavior of the correlator (7) vanishes regardless of the realization of the $\mathbb{Z}_{d}$ center symmetry. Of course, the minimal masses of mesons created by operators $\mathcal{A}$ or $\mathcal{B}$ grow with increasing quark mass $m_{q}$. Due to non-uniformity in the $m_{q}\to\infty$ and $r\to\infty$ limits, a non-zero string tension does emerge if one sends $m_{q}\to\infty$ first. In summary, we see that for ${n_{\rm f}}>0$ there is no relation between the presence of a non-zero string tension and the existence, or realization, of a center symmetry intertwined with flavor.

Conformal window.

Let $x\equiv{n_{\rm f}}/N$, and set to zero the common quark mass and temperature, $m_{q}=T=0$. If $x>\frac{11}{2}$, QCD becomes an infrared-free theory. For $x$ below some $x_{\chi}<\frac{11}{2}$, chiral symmetry is believed to be spontaneously broken. In the intermediate range of values $x\in(x_{\chi},\frac{11}{2})$, called the conformal window, QCD flows to a non-trivial infrared (IR) fixed point without chiral symmetry breaking. The value of $x_{\chi}$ has been the subject of intensive lattice investigations (see, e.g., Refs. Aoki et al. (2014); Rinaldi (2015); Appelquist et al. (2016); Fodor et al. (2009); Aoki et al. (2013a); Appelquist et al. (2014, 2012); Fodor et al. (2011); Aoki et al. (2012, 2013b)). The existence of an IR-conformal phase can be seen most easily in the Veneziano large $N$ limit of QCD, where $x$ is fixed along with the ’t Hooft coupling $\lambda\equiv g^{2}N$ as $N$ increases. If $\epsilon\equiv\frac{11}{2}-x\to 0^{+}$, perturbation theory self-consistently implies the existence of an IR fixed point with a parametrically small coupling Caswell (1974); Banks and Zaks (1982), $\lambda_{\rm IR}=\tfrac{64}{75}\pi^{2}\epsilon\ll 1$.

One may show that $\mathbb{Z}_{d}$ CFC symmetry is spontaneously broken in the conformal window, at least at large $N$. The Veneziano limit is taken through a sequence of values for which $d=\gcd({n_{\rm f}},N)$ is fixed and greater than 1, while the ratio $x={n_{\rm f}}/N$ approaches a non-zero limit. Hence, the Polyakov loop (1) remains an order parameter for the intertwined $\mathbb{Z}_{d}$ center symmetry.

The CFC realization may be determined by computing the quantum effective potential $V_{\rm eff}(\Omega)$. The loop expansion is controlled by the small value of $\lambda$ at all scales when $\epsilon\ll 1$, rather than the small size of $L$ compared to the inverse strong scale $\Lambda^{-1}$ as in the classic papers Gross et al. (1981); Weiss (1981). Hence, the following analysis is valid for any circumference $L$, including the $L\to\infty$ limit of interest.

Classically, $V_{\rm eff}(\Omega)$ is zero. Using standard methods Gross et al. (1981); Weiss (1981), at one loop one finds $V_{\rm eff}(\Omega)=V_{\rm g}(\Omega)+V_{\rm f}(\Omega)$ with gluon and fermion contributions given by

[TABLE]

The upper/lower sign refers to BCs (4a)/(4b). As required, $V_{\rm eff}$ is invariant under CFC symmetry. To determine the minima of $V_{\rm eff}$ note that $V_{\rm g}=O(N^{2})$ while, due to our imposition of flavor-twisted BCs, $V_{\rm f}=O(N^{-2})$. At large $N$, the minima of $V_{\rm eff}$ are entirely determined by the gluonic contribution $V_{\rm g}$, which favors coinciding eigenvalues, $\Omega\propto 1$. Consequently, when $\epsilon=\frac{11}{2}{-}x\ll 1$ the CFC symmetry is spontaneously broken at any $L$. On the other hand, at the pure Yang-Mill point, $x=0$, center symmetry is certainly expected to be unbroken at large $L$, and standard large $N$ counting arguments imply that the intertwined center symmetry should remain unbroken for sufficiently small $x$. Hence, there must be at least one transition at some $x=x_{\rm CFC}$ where the realization of the CFC symmetry changes. This point may or may not coincide with the point $x_{\chi}$ where the chiral symmetry realization changes. Logically possible phase diagrams are sketched in Fig. 1.

Introducing a non-zero quark mass or temperature gives a richer phase structure. With $\epsilon\ll 1$ and small quark mass, the theory develops a new strong scale, $\Lambda_{m}\sim m_{q}\,e^{-75/(8\epsilon^{2})}$. As $L$ increases and becomes comparable to $\Lambda_{m}^{-1}$ we expect a $\mathbb{Z}_{d}$ center-restoring phase transition. We also expect a CFC-restoring phase transition at a non-zero temperature $T_{*}\sim 1/L$ when $m_{q}=0$, similar to the large $N$ deconfinement transition in $\mathcal{N}\,{=}\,4$ super-Yang-Mills theory on $S^{3}\times S^{1}$ Witten (1998); Aharony et al. (2004).

Now consider $N\,{=}\,3$ and massless quarks. If the ${n_{\rm f}}=15$ IR fixed point is weakly coupled, as widely believed, then our above calculation applies and $\mathbb{Z}_{3}$ center symmetry is spontaneously broken at ${n_{\rm f}}=15$. At ${n_{\rm f}}\,{=}\,3$, lattice calculations Iritani et al. (2015) with boundary conditions (4a) are consistent with unbroken $\mathbb{Z}_{3}$ center symmetry when $L\Lambda\gg 1$. So for integer values of $x={n_{\rm f}}/3$, there must be a minimal value $2\leq x_{\rm CFC}\leq 5$ where the $\mathbb{Z}_{3}$ CFC symmetry first becomes spontaneously broken.

Dense quark matter. Consider the phase diagram of QCD with $N={n_{\rm f}}=3$ and a common quark mass $m_{q}$, as a function of the $U(1)_{Q}$ chemical potential $\mu$ and temperature $T$. Previously known symmetry principles only suggest the existence of a curve $T(\mu)$ in the $(T,\mu)$ plane below which lies a superfluid phase with spontaneously broken $U(1)_{Q}$ symmetry, leading to a hypothesis of continuity of quark matter and hadronic nuclear matter Schäfer and Wilczek (1999). Consideration of CFC symmetry implies the existence of additional phase structure when QCD is compactified with CFC-preserving BCs on a spatial circle large compared to other spatial scales.

First, consider the small $T$, small $\mu$ regime. Here lattice studies Iritani et al. (2015) imply that $\langle\text{tr}\,\Omega\rangle=0$ at large $L$. Next, consider high temperatures, $T\gg\mathrm{max}(\Lambda,\mu)$. Here, the dynamics on spatial scales large compared to $(g^{2}T)^{-1}$ are described by pure 3D YM theory Gross et al. (1981) which confines, so $\langle\text{tr}\,\Omega\rangle\,{=}\,0$ at high $T$. We expect this high-temperature region to be smoothly connected to the region near $T=\mu=0$. However, as we next discuss the CFC symmetry realization behaves non-trivially when $T\to 0$ with $\mu\gg\mathrm{max}(\Lambda,m)$. A simple phase diagram consistent with our results is sketched in Fig. 2.

High density QCD, $\mu\gg\Lambda$, is believed to be in a “color-flavor-locked” (CFL) color-superconducting phase Alford et al. (2008) when $T<T_{\rm CFL}$. The phase transition temperature $T_{\rm CFL}$ is comparable to the superconducting gap, $T_{\rm CFL}\sim\Delta\sim\mu\,g^{-5}e^{-(3\pi^{2}/\sqrt{2})/g}$. Electric and magnetic gluons develop Debye and Meissner static screening masses, respectively, both of order $g\mu$ in the CFL phase Son and Stephanov (2000); Rischke (2000). For $T_{\rm CFL}<T\lesssim g\mu$, low frequency magnetic fluctuations experience Landau damping. Consequently, for $T\lesssim g\mu$ the relevant gauge coupling is small, $g(\mu)\ll 1$, and cold dense quark matter is weakly coupled.

In typical gauge-dependent language, CFL superconductivity is driven by an expectation value for diquark operators, $\langle q^{a}_{i}C\gamma_{5}q^{b}_{j}\rangle\propto\epsilon^{abK}\epsilon_{ijK}$ Alford et al. (2008). The uncontracted flavor indices on the “condensate” might lead one to think that flavor permutation symmetry is broken, automatically implying accompanying spontaneous breaking of the CFC symmetry Kouno et al. (2016) when $x_{1}$ is compactified with BCs (4). But this gauge-dependent language is misleading. The true gauge-invariant order parameters for spontaneous breaking of chiral and $U(1)_{Q}$ symmetries, schematically $\langle\bar{q}C\gamma_{5}\bar{q}qC\gamma_{5}q\rangle$ and $\langle(qC\gamma_{5}q)^{3}\rangle$, are $SU({n_{\rm f}})_{V}$ singlets Alford et al. (2008). So the development of CFL superconductivity does not, ipso facto, imply spontaneous breaking of CFC symmetry in the circle compactified theory.

To study the CFC symmetry realization when $\mu\gg\Lambda$ and $T\ll g\mu$, we examine the one-loop effective potential for $\Omega$, which is the sum of gluonic and fermionic contributions, $V_{\rm eff}(\Omega)=V_{\rm g}(\Omega)+V_{\rm f}(\Omega)$. The loop expansion is controlled by $g(\mu)\ll 1$ and is applicable for all $L$. For $T<T_{\rm CFL}$, gluons have effective masses $m_{\rm g}\sim g\mu$ due to a combination of Debye screening and the Meissner effect from color superconductivity Son and Stephanov (2000); Rischke (2000). As with any one-loop holonomy effective potential contribution from massive adjoint bosons, we thus expect

[TABLE]

with coefficients $f_{n}>0$ which are exponentially small, $f_{n}\sim e^{-nm_{\rm g}L}$, when $m_{\rm g}L\gg 1$. At large $\mu$, $V_{\rm g}(\Omega)$ is highly suppressed compared to the $\mu=0$ result (8).

Fermion excitations near the Fermi surface are nearly gapless as $T\to 0$, up to non-perturbative corrections from quark pairing. Interactions are weak, so we are dealing with a nearly free Fermi liquid. In the cold dense limit, $V_{\rm eff}(\Omega)$ is completely dominated by the fermion contribution,

[TABLE]

where $\mathfrak{m}^{2}_{k}\equiv(2\pi kT+i\mu)^{2}+m_{q}^{2}$, and the upper/lower sign refers to BCs (4a)/(4b). (Derivation detailed in Supplemental Materials.) This result is manifestly invariant under the CFC symmetry, as required. To examine the realization of CFC symmetry, we work in the simplifying limit $m_{q}\ll\mu$ and focus on the regime $\mu L\gg 1$. If $TL\gg 1$, then the sum (Order parameters and color-flavor center symmetry in QCD) is dominated by the $k=\pm\frac{1}{2}$, $n=1$ terms, giving

[TABLE]

up to exponentially small corrections. The $e^{-\pi TL{n_{\rm f}}}$ factor arises from the lowest fermionic Matsubara frequency and our twisted boundary conditions. Alternatively, if $TL\to 0$ then the prefactor in (12) becomes $\pm({n_{\rm f}}^{2}\pi^{2}L^{3})^{-1}$. In either regime of $TL$, neglecting subdominant contributions, $V_{\rm eff}(\Omega)\propto\mathrm{Re}\,\text{tr}\,\Omega^{n_{\rm f}}$ with an amplitude which oscillates as a function of ${n_{\rm f}}\mu L$.

For ${n_{\rm f}}\,{=}\,N\,{=}\,3$, extrema of ${V}_{\rm f}(\Omega)$ fall into four categories: (a) one center-symmetric extremum at $\Omega=\textrm{diag}\,(1,e^{2\pi i/3},e^{4\pi i/3})$, where the $SU(3)$ gauge symmetry is “broken” down to $U(1)^{2}$ with the holonomy playing the role of an adjoint Higgs field; (b) three center-broken extrema with “residual” gauge group $U(1)^{2}$ where $\Omega=\textrm{diag}\,(e^{(2k-1)i\pi/3},e^{2ki\pi/3},e^{(2k+1)i\pi/3})$, $k=0,1,2$; (c) nine center-broken “$SU(2)\times U(1)$” extrema at $\Omega=\textrm{diag}\,(e^{ki\pi/9},e^{ki\pi/9},e^{-2ki\pi/9})$ with $k\bmod 6=2$, 3 or 4; (d) three center-broken “$SU(3)$” extrema, $\Omega=\textrm{diag}\,(e^{2ki\pi/3},e^{2ki\pi/3},e^{2ki\pi/3})$, $k=0,1,2$. These “$SU(3)$” extrema are also minima of $V_{\rm g}$.

The form (12) implies that the the locations of the minima of $V_{\rm eff}(\Omega)$ oscillate as a function of $\mu L$, as illustrated in the contour plots in Fig. 3 for two nearby values of $\mu L$. (Each plot shows a fundamental domain of the Weyl group of $SU(3)$, which acts by permuting the eigenvalues of $\Omega$.) There are quantum oscillations in the phase structure of cold dense QCD on a circle, with the minima of $V_{\rm eff}$ cycling through two inequivalent sets of local minima as $\mu L$ varies. These come in two groups within which $V_{\rm f}(\Omega)$ is degenerate. One group consists of the center-symmetric and three $SU(3)$ extrema. The other consists of the six $SU(2)\times U(1)$ extrema with $\Omega=\textrm{diag}\,(e^{ki\pi/9},e^{ki\pi/9},e^{-2ki\pi/9})$ with $k\bmod 6=2$ and $4$. (The remaining six extrema are always saddle points for $TL\gg 1$.)

At $T\,{=}\,0$ there are quantum phase transitions when the minimum energy state switches from one set of extrema to another, with associated jumps in the ground state degeneracy. (Similar behavior in other circle-compactified theories has been seen in Refs. Vshivtsev et al. (1998); Kanazawa et al. (2017).) There are an infinite number of phase transitions in the cold dense limit as $L$ increases and successive energy bands pass through the value of the chemical potential, with an accumulation point at $L=\infty$. Borrowing a term from the condensed-matter literature Fisher and Selke (1980, 1981), each point in the $(T,\mu)$ phase diagram for QCD where this phenomenon occurs can be called a multi-phase point 111Unlike the spin models discussed in Refs. Fisher and Selke (1980, 1981), our multi-phase points involve no fine-tuning, as they are obtained as limits of one of our two symmetry-preserving choices of compactification. . As we discuss below, this behavior is expected in a finite area domain of the $(T,\mu)$ phase diagram, so in fact we find a multi-phase region.

The small residual gluon contribution to $V_{\rm eff}$ favors configurations with clumped holonomy eigenvalues, lowering the energy of $SU(3)$ extrema relative to the center-symmetric point. Hence, we expect that all genuine minima of $V_{\rm eff}$ in this multi-phase region are associated with broken CFC symmetry, with $\langle\text{tr}\,\Omega\rangle\neq 0$ 222Higher order calculations of $V_{\rm f}$ are needed to confirm that the center-broken $SU(3)$ minima are favored, as an $\mathcal{O}(g^{2})$ two-loop fermion contribution could overwhelm the one-loop gluonic contribution when it is suppressed by Meissner screening. .

Putting everything together, we conclude that there must be some curve $T=T_{\rm CFC}(\mu)$ below which the CFC symmetry is spontaneously broken with oscillatory multiphase behavior. We lack a definitive calculation of $V_{\rm eff}(\Omega)$ valid for $T>T_{\rm CFL}$, but we expect that $T_{\rm CFC}(\mu)$ is $\mathcal{O}(g\mu)$, greatly exceeding $T_{\rm CFL}$ at large $\mu$. The $T_{\rm CFC}$ curve must end at some point $\mu_{c}$ on the $T=0$ axis. The simplest hypothesis is that $\mu_{c}$ coincides with $\mu_{n}\sim\Lambda$, the critical chemical potential needed to produce pressureless nuclear matter at $T=0$, as illustrated in Fig. 2.

Conclusions. We have shown that there are well-defined and non-trivial order parameters for quantum and thermal phase transitions in QCD, compactified on a circle, provided $\mathrm{gcd}\,({n_{\rm f}},N)>1$ with quarks having a common mass $m_{q}$. This is a consequence of the existence of color-flavor center symmetry, and has interesting implications for the phase structure of QCD as a function ${n_{\rm f}}/N$, $\mu$, $T$, and $m_{q}$.

There are many worthwhile extensions of our observations. Consideration of CFC symmetry may be helpful in studies of QCD behavior near the lower edge of the conformal window Kaplan et al. (2009). For applications to dense QCD, an explicit calculation of the one-loop gluon contribution to $V_{\rm eff}(\Omega)$ in the hard dense loop approximation would give a better estimate for the CFC symmetry restoration temperature $T_{\rm CFC}(\mu)$. The role of explicit $SU(3)_{V}$ symmetry breaking should be explored. Finally, it would be interesting to study local order parameters for CFC symmetry. At $\mu=0$ these order parameters violate $SU({n_{\rm f}})_{V}$ symmetry, and hence must vanish as one takes the $\mathbb{R}^{4}$ limit by the Vafa-Witten theorem Vafa and Witten (1984). But this theorem does not apply at finite $\mu$.

Acknowledgments. We are grateful to D. B. Kaplan, L. McLerran, S. Reddy, A. Roggero, M. Savage, and T. Schäfer for stimulating discussions. This work was supported, in part, by the U. S. Department of Energy via grants DE-FG02-00ER-41132 (A.C.), DE-FG02-04ER41338 (S.S.), DE-SC0013036 (M.Ü.), DE-FG02-00ER41132 (M.W.) and DE-SC0011637 (L.Y.). L. Yaffe thanks the University of Regensburg and the Alexander von Humboldt foundation and S. Sen thanks Los Alamos National Laboratory for their generous support and hospitality during completion of this work.